chash stringlengths 16 16 | content stringlengths 267 674k |
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Do electrons spontaneously jump from high orbitals to lower ones emitting photons?
Explaining this was was one of the key initial achievements of the Dirac equation.
Yes, but this is not predicted by the Schrödinger equation, you need to go to the Dirac equation.
A critical application of this phenomena is laser. |
f124fd4a824cd40b | 2.17. Schrödinger’s Equation
This example implements a complex PDE using the PDE. We here chose the Schrödinger equation without a spatial potential in non-dimensional form:
\[i \partial_t \psi = -\nabla^2 \psi\]
Note that the example imposes Neumann conditions at the wall, so the wave packet is expected to reflect off the wall.
pde schroedinger
from math import sqrt
from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph
grid = CartesianGrid([[0, 20]], 128, periodic=False) # generate grid
# create a (normalized) wave packet with a certain form as an initial condition
initial_state = ScalarField.from_expression(grid, "exp(I * 5 * x) * exp(-(x - 10)**2)")
initial_state /= sqrt(initial_state.to_scalar("norm_squared").integral.real)
eq = PDE({"ψ": f"I * laplace(ψ)"}) # define the pde
# solve the pde and store intermediate data
storage = MemoryStorage()
eq.solve(initial_state, t_range=2.5, dt=1e-5, tracker=[storage.tracker(0.02)])
# visualize the results as a space-time plot
plot_kymograph(storage, scalar="norm_squared")
Total running time of the script: ( 0 minutes 5.215 seconds)
Gallery generated by Sphinx-Gallery |
f2153369d99c88aa | Fall 2022 - PHYS 385 D100
Quantum II (3)
Class Number: 2019
Delivery Method: In Person
• Course Times + Location:
WMC 2532, Burnaby
• Exam Times + Location:
Dec 12, 2022
8:30 AM – 11:30 AM
AQ 5037, Burnaby
• Prerequisites:
MATH 252 or MATH 254; MATH 260; PHYS 255; PHYS 285 or ENSC 380 or CHEM 260. All prerequisite courses require a minimum grade of C-. Recommended Prerequisite: PHYS 211.
Stern-Gerlach experiments and the structure of quantum mechanics; operators; angular momentum and spin; Schrödinger equation and examples for time evolution; systems of two spin-½ particles; density operators; wave mechanics in one dimension including the double slit experiment, particle in a box, scattering in one dimension, tunnelling; one-dimensional harmonic oscillator; coherent states. Quantitative.
1. The Stern-Gerlach experiments and the spin of electron
2. Rotations and matrix mechanics: operators, eigenvalues, eigenstates
3. Angular momentum eigenstates using ladder operators
4. Commutators, and the uncertainty relations
5. Time evolution and the Schrödinger equation, spin precession, magnetic resonance
6. Two spin-1/2 particles, EPR paradox, Bell's inequality
7. Entanglement, quantum teleportation
8. Wave mechanics in one dimension: coordinate and momentum basis
9. Solutions to Schrödinger equation in 1D: free particle, particle in a box, scattering
10. Harmonic oscillator, coherent states
• Homework 30%
• Exams 70%
Required book:
"A modern approach to quantum mechanics" by John. S. Townsend, 2nd Edition
Department Undergraduate Notes:
Registrar Notes:
|
dbf57c19e1efd6f1 | Back to search
FRINATEK-Fri mat.,naturv.,tek
A grand-canonical framework for ground and excited state properties of molecules with electron number fluctuations
Alternative title: Eit storkanonisk rammeeverk for grunn- og eksitert-tilstands eigenskaper for molekyl med variasjon i elektrontal.
Awarded: NOK 7.6 mill.
In the search for obtaining electronic devices at ever smaller length scales, electronic components utilizing molecules as building blocks has become an topic of interest. In contrast to traditional macroscopic solid-state devices, molecules exhibit a plethora of fascinating and potentially useful phenomena in addition to the small-size advantage. In particular, an interesting feature is whether transport across a molecule can be initiated by absorption of light, so-called photoactivated transport. Theoretically, we can get information on how the distribution of electrons in molecules behave by finding an (approximate) solution, called the wave function, to the electronic Schrödinger equation for the molecule. Usually, we consider molecules or molecular systems with a fixed number of electrons, i.e., what we call electron conserving systems. The primary objective of this project is to develop a theoretical framework, using wave functions, to describe properties of molecules which are involved in electron transport. This requires going beyond standard electron conserving wave functions to consider wave functions where the number of electrons is not fixed. The developed theory will be able to also describe molecules which are photoexcited, i.e., which have absorbed energy in the form of light. The developments in the project will be aimed at applications to molecules involved in electron transport, and specfically how to use light to control such processes.
The overarching goal of the proposed project is establish a grand-canonical wave function based molecular electronic-structure theory for molecules with fluctuating number of electrons. Although the term grand-canonical is used, it is not considered in connection to equilibrium situations in statistical mechanics, but rather a formal device to achieve electron number fluctuations. The proposed grand-canonical framework will be achieved by exponential unitary transformations of a electron number conserving reference wave function. The parametrization allows for net flow of charge while still being unitary. The introduced parametrization through a unitary transformation preserves its analogy with standard electronic-structure methods of number conserving wave functions, and enables extension to the well-established framework of response theory. The approach proposed in the project will have significance for several areas of application, but in this three-year project I will focus on establishing the framework with applications to molecules involved in electron transport processes. A major theoretical challenge for transport across molecules is the comprehensive treatment of the interaction between the molecule and light. The description and prediction of photoinduced transport processes will enable a new playground for photoactivated control of transport properties through molecules. Hence, these processes serve as suitable and timely driving forces for the initial developments of the grand-canonical wave function proposed in this project.
Funding scheme:
FRINATEK-Fri mat.,naturv.,tek |
72e5c9c41a8f6c1c | A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. are conjugate to stochastic differential equations.[1]
Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician Kiyosi Itô, who introduced the concept of stochastic integral and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist Stratonovich, leading to a calculus similar to ordinary calculus.
The most common form of SDEs in the literature is an ordinary differential equation with the right hand side perturbed by a term dependent on a white noise variable. In most cases, SDEs are understood as continuous time limit of the corresponding stochastic difference equations. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral. Such a mathematical definition was first proposed by Kiyosi Itô in the 1940s, leading to what is known today as the Itô calculus. Another construction was later proposed by Russian physicist Stratonovich, leading to what is known as the Stratonovich integral. The Itô integral and Stratonovich integral are related, but different, objects and the choice between them depends on the application considered. The Itô calculus is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time. The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion on manifolds.
An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the Smoluchowski equation or the Fokker–Planck equation, an equation describing the time evolution of probability distribution functions. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of stochastic evolution operator.
In physical science, there is an ambiguity in the usage of the term "Langevin SDEs". While Langevin SDEs can be of a more general form, this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure,[2] leading to a N=2 supersymmetric model closely related to supersymmetric quantum mechanics. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e., (overdamped) Langevin SDEs are never chaotic.
Stochastic calculus
Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Itô stochastic calculus and the Stratonovich stochastic calculus. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003) and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down.
Numerical solutions
Numerical methods for solving stochastic differential equations include the Euler–Maruyama method, Milstein method and Runge–Kutta method (SDE).
Use in physics
In physics, SDEs have widest applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the dynamical systems theory to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.
There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:
where is the position in the system in its phase (or state) space, , assumed to be a differentiable manifold, the is a flow vector field representing deterministic law of evolution, and is a set of vector fields that define the coupling of the system to Gaussian white noise, . If is a linear space and are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case in which .
For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.[3] Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a stochastic difference equation. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a uniquely defined mathematical object that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.
In physics, the main method of solution is to find the probability distribution function as a function of time using the equivalent Fokker–Planck equation (FPE). The Fokker–Planck equation is a deterministic partial differential equation. It tells how the probability distribution function evolves in time similarly to how the Schrödinger equation gives the time evolution of the quantum wave function or the diffusion equation gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by Monte Carlo simulation. Other techniques include the path integration that draws on the analogy between statistical physics and quantum mechanics (for example, the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables) or by writing down ordinary differential equations for the statistical moments of the probability distribution function.[citation needed]
Use in probability and mathematical finance
The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different. It is also the notation used in publications on numerical methods for solving stochastic differential equations. This notation makes the exotic nature of the random function of time in the physics formulation more explicit. In strict mathematical terms, cannot be chosen as an ordinary function, but only as a generalized function. The mathematical formulation treats this complication with less ambiguity than the physics formulation.
A typical equation is of the form
where denotes a Wiener process (standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding integral equation
The equation above characterizes the behavior of the continuous time stochastic process Xt as the sum of an ordinary Lebesgue integral and an Itô integral. A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length δ the stochastic process Xt changes its value by an amount that is normally distributed with expectation μ(Xttδ and variance σ(Xtt)2 δ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function μ is referred to as the drift coefficient, while σ is called the diffusion coefficient. The stochastic process Xt is called a diffusion process, and satisfies the Markov property.
The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution. Both require the existence of a process Xt that solves the integral equation version of the SDE. The difference between the two lies in the underlying probability space (). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space.
An important example is the equation for geometric Brownian motion
which is the equation for the dynamics of the price of a stock in the Black–Scholes options pricing model of financial mathematics.
There are also more general stochastic differential equations where the coefficients μ and σ depend not only on the present value of the process Xt, but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process, X, is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values of X, the defining equation is called a stochastic delay differential equation.
Existence and uniqueness of solutions
As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in n-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2).
Let T > 0, and let
be measurable functions for which there exist constants C and D such that
for all t ∈ [0, T] and all x and y ∈ Rn, where
Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment:
Then the stochastic differential equation/initial value problem
has a P-almost surely unique t-continuous solution (tω) ↦ Xt(ω) such that X is adapted to the filtration FtZ generated by Z and Bs, s ≤ t, and
Some explicitly solvable SDEs[4]
Linear SDE: general case
Reducible SDEs: Case 1
for a given differentiable function is equivalent to the Stratonovich SDE
which has a general solution
Reducible SDEs: Case 2
for a given differentiable function is equivalent to the Stratonovich SDE
which is reducible to
where where is defined as before. Its general solution is
SDEs and supersymmetry
In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the differential forms on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological supersymmetry which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as chaos, turbulence, self-organized criticality etc. and the Goldstone theorem explains the associated long-range dynamical behavior, i.e., the butterfly effect, 1/f and crackling noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.
See also
1. ^ Imkeller, Peter; Schmalfuss, Björn (2001). "The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors". Journal of Dynamics and Differential Equations. 13 (2): 215–249. doi:10.1023/a:1016673307045. ISSN 1040-7294. S2CID 3120200.
2. ^ Parisi, G.; Sourlas, N. (1979). "Random Magnetic Fields, Supersymmetry, and Negative Dimensions". Physical Review Letters. 43 (11): 744–745. Bibcode:1979PhRvL..43..744P. doi:10.1103/PhysRevLett.43.744.
3. ^ Slavík, A. (2013). "Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters". Journal of Mathematical Analysis and Applications. 402 (1): 261–274. doi:10.1016/j.jmaa.2013.01.027.
4. ^ Kloeden 1995, pag.118
Further reading
• Adomian, George (1983). Stochastic systems. Mathematics in Science and Engineering (169). Orlando, FL: Academic Press Inc.
• Adomian, George (1986). Nonlinear stochastic operator equations. Orlando, FL: Academic Press Inc.
• Adomian, George (1989). Nonlinear stochastic systems theory and applications to physics. Mathematics and its Applications (46). Dordrecht: Kluwer Academic Publishers Group.
• Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3.
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1.
• Teugels, J. and Sund B. (eds.) (2004). Encyclopedia of Actuarial Science. Chichester: Wiley. pp. 523–527. {{cite book}}: |author= has generic name (help)
• C. W. Gardiner (2004). Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer. p. 415.
• Thomas Mikosch (1998). Elementary Stochastic Calculus: with Finance in View. Singapore: World Scientific Publishing. p. 212. ISBN 981-02-3543-7.
• Seifedine Kadry (2007). "A Solution of Linear Stochastic Differential Equation". Wseas Transactions on Mathematics. USA: WSEAS TRANSACTIONS on MATHEMATICS, April 2007.: 618. ISSN 1109-2769.
• P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations. Springer. ISBN 0-387-54062-8.
• Higham., Desmond J. (January 2001). "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations". SIAM Review. 43 (3): 525–546. Bibcode:2001SIAMR..43..525H. CiteSeerX doi:10.1137/S0036144500378302.
• Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM, ISBN 978-1-611976-42-7 (2021). |
b97bd8bb30ff1406 | 1. Introduction
1.1. Preface
I have a very bad memory. I am able to memorize quite a lot of things short term, but I am not able to remember most formulas from quantum mechanics over the long term (e.g. like over the summer). I don’t remember formulas for perturbation theory (neither time dependent or time independent), I don’t remember Feynman rules in quantum field theory, I don’t even remember the Dirac equation exactly (where the i should be, if there is m or m^2, …). The thing about quantum field theory is not that some particular steps are difficult, but that there are so many of them and one has to master all of them at once, in order to really “get it”.
I never got QFT, because once I mastered one part sufficiently, I forgot some other part and it took so much time to master that other part that I forgot the first part again. However, I was determined that I would get it. In order to do so, I realized I need to keep notes of things I understood, written in my own way. Then, when I relearn some parts that I forgot, it just takes me a few minutes to go over my reference notes to get into it quickly. My own style of understanding is that the notes should be complete (no need to consult external books), yet very short and getting directly to the point, and also with every single calculation carried out explicitly.
See also the preface to the QFT part.
If you want to study physics, learn math the physics way (as opposed to the usual mathematics way of a definition, theorem, proof, …). When I was beginning my undergrad physics studies (and even on a high school), I also had this common misconception, that I need to study math and understand every proof and then I’ll be somehow prepared for physics. I was very wrong. I used to study calculus by myself and then trying to learn the proofs, and Lebesgue integral and I was learning that from the mathematics books. At the university, I always did all my math exams first (as far as I remember, I always got A from those), hoping that would be a good start for the physics exams, but I always found out that it was mostly useless.
Now I know that the only way to study physics is to go and do physics directly and learn the math on the way as needed. The math section of this book reviews all the math, that is necessary for studying theoretical physics (graduate level).
There are actually quite a lot of good math books written by physicists as well as many excellent physics books, covering everything that I cover here. But I really like to have all the theoretical physics and the corresponding math explained in one book, and to keep it as short as possible. Also everyone has a bit different style and amount of rigor and I have not found a book that would perfectly suite my own style, thus I wrote one.
1.2. Introduction
The Theoretical Physics Reference is an attempt to derive all theoretical physics equations (that are ever needed for applications) from the general and special relativity and the standard model of particle physics.
The goals are:
• All calculations are very explicit, with no intermediate steps left out.
• Start from the most general (and correct) physical theories (general relativity or standard model) and derive the specialized equations from them (e.g. the Schrödinger equation).
• Math is developed in the math section (not in the physics section).
• Theory should be presented as short and as explicitly as possible. Then there should be arbitrary number of examples, to show how the theory is used.
• There should be just one notation used throughout the book.
• It should serve as a reference to any physics equation (exact derivation where it comes from) and the reader should be able to understand how things work from this book, and be ready to understand specialized literature.
This is a work in progress and some chapters don’t conform to the above goals yet. Usually first some derivation is written, as we understood it, then the mathematical tools are extracted and put into the math section, and the rest is fit where it belongs. Sometimes we don’t understand some parts yet, then those are currently left there as they are.
There are many excellent books about theoretical physics, that one can consult about particular details. The goal of this book (when completed) is to show where things come from and serve as a reference to any particular field, so that one doesn’t get lost when reading specialized literature.
Here is an incomplete list of some of the best books in theoretical physics (we only picked those that we actually read):
1. Landau, L. D.; Lifshitz, E. M: Course of Theoretical Physics
2. Richard Feynman: The Feynman Lectures on Physics
3. Walter Greiner: “Classical Theoretical Physics” series of texts
4. Herbert Goldstein: Classical Mechanics
5. J.D. Jackson: Classical Electrodynamics
6. Charles W. Misner, Kip S. Thorne, John Wheeler: Gravitation
7. Bernard Schutz: A First Course in General Relativity
8. Carrol S.: The Lecture Notes on General Relativity
9. J.J. Sakurai: Advanced Quantum Mechanics
10. Brown L. S.: Quantum Field Theory
11. Mark Srednicki: Quantum Field Theory
12. Claude Itzykson, Jean-Bernard Zuber: Quantum Field Theory
13. Zee A.: Quantum Field Theory in a Nutshell
14. Steven Weinberg: The Quantum Theory of Fields
15. L.H. Ryder: Quantum Field Theory
16. Jiří Hořejší: Fundamentals of Electroweak Theory
17. Michele Maggiore: A Modern Introduction to Quantum Field Theory
18. M.E. Peskin & D.V. Schroeder: An Introduction to Quantum Field Theory
19. J.W. Negele, H. Orland: Quantum Many-Particle Systems
20. X-G. Wen: Quantum Field Theory of Many-Body Systems
21. Dirac, P.A.M.: General Theory of Relativity |
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Latest Posts
The usual approach to the chemical bond is to "solve the Schrödinger equation", and this is done by attempting to follow the dynamics of the electrons. As we all know, that is impossible; the equation as usually presented requires you to know the potential field in which every particle moves, and since each electron is in motion, the problem becomes insoluble. Even classical gravity has no analytical solution for the three-body problem. We all know the answer – there are various assumptions and approximations made, and as Pople noted in his Nobel lecture, validation of very similar molecules allows you to assign values to the various difficult terms and you can get quite accurate answers for similar molecules.
However, you can only be sure of that if there are suitable examples from which to validate. So, quite accurate answers are obtained, but the question remains, is the output of any value in increasing the understanding of what is going on for chemists? In other words, can they say why A behaves differently to a seemingly similar B?
There is a second issue. Because validation and the requirement to obtain results equivalent to those observed, can we be sure they are obtained the right way? As an example, in 2006 some American chemists decided to test some programs that were considered tolerable advanced and available to general chemists on some quite basic compounds. The results were quite disappointing, even to the extent of showing that benzene was non-planar. (Moran, D. and five others. 2006. J. Amer. Chem. Soc. 128: 9342-9343.)
There is a third issue, and this seems to have passed without comment amongst chemists. In the state vector formalism of quantum mechanics, it is often stated that you cannot factorise the overall wave function. That is the basis of the Schrödinger cat paradox. The whole cat is in the superposition of states that differ on whether or not the nucleus has decayed. If you can factorise the state, the paradox disappears. You may still have to open the box to see what has happened to the cat, but the cat, being a macroscopic being, has behaved classically and was either dead or alive before you opened it. This, of course, is an interpretive issue. The possible classical states are "cat alive" (that has amplitude A) and "cat dead" (which has amplitude B). According to the state vector formalism, the actual state has amplitude (A B), hence thinking that the cat is in a superposition of states. The interesting thing about this is it is impossible to prove this wrong, because any attempt to observe the state collapses it to either A or B, and the "or" is the exclusive form. Is that science or another example of the mysticism that we accuse the ancients of believing, and we laugh at them for it? Why won't the future laugh at us? In my opinion, the argument that this procedure aids calculation is also misleading; classically you would calculate the probability that the nucleus had decayed, and the probability the rest of the device worked, and you could lay bets on whether the cat was alive or dead.
Accordingly, I am happy with factorizing the wave function. Indeed, every time you talk about a p orbital interacting with . . . you have factorized the atomic state, and in my opinion chemistry would be incomprehensible unless we do this sort of thing. However, I believe we can go further. Let us take the hydrogen atom, and accept that a given state has the action equal to nh associated with any state. We can factorise that (Schiller, R. 1962. Phys Rev 125 : 1100 – 1108 ) such that
nh = [(nr + ½) + ( l
+ ½)h
Here, while the quantum numbers count the action, they also count the number of radial and angular nodes respectively. What is interesting is the half quanta; why are they there? In my opinion, they have separate functions from the other quanta. For example, consider the ground state of hydrogen. We can rewrite (1) as
h = [( ½ ) + ( ½)]h (2)
What does (2) actually say? First there are no nodes. The second is the state actually complies with the Uncertainty Principle. Suppose instead, we put the RHS of (2) simply equal to 1. If we assign that to angular motion solely, we have the Bohr theory, and we know that is wrong. If we assign it to radial motion solely, we have the motion of the electron as lying on a line through the nucleus, which is actually a classical possibility. While that turns up in most text books, again I consider that to be wrong because it has zero angular uncertainty. You know the angular momentum (zero) and you know (or could know if you determined it) the orientation of the line. (The same reasoning shows why Bohr was wrong, although of course at the time he had no idea of the Uncertainty Principle.)
There is another good point about (2): it asserts the period involves two "cycles". That is a requirement for a wave, which must have a crest and a trough. If you have no nodes separating them, you need two cycles. Now, I wonder how many people reading this (if any??) can see what happens next?
Which gets me to a final question, at least for this post: how many chemists are actually happy with what theory offers them? Comments would be appreciated.
Posted by Ian Miller on Oct 22, 2017 9:45 PM BST
Following the Alternative interpretations theme, I shall write a series of posts about the chemical bond. As to why, and I hope to suggest that there is somewhat more to the chemical bond than we now consider. I suspect the chemical bond is something almost all chemists "know" what it is, but most would have trouble articulating it. We can calculate its properties, or at least we believe we can, but do we understand what it is? I think part of the problem here is that not very many people actually think about what quantum mechanics implies.
In the August Chemistry World it was stated that to understand molecules, all you have to do is to solve the Schrödinger equation for all the particles that are present. However, supposing this were possible, would you actually understand what is going on? How many chemists can claim to understand quantum mechanics, at least to some degree? We know there is something called "wave particle duality" but what does that mean? There are a number of interpretations of quantum mechanics, but to my mind the first question is, is there actually a wave? There are only two answers to such a discrete question: yes or no. De Broglie and Bohm said yes, and developed what they call the pilot wave theory. I agree with them, but I have made a couple of alterations, so I call my modification the guidance wave. The standard theory would answer no. There is no wave, and everything is calculated on the basis of a mathematical formalism.
Each of these answers raises its own problems. The problem with there being a wave piloting or guiding the particle is that there is no physical evidence for the wave. There is absolutely no evidence so far that can be attributed solely to the wave because all we ever detect is the particle. The "empty wave" cannot be detected, and there have been efforts to find it. Of course just because you cannot find something does not mean it is not there; it merely means it is not detectable with whatever tool you are using, or it is not where you are looking. For my guidance wave, the problem is somewhat worse in some ways, although better in others. My guidance wave transmits energy, which is what waves do. This arises because the phase velocity of a wave equals E/p, where E is the energy and p the momentum. The problem is, while the momentum is unambiguous (the momentum of the particle) what is the energy? Bohm had a quantum potential, but the problem with this is it is not assignable because his relationship for it did not lead to a definable value. I have argued that to make the two slit experiment work, the phase velocity should equal the particle velocity, so that both arrive at the slits at the same time, and that is one of the two differences between my guidance wave and the pilot wave. The problem with that is, it puts the energy of the system at twice the particle kinetic energy. The question then is, why cannot we detect the energy in the wave? My answer probably requires another dimension. The wave function is known to be complex; if you try to make it real, e.g. represent it as a sine wave, quantum mechanics does not work.
However, the "non-real" wave has its problems. If there is actually nothing there, how does the wave make the two-slit experiment work? The answer that the "particle" goes through both slits is demonstrably wrong, although there has been a lot of arm-waving to preserve this option. For example, if you shine light on electrons in the two slit experiment, it is clear the electron only goes through one slit. What we then see is claims that this procedure "collapsed the wave function", and herein lies a problem with such physics: if it is mysterious enough, there is always an escape clause. However, weak measurements have shown that photons go though only one slit, and the diffraction pattern still arises, exactly according to Bohm's calculations (Kocsis, S. and 6 others. 2011. Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer Science 332: 1170 – 1173.) There is another issue. If the wave has zero energy, the energy of the particle is known, and following Heisenberg, the phase velocity of the wave is half that of the particle. That implies everything happens, then the wave catches up and sorts things out. That seems to me to be bizarre in the extreme.
So, you may ask, what has all this to do with the chemical bond? Well, my guidance wave approach actually leads to a dramatic simplification because if the waves transmit energy that equals the particle energy, then the stationary state can now be reduced to a wave problem. As an example of what I mean, think of the sound coming from a church organ pipe. In principle you could calculate it from the turbulent motion of all the air particles, and you could derive equations to statistically account for all the motion. Alternatively, you could argue that there will be sound, and it must form a standing wave in the pipe, so the sound frequency is defined by the dimensions of the pipe. That is somewhat easier, and also, in my opinion, it conveys more information.
All of which is all very well, but where does it take us? I hope to offer some food for thought in the posts that will follow.
Posted by Ian Miller on Aug 28, 2017 12:19 AM BST
In a recent Chemistry World there was an item on chemistry in India, and one of the things that struck me was that Indian chemists seemed to be criticized because they published a very low proportion of the papers in journals such as JACS and Angewandte Chemie. The implication was, only the "best" stuff gets published there, hence the Indian chemists were not good enough. The question I want to raise is, do you think that reasoning is valid?
One answer might be that these journals (but not exclusively) publish the leading material, i.e. they lead the way that chemistry is taking in the future. When I started my career, these high profile journals were a "must read" because they were where papers that at least editors felt was likely to be of general interest or of practical interest to the widest number of chemists were published.
But these days, these sort of papers do not turn up. There may be new reactions, but they are starting to involve difficult to obtain reagents, and chemical theory has descended into the production of computational output. These prestige journals have moved on to new academic fields, which is becoming increasingly specialized, which increasingly needs expensive equipment, and which also needs a school that has been going for some time, so that the background experience is well embedded. There are exceptions, but they do not last, thus graphene was quite novel, but not for long. There are still publications involving graphene, but chemists working there have to have experience in the area to make headway. More importantly, unless the chemist is actually working in the area, (s)he will never touch something like graphene. I am certainly not criticizing this approach by the journals. Rather I am suggesting the nature of chemical research is changing, but I feel that in countries where the funding is not there to the same extent, chemists may well feel they might be more productive not trying to keep up with the Joneses.
Another issue is, by implication it is claimed that work published in the elite journals is more important. Who says? Obviously, the group who publish there, and the editorial board will, but is this so? There may well be work that is more immediately important, but to a modest sized subset of chemists working in a specific area. Now the chemist should publish in the journal that that subset will read.
My view is that chemistry has expanded into so many sub-fields that no chemist can keep up with everything. When I started research, organic chemists tended not to be especially interested in inorganic or physical chemistry, not because they were not important, but simply because they did not have the time. Now it has got much worse. I doubt there is much we can do about that, but I think it is wrong to argue that some chemistry that can only be done in very richly funded Universities is "better" or more important than a lot of other work that gets published in specialized journals. What do you think?
Posted by Ian Miller on Jul 3, 2017 3:23 AM BST
Some time ago now I published an ebook "Planetary Formation and Biogenesis", which started with a review including over 600 references, following which I tried analyzing their conclusions and tried to put them together to make a coherent whole. This ended up with a series of conclusions and predictions on what we might find elsewhere. It was in light of this I saw the article in the May edition of "Chemistry World". That article put up reasons to back some of the various thoughts as to where life started, but I found it interesting that people formed their views based on their chemical experience, and they tended to carry out experiments to support that hypothesis. That, of course, is fair enough, but it still misses what I believe to bee the key point, and that is, what is the most critical problem to overcome to get life started, and how hard is it to do?
The hardest thing, in my opinion, is not to make polymers. I know that driving condensation reactions forwards in water is difficult, but as Deamer pointed out in the article, if you can get a lipid equivalent, it is by no means impossible. No, in my opinion, the hardest thing to do is to make phosphate esters. Exactly how do you make a phosphate ester? As Stanley Miller once remarked, you don't start with phosphoryl chloride in the ocean. The simplest way is to heat a phosphate and an alcohol to about 200 degrees C. Of course, water will hydrolyse phosphate esters at 200 degrees C, so unless you drive off the water, which is difficult to do in an ocean, high temperature is not your friend because the concentration of water in the ocean always exceeds the concentration of phosphate or alcohol. You simply cannot do that around black smokers.
The next problem is, why did nature choose ribose? Ribose is not the only sugar that permits the formation of a duplex when suitably phosphated and bound to a nucleotide. Almost all other pentoses do it. So the question remains, why ribose? The phosphate ester is an important solubilizing agent for a number of biochemicals necessary for life but it invariably occurs bound to a ribose, which in turn is usually bound to adenine. The question then is, is this a clue? If so, why is it largely unnoticed? My conclusion was, ribose alone can form a phosphate ester on a primary alcohol group in solution because only ribose naturally has reasonable concentrations of itself in the furanose form.
It was not always unnoticed. There is a clearly plausible route, substantiated by experiment (Ponnamperuma, C., Sagan, C., Mariner, R., 1963. Synthesis of adenosine triphosphate under possible primitive earth conditions. Nature 199: 222-226.) that shows the way. What was shown here was that if you have a mixture of adenine, ribose and phosphate, and shine UV light that can be absorbed by the adenine, you make adenosine, and then phosphate esters, mainly at the 5 position of the furanose form, so you can end up with ATP, a chemical still used by life today. Why is that work neglected? Could it be that nobody these days goes back and reads the literature from 1963?
Why does this synthesis work? My explanation is this. You do not have to get to 200 degrees to form a phosphate ester. What you have to do is provide an impact between the alcohol group and phosphate equivalent to that expected at 200 degrees. If we think about the experiment described above, there is no way an excited electronic state of adenine can be delocalized into the ribose, so why is the light necessary?
My conclusion was that the excited state of the adenine can decay so that quite a considerable amount of vibrational energy is generated. That will help form the adenosine, but after that the vibrational energy will spread through the sugar. Now we see the advantage of the furanose: it is relatively floppy, and it will vibrate well, and even better, the vibrational waves will focus at C-5. That is how the phosphate ester is formed, and why ribose is critical. The pyranose forms are simply too rigid to focus the mechanical vibrations. Once you get adenosine phosphate, in the above experiment the process continued to make polyphosphates, but if some adenosine was also close by, it would start to form the polymer chain. Now, if that is true, then life must have started on the surface, either of the sea or on land. My view is the sea is more probable, because on land it is difficult to see where further biochemicals can come from.
Posted by Ian Miller on May 28, 2017 11:45 PM BST
One issue that has puzzled me is what role, if any, does theory play in modern chemistry, other than having a number of people writing papers. Of course some people are carrying out computations, but does any of their work influence other chemists in any way? Are they busy talking to themselves? The reason why this has struck me is that the latest "Chemistry World" has an article "Do hydrogen bonds have covalent character?" Immediately below is the explanation, "Scientists wrangle over disagreement between charge transfer measurements." My immediate reaction was, what exactly is meant by "covalent character" and "charge transfer"? I know what I think a covalent bond is, which is a bond formed by two electrons from two atoms pairing to form a wave function component with half the periodic time of the waves on the original free atom. I also accept the dative covalent bond, such as that in the BH3NH3 molecule, where two electrons come from the same atom, and where the resultant bond has a strength and length as if the two electrons originated from separate atoms. That is clearly not what is meant for the hydrogen bond, but the saviour is that word "character". What does that imply?
What puzzles me here is that on reading the article, there are no charge transfer measurements. What we have, instead, are various calculations based on models, and the argument is whether the model involves transfer of electrons. However, as far as I can make out, there is no observational evidence at all. In the BH3NH3 molecule, obviously the two electrons for the bond start from the nitrogen atom, but the resultant dipole moment does not indicate a whole electron is transferred, although we could say it is, and then sent back to form the bond. However, in that molecule we have a dipole moment of over 6 Debye units. What is the change of dipole moment in forming the hydrogen bond? If we want to argue for charge transfer, we should at least know that.
From my point of view, the hydrogen bond is essentially very weak, and is at least an order of magnitude less strong than similar covalent bonds. This would suggest that if there were charge transfer, it is relatively minor. Why would such a small effect not be simply due to polarization? With the molecule BH3NH3 it is generally accepted that the lone pair on the ammonia enters the orbital structure of the boron system, with both being tetrahedral in structure, more or less. The dipole moment is about 6 Debye units, which does not correspond to one electron fully transferring to the boron system. There is clear charge transfer and the bond is effectively covalent.
Now, if we then look at ammonia, do we expect the lone pair on the nitrogen to transfer itself to the hydrogen atom of another ammonia molecule to form this hydrogen bond? If it corresponded to the boron example, then we would expect a change of at least several Debye units but as far as I know, there is no such change of dipole moment that is not explicable in terms of it being a condensed system. The article states there are experimental data to support charge transfer, but what is it?
Back to my original problem with computational chemistry: what role, if any, does theory play in modern chemistry? In this article we see a statement such as the NBO method falls foul of "basis set superposition error". What exactly does that mean, and how many chemists appreciate exactly what it means? We have a disagreement where one is accused of focusing on energies, while they focus on charge density shifts. At least energies are measurable. What bothers me is that such arguments on whether different people use the same terminology differently is a bit like arguing about how many angels can dance on the head of a pin. What we need from theory is a reasonably clear statement of what it means, and a clear statement of what assumptions are made, and what part validation plays in the computations.
Posted by Ian Miller on Apr 24, 2017 12:54 AM BST
An interesting thing happened for planetary science recently: two papers (Nature, vol 541 (Dauphas, pp 521 – 524; Fischer-Gödde and Kleine, pp 525 – 527) showed that much of how we think planets accreted is wrong. The papers showed that the Earth/Moon system has isotope distributions across a number of elements exactly the same as that found in enstatite chondrites, and that distribution applied over most of the accretion. The timing was based on the premise that different elements would be extracted into the core at different rates, and some not at all. Further, the isotope distributions of these elements are known to vary according to distance to the star, thus Earth is different from Mars, which in turn is clearly different from the asteroid belt. Exactly why they have this radial variation is an interesting question in itself, but for the moment, it is an established fact. If we assume this variation in isotope distribution follows a continuous function, then the variations we know about have sufficient magnitude that we can say that Earth accreted from material confined to a narrow zone.
Enstatite chondrites are highly reduced, their iron content tends to be as the metal or as a sulphide rather than as an oxide, and they may even contain small amounts of silicon as a silicide. They are also extremely dry, and it is assumed that they were formed at a very hot part of the accretion disk because they contain less forsterite and additionally you need very high temperatures to form silicides.
In my mind, the significance of these papers is two-fold. The first is, the standard explanation that Earth's water and biogenetic material came from carbonaceous chondrites must be wrong. The ruthenium isotope analysis falsifies the theory that so much water arrived from such chondrites. If they did, the ruthenium on our surface would be different. The second is the standard theory of planetary formation, in which dust accreted to planetesimals, these collided to form embryos, which in turn formed oligarchs or protoplanets (Mars sized objects) and these collided to form planets must be wrong. The reason is that if they did collide like that, they would do a lot of bouncing around and everything would get well-mixed. Standard computer simulations argue that Earth would have formed from a distribution of matter from further out than Mars to inside Mercury's orbit. The fact that the isotope ratios are so equivalent to enstatite chondrites shows the material that formed Earth came from a relatively narrow zone that at some stage had been very strongly heated. That, of course, is why Earth has such a large iron core, and Mars does not. At Mars, much of the iron remained as the oxide.
In my mind, this work shows that such oligarchic growth is wrong and that the alternative, monarchic growth, which has been largely abandoned, is in fact correct. But that raises the question, why are the planets where they are, and why are there such large gaps? My answer is simple: the initial accretion was chemically based, and certain temperature zones favoured specific reactions. It was only in these zones that accretion occurred at a sufficient rate to form large bodies. That, in turn, is why the various planets have different compositions, and why Earth has so much water and is the biggest rocky planet: it was in a zone that was favourable to the formation of a cement, and water from the disk gases set it. If anyone is interested, my ebook "Planetary Formation and Biogenesis" explains this in more detail, and a review of over 600 references explains why. As far as I am aware, the theory outlined there is the only one that requires the results of those papers. So, every now and again, something good happens! It feels good to know you could actually be correct where others are not.
So, will these two papers cause a change of thinking. In my opinion, it may not change anything because scientists not directly involved probably do not care, and scientists deeply involved are not going to change their beliefs. Why do I think that? Well, there was a more convincing paper back in 2002 (Drake and Righter, Nature 416
: 39-44) that came to exactly the same conclusions. Instead of ruthenium isotopes, it used osmium isotopes, but you see the point. I doubt these two papers will be the straw that broke the camel's back, but I could be wrong. However, experience in this field shows that scientists prefer to ignore evidence that falsifies their cherished beliefs than change their minds. As a further example, neither of these papers cited the Drake and Righter paper. They did not want to admit they were confirming a previous conclusion, which is perhaps indicative they really do not wish to change people's minds, let alone acknowledge previous work that is directly relevant.
Posted by Ian Miller on Feb 5, 2017 9:41 PM GMT |
3c4ea86ea7b53ae2 | Theoretical chemistry
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{acid, form, water}
{theory, work, human}
Theoretical chemistry involves the use of physics to explain or predict chemical phenomena. In recent years, it has consisted primarily of quantum chemistry, i.e., the application of quantum mechanics to problems in chemistry. Theoretical chemistry may be broadly divided into electronic structure, dynamics, and statistical mechanics. In the process of solving the problem of predicting chemical reactivities, these may all be invoked to various degrees. Other "miscellaneous" research areas in theoretical chemistry include the mathematical characterization of bulk chemistry in various phases (e.g. the study of chemical kinetics) and the study of the applicability of more recent math developments to the basic areas of study (e.g. for instance the possible application of principles of topology to the study of electronic structure). The latter area of theoretical chemistry is sometimes referred to as mathematical chemistry.
Much of this may be categorized as computational chemistry, although computational chemistry usually refers to the application of theoretical chemistry in an applied setting, usually with some approximation scheme such as certain types of post Hartree-Fock, Density Functional Theory, semiempirical methods (such as PM3) or force field methods. Some chemical theorists apply statistical mechanics to provide a bridge between the microscopic phenomena of the quantum world and the macroscopic bulk properties of systems.
Theoretical attacks on chemical problems go back to the earliest days, but until the formulation of the Schrödinger equation by the Austrian physicist Erwin Schrödinger, the techniques available were rather crude and speculative. Currently, much more sophisticated theoretical approaches, based on Quantum Field Theory and Nonequilibrium Green Function Theory are in vogue.
Branches of theoretical chemistry
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0a32acfbe3eaa145 | Application of the Finite Difference Method to the 1-D Schrödinger Equation
by Trevor Robertson
University of Massachusetts Dartmouth
The time-dependent Schrödinger equation (TDSE) is a fundamental law in understanding the states of many microscopic systems. Such systems occur in nearly all branches of physics and engineering, including high-energy physics, solid-state physics, and semiconductor engineering, just to name a few. A robust and efficient algorithm to solve this equation would be highly sought-after in these respective fields. This study utilizes the well-known method for solving the TDSE, the finite difference method (FDM), but with an important modification to conserve flux and analyze the 1-D case given well-known potentials. Numerical results that agree with theoretical predictions are reported. [3] It becomes evident, however, that solving the TDSE still involves challenging problems of scaling to higher dimensions and refined grids. This study shows that it is a promising, intuitive, and accurate method for linear domains over lower dimensions with arbitrary potentials. [7]
The Schrödinger equation serves as the quantum analog to the classical cases of Newton’s laws of motion and conservation laws. On the macroscopic scale, the classical laws of Newton as well as the modern ideas of Einstein provide a sound understanding. However, delving into the microscopic realms, the laws governing interactions are dominantly quantum mechanical. Given a set of initial conditions, the Schrödinger equation can predict the dynamical processes of a particle as it propagates through space and time. The wavefunction Ψ(x,t) embeds the information of the particle system throughout its evolution.[3] Solving for Ψ(x,t) can allow for the extraction of any physically meaningful observables that can be measured in a lab, such as the flux. Take the particle in an infinite potential well, for example. Quantum mechanically, the wavefunction is a series of sinusoidal waves that are zero at the walls (boundaries). Classically, these waves correspond to fundamental modes of vibration on a string.
Solutions to the TDSE require a robust and efficient algorithm capable of dealing with a sufficient domain of propagation over reasonable time scales. The need for such stable and efficient methods is a growing trend both in physics and in general scientific computing. It is well known that FDM is such an algorithm capable of dealing with the TDSE. Other, more sophisticated, methods include algorithms such as the finite element method, spectral methods, and meshless alternatives.[4] Such alternatives have advantages over the standard finite difference method, which can be rigid or inefficient in its basic form.[7] It is also well known that the FDM is more difficult to apply with more arbitrary meshes of higher dimensions.[10] This bottleneck of the finite difference method must continue to be optimized for further application. The following study proposes an analysis of the 1-D TDSE using a modified approach of the FDM.
Solving the Schrödinger equation for arbitrary potentials is a valuable tool for extracting the information of a quantum system. In the following, an algorithm is presented that can extract probability density, as well as the expectation values of position, momentum, energy, and flux. As a test study, it should also be able to replicate results from numerically known potentials such as the simple harmonic oscillator, as well as potential wells and barriers.[4] Atomic units (a.u.) will be used throughout unless otherwise noted.
Computational Method
Since the wavefunction is composed of both real and imaginary parts, it is beneficial to deal with both components separately. Doing this simplifies the equations and creates a recursive relationship that can easily be integrated into the program.
〖(1.1) Ψ〗_n (t)=R_Ψn+iI_Ψn
This splitting of the wavefunction will be crucial in constructing a flux-preserving system compatible with FDM, and is the first advantage of our modified approach. In the following, subscripts will represent the given location of the operation (n), while superscripts will be the given time step this operation is carried over (m). Using the atomic units system, the TDSE can be formulated using the FDM as follows:[7]
(1.2) (dΨ_n (t))/dt=i/(2h^2 ) (Ψ_(n+1) (t)-2Ψ(t)+Ψ_(n-1) (t) )-iU_j (t)Ψ_j (t)
Converting a partial differential equation into a set of solvable ordinary differential equations reduces the system to an initial value problem. This is known as the method of lines.[5] Setting a domain of integration makes solving these ODEs straightforward, but it is important to recognize that as the evolution of the state occurs, the normalization of the wavefunction must also be preserved. Otherwise, physical properties of the system, such as flux and energy, can become unphysical. For this reason, we will use the leapfrog method, which is part of a family of symplectic solvers for dealing with numerical solutions of Hamiltonian systems.[5] This preserves the system to second order accuracy similar to other methods such as the midpoint method; but it is known from its Jacobian to preserve areas when going from one transformation space to another.[9] Now applying Equation (1.1) to this set of ODEs gives the following coupled relationship we can use to solve the TDSE.
(1.3a) (dR_Ψn)/dt=-1/(2h^2 ) I_(Ψn-1)+(1/h^2 +U_j ) I_j-1/(2h^2 ) I_(Ψn+1)→f_n ({I_( Ψ) },t)(1.3b) (dI_Ψn)/dt=-1/(2h^2 ) R_(Ψn-1)-(1/h^2 +U_j ) R_j+1/(2h^2 ) R_(Ψn+1)→ g_n ({R_( Ψ) },t)
Here the f_n and g_n denote that the right-hand sides are functions of purely imaginary or real parts of the wave function, respectively. The derivatives of both the real and imaginary parts of the system rely on one another, as is the applicability of leapfrog integration. This form of the coupled Schrödinger equation allows for the stepping forward of the TDSE in time, and is in a suitable format for symplectic integrators such as the leapfrog method.[7] Such a method is shown to conserve flux[7] and is a crucial modification to the standard FMD method in this study;it is thus a second advantage of the modified approach. From this evolution sequence the Schrödinger equation can be solved obtain any expectation value of the system can be obtained.
The quantity |Ψ(x,t)|^2 gives the probability of finding a particle. Given that a mesh was made between two boundary points (a, b), by integrating across all space with respect to that mesh one would expect the probability to be unity. This basic statement in probability states that if we impose a particle within space, if we scan all of space (the entire mesh) the particle will be found within that space. Otherwise there is a clear contradiction to our initial assumption of imposing a particle within that space. Given the probability distribution P_n^m (Eq 1.4a), expectation values (Eq 1.4b – 1.4f) can be obtained with a weighted average of appropriate observables as
(1.4a) P_n^m (x,t)= |Ψ_n^m |^2 (1.4b)〈x〉= 〈(Ψ_n^m )^* |x_n^m |(Ψ_n^m )〉= 〖∫_a^b dx[(R_Ψn^m )〗^2+(I_Ψn^m )^2] x_n^m (1.4c)〈p〉=
〈(Ψ_n^m )^* |p_n^m |(Ψ_n^m )〉=∫_a^b dx[R_Ψn^m (dI_Ψn^m)/dx- I_Ψn^m (dR_Ψn^m)/dx](1.4d)〈T〉= 〈(Ψ_n^m )^* |(p_n^m )^2/2m |(Ψ_n^m)〉=1/2 ∫_a^b dx[((dR_Ψn^m)/dx)^2+((dI_Ψn^m)/dx)^2 ](1.4e)〈U〉=
〈(Ψ_n^m )^* |U_n^m |(Ψ_n^m )〉= ∫_a^b dx〖[(R_Ψn^m )〗^2+(I_Ψn^m )^2]U_n^m (1.4f) J_n^m= [R_Ψn^m (dI_Ψn^m)/dx- I_Ψn^m (dR_Ψn^m)/dx]
where a and b are the respective left and right boundary points. Equations (1.4b – 1.4f) give the expectation values of position, momentum, kinetic energy, potential energy, and flux, respectively.
Although the algorithm can take any wavefunction and construct any potential, the following analysis will be focusing on the validation of the program. As is standard with partial differential equations such as the TDSE, boundary conditions as well as initial conditions must be specified in order to simulate an event.[6] The domain for the following analysis will be a span from -30 a.u. to 30 a.u. with 1024 grid points; this finds a balance between time efficiency as well as functional accuracy. This study uses an initial Gaussian wave packet with the starting position at -5 a.u.
〖(1.5)Ψ〗_m^0=(σ√π)^(-1/2) e^((-(x-x_0 )^2)/(2σ^2 )+ik_0 x); k_0=2π/λ x_0=Center of Gaussian; σ=width of gaussian
Relationships that contain a recursive relationship such as the FDM are known to suffer from stability problems depending on the set of parameters. Therefore, stability analysis must be performed to find suitable parameters. One such measure is the von Neumann stability analysis of a system.[7] Under the space discretized leapfrog method, the stability criterion yields
where h is the distance separation between two points on a mesh,[7] where λ is a coefficient dependent on the potential being used, and where 0<λ<1. For U=0, λ=1. For other potentials used in this study, the data show that the method is stable if λ=1/2.[7]
Another parameter that must be tested is how fast the initial wave packet can move within a certain time window. Given the mesh used for this analysis, the model begins to break down when k_0> 10. However, given a more refined mesh the algorithm can handle faster initial momentum of the wavefunction. This is an issue of resolution, as when the wavefunction gets too fast, the change in time (Δt) cannot keep up with the mesh size.[7] It can be shown that as the domain becomes more refined, the separation between points directly correlates to a higher resolution.
Results and Discussions
The first potential this study tests the algorithm with is the simple harmonic oscillator (SHO). This potential paints a very intuitive picture of the wave function’s evolution, similar to the classical case as we shall see. For the simple harmonic oscillator, the wave function will start with 0 initial momentum, making k_0= 0.
Fig 1: The integrated probability density (normalization) as a function of time.
Fig 2: The expectation value of position () for the simple harmonic oscillator as a function of time
Fig 3: The expectation value of momentum () for the simple harmonic oscillator starting as a function of time with zero inital given momentum.
Fig 4: The expectation values of kinetic engery (blue), potential energy (orange), and total energy (green) as a function of time.
Fig 5: The flux as a function of a space and time () for the simple harmonic oscillator.
Fig 6: A cross section of Fig 5 at point x = 0, the center of a simple harmonic oscillator, as a function of time.
The above figures were run for a sufficient time (typically 50,000 steps) to analytically extract all expectation values from Equations (1.4a-1.4f). They are for the simple harmonic oscillator with a potential of V(x)= 1/2 x^2. A fundamental statement of quantum mechanics is that if a particle is confined within a given region, the probability of finding the particle in that region is 1. For the SHO this is clearly the case (Fig 1) given the scale is +1 with a maximum deviation from one of 3.5e-5. The respective values for position (Fig 2), momentum (Fig 3), energies (Fig 4), and flux (Fig 5 and Fig 6) also seem to intuitively follow the expectation values of a Gaussian wave packet oscillating back and forth between two maximum potential values. Given that the initial conditions for this system were an initial position of -5 and no initial momentum, the figures follow this condition elegantly. Thus, the initial points of both the position and momentum graphs start at -5 and 0 respectively and follow the features of a typical oscillator.[5]
A slight fluctuation can be seen in the total energy from Fig 4, which should be constant because of energy conservation. The leapfrog method predicts such oscillations, but on a much smaller scale.[9] The leapfrog method conserves energy within a small range so it remains bounded.[8] Upon closer examination, it can be seen in Fig 4 that the amplitude of kinetic energy is less than that of the potential energy by a small amount, causing the total energy to fluctuate. Because the extraction of kinetic energy involves the derivative of the wavefunction (Eq 1.4d), it is more sensitive to inaccuracies in the wavefunction. These fluctuations can be attributed to the grid size, which is not yet sufficiently small. The investigation exhibits the convergence rate of the kinetic energy with respect to the grid size, and whether a more robust method of calculating the kinetic energy is needed.
Probability flux of the simple harmonic oscillator describes the particle flow through a given point. As expected, the surface of the flux (Fig 5) show oscillations between ± 5, which was the initial condition of this oscillator. Had the wavefunction started with some initial velocity (i.e k_0 ≠ 0) the evolution of the flux would begin to broaden before reaching constant oscillation again. When viewed across a cut at x=0 (Fig 6), the flux changes from positive to negative with equal areas under the peaks, consistent with the conservation of probability (Fig 1) in the periodic motion. This shows that our modified approach can reliably calculate the flux which gives rise to experimentally observable quantities such as scattering amplitudes.
The simple harmonic oscillator is a bound system regardless of the energy of the particle. In realistic potentials, particles are not always bound. The same method can also be applied to a different potential, namely the potential barrier. Rather than having a purely real wave function with zero initial momentum, the following wave function contains both real and imaginary initial conditions with a k_0=5.
Fig 7: The flux of an incident wave packet encountering a barrier
with potential 15 (a.u.). The flux is taken at x = 5.
Fig 8: The tranmission coefficient for the same potential as in Fig 7.
Figures 7 and 8 show the flux and transmission coefficient, respectively, for a potential barrier of width 3 and height 15. Given that flux must be taken over a particular region, the region the figures are expressing is when x = 5, directly after the barrier.
A common issue with all numerical solutions in open systems is how to treat boundary conditions when the solution reaches the outermost region of its domain.[7] This current algorithm uses a periodic boundary condition, meaning that once it hits one boundary, it moves instantaneously (wraparound) to the opposite side of the boundary. This is problematic for flux extraction because if left unchecked with a small domain, the wave function will wrap around and interfere with data collection. Towards the end of Fig 7 we can see another spike in the flux; this is due to the “background” of the wave function wrapping around and beginning to interfere with itself. One way to delay the onset of the spike is to make a sufficiently large domain such that the wraparound occurs well after the initial wave has passed. The disadvantage is the increased number of grid points, and hence increased computation time. Instead, we choose to maintain the domain size but to integrate the flux before the artificial spike arrives. This requires careful selection of the time window so it is robust and reproducible.
Fig 8 shows the integrated flux through this barrier. It represents the fraction of the incident wave transmitted through the barrier and is defined as the transmission coefficient. In this case roughly 0.07 or 7% of particles overcome this barrier. Because the height of the barrier (15 a.u.) is considerably larger than the center velocity of the wave packet (5 a.u.), the transmission coefficient is small, as expected. With increasing barrier height or width, the transmission is expected to become smaller because the process will occur mainly through tunneling.[3]
As stated earlier, because the flux and transmission coefficients relate directly to experimentally measurable data, these parameters are currently being calculated for other types of potentials. These potentials, depicted in Fig 9, may have complex shapes and contain multiple discontinuities. They present more stringent test cases of the robustness and accuracy of our modified FMD method.
Fig 9: Shapes of test potentials. Upper left: inverted quadratic well, Upper right: a standard potential well, and Bottom: a piecewise well-barrier potential.
This study has presented a modified finite difference method for solving the time-dependent Schrödinger equation. Two modifications are made: the separation of the real and imaginary parts of the wavefunction and the application of a norm-preserving symplectic integrator. When tested on the simple harmonic oscillator, the method is shown to be stable and can describe the motion accurately. The method preserves the normalization to a high degree of accuracy even with moderate grid sizes. Expectation values and probability flux can be computed efficiently. While the overall performance of the algorithm is high, it is found that the total energy has a slight fluctuation, indicating that energy conservation is not accurately computed. This can be attributed to the sensitivity in the calculation of kinetic energy which relies on the derivative of the wave function. Effort is underway to study more accurate ways of obtaining kinetic energy as well as the effect of grid size.
This study also applied the method to calculate the particle flux and transmission coefficients for a potential barrier. The results show that the transmission coefficient can be directly obtained from the wave function itself. However, boundary conditions can affect the actual value due to factors such as wraparound, so the window of integration must be carefully determined. A future goal is to perform calculations for other nontrivial potentials and compare with exact results where possible.
The ultimate goal is to apply the method to actual systems, for instance, in atomic reactions. This requires extending the modified method to higher (2D and 3D) dimensions. However, it will be challenging to scale the finite difference method to higher dimensions because of efficiency. The plan is to comparatively study the method discussed here with other approaches, including the finite element method and meshfree methods.[4, 11] The latter should have better scalability and be more suitable in higher dimensions. Thus, the modified method can serve as a useful benchmark for further studies.
A. Goldberg, H. M. Schey, and J. L. Schwartz. Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena. Am. J. Phys., 35:177–186, (1967).
C. R. Davis and H. M. Schey. Eigenvalues in quantum mechanics: a computer-generated film. Am. J. Phys., 40:1502–1508, (1972).
D.J. Griffiths and D.F. Schroeter, Introduction to Quantum Mechanics. (Cambridge University Press, Cambridge), 2018.
E. B. Becker, G. F. Carey, and J. T. Oden. Finite elements: an introduction. (Prentice Hall, Englewood Cliffs, NJ), 1981.
G. B. Arfken and H. J. Weber. Mathematical methods for physicists. (Academic Press, New York), 6th edition, 2005
G. R. Fowles and G. L. Cassiday. Analytical mechanics. (Thomson Brooks/Cole, Belmont, CA), 7th edition, 2004.
J. Wang, Computational Modeling and Visualization of Physical Systems with Python (John Wiley & Sons, Hoboken, NJ), 2016.
L. Verlet. Computer ‘experiments’ on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev., 159:98–103, (1967).
M. L. Boas. Mathematical methods in the physical sciences. (Wiley, New York), 3rd edition, 2006.
T. Liszka and J. Orkisz, Computers & Structures 11, 83 (1980).
J. Wang and Y. Hao. Meshfree computation of electrostatics and related boundary value problems. Am. J. Phys., 85:542–549, (2017).
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2abf196677aae61d | Introduction to Doug's RSt
Discussion of Larson Research Center work.
Moderator: dbundy
Posts: 134
Joined: Mon Dec 17, 2012 9:14 pm
Re: Introduction to Doug's RSt
Post by dbundy » Wed Oct 19, 2016 2:10 pm
Now that our RSt has a basis for calculating the discrete levels of energy transitions observed in Hydrogen, we need a scalar motion model of the atom that serves to explain the changes inducing the transistions, as does the vector motion model of LST theory. For right or wrong, whether it's the Bohr model of electron particles, orbiting a nucleus, or the Schrödinger model of electron waves, inhabiting shells around a nucleus, the LST's atomic model provides the LST community with a physical interpretation of the word "transition." However, this is much more difficult to achieve in a scalar motion model.
In our RSt, where the unit of elementary scalar motion from which higher combinations are derived, the S|T unit, is an oscillating volume of space and time, understanding what accounts for the observed atomic energy transitions in the combinations identified as atoms is not easy.
This is especially challenging given that, in our theory, the electron has no identity and consequently no properties, as an electron, until it is created in the disintegration of the atom, or in the process of its ionization. Fortunately, however, we have the scalar motion equation to help.
Recall, that the basic scalar motion equation is,
S|T = 1/2+1/1+2/1 = 4|4 num (natural units of motion),
And our basic energy, or inverse motion equation is,
T|S = 1/2+1/1+2/1 = 4|4 num (natural units of inverse motion),
Now, this terminology and notation will be a complete mystery to those who have not read the previous posts, so reading and understanding those posts first is a prerequisite for the study of what follows, and while we are on the subject, let me emphasize the tentative nature of all the conclusions presented thus far. It may be necessary to eat a lot of humble pie from time to time, during the development of our RSt, for several reasons, but if so, it won't be the first time. It has been said before and bears repeating: It takes courage to develop a physical theory, not to mention a new system of physical theory. Larson was an incredibly courageous man, as well as an intelligent and honest investigator. Perhaps those of us who try to follow his lead appreciate that fact more than most.
With that said, we have taken on the challenge of dealing with units of energy, with dimensions E=t/s, as well as motion, with dimensions v=s/t, and have dared to cross over the line of LST physics, which cannot brook the existence of entities over the speed of light, which are known as "tachyons," in that system. Nevertheless, the T units in our RSt are just such units, but because the dimensions of these units are actually the inverse of less than unit speed units, no known laws of LST physics are broken.
In retrospect, venturing into this unexplored realm of apparent over-unity, which seems so iconoclastic, appears to be the natural and compelling evolution of physical thought. So much so that one marvels that the world had to wait so long for the Columbus-like pioneer Larson to show science the way. However, incredible as it is, the entire LST community, save just a few, has no idea yet that our understanding of the nature of space and time has been revolutionized. They cannot, as yet, recognize that time is the inverse of space, even though it's as plain as the nose on your face, as soon as someone points out how it can be.
By the same token, the mathematics of the new system is just as iconoclastic. As we consider the basic scalar motion equation, n(S|T)=n4|n4, and its inverse, n(T|S)=n4|n4, and graph their simple magnitudes, we find that its also possible to formulate a basic scalar energy equation, where
S*T = n2.
In the previous posts above, I've explained how S|T units combine into entities identified with the observed first family of the LST's standard model (sm), and these combos combine into entities that are identified as protons and neutrons, which combine into elements of the periodic table, or Wheel of Motion:
The symbolic representation of the S|T units that, as preons, combine to form the fermions and bosons of our RSt, are a reflection of the S|T equation, making it possible to graphically represent them and their combinations as protons and neutrons along with their respective magnitudes of natural units of motion. On this basis, the S|T magnitudes for the proton, neutron and electron combos are:
P = 46|46 num,
N = 44|44 num,
E = 18|18 num
The magnitude of the Hydrogen atom (Deuterium isotope) is then the sum of these three:
H = 46+44+18 = 108|108 num.
At this point, however, representing the constituents of the atom as combos of S|T triplets (see previous posts above), becomes cumbersome and we need to condense the symbols from 20 triangles to 4 triangles, in the form of a tetrahedron, as shown below:
The top triangle of the tetrahedron is the odd man out for the nucleons, so that for the proton this is the down quark, but for the neutron it is the up quark.
The numbers in the four triangles are the net magnitudes of the quarks and the electron. So, the number of the down quark at the top of the proton is -1, because the magnitude of the inner term of its S|T equation is 2/1, (-2+1 = -1), whereas the number of its two up quarks is 2, because the inner terms of their S|T equations are both 3/5, (-3+5=2). For the neutron, the number of the single up quark at the top is 2, while the magnitude of the two down quarks below it is -1, given the inner terms of their S|T equations.
The inner term of the electron's S|T equation is 6/3, or -3, (-6+3=-3), so that the net motion of the three entities combined as the Deuterium atom balance out at 3-3 = 0, or neutral in terms of charge, as show in the graphic above.
In this way, each element of the Wheel could be represented by the numbers of its tetrahedron symbol, if there were some need to do so, but what is more useful is the S|T equation itself. Expanding the equation for Deuterium:
D = 27/54+27/27+54/27 = 108|108 num,
but factoring out 33, we get:
D = 33(1/2+1/1+2/1) = 108|108 num.
To take advantage of this factorization we can represent it by making the S|T symbols of the notation bold:
S|T = 33(1/2+1/1+2/1) = 108|108 num
D = S|T= 108; He = 2(S|T) = 216; Li = 3(S|T) = 324, etc.
This way, we can easily write the S|T equation for any element, X, given its atomic number, Z:
XZ = z(S|T)
However, while this should prove to be quite helpful as compact notation, there is still more to consider. Recall that the units on the world-line chart actually represent the expanding/contracting radius of an oscillating volume, and as such, its magnitude is the square root of 3, not 1. This factor expands the relative magnitudes involved considerably, which we will investigate more later on.
For now, I want to draw your attention to the scalar energy equation,
S*T = n2.
Recall that for the S*T unit,
S*T = 1/(n+1) * (n*n) * (n+1)/1 = n2,.
So when n = 1,
S*T = (1/2)*(1*1)*(2/1) = ((1/2)(2/1)(1*1)) = ((2/2)(1*1)) = 1*1 = 12,
but, if we invert the multiplication operation, we get:
S/T = (1/2)/(1/1)/(2/1) = ((1/2)(1/1))(1/2) = (1/2)(1/2) = 1/22,
which we want to do when we wish to view the S&T cycles, in terms of energy, so that:
E = hv ---> T/S = 1/S/T = n2,
where n is the number of cycles in a given S|T unit.
Now, this may seem to be contrived, and perhaps it is, especially when we invert the operation of the inner term of the S*T equation from division (n/n) to multiplication (n*n). However, if we don't invert it, the equation will always equal 1, while if we do invert it, then dividing 'T' cycles, by 'S' cycles (T/S) yields the correct answer, T/S = n2.
Now, this brings us to something else that needs to be clarified. In the chart showing the correlation between the quadratic equation of the S|T units and the line spectra of Hydrogen, given the Rydberg equation, the frequency of the S|T units is shown as decreasing with increasing energy, rather than increasing as it should. This is problematic to say the least, but I think it can be resolved, when we consider that the "direction" reversals of each S and T unit always remains at the 1/2 and 2/1 ratio, even though their combined magnitude is greater in the absolute sense; that is, while the space/time (time/space) ratio of 1/2, 2/4, 3/6, 4/8, ...n/2n, remains constant at n/2n = 1/2, the absolute magnitude of 2n - n increases, as n: 1, 2, 3, 4, ...2n-n.
Therefore, the length of the reversing "direction" arrows, shown in the graphic, as increasing in length, as the absolute magnitude increases, is an incorrect representation of the physical picture. The correct representation would show the number of arrows increasing, as S|T units are combined, with their lengths (i.e.their periods) remaining constant, so that, as the quadratic energy increases, the number of 1/2 periods in a given S|T combo increases. The frequency of the unit then is that of a frequency mixer, containing both the sum and difference frequencies of its constituent S|T units..
Of course, this is not exactly what is observed, but upon further investigation, we may be able to resolve the discrepancy. At least it is consistent with our theoretical development.
In the meantime, for the energy conversion of the S|T equation of the Hydrogen atom, where n = 1, with 33 factored out, we get:
S/T = (1/33n)2 = 1/272, and T/S = (33n)2 = 272 = 729,
when we put the 33 factor back in, so we get the actual number of 1/2 and 2/1 cycles, or S and T units, contained in the Hydrogen atom.
There are a lot of tantalizing clues to follow in the investigation of these equations and their relation to the conventions of the LST particle physics community, which uses units of electron volts for energy, dividing those units by the speed of light to attain units of momentum, and dividing them by the speed of light squared to attain units of mass. They even divide the reduced Planck's constant by eV to attain a unit of time, and that result times c to get the unit of space, according to Wikipedia:
Measurement--Unit---------SI value of unit
Energy----------eV--------------1.602176565(35)×10−19 J
Mass------------eV/c2-----------1.782662×10−36 kg
Momentum-----eV/c ------------5.344286×10−28 kg⋅m/s
Time-------------ħ/eV------------6.582119×10−16 s
Distance---------ħc/eV----------1.97327×10−7 m
It'll take a while to untangle these units and see how they correspond to the S|T units of motion, but in the meantime, we can use the progress achieved so far to analyze the periodic table of elements, showing why Larson's four, 4n2, periods define it, using the LRC model of the atom explained so far.
Posts: 134
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Re: Introduction to Doug's RSt
Post by dbundy » Thu Oct 20, 2016 10:12 am
Hi Jan,
Thanks for the comment. The math is so far over my head that I could never discuss it intelligently. You said,
Following Nehru's approach, the Schrodinger equation was assimilated into RS and spectroscopic calculations are possible right now without changes to the methods used today.
I'm not sure what you are referring to here. Who assimilated the wave equation into the RS? Where are these RS calculations using today's methods? I don't understand.
You also wrote:
to develop RS's own methods to calculate spectroscopy the connection between wave-function and the Larson's triplets has to be understood first. Have you dig into this so far?
The wave-function is based on 3D spherical harmonics, in other words, vector motion on a surface, while scalar motion is not any kind of vector motion. By definition, it is a change in scale, or size. If you are referring to the preon triplets as Larson's triplets in the LRC's RSt, then I would say that there is a remote connection that can be seen in the Lie algebras employed in QM, but, again, the disconnect is fundamental, because of the difference in the definitions of motion.
Using the wave equation, LST physicists are at a loss to understand the nature of quantum spin. By their own admission, they haven't a clue how to account for it, and when it comes to iso-spin, it's even worse. However, the root of their problem is, again, the definition of motion, which manifests itself in the fundamental understanding of the relation between numbers, geometry and physics.
When Larson changed our understanding of the nature of space and time, he changed everything, including our understanding of the nature of numbers, because the same principle of reciprocity that is not recognized as fundamental in LST physics is also not recognized as fundamental in LST algebra (meaning numbers). Thanks to Larson, we can now see the connection between 3D geometry and 3D numbers, and as Raul Bott so famously proved, Larson's postulate, that there are no physical phenomena beyond the third dimension, is established.
What this means is that the use of the modern algebra, while able to weave sophisticated and intricate, dare I say baroque, edifices out of fundamental concepts like magnitudes, dimensions and directions, cannot help us to get where we need to get because of errors of definition.
The clearest example I think I can point to is the vain attempt to use octonions in the string theory of QM. John Baez co-authored a 2011 article in Scientific American, entitled, The Strangest Numbers in String Theory, with the subtitle: "A forgotten number system invented in the 19th century may provide the simplest explanation for why our universe could have 10 dimensions."
He is referring to the attempt to use 7 imaginary numbers, with a real number, 8 numbers in all, to invent a 3D number system. The problem is, of course, the use of even 1 imaginary number, to compensate for the lack of understanding that Larson provides us, has taken us down the wrong path in understanding magnitudes, dimensions and directions. To be sure, it has been very fruitful, and, together with the concept called "real" numbers, has lead to Western society's undreamed of advances in technology.
Nevertheless, it has also led to the continuous/discrete impasse now plaguing the LST community, which string theory's attempt to go beyond three dimensions, was hoped to solve. However, nothing but a correction in the fundamental understanding of motion and numbers can do that. A non-pathological 1, 2 and 3D algebra is possible, but only if they are based on the correct understanding of points, lines, areas and volumes, generated by scalar motion over time (space).
The reason Baez et. al. think octonions are needed, is because the vector space of the Lie algebra associated with the 3D Lie group runs out of dimensions to use, after two dimensions. In other words, they can't use complex numbers in the SU(3) group, because it takes two dimensions for one complex number, (I'm ignoring the non-geometric meaning of "dimension," used by mathematicians.)
Now, Larson's reciprocal system, when applied to numbers, opens up a whole new world, where the three (four counting zero) dimensions of physical magnitudes, in two "directions" replace the foundation of modern vector algebra, based on imaginary numbers, with a scalar algebra, based on real (i.e. integer) numbers, which correspond to geometric points, lines, areas and volumes, generated over time (space).
These algebras do not lose the vital properties of 0D algebra, as they increase in dimension, because the dimensions of the unit itself change, going from 0 to 1 to 2 to 3 dimensions, in a completely different manner than the vector algebra does, which goes from real (0D), to complex (1D), to quaternions (2D), to octonions (3D), via imaginary numbers, losing the properties of algebra in the process.
In other words, unlike the vector algebras, our higher-dimensional scalar algebras are each as ordered, commutative and associative as our 0D scalar algebra. This is a huge change in the foundation of the mathematics employed in the two systems of theory.
Of course, that doesn't mean that we can't employ the LST algebras and calculus to advantage in the RST, to give us insight into the physics of vector motion that we can use in the development of the physics of scalar motion, but it means that we always have to understand the difference.
Posts: 134
Joined: Mon Dec 17, 2012 9:14 pm
Re: Introduction to Doug's RSt
Post by dbundy » Mon Nov 07, 2016 8:35 am
One of the LST challenges that Larson's RSt cannot meet is the diminishing size of the atom from the beginning of an energy level to the end, where it terminates in the noble element, So, not only does a viable physical theory have to be able to account for the line spectra of the elements, which organizes the 117 elements into the 4n2 periods we find, but it also has to account for this shrinkage in the diameter of the atom, as it gets more and more massive, within each period.
The LST theory accounts for it, by asserting that the orbiting electrons are pulled in towards the nucleus, as its mass increases, which seems very reasonable. However, there is no atomic nucleus in RST-based atoms, and also no orbiting electrons to pull closer in.
In the atom of Larson's RSt, there are two magnetic rotations and one electrical rotation (m1-m2-e (my notation, not his), and since the electrical rotation is one-dimensional, it takes n2 electrical rotations to equal one, two-dimensional m2 magnetic unit.
This structures the elements into the four periods quite nicely, without resorting to the spectral data at all. And, as it turns out, the LST's QM theory of spectral lines gets the number of elements in the periods wrong, as shown by Le Cornec's distribution of atomic ionization potentials.
The good news is that the LRC's RSt does appear to account for the decreasing size of the atom, as its mass increases in each period, even as we hope to unlock the mystery of the actomic spectra, as well, but more on the mass and size issue later.
At this point, the immediate challenge is how to account for the atomic spectra, given the scalar motion model that has no nucleus and no cloud of electrons surrounding it. In the LRC's RSt, the scalar motion model of the atom consists of combinations of motion called S|T units, which act as preons to the standard model (sm) of observed particles, namely the fermions, classified as quarks and leptons, and the bosons, as discussed previously in the above posts.
With the equation of motion, we can write the total motion of the Hydrogen atom as,
S|T = 27/54+27/27+54/27 = 108|108 num,
where the proton & neutron contribution of total motion is,
S|T = 21/42+21/24+48/24 = 90|90,
and the contribution of the electron is,
S|T = 6/12+6/3+6/3 = 18|18.
Assuming an electron absorbs a photon of lowest energy, we get:
e + γ = (6/12+6/3+6/3) + (3/6+3/3+6/3) = 9/18+9/6+12/6 = 30|30,
which is only 12 num higher than the electron, but the middle term, 9/6, is now 6-9 = -3. So, while the relative motion imbalance (the electrical charge) of the excited electron (as part of the atom) is unchanged, the scalar energy has increased by 3, a quantum jump in the energy (9x6 = 54, divided by 6x3 =18, given our scalar energy equation discussed above).
This is, in effect, equivalent to the quantum energy transition of the LST's Bohr model, where the ground state electron is excited to the next higher orbit, at least qualitatively speaking. It remains to be understood quantitatively, at this point, but it looks promising.
Nevertheless, our earlier analysis of the energy transitions were based on the sequence of single S|T units (12, 22, 32, ... n2), which worked out quite well, but here, the sequence has to be based on the assumption of basic boson triplets, as the unit, where the sequence is:
3, 6, 9, ...3n
so the photon motion equation is actually:
S|T = 3(1/2+1/1+2/1) = 3/6+3/3+6/3 = 12|12 num.
Hence, the magnitude of n quantum transitions is,
e + nγ = (6/12+6/3+6/3) + n(3/6+3/3+6/3),
So for n =
0, the middle term of e + nγ = 6/3 = -3, and energy = 6x3 = 18
1, the middle term of e + nγ = 9/6 = -3, and energy = 9x6 = 54
2, the middle term of e + nγ = 12/9 = -3, and energy = 12/9 = 108
3, the middle term of e + nγ = 15/12 = -3, and energy = 15x12 = 180
4, the middle term of e + nγ = 18/15 = -3, and energy = 18x15 = 270
in natural units of energy, we might say,
One would expect that this would lead to a very simple, Hydrogen-like explanation of the line spectra of the elements, but, of course, this is far from the case. In fact, the LST community's much hyped solution, using Schrödinger's wave equation, works only in principle, since they can't use the separation of variables technique to solve the equation analytically.
However, according to the work of amateur investigator Franklin Hu, "If you create a simple graph of the line
frequencies and intensities for Helium, a striking and predictable pattern appears which suggests that the spectra and the intensity can be calculated using simple formulas based on the Rydberg formula. This pattern also appears in lithium and beryllium."
Unfortunately, Hu lacks a suitable physical theory to explain the pattern, but we hope to supply that part.
Stay tuned for more developments coming soon.
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Joined: Thu Jan 17, 2013 1:36 am
Re: Introduction to Doug's RSt
Post by rossum » Tue Nov 08, 2016 2:59 am
Hello Doug,
thanks for answer and sorry for a my late answer, I have been extremely busy these days.
You wrote:
In my post I was referring to K.V.K. Nehru's article “Quantum Mechanics” As the Mechanics of the time region. In this article he says:
...[h]ence the Schrödinger equations can be admitted as legitimate governing
principles for arriving at the possible wave functions of an hypothetical particle of mass m traversing
the time region, with or without potential energy functions as the case may be.
Nehru's argumentation for the Schrodinger eq. in RS seems to me pretty solid, so I took it for generally accepted among RS community, though I'm aware I may be wrong at this point.
In the same article he proposed several potentials, namely
so it is (technically) possible to solve the Schrodinger equation for this potential and get both the spectral lines (as energy differences of the solutions) and their intensities (overlaps between the solutions). Although this potential removes the need for renormalization it is unfortunately incorrect: I analysed it and the solutions give incorrect energies for any combination of coefficients. The potential curve has simply a wrong shape. As a result e.g. chemical bonds would not break under high temperature etc.
What I really meant was that anyone can now calculate the spectra (even for molecules) using this Nehru's approach, but probably nobody except me tried.
You also wrote:
I don't quite understand: I would say, that on the contrary, nature of the spin is to certain point quite well understood: the best way to see the nature of the spin in modern physics is to use the Foldy-Wouthuysen's transformation of the Dirac Hamiltonian. In this way one gets a number of terms in which the magnetic field (intensity) generated by an electron is a result of the wavy nature of the "particle". The same is true for the composite particles like neutrons as quarks have charge (naively said). It all gets obscured only when relativity comes into play: people want to attribute the Lorentz invariance to the relativity instead of the nature of waves. (Waves in general are Lorentz invariant using their speed - a fact people tend to ignore or don't know). In short it is not possible to have "moving charge" or general electromagnetic wave without the magnetic component i.e. spin.
To the rest of your post: I have many comments but they are quite long for posting - maybe a skype talk would be more appropriate some day. One however I need to mention here is that string theory is definitely not a main-stream theory and it is far from being accepted. I think the whole theory is fundamentally flawed. For now the main-stream is quantum field theory (kind of opposite to the string theory).
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Joined: Mon Dec 17, 2012 9:14 pm
Re: Introduction to Doug's RSt
Post by dbundy » Tue Nov 15, 2016 10:45 am
Hi rossum.
Thanks for getting back to me on this. I knew you were referring to K.V. K.'s article, but he was never able to make any progress along the lines he suggested, but, as you write:
This is news to me. Have you posted the calculations somewhere? If so, please point me to them.
As far as the nature and origin of quantum spin goes, the LST community doesn't have a clue. All they know is that it is a conundrum yet to be solved. How can a particle with no spatial extent possess "intrinsic angular momentum?" It can't, but even if it could, the need to rotate its spin axis through 720 degrees to return the particle to its original state, is a complete mystery.
In the LRC's RSt, however, there is no spin axis, as the 3D oscillation is not a spherical wave, but a pulsation of volume, if you will, and the 720 degree cycle is easily explained.
But, again, the important thing for us to understand is how Larson has revolutionized the nature of space and time and the phenomenon of motion. Until we recognize that a repetitive change in scale constitutes motion and follow the consequences, the science of theoretical physics will ever be bound to the science of vector motion and mathematics will forever be hampered by imaginary numbers.
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Re: Introduction to Doug's RSt
Post by rossum » Thu Nov 24, 2016 10:28 am
Hello Doug,
dbundy wrote: This is news to me. Have you posted the calculations somewhere? If so, please point me to them.
No I didn't, but I can show you the problem here. On the following image you can see arbitrarily scaled Nehru's potential, the classical electrostatic potential (this one is usually used to calculate the hydrogen atom energies) and harmonic potential (usually used as an approximation in molecular dynamics etc.).
pot.png (6 KiB) Viewed 5497 times
As Nehru didn't write how to calculate the coefficients in his potential, but if there are some correct coefficients, it should be at least possible to fit them to experimental data. For this we could use the first and the last line in the Balmer series from the experiment, just to check the feasibility of this potential. However the lines in the Nehru's potential do not converge to certain value but rise to infinity instead. Note that in harmonic potential (i.e. x^2) the solutions are spaced equally whereas in classical potential (i.e. frac{1}{x}) converge to some value. Below are images graphically show the energies for these potentials.
Harmonic potential levels:
harmonic_levels.jpeg (26.48 KiB) Viewed 5497 times
Classical electrostatic potential levels:
levels.png (40.61 KiB) Viewed 5497 times
From the images above it should be obvious that the levels in the Nehru's potential will be closer to the harmonic potential than the classical potential and will diverge instead of converging. I even used an online differential equation solver to get the eigenfunctions in analytical form, but after seeing the result I didn't bother to calculate the eigenvalues (relative energies in atomic units). Here is the input and output from the
npot_diff.gif (1.46 KiB) Viewed 5497 times
npot_s.gif (4.48 KiB) Viewed 5497 times
Where U(a,b) is the Kummer's function of the second kind and L_n^m(a) are Laguerre function. Now the solution in the classical potential is N_{nl}e^{-\frac{\rho}{2} \rho^lL_{n-l-1}^{2l+1}(\rho)} whre L_{n-l-1}^{2l+1}(\rho)} is Laguerre polynomial \rho=r\frac{2}{na_l} and N_nl=\frac{2}{n^2}\sqrt{\frac{n-l-1}{[(n+l)!]^3}}. If we cut away everything we possibly could by adjusting constants and focus only on the "main quantum number" (i.e. set the Laguerre polynomial/function to be a constant) we see that in case of the classical potential the energy increases from a negative value to 0 but in case of Nehru's potential rises to infinity very similarly to the harmonic potential.
Although I did't calculate the energies for the hydrogen atom using Nehru's potential I was able to analyse the possible solutions and show that they must have in any case a wrong tendency. However this is relevant only to the part which was proposed to the 'electronic' part. I didn't analyse the rest of the potentials so they may or may not be correct.
Posts: 134
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Re: Introduction to Doug's RSt
Post by dbundy » Wed Nov 30, 2016 8:46 am
Thanks for the detailed explanation, rossum. I've been out of pocket for weeks now and don't know when I'll be able to reply, but will try to as soon as possible.
Posts: 134
Joined: Mon Dec 17, 2012 9:14 pm
Our RSt Goes Cosmic
Post by dbundy » Wed Feb 01, 2017 7:28 am
I finally have more time to devote to this discussion, and explanation of the LRC's RSt. The development of the topic to this point has been focused on the material sector of the theoretical universe, leading to rossum's post on the utility of applying the LST community's equations to obtain the atomic spectra.
As he indicates in the above post, this is a problematic approach, to say the least. We would like to find a scalar motion based solution, using the new scalar math, and I think we will eventually, as I have made some progress that is encouraging, along that line.
The trouble is, however, it's been months since I've been able to give it any attention and in the meantime, to my delight, a presentation of the work of Randell Mills and his energy generator, based on his Hydrino theory, which has a very interesting energy spectra, has been taken to the road. He first presented it in Washington, D.C., then in London, and this month, he will present it in California. It's very impressive (see:
I corresponded with Randell a year or so ago, discussing his hydrino theory somewhat, proffering some insight into the phenomenon in RST terms, but, of course he was not interested and I can certainly understand why. However, his theory invokes a new model of the electron, using the same spherical harmonics as QM does to derive the atomic orbitals we've discussed above, but he does so by turning the orbitals into electrons!
That's right. In his theory, the electron is treated as a two-dimensional surface of a sphere, surrounding the nucleus, called an "orbitsphere," with charges and currents flowing upon the surface, according to spherical wave equations. Fortunately, this approach eliminates the acceleration problem of orbiting electrons, as did the Bohr model, but without the position vs. momentum issue arising from the dual wave/particle concept of the electron, inherent in quantum mechanics.
This enables Mills to use the classical laws of Newton and the equations of Maxwell to solve the quantum mechanical experiments such as the dual slit phenomena giving rise to the particle/wave enigma. However, Mills' triumph of classical vs quantum mechanics is based on a photon and electron momentum interaction, which is necessarily problematic for "point-like" entities that actually do (must) have extent of some kind, and consequently cannot carry "charge" without flying apart, unless something ad hoc, like "Poincaré stresses," are postulated to hold them together.
It is the enigma of how elementary particles can exist, which have no extent (radius = 0), but yet can have something called "quantum spin" and angular momentum, which is the most fundamental mystery plaguing the LST community. If these "particles" can't exist in the theory, then arguing how their properties interact to produce observed phenomena is a little like putting the cart before the horse.
Nevertheless, the electron in Mills' theory, not only has extent, it changes form from a two-dimensional surface of a sphere, in its bound form, to a two-dimensional disk in its free form. Electron spin is then conceptualized as a disk flipping like a tossed coin.
The Hydrino theory enables Mills to postulate that many inverse excited states of 1/n exist for electrons in the atom, in addition to the known n excited states, found in the spectroscopy field. However, since these new states are viewed as fractional, rather than inverse states, the LST community rejects the idea categorically.
Of course, inverse states are readily acceptable to the RST community, and the experimental/engineering evidence indicating that they are real, is a highly motivating factor in our research.
The first thought is that the anti-Hydrogen atom of our RSt accounts for Mills' results. Recall that the standard model-like chart of S|T unit combos, includes the anti-particles (read "inverse particles") of the leptons and quarks, which combine to form the proton, neutron and electron, making up the Hydrogen atom.
Accordingly, these inverse versions of the the leptons and quarks combine to form inverse protons, neutrons and electrons, making up the inverse Hydrogen atom, as shown below:
In this image I just underlined, instead of overlined the particle labels to indicate the inverse nature of the particle, but, as can easily be seen, the inverse Hydrogen atom is formed from the inverse-quarks and the inverse-electron (positron) particles. Thus, the excited states of the positron in the inverse-Hydrogen atom would conform to the same calculations as shown for the Hydrogen atom, but in the inverse "direction."
We will follow the RST community's convention and designate these inverse entities as c (for cosmic sector) entities. We will also need to transform the "S|T Periodic Magnitudes" chart, used to graph the S|T combinations for Hydrogen excited states, into its c version, the "T|S Periodic Magnitudes" chart, as shown below:
This operation reverses the "direction" of the unit progression's plot, in order to show that the space and time oscillations of the S (red) units and the T (blue) units are progressing inversely.
While this graphical representation of the transformed ms S and T units into the cs S and T units is straightforward enough, it's not true for the equivalent transformation of the mathematics. Because our conventional mathematics treats inverse integers as fractions of a positive whole, confusion results when we try to use it for calculations in the cs.
Consequently, we have to modify the conventional mathematics once again, if we want consistent results. For instance, normally, if we want to express the difference between two inverse integers, such as 1/22 - 1/12, the result would be .25 - 1 = -.75 on our calculators. While this is the correct answer, given the fractional view of inverse integers, it's unsuitable for our purpose, where the correct answer is the inverse of 4-1 = 3 and that inverse is not a fraction of a positive unit, but three whole inverse units.
As shown in the graphic above, we will modify the conventional mathematics by coloring negative integers red and dispensing with the denominator above the numerator notation, just as is common practice for conventional, non-inverse, positive integers, where the denominator below the numerator is omitted.
On this basis, 4-1 = 3 is equivalent to (1/4) - (1/1) = 1/3 = -3 inverse units, not one third of 1 non-inverse unit. To be consistent, we could also color positive numbers blue and limit the plus and minus signs to indicating addition and subtraction operations only, but we won't normally do that, if the meaning is clear without it.
So, with this much understood, we can see that our RSt works out as nicely for the c sector as it does for the m sector, except there is a one obvious difference: The T|S units are superluminal; that is, they constitute the dreaded "tachyons" of the LST community, which are usually fatal for their theories.
However, they are an integral part of RST-based theories, like ours, because the fundamental definition of motion, as an inverse relation between space and time, defined in a universal space and time expansion, changes everything. Scalar motion can be motion in time as well as space, but it's not the motion of things, or vector motion.
In the case of our theory, i.e. the LRC's RSt, all physical entities consist of units of both space and time motion in combination. The difference between the S|T units of the ms and the T|S units of the cs is reflected in the number line of mathematics, as we have been discussing it: On one side of the unit datum, space/time ratios are the inverse of the other side, time/space ratios, and magnitudes increase in the opposite "direction." However this can be misleading, when we don't realize the reciprocal nature of ratios, and its effect on our perspective.
The bottom line is that increasing from 0 to 1/1 (light speed) in magnitude, in the ms, is no different than increasing from 0 to 1/1 in magnitude, in the cs, except for "direction." The increase from 0 to unit speed in the ms is in the "direction" of decrease from unit speed, in the cs and vice-versa:
0 --> 1/1 <-- 0
But when we don't understand this, the two, inverse, units of magnitude appear to us as two units of increasing magnitude:
0 --> 1/1 --> 2/1
and this leads us to mistakenly conclude that no superluminal (i.e. > 1/1) velocities are possible. They are possible all right, but only when the ratio of space and time is inverted, as preposterous as that may sound to the uniformed.
However, because Larson's new Reciprocal System of Physical Theory unveils the mystery of the true nature of space and time, it enables us to understand the exciting, life-changing phenomena Mills calls Hydrinos, where the 1/n2 magnitudes of atomic spectra become n2 magnitudes, with earth-shaking consequences.
Too bad we weren't fast enough to predict it, let alone produce it.
(More on this later)
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Our RSt Goes Cosmic
Post by dbundy » Thu Feb 02, 2017 11:03 am
It's really a shame that Randell Mills wasn't a student of Larson. He has developed a new general theory of physics under the LST, the "Grand Unified Theory of Classical Physics (GUT-CP)," which he claims, as the name implies, unifies the "forces" of physics, and even postulates a fifth "force."
However, as Larson insisted, the LST scientists ignore the fact that the definition of force eliminates the possibility of so-called autonomous forces, such as appear in the legacy system. Force, by definition, is a quantity of acceleration, and acceleration is a time (space) rate of change in the magnitude of motion.
This is a critical point to understand, but one which cannot be acknowledged, without destroying the foundation of the LST, the research program of which is to identify the fewest number of interactions (forces) among the fewest number of particles, which constitute physical reality.
This is typical of the challenges the LST community faces. They recognize that they are "stuck," and that they need a revolution in their understanding of the nature of space and time, but unless they can see what is hidden in plain sight, that time is the reciprocal of space, the definition of motion, and that physical reality in nothing but motion, they will never get "unstuck." It's a classic example of "you can't get there from here," type of crisis. They've painted themselves into a theoretical corner.
But the pitifully few current followers of Larson's work do not have the wherewithal of understanding or physical resources to conceive of, and conduct, a crucial experiment that would convince the LST community, in part or in whole, that space/time reciprocity is the key to understanding fundamental physical reality. It's the human nature of things, I suppose, but in the meantime, LST scientists like Mills come along and do remarkable things with the old system of theory.
Recall that in introducing the LRC's RSt, we hearkened back to Balmer and Rydberg to show the mysterious role of the number 4 in their breakthrough discoveries, and how that same number is fundamental in our RSt (S|T = 4|4).
But now, the insight it gives us into the Rydberg equation, goes to the heart of Mills work as well, because the empirically derived constant of Balmer's, that started it all, as re-configured by Rydberg, in his formula for the atomic spectra of Hydrogen, holds for the Hydrinos, too, but in inverted form.
The Rydberg formula,
1/λ = R(1/n12 - 1/n22),
which was inverted for convenience, can be re-inverted to give us a formula for the cosmic sector,
λ = 1/R(n12 - n22).
This is why the formula given for calculating the energy of Hydrino reactions is just 13.6 Ev times the difference between the 1/n2 "fractional" levels of the Hydrino. This so-called Rydberg energy is just the ionization energy of Hydrogen, the wavelength of which is the inverse of the R term in the formula, the Rydberg constant, which he obtained by dividing Balmer's constant by the number four.
But instead of explaining it this way, physicists write it as,
forcing us poor dummies to dig out the meat for ourselves, provided we can deal with their intimidation! Grrrr!
There is much more to say about this, but I want to stick to our proposition that the fractional values of n, in Mills' Hydrino theory, are actually the n values of inverse, or c-Hydrogen, as explained in the previous post. Of course, neither the point-like electron of the Bohr model, nor Mills' orbitsphere modification of it in his work, both of which are based on the vector motion of the LST community's theories, can be used in our work, but, at the same time, the scalar motion model of our RSt has to accommodate the experimental results of Mills' work.
In his SunCell, the Hydrinos are formed when atomic Hydrogen transfers energy to a catalyst, in what are called "resonant collisions." In this rare instance of atomic collision, the two atoms momentarily orbit one another, exchanging energy harmonically, like the two rods of a tuning fork. This extracts the energy from the Hydrino, shrinking the size of the orbit of its orbitsphere, in quantum decrements of 13.6 Ev times n2.
Of course, because the LST has no inverse cosmic sector, and because they have no notion of scalar motion, the vector motion magnitudes of their system are limited to c speed, and the velocity of the orbitsphere, surrounding the nucleus, is thus limited to c speed.
The interesting aspect of this situation, however, is that the size of the Hydrino is consequently reduced to a small fraction of stable (i.e. n = 1 or unit) Hydrogen, reaching a limit equal to 1/137 = α. Moreover, though I don't understand the logic behind it yet, theoretically, the nucleus of the Hydrino is transformed into a positron at this point!
This is very interesting for us, because, for the c-Hydrogen to form in our model, a transformation from m-particles to c-particles has to occur, and I have no idea how that could happen, but maybe there is a clue waiting for us in Mills' GUT.
(stay tuned)
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Re: Introduction to Doug's RSt
Post by Sun » Tue Jun 27, 2017 6:50 am
Hello Doug,
Thank you for your presentation.
Let me use a notation of a-c-b for my own convenient to represent your equation.
Am i correct that you assume everything starts from one net displacement, 1/2 and 2/1? Particles are consequences that combine variable numbers of 1/2 and 2/1 with variable numbers of 1/1? 1/1 represent for unit motion? a, c, b stand for the each dimension of motion?
How did you get 2S|T = 2/4 + 2/1 + 2/1? Why it is not S|T+SUDR=1/2+1/1+2/1+2/1?
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70edae3cc21259a1 | SUSU Scientist Improves Magnets for ‘Green’ Electric Motors
Magnets have been known to humanity since long ago. In Ancient Greece the ability of some rocks to draw chunks of iron had already been discovered. Contemporary scientists found out that not only materials on the base of iron (ferrites) possess magnetic properties. Nowadays among the most powerful magnets are samarium-cobalt (or samarium) magnets. Their principal feature is high temperature resistance and preserving of their initial magnetization intensity at temperatures up to 350°С.
Associate Professor from the Department of Computer Simulation and Nanotechnology of the SUSU Institute of Natural Sciences, Andrey Sobolev (Candidate of Physical and Mathematical Sciences) together with his colleagues from the M.N. Mikheev Institute of Metal Physics of the Ural Branch of Russian Academy of Sciences (Yekaterinburg) is conducting a research of samarium magnets. His article entitled “Simulation of structure and study of the influence of Cu-doped atoms on magnetic properties of alloys of Sm-Co system for high-temperature permanent magnets” allowed Andrey Sobolev to become one of the winners of the Beginning of Big Science contest which was held by SUSU within the frameworks of Project 5-100.
Андрей Соболев, доцент кафедры «Компьютерное моделирование и нанотехнологии» Института естественных и точных наук
Photo: Andrey Sobolev, Associate Professor of the Department of Computer Simulation and Nanotechnology of the Institute of Natural Sciences
Why does a magnet need constancy?
Magnet is a body which has its own magnetic field. Electron can be considered to be the most primitive and smallest magnet. Magnetic properties of all other magnets are determined by magnetic moments of electrons inside them. Permanent magnet is a product made of ferromagnetic material which is capable of preserving the residual magnetization after switching off the outer magnetic field.
“Nowadays permanent magnets find efficient use in many spheres of human life. Permanent magnets can be discovered in practically every apartment in various electronic and mechanical devices. They are used in medicine equipment and measurement apparatus, in various tools and in automobile industry, in direct current motors, acoustic systems, household electrical devices and in many other fields: radio engineering, instrumentation engineering, automation, telemechanics, etc. – neither of these fields can’t manage without the use of permanent magnets,” exemplifies Andrey Sobolev.
Magnetic properties are only specific to ferromagnetis, i.e. the substances which (at the temperature lower than the Curie point) are able to possess magnetization at the absence of the outer magnetic field. There are only three ferromagnetics in the Mendeleev’s table: ferrum, cobalt and nickel. Yet separately they possess quite poor magnetic properties. These properties increase if a ferromagnetic gets combined with one of rare-earth elements (for example, samarium or neodymium).
Вибрационный магнитометр
Photo: Vibration magnetometer
Samarium will solve the problem of import substitution
In contemporary production, the most popular are neodymium magnets. The originated in 1982 three-component alloy of neodymium, ferrum and boron (NeFeB) possesses an exceptional residual magnetization. Despite its small size, neodymium magnet generates a strong magnetic field; its coercitive force is very high, i.e. the magnet is resistant to demagnetizing. Self-cost of neodymium magnets is lower than of samarium-cobalt magnets, which predetermined their wide distribution in various fields of electric engineering, medicine, technology and industry.
“The problem is, the majority of neodymium mines are located in China. This country practically totally control the manufacture of ore possessing rare-earth metals. Though in Russia we have deposits of samarium. Therefore the use of samarium magnets can become a good strategy for import substitution,” assures the researcher.
Magnets with neodymium have one more sufficient disadvantage: when heated, they lose their magnetic properties. Compared to neodymium magnets, samarium magnets are capable of resisting higher temperatures. The maximum operating temperature of magnets made of samarium-cobalt alloy lies in the range from 250 to 350 Celsius degrees. Besides, samarium magnets compared to neodymium magnets are more resistant to corrosion and usually don’t require special coating. Thanks to the special corrosion resistance, precisely samarium magnets are used in strategic studies and military applications.
“Now there is a tendency to use electronic transport, electronic automobiles, electronic motors, wind-powered generators, i.e. the so-called ‘green’ energy. In such generators, temperatures are quite high and therefore neodymium magnets don’t work there because they lose their properties. In such cases, only samarium magnets can be used, as the samarium-cobalt alloy is perfectly fit for aggressive mediums and difficult exploitation conditions,” explains Andrey Sobolev.
Photo: Diffractometer
Research of samarium magnets by the Ural’s scientists
The objective of scientists of South Ural State University and the Institute of Metal Physics is in understanding the mechanism of permanent magnets. This will allow for improving their structure and properties. This requires special experiments which will be able to demonstrate what exactly should be done with these magnets to obtain the best magnetization. And even though such experiments were carried out multiple times, still experimental chemistry can’t answer the question of ‘how?’ i.e. explain the mechanism. For that, computer simulation comes to experimentalists’ aid.
“Computer simulation of materials with the use of the Theory of functional of electron density proposed by American scientists Hohenberg and Kohn is actively developing at the present time. This theory allows solving the system of Schrödinger equations which describes the behavior of separate electrons and describes the entire space in a whole. The colleagues from the Institute of Metal Physics conduct experimental part of the research; and our task is to determine magnets’ mechanism by computer simulation methods,” explains the SUSU scientist.
The method, developed by the team of scientists from the Ural universities, has already been tested on allied structures where cobalt was replaced with yttrium. The next stage will be replacing yttrium with samarium which is a heavier element. In the future, it is also planned to add copper in order to obtain better magnetization. This will allow obtaining more reliable magnets for industry and transport. In the upcoming year, results of the research are planned to be published in a high-ranked scientific journal entitled “Journal of magnetism and magnetic materials” included in Scopus Top 25%.
Olga Romanovskaya; photo by: Viktoria Matveychuk, collection of A.N. Sobolev, PressFoto (YayMicro/@delta_art)
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f8c0e0576cc4ceee | Most Important Scientific Discoveries – Chronological
I collected more than 17 lists of the greatest or most important scientific discoveries of all time and combined them into one list – here are the results, arranged chronologically into a timeline. You may notice there is some overlap with the Best Inventions lists as the line between ‘invention’ and ‘discovery’ is often a blurry one. I have provided some information on the nature of the discovery and the identities of the discovers. As with inventions, the discovery is often one link in a chain of scientific work that extends before and after the discovery in time, or is a collaboration (sometimes rivalry) among multiple discoverers. Also, for some reason, history sometimes identifies the discoverer as the person who first hypothesized the correct answer to a question, while in other cases, the credit goes to the person who confirmed the hypothesis by experiments or observations. I have decided to omit visual images for this list, although I may change my mind later. Where the narrative for one discovery mentions another discovery, I have placed it in boldface. This list includes every discovery on two or more of the 17+ lists. For a list of organized by rank (based on how many lists the discovery was on), go here.
33,000 BCE
Humans domesticated dogs from wild wolves over 30,000 years ago. The oldest remains of a dog associated with humans was found in a Siberian cave in Russia and dates to 33,000 BCE. A dog skeleton at a human site in Belgium dates to 29,7000 BCE. Archaeologists in the Czech Republic found a dog buried with a mammoth bone in its mouth that dates to 25,000-24,000 BCE. Joint burials of humans and their dogs have been found in Germany, dating to 15,000-14,000 BCE and Israel, from 12,000 BCE. The next animals to be domesticated were those associated with the development of agriculture: sheep in southwest Asia (11,000-9000 BCE); goats in Iran (11,000-8000 BCE); pigs in the Near East, China and Germany (9000-7000 BCE); cats in Cyprus and the Near East (8500-7500 BCE); cattle in India, the Middle East and North Africa (8000-7000 BCE); chickens in India and southeast Asia (6000 BCE); horses in the Eurasian steppes (4000-3000 BCE); llamas and alpacas in South America (3000 BCE); asses (3000 BCE); silk moths (3000 BCE); elephants in India (2000 BCE); red jungle fowl in India and Asia (2000 BCE); camels in Arabia (1500 BCE); and rabbits in Ancient Rome (100 BCE).
11,000 BCE
After humans spent many millennia as hunter-gatherers, they developed agriculture by humans gradually, at different times in different places. Evidence of humans exerting some control over wild grain is found in Israel in 20,000 BCE. There is evidence of planned cultivation and trait selection of rye at a Syrian site dating to 11,000 BCE. Lentils, vetch, pistachios and almonds found in Franchthi Cave in Greece may be evidence of early agriculture or a widespread trade in foodstuffs. The eight founder crops of agriculture (emmer wheat, einkorn wheat, barley, peas, lentils, bitter vetch, chickpeas and flax) were domesticated some time after 9500 BCE at various sites in the Levant (Syria, Lebanon, Palestine, Jordan, Cyprus and part of Turkey). The oldest known agricultural settlement is in Cyprus, dating from 9100-8600 BCE. Seedless figs are known from the Jordan Valley dating to 9300 BCE. Rice and millet were domesticated in China by 8000 BCE as was squash in Mexico. Farming was fully established along the Nile River by 8000 BCE after 2000 years of slow development beginning in about 10,000 BCE) and in Mesopotamia by 7000 BCE. The first evidence of agriculture in the Indus Valley dates from 7000-6000 BCE and in the Iberian peninsula from 6000-4500 BCE. Stone walls in Ireland dating to 5500 BCE are the earliest evidence of field systems. By 5500 BCE, the Sumerians had developed large-scale intensive cultivation of land, mono-cropping, organized irrigation and use of a specialized labor force. By 5000 BCE, humans in Africa’s Sahel region had domesticated rice and sorghum. Maize and cotton were domesticated in Mesoamerica between 3000 and 2700 BCE.
8700 BCE
The first metal that humans began to use and work was copper and the oldest evidence of metalworking is a copper pendant dating to 8700 BCE, found in northern Iraq. The earliest metalworking, referred to as ‘cold-work’, involved no heat. The earliest evidence that metalworkers were using heat to ease the production process dates from 8000 BCE in what is now eastern Turkey. Residents of Mehrgarh in what is now Pakistan worked copper beginning in 7000 BCE. The technique of smelting arose in places such as southeastern Iran and eastern Serbia, and by 6000 BCE, copper smelting was common in the Middle East. Copper working in North America is known from as early as 5000-4000 BCE in Wisconsin. The earliest gold artifacts date from 4450 BCE in Bulgaria. Mixing copper with other metals to create bronze first began about 2700 BCE.
7000 BCE
In 2009, scientists announced that they had found the oldest known woven cloth: twisted fibers of flax, some of them dyed, found in a cave in Georgia and dating to 36,000-30,000 BCE. Impressions of woven cloth found on a clay surface from Moravia in the Czech Republic date to 26,000 BCE. This early weaving was probably done by hand. The first evidence of a primitive weaving machine, or loom, dates to about 10,000 BCE. Some of the earliest loom-woven cloth, made of linen, was found wrapped around an axe handle at a site in Turkey and dates to 7000 BCE. Woven flax cloth dating to 5000 BCE was found in Egypt. It was made using a plain weave, 12 threads by 9 threads per centimeter. By 4400 BCE, Egyptian weavers were using two-beamed horizontal looms. By 2000 BCE, most cultures were using wool as the primary fiber for weaving. By 700 CE, weavers were using horizontal and vertical looms in Asia, Africa and Europe. At some point after 700 CE, Islamic civilizations invented the pit-treadle loom, which was soon found in Syria, Iran and East Africa. Christian weavers adopted the pit-treadle loom via Moorish Spain after 1177 and it soon became the standard in Europe. During the Middle Ages, weavers began using cotton and silk fibers to make cloth. In the 18th Century, advancements in technology transformed weaving, which had been a cottage industry, into factory-based manufacturing. Those technologies included John Kay (UK) invented the flying shuttle, invented by John Kay in 1733; the fully-automated loom created by Jacques de Vaucanson (France) in 1745; James Hargreaves’ (UK) spinning jenny, from 1764; the spinning mule, invented by Samuel Crompton (UK) between 1775-1779. Power weaving started (fitfully) with the work of Edmund Cartwright (UK) in 1785, but the first semiautomatic loom was not introduced until 1842. The Jacquard loom, invented by Joseph Marie Jacquard (France) in 1801, simplified the process of weaving of complicated patterns by using punch cards.
5000 BCE
The earliest plow, the ard, dug a furrow in the earth for planting, but did not turn the soil over. Although humans first pushed the plow, they eventually harnessed oxen and other draft animals to pull the plow across the fields. The wooden ard may have been invented in Mesopotamia and the Indus Valley between 6000 and 5000 BCE, about the same time the ox was domesticated. Other sources date the ard to 4500 BCE or 3500 BCE in Sumeria in southern Mesopotamia. Still others claim the ard arose in Ancient Egypt about 3000 BCE and then spread to Mesopotamia and the Indus Valley. The ard was known in China in 3000 BCE. The iron plow was invented in Egypt and Assyria in about 2300 BCE. The remains of the earliest wooden ard found in Europe, from Lavagone, Italy, date to about 2300 BCE. The earliest improvement to the plow was the coulter, a blade-like component that cut an edge in front of the plowshare to create a smoothly cut bank. The moldboard plow, which is first found in China around 500 BCE, was a major advance in agriculture. The moldboard is a piece attached to the plow that turns over the soil, allowing nutrients to be brought to the surface. Instead of merely digging a furrow to plant seeds, the plow now added to the productivity of the soil. As the invention spread all over the world, farmers experimented with the best shape for the moldboard. Moldboards were first seen in England in the late 6th Century CE. Thomas Jefferson (US) designed and built a modified moldboard for his plows in 1794. The traditional moldboard plow was no match for the American prairie, even when metal strips were fixed to the edges. The share couldn’t get through the sod and the roots, while the sticky soil clung to the moldboard, slowing down the job. American John Deere’s 1836 steel plow used a steel share and a polished iron moldboard and the results were magnificent. Over the years, he modified the design somewhat, particularly the shape of the moldboard.
4000 BCE
Civilizations in Mesopotamia, the Indus Valley, the Northern Caucasus and Central Europe all invented vehicles with round wheels of solid wood between 4000 and 3500 BCE. The earliest clear depiction of a wheeled vehicle was found in Poland and dates to 3500-3350 BCE. The oldest surviving wheel was found in the Ljubljana Marshes in Slovenia and dates to approximately 3250 BCE. Wheeled vehicles are found in the Indus Valley by 3000-2000 BCE. The spoke-wheeled chariot was invented in Russia and Kazakhstan some time between 2200 and 1550 BCE, and reached China and Scandanavia by 1200 BCE. Wire wheels and pneumatic tires were invented much later, in the mid-19th Century.
3500 BCE
A potter’s wheel is a machine used in the shaping of round ceramic ware. Before the invention of the potter’s wheel, potters made ceramics by coiling long threads of clay into the final shape, often placing the pot on a mat or large leaf to allow turning. The next development was the tournette, or slow wheel, which was invented about 4500 BCE in the Near East. The potter turned the tournette slowly by hand or foot while coiling a pot. The fast wheel was invented later, and operated on the flywheel principle. The potter wound up the wheel by kicking it or pushing it around with a stick. Fast wheels allowed potters to invent the throwing method of pottery making, in which lump of clay was placed on the center of the wheel and squeezed, lifted and shaped as the wheel turned. The oldest stone wheel ever found is from the Mesopotamian city of Ur in modern-day Iraq, and is dated to 3129 BCE but there is evidence that potter’s wheels were in use in the Indus Valley civilization by 3500 BCE. Flywheel potter’s wheels have been found in Ancient Egypt and China dating to 3000 BCE.
3200 BCE
Human communities had used pictures and primitive symbols for thousands of years before they developed proto-writing, which then evolved into the complex systems known as written languages. Sumerian archaic cuneiform script first appears around 3200 BCE, although the first true written texts do not appear until about 2600 BCE. The first Egyptian hieroglyphics date to about 3200 BCE; the Indus script of Ancient India dates to 3200 BCE; and Chinese characters date to 1600-1200 BCE, but there is debate about whether these were independent discoveries or derived from pre-existing scripts. The Phoenicians began to develop the first phonetic writing system between 2000 and 1000 BCE. Mesoamerican cultures developed writing systems independently, with the Olmecs of Mexico the earliest, beginning about 900-600 BCE.
The earliest evidence of sailing boats is an image on an artifact found in Kuwait dating to 5500-5000 BCE. By 3200 BCE, Sumerians in Mesopotamia were using square-rigged sailboats for trade. Ancient Egyptians and Phoenicians had significant knowledge of sail construction at least as far back as 3000 BCE. By 200 CE, Chinese boatbuilders were building multi-masted sailing junks that could carry 200 people.
3000 BCE
The history of iron working is incomplete due to the tendency of iron objects to corrode, so that even though iron is common in the Earth’s crust and in meteorites, very few ancient iron artifacts have survived. The oldest-known man-made iron objects were found in Iran and date to 5000-4000 BCE. They were made from iron-nickel meteorites, as were the earliest iron artifacts from Egypt and Mesopotamia, dating from 4000-3000 BCE, and China, from 2000-1000 BCE. The technique of smelting native iron to make wrought iron was discovered in Mesopotamia and Syria about 3000-2700 BCE, Anatolia (now Turkey) in 2500 BCE and India in 1800-1200 BCE. The Hittites in Anatolia dominated iron production in the area after they began working iron in bellows-aided furnaces called bloomeries in 1500-1200 BCE. Around 1500 BCE, smelted iron objects begin to appear more frequently in Mesopotamia, Egypt and Niger. By 1100-1000 BCE, iron smelting technology had spread to Greece, China and sub-Saharan Africa, marking the beginning of the Iron Age. Wrought ironworking reached central Europe in the 8th Century BCE and became common in Northern Europe and Britain after 500 BCE. Meanwhile, Chinese ironworkers first produced cast iron, which was cheaper to produce than wrought iron, in the 5th Century BCE. Cast iron production did not reach Europe until the Middle Ages. Cast iron technology advanced in 1709, when Abraham Darby (England) found he could make the process more efficient by using a coke-fired blast furnace. In 1783, Henry Cort (England) introduced the puddling process for refining iron ore, a major improvement in making wrought iron.
2700 BCE
The earliest-known evidence of an abacus counting tool comes from the Sumerians in about 2700 BCE. Abacus use then spread to Ancient Egypt, Persia (600 BCE) and Ancient Greece (400 BCE). The oldest abacus existing today was found on the Greek isle of Salamis and dates to 300 BCE. There is evidence of a Chinese abacus from the 2nd Century BCE. Ancient Romans used the abacus from the 1st Century BCE. Sources from the 1st Century CE refer to abacus use in India. Pope Sylvester II, who served from 999-1003 CE, brought Arabic numerals to Europe and reintroduced the Roman abacus, with improvements. The Chinese abacus migrated to Korea in 1400 and Japan in 1600. Late to the party, the Russians invented an abacus in the 17th Century.
2600 BCE
A lever is a machine consisting of a beam or rigid rod that pivots at a fixed hinge, or fulcrum, thereby amplifying an input force to provide a greater output force. Third Century Greek philosopher Archimedes first correctly stated the mathematical principle behind the lever. Pappus of Alexandria quotes Archimedes as saying of the lever, “Give me a place to stand, and I shall move the Earth with it.” Although there is no written evidence of levers prior to Archimedes, historians believe that the Ancient Egyptians must have had levers in order to construct the pyramids and other massive monuments weighing more than 100 tons in the 3rd Millenium BCE.
2400 BCE
Parchment is a material made from the skins of calves, sheeps, goats or other animals. Before the invention of paper, parchment was used to write on or as pages of a book, codex or manuscript. To make parchment, the animal skin was limed, scraped and dried under tension, but not tanned. The earliest known parchment documents, written on leather, come from Egypt and date to the 24th Century BCE. Assyrian and Babylonian parchment dates to 600 BCE. Herodotus mentions parchment as common in the 5th Century BCE. The term ‘parchment’ is derived from Pergamon, in Asia Minor (now Turkey), where animal skin was used as a writing material when papyrus was not available in either the 3rd or 2nd Century BCE.
1730 BCE
The first glass may have been produced as an accidental byproduct of metal-working. The earliest glass beads, found in Syria, Mesopotamia and Ancient Egypt, date to 2500 BCE. The first evidence of glassmaking technology comes from South Asia in 1730 BCE. A noticeable increase in glassmaking occurred in Egypt and Syria in the late Bronze Age (1550-1200 BCE). By the 15th Century BCE, glass was being produced in Western Asia, Crete, Egypt and Mycenaean Greece. Colorless glass is first found in the 9th Century BCE in Syria and Cyprus. The Chinese began making glass around 300 BCE. Glass blowing, a technique that significantly lowered the cost of glassmaking, was first discovered on the Syro-Judean coast in the 1st Century BCE. Alexandrian glass blowers learned to make clear glass around 100 CE, which allowed for the first glass windows.
1500 BCE
The notion that humans believed the Earth was flat until Christopher Columbus’s voyages in the 1490s is simply untrue. The earliest suggestion that the Earth is a sphere comes from the Rig Veda, the ancient Hindu scripture, which is believed to have been composed in India about 1500 BCE. The Ancient Greeks also believed that the Earth was a sphere, but it is not clear who has priority -Pythagoras in the 6th Century BCE or Parmenides or Empedocles in the 5th Century. Plato (Ancient Greece) asserted the roundness of the Earth in the early 4th Century BCE. Later in the 4th Century, Aristotle (Ancient Greece) reasoned that the Earth was a sphere because some stars are visible in the south that are not visible in the north, and vice versa. In 240 BCE, Eratosthenes (Ancient Greece) conducted an experiment that provided empirical evidence that the Earth’s surface was not flat, but curved.
1400 BCE
The first alphabet may have its origins in Proto-Sinaitic scripts found in the Sinai Peninsula dating to 1700 BCE, but the evidence is too sparse to be sure. The writing system of the Ugarits in Syria about 1400 BCE is the earliest definitive use of an alphabet. The Proto-Canaanite alphabet first appeared in Palestine/Israel in 1300 BCE; it was the precursor of the Phoenician alphabet, which arose about 1300 BCE. The Phoenician alphabet, in turn, was the precursor to many of the writing systems in use today and throughout history. It led directly to Aramaic, which led to Arabic and Hebrew, all of which, like the original Phoenician alphabet, contained only consonants, no vowels. In about 800 BCE, the Greeks transformed the Phoenician alphabet by converting several letters to vowels, and the resulting Greek alphabet became the basis for Latin, Cyrillic and Coptic scripts.
600 BCE
Crossbows were first used in China as weapons of war between 600 and 500 BCE. Greek soldiers began using crossbows between 500-400 BCE. Romans used crossbows in war and hunting starting between 50 and 150 CE. There is evidence of crossbow use in Scotland between the 6th and 9th Centuries CE. Crossbows with sights and mechanical triggers were developed in the early 11th Century. The Medieval invention of pushlever and ratchet drawing mechanisms allowed soliders and hunters to use crossbows on horseback. The Saracens invented composite bows, made from layers of different material, and the Crusaders adopted the design upon their return to Europe. By 1525, the military crossbow had mostly been replaced by firearms.
500 BCE
Early forms of the theory that all matter is composed of tiny particles called atoms were proposed by Indian philosophers of the Jain, Ajivika and Carvaka schools in the 6th Century BCE. Ancient Greek philosophers Leucippus and Democritus advocated atomism in c. 500 BCE, but it is not known if they developed the idea independently or whether they were influenced by the Indian philosophers. Epicurus (Ancient Greece) adopted a form of atomism in the 3rd Century BCE, and his ideas were promoted by Ancient Roman philosopher-poet Lucretius in the 1st Century BCE. Most early atomists presumed that all atoms were identical, but in the 2nd Century BCE, Kanada (India), founder of the Vaisheshika philosophy, proposed an atomic theory holding that different types of matter consisted of different kinds of atoms.
300 BCE
Euclid was a mathematician living in Alexandria, Egypt (then under Greek control), when he published his Elements in about 300 BCE, setting out the fundamentals of what is now called Euclidean geometry. Many of the axioms, postulates and proofs in the Elements were originally discovered by others, but Euclid’s achievement was to fit them all into a single rational system. After Euclid, Archimedes (Ancient Greece) developed equations for volumes and areas of various figures and Apollonius of Perga (Ancient Greece) investigated conic sections. Much later, in the 17th Century, René Descartes and Pierre de Fermat (France) developed analytic geometry, an alternative method that focused on turning geometry into algebra. Also in the 17th Century, Girard Desargues (France) invented projective geometry.
200 BCE
Although the invention of paper is traditionally attributed to Ts’ai Lun (China) in 105 CE, strong evidence indicates that the pulp process was developed in China 200-300 years earlier during the Han Dynasty. The first recipe may have included tree bark, cloth rags, hemp and fishing nets. The earliest use of paper was to wrap and pad delicate objects, such as mirrors. The use of paper for writing is first seen in the 3rd Century CE. Paper was used as toilet tissue from at least the 6th Century CE. In the Tang Dynasty (618-907 CE), paper was used to make tea bags, paper cups and paper napkins. In the Song Dynasty (960-1279 CE), paper was used to make bank notes, or currency. Paper was introduced into Japan between 280 and 610 CE. Meanwhile, in Mesoamerica, the Mayans independently developed a type of paper called amatl, made from tree bark, beginning in 5th Century CE. The Islamic world obtained the secret of papermaking from the Far East by the 6th Century, when it was being made in Pakistan. The knowledge had spread to Baghdad by 793 CE, to Egypt by 900 CE and to Morocco by 1100 CE. In Baghdad, an inventor discovered a way to make thicker sheets of paper, a crucial development. The first water-powered pulp mills were built in 8th Century Samarkand (modern-day Uzbekistan). In 1035, a traveler noted that Cairo market sellers were wrapping customers’ purchases in paper. The first European papermaking occurred in Toledo, Spain in 1085 CE following the expulsion of the Muslims. The first paper mill in France was established by 1190 CE. Arab merchants introduced paper into India in the 13th Century. The first definitive reference to a water-powered paper mill in Europe comes from Spain in 1282. Making paper using recycled fibers was costly, but an inexpensive technology did not arise until 1844, when Charles Fenerty (Canada) and F.G. Keller (Germany) independently developed a process for making paper out of wood pulp – essentially the same process used today.
150 BCE
In 1900, an ancient shipwreck off the Greek island of Antkythera produced a wooden box containing a bronze mechanism with at least 30 meshing gears. The device, known as the Antikythera Mechanism, was created between 150-100 BCE. It could predict astronomical positions, eclipses and even identify the date of the next Olympic Games. Scientists have described the instrument, which is the earliest-known complex gear mechanism, as an early analog computer.
206 CE
Magnetic compasses were invented after humans discovered that iron could be magnetized by contact with lodestone and once magnetized, would always point north. There is some evidence that the Olmecs, in present-day Mexico, used compasses for geomancy (a type of divination) between 1400-1000 BCE. The first confirmed compasses, made in China about 206 BCE, were also used for divination and geomancy and used a lodestone or magnetized ladle. The first recorded use of a compass for navigation is 1040-1044 CE, but possibly as early as 850 CE, in China; 1187-1202 in Western Europe and 1232 in Persia. Later (by 1088 in China), iron needles that had been magnetized by a lodestone replaced the lodestone or other large object as directional arm of the compass. In many early compasses, the iron needle would float in water. The first dry needle compasses are described in Chinese documents dating from 1100-1250. Another form of dry compass, the dry mariner’s compass, was invented in Europe around 1300, possibly by Flavio Gioja (Italy). Further developments included the bearing compass and surveyor’s compass (early 18th Century); the prismatic compass (1885); the Bézard compass (1902); and the Silva orienteering compass (Sweden 1932). Liquid compasses returned in the 19th Century, with the first liquid mariner’s compass invented by Francis Crow (UK) in 1813. In 1860, Edward Samuel Ritchie invented an improved liquid marine compass that was adopted by the U.S. Navy. Finnish inventor Tuomas Vohlonen produced a much-improved liquid compass in 1936 that led to today’s models.
322 CE
A stirrup is a light frame or ring that holds the foot of a person riding a horse or other animal. Stirrups, which are usually paired, are attached to the saddle by a strap. They assist the rider in mounting and provide support while riding. Stirrups greatly increase the rider’s ability to stay in the saddle and control the mount. Before metal stirrups, riders placed their feet under a girth or used a simple toe loop. Later, riders used a single stirrup as a mounting aid. Paired stirrups appeared after the invention of the treed saddle. Stirrups were invented in China in the early centuries of the common era. An image of a rider with paired stirrups was found in a tomb from the Jin Dynasty dating to about 322 CE and they became common in China during the 5th Century CE. The stirrup then apparently spread westward through the nomadic peoples of Central Eurasia (there are reports that Muslim cavalry wore stirrups in Persia in 694 CE) and eventually into Europe. There is evidence that stirrups reached Sweden by the 6th Century CE and a pair of stirrups was found in an 8th century tomb in Slovakia. Horse riders of the Frankish tribes in what is now France and Germany commonly used stirrups in the early 8th Century.
600 CE
The Babylonians (in what is now Iraq) invented a symbol for zero about 250 BCE, which they used to differentiate between magnitudes in their number system, as we use zeros to distinguish tens from hundreds and hundreds from thousands. Previously, the Sumerians had merely left a blank space to indicate an absence in a column of numbers. The Mayans of Mesoamerica came up with a similar marker for their calendars beginning in about 350 CE. The zero symbol evolved from a placeholder to a number with its own value and properties during the first centuries of the common era in India. For example, in 458 CE, a Jain cosmological work called the Lokavibhaga mentions zero and the decimal positional system. By about 600 CE, Indian mathematicians such as Brahmagupta fully grasped the importance of zero. Brahmagupta and others used small dots under numbers to show a zero placeholder, but they also viewed zero as having a value, which they called ‘sunya.’ Brahmagupta also showed that subtracting a number from itself equals zero. From India, the concept of zero traveled to China and to the Middle East by 773 CE, where Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī synthesized Indian arithmetic and showed how zero worked in algebraic equations. By the 9th Century CE, the Arabic numeral system had begun to include zero in a form similar to today’s empty oval shape. Zero finally reached Europe in the 12th Century.
Also known as the Indo-Arabic Counting System, the Hindu-Arabic Numeral System was the first counting system to include a zero and is the basis for most of subsequent mathematics. This positional decimal numeral system was invented in India, but there is much debate about the date. Some scholars believe there is evidence for a 1st Century CE date, while others say the earliest evidence is from the 3rd or 4th Century CE. All agree that the system was in use by 600 CE. The system began to spread elsewhere: Severus Sebokht mentions it in 662 CE in Syria and Muslim scholar al-Qifti cites an encounter between a Caliph and an Indian mathematics book in 776 CE. Persian mathematician Al-Khwarizmi gave a treatise on the system in an 825 CE book and Arab mathematician Al-Kindi did the same in 830 CE. Arabic numerals first appear in Europe in a 976 CE Spanish text. Italian mathematician Fibonacci sought to promote the system in a book published in 1202, but the system did not become standard in Europe until after the printing press was invented in the 1540s.
800 CE
Precursors to the windmill include the windwheel of Heron of Alexandria, a Greek engineer, and the prayer wheels that have been used in Tibet and China since the 4th Century. A windmill uses the power of the wind to create energy; the first windmills were used to mill grain. The first practical windmills were made in Persia in the 9th Century and had ‘sails’ that rotated in a horizontal plane, not vertically as we normally see in the West. This technology spread throughout the Middle East and Central Asia and later to China and India. A visitor to China in 1219 remarked on a horizontal windmill he saw there. Vertical windmills first appeared in an area of Northern Europe (France, England and Flanders) beginning in about 1175. The earliest type of European windmill was probably a post mill. The oldest known post mill, dating to 1191, was located in Bury St. Edmunds, England. By the late 13th Century, masonry tower mills were introduced; the smock mill was a 17th Century variation. Hollow-post mills arose in the 14th Century.
Historians believe that Chinese alchemists invented gunpowder in the 9th Century CE while looking for a chemical that would make them immortal. They soon found out the explosive potential for their discovery, which was used in creating many weapons, including rockets (10th Century CE), flamethrowers (1000) and bombs (1220). The Chinese had perfected the recipe by the mid-14th Century. The Mongols learned about gunpowder when they conquered China in the mid-13th Century and spread it throughout the world during their subsequent invasions. The Arabic empire obtained gunpowder in the mid-13th Century. The Mamluks used gunpowder-fueled cannons against the Mongols in 1260. In 1270, Syrian chemist Hasan al-Rammah described a method for purifying saltpeter in making gunpowder. Europeans first saw gunpowder used by the Mongols at the Battle of Mohi in what is now Hungary in 1241. The first known use of gunpowder used by Europeans in battle was during the 1262 siege of the Spanish city of Niebla by Castilian King Alfonso X. Roger Bacon (England) referred to gunpowder in a 1267 book. By 1350, cannons were a common sight in European wars. India had gunpowder technology from at least 1366 CE, if not earlier. In the late 14th Century, European powdermakers began adding liquid and ‘corning’ the powder, which improved performance significantly.
820 CE
Between 2000 and 1600 BCE, the Babylonians developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. They developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By the time of Plato in the 4th Century BCE, the Ancient Greeks had created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. The geometric work of the Greeks, typified in Euclid’s Elements (300 BCE), provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations. The Arithmetica, written by 3rd Century CE Alexandrian Greek mathematician Diophantus, deals with solving algebraic equations. Indian mathematician Brahmagupta’s Brahmasphutasiddhanta, from 628 CE, contains the first complete arithmetic solution to quadratic equations, including zero and negative solutions. In The Compendious Book on Calculation by Completion and Balancing, from 820 CE, Persian mathematician Muhammad ibn Mūsā al-Khwārizmī first established algebra as a mathematical discipline independent of geometry and arithmetic. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero. In the Treatise on Demonstration of Problems of Algebra, from 1070, Persian mathematician Omar Khayyám identified the foundations of algebraic geometry and found the general geometric solution of the cubic equation. In the late 12th and early 13th centuries, Persian mathematician Sharaf al-Dīn al-Tūsī found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. Other important early developers of algebra, who solved various cases of cubic, quartic, quintic and higher order polynomial equations using numerical methods, were: Mahavira (India, 850 CE); Al-Karaji (Persia, late 10th-early 11th century); Bhaskara II (India, 12th Century); and Zhu Shijie (China) who wrote Jade Mirror of the Four Unknowns in 1303. In the 13th century, Italian mathematician Fibonacci’s solution of a cubic equation was part of a European algebraic revival. The general algebraic solution of the cubic and quartic equations was developed in the mid-16th century. The work of François Viète (France) on new algebra at the close of the 16th Century was an important step towards modern algebra. In La Géométrie, published in 1637, René Descartes (France) invented analytic geometry and introduced modern algebraic notation. In the 17th Century, Kowa Seki (Japan) developed the idea of a determinant to solve systems of simultaneous linear equations using matrices, which Gottfried Leibniz (Germany) discovered independently soon thereafter. Gabriel Cramer (Switzerland) published work on matrices and determinants between 1728 and 1750. Joseph-Louis Lagrange (Italy/France) studied permutations in his 1770 paper Réflexions sur la résolution algébrique des equations, in which he introduced Lagrange resolvents. In an important 1799 work, Paolo Ruffini (Italy) developed the theory of permutation groups in the context of solving algebraic equations. In the 19th Century, mathematicians developed abstract algebra from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues. In an 1830 treatise, British mathematician George Peacock founded axiomatic thinking in arithmetic and algebra. Augustus De Morgan (UK) discovered relation algebra in his Syllabus of a Proposed System of Logic, from 1860. In the 1870s, American scientist Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley (UK) developed a noncommutative algebra of matrices.
976 CE
Prior to the invention of the mechanical clock, humans kept time using sundials (a type of shadow clock), hourglasses, water clocks and candle clocks. Chinese inventors improved on the water clock by adding escapements. Liang Lingzan and Yi Xing (China) designed and built a mechanized water clock with the first known escapement mechanism in 725 CE. Islamic scientists also made improvements on the water clock, including one given as a gift by Harun al-Rashid of Baghdad to Charlemagne in 797 CE. In 976 CE, Zhang Sixun (China) was the first to replace the water in his clock tower with mercury. In 1000, Pope Sylvester brought water clocks to Europe. In 1088, Su Song (China) further improved on Zhang’s design in his astronomical clock tower, nicknamed, ‘Cosmic Engine.’ The first geared water clock was invented by Arab engineer Ibn Khalaf al-Muradi in Spain in the 11th Century. There is some evidence of mechanical clocks that used falling weights instead of water in France in 1176 and England in 1198. Al-Jazari (Mesopotamia) built numerous clocks in the early 13th Century; there is also a reference to an Arabic mechanical clock in a 1277 Spanish book. There is also evidence of mechanical clocks in England in 1283 and 1292, as well as Italy and France. The oldest surviving mechanical clock is at Salisbury Cathedral (UK) and dates to 1386. Spring-driven clocks first appeared in the 15th Century. Clocks indicating minutes and seconds also begin to appear in the 15th Century. Jost Bürgi (Switzerland) invented the cross-beat escapement in 1584. Around the same time, the first alarm clocks were invented. The first pendulum clock was invented by Christiaan Huygens (The Netherlands) in 1656. A pendulum clock uses a weight that swings back and forth in a precise time interval, thus making this type of clock much more precise than previous designs. Galileo Galilei (Italy) had been exploring the properties of pendulums since 1602. He designed a pendulum clock in 1637, but died without completing it. Huygens, with the assistance of clockmaker Salomon Coster, designed and built a pendulum clock that realized Galileo’s dream.
984 CE
The law of refraction (also known as Snell’s law and the Snell–Descartes law) describes the relationship between the angle of incidence and the angle of refraction when light or other waves pass through a boundary between two different isotopic media. Although the law is named after Europeans, it was first described by Ibn Sahl (Persia) in his 984 CE book On Burning Mirrors and Lenses, written while he was at the Abbasid Court in what is now Iraq. Thomas Harriot (England) first rediscovered the law in 1602, but did not publish his findings. In 1621, Dutch astronomer Willebrord Snellius (known as Snell) rediscovered the law that bears his name, but did not publish in his lifetime. In his 1637 essay Dioptrics, René Descartes (France) independently derived the law using heuristic momentum conservation arguments in terms of sines and used it to solve a range of optical problems. Pierre de Fermat (France) arrived at the same solution in 1657 based solely on the principle of least time. The accusation that Descartes had seen Snell’s paper has been disproved. In his 1678 book Traité de la Lumiere, Christiaan Huygens (The Netherlands) showed how Snell’s law of sines could be explained by, or derived from, the wave nature of light, using what is now called the Huygens–Fresnel principle.
The Canon of Medicine, a five-book encyclopedia published in 1025 by Persian scientist, physician and philosopher Ibn Sina (often referred to by the Latinate form of his name, Avicenna) set out in a systematic way the medical knowledge and procedures known in the 11th Century. While the Canon relies primarily on medical theories dating back to Galen, Ibn Sina also adopts Aristotle’s explanations in some cases and drew from many other sources, including Chinese texts from the 4th and 7th centuries CE. In his introduction, Ibn Sina sets out his beliefs that medicine is a science, and that the physician must determine the causes of both health and disease before the body can be restored to health. The book contains specific instructions on diagnosis and treatments, including surgical procedures, and analyzes the efficacy of over 600 different drugs and herbal remedies. Originally written in Arabic, the Canon was translated into Latin by Gerard of Cremona in the 13th Century, which allowed it to become the premier textbook for European medical education in the medieval period.
Block printing was first invented in Japan in about 700 CE. Bi Sheng (China) invented movable type printing in 1040. He tried making the characters from wood but found that ceramics worked better. Choe Yun-ui (Korea) was the first to use metal for the type, in 1234. The technology did not spread to Europe. Johannes Gutenberg (Germany) invented movable type printing independently in 1440 or 1450.
The current view of the scientific method is that it is a way of investigating phenomena, obtaining new knowledge or correcting or assimilating prior knowledge that is based on empirical and measurable evidence, resting on certain rational principles. According to the Oxford English Dictionary, the scientific method involves “systematic observation, measurement and experiment, and the formulation, testing and modification of hypotheses.” The scientific method contrasts with the very influential method proposed by Aristotle of reasoning from first principles. Muslim scientists such as Jabir ibn Hayyan (721-815 CE) and Alkindus (801-873 CE) were among the first to use experiment and quantification to test theories. Use of the scientific method is clear in Arab scientist Ibn al-Haytham’s Book of Optics (1021) and the revised version of the Optics by Kamal al-Din al-Farisi in the early 14th Century. Abu Rayhan al-Biruni (Persia) used a quantitative scientific method in studying mineralogy, sociology and mechanics in the 1020s and 1030s. Ibn Sina (Avicenna) (Persia) set out a method using hypotheses in The Book of Healing, from 1027. In the 1220s, Robert Grosseteste (England) published a commentary of Aristotle’s Posterior Analytics in which he set out some aspects of the scientific method, including (1) to take particular observations to create a universal law, and then use the universal law to predict particular observations; and (2) the need to verify scientific principles through experimentation. Roger Bacon (England) followed up on Grosseteste’s work in his 1267 work, Opus Majus, which systematically set out the principles of the scientific method for the first time. The next major exponent of the scientific method was Francis Bacon (England), who sought to overturn the Aristotelian methods used in science education and practice by focusing on inductive reasoning and experimentation, especially in his Novum Organum of 1620. In the first half of the 17th Century, Galileo Galilei (Italy) promoted the scientific method in the face of Aristotelianism, by using observation, experiment, and inductive reasoning, and by changing his views based on the empirical findings. René Descartes (France) provided philosophical premises for the scientific method in 1637. In 1687, Isaac Newton (England) set out four rules of reasoning in science that embodied principles of the new scientific method. After David Hume attacked inductive reasoning, scientists and philosophers sought to rehabilitate scientific knowledge. These included Hans Christian Ørsted (Denmark) in 1811, John Herschel (UK) in 1831, and William Whewell (UK) in 1837 and 1840, and John Stuart Mill (UK) in 1843, and William Stanley Jevons (UK) in 1873 and 1877. Claude Bernard (France) applied the scientific method to medicine in 1865. Charles Sanders Peirce (US) articulated the modern scheme for testing hypotheses and the importance of statistical knowledge in science in 1878. Karl Popper proposed a revision of the scientific method in 1934 by stating that a scientific hypothesis must be falsifiable or it is not scientific. Not all scientists agreed with Popper, including Thomas Kuhn, in 1962, who noted that different scientists worked differently and falsifiability was not a methodology that scientists actually follow.
There are references going back as far as the 5th Century BCE to the use of lenses, jewels or water-filled globes to correct vision, but it was only after the Book of Optics, an 11th Century treatise by Arab scientist Ibn al-Haytham (also known as Alhazen), was translated into Latin in the 12th Century that the stage was set for the invention of true eyeglasses in Italy by an unnamed individual in 1286 who, according to a historian, was “unwilling to share them.” Alessandro di Spina (Italy) followed soon afterwards and “shared them with everyone.” The first eyeglasses used convex lenses to correct both farsightedness (hyperopia) and presbyobia. They were designed to be held in the hand or pinched onto the nose (pince-nez). Some have speculated that eyeglasses originated in India prior to the 13th Century. The earliest depiction of eyeglasses is Tommaso da Modena’s 1352 portrait of Cardinal Hugh de Provence. A German altarpiece from 1403 also shows the invention. The first glasses that extended over the ears were made in the early 18th Century. Concave lenses to cure myopia, or shortsightedness, were not developed until c. 1450.
A rainbow is a spectrum of light that appears in the sky, taking the form of a multicolored arc. It is caused by the reflection and refraction of light in illuminated above-ground water droplets. Aristotle (Ancient Greece) made the first recorded attempts to explain rainbows in the 4th Century BCE. In c. 65 CE, Seneca the Younger (Ancient Rome) devoted an entire book of his Naturales Quaestiones, to rainbows, noting that rainbows always appear opposite the sun. He speculated that rainbows might be caused by small prism-like rods of glass, or may be produced by a cloud shaped like a concave mirror. Persian scientist Ibn al-Haytham, in his 11th Century book On the Rainbow and Halo, also incorrectly proposed a form of the ‘cloud as concave mirror’ theory in which concentric circles form on an axis between the sun, the cloud and the eye of the viewer. Averroes agreed with al-Haytham. Ibn Sina (also known as Avicenna), writing in the 11th Century, thought instead that the rainbow forms in the thin mist between the cloud and the sun or observer and the cloud is only a background. Ibn Sina incorrectly concluded that the colors were merely a subject sensation in the eyes of the viewer. Sun Sikong and Shen Kuo (China) concluded independently in the mid-10th Century that rainbows were formed when sunlight encountered droplets of rain in the air. Persian astronomer Qutb al-Din al-Shirazi provided an accurate explanation for the rainbow in the late 13th Century, for which his student Kamal al-Din al-Farisi in his early late 13th or early 14th Century work Kitab Tanqih al-Manazir (The Revision of the Optics) conducted experiments and gave a more detailed mathematical explanation, in which a ray of sunlight was refracted twice by a drop of water, with one or more reflections occurring between the two refractions. After Ibn al-Haytham’s Book of Optics was translated into Latin in the late 12th or early 13th Century, the rainbow became an object of study by Robert Grosseteste (England, 13th Century) and Roger Bacon (England), who wrote in his 1268 Opus Majus of experiments with light shining through water droplets showing the colors of the rainbow and who first calculated the rainbow’s angular size. In 1307, Theodoric of Freiberg (Germany) gave an accurate theoretical explanation of primary and secondary rainbows with two refractions (upon ingress and egress) and one reflection (at the back of the drop). René Descartes (France) advanced the explanation of secondary rainbows in his 1637 Discourse on Method. Isaac Newton’s discovery in 1666 that white light was composed of all the colors of the spectrum, explained the rainbow’s colors. In the early 19th Century, Thomas Young (UK) explained supernumerary rainbows using a wave theory of light, while George Biddell Airy (UK) explained that the strength of the rainbow’s colors depends of the size of the water droplets. Gustave Mie (Germany), who discovered Mie scattering, provided a modern physical description of the rainbow in 1908.
Ockham’s Razor is a philosophical principle stating that, among competing hypotheses, the one with the fewest assumptions should be selected. Although the principle is traditionally attributed to William of Ockham (England, c. 1287-1347), it has a long history. In 4th Century BCE Greece, Aristotle wrote in his Posterior Analytics, “we may assume superiority (all things being equal) of the demonstration which derives from fewer postulates or hypotheses.” Ptolemy (Ancient Rome) stated in the 2nd Century CE, “We consider it a good principle to explain the phenomena by the simplest hypothesis possible.” Robert Grosseteste (England) wrote in his Commentary on Aristotle’s Posterior Analytics in c. 1217-1220, “That is better and more valuable which requires fewer, other circumstances being equal… For if one thing were demonstrated from many and another thing from fewer equally known premisses, clearly that is better which is from fewer because it makes us know quickly, just as a universal demonstration is better than particular because it produces knowledge from fewer premises. Similarly in natural science, in moral science, and in metaphysics the best is that which needs no premisses and the better that which needs the fewer, other circumstances being equal.” In the Summa Theologica (1265-1274) of Thomas Aquinas (Italy), he states, “it is superfluous to suppose that what can be accounted for by a few principles has been produced by many.” In the 13th Century work Vishnu Tattva Nirnaya, Indian philosopher Madhva wrote, “To make two suppositions when one is enough is to err by way of excessive supposition.” By the time of William of Ockham in the early 14th Century, the basic principle appears to have been common among philosophers. William of Ockham’s writings contain two different statements of a similar principle: “Plurality must never be posited without necessity”, in his Commentaries on Peter Lombard’s Sentences (c. 1320) and “It is futile to do with more things that which can be done with fewer”, in his Summa Logicae (c. 1323). The classic statement of the rule, “Entities are not to be multiplied unnecessarily” is now attributed to 17th Century Irish philosopher John Punch, although his formulation was slightly different. Subsequent iterations of Ockham’s Razor come from Sir Isaac Newton (England) in 1726 (“We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances”) and British philosopher Bertrand Russell in 1924 (“Whenever possible, substitute constructions out of known entities for inferences to unknown entities”).
Johannes Gutenberg (Germany) reinvented movable type printing in 1440 or 1450, independent of the Chinese invention of 1040. Gutenberg’s major innovation was to adapt the already-existing screw press to print his pages. He also created a special metal alloy for the type; invented a device for moving type quickly; and developed a new, superior ink. The result was the production of higher quality printing at a much faster pace. Offset printing was invented by Aloys Senefelder (Germany) in 1796. The cast iron printing press, which reduced the force needed and doubled the size of the printed area, was invented by Lord Stanhope (UK) in 1800. Between 1802 and 1818, Friedrich Koenig (Germany) created a steam-powered press with rotary cylinders instead of a flatbed. In 1843, Richard M. Hoe (US) invented a steam-powered rotary printing press. Linotype printing was invented by Ottmar Mergenthaler (US) in 1884.
For much of history, humans believed that they (and the Earth) were the center of the universe, with some exceptions. For example, in 270 BCE, Ancient Greek scientist Aristarchus of Samos suggested that the Earth and other planets revolved around the sun; his system received support from Seleucus of Seleucia (Ancient Greece) in the mid-2nd Century BCE. In the Middle Ages and long after, it was a tenet of Christian doctrine that the Earth was a stationary globe, around which the sun, the planets and the stars revolved, and it was heresy to say otherwise. Nevertheless, 15th Century German cleric Nicholas of Cusa wrote that the Earth was one of many similar celestial bodies, was not the center of the universe, and was not at rest. As early as 1514, astronomer and mathematician Nicolaus Copernicus (Poland) became convinced through his observations and mathematical calculations that the Earth and other planets revolved around the sun, not the other way around. He held off publishing his results until just before his death in 1543, for fear of reprisals. The Copernican model, which posited circular orbits, was revised by Johannes Kepler (Germany), who discovered in 1609 that the orbits of the planets were ellipses, one of his laws of planetary motion. Later in the 17th Century, Galileo Galilei (Italy) publicized telescopic observations that confirmed the heliocentric model and popularized the new view in his 1632 book Dialogue Concerning the Two Chief World Systems which led to his arrest and condemnation by the Roman Catholic Church.
The modern study of human anatomy began with physician Andreas Vesalius (Belgium), whose seven-volume 1543 treatise, De fabrica corporis humani, provided a detailed, well-researched and systematic study of the human body that corrected many errors of the past. Anatomical study has a long history before and after Vesalius. Ancient Egyptian treatises on anatomy date to 1600 BCE. Ancient Greek anatomists include Alcmaeon, Acron (480 BCE), Pausanias (480 BCE), Empedocles (480 BCE), Praxagoras (300 BCE?), Herophilus (280 BCE?); and Erasistratus (260 BCE?). Aristotle conducted empirical studies in the 4th Century BCE and began the study of comparative anatomy. A Greek living in the Roman Empire, Galen (2nd Century CE) was the first major anatomist. He was highly influential into the modern era, but performed few human dissections and propagated some serious errors. Italian physician Mondino de Luzzi performed the first human dissections since Ancient Greece between 1275 and 1326. In the late 15th Century, Leonardo da Vinci dissected approximately 30 human bodies and made detailed drawings, until the Pope ordered him to stop. In 1541, Giambattista Canano published illustrations of each muscle and its relation with the bones.
A supernova occurs when a star suffers a catastrophic explosion, causing it to increase greatly in brightness. The explosions of supernovae radiate enormous amounts of energy and normally expel most or all of the star’s contents at a velocity of 30,000 km/s, which sends a shock wave and an expanding shell of gas and dust (called a supernova remnant) into interstellar space. Supernovae generate much more energy than novae. There are two types of supernova: the first occurs when nuclear fusion suddenly reignites in a degenerate star due to accumulation of material from a companion star; the second occurs when a massive star undergoes sudden gravitational collapse. The first supernovae to be observed were those occurring in the Milky Way galaxy that were visible to the naked eye. Chinese astronomers observed a supernova in 185 CE. Chinese and Islamic astronomers described a supernova in 1006. A widely-seen supernova in 1054 created the Crab Nebula. Tycho Brahe (Denmark) described a supernova in Cassiopeia 1572 and Johannes Kepler (Germany) described one in 1604. The first supernova in another galaxy was seen in the Andromeda galaxy in 1885. Prior to 1931, supernovae were not distinguished from ordinary novae. Based on observations at Mt. Wilson Observatory, Walter Baade (Germany/US) and Fritz Zwicky (Switzerland/US) created a new category for supernovae, a term they began using in a series of 1931 lectures and announced publicly in 1933. In 1941, Zwicky and Rudolph Minkowski (Germany/US) developed the modern supernova classification scheme. In the 1960s, astronomers began to use supernova explosions as ‘standard candles’ to measure astronomical distances. More recently, scientists have been able to determine the dates and locations of supernovae that occurred in the past based on their aftereffects.
In 45 BCE, Sosigenes of Alexandria developed a calendar and presented it to Roman emperor Julius Caesar, who adopted it for the Roman Empire as the Julian Calendar. In the Julian calendar, each year consisted of 365 days divided into 12 months, with a leap year every four years, creating an average year of 365.25 days. Because a true solar year is slightly less than 365.25 days, the Julian calendar became out of sync with the seasons and religious holidays over the centuries. This led Pope Gregory XIII to revise the calendar in 1582 to skip three leap years every four centuries, which keeps the calendar in line with the seasons to this day. The average year is now 365.2425 days. Although some non-Western countries and religious groups maintain their own calendars, the Gregorian calendar is used universally for trade and international relations, including by the United Nations.
According to our current understanding, the size of the universe is unknown, but it may be infinite. The observable universe has a radius of 46 billion light years. The idea that the universe might be infinite goes back to ancient times, although the concept of a finite universe also has long roots. In Ancient Greece, Anaxagoras in the 5th Century BCE and Epicurus in the 3rd Century BCE, believed the universe was infinite. In the 15th Century, Nicholas of Cusa (Germany) proposed an infinite universe with an infinite number of stars and planets. Following both Nicholas of Cusa and Nicolaus Copernicus, Giordano Bruno (Italy) argued in 1584 in favor of an infinite universe where Earth and the other planets revolve around the sun, which is just one of an infinite number of stars, many of which have their own planets. In the 17th and 18th Centuries, Sir Isaac Newton (England), René Descartes (France), Immanuel Kant (Germany) and Johann Lambert (Switzerland) all proposed variations on the theme of a infinite steady state universe that is static but evolving. In 1917, Albert Einstein believed in a finite, static universe, although he had to invoke a cosmological constant to make the math work. The discovery of the expansion of the universe in 1929 and the proposal of the Big Bang theory in the 1930s led to various theories incorporating those new facts, with no unanimity on whether the universe is finite or infinite.
In the mid-16th Century, European scientists began to use experimentation to challenge Aristotle’s claim that heavier objects fall faster than light ones. Simon Stevin (Flanders), for example, showed in 1586 that two balls – one ten times heavier than the other – appeared to hit the ground at the same time when dropped 30 feet from a Delft church tower. In 1589-1590, while teaching at the University of Pisa, Galileo Galilei (Italy) not only performed similar experiments, but he also derived the mathematical equations to explain the phenomenon, as well as the acceleration of falling bodies and the phenomena of inertia and friction. He elaborated on his theories in 1634 and 1638 publications. The story that Galileo proved the theory by dropping balls from the Leaning Tower in Pisa is told by his pupil Vincenzo Viviani but may be apocryphal. Galileo preferred to experiment by rolling balls down an inclined board, which reduced air resistance. Galileo’s findings led to Isaac Newton’s law of universal gravitation.
Roger Bacon (England) first proposed the idea of a microscope in 1267, but it was not until about 1590 that two Dutch eyeglass makers, Hans Lippershey & Zacharias Jansenmade the first compound optical microscope. (‘Optical’ because it used visible light and lenses to magnify objects and ‘compound’ because it used multiple lenses, allowing for much greater magnification than the single lens, or simple optical microscope.) Galileo Galilei (Italy) developed a compound microscope in 1609, while Cornelius Drebbel (The Netherlands) created one in 1619. The importance of the microscope was made evident by Robert Hooke (UK), who published a book of drawings of his microscopic observations entitled Micrographia in 1665. Hooke’s book contained numerous scientific discoveries, including the first description of a biological cell.
The Chinese were aware around 1 CE that a magnet will align with north and south directions. About 200 CE, Chinese scientists discovered that magnetic north and true north were different. In the 16th Century, Georg Hartmann (Germany) and Robert Norman (England) independently discovered magnetic inclination, the angle between the magnetic field and the horizontal. In 1600, William Gilbert (England) published the results of his experiments using a small model of Earth, which led to his discovery that the Earth is a giant magnet, thus explaining why compasses point north. He also predicted accurately that the Earth has an iron core. Carl Friedrich Gauss (Germany) was the first to measure the Earth’s magnetic field in 1835. The true cause of the magnetic field was only discovered in the 20th Century, after the dominant theory – that the Earth is made of magnetic rocks – was disproved. In 1919, Sir Joseph Larmor (UK) proposed that a self-exciting dynamo could be the mechanism. W.M. Elsasser and Edward Bullard (UK) showed in the 1940s that the motion of a liquid core could produce a self-sustaining magnetic field.
Electricity is the name for a set of physical phenomena associated with the presence and flow of electric charge. One of the first to examine the phenomenon was Thales of Miletus (Ancient Greece), who studied static electricity in 600 BCE. It was not until the careful research of William Gilbert (England) in 1600 that electricity became a subject of scientific study. Gilbert also coined the Latin term ‘electricus’ from the Greek word for amber, which he rubbed to produce static electricity. The English words ‘electric’ and ‘electricity’ were derived by Thomas Browne in 1646. Otto von Guericke (Germany) made the first static electricity generator in 1660. Stephen Gray (England) discovered the conduction of electricity in 1729. The Leyden jar, the first capacitor, was invented independently in 1745 in Germany and The Netherlands. Henry Cavendish (England) measured conductivity of materials in 1747. Benjamin Franklin (US) discovered that lightning is a form of static electricity in 1752. Luigi Galvani (Italy) discovered the electrical basis of nerve impulses in 1786. Alessandro Volta (Italy) invented the electric battery in 1800.
The first telescopes were refractors because they used lenses to collect and magnify light. The earliest versions were made in 1608 by Hans Lippershey, Zacharias Jansen and Jacob Metius (The Netherlands). Galileo Galilei (Italy) built an improved refractor telescope in 1609. In 1655, Christiaan Huygens (The Netherlands) developed a compound eyepiece refractor based on a theory by Johannes Kepler (Germany). In 1668, Isaac Newton (UK) invented the first reflector telescope, which used a mirror instead of a lens to collect light. Laurent Cassegrain (France) improved on the reflector in 1672. Further improvements were made throughout the 18th Century.
After studying the astronomical observations of Tycho Brahe (Denmark), Johannes Kepler (Germany) derived three laws that determine the motion of the planets. He devised the first two laws in 1609: (1) The orbit of every planet is an ellipse with the sun at one of the two foci; and (2) A line joining a planet and the sun sweeps out equal areas during equal time intervals. In 1619, Kepler discovered a third law: (3) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In 1687, Sir Isaac Newton (England) showed that Kepler’s laws were consistent with classical mechanics.
In 1609 and 1610, Galileo Galilei (Italy) built a telescope and began a series of detailed observations of the heavens. In addition to providing support for the Copernican/Keplerian heliocentric model, he identified: four of the moons of Jupiter; the phases of Venus; sunspots; lunar mountains and craters; and masses of stars in the Milky Way ‘clouds.’ Galileo published his observations in Sidereus Nuncius (Starry Messenger) in 1610, which became a bestseller.
Although Chinese astronomer Gan De reportedly observed a moon orbiting Jupiter about 364 BCE, Galileo Galilei (Italy) is normally credited with discovering the four largest moons of Jupiter – Ganymede, Callisto, Io and Europa – through progressively stronger telescopes in 1609 and 1610. E.E. Barnard (US) discovered a fifth moon, Amalthea, in 1892. Using photographic telescopes, additional moons were discovered in 1904, 1905, 1908, 1914, 1938, 1951, and 1974. A 14th moon was discovered in 1975. The Voyager space probes found three more moons in 1979. Between 1999 and 2003, a team led by Scott S. Sheppard and David C. Jewitt (US) found 34 additional moons, most of them very small (averaging 1.9 miles in diameter) with eccentric orbits, probably captured asteroids. Since 2003, scientists have discovered 17 additional moons, bringing the 2014 total to 67.
Ancient Greek scientists had invented primitive thermoscopes, based on the principle that certain substances expanded when heated. Scientists in Europe, including Galileo Galilei (Italy) in c. 1593, developed more sophisticated thermoscopes in the 16th and 17th centuries. The thermometer was born in 1611-1613, when either Francesco Sagredo or Santorio Santorio (Italy) first added a scale to a thermoscope. Daniel Gabriel Fahrenheit (Netherlands/Germany/ Poland) invented an alcohol thermometer in 1709 and the first mercury thermometer in 1718. Each inventor used a different scale for his thermometer until Fahrenheit suggested the scale that bears his name in 1724. That scale was becoming the standard when, in 1742, Anders Celsius (Sweden) suggested a different scale. The two scales have been in competition ever since. William Thomson, Lord Kelvin (UK) developed the absolute zero scale, known as the Kelvin scale, in 1848.
The exponent that a fixed value, called the base, must be raised to in order to produce a number is called the logarithm of that number. (E.g., the logarithm of 1000 in base 10 is 3 because 1000 = 103.) Precursors to logarithms were invented by the Babylonians in 2000-1600 BCE, Indian mathematician Virasena in the 8th Century CE; and Michael Stifel (Germany) in a 1544 book. In 1614, John Napier of Merchiston (Scotland) announced the discovery of logarithms, which highly simplified multiplication and addition calculations. Henry Briggs (England) created the first table of logarithms in 1617. Joost Bürgi (Switzerland) discovered logarithms independently before Napier, but did not publish until 1620. Alphonse Antonio de Sarasa (Flanders) related logarithms to the hyperbola in 1649. Natural logarithms were first identified by Nicholas Mercator (Germany) in 1668, but John Speidell (England) had been using them since 1619. Swiss mathematician Leonhard Euler vastly expanded the theory and applications of logarithms in the 18th Century by using them in analytic proofs, expressing logarithmic functions using power series, and defining logarithms for negative and complex numbers.
The first submarine, the Drebbel, was designed by William Bourne and built by Cornelius Drebbel for James I of England in 1620; it was propelled by oars. The first military submarine, the Turtle, was built by David Bushnell (US) to aid the American Revolution in 1775. It used screws for propulsion. Subsequent human-powered subs were made and tested in France, Ecuador, Bavaria, Chile and both sides in the U.S. Civil War, leading to numerous drownings. The U.S. Navy’s first submarine, the French-designed Alligator, was the first to use compressed air and an air filtration system. The French Plongeur, launched in 1863, was the first submarine that did not rely on human power. A Spanish model from 1867, the Ictineo II, designed by Narcis Monturiol (Spain), was powered by a combustion engine that was air independent. Polish inventor Stefan Drzewiecki built the first electric-powered submarine in 1884. In 1896, Irish inventor John Philip Holland built the first submarine that used internal combustion engines on the surface and electric battery power underwater.
After years of careful study, physician William Harvey (England) first described the entire system by which the heart distributes the blood through the arteries and the blood returns to the heart via the veins as well as many other details of the circulatory system of humans and animals in his 1628 book De Motu Cordis (On the Motion of the Heart and Blood). Prior to Harvey, discoveries had been made by Galen (Ancient Greece/Ancient Rome) in the 2nd and 3rd centuries and especially Ibn al-Nafis (Syria) in 1242. Michael Servetus (Spain) also made important discoveries about pulmonary circulation that were published in 1553.
The invention of logarithms by John Napier (Scotland) in 1614 made multiplying easier and thus made calculators practical. In 1632, William Oughtred (England) invented the slide rule. The first mechanical calculator was invented by Blaise Pascal (France) in 1642. Gottfried Leibniz (Germany) made a multiplication machine in 1671 that did not improve on Pascal’s. Several machines were made in the 18th Century, including that of Poleni (Italy). The first commercial mechanical calculator was the Arithmometer of Thomas de Colmar (France), which was invented in 1820 but not marketed until 1851. Charles Babbage (UK) invented the difference machine in 1822 and a calculating machine in 1834-1835, which were programmable and precursors to the computer. Americans Frank S. Baldwin, Jay R. Monroe, and W. T. Ohdner all produced calculators in the second half of the 19th Century. Other machines included William Seward Burroughs’ from 1886, Felt and Tarrant’s (US) comptometer from 1887, and Otto Steiger’s Millionaire in 1894. James Dalton (US) introduced the Dalton Adding Machine in 1902, the first with push buttons. The Curta calculator, invented by Curt Herzstark (Austria) in 1948, was the last popular mechanical calculator. Casio (Japan) introduced the first all-electric calculator, the Model 14-A, in 1957, although it was built into a desk. British Bell announced its all-electronic desktop calculators – the ANITA Mk VII and Mk VIII – in 1961. The ANITAs were among the last to use vacuum tubes. The 1963 Friden EC-130 (US) used transistors. In 1964, Sharp (US) produced the CS-10A and Industria Macchine Elettroniche (Italy) announced the IME 84. Similar models followed from these and other companies, including Canon, Olivetti, SCM, Sony, Toshiba and Wang. The next development was the hand-held pocket calculator. In 1967, Jack Kilby, Jerry Merryman and James Van Tassel (US) at Texas Instruments made a prototype of the Cal Tech, although it was still too large to fit in a pocket. In the 1970s, manufacturers reduced size by switching from transistors to integrated circuits. The first microchip pocket calculators were the Sanyo Mini Calculator, the Canon Pocketronic (based on Kilby’s Cal Tech) and the Sharp micro Compet, all in 1970. Sharp brought out the EL-8 in 1971. Mostek (US) made the MK6010 the same year. Also in 1971, Pico Electronics and General Instrument collaborated on the Monroe Digital III, a single chip calculator. Busicom (Japan) made the first truly pocket-sized calculator, the 1971 LE-120A “Handy”, at 4.9 X 2.8 X 0.9 inches. The first US pocket-sized device was the Bowmar Brain from late 1971.
Atmospheric pressure, also known as air pressure, is the force exerted on a surface by the weight of the air above that surface in the atmosphere of the Earth. A barometer measures atmospheric pressure, which can forecast short-term changes in the weather. Evangelista Torricelli (Italy), a student of Galileo’s, discovered atmospheric pressure and invented the mercury barometer in 1643. Torricelli built on previous discoveries. In 1630, Giovanni Battista Baliani (Italy) conducted an experiment in which a siphon failed to work. Galileo Galilei (Italy) explained the result by noting that power of a vacuum held up the water, but that at a certain point the weight of the water was too much for the vacuum. René Descartes (France) designed an experiment to determine atmospheric pressure in 1631. Having read of Galileo’s ideas, Raffaele Magiotti and Gasparo Berti (Italy) devised an experiment between 1639 and 1641 in which Berti filled a long tube with water, plugged both ends, and stood the tube in a basin of water. Berti then unplugged the bottom of the tube. The result was that only some of the water flowed out, and the water in the tube leveled off at 10.3 meters, the same height Baliani observed in the siphon. Above the water in the tube was a space that appeared to be a vacuum. Torricelli analyzed the results from a different angle: instead of explaining the phenomenon with a vacuum, he chose to challenge common understanding and claim that the air itself had weight, and exerted pressure on the water. From this, he concluded that he could create a device that would measure the pressure of the atmosphere. By using mercury, which is 14 times heavier than water, he could use a tube only 80 centimeters long instead of 10.5 meters. He also discovered that the barometer measured different pressures on rainy days and sunny days. Blaise Pascal and Pierre Petit (France) repeated and perfected Torricelli’s experiment in 1646, showing that the liquid used did not change the results. Pascal had his brother-in-law, Florin Perier (France) perform another experiment which showed that the barometer (and therefore the air pressure) became lower as one increased in altitude, thus proving that the weight of the air was the cause of the barometer’s movements. In 1654, Otto von Guericke (Germany) demonstrated that a vacuum could exist, and he invented a pump that could create a vacuum. In 1661, Robert Boyle (Ireland) took advantage of the vacuum pump to discover Boyle’s Law.
Probability theory is a branch of mathematics that analyzes random phenomena. In the 16th Century, Gerolamo Cardano (Italy) took the first steps toward probability theory in his attempts to analyze games of chance. The next developments came from Pierre de Fermat and Blaise Pascal (France), who are considered to have originated probability theory in 1654. Christiaan Huygens (The Netherlands) published a book on probability in 1657. Books by Jacob Bernoulli (Switzerland) in 1713 and Abraham de Moivre (France) in 1718 developed the mathematical basis for probability theory. The fundamentals of probability and statistics were set down by Pierre-Simon Laplace (France) in a 1812 treatise. Richard von Mises (Austria-Hungary) made advances in the 20th Century, and modern probability theory was established by Andrey Nikolaevich Kolmogorov (USSR) and later Bruno de Finetti (Italy).
Galileo Galilei (Italy) observed Saturn’s rings through a telescope in 1610, but did not identify them as rings, but ears or a ‘triple form’. Galileo noted the disappearance of the rings when Saturn was oriented directly at the Earth in 1612 and their reappearance in 1613. Christiaan Huygens (The Netherlands), using a 50-power refracting telescope, definitively identified Saturn’s rings in 1655. In 1666, Robert Hooke (England) also identified the rings and noted that Saturn cast a shadow on the rings. Giovanni Domenic Cassini (Italy) noted in 1675 that Saturn had multiple rings with gaps between them. In 1787, Pierre-Simon Laplace (France) suggested that the rings consisted of many solid ringlets, a theory that James Clerk Maxwell (UK) disproved in 1859 by showing that solid rings would become unstable and break apart. Maxwell proposed instead that the rings were made of numerous small particles, which was experimentally confirmed by James Keeler (US) and Aristarkh Belopolsky (Russia) in 1895 using spectroscopy.
Boyle’s Law states that as the volume of a gas increases, the pressure of the gas decreases according to an inverse mathematical proportion. The relationship between pressure and volume of gases was first identified by Richard Towneley and Henry Power (UK), but it was Irish scientist Robert Boyle who conducted the experiments that confirmed the relationship and published his results in 1662 with a mathematical formula, the first to accompany a natural law. Boyle’s assistant Robert Hooke (UK) built the experimental apparatus. Edme Mariotte (France) independently reached the same result in 1676.
Robert Hooke (England) first used the word “cell” in 1665 to describe the compartments in a piece of cork he was examining under the microscope. He later identified cells in other organic materials. The importance of cells in biological systems would not be fully recognized until Theodor Schwann and Matthias Schleiden (Germany) proposed their cell theory in 1838-1839.
In the 13th Century, Roger Bacon (England) suggested that rainbows were produced the same way that light produced colors when passed through a glass or crystal. In 1666, Isaac Newton (England) discovered that visible white light is composed of a spectrum of colors. He made this discovery by studying the passage of light through a dispersive prism, which refracted the light into the colors of the rainbow: red, orange, yellow, green, blue and violet. He also found that the multicolored spectrum could be recomposed into white light by a lens and a second prism. He published his results in 1671.
Sir Isaac Newton (England) and Gottfried Leibniz (Germany) independently invented the infinitesimal calculus in the mid-17th Century. Newton appears to have priority over Leibniz, although the question of who was the inventor was the subject of much controversy at the time. An unpublished manuscript of Newton’s supports his claim to have been working on ‘fluxions and fluents’ as early as 1666. Leibniz began his work in 1674 and first introduced the concept of differentials in 1675, which he explained to Newton in a 1677 letter; Leibniz’s first publication on calculus using differentials was in 1684. Newton explained his geometrical form of calculus in his Principia of 1687, but did not publish his fluxional notation until 1693 and not fully until 1704. Precursors to Newton and Leibniz included Pierre de Fermat (France) in 1636, René Descartes (France) in 1637, Blaise Pascal (France) in 1654, John Wallis (England) in 1656, and Newton’s teacher Isaac Barrow (England) in 1669. Bonaventura Cavalieri (Italy) developed his method of indivisibles in the 1630s and 1640s, and computed Cavalieri’s quadrature formula. Evangelista Torricelli (Italy) extended this work to other curves such as the cycloid in the 1640s, and the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. In a 1668 book, James Gregory (England) published the first statement and proof of the fundamental theorem of the calculus, stated geometrically, and only for a subset of curves. Further developments came from Augustin Louis Cauchy (France) in 1821 and Karl Weierstrauss and Bernhard Riemann (Germany) in the 1850s.
While some early scientists, such as Leonardo da Vinci (Italy) in c. 1500 recognized that fossils were the remains of living things, this notion did not gain wide acceptance for many centuries. In 1665, Athanasius Kircher (Germany) suggested that giant fossil bones belonged to a race of giant humans. Also in 1665, Robert Hooke (England) looked at petrified wood through a microscope and suggested that it and fossil seashells were formed when living trees and shells were filled with water containing “stony and earthy particles.” In 1668, Hooke proposed that fossils told us about the history of life on Earth, a radical idea. Danish cleric Nicholas Steno is credited with first identifying the true nature of fossils. In 1667, he dissected a shark’s head and noticed that common fossils called tongue stones were actually shark’s teeth. Steno then began studying rock strata and published in 1669 a work that systematically disproved many of the prior theories about fossils (such as the theory that they grew inside of rocks like crystals). He proposed that fossils were the remains of living organisms that had become buried in layers of sediment, which had then hardened and formed horizontal layers of rock. One of the obstacles to acceptance of Steno’s theory was the existence of fossils of organisms that did not resemble any living creatures. More than a century later, in 1796, Georges Cuvier (France) definitively proved that extinction was a fact. The next advances came from William Smith (UK), who studied fossils in the different layers of rock and, between 1799 and 1819, proposed the law of superposition and the principle of faunal succession, which would allow scientists to compare fossils from different areas.
In 1669, German scientist Hennig Brand isolated a substance from evaporated urine that he called ‘cold fire’ because it glowed in the dark; Brand did not realize it at the time, but he had discovered a new chemical element, phosphorus. Phosphorus is a nonmetallic element with the atomic number 15; it is not found naturally as a pure substance, but is always found in compounds. It was the 13th element to be discovered, but the first since ancient times and the first by an identified individual. Brand sold the secret of his method to Johann Daniel Kraft, Kunckel von Lowenstern and Gottfried Leibniz (Germany). In 1737, someone sold the information to the Academy of Sciences in Paris. In 1769, Carl Wilhelm Scheele and Johann Gottleib Gahn (Switzerland) showed that it was possible to obtain phosphorus from bone ash. In 1777, Antoine Lavoisier (France) recognized that phosphorus was a chemical element.
In the 1670s, Antonie van Leeuwenhoek (The Netherlands), used microscopes of his own design to observe pond water and became the first person to see living organisms that were too small to see with the naked eye. In doing so, he began the field of microbiology. In 1674, Leeuwenhoek first observed the one-celled and multi-celled freshwater creatures formerly labeled the Infusoria, most of which are now categorized as members of the Protista kingdom. Leeuwehnhoek discovered bacteria in 1676. The discovery of these microscopic organisms, or microorganisms (which Leeuwenhoek called ‘animalcules’), led to many further revelations. In 1768 Italian priest and scientist Lazzaro Spallanzani discovered that boiling a liquid containing microorganisms killed them, thus sterilizing the liquid. French scientist Louis Pasteur’s experiments in the 1860s showed that microorganisms did not spontaneously generate in sterilized liquids and that they came from elsewhere; it also helped to prove the germ theory of disease. Beginning in 1876, Robert Koch (Germany) identified the microorganisms that caused specific diseases.
Light is electromagnetic radiation that is visible to the human eye. Scientists now generally accept that light has wavelike and particle-like qualities, but the different aspects of light were discovered over a period of 225 years: (1) light is a beam of particles (1675); (2) light is a wave (1678); light is an electromagnetic force (1862); light has a dual wave-particle nature (1900). In the 5th Century BCE, Empedocles (Ancient Greece) believed our eyes emit beams of light that cause sight. In 300 BCE, Euclid (Ancient Greece) questioned the beam from the eye theory, saying it could only be true if the speed of light was infinite. Lucretius (Ancient Rome) in 55 BCE supposed that light consisted of atoms moving from the sun to the Earth. In the 2nd Century CE, Ptolemy (Ancient Greece/Ancient Rome) was one of the first to attempt an explanation of the refraction of light. Hindu philosophers in India during the early centuries of the common era proposed a particle theory of light, but Indian Buddhists in the 5th and 7th centuries CE suggested that light was composed of atom-like flashes of energy. In 1604, Johannes Kepler found that the intensity of a light source varies inversely with the square of one’s distance from that source. René Descartes (France) theorized in 1637 that light was a mechanical property of the luminous body and the medium transmitting the light. The modern particle theory of light was proposed by Pierre Gassendi and published after his death in the 1660s. Isaac Newton developed Gassendi’s particle theory in 1675, with a final version of the theory published in 1704, stating that corpuscles of light were emitted from a source in all directions. Newton also explained diffraction, polarization and (incorrectly) refraction. Further work on polarization of light was done by Étienne-Louis Malus in 1810 and Jean-Baptiste Biot in 1812. Although Newton’s particle theory held sway for at least a century, others found that light had wavelike properties. Robert Hooke (England) invoked a wave theory of light to explain the origin of colors in 1665 and expanded on the theory in 1672. Christiaan Huygens (The Netherlands) developed a mathematical wave theory of light in 1678. The wave theory predicted interference patterns, which were proven by Thomas Young (UK) in 1801. In 1746, Leonhard Euler (Switzerland) argued that wave theory provided a better explanation for diffraction than particle theory. Augustin-Jean Fresnel (France) developed a separate wave theory in 1817, which received support from Siméon Denis Poisson (France). Measurements of the speed of light in 1850 supported the wave theory. Wave theory suffered a non-fatal blow in 1887. Huygens had proposed that light waves were propagated by a luminiferous aether, but the Michelson-Morley experiment in 1887 proved that the aether did not exist. Meanwhile, as the result of experiments performed in 1845-1847, Michael Faraday (UK) suggested that light was a form of electromagnetic wave, which could be propagated in a vacuum. In 1862 and then in 1873, James Clerk Maxwell (UK) took the results of Faraday’s experiments and provided a mathematical basis for the conclusion that light, electricity and magnetism were all forms of the same wave force he called electromagnetism. Heinrich Hertz (Germany) provided experimental confirmation of Maxwell’s theory by propagating electromagnetic, or radio waves in his laboratory in 1886-1887. In 1900, Max Planck (Germany) proposed that light and other electromagnetic radiation consisted of waves that could gain and lose energy only in finite amounts or quanta. Albert Einstein’s 1905 paper on the photoelectric effect suggested that quanta were real, and Arthur Holly Compton (US) in 1923 showed that certain behavior of X-rays could be explained by particles, but not waves. In 1926, Gilbert N. Lewis (US) named the electromagnetic quanta ‘photons.’ By the mid-20th Century, the scientific community had accepted that light was the visible form of the electromagnetic force (which in turn is part of the electroweak force) and that it had the qualities of both a wave and beam of ‘particles’, that is, discrete packets of energy called photons.
Most early scientists including Aristotle (Ancient Greece, 4th Century BCE), believed that light did not move, or, if it did, it moved at infinite speed. Empedocles (Ancient Greece, 5th Century BCE) was the first to claim that light traveled at a finite speed. Alhazen (Ibn al-Haytham) concluded in his 1021 Book of Optics that light had a finite speed that varied depending on the medium. In the 13th Century, Roger Bacon argued that light traveled through air at a finite speed. Witelo (Poland, 13th Century) suggested that light traveled at infinite speed in a vacuum, but slowed down in air. Johannes Kepler believed light’s speed was infinite, as did Descartes, who claimed that if the speed of light were finite, a lunar eclipse would be out of alignment. Pierre de Fermat assumed that light traveled at a finite speed, the more dense the medium, the slower it traveled. Unsuccessful attempts to measure the speed of light were made by Isaac Beeckman (The Netherlands) in 1629 and Galileo Galilei (Italy), who proposed it in 1638; it was carried out after his death in 1667. Ole Rømer (Denmark) determined in 1676 that light travels at a finite speed, by noting that the period of Jupiter’s innermost moon (Io) appeared shorter when the Earth was approaching Jupiter than when receding from it. From his observations, he calculated that light took 22 minutes to travel from one end of the Earth’s orbit to the other. Christiaan Huygens (The Netherlands) used Rømer’s results to calculate the speed of light as 220,000 kilometers/second. In 1704, Isaac Newton (England), using Rømer’s and Huygens’ results, calculated the time for light to travel from the sun to the Earth as “seven or eight minutes” (actual time: 8 minutes, 19 seconds). James Bradley (England) discovered the aberration of light phenomenon in 1729 and adjusted the calculation of the sun-earth time to 8 minutes, 12 seconds. Hippolyte Fizeau (France) made a calculation of 313,300 km/s in 1849 without using astronomical measurements. Albert Michelson and Edward Morley (US) conducted an experiment in 1887 that measured light at 185,000 miles per second. A 1928 experiment by Michelson refined the speed of light to 186,284 miles per second. The current figure is 186,282 miles per second.
One of the oldest and most common forms of life on Earth, bacteria are usually a few micrometers long. Antonie van Leeuwenhoek (The Netherlands) first observed bacteria in 1676, using a single-lens microscope. After 1773, Otto Frederik Müller distinguished two types of bateria: bacillum (rod-shaped) and spirillum (spiral). Christian Gottfried Ehrenberg (Germany) coined the term ‘bacterium’ in 1828 to describe certain rod-shaped bacteria. Robert Koch (Germany) identified the bacteria that cause anthrax (1875), tuberculosis (1882) and cholera (1883). In 1977, Carl Woese (US) recognized that, based on their ribosomal RNA, some organisms formerly considered bacteria were so genetically different that they belonged to another domain or kingdom, the Archaea.
Binary numbers are numbers expressed in a binary or base-2 numeral system, which normally represents numeric values with the symbols zero and one. A binary code is text or computer processor instructions using the binary number system. An early form of binary system is used in the ancient Chinese book, the I Ching (2000 BCE?). Between the 5th and 2nd Centuries BCE, Indian scholar Pingala invented a binary system. Shao Yong (China) developed a binary system for arranging hexagrams in the 11th Century. Traditional African geomancy such as Ifá used binary systems and French Polynesians on the island of Magareva used a hybrid binary-decimal system before 1450. Francis Bacon invented an encoding system in 1605 that reduced the letters of the alphabet to binary digits. Gottfried Leibniz (Germany), who was aware of the I Ching, invented the modern binary number system and presented it in his 1679 article Explication de l’Arithmétique Binaire. In 1875, Émile Baudot (France) added binary strings to his ciphering system. In 1937, Claude Shannon (US), in his MIT master’s thesis, first combined Boolean logic and binary arithmetic in the context of electronic relays and switches. He showed that relay circuits, being switches, resembled the operations of symbolic logic: two relays in series are and, two relays in parallel are or, and a circuit which can embody not and or can embody if/then. This last meant that a relay circuit could make a choice. Since switches are either on or off, binary mathematics was therefore possible. George Stibitz (US), at Bell Labs, demonstrated a relay-based computer in 1937 that calculated using binary addition. Stibitz and his team made a more complex version called the Complex Number Computer in 1940. Common binary coding systems include ASCII (American Standard Code for Information Interchange) and BCD (binary-coded decimal).
The law of universal gravitation holds that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Sir Isaac Newton (England) articulated the law in the first book of his Philosophiae Naturalis Principia Mathematica, which was presented to the Royal Society in 1686. The law was based in part on Galileo Galilei’s law of falling bodies, which summarized the results of Galileo’s 1589-1590 experiments. Previous scientists had proposed theories of gravity that contained some of the elements of Newton’s law, particularly the theory proposed by Robert Hooke (England) in 1674-1679 (in fact, Hooke accused Newton of stealing the law from him). Newton’s law was superseded by Einstein’s theory of general relativity in 1916.
In his 1687 book Philosophiae Naturalis Principia Mathematica, Sir Isaac Newton (England) established the three laws of motion – (1) the law of inertia; (2) the law of acceleration; and (3) the law of action and reaction – and derived the mathematical basis for the laws. These laws and Newton’s law of universal gravitation formed the basis for the science of physics for more than 200 years. In the early 20th Century, classical mechanics was displaced by the theories of relativity and quantum mechanics, but Newton’s laws still accurately explain the behavior of most objects in most environments familiar to humans.
Humans have been using the steam from boiling water to do mechanical work since ancient times, but practical designs only arrived in the 17th Century. Jerónimo de Ayanz y Beaumont (Spain) patented a steam engine in 1606 for removing water from mines. In 1679, Denis Papin (France) developed a steam digester, a precursor to the steam engine. Briton Thomas Savery’s pistonless steam pump of 1698 was the first practical steam engine design. In 1712, Thomas Newcomen (England) created a piston-driven “atmospheric-engine” that proved to be the first commercially viable steam engine. In 1725, Savery and Newcomen built a steam engine for pumping water from collieries. Between 1765 and 1774, James Watt (England) improved on the Newcomen engine by making it condensing and double acting, which hugely increased its efficiency. A high pressure steam engine was developed by Oliver Evans (US) in 1804. Further improvements followed throughout the 19th Century.
While comets have been observed since ancient times, the first modern scientific theory of comets was developed by Tycho Brahe (Denmark), who measured the parallax of the Great Comet of 1577 and determined that it must exist outside the Earth’s atmosphere. Sir Isaac Newton (England) demonstrated the orbit of the comet of 1680 in his Principia of 1687. In 1705, Edmund Halley (England) analyzed 23 appearances of comets between 1337 and 1698 and concluded that three of the appearances were the same comet, which he predicted would return in 1758-1759. (Three French mathematicians further refined the date.) When the comet returned as scheduled, it was named Halley’s Comet. Scientists of the 17th, 18th and 19th centuries proposed various theories for the composition of comets, but it was Fred Lawrence Whipple (US) who suggested in 1950 that comets were made of ice mixed with dust and rock – the ‘dirty snowball’ theory. A number of observations appeared to confirm this view, but in 2001, high resolution images of Comet Borrelly by a US probe showed no ice, only a hot, dry, dark surface. Another US probe, which crashed into Comet Tempel 1 in 2005, found that most of the ice is beneath the surface.
The flying shuttle was invented by John Kay (England) in 1733. Prior to Kay’s invention, shuttles containing the thread were thrown or passed through the threads of a loom by hand, and a single weaver could only work on a fabric that was as wide as her arms’ reach, unless another weaver was seating beside her to whom she could pass the shuttle. The flying shuttle was mounted on wheels in a track, and the weavers used paddles to shoot the shuttle from side to side when the weaver jerked a cord. A single weaver using a flying shuttle could weave fabrics of any width more quickly than two weavers could using the old system. The flying shuttle was a step in the direction of automatic machine looms, which would change weaving from a cottage industry to mass production in factories. Unfortunately, John Kay received no respect for his innovation. In 1753, angry weavers raided Kay’s establishment and destroyed his looms. Later, after the weavers recognized the benefit of the flying shuttle, they adopted it themselves but refused to pay Kay any royalties. His attempts to defend his patent eventually bankrupted him.
Aristotle (Ancient Greece) was the first to systematically classify living things into categories in the 4th Century BCE when he introduced the concepts of genus and species. In 1552, Konrad Gesner (Switzerland) developed a system that distinguished genus from species and order from class. Early classification systems were developed by Andrea Caesalpino (Italy) in 1583; John Ray (UK) in 1686; Augustus Quirinus Rivinus (Germany) in 1690; and Joseph Pitton de Tournefort (France) in 1694. It was Carolus Linnaeus (Sweden) who, beginning in 1735, developed the modern system of taxonomy for living organisms by establishing three kingdoms divided into classes, orders, families, genera and species. Linnaeus classified animals and plants based on their physical characteristics. He also adopted the binomial system of naming each organism with a unique combination of genus and species names. Since the 1960s, biologists have adapted Linnaean taxonomy to include evolutionary relationships, instead of relying on physical characteristics only.
According to the kinetic theory of gases, a gas consists of a large number of atoms or molecules in constant, random motion that constantly collide with each other and the walls of a container. Lucretius (Ancient Rome) proposed in 50 BCE that objects were composed of tiny rapidly moving atoms that bounced off each other. Daniel Bernoulli (Switzerland) proposed the kinetic theory of gases in 1738. He proposed that gas pressure is caused by the impact of gas molecules hitting a surface and heat is equivalent to the kinetic energy of the molecules’ motion. Other advocates of the kinetic theory included: Mikhail Lomonosov (Russia, 1747), Georges-Louis Le Sage (Switzerland, ca. 1780, published 1818), John Herapath (UK, 1816)John James Waterston (UK, 1843), August Krönig (Germany, 1856) and Rudolf Clausius (Germany, 1857). James Clerk Maxwell (UK) formulated the Maxwell distribution of molecular velocities in 1859, and Ludwig Boltzmann (Austria) formulated the Maxwell-Boltzmann distribution in 1871. In papers on Brownian motion, Albert Einstein (Germany, 1905) and Marian Smoluchowski (Poland, 1906) made testable predictions based on kinetic theory.
The Leyden jar is the prototype electrical condenser and the first capacitor, which could store static electric charge. It was invented independently in 1745 by German cleric Ewald Georg von Kleist and in 1746 by Dutch scientists Pieter van Musschenbroek and Andreas Cunaeus at the University of Leyden, The Netherlands. Leyden jars were used in many early experiments on electricity. Daniel Gralath (Poland) was the first to join multiple Leyden jars to each other in parallel to increase the stored charge, a formation for which Benjamin Franklin (US) coined the term ‘battery.’
William Cullen (Scotland) invented artificial refrigeration at the University of Glasgow in 1748. Oliver Evans (US) created the vapor-compression refrigeration process in 1805. Jacob Perkins (US) took Evans’s process and built the first actual refrigerator in 1834. John Gorrie (US) invented the first mechanical refrigeration unit in 1841. Further improvements were made by Alexander Twining (US) in 1853; James Harrison (Scotland/Australia) in 1856; Ferdinand Carré (France) in 1859; Andrew Muhl (France/US) in 1867 and Carl von Linde (Germany) in 1895. Electrolux produced the first electric refrigerator in 1923.
According to the law of conservation of mass, the mass of any system that is closed to all transfers of matter and energy must remain constant over time. The law has ancient roots. The Jains in 6th Century BCE India believed that the universe and its constituents cannot be created or destroyed. In Ancient Greece, Empedocles said in the 5th Century BCE that nothing can come from nothing and once something exists it can never be completely destroyed, a belief echoed by Epicurus in the 3rd Century BCE. Persian philosopher Nasir al-Din al-Tusi stated a version of the law in the mid-13th Century. The first modern scientific statement of the law came from Mikhail Lomonosov (Russia) in 1748. Although Antoine Lavoisier (France) is often credited with discovering the law in 1774, precursors (in addition to Lomonosov) include Jean Rey (France, 1583-1645), Joseph Black (Scotland, 1728-1799) and Henry Cavendish (UK, 1731-1810).
A lightning rod is a metal rod or other object mounted on top of a building or other elevated structure that is electrically bonded using a wire or electrical conductor to connect with a ground through an electrode, in order to protect the structure if lightning hits it. For thousands of years, builders in Sri Lanka have protected buildings from lightning by installing metal tips made of silver or copper on the highest point. The Leaning Tower of Nevyansk in Russia, which was built between 1721 and 1745, is crowned with a metal rod that is grounded and pierces the entire building, but it is not known whether it was intended as a lightning rod. Benjamin Franklin (US) invented a lightning rod in 1749. Prokop Diviš (Bohemia) independently invented the grounded lightning rod in 1754.
Lightning was in the air in the late 1740s and early 1750s. Benjamin Franklin (US) listed a dozen analogies between lightning and electricity in his notebooks in 1749. Similar speculation by Jean Antoine Nollet (France) led to a contest on the topic, which was won in 1750 by Denis Barbaret (France), who said lightning was caused by the triboelectric effect. Jacques de Romas (France) proposed a similar theory in a 1750 memoir; he also claimed to have suggested a test of the theory using a kite. In 1752, Franklin proposed to test the theory by using rods to attract lightning to a Leyden jar. The experiment was carried out by Thomas-François Dalibard in May 1752 and by Franklin himself in June 1752, but using a kite instead of a rod. He attached a key to the kite string, which was connected to a Leyden jar. Although the kite was not struck by lightning, static electricity was conducted to the key, and Franklin felt a shock when he moved his hand near the key. Georg Wilhelm Richmann (Germany/Russia) was killed by electrocution while attempting to recreate the experiment in St. Petersburg in 1753.
A marine chronometer is a clock that is accurate enough to be a portable time standard, which can be used to determine longitude by using celestial navigation. Until the 18th Centuries, navigators were only able to determine the latitude of a ship at sea, not its longitude. Gemma Frisius (The Netherlands) suggested in 1530 that a highly accurate clock could be used to calculate longitude. In the 17th Century, Galileo Galilei (Italy), Edmund Halley (England), Tobias Mayer (Germany) and Nevil Maskelyne (England) proposed observations of astronomical objects as the solution, but the deck of a ship at sea proved too unstable for accurate measurements. Recognizing that his pendulum clock would not be effective at sea, Christiaan Huygens (The Netherlands) invented a chronometer in 1675 with a balance wheel and a spiral spring, but it proved too inaccurate in nautical conditions. Similar problems plagued the chronometers made by Jeremy Thacker (England) in 1714 and Henry Sully (France) in 1716. In 1714, the British government offer a large cash reward for anyone who could invent an accurate chronometer. John Harrison (England) submitted versions in 1730, 1735 and 1741, although they were all sensitive to centrifugal force. A 1759 version, with a bi-metallic strip and caged roller bearings, was even more accurate, but it was the much smaller 1761 design that won Harrison the £20,000 prize in 1765. French clockmaker Pierre Le Roy’s 1766 chronometer, with a detente escapement, temperature-compensated balance and isochronous balance spring, was the first modern design. Thomas Earnshaw and John Arnold developed an improved version with Le Roy’s innovations in 1780, which led to the standard chronometer used for many years afterwards.
Latent heat is the energy released or absorbed by an object or thermodynamic system during a constant-temperature process, particularly when that object or system undergoes a change in state. Jean André Deluc (France) conducted experiments on the melting of ice in 1754-1756 in which he showed that the temperature of the ice stopped rising at the moment the ice began to melt. Deluc theorized that the additional heat was used to convert the ice into water. Joseph Black (England) introduced the term ‘latent heat’ in 1762 in the context of calorimetry and distinguished it from sensible heat. Both are forms of heat, but sensible heat is detectable by a thermometer, while latent heat is not. In the mid-19th Century, James Prescott Joule (UK) analyzed latent heat as a form of potential energy, that is, the energy of interaction in a configuration of atoms or molecules, and sensible heat as a form of thermal energy, that is, the energy indicated by the thermometer.
The spinning jenny is the multi-spindle spinning machine invented by James Hargreaves (England) in 1764 that allowed a group of eight spindles to be operated together. Dependent on the recently-invented flying shuttle, the spinning jenny held more than one ball of yarn so it could make more yarn in a shorter time, thus reducing cost and increasing productivity. The technology was replaced in about 1810 by the spinning mule.
In the mid-18th Century, the English textile industry was growing, and its machines were becoming faster. The flying shuttle had doubled loom speed, and the invention of the spinning jenny in 1764 had also increased speed and production. John Kay and Thomas Highs (England) had designed a new machine called the spinning frame, which produced a stronger thread than the spinning jenny. The spinning frame used the draw rollers invented by Lewis Paul to stretch the yarn. In 1769, Sir Richard Arkwright (England) asked John Kay to produce the spinning frame for him. Because the spinning frame was too large to be operated by hand, Arkwright experimented with other power sources, trying horses first and then switching to the water wheel. Unlike the spinning jenny, which was inexpensive but required skilled labor, the spinning frame required considerable capital outlay but little skill to operate.
Carl Wilhelm Scheele (Sweden) was the first to create oxygen gas and identify it as a separate element in 1772, although he did not publish his discovery until 1777. Joseph Priestley (UK) also isolated oxygen in 1774 and published in 1775. Prior to these discoveries, 17th Century scientists Robert Boyle (Ireland) determined that air was necessary for combustion and John Mayow (England) discovered that a portion of the air was necessary for combustion and respiration. Research in the 17th and 18th centuries was slowed by the phlogiston theory, which held that when a substance burned it released phlogiston into the air, and the reason some substances burned more completely than others was that they consisted of a higher proportion of phlogiston. Although most historians dispute claims by Antoine Laurent Lavoisier (France) that he also discovered oxygen in 1774, Lavoisier did discover the nature of combustion and conducted important experiments on oxidation. His work also definitely disproved the phlogiston theory.
Combustion is a sequence of exothermic chemical reactions between a fuel and an oxidant that is accompanied by the production of heat and the conversion of chemical species. The release of heat can produce light in the form of glowing or flames. Modern scientific attempts to determine the nature of combustion began in 1620, when Francis Bacon (England) observed that a candle flame has a structure. At about the same time, Robert Fludd (England) described an experiment in a closed container in which he determined that a burning flame used up some of the air. Otto von Guericke (Germany) demonstrated in 1650 that a candle would not burn in a vacuum. Robert Hooke (England) suggested in 1665 that air had an active component that, combined with combustible substances when heated, caused flame. Antoine-Laurent Lavoisier (France) was the first to give an accurate account of combustion when in 1772 he found that the products of burned sulfur or phosphorus outweighed the initial substances, and he proposed that the additional weight was due to the combining of the substances with air. Later Lavoisier concluded that the part of the air that had combined with the sulfur was the same as the gas released when English chemist Joseph Priestley heated the metallic ash of mercury, which was the same as the gas described by Carl Wilhelm Scheele (Sweden) as the active fraction of air that sustained combustion. Lavoisier named the gas found by Priestley and Scheele, “oxygen.”
Flemish scientist Jan van Helmont discovered in the mid-17th Century that the mass of the soil used by a plant changed very little as the plant’s mass increased. He hypothesized that the additional mass came from the added water. In 1774-1777, Joseph Priestley (England) published the results of experiments in which he burned a candle in a sealed jar, it quickly stopped burning, and that a mouse trapped in a jar would soon stop breathing, but he found that if he added a plant to the jar, both mouse and candle would continue to flourish. Priestley concluded that plants make and absorb gases. Following up on Priestley’s experiments, Jan Ingenhousz (The Netherlands) discovered that when light is present, plants give off bubbles from their green parts, which he identified as oxygen, but not in the shade, and that it was the oxygen that revived Priestley’s mouse. He also discovered that plants give off carbon dioxide in the dark, but that the amount of oxygen given off in the light is greater than the amount of carbon dioxide given off in the dark. In 1796, Jean Senebier (Switzerland) confirmed Ingenhousz’s finding that plants release oxygen in the light, and also found that they consume carbon dioxide in the light. Calculations by Nicolas-Théodore de Saussure (Switzerland) in the late 1790s showed that the increase in the plant’s mass was due to both carbon dioxide and water. Charles Reid Barnes (US) proposed the term ‘photosynthesis’ in 1893. In 1931, Cornelis Van Niel (The Netherlands/US) studied the chemistry of photosynthesis and demonstrated that photosynthesis is a light-dependent reaction in which hydrogen reduces carbon dioxide. Also in the 1930s, scientists proved that the oxygen liberated in photosynthesis comes from water.
The spinning mule was a machine used to spin cotton and other fibers between the late 18th Century and the beginning of the 20th Century. After the invention of the more productive flying shuttle loom in 1733, traditional spinners could no longer supply enough thread and the industry sought faster technologies. Beginning in 1764, the spinning jenny allowed eight spindles to operate together, which increased the amount of thread available. Samuel Crompton (England) invented the spinning mule in 1779 but due to lack of funds, he did not obtain a patent. Crompton’s original wooden machine had 48 spindles and could produce a pound of thread a day. Shortly thereafter, Henry Stones (England) made an improved spinning mule with toothed gearing and metal rollers. Other advancements were made by William Kelly (Scotland) in 1790; Wright of Manchester; John Kennedy (England) in 1793; and William Eaton (England) in 1818. A major innovation was the self-acting or automatic spinning mule, invented by Richard Roberts (England) in 1825, with an improved version in 1830. By the end of the 19th Century, a typical spinning mule was 150 feet long, had up to 1320 spindles, and moved back and forth a distance of five feet four times a minute. Modern versions of the spinning mule are still used to make yarn from fine fibers such as cashmere, merino and alpaca.
Uranus, the seventh planet from the sun, had been recognized possibly as early as 188 BCE and also by John Flamsteed (England) in 1690 and Pierre Lemonnier (France) between 1750 and 1769, but it was not identified as a planet due to its dimness and slow orbit. William Herschel (England) first observed Uranus in March 1781, although he first identified it as a comet. When Anders Johan Lexell computed the object’s orbit, he concluded it was a planet, not a comet, as did Johann Elert Bode (Germany). Herschel acknowledged that he had discovered a new planet in 1783. Herschel suggested the name Georgium Sidus, after King George III, but it was Bode’s suggestion of Uranus, the father of Saturn, that eventually won out.
The ability to raise small unmanned balloons into the air using hot air was known in China from the 3rd Century CE. Jacques and Joseph Montgolfier (France) built the first hot-air balloons capable of carrying human passengers in France in the late 18th Century. They tested their design first with no passengers on June 4, 1783, then on September 19, 1783 with a sheep, a duck and a rooster, who survived an eight-minute flight. Then, on November 21, 1783, French scientist Pilâtre de Rozier and the Marquis d’Arlandes, an Army officer, climbed aboard a Montgolfier balloon to make the first untethered manned flight. They traveled for 25 minutes, covered a distance of five miles and attained an altitude of 3,000 feet before safely landing. Among those in the audience were King Louis XVI and Benjamin Franklin.
Leonardo da Vinci (Italy) designed the first parachute between 1470 and 1485, but there is no evidence that he or anyone else actually built one to see if it would work. Leonardo’s plan required a fixed triangular frame made of wood and covered with linen. It wasn’t until 2000 that the design was put to the test – Adrian Nicholas (UK) built a prototype using only materials available in 15th Century Milan and jumped out of a hot-air balloon at 10,000 feet. Fortunately for all involved, it worked. Louis-Sébastien Lenormand (France) made the first modern parachute, which consisted of linen stretched over a wooden frame. He demonstrated the invention in 1783 by jumping off the tower of the Montpellier observatory. In 1785, Jean-Pierre Blanchard (France) showed that a parachute could be used to disembark from a hot-air balloon. Blanchard made the first folded silk parachute in the late 1790s. In 1797, André Garnerin (France) was the first to make a jump using the silk design. Garnerin also invented the vented parachute, which had a more stable descent. The first person to make a parachute jump from an airplane was either Grant Morton (US) in 1911 or Albert Berry (US) in 1912. (Some have claimed that Morton’s jump took place in 1912, after Berry’s.)
Benjamin Franklin (US) invented bifocals to solve the problem of switching between two types of glasses – one for seeing close-up and the other for distance. He refers to the bifocals in a 1784 letter, but there is evidence that he invented them many years before, perhaps as early as 1760. At first, bifocals were made by fitting two half lenses in the frame. Louis de Wecker (Germany) invented a method for fusing the lenses in the late 19th Century, which was patented by John L. Borsch, Jr. (US) in 1908. The first progressive lens was patented in 1907 by Owen Aves (UK), but it was never produced commercially. After intermediate developments by H. Newbold (UK) in 1913 and Stewart Duke-Elder (UK) in 1922, Irving Rips (US), of Younger Optics, created the first commercially successful seamless bifocal in 1955. The first modern progressive lenses were developed by Bernard Maitenaz (France), patented in 1953 and offered commercially in 1959.
Although historians do not believe that ancient people knew that fingerprints are unique to the individual, they had some understanding that they were useful in identification as far back as the 2nd Millenium BCE, when the Babylonians used them as signatures. To avoid forgeries, contracting parties pressed their fingerprints into the clay tablet that the contract was written on. The Code of Hammurabi, from the 18th Century BCE, directed officials to obtain the fingerprints of everyone they arrested. By 246 BCE, it was the practice of Chinese officials to place their fingerprints on the clay used to seal documents. Parties to Chinese contracts written on paper placed their handprints on the documents. In the Qin Dynasty (221-206 BCE), Chinese officials took fingerprints (and also hand prints and foot prints) as evidence from a crime scene. Also in China, fingerprints were used as authentication by 650 CE. Sometime before 851 CE, Chinese merchants began using fingerprints to authenticate loans. By 702 CE, illiterate petitioners seeking a divorce in Japan could sign the papers with a fingerprint. Thumbprints were used to authenticate government documents in Persia in the early 14th Century. It was not until 1788 that German anatomist Johann Christoph Andreas Mayer recognized that fingerprints were unique to every individual. In 1858, Sir William James Herschel (UK) began using fingerprints on contracts and deeds; he also registered government pensioners’ fingerprints to prevent fraud and fingerprinted prisoners upon sentencing. Scottish surgeon Henry Faulds published a paper in 1880 discussing the use of fingerprints for identification; he also established the first classification of fingerprints, which was further developed by Francis Galton (UK), a cousin of Charles Darwin’s, in a series of articles and books between 1888 and 1895. Croatian-born Argentine police chief Juan Vucetich developed a system of recording fingerprints in 1892 by adapting the anthropometric system of Alphones Bertillion (France). In 1897, two men working in the Calcutta Anthropometric Bureau, Azizul Haque and Hem Chandra Bose (India), while working under Sir. Edward Richard Henry (Ireland/UK), developed the Henry Classification System for classifying fingerprints, which was eventually adopted throughout the United Kingdom. Fingerprinting was introduced in the US by Henry P. DeForrest in 1902 (New York Civil Service) and Joseph A. Faurot in 1906 (New York City Police Department).
A cotton gin separates the cotton seeds from the fibers, a task previously done by hand. Primitive labor-intensive gins had been invented in India (5th Century CE) and elsewhere, but American Eli Whitney’s 1793 hand-powered cotton gin was the first mechanical cotton gin that efficiently separated fibers and seeds from large amounts of cotton. Whitney’s invention revolutionized the U.S. cotton industry and led to the growth of slave labor in the South. Modern cotton gins are automated and much more productive than Whitney’s original.
The Rosetta Stone is a block of granodiorite measuring 3 ft., 9 in. tall and 2 ft. 4 in. wide that contains a 196 BCE decree from Ancient Egyptian King Ptolemy inscribed in three languages: Egyptian hieroglyphics, Egyptian demotic script and Ancient Greek. The Rosetta Stone was the key to deciphering Egyptian hieroglyphics. The Stone was found near the town of Rashid (Rosetta) in the Nile Delta by French soldier Pierre-François Bouchard in 1799 but became the property of the British as a result of the Capitulation of Alexandria. The Rosetta Stone has been on display at the British Museum in London since 1802. Jean-François Champollion (France) published the first translation of the Rosetta Stone hieroglyphics in 1822.
Although there is some evidence of primitive batteries from the first centuries of the common era in Mesopotamia and India, the modern precursor to the electric battery was the Leyden Jar, which was invented in 1745-1746. Benjamin Franklin coined the term ‘battery’ to describe a set of linked Leyden jars he used for experiments. Then, in 1791, Italian scientist Alessandro Volta published the results of experiments showing that two metals joined by a moist intermediary could create electric energy. In 1800, Volta used this principle to create the first true battery, called a voltaic pile. Over the next century many scientists developed Volta’s invention further: William Cruickshank (UK) invented the trough battery in 1800; William Sturgeon (UK) improved upon the design in 1835; John Daniell (UK) invented the Daneill cell in 1836; Golding Bird (UK) invented the Bird cell in 1837; John Dancer (UK) invented the porous pot Daniell cell in 1838; William Grove (Wales) invented the Grove cell in 1844; Gaston Planté (France) invented the lead-acid battery in 1859; Callaud (France) created the gravity cell in the 1860s; Johann Poggendorff (Germany) created the Poggendorff cell; Georges Leclanché (France) invented the Lelanché cell in 1866; and the first dry cells were invented independently by Carl Gassner (Germany), Frederick Hellesen (Denmark) and Yai Sakizo (Japan) in 1886-1887.
Infrared is energy that lies on a portion of the spectrum of electromagnetic radiation that is invisible and has longer wavelengths than visible light. Most thermal radiation emitted by objects near room temperature is infrared. French scientist Émilie du Châtelet predicted the existence of infrared radiation in 1737. William Herschel (Germany/UK) discovered infrared radiation in 1800, although he referred to it as ‘calorific rays.’ American inventor Theodore Case created an infrared signaling system for the US military in 1917. In 1945, Germany developed the Zielgerät 1229, the first portable infrared device for military use. In the 1950s, Paul Kruse at Honeywell and scientists at Texas Instruments recorded the first infrared images.
Denis Papin (France) made a ship powered by his steam engine, mechanically linked to paddles, in 1704, although it did not create sufficient pressure to be practical. Jonathan Hulls (England) received a patent for a Newcomen steamboat in 1736, but there is little evidence of any real success. William Henry (US) built several steamboats in 1763 and after but had little success with them. Marquis Claude de Jouffroy (France) made the first working steam-powered ship in 1783, the paddle steamer Pyroscaphe, which worked for 15 minutes. John Fitch (US) and William Symington (Scotland) made similar boats in 1785. Symington and Patrick Miller (Scotland) made a boat with manually-cranked paddle wheels between double hulls in 1785, with a successful try-out in 1788. Using Symington’s design, Alexander Hart (UK) built and launched a successful steamboat in 1801. The same year, Symington designed a second steamboat with a horizontal steam engine linked directly to a crank, the Charlotte Dundas, which was built by John Allan and the Carron Company. Its maiden voyage was in 1803. The same year, Robert Fulton (US) observed the Charlotte Dundas and, with engineer Henry Bell (UK) designed his own steamboat, which he sailed on the Seine in 1803. Fulton then brought the boat to the US where as the North River Steamboat (later the Clermont), it carried passengers between New York City and Albany, New York in 1807. Other names in the steamboat saga include: J.C. Perier (France), 1775; James Rumsey (US), 1787; and Oliver Evans (US), 1804.
In 1804, Richard Trevithick’s first steam locomotive pulled a train containing 10 tons of iron and 70 passengers in five cars approximately nine miles near Merthyr Tydfil in Wales. The first commercially successful steam locomotives were built by Matthew Murray (UK) in 1812 (Salamanca); and Christopher Blackett & William Hedley (UK) in 1813 (Puffing Billy). George Stephenson (UK) improved on Trevithick’s and Hedley’s designs by adding a multiple fire tube boiler in 1814 with the Blücher and again in 1825 with the Locomotion and in 1929 with The Rocket. The largest steam-powered locomotive was the Union Pacific’s Big Boy (US) of 1941. Steam locomotives were gradually phased out in the first half of the 20th Century, to be replaced by diesel and electric locomotives.
In 1808, John Dalton (UK) theorized that all matter was made of very small indivisible particles called atoms, that each element is made of different atoms, that each element’s atoms are identical, that atoms combine to make chemical compounds and are combined, separated or rearranged in chemical reactions.
Although the notion that biological organisms change over time has ancient roots, the first fully-developed theory of evolution, or transmutation of species, was proposed in 1809 by Jean-Baptiste Lamarck (France) in his Zoological Philosophy. Early formulations of the idea of evolution come from Epicurus (Ancient Greece) in the 3rd Century BCE; Lucretius (Ancient Rome) in the 1st Century BCE; Augustine of Hippo in the 4th Century CE; and Ibn Khaldun (Tunisia) in 1377. More sophisticated concepts of evolution, with or without divine intervention, came from Gottfried Leibniz (Germany) in the early 18th Century; Benoît de Maillet (France) in 1748; and Pierre Louis Maupertuis (France) in 1751. Charles Bonnet (Switzerland) first used the term evolution to refer to species development in 1762. Between 1749 and 1788, G. L. L. Buffon (France) suggested that each species is just a well-marked variety that was modified from an original form by environmental factors. In 1753, Denis Diderot (France) wrote that species were always changing through a constant process of experiment where new forms arose and survived or not based on trial and error. James Burnett, Lord Monboddo (England), suggested between 1767 and 1792 that man had descended from apes and that organisms had transformed their characteristics over long periods of time in response to their environments. In 1796, Charles Darwin’s grandfather, Erasmus Darwin, published Zoönomia, which proposed that “all warm-blooded animals have arisen from one living filament”, a theme he developed in his 1802 poem Temple of Nature. The mechanism for evolution was a source of much controversy. Lamarck proposed that organisms acquired new characteristics during their lifespans (such as longer necks from stretching to reach food on trees), which they then passed down to their offspring. He also believed in spontaneous generation of species. Many scientists rejected these ideas. Prominent evolutionists in the years after Lamarck included Étienne Geoffroy Saint-Hilaire (France), Robert Grant (UK) (whose pupils included a young Charles Darwin), Robert Jameson (UK); and Robert Chambers (UK), whose anonymous Vestiges of the Natural History of Creation proposed that evolution was progressively leading to better and better organisms. It was not until 1858-1859 that Charles Darwin and Alfred Russell Wallace provided a convincing mechanism for evolution: natural selection. In the years after Darwin, developments in genetics, molecular biology and paleontology have brought about many changes to the field now known as evolutionary biology.
The atomists of Ancient Greece theorized that different atoms connected to one another in different ways depending on the substance involved. Iron atoms, they supposed, had hooks to connect to other iron atoms, while water atoms were slippery. When the atom theory saw a resurgence in the 17th Century, Pierre Gassendi (France) adopted some of the Ancient Greek ideas. Isaac Newton (England) on the other hand, suggested in 1704 that particles attract one another by a force that is strong at short distances. Irish chemist Robert Boyle first discussed the concept of the molecule in his 1661 treatise, The Sceptical Chymist, in which he suggested that matter is made of clusters of particles or corpuscles of various shapes and sizes and that chemical reactions rearrange those clusters. In 1680, Nicolas Lemery hypothesized that acidic substances had points, while alkalis had pores, and the points locked into the pores to create Boyle’s clusters. In 1738, Daniel Bernoulli (Switzerland) proposed his kinetic theory of gases, which presumed that gases consist of great numbers of clusters of atoms. William Higgins (Ireland) proposed a theory describing the behavior of clusters of ultimate particles in 1789. John Dalton (UK) published in 1803 the atomic weight of the smallest amount of certain compounds. Italian chemist Amedeo Avogadro published a paper in 1811 that coined the word “molecule”, although he used it to refer to both molecules and atoms. Later, in setting out Avogadro’s Law, Avogadro distinguished between atoms and molecules for the first time. Jean-Baptiste Dumas (France) built on Avogadro’s findings in 1826 and Marc Antoine Auguste Gaudin (France) clearly stated the implications of the molecular hypothesis in 1833, where he suggests molecular geometries and molecular formulas that are consistent with atomic weights. In 1857-1858, German chemist Friedrich August Kekulé proposed that every atom in an organic molecule was bonded to every other atom, and he showed how carbon skeletons could form in organic molecules. At about the same time, Archibald Couper (UK) developed a theory of molecular structure complete with a new form of notation very similar to that used today. In 1861, Joseph Loschmidt (Austria) self published a booklet with a number of new molecular structures. August Wilhelm von Hofmann (Germany) made the first stick and ball models of molecules in 1865. Summing up the knowledge gained so far, James Clerk Maxwell (UK) published an article in 1873 entitled, ‘Molecules’ in which he defined a molecule as “the smallest possible portion of a particular substance.”
In 1811, Amedeo Avogadro (Italy), set out the law that equal volumes of all gases at the same temperature and pressure contain the same number of molecules. This law had the effect of reconciling Joseph Louis Gay-Lussac’s 1808 law on volumes and combining gases with John Dalton’s atomic theory.
Hans Christian Ørsted (Denmark) noticed an interaction between electricity and magnetism in 1820, but it was French scientist André-Marie Ampère’s follow-up experiments that demonstrated the unity of electricity and magnetism. Beginning in 1831, Michael Faraday (England) discovered electromagnetic induction, diamagnetism and electrolysis and invented the first current-generating electric dynamo. Joseph Henry (US) discovered induction at about the same time. James Clerk Maxwell (England) linked electricity, magnetism and light mathematically 1861-1862. In 1866, Werner von Siemens (Germany) invented an industrial generator that didn’t need external magnetic power. In 1882, Thomas Edison (US) built the first large-scale electrical supply network, which provided 110 volts of direct current (DC) to 59 homes in Manhattan. In the late 1880s, George Westinghouse (US) set up a rival system using alternating current (AC), using an induction motor and transformer invented by Nikola Tesla (Serbia/US). AC eventually prevailed over DC. Another key invention was Sir Charles Parsons’ steam turbine, from 1884, which provides the mechanical power that creates most of the world’s electricity.
Hans Christian Ørsted (Denmark) set the stage for the invention of the electromagnet in 1820, when he discovered that electric currents create magnetic fields. The first electromagnet was invented by William Sturgeon (UK) in 1824; he wrapped a bare copper wire around a varnished piece of iron and then ran an electric current through the wire, which magnetized the iron. Joseph Henry (US) improved on Sturgeon’s design beginning in 1827, using insulated wire to create much more powerful magnets, including one that could support more than 2000 pounds.
The second law of thermodynamics states that the entropy of an isolated system never decreases, because isolated systems always evolve toward thermodynamic equilibrium, which is a state with maximum entropy. The earliest statement of the law was by Sadi Carnot (France) in 1824, who, while studying steam engines, postulated that no reversible processes exist in nature. Beginning in 1850, Rudolph Clausius (Germany) set out the first and second laws of thermodynamics, although it is his 1854 formulation that was most highly regarded: “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.” William Thomson, Lord Kelvin (UK) reformulated the second law in 1851 as: “It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.”
Concrete is a composite material composed of coarse granular aggregate embedded in a matrix of cement or other binder that fills the space among the aggregate particles and glues them together. There is evidence that ancient humans used forms of concrete as long ago as 1800 BCE. The Ancient Romans, in particular, built extensively with concrete. The Pantheon contains the largest unreinforced concrete dome in the world. The next advance in concrete technology didn’t arrive until 1793, when John Smeaton (UK) developed a new more effective way to produce lime for cement. In 1824, Joseph Aspdin (UK) ushered in the world of modern concrete when he developed Portland cement, which is still the cement used in today’s concrete. The next step was taken in 1867, when Joseph Monier, a French gardener, invented reinforced concrete.
The Carnot cycle is a theoretical thermodynamic cycle that is the most efficient cycle for converting a given amount of thermal energy into work, or conversely, for creating a temperature difference by doing a given amount of work. Nicolas Léonard Sadi Carnot (France) proposed the Carnot cycle in 1824, when he also proposed the Carnot engine, a hypothetical machine that converts heat into work or vice versa. In the 1850s, Rudolf Clausius (Germany) and William Thomson, Lord Kelvin (UK) realized that the Carnot engine converts only part of the heat into work. A sizable amount of remaining heat is given off to the cold reservoir and the first law of thermodynamics states that the remaining heat plus delivered work are equal to the input heat. Although Carnot’s ideas were in contradiction to the first law, nevertheless they proved to be very useful to Clausius and Thomson, when they postulated the second law of thermodynamics.
Joseph Nicéphore Niépce (France) made the first photograph on a piece of bitumen-covered metal in 1826. His photographic method required an exposure of eight hours or more, and the final image was only viewable when held at an angle. In 1835 William Talbot (UK) created the calotype, a photographic method that created a negative image from which multiple positive prints could be made. In 1837, Louis-Jacques-Mandé Daguerre (France) invented the daguerrotype process, which created much more detailed images in a much shorter time. Unfortunately, there was no way to make multiple copies of the daguerrotypes. The wet-plate collodion process also saw a brief period of popularity in the 19th Century. Alexandre Becquerel and Claude Niepce de Saint-Victor (France) produced the first color images between 1848–1860. John Carbutt (US) produced the first commercially successful celluloid film in 1888. Also in 1888, George Eastman introduced the hand-held Kodak camera with roll film. Kodak also introduced the first commercial color film with three emulsion layers, Kodachrome, in 1935. In the late 20th Century, digital photography became the standard, replacing film.
There are references in China to the use of early matches made with sulfur at the end of a small stick of pinewood from 577 CE, 950 CE and 1270 CE. The first modern match was invented in 1805 by Jean Chancel (France). It was ignited by dipping the tip in an asbestos bottle filled with sulfuric acid, making it expensive, dangerous and inconvenient. Samuel Jones (UK) used a similar approach with his Promethean Match of 1828, which consisted of a small glass capsule containing sulfuric acid colored with indigo and coated with potassium chlorate, wrapped in a roll of paper. Crushing the capsule with pliers created a flame. A crude friction match invented by François Derosne (France) in 1816 required the user to scrape the head inside a phosphorus-coated tube. John Walker, an English chemist, invented the first true friction match in 1826. It was lit by pulling it through a folded piece of sandpaper. Unfortunately, his design was liable to create flaming balls that lit clothing and carpets on fire. Improvements were made by Scottish inventor Isaac Holden in 1829 and a version of Holden’s match was patented by Samuel Jones (UK). Charles Sauria (France) improved the chemical formula in 1830 by adding white phosphorus, which turned out to be poisonous. János Irinyi (Hungary) invented a noiseless match in 1836. In 1844, Gustaf Erik Pasch (Sweden) invented the specially-designed striking surface for safety matches that could not ignite on their own. Austrian chemist Anton Schrötter von Kristelli discovered in 1850 that heating white phosphorus created an allotropic red form that was not poisonous, although it was more expensive. Johan Edvard and Carl Frans Lundström (Sweden) began selling safety matches in 1850-1855. Henri Savene and Emile David Cahen, two French chemists, found that phosphorus sesquisulfide was a safe substitute for white phosphorus in 1898. Albright and Wilson (UK) began selling phosphorus sesquisulfide matches in 1899.
Ohm’s Law states that the ratio of the potential difference between the ends of a conductor and the current flowing through it is constant, and is the resistance of the conductor. (Alternately, the current through a conductor between two points is directly proportional to the potential difference across the two points.) The law establishes the relationship between strength of electric current, electromotive force, and circuit resistance. Henry Cavendish (England) arrived at a formulation of Ohm’s Law in 1781 but he did not communicate his results at the time. Georg Ohm (Germany) conducted experiments on resistance in 1825 and 1826 and published his results, including a more complicated version of Ohm’s Law, in 1827.
Dutch scientist Herman Boerhaave discovered urea in urine in 1727. In 1828, Friedrich Wöhler (Germany) synthesized urea by treating silver cyanate with ammonium chloride. This was the first artificial synthesis of an organic compound from inorganic materials. It had important consequences for organic chemistry and also tended to provide evidence against vitalism, the notion that living organisms are fundamentally different from inanimate matter.
André-Marie Ampère (France) invented the solenoid in 1820. Peter Barlow (UK) invented Barlow’s wheel, an early homopolar motor, in 1822. Ányos István Jedlik (Hungary) made the first commutated rotary electromagnetical engine in 1828. William Sturgeon (UK) made a commutated rotating electric machine in 1833. Joseph Saxton (US) made a magneto-electric machine in 1833. Thomas Davenport (US) created a battery-powered direct current motor in 1834. Moritz von Jacobi (Germany/Russia) made a 15-watt rotating motor in 1834 and the first useful rotary electrical motor in 1839. Solomon Stimpson (US) made a 12-pole electric motor with segmental commutator in 1838. Truman Cook (US) made the first electric motor with PM armature in 1840. Paul-Gustave Froment (France) made the first motor translated linear electromagnetic piston’s energy to wheel’s rotary motion in 1845. Zénobe Gramme (Belgium) made the first anchor ring motor in 1871. Galileo Ferraris (Italy) made the first AC commutatorless induction motor with two-phase AC windings in space quadrature in 1885. Nikola Tesla (Serbia/US) made three different two-phase four-stator-pole motors, including a synchronous motor with separately excited DC supply to rotor winding in 1886-1889.
Before the 18th Century, most scientists believed that the Earth was very young and that it was shaped by catastrophic processes. In the mid to late 1700s, some geologists challenged the prevailing theory. In 1759, Mikhail Vasilevich Leomonsov (Russia) suggested that the Earth’s topography is result of very slow natural activity, including uplift and erosion. Beginning in 1785, James Hutton (Scotland) proposed Plutonism (to contrast with Neptunists who believed that the Biblical Flood was the cause of much geology), which stated that the Earth formed through the gradual solidification of molten rock at a slow rate, by the same processes, particularly erosion and volcanism, that occur today. The result of this theory was that the Earth must be millions of years old. The study of fossils in rock layers by William Smith (UK) in the 1790s and independently by French scientists Georges Cuvier and Alexandre Brogniart in 1811 introduced the notion of stratigraphy for determining the relative age of different rocks. Following in Hutton’s footsteps, Scottish geologist Charles Lyell’s Principles of Geology (first volume published in 1830) set out the voluminous evidence for uniformitarianism over catastrophism. Charles Darwin read the volumes of Lyell’s Principles while serving as naturalist on the Beagle in the 1830s and uniformitarianism ultimately provided the geological basis for his theory of evolution by natural selection.
Thomas Saint (UK) received a patent for a sewing machine in 1790, but no one has ever found a working model of the machine. (In 1874, William Newton Wilson (UK) built a working machine using Saint’s drawings – with a few adjustments). Austrian tailor Josef Madersperger independently created a working sewing machine in 1814. French tailor Barthélemy Thimonnier developed an improved model that was patented in 1830 and used for the first time in manufacturing. By 1841, Thimonnier had 80 machines in his factory, but a mob of French tailors, afraid of the effect the new machine would have on their profession, raided the factory and destroyed everything. American Walter Hunt invented the first eye-pointed needle and double-lock stitch sewing machine in 1832. American Joseph Greenough patented a sewing machine in 1842. American Elias Howe patented an improvement on Hunt’s design in 1845. In 1851, Isaac Singer (US) combined elements of Thimonnier’s, Hunt’s and Howe’s machines and obtained a patent for the first practical sewing machine for domestic use, although after Howe sued him, he had to pay Howe under a license. In 1857, American James Gibbs patented the first chain-stitch single-thread sewing machine. Singer patented the Vibrating Shuttle sewing machine in 1885 and produced the first electric machines in 1889.
Non-Euclidean geometries arise when one of two Euclidean axioms is not followed. If the postulate that two parallel lines can never meet is ignored, then hyperbolic geometry and elliptic geometry are possible. If the metric requirement (which states that the shortest distance between two points is a straight line) is relaxed, then kinematic geometries arise. Among the earliest to challenge Euclid’s parallel postulate and explore hyperbolic and elliptic geometries were Ibn al-Haytham (Iraq/Egypt, 11th Century); Omar Khayyám (Persia, 12th Century) and Nasir al-Din al-Tusi (Persia, 13th Century). Other scholars worked with the parallel postulate and arrived at non-Euclidean geometries which they either ignored or attempted to disprove. The true birth of non-Euclidean geometry occurred in the 19th Cerntury. Carl Friedrich Gauss (Germany) in 1813, and Ferdinand Karl Schweikart (Germany) in 1818, independently worked out the central premises of non-Euclidean geometries, but both failed to publish. In 1830, János Bolyai (Hungary) and Nikolai Ivanovich Lobachevsky (Russia) separately published treatises on hyperbolic geometry, which is now known as Bolyai-Lobachevskian geometry. In 1854, Bernhard Riemann created Riemannian geometry, which includes elliptic geometry and is considered non-Euclidean because it lacks parallel lines. Arthur Cayley (UK) developed a method for defining the distance between points inside a conic, which was applied to non-Euclidean geometries by Felix Klein (Germany) in 1871 and 1873.
In electromagnetic induction, an electromotive force is produced across a conductor when it is exposed to an electromagnetic field. Michael Faraday (UK) and Joseph Henry (US) independently discovered this phenomenon in 1831. Because Faraday published his results first, he is usually credited with the discovery. James Clerk Maxwell later devised the mathematical principle underlying electromagnetic induction, naming it Faraday’s Law. Electromagnetic induction is the principle underlying electrical generators, transformers and many other electrical machines.
Michael Faraday (UK) created the first disk generator in 1831. Hippolyte Pixii (France) made the first alternating current generator in 1832 and the first oscillating direct current generator in 1833. Charles Wheatstone (UK) created a magneto-electric generator in 1840. Anyos Jedlik (Hungary) created electromagnetic rotating devices between 1852-1854. Werner von Siemens (Germany) made a generator with double-T armature and slots windings in 1856. Charles Wheatstone (UK), Werner von Siemens (Germany) and Samuel Alfred Varley (UK) independently invented the dynamo-electric machine (dynamo) in 1866-1867. Zénobe Gramme (Belgium) made the first anchor ring motor in 1871. J.E.H. Gordon (UK) invented an alternating current generator in 1882. William Stanley, Jr. (US), of Westinghouse Electric, demonstrated an alternating current generator in 1886. In 1891, Sebastian Ziani de Ferranti and Lord Kelvin (UK) invented the Ferranti-Thompson Alternator. Also in 1891, Nikola Tesla (Serbia/US) patented a high-frequency alternator.
In ancient times, dinosaur fossils were explained as the bones of a giant race of humans that had vanished from the Earth. More scientific approaches came in the 19th Century. In 1808, Georges Cuvier (France) identified a German fossil as a giant marine reptile that would later be named Mosasaurus. He also identified another German fossil as a flying reptile, which he named Pterodactylus. Cuvier speculated, based on the strata in which these fossils were found, that large reptiles had lived prior to what he was calling “the Age of Mammals.” Cuvier’s speculation was supported by a series of finds in Great Britain in the next two decades. Mary Anning (UK) collected the fossils of marine reptiles, including the first recognized ichthyosaur skeleton, in 1811, and the first two plesiosaur skeletons ever found, in 1821 and 1823. Many of Anning’s discoveries were described scientifically by the British geologists William Conybeare, Henry De la Beche, and William Buckland. Anning first observed that stony objects known as “bezoar stones”, which were often found in the abdominal region of ichthyosaur skeletons, often contained fossilized fish bones and scales when broken open, as well as sometimes bones from small ichthyosaurs. This led her to suggest to Buckland that they were fossilized feces, which he named coprolites. In 1824, Buckland found and described a lower jaw that he believed belonged to a carnivorous land-dwelling reptile he called Megalosaurus. That same year Gideon Mantell (UK) realized that some large teeth he had found in 1822 belonged to a giant herbivorous land-dwelling reptile, he named Iguanodon, because the teeth resembled those of an iguana. Mantell published an influential paper in 1831 entitled The Age of Reptiles in which he summarized the evidence for an extended time during which the Earth had teemed with large reptiles. He divided that era, based in what rock strata different types of reptiles first appeared, into three intervals that anticipated the modern periods of the Triassic, Jurassic, and Cretaceous. In 1832 Mantell found a partial skeleton of an armored reptile he would call Hylaeosaurus. In 1841 the English anatomist Richard Owen created a new order of reptiles for MegalosaurusIguanodon, and Hylaeosaurus, which he called Dinosauria.
Nearly all cells of living organisms have an organelle called the nucleus that contains the cell’s DNA in chromosomes. Antonie van Leeuwenhoek (The Netherlands) observed nuclei, which he called ‘lumen’, in the red blood cells of salmon in 1719. Franz Bauer (Austria) described cell nuclei in more detail in 1804. More detailed still was the 1831 report of Robert Brown (UK) regarding what he called the areola, or nucleus, in the cells of orchids. Matthias Schleiden (Germany) called the nucleus the ‘cytoblast’ because he believed that the organelle played a role in generating cells. Oscar Hertwig showed in 1877-1878 that during fertilization of an egg, the nucleus of the sperm fuses with the egg’s nucleus. It took the discovery of mitosis and the rediscovery of Mendel’s laws of heredity in the 20th Century before scientists understood the true importance of the nucleus.
A reaper is a farming tool that cuts and gathers crops at harvest. Either the Celts or the Romans invented a mechanical reaper but the machine was forgotten after the fall of the Roman Empire. Thomas Dobbs (UK) invented a reaping machine in 1814. Patrick Bell (UK) created a reaper with a revolving reel, cutting knife and canvas conveyor in 1828. Robert McCormick (US) attempted to invent a horse-drawn reaper for small grain crops but was unable to complete the project so, in 1831, he handed it over to his son Cyrus, who perfected and tested a new reaper the same year. Obed Hussey (US) invented and patented a commercially successful horse-drawn reaper in 1833. McCormick did not patent his reaper until 1834. The McCormick and Hussey reapers competed until 1858, when Hussey finally agreed to sell McCormick the rights to his superior cutter-bar mechanism. After 1872, the reaper was generally replaced by the reaper-binder, which reaped the crop and bound it into sheaves. This in turn was replaced by the swather, and the swather by the combine harvester.
Enzymes are macromolecular biological catalysts, usually proteins, that greatly accelerate the rate and specificity of thousands of metabolic reactions. In 1833, Anselme Payen, with Jean-François Persoz (France) discovered the first enzyme, diastase, which they isolated from barley malt. Jon Jakob Berzelius (Sweden) studied enzymes in 1835 and was the first to describe their action as ‘catalytic.’ Wilhelm Kühne (Germany) first used the term ‘enzyme’ in 1877. In 1897, Eduard Buchner discovered that yeast extracts could ferment sugar in the absence of living yeast cells and he identified the enzyme responsible as ‘zymase.’ James B. Sumner (US) at Cornell was able to isolate and crystallize the enzyme urease from the jack bean in 1926 and for catalase in 1937. John H. Northrop and Wendell M. Stanley (US) invented a precipitation technique that allowed them to isolate pepsin in 1930, and later trypsin and chymotrypsin. Using X-ray crystallography, a team led by David Chilton Phillips discovered the structure of lysozyme in 1965.
A revolver is a repeating handgun with a revolving cylinder containing multiple chambers and at least one barrel for firing. The first guns with multichambered cylinders that revolved to feed one barrel were made in Europe in the late 16th Century. The first flintlock revolver was made by Elisha Collier (US) in 1814. The first percussion cap revolver was invented by Francesco Antonio Broccu (Italy) in 1833. Samuel Colt (US) made a similar revolver in 1835 but, unlike Broccu, he patented it. The first cartridge revolvers were made by Smith & Wesson (US) in 1856.
The first step towards an electrical telegraph was taken by Benjamin Franklin (US) in 1750, when he created a device that sent an electrical signal across a conductive wire that was registered at a remote location. An electrochemical telegraph was created by Spanish scientist Francisco Salva Campillo in 1804; an improved version was made by Samuel von Sömmering (Germany) in 1809. The messages could be transmitted a few kilometers and would release a stream of bubbles in a tube of acid, which had to be read to determine the letter or number. Pavel Schilling (Estonia) created an electromagnetic telegraph in 1832, but it was Carl Friedrich Gauss and Wilhelm Weber (Germany) who built the first electromagnetic telegraph used for regular communication, in 1833. David Alter (US) invented the first American electric telegraph in 1836. The first commercial electrical telegraph was created by William Cooke and Charles Wheatstone (UK); it was patented and demonstrated in 1837 and put in operation in 1839. Edward Davy (UK) built his own telegraph system in 1837. Samuel Morse (US) independently invented his own electrical telegraph in 1837, along with Morse code. Morse’s system quickly spread throughout the US.
As with many scientific discoveries, the concept of ice ages developed slowly over time. The first inklings of a theory were provided by scientists and others who explained the presence of large erratic boulders and moraines by suggesting that glaciers had placed them in their current locations in the past. These included Pierre Martel (Switzerland) in 1744; James Hutton (Scotland) in 1795; Jean-Pierre Perraudin (France) in 1815; Göran Wahlenberg (Sweden) in 1818; Ignaz Venetz (Switzerland) in 1829; Johann Wolfgang von Goethe (Germany); and Ernst von Bibra (Germany) in 1849-1850. In 1824, Jens Esmark (Denmark/Norway) proposed that changes in climate caused a sequence of worldwide ice ages. Robert Jameson (Scotland) accepted and promoted Esmark’s ideas, as did Albrecht Reinhard Bernhardi (Germany), who speculated in 1832 that former polar ice caps may have reached the temperate zones. Momentum began to build when Venetz convinced Jean de Charpentier (Switzerland/Germany) of his glaciation theory and de Charpentier presented a paper in 1834. German botanist Karl Friedrich Schimper gave lectures in Munich in 1835-1836 in which he proposed that erratic boulders were the result of periods of global obliteration, when the climate was cold and water was frozen. Schimper spent the summer of 1836 in the Swiss Alps with de Charpentier and Louis Agassiz (Switzerland), during which time Agassiz became convinced of the glaciation theory. In 1836-1837, Agassiz and Schimper developed a theory of a sequence of glaciations and in 1837, Schimper coined the term ‘ice age.’ The reception of the scientific community was cool, so Agassiz set out to collect more data to support the theory, which he published in 1840. Widespread acceptance of the theory did not come until 1875, when James Croll (UK) became the first to propose a convincing mechanism to explain the ice ages. In his book Climate and Time, in their Geological Relations, Croll hypothesized that cyclical changes in the Earth’s orbit could have triggered the growth of the glaciers. The orbital changes were later proven experimentally.
Cell theory holds that (1) All living organisms are composed of one or more cells; (2) The cell is the most basic unit of life; and (3) All cells arise from pre-existing, living cells. German biologist Matthias Schleiden proposed that all plants are composed of cells or the products of cells and that cells are the basic form of life. Theodor Schwann (Germany) made a similar claim for animals. Schleiden’s theory also adopted the theory of Barthelemy Dumortier (Belgium) that cells were created by a crystallization process either from other cells or from outside. This portion of the theory was refuted by Robert Remak (Poland/Germany), Rudolf Virchow (Germany), and Albert Kolliker (Switzerland) in the 1850s. In 1855, Virchow proposed a third premise of cell theory, that all cells arise only from pre-existing cells.
A stellar parallax is the apparent shift of position of a nearby star against the background of distant objects that is made possible by the movement of the Earth in its orbit. Once a stellar parallax is measured, the distance to the star can be determined using trigonometry. Stellar parallax is so difficult to detect that some scientists argued that it did not exist. For example, James Bradley tried but could not measure stellar parallaxes in 1729. Then, in 1838, Friedrich Bessel (Germany) measured the stellar parallax for the star 61 Cygni using a Fraunhofer heliometer. This discovery was closely followed by Thomas Henderson (Scotland) for the star Alpha Centuri in 1839, and Friedich von Struve (Germany) for the star Vega in 1840.
Vulcanization is a chemical process that converts natural rubber into more durable materials by adding sulfur or other curatives or accelerators, which modify the polymer by forming bridges between individual polymer chains. Vulcanized rubber is less sticky than natural rubber and has superior mechanical properties. Natural latex rubber from rubber trees was known to Mesoamericans since ancient times. As early as 1600 BCE, Mesoamericans created processed rubber by mixing the latex with the juice of a local vine. Unprocessed rubber was used for some products in Europe and America prior to vulcanization, but was of limited practicality. According to Charles Goodyear (US), he discovered vulcanization in 1839 through a series of experiments. Thomas Hancock (UK), who may have seen Goodyear’s early samples, was the first to patent vulcanization in the UK in 1844. Goodyear received a US patent the same year. An important development occurred in 1905, when George Oenslager (US) at Goodrich discovered that addition of thiocarbanilde accelerated the sulfur-rubber reaction and reduced consumption of energy.
Ozone is an inorganic molecule made of three oxygen atoms that is a pale blue gas at room temperature with a sharp, pungent smell. Ozone is an allotrope of oxygen. Christian Friedrich Schönbein (Switzerland/Germany) isolated ozone and identified it as a distinct compound in 1840. Jacques-Louis Soret (Switzerland) determined the chemical formula for ozone in 1865. In 1913, French physicists Charles Fabry and Henri Buisson discovered a layer of ozone in the atmosphere. This ozone layer absorbs 97-99% of dangerous medium-frequency ultraviolet light from the sun. In 1930, Sydney Chapman (UK) determined the photochemical process that created the ozone layer. David Bates and Marcel Nicolet demonstrated in the 1950s that free radicals reduced the amount of ozone in the atmosphere. Paul Crutzen (The Netherlands) showed in 1970 that human activity (such as use of nitrogen fertilizers) could reduce the ozone in the atmosphere. Frank Sherwood Rowland (US) and Mario J. Molina (Mexico/US) hypothesized in 1974 that organic halogen compounds such as CFCs could deplete the ozone layer; the theory was confirmed by experiment within three years. This led to the banning of CFCs in aerosol spray cans in 1978.
Ancient physicians used various herbs, including Solanum, opium and coca, to induce unconsciousness and/or relieve pain in their patients. Alcohol was also used. There is some evidence that the Arabs used an inhaled anesthetic. In the late 12th Century, in Salerno, Italy, physicians used a ‘sleep sponge’ soaked in a solution of opium and various herbs, which was held under the patient’s nose. The sleep sponge was used by Ugo Borgognoni and his son Theodoric (Italy) in the 13th Century. In 1275, Spanish physician Raymond Lullus invented what would later be called ether. He and Swiss physician Paracelsus experimented with animals but not humans. In 1772, Joseph Priestley (UK) discovered nitrous oxide, or laughing gas, and in 1799, British chemist Humphry Davy discovered the gas’s anesthetic properties by experimenting on himself and his friends. Morphine was discovered in 1804 by Friedrich Sertürner (Germany) but it was only widely used as an anesthetic after the invention of the hypodermic syringe. In 1842, American physician Crawford Long became the first to use ether as an anesthetic for human surgery – he removed two small tumors from James Venable, one of his students, in a painless procedure. The operation was not publicized until 1849. In a widely publicized event, Boston dentist William Morton administered inhaled ether to a patient in Massachusetts General Hospital in 1846, after which a surgeon painlessly removed a tumor. Ether was eventually replaced by other chemicals due to its flammability. Cocaine, which was first identified in 1859, became the first effective local anesthetic in 1884 when Austrian physician Karl Koller used it during eye surgery.
According to the law of conservation of energy, the total energy of a system is conserved because it does not change over time; energy cannot be created or destroyed, although it can change form. German chemist Karl Friedrich Mohr gave one of the first statements of the law in 1837. The key concept that heat and mechanical work are equivalent was first stated by Julius Robert von Mayer (Germany) in 1842. James Prescott Joule (UK) reached the same conclusion independently in 1843, as did Ludwig A. Colding (Denmark). In 1844, William Robert Grove (Wales) suggested that mechanics, heat, light, electricity and magnetism were all manifestations of a single force, a notion he published in 1846. Drawing on the work of Joule and others, Hermann von Helmholtz (Germany) reached similar conclusions to Grove in an 1847 book, which led to the wide acceptance of the idea. In 1850, William Rankine (Scotland) first coined the phrase ‘law of conservation of energy.’
The Doppler effect is the apparent change in the frequency of a wave that occurs when an observer is moving relative to the source of the wave (or when the source of the wave is moving relative to the observer). For example, as a wave-emitting object approaches an observer, the observer receives the waves at a higher frequency than the waves actually being emitted; when the object recedes from the observer, the waves are received at a lower frequency. The Austrian physicist Christian Doppler first proposed the effect in 1842. Buys Ballot (The Netherlands) confirmed Doppler’s hypothesis for sound waves in 1845. Hippolyte Fizeau independently discovered the phenomenon in electromagnetic waves in 1848. John Scott Russell (UK) confirmed the Doppler effect in a series of experiments published in 1848.
Some ancient civilizations had relatively sophisticated sanitation systems. The Indus Valley Civilization (3300-1300 BCE) had flush toilets, a public water supply, and a sewerage system with drainage channels and street ducts. The Minoans (2700-1450 BCE) on the island of Crete had a sophisticated water intake and drainage system, with flush toilets. The Ancient Romans had indoor plumbing and a series of lead pipes for bringing water in and out of homes and public buildings. The Mayans of the Classic Period (800-1100 CE) had indoor plumbing and pressurized water. Modern sanitation as we know it did not begin until the mid-19th Century. The first true modern sewer system was built by W. Lindley (UK) in Hamburg, Germany in 1842, after a fire destroyed much of the city. In London, John Gibb (UK) first began purifying water using a sand filter in 1804. James Simpson (UK) installed a improved filtration system for the city of London in 1829. After an outbreak of cholera in 1854, concerns about water safety increased and beginning in 1858, Joseph Bazalgette began a comprehensive sewer project for London, which opened in 1865, but was not completed until 1875.
Neptune is the eighth and farthest planet from the sun. It is the fourth largest planet by diameter and the third largest by mass. In 1821, French astronomer Alexis Bouvard published tables of the orbit of Uranus that contained significant discrepancies, which led to the prediction of another planet. In 1835, Jean Élix Benjamin Valz, Friedrich Bernhard Gottfried Nicolai, and Niccolo Cacciatore, each independently, conjectured that a trans-Uranian planet caused the otherwise inexplicable discrepancies in the historical record of the orbits of both Halley’s comet and Uranus. Using Bouvard’s tables, both Urbain Jean Joseph Le Verrier (France) and John Couch Adams (UK), working independently, calculated the location where the new planet should be found in 1846. On September 23, 1846, German astronomer Johann Gottfried Galle, with the assistance of Heinrich Louis d’Arrest, observed the new planet within one degree of the predicted location.
George Boole (UK) developed Boolean algebra and Boolean logic in books published in 1847 and 1854. Boolean logic and algebra have been fundamental in the development of digital electronics and is used in set theory and statistics. Many are familiar with it as the basis for computer database search engines.
Absolute zero is the lower limit of the thermodynamic temperature scale. It is the state at which the enthalpy and entropy of a cooled ideal gas reaches its minimum value of zero. Absolute zero is −273.15° Celsius, −459.67° Fahrenheit and 0° Kelvin. Robert Boyle (Ireland) was one of the first to propose the idea of an absolute zero, or primum frigidum, in 1665. Eighteenth Century scientists accepted the idea of absolute zero and tried to calculate it. Guillaume Amontons (France) calculated –240° C in 1702; Johann Heinrich Lambert (Switzerland) arrived at –270° C in 1779. Despite these relatively accurate numbers, other scientists proposed a variety of absolute temperatures. In 1780, Pierre-Simon Laplace and Antoine Lavoisier (France) thought it must be at least –600° C, while John Dalton (UK) in 1808 suggested absolute zero was –3000° C. In 1848, William Thomson, Lord Kelvin (UK), arrived at the temperature for absolute zero that is still recognized today: -273.15° C. Kelvin’s scale was independent of the properties of any particular substance and based on Carnot’s theory of the motive power of heat.
The first law of thermodynamics is derived from the law of conservation of energy, as applied to thermodynamic systems. The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed. The first law of thermodynamics states that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. The first law of thermodynamics was first stated by Rudolf Clausius (Germany) in 1850. The principles were developed by William Rankine (Scotland) in the 1850s. The law was conceptually revised by George H. Bryan (UK) in 1907 to state, “When energy flows from one system or part of a system to another otherwise than by the performance of mechanical work, the energy so transferred is called heat.” Max Born (Germany/UK) revised this reformulation in 1921 and 1949.
While most ancient scientists believed that the Earth was stationary, some suggested that the apparent movement of the sun, moon, and planets was the result of the Earth rotating on its axis. The first was Ancient Greek scientist Philolaus in the 5th Century BCE, followed by Hicetas, Heraclides and Ecphantus in the 4th Century BCE. In the 3rd Century, Aristarchus of Samos proposed that the Earth rotated on its axis and revolved around the sun. In the opposite camp were Aristotle (4th Century BCE) and Ptolemy (2nd Century CE), who suggested a rotating Earth would create horrific winds. In 499 CE, Indian astronomer Aryabhata theorized that the Earth rotated each day. William Gilbert (England) supported the idea of a rotating Earth in 1600 and in 1687, Sir Isaac Newton (England) calculated the extent that the poles would flatten and equator would bulge if the Earth was rotating. Seventeenth Century scientists also recognized that, if the Earth was rotating, there should be a slight deflection (the Coriolis effect) to falling bodies. Attempts to measure any such effect by Giovanni Riccioli (Italy) and Robert Hooke (England) in the 17th Century failed, but Giovanni Battista Guglielmini (Italy) in 1791-1792, Johann Friedrich Benzenberg (Germany) in 1802 and 1804, and Ferdinand Reich (Germany) in 1831 all found small deviations that supported the rotation hypothesis. Newton’s predictions about the flattening of the Earth’s poles was proven by Pierre Louis Maupertuis (France) in 1736-1738. In 1851, French physicist Léon Foucault definitively demonstrated the rotation of the Earth by his pendulum, which slowly turned with the Earth, with the rate depending on the latitude where the pendulum is located.
An elevator is mode of vertical transportation that moves people or goods between floors or decks of a building, vessel or other structure, often by electric motors. Archimedes reportedly built an elevator in 236 BCE. Elevators made with a cab on hemp rope and powered by humans or animals are referred to in ancient documents and may have been installed at the Sinai Monastery in Egypt. Islamic scholar al-Muradi mentioned a machine that lifted a battering ram up the side of a fortress in 1000. Louis XV (France) had an early elevator called a ‘flying chair’ built in the Palace of Versailles in 1743. The first screw drive elevator was made by Ivan Kulibin (Russia) and installed in the Winter Palace in St. Petersburg in 1793. British architects Burton and Hormer created a steam-powered ‘ascending room’ in 1823 London as a tourist attraction. Frost & Stutt (UK) build a steam-driven, belt-driven, counterweighted elevator called the “Teagle” in 1835. In 1845, Italian architect Gaetano Genovese created a ‘flying chair’ for the Royal Palace of Caserta. Sir William Armstrong’s hydraulic crane of 1846 was much stronger than the steam-powered elevators, using a water pump, a plunger, counterweights and balances. Standing rope control was invented by Henry Waterman (US) in 1850. Elisha G. Otis (US) invented the safety elevator in 1852, which prevented the fall of the cab if the cable broke. He publicly demonstrated it in 1854 and the first one was installed in New York City in 1857. A device for safely opening and closing elevator doors was developed by J.W. Meaker (US) in 1874. The first electric elevator was made by Werner von Siemens (Germany) in 1880. Between 1892 and 1895, Frank Sprague (US) developed a number of improvements for electric elevators, including floor control, automatic controls, acceleration control and safeties.
A gyroscope is designed to measure or maintain orientation using angular momentum principles. Gyroscopes developed from the child’s toy known as the top. British sea captain John Serson invented the ‘whirling speculum’ using a top in 1743 to locate the horizon in bad weather. Johann Bohnenberger (Germany) created the ‘machine’ in 1817 using a rotating massive sphere. American Walter R. Johnson’s 1832 device used a rotating disc. When Léon Foucault (France) learned of the gyroscope, he realized its potential to prove that the Earth rotated on its axis. In 1852, he designed and built a device, which for the first time he called a gyroscope, that included a rotating wheel inside a supporting ring so that axis of the spinning wheel moved independently of the ring. The supporting ring moved with the Earth, while the spinning wheel remained still, showing that the Earth was rotating.
Aspirin is acetylsalicylic acid. In ancient times, plants containing salicylate, such as willow, were used to prepare medicines. There are references to it in Egyptian manuscripts from between 2000 and 1000 BCE and Hippocrates mentions salicylic tea to reduce fever in 400 BCE. Willow bark extract was a common remedy in the 18th and early 19th centuries, after which pharmacists began to experiment with and prescribe chemicals related to salicylic acid. French chemist Charles Frédéric Gerhardt first produced acetylsalicylic acid in the lab in 1853. A pure form of the chemical was synthesized by Felix Hoffman (Germany), a chemist with the Bayer company, in 1897 and it was soon marketed all over the world. Sales rose after the flu epidemic of 1918, but dropped after the introduction of acetaminophen in 1956 and ibuprofen in 1962. Aspirin sales once again increased in the last decades of the 20th Century, when scientists discovered the anti-clotting benefits of aspirin.
Celluloid is a chemical compound made from nitrocellulose and camphor with added dyes and other agents. It is considered the first thermoplastic. Alexander Parkes (UK) created the first celluloid, called Parkesine, in 1855, although his business soon went bankrupt. Daniel Spill (UK), who had worked with Parkes, formed the Xylonite Company with the intention of taking over Parkes’ patents and making Parkesine under the name Xylonite. John Wesley Hyatt (US) also claimed to have acquired Parkes’ patent in the 1860s and began experimenting with the intention of making billiard balls. Isaiah Hyatt (US), brother of John Wesley, dubbed the new product celluloid in 1872. A patent dispute between the Hyatts and Spill between 1877 and 1884 resulted in a ruling that both men could continue to made celluloid. In 1888 and 1889, celluloid was adapted for use as photographic film, and all movie and photography films were made of celluloid until acetate film replaced it in the 1950s. The biggest drawback of celluloid film was its extremely high flammability and it was eventually replaced by other materials.
Before the 19th Century, manufacturing steel was a slow, expensive process that required carbon-free wrought iron as the main ingredient. As a result, it was impossible to produce mass quantities of steel. In 1740, Benjamin Hunstman (UK) developed the crucible technique, which increased the cost and duration of the process but increased quality. Beginning in 1847, William Kelly (US) began to experiment with reducing carbon content by blowing air through the molten iron. In 1856, Henry Bessemer (UK) independently invented a similar process that greatly improved the purity of the finished product on a mass production level. Shortly afterwards, Robert Mushet (UK) improved Bessemer’s process, creating a more malleable final product. In 1878, Sidney Thomas (UK) designed a way to reduce phosphorus residue in the Bessemer process, increasing the quality of the steel. By the late 20th Century, the Bessemer process had been replaced by the basic oxygen process, which allowed better control of the chemistry.
William Henry Perkin (UK) created the first synthetic organic dye in 1956, when, in an attempt to synthesize quinine, he made mauveine instead. Synthetic dyes, many of them originally derived from coal tar, quickly replaced the traditional natural dyes. Many synthetic dyes are made from aniline or chrome. Synthetic dyes are generally harmless when set into fabrics and other substances, but when released into the environment, the chemicals in dyes – which include mercury, lead, chromium, copper, cadmium, toluene and benzene – can have harmful effects.
While fermentation has been used by humans to make fermented beverages and foods since at least 7000 BCE, the scientific explanation for the process only became understood in the 19th Century. In 1837 and 1838, Theodor Schwann (Germany), Charles Cagniard de la Tour (France), and Friedrich Traugott Kützing (Germany), working independently, concluded that fermentation was caused by yeast, a living organism. In 1857, Louis Pasteur (France) demonstrated that lactic acid fermentation is carried out by living bacteria. In 1897, Eduard Buechner (German) isolated the enzyme in yeast that caused fermentation.
While most scientists believed that biological organisms had undergone evolution over time, no one had been able to provide a convincing evolutionary mechanism. After returning from his five-year voyage to South America, the South Seas and Australia in 1836, and reading Thomas Malthus’s works on population growth, Charles Darwin (UK) came to believe that (1) all species contained individual variations; (2) some of the variations were more advantageous than others and (3) given limits on population growth, those individuals with the more advantageous variations would be more likely to survive and reproduce. The result of such a system, over a long period of time, would be the generation of new species. Although Darwin formulated the theory by 1838, he feared the reaction of scientists and the public and spent the next 20 years collecting evidence to support his conclusions. He drafted a comprehensive essay on the matter in 1844, but did not publish it. In 1858, Darwin learned that another biologist, Alfred Russell Wallace (UK), had reached nearly identical conclusions in an essay. Wallace’s paper was presented to the Royal Society in 1858 along with excerpts from Darwin’s 1844 essay. In 1859, Darwin published The Origin of Species, which set out the evidence behind his theory. The theory of evolution by means of natural selection is the fundamental premise of the modern science of biology.
Beginning in 1859, Louis Pasteur (France) conducted a series of experiments that proved the connection between disease and microorganisms, or germs. Pasteur’s discovery, which he published in 1862, revolutionized medicine and eventually had a significant impact on human mortality rates. Scientists whose prior work led to Pasteur’s discovery include: Girolamo Fracastoro (Italy), who proposed a germ theory in 1546; Agostino Bassi (Italy), who conducted crucial experiments in 1808-1813; Ignaz Semmelweis (Hungary), who conducted clinical studies of disease in 1847; and John Snow (UK), who studied public health response to disease outbreaks in 1854-1855. Following up on Pasteur’s findings, in 1884, Robert Koch (Germany) articulated a four-part test for determining if disease is caused by microorganisms and also identified the bacteria that cause cholera, tuberculosis and anthrax.
Joseph Fraunhofer (Germany) discovered in 1814 that the sun’s light was not distributed evenly throughout the spectrum, but no one pursued this finding at the time. In the 1850s, Gustav Kirchhoff (Poland) began systematically studying the colors made by different elements when placed in flame, known as their atomic light signatures. In 1859, Kirchhoff joined with Robert Bunsen (Germany); together, they placed different substances in flame (using a new burner invented by Bunsen that produced little interfering light), placed the light from the flames through a prism and noted the spectrum. The result was the identification of the unique spectrum for each known element, and the discovery of two previously-unknown elements: cesium and rubidium. Kirchhoff and Bunsen also performed spectroscopy on the light of the sun, which created a new tool for astronomers to understand the make-up of stars.
Digging or drilling for underground oil dates back to the 4th Century CE in China, where drill bits were attached to bamboo poles to dig wells of up to 800 feet deep. People in Arabian countries and Persia dug for oil as far back as the 9th Century. Also from the 9th to the 16th centuries, those living near Baku, in modern-day Azerbaijan, hand dug holes of up to 115 feet. Also in Baku, the first offshore drilling began in 1846. The first recorded land-based commercial oil well was begun in Oil Springs, Ontario in 1858. But it was American Edwin Drake’s drilling operation in Titusville, Pennsylvania in 1859 that was the first oil well using modern principles. One of Drake’s key innovations was the drive pipe – he drove a cast iron pipe into the ground and then lowered the drill through the pipe, thus preventing the hole from collapsing.
It took a long time for spontaneous generation – the idea that living things could arise from non-living matter – to die. Francesco Redi (Italy) proved in 1668 that maggots did not spontaneously generate from rotten meat – they were hatched from tiny eggs laid by flies. Lazzaro Spallanzani (Italy) conducted an experiment in 1768 that supported Redi’s conclusion and contradicted the 1745 experiment of John Needham (England) that seemed to support spontaneous generation. Louis Pasteur (France) put the final nails in spontaneous generation’s coffin in 1859 with an experiment in which no life grew in a sterile flask for a year until the neck of the flask was removed and microorganisms had access to the liquid inside. John Tyndall (UK) conducted further investigations in 1875-1876 to support Pasteur’s work and dispel any lingering objections to his conclusion, although his experiments were plagued by contamination by airborne bacterial spores.
The history of the internal combustion engine (ICE) is long and complex. Components of the system were invented as long ago as the 3rd Century CE. In the 17th Century, Christiaan Huygens (The Netherlands) created a rudimentary ICE piston engine when he used gunpowder to drive water pumps for the Versailles palace gardens. In the 1780s, Alessandro Volta (Italy) built a toy pistol, in which an electric spark exploded a mix of air and hydrogen, firing a cork. In 1791, John Barber (UK) received a patent for a turbine. In 1794, Robert Street (UK) built the first compressionless engine. In 1807, Nicéphore Niépce (France) powered a boat with an ICE, the Pyréolophore, fueled by moss, coal dust and resin. In 1807, Swiss engineer François Isaac de Rivaz built an ICE powered by a mix of hydrogen and oxygen, and ignited by an electric spark. In 1823, Samuel Brown patented the first industrial ICE, a compressionless model. Nicolas Léonard Sadi Carnot (France) established the theoretical basis for idealized heat engines in 1824. In 1826, Samuel Morey (US) received a patent for a compressionless ICE. In 1833, Lemuel Wellman Wright (UK) invented a table-type gas engine with a double acting gas engine and, for the first time, a water-jacketed cylinder. In 1838, William Barnett (UK) received a patent for the first machine with in-cylinder compression. Between 1853 and 1857, Eugenio Barsanti and Felice Matteucci (Italy) invented and patented an engine using the free-piston principle that was possibly the first 4-cycle engine. In 1856, Pietro Benini (Italy) built an engine that supplied five horsepower. Later, he developed more powerful engines with one or two pistons. In 1860, Jean Joseph Etienne Lenoir (Belgium) produced and sold the first two-stroke gas-fired ICE with cylinders, pistons, connecting rods, and flywheel – he is recognized as the inventor of the ICE. In 1861, Alphonse Beau de Rochas (France) received the first patent for a four-cycle engine. In 1862, German inventor Nikolaus Otto built and sold a four-cycle free-piston engine that was indirect-acting and compressionless. Alphonse Beau de Roche (France) set out the ideal operating cycle for a four-stroke ICE in 1862. In 1865, Pierre Hugon (France) created the Hugon engine, similar to the Lenoir engine, but with better economy, and more reliable flame ignition. In 1867, Nikolaus Otto and Eugen Langen (Germany) introduced a free piston engine with less than half the gas consumption of the Lenoir or Hugon engines. In 1870, Siegfried Marcus (Austria) put the first mobile gasoline engine on a handcart. In 1872, American George Brayton invented Brayton’s Ready Motor, which used constant pressure combustion, and was the first commercial liquid fueled ICE. In 1876, Nikolaus Otto, working with Gottlieb Daimler and Wilhelm Maybach (Germany), began developing and patenting the four-cycle engine. In 1878, Dugald Clerk (UK) designed the first two-stroke engine with in-cylinder compression. In 1879, Karl Benz (Germany), working independently, received a patent for a two-stroke gas ICE using De Rochas’s four-stroke design. In 1885, Benz designed and built a four-stroke engine to use in an automobile. In 1882 James Atkinson (UK) invented the Atkinson cycle engine, which had one power phase per revolution together with different intake and expansion volumes. In 1884, British engineer Edward Butler constructed the first gasoline ICE. Butler also invented the spark plug, ignition magneto, coil ignition and spray jet carburetor. Rudolf Diesel (Germany) invented the diesel engine in 1892 and Felix Wankel (Germany) invented the rotary engine in 1956.
In 1718, James Puckle (UK) designed and patented a weapon that could fire nine rounds before reloading, but it was not a true machine gun. In 1777, Joseph Belton (US) created a gun that could fire 20 shots in five seconds automatically, but it was too expensive to be commercially viable. In the 19th Century, a number of multi-shot weapons appeared, including volley guns, double barreled pistols and pepperbox pistols, but all were only semiautomatic. In 1861, Wilson Agar (US) invented the Agar Gun, an automatic loading single barrel gun that used a hand crank for firing. In 1862, Richard Jordan Gatling (US) invented the Gatling Gun, the first weapon with controlled, sequential fire with automatic loading. It had prepared cartridges and a hand-operated crank and was used widely into the early 20th Century. The next development was the Nordenfelt Gun, which was designed by Helge Palmcrantz (Sweden) in 1873. William Gardner (US) invented the Gardner Gun in 1874. A major design improvement came from Hiram Maxim (US/UK), who built the Maxim Gun, the first self-powered machine gun, in 1884. After Vickers Ltd. (UK) bought the Maxim Gun, it revised the gun to create the Vickers machine gun in the early 20th Century. Numerous developments continued through the 20th Century.
In the early 19th Century, a number of scientists working with pea plants noticed the segregation of a recessive trait, one of the key elements of the laws of heredity, but unfortunately none of these early scientists kept records of later generations, severely limiting the benefit of their work. Augustinian friar Gregor Mendel (Silesia) performed a comprehensive series of experiments on numerous generations of pea plants between 1856 and 1863 that allowed him to develop the basic rules of heredity and inheritance, including the existence of dominant and recessive traits, which would form the basis of the modern science of genetics. Mendel presented the results of his work in a paper he read at meetings of the Natural History Society of Brno, Moravia in 1865, which was published in 1866. Because the work was perceived to be about hybridization and not inheritance, it did not receive wide distribution, and most scientists, including Charles Darwin (UK), never learned of it. It was only after 1900, when other scientists, particularly Hugo de Vries (The Netherlands), Carl Correns (Germany), Erich von Tschermak (Austria) and William Jasper Spillman (US), independently rediscovered Mendel’s work, that his importance to science was appreciated.
Entropy is the measure of the ways in which a thermodynamic system can be arranged, that is, of the disorder of the system. The second law of thermodynamics states that the entropy of an isolated system does not decrease but tends to evolve towards maximum entropy, also known as thermodynamic equilibrium. Precursors to the discovery of entropy included Lazare Carnot (France), who suggested that natural processes lead to the dissipation of useful energy in 1803. His son Sadi Carnot (France) found in 1824 that in all heat engines, when heat falls through a temperature difference, work can be produced. Rudolph Clausius (Germany) disagreed with Carnot that no change occurs in the working body, holding that it violated the first law of thermodynamics. Instead, Clausius proposed in 1865 what he called entropy, which was the transformation-content or dissipative energy use of a thermodynamic system (or working body) during a change of state. In 1877, Ludwig Boltzmann (Austria) proposed a probabilistic way to measure the entropy of a gas by defining entropy to be proportional to the logarithm of the number of microstates such a gas could occupy. In the 1870s, Boltzmann, Josiah Willard Gibbs (US) and James Clerk Maxwell (UK) provided a statistical basis for entropy. In 1909, Constantin Carathéodory (Greece) linked entropy with a mathematical definition of irreversibility, in terms of trajectories and integrability.
An antiseptic is a substance applied to living tissue or skin to kill microbes and reduce the possibility of infection or sepsis. Sumerian clay tablets from 2150 BCE and writings of Hippocrates (c. 400 BCE) and Galen (c. 130-200 CE) all advocate the use of antiseptic agents. In the early 13th Century, Italian surgeons Hugh of Lucca and Theodoric of Lucca disregarded Galen’s view that pus was beneficial and cleaned pus out of wounds, then used wine to clean the wound and prevent infection. In an 1843 paper that was reissued in 1855, physician Oliver Wendell Holmes (US) advocated cleaning of medical instruments to prevent the spread of puerperal fever. In 1861, Ignaz Semmelweis (Austria) recommended that the physician wash his hands in chlorine solution before assisting in childbirth. While serving in the Confederate Army in the American Civil War in the early 1860s, George H. Tichenor (US) used alcohol on wounds. The adoption of antiseptic practices only became mainstream after British surgeon Joseph Lister’s 1867 paper, Antiseptic Principle of the Practice of Surgery, where he advised the use of carbolic acid to create a sterile surgical environment.
For many years, the most powerful explosive was gunpowder (black powder). In 1847, Italian chemist Ascanio Sobrero synthesized nitroglycerine, which was more explosive than gunpowder, but very dangerous to handle. Numerous explosions, deaths and consequent bans on the use of nitroglycerine created the need for an alternative. In 1867, Alfred Nobel (Sweden), whose German company sold a nitroglycerin-gunpowder mixture, invented dynamite, in which nitroglycerine was combined with inert absorbents such as diatomaceous earth, making it much safer to handle. Nobel packaged his dynamite in long, round cylinders so it could be inserted more easily into rock crevices.
Henry Mill (UK) received a patent for a typewriter-like machine in 1714. Pellegrino Turri (Italy) invented a primitive typewriter (and carbon paper) in 1808. In 1829, William Austin Burt (US) patented the Typographer, which was too slow to be commercially viable. Charles Thurber invented the Chirographer in 1845. In 1855, Giuseppe Ravizza (Italy) created a prototype typewriter that let the user see the writing as it was typed. In 1861, Francisco João de Azevedo, a Brazilian priest, built a typewriter. John Pratt (US) made a typewriter he called the Pterotype in 1865 that was featured in Scientific American magazine. In the same year, Rasmus Malling-Hansen (Denmark) invented the Hansen Writing Ball, which in 1870 became the first commercially sold typewriter. Peter Mitterhofer (Austria) developed a fully functioning prototype typewriter in 1867. The first commercially successful typewriter was invented by Christopher Latham Sholes (US), with the help of Carlos Glidden and Samuel Soule, in 1868. Its QWERTY keyboard layout was presumably created to avoid common letters from getting stuck together. Sholes sold the patent to Remington & Sons in 1873. Various other designs were patented and sold in the years following. While electric typewriters were invented as early as 1902 (arguably even earlier, with Thomas Edison’s 1870 Universal Stock Ticker), Remington began selling the first commercially viable electric typewriter, the Remington Rand, in 1925, based on a 1914 design by James Smathers (US). IBM introduced the first typewriter with proportional spacing in 1941. In 1961, IBM debuted the IBM Selectric, which replaced the typebars with a typeball. Electronic typewriters came into their own with Xerox’s model in 1981. In the late 1980s and 1990s, the computer gradually made typewriters obsolete.
Dmitri Mendeleev (Russia) discovered in 1869 that the elements could be arranged according to their atomic numbers and chemical properties into a table. It was then possible to derive relationships between the properties of the elements and to predict the existence, nature and properties of then-unknown elements. Mendeleev’s periodic table is essentially the same as the one in use today. Prior to Mendeleev’s discovery, other scientists made attempts to define the nature of an element, and to catalogue and categorize the known elements. These included: Robert Boyle (UK) who defined an element in 1661 as “a substance that cannot be broken down into a simpler substance by a chemical reaction”; Antoine-Laurent de Lavoisier (France), who made a list of elements in 1789; Johann Wolfgang Döbereiner (Germany), who made one of the first attempts to classify the elements into groups in 1829; geologist Alexandre-Emile Béguyer de Chancourtois (France) who first noticed that similar elements occur at regular intervals when ordered by their atomic weights and made an early version of the periodic table in 1862-1863; and chemist John Newlands, who classified the 56 known elements into 11 groups based on their physical properties in 1865.
Nucleic acid are polymeric macromolecules that are essential for life as we know it. The nucleic acids include DNA (deoxyribonucleic acid) and RNA (ribonucleic acid). Each nucleic acid consists of (1) a five-carbon sugar; (2) a phosphate group; and (3) a nitrogenous base. Swiss biologist Johannes Friedrich Meischer first isolated and identified a nucleic acid (DNA) from the nuclei of white blood cells in 1869.
The Crookes Tube was preceded by the Geissler Tube, which, although intended to be a vacuum, contained a significant amount of gas. A co-worker of William Crookes (UK), Charles A. Gimingham (UK), created an improved Sprengel mercury vacuum pump that allowed Crookes to create a much better vacuum in his tubes. Crookes Tubes were instrumental in the discovery of cathode rays (1869), X-rays (1895), and the electron (1897).
Cathode rays are now understood to be streams of electrons observed in vacuum tubes to which two electrodes (the negative cathode and positive anode) are attached and a voltage is applied. Early vacuum tubes still contained so much gas that electrons collided with the gas and made it glow (leading to the invention of neon lights). In the late 1860s and early 1870s, William Crookes (UK) invented the Crookes Tube, which contained almost no gas, so the inside of the tube was dark and the electrons struck the back of the tube near the anode instead. In 1869, Johann Hittorf (Germany) realized that the anode was casting a shadow on the back wall of the tube, meaning that something was traveling in straight lines from the cathode. In 1876, Eugen Goldstein (Germany) coined the term ‘cathode rays’ for Hittorf’s beams. In the next decades, scientists debated about the nature of cathode rays. Crookes and Arthur Schuster (Germany/UK) believed they were electrically-charged atoms; Goldstein, Eilhard Wiedermann and Heinrich Hertz (Germany) believed they were a new form of electromagnetic radiation. Experiments by Philipp Lenard (Germany) eventually led to J.J. Thomson’s discovery that cathode rays (and electricity generally) consisted of a beam of electrons, the first subatomic particle discovered.
The telephone evolved from the telegraph. Numerous inventors sought to develop acoustic telegraphy, to send sound waves over the electrical wires. Antonio Meucci (US), an Italian immigrant, created a voice communication device about 1854 that he described to the U.S. Patent Office in an 1871 caveat. A precursor to the telephone was created by Johann Philipp Reis (Germany) in 1860 – it could transmit music and speech, although usually indistinctly. There is some evidence that Innocenzo Manzetti (Italy) may have created a telephone in 1864. In 1870, Cromwell Varley (UK) created a machine that could transmit sounds, but not speech. Poul la Cour (Denmark) made a similar machine in 1874. In 1875, Elisha Gray (US) invented a tone telegraph that could transmit musical notes. Gray filed a patent caveat for a telephone with a water transmitter on the same day in 1876 that Alexander Graham Bell (Scotland/Canada/US) filed a patent application for a similar device. In future models, however, Bell did not use the water transmitter. Thomas Edison’s invention of the carbon grain transmitter, or microphone, further improved the telephone.
Although Charles Wheatstone (UK) first used the term ‘microphone’ to describe an 1827 invention consisting of two rods that amplified sounds to both ears (a type of stethoscope), today’s electric microphone evolved from the telephone transmitter. In 1861, Johann Philipp Reis (Germany) created an early telephone that included a primitive microphone using a metallic strip attached to a vibrating membrane. American inventor Elisha Gray’s design consisted of a diaphragm attached to a conductive rod in an acid solution, which was very similar to Alexander Graham Bell’s 1876 telephone’s microphone. The carbon microphone was invented by David Edward Hughes (UK/US), Emile Berliner (Germany/US) and Thomas Edison (US) independently in 1876-1878. Berliner saw Bell’s telephone in 1876 and invented an improved microphone the same year. Edison patented his microphone in 1877 and was eventually credited with the invention. Hughes created his microphone in 1878 or earlier, but failed to seek a patent; many historians believe that Hughes was actually first. Hughes also revived Wheatstone’s term ‘microphone’ to describe his invention. By 1886, Edison had created the carbon-button microphone. E.C. Wente (US) of Bell Labs invented the first condenser or capacitor microphone in 1916 and in 1923, Captain H.J. Round (UK) invented the moving coil microphone. Before 1925, Georg Neumann (Germany) created the Marconi-Reisz ‘transverse-current’ carbon microphone. In 1930, Harry F. Olson (US) invented the ribbon microphone. The first shotgun microphone was produced in 1963 by Electro-Voice (US). James West and Gerhard Sessler (US), at Bell Labs, invented the electret microphone in 1964.
The phonograph had many precursors. Édouard-Léon Scott de Martinville (France) invented the phonautograph in 1857 – it could record sound as lines on paper, but could not reproduce the sounds. Charles Cross (France) invented the paleophone in 1877, which had the capacity to both record and play sounds. Thomas Edison invented the first true phonograph in late 1877. The first model embossed sounds on a tin foil cylinder; a later device used a wax-covered cardboard cylinder. In 1886, Chichester Bell and Charles Sumner Tainter (US) invented the Graphophone, which engraved recordings on wax coated cylinders. German-born Emile Berliner (US) invented the Gramophone in 1887 – it traced a spiral on a zinc disc coated with beeswax. Discs were first offered to the public in 1892, and by 1908 had become the dominant format. Edison began producing discs in 1912 and ended cylinder production in 1929. The first discs were made of hard rubber; in 1895, Berliner switched to shellac; more flexible vinyl discs became the standard during World War II. Throughout the 20th Century, various improvements and changes occurred (e.g., 78 rpm led to 45 rpm singles and then 33 1/3 rpm long playing records, or LPs) until the 1980s, when the compact disc became the dominant format for listening to recorded music for a period of time. In the 21st Century, many if not most listen to music in various digital formats, such as mp3 files, on computers and other electronic devices.
Humphry Davy (UK) invented an arc lamp in 1801 but it was not very bright and did not last very long. James Bowman Lindsay (Scotland) invented an incandescent electric light in 1835 but failed to pursue it. Others who produced light bulbs were Warren de la Rue (UK) in 1840; Frederick de Moleyns (UK) in 1841; John W. Starr (US) in 1845; Jean Eugène Robert-Houdin (France) in 1851; Joseph Swan (UK) in 1860; Alexander Lodygin (Russia) in 1872; and Henry Woodward and Mathew Evans (Canada) in 1874. A.E. Becquerel (France) invented a fluorescent lamp in 1867. In 1878, Joseph Swan and Charles Stearn (UK) developed a very effective light bulb using a carbon rod from an arc lamp, but it was not commercially viable due to the high current required and short lifetime. Swan switched to a carbon filament by 1880 and began installing light bulbs in British homes. Thomas Edison began experimenting with light bulbs in 1878 and tested a long-lasting carbon filament bulb in 1879. He also began installing light bulbs in 1880. Lewis Latimer, an Edison employee, made further improvements between 1880 and 1882. Meanwhile Hiram Maxim and William Sawyer (US) set up a rival company to Edison. In 1897, Walther Nernst (Germany) made an incandescent bulb that did not require a vacuum. Carl Auer von Welsbach (Austria) made the first commercial metal filament lamp in 1898. Frank Poor (US) also made improvements in 1901. In 1903, Willis Whitney (US) made a metal-coated carbon filament that did not blacken the bulb and in 1915 Irving Langmuir invented a tungsten filament. Peter Cooper Hewitt made the first mercury vapor lamp in 1903 and Georges Claude (France) invented the neon light bulb in 1911.
Hugo von Mohl (Germany) described mitosis in 1839, including the appearance of cell plate between daughter cells during cell division. Carl Nageli (Germany) observed cell division and chromosomes in 1842, but he thought it was an anomaly. Walther Flemming (Germany) used aniline dyes to study salamander embryos beginning in 1879. Flemming made first accurate counts of chromosomes and observed longitudinal splitting of chromosomes. His 1882 book on cell division was seminal. Additional work was done by Edouard Van Beneden (Belgium) and Eduard Strasburger (Poland/Germany), who identified chromosome distribution during mitosis. In 1888, Heinrich Wilhelm Gottfried von Waldeyer-Hartz (German) coined the term ‘chromosome’ to identify what Flemming had described.
In 1882, Thomas Edison (US) built the first electrical supply network, which provided 110 volts of direct current to 59 homes in Manhattan. In the late 1880s, George Westinghouse (US) set up a rival system using alternating current, using an induction motor and transformer invented by Nikola Tesla (Serbia/US). AC eventually prevailed over DC.
Some scientists believe that tuberculosis has affected humans for 40,000 years. Evidence of tuberculosis infection was found in human remains from 9,000 years ago in the eastern Mediterranean, even though specimens of the tuberculosis bacterium from human skeletons in Peru and Africa indicate that its DNA is less than 6,000 years old. Signs of the disease were discovered in Egyptian mummies from 3000-2400 BCE. Tuberculosis (also called consumption or phthisis) is mentioned in texts in ancient India, China and Greece. In 1810, French physician Gaspard Laurent Bayle studied 900 corpses and identified six types of tuberculosis. René Laennec (France) studied the disease from 1816-1826, eventually dying from it, and invented the stethoscope to identify respiratory symptoms. In the 1820s, Pierre Charles Alexandre Louis (France) followed up on Laennec’s work using numerical analysis. Jean Antoine Villemin (France) demonstrated in 1869 that the disease was contagious. In 1882, Robert Koch (Germany) identified Mycobacterium tuberculosis as the cause of the disease. Albert Calmette and Camille Guérin (France) developed a vaccine in 1906. The antibiotic streptomycin, discovered by Albert Schatz (US) in 1943, was found to be effective against tuberculosis in a 1946-1947 double-blind, placebo-controlled trial at the Medical Research Council Tuberculosis Unit (UK).
A fan is a machine that creates air flow, usually by means of a rotating arrangement of vanes or blades. In the late 17th or early 18th Century, the English architect Sir Christopher Wren applied an early ventilation system in the Houses of Parliament that used bellows to circulate the air. John Theophilus Desaguliers, a British engineer, demonstrated a successful use of a fan system to draw out stagnant air from the coal mines in 1727 and soon afterwards he installed a similar apparatus in Parliament. Eighteenth Century civil engineer John Smeaton, and later John Buddle installed reciprocating air pumps in the mines in northern England. Steam power allowed the practical use of fans for ventilation. David Boswell Reid, a Scottish physician, installed four steam-powered fans in the ceiling of St George’s Hospital in Liverpool in the 19th Century; the pressure produced by the fans forced the incoming air upward and through vents in the ceiling. William Brunton (UK) designed a steam driven fan with a radius of six meters in 1849 for the Gelly Gaer Colliery in South Wales. Technological improvements were made by James Nasmyth (UK), Theophile Guibal (France) and J. R. Waddle. In 1882, Schuyler Skaats Wheeler (US) invented a fan powered by electricity, which was commercially marketed by the American firm Crocker & Curtis electric motor company. Philip Diehl (Germany/US) introduced the electric ceiling fan in 1882. Heat-convection fans fueled by alcohol, oil, or kerosene were common around the turn of the 20th century. In the 1920s, industrial advances allowed steel fans to be mass-produced in different shapes, bringing fan prices down and allowing more homeowners to afford them. In the 1960s, central air conditioning reduced the need for fans. In 1998, Walter K. Boyd (US) invented the high-volume low-speed (HVLS) ceiling fan, a slow moving fan eight feet in diameter. Due to its size, the fan moved a large column of air down and out 360 degrees and continuously mixed fresh air with stale air. They are used in many industrial settings, because of their energy efficiency.
The first known device for smoothing the wrinkles in fabric was a metal pan filled with hot water, used in China sometime after 100 BCE. By the 17th Century, the first flat irons consisted of thick slabs of cast iron with a handle that were heated in a fire. Later, a hollow iron that could be filled with hot charcoals was invented. By the late 19th Century, there were irons that were heated by liquid fuels (kerosene, e.g.) and gas. An electric iron using a carbon arc was invented in France in 1882, but was very dangerous. A much safer electric iron was created by Henry W. Seely (US) in 1882, although it had no thermostat to control the temperature. The first thermostatically controlled iron arrived in the 1920s. The first steam iron was also invented in the 1920s, by Thomas Sears (US).
The bacterium that causes cholera is Vibrio cholerae, which infects the small intestine and causes diarrhea and vomiting. Transmission occurs by eating food or water that has been contaminated by the feces of an infected person. Cholera first arose in the Indian subcontinent and has spread through pandemics, with the first from 1817-1824, the second from 1827-1835, the third from 1839-1856, and the fourth from 1863-1875, killing millions. Filippo Pacini (Italy) isolated the bacterium in 1854 but its nature was not known. The same year, John Snow (UK) connected cholera with contaminated drinking water. After studying cholera during an outbreak in Egypt, then moving to an outbreak in India, Robert Koch (Germany) identified Vibrio cholerae as the cause of cholera in 1883. Cholera vaccines were developed by Jaume Ferran i Clua (Spain) in 1885 and Waldemar Haffkine (Russia) in 1892. In the 1940s and 1950s, Robert Allan Phillips (US) and the US Naval Medical Research Unit 2 conducted extensive research on cholera prevention and treatment.
The earliest known studies of the metabolism of living organisms were conducted in the 13th Century by Arab physician Ibn al-Nafis, who wrote in 1260 that “the body and its parts are in a continuous state of dissolution and nourishment, so they are inevitably undergoing permanent change. In 1614, Italian scientist Santorio Sanctorius published the results of experiments in which he weighed himself before and after various activities. He concluded that most of the food he ate was lost through insensible perspiration. By synthesizing urea from inorganic elements in 1828, German chemist Friedrich Wöhler showed that the same chemistry applied to both living and non-living realms. In the 1850s, Louis Pasteur (France) discovered that fermentation was a chemical reaction that was catalyzed in the cells of living yeast. Max Rubner (Germany) made many advances in the study of metabolism. In 1873, he identified the isodynamic law of calories (which he revised in 1902). In 1883, Rubner introduced the surface hypothesis, which holds that the metabolic rate of birds and mammals maintaining a steady body temperature is proportional to the body surface area of the organism. In 1897-1899, German chemist Eduard Buchner isolated the fermentation enzyme in yeast, thus beginning the science of enzymology. Hans Krebs (Germany/UK) and Kurt Henseleit identified the chemical reactions of the urea cycle in 1932 and Krebs identified the citric acid (or Krebs) cycle in 1937. Krebs and Hans Kornberg (Germany/UK) discovered the glyoxylate cycle in 1957.
Before the modern motorcycle came the steam-powered cycle. Pierre Michaux, his son Ernest, and Louis-Guillaume Perreaux (France) built the single-cylinder Michaux-Perreaux steam velocipede, possibly as early as 1867 but no later than 1871. American Sylvester Roper created a twin-cylinder steam velocipede in either 1867 or 1868. In 1881, Lucius Copeland (US) attached a steam engine to an American Star high-wheeler bicycle. The three-wheeled Butler Petrol Cycle, which used an internal combustion engine fueled by gasoline, was introduced by Edward Butler (UK) in 1884, although it was not commercially available until 1888. In 1885, Gottleib Daimler and Wilhelm Maybach (Germany) built the Petroleum Reitwagen, a two-wheeled prototype. In 1887, Copeland began selling a three-wheeled “Moto-Cycle.” Hildebrand & Wolfmüller (Germany) produced several hundred motorcycles beginning in 1894. Excelsior Motor Company (UK) began producing a motorcycle in 1896, while the first US motorcycle, the Orient-Aster, began production in 1898, in Charles Metz’s factory in Waltham, Massachusetts. Royal Enfield (UK) put out its first motorcycle in 1901, and Triumph followed in 1902. Also in 1901, the Indian Motorcycle Manufacturing Company (India) began operations, producing the Indian Single (with a US-made engine). US manufacturer Harley-Davidson made its first motorcycles in 1903.
The historical progression of pens is as follows: quill pens (6th -19th Century) were followed by dip pens (early-late 19th Century), which were followed by fountain pens (late 19th-mid-20th Century), which were followed by ballpoint pens (mid-20th Century-present). A fountain pen is a pen with a nib that has an internal reservoir of water-based ink. The earliest reference to a fountain pen is from 973 CE, when Qadi al-Nu’man al-Tamimi wrote that a pen with ink in a reservoir and a nib was made by order of Ma’ad al-Mu’izz, the caliph of the Mahgreb in North Africa. In 1636, Daniel Schwenter (Germany) described a fountain pen made with two quills and a cork. References to fountain pens include a 1663 quote from Samuel Pepys (England), a late 17th Century English notation and a 1734 American reference. The oldest existing fountain pen was made by Nicolas Bion (France) and dates to 1702. The term fountain pen was common from the early 18th Century. American Peregrin Williamson received a patent for a fountain pen in 1809 and John Scheffer (UK) obtained a patent in 1819. Josiah Mason (UK) invented an improved nib in 1828 and in 1830, William Joseph Gillott, William Mitchell and James Stephen Perry (UK) developed a better way to manufacture in expensive steel nibs. Meanwhile, in France, Romanian inventor Petrache Poenaru introduced a new fountain pen in 1827. John Jacob Parker (US) patented a self-filling fountain pen in 1831. Azel Storrs Lyman (US) invented a pen with a combined holder and nib in 1848. In the 1850s, improved fountain pens were made with iridium-tipped gold nibs, hard rubber parts and free-flowing, sediment-free ink. In the 1870s, Duncan MacKinnon (Canada/US) and Alonzo Cross (US) made the first stylographic pens. Mass production of fountain pens began in the 1880s with Waterman and Wirt (US). Lewis Waterman (US) patented what some have called the first practical fountain pen in 1884; in his design, ink was fed to the nib by gravity and air being drawn into the reservoir to allow a constant flow of ink without flooding. Fountain pens continued to dominate the market until the 1950s, when improved ballpoint pens outcompeted them.
The first automobiles powered by internal combustion engines used gases instead of gasoline. Samuel Brown (UK) used hydrogen to fuel his vehicle in 1826. John Joseph Etienne Lenoir (Belgium) also used hydrogen, then coal gas, to power his Hippomobile in 1860. In 1870, Siegfried Marcus (Austria) used liquid fuel to propel a handcart, known as The Marcus Car. He developed a more sophisticated four-seat vehicle in 1888-1889. Edouard Delamare-Debouttevile (France) built a gas-powered automobile in 1884. German inventor Karl Benz made his first automobile in 1885 – the first with a practical high-speed internal combustion engine – and started production in 1888. Gottlieb Daimler and Wilhelm Maybach (Germany) designed and built the first true automobile (not a carriage with a motor) from scratch in 1885. John William Lambert (US) built a three-wheeler in 1891, the same year that Henry Nadiq (US) built a four-wheeler. René Panhard and Emile Lavassor (France) built the first automobile with a spray carburetor in 1891. Charles Duryea tested the first US gasoline-powered automobile in Massachusetts in 1893. In the UK, an early automobile was built by Frederick William Lanchester in 1895. In Quebec, Canadian George Foss built a single-cylinder gasoline car in 1896. Many developments followed.
Hippolyte Pixii (France) created a crude form of alternating current (AC) in 1832 when he designed and built the first alternator. In 1879, Walter Baily (UK) demonstrated a battery operated polyphase motor aided by a commutator. Marcel Deprez (France) described a similar motor, with a rotating magnetic field produced by a two-phase AC system of currents, in an 1880 paper. In 1881, Lucien Gaulard (France) and John D. Gibbs (UK) demonstrated an AC transformer in London, which they sold to Westinghouse. Elihu Thomson (UK) built an AC motor in 1886 by using the induction-repulsion principle. In 1887, Charles Schenk Bradley patented a two-phase AC power transmission with four wires. Working independently, Galileo Ferraris (Italy), in 1885, and Nikola Tesla (Serbia/US), in 1887, built commutatorless AC induction motors. Ferraris made a single phase motor, while Tesla made a two-phase motor. In 1890, Mikhail Dolivo-Dobrovolsky (Russia) made the first three-phase induction motor. In 1891, he combined the motor with a three-phase generator and transformer to create the first three-phase AC system. Charles Eugene Lancelot Brown (Switzerland) further developed the three-phase motor design. Other three-phase AC systems were developed by Friedrich August Haselwander (Germany) and Jonas Wenstrom (Sweden).
In 1867, James Clerk Maxwell (UK) predicted the existence of radio waves, electromagnetic waves that are radiated by charged particles as they accelerate. Heinrich Hertz (Germany) proved the existence of radio waves by generating them experimentally in his laboratory in 1887. He also showed that the radio waves traveled at the speed of light.
The Ancient Greeks believed that the gods did not breathe air, but aether, a pure essence that filled up the space where they lived. Plato believed aether was ‘the most translucent kind’ of air, but Aristotle thought it was the fifth element (or quintessence), after air, water, fire and earth, and it did not follow the rules that applied to other substances. According to Aristotle, the sun, moon, planets and stars were held in circular orbits by spheres made of crystallized aether. By the Middle Ages, scholars believed that celestial bodies traveled in dense aether, while ’empty’ space was filled with aether that was ‘subtler than light.’ Beginning in the late 17th Century, the notion of aether was revived to explain certain scientific phenomena. So, proponents of the wave theory of light, such as Christiaan Huygens (The Netherlands) in 1678, invoked luminiferous aether as the medium to propagate the waves, just as air propagated sound waves. Sir Isaac Newton (England) was a big fan of aether: he invoked it to support his particle theory of light in 1675, and also to explain how gravity operated in the Principia in 1687. Similarly, Johann Bernoulli (Switzerland), a proponent of the ‘light is particles’ theory, also called upon the aether in 1736 as the medium in which the particles traveled. A type of aether composed of tiny unseen particles formed the basis for Le Sage’s theory of gravitation, proposed by Nicolas Fatio de Duillier in 1690 and Georges-Louis Le Sage in 1748. By the 19th Century, the luminiferous aether theory was predominant, yet no one had been able to confirm that aether existed. It was hypothesized that the moving Earth would drag the aether either partly (Augustin-Jean Fresnel, France 1818) or completely (George Gabriel Stokes, Ireland/UK, 1844) as it revolves around the sun and scientists began to design experiments to detect the aether wind. An 1887 experiment by Albert A. Michelson and Edward W. Morley (US) conclusively found no stationary aether, although it left open the possibility of the less-popular theory that the Earth completely dragged the aether along with it. Further experiments between 1893 and 1935 by numerous scientists with ever more sophisticated equipment failed to turn up any evidence for aether. The last gasp of the aether theory came from Hendrik Lorentz (The Netherlands) in 1892-1895, who developed a theory of a completely motionless aether, which eventually morphed into a variation of Einstein’s special theory of relativity, which assumed no aether whatsoever.
George Eastman (US) invented the first roll film in 1884, which eventually replaced photographic plates. In 1888 he invented the portable Kodak camera, the first camera to use roll film, which made thousands into amateur photographers. It was also one of the first cameras that did not require a tripod, thus increasing mobility.
A mitochondrion is a membrane-bound organelle found in most eukaryotic cells, that generates most of a cell’s supply of adenosine triphosphate and is also involved in signaling, cell differentiation, cell death, and cell growth. The majority view supports the theory of endosymbiosis, that mitochondria were originally prokaryotic cells (related to Rickettsia bacteria) that became endosymbionts living inside the eukaryotic cells. In the 1850s, Albert von Kölliker (Switzerland) described granules in the sarcoplasm of muscle cell nuclei that were later identified as mitochondria. Gustaf Retzius (Sweden) named these granules sarcosomes in 1890. In 1894, Richard Altmann (Germany) identified that mitochondria, which he called bioblasts, were organelles. Carl Benda (Germany) coined the term mitochondria for the organelles in 1898. Friedrich Meves observed mitochondria in plants in 1904. In 1908, Meves and Claudius Regaud (France) suggested that mitochondria contain proteins and lipids. Benajamin F. Kingsbury linked mitochondria and respiration in 1912. Early evidence of the respiratory function of mitochondria was made by Otto Heinrich Warburg and Heinrich Otto Wieland (Germany) in 1913, but the actual respiratory chain was not revealed until 1925, when David Keilin (USSR) rediscovered cytochromes. In 1957, Philip Siekevitz (US) described mitochondria as the ‘powerhouse of the cell.’ In 1967, scientists discovered that mitochondria contained ribosomes and had their own DNA. A map of the mitochondrial genome was completed in 1976. As early as 1918, Paul Portier became convinced that mitochondria were direct descendents of bacteria. Ivan Wallin (US) proposed that mitochondria had an endosymbiotic origin in the 1920s, but the theory was ignored at the time. In a 1967 paper, Lynn Margulis (US) advanced the endosymbiotic theory, which she elaborated on in a 1981 book.
Prior to motion pictures, numerous devices were invented to take advantage of the phenomenon of persistence of vision, in which the brain continues to perceive an image for a short period after it has been removed, which allows a series of still images to create the illusion of motion. In the late 1870s, photographic technology had advanced enough to capture moving objects, as demonstrated by Eadweard Muybridge’s photos of human and animal locomotion. In 1882, Etienne-Jules Marey (France) invented a camera that was a precursor to the movie camera that could take 12 pictures per second and eventually 30 frames per second. The first movie camera, which could rapidly photograph a series of images, was invented by Louis Le Prince (France/UK) in 1888. William Friese-Greene (UK) invented a movie camera in 1889, with a public demonstration in 1890, but its 10 frames per second rate and unreliability were serious drawbacks. George Eastman invented celluloid roll film in 1889, which would become the film used to make movies. In late 1890, Thomas Edison and his assistant William Dickson (US) invented the Kinetographic Camera, a motor-driven movie camera that became the first commercially successful movie camera after it was introduced in 1892. Dickson and Edison also invented the Kinetoscope, which allowed individual viewers to watch motion pictures through a peephole. The Kinetoscope was demonstrated publicly on May 20, 1891. Dickson and Edison produced an improved version in 1892 and debut it at the Chicago World’s Fair in 1893. Edison’s movie studio, the Black Maria, opened in February 1893 in West Orange, New Jersey. Early films include Fred Ott’s Sneeze (1894), Carmencita (1894), Annabelle Butterfly Dance (1894) (one of the first color tinted films) and The Kiss (1896). Georges Demenÿ (France) built his Beater Movement camera in 1893. Polish inventor Kazimierz Prószyński made the Pleograph, which combined camera and projector, in 1894. Charles Moisson (France) made the Domitor camera for Auguste and Louis Lumière (France) in 1894. Edison released the Kinetoscope for commercial use in 1894. Using his Phantoscope projector, inventor Charles Francis Jenkins (US) projected a motion picture he filmed and hand tinted with color onto a screen for an audience in Richmond, Indiana on June 6, 1894. Jenkins and Thomas Armat (US) improved the Phantoscope and demonstrated it at two exhibitions in the last half of 1895. After a patent dispute, the Phantascope was sold to Edison, who renamed it the Vitascope. Inspired by Edison, the Lumières invented the Cinematographe in 1895 – a combination movie camera (hand cranked), film printer and projector. They demonstrated the system in their basement on March 22, 1895 with their first film, Workers Leaving the Lumiere Factory. On December 28, 1895, the Lumières presented the first public, commercial exhibition of projected motion pictures to a paying audience in Paris’s Salon Indien at the Grand Café. The ten short films in the program included The Gardener, or The Sprinkler Sprinkled, the first comedy. Perhaps the Lumieres’ most famous short film is Arrival of a Train at La Ciotat (1895), which reportedly frightened some moviegoers. The first theatrical exhibition of Edison’s Vitascope projector occurred at Koster and Bial’s Music Hall in New York City on April 23, 1896. French filmmaker Georges Melies’ A Trip to the Moon of 1902 introduces special effects to the movies. In 1903, Edison employee Edwin S. Porter (US) makes the 12-minute film The Great Train Robbery, the first Western and the most sophisticated motion picture to date, with 14 shots cutting between simultaneous events. The first permanent theater dedicated to showing motion pictures was The Nickelodeon, which opened in Pittsburgh, Pennsylvania in 1905. The first two-reel film is D.W. Griffith’s Enoch Arden, from 1911. The first sound film is The Jazz Singer, from 1927. The Production Code was introduced in Hollywood in 1930 to oversee morals in movies.
Argon is a chemical element with the atomic number 18. It is a gas at room temperature and is the third most common component of the Earth’s atmosphere, after nitrogen and oxygen. It is one of the noble gases, so-called because they do not interact with other elements. In 1785, Henry Cavendish conducted experiments on the contents of air. He was able to determine the percentage of phlogisticated air (later found to be nitrogen) and dephlogisticated air (later found to be oxygen), but he noted a small residue of gas that he assumed was the result of an error. In fact, Cavendish had found argon, without realizing it. In 1882, H.F. Newall and W.N. Hartley, working independently, observed unidentified lines in the color spectrum of air, but were unable to determine what element was responsible – they, too, had encountered argon. It was only in 1894 that Sir William Ramsay and Lord RayLeigh (UK), at University College London, by recreating Cavendish’s experiments and painstakingly removing each known substance from air that they were able to isolate argon, the first of the noble gases to be discovered.
Researchers first noticed unidentified rays emanating from experimental discharge tubes called Crookes tubes around 1875. In 1886, Ivan Pulyui (Ukraine/Germany) discovered that sealed photographic plates darkened when exposed to Crookes tubes. Nikola Tesla (Serbia/US) began experimenting with the rays in 1887. Fernando Sanford (US) generated and detected the rays in 1891. Wilhelm Röntgen (Germany) began studying the rays in 1895 and announced their existence (giving them the name ‘X-rays’) in a scientific paper. Röntgen was the first to recognize the medical use of X-rays when he X-rayed his wife’s hand. In 1896, Thomas Edison (US) invented the flouroscope for X-ray examinations. In the same year, John Hall-Edwards (UK) was the first physician to use X-rays under clinical conditions. Problems with the cold cathode tubes used to generate X-rays led to the invention of the Coolidge tube, by William D. Coolidge (US) in 1913.
James Clerk Maxwell (Scotland) established the mathematical basis for propagating electromagnetic waves through space in a paper published in 1873. David E. Hughes (Wales/US) was probably the first to intentionally send a radio signal through space in 1879 using his spark-gap transmitter, although the achievement was misunderstood at the time. In 1880, Alexander Graham Bell and Charles Sumner (US) invented the photophone, a wireless telephone that transmitted sound on a beam of light. In 1885, Thomas Edison (US) invented a method of electric wireless communication between ships at sea. In 1886, Heinrich Hertz (Germany) conclusively demonstrated the transmission of electromagnetic waves through space to a receiver. Édouard Branly (France) improved the receiver device in 1890. In 1892, Nikola Tesla (Serbia/US) invented the Tesla coil, which generated alternating current electricity; in 1893 Tesla developed a wireless lighting device and in 1898 he demonstrated a remote controlled boat. In 1894, Sir Oliver Lodge (UK) improved Branly’s receiver, calling it a coherer, and demonstrated a radio transmission in 1894. In the same year, Lodge showed the reception of Morse code signals by a wireless receiver. Also in 1894, Jagadish Chandra Bose (India) demonstrated transmission of radio waves over distance; Bose developed an improved transmitter and receiver in 1899. Guglielmo Marconi (Italy/UK) read Lodge’s and Tesla’s papers in 1894 and built his first radio devices in early 1895. By the end of 1895, he had developed a device that could transmit radio waves 1.5 miles. In 1896, Marconi moved to England, where he presented his device to Sir William Preece at the British Telegraph Service. By 1897, Marconi had patented his device and started his own wireless business, which established radio stations at various locations. In 1899, Marconi sent radio waves across the English Channel; he sent the first transatlantic message, possibly as early as 1901. Alexander Popov (Russia) built and demonstrated improved versions of both the transmitter and receiver, first in May 1895 for a scientific group and then a public display in March 1896. There is some evidence that Popov set up a radio transmitter with two-way communication between a naval base and a battleship in 1900. Beginning in 1899, Ferdinand Braun (Germany) made significant improvements to the design of wireless devices, including inventing the closed circuit system and increasing the distance the signals would carry. Roberto Landell de Moura, a Brazilian priest and scientist, invented a radio in 1900 that could transmit a distance of eight kilometers. In 1904, Sir John Fleming (UK) invented the vacuum electron tube, which became the basis for radio telephony. Lee de Forest (US) invented the triode amplifying tube in 1906. The regenerative circuit, which allowed long-distance sound reception was invented by Edwin H. Armstrong (US) in 1912. Armstrong also discovered frequency modulation, or FM radio, in 1933.
An airship (also known as a dirigible) is a lighter-than-air aircraft that can navigate through the air under its own power. There are three types: (1) a non-rigid airship consists of a gas-filled envelope; (2) a semi-rigid airship is a pressurized gas balloon or envelope attached to a lower metal keel; and (3) a rigid airship has an internal frame and gas-filled bags. In 1670, Francesco Lana de Terzi (Italy) designed an ‘Aerial Ship’ supported by four copper spheres from which air was evacuated. The design was unsound and it was never built. A more practical design was proposed by Jean Baptiste Marie Meusnier (France) in 1783: a 260-foot long ship with internal balloons to regulate lift, attached to a carriage that doubled as a boat in the unlikely event of a water landing. When Jean-Pierre Blanchard (France) fitted a hand-powered propeller to a hot-air balloon in 1784, he created the first powered airship; he used wings for propulsion and a tail for steering to navigate a balloon across the English Channel in 1785. A design of a balloon with a steam engine driving twin propellers was proposed by William Bland (Australia) in 1851. Henry Giffard (France) was the first to make an engine-powered flight when he flew 17 miles in 1852. In 1872, Dupuy de Lome (France) designed and flew a balloon driven by a large propeller turned by eight men. The same year, Paul Haenlein (Germany) flew an airship with an internal combustion engine that ran on coal gas. Charles F. Ritchel (US) created a hand-powered one man rigid airship in 1878. Gaston Tissandier (France) made the first electric-powered flight in 1883, using a 1.5 horsepower Siemens electric motor. Fully controllable free flight was achieved by Charles Renard and Arthur Constantin Krebs (France) in La France in 1884, using an 8.5 hp electric motor and 435 kilogram battery. The Campbell Air Ship, designed and built by Peter C. Campbell in 1888, was lost at sea in 1889. Frederich Wölfert (Germany) built three airships in 1888-1897 powered by gasoline engines, the last of which caught fire in flight and killed both occupants. Augusto Severo de Albuquerque Maranhão (Brazil) designed and built semi-rigid airships in 1894 and 1902. In 1895, Count Ferdinand von Zeppelin (Germany) patented a rigid airship that combines balloon air cells with a structural framework. An aluminum airship was built by David Schwarz (Hungary) in 1897. The Luftschiff Zeppelin LZ1, a rigid airship, flew in July 1900. An improved LZ2 was built in 1906. The Zeppelin airships held the crew and engines in a gondola that hung beneath the hull driving propellers attached to the sides of the frame. Alberto Santos-Dumont (Brazil/France) designed 18 balloons and dirigibles beginning in 1901 with the Number 6, which flew around the Eiffel Tower. Thomas Scott Baldwin (US), built and flew airships beginning in 1904, and created the first US military airship in 1908. Walter Wellman (US) and Melvin Vaniman (US) unsuccessfully attempted airship flights to the North Pole in 1907 and 1909 and across the Atlantic in 1910 and 1912. An innovative new three-lobed design was proposed by Leonardo Torres Quevedo (Spain) in 1902; he and Captain A. Kindelan built the Espana in 1905, then designed an improved version in 1909, which was mass produced in 1911. Hans Gross (Germany) developed one of the first successful semi-rigid airships in 1907. Airships were used as bombers in World War I. Goodyear (US) launched its first helium-filled blimp in 1925. In the 1930s, rigid airships were used for luxury passenger transport until the world’s largest passenger airship, the Hindenburg, caught fire and burned in New Jersey in 1937, killing 36.
Psychoanalysis is a series of techniques intended to cure mental and emotional disturbances. The premises of psychoanalytic theory include: (1) psychological development is determined by genetic inheritance and early childhood experiences; (2) attitude, mannerism, experience and thought are largely influenced by unconscious irrational drives; (3) the mind resists attempts to become aware of the irrational drives through defense mechanisms; (4) mental and emotional disturbances such as neuroses and mental illness are the result of conflicts between the conscious and unconscious mind; and (5) to liberate the self from the harmful effects of the unconscious mind, the psychoanalyst must assist the patient to bring the unconscious material into the conscious mind. The foundation was laid for psychoanalysis in the 1880s, when Austrian physician Josef Breuer began his ‘talking cure’ with a patient named Anna O. Breuer. Breuer discussed the case and the theory behind it with his protege Sigmund Freud, and together they wrote Studies on Hysteria, which set out many of the principles of psychoanalysis, in 1895. Freud continued to develop the theory in a series of publications between 1900 and 1925. Other contributors were Austrians Otto Rank (1924), Robert Waelder (1936) and Freud’s daughter Anna (1936). Later theorists included Heinz Hartmann (Austria/US), Karen Horney (Germany/US), Charles Brenner (US), Erik Erikson (Germany/US), Heinz Kohut (Austria/UK), Jacques Lacan (France), Harry Stack Sullivan (US), Robert Langs (US), Stephen Mitchell (US) and Robert Stolorow (US), who have taken Freud’s original theory in many different directions.
Radioactivity, also known as radioactive decay, or nuclear decay, occurs when unstable atoms in a particular element emit either alpha particles, beta particles or gamma rays from their nuclei. In the process of emitting radiation, the atom changes from one element to another. Henri Becquerel (France) discovered the radioactivity of uranium in 1896; he recognized the phenomenon was different from the recently discovered X-rays. In 1898, Marie and Pierre Curie (France) identified two more radioactive elements – radium and polonium. Ernest Rutherford (NZ/UK) identified two types of radiation – the alpha and beta rays – in 1899. Pierre Curie classified alpha and beta particle radiation in 1900. Paul Ulrich Villard (France) discovered a third type of radiation in 1900, which Rutherford called gamma rays. The dangerous effects of radiation exposure to humans were not identified until much later. Marie Curie herself died of an illness that was probably related to her frequent exposure to radioactivity.
In the mid-19th Century Richard Laming suggested that atoms consist of a core surrounded by small charged particles. George Johnstone Stoney (Ireland) proposed in 1874 that electricity consisted of charged ions, with a measurable charge. Hermann von Helmholz (Germany) suggested in 1881 that the positive and negative charges were divided into basic parts and both were atoms of electricity. In 1891, Stoney coined the name ‘electron’ for the fundamental unit of electricity. Experiments leading up to the discovery of the electron began with German physicist Johann Wilhelm Hittorf’s conductivity work in 1869; the discovery of cathode rays by Eugen Goldstein (Germany) in 1876; and the development of a high vacuum cathode ray tube by Sir William Crookes (UK) in the 1870s. Arthur Schuster (Germany/UK) performed cathode ray experiments that allowed him to estimate the charge-to-mass ratio of the electron. In 1896, J.J. Thomson, with John S. Townsend and H.A. Wilson (UK), performed a series of experiments that conclusively identified the cathode ray ‘electron’ as a particle with a definite mass and a negative charge and that electrons produced in different contexts (heating, illumination, radioactivity) were identical. George Fitzgerald (Ireland) proposed the name ‘electron’ for Thomson’s particle. In 1900, Henri Becquerel (France) showed that beta rays emitted by radioactive elements were electrons. The charge of the electron was measured more carefully by Robert Millikan and Harvey Fletcher (US) in a 1909 experiment the results of which were published in 1911.
Around the beginning of the 20th Century, physicists began to explore certain phenomena that did not appear to follow the rules of classical mechanics, leading to the development of what is known as quantum theory, or old quantum theory, which was essentially replaced by quantum mechanics in about 1925. In 1900, German physicist Max Planck explained the results of his studies of light emission and absorption by theorizing that light and other forms of electromagnetic energy could only be emitted in quantized form, in what would later be known as photons. In 1905, Albert Einstein (Germany) postulated the light is made of individual quantum particles in order to explain the photoelectric effect identified by Heinrich Hertz in 1887. Einstein also used quantum principles to explain the specific heat of solids. In 1913, Niels Bohr (Denmark) revised the model of the atomic structure to explain the atomic spectra by incorporating quantum energy states into the electron orbits. In the following years Arnold Sommerfeld (Germany) further developed the quantum theory.
A tractor is a powered vehicle designed to pull heavy equipment at slow speeds, usually in an agricultural setting. Richard Trevithick (UK) built a semi-portable steam-powered ‘barn engine’ in 1812, which drove a threshing machine. William Tuxford (UK) invented the first ‘portable engine’ – a steam engine on wheels – in 1839. In 1859, Thomas Aveling (UK) made the first self-propelled traction engine. The 1860s saw the first steam-powered plowing engines or steam tractors, which continued to be made well into the 20th Century. One of those who experimented with steam traction engines was American Benjamin Holt, whose first attempt was an immense 48,000 pound steam tractor called Old Betsy in 1890. American John Froelich made the first gasoline-powered tractor in 1892 in Iowa by mounting a Van Duzen single-cylinder gas engine on a Robinson engine chassis. In 1894, William Paterson (US) made an experimental gas traction engine for J.I. Case, but it was never sold commercially. In the UK, Herbert Ackroyd designed and Richard Hornsby & sons built the Hornsby-Akroyd Patent Safety Oil Traction Engine in 1896. The first one sold in 1897. Americans Charles W. Hart and Charles H. Parr formed the Hart-Parr Gasoline Engine Company in 1897 to develop and sell gas traction engines. Hart and Parr rejected the name ‘gas traction engine’ and instead combined ‘traction’ and ‘power’ to form ‘tractor.’ Their model No. 1 went on the market in 1901. Also in 1901, Dan Albone (UK) made the first successful light-weight gas-powered tractor, a three-wheeled model he called the Ivel Agricultural Motor. In 1904, Benjamin Holt demonstrated the first successful tractor using crawler-type treads, which was later adapted by the British Army to build the first tank. Holt’s invention evolved into the Caterpillar tractor in 1925. Henry Ford (US) made a gas-powered tractor that he called an “automobile plow” in 1907, but sales did not take off until 1917, when Ford brought out the low-priced Fordson tractor; by 1923, the Fordson had 77% of the U.S. market. The Saunderson Tractor and Implement Co. (UK) produced a four-wheeled tractor in 1908 and its models dominated the British market.
During experiments with blood transfusion, Karl Landsteiner (Austria) identified the ABO blood group in 1901. Alfred von Decastello and Adriano Sturli (Austria) identified the AB blood type in 1902. Czech physician Jan Jansky discovered the four basic blood groups independently and published the finding in a little-noticed 1907 paper. William Lorenzo Moss (US) made similar discoveries, which were published in 1910. In 1910-1911, Ludwik Hirszfeld (Poland) and Emil von Durgern (Germany) discovered that ABO blood groups are inherited. Felix Bernstein (Germany) determined the chromosomal basis for blood groups in 1924. In 1937, Landsteiner, together with Alexander Wiener (US) identified the Rhesus group. In 1945, Robin Coombs, Arthur Mourant and Rob Race (UK) developed the Coombs blood test. At present, 33 human blood group systems have been identified, and more than 600 blood group antigens.
Scientists currently believe the Earth’s atmosphere is made up of the following layers: (1) troposphere (0 to 7 miles) (the tropopause is located at the top of the troposphere); (2) stratosphere (7-31 miles); (3) mesophere (31-50 miles); (4) thermosphere (50-440 miles) and (5) exosphere (440 miles and up). The ozone layer is located in the stratosphere, usually between 9.3-21.7 miles. The ionosphere includes the mesosphere, the thermosphere and part of the exosphere (31-621 miles). Léon Philippe Teisserenc de Bort (France) discovered that atmosphere is divided into troposphere and stratosphere; German scientist Richard Assmann reached the same conclusion independently. Both de Bort and Assmann announced their discoveries in 1902. Oliver Heaviside (UK) proposed the existence of the ionosphere, a conducting layer of the atmosphere in 1902. Also in 1902, Arthur Edwin Kennelly (Ireland/US) discovered some of the radio and electrical properties of the ionosphere. Robert Watson-Watt (UK) coined the term ionosphere in 1926. Edward V. Appleton (UK) confirmed the existence of the ionosphere in 1927. Lloyd Berkner (US) measured the ionosphere’s height and density in the 1950s. Charles Fabry and Henri Buisson (France) discovered the ozone layer in 1913. G.M.B. Dobson (UK) studied the ozone layer and set up a worldwide network of ozone monitoring stations between 1928 and 1958.
Hormones are signaling molecules produced by the glands of living organisms that are transported to distant target organs by the circulatory system in order to regulate physiology and behavior. In 1894, George Oliver and Eduard Albert Sharpey-Schaeffer (UK) demonstrated the effect of an extract of the adrenal gland, that is to say, the hormone adrenaline, which contracted blood vessels and muscles and raised blood pressure. In 1902, Ernest Starling and William Bayliss (UK) discovered secretin, which upon stimulation was released from the duodenum and carried to the pancreas, where it stimulated the pancreas to release digestive juices into the intestine. In 1905, Starling and Bayliss coined the term ‘hormone’ to describe secretin and similar substances. Edward C. Kendall isolated the thyroid hormone thyroxin in 1915. The same year, Walter Bradford Cannon demonstrated the close connection between endocrine glands and emotions.
Air cooling techniques have existed since Ancient Egypt, when people hung reeds in windows and moistened them with trickling water – the evaporating water cooled the air blowing through the window. Ancient Roman houses had cool water circulating through the walls. Ding Huan (China) invented a manually-powered rotary fan in 180 CE, and by 747 CE, there are references to water-powered fan wheels in China. Medieval Persians used cisterns and wind towers to cool their buildings. Cornelius Drebbel (The Netherlands) developed an evaporation-based cooling system in the 17th Century. Benjamin Franklin and John Hadley (US) conducted important evaporation experiments in 1758. In 1820, Michael Faraday (UK) discovered the cooling power of compressed, liquefied ammonia. In 1842, American physician John Gorrie created ice using compression and then used fans to circulate the cool air, but financial woes prevented him from developing the invention. In 1851, James Harrison (Australia) developed an ice-making machine. Willis Carrier (US) invented the first modern electrical air conditioner in 1902 in order to control temperature and humidity in a printing plant. Stuart Cramer (US) developed a similar machine in 1906 for a textile mill. Cramer coined the term ‘air conditioning’, which Carrier adopted. An air conditioning unit was installed in the home of Charles Gates (US) in 1914. Thomas Midgley, Jr. (US) invented Freon, the first non-flammable, non-toxic refrigerant, in 1928. (Unfortunately, Freon and other chlorofluorocarbon gases destroy the ozone layer and are being phased out.) In 1931, H.H. Schultz and J.Q. Sherman (US) created a very expensive individual room air conditioner. The DuBose house in Chapel Hill, NC (US) became fully air conditioned in 1933. Packard introduced the first air conditioned automobile in 1939. In 1945, Robert Sherman (US) invented an affordable, portable, in-window air conditioner. In the 1970s, central air conditioning was developed.
Classical conditioning is a type of learning in which a conditioned stimulus is paired with an unconditioned stimulus, which leads to an unlearned reflex response. After the pairing is repeated to the organism, it begins to exhibit the reflex response in the presence of the conditioned stimulus without the unconditioned stimulus. In Russian scientist Ivan Pavlov’s famous experiments in the early 20th Century, he noticed that dogs salivated in the presence of meat. He began to ring a bell (or use other stimuli) whenever the meat was brought to them. Eventually, the dogs would salivate upon the stimulus, even when no meat was present. Pavlov announced his results in 1903 at conferences in Sweden and Spain, and he is generally credited with the discovery. American psychologist Edwin Twitmyer independently discovered the conditioned reflex in 1902, when he associated the use of a hammer to induce the knee jerk reflex with the sound of a bell. Eventually his human subjects would jerk their knees at the sound of the bell, without the hammer. Twitmyer’s discovery was barely noticed when he presented it at a conference in 1904. Russian psychologist Vladimir Bekhterev set out a rival theory of conditioned reflexes in a 1903 book. American psychologist John B. Watson used Pavlov’s experiments as the basis for his new behaviorist model of psychology in 1913. His controversial 1921 “Little Albert” experiment involved conditioning a human infant to associate a white rat with a frightening noise, so that eventually he feared the rat.
Humans have tried to fly since ancient times. Abbas Ibn Firnas (Berber/Andalusia) built a glider in the 9th Century; Eilmer of Malmesbury (UK) tried it in the 11th Century; and Leonardo da Vinci (Italy) designed a man-powered aircraft in 1502. Sir George Cayley (UK) designed fixed-wing airplanes from 1799 and built models from 1803. He built a successful glider in 1853. In 1856, Jean-Marie Le Bris (France) took the first powered flight when a horse pulled his glider, the Albatross, across a beach. John J. Montgomery (US) made a controlled flight in a glider in 1883, as did Otto Lilienthal (Germany), Percy Pilcher (UK) and Octave Chanute (France/US) about the same time. Between 1867 and 1896, Lilienthal made numerous heavier-than-air glider flights. Clément Ader (France) built a steam-powered airplane in 1890 and may have flown 50 meters in it. Hiram Maxim (US/UK) built an airplane powered by steam engines in 1894 that had enough lift to fly, but was uncontrollable and never actually flew. Lawrence Hargrave (Australia) experimented with box kites and rotary aircraft engines in the 1890s. In 1896, American Samuel Pierpont Langley’s Aerodrome No. 5 made the first successful sustained flight of an unmanned, engine-driven heavier-than-air craft, but his attempts at manned flight in 1903 did not succeed. There is some evidence that Gustave Whitehead (Germany/US) flew his Number 21 powered monoplane at Fairfield, Connecticut (US) in 1901, two and a half years before the Wright Brothers, but the matter is subject to debate. Most believe that Orville and Wilbur Wright (US) accomplished “the first sustained and controlled heavier-than-air powered flight” (FAI) on December 17, 1903 at Kill Devil Hills, North Carolina. By 1905, the third version of the Wright Brothers’ airplane was capable of fully controllable, stable flight for substantial periods. Traian Vuia (France) flew in a self-designed, fully self-propelled, fixed wing aircraft with a wheeled undercarriage in 1906. Jacob Ellehammer (Denmark) also flew a monoplane in 1906. In 1906, Alberto Santos Dumont (Brazil) flew 220 meters in less than 22 seconds, without the assistance of a catapult. In 1908-1910, Dumont designed a number of Demoiselle airplanes that were well received. In 1908 and 1909, Louis Blériot (France) designed airplanes that were improvements over earlier models. The first jet aircraft was the German Heinkel He 178, first tested in 1939, followed by the Messerschmitt Me 262 in 1943. The first aircraft to break the sound barrier was the Bell X-1, in 1947. The first jet airliner was the de Havilland Comet, introduced in 1952. The first widely successful commercial jet was the Boeing 707, which arrived in 1958. The Boeing 747 was the largest passenger jet from 1970 until 2005, when it was surpassed by the Airbus A380.
Albert Einstein (Germany) developed the special theory of relativity to correct Newton’s laws of classical mechanics, which do not accurately explain phenomena at velocities near the speed of light. The theory explains how objects behave when moving at a constant speed relative to each other. Einstein relied on the principles that (a) the laws of physics remain the same despite one’s frame of reference; and (b) the speed of light is the same to all observers. Under the theory, space and time are two aspects of the same phenomenon, giving us four dimensions instead of three. A key implication (which has been proven many times by experiment) is that time slows down as acceleration increases.
German physicist Albert Einstein’s famous equation ‘E = mc2’ states the physical law that matter and energy are two forms of the same substance, that one can be converted to the other, and that the amount of energy produced by converting (i.e., destroying) even a small amount of mass is enormous, as it is proportional to the square of the speed of light (186,000 miles per second). A number of precursors led up to Einstein’s revolutionary equation. In 1717, Sir Isaac Newton (England) wondered whether particles of mass and particles of light might be converted, one into the other. Emanuel Swedenborg (Sweden) speculated in 1734 that matter was made of points of potential motion. Numerous physicists at the end of the 19th and beginning of the 20th centuries sought to understand how an electromagnetic field affects the mass of a charged particle. Einstein first introduced a mass-energy equivalence equation in his 1905 paper on special relativity; it was later reduced to the famous form of E = mc2. The equivalence of mass and energy has been experimentally proven in both directions. In 1932, John Cockcroft and E.T.S. Walton (UK) broke apart an atom, releasing energy, and found that the total mass of the fragments had decreased slightly. In 1933, Irène and Frédéric Joliot-Curie (France) detected the conversion of energy into mass when they photographed a photon (a quantum of electromagnetic energy) converting into two subatomic particles. Perhaps the most famous demonstration of Einstein’s equation is the force released by atomic weapons.
Brownian motion refers to the random movements of particles suspended in a liquid or gas fluid that result from collisions with smaller atoms or molecules in the fluid. Dutch scientist Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, an early example of Brownian motion. The official discovery of Brownian motion took place in 1827, when Scottish botanist Robert Brown noted the unusual random movements of pollen grains suspended in water. Thorvald N. Theile (Denmark) provided the mathematical underpinnings of Brownian motion in 1880, and Louis Bachelier (France) used the model of Brownian motion to explain the stochastic processes of economic markets in a 1900 thesis. In 1905, Albert Einstein (Germany) explained Brownian motion as the result of the larger particle (e.g., pollen grain) being moved by individual molecules of the fluid in which it is suspended (e.g., water). Einstein’s explanation proved definitively that atoms and molecules exist. The predictions of Einstein’s paper were verified experimentally in 1908 by Jean Perrin (France).
The photoelectric effect refers to the phenomenon that many metals emit electrons when light shines on them. Heinrich Hertz (Germany) discovered the photoelectric effect in 1887. In 1905, Albert Einstein (Germany) explained that the results of experiments measuring the photoelectric effect could be explained if light energy was carried in discrete quantized packets, or quanta. Einstein’s explanation lent support to quantum theory.
Radiometric dating (also called radioactive dating) is a technique used to date materials by comparing the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates. Ernest Rutherford (NZ/UK) first proposed the possibility of radiometric dating in 1905. Following up on Rutherford’s suggestion, Bertram Boltwood (US) demonstrated that radiometric dating was possible in 1907. Radiometric dating has permitted scientists to date rocks and fossils and determine the age of the Earth. Scientists use a specific form of radiometric dating called radiocarbon dating to find the age of organic materials less than 50,000 years old.
The third law of thermodynamics states that the entropy of a perfect crystal, at absolute zero Kelvin, is exactly equal to zero. Walter Nernst (Germany) first formulated the law between 1906 and 1912, when he stated, “It is impossible for any procedure to lead to the isotherm T = 0 in a finite number of steps. The entropy of ordered solids reaches zero at the absolute zero of temperature.” Gilbert N. Lewis and Merle Randall (US) proposed an alternative version of the law in 1923: “If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.” A later formulation of the third law, known as the Nernst-Simon statement, is: “The entropy change associated with any condensed system undergoing a reversible isothermal process approaches zero as temperature approaches 0 K, where condensed system refers to liquids and solids.”
Henry Cavendish (UK) calculated the density of the Earth in 1798 as 5.48 times that of water (close to the modern value of 5.53). Exactly a century later, Germany seismologist Emil Wiechert noted that the Earth could not be made entirely of rock because it was denser than rock. He calculated that the Earth’s density could be explained if, like a meteorite, the Earth had a core made of nickel and iron. In 1906, seismologist Richard Dixon Oldham confirmed the existence of the Earth’s core while measuring the speed of earthquake waves and finding that they accelerated through the Earth until they reached a certain depth, after which they slowed down. Beno Gutenberg (Germany) determined the exact diameter of the core in 1909. In 1926, Harold Jeffreys (UK) proved that the core was in the liquid state. Then in 1936, in an attempt to interpret some unusual experimental results, Danish seismologist Inge Lehmann correctly hypothesized that inside the liquid core lay a denser inner core, which was later determined to be solid. In 1946, German-born physicist Walter Elsasser proposed that the liquid outer core was a dynamo that generated the Earth’s magnetic field.
Prior to creating the first synthetic plastic, Bakelite, chemists had made plastic substances with nitrocellulose mixed with other materials. Alexander Parkes (UK) invented the partially-synthetic Parkesine in 1856, a thermoplastic celluloid based on nitrocellulose treated with solvents. John W. Hyatt modified Parkesine to create Celluloid in 1869. In an effort to find a substitute for shellac, Leo Baekeland, a Belgian-born chemist working in the US, invented Bakelite, the first completely synthetic plastic, in 1907. The chemical name of Bakelite is polyoxybenzylmethylenglycolanhydride. In 1922, Hermann Staudinger (Germany) set out the theoretical background of macromolecules and polymerization on which the modern plastics industry rests. In 1958, Robert Banks and Paul Hogan (US) invented polypropylene and devised a low-pressure method for producing high-density polyethylene.
For centuries, humans washed clothes and other textiles by hand. A UK patent was issued in 1691 for a Washing and Wringing Machine. A drawing of an early washing machine appears in a British magazine in 1752. Jacob Schäffer (Germany) invented a hand-driven washing machine in 1766. Rogerson (UK) invented a machine in 1780 and Henry Sidgier (UK) created a machine with a rotating drum in 1782. In 1787, Edward Beetham and Thomas Todd (UK) successfully promoted their own washing machine. By 1790, Beetham was promoting a portable washing mill designed by James Wood (UK). Kendall (UK) offered a rival invention in 1791. John Turnbull (Canada) invented a washing machine with a wringer in 1843. In 1851, James King (US) patented a hand-powered washing machine with a drum. Commercial laundry machines advanced more quickly than those for domestic use. Steam-driven commercial machines were sold in the mid-1850s in the US and UK. Hamilton Smith (US) patented a rotary washing machine in 1858. Richard Lansdale (UK) demonstrated a rotary washing machine with rollers for wringing/mangling in 1862. William Blackstone (US) built a hand-driven washing machine for his wife’s birthday in 1874. Margaret Colvin (US) invented the Triumph Rotary Washer in 1876. Some believe that Louis Goldenberg (US) at Ford Motor Co. invented the first electric washing machine in about 1900. Newspaper advertisements for electric washing machines appear as early as 1904. There is evidence that J.T. Winans (US) designed an electric washing machine that was produced by the 1900 Company in 1907. In 1908, Alva Fisher (US) invented an electric washing machine called The Thor that was manufactured by the Hurley Machine Co. The Thor was a drum-type machine with a galvanized tub and an electric motor. In 1911, the Upton Machine Co. (later Whirlpool) introduced an electric motor-driven wringer washer. Washing machines with spin dryers instead of wringer/manglers were introduced in the 1930s. Bendix Corporation introduced the first automatic washing machine in 1937. General Electric introduced a top loading automatic washing machine in 1947. Schulthess Group (Europe) produced an automatic washing machine in 1951. The first microchip-controlled automatic washing machines were introduced in 1978. In 1994, Staber Industries released the System 2000 machine, the only top-loading, horizontal axis washer made in the US. Fisher & Paykel (NZ) introduced SmartDrive washing machines in 1998. Maytag released a water-efficient top-loader in 2003. Sanyo made a drum-type machine with Air Wash in 2007.
The mantle of the Earth is a layer between the crust and the outer core. The mantle is a silicate rocky shell about 1800 miles thick that constitutes about 84% of the Earth’s volume. The mantle is divided into four layers: (1) lithosphere; (2) asthenosphere; (3) upper mantle; and (4) lower mantle. Although it is mostly solid, over periods of geologic time it behaves like a viscous liquid. In 1909, Andrija Mohorovičić (Croatia) discovered that there is a sudden increase in seismic activity at the top of the mantle; the boundary of such activity is known as the Mohorovičić discontinuity or “Moho”.
The Burgess Shale is a series of fossil-bearing rock formations in the Canadian Rockies of British Columbia. The fossils date to the Middle Cambrian Period (505 million years old) and contain many unusual and unique life forms, many preserved with impressions of soft body parts. The first person to notice the Burgess Shale was Richard McConnell of the Geological Survey of Canada in 1886. His discovery came to the attention of American paleontologist Charles D. Walcott, who first explored the area in 1907 but did not discover the main fossil-bearing area until his 1909 visit. By 1910, Walcott had opened a quarry. He returned each year until 1913, and again in 1917, 1919, 1921 and 1924. He brought back 65,000 specimens on 30,000 rock slabs to the Smithsonian before his death in 1927. Harvard professor Percy Raymond (US) began collecting fossils from the area in 1924 and into the 1930s. British scientist Harry B. Whittington returned to the Burgess Shale in the 1960s and his team reexamined Walcott’s original fossils. They determined that many of the fossils were previously unknown types of animals and some belonged to entirely new phyla. UNESCO named the Burgess Shale a World Heritage site in 1981.
Through his experiments with fruit flies (Drosophila melanogaster), biologist Thomas Hunt Morgan (US) proved that genes are carried on chromosomes and are the mechanical basis for heredity. In so doing, Morgan established the modern science of genetics.
In 1903, Dutch botanist Hugo DeVries suggested that the mechanism driving evolution was the mutation of genes; he even proposed that a single mutation in one gene might create an entirely new species. The work of Thomas Hunt Morgan (US) with fruit flies fleshed out the role of mutations in evolution. Morgan identified specific mutations in the flies and found that they changed traits, not species. The first mutation he discovered was a fly with white eyes instead of red ones. He found that the mutation appeared in future generations in proportions consistent with Mendel’s rules of inheritance. The science of genetics had been born.
Rejecting J.J. Thompson’s ‘plum pudding’ model of the atom, Ernest Rutherford (NZ/UK) proposed a model of the atom in 1911 that some referred to the solar system model, with nuclear sun orbited by electron planets. Rutherford’s proposal was based in part on the 1909 experiments by Hans Geiger and E. Marsden, in his lab, who scattered alpha particles with thin films of heavy metals, providing evidence that atoms possessed a discrete nucleus. In Rutherford’s model, electrons with very low mass orbited a very small charged nucleus, which contained most of the atom’s mass. Niels Bohr revised the model in 1913 to be consistent with quantum theory. His electrons had fixed orbits and could only jump from one orbit to another. Arnold Sommerfeld further revised the model to incorporate elliptical electron orbits about 1920.
Superconductivity is a phenomenon in which certain materials, when cooled below a critical temperature experience zero electrical resistance and expulsion of magnetic fields. Heike Kamerlingh Onnes (The Netherlands) discovered lack of electrical resistance in liquid helium in 1911. In 1933, Fritz Walther Meissner and Robert Ochsenfeld (Germany) discovered that substances undergoing superconductivity expelled their magnetic fields, which became known as the Meissner Effect. In 1935, Fritz and Heinz London (Germany) developed a mathematical explanation for superconductivity. Lev Landau and Vitaly Ginzburg (USSR) proposed a phenomenological theory of superconductivity in 1950. John Bardeen, Leon Cooper and John Scheiffer (US) developed a complete microscopic theory of superconductivity (the BCS theory) in 1957. The Landau-Ginzburg and BCS models were reconciled through the work of N.N. Bogolyubov (USSR) in 1958 and Lev Gor’kov (USSR) in 1959.
That certain diseases were caused by the lack of particular nutrients was known by the Ancient Egyptians. In 1747, Scottish physician James Lind discovered that citrus fruits prevented scurvy. Deprivation experiments allowed late 18th and early 19th century scientists to identify a lipid from fish oil, called ‘antirachitic A’ that cured rickets. In a series of experiments with mice in 1881, Nikolai Lunin (Russia) found that an unidentified natural component of milk prevented scurvy. Takaki Kanehiro (Japan) performed an experiment on Japanese naval crews showing that a diet of white rice lacked a nutrient that prevented beriberi. In 1897, Christiaan Eijkman (The Netherlands) showed that unpolished white rice led to beriberi in chickens, while polished rice prevented it. In 1907, Norwegian physicians Axel Holst and Theodor Frølich conducted a series of experiments with guinea pigs that set the stage for the discovery of ascorbic acid, or vitamin C. In 1910, Umetaro Suzuki (Japan) became the first scientist to isolate a vitamin complex, which he called aberic acid (later Orizanin and ultimately identified as vitamin B1, or thiamin) but the discovery received little attention. Frederick Hopkins (UK) conducted a series of experiments that led him to the conclusion in 1912 that some foods contained what he called ‘accessory factors’ that were necessary for functioning. Casimir Funk (Poland) independently repeated Suzuki’s results in 1912, calling the micronutrients a “vitamine” for vital amines, although the name was shortened to vitamin when it became clear that not all vitamins were amines. Elmer V. McCollum and M. Davis (US), discovered vitamin A in 1912–1914. McCollum also discovered vitamin B in 1915-1916. Sir Edward Mellanby (US) discovered vitamin D in 1920, while McCollum also isolated vitamin D in 1922. Also in 1922 Sir Herbert McLean Evans (US) discovered vitamin E. D.T. Smith and E.G. Hendrick (US) discovered vitamin B2 (riboflavin) in 1926. Henrik Dam (Denmark) and Edward Adelbert Doisy (US) discovered vitamin K in 1929. Paul Karrer (Switzerland) determined the structure for beta-carotene, the precursor of vitamin A, in 1930. Between 1928 and 1932, a Hungarian team led by Albert Szent-Györgyi and Joseph L. Svirbely, and an American team led by Charles Glen King, first identified and isolated vitamin C. The discovery was confirmed by Karrer and Norman Haworth (UK). Vitamin C was the first vitamin to be synthesized in the laboratory, by Haworth and Edmund Hirst in 1933-1934, and independently by Tadeus Reichstein (Poland) in 1933.
Early work with the symmetry of crystals was done by Nicholas Steno (Denmark) in 1669; René-Just Haüy (France) in 1784 and 1801; William Hallowes Miller (UK) in 1839; William Barlow (UK) in 1894 and others. Paul Peter Ewald (Germany/UK) and Max von Laue (Germany) raised the idea that crystals could be used as a diffraction grating for X-rays in 1912 and the same year, Von Laue performed the first X-ray diffraction using a copper sulfate crystal. William Henry Bragg (UK) and his son William Lawrence Bragg (Australia/UK) followed up on Von Laue’s experiments in 1912-1913 to determine the structures of molecules and minerals. Ralph Walter Graystone Wyckoff (US) used X-ray crystallography to determine the structures of sodium nitrate and caesium dichloroiodide in 1919. Dorothy Hodgkin (UK) used X-ray crystallography to determine the three-dimensional structures of cholesterol (1937), penicillin (1946), vitamin B12 (1956), and insulin (1969). In addition to X-ray crystallography, other methods of X-ray diffraction include powder diffraction, SAXS and X-ray fiber diffraction. Powder diffraction was invented by Peter Debye (The Netherlands/US) and Paul Scherrer (Switzerland) in 1916, and, independently, Albert Hull (US), in 1917. Rosalind Franklin (UK) used X-ray fiber diffraction in 1952 to take a photograph that helped determine the double helix structure of DNA.
Cosmic rays consist of very high-energy radiation, usually originating outside our solar system and consisting mainly of high-energy protons and atomic nuclei. When they hit the Earth’s atmosphere, the primary cosmic rays produce secondary radiation made mostly of electrons, photons and muons. Prior to the discovery of cosmic rays, most scientists believed that radiation in the Earth’s air and water came from radioactivity in the Earth itself, meaning that ionization levels should decrease with altitude. The first experimental indication of cosmic rays was German scientifist Theodor Wulf’s finding that radiation levels were higher at the top of the Eiffel Tower than at the bottom, although his finding was not generally accepted. Domenico Pacini (Italy) showed in 1911 that underwater ionization rates were less than above ground, which led him to hypothesize that some of the radiation must be coming from outside the Earth. In 1912, Victor Hess (Germany/US) rose in a hot-air balloon to 5300 meters and found that ionization rates increased by four times from ground level. To rule out the sun as the source of the radiation, Hess made a balloon flight during a solar eclipse. Over the next two years, Werner Kolhörster confirmed Hess’s results. In the 1920s, Robert Millikan coined the term ‘cosmic rays’ and suggested they were mainly photons. In 1927, J. Clay discovered that cosmic ray intensity varied with latitude, meaning they were particles that could be deflected by the Earth’s magnetic field, and not photons. In 1935, Albert W. Stevens and Orvil J. Anderson ascended in a balloon into the stratosphere to obtain the first visual evidence of cosmic rays: tracks on photographic plates. By the 1940s, scientists knew that protons made up the majority of particles in cosmic rays. From unmanned balloons sent near the top of the atmosphere in 1948, scientists learned that 10% of the particles were helium nuclei and 1% were nuclei of heavier elements. The cause of cosmic rays is still unknown. In 1934, Baade and Zwicky suggested they originated in supernovae, a theory which received significant support in 2009. In 2007, James Cronin (US) and Alan Watson (UK) announced partial results of a study showing that many cosmic rays originated in active galactic nuclei, which are thought to harbor supermassive black holes.
Elements of the assembly line include division of labor, interchangeable parts, and a moving, linear start-to-finish assembly process. A very early example of a division of labor comes from China in the 3rd Century BCE, when workers created the Terracotta Army for the tomb of Chinese Emperor Qin Shi Huangdi. Another early example was the Venetian Arsenal (Italy) in the early 16th Century, which employed 16,000 workers and used standardized parts to build ships. The notion of interchangeable parts was championed in the mid-18th Century by Honoré Blanc (France), who inspired Eli Whitney (US) to use some of Blanc’s ideas in making muskets in 1798. Oliver Evans (US) built an automatic flour mill in 1785 using conveyors, elevators and other devices. An early example of a linear and continuous assembly process was Porstmouth Block Mills (UK), built by Marc Isambard Brunel (France/UK) between 1801 and 1803. Another assembly line factory was the Bridgewater Foundry, built by James Nasymth and Holbrook Gaskell (UK) in 1836. Starting in 1867, Chicago meatpackers began to use assembly lines in which workers would stand at fixed stations and a pulley system would move the meat along the line. Ransom Olds (US) built a modern assembly line in 1901 to mass-produce the Oldsmobile Curved Dash automobile. The assembly line idea was brought to Henry Ford by his employee William “Pa” Klann, after visiting a Chicago slaughterhouse, and was implemented by a team of Ford employees. The Ford assembly line to build the Model T began operating on December 1, 1913. Soon, the assembly line method had spread throughout automobile manufacturing.
The atomic number of a chemical element (Z) is equal to the number of protons in its nucleus. Each element has a unique atomic number and each atomic number identifies a unique element. In an atom with no charge, the atomic number also indicates the number of electrons in the atom. When Dmitri Mendeleev (Russia) created his periodic table in 1869, before atomic number was understood, he used atomic weight to organize the elements, although he made some exceptions (e.g., putting tellurium ahead of iodine), which took account of elements in which the atomic number was not half the atomic weight. According to Ernest Rutherford’s 1911 theory of the atomic structure, he focused on the positively charged nucleus and the negatively charged electrons. He hypothesized (incorrectly) that the atomic weight equaled twice the number of electrons, if each electron weighed as much as a hydrogen atom (i.e., a proton). Following up on Rutherford, Antonius van den Broek (The Netherlands) suggested in 1911 that the number of electrons was exactly equal to the element’s place in the periodic table, thus anticipating the concept of atomic number. Niels Bohr (Denmark) used van den Broek’s theory in his 1913 model of the atom, where he predicted that the frequency of atomic spectra should be proportional to the square of Z, the atomic number. After discussions with Bohr, Henry Moseley (UK) obtained spectra for elements from aluminum (Z = 13) to gold (Z = 79) and found that the results confirmed Bohr’s prediction, thus providing conclusive evidence that atomic number, not atomic weight, was the defining characteristic of a chemical element.
Isotopes of a chemical element all have the same number of protons in their nuclei (and therefore the same atomic number) but different numbers of neutrons (and therefore varying atomic masses). Some isotopes of an element have varying chemical properties and some are radioactive. In 1902, Ernest Rutherford and Frederick Soddy (UK) set the stage for the discovery of isotopes with their study of the way in which radioactive decay changed one element into another, eventually reaching a stable element, as uranium eventually decayed into lead. Work by H.N. McCoy and W.H. Ross (US) in 1907 further explored the nature of decay products and devised a method for separating them. In 1913, Soddy predicted the existence of isotopes in 1913 based on the results of experiments on the radioactive decay of uranium to lead – even though there were only 11 elements in the decay chain, Soddy found 40 separate decay products, multiple ‘elements’ to occupy the same space in the periodic table. Margaret Todd (UK) suggested the Greek term ‘isotope’ meaning ‘in the same place’ to Soddy. Also in 1913, Polish-American chemist Kazimierz Fajans reached essentially the same conclusion as Soddy. J.J. Thomson (UK) found the first evidence of multiple isotopes for a stable, non-radioactive element (the inert gas neon) in 1913. In 1914, Theodore William Richards (US) found that different radioactive forms of the same element had different atomic weights. In 1919, Francis W. Aston (UK) used a mass spectrograph to identify multiple isotopes for a number of stable elements. He also formulated the whole number rule, which states that the atomic masses of isotopes are integers and a deviation from an integral atomic mass is usually the result of a mixture of isotopes. Harold Urey and G.M Murphy discovered deuterium, an isotope of hydrogen, in 1931.
With the general theory of relativity, Albert Einstein (Germany/US) amended Newton’s law of universal gravitation to explain that the gravitational ‘pull’ of an object is best understood not as a force but as a warp in the curvature of space-time caused by the object’s mass. In 1919, during an eclipse, Arthur Eddington and Frank W. Dyson (UK) measured the bending of starlight by the gravitational pull of the sun, thus confirming Einstein’s theory. The general theory of relativity makes many predictions, which have received experimental confirmation, such as the expanding universe, and the existence of black holes and gravitational waves.
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electrostatic force of attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction. In 1704, Sir Isaac Newton (England) proposed that “particles attract one another by some force, which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect.” In 1801, Jöns Jakob Berzelius (Sweden) developed a theory of chemical bonding that emphasized the electronegative and electropositive character of the combining atoms. By the mid 19th century, Edward Frankland (UK), F.A. Kekulé (Germany), A.S. Couper (UK), Alexander Butlerov (Russia), and Hermann Kolbe (Germany), developed the theory of valency (originally, ‘combining power’), which held that compounds joined due to an attraction of positive and negative poles. In 1916, American chemist Gilbert N. Lewis developed the modern concept of the electron-pair bond, in which two atoms may share one to six electrons, thus forming the single electron bond, a single bond, a double bond, or a triple bond. According to Lewis, “An electron may form a part of the shell of two different atoms and cannot be said to belong to either one exclusively.” Also in 1916, Walther Kossel (Germany) put forward a theory that assumed complete transfers of electrons between atoms, and was thus a model of ionic bonds. Both Lewis and Kossel structured their bonding models on that of Abegg’s rule (1904). In 1927, Danish physicist Oyvind Burrau was the first to describe a simple chemical bond in mathematically complete quantum terms. Walter Heitler (Germany) and Fritz London (Germany/US) invented a more practical approach in 1927, which is now called valence bond theory. In 1929, Sir John Lennard-Jones (UK) introduced the linear combination of atomic orbitals molecular orbital method (LCAO) approximation.
A black hole is a region of spacetime, the gravitational pull of which is so strong that nothing, not even electromagnetic radiation, can escape it. The boundary of the region from which there is no escape is called the event horizon. According to the general theory of relativity, a mass that is sufficiently compact will deform spacetime enough to form a black hole. Some scientists believe supermassive black holes lie at the center of many galaxies, including the Milky Way. John Michell (UK) in 1783 and Pierre-Simon Laplace (France) in 1796 both suggested that some objects might have such strong gravitational fields that light could not escape. In 1916, soon after Albert Einstein published his general theory of relativity, Karl Schwarzschild (Germany) was the first to show mathematically that Einstein’s theory predicted black holes under certain conditions. Johannes Droste followed up Schwarzschild’s findings in 1916-1917, finding that Schwarzschild’s solution to general relativity created a singularity (where some terms became infinite) at a point known as the Schwarzschild radius, which defines the event horizon. Arthur Eddington (UK) showed in 1924 that the singularity disappeared after a change of coordinates. Subrahmanyan Chandrasekhar (India) showed in 1931 that stars and other objects above a certain mass (1.4 solar masses) were inherently unstable and would eventually collapse. In 1939, Robert Oppenheimer (US) and others predicted that neutron stars larger than three suns would collapse into black holes. In 1958, David Finkelstein (US) was the first to describe a black hole as a region of space from which nothing could escape. Important theoretical discoveries about the nature of black holes were made by Roy Kerr (NZ) in 1963, Ezra Newman (US) in 1965, Werner Israel (Germany/South Africa/Canada), Brandon Carter (Australia) and David Robinson. The term ‘black hole’ was first used by journalist Ann Ewing in 1964; John Wheeler used the term in a 1967 lecture. Roger Penrose and Stephen Hawking (UK) showed in the late 1960s that singularities appear in generic solutions of general relativity. In the early 1970s, Hawking, Carter, James Bardeen (US) and Jacob Bekenstein (Mexico/Israel) formulated black hole thermodynamics and Hawking showed in 1974 that black holes should give off black body radiation. Black holes cannot be detected directly, but indirect evidence exists. The first indirect evidence of a black hole in an X-ray binary system, Cygnus X-1, was discovered by Charles Thomas Bolton (US) and Louise Webster and Paul Murdin (UK), working independently, in 1972. Numerous other candidates have since been found.
A tank is a tracked, armored fighting vehicle with offensive and defensive capabilities. Leonardo da Vinci sketched a proto-tank in the late 15th Century. French captain Levavasseur designed a tracked armored vehicle in 1903 but the French Army abandoned the project in 1908. H.G. Wells imagined tanks in his 1903 story “The Land Ironclads.” In 1911, Austrian engineering officer Günther Burstyn proposed a tank design, as did Australian civil engineer Lancelot de Mole. Both were rejected. Beginning in 1904, British and US companies began making tracked, crawler-type tractors. Vasily Mendeleev (Russia) designed a heavy tank between 1911 and 1915 but it was too expensive to build. When World War I broke out, the British Army began using Caterpillar tractors made by Benjamin Holt (US) to transport supplies and artillery in difficult terrain. In 1914-1915, the French designed the Boirault machine, but it was a failure. Other efforts – the Frot-Laffly, tested in March 1915 and the electric Aubriot-Gabet “Fortress” – also tanked. The French then designed two different tanks based on the Holt tractor and successfully tested them in 1915, ordered mass production in 1916 and began operating them in 1916 and 1917. In the UK, Ernest Swinton proposed the idea of using Holt’s Caterpillar tractors to make a tank in 1914, but the tank idea was shelved until Winston Churchill, on Swinton’s advice, revived it a year later. The UK made numerous attempts of various designs in 1915 without success until January 1916 when a rhomboidal design by Walter Gordon Wilson passed the test, leading to the delivery of the first Mark I tanks in August 1916. Aleksandr Porokhovschikov (Russia) built the Vezdekhod proto-tank in 1915 but it was not pursued, nor was Lebedenko’s Tsar Tank, which failed tests in 1915. The British Army was the first to use tanks in warfare when it deployed Mark I tanks at the Battle of the Somme on September 15, 1916. The French introduced their two types of tanks later in the war, and introduced the Renault FT in 1918. Germany brought in the A7V in 1918. The US Army produced the Mark VIII Liberty and the M1917 in 1918. In the 1920s, France developed the Char B1 bis. The Soviet Union deployed the KV-1 in 1939 and the T-34 in 1940. In the 1930s, Germany had the Panzer I and II; after 1940, it developed the Panzer III and IV; the Tiger I arrived in 1942 and the Panther in 1943. The US developed the M4 Sherman tank in 1942. The UK brought in the Centurion in 1945; the Soviets put the T-54/55 in service in 1946. The US introduced the M48 Patton in 1951. In 1979, West Germany developed the Leopard 2 and Israel made the Merkava. The US started the M1 Abrams in 1980. Former Soviet states built the T-90 in 1992 and the T-84 in 1999. Ukraine developed the T-84-12 Oplot. Italy made the C1 Ariete in 1995. Russia made the Black Eagle in 1997. The UK introduced the FV4034 Challenger 2 in 1998.
The proton is a subatomic particle with a positive electric charge. Every atomic nucleus includes one or more protons. The number of protons in an atom’s nucleus is its atomic number. In 1815, William Prout (UK) suggested that all atoms are composed of one or more hydrogen atoms. In the Standard Model, the proton is a hadron composed of three quarks. In 1886, Eugen Goldstein discovered positively charged particles that were produced from gases, although the different values of charge-to-mass ratio prevented Goldstein from reducing the theory to a single particle. In 1898, Wilhelm Wien (Germany), while studying streams of ionized gas, identified a positive particle equal in mass to the hydrogen atom. Ernest Rutherford (NZ/UK) conducted experiments in 1917 (and announced in 1919) that showed the hydrogen nucleus was present in other nuclei. Rutherford named the particle in the hydrogen nucleus the ‘proton’ (a combination of Prout’s name and Prout’s term ‘protyle’) in 1920.
In 1918, American scientist Florence Sabin’s studies of chicken embryos led her to discover the origin of the entire circulatory system in the development process: arteries, veins, all three types of blood cells, and the heart. Specifically, the blood cells arise from the cells making up the inner wall of the proto-arteries. These findings proved invaluable in the understanding and treatment of certain diseases.
Stellar nucleosynthesis is the process by which fusion reactions inside stars create the heavier elements found in the universe, which are then distributed throughout space when the star explodes. Scientists believe that the Big Bang alone only created hydrogen, helium and a few of the lighter elements, while the remainder were created in stars or in exploding stars. The idea was first proposed by Arthur Eddington (UK) in the 1920s. Fred Hoyle (UK) developed the theory in the late 1940s. In 1951, Ernst Öpik (Estonia/Northern Ireland) and, independently, the following year, Edwin E. Salpeter (Austria/Australia/US) explained how helium could become carbon inside the cores of red giant stars through the triple alpha process. In 1957, Hoyle’s team of physicists published a paper (known as the B2FH paper) that organized nucleosynthesis into complementary nuclear processes and explained the synthesis of heavy elements in detail.
In a series of experiments beginning in 1869 in Germany, scientists identified the islets of Langerhans in the pancreas and determined that these islets secreted a substance that controlled blood sugar levels. Absence of this secretion caused diabetes mellitus. Early attempts to treat diabetes with general pancreatic fluids had had mixed results. Frederick Banting (Canada), working with medical student Charles Best (Canada), finally isolated and extracted the substance, now known as insulin, in 1921. James Collip was instrumental in developing a purified extract. The first successful treatment of a human diabetic occurred in 1922. Later the same year, Eli Lilly and Co. developed a method for producing large quantities of insulin. In 1923, Banting and John Macleod were able to purify insulin for use in humans. Frederick Sanger (UK) identified the molecular structure of insulin in the 1950s. In the early 1960s, Panayotis Katsoyannis (US) and Helmut Zahn (Germany) independently invented the first synthetic insulin, but it was not specifically designed for humans. Scientists in China synthesized insulin in 1966. In 1977, a team of scientists (Arthur Riggs, Keiichi Itakura and Herbert Boyer) created the first genetically engineered synthetic ‘human’ insulin. It went on the market in 1982 as Humulin.
Neurotransmitters are chemicals found in the nervous systems of living organisms that transmit signals across a synapse from one neuron to another. Prior to their discovery, most scientists believed neurons communicated exclusively through electric impulses. In the late 19th and early 20th Century, Santiago Ramón y Cajal (Spain) discovered a gap between neurons known as the synaptic cleft, which suggested that some neuronal communication took place via chemicals. T.R. Elliott (US) suggested in 1904 that adrenaline acted as a neurotransmitter in nerves, helping the nerve signal across the synapse. Otto Loewi (Germany) conducted experiments on the vagus nerve of a frog in 1921 that provided direct evidence that neurons communicate by releasing chemicals. Loewi also identified the first known neurotransmitter, acetylcholine. Ulf von Euler (Sweden) discovered the neurotransmitter norepinephrine in 1946 and Arvid Carlsson (Sweden) disovered dopamine in the 1950s. Vittorio Erspamer discovered seratonin in the 1930s; Irvine Page rediscovered it and named it in 1948, and Betty Twarog and John Welsh, identified seratonin as a neurotransmitter in 1952 and 1954, respectively.
In 1914, Arthur Stanley Eddington (UK) hypothesized that spiral nebula were actually distant galaxies. In 1924, Edwin Hubble (US) conclusively proved that the Milky Way is just one of many millions of galaxies in the universe. Precursors to Hubble included Thomas Wright (UK), who speculated in 1750 that the Milky Way was a flattened disk of stars (a galaxy) and that some nebulae might be separate galaxies. Lord Rosse (Ireland/UK) in 1845 detected individual stars in some nebulae. Vesto Slipher (US) studied nebulae and detected red shifts in 1912. Heber Curtis (US) found evidence to support independent galaxies in 1917. In 1922, Ernst Öpik (Estonia) proved Andromeda Galaxy is separate from the Milky Way.
In 1924, Louis de Broglie (France) used Einstein’s special theory of relativity as the basis for a theory that particles can exhibit the characteristics of waves, and vice versa. Based on de Broglie’s theory of matter waves, German physicists Werner Heisenberg, Max Born and Pascual Jordan created matrix mechanics and Austrian physicist Erwin Schrödinger invented wave mechanics and the Schrödinger equation in 1925, which set out the principles of quantum mechanics. Further developments were Heisenberg’s uncertainty principle, of 1927, and the Dirac equation, proposed by Paul Dirac in 1928, which describes the electron’s wave function, predicts electron spin and the existence of the positron. John von Neumann (Hungary) formulated the mathematical basis for quantum mechanics in 1932.
Prior to Charles Darwin’s The Origin of Species, skulls of Neanderthals, a close relative of modern man, had been discovered in Belgium (1829), Gibraltar (1848) and the Neander Valley in Germany (1856). Eugène Dubois (The Netherlands) discovered a fossil skeleton of “Java Man”, now called Homo erectus, in Java in 1891. It was Australian scientist Raymond Dart’s 1924 discovery (and 1925 publication) of a new species of hominid, Australopithecus africanus, in South Africa, that convinced many in the scientific community that humans had evolved. Subsequent discoveries have included British scientist Louis Leakey’s 1964 discovery of Homo habilis; American Donald Johanson’s discovery of of an almost complete skeleton of Australopithecus afarensis, known as “Lucy” in Ethiopia in 1974; British scientist Mary Leakey’s 1978 discovery of 3.5 million year old fossilized human footprints; and the discovery of a 1.6 million year old Homo erectus skeleton in 1984 by Richard Leakey (UK) and Alan Walker (UK). In 1994, Meave Leakey (UK) discovered Australopithecus anamensis, which lived in Kenya and Ethiopia about 4 million years ago. Tim White (US) discovered the 4.2 million year old Ardipithecus ramidus in 1995. Martin Pickford (UK) and Brigitte Senut found a bipedal hominid they named Orrorin tugenensis from 6 million years ago, In 2001, Michel Brunet (France) claimed to have found a skull of a bipedal hominid that was 7.2 million years old, Sahelanthropus tchadensis. Since the 1960s, much of the study of human evolution has been conducted through analysis of the DNA of living humans and apes.
According to the Pauli exclusion principle, no two electrons in the atom can be in the same quantum state; in other words, two electrons must have opposite spin, thus cancelling each other, and there can be no more than two in the same orbital. Leading up to Austrian physicist Wolfgang Pauli’s articulation of the principle in 1925, there were a number of precursors. In 1916, Gilbert N. Lewis stated that the atom tends to hold an even number of electrons in the shell and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube. In 1919, Irving Langmuir (US) suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable “closed shells.” Pauli tried to explain these empirical findings as well as the results of experiments on the Zeeman effect in atomic spectroscopy and in ferromagnetism. A 1924 paper by Edward Stoner (UK) pointed out that for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin.
Modern rocketry was born in 1926 when Robert H. Goddard (US) launched the first liquid fuel rocket in Auburn, Massachusetts. His invention led to the V-2 and the ICBM missiles as well as the rockets that sent satellites into orbit, men to the moon, and probes into deep space. The first rockets, fueled by gunpowder, were made by the Chinese in the 13th Century for war and fireworks. They spread to the Mongols, who then brought them to Europe and the Muslim world, including the Ottoman Empire, in the 13th, 14th and 15th centuries. The Kingdom of Mysore in southern India in the 1780s and 1790s developed an artillery rocket that used iron cylinders to contain the combustible element, which significantly improved range. William Congreve (UK) adapted the Mysore rocket to create the Congreve rocket. In 1844, William Hale (UK) altered the design of the Congreve rocket to improve its accuracy significantly.
Television had many inventors. Abbe Giovanna Caselli (Italy) transmitted the first still image over an electronic wire in 1862. In 1877, George Carey (US) designed a machine that would use selenium to allow people to see electrically-transmitted images; by 1880, he had built a primitive system with light-sensitive cells. In 1884, Paul Nipkow (Germany) sent images over wires with 18 lines of resolution using a rotating metal disk (mechanical model). In 1906, Lee De Forest (US) invented the Audion vacuum tube, which could amplify electronic signals. In the same year, Boris Rosing (Russia) combined a cathode ray tube with Nipkow’s disk to make a working television. In 1907, Rosing begin developing an electronic scanning method of reproducing images using a cathode ray tube (electronic model). In 1908, Alan Campbell-Swinton (UK) described how a cathode ray tube could be used as a transmitting and receiving device in a television system. In 1909, Georges Rignoux and A. Fournier (France) demonstrated instantaneous transmission of images in a mechanical system. In 1911, Rosing and Vladimir Zworykin (Russia) developed a mechanical/electronic system that transmitted crude images. In 1923, Zworkin (now in the US) patented a TV camera tube, the iconoscope, and later the kinescope, or receiver, although a 1925 demonstration was unimpressive. On March 25, 1925, John Logie Baird (Scotland) demonstrated transmission of silhouette images. In May, 1925, Bell Labs transmitted still images. On June 13, 1925, Charles Francis Jenkins (US) transmitted the silhouette image of a moving toy over a distance of five miles. Also in 1925, Zworkin patented a color TV system. On December 25, 1925, Kenjiro Takayanagi (Japan) demonstrated a mechanical/electronic system with 40 lines of resolution. In the USSR, Leon Theremin developed a series of increasingly higher resolution television systems, from 16 lines in 1925 to 100 lines in 1927. On January 26, 1926, Baird demonstrated a system with 30 lines of resolution, running at five frames per second, showing a recognizable human face. Also in 1926, Kálmán Tihanyi (Hungary) solved the problem of low sensitivity to light in television cameras through charge-storage. In 1927, Philo Farnsworth patented the Image Dissector, the first complete electronic television system. On April 7, 1927, Herbert Ives and Frank Gray of Bell Labs (US) demonstrated a mechanical television system that produced much higher-quality images than any prior system. Charles Jenkins received the first television station license in 1928. In 1929, Zworkin demonstrated both transmission and reception of images in an electronic system. After a series of improvements to his design, Farnsworth transmitted live human images in 1929. In 1931, Jenkins invented the Radiovisor and began selling it as a do-it-yourself kit. Manfred von Ardenne (Germany) demonstrated a new type of system in 1931. Farnsworth gave a public demonstration of an all-electronic TV system, with a live camera, on August 25, 1934. The BBC began the first public television service on November 2, 1936 with 405 lines of resolution. In 1937, the BBC used new equipment that was far superior to prior systems. In 1940, Peter Goldmark invented a mechanical color TV system with 343 lines of resolution. In 1941, the US adopted a 525-line standard. In 1943, Zworkin developed an improved camera tube that allowed recording of night events. In 1948, USSR began broadcasting at 625-lines of resolution, which was eventually adopted throughout Europe. Cable television was introduced in 1948 to bring television to rural areas. Videotape broadcasting was introduced in 1956 by Ampex. In 1962, the launching of the Telstar satellite permitted international broadcasting. Color televisions began to outnumber black & white TVs in the 1970s. Satellite television began in 1983. High definition TV appeared in 1998. Analog broadcast TV ended on June 12, 2009, leaving digital television.
The uncertainty principle holds that there is a mathematically-determined fundamental limit to knowing precisely and simultaneously certain pairs of physical properties of a particle, known as complementary variables, such as position and momentum. Werner Heisenberg (Germany) first articulated the uncertainty principle in 1927 by stating that the more precisely a particle’s position is determined, the less precisely its momentum can be known, and vice versa. The uncertainty is sometimes confused with the observer effect, which states that measurements of certain systems cannot be made without affecting the system.
People in ancient Egypt, China and Mesoamerica used molds to treat infected wounds. The germ theory of disease, propagated by Louis Pasteur (France) in the mid-19th Century, sparked the search for antibiotic agents. In 1871, Joseph Lister discovered that bacteria would not grow in mold-infected urine. John Tyndall (Ireland/UK) noted fungal inhibition of bacteria in 1875. In 1887, Louis Pasteur and Jules-François Joubert (France) demonstrated the antibiotic effect. In 1895, Italian physician Vincenzo Tiberio noted that the Penicillium mold killed bacteria. In the 1890s, Rudolf Emmerich and Oscar Löw (Germany) created an antibiotic but it often failed. In 1904 Paul Ehrlich (Germany) sought the ‘magic bullet’ against syphilis and systematically tested hundreds of substances before finding Salvorsan in 1909. In 1928, Alexander Fleming (Scotland) discovered that a mold, Penicillium notatum, destroyed bacterial colonies. After years of research following up on Fleming’s discovery, Howard Florey (Australia/UK), Norman Heatley (UK), Ernst Chain (Germany/UK) and Andrew J. Moyer (US) developed a method of manufacturing penicillin as a drug in 1942. Dorothy Hodgkin (UK) discovered the structure of the penicillin molecule in 1943. Penicillin proved to be effective against many serious diseases caused by bacterial infections. In 1932, German scientists at Bayer (Josef Klarer, Fritz Mietzsch and Gerhard Domagk) synthesized and tested the first sulfa drug, Prontosil. In 1939, Rene Dubos (France/US) created the first commercially manufactured antibiotic – tyrothricin – although it proved too toxic for systemic usage. In 1943, Selman Waksman (US), derived stretomycin from soil bacteria. In 1955, Lloyd Conover (US) patented tetracycline. In 1957, Nystatin was patented. SmithKline Beecham patented the semisynthetic antibiotic amoxicillin in 1981; it was first sold in 1998. A major concern throughout the history of antibiotics is the development of antibiotic resistant strains of bacteria, which is such a significant problem that researchers are looking for alternatives to antibiotic treatments.
In 1912, American astronomer Vesto Slipher obtained spectrograms of the Andromeda Nebulae, M31, which all showed clear evidence of a Doppler redshift. By 1914, he had measured 15 more Doppler shifts of galaxies (then called nebulae), all but three toward red. Georges LeMaitre (Belgium) first proposed that the universe was expanding in 1927. Edwin Hubble (US) obtained the first direct evidence that the universe is expanding in 1929 by measuring the redshifts of galaxies. Hubble also devised the Hubble constant – a measure of the rate at which the universe is expanding.
Quantum electrodynamics is a theory that describes how light and matter interact, using both quantum mechanics and the special theory of relativity. In the 1920s, Paul Dirac (UK) set out the first formulation of a quantum theory describing the interaction of radiation and matter when he computed the coefficient of an atom’s spontaneous emission. Enrico Fermi (Italy) and others then contributed to the theory in the 1930s. In 1937 and 1939, Felix Bloch (Germany/US) , Arnold Nordsieck and Victor Weisskopf (Austria/US) discovered that it might not be possible to perform Dirac’s computation on all processes involving photons and charged particles. The invention of the renormalization procedure by Hans Bethe (Germany/US) in 1947 was a key breakthrough. After papers by Sin-Itiro Tomonaga (Japan), Richard Feynman (US), Julian Schwinger (US), and Freeman Dyson (US) in 1946-1950, it became possible to get fully covariant formulations that were finite at any order in a perturbation series.
In 1931, Georges Lemaître (Belgium) proposed that the universe began with the explosion of a ‘primeval atom.’ In 1947-1949, George A. Gamow (USSR/US) developed Lemaître’s idea into a comprehensive scientific theory. In 1965, Arno A. Penzias and Robert W. Wilson (US) detected cosmic microwave background radiation, which provided the first evidence to support the Big Bang theory. Since that time, the Big Bang has become the central premise of cosmology.
A cyclotron is a particle accelerator in which charged particles accelerate outwards along a spiral path. A rapidly varying electric field accelerates the particles and a static magnetic field holds the particles to a spiral trajectory. The idea of the cyclotron came to both Leó Szilárd (Hungary) and Ernest O. Lawrence (US). Szilárd applied for a patent in 1929. Lawrence had precedence; his cyclotron, which he built with student Stanley Livingston, began operating at the University of California at Berkeley in 1932. The first European cyclotron was proposed in 1932 by George Gamow and Lev Mysovskii (USSR) and began operating at the Radium Institute in Leningrad in 1937. In Nazi Germany, Walther Bothe and Wolfgange Gentner created a cyclotron in Heidelberg, where it began running in 1943. Some of the largest cyclotrons are those at the RIKEN laboratory in Japan and TRIUMF at the University of British Columbia in Vancouver, Canada.
Dueterium (also known as heavy hydrogen) is a rare isotope of hydrogen that contains one proton and one neutron in the nucleus. The more common isotope, with a single proton in the nucleus, makes up 99.98% of naturally occurring hydrogen. Harold Urey (US), assisted by George Murphy and Ferdinand Brickwedde, discovered and isolated deuterium for the first time in 1931 (although it was only the discovery of the neutron in 1932 that explained the chemical nature of deuterium). Urey prepared samples of water containing deuterium, referred to as ‘heavy water.’ Gilbert N. Lewis (US) produced the first pure heavy water in 1933. Heavy water was later used in nuclear reactors and in nuclear weapons.
Numerous different astronomical phenomena produce radio waves, including stars, galaxies, quasers, pulsars and even the Big Bang, through the cosmic microwave background radiation. American physicist and radio engineer Karl Jansky gave birth to radio astronomy in 1931 when he discovered a strong radio signal emanating from the center of the Milky Way galaxy. Amateur American astronomer Grote Reber followed up on Jansky’s discovery by conducting the first sky survey of radio signals between 1938 and 1943. In 1942, British Army officer James S. Hey was the first to detect radio waves coming from the sun. In 1946, British astronomers Martin Ryle and D. Vonberg build the first astronomical radio interferometer using aperture synthesis. Also in 1946, Hey, S. J. Parsons, and J. W. Phillips announced their discovery of a discrete radio source from the direction of the constellation Cygnus A. John R. Shakeshaft (UK) published the sky survey known as the Second Cambridge Catalogue of Radio Sources (2C) in 1955. A more accurate survey of radio sources, known as 3C, followed in 1959; it was revised by A.S. Bennett (UK) in 1962 (3CR) and by R.A. Laing, J.M. Riley and M.S. Longair (UK) in 1983 (3CRR).
Sulfonamide (also known as sulphonamide) is the basis for several groups of drugs, some of which are antibacterial. The first antibacterial sulfa drug was Prontosil, which has the chemical name sulfonamidochrysoidine. Although Paul Gelmo (Austria) had synthesized the chemical in 1909, he did not pursue his findings. Josef Klarer and Fritz Mietzsch (Germany) synthesized it at Bayer, and in 1932, Gerhard Domagk (Germany) discovered it was effective in treating bacterial infections in mice. Results of clinical human studies were published in 1935, but it was treatment of Franklin Delano Roosevelt’s bacterial infection in 1936 that led to widespread acceptance of the drug. In 1935, scientists at the Pasteur Institute (France) discovered that Prontosil is metabolized to sulfanilamide, a much simpler molecule, which, as Prontalbin, soon replaced Prontosil. The chemical nature of sulfanilamide made it easy for chests to link it to other molecules, which led to hundreds of sulfa drugs.
Ernest Rutherford proposed the existence of the neutron in 1920 to explain the disparity between the atomic number of an atom’s nucleus (i.e., the number of positively-charged protons) and the atomic mass. Some scientists believed the answer was that there were electrons in the nucleus whose negative charge canceled out some of the proton charge. But Viktor Ambartsumian and Dmitri Ivanenko (USSR) proved in 1930 that electrons could not exist in the nucleus and there must be neutral particles present. Walther Bothe and Herbert Becker (Germany) discovered unusual radiation in 1931, a result that was pursued in 1932 by Irène Joliot-Curie and Frédéric Joliot (France). Following up on the strange radiation found by the German and French scientists, James Chadwick (UK) in 1932 definitively identified the neutron, an uncharged particle approximately the same mass as the proton. The discovery of the neutron was a key in the development of nuclear reactors and atomic weapons.
The positron (also known as the antielectron) is the antimatter counterpart of the electron that is part of the Standard Model. Paul Dirac (UK) suggested in 1928 that electrons can have both positive and negative charge. In a follow-up paper in 1929, Dirac suggested that the proton might be the negative energy electron. Robert Oppenheimer strongly disagreed with Dirac’s suggestion, which led Dirac in 1931 to predict the existence of an anti-electron with the same mass as an electron, which would be annihilated upon contact with an electron. Ernst Stueckelberg (Germany) and Richard Feynmann (US) developed the theory of the positron, while Yoichiro Nambu (Japan/US) applied the theory to all matter-antimatter pairs of particles. The first to observe the positron was Dmitri Skobelsyn (USSR) in 1929. The same year, Chung-Yao Chao (China/US) conducted similar experiments but results were inconclusive. Carl D. Anderson (US) is acknowledged to have discovered the positron on August 2, 1932; he also coined the word.
The strong interaction is the mechanism by which the strong nuclear force works. The strong nuclear force only operates at a distance of a femtometer (10‾15 meters) but it is the strongest force, 137 times stronger than magnetism. The strong nuclear force, which is carried by gluons, holds protons and neutrons together in the nucleus and binds quarks into hadrons. Most of the mass-energy of protons and neutrons consists of the strong force field energy. The model of the atom prior to the 1970s contained a number of contradictions; according to the existing physics, the positive charges of the protons should cause the nucleus to fly apart, which did not occur. Scientists then hypothesized a new force, the strong force, that held the protons and neutrons together. When the Standard Model was developed, it became clear that the strong interaction causes quarks with unlike color charge to attract one another. In about 1932, Eugene Wigner (Hungary/US) and Werner Heisenberg (Germany) independently theorized that protons and neutrons were held together by a force separate from the electromagnetic force. Hideki Yukawa (Japan) attempted an early hypothesis in 1934 (1934-1935). Major discoveries were made by Murray Gell-Mann (US) and Yuval Ne’eman (Israel) in 1962 and Gell-Mann and George Zweig (US) in 1964.
The weak interaction is the mechanism responsible for the weak nuclear force, one of the four basic forces of nature, along with electromagnetism, gravity and the strong nuclear force. The weak interaction is responsible for radioactive decay and nuclear fusion of subatomic particles; caused by emission or absorbtion of W and Z bosons. Fermions also interact through weak interaction. Ernest Rutherford (NZ/UK) proposed the weak nuclear force in 1899 to explain beta decay of radioactive elements. Enrico Fermi (Italy) first suggested the existence of the weak interaction in 1933 in explaining beta decay. Fermi thought it was a force with no range, dependent on contact. In 1956, Clyde Cowan and Frederick Reines (US) showed that electrons and antineutrinos were released in beta decay. The same year, Tsung-Dao Lee and Chen Ning Yang (China/US) predicted that the weak force did not follow parity, the symmetry of the other forces. In 1968, Sheldon Glashow (US), Abdus Salam (Pakistan) and Steven Weinberg (US) proved that the weak interaction and electromagnetism were two aspects of the same force, now known as the electroweak force. W and Z bosons were first experimentally detected by Carlo Rubbia (Italy) and Simon van der Meer (The Netherlands) in 1983. It is now believed that the weak force is a non-contact force with a finite range.
Dark matter is a substance that scientists have proposed to explain certain gravitational effects in the universe. Although there is significant indirect evidence for the existence of dark matter, it has not been directly observed or detected. According to the dark matter hypothesis, it cannot be seen with telescopes and does not appear to emit or absorb electromagnetic radiation (including light) at any significant level. Some have suggested that it may be composed of an undiscovered subatomic particle. According to the most recent estimate, the known universe contains 4.9% ordinary matter; 26.8% dark matter and 68.3% dark energy. Jan Oort (The Netherlands) first proposed the existence of unseen matter in 1932 to explain the orbital velocities of stars in the Milky Way. Fritz Zwicky (Switzerland/US) suggested the existence of what he called ‘dark matter’ in 1933 to explain what appeared to be missing mass in measuring the orbital velocities of galaxies in clusters. In 1973, Jeremiah Ostriker and James Peebles (US) calculated mathematically that galaxies would collapse if they only contained the mass we can see. They proposed that an additional mass of three to 10 times the size of the visible mass was necessary to explain the observed shapes of the galaxies. At about the same time, Kent Ford and Vera Rubin, using new photon detectors, found that the movement of hydrogen clouds in the Andromeda galaxy could not be explained if the majority of the galaxy’s mass was contained in the visible matter, but only if it was contained in invisible matter that existed outside the visible edge of the galaxy. In 2013, a team of scientists said they had discovered a weakly-interacting massive particle (WIMP) that could make up dark matter. In 2014, NASA’s Fermi Gamma-ray Space Telescope recorded high-energy gamma-ray light emanating from the center of the Milky Way that confirmed a prediction about dark matter.
An ecosystem is a community of living organism in conjunction with the non-living components of their environment, interacting as a system. The biotic and abiotic components of an ecosystem are linked through nutrient cycles and energy flows. In 1924, Alfred Lotka, in Elements of Physical Biology, compared the global eco-system to “a great world engine” in which “plants and animals act as coupled transformers of energy” in “the mill-wheel” that is driven by “solar energy.” Arthur Tansley (UK) coined the term ‘ecosytem’ in a 1935 publication in which he emphasized the transfers of materials between organisms and their environment. Early developers of the ecosystem concept were G. Evelyn Hutchinson (UK), Raymond Lindeman (US) and Howard T. and Eugene P. Odum (US), who developed a systems approach to studying ecosystems.
A virus is a very small infectious agent that replicates only inside the living cells of other organisms. In the mid-late 19th Century, when Louis Pasteur (France) could not find a bacterial cause for rabies, he hypothesized that the disease might be caused by a pathogen too small to be seen by a microscope. In 1884, Charles Chamberland created a filter with holes smaller than bacteria, which would become essential for studying viruses. Dmitri Iosefovich Ivanovsky (Russia) used the Chamberland filter to determine that the cause of the tobacco mosaic disease was a pathogen smaller than a bacterium, which he announced in an 1882 article. Martinus Beijerinck (The Netherlands) arrived at similar results in 1898, and he coined the term ‘virus’ to describe the unseen pathogen. Also in 1898, Friedrich Loeffler and Paul Frosch (Germany) determined that foot and mouth disease in animals was caused by a virus. In 1914, Frederick Twort (UK) discovered the first bacteriophage, a type of virus that infects bacteria; he published his results in 1915 but they were ignored. French-Canadian microbiologist Félix d’Herelle discovered bacteriophages indendently, and announced his discovery in 1917. Ernst Ruska and Max Knoll (Germany) made the first electron micrographs of viruses in 1931. In 1935, Wendell Stanley (US) succeeded in crystallizing the tobacco mosaic virus in 1935, proving it was a particle, not a fluid, and that is was made largely of protein. Electron micrographs of the tobacco mosaic virus were made in 1939 and X-ray crystallography was performed on it by Bernal and Fankuchen in 1941. Rosalind Franklin (UK) discovered the full structure of the tobacco mosaic virus in 1955. Over 2,000 species of virus have been identified as of 2014.
In 1886, Heinrich Hertz (Germany) showed that radio waves could be bounced off solid objects. In 1897, while testing an early radio communication device (the spark-gap transmitter) between two ships at sea, Alexander Popov (Russia) noted that the passage of a third ship caused interference in the signal. Christian Hülsmeyer (Germany) was the first to use radio waves to detect the presence of distant objects in a 1904 experiment in dense fog, but the device could not measure the distance to the object. In 1922, U.S. Navy scientists Albert Taylor and Leo Young noticed that that ships reflected radio signals. In 1930, Taylor and Young, with Lawrence Hyland (US), detected a plane using the same method, but without information about distance or speed. The team, with new member Robert Page (US), then developed a pulse-interference device and successfully used in to identify the range and speed of an airplane in December 1934. Earlier in 1934, a team of German scientists led by Rudolf Kühnhold used Doppler-beat interference to detect ships and airplanes, including their range. On January 3, 1934, Russian scientists M.M. Lobanov and Y.K. Korovin detected an airplane at 600 meters range and 100-150 meters altitude using a Doppler signal; later the same year, the Bistro device was introduced. Maurice Ponte, in France, had developed a short wavelength device that did not measure distance. Meanwhile, by 1935, the Germans had developed a much more accurate pulse-modulated system. In 1935, Robert Watson-Watt, Arnold “Skip” Wilkins and Edward Bowen (UK) demonstrated a device that could detect radio waves reflected off a flying airplane 17 miles away and determine its range. By 1936, the U.S. Navy had a prototype radar system that could detect aircraft at 25 miles distance. Also by 1936, German companies Lorenz and Telefunken had developed accurate radar systems. The U.S. Army developed its own radar system by 1937. By 1938, the U.S. Navy system could detect aircraft at 100 miles and the first radar was placed on an American ship. In 1939, the USSR developed a radar system capable of determining range and velocity. In 1940, the word ‘radar’ was coined from the phrase ‘Radio Detection and Ranging.’ In 1940 John Randall and Harry Boot (UK) invented the cavity magnetron, which made short wavelength radar a reality. Robert Page invented monopulse radar in 1943. Luis Alvarez (US) invented phased-array radar during World War II. Goodyear Aircraft Corp. (US) invented synthetic-aperture radar in the early 1950s.
In 1865, an artificial fiber made from cellulose was used to make acetate. Sir Joseph Swan (UK) invented the first artificial fiber in about 1883 by chemically modifying fibers from tree bark to create a cellulose liquid, from which the fibers were drawn. Swan displayed fabrics made from his material at an 1885 exhibition. Hilaire de Chardonnet (France) produced an artificial silk in the late 1870s from nitrocellulose. He displayed products made from the artificial fiber at an 1889 exhibition, but the material was extremely flammable and was not successful. Arthur D. Little (US) reinvented acetate from cellulose in 1893. In 1894, Charles Frederick Cross, with Edward John Bevan and Clayton Beadle (UK), produced an artificial fiber from cellulose that they called viscose. Courtaulds Fibers (UK) produced viscose commercially in 1905 and in 1924 renamed it rayon. Camille and Henry Dreyfus (Switzerland) used acetate to make motion picture film and other products beginning in 1910. The Celanese Company used acetate to make textiles beginning in 1924. The first completely synthetic fiber, not based on naturally-occurring cellulose, was nylon, which was invented by Wallace Carothers (US) at DuPont in 1935. In 1938, Paul Schlack of I.G. Farben in Germany invented another form of nylon. DuPont began commercial production of nylon for use in women’s stockings as well as parachutes and ropes, among other things, in 1939. Polyester was invented by John Rex Whinfield and James Tennant Dickson (UK) at the Calico Printers’ Association in 1941, who patented their first polyester fiber as Dacron. Also in 1941, DuPont introduced acrylic, a new synthetic fiber that resembled wool, under the brand name Orlon.
In the late 19th Century, pens were invented in which ink was placed in a thin tube with a tiny ball at the end. The ink clung to the ball and the ball spun as the pen wrote. In 1888, John J. Loud (US) invented and patented a ballpoint pen for writing on leather, but it didn’t work well on paper and the patent expired. Over 300 ballpoint pen patents were filed between 1890 and 1930, but each one had problems and none was produced commercially. In 1935, László and György Bíró (Hungary) developed a much more effective ink formula and ball-socket mechanism, which they patented in 1938. They moved to Argentina in 1943 and further improved the design, which they manufactured. Because the new pens worked at high altitudes, the British government obtained a license from the Biros for RAF aircrews. In 1945, Eversharp Co. and Eberhard Faber Co. obtained the license from Biro to sell in the US. At about the same time, Milton Reynolds (US) bought a Biro pen in Argentina and modified it enough to obtain a US patent. He introduced his version, the Reynolds Rocket, at Gimbels in New York City on October 29, 1945, where he sold 10,000 pens the first day. In the UK, the Miles-Martin Pen Company markets ballpoint pens in December 1945. In 1949, Patrick J. Frawley and Fran Seech (US) began working on improvements to the ballpoint pen and by 1950 had developed a pen with a retractable tip and no-smear ink called the Papermate. They continued to introduce improved ink formulas in the 1950s. Marcel Bich (France) introduced a Biro-design pen to the US in 1950 and an improved version with a clear-barrel in 1952. The brand name was changed to BIC in 1953. In 1954, Parker Pens released The Jotter, which was more reliable and convenient than earlier models. In 1957, Parker introduced a pen using a tungsten carbide textured ball bearing. By 1960, BIC dominated the ballpoint pen market.
Charles Babbage (UK) is the grandfather of computer science. Beginning in the 1810s, he developed a theory of computing machines, which he put into practice in progressively more complex designs. Babbage’s 1837 proposal for an Analytical Engine would possibly have been the first true computer, had it actually been built. It had expandable memory, an arithmetic unit, logical processing abilities, and the ability to interpret a complex programming language. Ada Lovelace (UK), who worked with Babbage, furthered his work by designing the first computer algorithm and by predicting a computer that would not only perform mathematical calculations but manipulate symbols of all kinds. Kurt Gödel (Austria/US) established the mathematical foundations of computer science in 1931 with his incompleteness theorem, which showed that every formal system contained limits to what could be proved or disproved within it. If Babbage was the grandfather, Alan Turing (UK) was the father of computer science. In 1936, Turing (along with American Alonzo Church), formalized an algorithm containing the limits of what can be computed, as well as a purely mechanical model for computing. The Church-Turing thesis of the same year states that, given sufficient time and storage space, a computer algorithm can perform any possible calculation. Turing introduced the ideas of the Turing machine and the Universal Turing machine (which can simulate any other Turing machine) in 1937. Turing machines are not real objects but mathematical constructs designed to determine what can be computed by any proposed computer.
Ancient Greek scientist Archytas is reputed to have invented an artificial, self-propelled flying device that flew 200 meters propelled by a jet of steam between 400 and 350 BCE. In 150 BCE, Hero of Alexandria described a device called an aeolipile that used steam to cause a sphere to spin rapidly on its axis. Chinese engineers invented rockets in the 13th Century and Lagari Hasan Çelebi, of the Ottoman Empire, reputedly launched himself into the air on a homemade rocket in 1633. John Barber (UK) patented a turbine design in 1791. Charles Parsons (UK) invented the steam turbine in 1884. In 1903, Aegidius Elling (Norway) built the first gas turbine with a centrifugal compressor. Between 1903 and 1906, Armengaud and Lemale (France) built an inefficient gas turbine engine. Hans Holzwarth (Germany) began work on an explosive cycle gas turbine in 1908 and reached 13% efficiency by 1927. In 1908, René Lorin (France) patented a ramjet engine, which was modified by Georges Marconnet (France) in 1909 to create the pulsejet. In 1910, Henri Coandă (Romania) built and briefly flew the Coandă-1910, the first motorjet. Sanford Alexander Moss (US) began work on turbochargers at General Electric in 1917. In 1921, Maxime Guillaume (France) designed the first axial-flow turbine engine. In a seminal 1926 paper, Alan Arnold Griffith (UK) explained how jet engines are possible. A single-shaft turbocompressor based on Griffith’s theory was tested in the UK in 1927. Frank Whittle (UK) presented a jet engine design to the UK Air Force in 1928 but it was rejected. He submitted a patent for the design in 1930. Also in 1930, Paul Schmidt (Germany) patented a pulsejet engine. In 1931, Secondo Campini (Italy) patented a motorjet engine. In 1934, Hans Von Ohain (Germany) patented a jet propulsion engine. In April 1937, Whittle bench tested an engine with a single-stage centrifugal compressor coupled to a single-stage turbine, a prototype of the turbojet engine. In September 1937, Von Ohain and Ernst Heinkel (Germany) bench tested a jet engine. Also in 1937, György Jendrassik (Hungary) designed and built the first working turboprop engine, although it was never installed in a plane. Heinkel built an airplane to test Von Ohain’s engine – the Heinkel He178 – which flew for the first time on August 27, 1939. Von Ohain then improved his design and flew it in the He S.8A aircraft on April 2, 1941. On May 15, 1941, the Pioneer aircraft flew with a Whittle engine (the W1) for the first time. A centrifugal jet engine designed by Frank Halford (Scotland) called the de Havilland Goblin flew in 1942. Anselm Franz (Austria) improved on the centrifugal jets by creating the axial-flow compressor, with the first test in 1940. His engine was used in the Messerschmitt Me 262 in 1942. The first axial-flow engine in the UK, the Metrovick F.2, was tested in 1941 and flown in 1943. By the 1950s, almost all combat aircraft used jet engines. In 1952, the first commercial jet airliner, the de Havilland Comet, entered the market, and by the 1960s, almost all large civilian aircraft were jet-powered. In the 1970s, the high bypass jet engine increased fuel efficiency beyond that of piston and propeller engines.
The Krebs Cycle (also known as the citric acid cycle) is the series of chemical reactions used by aerobic organisms to obtain energy by oxidizing acetate from carbohydrates, fats and proteins into carbon dioxide and adenosine triphosphate (a source of energy). Several components of the cycle were discovered by Albert Szent-Györgyi (Hungary) in the early 1930s, but the cycle itself was identified by Hans Adolf Krebs (Germany/UK) in 1937. Because the Krebs Cycle is so central to biochemisty, some scientists believe that it was one of the first elements of living cells and may even have existed before life originated.
Considered a ‘living fossil’ at the time of its discovery in 1938, the coelacanth is a member of an ancient group of fish long thought to be extinct since the end of the Cretaceous period, 65 million years ago. The first fossil coelacanth (genus Latimeria) was identified and described by Louis Agassiz in 1839. The coelacanth is considered a transitional species between fish and the first land animals and is related to lungfish and other lobe-finned fishes. In December 1938, a South African fisherman named Hendrick Goosen found an unusual fish in his catch while fishing in the Indian Ocean near the mouth of the Chalumna River. Museum creator Marjorie Courtenay-Latimer identified the fish as unique, and Rhodes University ichthyologist J.L.B. Smith recognized that this was a great discovery. The first coelacanth found is now the West Indian Ocean coelacanth (Latimeria chalumnae). A second species was later discovered, the Indonesian coelacanth (Latimeria menadoensis). Coelacanths are deep-sea dwellers that can grow up to 6.5 feet long and nearly 200 pounds and live up to 60 years. Both species are endangered.
In induced nuclear fission, the nucleus of an atom is split by bombarding it with a subatomic particle, often a neutron. The fission process usually releases free neutrons and protons (in the form of gamma rays) and a very large amount of energy. In 1917, Ernest Rutherford (NZ/UK) used alpha particles to convert nitrogen into oxygen, the first nuclear reaction. Ernest Walton and John Cockcroft used artificially accelerated protons to split the nucleus of a lithium-7 atom into two alpha particles. In 1934, Enrico Fermi (Italy) and his team, bombarded uranium with neutrons, but concluded the experiments created new elements with atomic numbers higher than uranium. (established that slow neutrons worked better than fast ones in bringing about a fission reaction. (while improving on the Curie-Joliet artificial radiation technique). In 1934, Ida Noddack suggested that Fermi’s experiments had actually broken the nucleus into several large fragments. After reading of Fermi’s results, Otto Hahn, Fritz Strassman (Germany) and Lise Meitner (Austria) began performing similar experiments until Meitner, a Jew, was forced to flee to Sweden. In December 1938, Hahn and Strassman proved that bombarding uranium nuclei with neutrons had created barium, an element with 40% less atomic mass than uranium. In 1939, Meitner and Otto Robert Frisch, explained Hahn and Strassmann’s results as evidence that they had split the uranium nucleus, and coined the term ‘fission’ to describe the reaction. Firsch confirmed this theory experimentally in January 1939. Also in January 1939, a team at Columbia University, including Erico Fermi, replicated the nuclear fission experiment.
Global warming refers to a rise in the average temperature of the Earth’s climate system in recent years. Because 90% of the recent increase in temperature has been absorbed by the oceans, global warming is often used to refer to the average temperture of the air and sea at the Earth’s surface. Climate change, in this context, refers to changes in the climate, including temperature, caused by human activities. The major human activities influencing climate change are fossil fuel combustion, which sends gaseous emissions into the atmosphere, aerosols, carbon dioxide released by cement manufacture, land use, ozone depletion, animal agriculture and deforestation. While humans have long speculated how their activity affects the climate (e.g., 19th Century Americans debated whether cutting down trees might affect rainfall), the modern science of climate change began in 1896, when Svante Arrhenius (Sweden) predicted the ‘greenhouse effect’ – as humans burned fossil fuels, they would add carbon dioxide to the atmosphere, wich would raise the temperature. Arrhenius was not concerned about his conclusions, however, because he believed the warming would take thousands of years and would benefit humanity. In 1938, Guy Steward Callendar (Canada/UK) demonstrated the global land temperature had increased over the past 50 years. Callendar published 35 scientific articles between 1938 and 1964 developing the theory that the increase in global temperature was caused by rising carbon dioxide levels. Canadian physicist Gilbert Plass and others expanded on Callendar’s work in the 1950s and 1960s. Hans Suess (Austria/US) in 1955 and Roger Revelle (US) in 1957 discovered that the oceans only had a limited ability to absorb carbon dioxide from fossil fuel combustion. In 1961, Charles David Keeling (US) published detailed, comprehensive measurements showing that the amount of carbon dioxide in the atmosphere was rising. In the following years, collection of data continued, showing both carbon dioxide increases and temperature rise and mathematical models of the climate were developed. In 1967, Syukoro Manabe (Japan) and Richard Wetherald used a computer to make the first detailed calculation of the greenhouse effect incorporating convection and in 1975, the developed a roughly accurate three-dimensional model of the global climate that showed the doubling the carbon dioxide in the atmosphere would lead to a 2° C rise in global temperature.
Nuclear fusion occurs when two or more atomic nuclei collide at high speed and join to form the nucleus of a different element, usually releasing energy in the process. Arthur Eddington (UK) suggested in 1920 that stars obtain their energy by fusing hydrogen into helium. In 1929, Robert Atkinson (UK) and Fritz Houtermans (The Netherlands/Austria /Germany) predicted that fusing small nuclei would release large amounts of energy. Mark Oliphant (Australia) obtained fusion of hydrogen isotopes in 1932. In 1938, Hans Bethe (Germany/US) explained how nuclear fusion provided the source of energy in stars. In 1951, the US carried out a test of nuclear fusion, and in 1952 it tested a hydrogen bomb, which was based on fusion.
A helicopter is an aircraft that moves by means of rotors, which allows it to take off and land vertically, hover and fly backward, forward and laterally. Bamboo flying toys, which use a spinning rotor to rise vertically, were invented in China about 400 BCE. In the 1480s, Leonardo da Vinci (Italy) designed an ‘aerial screw’ for vertical flight. Mikhail Lomonosov (Russia) adapted the Chinese toy by adding a spring mechanism and proposed his machine as a way to raise meteorological instruments into the air. Christian de Launoy and Bienvenu (France) also built a prototype machine based on the Chinese toy in 1783, which they continued to improve. French inventor Gustave de Ponton d’Amécourt was the first to coin the word ‘helicopter’ but his steam-powered prototype did not lift off the ground. In 1870, Alphonse Pénaud (France) created a coaxial rotor toy powered by a rubber band that inspired the young Wright brothers. Enrico Forlanini (Italy) created a steam-powered unmanned helicopter that rose 40 feet vertically and hovered for 20 seconds. Emmanuel Dieuaide also made a steam version connected by a hose to a ground-based boiler. Thomas Edison (US) was the first to design a helicopter powered by an internal combustion engine in 1885 but it and later designs never flew. Ján Bahýľ (Slovakia) built a helicopter with an internal combustion engine that rose 1.6 feet in 1901; a revised version rose 13 feet and flew 4,900 feet in 1905. The first manned, but tethered helicopter flight took place in 1907, of the Gyroplane No. 1, made by Jacques and Louis Breguet (France), which rose two feet for one minute. Paul Cornu (France) made an untethered manned flight of his helicopter in 1907, rising one foot and staying aloft 20 seconds; the machine eventually achieved an altitude of 6.5 feet. Danish inventor Jacob Ellehammer’s helicopter made several free take-offs between 1912 and 1916. Raúl Pateras-Pescara de Castelluccio (Argentina/UK) built a series of three helicopters beginning in 1924 that used cyclic pitch and autorotation; the third version could fly for up to 10 minutes. Étienne Oehmichen (France) set a record in 1924 when his quadrotor helicopter flew 1,181 feet. Dutch engineer Albert Gillis von Baumhauer invented cyclic and collective controls in 1925. American Arthur M. Young invented the stabilizer bar in 1928. In 1928, Oszkár Asbóth (Hungary) built and flew a helicopter that remained aloft for 53 minutes. Corradino d’ Ascanio (Italy) made an influential coaxial helicopter, the D’AT3, in 1930 which held speed and altitude records. Boris N. Yuriev and Alexei M. Cheremukhin (USSR) constructed and flew the TsAGI 1-EA single rotor helicopter, which reached an altitude of 1,985 feet in 1932. Nicolas Florine (USSR/Belgium) built a twin tandem rotor helicopter that rose to 20 feet and stayed aloft for eight minutes in 1933. Built in 1933, the Bréguet-Dorand Gyroplane Laboratoire set distance records of 1,600 feet in 1935, a height record of 520 feet and duration record of 62 minutes in 1936. Also in 1936, Heinrich Focke (Germany) made and flew the first practical transverse twin-rotor helicopter, the Focke-Wulf Fw 61. During World War II, the German military used the Foche Achgelis Fa 223 Drache twin-rotor helicopter and the FI 282 Kolibri synchopter, mady by Anton Flettner. In America, Russian-born Igor Sikorsky built the first practical single lifting-rotor helicopter, the VS-300, in 1939. W. Lawrence LePage (US) built the XR-1, which was based on Foche’s Fw 61, in 1941. In 1942, Sikorsky upgraded the VS-300 into the R-4, which became the first large-scale mass-produced helicopter. Arthur Young joined Bell Aircraft in 1941, where he built Bell Models 30 (1942) and 47 (1945).
Polyethylene is the most common type of plastic in use today and is used to make plastic bags, plastic bottles and many other items. German chemist Hans von Pechmann was the first to synthesize polyethylene, albeit accidentally, while heating diazomethane in 1898. His colleagues Eugen Bamberger and Friedrich Tschirner (Germany) analyzed the resulting substance and named it polymethylene. In 1933, Eric Fawcett and Reginald Gibson (UK) of ICI, accidentally synthesized polyethylene using a very high pressure method that was industrially practical (unlike von Pechmann’s). ICI chemist Michael Perrin was able to reproduce the synthesis in 1935 and industrial production of the plastic began in 1939. Its first use was for insulation of radar cables during World War II. Large scale production began at Bakelite and DuPont in 1944. In 1951, Robert Banks and J. Paul Hogan (US) at Philips Petroleum discovered a catalyst that allowed synthesis of polyethylene at milder temperatures and pressures. In 1953, Karl Ziegler (Germany) and Giulio Natta (Italy) discovered a catalyst that worked in milder conditions but was more expensive. Another catalytic system, using soluble catalysts, was invented by Walter Kaminsky (Germany) and Hansjörg Sinn in 1976. Polyethylene now comes in two basic types: (1) hard, or HDPE or (2) soft, or LDPE.
Neptunium is the chemical element with an atomic number of 93, one more than uranium, making it the first transuranic element. It is a radioactive actinide metal that is pyrophoric (meaning it may burst into flames at room temperature). It is silver in color but quickly oxidizes in air, creating a tarnish. There are three allotropic forms of the element and several isotopes, of which neptunium-237 is the most stable. Neptunium has a half life of about 80 million years, which means that any neptunium that existed at the time of the Earth’s formation should have already decayed into stable elements. Trace amounts of neptunium are found in rocks containing uranium, which can be transformed into neptunium through neutron capture reactions and beta decay. Neptunium is also produced as a by-product of neutron irradiation of uranium in nuclear reactors. Edward M. McMillan and Philip H. Abelson (US) first synthesized neptunium at Berkeley Radiation Laboratory in 1940.
In the 1930s, while experimenting with the genes for eye color in fruit flies, George Beadle and Boris Ephrussi concluded that each gene was responsible for an enzyme acting in the metabolic pathway of pigment synthesis. In 1941, George Wells Beadle and Edward Lawrie Tatum (US), using the bread mold Neurospora crassa, published the assertion that genes control cells by controlling the specificity of enzymes, i.e., one gene controls one enzyme so a mutation in a gene will change the enzymes available, causing the blockage of a metabolic step. With modifications, the one gene-one enzyme hypothesis remains essentially valid.
A nuclear reactor initiates and controls a sustained nuclear chain reaction. Heat from nuclear fission occurring in a nuclear reactor is used to generate electricity and propel ships. In 1933, Hungarian-American scientist Leó Szilárd recognized that neutron-caused nuclear reactions ould lead to a nuclear chain reaction. In 1934, Szilárd filed the first patent application for the idea of a nuclear chain reaction using neutrons bombarding light elements. After the discovery of nuclear fission of uranium in 1938, Szilárd and Enrico Fermi (Italy) confirmed experimentally in 1939 Otto Hahn and Fritz Strassmann’s prediction that nuclear fission released several neutrons, which were then available to bombard other nuclei. Also in 1939, Francis Perrin (France) and Rudolph Peierls (Germany/US) independently worked out the ‘critical mass’ of uranium needed to sustain the reaction. In 1939, Leo Szilard proposed that a nuclear chain reaction would work best by stacking alternate layers of graphite and uranium in a lattice, the geometry of which would define neutron scattering and subsequent fission events. In 1942, Ernico Fermi and his team at the University of Chicago (including Szilárd) created the first controlled, self-sustaining nuclear chain reaction (the first nuclear reactor) from ‘piles,’ using Szilard’s lattice of uranium and graphite. (The term ‘reactor’ has since replaced ‘piles’.) A number of nuclear reactors were built by the U.S. military beginning in 1943 as part of the Manhattan Project to build a nuclear weapon. The first nuclear reactor for civilian use was launched in June 1954 in the USSR.
In a groundbreaking 1928 experiment, Frederick Griffith found a ‘transforming’ principle that could change one type of bacteria to another. Over the next 15 years, scientists at the Rockefeller Institute for Medical Research in New York, sought to isolate the transformative substance. In 1944, Oswald Avery, Colin MacLeod and Maclyn McCarty (US) published their surprising results: the substance that contained the genetic information was DNA (deoxyribonucleic acid). In 1952, Alfred Hershey and Martha Chase followed up on their experiment, confirming the results.
Between 1833 and 1837, Charles Babbage (UK) used a punch card system to design an analytical engine that, if ever completed, would have been the first programmable computer. (In 1843, Per Georg and Edward Schulz, of Sweden built a working model of Babbage’s difference engine, from 1822.) Late in the 1880s, Herman Hollerith (US) used punch cards on a machine that could store and read the data contained on them by using a tabulator and a key punch machine. The machine was used to tabulate the 1890 U.S. Census. Hollerith’s company eventually became IBM. In the first half of the 20th Century, a number of analog computers were developed, usually for specific purposes. They included the Dumaresq (John Dumaresq, UK, 1902); Arthur Pollen’s fire-control system (UK, 1912); the differential analyzer (H.L. Hazen and Vannevar Bush/MIT, US, 1927); the FERMIAC (Enrico Fermi, Italy/US, 1947); MONIAC (US, 1949); Project Cyclone (Reeves, US, 1950); Project Typhoon (RCA, US, 1952); and the AKAT-1 (Jacek Karpiński, Poland, 1959). In 1909, Percy Ludgate, of Ireland, apparently unaware of Babbage’s work, independently designed a programmable mechanical computer. In 1936, Alan Turing (UK) published a paper that described the Turing Machine – the theoretical basis for all modern computers. John von Neumann (Hungary/US) invented a computer architecture based on Turing’s theory. In a 1937 MIT master’s thesis, Claude Shannon (US) showed how electronic relays and switches can realize the expressions of Boolean algebra. In 1937, George Stibitz (US), of Bell Labs, invented and built the first relay-based calculator to use binary form – the Model K. Starting in 1936, Konrad Zuse (Germany) built a series of progressively more complex programmable binary computers with memory: the Z1 (1938) never worked reliably, but the Z3 (May 1941) is considered by some the first working programmable fully automatic modern computer that meets the criteria for Alan Turing’s “universal machine.” In 1939, John V. Atanasoff and Clifford E. Berry (US) at Iowa State created the Atanasoff-Berry Computer, which was electronic and digital but not programmable. In 1940, George Stibitz and his team produced and demonstrated their Complex Number Calculator. In 1943, Max Newman, Tommy Flowers and others (UK) built the Mk I Colossus, a computer designed to break the German encryption system, building on 1941 work by Britons Turing and Gordon Welchman (who in turn built on 1938 work by Marian Rejewski, of Poland). Some consider Colossus to be the world’s first electronic programmable computing device. The improved Mk II Colossus followed in 1944. Also in 1944, the Harvard Mark I began operation, after being built at IBM’s Endicott labs by a team headed by Howard Aiken, starting in 1939. Beginning in 1943, the U.S. Government sponsored the development of ENIAC under the lead of John Mauchly and J. Presper Eckert (US) at the University of Pennsylvania. When it began operating at the end of 1945, ENIAC met all of Alan Turing’s criteria for a true computer. Also in 1945, Konrad Zuse developed the Z4, which also met Turing’s criteria. Improvements to ENIAC in 1948 made it possible to execute stored programs set in function table memory. Frederic C. Williams, Tom Kilburn and Geoff Tootill (UK) at Victoria University of Manchester, built the Manchester Small-Scale Experimental Machine, or “Baby” in 1948, the first stored-program computer. Baby led to the Manchester Mark 1, which became operational in 1949. The Mark 1, in turn, led to the first commercial computer, the Ferranti Mark 1, in 1951. Maurice Wilkes (UK) at Cambridge developed the EDSAC in 1949. Not to be outdone, Australians Trevor Pearcey and Maston Beard built CSIRAC in 1949. Another commercial computer was the LEO I, made by J. Lyons & Co. (UK) in 1951. Also in 1951, the U.S. Census Bureau purchased a UNIVAC I (essentially a variation of ENIAC using a new metal magnetic tape) from Remington Rand. After years of delays, EDVAC, Eckert and Mauchly’s follow-up to ENIAC, began operations in 1951 at the Ballistics Research Lab. IBM began marketing the 701, its first mainframe computer, in 1952. In 1954, IBM released the IBM 650, a smaller, more affordable computer. Maurice Wilkes (UK) invented microprogramming in 1955. In 1956, IBM introduced the first hard disk drive – it could store five megabytes of data. Beginning about 1953, transistors began replacing vacuum tubes in computers. The invention of the integrated circuit, or microchip, led to the invention of the microprocessor in the late 1960s.
The road to the atom bomb began in 1934, when Hungarian scientist Leo Szilárd proposed the idea of bombarding radioactive atoms with neutrons to form a nuclear chain reaction, an idea he patented and then transferred to the British Admiralty so it would be kept secret. In 1938, Otto Hahn and Fritz Strassmann split the uranium atom, a fact explained and confirmed by Lise Meitner and Otto Robert Frisch in January 1939. Meitner and Frisch named the process fission. Scientists at Columbia University repeated the experiment in January 1939. In August 1939, fearing that the Germany would produce a fission-based weapon, Albert Einstein and other scientists wrote a letter of warning to US President Franklin Roosevelt , who responded by setting up a committee to study the matter, which only received significant funding after the US entered World War II in December 1941. In 1940 and 1941, the British conducted research into uranium and potential weapons. The US research did not begin in earnest until September 1942, with the start of the Manhattan Project, led by General Leslie Groves, which took over the British research Physicist Robert Oppenheimer (US) led the Manhattan Project’s science team. In addition to the Los Alamos laboratory, an Oak Ridge, Tennessee facility produced the rare uranium-235 isotope needed for a chain reaction. The Project also used plutonium-239, a byproduct of uranium-238, as a basis for a fission weapon. The Manhattan Project ultimately produced two types of fission bombs: a uranium-235 gun-type weapon (“Little Boy”) and a plutonium-239 implosion-type bomb (“Fat Man”). The first atomic weapon – a plutonium implosion bomb – was detonated at Los Alamos, New Mexico on July 16, 1945, releasing the equivalent of 19 kilotons of TNT. On August 6, 1945, the US dropped a uranium gun-type bomb on Hiroshima, Japan. On August 9, 1945, the US dropped a plutonium implosion-type bomb on Nagasaki, Japan. The two bombings resulted in the deaths of approximately 200,000 people, mostly civilians. The USSR tested its first fission bomb on August 29, 1949. The US began developing the much more powerful thermonuclear or hydrogen bomb, which uses fission to create a fusion reaction, in 1950 and first tested a bomb in 1952, releasing energy equal to 10.4 megatons of TNT. The USSR followed with its first thermonuclear bomb test on August 12, 1953.
A communications satellite works by receiving information from Earth and relaying that information back to a different location on Earth. The United States military attempted to use the Earth’s moon as a natural communication satellite in the 1950s, under the Moon Relay program. The first American satellite was Project Score, launched in 1958, which contained a tape recorder to store and forward voice messages. Echo 1 was a metallic balloon 100 feet across that was launched on August 12, 1960 and rose 1,000 miles into the atmosphere. Echo 1 was a passive satellite and acted as a large mirror that reflected signals back to Earth. Courier 1B, the first active repeater satellite, was also launched in 1960. The first active, direct relay communications satellite was AT&T’s Telstar, which was launched in July 1962. Relay 1, launched in December 1962, was the first satellite to broadcast across the Pacific. Syncom 2, launched in July 1963, revolved around the Earth once a day at constant speed, although it also moved north and south. The idea of a geostationary satellite that revolves around the Earth at the Earth’s velocity and thus remains fixed over a point on the globe, was proposed by science fiction writer Arthur C. Clarke in a 1945 article drawing on the work of Konstantin Tsiolkovsky (USSR) and Herman Potočnik (Slovenia). A ground antenna can aim at a geostationary satellite without having to track it, which saves money. The first geostationary satellite was Syncom 3, launched in August 1964. Intelsat I followed in 1965. Canada launched its first geostationary satellite, Anik A1 (Telesat Canada), in November 1972. The US sent up Westar 1 (Western Union) in April 1974. Many hundreds of communications satellites are now orbiting the Earth.
A transistor is a device made of semiconductor material that amplifies and switches electronic signals and electrical power. The precursor to the transistor was the vacuum-tube triode, or thermionic valve, first created in 1907 by Lee De Forest (US). Julius Edgar Lilienfeld (Austria/Hungary) patented a field-effect transistor in 1925, but his work was ignored at the time. (Years later, William Shockley and Gerald Pearson (US) at Bell Labs made a functional device using Lilienfeld’s design.) German physicist Oskar Heil patented a field-effect transistor in 1934. In the mid-1940s, John Bardeen and Walter Brattain (US) built a semiconducting triode for use in military radar equipment. After the end of World War II, Schockley, Bardeen and Brattain worked on using semiconductors to replace vacuum tubes in electrical systems. In December 1947, they created a germanium point-contact transistor – the first solid-state electronic transistor. In June 1948, Shockley designed a grown-junction transistor; a prototype was built in 1949. German physicists Herbert F. Mataré and Heinrich Welker invented a transistor they called the transistron in August 1948. In 1950, Shockley developed a bipolar junction transistor. Morgan Sparks (US) at Bell Labs made the new transistor into a useful device. General Electric and RCA produced an alloy-junction transistor – a type of bipolar junction transistor – in 1951. By 1953, transistors were being used in products such as hearing aids and telephone exchanges. Dick Grimsdale (UK) built the first transistor computer in 1953. Also in 1953, Philco (US) invented the first surface-barrier transistor. In the early 1950s, Bell Labs also produced the first tetrode and pentode transistors. Around the same time, the spacistor was created, but it was soonobsolete. In 1954, two teams working independently invented the first silicon transistor: Morris Tanenbaum (US) at Bell Labs and Gordon Teal (US) at Texas Instruments. Also in 1954, Bell Labs produced the first diffusion transistor, while in 1955, Bell made the first diffused silicon mesa transistor, which was developed commercially by Fairchild Semiconductor (US) in 1958. Also in 1955, Tanenbaum and Calvin Fuller invented a much improved silicon transistor. The first gallium-arsenide Schottky-gate field-effect transistor was invented by Carver Mead (US) in 1966.
Radiocarbon dating uses a radioactive isotope carbon, known as carbon-14, to determine the age of an organic object. It relies on the fact that radioactive carbon in an organism begins to decay at a predictable rate, starting at the time of death. Radiocarbon dating is normally accurate for objects that are 50,000 years old or younger. In a series of experiments beginning in 1939, Willard F. Libby investigated isotopes of elements in organic material, including carbon-14. A 1939 paper by W.E. Danforth and S.A. Korff on carbon-14 sparked Libby’s idea that radiocarbon dating might be possible. In 1946, Libby proposed that living matter might include carbon-14 and went on to discover carbon-14 in organic methane. The suggestion of using carbon-14 as a way to date organic materials came in a 1947 paper by Libby and others. Libby and James Arnold announced in 1949 that they had used carbon-14 to date wood samples from the tombs of two Ancient Egyptian kings to 2800 BCE, plus or minus 250 years, which was consistent with independent dates of 2625 BCE plus or minus 75 years. In the years following, scientists improved and refined the accuracy of radiocarbon dating.
Holography emerged from X-ray microscopy and photographic technology. In Max von Laue (Germany) showed X-ray diffraction through a crystal lattice of copper sulfate in 1912. In 1913, William Henry Bragg and William Lawrence Bragg (UK) developed a law of diffraction, which allowed scientists to make diffraction gratings that would control the angle of deflected light and separate different wavelengths of light. In 1947, Dennis Gabor (Hungary/UK) was working on improving the design of the electron microscope at Thomson-Houston (BTH) Company in Rugby, England when he used a reference beam to encode one wave while superimposing it with another, thereby recording the interference pattern, a process known as double diffraction. Although the resulting images were two-dimensional and could only be seen when looking along the axis of the light beam, they marked the beginning of holography. Gabor was awarded the Nobel Prize in Physics in 1971 “for his invention and development of the holographic method”. The technique as originally invented is still used in electron microscopy, where it is known as electron holography. Optical holography did not advance until the development of the laser in 1960. In 1962, Emmett Leigh and Juris Upatnieks (US) at the University of Michigan duplicated Gabor’s work using a laser and an off-axis reference beam method adapted from their work with side-reading radar. The result was the first laser transmission hologram of 3-D objects, made in 1964. The only way to see the image was by viewing it with laser light. Also in 1962, Yuri Denisyuk (USSR) produced a white light reflection hologram that could be viewed under incandescent lights. By 1965, several groups of scientists had produced off-axis reflection holograms. Early holograms used silver halide photographic emulsions as the recording medium. They were not very efficient as the grating absorbed much of the incident light. Various methods of converting the variation in transmission to a variation in refractive index (known as “bleaching”) were developed which enabled the production of much more efficient holograms. In 1967, T.A. Shankoff and Keith Pennington (US) began using a dichromated gelatin as a medium for recording holograms, which made it possible to record a hologram on any clear, non-porous surface. Larry Siebert of the Conductron Corporation made the first hologram of a human being in 1967. In 1968, Stephen Benton at Polaroid developed white-light transmission holography, and he invented the first rainbow hologram in 1969. In 1977, Benton used an achromatic geometry to recombine the spectrum and make black and white holograms. Holograms have been used in scientific research, fashion and art. Since the 1980s, holograms have been used to protect against counterfeiting.
Information Theory is a branch of applied mathematics, electrical engineering, and computer science that involves the quantification of information. Information theory was developed by Claude E. Shannon (US) in 1948 to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception, information theory has expanded and has been applied in numerous contexts.
Also known as transposons or jumping genes, transposable elements (TEs) are sequences of DNA that can change position within the genome. In a series of experiments with maize beginning in 1944, Barbara McClintock at Cold Spring Harbor Laboratory discovered in 1948 that certain parts of the chromosomes had switched positions, which disproved the common belief that genes had fixed positions. McClintock also showed that TEs sometimes reversed earlier mutations and may be responsible for turning genes on and off. McClintock reported her findings in a series of reports and articles between 1950 and 1953, but her work was largely ignored until after TEs were independently discovered in bacteria in 1967. In 1996, Philip SanMiguel estimated that TEs make up a large proportion of the genome of eukaryotic organisms, 50% of human genes and up to 90% of maize genes.
Also known as the birth control pill, or just “The Pill”, the combined oral contraceptive pill includes a combination of estrogen (estradiol) and progestogen (progestin). Margaret Sanger (US) began advocating for access to birth control in 1914. In 1916, she opened the first US birth control clinic, which led to her arrest. In 1921, Sanger formed the American Birth Control League, which later became Planned Parenthood. In the following decades, Sanger fought to overturn laws banning birth control methods and sought to educate the public about birth control. By the 1930s, scientists knew that high doses of certain steroid hormones inhibited ovulation in rabbits. European scientists had synthesized hormones but they were too expensive to import elsewhere. In 1939, Russell Marker (US) at Penn. State University, learned to synthesize progesterone from sarsaparilla, and then Mexican yams. In 1944, he and two partners started Syntex in Mexico and began producing synthetic steroid hormones. In 1951, Gregory Pincus (US) , a hormone specialist, attended a dinner with Sanger and Planned Parenthood medical director Abraham Stone. They urged Pincus to research the birth control possibilities of hormones. In 1951, Pincus’s colleague Min Chueh Chang (China/US) repeated a 1937 experiment showing that progesterone suppressed ovulation in rabbits. Also in 1951, Carl Djerassi (Austria/US), Luis Miramontes (Mexico) and George Rosenkranz (Hungary/Mexico) at Syntex synthesized the first orally highly active progestin, norethindrone. Further research was stalled by lack of funding support for this controversial area. Then, Sanger found a donor – Katherine Dexter McCormick – who contributed large sums beginning in 1953. Pincus brought in Harvard gynecologist John Rock (US) to do clinical research with women. Rock had been using progesterone and estrogen with infertility patients since 1952. Meanwhile, Frank B. Colton (US) at Searle had synthesized two orally highly active progestins in 1952 and 1953. John Rock started a clinical trial of three different oral progestins in 1954 and concluded that Colton’s norethynodrel worked best. After more experimentation, Pincus and Rock decided they should add a small amount of estrogen, to prevent bleeding. They named the resulting pill Enovid. Trials were conducted by Edris Rice-Wray and Edward T. Tyler (US) in 1956. The results indicated that they could reduce the estrogen content by a third. The FDA approved Enovid in 1957, but only for menstrual disorders. The FDA approved a 10 mg dose of Enovid for contraception in 1960 and a 5 mg dose in 1961. Legal barriers still existed, however, until Supreme Court decisions in 1965 (married women) and 1972 (unmarried women). In 1969-1970, concerns were raised that The Pill was unsafe and increased the risk of certain health conditions. In the 1980s, the modern low dose, two- and three-phase birth control pills became available. The first birth control pill outside the US, Anovlar, was released by Schering (Germany) in 1961, first in Australia and then in West Germany. The UK tested both Enovid and Anovlar in 1960 and approved them for contraception in 1961-1962. The Pill became legal in France in 1967. The Pill was not approved in Japan until 1999. Japanese doctors lobbied against The Pill, fearing reduced condom use and higher rates of sexually transmitted diseases. In 2003, the first continuous birth control pill, which suppress periods and provides birth control, was approved.
Poliomyelitis (often called polio) is an acute infectious disease carried by virus and spread from person to person. Polio, which can cause paralysis, was first identified by Jakob Heine in 1840. Karl Landsteiner identified the pathogen, poliovirus, in 1908. Failed early attempts to create a vaccine for polio wee made by Maurice Brodie and John Kolmer, working independently, in 1936. John Enders’ and Thomas Weller’s successful cultivation of human poliovirus in the laboratory in 1948 was a significant step for vaccine research. Jonas Salk developed a vaccine using inactivated (i.e., dead) poliovirus in 1952, which was approved and released in 1955. Albert Sabin used live but attenuated poliovirus to create a second, oral vaccine in 1957, which was licensed for use in 1962. The two vaccines have eliminated polio from most of the world.
Friedrich Miescher (Switzerland) first isolated DNA (deoxyribonucleic acid) in 1869. In 1928, experiments by Frederick Griffith (UK) showed that traits could be transferred from one type of organism to another. In 1943, Oswald Avery (Canada/US), Colin MacLeod (Canada/US) and Maclyn McCarty (US) identified DNA as the genetic material. In 1952, Rosalind Franklin (UK) and Raymond Gosling (UK) created an x-ray diffraction image of DNA that was then used by James Watson (US) and Francis Crick (UK) to determine the double helix structure of DNA in 1953. Experiments in 1953 by Maurice Wilkins (NZ/UK) confirmed the structure. The double helix structure, with paired bases forming the rungs between the two strands, perfectly explains how DNA replicates during mitosis.
While no one has yet definitively determined how life began, a number of theories have been proposed and at least one famous experiment conducted. In 1922 and 1924, Alexander Oparin (USSR) suggested that life could have arisen from basic organic chemicals in the Earth’s primordial ocean given a strongly reducing atmosphere (methane, ammonia, hydrogen and water vapor) and the forces of natural selection. J.B.S. Haldane (UK) made similar proposals in 1926 and 1929, in which he suggested that an ‘oily film’ would have enclosed self-reproducing molecules, creating the first cells. Both Oparin and Haldane suggested that complex organic molecules might begin to self-reproduce while still inanimate. The first experiment to test the Oparin-Haldane theory was conducted by Stanley Miller and Harold Urey (US) in 1953. They simulated an early Earth atmosphere and ocean by placing liquid water, methane, ammonia and hydrogen in a sealed container with a pair of electrodes. They heated the water to induce evaporation, and fired sparks between the electrodes to simulate lightning, then cooled the environment to allow the products in the atmosphere to condense. The result was the production of many organic compounds, including all the amino acids needed to make proteins, and sugars. No nucleic acids were created. Many others have followed up the experiment. In 1961, Joan Oró was able to create a nucleotide base from hydrogen cyanide and ammonia in water. As scientists learn more about the early Earth’s atmosphere and other conditions, revised experiments have been conducted.
German physicist Werner Jacobi at Siemens AG designed the first integrated transistor amplifier in 1949. In 1952, Geoffrey Dummer (UK) suggested that a variety of standard electronic components could be integrated in a monolithic semiconductor crystal. In 1956, Dummer built a prototype integrated circuit. In 1952, American Bernard Oliver invented a method of manufacturing three electrically connected planar transistors on one semiconductor crystal. Also in 1952, Jewell James Ebers (US) at Bell Labs created a four-layer transistor, or thyristor. William Shockley (US) simplified Ebers’s design to a two-terminal, four-layer diode, but it proved unreliable. Harwick Johnson (US) at RCA patented a prototype integrated circuit in 1953. In 1957, Jean Hoerni (Switzerland/US) at Fairchild Semiconductor proposed a planar technology of bipolar transistors. Three breakthroughs occurred in 1958: (1) Jack Kilby (US) at Texas Instruments patented the principle of integration and created the first prototype integrated circuits: (2) Kurt Lehovec (Czech Republic/US) of Sprague Electric Co. invented a method of isolating components on a semiconductor crystal electrically; and (3) Robert Noyce (US) of Fairchild Semiconductor invented aluminum metallization – a method of connecting integrated circuit components. Noyce also adapted Hoerni’s planar technology as the basis for an improved version of insulation. Hoerni made the first prototype of a planar transistor in 1959. Jay Last and others at Fairchild built the first operational semiconductor integrated circuit on September 27, 1960. Texas Instruments announced its first integrated circuit in April 1960, but it was not marketed until 1961. Texas Instruments sued Fairchild in 1962 based on Kilby’s patent and the parties settled in 1966 with a cross-licensing agreement. The first integrated circuits with transistor-transistor logic instead of resistor-transistor logic were invented by Tom Long (US) at Sylvania in 1962. In 1964, both Texas Instruments and Fairchild replaced the resistor-transistor logic of their integrated circuits with diode-transistor logic, which was not vulnerable to electromagnetic interference. In 1968, Italian physicist Federico Faggin developed the first silicon gate integrated circuit with self-aligned gates. The same year, Robert H. Dennard (US) invented dynamic random-access memory, a specialized type of integrated circuit. Also in the late 1960s, medium scale integration (MSI), in which each chip contained hundreds of transistors, was introduced. The specialized integrated circuit known as a microprocessor was introduced by Intel in 1971. Large-scale integration (LSI), which arrived in the mid-1970s, brought chips with tens of thousands of transistors each. Ferranti (Italy) introduced the first gate-array, the Uncommitted Logic Array (ULA) in 1980, which led to the creation of application-specific integrated circuits (ASICs). Very large-scale integration (VLSI) brought chips with hundreds of thousands of transistors in the 1980s and several billion transistors as of 2009.
A laser (Light Amplification by Stimulated Emission of Radiation) emits light through optical amplification based on the stimulated emission of electromagnetic radiation. Albert Einstein (Germany) established the theoretical basis for lasers and masers in a 1917 paper. Other aspects of the science were developed by Rudolf Ladenburg (Germany) in 1928; Valentin Fabrikant (USSR) in 1939; Willis E. Lamb and R.C. Retherford (US) in 1947 (stimulated emission) and Alfred Kastler (France) in 1950 (optical pumping). Charles Hard Townes, with students James Gordon and Herbert Zeiger (US) created the first microwave amplifier, or maser, in 1953, although it was incapable of continuous output. In the USSR, Nikolay Basov and Aleksandr Prokhorov had solved the continuous output problem using a quantum oscillator in 1952, but results were not published until 1954-1955. In 1957, Townes and Arthur Leonard Schawlow (US), at Bell Labs, began working on an infrared laser, but soon changed to visible light, for which they sought a patent in 1958. Also in 1957, Columbia University grad student Gordon Gould, after meeting with Townes, began working on the idea for a “laser” using an open resonator. Prokhorov independently proposed the open resonator in 1958. In 1959, Gould published the first paper using the term LASER (light amplification by stimulated emission of radiation) and filed for a patent the same year. The U.S. Patent Office granted Townes’ and Schawlow’s patent and denied Gould’s in 1960. The first working laser was created by Theodore Maiman (US) in 1960, but it was only capable of pulsed operation. Also in 1960, Ali Javan (Iran/US), William Bennett and Donald Herriott (US) made the first gas laser. In 1962, Robert N. Hall (US) invented the first laser diode device. The same year, Nick Holonyak, Jr. (US) made the first semiconductor laser with a visible emission, although it could only be used in pulsed-beam operation. In 1970, Zhores Alferov (USSR), Izuo Hayashi (Japan) and Morton Panish (US) independently developed room-temperature, continual-operation diode lasers. In 1987, following years of patent litigation, a Federal judge ordered the U.S. Patent Office to issue patents to Gordon Gould for the optical pump and gas discharge lasers.
While pursuing the work of Kristian Birkeland (Norway) on auroras, Carl Størmer (Norway) proposed that particles might be trapped within the Earth’s magnetic field, and he worked out the orbits of these trapped particles between 1907 and 1913. To test the theory, American scientist James Van Allen placed Geiger–Müller tube experiments on three 1958 satellites: Explorer 1, Explorer 3 and Pioneer 3. The experiments confirmed the existence of two radiation belts containing trapped particles. The inner belt consists mostly of energetic protons, which are the product of the decay of neutrons created by cosmic ray collisions in the upper atmosphere. The outer belt consists mostly of electrons; they are injected from the geomagnetic tail after geomagnetic storms and are energized through wave-particle interactions. Two Van Allen probes were launched by NASA in 2012. They discovered a temporary third Van Allen belt.
While man has been observing the Earth’s moon since ancient times, and Galileo Galilei made the first telescopic observations in 1610-1612, physical exploration of the moon began on September 14, 1959, when the USSR’s unmanned probe Luna 2 made a hard landing on the moon’s surface. Luna 3 photographed the far side of the moon for the first time on October 7, 1959. Luna 9 made a soft landing on the moon and sent the first pictures from the moon’s surface on February 3, 1966. Frank Borman, James Lovell and William Anders, in Apollo 8 (US), became the first humans to enter lunar orbit and see the far side of the moon on December 24, 1968. Neil Armstrong and Edwin “Buzz” Aldrin in Apollo 11 (US) landed on the moon on July 20, 1969. The next day, Armstrong became the first man to walk on the moon. US astronauts Edwin Aldrin and Neil Armstrong land on the Moon. The US sent a total of six manned missions to the moon between 1969 and 1972. There were 59 unmanned missions by the US or USSR between 1959 and 1976. Three Luna probes and six Apollo missions returned to Earth with moon rock samples. Japan sent probes into the moon’s orbit in 1990 and 2007. NASA and the Ballistic Missile Defense Organization launched orbiters in 1994 and 1998. A European Space Agency probe began orbiting the mooon in 2004, then intentionally crashed in 2006. China send an orbital probe in 2007; it was intentionally crashed on the moon’s surface in 2009. A rover from a second Chinese orbiter soft-landed on the moon on December 14, 2013. India sent an orbiter to the moon in 2008 and landed an impact probe on November 14, 2008. Many other orbiters and landers are planned for the future.
The idea of humans traveling outside the Earth’s atmosphere has a long history in literature, but the first scientific proposal came from Konstantin Tsiolkovsky (Russia) in 1903. A 1919 paper by Robert Goddard (US) combining the de Laval nozzle with rockets using liquid fuel was the first practical proposal, and influenced Hermann Oberth and Wernher Von Braun (Germany). The unmanned German V-2 rocket was the first to reach space in 1944. The first unmanned satellite was placed into orbit by the USSR in 1957. The first man to leave the atmosphere and orbit the Earth was Yuri Gagarin (USSR) on April 12. 1961. On May 5, 1961, the US sent Alan Shepherd into a suborbital flight, while John Glenn became the first American to orbit the Earth on February 20, 1962. The USSR put a number of men and the first woman (Valentina Tereshkova) in space in 1962 and 1963. Alexei Leonov (USSR) made the first spacewalk on March 8. 1965. On July 20, 1969, an American crew landed on the moon.
Seafloor spreading is a process in which new oceanic crust if formed through volcanic activity at mid-ocean ridges and then gradually moves away from the ridge. Seafloor spreading helps to explain continental drift in the theory of plate tectonics. Between 1947 and 1953, William Maurice Ewing, Bruce Heezen, David Ericson and Marie Tharp (US) at Lamont Geological Observatory collected enormous amounts of data about the nature of the seafloor and in so doing discovered the global mid-ocean ridge and its volcanic nature, particularly a rift valley that ran down the center of the ridges. In 1960, Harry Hess (US), with the help of Robert Dietz (US), hypothesized that the seafloor was spreading, and that was the cause of continental drift. In 1962, Hess furher proposed that volcanic activity along the mid-ocean ridges caused seafloor spreading. Evidence of seafloor spreading came in 1963 from Frederick Vine and Drummond Matthews (UK) who discovered that as the crust along the mid-ocean ridges consisted of parallel crust with alternating magnetic polarity. (Although it was not until 1966 that scientists proved the Earth’s magnetic field periodically reversed itself.) This showed that new crust was created at the ridges and then spread out from the ridges over time. In 1965, J. Tuzo Wilson (Canada) combined the continental drift and seafloor spreading hypotheses into plate tectonics theory.
Quasars (short for quasi-stellar radio sources) belong to a class of objects called active galactic nuclei that are very luminous sources of electromagnetic energy with a high redshift. Most scientists now believe that a quasar is the compact region in the center of a galaxy that surrounds a supermassive black hole and that the quasar’s energy comes from the mass falling onto the accretion disc around the black hole. Allan Sandage (US) and others discovered the first quasars in the early 1960s. In 1960, a radio source named 3C 48 was tied to a visible object. John Bolton (UK/Australia) observed a very large redshift for the object but his claim was not accepted at the time. In 1962, Bolton and Cyril Hazard identified another such radio source, 3C 273. In 1963, Marten Schmidt (The Netherlands) used their measurements to identify the visible object associated with the radio source and obtain an optical spectrum, which showed a very high redshift. (37% of the speed of light). Hong-Yee Chiu (China/US) first used the term quasar to describe the new type of object in a 1964 article. Scientists debated the distance of quasars until the 1970s, when the mechanisms of black hole accretion discs were discovered. In 1979, images of a double quasar provided the first visual evidence of the gravitational lens effect predicted by Einstein’s general theory of relativity.
Murray Gell-Mann and George Zweig independently proposed the quark model, also known as quantum chromodynamics in 1964. They suggested that there were three types of quarks (up, down and strange) and that all hadrons were composed of combinations of quarks and antiquarks. Sheldon Lee Glashow and James Bjorken proposed a fourth quark in 1965: charm. Experiments by Jerome Friedman, Henry Kendall, and Richard Taylor (US) in 1968 using the Stanford Linear Accelerator eventually revealed the existence of the up, down and strange quarks. In 1973, Makoto Kobayashi and Toshihide Maskawa (Japan) proposed two more quarks: top and bottom. The charm quark was observed in 1974 by Burton Richter and Samuel Ting. In 1977, the bottom quark was observed by Leon Lederman. A team at Fermilab found the top quark in 1995.
A pulsar (short for ‘pulsating star’) is a highly magnetized, rotating neutron star that emits a beam of electromagnetic radiation. On November 28, 1967, Antony Hewish and Jocelyn Bell Burnell (UK) became the first scientists to observe a pulsar, which had a pulse period of 1.33 seconds . Walter Baade (Germany/US) and Fritz Zwicky (Switzerland/US) had predicted neutron stars in 1934, and in early 1967, Franco Pacini (Italy) suggested that a rotating neutron star with a magnetic field would emit radiation. In 1968, David Staelin, Edward C. Reifenstein III and Richard Lovelace (US) discovered a pulsar in the Crab Nebula with a 33 millisecond pulse period and a rotation speed of 1,980 revolutions per minute. Joseph Hooton Taylor, Jr. and Russell Hulse (US) discovered the first pulsar in a binary system in 1974. A team led by Don Backer (US) discovered the first millisecond pulsar, with a rotation period of 1.6 milliseconds, in 1982.
The development of the Internet was a complex, many-faceted process. It is impossible to identify one person who invented the Internet, and it is very difficult to choose a point in time when the Internet was invented but there is a rational basis for choosing 1969 (see below). Leonard Kleinrock’s (US) July 1961 paper on packet switching theory at MIT was an early precursor to the Internet, as was a series of “Galactic Network” memos by J.C.R. Licklider (US), also at MIT, in August 1962. In October 1962, Licklider became the first computer research program head at DARPA (Defense Advanced Research Projects Agency), where he convinced Ivan Sutherland, Robert Taylor and MIT’s Lawrence G. Roberts (US) of the importance of networking. Kleinrock convinced Roberts of using packets rather than circuits. The first wide-area computer network was built in 1965 when Roberts and Thomas Merrill (US) connected the TX-2 computer in Massachusetts to the Q-32 in California. In 1966, Roberts went to DARPA, where he developed the computer network concept and put together a plan for the ARPANET, which he published in 1967. Parallel research on networks had been going on at RAND (1962-1965) (esp. Paul Baran (US)) and National Physical Laboratory (NPL) (1964-1967) (esp. Donald Davies and Roger Scantlebury (UK)). After Roberts and DARPA refined the ARPANET’s specifications, in 1968, they chose Frank Heart’s (US) team at Bolt Beranek and Newman to built the packet switches, called Interface Message Processors (IMPs). Robert Kahn (US) at BBN; Howard Frank (US) at Network Analysis Corp.; and Kleinrock, played significant roles. In September 1969, BBN installed the first IMP at UCLA, which became the first node. Doug Engelbart’s Stanford Research Institute (SRI) provided the second node. The first message was sent between UCLA and SRI in October 1969. Four computers were linked by the end of 1969 and many more joined in the next few years. In December 1970, S. Crocker (US) and his Network Working Group finished the ARPANET’s initial host-to-host protocol, the Network Control Protocol (NCP). Also in 1970, NPL started the Mark I network. In 1971, the Merit Network and Tymnet networks became operational. Kahn successfully demonstrated the ARPANET at a conference in October 1972. Also in 1972, Louis Pouzin in France began an Internet-like project called Cyclades, that was based on the notion that the host computer, not the network, should be responsible for data transmission. Cyclades was eventually shut down, but the Internet eventually adopted its basic principle. The first trans-Atlantic transmission occurred in 1973, to University College of London. In 1974, a proposal was made to link ARPA-like networks into a larger inter-network that would have no central control. Also in 1974, the International Telecommunication Union developed X.25 packet switching network standards. The PC modem was invented by Dennis Hayes and Dale Heatherington in 1977. The first bulletin board system was invented in 1978. Usenet was invented in 1979 by Tom Truscott and Jim Ellis (US) and CompuServe was launched the same year. In 1981, the National Science Foundation created CSNET, the Computer Science Network, which linked to ARPANET. In 1982, the TCP/IP protocol suite, invented by Vinton Cerf (US), was formalized. ARPANET computers were required to switch from the NCP protocol to the TCP/IP protocols by January 1, 1983. In 1984, the system of domain names was adopted – the first .COM domain name was registered in 1985. In 1986, NSF created NSFNET, which was linked with ARPANET. In 1988, Internet Relay Chat was first introduced. America Online (AOL) was launched in 1989. In 1990, ARPANET was decommissioned in favor of NSFNET. NSFNET was decommissioned in 1995 when it was replaced by networks operated by several commercial Internet Service Providers.
According to string theory, all elementary particles are actually made of vibrating one-dimensional objects called strings. String theory purports to unite all four basic forces in one explanatory framework. String theory requires multiple spatial dimensions; one version of the theory requires 11 dimensions. While some physicists have embraced it, others criticize it because it is difficult (some say impossible) to test its predictions. A precursor to string theory was S-matrix theory, proposed by Werner Heisenberg (Germany) in 1943. Some physicists expanded on the theory in the 1950s, particularly Tullio Regge (Italy), Geoffrey Chew and Steven Frautschi (US). The theory eventually developed into the dual resonance model of Gabriele Veneziano in 1968. The scattering amplitude that Veneziano predicted was essentially a closed vibrating string. Then in 1970, Yochiro Nambu (Japan/US), Holger Bech Nielsen (Denmark) and Leonard Susskind (US) proposed a theory that represented nuclear forces as one-dimensional vibrating strings. John H. Schwarz (US) and Joel Scherk (France) proposed bosonic string theory in 1974. Michael Green (UK) and John Schwarz proposed the existence of supersymmetric strings, or superstrings, in the early 1980s. Between 1984 and 1986, a number of scientific discoveries occurred that have been termed the first superstring revolution. These discoveries resulted in a number of rival versions of the theory. In 1994, Edward Witten (US) suggested that the five different versions of string theory were all different limits on an 11-dimensional theory he called M-theory, an announcement that led to the second superstring revolution between 1994 and 1997. Chris Hull and Paul Townsend (UK) played important roles in this phase. Some scientists believe that the Large Hadron Collider at CERN may be able to produce enough energy to provide experimental evidence for string theory.
Domestication of plants and animals through selective breeding and hybridization has been occurring since at least 9,000 BCE. Charles Darwin explained in 1859 that the principles underlying selective breeding also occur through natural selection. Gregor Mendel (Silesia) discovered the principles of genetics in 1865. In 1910, T.H. Morgan showed that genes are carried on chromosomes in the cell’s nucleus. In 1927, H.J. Muller first used x-rays to create genetic mutations in plants. Frederick Griffith (UK) made early discoveries of the chemistry of genetics in 1928. Barbara McClintock and Harriet Creighton showed direct physical recombination in DNA in 1931. In 1941, Edward Tatum and George Beadle (US) determined that the genetic substance coded for proteins. Oswald Avery, Colin MacLeod and Maclyn McCarty (US) identified the genetic substance as DNA in 1944. In 1953, James Watson (US) and Francis Crick (UK) identified the double helix structure of DNA. In 1967, scientists discovered DNA ligases, which could join pieces of DNA together. In the late 1960s, Stewart Linn and Werner Arber discovered restriction enzymes. In 1970, Hamilton Smith (US) used restriction enzymes to target DNA at a specific location and separate the pieces. Also in 1970, Morton Mandel and Akiko Higo (US) inserted a bacteriophage virus into the DNA of the E. coli bacteria. In 1972, Paul Berg (US) created the first recombinant DNA molecules. Also in 1972, Herbert Boyer and Stanley Cohen (US) inserted recombinent DNA into bacterial cells using a technique called DNA cloning. They then created the first genetically modified organism by inserting a gene for resistance to an antibiotic into bacteria that had no such gene, making the bacteria resistant. Later, they placed a frog gene into a bacterial cell. In 1973, Stanley Cohen, Annie Chang and Herbert Boyer created a genetically modified DNA organism. In 1974, Rudolf Jaenisch (Germany/US) inserted foreign DNA into a mouse. Beginning in 1976, recombinant DNA research has been subject to regulation in the US. Frederick Sanger (UK) developed a way to sequence DNA in 1977. In 1979, scientists were able to modify bacteria to produce human insulin. In 1981, Frank Ruddle (US), Frank Constantini and Elizabeth Lacy (UK) were able to pass new genes into subsequent generations by inserting foreign DNA into a mouse embryo. In 1983, Michael Bevan, Richard Flavell (UK) and Mary-Dell Chilton (US) inserted new genetic material into a tobacco plant – the first genetically modified plant. In 1983, Kary Mullis (US) identified the polymerase chain reaction, which amplified small sections of DNA. In 1984, mice were genetically modified to predispose them to cancer. In the late 1980s, electroporation – the use of electricity to make a cell membrane more porous – increased scientists’ ability to insert foreign DNA into cells. In 1989, Mario Capecchi (US), Martin Evans (UK) and Oliver Smithies (UK/US) were the first to manipulate a mouse’s DNA to turn off a gene. After the discovery of microRNA in 1993, Craig Mello and Andrew Fire (US) were able to silence genes in mammalian cells in 2002 and in an entire mouse in 2005. The first of many commercial enterprises featuring genetic engineering was Genentech, founded by Herbert Boyer and Robert Swanson (US) in 1976. The release of GMOs into the environment has been a focus of protests around the world.
In 1961, MIT introduced an internal computer network called the Compatible Time-Sharing System (CTSS). CTSS included a program written by T. Van Vleck that allowed users to send “mail” messages to each other. In the 1960s and 1970s, many organizations with mainframes or networks had their own internal e-mail systems. In 1971, in response to a memo by Dick Watson of SRI Int’l, Ray Tomlinson of BBN wrote an e-mail program (based on SNDMSG and CPYNET) and sent the first e-mail on the ARPANET network. He was the first to use the @ symbol to separate the name of the user and the user’s computer. In 1971-1972, Abhay Bhushan developed FTP (File Transfer Protocol), and then MLFL (mail file). Also in 1972, Bob Clements, Ken Pogran and Mike Padlipsky made additional improvements. In 1972, Lawrence Roberts wrote an e-mail management program that improved on Tomlinson’s e-mail, called RD. Barry Wessler then improved on RD, creating NRD. In 1973, Martin Yonke took SNDMSG and NRD and recoded them into the improved WRD (later BANANARD). In 1975, John Vittal created the first all-inclusive e-mail program, MSG, which included the commands Move, Answer (now Reply), and Forward. Also in 1975, Dave Farber, Dave Crocker, Steve Tepper and Bill Crosby created MH, an e-mail program for the Unix operating system. In 1977, Crocker, Vittal, Kenneth Pongran and D. Austin Henderson attempted to standardize various e-mail formats over the ARPANET. In 1978, Crocker and Farber began work on MMDF(Multi-purpose Memo Distribution Facility) to allow dial-up relay of e-mail. Ed Szurkowski, Doug Kingston, Craig Partridge and Steve Kille followed up on the project. Kevin MacKenzie suggested emoticons for the first time in 1979. In 1982, Crocker revised the standard to include the syntax for domain names. Also in 1982, EUnet was created to provide e-mail in Europe. In the early 1980s, Eric Allman created the delivermailand sendmail programs. In 1988, Vinton Cerf arranged for MCI to become the first commercial e-mail carrier to connect to the Internet. CompuServe followed in 1989. America Online and Delphi began connecting their e-mail systems to the Internet in 1993.
Also known as Standard Model of Quantum Field Theory and the Standard Model of Particle Physics, the Standard Model, which is the result of the work of many scientists over the period of 1970-1973, summarizes the forces and particles that make up the universe. According to the Standard Model, there are three classes of elementary particles: fermions, gauge bosons, and the Higgs boson. There are 12 fermions, all of which have spin ½; they include six leptons (including electrons, muons, and tauons and their neutrino counterparts), and six quarks (including up, charm, and top and their charge complements, down, strange, and bottom). Leptons and quarks interact by exchanging generalized quanta, particles of spin 1. Bosons, which have spin 1, are particles involved in the transmission of forces and include gluons, which carry the strong force that binds quarks together. Thus bound together, the quarks form hadrons. The proton and the neutron that make up atomic nuclei are hadrons. Bosons also include photons, which carry the electroweak force and attract electrons to orbit the nuclei. Other weak interactions are carried by the W , W+, and Z particles. Additional forces are carried by gravitons and Higgs bosons. The combination of the electromagnetic force and the weak interaction into the electroweak force by Sheldon Glashow (US) and others in 1961 paved the way for the Standard Model. The muon neutrino was first detected in 1962. In 1964, Murray Gell-Mann and George Zweig propose that hadrons are made of quarks. Steven Weinberg (US) and Abdus Salam (Pakistan) incorporated the Higgs mechanism into the electroweak theory in 1967. Experimental confirmation of the electroweak theory came in 1973 when the supercollider at CERN detected the neutral weak currents that were predicted to result from Z boson exchange. The Standard Model’s explanation of the strong interaction received experimental confirmation in 1973-1974 when it was shown that hadrons are composed of fractionally-charged quarks. In 1983, Carlo Rubbia discovered the W and Z bosons. In 1995, the last undiscovered quark, the top quark, was discovered. The tau neutrino was detected in 2000 at Fermilab. The Higgs boson was finally discovered in the Large Hadron Collider at CERN in 2012.
A hydrothermal vent is a fissure in the planet’s surface from which geothermally heated water issues. Hydrothermal vents are found near volcanic activity, in ocean basins and hotspots and in areas where tectonic plates are moving apart. Deep sea hydrothermal vents often form large features called black smokers. Although they have no access to sunlight, some hydrothermal vents are biologically active and host dense and complex communities based on chemosynthetic bateria and archaea. A deep water survey of the Red Sea in 1949 revealed hot brines that could not be explained. In the 1960s, the hot brines and muds were confirmed and found to be coming from an active subseafloor rift. No biological activity was found in the highly saline environment. A team from Scripps Intitution of Oceanography led by Jack Corliss (US) found the first evidence of chemosynthetic biological activity surrounding underwater hydrothermal vents that formed black smokers along the Galapagos Rift in 1977; they returned in 1979 to use Alvin, a deep-water research submersible, to observe the hydrothermal vents directly. Peter Lonsdale published the first paper on hydrothermal vent biology in 1979. Neptune Resourse NL discovered a hydrothermal vent off the coast of Costa Rica in 2005. Among the deepest hydrothermal vents are the Ashadze hydrothermal field on the Mid-Atlantic Ridge (-4200 meters), a vent at the Beebe site in the Cayman Trough (-5000 meters), discovered in 2010 by scientists from NASA and Woods Hole Oceanographic Institute; and a series of hydrothermal vents in the Caribbean found in 2014 (-5000 meters). By 1993, more than 100 species of gastropods had been found in hydrothermal vent communities. Scientists have discovered 300 new species at hydrothermal vents including the Pompeii worm, discovered by Daniel Desbruyères and Lucien Laubier (France) in 1980 and Craig Gary (US) in 1997 and the scaly-foot gastropod in 2001.
Between 1948 and 1957, John Lentz (US) developed the Personal Automatic Computer at Columbia University; it later became the IBM 610, which sold for $55,000. Only 180 were made. (Minicomputers such as the LINC (1962) and PDP-8 (1965) and similar models from DEC, Data General and Prime could also be classified as personal computers, although they were refrigerator-sized and very costly.) In 1965, Olivetti (Italy) introduced the Programma 101 – the first commercially produced desktop computer. Olivetti sold 44,000 units at $3,200 each. Victor Glushov (USSR) produced the MIR from 1965-1969. John Blankenbaker (US) of Kenbak Corp. invented the Kenbak-1 in 1970, but only 40 were made. Also in 1970, CTC (now Datapoint) created the Datapoint 2200 – the first machine to resemble modern p.c.’s. Polish scientist Jacek Karpiński and his team developed the K-202 in 1971-1973, the first 16-bit non-kit desktop, but only 30 were sold. R2E (France) made the Micral N in 1973. Also introduced in 1973 was the Xerox Alto, which had a mouse and a graphical user interface. The Mark-8 was a 1974 microcomputer build-it-yourself kit. IBM brought out the IBM 5100 in 1975. The Altair 8800, created by MITS (US) in 1975, was a very popular and inexpensive kit that spawned many imitators. In 1976, Gary Ingram and Bob Marsh (US) at Processor Technology Corporation, designed the Sol-20 Personal Computer, the first all-in-one p.c., which sold well from 1976-1979. Three important personal computers were released in 1977: (1) the Commodore PET, created by Chuck Peddle (US), sales less than one million units; (2) the Apple II, created by Steve Wozniak/Apple (US), 2.1 million sold by 1985; and (3) the TRS-80/Model I, created by Tandy/Radio Shack, sales of 1.5 million units by 1981. The IBM 5100 was announced in 1978 and withdrawn in 1982. Atari released its first home computers, the 400 and 800, in 1978-1979. The same year, Texas Instruments made the TI-99/4A home computer. In 1980, two UK companies released home computers: the Sinclair ZX80, by Science of Cambridge and the Acorn Atom by Acorn Computers. Also in 1980, Commodore released the VIC-20. In 1981, Xerox introduced the Xerox Star workstation, with many modern features. The IBM PC was also introduced in 1981, as was the BBC Micro. In 1982, the Commodore 64 was introduced – it would sell 17 million units. The most popular personal computer in Japan was NEC’s PC-9801, which was released in 1982. IBM followed up the PC with the XT in 1983 and the PC/AT in 1984. Many companies created clones of the IBM products. In 1983, Apple brought out the Lisa, a mass-marketed microcomputer with a graphical user interface but it was too slow and expensive to succeed. Apple’s 1984 design, the mouse-driven Macintosh, was much more successful.
Prior to the 1960s, biologists classified living organisms into two domains: the Bacteria and the Eukaryota. Classification was based on biochemistry, morphology and metabolism. All bacteria were prokaryotes because they lacked a nucleus or any other organelles with a cell membrane. In 1965, Linus Pauling (US) and Emile Zuckerlandl (Austria/France) proposed a new basis for classification based on genetic relationships, calling it phylogenetics. In 1977, Carl Woese and George E. Fox (US) analyzed the ribosomal RNA of organisms classified as bacteria and found that there were two very distinct groups: the traditional bacteria and the Archaea. Originally scientists classified the two groups as separate kingdoms or subkingdoms called Archaebacteria and Eubacteria of a larger Prokaryote domain. Later, Woese proposed that there were three domains of living organisms: the Eukarya (divided into three kingdoms: animals, plants and fungi), the Bacteria and the Archaea. Archaea are similar in appearance to bacteria, but their genes and certain metabolic pathways (such as enzymes) are closer to those of eukaryotes. They were once thought to be restricted to extreme environments, but have been discovered in a wide variety of habitats.
In 1956, Élie Wollman and Francois Jacob (France) published a first, rudimentary genetic map of the E. coli chromosome. Walter Fiers (Belgium) and his team at University of Ghent sequenced the nucleotides of the genome of bacteriophage MS2, the genetic material of which is RNA, not DNA, in 1972 and 1976. In 1977, Frederick Sanger and his team sequenced the genome of bacteriophage ΦX174. Scientists at the Medical Research Council (UK) sequenced the Epstein-Barr virus in 1984. In 1995, J. Craig Ventner, Hamilton Smith and others at the Institute for Genomic Research (US) published the first complete nucleotide sequence of a free-living organism, the bacterium Haemophilus influenzae. Ventner and Celera Genomics (US) sequenced the genome of the fruit fly, Drosophila melanogaster, in 2000. Between 2000 and 2003, the Human Genome Project and Celera Genomics both sequenced the human genome.
At the end of the Cretaceous Period, 66 million years ago, a mass extinction eliminated 75% of all animal and plant species, including the dinosaurs. Although many hypotheses have been made to explain this mass extinction (one of several in Earth’s history), the predominant theory is that of Luis Alvarez (US), who proposed in 1980 that an asteroid impact resulted in the extinctions. In 1980, Alvarez, a physicist, his son geologist Walter Alvarez and chemists Frank Asaro and Helen Michel (US) reported that the sedimentary rocks at the border between the Cretaceous and Paleogene (then called Tertiary) periods contained an abnormally high amount of the rare element iridium, which is common in asteroids and comets. They suggested an asteroid impact occurred about 66 million years ago. The theory has been supported by additional evidence, including the finding of rock spherules formed by the impact and shocked minerals from intense pressure. The presence of thicker sedimentary layers and giant tsunami beds in southern US and Central America supported the idea that the asteroid impact site was nearby, a prediction confirmed by the discovery of a giant crater (110 miles in diameter) at Chicxulub along the coast of the Yucatan in Mexico in 1990, based on 1978 work by geophysicist Glen Penfield.
According to inflation theory, the universe underwent an exponential expansion of space from 10 ‾36 seconds after the Big Bang to between 10 ‾33 and 10 ‾32 seconds. Inflation theory purports to explain the origin of the large-scale structure of the universe. The origins of inflation theory go back to 1917, when Albert Einstein invoked the cosmological constant to prove that the universe was static. At about the same time, Dutch scientist Willem de Sitter, analyzing general relativity, discovered a formula that described a highly symmetric inflating empty universe with a positive cosmological constant. Some believe that inflation theory was proposed by Erast Gliner (USSR) in 1965, who was not taken seriously at the time. In the early 1970s, Yakov Zeldovich (USSR) noted that the Big Bang model had serious problems with flatness and horizon. Vladimir Belinski and Isaak Khalatnikov (USSR) and Charles Misner (US) tried to solve the problems. Sidney Coleman (US) study of false vacuums in the late 1970s raised important questions for cosmology. In 1978, Zeldovich drew attention to the monopole problem, a version of the horizon problem. In 1979, Alexei Starobinsky (USSR) predicted that the early universe went through a de Sitter phase, or inflationary era. In January 1980, Alan Guth (US) proposed scalar driven inflation to solve Zeldovich’s problem of the nonexistence of magnetic monopoles. In October 1980, Demosthenes Kazanas (Greece/US) suggested that exponential expansion might eliminate the particle horizon. Martin Einhorn (US) and Katsuhiko Sato published a model similar to Guth’s in 1981. Guth’s theory and other early versions of inflation had a problem: bubble wall collisions. Andrei Linde (USSR/US) solved the problem in 1981, as did Andreas Albrecht and Paul Steinhardt (US), independently, with new inflation, or slow-roll inflation. Linde revised the model in 1983, calling the new version chaotic inflation. Numerous scientists worked on calculating the tiny quantum fluctuations in the inflationary universe that led to the structure we see today, particularly at a 1982 workshop at Cambridge University. Predictions of inflation theory were experimentally confirmed in 2003-2009 by the Wilkinson Microwave Anisotropy Probe’s findings of the flatness of the universe. The first direct evidence of gravitational waves, announced by Harvard-Smithsonian Center astronomers on March 17, 2014, provides additional support for inflation.
The human immunodeficiency virus (HIV) causes acquired immunodeficiency syndrome (AIDS), a highly fatal disease that cripples the immune system, allowing opportunistic infections and cancers to wreak havoc. AIDS was first observed in the US in 1981 in patients with a rare form of pneumonia and later a rare skin cancer called Kaposi’s sarcoma. In May 1983, a French research group led by Luc Montagnier (with Françoise Barré-Sinoussi) isolated a new retrovirus they called LAV (lymphadenopathy-associated virus), that appeared to be the cause of AIDS. In May 1984, an American team led by Robert Gallo discovered the same virus, which they named HTLV-III (human T lymphotropic virus type III). By March 1985, it was clear that LAV and HTLV-III were the same virus and in May 1986, the International Committee on Taxonomy of Viruses named the virus discovered by both groups HIV, for human immunodeficiency virus. Further study indicated that two types of HIV originated in primates in west-central Africa and transferred to humans in the early 20th Century.
A fullerene is a molecule made entirely of carbon in the form of a hollow sphere, ellipsoid, tube or certain other shapes. Spherical fullerenes are also known as buckyballs. Cylindrical fullerenes are called carbon nanotubes or buckytubes. Sumio Iijima (Japan) had predicted the existence of the C 60 molecule (which became the first fullerene) in 1970 and identified it in a electron micrograph in 1980. R.W. Henson (UK) had proposed the structure of C 60 in 1970 and made a model of it, but his results were not accepted. In 1973, Professor Bochvar (USSR) made a quantum-chemical analysis of C 60’s stability and calculated its electronic structure, but the scientific community rejected his conclusions. In 1985, Harold Kroto (UK) and Americans Richard Smally, Robert Curl, James Heath and Sean O’Brien and at Rice University, in the course of experiments designed to mimic carbon clusters, discovered and prepared C-60, the first fullerene, which they named buckminsterfullerene, by firing an intense pulse of laser light at a carbon surface in the presence of helium and then cooling the gaseous carbon to near absolute zero.
In 1980, Tim Berners-Lee (UK), working at CERN in Switzerland, built a personal database of people and software models called ENQUIRE, that used hypertext. In March 1989, Berners-Lee proposed a large hypertext database with typed links. He began implementing his proposal on a NeXT workstation, calling it the World Wide Web. Berners-Lee’s collaborator Robert Cailliau (Belgium) rewrote the proposal in 1990. By Christmas 1990, Berners-Lee had created the HyperText Transfer Protocol (HTTP), the Hypertext Markup Language (HTML), the first Web browser, the first HTTP server software, the first Web server and the first Web pages. Nicola Pellow (UK) created Line Mode Browser, that allowed the system to run on any computer. In January 1991, the first non-CERN servers came online. The Web became publicly available after August 23, 1991. The first American Web server was established at the Stanford Linear Accelerator Center by Paul Kunz and Louise Addis (US) in September 1991. In 1993, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, under the lead of Marc Andreessen (US), introduced the Mosaic graphical web browser, which evolved into Netscape Navigator in 1994. Also in 1993, Microsoft released Cello, written by Thomas R. Bruce (US) at Cornell, a browser for Microsoft Windows. As of September 2014, there were 3.2 billion web pages on the World Wide Web.
The US (with assistance from Europe) launched the Hubble Space Telescope into low Earth orbit in April 1990 with a 2.4 meter (7.9 ft.) mirror. After an adjustment to its mirror in 1993, it has been able to observe distant space objects in the ultraviolet, visible and infrared spectra. Its images have helped determine the rate of expansion of the universe (the Hubble constant); identify that the expansion of the universe is accelerating; locate black holes in the center of galaxies; create deep field images of distant galaxies. Over 9,000 papers based on Hubble data have been published in peer-reviewed journals. It was still operating in August 2014.
The first hollow, nanometer-size tubes composed of graphitic carbon were probably discovered in the USSR in the early 1950s. L.V. Radushkevich and V.M. Lukyanovich published the first electron microscope images of 50 nanometer diameter tubes in a Soviet publication in 1952. In 1976, Oberlin, Endo and Koyama produced hollow carbon fibers with nanometer-scale diameters and a single wall of graphene referred to as a single-walled nanotube. John Abrahamson presented evidence of carbon nanotubes at a 1979 conference. In 1981, Soviet scientiest conducted thermocatalytical disproportionation of carbon monoxide, which produced carbon multi-layer tubular crystals formed by rolling graphene layers into cylinders. Howard Tennett of Hyperion Catalysis obtained a patent in 1987 to produce cyclindrical discrete carbon fibrils. In 1991, Sumio Iijima of NEC discovered multi-walled carbon nanotubes in arc-burned graphite rods, and then discovered single walled carbon nanotubes and methodes to produce them by adding transition metal catalysts to the carbon in an arc discharge. At about the same time, Mintmire, Dunlap and White predicted that single-walled carbon nanotubes would have remarkable conducting properties.
In 1637, Pierre de Fermat scrawled a theorem in the margin of Arithmetica, by 3rd Century CE mathematician Diophantus. Fermat stated that no three positive integers a, b and c can satisfy the equation an + bn = cfor any integer value of n greater than 2 but did not provide a proof. Mathematicians have attempted to prove the theorem for centuries, until it was solved by British mathematician Andrew Wiles in 1994.
In 1584, Giordano Bruno speculated that other stars had planets circling them. A number of claims of discovery of planets around other stars in the 19th and 20th centuries have been discredited. A 1988 claim by Canadian astronomers Bruce Campbell, G. A. H. Walker, and Stephenson Yang that they had discovered a planet orbiting the star Gamma Cephei was tentative at the time but confirmed in 2003 after advances in technology. In 1992, Aleksander Wolszczan (Poland) and Dale Frail (Canada/US) discovered two Earth-sized planets orbiting the pulsar PSR 1257+12, which is generally considered the first definitive detection of exoplanets, or extrasolar planets. The team found a third planet in 1994. In 1995, Michel Mayor and Didier Queloz (Switzerland) observed a giant planet in a four-day orbit around 51 Pegasi, the first detection of a planet circular a standard, or main sequence star. In 2009, the US launched the Kepler Space Telescope, which has the mission of discovering Earth-like extrasolar planets. As of August 14, 2014, 1815 confirmed exoplanets have been found in 1130 planetary systems, of which 466 are multiple planetary systems. The smallest planet known is about twice the size of the moon, while the largest is 29 times the size of Jupiter. Some planets are so near their star they their orbits take only a few hours, while some are so far out that they take thousands of years to complete one orbit. In March 2014, the Kepler space telescope identified the first exoplanet that is similar to Earth in size with an orbit within the habitable zone (i.e., the area that would support life) of another star. The planet is Kepler-186f and the star is a red dwarf (Kepler 186) about 500 light years from Earth.
Cloning (also known as reproductive cloning) is a biotechnological technique in which scientists create biological organisms from the DNA of another organism, usually by taking the DNA from an adult somatic cell and transferring it to an egg cell from which the DNA has been removed. In 1924, German embryologists Hans Spemann and Hilde Mangold first discovered embryonic induction, in which parts of the embryo direct the development of groups of cells into particular tissues or organs. Spemann and Mangold performed the first somatic-cell nuclear transfer in 1928. Somatic-cell nuclear transfer is one form of cloning; another form is embryo-splitting, which uses an existing embryo to create artificial identical twins. Robert Priggs and Thomas J. King (US) cloned northern leopard frogs in 1952 using nuclear transfer of embryonic cells In 1958, John Gurdon (UK) further advanced the work of Priggs and King. In 1963, Tong Dizhou (China) cloned a carp. Soviet scientists Chaylakhyan, Veprencev, Sviridova and Nikitin cloned a mouse in 1986. A sheep was cloned by Steen Willadsen (Denmark) using early embryonic cells in 1984. In 1995, scientists at the the Roslin Institute in Edinburgh, Scotland cloned two sheep from differentiated embryonic cells in 1995. Ian Wilmut and his team at the Roslin Institute cloned a sheep (named Dolly) from a somatic cell in 1996 (although not announced until 1997) – this was the first mammal cloned from an adult somatic cell. Scientists have also cloned: cow (1997); rhesus monkey (1999, 2007); pig (2000); cat (2001); gaur (2001); goat (2001); mule (2003); horse (2003); deer (2003); rabbit (2003); ferret (2004); dog (2005); fruit fly (2005); wolf (?), water buffalo (2005); camel (2009); zebrafish (2009); and Pyrenean ibex (extinct) (2009). Humans have not yet been cloned; some nations have laws prohibiting human cloning.
While scientists have known since 1929 that the universe is expanding, they believed that the rate of expansion was constant. In 1998, Saul Perlmutter (US), leader of the Supernova Cosmology Project, along with Gerson Goldhaber (Germany/US), Rich Muller and Carl Pennypacker (US), and Brian P. Schmidt and Adam G. Riess (US), heads of the High-Z Supernova Search Team, discovered that the expansion of the universe is accelerating, through observations of Type Ia supernovae. This finding has since been corroborated by: the cosmic microwave background radiation, the large scale structure of the universe, the size of baryon acoustic oscillations, measurements of the age of the universe, X-ray properties of galaxy clusters and Hubble constant data. Most models proposing explanations of the accelerating expansion invoke dark energy.
Discussions began in the US in 1984 to sequence the entire human genome. The Human Genome Project began in 1986 through the National Institutes of Health and the Department of Energy, but the actual project did not begin until 1990. Researchers from all over the world determined the genetic sequencing of human DNA from approximately 270 individuals. A first draft was announced in 2000, while the project was declared finished in 2003. Craig Venter (US) and Celera Genomics ran a parallel sequencing project. In 2001, Ventner (US) and Francis Collins (US) of the Human Genome Project, jointly published their decoding of the human genome. Celera used a rapid sequencing progress that used the automatic sequencer ABI PRISM 3700 DNA Analyzer developed by Michael Hunkapiller. Celera’s assembling of the fragments of the genome into a complete sequence depended on computer programs developed by Phillip Green.
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435665a4bb7903f7 | Credits: 4 Contact Lucture Hours: 72
Unit 1: Structural Aspects and Bonding (18 hours)
Classification of complexes based on coordination numbers and possible geometries. Sigma and pi bonding ligands such as CO, NO, C, R3P, and Ar3P. Stability of complexes, thermodynamic aspects of complex formation-Irving William order of stability, chelate effect.
Splitting of d orbitals in octahedral, tetrahedral, square planar, square pyramidal and triagonal bipyramidal fields, LFSE, Dq values, Jahn Teller (JT) effect, theoretical failure of crystal field theory, evidence of covalency in the metal-ligand bond, nephelauxetic effect, ligand field theory, molecular orbital theory-M.O energy level diagrams for octahedral and tetrahedral complexes without and with π-bonding, experimental evidences for pi-bonding.
Unit 2: Spectral and Magnetic Properties of Metal Complexes (18 Hrs)
Electronic Spectra of complexes-Term symbols of dn system, Racah parameters, splitting of terms in weak and strong octahedral and tetrahedral fields. Correlation diagrams for dn and d10-n ions in octahedral and tetrahedral fields (qualitative approach), d-d transition, selection rules for electronic transition-effect of spin orbit coupling and vibronic coupling.
Interpretation of electronic spectra of complexes-Orgel diagrams, demerits of Orgel diagrams, Tanabe-Sugano diagrams, calculation of Dq, B and β (Nephelauxetic ratio) values, spectra of complexes with lower symmetries, charge transfer spectra, luminescence spectra.
Magnetic properties of complexes-paramagnetic and diamagnetic complexes, molar susceptibility, Gouy method for the determination of magnetic moment of complexes, spin only magnetic moment. Temperature dependence of magnetism-Curie’s law, Curie-Weiss law. Temperature Independent Paramagnetism (TIP), Spin state cross over, Antiferromagnetism-inter and intra molecular interaction. Anomalous magnetic moments.
Elucidating the structure of metal complexes (cobalt and nickel complexes) using electronic spectra, IR spectra and magnetic moments.
Unit 3: Kinetics and Mechanism of Reactions in Metal Complexes (18 Hrs)
Thermodynamic and kinetic stability, kinetics and mechanism of nucleophilic substitution reactions in square planar complexes, trans effect-theory and applications.
Kinetics and mechanism of octahedral substitution- water exchange, dissociative and associative mechanisms, base hydrolysis, racemization reactions, solvolytic reactions (acidic and basic).
Electron transfer reactions: outer sphere mechanism-Marcus theory, inner sphere mechanism-Taube mechanism.
Unit 4: Stereochemistry of Coordination Compounds (9 Hrs)
Geometrical and optical isomerism in octahedral complexes, resolution of optically active complexes, determination of absolute configuration of complexes by ORD and circular dichroism, stereoselectivity and conformation of chelate rings, asymmetric synthesis catalyzed by coordination compounds,
Linkage isomerism-electronic and steric factors affecting linkage isomerism. Symbiosis-hard and soft ligands, Prussian blue and related structures, Macrocycles-crown ethers.
Unit 5: Coordination Chemistry of Lanthanides and Actinides (9 Hrs)
General characteristics of lanthanides-Electronic configuration, Term symbols for lanthanide ions, Oxidation state, Lanthanide contraction. Factors that mitigate against the formation of lanthanide complexes. Electronic spectra and magnetic properties of lanthanide complexes. Lanthanide complexes as shift reagents.
General characteristics of actinides-difference between 4f and 5f orbitals, comparative account of coordination chemistry of lanthanides and actinides with special reference to electronic spectra and magnetic properties.
F.A. Cotton, G. Wilkinson, Advanced Inorganic Chemistry: A Comprehensive Text, 3rd Edn., Interscience,1972.
J.E. Huheey, E.A. Keiter, R.A. Keiter, Inorganic Chemistry Principles of Structure and Reactivity, 4th Edn., Pearson Education India, 2006.
K.F. Purcell, J.C. Kotz, Inorganic Chemistry, Holt-Saunders, 1977.
F. Basolo, R.G. Pearson, Mechanisms of Inorganic Reaction, John Wiley & Sons, 2006.
B.E. Douglas, D.H. McDaniel, J.J. Alexander, Concepts and Models of Inorganic Chemistry, 3rd Edn., Wiley-India, 2007.
R.S. Drago, Physical Methods in Chemistry, Saunders College, 1992.
B.N. Figgis, M.A. Hitchman, Ligand Field Theory and its Applications, Wiley-India, 2010.
J.D. Lee, Concise Inorganic Chemistry, 4th Edn., Wiley-India, 2008
A.B.P. Lever, Inorganic Electronic Spectroscopy, 2nd Edn., Elsevier, 1984.
S. Cotton, Lanthanide and Actinide Chemistry, John Wiley & Sons, 2007.
T. Moeller, International Review of Science: Inorganic Chemistry, Series I, Vol VII, Butterworth, 1972.
Credit: 4 Contact Lecture Hours: 72
Unit 1: Review of Organic Reaction Mechanisms (9 Hrs)
Review of organic reaction mechanisms with special reference to nucleophilic and electrophilic substitution at aliphatic carbon (SN1, SN2, SNi, SE1, SE2, addition-elimination and elimination-addition sequences), elimination (E1 and E2) and addition reactions (regioselectivity: Markovnikov’s addition-carbocation mechanism, anti-Markovnikov’s addition-radical mechanism). Elimination vs substitution.
A comprehensive study on the effect of substrate, reagent, leaving group, solvent and neighbouring group on nucleophilic substitution(SN2 and SN1) and elimination (E1 and E2) reactions.
Unit 2: Chemistry of Carbanions (9 Hrs)
Formation, structure and stability of carbanions. Reactions of carbanions: C-X bond (X = C, O, N) formations through the intermediary of carbanions. Chemistry of enolates and enamines. Kinetic and Thermodynamic enolates- lithium and boron enolates in aldol and Michael reactions, alkylation and acylation of enolates.
Nucleophilic additions to carbonyls groups. Named reactions under carbanion chemistry-mechanism of Claisen, Dieckmann, Knoevenagel, Stobbe, Darzen and acyloin condensations, Shapiro reaction and Julia elimination. Favorski rearrangement.
Ylids: chemistry of phosphorous and sulphur ylids – Wittig and related reactions, Peterson olefination.
Unit 3: Chemistry of Carbocations (9 Hrs)
Formation, structure and stability of carbocations. Classical and non-classical carbocations.
C-X bond (X = C, O, N) formations through the intermediary of carbocations. Molecular rearrangements including Wagner-Meerwein, Pinacol-pinacolone, semi-pinacol, Dienone-phenol and Benzilic acid rearrangements, Noyori annulation, Prins reaction.
C-C bond formation involving carbocations: oxymercuration, halolactonisation.
Unit 4: Carbenes, Carbenoids, Nitrenes and Arynes (9 Hrs)
Structure of carbenes (singlet and triplet), generation of carbenes, addition and insertion reactions.
Rearrangement reactions of carbenes such as Wolff rearrangement, generation and reactions of ylids by carbenoid decomposition.
Structure, generation and reactions of nitrene and related electron deficient nitrene intermediates.
Hoffmann, Curtius, Lossen, Schmidt and Beckmann rearrangement reactions.
Arynes: generation, structure, stability and reactions. Orientation effect-amination of haloarenes.
Unit 5: Radical Reactions (9 Hrs)
Generation of radical intermediates and its (a) addition to alkenes, alkynes (inter and intramolecular) for C-C bond formation – Baldwin’s rules (b) fragmentation and rearrangements-Hydroperoxide: formation, rearrangement and reactions. Autooxidation. Named reactions involving radical intermediates: Barton deoxygenation and decarboxylation, McMurry coupling.
Unit 6: Chemistry of Carbonyl Compounds (9 Hrs)
Reactions of carbonyl compounds: oxidation, reduction (Clemmensen and Wolf-Kishner), addition (addition of cyanide, ammonia, alcohol) reactions, Cannizzaro reaction, addition of Grignard reagent. Structure and reactions of α, β- unsaturated carbonyl compounds involving electrophilic and nucleophilic addition-Michael addition, Mannich reaction, Robinson annulation.
Unit 7: Concerted reactions (18 Hrs)
Classification: electrocyclic, sigmatropic, cycloaddition, chelotropic and ene reactions. Woodward Hoffmann rules-frontier orbital and orbital symmetry correlation approaches-PMO method.
Highlighting pericyclic reactions in organic synthesis such as Claisen, Cope, Wittig, Mislow-Evans and Sommelet-Hauser rearrangements. Diels-Alder and Ene reactions (with stereochemical aspects), dipolar cycloaddition(introductory).
Unimolecular pyrolytic elimnination reactions: cheletropic elimination, decomposition of cyclic azo compounds, β-eliminations involving cyclic transition states such as N-oxides, acetates and xanthates.
Problems based on the above topics.
R. Bruckner, Advanced Organic Chemistry: Reaction Mechanism, Academic Press, 2002.
F.A. Carey, R.A. Sundberg, Advanced Organic Chemistry, Part B: Reactions and Synthesis, 5th Edn., Springer, 2007.
W. Carruthers, I. Coldham, Modern Methods of Organic Synthesis, Cambridge University Press, 2005.
J. March, M.B. Smith, March’s Advanced Organic Chemistry: Reactions, Mechanisms, and Structure, 6th Edn., Wiley, 2007.
A. Fleming, Frontier Orbitals and Organic Chemical Reactions, Wiley, 1976.
S. Sankararaman, Pericyclic Reactions-A Text Book, Wiley VCH, 2005.
R.T. Morrison, R.N. Boyd, S.K. Bhatacharjee, Organic Chemistry, 7th Edn., Pearson, 2011.
J. Clayden, N. Greeves, S. Warren, P. Wothers, Organic Chemistry, Oxford University Press, 2004.
Unit 1: Approximate Methods in Quantum Mechanics (9 Hrs)
Credit: 4 Contact Lecture Hours: 72
Many-body problem and the need of approximation methods, independent particle model. Variation method, variation theorem with proof, illustration of variation theorem using the trial function x(a-x) for particle in a 1D-box and using the trial function e-ar for the hydrogen atom, variation treatment for the ground state of helium atom.
Perturbation method, time-independent perturbation method (non-degenerate case only), first order correction to energy and wave function, illustration by application to particle in a 1D-box with slanted bottom, perturbation treatment of the ground state of the helium atom. Qualitative idea of Hellmann-Feynman theorem.
Hartree Self-Consistent Field method. Spin orbitals for many electron atoms-symmetric and antisymmetric wave functions. Pauli’s exclusion principle. Slater determinants. Qualitative treatment of Hartree-Fock Self-Consistent Field (HFSCF) method. Roothan’s concept of basis functions, Slater type orbitals (STO) and Gaussian type orbitals (GTO), sketches of STO and GTO.
Unit 2: Chemical Bonding (18 Hrs)
Schrödinger equation for molecules. Born-Oppenheimer approximation. Valence Bond (VB) theory, VB theory of H2 molecule, singlet and triplet state functions (spin orbitals) of H2. Molecular Orbital (MO) theory, MO theory of H2+ ion, MO theory of H2 molecule, MO treatment of homonuclear diatomic molecules Li2, Be2, N2, O2 and F2 and hetero nuclear diatomic molecules LiH, CO, NO and HF. Bond order. Correlation diagrams, non-crossing rule. Spectroscopic term symbols for diatomic molecules. Comparison of MO and VB theories. Hybridization, quantum mechanical treatment of sp, sp2 and sp3 hybridisation. Semiempirical MO treatment of planar conjugated molecules, Hückel Molecular Orbital (HMO) theory of ethene, allyl systems, butadiene and benzene. Calculation of charge distributions, bond orders and free valency.
Unit 3: Applications of Group Theory in Chemical Bonding (9 Hrs)
Applications in chemical bonding, construction of hybrid orbitals with BF3, CH4, PCl5 as examples. Transformation properties of atomic orbitals. Symmetry adapted linear combinations (SALC) of C2v, C2h, C3, C3v and D3h point groups. MO diagram for water and ammonia.
Unit 4: Computational Chemistry (18 Hrs)
(The units 4 and 5 have been designed to expose the students to the field of computational chemistry, which has emerged as a powerful tool in chemistry capable of supplementing and complementing experimental research. The quantities which can be calculated using computational methods, how to prepare the input to get these results and the different methods that are widely used to arrive at the results are introduced here. Detailed mathematical derivations are not expected. Though computer simulations form an important part of computational chemistry, they are not covered in this syllabus.)
Introduction: computational chemistry as a tool and its scope.
Potential energy surface: stationary point, transition state or saddle point, local and global minima.
Molecular mechanics methods: force fields
bond stretching, angle bending, torsional terms, non-bonded interactions, electrostatic interactions. Mathematical expressions. Parameterisation from experiments or quantum chemistry. Important features of commonly used force fields like MM3, MMFF, AMBER and CHARMM.
Ab initio methods: A review of Hartee-Fock method. Basis set approximation. Slater and Gaussian functions. Classification of basis sets – minimal, double zeta, triple zeta, split valence, polarization and diffuse basis sets, contracted basis sets, Pople style basis sets and their nomenclature, correlation consistent basis sets.
Hartree-Fock limit. Electron correlation. Qualitative ideas on post Hartree-Fock methods variational method, basic principles of Configuration Inetraction(CI). Perturbational methodsbasic principles of Møller Plesset Perturbation Theory.
General introduction to semiempirical methods: basic principles and terminology.
Introduction to Density Functional Theory (DFT) methods: Hohenberg-Kohn theorems. Kohn-Sham orbitals. Exchange correlation functional. Local density approximation. Generalized gradient approximation. Hybrid functionals (only the basic principles and terms need to be introduced).
Model Chemistry-notation, effect on calculation time (cost).
Comparison of molecular mechanics, ab initio, semiempirical and DFT methods.
Unit 5: Computational Chemistry Calculations (9 Hrs)
Molecular geometry input-cartesian coordinates and internal coordinates, Z-matrix. Z-matrix of: single atom, diatomic molecule, non-linear triatomic molecule, linear triatomic molecule, polyatomic molecules like ammonia, methane, ethane and butane. General format of GAMESS / Firefly input file. GAMESS / Firefly key word for: basis set selection, method selection, charge, multiplicity, single point energy calculation, geometry optimization, constrained optimization and frequency calculation.
Identifying a successful GAMESS/ Firefly calculation-locating local minima and saddle points, characterizing transition states, calculation of ionization energies, Koopmans’ theorem, electron affinities and atomic charges.
Identifying HOMO and LUMO-visualization of molecular orbitals and normal modes of vibrations using suitable graphics packages.
I.N. Levine, Quantum Chemistry, 6th Edn., Pearson Education, 2009.
D.A. McQuarrie, Quantum Chemistry, University Science Books, 2008.
R.K. Prasad, Quantum Chemistry, 3rd Edn., New Age International, 2006.
F.A. Cotton, Chemical Applications of Group Theory, 3rd Edn., Wiley Eastern, 1990.
V. Ramakrishnan, M.S. Gopinathan, Group Theory in Chemistry, Vishal Publications, 1992.
A.S. Kunju, G. Krishnan, Group Theory and its Applications in Chemistry, PHI Learning, 2010
E.G. Lewars, Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics, 2
nd Edn., Springer, 2011.
J.H. Jensen, Molecular Modeling Basics, CRC Press, 2010.
F. Jensen, Introduction to computational chemistry, 2nd Edn., John Wiley & Sons, 2007.
A. Leach, Molecular Modelling: Principles and Applications, 2nd Edn., Longman, 2001.
J.P. Fackler Jr., L.R. Falvello (Eds.), Techniques in Inorganic Chemistry: Chapter 4, CRC Press, 2011.
K.I. Ramachandran, G. Deepa, K. Namboori, Computational Chemistry and Molecular Modeling: Principles and Applications, Springer, 2008.
A. Hinchliffe, Molecular Modelling for Beginners, 2nd Edn., John Wiley & Sons, 2008.
C.J. Cramer, Essentials of Computational Chemistry: Theories and Models, 2nd Edn., John Wiley & Sons, 2004.
D.C. Young, Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems, John Wiley & Sons, 2001.
Molecular Mechanics:
Ab initio, semiempirical and dft:
Firefly / PC GAMESS
available from
WINGAMESS available from Graphical User Interface (GUI):
Gabedit available from
2. wxMacMolPlt available from
Avogadro from
Credit: 3 Contact Lecture Hours: 54
Unit 1: Foundations of Spectroscopic Techniques (9 Hrs)
Origin of spectra: origin of different spectra and the regions of the electromagnetic spectrum, intensity of absorption, influencing factors, signal to noise ratio, natural line width, contributing factors, Doppler broadening, Lamb dip spectrum, Born Oppenheimer approximation, energy dissipation from excited states (radiative and non radiative processes), relaxation time.
Microwave spectroscopy: principal moments of inertia and classification (linear, symmetric tops, spherical tops and asymmetric tops), selection rules, intensity of rotational lines, relative population of energy levels, derivation of Jmax , effect of isotopic substitution, calculation of intermolecular distance, spectrum of non rigid rotors, rotational spectra of polyatomic molecules, linear and symmetric top molecules, Stark effect and its application, nuclear spin and electron spin interaction, chemical analysis by microwave spectroscopy.
Infrared spectroscopy: Morse potential energy diagram, fundamentals, overtones and hot bands, determination of force constants, diatomic vibrating rotator, break down of the Born-Oppenheimer approximation, effect of nuclear spin, vibrational spectra of polyatomic molecules, normal modes of vibrations, combination and difference bands, Fermi resonance, finger print region and group vibrations, effect of H-bonding on group frequency, disadvantages of dispersive IR, introduction to FT spectroscopy, FTIR.
Raman spectroscopy: scattering of light, polarizability and classical theory of Raman spectrum, rotational and vibrational Raman spectrum, complementarities of Raman and IR spectra, mutual exclusion principle, polarized and depolarized Raman lines, resonance Raman scattering and resonance fluorescence.
Electronic spectroscopy: term symbols of diatomic molecules, electronic spectra of diatomic molecules, selection rules, vibrational coarse structure and rotational fine structure of electronic spectrum, Franck-Condon principle, predissociation, calculation of heat of dissociation, Birge and Sponer method, electronic spectra of polyatomic molecules, spectra of transitions localized in a bond or group, free electron model, different types of lasers-solid state lasers, continuous wave lasers, gas lasers and chemical laser, frequency doubling, applications of lasers, introduction to UV and X-ray photoelectron spectroscopy.
Unit 2: Resonance Spectroscopy (27 Hrs)
NMR spectroscopy : interaction between nuclear spin and applied magnetic field, nuclear energy levels, population of energy levels, Larmor precession, relaxation methods, chemical shift, representation, examples of AB, AX and AMX types, exchange phenomenon, factors influencing coupling, Karplus relationship.
FTNMR, second order effects on spectra, spin systems (AB, AB2), simplification of second order spectra, chemical shift reagents, high field NMR, double irradiation, selective decoupling, double resonance, NOE effect, two dimensional NMR, COSY and HETCOR, 13C NMR, natural abundance, sensitivity, 13C chemical shift and structure correlation, introduction to solid state NMR, magic angle spinning.
EPR spectroscopy: electron spin in molecules, interaction with magnetic field, g factor, factors affecting g values, determination of g values (g and g), fine structure and hyperfine structure, Kramers’ degeneracy, McConnell equation.
An elementary study of NQR spectroscopy.
Mossbauer spectroscopy: principle, Doppler effect, recording of spectrum, chemical shift, factors determining chemical shift, application to metal complexes, MB spectra of Fe(II) and Fe(III) cyanides.
C.N. Banwell, E.M. McCash, Fundamentals of Molecular Spectroscopy, 4th Edn., Tata McGraw Hill, 1994.
G. Aruldhas, Molecular Structure and Spectroscopy, Prentice Hall of India, 2001.
P.W. Atkins, Physical Chemistry, ELBS,1994
R.S. Drago, Physical Methods in Inorganic Chemistry, Van Nonstrand Reinhold, 1965.
K.J. Laidler, J.H. Meiser, Physical Chemistry, 2nd Edn., CBS, 1999.
W. Kemp, NMR in chemistry-A Multinuclear Introduction, McMillan, 1986.H. Kaur, Spectroscopy, 6 th Edn., Pragati Prakashan, 2011.
H. Gunther, NMR Spectroscopy, Wiley, 1995.
D.A. McQuarrie, J.D. Simon, Physical Chemistry: A Molecular Approach, University Science Books, 1997.
D.N. Sathyanarayan, Electronic Absorption Spectroscopy and Related Techniques, Universities Press, 2001.
D.N. Sathyanarayana, Vibrational Spectroscopy: Theory and Applications, New Age International, 2007
D.N. Sathyanarayana, Introduction To Magnetic Resonance Spectroscopy ESR,NMR,NQR, IK International, 2009.
Credit: 3 Contact Lab Hours: 54+54=108
Separation and identification of two less familiar metal ions such as Tl, W, Se, Mo, Ce, Th, Ti, Zr, V, U and Li. Anions which need elimination not to be given. Minimum eight mixtures to be given.
Colorimetric estimation of Fe, Cu, Ni, Mn, Cr, NH4+, nitrate and phosphate ions.
Preparation and characterization complexes using IR, NMR and electronic spectra.
Tris (thiourea)copper(I) complex
Potassium tris (oxalate) aluminate (III).
Hexammine cobalt (III) chloride.
Tetrammine copper (II) sulphate.
Schiff base complexes of various divalent metal ions.
A.I. Vogel, G. Svehla, Vogel’s Qualitative Inorganic Analysis, 7th Edn., Longman,1996.
A.I. Vogel, A Text Book of Quantitative Inorganic Analysis, Longman, 1966.
I.M. Koltoff, E.B. Sandell, Text Book of Quantitative Inorganic analysis, 3rd Edn., McMillian, 1968.
V.V. Ramanujam, Inorganic Semimicro Qualitative Analysis, The National Pub.Co., 1974.
Credit: 3 Contact Lab Hours: 54+54=108
General methods of separation and purification of organic compounds such as:
Solvent extraction
Soxhlet extraction
Fractional crystallization
TLC and Paper Chromatography
Column Chromatography
Membrane Dialysis
Separation of Organic binary mixtures by chemical/solvent separation methods
Separation of organic mixtures by TLC
Separation/ purification of organic mixtures by column chromatography
Drawing the structures of organic molecules and reaction schemes by ChemDraw, Symyx Draw and Chemsketch. Draw the structures and generate the IR and NMR spectra of the substrates and products in the following reactions:
Cycloaddition of diene and dienophile (Diels-Alder reaction)
Oxidation of primary alcohol to aldehyde and then to acid
Benzoin condensation
Esterification of simple carboxylic acids
Aldol condensation
A.I. Vogel, A Textbook of Practical Organic Chemistry, Longman, 1974.
A.I. Vogel, Elementary Practical Organic Chemistry, Longman, 1958.
F.G. Mann, B.C Saunders, Practical Organic Chemistry, 4th Edn., Pearson Education India, 2009.
R. Adams, J.R. Johnson, J.F. Wilcox, Laboratory Experiments in Organic Chemistry, Macmillan, 1979.
Credit: 3 Contact Lab Hours: 72+72 =144
(One question each from both parts A and B will be asked for the examination)
Part A
Verification of Freundlich and Langmuir adsorption isotherm: charcoal-acetic acid or charcoal-oxalic acid system.
Determination of the concentration of the given acid using the isotherms.
Phase diagrams
Construction of phase diagrams of simple eutectics.
Construction of phase diagram of compounds with congruent melting point: diphenyl amine-benzophenone system.
Effect of (KCl/succinic acid) on miscibility temperature.
Construction of phase diagrams of three component systems with one pair of partially miscible liquids.
Distribution law
Distribution coefficient of iodine between an organic solvent and water.
Distribution coefficient of benzoic acid between benzene and water.
Determination of the equilibrium constant of the reaction KI + I2 ↔ KI3
IV. Surface tension
Determination of the surface tension of a liquid by
Capillary rise method
Drop number method
Drop weight method
Determination of parachor values.
Determination of the composition of two liquids by surface tension measurements
Part B
Computational chemistry experiments
Experiments illustrating the capabilities of modern open source/free computational chemistry packages in computing single point energy, geometry optimization, vibrational frequencies, population analysis, conformational studies, IR and Raman spectra, transition state search, molecular orbitals, dipole moments etc.
Geometry input using Z-matrix for simple systems, obtaining Cartesian coordinates from structure drawing programs like Chemsketch.
J.B. Yadav, Advanced Practical Physical Chemistry, Goel Publishing House, 2001.
G.W. Garland, J.W. Nibler, D.P. Shoemaker, Experiments in Physical Chemistry, 8th Edn., McGraw Hill, 2009.
GAMESS documentation available from: |
80a18874c3014c62 | All-optical attoclock for imaging tunnelling wavepackets
All-optical attoclock for imaging tunnelling wavepackets
Photo-electron versus all-optical attoclock. (a) The photo-electron attoclock. The angle-resolved photo-electron spectrum generated by the driving field (red line) reveals attosecond delays and deflections of the electronic wavepacket shifting its maximum by ϕτ, interpreted as an effective delay τ = ϕτ/ω0. b, Probing of this delay by, instead of electrons, the zeroth-order Brunel radiation in a two-colour field (magenta line). Such harmonic generated in gas or solid (blue box) can be selected by a low-pass filter (brown box). It is nearly linearly polarized (brown line) and rotated, having the same effective τ as in a. The inset shows schematically the harmonic response spectrum, including the pump (blue) and harmonic of interest (red). c, Polarization state of the zeroth harmonic for the equivalent intensity ϵ0c|Emax|2/2=150 TW cm−2, obtained from TDSE simulations with Yukawa potential, Coulomb potential and the simple-man classical Drude model. d, Effective delay τ as a function of intensity obtained from the polarization rotation of the zeroth harmonic as determined by TDSE simulations and compared with the results of the photo-electron attoclock: the centre-of-mass position of the electronic wavepacket (TDSE-CM), as well as the simulations (where the plane z = 0 instead of centre of masses was analysed) and the experiment. Credit: Nature Physics, https://doi.org/10.1038/s41567-022-01505-2
Physicists can study the possible time delays of light-induced tunneling of an electron from an atom after conducting measurements of time delays when cold atoms tunnel through an optically created potential barrier. In a new report now published in Nature Physics, Ihar Babushkin and a research team in Germany, complemented photo-electron detection in laser-induced tunneling by measuring light emitted by the tunneling electron, known as Brunel radiation. Based on combined single and two-color driving fields, they identified all-optical signatures of reshaping tunneling wave-packets as they emerged from the tunneling barrier and moved away from the core. This reshaping led to an effective time-delay and time-reversal symmetry of the ionization process, described in theory, for experimental observation. The all-optical detection method can facilitate time-resolved measurements of optical-tunneling in condensed matter systems at the attosecond time-scale.
Attosecond science
Attosecond science is a revolutionary technology, which combines optical and collision science to greatly extend the reach of each. The possibility of tunneling an electron through the potential barrier created by an oscillating and the binding potential of the core is a fundamental resource in attosecond science. The phenomenon is at the heart of high harmonic generation and high harmonic spectroscopy. High harmonic generation is associated with radiative recombination based on the return of the laser-driven electron to the parent ion. But even when the electron does not return to the core, the setup emitted high harmonic radiation, referred to as the Brunel radiation or Brunel harmonics. The process is associated with bursts of current triggered by laser-induced tunneling, ubiquitous in atoms, molecules and solids. In this work, Babushkin et al. showed how Brunel harmonics generated in elliptically polarized single- and two-color laser fields provided a detailed picture of light induced tunneling of an electron. The described approach to imaging ionization dynamics distinctly differed from existing attoclock approaches based on photo-electron detection. The method allowed the introduction of a complementary, all-optical measurement protocol to establish extended measurements of tunneling dynamics in bulk solids.
All-optical attoclock for imaging tunnelling wavepackets
Extracting photo-ionization information from higher-order Brunel harmonics for the hydrogen atom. (a) A single-colour elliptically polarized pump (red line) produces the third Brunel harmonic (magenta line), with the polarization state (ellipticity and polarization direction) encoding the sub-cycle dynamics of the ionization process. Inset shows schematically the harmonic response spectrum, including the pump (blue) and harmonic of interest (red). (b) Polarization of the third harmonic for the single-colour driving pulse (|Emax|=36.3 GV m−1, ϵ = 0.6, I = 280 TW cm−2) calculated using TDSE with Coulomb and Yukawa potentials as well as simple-man Drude model (denotations as in Fig. 1c). Inset shows the attoclock mapping M and its inverse. (c) Optical reconstruction of the ionization dynamics (Wopt(t), solid red line) compared with the reconstruction from the photo-electron spectrum (We(t), dot-dashed red line) for the Coulomb potential. The estimation of error is given in Methods. Optical reconstruction for the Yukawa potential (WY(t), blue line) is also presented. Dot-dashed black line shows the electron wavepacket distribution P(ϕ). The attoclock delay ω0τ given by the position of the center of mass (CM) of We(t) ≈ Wopt(t) is also indicated (vertical dotted red line). Dashed grey line shows the field Ex for reference. d, The positions of the maxima of the electronic spectra P(ϕ) (black markers), their CM (blue markers) and the effective delays τ=M(ϕ) reconstructed from the photo-electron spectra (red markers) for different full-width at half-maximum pulse durations and ellipticities ϵ of the driving field. Horizontal red line shows the attoclock delay extracted optically using the two-colour configuration for the corresponding peak intensity. Asymmetry of the wavepacket reveals itself from the different positions of the maximum and CM of the photo-electron spectra (traced by blue and red dotted arrows for an exemplary case of ϵ = 0.5). Credit: Nature Physics, https://doi.org/10.1038/s41567-022-01505-2
Physical principle and theoretical analysis
The scientists validated the central idea behind the all-optical attoclock by determining vectorial properties of the emitted light, determined by the vectorial properties of the current generated by the tunneling electron to reflect the tunneling dynamics. The team considered two field arrangements, in the first they combined an intense circularly polarized infra-red pump with its co-rotating second harmonic to generate a total electric field with a reference direction for the optical attoclock. In the second arrangement the reference direction was provided by the major axis of the single-color elliptically polarized driving field. The team began with the first arrangement where the nonlinear response contained even and odd harmonics; with a signal dominated by Brunel radiation. For instance, the team injected a classical free electron by strong-field ionization into the atomic continuum with some velocity to accelerate in the laser field and potential of the core. Babushkin et al. verified the outcomes using the ab initio time-dependent Schrödinger equation (TDSE) simulations to compute the radiated field.
All-optical attoclock for imaging tunnelling wavepackets
Experimental reconstruction of sub-cycle ionization dynamics in Helium (He) compared with theoretical simulations. Credit: Nature Physics, https://doi.org/10.1038/s41567-022-01505-2
Imaging ionization dynamics and outlook
During the experiments, the team confirmed the predicted rotation of the polarization ellipse of the nonlinear response using experimental measurements with the setup. Babushkin et al. accomplished this using an 800-nm, 43-femtosecond-long, elliptically polarized pump pulse focused into a plasma spot for third harmonic generation to carefully separate and detect polarization components. The scientists compared the experimentally measured intensity-dependent parameters of the polarization ellipse with TDSE (time-dependent Schrödinger equation) simulation results to show good agreement between the experiment and simulation.
All-optical attoclock for imaging tunnelling wavepackets
Experimental setup and investigation of errors. Experimental setup for the investigation of the polarization rotation of the third harmonic in Helium. The λ/4 plate is an achromatic plate extending from 600 to 1200 nm. The used UV polarizing beam splitter is an α-BBO Glan-Laser polarizer (ThorLabs GLB10-UV). The SiC UV photodiode is highly insensitive to other radiation frequencies than UV. The chamber was typically filled with He at 1.3 Bar pressure. All three detectors were connected to boxcar integrators triggered by the output of the regenerative amplifier. Credit: Nature Physics, https://doi.org/10.1038/s41567-022-01505-2
In this way, Ihar Babushkin and colleagues established a firm quantitative link between photo-electron spectra in strong-field ionization. They measured Brunel radiation generated by electrons on their way to the continuum to reveal the reshaping of electron wave packets during laser-induced tunneling. Based on Brunel harmonics imaging, the team reshaped mapping onto effective ionization delays, where Brunel harmonics in the terahertz and ultraviolet regions contained signatures of attosecond and sub-angstrom-scale electron dynamics. The researchers credited the origin of ionization asymmetry to the dynamics of the electron wave packet during and after tunneling for high intensities or saturation effects. The study provides promising capability to image and explore attosecond-scale wave packet reshaping in systems where photo-electron detection wasn't readily available. Such systems include bulk solids, where the detection of light is much simpler compared to the detection of electrons. Babushkin et al. expect the Brunel harmonics of yet higher order to allow the resolution of electron dynamics even closer to the core. The outcome will have impact beyond physics, to influence chemistry, biology and future technologies.
Explore further
Decoding electron dynamics
More information: Ihar Babushkin et al, All-optical attoclock for imaging tunnelling wavepackets, Nature Physics (2022). DOI: 10.1038/s41567-022-01505-2
R. E. F. Silva et al, Topological strong-field physics on sub-laser-cycle timescale, Nature Photonics (2019). DOI: 10.1038/s41566-019-0516-1
Journal information: Nature Physics , Nature Photonics
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Citation: All-optical attoclock for imaging tunnelling wavepackets (2022, March 7) retrieved 30 June 2022 from https://phys.org/news/2022-03-all-optical-attoclock-imaging-tunnelling-wavepackets.html
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The Green Zone: Confidence
A key clue that the observable patch of the universe was once hot and dense, and cooling as it expanded — i.e. that there was a Hot Big Bang period (preceded perhaps by cosmic inflation, but we don’t need to focus on that yet) — is the presence in the universe of the cosmic microwave background [CMB]. From all parts of the sky, an easily detectable sea of electromagnetic radiation in the form of microwaves (photons with energy about 1000 times less than photons of visible light) is streaming across the universe, and onto the Earth. Not only is the CMB a broad clue as to the universe’s past, it is also a key tool for scientists to use in finding even more precise and focused clues. Much of the history of the observable patch is imprinted in the details of the CMB.
A prediction of the Hot Big Bang is that the microwaves are just a sign of past heat, and of nothing else. Specifically, if you measure the microwave photons’ energy, and plot the amount of energy coming from a patch of sky due to photons of a particular frequency, you should see the same plot (a “black-body spectrum”, as it is called) as you would see for photons emitted by a cold piece of ice or rock or metal. [Yes, ice at 2.7 degrees C above absolute zero does glow in microwaves.] The fact that the COBE satellite’s data agrees closely with the prediction of a black-body spectrum (shown in the figure) is overwhelming evidence that there was a Hot Big Bang going on at the time that atoms first formed, when the observable patch of the universe became transparent to light — a time now believed to be about 380,000 years after the Hot Big Bang began.
Amount of energy in CMB photons as a function of the photons' frequency. Data from the COBE satellite is the red crosses; the green curve is the prediction of a black-body spectrum.
Amount of energy in CMB photons as a function of the photons’ frequency. Data from the COBE satellite is the red crosses; the green curve is the prediction of a black-body spectrum.
The most powerful evidence that a simple view of the Hot Big Bang is correct even earlier is the success of the theory (i.e. equations) for nucleosynthesis: the formation of atomic nuclei for the lightest atoms during the Hot Big Bang. (Note the atoms themselves formed hundreds of thousands of years later, when the temperature was cool enough for the nuclei to capture electrons.) The equations of the Hot Big Bang include
These equations can be used to calculate how abundant certain light atoms should be, if there was a Hot Big Bang. Specifically, one can compute the ratios of how much hydrogen (normal and heavy [“deuterium”]), helium (normal [“helium-4”] and light [“helium-3”]), and lithium should be present in pristine parts of the observable patch, if the atomic nuclei of those elements were mainly forged during the cooling stages of the Hot Big Bang. If the elements other than hydrogen were only formed in stars, then there should be very little helium and virtually no deuterium in the observable patch of the universe. (Deuterium is actually destroyed in stars.) But in a cooling Hot Big Bang, a lot of helium — about 1/4 of the amount of hydrogen — and a small but measurable amount of deuterium should be produced.
These predictions of the Hot Big Bang depend in great detail on our understanding of particle physics and gravity — of the weak nuclear force, the strong nuclear force, the electromagnetic force, the gravitational force (in Einstein’s version of it) and on the properties of neutrinos, electrons, photons, protons and neutrons. And the numbers work! The amount of helium, relative to ordinary hydrogen, is predicted to be in the 20 – 25% range; deuterium is predicted to be in the range of one part in 10,000 to 100,000, helium-3 a factor of 10 smaller than that, and lithium in the part per 10 billion range. The data agree!
Data showing the prediction for various element abundances (yellow band), data (colored horizontal bands), and predictions (solid colored curves). The fact that the colored curves, colored horizontal bands and the vertical yellow band all meet -- except lithium, which is slightly off but not too far --- indicates that the Hot Big Bang occurred. That Lithium is a bit low has been studied extensively and might mean that a bit of particle physics is missing from the theory. From http://www.einstein-online.info/spotlights/BBN/ ; Adapted from an image by E. Vangioni, Institut d'Astrophysique de Paris]
Data showing data (colored horizontal bands) and predictions from the Hot Big Bang (solid colored curves) for the abundances of helium-4, deuterium, helium-3 and lithium relative to hydrogen. The quantity on the horizontal axis is essentially the amount of ordinary matter in the observable patch of our universe. The fact that the vertical yellow band meets the colored curves and the corresponding colored horizontal bands — except lithium, where the data is slightly too low — strongly indicates that the Hot Big Bang really occurred. (That Lithium is a bit low has been studied extensively and might mean that a bit of particle physics is missing from the theory; or it might be a problem with the actual measurement of Lithium abundance.) From http://www.einstein-online.info/spotlights/BBN/ ; Adapted from an 2007 image by E. Vangioni, Institut d’Astrophysique de Paris.
That these predictions agree with observations of the distant cosmos (see the figure below) provides a strong argument that the Hot Big Bang really occurred, and was originally hot enough to break apart nuclei. That means that temperatures were initially at least hot enough that the typical particle had an energy of about 0.001 GeV, which would have happened a few minutes after the Hot Big Bang began. More precise measurements even help us understand how much ordinary matter there is in the observable patch of the universe, with results that agree with other measurements of the same quantity.
For this reason, I’ve marked the nucleosynthesis period and what follows it in green, to emphasize that we have strong observational evidence that the Hot Big Bang did occur, and was so hot — hot enough to destroy atomic nuclei — that the atomic nuclei around us today had to form from scratch as the temperature cooled.
The Yellow Zone: Extrapolating a Bit Using Particle Physics
However, for times before nucleosynthesis, the arguments are weaker. We don’t have much direct evidence for what was taking place. But we have a lot of knowledge about particle physics, up to energies of a few hundred GeV, thanks to the Large Hadron Collider [LHC] and its predecessors over the last few decades. So if we assume that the Hot Big Bang started at temperatures so high that a typical particle would have had an energy of a few hundred GeV, we can calculate the observable patch’s properties from that temperature downward. At such a high temperature,
As the temperature cooled, we can calculate (now that we know the Higgs particle’s mass, and if we assume there aren’t any lightweight particles that we don’t know about) that first the Higgs field would have turned “on”, making the weak nuclear force henceforth weak; and then, a bit later, the strong nuclear force would have become very strong, so that quarks and gluons and anti-quarks would have been trapped henceforth and forever into protons, neutrons, anti-protons and anti-neutrons, as well as other very short-lived hadrons.
However, we can’t yet directly confirm by cosmological observation that this is true. It’s not entirely impossible that there was something odd and unknown about the universe that makes some of these conclusions premature. Someday we may be able to be more certain, but not yet. So in this time period, starting just before the Higgs field turns on and going until nucleosynthesis, we have a highly educated guess as to what was happening, though no direct experimental checks. That’s why I’ve marked this time period in yellow.
The Orange Zone: What Started the Hot Big Bang?
Even earlier? Well, at even higher temperatures we don’t know the particle physics with confidence; we’d need particle accelerators more powerful than the LHC. To figure out what happened in the Big Bang in detail, we have to make assumptions about the particles available, without any way to check them. But still, we are not without information, because we do have an experimental probe: the CMB itself.
It is a remarkable fact that the CMB is incredibly uniform, to one part in 100,000. That’s already information, and explaining why it’s true is one of the arguments in favor of cosmic inflation.
But there’s much, much more information in the imperfections in that uniformity. The non-uniformities, which were discovered by the COBE satellite, have been studied in detail by the WMAP and Planck satellites, along with many other experiments from the ground. And it was in studying the non-uniformities in the CMB and their tiny bit of polarization that BICEP2 made the big (but unconfirmed now discredited) discovery that was announced last week (here’s some background information on that discovery, in the form of FAQs, that you may find useful).
The photons in the CMB are astonishingly sensitive. There are very few things that affect them, and fortunately, most of the things that do are relics from the extremely early periods of the observable patch of the universe. It’s an unexpected gift for scientists that this is true! The CMB photons were created about 380,000 years after the Hot Big Bang, and yet they give us insight into the entire period before that, all the way back to a period just before the Hot Big Bang!!! (They also can give us important insights into what happened after 380,000 years, too.) To me, this incredibly sensitivity of the CMB is one of the most amazing natural phenomena in all of science… and it has been the key to progress in cosmology in recent years.
Detailed measurements of the CMB have increasingly given us insight into the zone marked in orange, where we don’t have any particle physics data — just guesses, and calculations based on those guesses. The reason insights have been possible is that our calculations rely on very general properties of the Hot Big Bang and the pre-Hot Big Bang period to make predictions. What I mean by this is that these calculations begin with very simple assumptions about the patterns of non-uniformities that were present at the start of the Hot Big Bang, and then we check if these assumptions give predictions for non-uniformities in the CMB that agree with data. They do! This is shown in the figure below, where predictions of the Hot Big Bang theory, with these simple assumptions about the starting point, are compared with data from the Planck satellite. (Similar but less precise measurements were made before Planck, by WMAP and a number of ground-based experiments.)
Data (red dots) from the Planck satellite, showing the average size of non-uniformities on different angular scales on the sky. The solid curve is the prediction of the current standard cosmological model, which assumes a very simple type of non-uniformity, of the sort that cosmic inflation would produce.
Data (red dots) from the Planck satellite, showing the average size of non-uniformities on different angular scales on the sky. The solid curve is the prediction of the current standard cosmological model, which assumes a very simple type of non-uniformity, of the sort that cosmic inflation would produce. The agreement is remarkable, and strongly supports the notion of a Hot Big Bang, but by itself does not confirm there was a period of inflation before the Hot Big Bang.
Thus, simple assumptions about the non-uniformities at the start of the Hot Big Bang seem to be right, more or less. This lends support to the hypothesis of cosmic inflation, which can produce simple non-uniformities. But it’s not convincing, because one can imagine simple non-uniformities arising in some other way. There is at least one alternative — the ekpyrotic universe — that is also largely consistent with this data, and there might have been others that no one has thought of. Meanwhile, there were many qualitatively different forms that cosmic inflation might take, some of which could have been consistent with universes quite different from those with a simple Big Bang. For example, the data has been consistent with universes that, at temperatures a bit higher than reached in the yellow zone in the figure, have more spatial dimensions than the three we’re used to. So the number of possible options in the orange zone has been very large, and the degree of speculation involved has been very high.
Data from BICEP2, black dots, is compared with the prediction from a model of cosmic inflation (upper dashed line). The prediction is the sum of the solid line (an effect from gravitational lensing in the last few billion years) and the lower dashed line (the effect of gravitational waves from inflation with a very large amount of dark energy.) Had inflation not occurred, or had the amount of dark energy during inflation been small, the dots would have followed the solid "lensing" curve instead of the upper dashed line. See this post for more comments.
Data from BICEP2, black dots, is compared with the prediction from a model of cosmic inflation (upper dashed line). The prediction is the sum of the solid line (an effect from gravitational lensing in the last few billion years) and the lower dashed line (the effect of gravitational waves from inflation, if inflation involved a very large amount of dark energy.) Had inflation not occurred, or had the amount of dark energy during inflation been small, the dots would have followed the solid “lensing” curve instead of the upper dashed line. An alternative interpretation, that this is an effect of galactic dust, is supported by 2015 data.
This may have changed with BICEP2s new measurement, which gives unfortunately did not give powerful new evidence for cosmic inflation through inflation’s generation of gravitational waves… assuming the measurement itself, and the interpretation of the measurement, both hold up over time. [The interpretation did not.] Since a measurable amount of gravitational waves can only be produced, as far as we know, if a period of inflation occurred and was driven by a very large amount of “dark `energy’ “, BICEP2’s result may rule out all known alternatives to inflation, and significantly narrow down the options for how inflation may have occurred, excluding many of the more radical options as well as many simple ones. As such, it may increase confidence in the inflation hypothesis to change the color of the orange zone to yellow, or even, as more measurements come in and become more precise, to green. [But it did not.]
However, let’s not jump to conclusions until the measurement is confirmed with more data and by other observing teams. [You see how important it is that I write these cautionary caveats in my articles. I have seen many measurements come and go; the fraction that stick is much less than half.]
The Red Zone: Scientific Guesswork
But BICEP2 can really only tell us about the late stage and exit from inflation. If we go deep into the inflationary period and try to go earlier, into the red zone in the figure, we face two problems:
• we don’t have any experimental probes of this early time, and
• we don’t know which particle physics and/or gravity equations we should be using.
We do not yet have any cosmological observations that access this time period; the CMB photons bring us back to the late stage of inflation, but not further. We cannot infer much yet from CMB measurements about the “inflaton” field (and its particle) that is supposed to play the dominant particle-physics role during inflation, and we have no idea what other types of fields it may interact with, so it is hard to be sure what type of process got inflation started. Meanwhile, theoretical physics doesn’t help much either. We can try assuming that string theory is the true theory of quantum gravity in our world, but if we do that, we still find the equations are hard to solve and give, at best, a variety of possibilities. And trying to work in greater generality, without assuming string theory, doesn’t help; the calculations are still ambiguous and difficult. This doesn’t stop theoretical physicists from making educated, scientific guesses — i.e., speculations backed with concrete equations and calculations. But until we get some data that allows us to distinguish the many possibilities, we can’t say for sure what was actually happening back then.
So if people tells you that the universe started in such and such a way, perhaps “with a singularity” or “with a quantum fluctuation out of nothing” or “in the Big Crunch (i.e. the collapse) of a previous phase of the universe”, remember that they’re telling you about the red zone. They’re neglecting to tell you that what they’re saying is pure theory, with neither an experiment to back it up nor a clear theoretical reason to believe their suggestion is unique and preferable over someone else’s alternative. Only a bit later in cosmic history, once we focus on the late stages of inflation, and forward in time from there to nucleosynthesis, do we have both data (cosmological observation and particle physics collisions) and reasonably reliable theory (Einstein’s theory of gravity plus the Standard Model of particle physics). Our confidence grows as time moves forward, the observable patch of the universe cools, and physics becomes of a sort that we’ve tested in numerous experiments already.
From this, I hope that you can see that the Big Bang Theory really isn’t a single, undifferentiated structure. The most reliable part of the theory is that there was, at one point, a Hot stage of the Big Bang. (Some people call that the Big Bang, in fact.) If BICEP2’s measurement is accurate, and correctly interpreted, then the reliable portion of the theory may then include a period of inflation that preceded the Hot Big Bang. (Some people would call “inflation plus the Hot Big Bang” the “Big Bang”.) But anything before inflation is not in the least reliable… in particular, the notion that the universe’s heat and density increased to the extent that Einstein’s equations have a singularity, presumably indicating that they’re not sufficient to describe what was going on, is an assumption. So that part of the Big Bang Theory may not survive the test of time. You should be prepared, as scientists are, to let it go. After all, from 1960-1980 people had a vision of the early Big Bang that didn’t include something like inflation; their version was significantly adjusted as knowledge was gained, so why shouldn’t ours be?
Moreover, it is quite possible that the start of the universe, which people often refer to as the ultimate Big Bang, may not have had anything “Bang-ish” or even Big about it. Yes, we might even need to let go someday of the idea that the universe truly began with a Big Bang. The scientific community won’t have any problem with this, because many of us have never taken that idea very seriously in the first place. But what about the rest of our society, which seems to hold to this name and notion far harder than do scientists themselves, to the point of (egad!) naming a television show after it?! The loss may be more shocking than necessary…
255 responses to “Which Parts of the Big Bang Theory are Reliable, and Why?
1. Matt, what is more complicated – description of the Universe or making a weather forecast?
• How precise do you want either of those? It is a simple matter to tell you that next winter will be colder than summer on average. And I can say ‘The universe has stars in it’ pretty easily too. But it’s impossible to describe the position of all the atoms in a single cloud or nebula.
2. IMO when one takes into account; that law of thermodynamics that energy only changes form and is neither destroyed nor created and that absolute zero is a tempature never obtained, with statistics and probabilities, combined with extrapolation of a space/time graph clearly shows beyond a doubt, there has been an infinite number of Big Bang Cycles, and will be an infinite number of Big Bang Cycles.
• What would reverse the accelerating expansion of the visible universe?
• What would need to happen is the ‘dark energy’ would need to ‘reverse’ that is the current negative pressure that is driving expansion would have to become zero or positive. At present we cannot tell whether this will happen naturally, will never happen or will get worse (the ‘big rip’ scenario.)
• With your terms, that negative pressure is somehow getting bigger (accelerating expansion), how’s that possible? And in the context of your future answer how things will change in future so that the phenomenon gets reversed?
• The questions you ask are deep indeed and we still do not have good answers to them. In regards to your second question the only answer we have is ‘somehow it has to change’ Since we don’t know what dark energy is we don’t know what it can do or how.
The second question is likewise unanswered; we know that after inflation the universe’s expansion slowed due to gravity, but at some time around 7 billion years ago it started to accelerate again. it’s important to remember too that things get ‘worse’ even if the negative pressure remains the same.
The reason is that dark energy is a property of space and makes more space. That is, it makes more of itself. Take a small universe with matter and radiation in equal quantities and 1/10th as much dark energy. Each then has some ‘energy density’ the amount of stuff diided by the volume it occupies. This controls how much affect it has on space. After some time the universe’s volume will double. The matter’s energy density will halve (divide by 2x volume), the radiation’s will be 1/16th (Divide by 2x volume, *plus* its frequency is stretched double which gives it 1/8th the energy.) but the dark energy’s density will be the same. (So there’s now twice as much of it.) It now has more effect than the radiation and after a few more doublings, it will out-effect the matter.
• Richard Bauman
Are you saying points in space double, and then double again ?
• During inflation space ‘makes more of itself’ So any two objects in that space continually find more space between them. And the more space there is between them the more ‘new’ space appears. The end result of this is that the two objects see themselves as moving apart faster and faster. At the time of inflation the universe did indeed double in size repeatedly.
• What sort of cosmological experiment would allow you to confirm some of the yellow region? Is there anything?
• Dang it, that was supposed to be a top-level question. Not sure how it wound up as a reply to you. Sorry.
• We have most of what we (currently) think we can get relating to the yellow region via astrological observations, most frontiers lie with the LHC and similar setups. But as BICEP2 has shown astronomy still has a big role to play regarding earlier eras (And in fact alter eras such as determining exactly when the expansion of the universe began to accelerate.)
3. Matt said
I am writing to suggest that perhaps “pure theory” should be referrred to as conjecture or hypothesis since “theory” has a specific meaning in theoretical physics and science in general. Just a thought.
4. Thank you for this excellent clear presentation on the cosmic origin in this chaotic sea of speculative bubbles spun out by charlatans and confused physicists.
Would you point out or tell us the region we might be able to probe with the proposed 100 TeV collider if it were to be built, in your chart? Thanks.
5. Are we in this stage have the data necessary and suffiecent to say that Inflation IS the only option or another routs are possible without Inflation ?
• I mean is it possible in principle for some mechanism generating the universe at HBB to exist without any thing before HBB .?
• At the present time there is one ‘big’ alternative to the BB, the ekpyrotic universe (In which case the universe could be infinite in age.) If the current data holds up however it will become heavily disfavored (but possibly not ruled out.) Then of course there may be alternatives we have not thought of.
At present inflation is the ‘best educated guess’ for what preceded the HBB. We have no theories at all that postulate the universe starting with a ‘plain’ HBB; this was the consensus until a few decades back, but as noted in the article various properties of the universe have forced us to abandon that simple approach.
• Then , Am I right if i say that as per now there is no unique inference that we can deduce from all the data we have concerning the begining of the universe ?
Thanks Kudzu
• *If* BICEP2’s results are correct the ekpyrotic universe will be *almost* ruled out, but I don’t believe it will entirely be dismissed. I think this is similar to how supersymmetry is always said to be ‘ruled out’ by the LHC.) and even an inflating universe may be an infinitely old one.
At the present moment, even with BICEP2 we have only guesses as to the ‘start’ of the universe and we can’t even be sure there was a start at all.
• redmudislander
All of a sudden you can compose complete sentences in English?
You must now be more than one person.
• But Kudzu , according to Nature magazine , the Bicep2 proved that the Ekpyrotic assumption is wrong and doctor paul agree .
• The faults are still with us. So it is important to check the reported results and then, if correct, solve the problems (or find a better theory that matches what we observe).
Paul Steinhardt
• This is what Dr. Steinhardt say about inflation in a correspondance with him yesterday ……inflation is a failure ,,,this is what I understood.
• He does make the case that inflation is a failure. For a good reference I recommend his “The Inflation Debate: Is the theory at the heart of modern cosmology deeply flawed?” Scientific American, April 2011, p 36-43.
6. I have a question about the end of inflation. I get that there’s a phase transition of some sort where the energy of the inflaton field is converted into particles. Was this process instant? that is to say, did it happen “everywhere” at the same time?
Naively, I imagine it starting at a nucleation point and spreading out to encompass the observable patch we live in. Is that possible? Could that process leave an observable imprint?
Alternatively, are we talking about something more like the pressure in a gas? there’s some property of the inflaton field that’s broadly the same everywhere, less quantum fluctuations, and when it reaches a critical value – bang. If that’s true, is there any reason to believe there would be regions of space outside our horizon where inflation is ongoing? That’s what I’ve understood from some of the “multiverse” models being bandied about.
• For clarity (I wish you could edit posts) with the gas analogy, I was visualising whatever this quantity is to be changing over time, as the universe inflates.
• This is actually an interesting problem. In the theory of ‘eternal inflation’ inflation stops randomly at various points in space, spreading out as a sphere at the speed of light. Since inflating space expands much faster this means that the universe ends up consisting nearly entirely of inflating space with infinitely isolated bubbles of ‘our kind’ of space appearing in it. This of course is a multiversal theory.
Or the process could have been instantaneous across ‘all’ space or some weird hybrid. We have theories but no way to know just yet.
• Conversion from energy to particles was instant (revolution) everywhere or only in our patch evolved by natural selection ?
If we look at first fig’s timeline, todays time and inflation time have same darkness of temperature. If it reaches a critical value – again bang – from mass energy to vacuum energy ?
• Actually the inflation era was vastly colder than our current one. It is highly unlikely that simple dropping temperature will ‘kick off’ a bout of inflation. (Especially since we don’t know what started the first bout.) It is possible that in a ‘big rip’ scenario we will see a second round of inflation as the universe’s expansion spirals out of control, but details are too hazy at present.
7. Matt,
I do not understand why the universe cooled after the Hot Big Bang. Energy is conserved, so where did all the heat go? Is the cooling due to the still – if only much slower – ongoing inflation of the universe? And if the inflation had stopped completely after the Hot Big Bang, would the universe still be hot today?
• The heat did not ‘go’ anywhere; the total amount of energy in the universe is the same now as at the HBB (Or at least we assume it is.) Three things have however happened to this energy. The first is that it has been diluted; as space expands there is more ‘room’ for a given amount of energy to occupy. The sun is very hot at its surface but if you are a billion miles away it seems colder 9though the energy of the few photons you receive is still the same.)
The second thing is that entropy has increased. A high energy photon from the sun will hit the earth then be remitted as several lower energy photons (in the infrared\microwave spectrum.) The same total energy is there but less ‘hot’.
Finally he expansion of space itself ‘stretches’ things moving through it like photons, taking them from a higher to lower energy. (I don’t know where that energy goes myself, but I’m told it all works out.)
• As some would say, “morally” the universe is zero energy. (E.g. that is what guys like Linde and Siegel@Starts With A Bang use. There are several ways to derive that, say Siegel uses potential energy of gravity = energy of everything else, so it is still a hypothesis.) The energy doesn’t go anywhere (where should it go?), it is just converted.
As in any thermodynamic system it amounts to phase changes of internal free energy. (Like how an engine works on combustion of fuel.) E.g. the universe started out in equilibrium (very cold) as I understand it, then got matter out of the matter/antimatter symmetry breaking so the disequilibrium that enables us.
8. You have made mention of the ‘inflation particle’. I realize in general this is just shorthand for ‘the mechanism that causes inflation’ but are there any proposed models that include such a particle and remains consistent with the Standard Model?
9. Matt, your articles have been so helpful in getting the beginning of a grip on what people are talking about with eternal inflation. I have a mental model now of it, derived from the very nice article by Guth you posted last time, which leads me to a question. If you don’t mind, could you tell me whether the following analogy, with two dimensions fewer than what we observe, has any validity, and whether the question at the end is well-motivated: before the birth of our observable universe we have something analogous to a rubber strip, being stretched, endlessly and very fast. At some point one tiny area of the expanding rubber strip “converts”, or phase-flips, or something like that, and is no longer the same “stuff” as the space-time foam. This tiny converted area continues to expand–into our pocket universe or one like it–but much less quickly. Outside our phase-flipped area, the rubber band continues to be stretched, unimaginably quicker than our now converted area, and from time to time other bubbles like ours will nucleate, meaning they’ll exit the hyperfast expansion and continue their own expansion at a slower rate (not necessarily identical to ours). If that is not too misleading an account of the scenario of eternal inflation, my question is: since any pocket universes are being driven apart from one another by the incessant stretching of the rubber strip faster than they are expanding themselves internally, does it follow that there can be no danger of expanding bubbles within the space-time foam eventually intersecting?
10. thetasteofscience
Reblogged this on thetasteofscience and commented:
A masterful discussion of what happened in the cosmos even before The Big Bang Theory hit television 🙂
11. Thanks for the great post. I have a doubt: when people say that inflation happened 10^-35 seconds after the big bang, what do they mean?
12. kashyap vasavada
Matt: Excellent review. I have two questions: (1) during reheating after inflation did the temp go up to 10^29 K (10^16 GeV) or more?. So the quarks and leptons were produced during reheating or they were already there from inflaton field? (2) I guess most people have problem with the concept that space expansion does not require any energy. Alan Guth says it is ultimate free lunch. Also many people say in GR energy is not concerned. The problem is that nowhere else we have seen non conservation of energy, not even in quantum mechanics, except for uncertainty fluctuations. That must be the hang up. What do you think? BTW can this energy come from another universe which is collapsing?
• One way to think about (2): conservation laws exist because of symmetries, as first figured out by Emmy Noether (http://4gravitonsandagradstudent.wordpress.com/2013/04/19/theres-something-about-symmetry/). So the reason momentum in a line is conserved is because space is the same if you move in a straight line, but when space isn’t the same (say, because there’s a planet there) then momentum isn’t conserved, so a spaceship moving in a straight line falls towards the planet. Similarly, the conservation of energy is due to symmetry in _time_. If the universe changes over time (say, with inflation) then energy will not be conserved. That’s what people mean when they say in GR energy is not concerned: if you change the shape of space and time, you change the rules for conservation of energy.
• Now this is something you seem to have a greater grasp of than I do.
So energy conservation ‘works’ only if my system is the same at one time vs another time? I assume there is something more subtle to it since *everything* is changing all the time.
My main question to you is the CMB; these photons were originally much higher in energy but the expansion of space has ‘stretched’ them. Did the energy ‘go’ somewhere or is it simply that under these conditions the total energy in these photons will decrease?
• It’s not that the system needs to be the same as a whole, rather it’s the “background” that needs to be the same. If you’re looking at the Earth by itself, then it appears that energy isn’t conserved, because the Sun is changing and you’ve put the Sun outside your system. But if you include both, then energy is conserved.
You definitely can try to include changes in space-time in your system instead of your background, and conserve energy that way. The problem is that there’s no clean way to separate which part of spacetime is background and which is the change, which is why this isn’t generally the way we think about it.
This is why there isn’t a clean way to answer your question about the CMB. You can think of the energy as having “gone into” spacetime, or as the photons just losing energy, depending on how you frame the question.
• All the particles (Quarks, gluons, leptons and what have you.) are all produced *from* the inflaton field when it ‘flips off’; they appear already high energy (‘hot’), I believe as high energy as the dark energy that drives inflation or less. So ‘reheating’ is actually just particle production.
• kashyap vasavada
@ Kudzu:Let us see what Matt says. But according to elementary physics, heat is essentially kinetic energy of the particles. So my inclination is that the particles were produced sometime during inflation and then potential energy of inflaton was converted into kinetic energy of particles. But I am not sure!
• The important thing to remember about particles is that they don’t have to be created with a single ;rest mass’ energy. (Otherwise massless photons would never be created!) Nearly all particles will be created moving at some speed, with some ‘heat’. Take for example beta decay radiation, electrons are emitted with any kinetic (‘heat’) energy from ‘none’ to some maximum value. (See the 5th paragraph of this article http://en.wikipedia.org/wiki/Beta_decay )
So at the end of inflation many particles would be produced already ‘hot’ with a great deal of energy.
• Photons have rest mass – proof photo electric effect ?
• kashyap vasavada
@Kudzu: Well, yes it is possible. But I remember to have asked Matt previously a question. He did insist that inflationary expansion has to lead to cooling. No choice. That is why they call subsequent heating as REHEATING! So if particles produced were already hot they would have to cool off. Also remember, before Higgs, these were massless particles! They were moving at c all the time.
• No, photons most definitely do not have a rest mass. This is something that all of physics agrees on. The photoelectric effect is a result of the *energy* of a photon being given to an electron and is a good proof that light is particle-like (or quantized.)
If photons did have a rest mass they couldn’t move at light speed and we could stop them, put them in a box and look at them.
• Kudzu, photons have been stopped, see:
And a photon at rest relative to you wouldn’t be something you could detect directly; it would have to be infered.
I’m of the school of thought that if it moves, and isn’t a shadow, it must have mass. My understanding of relativity theory does not prohibit something with mass moving at, or above, the speed of light in your reference frame. If something were accelerated beyond the speed of light away from you it would disappear from your universe. Black holes do this.
• Thank you Mr.Kudzu, gauge invariance is a fuzzy. Cannot grab, but make us to “feel”, dislocating ?
• During inflation the expansion of space both dilutes and removes the energy in the universe. This includes massless particles such as photons. Microwaves are ‘cooler’ than visible light for example.) So yes by necessity it cools the universe. The term ‘reheating’ though is perhaps not the best one to use here as it is not matter that is being reheated but the universe itself.
You can think of the process rather like a red hot piece of metal. This is converting the kinetic energy of its atoms into photons with a range of energies. (Google ‘black body’ for details) The photons are already as ‘hot’ as the metal; as the metal cools the photons it emits have less energy (They go deeper red and eventually IR and microwave.) Notably it does not emit a single color (energy) of photon like a laser nor does it produce the ‘coldest’ photons possible (which would be infinitely cold!)
The case of dark energy is the same; at that time most particles as you note were massless so as the energy decayed it would emit all sorts of particles each with a spectrum of energies just like a hot piece of metal. But in this case the amount of energy being converted is much greater, dark energy was very ‘hot’. (Exactly how ‘hot’ is still up for debate.)
If this was a two step process we have not one but two problems to solve. Firstly, ‘why would the dark energy field behave differently from all other fields where similar things happen?’ and secondly ‘where does the energy come from to heat all the cold particles?’
• kashyap vasavada
@Kudzu: Obviously there may be a number of models and lot of ambiguities. But my understanding is that reheating actually comes from the potential energy of inflaton becoming minimum and converting into kinetic energy. That part I am reasonably sure, although I could be wrong!! As an elementary example a falling ball loses P.E. and gains K.E. My main question was that at what stage inflaton materializes particles.
• The conversion of dark energy to other forms of energy occurred at the end of inflation. According to Professor Strassler’s article (published earlier on this site.) the inflaton field started to oscillate, in his words ‘become larger and smaller in turns’ which is equivalent to having a lot of inflaton particles. (We see similar effects if you say, take a magnet and wiggle it about near a wire. The oscillating electromagnetic field produced dumps its energy into the electrons present, making a current flow.) So it is not so much the energy becoming a minimum as the field becoming unstable.
13. Pingback: Reliability Big Bang | Tyranosopher Overflow
14. Ken Ziebarth
Matt: When inflation was first hypothesized, and at least into the last decade, Stephen Hawking objected that it violates the general co-variance requirement of General Relativity because a specific reference frame is chosen in which one dimension of space-time becomes that along which the inflation of the other three expand. Is this now considered a serious issue?
• Torbjörn Larsson, OM
The universe expands, so inflation can happen.
Hawking’s objection seems directed toward the speculative part of if and how inflation started.
15. A question about “times” and “durations” in the very early universe.
Presumably a very dense universe would have extraordinarily high gravitational fields, which would lead to extreme time dilation effects.
So when we say some epoch lasted 1 second or 1E-30 seconds, in which frame of reference? Us in the present epoch, or some imaginary observer in the early universe?
• We actually don’t have to worry too much about this; the HBB wasn’t particularly incredibly dense, had it been then the universes expansion would have been severely curtailed or even reversed. What is more of a problem is our simple lack of knowledge. Inflation would have involved a nearly completely empty volume of space, but we are unsure as to how long it lasted and anything before that is theoretical guesswork.
16. In my lifetime I have seen the age of the universe revised several times. Even Hubble finally realised that the age he had arrived at through observation had to be wrong as evidence appeared that the earth was older than it. Now I accept that’s the way science works, more measurements are made and results improved. Are we so sure on the current age as most people seem to be or is there something yet to be discovered that will jump up and bite us in the ass.
• For what it’s worth, the estimates of the age of the universe have been converging (error bars narrowing) over the last twenty years. People now quote a third significant digit (13.8 Gy), while in the mid-90s it was more like “12-15 Gy”. (Actually, the estimate derived from the observed expansion of the universe was more like 8-15 Gy, but we knew the low end was wrong because there are old stars aged 12 Gy or more.)
• Yes but scientists have merely measured those parameters that they believe indicate the age of the universe more accurately. My question really is are those parameters the only ones or are we missing something.
• We can be quite confident that the parameters we’re measuring are them ‘right’ ones *from the start of the hot big bang* since we’re pretty much looking at *all* of the parameters we can think of. They fit nicely into our current theories and it’s very hard indeed to imagine how something may be otherwise yet still rpeserve what we know to be true.
For example, to change the age from 13.8 to 13.9Gy we need to insert 100 million years into the universe somewhere. There is little room for a ‘smooth’ insertion (Where nothing abrupt happens. Say the universe expands slightly slower than we think it does now.) because this would produce significant deviations from what we have already measured (Such as the ages of the oldest stars, galaxy structures and so on.)
Instead it would need to be something ‘abrupt’ (Say a sudden stop in expansion followed by a bit of ‘catch up’ later.) which is rather unnatural and begs the question of what mechanism could be responsible. Nothing is ever ruled out in science but it seems quite unlikely that our current estimate is significantly wrong.
18. Another very nice post Matt. Don’t know where you find the time or energy. But no mention of antimatter?! Where is all the antimatter? Isn’t that still a mystery?
• Well he does mention anti-quarks, anti-protons and anti-neutrons, so it’s there. But I don’t think it’s a relevant detail in this very broad brush treatment of the topic. (I do believe antimatter-matter imbalance would be sorted out far before nucleosynthesis and maybe before the Higgs field turns on, in the speculative zone.)
• Yes, you are right it is mentioned, but then just left hanging: “…so that quarks and gluons and antiquarks would have been trapped henceforth and forever into protons, neutrons, antiprotons and antineutrons…”
To his credit Matt does say: “It’s not entirely impossible that there was something odd and unknown about the universe that makes some of these conclusions premature.”
However I disagree with your notion that this is an irrelevant detail here. I think, and I know I’m not alone in this, that it is one of the universes greatest unsolved mysteries, so shouldn’t it at least be noted as such?
So I say again: Where is all the antimatter?
(And anyone replying to this post – Please don’t offer the standard ‘explanation’ concerning differing decay rates. I know all about it – those rates are woefully inadequate, by many orders of magnitude, to produce the matter dominated universe. So save it – I’m looking for a more creative solution).
• Particles are their own antiparticles in right circumstances, and that’s a creative solution 😉
• I agree with you that this is indeed a big mystery and worth an entire post of its own (maybe several.) but I can see why it was not much commented upon; even in the ‘green zone’ there are many things we don’t know well if at all. Dark matter too is given a wide berth and I’m sure some astronomers here would have liked a mention of star and galaxy formation. There’s a lot to cover and it may have been simple oversight or deliberate choice just to save space.
• kashyap vasavada
@ S. Dino: The last I heard was that people have found larger CP violation in heavy quarks (b, bbar) than K meson decays. So it is not unlikely that at extremely high energies, there was a very high CP violation which would explain matter-antimatter asymmetry. But at this point nobody has pinned it down.
• My solution:
There are three dimensions for the space, one for the causality and one for the duality, in total 5 dims. The duality dimension is curled and interaction spacetime cycles between matter and antimatter. We see it always as matter although there is a ticking rhythm of magnetic monopoles in elementart particles (just like Higgs mechanism) creating inertia and balancing the dynamic vacuum energy and the mass energy. The phase of the matter duality state is in coherence with a causal distance in accordance with the speed limit of c (in the same coordinate system with neutrinos – see the oscillation phase difference between neutrinos and electromagnetic fields).
So at a proper distance matter change to antimatter in respect to the common background coordinate system – the distance could be the longest distance in the unverse or something like that.
The idea demands a radical definition of dynamic vacuum energy and it’s possible we have two nested net structures: one for matter and other for antimatter both interacting with light in a same way so that we cannot see any diffences getting radiation from galaxy or from antigalaxy. The potential relation to the local vacuum energy field has the same value and the same math in respect to the excited states of particles. Light propagates via both of duality dimensions phase, particles are divided to prpagate either via matter-phase or via antimatter-phase.
If you got some reasonable thoughts, good – if not, it’s not very odd, I’m in trouble to be understood even in my own native language…
• “he” ? I was under the impression that “Kudzu” = Matt.
19. Tony (Racz) Rotz
There is a saying. “Imagination is the mother of invention”. I say speculation is a necessary ingredient to research, it fires the urge explore new frontiers, even in the red zone. Even if eventually proven wrong.
20. Tony (Racz) Rotz
I’ve read where they are using string theory math in condensed matter labs.
• If the universe get more cold, we will be in condensed matter ?
• We already are condensed matter; solids and liquids (the stuff of which we are made.) is condensed matter. Our existence now is indeed only possible because the universe now is quite cool on average.
• Thank you……, I mean, the Meissner effect. Below the transition temperature, a massless gauge field can aquire a mass in the presence of a coupling to a spontaneously broken field. A concrete realisation of this occurs in superconductors. This is like the photon acquires a mass.
The universe we live is a kind of cosmological Meissner phase, formed in the early universe (vortices).
This low-temperature expression is completely analogous to the one in the high-temperature phase for the correlation function.
So below “0”K, there will be decrease in entropy – making a “Dark-energy star” ? – based on the physics of superfluids, avoiding a singularity. which is analogous to “blobs of liquid condensing spontaneously out of a cooling gas” ?
• Aaah. I see. I don’t think we will ever get below 0K (even inflation didn’t manage that.) but the phenomena you mention are interesting. I am unqualified to answer but I am sure others will speculate.
21. Why is it that we think there was a time when the Higgs field expectation value was zero?
• As you are aware all fields have a lowest energy vev. But they can have other values, it just takes more energy. We can visualize this by plotting the ‘field’ on a 2D graph (one axis real, one axis imaginary, the details are beside the point here.) and get various results. For most fields we end up with a ‘well’ centered at 0. If we add energy the value of a field can ‘climb’ the well but never leave it.
The Higgs field however has its lowest energy point at 246Gev. This results in a ‘sombrero’ plot, like the well plot where someone has pushed up the middle. At its lowest energy the Higgs field lies in the ‘brim’ of the sombrero and again raising the energy lets it climb the walls of the sombrero.
But for various reasons the energy at 0,0 cannot be infinite; it is a hill with a top. And as such as energy is added the Higgs field can ‘climb to the top’ of this hill and get stuck at a value of 0. Since the conditions of the early universe had sufficient energy for this, this is what we believe happened.
22. Hi prof Matt
You says about 380,000 years, but if we know that our time reference were created in earth’s translation bases, how to say something about a very far past before this existence? Is it correct that observation calculus? Is there any other reference? How can we talk about billion years before earth’s existence?
• The physical laws that we use to make sense of our measurements originated at least early on in our universe. As such we can use these to determine earlier times. As an example, we know light travels at a constant speed and gets dimmer the further from a source an observer is. S things far away are dimmer and take longer to see. We can see light from some objects that we can calculate was emitted before the earth was formed and that therefore the universe has to be at least old enough for those objects to exist.
If the laws of the universe were so impermanent as to change during the time we were here the universe would be a very strange place indeed, and likely too inhospitable for life or even planets to form.
• Thanks, but I think that you dont had assimilated my question: You talk about constant speed and this is my exactly question about. How can you says about a “constant” term? It is clear that in those moment, light speed was impossible to had the same proprieties like now. If you says yes, how you can fundament it? If not, our calculus about everything at that moment (1/10³³ sec or less) are no fundamented
• Right. So your question then boils down to ‘how do we know the universe now works like it did before we were here?’
Basically it’s very hard for it NOT to. Things of course change; the universe is hotter and smaller going back for example, but even then usually in a smooth and predictable manner. And many fundamental things upon which we base our calculations are very hard to tweak.
Take the speed of light. You might well say ‘What if it moved faster a billion years ago?’ But the speed of light isn’t just a lone fact; it affects many things, the size and energy levels of atoms, doppler shifts, the very stability of matter. Any change would be ridiculously easy to detect, the universe’s basic structure would have been messed with, planets might not form, or stars, or life.
In fact scientists have been looking, all the time, for things like you suggest. Recently the ‘fine structure constant’ has been checked and found to be quite well… constant. As for ‘1/10³³ ‘, we *don’t* know what happened then, we have some nice guesses but anything before a few seconds past the HBB scientists are busy asking your question.
So far yes, it’s always possible that we are deceived and that somehow the universe didn’t work the way we think ‘back then’; but we have no evidence it until a few seconds after the HBB and no reason to expect it to.
• What, pray tell, are you talking about?
For instance, what does ”earth’s translation bases” refer to?
Are you aware that the statement/potential question ”Is it correct that observation calculus?” makes no sense? It is not a complete sentence. Neither is it a complete thought.
Finally the question, ”How can we talk about billion years before earth’s existence?” We can easily look into the sky and see things that are billions of years older than the duration of the Earth’s existence, and it’s not a problem to talk about them, so . . . What are you asking?
• Notacrackpot
When Matt’s away, the Crackpots come out to play.
• Steve Donaldson
@Michael, I don’t know if Edward is a “crackpot” or just someone with less than perfect English, sincerely trying to understand what we know about the history of the Universe. Giving him the benefit of the doubt, my guess is that he was asking the following. If we define a year as the time it takes for the Earth to make one trip around the Sun (”earth’s translation bases”?), then what does it mean to talk about so and so many years, before there was an Earth or even Sun? It is actually a pretty insightful question. Fortunately there is a pretty straightforward answer. Yes, if we are to speak of time measurements spanning the vast history of the universe, we need to use a time standard that also spans that whole period. Well, we know light existed very near the beginning of the Universe and we know a lot about how light vibrates. For instance, the specific light from a specific excited state of Hydrogen will vibrate at a very specific number of times per second. Thus we can define a second based on light from Hydrogen and we can define a year as being a specific number of seconds. Now our definition of a year no longer depends on the Earth revolving around the Sun. I am not an astrophysicist or cosmologist, so I don’t know exactly how they define their time units, but you get the idea of how the measurement of time does not need to be in reference to Earth any more. I hope that helps. (P.S. Kudos to Kudzu for also trying to help.)
• redmudislander
@ Steve Donaldson:
I don’t know if Edward is a *crackpot* either, however, I didn’t call him one. Careful reading above will show that a contributor with the stated name of “Notacrackpot” was the entity who so cavalierly spewed out this term. Not me.
23. What does “the Higgs Field turns on” signify? Is it the point at which it assumes a non-zero average value, or does it mean ‘comes into existence?’
In general, how do quantum fields arise initially? Can they exist without “real” particles’, and without quantum fluctuations? The description of the inflaton field made it sound like it may exist with few (no?) particles. (From your post on inflation: “dark energy (really a combination of energy and negative pressure) is never associated with particles. . . . Basically, the dark energy is stored in a field (the inflaton) and at the end of inflation the inflaton field starts oscillating, becoming rapidly larger and smaller by turns. . . . This is like having something like a laser made from inflaton particles. The inflaton particles, however, have a large mass and are unstable. They will decay into other particles, which may in turn decay into yet others.”)
Could you please clarify this for me?
• The Higgs field exists from ‘the start’, like all fields in nature. They are in a way intrinsic to space in our universe. When the Higgs field ‘turns on’ its vev changes from zero (like most fields) to its current value. (An odd situation indeed.)
Fields can indeed exist without particles in the same way air (a sound field) can exist without any noise in it. As to how they arise, that is similar to asking how space starts.
The inflaton field doesn’t have any particles worth mentioning (If it had any in it they would quickly be diluted to nothing.) When it ‘flips’ there is something particlish (How much so is up for debate.) before its energy is converted to more usual particles.
• @Kudzu
Thanks for that. You’ve answered the question that was really behind my original question: Are all the fields described in quantum theory present before the hot big bang (i.e., during the inflationary period), or do they come into existence later.
• Kudzu, air is a field of particles that can transmit sound, which are compressions and ratifications of the field particles.
In my mind, fields are always full of particulate entities that have mass.
• Typo. Meant rarefactions, not ratifications.
• While there are numerous problems with the idea the biggest problem is the most obvious; had I been Matt I would have seen it instantly, but I am not that smart. On further thinking problem 3 becomes more meta. If fields are composed of massive particles then what we call particles currently…aren’t particles. There would be no photons only ‘light field particles’; no protons, only ‘proton field particles’ If this is so… what IS a particle? It can’t be a wave in a field because that would lead to an infinite stack of more fundamental ‘field particles’
So the question I must ask is ‘What ARE the massive particles that make up your fields’? They must ‘really’ be something different from what we call particles ordinarily, they must behave in a different way or arise from a different mechanism.
• There are some problems with that idea and indeed realizing that there were fields that were not made of particles was a big discovery in science. (You may recall that light was believed to move through a field of ‘ether’ at the dawn of the century.)
The first problem is the mass of these particles; when you have any other mass in the field (say a lump of rock or earth) it starts attracting and compressing the field. Air for example is denser at sea level than near space. Massive particle fields that permeate the universe would thus feed things like black holes or even ordinary stars with incredible amounts of mass.
Secondly a particulate field by its very construction has places where it ‘runs out’; there is no air in space, no ocean above sea level. Yet we have not seen any evidence of places where say light or gravity can’t go because there are no ‘field particles’ there.
Thirdly if a field is made of particles then we should be able to detect them in the way we can detect air molecules. Fourthly a field of massive particles alters how waves move through it, we would see light speed change depending on how we move through that field. There are about a dozen other problems.
The concept of a field almost as a part of space, not being composed of a particulate substance being ‘smooth’ and continuous and throughout all space is a tricky one sometimes to grasp but was a great advance to science,
• Vincent Sauve
Kudzu, I’m not sure that the problems you list are significant.
Problem 1, Attraction and Compression: Why is this a problem? The field in space is dilute. Maybe the compression can be the explanation for the “gravitational” bending of light, with more bending the closer to the large mass? Feeding stars and black holes with dilute field particles? I’m not seeing that as a problem. Stars may be generators of field particles.
P2, Field running out: I don’t know under what circumstance such particulate mass fields should run out. Neutrinos don’t run out as far as I know. I’m not saying that neutrinos might be a field source for electromagnetic field effects, but if they have mass and can be everywhere then I don’t see a problem here.
P3, Detecting field particles: Inferred by the way light behaves just as we infer many things in particle physics.
P4, Light speed change depending on how we move through field: Are we referring to the Michelson–Morley experiment or the Doppler effect? The null result of the M-M experiment can be explained by the earth dragging the local field along. The Doppler effect can be explained the same way and thus light from the edge of a galaxy that is rotating toward us is blue shifted relative to the opposite edge that is moving away from us. Yes, that implies that light moves toward us somewhat faster from the blue shifted side and somewhat slower from the redshifted side. If we are able to measure that light in our earth-bound equipment we would measure it as moving at the normal speed of light because the local nearby field that it interacts with first would interfere with the raw original speed, i.e., we detect the wavelength energy shift but not any difference in speed because after the local field interactions which include our atmosphere, windows, lenses, other equipment, etc. the light leaves these things at the normal velocity relative to us.
What are the other objections to a particulate field in space that has mass?
• What I should rather have said in my reply to Problem 1 is that maybe the attraction and compression of field mass particles can be an explanation for the excess bending of light as in a possible role in explaining why astronomers think there is unseen dark matter.
• Kudzu, this reply is to your statement of March 31, 2014 at 7:14 AM,
I don’t follow your reply. Logically it is like saying the particles of air or water aren’t particles. Air and water transmit sound but sound isn’t transmitted without particles. It seems many in physics want to say that light is not a massive particle, or a collection of particles, (or the result of a collection of relatively moving particles?) and doesn’t need a field of particles to be transmitted. I could grant that photons may only be a disturbance of a field that has massive particles such as a vacuum field of electron-positrons, but at least there is something with mass involved.
The following paragraph is from http://en.wikipedia.org/wiki/Vacuum_state:
“Photon-photon interaction can occur only through interaction with the vacuum state of some other field, for example through the Dirac electron-positron vacuum field; this is associated with the concept of vacuum polarization. — Jauch, J.M., Rohrlich, F. (1955/1980). The Theory of Photons and Electrons. The Relativistic Quantum Field Theory of Charged Particles with Spin One-half, second expanded edition, Springer-Verlag, New York, ISBN 0–387–07295–0, pages 287–288.”
Kudzu, for me this is analogous to a room of air molecules where we are not attuned well to hearing the background fluctuations because we are normally hearing things that are much louder. In this analog a photon is like a sound pulse in a material field (for light–of electron and positrons?). Light then has wave and interference patterns in the same way as sound in air and water.
In my epistemology, my theory of knowledge, a source of energy must have to do with particles that possess inertia natively, without any cause, and occupies space to the exclusion of other particles. Otherwise, nothing can make sense. To me it is best to try to understand nature from that starting point. No one, regardless of your personal epistemology, can know everything about how the universe works. We just are not the kind of animals that have the intellectual facilities and life-span for that, but we have come very far.
24. Reblogged this on Patrice Ayme's Thoughts and commented:
I long held that there was no proof whatsoever that the universe was 13.7 billion year old, as all too many Big Bang theorists have long claimed, right and left, and all over the Main Stream Media They have exclusive access to.
Now I am happy to report that a main stream physicist, the very honorable professor Matt Strassler supports this point of view in an excellent article:
Professor Strassler’s broad reasoning is exactly the one I long put forward: the equations and the experimental date we have break down at very high energies, so we cannot use them to extrapolate at such energies (something similar happens with gravitation: we have no proof it holds beyond the Solar System… and some hints that it does not).
I guess I will have to get more subtle with my own, much older “Universe: 100-billion years old?” . I proceeded to thank professor Strassler:
Unfortunately, the great professor Strassler then removed by comment from his own blog (scientists’ ways are exclusive, and do not have to do with the scientific process). In any case, here is professor Strassler’s excellent post, which demonstrates, in fascinating detail, the broad point I made previously:
• Patrice Ame:
It seemed clear to me the start of the Hot Big Bang epoch could be estimated to within ~1 second — see the timeline in the Green part of the diagram.
It’s the “before Inflation” epoch that doesn’t have any theory or data to back it up.
So when you opine the universe may be “100 billion yrs old”, how are _you_ defining “age of universe”?
And if you have some equations or theory to support this, maybe you should publish it at Arxiv or something ….
• TomH: I have a problem with time. At this point I distinguish two notions of time, one of which I called holonomic time, because it’s locally, but not punctually, defined. I can’t see how holonomic time could be defined during inflation, let alone being equal to punctual time (normal space-time physics makes no distinction between two different of time). Thus that “second” could well last an eternity.
I will put more details on my site.
• No evidence our equations of gravity work outside the solar system? Are you mad? Don’t answer that. Instead, please follow the proper path for overturning the mainstream cosmology:
1) Understand the existing theories and evidence behind them
2) Revolutionize physics.
No skipping steps!
• Anon: OK, let’s apply your own recommendations, to… you. You have been skipping steps from the raw evidence.So learn this:
1) the rotation curves of galaxies do not work. Galaxies do not rotate according to the 17 C gravitation theory (attributed by Isaac Newton to a Frenchman, let it be said in passing). This is “explained” by, and “evidence” for, Dark Matter.
2) Strictly speaking the same situation is true for galactic clusters. That, again, is “explained” by Dark Matter.
3) I will not insult you by mentioning Dark Energy, a significant complication you probably heard of.
Let’s forget for an instant the added hypothesis of “Dark Matter”. Strictly speaking, in first approximation, the fact is, that, the direct experimental evidence is: gravity according to 1/dd, does not work outside the SS.
Dark Matter is just an interpretation of the raw data, to save 1/dd. (BTW, I am ready to believe in DM, because, you guessed it, I have an implausible explanation… completely outside of the SM, or any SUSY…).
I understand that, the way I put it, I am violating cosmological Newton, thus cosmological “General Relativity”, and that there is solid evidence for both in the Solar System, and that the instinct of physicists is to extrapolate known laws. But the whole spirit of denouncing, as professor Strassler just did, the red zone non sense, is to point out that the general method of extrapolating known laws works only so far.
• I’m familiar with the issue of galaxy rotation, thanks, but that’s not what you said. You said there was no evidence our theory of gravity worked “outside the solar system”, which is utter nonsense. The number of observations of phenomenon outside the solar system that are in precise accordance with General Relativity are multitudinous, including predictions of energy lost through gravitational radiation. Your statement was ridiculously out of touch with the observational status of GR.
I could go on and on about how the evidence supports the idea that General Relativity is correct and that there is unseen mass, far better than it supports the idea that GR is incorrect. While hardly settled, it’s hardly up in the air as anyone’s game either. Certainly the statement that there is no evidence that gravity works at those scales is false. But far more important is to clear up this nonsense about their not being evidence that it works anywhere outside the solar system.
• Anon: We can use only so many words: check the Incompleteness Theorems in mathematical logic. If you decide to make a ridiculous over-interpretation of what I said, I feel sorry for you, and beg forgiveness. I know about rotating neutron stars and the support they provide for gravitational waves (a necessary sub-logic of GR that I am very comfortable with).
• Yes, how ridiculous of me to interpret “outside the solar system” to mean “outside the solar system”. And using Godel to justify your inability to express yourself clearly, that’s definitely not ridiculous.
I guess it would also be a ridiculous over-interpretation to take “no proof” to mean “no proof”. I’ll assume you also know that in addition to galaxy rotation curves, galaxy cluster behavior including collisions, there’s also the power scale of the large-scale structure of the universe and the CMBR all of which GR+DM explains very well with the same DM fraction, and which alternative gravity theories do terribly at. I’ll assume what you meant to say was “There’s actually lots of evidence (not proof because this isn’t math) that GR works at the relevant scales, but I feel there’s enough wiggle room for some future alternate gravity theory to possibly explain these things better even though none have come close yet.”
Fair enough!
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26. The only reliable portion of theory is this one, which we already understand both from formal perspective (i.e. we already have a reliable mathematical regression verified with experiments), both from intuitive perspective of physical analogies. The epicycle model failure teaches us, just the formal agreement with data is not enough, no matter how good such an agreement can be.
From this perspective the whole Big Bang theory still holds water, because the mainstream physics has absolutely no idea, why the Universe should explode, why this explosion should suddenly stop for to continue with another explosion (inflation) in another moment, which subsequently stopped again and it has been replaced with accelerated expansion of Universe. The proposal and explanation of such a scenario actually really requires a pretty huge physical fantasy.
27. Matt is right to spell out the zones of confidence and speculation. Regarding the green zone of high confidence, I would suggest some caution for more reasons than the statement below.
The following is one of three paragraphs of the physicist Anthony Peratt, in reply to some critics of his article “Not With a Bang.” This excerpt below appeared in The Sciences July/August 1990, page 12:
Ralph Alpher and Robert Herman attempt to perpetuate the twofold myth that only Big Bang advocates (themselves) predicted the cosmic microwave background and that only the Big Bang produces a blackbody microwave background. In fact, the discovery of the microwave background might have come as early as 1940, when the astronomer Andrew McKellar interpreted a 2.3-degree-Kelvin temperature for interstellar molecules; according to the pioneering radio astronomer Grote Reber, however, the cosmologists of that period were not interested in microwaves. In 1953 Erwin Finlay-Freundlich, a critic of the big bang, calculated a blackbody temperature for the universe of 2.3 degrees Kelvin, within 16 percent of COBE’s 1990 measurement. In the same year the Noble laureate Max Born, another big bang critic, verified Finlay-Freundlich’s prediction and suggested radio astronomy measurements at fifteen centimeters–precisely the wavelength at which data were obtained little more than a decade later.
28. IMO it’s time to return to tired light model represented with analogy with scattering of ripples at the water surface. The older tired light models considered, that the light was scattered with tiny but stable particles, whereas we know today, that the light would be scattered with quantum density fluctuations of vacuum, which manifest itself with CMBR photons with 2-spin component, which are temporal but larger than the wavelenght of visible light in general Such an emergent scattering would lead in formal agreement with inflation and Big Bang scenario and it doesn’t suffer with various paradoxes, which are common for Big Bang cosmology.
• Tired light is an interesting possibility which could oppose the accelerating expanding universe even into an contracting universe into a big crunch! so;We should postulate, that the Big bang was the splitting and evaporation of the big crunch black hole into what I call a black hole splitting fractal inflation creating the Lyman Alpha tree structure. What we observe in the B-Modes of the BICEPS 2 image should be the END of the Fractal splitting black hole inflation process at the rim of our own Universe.
29. Curious George
I like this article a lot. It shows clearly a decreasing confidence in our knowledge as we move back in time. Let me quote an inflation theory from John Preskill’s “Inflation on the back of an envelope”
Of course, mathematically there is much more to it, but favorite fairy tales of my young years started in the same spirit.
• James Gallagher
Very cute, I like it:
But let us not forget, in 1919 Eddington had fewer than BICEP2’s 4 data points when he confirmed GR – and THAT also made the front pages.
• Curious George
James – the “proofs” that we see are increasingly indirect. Nothing wrong about it; direct measurements have mostly been done decades ago. But an inflaton, dark energy, and dark matter are not GR constructs – we postulate them in addition to GR to make sense of observations after postulating the Big Bang. I am not very happy having to postulate four different things. Nice that two postulates support each other, though.
” …turned on …” What does “on” mean and was it created by quantum fluctuations? By that I mean was there (Higgs and/or gravitational field) fundamental resonance when the “universe” was a certain size and density?
Is the universe a spherical spring, an expanding and contracting shell?
• ‘On’ in this case is its vev going from 0 everywhere to its current value. This wasn’t a random process initiated by some quantum spark but rather a simple result of the fact that having the Higgs vev at 0 takes more energy. When the universe was no longer hot enough to keep the Higgs vev at 0 it settled into its current value. If you were to energize a volume of space enough the Higgs vev there could be made to go back to, at least for a short time.
31. For those with a little more math background, here is a link to a series of 4 lectures that take you from beginning cosmology to inflation. Coupled with Matt’s wonderful explanations, it starts to gel.
32. robotraptorjesus
Maybe a stupid question, but in reference to the “red zone” in the chart at the top, how could we infer that it was extremely cold? Since everything is speculative about that era, wouldn’t the temperature be also?
• The temperature IS speculative but it is firmly tied to the speculation of inflation; since space is expanding incredibly rapidly anything in it (Any energy and ‘heat’) is diluted to nothing so the space is at a temperature near absolute zero. Thus this is only true if inflation is true and happens the way we think it did. If we live in an ekpyrotic universe for example there is no inflation and thus no time of extreme cold.
33. When you build a house, you always begin with foundations… This is the major problem of physics… That’s why imagination is necessary.
Nice work!
34. Robo
Whatever temperature it happened to be beforehand, during the expansion the energy is dispersed across the greater volume thus dropping the temperature. Much like when you spray a can of deodorant , it feels cold to you because of the gas expansion.
35. The key word here is TEMPERATURE. What are the microscopic elements of the vacuum that give rise to the macroscopic phenomenon of temperature?If we identify these elements then the problem of quantum gravity is as good as solved.I think Verlinde is onto something with his work on the origins of gravity.
36. Pingback: BIGBANG 2014 : Doorbraak INFLATIE THEORIE | Tsjok's blog
37. Could anyone offer empirical evidence that would falsify the idea that virtually infinitesimal regions in the deep interior of a Type-II supernova event undergo space-time EXPANSION.
Given our lack of understanding of supernova events, I do not think the idea cited above can be falsified at present.
Dr. David Arnett, who has developed one of the newest and most sophisticated SN models says of its inability to reproduce all observed phenomena: “Perhaps what we need is a more sophisticated notion of what an explosion is to explain what we are seeing.”
Are we missing an important alternative to the standard cosmological paradigm simply because it conflicts with the standard cosmological liturgy?
• Develop an SN model that includes space-time expansion, and see if it matches the data better. “Our current models aren’t perfect… space-time expansion?” is just a stab in the dark until it’s shown that it actually helps explain supernovas.
Calling the ideas that currently have the most evidence behind them a “liturgy” does nothing to help your case. It shows more of a desire to win a debate through narrative framing than to discover reality through science.
• Robert L. Oldershaw
I think the word liturgy quite accurately describes the situation in some areas of cosmology and particle physics where just-so stories are substituted for evidence-based understanding because there is just not a lot of empirical evidence to guide us.
The problem then, as de Vaucouleurs so aptly put it, is that the dogged repetition of the just-so story transforms it into “common sense” or “how we should think” or “the best science”, or worst: “the only scientific way to see things”.
The initiates gradually become evermore closed-minded to alternative paradigms. It is they who insist on winning the debate because their liturgy is the only possible liturgy that makes sense to them. Alternative ideas are anathema.
And feel free to come out from behind the “anon” and into the open.
• I don’t personally see this as much of a problem with things like the standard model. Most physicists are deeply troubled by the fact that there are various parameters that ‘just are’ in various models. Notably the Higgs mechanism doesn’t ‘solve’ the problem of the various masses of particles but merely replaces the question of why asses differ with why the interactions of various particles with the Higgs field differ.
This is why so many look for ‘new physics’ beyond the various models we have so far. It is why we look for supersymmetry and evidence that the Higgs is not the simplest standard model variant. It is why we have pursued string theory for so long; our current models work well, annoyingly well in fact and many feel progress will only be made if we can find something that makes them break down. But alternatives are hard to come by.
• Liturgy… Liturgy… Liturgy… Mainstream cosmology is liturgy… Particle physics is liturgy…
You do seem to repeat that a lot. It must bring you great comfort. Almost like a…
38. Lest there be any confusion about what I am proposing, the object that is hypothesized to undergo the putativeType-II supernova Bang would be a Metagalactic Scale B star with an initial radius on the order of 10^47 cm.
The observable universe would be an infinitessimal region deep within the interior of this event.
39. “As the temperature cooled, we can calculate (now that we know the Higgs particle’s mass, and if we assume there aren’t any lightweight particles that we don’t know about) that first the Higgs field would have turned “on”..” Could you explain how and why “Higgs field” can turn on as the temperature cools? Why is the kinetic energy of a particle(s) (or the temperature) is related to if they acquire mass via Higgs field or not?
• The Higgs field ‘turns on’ because being ‘on’ takes less energy than being ‘off’. This is the opposite of most fields which take less energy to be ‘off’ As such in the early universe there was enough energy to keep the Higgs field’s value at any desired level. But at a certain point there was no longer enough energy to do so and it settled into its current universally high value.
The same thing happened to other fields except their preferred value is nothing. It is possible to reverse these transitions, at least for a short time, by pumping a lot of energy into a small volume of space. This is how we hope to learn a lot about the early universe.
• Robert L. Oldershaw
Do you have a Higgs field meter with which you measure whether its on or off?
Is any of your story scientifically testable, or is this just theoretical hand-waving?
• Indeed we do, do not forget that the Higgs field gives many particles their mass as well as being responsible for electroweak symmetry breaking. The standard model of which the Higgs mechanism is a part has been fantastically (some would say infuriatingly) successful. We even have a precise value for the Higgs vev of 246Gev. Discerning this value is not quite as simple as pushing a button on some detector and Matt is better qualified to give you the details but suffice to say we’re very confident that it’s ‘on’.
• Robert L. Oldershaw
I do not feel the slightest scientific compulsion to regard the Higgs mechanism as anything but a Ptolemaic just-so story.
• Torbjörn Larsson, OM
@RLO: Except of course that LHC _did_ find a Higgs with the expected prioperties (so far). So the mechanism is considered more or less acceptable right now.
• “The Higgs field ‘turns on’ because being ‘on’ takes less energy than being ‘off’. This is the opposite of most fields which take less energy to be ‘off’” -> If this is the case, could you explain why Higgs field has this property or behaves this way while the other fields do not? Why/how Higgs field gives mass to (interact with) only certain elementary particles (electron, quark, neutrino etc.) but not to the others(photon, gluon etc.) at current relatively low temperature? What is the mechanism?
40. Matt: a nice read. But I wasn’t sure where inflation fits. I was looking out for that because I’m thinking there are some big issues with inflation. Like it’s a solution to a problem that never really existed.
41. Inflation is a problem not needed in a non-expanding infinite universe. But due to historical cultural preferences for a beginning, cosmology has taken the road down a path that is consonant with creationism, and now it is very difficult to climb out and away from where that road led acedemia. I’m sorry if those of you in academia strongly disagree, but that’s how my decades of study has led me to see things in current western cosmology. Our flat, homogenous (at aprox. 300 million light year scale) isotropic universe as observed, best fits a non-expanding infinite cosmology just as Hubble went to his grave prefering.
• That is an interesting viewpoint, do you have any explanation for the fact that galaxies largely appear to be moving away from us or the origin of matter? Something must be inserting new useable energy into the universe or everything would have run down by now. I would be interested indeed in a proper steady state theory.
• Galaxies do not appear to be moving away from us. They are inferred to be moving away (all but some in our local group) due to the interpretation that the cosmological redshift is essentially Doppler-like. Neither Edwin Hubble nor I favor that interpretation. There is a redshift that is accounted for by Doppler effects. Yet, their is another component in cosmology that causes a redshift. There are many tired-light theories, but they haven’t been given much attention by academia due, I think, to the entrenchment with the standard creation paradigm.
• An infinite non-expanding universe wouldn’t “run down.” Gravity is the negation of increasing entropy. The only problem I haven’t yet figured out is how matter-energy is recycled from black holes.
Matter-energy doesn’t need a causal explanation.
• could you reference any peer reviewed papers detailing your theories
• Please be specific and pick one point. But generally I don’t have any theories. I mostly just point out the flaws in what many have mis-learned. Science is like that. It is not about beliefs, unless they are really well grounded in physical tests and very solid interpretations.
42. What about the temperature of the Higgs field? After correcting the CMB for the Higgs structure’s temperature there is little CMB left for the “big bang.” And the polarization patterns also disappear.
• How so? I can’t say I have heard of this. Are you talking about the current ‘temperature’ of the Higgs field or something tod o with the early universe? Do you have links to anything that discusses this?
43. Doug McDonald
Matt makes a statement that at high temperatures quarks and gluons were free. I thought that the requirement was that both temperate and density were high. The density required being high enough that any given quark interacts through gluons to multiple other quarks, each of which interacts to yet more of them, like the interactions of atoms in a liquid. Comment?
• kashyap vasavada
@Doug McDonald: The fact that quarks and gluons are free at high temp (high energy) has nothing to do with density. This is called asymptotic freedom i.e coupling constant which determines interaction goes to zero at high enough energy. You may want to google for “asymptotic freedom”
• Doug McDonald
I thought I understood asymptotic freedom. But I didn’t realize that
it permitted free electric charges of 1/3 or 2/3, also free color charges.
I took the word “free” to mean “unconfined”. Is he (Strassler)
using the work “free” to mean “asymptotically free”? I took it
to mean unconfined (i.e. not in color singlets).
• kashyap vasavada
@ Doug McDonald: I believe, at these energies, the two words, free and unconfined may mean same, with no coupling between them. The moment there is coupling, they appear in the form of hadrons (protons ,neutrons etc). So no one has seen free quarks or gluons. Even in high energy experiments, the evidence for quarks and gluons is in the form of jets of large number of hadrons moving in close directions or heavy quarks decaying into known particles. No one has seen any fractional charge.
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45. I would like to understand more about how information about the structure of the early universe can be gleaned from the CMB
How can physicists tell from a bunch of microwave noise that its constituent photons originate from some particular place and time?
I am guessing that you could detect something like red-shifted atomic spectra and deduce an age.:
-> a redshift implies a speed (Doppler effect), a speed implies a distance (Hubbles’ law), and a distance implies an age (speed of light)
So I imagine you would need to start by finding a signal in the noise which matches e.g. “red-shifted hydrogen atom spectrum”. but given the amount of rubbish that must e floating around out there I’m surprised that any such signal is still detectable
• I probably have a so many questions I should spend a week writing them down and structuring them somehow…
Is the dark energy during inflation the same kind of balancing act as now; with bosons adding their mass^4 and fermions removing theirs? Why didn’t we have a huge jump in some direction when the Higgs field switched on and gave masses to particles that were massless before?
If the energy density at the end of inflation was roughly that of GUT symmetry breaking, shouldn’t stuff that generically happens in GUTs like magnetic monopoles have been created then, rather than at the start of inflation so they had time to get diluted? Or did inflation start at GUT symmetry breaking and remain constant until… what exactly happened? A bunch of particles were created to even out the cosmological constant but they aren’t related to either GUT or electroweak symmetry breaking?
Also; unitless constants at low energies like the fine structure constant aren’t expected to have a simple explanation, but how about the ratio of Planck scale to inflaton scale? Are there inflation scenarios that give an exact number?
• Oops. This wasn’t intended as a reply. Sorry about that.
• There is no ‘balancing act’ in the universe now; dark energy is in fact not at all balanced by the matter and radiation in our universe, which is why its expansion is accelerating. Likewise during inflation is massively overwhelmed anything in the early universe, the entirely of all the matter and radiation in our universe now is believed to have been created from dark energy at the end of inflation. Interestingly the Higgs field *did* change the pressure dynamics of the universe since it converted ‘radiation-like’ massless particles into ‘matter like’ massive ones, though the details are rather complex.
We are still not entirely sure as to how much energy was available at the end of inflation (though we now have a surprisingly high upper bound.) And our understanding of GUT-like phenomena is still very incomplete. It is quite possible that all sorts of high energy phenomena, from magnetic monopoles to cosmic strings were created in the early universe and we are working on their detection and theories.
• kashyap vasavada
@Pat Ryan: All of this is based on multipole, spherical harmonics analysis, together with some Fourier analysis of the microwave spectra.Technical articles are available on web if you search in google.
• kashyap vasavada
@Pat Ryan: All of this is based on multipole, spherical harmonics analysis, together with some Fourier analysis of the microwave spectra.Technical articles are available on web if you search in google. Sorry. This reply appeared at a different place at first.
46. To Matt. /Kudzu ?? :
Are you saying now that what you said before about a small patch of extended expance of space started to inflate is no longer represent but a pre-speculation status ? Or Not even wrong status ?
• I am not Matt, he answers far better than I could ever hope to.
At the present time we do not know anything about the pre-inflation universe. We do not know if the entire universe inflated or a small patch of it or whether those questions even make sense. Theories about that time are speculative and may not be confirmed for a long time, but should still be considered scientific.
• Robert L. Oldershaw
A little better, but I would say our ignorance extends up until the generation of the CMB. After that we have enough empirical evidence to produce something more than just-so stories.
47. The polarization of the CMB is further evidence our Universe spins about a preferred axis.
It’s not the Big Bang; it’s the Big Ongoing.
• That’s nuts. If the universe spins it would be flat like a spiral galaxy. No observation shows anything like that.
• There are a few problems with that reasoning. Firstly there is no ‘big spiral’ in the CMB. We (apparently) see a lot of small ‘spirals’ at various points in the sky but not ‘universal spiral’. (If it were present we should have easily detected it by now.) Secondly if the universe were spinning (That is the space within it were moving in such a way.) then it would have a far greater effect on the CMB’s temperature then we see.
There are of course many theories that postulate a universe of infinite age, you may wish to look into ‘eternal inflation’.
48. Robert L. Oldershaw
Here is how the illusion of highly accurate definitive predictions is created by model-building in theoretical physics.
In the early phase the theory/model makes a genuine prediction before the initial observations are made. Unfortunately, the genuine prediction usually compares rather poorly with the subsequent observations. For example, someone pointed out that the original prediction for the temperature of the cosmic background radiation (CMB) was off by 400%.
Early failure is no problem in model-building. In the middle phase of model-building one just “adjusts” and tinkers with the model until it “predicts” the previous observational results. When the next round of observations become available, the adjusted “predictions” do much better than the initial predictions, but further adjustments to the model and its “predictions” are required. The iterative process of observation, adjusted “predictions”, new observations, further adjustments to the “predictions, goes on until the observations become ever more consistent and change by increasingly tiny amounts.
Now comes the final phase wherein the predictions have been repeatedly adjusted to fit the increasingly refined observations. The model–builders say: “ See! We made highly detailed predictions and the observational data confirm our predictions perfectly – it’s uncanny how good the fit is!”
The problem is that only the initial mediocre/ poor prediction is genuine. The subsequent “adjusted predictions” are not definitive predictions by any scientific definition. At best, these pseudo-predictions and are highly massaged retrodictions, which have far less scientific credibility.
And that’s the truth of pseudo-predictions in the pseudo-science era.
• “And that’s the truth of pseudo-predictions in the pseudo-science era”. I am an outsider but even I get upset by this kind of assertions. In science, people also learn from trials and errors. Why do you find that so problematic? If you think that finding a good model with a relatively small number of initial assumptions is trivial, do it better if you can! Nobody stops you. It would be much healthier than spitting on other people’s work.
• Robert L. Oldershaw
I am not rejecting model-building as a tool of science.
Sometimes we need this tool to make progress.
I am condemning the treatment of model-building retrodictions as if they were definitive predictions. There is a very important distinction here.
Definitive predictions and testing define the scientific method.
Model-building runs the risk of descent into the Ptolemaic method, i.e., keep adding epicycles until the model fits the data, and never consider alternative approaches.
• I feel that some of what you say is right. Namely we may be in a time when science is adding epicycles. However I think it is unrealistic of you to expect otherwise. If you read: ‘The Structure of Scientific Revolutions’, you will find that ‘adding epicycles’ is part of Normal Science. What would you suggest as an alternative? A hodgepodge of theories where no one gains ascendance over the others? Or perhaps (and as hard as this is, please don’t take this personally) only yours does?
There are many theories out there, only a few are recognized as keepers by mainstream science. Those theories like Special & General Relativity, Quantum theory form the pillars of modern physics.
Now it is possible that dark matter, dark energy, and now inflation are telling us that something is very wrong with General Relativity (GR). However, 1) until an EXPERIMENT blows a decisive hole in GR, or 2) until the 21st Century’s Einstein comes along with a wonderful theory that neatly explains all these things or more likely (1) and then (2) you should not expect anything but epicycles.
• Vincent Sauvé
As I see things the issues of the hypotheses of dark energy, dark matter and inflation stand separately from general relativity. GR is rather simple and strong at its core like a sturdy dinning table. The plates being placed on that table may or may not be edible.
49. I completely agree with Robert here. I’ve seen the same thing going on over many years of observing the field. I’ve also read from commentators who have pointed this out, but as I recall those were the critics of the BB, not the supporters.
Another aspect is how ugly the model building became to fit their wrong ideas to the observations. Very few seem attuned to the problem. I can image that real professionals have encountered those thoughts from time to time but probably lacked the wherewithal to change the momentum of the field, or to challenge their professors, or the field leaders.
50. Where a heck is Matt? You are needed here! All kinds of wild speculation are floating around. Please, come back asap!
51. Steve Donaldson
Hello Matt,
The way you distinguish between popular misconceptions about the BBT that have have become entrenched in our culture and what is actually known based on evidence, is very helpful. Thank you.
Could you similarly say something about the concept of entropy? One popular conception is that entropy is the driving force of the universe and indeed is responsible for the BB and the expansion ever since. But entropy is not really a force is it? So how can it be the driving force of the universe? Also, at one time it was popular to say that the universe would end in “heat death” due to the ever increasing entropy. Could you explain what is right and what is wrong about this popular conceptions?
• Edwin Steiner
I also hope Matt will address these good (and big!) questions. Maybe I can answer a small part: Yes, entropy is not a force itself. However, the second law of thermodynamics, which says that in a closed (isolated) system entropy will always increase until equilibrium is reached, can cause effects that appear as if a force was acting. One example is osmotic pressure.
The conventional explanation for the second law is that entropy is a measure for the probability of a state – or for the number of possible states that look the same when seen through smudgy glasses – and that a system with a large number of particles is just enormously more likely to change from one state to a more generic state (higher entropy) than to a more special state (lower entropy). Thus entropy (almost) always increases.
What does this mean for the universe? If the universe can be regarded as a closed thermodynamical system and if it makes sense to define an entropy of the whole universe and if gravity or unknown physics do not change the story, then we can expect the universe to reach thermal equilibrium in a very very far future and thus become tremendously boring.
I have no idea about those “if”s. Entropy is one of the most subtle concepts in physics and it has bothered some of the greatest giants of theoretical physics for ages. I have the feeling that not all has been said and done about it. BTW, I was not aware that there is a popular conception that “entropy is the driving force of the universe”.
• It is very easy to think of many situations where gravity leads to more ordered states. The second law of thermodynamics never included scales where gravity is important.
The concept of time came from thermodynamics, without that, no relativity.
“Time is just an illusion. Einstein told us that.” “What quantum physicists and Einstein tell us is that everything is happening simultaneously.” ding kinematics and “inertia” of momentum, the arrow of time: (1) It is vividly recognized by consciousness.
(3) It makes no appearance in physical science except in the study of organisation of a number of individuals. Here the arrow indicates the direction of progressive increase of the random element. – Sir Arthur Eddington.
• Veeramohan, I disagree with what you wrote and/or Sir Arthur Eddington. The concept of time comes from the motion of matter, not thermodynamics. Relativity can exist independently from thermodynamics. And Einstein had a good definition for time. Please read original sources for more reliable facts. Here is what Einstein wrote on the subject:
The measurement of time is effected by means of clocks. A clock is a thing which automatically passes in succession through a (practically) equal series of events (period). The number of periods (clock-time) elapsed serves as a measure of time. The meaning of this definition is at once clear if the event occurs in the immediate vicinity of the clock in space; for all observers then observe the same clock-time simultaneously with the event (by means of the eye) independently of their position. Until the theory of relativity was propounded it was assumed that the conception of simultaneity had an absolute objective meaning also for events separated in space.
This assumption was demolished by the discovery of the law of propagation of light. For if the velocity of light in empty space is to be a quantity that is independent of the choice (or, respectively, of the state of motion) of the inertial system to which it is referred, no absolute meaning can be assigned to the conception of the simultaneity of events that occur at points separated by a distance in space. Rather, a special time must be allocated to every inertial system. If no co-ordinate system (inertial system) is used as a basis of reference there is no sense in asserting that events at different points in space occur simultaneously. It is in consequence of this that space and time are welded together into a uniform four-dimensional continuum. –Einstein, A., 1992, Fadiman, C., general editor, “Albert Einstein On Space-Time,” The Treasury of the Encyclopedia Britannica, (Viking Penguin, a division of Penguin Books USA Inc., New York, NY), pp. 371-383.
• Thank you Vincent Sauvé, propagation involves invariant sequences between events, which become central for the understanding of time.
At free fall, kinetic energy is increased – means increase in temperature. More heavy, more lowest energy level. Massless (like in inflation) at most lowest energy level (more cold) – then got “reheated”. Which maintain its kinetic energy at the maximum level – what we call as spacetime. The energy densities whose potential energy not fully converted to kinetic energy forms particles, which makes “invariant” sequences between events – making speed of light as reference point ?
The axioms of thermodynamics (like kinematic geometry) were extrapolated to relativistic equations to define the dynamics of time ?
The non-zero value of the cosmological constant (uncertanity principle) represents the point of “reheating”?
The energy density of the vacuum of space is maintained by moment of “inertia”- otherwise the energy allowed by Heisenberg uncertainty principle to briefly decay into particles and antiparticles and then annihilate without violating physical conservation laws – will become Dark-energy star ?
52. sorry spelling mistakes.. at the end… ??
53. Dark Energy is purely vacuum energy, but not purely – what else is it dependent on? It drives inflation and the accelerated expansion of the universe. It does not interact with matter at all. Right? What else?
• kashyap vasavada
@Margot: Since Matt may be busy these days, let me try to answer some of your questions, the way I understand them. Of course, experts are welcome to correct my answers. According to the current model, there was a field called inflaton which gave rise to exponential expansion in first 10^(-35 to -37) sec or so. This increased the size of our observable universe by a factor of 10 ^(90) or so, starting from an extremely small size like 10^(-30) cm or so. All these numbers are tentative. During the expansion or after the expansion (it is not clear to me) whole bunch of particles , quarks, leptons etc were produced by quantum fluctuation. The expansion cooled off the universe to perhaps near absolute zero (?) . Then or perhaps while inflation was going on the potential energy of inflaton was converted into kinetic energy of particles . That process is called reheating.Since that time the universe is expanding at a much slower rate (non exponential, at rate of some power of time). Some time during the first second Higgs field was turned on which gave rise to masses for quarks and leptons.During the last 5-7 Billion years , there has been a new effect called cosmological constant which gives rise to accelerated expansion. This constant is present in Einstein’s equation. I have been trying to find out if people think that this constant is somehow related to the original inflaton from this and other physics blogs. But it seems that in case there is a connection, no one knows at this point. BICEP2 may have verified existence of inflation and the prime cause inflaton. But it remains to be confirmed. Cosmological constant has been verified and three scientists won Nobel prize for that discovery.
• Thanks, Kashyap. I knew some parts of your answer already, but good to hear them again. So coming back to dark energy – it is in part vacuum energy, in part inflaton field energy? Yes, I know that dark energy is the cosmological constant which Einstein to his own dismay needed to add to his equations. I know it’s also called quintessence. I guess since Matt and others here are reluctant to go beyond the inflaton field or quantum jitter maybe as still acceptable and talk about the quantum vacuum state, my question might not be answerable. Most of what you wrote I knew, Kashyap. Matt explained it quite well:-) in his history of the universe.
Dark energy fills the universe to two thirds, I read. Such a mysterious, in my eyes, power!! I wanted to learn more about it.
By the way, you have probably all heard about the new arxiv paper by James B. Dent, Lawrence M. Krauss and Harsh Mathur arxiv.org/abs/1403.5166 : Killing the Straw Man: Does BICEP Prove Inflation?
“However, while there is little doubt that inflation at the Grand Unified Scale is the best motivated source of such primordial waves, it is important to demonstrate that other possible sources cannot account for the current BICEP2 data before definitely claiming Inflation has been proved. ” – Dent, Krauss, and Mathur (arXiv:1403.5166 [astro-ph.CO])
• kashyap vasavada
@Margot: Yes. There are lots of controversies about interpretation of BICEP2 and even that experiment has to be confirmed by BICEP3 or Planck etc. But this is all part of how science progresses and the controversies are very healthy. There is more agreement about cosmological constant (CC) than about inflatons. We know about CC (it could be a quintessence field also, dark energy) from well established analysis of well established experimental data using CMB and lots of other research. The main reason is that we are talking about what is happening *now* rather than in the first second. I suppose in the beginning there was vacuum and inflaton field turned on. So that makes it vacuum energy or dark energy. Most scientists may not want to speculate about whether there was any other energy. There was this inflaton field obeying certain equation and there was potential energy associated with it. But that is it as far as anyone wants to say anything about it!!! Let us see if Matt says something about these things in next few days.
• As far as I understand, quantum (vacuum/) fluctuations, i.e. dark energy, produced fluctuations or density perturbations in the inflaton and these small density perturbations grew via gravitational instability to form the large-scale structures observed in the late universe. Is that correct? Quantum fluctuations during inflation induce a non-zero variance for fluctuations in all light fi elds (like the inflaton or the metric perturbations). ”Light” here means…..? Read this in TASI Lectures on Inflation by Daniel Baumann, 2009 (rev. 2012)
• kashyap vasavada
@Margot: This is reply to your latest comment, but there,there was no reply button. Well, I am not qualified to criticize Daniel Baumann!!! So we have to wait for someone more qualified to say if what Baumann says is right!
• I might even not have expressed what he says correctly….. that could have happened as well. But I would understand that.
Description of ancient subjective science of consciousness:
From the unmanifest, totally abstract transcendental field of Being, the first manifestation of creation is the self-illuminant effulgence of life. The second step in the process of manifestation is the rise of vibration. Just before the beginning of action, just before the beginning of the subtlest vibration, in that self-illuminant state of existence, the self-illuminant effulgence of life, lies the source of creation, the storehouse of limitless energy.
• May Indra knock some sense into Margot with his thunderbolt.
May Agni burn the “subjective science of consciousness” nonsense to ashes and may Vayu blow the ashes away.
54. Robert L. Oldershaw
The latest Planck results indicate a slower accelerated expansion than was previously determined from supernova data.
The Planck mission also confirmed a surprising dipole anisotropy in the CMB temperature fluctuations.
The cosmological constant is only one of several theoretical explanations for the late period of accelerated expansion.
Reality is so much more complicated and uncertain that the just-so stories of the currently fashionable liturgy.
• kashyap vasavada
@Robert L. Oldershaw. I am just repeating what seem to be some kind of peer reviewed theoretical models. If you have rival models, then let them go through peer reviewed process. BTW, dipole anisotropy is no mystery. It is due to the motion of solar system with respect to CMB.
55. Robert L. Oldershaw
(1) Done, many times over.
(2) Newest Planck results are not well understood. Go to arxiv.org or more information on these observational results and their implications. The dipole anisotropy I am referring to is in addition to the old motion of the Solar System Doppler anisotropy.
56. Steve Donaldson
@Edwin Steiner, thanks for your response. By “driving force of the universe” I mean that entropy was at one time said to be the reason why things happen. Is was said to be responsible for “the arrow of time” and thus the ultimate explanation of cause and effect. Cause precedes effect and the effect is that entropy and disorder must always increase over time. The concept of entropy, and its inevitable increase, are as engrained in our popular culture as the concept of the Big Bang itself. It was, and I think still is, a big deal for (some?) cosmologists. I get the impression that for particle physicists, entropy is not such a big deal. I was hoping to find out what the physicists of today think of entropy. What is its proper role in cosmology and the Big Bang Theory? Specifically, should we say the universe is expanding because the second law of thermodynamics is true, or should we say the second law of thermodynamics is true because the universe is expanding?
• This is actually an interesting area of research. By and large the interactions that are studied by particle physicists are time symmetric; they look the same if time is run backwards. (Velocities are reversed of course, but otherwise…) But on large scales the universe is *not* time symmetric. (A smashing cup looks very different to one suddenly forming out of shattered fragments.)
The problem quickly gets bogged down in defining time itself and why it proceeds only in one direction. (The universe is expanding? You mean it gets bigger over time? Why can’t we reverse time and say it gets smaller instead?) An interesting discussion of this for laymen is Sixty Symbol’s ‘Arrow of time’ video: https://www.youtube.com/watch?v=9VFGuupXwng
• Steve Donaldson
@Kudzu, Thanks for the video link of Sean Carroll talking about the arrow of time. I see the meme of entropy defining the arrow of time is still alive and well. However, I am not sure this kind of talk is serving the interest of science. In my opinion, this is an example of how a simple concept can be made more complicated than it is and in the process make science seem more mysterious (or even mystical) than is warranted. Sure, it is great entertainment, but I think the mission of science ought to be to clarify, not mystify. That is what I find so valuable about Matt’s articles such as this one on the BBT – he separates the entertaining hype from the hard science.
We all know what time is. It is what distinguishes yesterday from tomorrow. As the questioner in the video put it, time flows from the past to the future. Sean was making the argument that on the microscopic level, there is no direction of time, while on the macroscopic level there is a direction of time. This is a misunderstanding of what time is. By definition, time has direction on all levels – microscopic and macroscopic. The fact that the equations for microscopic processes are symmetrical with respect to the time variable and the equations for macroscopic processes are asymmetrical does not mean time can reverse itself for the former and not reverse itself latter. Time never reverses itself, by definition. Processes can reverse themselves, but time cannot. Witnessing an omelette unscramble itself and assemble itself back into an egg, would not be witnessing time running backwards. It would be witnessing time running forward and a process running backwards. Time is a prerequisite for process.
I think it is time for talk of entropy to be removed from cosmology.
• Edwin Steiner
Regarding “the mission of science ought to be to clarify, not mystify”: Yes, but there is profound clarity and false clarity. If you claim that time or entropy are simple concepts, I think you are going for false clarity.
We all have the perception of time passing, but that does not mean we know what time is (by which I mean how to precisely describe it and which role it plays in a fundamental description of nature). In your comment you seem to define time by this mental time arrow of ours. But where does this sense of direction of time originate if our consciousness is an emergent phenomenon based on fundamentally time-reversible processes?
The connection between the increase of entropy and the mental arrow of time is more than a meme. As far as I know it can be shown under some rather generic assumptions that storage of information is always accompanied by an increase of entropy. If you accept that storing and retrieving information is part of what makes up our consciousness, the mental arrow of time must be identical with the thermodynamic one. (If you do not think of consciousness as an emergent phenomenon on the other hand, then you face the question why these two arrows are aligned and how our mental time is related to the physical time.)
As cosmology deals with a huge physical system with an enormous number of constituents, statistical mechanics and thus entropy naturally come into play. Even if it may not always be clear how to define or apply it, the concept of entropy will not be removed from cosmology any time soon, I think.
• Edwin, I provided some quotes with solid definitions for time and space and explained where we should see time and space in absolute terms and in relative terms in an essay I wrote. If you are interested please see:
https://sites.google.com/site/bigbangcosmythology/time. There is nothing mathematical in it and I’ve made it very easy to follow.
• Steve Donaldson
Edwin, I agree that one can err on both sides of the clarity target. As Einstein is said to have said, “Everything should be made as simple as possible, but not simpler.” However, I still hold that talk of entropy in cosmology is erring on the side of introducing more confusion than clarity, at least in popular media. I believe the term invites undue philosophical interpretation and unfounded conclusions in much the same way that talk of “the singularity at the beginning of the Big Bang” does. For instance, one might cite the second law of thermodynamics as a reason why the universe is expanding and why it must expand forever and why it can never reverse itself back into a “Big Crunch”. I think this kind of reasoning is a misuse and a misunderstanding of thermodynamics. An authoritative article separating what is true and useful about the concept of entropy and what is misleading and misapplication, would be very helpful. The interest of science would be well served if we could clean up the messy thinking the concept of entropy has spawned.
• Vincent Sauvé
I agree. This would be a good subject for Matt to address someday.
• Have you read Sean Carroll’s “From Eternity to Now”? It’s the most thorough examination of the question of entropy as it relates to cosmology that I’ve come across (though you may or may not agree with his tentative conclusions).
• Steve Donaldson
@papanca2, I have looked at but not read Sean Carroll’s “From Eternity to Now”. Best I can tell at a glance, he has reiterated all the popular memes concerning time, entropy, relativity, and cosmology. What put me off from reading the book is that I could not agree with his very first premise as stated on page two of the prologue, “The most mysterious thing about time is that it has direction; the past is different from the future.” My immediate response was, no, that is not a mystery of time. That is the definition of time. If the past were not different from the future, what would it mean to speak of time. The fact that time has direction is natural and expected. For time not to have direction would be more than mysterious – it would be meaningless. Asking why time has direction is like asking why counting numbers increase in the positive direction.
The error, as I see it, is that Carroll is conflating time with events and processes that happen in time (a process being a sequence of events ordered in time). The proper question to ask is 1) why do events that happen, happen, and 2) why do they happen in the order that they happen. This is precisely what all of science, in one form or another, is tasked to find out. The answers are coming. The answers are coming and they are coming with a deeper and deeper understanding of how nature works on a more and more fundamental level.
As for the apparent conflict between the laws governing the ultra-small and the laws of thermodynamics which govern the larger, I think it is fair to say that thermodynamics is an incomplete theory. It does not provide a complete picture. It is this incompleteness, not some mystery of time, that is the true source of the apparent contradiction.
• @Steve Donaldson: Thanks for your thoughtful reaction to Carroll’s book (even though you admit to not having read it!) I don’t know enough to be familiar with “all the popular memes concerning time, entropy, relativity, and cosmology.” Do I understand that you view time as something “real”, in a sense “absolute” (a la Smolin)? I used to feel as I think you do, that past is past and future is future. But what does one make of the problem of simultaneity for different observers?
I like the distinction you make between the omelette “unscrambling” itself as an example of a “process running backwards” and not of time running backwards. That has been my intuitive reaction to such examples of processes (not time) reversing themselves.
• Steve Donaldson
@papanca2: Yes, I think it is safe to say time is as real as the Universe itself. And yes I am a big fan of Lee Smolin. I have read his first three books and I am working on his most recent, “Time Reborn”. I don’t agree with everything he says, but by and large I think he is on the right track. As for relativity, that is another whole can of worms, that I am not sure I want to open right here. Let me just note that Einstein was not suggesting time was an illusion or that it could run backwards. He took a very practical approach, limiting talk of time to time that can actually be measured. It is the measurement of time (which is a process) that can be affected by your frame of reference. So yes, clocks can and do run faster or slower in different frames of reference in different circumstances and the equations that predict those time differences have been extensively tested and found to be true. No argument here.
57. I agree with Matt about being prudent with the BICEP2 results. For example, why did PLANCK2013 NOT SEE THESE RESULTS?
• Torbjörn Larsson, OM
It may have. The full polarization data isn’t released yet, now delayed into October/November due to difficulties.
There are papers on arxiv that pick up the signal from WMAP and Planck by some non-standard procedures FWIW.
58. Robert L. Oldershaw
The Eternal Inflation multiverse introduces two important concepts into cosmology. Firstly, it finally recognizes that our observable universe is probably only a drop in the infinite ocean of the Universe. Secondly, it proposes that the structure of the infinite Universe is fundamentally fractal.
At least, that’s what Linde has said for a decade. Of course, fractal structures can come in a large variety of different forms, but a crude start is still a start. Better than pacing forever in a cage whose bars are merely an illusion caused by a lack of imagination.
59. kashyap vasavada
Interesting debate about direction of arrow of time, 2nd law of thermodynamics and entropy increase. As far as I can tell, no one (including Sean Carroll) has solved the problem. Why a small (or large) fireball of immensely high temperature and high density can have entropy smaller than today defies explanation.And in case of big crunch whether entropy will decrease again also is not clear. BTW, time reversal invariance of microscopic laws of physics has nothing to do with your philosophical belief whether time flows or not. It is merely a mathematical operation t -> -t in the equations and subsequent invariance. Yes. It would be nice if Matt writes a blog on this.
• The expansion of the universe generates the arrow of time i.e. increases the degrees of freedom (entropy)
• Even hot atoms traped in a container of dimensions comparable to the number of atoms is ordered i.e less entropy
• kashyap vasavada
@Stuart: Your first answer will have problem with time reversal invariance of microscopic laws. After all in the spirit of physics, macroscopic laws should follow from microscopic laws. About the second (less entropy), are you sure this works out mathematically? You need believable size of the container and the correct number of atoms. Do you have a reference?
• The early universe although in a high energy state is dense and compact leaving little or no room for particles to randomly move. As expansion comences more room is created and entropy increases. I do not see a violation of time reversal invariance.
• PS for reference read about negative temperature or population inversion. I also have a paper accepted for publication in the peer reviewed journal IJGMMP on Quantum gravity, DE and DM which may help clarify issues regarding the arrow of time and inflation.
• kashyap vasavada
@Stuart: OK. Please provide a link to your paper.
• @kayshap The paper is yet to appear online will do so as soon as it does but you find it behind a paywall £20 high.
• Quote
”Quantum Cosmology offers a unique explanation of how all the change displayed by Nature has its basis in the unchanging, self-referral state of the Unified Field, by showing how the quantum-mechanical correlations between observer and observed give rise to the emergence of a sequentially evolving universe.
Quantum Cosmology starts its investigations with a formula called the wave function of the universe, which actually describes both the quantum regime of Cosmology where the classical universe has not yet emerged out of the Unified Field, and simultaneously the emergence of a classical universe. The actual formula of the wave function of the universe does not contain the notion of time and thus can be understood to be beyond any change in time – and this just signifies the completely unchanging nature of the Unified Field.
The wave function of the universe can be seen as the formula which describes how all possible states of an infinite number of vibrational modes of the unified Field –also called degrees of freedom of the Unified Field – are coexisting to form a kind of a quantum-mechanical state of superposition. Simultaneously, from a different viewpoint, the wave function of the universe expresses the quantum-mechanical correlation between selected sets of these vibrational modes of the Unified Field – and in Quantum Cosmology this is the origin of the interplay between observer, observed and observer-observed-relationship. One aspect of the infinite number of vibrational modes of the unified field assumes the role of the object of observation, and another part the role of the observers (Information Gathering and Utilizing Systems, IGUS. The observer-observed relationship is the n quantified through the quantum-mechanical correlations contained within the wave function of the universe. There is however no fixed assignment that invariably determines, which is the observer and which is the observed, rather there is an infinite number of different ways to make suitable selections.
The conceptual emergence of a classical universe evolving in time out of this unmanifest state of the Unified Field is now achieved by a further step.
Sequence of states of the object of observation which are directly step-by-step correlated to a corresponding sequence of state of characteristic aspects of the observer. The conceptual location of such a sequence of states gives rise to the notion of time and thereby creates the impression of a universe which evolves and changes in time.
The emergence of the evolving universe is only a conceptual notion that results from a kind of ignorance through which the totality of the infinite dynamism of the Unified Field is forgotten in favor of isolated aspects of this dynamism –as formalized in Physics by so-called conditional observations. In particular, the real infinite quantum-mechanical correlations between observer and observed is neglected or reduced to such a degree that the impression of an independent, objective universe can arise (appearance of a snake in a string).
Quantum Cosmology furthermore confirms that the whole universe emerges out of the Unified Field through its own eternal process of self-observation.
(…)creation arises from and always remains submerged within the eternal silence of the unified Field. The display of this phenomenon is in the eternal galactic dynamism within the eternal empty space of the universe.”
60. •”However, quintessence and phantom fields are still more problematic; therefore the explanation based on the dynamic quantum vacuum could be the more simple and natural one,” Sola said.
• “What we think is happening is a dynamic effect of the quantum vacuum, a parameter that we can calculate,” explained the researcher. The concept of the quantum vacuum has nothing to do with the classic notion of absolute nothingness. “Nothing is more ‘full’ than the quantum vacuum since it is full of fluctuations that contribute fundamentally to the values that we observe and measure,” Solà pointed out.
• Spyros Basilakos, Joan Sola. “Effective equation of state for running vacuum: “mirage” quintessence and phantom dark energy”. Monthly Notices of the Royal Astronomical Society 437(4), February 2014. DOI:10.1093/mnras/stt2135.
• @Margot the value of the cosmological constant has been calculated from Quantum gravity to be lambda=3(E/hc)^2 where E =hH
here H is the Hubble constant 2.3 x10^-18 s^-1
and h Planck constant
The cosmic expansion is a result of the emission of a graviton of least energy E=hH by spacetime
• Margot, “the Self” or consciousness were the sedimentary sediments of adopted informations”. Like RNA, the oldest data storage medium. These are the past leftovers of Evolution. If you remove aside those informations, there is selfless consciousness. The dynamics of those informations increases the entropy, contradictions and conflicts.
“Storage of informations is always accompanied by an increase of entropy. Stroring and retrieving information is part of what makes up our consciousness”.
The fundamental law of quantum mechanics is that the “Evolution” is linear, meaning that if state A turns into A’ and B into B’ after 10 secondy, then after 10 seconds the superposition Ψ turns into a mixture of A’ and B’ with the same coefficients as A and B. Quantum superposition, a property of Schrödinger equation.
But “The vacuum expectation value” is bedlam here, not having universal arrow of time – the arrow of time is inertial. But there may be also entangled part of vacuum expectation value, whose “amplitude interferes” and allows energy to briefly decay into particles and antiparticles and then annihilate without violating physical conservation laws.
Einstein envisaged the rest mass and cosmological constant for a static universe. I could not understand its reverse evolutionary cauchy? http://arxiv.org/abs/1311.0700
“Their developments backwards in time induce a set of standard Cauchy data on space-like slices for the Einstein-massive-scalar field equations which is open in the set of all Cauchy data for this system”.
So information and entropy is evolutionary only in some conditions (relative) like “un-naturalness” ?
• @veeramohan Consciousness is the Self (Atma). Self is no information. But otherwise I agree. Consciousness is only the screen on which information (events) appears. But in itself the screen is untouched by it. The events are different in different states of consciousness. In the end those events come about by forgetting the totality of the infinite dynamism of the Unified Field in favor of isolated aspects of this dynamism –as formalized in Physics by so-called conditional observations. Events thus are conditional operations in reality, a certain perspective which blends out or ignores relations. A mistake of the intellect – which is exposed when consciousness is fully developed.
61. I have tried to figure out the time schedule:
Let´s say the Very First Beginning (the Mother of all Beginnings=MAB) occurred at 12.00 noon (whether it is true or not, we simply do not know). Time between noon and one o´clock unknown.
At one o´clock pm a tiny part of the universe started to inflate and expanded with exponential speed (for the reasons we do not know for sure). The expansion was so rapid that it soon cooled out the universe to an extreme coldness.
All the fields of the universe have been present since the MAB. So at this low temperature the Higgs field must have been “on”. Question: were there in the first place any particles present at this time? If yes, all the particles (the ones that feel the Higgs field) should have mass? But how is this possible, as there was no symmetry braking of electroweak interaction at this stage?
At half past one the inflaton field started to slow down and thus the temperature got higher. The energy which the inflaton field lost at this stage, was turned to the birth (?) and kinetic energy of different particles. As the clock approached two, the universe (our part of it) was heated up to a temperature corresponding energy level of 10^16 GeV. Now around half past one the Higgs field got enough of energy in order to turn “off” (it needs energy in order to be “off”, zero value).
At two o´clock the inflation (or at least the most of it) stopped (all the energy, or almost all of it, had gone to the birth of particles). The universe again cooled very rapidly. Because the Higgs field was now again “on”, it resulted in the symmetry braking of the electroweak interaction (and hence some particles remained massless and some got restmass).
Not all of the inflaton field (dark energy) was gone but is working even today and expanding the universe. Although the energy of this field is very tiny it has won (because of expansion) over gravitation some 5-7 billion years ago, and hence the accelerating speed of expansion.
• Vincent Sauvé
Is this really any better than the cosmological creation stories we told each other hundreds and thousands of years ago? In my humble opinion today’s stories are not a real advancement over any of the thousands of other stories we have created over the generations. Yet, every generation thinks their story is the right one.
I recall years ago informing a women up the hill at the Chabot Space and Science Center here in Oakland California (where I would commonly bring my telescope to share the views) of my alternative views on cosmology. She then expressed her sadness that my view of an infinite non expanding universe, if correct, would be a let down from the point of view of a great story. She really was preferring a great creation story over my possibility.
• kashyap vasavada
@Vincent Sauvé: Well! As for which story is better(!) we can point to the success of physics’ method of making mathematical models and comparison with experiments. This has been working since the time of Galileo and Newton. Matt has emphasized this point repeatedly. Any one can sit in his/her room and scribble some wild ideas. What makes them acceptable is the peer review system and eventual confirmation by experiments. There may be lot of confusion from time to time. Beginning of 20th century was such a time. Current confusion is somewhat similar to that period. Eventually the best theory will win. I do not have any doubt about it. The cell phone in your pocket (if nothing else) is an indication of success of scientific method. It may be harsh to say but one has to say that if some theoretical idea cannot be put in a mathematical form it can be safely ignored!
• Well said kashyap 🙂
• Kashyap, I love the scientific method. At issue is what is being speculated by many in this thread doesn’t really fall into the category of science. And some people here can’t even form a real sentence or a thought. Throwing around physics jargon is not science. Theoretical ideas must be grounded to real physics, not to mathematical make-believe.
While my opportunities to observe the behavior of scientists mainly comes from what I’ve learned by way of books, I have observed engineers in action and have witnessed colossal mistakes, including biases that turned out to be wrong that led to wastes of time and products that weren’t their best. In the field of smartphone technology we are all amazed at how great they have become, but let’s not ignore the fact that the ones that make it to market are the result of many trials, and the filtering out of failures, and the filtering in of what the public likes. Cosmology is also like that, and the public likes a good story, and if it is dressed in the clothing of science all the better in this new world of high technology.
62. @Stuart – I liked the definition for dark energy of being a ”dynamical quantum vacuum energy”.
• In my work on quantization of spacetime, gravity and the quantum vacuum I find that the graviton is inextricably linked with an element of the quantum vacuum. This graviton is different from the one discribed by the standard model.The wave function of this graviton is a wave packet that spreads in quantized time evolution steps by emitting a low energy graviton of energy E=hH.This is the dynamical aspect of the quantum vacuum in my theory.
• Moreover Hubble’s law and MoND can be derived from this dynamical aspect of my theory.
• Sounds wonderful! Unfortunately I am not in the situation to comprehend in detail what you have written. Did you read the paper by Basilakos and Sola? Did they define dynamic vacuum in a similar way?
63. Matt or Kunzu,
In my schedule speculation I made some questions (look above). Is this my schedule, of what “has” happened, the right one as understood today?
• At the present time we do not know if there was a ‘mother of all beginnings’ even one ‘infinitely long ago’. But this can be ignored.
We are also not *completely* sure that during inflation fields worked as they do now; the inflaton field would have (as yet) unknown effects. In the same way the fact the Higgs field is now ‘on’ affects the very basics of our universe the inflaton field being ‘on’ during this period also affects things.
We are not sure whether there were any particles at this time; that would depend on the state of the universe before inflation. If there were they wouldn’t matter as they are so quickly diluted.
The situation of these particles is interesting. Inflation is an incredibly short process; many particle interactions or decays would simply not have time to occur. Also since space itself is expanding faster than light all particles are totally isolated, they will not be able to interact with any other particle. This includes interacting with the Higgs field and in any case mass is a rather tricky concept when dealing with a particle in this situation.
At the core of your question is a misconception; electroweak symmetry breaking is not a one way street, an irreversible transition. It is more like water freezing; it reverses at high enough energy. As such it is quite possible for the symmetry to be broken during inflation then to reform at the end of inflation before breaking again. It may even be possible one day for humans to collide exceedingly high energy particles and watch electroweak symmetry reform in the lab.
Interestingly you state as much when you say ‘Now around half past one the Higgs field got enough of energy in order to turn “off”’ so I am not entirely sure on your understanding of this subject. You are correct in saying that the energy of the inflaton field turns into both kinetic energy and the birth of particles. The precise temperature cannot be larger than 10^16 GeV, but it is possible for it to be *lower* for various reasons. Inflation ‘stops’ at this time, not after.
The exact relationship between the current dark energy and that if inflation is not entirely clear and we will need more information on both before the link becomes apparent.
• /It may even be possible one day for humans to collide exceedingly high energy particles and watch electroweak symmetry reform in the lab./–
At the technical level the spontaneous symmetry breaking means just that the lowest energy configuration, the vacuum, does not have the symmetry of the Lagrangian. A particular vacuum to be realised in nature is chosen spontaneously. In the sense, that there can be another vacuum configuration chosen.
Here comes the masslessness of photons. Masslessness was realized at lowest possible potential energy and highest possible kinetic energy. the possible more potential energy is “heated (decrease in entropy) up” (don’t know why?) – thus forms the reference frame to define mass-energy (or mass).
The angular momentum (mass x velocity) “c^2” in E = mc^2 represent this potential energy (or rest mass).
The term “Spontaneos symmetry breaking’ is used in statistical physics for the change of symmetry of the ground state of a system due to the change of its TEMPERATURE, i.e., not due a change of a parameter in the Hamiltonian.
As far as cosmological SSB goes, many effects are clearly due to virtual energy borrowed from the quantum foam, tipping a precipitation down one particular way and at a particular time rather than down another route.
Gauge fields spontaneously interact with scalars because they are not singlets.
A good example from our daily experience is a round dinner table. At some point you have to make the choice whether you take the glass to your left or to your right. Once someone selects one, all others at the table have to follow. The left-right symmetry is broken. 🙂
So the mass in momentum (mass x velocity) is removed by high energy collision – means, the symmetry of lowest energy level of gauge fields will be retained, like masslessness of photon field – creating “new space” by leftover velocity – like in dark energy star ?
• OK Kudzu, thanks a lot for your kind answer. Seems there are a lot of uncertain things prevailing
OK thanks. So the electroweak symmetry breaking might have happened twice.
While the lowest value of the Higgs field is nonzero, it needs energy in order to be zero. At most of the inflation time the Higgs field was “on” (cold) and the particles had mass, but at the end of inflation the field turned “off “ (hot) and the particles turned to be massless, and then again at electroweak symmetry breaking “on” (cold) and the particles gained again mass and that situation has stayed “on” since then.
But as you say, in spite of the Higgs field being “on” or “off” the particles had no chance to interact with the field during inflation. The question of mass at this stage is tricky. Might there have been other alternative ways for particles to get mass (inflaton field itself)? Or is the question of particle masses totally irrelevant here?
• There are many, many theoretical possibilities covering the period of inflation and what they do to particles. The problem with questions about this time period is twofold, firstly we don’t know everything (or even very much at all) about this time so any answers we give are only best guesses, if that.
And secondly a lot of things we take for granted in everyday life become rather more difficult to define and describe under such conditions. For example, to define a distance we need to be able to measure it, but these particles cannot be measured during this time so do questions of distance and speed mean anything? Or are they the same as asking what the color green sounds like?
64. What does the Higgs field and a Catholic priest have in common: they both give mass.
• James Gallagher
I’m a big fan of electrons, but positrons, well, that’s another matter
• If both of its polarization had high energy like gamma rays, they travel in opposite direction and one of them enter dark matter, then they were entangled. The chaos inside dark matter also affect another one. This will disturb the harmony of vacuum energy, creating the VEV to “pull” other energy densities to lowest energy level – making spontaneous symmetry breaking. So the rest mass constancy inside our 3D spacetime could be changed by very high energy gamma rays.
• Space expansion is evaporation of dark matter into dark energy (like black hole evaporarion – takes billions of years).
Changing of mass energy into vacuum energy like dark enetgy star – creating “new space”, leaving the movementum (responsible for mass) to create volume for accommodating the new space.
65. Robert L. Oldershaw
Someone commented recently that we are in the era of data-driven cosmology and that the standard model agrees with “ALL” of the data.
A few facts might be helpful here.
In model-building, as opposed to theories of principle, the model fits the data virtually by the definition and modus operandi of model-building. The model is constructed to FIT the data, and ad hoc modifications (like epicycles) are added whenever new data “require” modifying the model.
That does not by any stretch of the imagination mean that the model accurately represents how nature actually works. Think of the Ptolemaic “universe”; it fit the data quite well, but it was complete rubbish CONCEPTUALLY.
The only scientific way to demonstrate that one’s model accurately represents nature is through the predictions/testing steps of the scientific method.
In light of this science 101, the Big Bang is not on very solid grounds. Global expansion is on strong grounds, but there is no justification for an acausal “beginning to the entire Universe”; there is no explanation of what the inflaton field is; there is no coherent and tested explanation for how the temperature could be very cold at the “beginning” and then suddenly become very hot for the hot Big Bang; extrapolating from the polarization of the CMB observed NOW to the just-so story about what was happening at 10^-35 sec is laughably speculative; there is no well-accepted explanation of the “recent” accelerated expansion, or Planck’s new unexpectedly slower quantification of it; dark matter (i.e., nearly all of the matter of the Universe) continues to be a complete mystery; the Planck mission recently confirmed an unexpected directional anisotropy for the observable universe; … . Should I go on for a few more pages?
Let’s try to show less hubris and more scientific humility about what we know and what we do not know. Let’s not be: “Often wrong, but never in doubt”. Let’s be scientific.
Robert L. Oldershaw
• Robert, thank you for the post about model building. I am in agreement with you on that. Where I don’t necessarily agree (from one BB critic to another) is the statement that global expansion is on strong grounds. Do you happen to have a list handy of reasons why global expansion should be considered to be on strong grounds? If you do, I would like to provide a rebuttal.
• Fixed ideas are the enemy of scientific progress.
A balance of open-mindedness and skepticism is the ally of scientific progress.
• I agree with you— The big bang, as presented in the SM, is a house of cards balanced on the point of a pin. The big bang notion has inspired a lot of clever ideas and encouraged the development of useful math techniques; while at the same time granting validation to a number of rather silly schemes to accommodate a big bang with scientific observation.
66. Quote: ”By virtue of being awareness, transparent to itself, consciousness emerges from within its pure potentiality and, curving back on to itself, establishes an ‘observer-observed’ relationship within its own structure. This process of consciousness becoming aware of itself creates an unmanifest space-time geometry within the field of consciousness. The unmanifest space-time curve within the field of consciousness is at the source of space-time curvature, which Einstein’s general theory of relativity shows to be the basis of all objective creation.”
from some of the information you provide here. Please
let me know if this alright with you. Regards!
68. In AWT the red shift is the result of light scattering at the density fluctuations of vacuum and the Big Bang did never happen. After all, the FLRW metric is inverse metric to black hole and we aren’t saying, the inflation happens at the event horizon. It’s all stationary. The omnipresent inflation brings Einstein expansion paradox and it can be explained with light wave scattering together with dark energy. This model brings testable predictions, by which it can be tested (like the dependence of red-shift to wavelength of light). And it remains fully physical and it doesn’t bring any ad-hoced assumptions about past of Universe.
69. Professor Strassler’s articles are always fantastically good, but they always leave unanswered questions AND generate new questions that he does not have time to answer (we all need to work sometimes for a living from our day job). Some of the unanswered questions I have (as, in an inflating universe with only a tiny speck of mass, where did a universe’s amount of mass come from?) are neatly answered by Max Tegmark in a free excerpt from his book, “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality”, in which he gives an explanation of inflation. You may read the free excerpt at http://space.mit.edu/home/tegmark/pdf/inflation_excerpt.pdf . It has also motivated me to buy his book! Now, when will Matt’s book come out?
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71. “the observable patch of our universe, which may be a tiny fraction of the universe” this “may be” always irritates me and for this reason: as the astronomic community became aware recently, there are minor planets beyond Pluto which have orbits that seem to be disturbed in a way that suggests “out there” is another giant (maybe Jupiter size) planet, as yet unobservable. Unobservable as it may be, it leaves traces that show in the behavior of nearer objects. Now if the universe were indeed larger than our “observable ‘patch'” – and IF there were big objects / dark matter / dark energy, galaxies, black holes etc. etc. just beyond “observable” surely they must interact with what is still observable in such a way as to leave traces just as the potential second Jupiter leaves on observable objects in our solar system? And is this or is it not, reflected in some of the branches of inflation theory?
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517e4f6f0bdacc01 | Saturday, May 15, 2021
Quantum Computing: Top Players 2021
[This is a transcript of the video embedded below.]
Quantum computing is currently one of the most exciting emergent technologies, and it’s almost certainly a topic that will continue to make headlines in the coming years. But there are now so many companies working on quantum computing, that it’s become really confusing. Who is working on what? What are the benefits and disadvantages of each technology? And who are the newcomers to watch out for? That’s what we will talk about today.
Quantum computers use units that are called “quantum-bits” or qubits for short. In contrast to normal bits, which can take on two values, like 0 and 1, a qubit can take on an arbitrary combination of two values. The magic of quantum computing happens when you entangle qubits.
Entanglement is a type of correlation, so it ties qubits together, but it’s a correlation that has no equivalent in the non-quantum world. There are a huge number of ways qubits can be entangled and that creates a computational advantage - if you want to solve certain mathematical problems.
Quantum computer can help for example to solve the Schrödinger equation for complicated molecules. One could use that to find out what properties a material has without having to synthetically produce it. Quantum computers can also solve certain logistic problems of optimize financial systems. So there is a real potential for application.
But quantum computing does not help for *all types of calculations, they are special purpose machines. They also don’t operate all by themselves, but the quantum parts have to be controlled and read out by a conventional computer. You could say that quantum computers are for problem solving what wormholes are for space-travel. They might not bring you everywhere you want to go, but *if they can bring you somewhere, you’ll get there really fast.
What makes quantum computing special is also what makes it challenging. To use quantum computers, you have to maintain the entanglement between the qubits long enough to actually do the calculation. And quantum effects are really, really sensitive even to smallest disturbances. To be reliable, quantum computer therefore need to operate with several copies of the information, together with an error correction protocol. And to do this error correction, you need more qubits. Estimates say that the number of qubits we need to reach for a quantum computer to do reliable and useful calculations that a conventional computer can’t do is about a million.
The exact number depends on the type of problem you are trying to solve, the algorithm, and the quality of the qubits and so on, but as a rule of thumb, a million is a good benchmark to keep in mind. Below that, quantum computers are mainly of academic interest.
Having said that, let’s now look at what different types of qubits there are, and how far we are on the way to that million.
1. Superconducting Qubits
Superconducting qubits are by far the most widely used, and most advanced type of qubits. They are basically small currents on a chip. The two states of the qubit can be physically realized either by the distribution of the charge, or by the flux of the current.
The big advantage of superconducting qubits is that they can be produced by the same techniques that the electronics industry has used for the past 5 decades. These qubits are basically microchips, except, here it comes, they have to be cooled to extremely low temperatures, about 10-20 milli Kelvin. One needs these low temperatures to make the circuits superconducting, otherwise you can’t keep them in these neat two qubit states.
Despite the low temperatures, quantum effects in superconducting qubits disappear extremely quickly. This disappearance of quantum effects is measured in the “decoherence time”, which for the superconducting qubits is currently a few 10s of micro-seconds.
Superconducting qubits are the technology which is used by Google and IBM and also by a number of smaller companies. In 2019, Google was first to demonstrate “quantum supremacy”, which means they performed a task that a conventional computer could not have done in a reasonable amount of time. The processor they used for this had 53 qubits. I made a video about this topic specifically, so check this out for more. Google’s supremacy claim was later debated by IBM. IBM argued that actually the calculation could have been performed within reasonable time on a conventional super-computer, so Google’s claim was somewhat premature. Maybe it was. Or maybe IBM was just annoyed they weren’t first.
IBM’s quantum computers also use superconducting qubits. Their biggest one currently has 65 qubits and they recently put out a roadmap that projects 1000 qubits by 2023. IBMs smaller quantum computers, the ones with 5 and 16 qubits, are free to access in the cloud.
The biggest problem for superconducting qubits is the cooling. Beyond a few thousand or so, it’ll become difficult to put all qubits into one cooling system, so that’s where it’ll become challenging.
2. Photonic quantum computing
In photonic quantum computing the qubits are properties related to photons. That may be the presence of a photon itself, or the uncertainty in a particular state of the photon. This approach is pursued for example by the company Xanadu in Toronto. It is also the approach that was used a few months ago by a group of Chinese researchers, which demonstrated quantum supremacy for photonic quantum computing.
The biggest advantage of using photons is that they can be operated at room temperature, and the quantum effects last much longer than for superconducting qubits, typically some milliseconds but it can go up to some hours in ideal cases. This makes photonic quantum computers much cheaper and easier to handle. The big disadvantage is that the systems become really large really quickly because of the laser guides and optical components. For example, the photonic system of the Chinese group covers a whole tabletop, whereas superconducting circuits are just tiny chips.
The company PsiQuantum however claims they have solved the problem and have found an approach to photonic quantum computing that can be scaled up to a million qubits. Exactly how they want to do that, no one knows, but that’s definitely a development to have an eye on.
3. Ion traps
In ion traps, the qubits are atoms that are missing some electrons and therefore have a net positive charge. You can then trap these ions in electromagnetic fields, and use lasers to move them around and entangle them. Such ion traps are comparable in size to the qubit chips. They also need to be cooled but not quite as much, “only” to temperatures of a few Kelvin.
The biggest player in trapped ion quantum computing is Honeywell, but the start-up IonQ uses the same approach. The advantages of trapped ion computing are longer coherence times than superconducting qubits – up to a few minutes. The other advantage is that trapped ions can interact with more neighbors than superconducting qubits.
But ion traps also have disadvantages. Notably, they are slower to react than superconducting qubits, and it’s more difficult to put many traps onto a single chip. However, they’ve kept up with superconducting qubits well.
Honeywell claims to have the best quantum computer in the world by quantum volume. What the heck is quantum volume? It’s a metric, originally introduced by IBM, that combines many different factors like errors, crosstalk and connectivity. Honeywell reports a quantum volume of 64, and according to their website, they too are moving to the cloud next year. IonQ’s latest model contains 32 trapped ions sitting in a chain. They also have a roadmap according to which they expect quantum supremacy by 2025 and be able to solve interesting problems by 2028.
4. D-Wave
Now what about D-Wave? D-wave is so far the only company that sells commercially available quantum computers, and they also use superconducting qubits. Their 2020 model has a stunning 5600 qubits.
However, the D-wave computers can’t be compared to the approaches pursued by Google and IBM because D-wave uses a completely different computation strategy. D-wave computers can be used for solving certain optimization problems that are defined by the design of the machine, whereas the technology developed by Google and IBM is good to create a programmable computer that can be applied to all kinds of different problems. Both are interesting, but it’s comparing apples and oranges.
5. Topological quantum computing
Topological quantum computing is the wild card. There isn’t currently any workable machine that uses the technique. But the idea is great: In topological quantum computers, information would be stored in conserved properties of “quasi-particles”, that are collective motions of particles. The great thing about this is that this information would be very robust to decoherence.
According to Microsoft “the upside is enormous and there is practically no downside.” In 2018, their director of quantum computing business development, told the BBC Microsoft would have a “commercially relevant quantum computer within five years.” However, Microsoft had a big setback in February when they had to retract a paper that demonstrated the existence of the quasi-particles they hoped to use. So much about “no downside”.
6. The far field
These were the biggest players, but there are two newcomers that are worth having an eye on.
The first is semi-conducting qubits. They are very similar to the superconducting qubits, but here the qubits are either the spin or charge of single electrons. The advantage is that the temperature doesn’t need to be quite as low. Instead of 10 mK, one “only” has to reach a few Kelvin. This approach is presently pursued by researchers at TU Delft in the Netherlands, supported by Intel.
The second are Nitrogen Vacancy systems where the qubits are places in the structure of a carbon crystal where a carbon atom is replaced with nitrogen. The great advantage of those is that they’re both small and can be operated at room temperatures. This approach is pursued by The Hanson lab at Qutech, some people at MIT, and a startup in Australia called Quantum Brilliance.
So far there hasn’t been any demonstration of quantum computation for these two approaches, but they could become very promising.
So, that’s the status of quantum computing in early 2021, and I hope this video will help you to make sense of the next quantum computing headlines, which are certain to come.
I want to thank Tanuj Kumar for help with this video.
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2. It is not my speciality but, like you, I think it is an important new discipline in modern sciences. Presumably we will have to make enorm progresses in controlling the fabrication of superconducting devices at room temperature or and of cooling systems. Once again the technological progresses are meaningful only if they bring advantages, for example in chemistry and-as consequence- in biology and then in medicine (bioengeneery). Otherwise, if the long range purpose is only a military suppremacy... We are actually facing more important and immediate problems, I think !
1. There is a challenge that involves China on this. It is interesting how China has become the bogie-man these days. Their government has been playing dishonestly for many years, but American companies were making profits anyway, so nobody paid attention. It took t'Rump to raise the alarm, but his way of doing this was hopelessly wrong. With all his stuff about Kung-Flu and China-virus we now have a sickening east-Asian hate thing going on.
China has its sights on gaining a monopoly on as many technological areas as possible. They have a lot of world outside their sphere of influence to surpass. Whether they succeed and if the US and EU (UK if there is such a thing before long) rise to the challenge is to be seen.
Russia is also a bit of a challenge, but their military developments are being built on a weak national and economic basis. From the white Tsars, to the red Tsars and now the blue-grey Tsars this pattern has happened repeatedly.
China wants to master a global quantum internet. EPR and other entanglements W, GHZ etc will probably be implemented on fiber optic and U-verse. They may succeed, and the logo image of Huawei looks a bit like a sliced up apple.
It is too bad in a way that this all feeds into power games and militarization.
2. If I am not mistaking Roger Penrose in the Emperors New Mind supposes that the Human Brain might be another type of quantum computer, quite advanced one indeed.
3. While Penrose has written extensively on this and related topics, it's all philosophic speculation with not one whit of actual evidence behind it.
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5. Penrose's idea of humans performing quantum computations is not likely right. For one thing, bounded quantum polynomial space is a subset of PSPACE, which is the set of algorithms that satisfy the Church-Turing thesis --- well modulo oracle inputs. This most likely means the human brain does not perform self-referential loop calculations that skirt the limits set by Godel and Turing.
3. The D-wave computer is an annealing machine. This is a sort of quantum version of a neural network. It is an optimizing system.
Quantum computers are based on linear algebra on a complex field. As such quantum computers only really solve linear algebra problems. The Shor algorithm is a Fourier transform method. A lot of mathematics, and by extension physics, involves linear algebra. Many mathematical theorems are solved by transforming the problem into linear algebra, where the methods are well known.
The quantum computer will creep into the computing world slowly at first and in time will assume some level of importance. There are other architectures that will also assume more importance, artificial neural nets, spintronics, and others. As computing most probably must conform to the Church-Turing thesis it is likely though these systems will be supplementary to a standard von-Neumann computer, such as what we have.
1. Hi Lawrence,
Perhaps neural nets and other AI computing will be among the specialist applications that quantum computing will be used for, with hybrid systems made to increase efficiency.
And WRT China, I think where they lack advantage is in collaboration and information exchange internationally; same with Russia. I personally think the greatest leaps will come with cross-pollination of methods and ideas.
4. If you have some spare time and enjoy a quality Sci-Fi you may find recent "DEVS" series quite enjoying..Long story short a billioner builds a working quantum computer capable of emulating physical Reality; the story is smart, intriguing and the ending is quite unexpected.
5. ion traps - shouldn't be positive charge (instead of negative
1. Yes, sorry, I have fixed that in the text. Can't fix it in the video. It's in the info.
6. Interesting video. So how long do you think before Shor's Algorithm will be run for non trivial cases?
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8. Is Shor's Algorithm the fastest algorithm, or is the mindset on what is logically fastest wrong? I think fixing that issue would be necessary for understanding if quantum or regular computers will be the fastest. Only example I have that can produce evidence of screwing up the notion of run time is Simple algorithm I figured out years ago. Take N odd, and instead of trying to find PQ as a rectangle like kids would do with unit blocks, you find a unit block trapazoid it converges to and then that shape can be broken into a rectangle. While it can be simplified in computer speak, geometry of it is just first try to make a right triangle with the square blocks. If the blocks make a right triangle then break the triangle if its even height in half and flip around and you got a rectangle, which gives two factors. If its odd break off the nose and flip it up and you got a rectangle, which gives factors. But likely there remainder blocks at bottom of triangle, so you take rows off the top and throw them on the bottom till there is no remainder, you converge to a block trapezoid. The run time for this is bizaree. If one takes N odd, and trying to find p or q, but say takes 3N, it can actually often does converge faster.. it can pop out p or q or 3p or 3q or pq. This is because of the geometry, because it will tend to converge to the largest factor first below the initial triangle height. That why increasing the size it can converge faster. Shor's algorithm fits this nice notion of Log N convergence... not this chaos algorithm convergence time of periodic unknown. Its runtime is like rolling a dice and sometimes it will be instant, not matter how large the key size of something like RSA. That is why I think its important for the discussion of quantum computers verses regular computers that the computer science notion of Run Time itself be challenged with evidence like the chaos algorithm above has produced. If we live in an evidence based system, and there out of the blue pops up evidence that run time itself may need core logic retuning, then that needs to be done. I really should publish the algorithm and the data charts for how it defies the notion of runtime. But I can say with evidence that Shor's algorithm while nice and pretty, these screwy unexplored functions at times can beat it. And if a set of these chaos algorithms can be put together and ran at the same time, there possible a high density of constant run time convergence where N size is relatively unimportant. Entire class of algorithms that are unexplored. The beauty of math has no boundaries.
1. Shor's algorithm is the fastest polynomial time quantum algorithm. Non-quantum integer factorization algorithms are exponential so incomparably slower. That's why RSA and majority of all our cryptosystems are in danger if a ~1M qubits size working quantum computer goes online...
2. FWIW, Wikipedia reports that there are multiple cryptographic systems that are not breakable by the known quantum computer algorithms, and that several of these date back to the last century*. It'll be a minor irritation to have to switch to a different system, but my understanding is that software that implements quantum-safe cryptography has already been written and is ready to use.
*: At least two (from the 1970s) predated Shor's Algorithm, and a couple more were invented after that.
9. I am surprised that when a quantum computer needs about a million qubit, Google already declared quantum supremacy with only 53 qubit. If this claim is correct then, with a million qubit, the quantum computer will be really fantastic. Do you agree?
1. FWIW, the "problem" that Google "solved" in achieving "quantum supremacy" was simulating quantum gates. It's not particularly surprising (to me, anyway) that quantum gates are good at simulating quantum gates, but, whatever. The main spokesperson for quantum computing has been very specific in stating that this "problem" and it's "solution" are unrelated to anything anyone would ever want or need to do with a computer, but he insists that (a) it's true that quantum supremacy has been achieved, and (b) it's really good that someone found something to actually do with current quantum computers.
As I said, "whatever".
I will admit, though, that as a 1970s/1980s generation Comp. Sci. type, I'm surprised how few things other than Shor's algorithm have been found that can be done with quantum computers. According to a recent blog post at a Comp. Sci. blog, there's really only one other algorithm. And it's been 25 years.
10. The algorithms which supposedly demonstrate "quantum speedup" tend to have caveats, for example the quantum Fourier transform part of Shor's algorithm would scale well but the exponentiation part which is required to load your number (and test integer) into the QFT is much heavier. There are algorithms which supposedly demonstrate that an oracle can be interrogated once in order to obtain all the information about it, but the oracle is necessarily part of your quantum circuit so you already knew how to program it. They tend to rely on a conditional-NOT gate kicking its phase back to the control qubit if your control and target qubits are not in pure |0> or |1>.
(Generally, even the problem of how to make the most efficient transpilation of a quantum circuit onto real hardware isn't even solved.)
But "entanglement" isn't always necessary, but rather superposition.
(I make material for semiconductor qubits by the way, but I'm not affiliated with TU Delft).
1. For oracle problems, if you put your oracle on one half of the computer chip and the algorithm circuit on the other half, I really don't see why this wouldn't demonstrate quantum speedup.
11. Quantum computing is a scam to pump up stock, the base physics of it is flood/wrong, they will use specialied hardware (ex: Cuda Cores) with AI to get certain calculations done then will call it quantum computer.
Let's look at other things like Light Computing and see what goes on there.
1. Lee, I agree entirely. The 'ideal' they still aim for remains an infinite distance away because the theory behind it is wrong. The Swiss banks had the good sense to turn down Anton Zeilingers proposal for quantum cryptographic security because is was founded on Poppers 'mud'. Those throwing millions into trying to develop true quantum computers may also one day see through the hype. Uncertainty has a real fundamental physical cause (SpringerNature paper imminent) and I suggest it can't be overcome.
2. Hi Lee and Peter,
so you both think Dr. Hossenfelder is mistaken about the current developments, or what?
3. C Thompson,
Those are guys who don't understand how quantum computing works and who also haven't noticed, apparently, that as a matter of fact it does work, and that we know -- again, for a fact -- that the theory behind it is correct (in the parameter range tested, etc etc).
World's full with people who have strong opinions on things they know very little about.
4. Dr. Hossenfelder,
Indeed. I wondered what they thought made them better-informed about quantum computing that you supposedly missed with your well-researched and comprehensive summary, and why they saw fit to comment thusly on this blog, with no evidence to back up their claims, especially Lee's comment.
5. C Thompson,
Yes, you are asking very good questions...
12. 1000000 qubits? Still a lot of questions.
I think I can show even this would live up to the hype with the following thought experiment:
For the sake of discussion, lets imagine we want to break an encryption which uses 100 digit prime numbers for keys. So you would have factor a 200 digit number into a couple of 100 digit primes to break the code.
Now this is going to be pretty tough. Consider how many 100 digit prime numbers there are, and that defines the space you have search with your quantum code breaker in order to find your answer.
Now remember that getting "close" with some kind of refinement process isn't going to hack it. The nature of beast is such that you either find the answer or you don't. What is more, you are constrained by your search because you have to deal with the whole search space directly as a whole, because the information held by your superimposition is only held holistically. You can't look at just a part of the space and hope to find your answer.
So just how big is our search space? A reasonable approximation for discussion purposes is the set of all 100 digit integers. So lets get a handle on just how large this search space is on an intuitive level.
Lets look for our needle in a haystack by getting an idea how large the haystack is. Assume a 0.5 mm x 0.5 mm x 3 cm as our needle size. Now what is the size of the haystack?
The volume of our needle would be 7.5 mm^3 and the volume of our haystack would be 7.5 x 10^100 mm^3
Converting 1 light-year to millimeters we have:
(1 lightyear)x365.35x24x60x60x186000x5280x12x2.54x10 = 9.45x10^18 mm
or (1 lightyear) / 9.45x10^18 = 1 mm
Which means 1 cubic lightyear / (8.43x10^56) = 1 cubic millimeter
So our haystack is (7.5x10^100 / 8.43x10^56) cubic lightyears
or 8.897x10^43 cubic lightyears
Assuming the diameter of the Universe is a cube 93 billion lightyears on a side, then the volume of the Universe is about 8.04x10^32 cubic lightyears.
So 8.897x10^43 / 8.04x10^32 = a haystack about 111 billion times as big as the whole of the known Universe.
In other words we need to find and isolate 1 needle in a haystack over 100 billion times as big as our whole Universe.
Somehow I doubt we will ever achieve such a feat even with a million qubit quantum computer because our error rate would have to be low enough to distinguish that one prime number "needle" from all other 100 digit prime numbers without error.
13. Dear John, I think your estimation is incorrect here. 1000000 qubit would roughly size up to 2^1000000 which is incomparably bigger to whatever size you mention above.
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7a21c833eada5cbc | FermiNet: Quantum Physics and Chemistry from First Principles
October 19, 2020
FermiNet: Quantum Physics and Chemistry from First Principles
October 19, 2020
In an article recently published in Physical Review Research, we show how deep learning can help solve the fundamental equations of quantum mechanics for real-world systems. Not only is this an important fundamental scientific question, but it also could lead to practical uses in the future, allowing researchers to prototype new materials and chemical syntheses in silico before trying to make them in the lab. Today we are also releasing the code from this study so that the computational physics and chemistry communities can build on our work and apply it to a wide range of problems. We’ve developed a new neural network architecture, the Fermionic Neural Network or FermiNet, which is well-suited to modeling the quantum state of large collections of electrons, the fundamental building blocks of chemical bonds. The FermiNet was the first demonstration of deep learning for computing the energy of atoms and molecules from first principles that was accurate enough to be useful, and it remains the most accurate neural network method to date. We hope the tools and ideas developed in our AI research at DeepMind can help solve fundamental problems in the natural sciences, and the FermiNet joins our work on protein folding, glassy dynamics, lattice quantum chromodynamics and many other projects in bringing that vision to life.
A Brief History of Quantum Mechanics
Mention “quantum mechanics” and you are more likely to inspire confusion than anything else. The phrase conjures up images of Schrödinger’s cat, which can paradoxically be both alive and dead, and fundamental particles that are also, somehow, waves. In quantum systems, a particle such as an electron doesn’t have an exact location, as it would in a classical description. Instead, its position is described by a probability cloud - it’s smeared out in all places it’s allowed to be. This counterintuitive state of affairs led Richard Feynman to declare: “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” Despite this spooky weirdness, the meat of the theory can be reduced down to just a few straightforward equations. The most famous of these, the Schrödinger equation, describes the behavior of particles at the quantum scale in the same way that Newton’s laws describe the behavior of objects at our more familiar human scale. While the interpretation of this equation can cause endless head-scratching, the math is much easier to work with, leading to the common exhortation from professors to “shut up and calculate” when pressed with thorny philosophical questions from students.
These equations are sufficient to describe the behavior of all the familiar matter we see around us at the level of atoms and nuclei. Their counterintuitive nature leads to all sorts of exotic phenomena: superconductors, superfluids, lasers and semiconductors are only possible because of quantum effects. But even the humble covalent bond - the basic building block of chemistry - is a consequence of the quantum interactions of electrons. Once these rules were worked out in the 1920s, scientists realised that, for the first time, they had a detailed theory of how chemistry works. In principle, they could just set up these equations for different molecules, solve for the energy of the system, and figure out which molecules were stable and which reactions would happen spontaneously. But when they sat down to actually calculate the solutions to these equations, they found that they could do it exactly for the simplest atom (hydrogen) and virtually nothing else. Everything else was too complicated.
The heady optimism of those days was nicely summed up by Paul Dirac:
Paul Dirac, 1929
Many took up Dirac’s charge, and soon physicists built mathematical techniques that could approximate the qualitative behavior of molecular bonds and other chemical phenomena. These methods started from an approximate description of how electrons behave that may be familiar from introductory chemistry. In this description, each electron is assigned to a particular orbital, which gives the probability of a single electron being found at any point near an atomic nucleus. The shape of each orbital then depends on the average shape of all other orbitals. As this “mean field” description treats each electron as being assigned to just one orbital, it is a very incomplete picture of how electrons actually behave. Nevertheless, it is enough to estimate the total energy of a molecule with only about 0.5% error.
Figure 1 - Atomic orbitals. The surface denotes the area of high probability of finding an electron. In the blue region the wavefunction is positive, while in the purple region it is negative.
Unfortunately, 0.5% error still isn’t enough to be useful to the working chemist. The energy in molecular bonds is just a tiny fraction of the total energy of a system, and correctly predicting whether a molecule is stable can often depend on just 0.001% of the total energy of a system, or about 0.2% of the remaining “correlation” energy. For instance, while the total energy of the electrons in a butadiene molecule is almost 100,000 kilocalories per mole, the difference in energy between different possible shapes of the molecule is just 1 kilocalorie per mole. That means that if you want to correctly predict butadiene’s natural shape, then the same level of precision is needed as measuring the width of a football field down to the millimeter.
With the advent of digital computing after World War II, scientists developed a whole menagerie of computational methods that went beyond this mean field description of electrons. While these methods come in a bewildering alphabet soup of abbreviations, they all generally fall somewhere on an axis that trades off accuracy with efficiency. At one extreme, there are methods that are essentially exact, but scale worse than exponentially with the number of electrons, making them impractical for all but the smallest molecules. At the other extreme are methods that scale linearly, but are not very accurate. These computational methods have had an enormous impact on the practice of chemistry - the 1998 Nobel Prize in chemistry was awarded to the originators of many of these algorithms.
Fermionic Neural Networks
Despite the breadth of existing computational quantum mechanical tools, we felt a new method was needed to address the problem of efficient representation. There’s a reason that the largest quantum chemical calculations only run into the tens of thousands of electrons for even the most approximate methods, while classical chemical calculation techniques like molecular dynamics can handle millions of atoms. The state of a classical system can be described easily - we just have to track the position and momentum of each particle. Representing the state of a quantum system is far more challenging. A probability has to be assigned to every possible configuration of electron positions. This is encoded in the wavefunction, which assigns a positive or negative number to every configuration of electrons, and the wavefunction squared gives the probability of finding the system in that configuration. The space of all possible configurations is enormous - if you tried to represent it as a grid with 100 points along each dimension, then the number of possible electron configurations for the silicon atom would be larger than the number of atoms in the universe!
This is exactly where we thought deep neural networks could help. In the last several years, there have been huge advances in representing complex, high-dimensional probability distributions with neural networks. We now know how to train these networks efficiently and scalably. We surmised that, given these networks have already proven their mettle at fitting high-dimensional functions in artificial intelligence problems, maybe they could be used to represent quantum wavefunctions as well. We were not the first people to think of this - researchers such as Giuseppe Carleo and Matthias Troyer and others have shown how modern deep learning could be used for solving idealised quantum problems. We wanted to use deep neural networks to tackle more realistic problems in chemistry and condensed matter physics, and that meant including electrons in our calculations.
There is just one wrinkle when dealing with electrons. Electrons must obey the Pauli exclusion principle, which means that they can’t be in the same space at the same time. This is because electrons are a type of particle known as fermions, which include the building blocks of most matter - protons, neutrons, quarks, neutrinos, etc. Their wavefunction must be antisymmetric - if you swap the position of two electrons, the wavefunction gets multiplied by -1. That means that if two electrons are on top of each other, the wavefunction (and the probability of that configuration) will be zero.
This meant we had to develop a new type of neural network that was antisymmetric with respect to its inputs, which we have dubbed the Fermionic Neural Network, or FermiNet. In most quantum chemistry methods, antisymmetry is introduced using a function called the determinant. The determinant of a matrix has the property that if you swap two rows, the output gets multiplied by -1, just like a wavefunction for fermions. So you can take a bunch of single-electron functions, evaluate them for every electron in your system, and pack all of the results into one matrix. The determinant of that matrix is then a properly antisymmetric wavefunction. The major limitation of this approach is that the resulting function - known as a Slater determinant - is not very general. Wavefunctions of real systems are usually far more complicated. The typical way to improve on this is to take a large linear combination of Slater determinants - sometimes millions or more - and add some simple corrections based on pairs of electrons. Even then, this may not be enough to accurately compute energies.
Figure 2 - Illustration of a Slater determinant. Each curve is a slice through one of the orbitals from Figure 1. When electrons 1 and 2 swap positions, the rows of the Slater determinant swap, and the wavefunction is multiplied by -1. This guarantees that the Pauli exclusion principle is obeyed.
Deep neural networks can often be far more efficient at representing complex functions than linear combinations of basis functions. In the FermiNet, this is achieved by making each function going into the determinant a function of all electrons (1). This goes far beyond methods that just use one- and two-electron functions. The FermiNet has a separate stream of information for each electron. Without any interaction between these streams, the network would be no more expressive than a conventional Slater determinant. To go beyond this, we average together information from across all streams at each layer of the network, and pass this information to each stream at the next layer. That way, these streams have the right symmetry properties to create an antisymmetric function. This is similar to how graph neural networks aggregate information at each layer. Unlike the Slater determinants, FermiNets are universal function approximators, at least in the limit where the neural network layers become wide enough. That means that, if we can train these networks correctly, they should be able to fit the nearly-exact solution to the Schrödinger equation.
Fig 3 - Illustration of the FermiNet. A single stream of the network (blue, purple or pink) functions very similarly to a conventional orbital. The FermiNet introduces symmetric interactions between streams, making the wavefunction far more general and expressive. Just like a conventional Slater determinant, swapping two electron positions still leads to swapping two rows in the determinant, and multiplying the overall wavefunction by -1.
We fit the FermiNet by minimising the energy of the system. To do that exactly, we would need to evaluate the wavefunction at all possible configurations of electrons, so we have to do it approximately instead. We pick a random selection of electron configurations, evaluate the energy locally at each arrangement of electrons, add up the contributions from each arrangement and minimise this instead of the true energy. This is known as a Monte Carlo method, because it’s a bit like a gambler rolling dice over and over again. While it is approximate, if we need to make it more accurate we can always roll the dice again. Since the wavefunction squared gives the probability of observing an arrangement of particles in any location, it is most convenient to generate samples from the wavefunction itself - essentially, simulating the act of observing the particles. While most neural networks are trained from some external data, in our case the inputs used to train the neural network are generated by the neural network itself. It’s a bit like pulling yourself up by your own bootstraps, and it means that we don’t need any training data other than the positions of the atomic nuclei that the electrons are dancing around. The basic idea, known as variational quantum Monte Carlo (or VMC for short), has been around since the ‘60s, and it is generally considered a cheap but not very accurate way of computing the energy of a system. By replacing the simple wavefunctions based on Slater determinants with the FermiNet, we have dramatically increased the accuracy of this approach on every system we’ve looked at.
Fig 4 - Simulated electrons sampled from the FermiNet move around the bicyclobutane molecule.
To make sure that the FermiNet really does represent an advance in the state of the art, we started by investigating simple, well-studied systems, like atoms in the first row of the periodic table (hydrogen through neon). These are small systems - 10 electrons or fewer - and simple enough that they can be treated by the most accurate (but exponential scaling) methods. The FermiNet outperforms comparable VMC calculations by a wide margin - often cutting the error relative to the exponentially-scaling calculations by half or more. On larger systems, the exponentially-scaling methods become intractable, so instead we use the “coupled cluster” method as a baseline. This method works well on molecules in their stable configuration, but struggles when bonds get stretched or broken, which is critical for understanding chemical reactions. While it scales much better than exponentially, the particular coupled cluster method we used still scales as the number of electrons raised to the seventh power, so it can only be used for medium-sized molecules. We applied the FermiNet to progressively larger molecules, starting with lithium hydride and working our way up to bicyclobutane, the largest system we looked at, with 30 electrons. On the smallest molecules, the FermiNet captured an astounding 99.8% of the difference between the coupled cluster energy and the energy you get from a single Slater determinant. On bicyclobutane, the FermiNet still captured 97% or more of this correlation energy - a huge accomplishment for a supposedly “cheap but inaccurate” approach.
Fig 5 - Graphic depiction of the fraction of correlation energy that the FermiNet captures on molecules. The purple bar indicates 99% of correlation energy. Left to right: lithium hydride, nitrogen, ethene, ozone, ethanol and bicyclobutane.
While coupled cluster methods work well for stable molecules, the real frontier in computational chemistry is in understanding how molecules stretch, twist and break. There, coupled cluster methods often struggle, so we have to compare against as many baselines as possible to make sure we get a consistent answer. We looked at two benchmark stretched systems - the nitrogen molecule (N2) and the hydrogen chain with 10 atoms, (H10). Nitrogen is an especially challenging molecular bond, because each nitrogen atom contributes 3 electrons. The hydrogen chain, meanwhile, is of interest for understanding how electrons behave in materials, for instance predicting whether or not a material will conduct electricity. On both systems, coupled cluster did well at equilibrium, but had problems as the bonds were stretched. Conventional VMC calculations did poorly across the board. But the FermiNet was among the best methods investigated, no matter the bond length.
We think the FermiNet is the start of great things to come for the fusion of deep learning and computational quantum chemistry. Most of the systems we’ve looked at so far are well-studied and well-understood. But just as the first good results with deep learning in other fields led to a burst of follow-up work and rapid progress, we hope that the FermiNet will inspire lots of work on scaling up and many ideas for new, even better network architectures. Already, since we first put our work on arXiv last year, other groups have shared their approaches to applying deep learning to first-principles calculations on the many-electron problem. We have also just scratched the surface of computational quantum physics, and look forward to applying the FermiNet to tough problems in material science and condensed matter physics as well. Mostly, we hope that by releasing the source code used in our experiments, we can inspire other researchers to build on our work and try out new applications we haven’t even dreamed of.
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f5841c7b6ef4b68a | Saturday, May 10, 2014
What Is Time?
Pairs of light photons called biphotons
Monday, April 7, 2014
Unification of charge and gravity force
The nothing of empty space seems to be what makes up most of the universe and is the single most important common intuition of mainstream science. Sources move around in the emptiness of space over time and observers keep track of those sources and predict their futures in space and time. However, making the nothing of empty space into an source in effect creates something out of nothing and so there are well-recognized limitations for notions of continuous space and time.
Space simply does not include all sources in the universe and the notions of space and time do not apply to ether very large or very small sources. The space inside of a black hole, for example, has no meaning despite a black hole occupying a well determined volume in the universe. Likewise the microscopic space called the Planck volume at the center of all particles likewise has no meaning either. The space inside of an electron likewise has no meaning.
Mainstream science describes the universe from the foundational notions of mass, empty space, and time and science defines these fundamental quantities or axioms only by the rules for their measurement. A one kg bar of an iridium platinum alloy defines mass and the wavelength and frequency of light defines both length and time. Thus science defines its axioms as continuous and infinitely divisible even though science knows that quantum reality is discrete.
Therefore there is a simpler way of describing the universe beginning from discrete matter and discrete action instead of continuous matter, space, and time. The discrete nature of matter and action in the universe finally includes the notion of quantum phase and thereby unifies charge, gravity, and consciousness into one single foundation. But there science must significantly change its foundation principles of physical reality and in particular, continuous matter, space, and time all emerge from discrete matter and action
Mainstream science has not yet succeeded in unifying charge and gravity due to the limitations of the notions of continuous space, motion, and time. Furthermore, the nature of consciousness is a fundamental part of objective reality since it is by subjective notions of reality by many observers that science comes to agree on an objective reality of sources.
Given a proportionality between matter and action, dimensionless ratios of either discrete time delay or discrete space emerge. Time and space both emerge as ways for observers to keep track of sources and predict their futures and discrete time and space naturally emerge from discrete matter and action. All spatial separations are equivalent to time delays and all motion is equivalent to changes in mass by the matter-energy equivalence principle. The alternate coordinates of discrete aether and discrete action augment the standard notions of continuous space, motion, and time in the aethertime scheme that finally unites charge and gravity forces as a common aether decoherence rate. The now compatible forces are both part of the universal quantum force of aether decoherence. Discrete aether and action keeps track of the objects of physical reality and augment the more limited notions of continuous space and motion and time for tracking those same objects.
Instead of the four dimensions of spacetime for Einstein's general relativity, a discrete aether universe has two dimensions for each of matter and action; amplitude and phase. For quantum action there are many possible paths for an object while there is only one possible future for GR action along its geodesic. Within the limitations of continuous space and motion, the actions of objects represent spatial displacements as well as moments of time. But what is not very clear is that a spatial displacement along the dimension of motion and time evolution are both essentially identical with the discrete aether dimension of action.
The equivalence of time and space as proportional displacements means that the continuous time and space of spacetime emerge from the discrete action of aether. Although quantum action is usually based on the conjugates of continuous space and momentum, the corresponding aethertime conjugates are discrete aether and action. Discrete aethertime provides a common quantum basis for both charge and gravity forces even while aethertime is consistent with the principles of both relativity and quantum action.
Aethertime therefore unifies gravity with quantum action by using that second time dimension, the aether decoherence time that we already know exists outside of the atomic time of our mind. This involves time and matter that augments space or alternatively, space and matter as emergent time. Even though our intuition as well as mainstream science use the whiteboard of spacetime for predicting action, predicting the future of an object based on just matter and action is also possible. It is very difficult to imagine that space and time are a result of action of matter and action instead of spacetime as a place for action to occur. The key to a quantum gravity is in the realization that continuous time and space are just two different representations for the same reality of discrete aether and action.
Space and time emerge from the discrete actions of objects and therefore space and time are all mixed up with discrete aether and action. In the common parlance of mainstream science, space and time emerge from the actions that occur with discrete aether and continuous time emerges from the discrete time delays of objects. Given the fact of a constant speed of light, time dilation and matter changes that occur depend on the frame of reference. Projecting spatial displacements from discrete changes in time delay and matter results in the dilation of space as well. Since all prediction of relativistic action occurs with discrete matter and time delay, the 4-space for predictions of GR remain valid within the limits of continuous space and time.
In aethertime, photon and other matter exchange explains all force and the primal force is aether decoherence. A dipole photon exchange particle binds electrons to protons together into atoms and molecules and a mono-quadrupole biphoton exchange particle binds neutral atoms to each other in the universe as gravity force. This notion, in a nutshell, describes the way that discrete aether and quantum action unify charge and gravity forces.
Light is a rather unusual matter wave made up of photons created at the CMB entangled with the photon exchange particle that binds charges together as well. An exchange of a photon between an electron and nucleus represents the charge force that stabilizes an atom, which is the basis of quantum electrodynamics and is well accepted by science. The entanglement of bonding photons with those emitted photon at CMB creation is the entangled biphoton that is gravity.
In discrete aethertime, it is actually coherent mono-quadrupoles of biphotons that are the exchange particles for gravity force and that is certainly not yet a common understanding in science. Most ideas of quantum gravity in science invent a new exchange particle called a graviton, but gravitons or any other gravity particle simply cannot exist in the current conundrum of continuous space and time. In contrast, the simple notion of biphoton discrete exchange determines gravity force with a really appealing symmetry that a complementary photon pair exchange that binds atoms to each other in the universe.
The figure shows two hydrogen atoms that are each bound by a photon exchange between their respective electron and proton. In the process, each atom must lose that equivalent photon energy as a photon to the universe. This emitted photon is really both a wave and a particle and is in an exchange that binds each atom to the boson matter of the universe.
In order for an atom to form from an electron and proton, it must radiate its binding energy as a complementary photon and that emitted photon is equivalent to the atom’s binding energy, which is the Rydberg energy for hydrogen. There actually can be and are many photon emissions and absorptions of various energies and so this description just simplifies the more complex chaos of many events into one single entangled event.
The Rydberg photons emitted from a pair of hydrogens exchange with the universe and that exchange force is part of what binds each hydrogen atom to the matter of the universe with an orbit that is coherent with the orbit of the electron around the proton. The photon exchange from two such atoms binds each atom to the universe mass and the shrinkage of the universe about those atom’s center of mass represents what we interpret as the binding force of gravity between these two particles.
The two emitted photons represent coherent polarization phases between the two atoms. Charge polarization still dominates the forces between two atoms and it is only when a large matter accretion reaches about 1e39 atoms that gravity force is equivalent to charge force for two bodies. Two such bodies exchange large amounts of light with the universe and with each other as well and that exchange results in what we call gravity action.
Since there are two complementary spin = 1 photons, each gravity state is a multiplet of spin = {0, +/-2}. Although the gravity of general relativity dilates space and time, there is still just one future for the determinate paths of gravity action under general relativity, and this causality is consistent with our intuition. There are many strange results of general relativity having to do with simultaneity and frame of reference, but objects far away from a gravity action still do not affect a local action very much.
The phase coherence of quantum gravity provides many more possible futures as compared to the determinate geodesic paths for the gravity action of general relativity. Quantum electrodynamics fills space with an infinity of vacuum oscillators as a means of moving charge and light around. Therefore a key step called renormalization ratios out that infinity and focuses only on those futures that are most likely. Aethertime's quantum gravity is based on a very large, but finite, number of boson particles in the universe and so also depends on a kind of renormalization to focus on those futures of quantum gravity action that are most likely.
Of course, this schematic is a gross oversimplification of the difference between charge and gravity forces. There are quantum exchange and phase effects between two hydrogen atoms that actually forms a molecular bond that is very much stronger than gravity force. Then, hydrogen molecules bond to each other as a liquid or solid with dipole-induced dipoles, higher multipoles, including further quantum exchange forces. Once an object of hydrogen has accreted sufficient mass through all of this charge force bonding, the center of the accretion begins to show the effects of gravitational compression of biphoton exchange and the concomitant heating.
The heat generated by gravitational compression is equivalent to the gravitational binding states of that object as well as its charge binding states. The light emitted as a result of the object's gravitational heat represents the gravity bonding states of the object and correspondingly the gravity force between that object and other objects. All of the photon pairs emitted by an object contribute to the quadrupole that we call gravity force.
What this all means is that gravity force and charge force are both due to the same decoherence rate of universe matter. Charge force represents the inner decoherence of dipole force on the dimensions of the atom and electron while gravity force represents the outer decay and the dimensions of the folded quadrupole universe and its fundamental boson, the gaechron, mae. Just like all of the theories of quantum gravity, there is a smallest scale of matter or energy called the Planck scale and the gaechron mass is just the result of the matter spectrum from a Fourier transform of the universe time pulse.
Sunday, March 23, 2014
Gravity Waves at the Cosmic Microwave Background
Big Bang blunder bursts the multiverse bubble
03 June 2014
Oh dear, easy come, easy go...
Saturday, March 15, 2014
The Neural Sound of Music
Saturday, March 1, 2014
Aware Matter as Consciousness
Unlike linear computers that store and retrieve data packets composed of bytes, aware matter packets carry information as superimposed coherent bilateral aware matter states at 64 independent frequencies or colors. This is like the information content of a modulated 64 color laser with 16 levels of intensities for each color, but with a fundamental color mode of just 1.6 Hz.
Aware Matter Spectra as EEG Waves
It turns out that one particular mode, the alpha EEG mode at 11 Hz, is particularly prominent in the EEG. Evidently this mode is associated with a 7-mer aware matter of the eye. In the figure, the foveal cone is the most sensitive part of the retina and its basic symmetry is the 7-mer and 7 x 1.6 = 11 Hz.
Aware Matter Packets and Internet Packets
Aware Matter as Sleep
Aware Matter as a Quantum Fluid
Ea = 2.9e-13 J or 1.8 MeV
Ña = aware matter action constant, ma / 2π / f
t = time, s
fa = aware matter object frequency, Hz
ya = aware matter wavefunction
ya with dot = time derivative of ya
Sterile Neutrinos...Cousins or Siblings or Self?
Sunday, February 23, 2014
What is Action?
Action completes the trimal of matter and time and action’s simplest and really only true definition is as a product of matter and time. Although we associate action with the integration of object motion through space, object motion through space is simply the way that we imagine action in the universe. The universe is full of an equivalence of matter objects that are gaining and losing mass relative to the universe outside of that object. We call a change in matter from some rest frame motion and we see comoving and countermoving objects as increases in the mass of each object as its relative motion increases towards or away from a given frame of reference. We assign that change in mass to the kinetic energy of the object in motion through space because of the equivalence of energy and mass.
But we can equivalently describe relative motion as a change in an object mass and then project that mass change into a motion through space. In fact, all objects in the universe are shrinking and comoving at the speed of light and that action is equivalent to each object’s proper mass. Thus objects get their proper masses from the primal action of the shrinking universe and objects alter their motion by changing their mass. There is no inaction in the universe since all action drives all matter.
The rate of change of the universe mass in time is the primal action constant, mdot, and that matter decay that determines all force. The potential energy of an object that is subject to a force is equivalent to an object’s change in mass over time. In other words, while kinetic energy or relative motion is a step change of an object’s mass, potential energy or force or acceleration represents a continuous change in object mass over time, and it is the integration of matter over time that is the definition of action.
We associate action with motion through space but it is only with both step and continuous changes in matter that we can project that space. Motion necessarily involves both kinetic and potential energies. Since matter, by definition, is never without gravity force, the change in matter that is gravity is always present in the universe. Objects are always exchanging radiation and atoms with other objects, but the primal dimensions of matter, time, and action are what determine motion and motion is how we project a Cartesian space all around us.
We do not really need the a priori empty void of space to journey from one object to another. It is clear that a journey from one object to another does take time and that time can be no shorter than speed of light. The speed of light is how we can think of time as a means to separate objects in our mind. Even though time only has one dimension while space has three, there are two other primitive dimensions of matter time, matter and phase, that project Cartesian displacement.
In a universe of two counterrotating hydrogens, there is only one world time line and that world line is then equivalent to time. The two objects are trapped in a perpetual ballet of gravity and charge and ionization and recombination and photon absorption and emission. We only need to consider other world lines once there are other objects in our universe, for example, the universe itself. We assign a gaechron amplitude from the matter spectrum of a world line to a dimension orthogonal to time, m, and we further associate a phase, θ, to describe the rotation of m around t, the phase relationship between matter and time. Now with these three dimensions of matter, time, and phase, we have a basis for projecting all three Cartesian dimensions from the primal dimensions of matter, time, and action.
So unlike the approach of relativity, which begins with the axiom of three Cartesian dimensions and adds a time axiom as a fourth spatial dimension, matter time begins with the three primitive dimensions of matter, time, and phase from which matter time projects the three Cartesian displacements. The approach of matter time still means that time dilation occurs with velocity and so spatial dilation also occurs and mass increases with velocity as well, all in accord with relativity and Lorentz invariance.
As opposed to the relativity of space time, space is a result of action in matter time that is a very convenient and useful projection of our minds from a primitive quantum reality. All action derives from matter and its change in time and the proportionality between an object’s change in matter with time to the object’s matter is the Schrödinger equation. Both charge and gravity force derive from the exchange of gaechron and the action of gaechron is what we project as space. It is important to note that gaechron action does not fill space because there is actually no space to fill. Space is a projection of matter action and is not independently necessary for predicting matter action.
Space is therefore very much a timelike projection of our mind's mathematical models and time is the differential of action with matter. Correspondingly, there is a matter and an action that defines space as well, such as the footsteps that are the action of a journey. We imagine objects separated by the empty void of space, but that void has a distance as an integration of matter over time just as objects separated in time have both the matter of a moment and an integration of those matter moments as action. We do not think of time as moments separated by timelessness and so we should not think of space as an empty void between objects.
In other words, there is always a time distance between objects even though those times might be very long and even cosmic. There is always a matter exchange among objects and no object is truly isolated or constant. And all objects are under action and there is no true inaction in the universe.
Time is axiomatic and is the differential of action with matter while space is just a projection of time as the differential of action with matter. Dividing the action of a journey by the matter of a footstep gives us a distance in time. Dividing the action of a journey with the moment of a footstep gives us a distance as matter, which we interpret as Cartesian distance. We project an empty void of space between two objects as a convenient way to separate objects in time but we do not project an empty void of time, a timeless eternity, between two moments of time.
As Einstein showed with his relativity, time is a spatial dimension and that allowed us to better understand the universe. What Einstein did not show, though, was that there was a simpler reality that projects space as a result of action. Since space is a projection of matter action in time, convolving time with space results in the very complex mathematics of general relativity for gravitational force that has thus far resisted any unification with quantum mechanics or with charge force.
Nevertheless, the principles of general relativity are perfectly useful given the limited realm of their application just as are the principles of quantum mechanics useful in their limited realm. But general relativity does not provide a complete description of action for the universe. In particular, dark matter and dark energy are straightforward manifestations of a quantum gravity. A quantum exchange coupling among the matter decays of stars and galaxies and the decay of the universe provides an additional term to the gravitational virial equation.
The coupling of the matter decay of a star with the decay of its space results in a force. These forces among stars are part of the fabric of the universe and result in resonances called matter waves that are concerted and cyclic variations in gravity and charge forces as well as in the masses of objects. By changing the density and polarizability of matter, matter waves also affect the convection of gravitationally compressed plasma in stars and magma in planets and matter waves also affect the nuclear weak force as well. The cycles of matter waves in our local star neighborhood seem to determine solar cycles as well as cycles of earth’s magmatic activity while our sun’s journey through galaxy matter waves determine the cycles of ice ages as well as other geologic ages.
What is Matter?
We easily describe what matter is like since matter is just the stuff that makes up all objects and so each object has a single dimension of mass. Objects are made of matter and that matter is finitely divisible into the atoms, electrons, protons, and neutrons of our microscopic universe. Unlike the equally intuitive notion of space, though, matter does not suffer from being infinitely divisible. The hard stop for matter is the electron, which is indivisible, and the quark pair, since a quark pair along with its gluon particle exchange would take the energy of the universe to separate.
Both protons and neutrons are made of three quarks, or really two quark pairs and bonding gluons, that is it as far as matter is concerned. In matter time, the universe is mostly boson matter and the smallest boson particle is the gaechron and gaechron are very much smaller than other matter particles. But even atoms are very small and their numbers are very large. A kilogram of hydrogen is 6e26 atoms and matter is therefore a virtual infinity of particles.
Although we experience matter as the single dimension of intensity or amplitude squared, objects actually exist as matter wave amplitudes that have both phase and oscillation of their amplitude. This means that a particle can exist as matter wave amplitude among any number of world timelines along that matter wave, but that particle will only be realized as intensity on one particular timeline. Our universe is mostly space with only a relatively small amount of fermionic matter, like hydrogen, on the order of one atom of hydrogen per cubic meter of space. However, in matter time most of the matter in the universe is bosonic and is not in the form of fermions. In fact, there is about eleven million times more bosonic than fermionic matter in the universe and so it turns out that shrinking bosonic matter largely drives force and action and force and action are how the universe evolves.
The small amount of baryonic matter, the protons and neutrons of fermionic matter, stands in contrast to the overwhelming amount of bosonic matter. So where are the bosons hiding? In plain sight of course, or maybe plainly out of sight. Although it is tempting to imagine that space is filled with a quantum boson foam from which fermions seethe into and out of existence, that implies that space has an existence independent of the action of matter in time. It is much better to assume space is a projection of matter action and that there is a universal matter spectrum that describes all of the possibilities of objects as matter waves.
Our universe is both a pulse of matter in time as well as a spectrum of the possibilities of matter waves, which is the Fourier transform of the universe matter pulse. However, our universe is not actually made up of the empty void of nothing that we call space. Rather that empty void of nothing that we call space is just a projection of the actions of objects in time and it is matter action that actually separates objects.
Each of time and matter are complex amplitudes with a common phase, but matter and time are also related to each other by the Schrödinger equation. This relationship imposes a quantum phase differential between matter and time, π/2, that is the basis for orthogonality between matter and time as well as the basis of the right angle of Euclidean geometry that matter time projects as space. The conjugate coordinates {m, t} along with the action of the Schrödinger equation provide the basic dimensions of reality that then project a Cartesian displacement that is the right angle of Euclidean geometry.
In the early universe, forces were vanishingly small and matter was an equilibrium of bosons and fermions since there was not yet enough force to condense or freeze bosons into fermions. As the universe pulse collapsed, forces increased and when matter’s rate of change, force, reached a threshold of mp/me, the ratio of proton and electron masses, a fraction of matter froze out from the boson sea as the light elements of hydrogen, deuterium, helium, and other isotopes. Each boson condensate formed into fermions as pairs of atoms with complementary angular momentum.
The same charge force that bound rotating electrons and protons also bound their rotating neutral atoms to themselves with gravity, but in the folded universe, gravity forces were very much smaller than charge forces. The very much weaker gravity force condensed rotating hydrogen atoms into rotating planets and stars that fused hydrogen into heavier elements up to iron. Photon and neutrino radiation not only provides the light and warmth of the heavens, but that radiation also results in star matter decay over and above the decay of space. The coupling of star decay with spatial decay then provides an extra force that transfers angular momentum from inner to outer stars in a galaxy.
Rotating stars cluster into rotating elliptical and spiral disks called galaxies, which are fueled both by the fire of the stars as well as by the angular momentum of the atom. Ever more massive accumulations of matter yield the heavier elements as well as neutron stars, magnetars, and finally, massive rotating boson stars known as supermassive black holes. Boson stars represent the ultimate destiny of all matter in the shrinking universe with an ultimate dephasing of all matter. |
85aec04957fd680a | Cycle stability - Chaos: Classical and Quantum
Cycle stability - Chaos: Classical and Quantum
Chapter 5Cycle stabilityTOPOLOGICAL FEATURES of a dynamical system –singularities, periodic orbits,and the ways in which the orbits intertwine– are invariant under a generalcontinuous change of coordinates. Surprisingly, there also exist quantitiesthat depend on the notion of metric distance between points, but nevertheless donot change value under a smooth change of coordinates. Local quantities suchas the eigenvalues of equilibria and periodic orbits, and global quantities suchas Lyapunov exponents, metric entropy, and fractal dimensions are examples ofproperties of dynamical systems independent of coordinate choice.We now turn to the first, local class of such invariants, linear stability of periodicorbits of flows and maps. This will give us metric information about localdynamics. If you already know that the eigenvalues of periodic orbits are invariantsof a flow, skip this chapter.fast track:chapter 7, p. 1215.1 Stability of periodic orbitsAs noted on page 40, a trajectory can be stationary, periodic or aperiodic. Forchaotic systems almost all trajectories are aperiodic–nevertheless, equilibria andperiodic orbits turn out to be the key to unraveling chaotic dynamics. Here wenote a few of the properties that make them so precious to a theorist.An obvious virtue of periodic orbits is that they are topological invariants: afixed point remains a fixed point for any choice of coordinates, and similarly aperiodic orbit remains periodic in any representation of the dynamics. Any reparametrizationof a dynamical system that preserves its topology has to preservetopological relations between periodic orbits, such as their relative inter-windings94
and knots. So the mere existence of periodic orbits suffices to partially organizethe spatial layout of a non–wandering set. No less important, as we shall nowshow, is the fact that cycle eigenvalues are metric invariants: they determine therelative sizes of neighborhoods in a non–wandering set.We start by noting that due to the multiplicative structure (4.44) of Jacobianmatrices, the Jacobian matrix for the rth repeat of a prime cycle p of period T p isJ rT p(x) = J T p( f (r−1)T p(x)) · · · J T p( f T p(x))J T p(x) = J p (x) r , (5.1)where J p (x) = J T p(x) is the Jacobian matrix for a single traversal of the primecycle p, x ∈ M p is any point on the cycle, and f rT p(x) = x as f t (x) returns to xevery multiple of the period T p . Hence, it suffices to restrict our considerations tothe stability of prime cycles.fast track:sect. 5.2, p. 995.1.1 Floquet vectorsWhen dealing with periodic orbits, some of the quantities already introduced inheritnames from the Floquet theory of differential equations with time-periodiccoefficients. Consider the equation of variations (4.2) evaluated on a periodic orbitp,˙ δx = A(t) δx , A(t) = A(x(t)) = A(t + T p ) . (5.2)The T p periodicity of the stability matrix implies that if δx(t) is a solution of (5.2)then also δx(t + T p ) satisfies the same equation: moreover the two solutions arerelated by (4.6)δx(t + T p ) = J p (x) δx(t) . (5.3)Even though the Jacobian matrix J p (x) depends upon x (the ‘starting’ point ofthe periodic orbit), we shall show in sect. 5.2 that its eigenvalues do not, so wemay write for its eigenvectors e ( j) (sometimes referred to as ‘covariant Lyapunovvectors,’ or, for periodic orbits, as ‘Floquet vectors’)J p (x) e ( j) (x) = Λ p, j e ( j) (x) , Λ p, j = σ ( p j) j)λ(ep T p. (5.4)where λ ( p j) = µ ( p j) ± iω ( p j) and σ ( pj) are independent of x. When Λ p, j is real, we docare about σ ( pj) = Λ p, j /|Λ p, j | ∈ {+1, −1}, the sign of the jth Floquet multiplier.invariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
Figure 5.1: For a prime cycle p, Floquet matrixJ p returns an infinitesimal spherical neighborhood ofx 0 ∈ M p stretched into an ellipsoid, with overlap ratioalong the eigdirection e (i) of J p (x) given by the Floquetmultiplier |Λ p,i |. These ratios are invariant undersmooth nonlinear reparametrizations of state space coordinates,and are intrinsic property of cycle p.x + Jpδxx 00+ x δIf σ ( j)p = −1 and λ ( j)p 0, the corresponding eigen-direction is said to be inverse section 7.2hyperbolic. Keeping track of this by case-by-case enumeration is an unnecessarynuisance, so most of our formulas will be stated in terms of the Floquet multipliersΛ j rather than in the terms of the multiplier signs σ ( j) , exponents µ ( j) and phasesω ( j) .Expand δx in the (5.4) eigenbasis, δx(t) = ∑ δx j (t) e ( j) , e ( j) = e ( j) (x(0)) .Taking into account (5.3), we get that δx j (t) is multiplied by Λ p, j per each period∑∑δx(t + T p ) = δx j (t + T p ) e ( j) = Λ p, j δx j (t) e ( j) .jjWe can absorb this exponential growth / contraction by rewriting the coefficientsδx j (t) asj)λ(δx j (t) = ep t u j (t) , u j (0) = δx j (0) ,with u j (t) periodic with period T p . Thus each solution of the equation of variations(4.2) may be expressed in the Floquet form∑δx(t) =jj)λ(ep t u j (t) e ( j) , u j (t + T p ) = u j (t) . (5.5)The continuous time t appearing in (5.5) does not imply that eigenvalues of theJacobian matrix enjoy any multiplicative property for t rT p : λ ( p j) = µ ( p j) ± iω ( pj)refer to a full traversal of the periodic orbit. Indeed, while u j (t) describes thevariation of δx(t) with respect to the stationary eigen-frame fixed by eigenvectorsat the point x(0), the object of real interest is the co-moving eigen-frame definedbelow in (5.13).5.1.2 Floquet matrix eigenvalues and exponentsThe time-dependent T-periodic vector fields, such as the flow linearized aroundthe periodic orbit, are described by Floquet theory. Hence from now on we shallinvariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
x’x=x(T)Figure 5.2: An unstable periodic orbit repels everyneighboring trajectory x ′ (t), except those on its centerand unstable manifolds.x’(T)refer to a Jacobian matrix evaluated on a periodic orbit as a Floquet matrix, to itseigenvalues Λ p, j as Floquet multipliers (5.4), and to λ ( p j) = µ ( p j) + iω ( pj) as Floquetor characteristic exponents. We sort the Floquet multipliers {Λ p,1 , Λ p,2 , . . ., Λ p,d }of the [d×d] Floquet matrix J p evaluated on the p-cycle into sets {e, m, c}expanding: {Λ} e = {Λ p, j : ∣ ∣ ∣Λp, j∣ ∣∣ > 1}marginal: {Λ} m = {Λ p, j : ∣ ∣ ∣Λp, j∣ ∣∣ = 1} (5.6)contracting: {Λ} c = {Λ p, j : ∣ ∣ ∣Λp, j∣ ∣∣ < 1} .and denote by Λ p (no jth eigenvalue index) the product of expanding Floquetmultipliers∏Λ p = Λ p,e . (5.7)eAs J p is a real matrix, complex eigenvalues always come in complex conjugatepairs, Λ p,i+1 = Λ ∗ p,i, so the product (5.7) is always real.The stretching/contraction rates per unit time are given by the real parts ofFloquet exponentsµ (i)p = 1T pln ∣ ∣ ∣Λp,i∣ ∣∣ . (5.8)The factor 1/T p in the definition of the Floquet exponents is motivated by itsform for the linear dynamical systems, for example (4.16), as well as the fact thatexponents so defined can be interpreted as Lyapunov exponents (17.33) evaluatedon the prime cycle p. As in the three cases of (5.6), we sort the Floquet exponentsλ = µ ± iω into three sets section 17.3expanding: {λ} e = {λ (i)pmarginal: {λ} m = {λ (i)pcontracting: {λ} c = {λ (i)p: µ (i)p > 0}: µ (i)p = 0}: µ (i)p < 0} . (5.9)invariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
A periodic orbit p of a d-dimensional flow or a map is stable if real partsof all of its Floquet exponents (other than the vanishing longitudinal exponent,explained in sect. 5.2.1) are strictly negative, µ (i)p < 0. The region of system parametervalues for which a periodic orbit p is stable is called the stability windowof p. The set M p of initial points that are asymptotically attracted to p as t → +∞(for a fixed set of system parameter values) is called the basin of attraction of p. Ifall Floquet exponents (other than the vanishing longitudinal exponent) are strictlypositive, µ (i) ≥ µ min > 0, the cycle is repelling, and unstable to any perturbation.If some are strictly positive, and rest strictly negative, −µ (i) ≥ µ min > 0, the cycleis said to be hyperbolic or a saddle, and unstable to perturbations outside its stablemanifold. Repelling and hyperbolic cycles are unstable to generic perturbations,and thus said to be unstable, see figure 5.2. If all µ (i) = 0, the orbit is said to beelliptic, and if µ (i) = 0 for a subset of exponents (other than the longitudinal one),the orbit is said to be partially hyperbolic. Such orbits proliferate in Hamiltonianflows. section 7.3If all Floquet exponents (other than the vanishing longitudinal exponent) ofall periodic orbits of a flow are strictly bounded away from zero, the flow is saidto be hyperbolic. Otherwise the flow is said to be nonhyperbolic.Example 5.1 Stability of cycles of 1-dimensional maps: The stability of a primecycle p of a 1-dimensional map follows from the chain rule (4.51) for stability of the n p thiterate of the mapΛ p =dn∏p −1f n p(x 0 ) = f ′ (x m ) , x m = f m (x 0 ) . (5.10)dx 0m=0Λ p is a property of the cycle, not the initial periodic point, as taking any periodic pointin the p cycle as the initial one yields the same Λ p .A critical point x c is a value of x for which the mapping f (x) has vanishingderivative, f ′ (x c ) = 0. A periodic orbit of a 1-dimensional map is stable if∣∣Λ p∣ ∣∣ =∣ ∣∣ f ′ (x np ) f ′ (x np −1) · · · f ′ (x 2 ) f ′ (x 1 ) ∣ ∣ ∣ < 1 ,and superstable if the orbit includes a critical point, so that the above product vanishes.For a stable periodic orbit of period n the slope Λ p of the nth iterate f n (x) evaluatedon a periodic point x (fixed point of the nth iterate) lies between −1 and 1. If ∣ ∣ ∣Λp∣ ∣∣ > 1,p-cycle is unstable.Example 5.2 Stability of cycles for maps: No matter what method we use todetermine the unstable cycles, the theory to be developed here requires that their Floquetmultipliers be evaluated as well. For maps a Floquet matrix is easily evaluatedby picking any periodic point as a starting point, running once around a prime cycle,and multiplying the individual periodic point Jacobian matrices according to (4.52). Forexample, the Floquet matrix M p for a Hénon map (3.19) prime cycle p of length n p isgiven by (4.53),1∏ ( )−2axk bM p (x 0 ) =1 0, x k ∈ M p ,k=n pinvariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
and the Floquet matrix M p for a 2-dimensional billiard prime cycle p of length n p1∏ (M p = (−1) n p 1 τk0 1k=n p) (1 0r k 1)follows from (8.11) of chapter 8 below. The decreasing order in the indices of theproducts in above formulas is a reminder that the successive time steps correspondto multiplication from the left, M p (x 1 ) = M(x np ) · · · M(x 1 ). We shall compute Floquetmultipliers of Hénon map cycles once we learn how to find their periodic orbits, seeexercise Floquet multipliers are invariantThe 1-dimensional map Floquet multiplier (5.10) is a product of derivatives overall points around the cycle, and is therefore independent of which periodic pointis chosen as the initial one. In higher dimensions the form of the Floquet matrixJ p (x 0 ) in (5.1) does depend on the choice of coordinates and the initial pointx 0 ∈ M p . Nevertheless, as we shall now show, the cycle Floquet multipliersare intrinsic property of a cycle in any dimension. Consider the ith eigenvalue,eigenvector pair (Λ p,i , e (i) ) computed from J p evaluated at a periodic point x,J p (x) e (i) (x) = Λ p,i e (i) (x) , x ∈ M p . (5.11)Consider another point on the cycle at time t later, x ′ = f t (x) whose Floquetmatrix is J p (x ′ ). By the group property (4.44), J T p+t= J t+T p, and the Jacobianmatrix at x ′ can be written either asJ T p+t (x) = J T p(x ′ ) J t (x) = J p (x ′ ) J t (x) ,or J t (x) J p (x). Multiplying (5.11) by J t (x), we find that the Floquet matrix evaluatedat x ′ has the same Floquet multiplier,J p (x ′ ) e (i) (x ′ ) = Λ p,i e (i) (x ′ ) , e (i) (x ′ ) = J t (x) e (i) (x) , (5.12)but with the eigenvector e (i) transported along the flow x → x ′ to e (i) (x ′ ) =J t (x) e (i) (x). Hence, in the spirit of the Floquet theory (5.5) one can define timeperiodicunit eigenvectors (in a co-moving ‘Lagrangian frame’)e ( j) j)−λ((t) = ep t J t (x) e ( j) (0) , e ( j) (t) = e ( j) (x(t)) , x(t) ∈ M p . (5.13)J p evaluated anywhere along the cycle has the same set of Floquet multipliers{Λ p,1 , Λ p,2 , · · · , 1, · · · , Λ p,d−1 }. As quantities such as tr J p (x), det J p (x) dependinvariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
only on the eigenvalues of J p (x) and not on the starting point x, in expressionssuch as det ( 1 − J r p (x)) we may omit reference to x,det ( 1 − J r p)= det(1 − Jrp (x) ) for any x ∈ M p . (5.14)We postpone the proof that the cycle Floquet multipliers are smooth conjugacyinvariants of the flow to sect. Marginal eigenvaluesThe presence of marginal eigenvalues signals either a continuous symmetry of theflow (which one should immediately exploit to simplify the problem), or a nonhyperbolicityof a flow (a source of much pain, hard to avoid). In that case (typicalof parameter values for which bifurcations occur) one has to go beyond linearstability, deal with Jordan type subspaces (see example 4.4), and sub-exponentialgrowth rates, such as t α . chapter 24For flow-invariant solutions such as periodic orbits, the time evolution is itselfa continuous symmetry, hence a periodic orbit of a flow always has a marginalFloquet multiplier:As J t (x) transports the velocity field v(x) by (4.7), after a complete periodexercise 5.1J p (x) v(x) = v(x) , (5.15)so for a periodic orbit of a flow the local velocity field is always has an eigenvectore (‖) (x) = v(x) with the unit Floquet multiplier,Λ p,‖ = 1 ,λ (‖)p = 0 . (5.16)exercise 6.3The continuous invariance that gives rise to this marginal Floquet multiplier is theinvariance of a cycle (the set M p ) under a translation of its points along the cycle:two points on the cycle (see figure 4.3) initially distance δx apart, x ′ (0) − x(0) =δx(0), are separated by the exactly same δx after a full period T p . As we shall seein sect. 5.3, this marginal stability direction can be eliminated by cutting the cycleby a Poincaré section and eliminating the continuous flow Floquet matrix in favorof the Floquet matrix of the Poincaré return map.If the flow is governed by a time-independent Hamiltonian, the energy is conserved,and that leads to an additional marginal Floquet multiplier (we shall showin sect. 7.3 that due to the symplectic invariance (7.19) real eigenvalues come inpairs). Further marginal eigenvalues arise in presence of continuous symmetries,as discussed in chapter 10 below.invariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
5.3 Stability of Poincaré map cycles(R. Paškauskas and P. Cvitanović)If a continuous flow periodic orbit p pierces the Poincaré section P once, thesection point is a fixed point of the Poincaré return map P with stability (4.57)(Jˆi j = δ ik − v )i U kJ k j , (5.17)(v · U)with all primes dropped, as the initial and the final points coincide, x ′ = f T p(x) =x. If the periodic orbit p pierces the Poincaré section n times, the same observationapplies to the nth iterate of P.We have already established in (4.58) that the velocity v(x) is a zero eigenvectorof the Poincaré section Floquet matrix, Jˆv = 0. Consider next (Λ p,α , e (α) ),the full state space αth (eigenvalue, eigenvector) pair (5.11), evaluated at a periodicpoint on a Poincaré section,J(x) e (α) (x) = Λ α e (α) (x) , x ∈ P . (5.18)Multiplying (5.17) by e (α) and inserting (5.18), we find that the full state spaceFloquet matrix and the Poincaré section Floquet matrix Jˆhave the same Floquetmultiplierˆ J(x) ê (α) (x) = Λ α ê (α) (x) , x ∈ P , (5.19)where ê (α) is a projection of the full state space eigenvector onto the Poincarésection:(ê (α) ) i =(δ ik − v )i U k(e (α) ) k . (5.20)(v · U)Hence, ˆ J p evaluated on any Poincaré section point along the cycle p has the sameset of Floquet multipliers {Λ p,1 , Λ p,2 , · · · Λ p,d } as the full state space Floquet matrixJ p , except for the marginal unit Floquet multiplier (5.16).As established in (4.58), due to the continuous symmetry (time invariance) ˆ J pis a rank d −1 matrix. We shall refer to any such rank [(d −1− N)× (d −1− N)]submatrix with N − 1 continuous symmetries quotiented out as the monodromymatrix M p (from Greek mono- = alone, single, and dromo = run, racecourse,meaning a single run around the stadium). Quotienting continuous symmetries isdiscussed in chapter 10 below.invariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
5.4 There goes the neighborhoodIn what follows, our task will be to determine the size of a neighborhood of x(t),and that is why we care about the Floquet multipliers, and especially the unstable(expanding) ones. Nearby points aligned along the stable (contracting) directionsremain in the neighborhood of the trajectory x(t) = f t (x 0 ); the ones to keep aneye on are the points which leave the neighborhood along the unstable directions.The sub-volume |M i | = ∏ ei∆x i of the set of points which get no further awayfrom f t (x 0 ) than L, the typical size of the system, is fixed by the condition that∆x i Λ i = O(L) in each expanding direction i. Hence the neighborhood size scalesas ∝ 1/|Λ p | where Λ p is the product of expanding Floquet multipliers (5.7) only;contracting ones play a secondary role.So the dynamically important information is carried by the expanding subvolume,not the total volume computed so easily in (4.47). That is also the reasonwhy the dissipative and the Hamiltonian chaotic flows are much more alike thanone would have naively expected for ‘compressible’ vs. ‘incompressible’ flows.In hyperbolic systems what matters are the expanding directions. Whether thecontracting eigenvalues are inverses of the expanding ones or not is of secondaryimportance. As long as the number of unstable directions is finite, the same theoryapplies both to the finite-dimensional ODEs and infinite-dimensional PDEs.RésuméPeriodic orbits play a central role in any invariant characterization of the dynamics,because (a) their existence and inter-relations are a topological, coordinateindependentproperty of the dynamics, and (b) their Floquet multipliers form aninfinite set of metric invariants: The Floquet multipliers of a periodic orbit remain section 6.6invariant under any smooth nonlinear change of coordinates f → h ◦ f ◦ h −1 . Letus summarize the linearized flow notation used throughout the ChaosBook.Differential formulation, flows:ẋ = v ,˙ δx = A δxgoverns the dynamics in the tangent bundle (x, δx) ∈ TM obtained by adjoiningthe d-dimensional tangent space δx ∈ TM x to every point x ∈ M in the d-dimensionalstate space M ⊂ R d . The stability matrix A = ∂v/∂x describes theinstantaneous rate of shearing of the infinitesimal neighborhood of x(t) by theflow.Finite time formulation, maps: A discrete sets of trajectory points {x 0 , x 1 , · · · ,x n , · · ·} ∈ M can be generated by composing finite-time maps, either given asx n+1 = f (x n ), or obtained by integrating the dynamical equations∫ tn+1x n+1 = f (x n ) = x n + dτ v(x(τ)) , (5.21)t ninvariants - 2dec2009 ChaosBook.org version13, Dec 31 2009
for a discrete sequence of times {t 0 , t 1 , · · · , t n , · · ·}, specified by some criterion suchas strobing or Poincaré sections. In the discrete time formulation the dynamics inthe tangent bundle (x, δx) ∈ TM is governed byx n+1 = f (x n ) , δx n+1 = J(x n ) δx n , J(x n ) = J t n+1−t n(x n ) ,where J(x n ) = ∂x n+1 /∂x n = ∫ dτ exp (A τ) is the Jacobian matrix.Stability of invariant solutions: The linear stability of an equilibrium v(x E Q) =0 is described by the eigenvalues and eigenvectors {λ ( j) , e ( j) } of the stability matrixA evaluated at the equilibrium point, and the linear stability of a periodic orbitf T (x) = x, x ∈ M p ,J p (x) e ( j) (x) = Λ p, j e ( j) (x) , Λ p, j = σ ( p j) j)λ(ep T p,by its Floquet multipliers, vectors and exponents {Λ j , e ( j) }, where λ ( p j) = µ ( j)For every continuous symmetry there is a marginal eigen-direction, withiω ( j)pp ±Λ p, j = 1, λ ( pj) = 0. With all 1 + N continuous symmetries quotiented out (Poincarésections for time, slices for continuous symmetries of dynamics, see sect. 10.4)linear stability of a periodic orbit (and, more generally, of a partially hyperbolictorus) is described by the [(d-1-N) × (d-1-N)] monodromy matrix, all of whoseFloquet multipliers |Λ p, j | 1 are generically strictly hyperbolic,M p (x) e ( j) (x) = Λ p, j e ( j) (x) , x ∈ M p /G .We shall show in chapter 11 that extending the linearized stability hyperboliceigen-directions into stable and unstable manifolds yields important global informationabout the topological organization of state space. What matters most arethe expanding directions. The physically important information is carried by theunstable manifold, and the expanding sub-volume characterized by the product ofexpanding Floquet multipliers of J p . As long as the number of unstable directionsis finite, the theory can be applied to flows of arbitrarily high dimension.in depth:appendix B, p. 751fast track:chapter 9, p. 142CommentaryRemark 5.1 Floquet theory. Study of time-dependent and T-periodic vector fields isa classical subject in the theory of differential equations [5.1, 5.2]. In physics literatureFloquet exponents often assume different names according to the context where the theoryis applied: they are called Bloch phases in the discussion of Schrödinger equationwith a periodic potential [5.3], or quasi-momenta in the quantum theory of time-periodicHamiltonians.exerInvariants - 13jun2008 ChaosBook.org version13, Dec 31 2009
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Prognosis of Site-Selective Chemical Reactivity for Organic Molecules
prognosisofsiteselectivechemicalThe goal of this project is to create a collection of fast and reliable algorithms to make a prognosis of the reactivity of organic molecules from their structure employing quantum chemistry calculations. Traditionally, computational analysis of possible reaction pathways requires working with large datasets.
There are some general-purpose workflow engines that allow users to organize and schedule different tasks using a graphical user interface. However, quantum chemistry calculations yield results, which cannot be used by most organic chemists directly. The output of the calculations must be translated into language or formalism to point out their chemical relevance. Although there are billions of reactions involved, only a limited number of factors or reactivity principles exist, which apply to the vast majority of chemical reactions. For example, factors such as “Acidity”, “Basicity”, “Lewis basicity”, are important reactivity descriptors for the largest number of reactions. The main idea of this project approach is to automate analysis of proposed reaction pathways by the calculating their key parameters. We strongly believe that the project will help chemists to understand various reaction mechanisms and to discover new reactions.
Algorithms for Optimization of Heterogeneous Catalysts
algorithmsforoptimazationofheteroThe development of efficient algorithms of computational catalytic design with minimal human intervention and optimal computational expenses represents one of the main challenges of present-day theoretical chemistry and physics. We are working on a novel approach for computational screening heterogeneous catalysts with variable-composition simulations of the material. At the core of our algorithm is an evolutionary algorithm which incorporates “learning from history” done through selection of the low-energy structures and high-catalytic activity to become parents of the new generation. Combining it with automated algorithms for screening catalytic activity of heterogeneous catalysts provides systematic and exhaustive tools for screening a set of chemically varied complex compounds. With the proper choice of the descriptor of catalytic activity, all relevant parameters can be automatically analyzed and the most promising materials identified. The method has several innovative characteristics that allows for its application to probe complex materials as it is automated and requires minimum human intervention. We are working on several interesting applications.
Selected Publications |
877f2862b2f150dc | Category: thermodynamics
No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time
“As far as we can tell, the second law of thermodynamics is true: entropy never decreases for any closed system in the Universe, including for the entirety of the observable Universe itself. It’s also true that time always runs in one direction only, forward, for all observers. What many don’t appreciate is that these two types of arrows — the thermodynamic arrow of entropy and the perceptive arrow of time — are not interchangeable.
During inflation, where the entropy remains low and constant, time still runs forward. When the last star has burned out and the last black hole has decayed and the Universe is dominated by dark energy, time will still run forward. And everywhere in between, regardless of what’s happening in the Universe or with its entropy, time still runs forward at exactly that same, universal rate for all observers.
If you want to know why yesterday is in the immutable past, tomorrow will arrive in a day, and the present is what you’re experiencing right now, you’re in good company. But thermodynamics, interesting though it may be, won’t give you the answer. As of 2019, it’s still an unsolved mystery.”
No matter who you are, where you are, or what you’re doing, you’ll always perceive time running forward, from your frame of reference, at exactly the same rate: one second-per-second. The fact that this is true has led many to speculate as to what the cause of time’s arrow might be, and many, having noticed that entropy never decreases in our Universe, place the blame squarely on thermodynamics as the root of our arrow of time.
But that’s almost certainly not the case, and we can demonstrate that fact in a number of ways, including by decreasing entropy in a region and noting that time still moves forwards. The perceived arrow of time is still a mystery.
Physics, Not Genetics, Explains Why Flamingos Stand On One Leg
“Compared to a flamingo in the water that stands on one leg, an identical flamingo with two legs in the water will lose somewhere between 140-170% the total body heat that the flamingo on one leg loses. That means the flamingo that does learn the preferred behavior — standing on one leg — is free to spend more time in the water: more time feeding, grooming itself, scouting the waters, etc.
Flamingos are pretty weird birds. They have unusually long and skinny legs and necks; their beaks are inverted from most birds; their mating dances only occur in enormous groups; and they range in color from a pale white to a deep pink, orange, or even red. But the defining property of a flamingo, at least to most humans, is that they stand on one leg.
Why would it benefit a flamingo to stand on one unstable leg, rather than two stable ones? Physics, not genetics, explains this flamingo behavior. Come understand the reason today.
The bottom line of every thermodynamics conference.
The bottom line of every thermodynamics conference.
Sunday 5.5.19, 9:00 am. The best moment of the day when you’re arriving at the library for doing some thermodynamics and you’re still alone.
We Still Don’t Understand Why Time Only Flows Forward
“It’s true that entropy does explain the arrow of time for a number of phenomena, including why coffee and milk mix but don’t unmix, why ice melts into a warm drink but never spontaneously arises along with a warm beverage from a cool drink, and why a cooked scrambled egg never resolves back into an uncooked, separated albumen and yolk. In all of these cases, an initially lower-entropy state (with more available, capable-of-doing-work energy) has moved into a higher-entropy (and lower available energy) state as time has moved forwards. There are plenty of examples of this in nature, including of a room filled with molecules: one side full of cold, slow-moving molecules and the other full of hot, fast-moving ones. Simply give it time, and the room will be fully mixed with intermediate-energy particles, representing a large increase in entropy and an irreversible reaction.”
Why does time flow forwards and not backwards, in 100% of cases, if the laws of physics are completely time-symmetric? From Newton’s laws to Einstein’s relativity, from Maxwell’s equations to the Schrödinger equation, the laws of physics don’t have a preferred direction. Except, that is, for one: the second law of thermodynamics. Any closed system that we look at sees its entropy only increase, never decrease.
Could this thermodynamic arrow of time be responsible for what we perceive as the forward motion of time? Interestingly enough, there’s an experiment we can perform: isolate a system and perform enough external work on it to force the entropy inside to *decrease*, an “unnatural” progression of entropy. What happens to time, then? Does it still run forward?
Find out the answer, and learn whether thermodynamics has anything to do with the passage of time or not!
Heat sinks
If you have explored the interior of your CPU then you might have noticed that there are these horizontal metal plates (called fins) many a times with a fan on top of the central or graphic processors.
They are called heat sinks/heat exchangers and are used to dissipate the heat generated by the processor to the surrounding.
The reason why they work is that according to the Fourier’s law, the
heat dissipated is directly proportional to the cross sectional area.
And adding protrusions to the surface increases the net cross section area for exchanging the heat with the surrounding.
Cooking with a computer
In order to demonstrate the extent to which the processor would heat up, let’s remove the heat sink and place a piece of meat on it.
At such high temperatures where cooking a piece of meat becomes possible on a processor, you can be damn sure that the probability of the survival of a computer running without a heat sink is just 0. **
Have a great day!
Stegosaurus and its huge ‘fins’
** for all practical intents and purposes, not merely for testing
If you’ve ever popped open a chilled bottle of champagne, you’ve probably witnessed the gray-white cloud of mist that forms as the cork flies. Opening the bottle releases a spurt of high-pressure carbon dioxide gas, although that’s not what you see in the cloud. The cloud consists of water droplets from the ambient air, driven to condense by a sudden drop in temperature caused by the expansion of the escaping carbon dioxide. Scientifically speaking, this is known as adiabatic expansion; when a gas expands in volume, it drops in temperature. This is why cans of compressed air feel cold after you’ve released a few bursts of air.
If your champagne bottle is cold (a) or cool (b), the gray-white water droplet cloud is what you see. But if your champagne is near room temperature ( c ), something very different happens: a blue fog forms inside the bottle and shoots out behind the cork. To understand why, we have to consider what’s going on in the bottle before and after the cork pops.
A room temperature bottle of champagne is at substantially higher pressure than one that’s chilled. That means that opening the bottle makes the gas inside undergo a bigger drop in pressure, which, in turn, means stronger adiabatic expansion. Counterintuitively, the gas escaping the warm champagne actually gets colder than the gas escaping the chilled champagne because there’s a bigger pressure drop driving it. That whoosh of carbon dioxide is cold enough, in fact, for some of the gas to freeze in that rushed escape. The blue fog is the result of tiny dry ice crystals scattering light inside the bottleneck.
That flash of blue is only momentary, though, and the extra drop in temperature won’t cool your champagne at all. Liquids retain heat better than gases do. For more, on champagne physics check out these previous posts. (Image and research credit: G. Liger-Belair et al.; submitted by David H.) |
cfb6d7ce22608d53 |
You might be familiar with the Schrödinger’s cat thought experiment, where the eponymous feline in a box can be both alive or dead at the same time, often used to illustrate the multi-state paradox of quantum mechanics.
Well, now scientists have managed to apply that theory to huge molecules made up of 2,000 atoms.
Quantum superposition has been tested countless times on smaller systems, with physicists successfully showing that individual particles can be in two places at one time. But this type of experiment hasn’t been carried out at this scale before.
What the experiment does is allow scientists to refine the hypotheses of quantum mechanics and understand more about how this particularly mind-bending branch of physics actually works – and how the laws of quantum mechanics join up with the more traditional, larger scale, classical laws of physics.
“Our results show excellent agreement with quantum theory and cannot be explained classically,” state the researchers in their published paper.
In particular, the new study involves the Schrödinger equation (yes, him again), which describes how even single particles can also act as waves in multiple places at once, interfering with each other just like ripples on a pond.
To test this, the scientists set up a double-slit experiment - an experiment that’s very familiar to quantum physicists.
Traditionally, it involves projecting individual particles of lights (photons) through two slits. If the photons acted simply as particles, the resulting projection of light on the other side would simply show one band. But in reality, the light projected on the other side shows an interference pattern – multiple bands that interact, showing that light particles can also act as waves.
BTezp48v5dHJs6Tw7jHHnB 650 80(Johannes Kalliauer/Wikimedia, CC-BY-SA 3.0)
It effectively seem as if the photons are in two places at once, just like Schrödinger’s cat. But as most of us are aware, the cat is only in two states while it remains unobserved. As soon as the box is open, it’s either confirmed as being alive or dead, not both.
It’s the same with photons. As soon as the light is measured or observed directly, this superposition disappears and the state of the photon is locked in. This is one of the conundrums at the heart of quantum mechanics.
This same double-slit experiment has been done with electrons, atoms, and smaller molecules. And now physicists show it applies to massive molecules, too.
In this take on the double-slit experiment, the team was able to use these heavy molecules, made up of as many as 2,000 atoms, to create quantum interference patterns, as if they were behaving as waves and being in more than one place.
The molecules were known as “oligo-tetraphenylporphyrins enriched with fluoroalkylsulfanyl chains”, and some were more than 25,000 times the mass of a hydrogen atom.
But as molecules get bigger, they also get less stable, and the scientists were only able to get them interfering for seven milliseconds at a time, using a newly designed piece of equipment called a matter-wave interferometer (designed to measure atoms along different paths).
Even factors like the Earth’s rotation and gravitational pull had to be factored in. It was worth the effort though – we now know these giant molecules can be in two places at once, as well as much smaller atoms.
As quantum mechanics traditionally comes into play on very small scales, and classical physics on larger scales, the bigger the molecules we can get working with the double slit experiment, the closer we get to that quantum-classical boundary line. A previous record for this kind of study involved molecules up to 800 atoms in size.
“Our experiments show that quantum mechanics, with all its weirdness, is also amazingly robust, and I’m optimistic that future experiments will test it on an even more massive scale,” says physicist Yaakov Fein, from the University of Vienna in Austria.
The research has been published in Nature Physics.
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4dacabd7edf82d69 | Structural Biochemistry/Organic Chemistry/Reagents
A reagent is an inorganic or small organic molecule that helps the reactant react in a chemical reaction.
List of Reagents, Its Uses and InformationEdit
1) AIBN [azobis (isobutyronitrile)] is used for radical initiator. AIBN is a white acicular crystal,which is insolvable in water,solvable in organic solvent such as methyl alcohol,ethanol,acetone,ethyl ether and light petroleum etc.lce point of pure product is 105 degree Celsius. The product is decomposed flashily and it releases nitrogen gas in the position of melting. It decomposes slowly under ordinary temperature,which should be stored under 20 degreee Celsius. AIBN is maily used as polymerization initiator of monomer such as chloroethylene,vinyl acetate,acrylonitrile,etc. Also, it is used as blowing agent for PVC,polyalkene,polyurethane,polyvinyl alcohol,acrylonitrile/butadiene copolymer,chloroethylene copolymer,acrylonitrile/ butadiene/styrene copolymer,polyisocyanate,polyvinyl acetate,polyamide and polyester,etc. Moreover,it is also used in other organic synthesis.
2) AlCl3 (aluminum trichloride) is used for Lewis acid catalyst. It is a yellowish or grayish-white, crystalline powder with a sharp oodor. It is used as a chemical intermediate for Aluminum compounds, as a catalyst for cracking petroleum, in preserving wood, and in medications, disinfectants, cosmetics, photography and textiles.
3) BF3 (boron trifluoride) is used for Lewis acid catalyst of chemical reactions. It is a colorless gas with a pungent odor. It reacts readily to form coordination complexes with molecules having at least one pair of unshared electrons.
4) BH3 (borane) is used for hydroboration. Borane-lewis base complexes are often found in literature. Borane-tetrahydrofuran (BTHF) and borane-dimethyl sulfide (BMS, DMSB) are often used as a borane source. Both reagents are available in solution (e.g. 1 M in THF), and are therefore easier to handle than diborane. Volatility and flammability are always a drawback. BMS is more stable than BTHF but has an unpleasant odor.
5) Br2 (bromine) is used for radical bromination and dibromination. Bromine compounds are used as pesticides, dyestuffs, water purification compounds, and as a flame-retardants in plastics. 1,2-dibromoethane is used as an anti-knock agent to raise the octane number of gasoline and allow engines to run more smoothly. This application has declined as a result of environmental legislation. Potassium bromide is used as a source of bromide ions for the manufacture of silver bromide for photographic film.
7) CHCl3 (chloroform) is used for polar, nonflammable solvent. It is also a highly volatile, clear, colourless, heavy, and highly refractive.
8) CH2Cl2 (dichloromethane) is used for polar, nonflammable solvent. Chloroform has a relatively narrow margin of safety and has been replaced by better inhalation anesthetics. In addition, it is believed to be toxic to the liver and kidneys and may cause liver cancer. Chloroform was once widely used as a solvent, but safety and environmental concerns have reduced this use as well. Nevertheless, chloroform has remained an important industrial chemical.
9) CH2I2 (diiodomethane) is used for Simmons-Smith cyclopropanation. It is a colorless liquid. It decomposes upon exposure to light liberating iodine, which colours samples brownish.
10) CH2N2 (diazomethane) is used for making methyl esters from acid and cyclopropanation. It is not only toxic but also potentially explosive.
11) DIBAL (diisobutylaluminum) is used for selective reduction of esters, amides, and nitriles to aldehydes.
12) Dicycolhexylborane is used for hydroboration of alkyne derivatives and anti-Markovnikov hydration.
13) Dioxane is used for good solvent for dissolving water and organic substrates. It is a colorless liquid with a faint sweet odor similar to that of diethyl ether. It is classified as an ether.
14) DMD (dimethyldioxirane) is used for epoxidation of alkenes. It is the most commonly used dioxirane in organic synthesis, and can be considered as a monomer of acetone peroxide.
15) DMF (dimethylformamide) is used for polar aprotic solvent. This colourless liquid is miscible with water and the majority of organic liquids. DMF is a common solvent for chemical reactions.
16) DMSO (dimethylsulfoxide) is use for polar aprotic solvent. This colorless liquid is an important polar aprotic solvent that dissolves both polar and nonpolar compounds and is miscible in a wide range of organic solvents as well as water.
17) Et2O (diethyl ether) is used for medium polarity solvent. It is a colorless, highly volatile flammable liquid with a characteristic odor.
18) FeBr3 (iron tribromide) is used for Lewis acid catalyst in the halogenation of aromatic compounds.
19) H2 (hydrogen) is used for hydrogenation and reduction of nitro. Hydrogen is the only element that can exist without neutrons. Hydrogen’s most abundant isotope has no neutrons. Hydrogen forms both positive and negative ions. It does this more readily than any other element. It is the most abundant element in the universe. Hydrogen is the only atom for which the Schrödinger equation has an exact solution. Moreover, it reacts explosively with the elements oxygen, chlorine and fluorine: O2, Cl2, F2.
20) H2O2 (hydrogen peroxide) is used for oxidative workup of hydroboration. It is used to help stop infection in cuts or scrapes, we can use it as a mouthwash when diluted with water, and it is also used to bleach hair.
21) Hg(OAc)2 (mercuric acetate) is used for oxymercuriation. Mercuric Acetate can affect in breating and by passing through skin. Also, mercuric acetate should be handled as a teratogen with extreme caution. Mercury poisoning can cause "shakes", irritability, sore gums. It increased saliva, personality change and permanent brain or kidney damage. Mercury accumulates in the body.
22) HgSO4 (mercuric sulfate) is used for Markovnikov hydration of alkynes. It is an odorless solid that forms white granules or crystalline powder. In water, it separates into an insoluble sulfate with a yellow color and sulfuric acid.
23) HIO4 (metaperiodic acid) is used for oxidative cleavage of 1,2-diols. In dilute aqueous solution, periodic acid exists as discrete hydronium and metaperiodate ions.
24) HMPA (hexamethylphosphoramide) is used for preventing aggregation (polar aprotic solvent). It is a phosphoramide having the formula [2N]3PO.
25) K2Cr2O7/H2SO4 (potassium dichromate) is used in oxidation of alcohols. (Jones Reagent)
26) LAH (lithium aluminum hydride) is used for very strong hydride source and reduces esters to alcohols. It is an inorganic compound with the chemical formula LiAlH4.
27) LiAl(Ot-Bu)3H [lithium tri(t-butoxy) aluminum hydride] is used for modified hydride source and reduces acid chlorides to aldehydes.
28) LDA (lithium diisopropylamide) is used for strong, hindered base.
29) Lindlar's catalyst is used for reducing alkynes to cis-alkenes.
30) mCPBA (m-chlroperbenzoic acid) is used for epoxidation of alkenes.
31) MnO2 (manganese dioxide) is used for selective oxidation of allylic alcohols.
32) MsCl [methanesulfonyl chloride(mesyl chloride)] is used for converting hydroxyl to a good LG.
33) NaBH4 (sodium borohydride) is used for mild source of hydride. It is an inorganic compound.
34) NaBH3CN (sodium cyanoborohydride) is used for reductive amination and hydride source stable to mild acid.
35) NaNO2 (sodium nitrite) is used for diazotization of amines. Sodium nitrite is a salt and an anti-oxidant that is used to cure meats like ham, bacon and hot dogs. Sodium nitrite serves a vital public health function: it blocks the growth of botulism-causing bacteria and prevents spoilage. It is also gives cured meats their characteristic color and flavor. Also, USDA-sponsored research indicates that sodium nitrite can help prevent the growth of Listeria monocytogenes, an environmental bacterium that can cause illness in some at-risk populations.
36) NBS (N-bromosuccinimide) is used for bromine surrogate. It is a brominating and oxidizing agent that is used as source for bromine in radical reactions. For example: allylic brominations and various electrophilic additions. The NBS bromination of substrates such as alcohols and amines, followed by elimination of HBr in the presence of a base, leads to the products of net oxidation in which no bromine has been incorporated.
37) n-BuLi is used for strong base.
38) NCS (N-chlorosuccinimide) is used for chlorine surrogate.
39) O3 (ozone) is used for oxidative cleavage of alkenes. It is a triatomic molecule, consisting of three oxygen atoms. It is an allotrope of oxygen that is much less stable than the diatomic allotrope, breaking down with a half life.
40) OsO4 (osmium tetroxide) is used for dihydroxylation of alkenes.
41) PCC (pyridinium chlorochromate) is used for selective oxidation of primary alcohols to aldehydes.
42) PPH 3 (triphenylphosphine) is used for making Wittig reagents.
43) SOCl2 (thionyl chloride) is used for converting alcohols to alkyl chlorides.
44) THF (tetrahydrofuran) is used for medium polarity solvent.
45) pTsCl [p-toluenesulfonyl chloride (tosyl chloride)] is used for converting hydroxyl to a good LG.
46) pTsOH [p-toluenesulfonic acid (tosic acid)] is used for oragnic-soluble source of strong acid.
47) Zn(Hg) (zinc amalgam) is used for Clemmensen reduction (with HCl).
48) Jones Reagent (CrO3, H2SO4, H2O) is a solution of chromium trioxide in diluted sulfuric acid that can be used safely for oxidations of organic substrates.
49) SOCL2 - forms alkyl chlorides from alcohols
50) Clemmensen Reduction (Zn(Hg), HCl) - removes a ketone and replaces it with hydrogens.
51) Grinard reagents (R-Mg-X) - an organometalic chemical reaction where an alkyl-magnesium halide is added to a carbonyl group in an aldehyde or ketone. |
f4ee3824c9ddbf90 | Electronic structure and time-dependent description of rotational predissociation of LiH
View graph of relations
The adiabatic potential energy curves of the 1Σ+ and 1Π states of the LiH molecule were calculated. They correlate asymptotically to atomic states, such as 2s + 1s, 2p + 1s, 3s + 1s, 3p + 1s, 3d + 1s, 4s + 1s, 4p + 1s and 4d + 1s. A very good agreement was found between our calculated spectroscopic parameters and the experimental ones. The dynamics of the rotational predissociation process of the 11Π state were studied by solving the time-dependent Schrödinger equation. The classical experiment of Velasco [Can. J. Phys., 1957, 35, 1204] on dissociation in the 11Π state is explained for the first time in detail.
Original languageEnglish
JournalPhysical Chemistry Chemical Physics
Issue number30
Pages (from-to)19777-19783
Number of pages7
Publication statusPublished - 2017
CitationsWeb of Science® Times Cited: No match on DOI
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ID: 133318959 |
1d1a90b906d17da8 | Comment by dynamically_linked on The Magnitude of His Own Folly · 2008-09-30T14:56:50.000Z · score: 7 (7 votes) · LW · GW
Eliezer, after you realized that attempting to build a Friendly AI is harder and more dangerous than you thought, how far did you back-track in your decision tree? Specifically, did it cause you to re-evaluate general Singularity strategies to see if AI is still the best route? You wrote the following on Dec 9 2002, but it's hard to tell whether it's before or after your "late 2002" realization.
I for one would like to see research organizations pursuing human intelligence enhancement, and would be happy to offer all the ideas I thought up for human enhancement when I was searching through general Singularity strategies before specializing in AI, if anyone were willing to cough up, oh, at least a hundred million dollars per year to get started, and if there were some way to resolve all the legal problems with the FDA.
Hence the Singularity Institute "for Artificial Intelligence". Humanity is simply not paying enough attention to support human enhancement projects at this time, and Moore's Law goes on ticking.
Aha, a light bulb just went off in my head. Eliezer did reevaluate, and this blog is his human enhancement project!
Comment by dynamically_linked on Whither Moral Progress? · 2008-07-16T11:24:15.000Z · score: 2 (2 votes) · LW · GW
My view is similar to Robin Brandt's, but I would say that technological progress has caused the appearance of moral progress, because we responded to past technological progress by changing our moral perceptions in roughly the same direction. But different kinds of future technological progress may cause further changes in orthogonal or even opposite directions. It's easy to imagine for example that slavery may make a comeback if a perfect mind control technology was invented.
Comment by dynamically_linked on Fundamental Doubts · 2008-07-14T09:23:45.000Z · score: 0 (0 votes) · LW · GW
Aaron, statistical mechanics also depend on particle physics being time-reversible, meaning that two different microstates at time t will never evolve to the same microstate at time t+1. If this assumption is violated then entropy can decrease over time.
Is there some reason why time-reversibility has to be true?
If we can imagine a universe where entropy can be made to decrease, then living beings in it will certainly evolve to take advantage of this. Why shouldn't it be the case that human beings are especially good at this, and that is what they are being used for by the machines?
Comment by dynamically_linked on Is Morality Given? · 2008-07-07T04:06:00.000Z · score: 0 (0 votes) · LW · GW
Constant, if moral truths were mathematical truths, then ethics would be a branch of mathematics. There would be axiomatic formalizations of morality that do not fall apart when we try to explore their logical consequences. There would be mathematicians proving theorems about morality. We don't see any of this.
Isn't it simpler to suppose that morality was a hypothesis people used to explain their moral perceptions (such as "murder seems wrong") before we knew the real explanations, but now we find it hard to give up the word due to a kind of memetic inertia?
Comment by dynamically_linked on Is Morality Given? · 2008-07-06T22:58:06.000Z · score: 0 (0 votes) · LW · GW
For those impatient to know where Eliezer is going with this series, it looks like he gaves us a sneak preview a little more than a year ago. The answer is morality-as-computation.
Eliezer, hope I didn't upset your plans by giving out the ending too early. When you do get to morality-as-computation, can you please explain what exactly is being computed by morality? You already told us what the outputs look like: "Killing is wrong" and "Flowers are beautiful", but what are the inputs?
Comment by dynamically_linked on Is Morality Given? · 2008-07-06T20:49:05.000Z · score: 0 (0 votes) · LW · GW
Constant wrote: So one place where one could critique your argument is in the bit that goes: "conditioned on X being the case, then our beliefs are independent of Y". The critique is that X may in fact be a consequence of Y, in which case X is itself not independent of Y.
Good point, my argument did leave that possibility open. But, it seems pretty obvious, at least to me, that game theory, evolutionary psychology, and memetics are not contingent on anything except mathematics and the environment that we happened to evolve in.
So if I were to draw a Bayesian net diagram, it would look it this:
math --- --- game theory ------------
\ / \
--- evolutionary psychology - moral perceptions
/ \ /
environment -- --- memetics ---------------
Ok, one could argue that each node in this diagram actually represents thousands of nodes in the real Bayesian net, and each edges is actually millions of edges. So perhaps the following could represent a simplification, for a suitable choice of "morality":
\ / \
-- morality -- evolutionary psychology --- moral perceptions
/ \ /
Before I go on, do you actually believe this to be the case?
Comment by dynamically_linked on Is Morality Given? · 2008-07-06T10:39:56.000Z · score: 2 (4 votes) · LW · GW
And to answer Obert's objection that Subhan's position doesn't quite add up to normality: before we knew game theory, evolutionary psychology, and memetics, nothing screened off our moral perceptions/intuitions from a hypothesized objective moral reality, so that was perhaps the best explanation available, given what we knew back then. And since that was most of human history, it's no surprise that morality-as-given feels like normality. But given what we know today, does it still make sense to insist that our meta-theory of morality add up to that normality?
Comment by dynamically_linked on Is Morality Given? · 2008-07-06T10:14:51.000Z · score: 9 (10 votes) · LW · GW
But we already know why murder seems wrong to us. It's completely explained by a combination of game theory, evolutionary psychology, and memetics. These explanations screen off our apparent moral perceptions from any other influence. In order words, conditioned on these explanations being true, our moral perceptions are independent of (i.e. uncorrelated with) any possible morality-as-given, even if it were to exist.
So there is a stronger argument against Obert than the one Subhan makes. It's not just that we don't know how we can know about what is right, but rather that we know we can't know, at least not through these apparent moral perceptions/intuitions.
Comment by dynamically_linked on Is Morality Preference? · 2008-07-05T03:31:21.000Z · score: 2 (2 votes) · LW · GW
Why is it a mystery (on the morality-as-preferences position) that our terminal values can change, and specifically can be influenced by arguments? Since our genes didn't design us with terminal values that coincide with its own (i.e., "maximize inclusive fitness"), there is no reason why they would have made those terminal values unchangeable.
We (in our environment of evolutionary adaptation) satisfied our genes' terminal value as a side-effect of trying to satisfy our own terminal values. The fact that our terminal values respond to moral arguments simply means that this side-effect was stronger if our terminal values could change in this way.
I think the important question is not whether persuasive moral arguments exist, but whether such arguments form a coherent, consistent philosophical system, one that should be amenable to logical and mathematical analysis without falling apart. The morality-as-given position implies that such a system exists. I think the fact that we still haven't found this system is a strong argument against this position.
Comment by dynamically_linked on The Bedrock of Fairness · 2008-07-03T09:50:55.000Z · score: 6 (6 votes) · LW · GW
Why doesn't Zaire just divide himself in half, let each half get 1/4 of the pie, then merge back together and be in possession of half of the pie?
Or, Zaire might say: Hey guys, my wife just called and told me that she made a blueberry pie this morning and put it in this forest for me to find. There's a label on the bottom of the plate if you don't believe me. Do you still think 'fair' = 'equal division'?
Or maybe Zaire came with his dog, and claims that the dog deserves an equal share.
I appreciate the distinction Eliezer is trying to draw between the object level and the meta level. But why the assumption that the object-level procedure will be simple?
Comment by dynamically_linked on What Would You Do Without Morality? · 2008-06-29T21:48:00.000Z · score: 0 (0 votes) · LW · GW
Notice how nobody is willing to admit under their real name that they might do something traditionally considered "immoral". My point is, we can't trust the answers people give, because they want to believe, or want others to believe, that they are naturally good, that they don't need moral philosophies to tell them not to cheat, steal, or murder.
BTW, Eliezer, I got the "enemies list" you sent last night. Rest assured, my robot army will target them with the highest priority. Now stop worrying, and finish that damn proof already!
Comment by dynamically_linked on What Would You Do Without Morality? · 2008-06-29T11:02:18.000Z · score: 0 (0 votes) · LW · GW
Seriously, most moral philosophies are against cheating, stealing, murdering, etc. I think it's safe to guess that there would be more cheating, stealing, and murdering in the world if everyone became absolutely convinced that none of these moral philosophies are valid. But of course nobody wants to publicly admit that they'd personally do more cheating, stealing, and murdering. So everyone is just responding with variants of "Of course I wouldn't do anything different. No sir, not me!"
Except apparently Shane Legg, who doesn't seem to mind the world knowing that he's just waiting for any excuse to start cheating, stealing, and murdering. :)
Comment by dynamically_linked on What Would You Do Without Morality? · 2008-06-29T11:01:46.000Z · score: 1 (1 votes) · LW · GW
Eliezer, I've got a whole set of plans ready to roll, just waiting on your word that the final Proof is ready. It's going to be bloody wicked... and just plain bloody, hehe.
Comment by dynamically_linked on Timeless Causality · 2008-05-30T20:22:37.000Z · score: 2 (2 votes) · LW · GW
Nick, here's what Judea Pearl wrote on this topic. On page 59 of his book:
This suggests that the consistent agreement between physical and statistical times [i.e., the direction of time and the direction of causality] is a byproduct of the human choice of linguistic primitives and not a feature of physical reality. ... Pearl and Verma (1991) speculated that this preference represents survival pressure to facilitate prediction of future events, and that evolution has evidently ranked this facility more urgent than that of finding hindsighted explanation for current events.
Eliezer wants to go from timeless physics to causality, to computation, to anticipation. He admits being unsure about the latter two steps, but even the first step doesn't seem to work. And besides, timeless physics (and relational physics, which timeless physics builds on top of) itself is highly speculative and problematic. Is the intention to actually convince us of the correctness of these ideas, or just to make us "think outside the box" and realize that these possibilities exist?
Comment by dynamically_linked on Timeless Causality · 2008-05-29T23:16:40.000Z · score: 0 (0 votes) · LW · GW
RI, what if I wanted to buy two windows such that one is twice the mass of the other. Is that still cheating?
Nick, how would you transform my causal hypothesis (in the comment above) with intramoment dependencies into one without?
Comment by dynamically_linked on Timeless Causality · 2008-05-29T22:13:21.000Z · score: 4 (4 votes) · LW · GW
This definition of causality doesn't seem to work, since the universe clearly doesn't generate future values independently of each other. Consider the following story:
On Monday I decide to buy 2 windows of the same mass. Suppose I want to buy the biggest windows I can afford, and I have money in two bank accounts that I can use for this purpose. On Tuesday a couple of cute little vandals break both of my windows. Some of the glass falls inside my home, and rest outside. Now let:
L1 = how much money I had in bank 1 L2 = how much money I had in bank 2 M1 = mass of window 1 M2 = mass of window 2 R1 = mass of glass that fell inside my home R2 = mass of glass that fell outside my home
Intuitively it seems pretty obvious that the arrow of causality runs from left to right, but if you use the definition Eliezer gave, you'd get the opposite result. Quoting Eliezer:
if we see:
P(M2|L1,L2) ≠ P(M2|M1,L1,L2) P(M2|R1,R2) = P(M2|M1,R1,R2)
Then we can guess causality is flowing from right to left.
Well, P(M2|L1,L2) ≠ P(M2|M1,L1,L2) because M2 depends on the price of glass as well as L1 and L2, but knowing M1 gives us the precise value of M2 (remember that I wanted to buy 2 windows of the same mass). P(M2|R1,R2) = P(M2|M1,R1,R2) since M2=(R1+R2)/2 and M1 doesn't give any more information on top of that.
Comment by dynamically_linked on Timeless Physics · 2008-05-28T05:08:04.000Z · score: 4 (4 votes) · LW · GW
I went back to the beginning of this series of posts, and found this introduction:
I think I must now temporarily digress from the sequence on zombies (which was a digression from the discussion of reductionism, which was a digression from the Mind Projection Fallacy) in order to discuss quantum mechanics. The reasons why this belongs in the middle of a discussion on zombies in the middle of a discussion of reductionism in the middle of a discussion of the Mind Projection Fallacy, will become apparent eventually.
Eliezer, would you mind telling us the reasons now, instead of having them become apparent eventually? I ask this because I'd like to know, if I detect some error or confusion in the posts or comments, whether it's central to your eventual point, or if it's just an inconsequential nit. Do you actually need Barbour's timeless physics to make your point, or would the standard block universe do? I'd like to skip explaining the difference between the two if the difference doesn't really matter. I mean we're not here to learn about some speculative physics for its own sake...
Comment by dynamically_linked on Timeless Physics · 2008-05-28T02:12:29.000Z · score: 2 (2 votes) · LW · GW
Barbour is proposing something quite different from the block universe. I'm not sure if Eliezer is missing the point, or just not carrying it across. Barbour is speculating that if we solve the Wheeler-DeWitt equation, we'll get a single probability distribution over the configuration space of the universe, and all of our experiences can be explained using this distribution alone. Specifically, we don't need a probability distribution for each instant of time, like in standard QM.
I think Eliezer's picture with the happy faces is rather misleading, if it's suppose to represent Barbour's idea. I'd fix it by getting rid of the arrows, jumble the faces all around so that there is no intrinsic time-like ordering between them, and then attach a probability to each face that together add up to less than 1.
Steve, thanks for the paper link. Parity violation clearly represents a big problem to relational physics, and I'm glad I'm not the only one who noticed. :)
Comment by dynamically_linked on Timeless Physics · 2008-05-27T20:42:59.000Z · score: 2 (2 votes) · LW · GW
This abstract of one of Barbour's papers may be helpful for those wondering (like me) how exactly Barbour was proposing to get rid of "t":
Abstract. A strategy for quantization of general relativity is considered in the context of the timelessness' of classical general relativity discussed in the preceding companion paper. The Wheeler--DeWitt equation (WDE) of canonical quantum gravity is interpreted as being like a time-independent Schrödinger equation for one fixed energy, the solution of which simply gives, once and for all, relative probabilities for each possible static relative configuration of the complete universe. Each such configuration is identified with a possible instant of experienced time. These instants are not embedded in any kind of external or internal time and, if experienced, exist in their own right. The central question is then: Whence comes the appearance of the passage of time, dynamics, and history? The answer proposed here is that these must all becoded', in the form of what appear to be mutually consistent `records', in the individual static configurations of the universe that are actually experienced. Such configurations are called time capsules and suggest a new, many-instants, interpretation of quantum mechanics. Mott's explanation of why -particles make straight tracks in Wilson cloud chambers shows that the time-independent Schrödinger equation can concentrate its solution on time capsules. This demonstrates how the appearance of dynamics and history can arise in a static situation. If it can be shown that solutions of the Wheeler--DeWitt equation are spontaneously and generically concentrated on time capsules, this opens up the possibility of an explanation of time at a very deep level: the timeless wavefunction of the universe concentrates the quantum mechanical probability on static configurations that are time capsules, so that the situations which have the highest probability of being experienced carry within them the appearance of time and history. It is suggested that the inescapable asymmetry of the configuration space of the universe could play an important role in bringing about such concentration on time capsules and be the ultimate origin of the arrow of time.
Comment by dynamically_linked on Relative Configuration Space · 2008-05-26T23:05:28.000Z · score: 4 (4 votes) · LW · GW
I don't have a copy of Barbour's book. Maybe someone who does can check what it says about parity violation? (Never mind, I just did an Amazon search inside the book, and it contains no mention of "parity" or "chirality".)
Anyway, my understanding is that parity violation means that reversing left and right of the entire universe would not give you the same internal experience. If this is hard to imagine, suppose that the laws of physics were such that right-handed DNA works the same as in our universe, but left-handed DNA is 10% less stable. (This actually seems to be the case in our own universe, but the effect is much smaller. See Reversing left and right of the entire universe would mean that our mutation rate suddenly increases by 10%, and the mutation rate of some aliens with left-handed DNA suddenly decreases by 10%. This kind of law of physics would be impossible to formulate with a relative configuration space.
Even if Barbour does handle this problem somehow, I think making certain types of physics impossible to imagine is not such a great idea. What if it turns out that we need those types of physics to describe our universe?
Comment by dynamically_linked on Relative Configuration Space · 2008-05-26T17:52:04.000Z · score: 5 (5 votes) · LW · GW
But if you could learn to visualize the relative configuration space, then, so long as you thought in terms of those elements of reality, it would no longer be imaginable that Mach's Principle could be false.
If one learned to think only in terms the relative configuration space, it would also become impossible to imagine that parity violation could be possible, since the left-hand and right-hand versions of a system have the same relative distances. Yet the weak nuclear force does violate parity.
Comment by dynamically_linked on Many Worlds, One Best Guess · 2008-05-11T22:27:09.000Z · score: 2 (2 votes) · LW · GW
Eliezer, I think your (and Robin's) intuition is off here. Configuration space is so vast, it should be pretty easy for a small blob of amplitude to find a hiding place that is safe from random stray flows from larger blobs of amplitude.
Consider a small blob in my proposed experiment where the number of 0s and 1s are roughly equal. Writing the outcomes on blackboards does not reduce the integrated squared modulus of this blob, but does move it further into "virgin territory", away from any other existing blobs. In order for it to be mangled by stray flows from larger blobs, those stray flows would somehow have to reach the same neighborhood as the small blob. But how? Remember that in this neighborhood of configuration space, the blackboards have a roughly equal number of 0s and 1s. What is the mechanism that can allow a stray piece of a larger blob to reach this neighborhood and mangle the smaller blob? It can't be random quantum fluctuations, because the Born probability of the same sequence of 0s and 1s spontaneously appearing on multiple blackboards is much less than the integrated squared modulus of the small blob. To put it another way, by the time a stray flow from a larger blob reaches the small blob, its amplitude would be spread much too thin to mangle the small blob.
Comment by dynamically_linked on Many Worlds, One Best Guess · 2008-05-11T18:14:27.000Z · score: 0 (1 votes) · LW · GW
Robin, can you offer some intuitive explanation as to why defense against world mangling would be difficult? From what I understand, a larger blob of amplitude (world) can mangle a smaller blob of amplitude only if they are close together in configuration space. Is that incorrect? If those "secure storage facilities" simply write the quantum coin toss outcomes in big letters on some blackboards, which worlds will be close enough to be able to mangle the worlds that violate Born's rule?
Comment by dynamically_linked on Many Worlds, One Best Guess · 2008-05-11T17:09:28.000Z · score: 1 (1 votes) · LW · GW
Robin Hanson suggests that if exponentially tinier-than-average decoherent blobs of amplitude ("worlds") are interfered with by exponentially tiny leakages from larger blobs, we will get the Born probabilities back out.
Shouldn't it be possible for a tinier-than-average decoherent blobs of amplitude to deliberately become less vulnerable to interference from leakages from larger blobs, by evolving itself to an isolated location in configuration space (i.e., a point in configuration space with no larger blobs nearby)? For example, it seems that we should be able to test the mangled worlds idea by doing the following experiment:
1. Set up a biased quantum coin, so that there is a 1/4 Born probability of getting an outcome of 0, and 3/4 of getting 1.
2. After observing each outcome of the quantum coin toss, broadcast the outcome to a large number of secure storage facilities. Don't start the next toss until all of these facilities have confirmed that they've received and stored the previous outcome.
3. Repeat 100 times.
Now consider a "world" that has observed an almost equal number of 0s and 1s at the end, in violation of Born's rule. I don't see how it can get mangled. (What larger blob will be able to interfere with it?) So if mangled worlds is right, then we should expect a violation of Born's rule in this experiment. Since I doubt that will be the case, I don't think mangled worlds can be right.
Comment by dynamically_linked on Argument Screens Off Authority · 2007-12-15T03:48:34.000Z · score: 0 (0 votes) · LW · GW
Has anyone read Learning Bayesian Networks by Richard E. Neapolitan? How does it compare with Judea Pearl's two books as an introduction to Bayesian Networks? I'm reading Pearl's first book now, but I wonder if Neapolitan's would be better since it is newer and is written specifically as a textbook.
Comment by dynamically_linked on When None Dare Urge Restraint · 2007-12-10T03:45:00.000Z · score: 0 (0 votes) · LW · GW
Eliezer, the US killed at least a million Japanese in World War 2, while the attack at Pearl Harbor killed less than 2500. Maybe it is true that the US response to 9/11 is "greater than the appropriate level, whatever the appropriate level may be" but I don't think you have showed that to actually be the case.
Comment by dynamically_linked on Truly Part Of You · 2007-11-21T20:46:27.000Z · score: 0 (0 votes) · LW · GW
Comment by dynamically_linked on Evolving to Extinction · 2007-11-17T01:25:21.000Z · score: 0 (0 votes) · LW · GW
The issue is replication with variation and the necessary historical consequences of this.
Evolution requires more than replication with variation. It needs differential replication with variation.
There is therefore no way to avoid the consequences of evolution: they are not biological consequences, but consequences of the laws of physics and logic. There is no way around them.
I can think of a couple of potential ways to avoid the consequences of evolution, by attacking the "differential" part.
1. The Singleton.
2. Some other method for achieving absolute security and property rights. For example a completely impenetrable shield. Or having automatic fail-proof self-destruct mechanisms built into everything to make it pointless for anyone to try to appropriate other people's property. |
c98a0341cf9a354b | Take the 2-minute tour ×
My question is in the title. I do not really understand why water is not a superfluid. Maybe I make a mistake but the fact that water is not suprfluid comes from the fact that the elementary excitations have a parabolic dispersion curve but for me the question remain. An equivalent way to ask it is: why superfluid helium is described by Gross-Pitaevsky equation and it is not the case for water?
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Recent work actually suggests that water may have a superfluid liquid phase – user20145 Jan 23 '13 at 15:24
@x you have to substantiate this claim by a reference or link and a quote, at least from the abstract. – anna v Jan 23 '13 at 15:31
2 Answers 2
You refer to the Landau criterion for superfluidity (there is a separate question whether this is really the best way to think about superfluids, and whether the Landau criterion is necessary and/or sufficient). In a superfluid the low energy excitations are phonons, the dispersion relation is linear $E_p\sim c p$, and the critical velocity is non-zero. In water the degrees of freedom are water molecules, the dispersion relation is quadratic, $E_p\sim p^2/(2m)$, and the critical velocity is zero.
The Gross-Pitaevskii equation applies (approximately) to Helium, because in the superfluid phase there is a single particle state which is macroscopically occupied. The GP equation describes the time evolution of the corresponding wave function. In water there are no macroscopically occupied states. You can try to solve the full many-body Schroedinger equation, but at least approximately this problem reduces to cassical kinetic theory.
I think the best criterion for superfluidity is irrotational flow: The non-classical moment of inertia, quantization of circulation, and persistent flow in a ring. Again, these don't appear in water because there is no spontaneous symmetry breaking, and no macroscopically occupied state.
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So now my question is why there is no macroscopically occupied state for water ans there is one for helium? In general we don't try to solve Schrödinger equation for helium in order to obtain GP equation, isnt'it? And how can I obtain a classical kinetic equation for water starting from Schrödinger ? – PanAkry Sep 24 '12 at 6:56
A rough criterion is the condition for Bose condensation in an ideal gas, $n\lambda^3\sim 1$, where $n$ is the density and $\lambda$ is the thermal wave length. Note that your question is in some sense backwards: Helium is the exception, water is the rule. Most ordinary fluids solidify instead of becoming superfluid at low $T$. – Thomas Sep 24 '12 at 12:38
Because water is liquid at much too high a temperature. Helium is only superfluid near absolute zero. To have a superfluid, you need the quantum wavelength of the atoms given the environmental decoherence to be longer than the separation between the atoms, so they can coherently come together.
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Your Answer
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4ec03126b82d3a18 | RE: [asa] The whole of reality
From: Gregory Arago <>
Date: Thu May 28 2009 - 18:04:55 EDT
Hi Moorad,
Someone (perhaps in North Carolina) wrote:
<I do not define science via the human <detector> but by means of purely physical devices.>
My question: who are YOU then that <defines> science? Are you not a <detector> or observer> yourself? How can one <define science> using <purely physical devices>?
The issue is one of distingushing <purely physical devices> from an attempt at being totally <objective>, when that is impossible. Human beings are not robots!
Moorad also wrote:
<Spirit is a supernatural concept.>
Is it not also a <non-physical> concept? It would be helpful for you to distinguish between the non-physical and the super-natural.
I would prefer the term <supra-natural>, but perhaps that,s just me.
Instead of just three terms to view <the whole of reality>, perhaps Terry,s suggestion of a richer ontology might be helpful here. Do you know the work of Herman Dooyeweerd, Moorad? He is a surprisingly holistically-oriented western thinker. You can find out more about him here: here:
"for everyone engaged in scientific research proceeds from definite presuppositions, from certain basic convictions, and is tied to all kinds of conceptions and insights in his [sic] inner being. And the less a scientist is conscious of this inescapable attachment to certain presuppositions, the more is he chained to them and the more strongly is he dominated by them." - C. Veenhof (from the Dooyeweerd site)
--- On Fri, 5/29/09, Alexanian, Moorad <> wrote:
From: Alexanian, Moorad <>
Subject: RE: [asa] The whole of reality (subject name changed)
To: "Bill Powers" <>
Cc: "Gregory Arago" <>, "Cameron Wybrow" <>, "" <>
Received: Friday, May 29, 2009, 1:04 AM
The reason science can only investigate the physical aspect of nature is on how science collects data, which is by means of purely physical devices. It would be miraculous if purely physical devices reacted or recorded nonphysical or supernatural “data.” I find easier to deal with the concept of physical than material. Material carries a lot more baggage as a term and I do not know how to define it operationally.
What is real is the physical data. How the data is interpreted is where theory comes in whereby constituents are postulated and further corroborated experimentally. That is in essence what science is.
One real does not need direct human sensible experience except to read the measuring devices and thus obtain the data. Note that all phenomena are not physical. Love is a human phenomenon that goes beyond its physical manifestation where even the supernatural comes into play.
Humans as “detectors” can experience nonphysical and supernatural phenomena. That is why I do not define science via the human “detector” but by means of purely physical devices.
The notion of empirical is equivocal since includes purely physical devices as well as humans ad detectors. There is nothing wrong with definitions. That is what makes human communication meaningful and unambiguous.
If one can classify phenomena into clear differences, then that ought to clarify the different kinds of knowledge we use to study the whole of reality, that is the seat of all phenomena. Photons are very physical but they are not ghosts.
Spirit is a supernatural concept. It is an element of what makes humans in the image of God. Therefore, science can never tell us of what a human being is. One needs knowledge of God and more so His Son, Jesus the Christ.
It may be so that all our conception of reality and what underlies it may change in the future. However, as of now, we have to make the best our brains allow us to comprehend.
From: Bill Powers []
Sent: Thursday, May 28, 2009 4:19 PM
To: Alexanian, Moorad
Cc: Gregory Arago; Cameron Wybrow;
Subject: Re: [asa] The whole of reality (subject name changed)
Just some quick questions.
Why do we say that science can only investigate the physical?
Is there a difference between the material and the physical?
Are some immaterial things physical?
It is not at all clear why we presume that the spiritual and the physical
cannot interact.
Science all the time investigates the invisible, even in principle
invisible (quarks). When it does so is it utterly clear that it is
investigating the physical?
Science studies phenomena; and what are phenomena?
Ultimately, in all cases, phenomena must be capable of human sensible
Complex instruments, in conjunction with complex theory of the invisible,
are employed in enabling us to make of the invisible phenomena.
Are all phenomena physical?
Why presume that because we can make of the invisible something that
influences our senses that it is physical?
Are all such invisible entities physical by definition, or is it an
empirical conclusion.
It seems that it is definitional.
It is perhaps better to drop notions such as physical, immaterial, and
nonphysical, and speak instead of just the phenomena and the stories we
concoct about the phenomena. Does it really add anything to call
photons physical, although they are immaterial? What is spirit and how do
we distinguish it from the physical? This is by no means clear. What is
called physical today in the early days of science was considered a type
of spirit.
On Thu, 28 May 2009,
Alexanian, Moorad wrote:
> Gregory,
> The reason science cannot study the nonphysical and the supernatural is
that by definition those two sets are the relative complement of the purely
physical elements of the physical set.
This is so since science has been defined by its subject matter, which is
data that can be collected, in principle, by purely physical devices.
In other words, purely physical devices cannot detect thoughts and other
mental concepts, self, etc. nor the supernatural.
> The physical and the supernatural sets do overlap, but are not the same.
For instance, humans are elements of the union of the physical, nonphysical,
and supernatural sets. This is the reason I take the supernatural as being
part of Nature because humans are part of Nature. In addition, the creation
of man in the image of God forces us to make the supernatural an aspect of
> Knowledge, the number pi, mental abstractions, etc., are nonphysical but
certainly not supernatural. I think, as C.S. Lewis indicates, reasoning is
indeed supernatural. However, God is Supernatural but, as Creator, is not in
Nature. Of course, the incarnation is a deliberate invasion of God himself
into His creation.
> Different kinds of knowledge study different aspects of the whole of reality. For instance, to study only the physical aspect of man does not tell us who man truly is. The claim that it does, is reductionism at its worst. This is my qualm with evolutionary theory, which will eventual base all on genetic coding, which is purely physical.
> I am here merely indicating what my thoughts are regarding what is real. Future research may prove some aspects of this wrong. However, I doubt it.
> I am attempting to order the different kinds of knowledge, which are defined by their subject matters, and integrate them so that we truly deal with the whole of reality. This must be accomplished without any sort of reductionism.
> I hope I have answered all your questions Gregory. If not, keep on asking. Some of this material can be found in my website:
> Moorad
> ________________________________
> From: Gregory Arago []
> Sent: Tuesday, May 26, 2009 12:49 PM
> To: Bill Powers; Alexanian, Moorad
> Cc: Cameron Wybrow;
> Hi Bill and Alexanian,
> I suppose the thread has turned a bit off topic, since nobody,s talking about Behe any more. But then again, that,s often when the fun begins on ASA, when people diverge from often travelled pathways. : - )
> As with Bill, I agree that that what Moorad is proposing is appealing. The triad of physical/non-physical/supernatural means that <science> cannot study the <non-physical> or the <supernatural>. It also restricts the <physical> and the <supernatural> from overlapping. And it claims that <science> only studies one-third of what constitutes human beings, i.e. as an <entity>, which is what Moorad calls them/us.
> I can,t help but suggest that Moorad,s triad echos the language of <positive science>, like what a zoologist would speak, and not the language of <reflexive science>, as an anthropologist would speak. But perhaps that is part of his legitimization strategy.
> I wonder how Moorad distinguishes between what is <non-physical> and what is <supernatural> given that if we *are* (as a fact) created imago Dei, the <non-physical> aspects of humanity would be presumably available also in/to the <supernatural>. I also wonder if Moorad,s triad is reasonable or logical or mystical given that it opposes two different base concepts. Why not <natural>, <non-natural> and <supernatural> instead? Why not <superphysical> instead of <supernatural>? Perhaps he,ll address these questions here or in a new thread.
> How does he distinguish the <natural> from the <physical.> (Or is that not important?) For example, <physics>, as a scientific and academic discipline, is typically categorized as a <natural science.> Is he taking offence that <natural> is typical a <larger> or <wider> category than <physical> and thus trying to simplify his definition of <science>?
> And then what about all of those <sciences>, i.e. as many people call them, that do not particularly study <physical> things? Does his perspective disqualify them as <science> or devalue their contribution to human (self-)knowledge? Or does his position actually uplift those fields because they study human beings, which are partly <supernatural> entities? And what about all of the human-social scientists who don,t think that there is anything <supernatural> about human beings? Are they contradictory in their own disciplines?
> It also doesn't seem to me that Moorad has answered Bill,s question, or at least not directly. Bill asked: <The materialist will argue that if ,every, behavior can be accounted for by a physical process that the living are nothing but physical. What would you say to that?>
> Moorad answered: <To the materialist I would say, go tell your wife, husband, children, friends, etc. that they are nothing but a complicated solution of the Schrödinger equation. Let us see how they take that.>
> The materialist, as you know, Moorad, can argue for <non-physical> things just as easily as the person who believes in spiritual reality. I think this is partly what was behind Bill,s question. Aren't there various <levels> of explanation, which are available even to materialists, Moorad? Or is it just something simple like <vulgar materialism> and not something more sophisticated like <dialectical materialism> that you would argue this way against? One could just as easily point the figure at <mechanistic> thinkers in our age of electricity and computers (i.e. machines).
> Indeed, there are those in science who think <consciousness> will one day be explained via physical or material processes. How do you respond to them? Is it merely fantasy? Is the <power> of <science> blown way out of proportion (no pun intended given the DPRKs recent posturing on the Korean Penninsula) to what is most important in people,s lives? Are you <promoting> a humanisation of <science> or rather greater relevance for whatever fields study the <supernatural> in human existence, to contribute to our self-community knowledge? I,d sure appreciate your insights, Moorad, as I think you offer a unique view amongst the ASA listserve community.
> I think Moorad,s position can help to <put science in its place>, to <situate> it or draw boundaries around it, so to speak. But I worry that by limiting <science> to merely physical things, he,ll lose the strongest weapon available against scientism. The uniqueness of human-social scholarship conveys something that <science> as Moorad considers it can never address. But don,t trust me on the <never>, folks, just because I,m on your side working in a (roughly 2/3 of the academy) realm that is predominantly against us.
> Gregory
> --- On Tue, 5/26/09, Alexanian, Moorad <> wrote:
> From: Alexanian, Moorad <>
> To: "Bill Powers" <>
> Cc: "Cameron Wybrow" <>, "" <>
> Received: Tuesday, May 26, 2009, 7:14 PM
> Bill,
> A human is a physical/nonphysical/supernatural entity. Also, life cannot be characterized in purely physical terms. I totally reject physicalism; however, the subject matter of science is data that can be collected, in principle, with the aid of purely physical devises.
> To the materialist I would say, go tell your wife, husband, children, friends, etc. that they are nothing but a complicated solution of the Schrödinger equation. Let us see how they take that. Also, let the materialist live by what he/she preaches by not using words that cannot be characterized in terms of the purely physical. For instance, do not use the words like love, kindness, sin, etc. Let us face it, if a materialist description of him/her was realized, then he/she would be reduced to a pile of useless chemicals, viz. no life, no consciousness not self at all, the original dirt.
> Moorad
> ________________________________
> To unsubscribe, send a message to with
To unsubscribe, send a message to with
Received on Thu May 28 18:05:16 2009
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1. Quantum theory: introduction and principles = c 1.1 The failures of classical physics 1.2 Wave-particle duality 1.3 The Schrödinger equation 1.4 The.
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Presentation on theme: "1. Quantum theory: introduction and principles = c 1.1 The failures of classical physics 1.2 Wave-particle duality 1.3 The Schrödinger equation 1.4 The."— Presentation transcript:
1 1. Quantum theory: introduction and principles = c 1.1 The failures of classical physics 1.2 Wave-particle duality 1.3 The Schrödinger equation 1.4 The Born interpretation of the wavefunction 1.5 Operators and theorems of the quantum theory 1.6 The Uncertainty Principle
2 1.1 The failures of classical physics A. Black-body radiation at max Observations: 1) Wien displacement law: 2) Stefan-Boltzmann law: ( = E/V)
3 Equipartition of the energy: Per degree of freedom: average energy = kT (26 meV at 25°C). The total energy is equally “partitioned” over all the available modes of motion. Rayleigh and Jeans used the equipartition principle and consider that the electromagnetic radiation is emitted from a collection of excited oscillators, which can have any given energy by controlling the applied forces (related to T). It led to the Rayleigh-Jeans law for the energy density as a function of the wavelength. Tentative explanation via the classical mechanics It does not fit to the experiment. From this law, every objects should emit IR, Vis, UV, X-ray radiation. There should be no darkness!! This is called the Ultraviolet catastrophe.
4 Introduction of the quantization of energy to solve the black-body problem Max Planck: quantization of energy. E = n h only for n= 0,1,2,... h is the Planck constant Cross-check of the theory: from the Planck distribution, one can easily find the experimental Wien displacement and the Stefan-Boltzmann law. the quantization of energy exists!
5 C. Atomic and molecular spectra Excitation energy Photon emission h =hc/ Fe Photon absorption The emission and absorption of radiation always occurs at specific frequencies: another proof of the energy quantization. NB: wavenumber
6 1.2 Wave-particle duality A.The particle character of electromagnetic radiation: The photoelectric effect The photon h ↔ particle-like projectile conservation of energy ½mv 2 = h - = metal workfunction, the minimum energy required to remove an electron from the metal to the infinity. Threshold does not depend on intensity of incident radiation. NB: The photoelectron spectroscopy (UPS, XPS) is based on this photoelectric effect. e - (E k ) h metal
7 B. The wave character of the particles: Electron diffraction Diffraction is a characteristic property of waves. With X-ray, Bragg showed that a constructive interference occurs when =2d sin . Davidsson and Germer showed also interference phenomenon but with electrons! d V Particles are characterized by a wavefunction An appropriate potential difference creates electrons that can diffract with the lattice of the nickel A link between the particle (p=mv) and the wave ( ) natures
8 1.3 The Schrödinger Equation From the wave-particle duality, the concepts of classical physics (CP) have to be abandoned to describe microscopic systems. The dynamics of microscopic systems will be described in a new theory: the quantum theory (QT). A wave, called wavefunction (r,t), is associated to each object. The well-defined trajectory of an object in CP (the location, r, and momenta, p = m.v, are precisely known at each instant t) is replaced by (r,t) indicating that the particle is distributed through space like a wave. In QT, the location, r, and momenta, p, are not precisely known at each instant t (see Uncertainty Principle). In CP, all modes of motions (rot, trans, vib) can have any given energy by controlling the applied forces. In the QT, all modes of motion cannot have any given energy, but can only be excited at specified energy levels (see quantization of energy). The Planck constant h can be a criterion to know if a problem has to be addressed in CP or in QT. h can be seen has a “quantum of an action” that has the dimension of ML 2 T -1 (E= h where E is in ML 2 T -2 and is in T -1 ). With the specific parameters of a problem, we built a quantity having the dimension of an action (ML 2 T -1 ). If this quantity has the order of magnitude of h ( 10 -34 Js), the problem has to be treated within the QT.
9 Hamiltonian function H = T + V. T is the kinetic energy and V is the potential energy. correspondence principles are proposed to pass from the classical mechanics to the quantum mechanics Classical mechanics Quantum mechanics Schrödinger Equation
10 The Schrödinger Equation (SE) shows that the operator H and iħ / t give the same results when they act on the wavefunction. Both are equivalent operators corresponding to the total energy E. In the case of stationary systems, the potential V(x,y,z) is time independent. The wavefunction can be written as a stationary wave: (x,y,z,t)= (x,y,z) e -i t (with E=ħ ). This solution of the SE leads to a density of probability | (x,y,z,t)| 2 = | (x,y,z)| 2, which is independent of time. The Time Independent Schrödinger Equation is: or The Schrödinger equation is an eigenvalue equation, which has the typical form: (operator)(function)=(constant)×(same function) The eigenvalue is the energy E. The set of eigenvalues are the only values that the energy can have (quantization). The eigenfunctions of the Hamiltonian operator H are the wavefunctions of the system. To each eigenvalue corresponds a set of eigenfunctions. Among those, only the eigenfunctions that fulfill specific conditions have a physical meaning. NB: In the following, we only envisage the time independent version of the SE.
11 1.4 The Born interpretation of the wavefunction Example of a 1-dimensional system Physical meaning of the wavefunction: probability of finding the particle in an infinitesimal volume d =dxdydz at some point r is proportional to | (r)| 2 d | (r)| 2 = (r) * (r) is a probability density. It is always positive! wavefunction may have negative or complex values Node
12 A. Normalization Condition The solution of the differential equation of Schrödinger is defined within a constant N. If is a known solution of H =E , then =N is a also solution for the same E. H =E H(N )= E(N ) N(H )=N(E ) H =E The sum of the probability of finding the particle over all infinitesimal volumes d of the space is 1: Normalization condition. We have to determine the constant N, such that the solution =N of the SE is normalized. B. Other mathematical conditions (r) ; r if not: no physical meaning for the normalization condition (r) should be single-valued r if not: 2 probability for the same point!! The SE is a second-order differential equation: (r) and d (r)/dr should be continuous
13 C. The kinetic energy and the wavefunction T The kinetic energy is then a kind of average over the curvature of the wavefunction: we get a large contribution to the observed value from the regions where the wavefunction is sharply curved ( 2 / x 2 is large) and the wavefunction itself is large ( * is large too). We can expect a particle to have a high kinetic energy if the average curvature of its wavefunction is high. Example: the wave function in a periodic system: electrons in a metal
14 1.5 Operators and principles of quantum mechanics A. Operators in the quantum theory (QT) An eigenvalue equation, f = f, can be associated to each operator . In the QT, the operators are linear and hermitian. Linearity: is linear if: (c f)= c f (c=constant) and (f+ )= f+ NB: “c” can be defined to fulfill the normalization condition Hermiticity: A linear operator is hermitian if: where f and are finite, uniform, continuous and the integral for the normalization converge. The eigenvalues of an hermitian operator are real numbers ( = * ) When the operator of an eigenvalue equation is hermitian, 2 eigenfunctions (f j, f k ) corresponding to 2 different eigenvalues ( j, k ) are orthogonal.
15 B. Principles of Quantum mechanics 1. To each observable or measurable property of the system corresponds a linear and hermitian operator , such that the only measurable values of this observable are the eigenvalues j of the corresponding operator. f = f 2. Each hermitian operator representing a physical property is “complete”. Def: An operator is “complete” if any function (finite, uniform and continuous) (x,y,z) can be developed as a series of eigenfunctions f j of this operator. 3. If (x,y,z) is a solution of the Schrödinger equation for a particle, and if we want to measure the value of the observable related to the complete and hermitian operator (that is not the Hamiltonian), then the probability to measure the eigenvalue k is equal to the square of the modulus of f k ’s coefficient, that is |C k | 2, for an othornomal set of eigenfunctions {f j }. Def: The eigenfunctions are orthonormal if NB: In this case:
16 4. The average value of a large number of observations is given by the expectation value of the operator corresponding to the observable of interest. The expectation value of an operator is defined as: 5. If the wavefunction =f 1 is the eigenfunction of the operator ( f = f), then the expectation value of is the eigenvalue 1. For normalized wavefunction See p305 in the book
17 1.6 The Uncertainty Principle 1. When two operators are commutable (and with the Hamiltonian operator), their eigenfunctions are common and the corresponding observables can be determined simultaneously and accurately. 2. Reciprocally, if two operators do not commute, the corresponding observable cannot be determined simultaneously and accurately. If ( 1 2 - 2 1 ) = c, where “c” is a constant, then an uncertainty relation takes place for the measurement of these two observables: where Uncertainty Principle
18 Example of the Uncertainty Principle 1. For a free atom and without taking into account the spin-orbit coupling, the angular orbital moment L 2 and the total spin S 2 commute with the Hamiltonian H. Hence, an exact value of the eigenvalues L of L 2 and S of S 2 can be measured simultaneously. L and S are good quantum numbers to characterize the wavefunction of a free atom see Chap 13 “Atomic structure and atomic spectra”. 2. The position x and the momentum p x (along the x axis). According to the correspondence principles, the quantum operators are: x and ħ/i( / x). The commutator can be calculated to be: The consequence is a breakdown of the classical mechanics laws: if there is a complete certainty about the position of the particle ( x=0), then there is a complete uncertainty about the momentum ( p x = ). 3. The time and the energy:If a system stays in a state during a time t, the energy of this system cannot be determined more accurately than with an error E. This incertitude is of major importance for all spectroscopies: see Chap 16, 17, 18
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52b984b0543cc834 | nonrelativistic quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures were drawn from functional analysis, a research area within pure mathematics that developed in parallel with, and was influenced by the needs of, quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues (more precisely: as spectral values (point spectrum plus absolute continuous plus singular continuous spectrum)) of linear operators in Hilbert space.
This formulation of quantum mechanics continues to be used today. At the heart of the description are ideas of quantum state and quantum observable which, for systems of atomic scale, are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of quantum observables.
Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations; probability theory was used in statistical mechanics. Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld-Wilson-Ishiwara quantization rule, which was formulated entirely on the classical phase space.
History of the formalism
The "old quantum theory" and the need for new mathematics
In the decade of 1890, Planck was able to derive the blackbody spectrum which was later used to solve the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of radiation with matter, energy could only be exchanged in discrete units which he called quanta Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Planck's constant in his honour.
In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which are called photons.
In 1913, Bohr calculated the spectrum of the hydrogen atom with the help of a new model of the atom in which the electron could orbit the proton only on a discrete set of classical orbits, determined by the condition that angular momentum was an integer multiple of Planck's constant. Electrons could make quantum leaps from one orbit to another, emitting or absorbing single quanta of light at the right frequency.
All of these developments were phenomenological and flew in the face of the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld-Wilson-Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.
In 1923 de Broglie proposed that wave-particle duality applied not only to photons but to electrons and every other physical system.
The situation changed rapidly in the years 1925-1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger and Werner Heisenberg and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas.
The "new quantum theory"
Erwin Schrödinger's wave mechanics originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle duality.
To be more precise: already before Schrödinger the young student Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics, i.e. the essential breakthrough. But Schrödinger's wave mechanics was created independently, was uniquely based on de Broglie's concepts, less formal and easier to understand, visualize and exploit. Originally the equivalence of Schrödinger's theory with that of Heisenberg was not seen; to show it, was also an important accomplishment of Schrödinger himself, performed in 1926, some months after the first publication of his theory:
Schrödinger proposed an equation (now bearing his name) for the wave associated to an electron in an atom according to de Broglie, and explained energy quantization by the well-known fact that differential operators of the kind appearing in his equation had a discrete spectrum. However, Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space.
It was Max Born, who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen, who then became the "father" of the Copenhagen interpretation of quantum mechanics which held until the Many Worlds Interpretation which resolved its many paradoxes.
With hindsight, Schrödinger's wave function can be seen to be closely related to the classical Hamilton-Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. I.e., the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, where one uses Poisson brackets.
Werner Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, being certainly very radical in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even knowing that his "index-schemes" were matrices. In fact, in these early years linear algebra was not generally known to physicists in its present form.
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches is generally associated to Paul Dirac, who wrote a lucid account in his 1930 classic Principles of Quantum Mechanics, being the third, and perhaps most important, person working independently in that field (he soon was the only one, who found a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in all kind of generalizations of the field. Concerning quantum mechanics, Dirac's method is now called canonical quantization.
Later developments
The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics.
Mathematical structure of quantum mechanics
Postulates of quantum mechanics
The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.
• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product scriptstyle langlephimidpsirangle. Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.
• The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily (supersymmetry is another matter entirely).
• Physical observables are represented by densely-defined self-adjoint operators on H.
The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector scriptstyle left|psirightranglein H is
langlepsimid Amidpsirangle
By spectral theory, we can associate a probability measure to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.
More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator rho normalized to be of trace 1. The expected value of A in the state rho is
If rho_psi is the orthogonal projector onto the one-dimensional subspace of H spanned by scriptstyle left|psirightrangle, then
operatorname{tr}(Arho_psi)=leftlanglepsimid Amidpsirightrangle
Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.
Superselection sectors. The correspondence between states and rays needs to be refined somewhat to take into account so-called superselection sectors. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.
Pictures of dynamics
The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If left|psileft(tright)rightrangle denotes the state of the system at any one time t, the following Schrödinger equation holds:
ihbarfrac{d}{d t}left|psi(t)rightrangle=Hleft|psi(t)rightrangle
where H is a densely-defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and hbar is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U(t): HH such that
for all times s, t. The existence of a self-adjoint Hamiltonian H such that
U(t)=e^{-(i/hbar)t H}
is a consequence of Stone's theorem on one-parameter unitary groups. (It is assumed that H does not depend on time and that the perturbation starts at t_0=0; otherwise one must use the Dyson series, formally written as U(t)={mathcal{T}}, ,exp{-(i/hbar )intlimits_{t_0}^t ,rm dt', H(t')}, where {mathcal{T}} is Dyson's time-ordering symbol.)
• The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:
left|psirightrangle = left|psi(0)rightrangle
A(t) = U(-t)AU(t). quad
langlepsimid A(t)midpsirangle=langlepsi(t)mid Amidpsi(t)rangle
and that the time-dependent Heisenberg operators satisfy
ihbar{dover dt}A(t) = [A(t),H].
This assumes A is not time dependent in the Schrödinger picture. Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.
• The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:
ihbarfrac{d }{dt}left|psi(t)rightrangle ={H}_{rm int}(t) left|psi(t)rightrangle
ihbar{d over d t}A(t) = [A(t),H_{0}].
The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. H=H_0+V, in the interaction picture it does, at least, if V does not commute with H_0, since H_{rm int}(t)equiv e^{{(i/hbar})tH_0},V,e^{{(-i/hbar})tH_0}. So the above-mentioned Dyson-series has to be used anyhow.
The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stone-von Neumann theorem states all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. This is related to quantization and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework.
The quantum harmonic oscillator is an exactly-solvable system where the possibility of choosing among more than one representation can be seen in all its glory. There, apart from the Schrödinger (position or momentum) representation one encounters the Fock (number) representation and the Bargmann-Segal (phase space or coherent state) representation. All three are unitarily equivalent.
Time as an operator
The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H-E, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H-E (this requires the use of a rigged Hilbert space and a renormalization of the norm).
This is related to quantization of constrained systems and quantization of gauge theories. It is also possible to formulate a quantum theory of "events" where time becomes an observable(see D. Edwards ).
In addition to their other properties all particles possess a quantity, which has no correspondence at all in conventional physics, namely the spin, which is some kind of intrinsic angular momentum (therefore the name). In the position representation, instead of a wavefunction without spin, psi = psi(mathbf r), one has with spin: psi =psi(mathbf r,sigma), where sigma belongs to the following discrete set of values: sigma in{ -Scdothbar , -(S-1)cdothbar , ... ,+(S-1)cdothbar ,+Scdothbar}. One distinguishes bosons (S=0 or 1 or 2 or ...) and fermions (S=1/2 or 3/2 or 5/2 or ...)
Pauli's principle
The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have
psi (,..., ;,mathbf r_i,sigma_i,;, ...,;mathbf r_j,sigma_j,;,...) stackrel{!}{=}(-1)^{2S}cdot psi (,..., ;,mathbf r_j,sigma_j,;, ...,;mathbf r_i,sigma_i,;,...)
i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor (-1)^{2S} which is +1 for bosons, but (-1) for fermions. Electrons are fermions with S=1/2; quanta of light are bosons with S=1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d=2 one can construct entities where (-1)^{2S} is replaced by an arbitrary complex number with magnitude 1 (-> anyons).
The problem of measurement
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of measurement. The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following:
• Let A have spectral resolution
A = int lambda , d operatorname{E}_A(lambda),
where EA is the resolution of the identity (also called projection-valued measure) associated to A. Then the probability of the measurement outcome lying in an interval B of R is |EA(B) ψ|2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure
langle psi mid operatorname{E}_A psi rangle.
• If the measured value is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state EA(B) ψ. If the measured value does not lie in B, replace B by its complement for the above state.
For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λi, with corresponding eigenvectors ψi. The projection-valued measure associated with A, EA, is then
operatorname{E}_A (B) = | psi_irangle langle psi_i|,
where B is a Borel set containing only the single eigenvalue λi. If the system is prepared in state
| psi rangle ,
Then the probability of a measurement returning the value λi can be calculated by integrating the spectral measure
langle psi mid operatorname{E}_A psi rangle
over Bi. This gives trivially
langle psi| psi_irangle langle psi_i mid psi rangle = | langle psi mid psi_irangle | ^2.
| psi_irangle langle psi_i| ,
by a finite set of positive operators
F_i F_i^* ,
whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ1 ... λn} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λi. Instead of collapsing to the (unnormalized) state
| psi_irangle langle psi_i |psirangle ,
after the measurement, the system now will be in the state
F_i |psirangle. ,
Since the Fi Fi* 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.
The same formulation applies to general mixed states.
The relative state interpretation
List of mathematical tools
The main tools include:
See also: list of mathematical topics in quantum theory.
• T.S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912, Clarendon Press, Oxford and Oxford University Press, New York, 1978.
• D. Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, 42 (1979),pp.1-70.
• R. Jost, The General Theory of Quantized Fields, American Mathematical Society, 1965.
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin 1964 (Reprinted by Princeton University Press)
• M. Reed and B. Simon, Methods of Mathematical Physics, vols 1-IV, Academic Press 1972.
• Nikolai Weaver, "Mathematical Quantization", Chapman & Hall/CRC 2001.
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Open Access Nano Express
The effects of porosity on optical properties of semiconductor chalcogenide films obtained by the chemical bath deposition
Yuri V Vorobiev1*, Paul P Horley2, Jorge Hernández-Borja1, Hilda E Esparza-Ponce2, Rafael Ramírez-Bon1, Pavel Vorobiev1, Claudia Pérez1 and Jesús González-Hernández2
Author Affiliations
1 CINVESTAV-IPN Unidad Querétaro, Libramiento Norponiente 2000, Fracc. Real de Juriquilla, Querétaro, Qro, CP 76230, México
2 CIMAV Chihuahua/Monterrey, Avenida Miguel de Cervantes 120, Chihuahua, Chih, CP 31109, México
For all author emails, please log on.
Nanoscale Research Letters 2012, 7:483 doi:10.1186/1556-276X-7-483
Received:16 April 2012
Accepted:4 August 2012
Published:29 August 2012
© 2012 Vorobiev et al.; licensee Springer.
This paper is dedicated to study the thin polycrystalline films of semiconductor chalcogenide materials (CdS, CdSe, and PbS) obtained by ammonia-free chemical bath deposition. The obtained material is of polycrystalline nature with crystallite of a size that, from a general point of view, should not result in any noticeable quantum confinement. Nevertheless, we were able to observe blueshift of the fundamental absorption edge and reduced refractive index in comparison with the corresponding bulk materials. Both effects are attributed to the material porosity which is a typical feature of chemical bath deposition technique. The blueshift is caused by quantum confinement in pores, whereas the refractive index variation is the evident result of the density reduction. Quantum mechanical description of the nanopores in semiconductor is given based on the application of even mirror boundary conditions for the solution of the Schrödinger equation; the results of calculations give a reasonable explanation of the experimental data.
polycrystalline films; chalcogenide materials; nanopores; quantum confinement in pores
Chemical bath deposition (CBD) is a cheap and energy-efficient method commonly used for the preparation of semiconductor films for sensors, photodetectors, and solar cells. It was one of the traditional methods to obtain chalcogenide semiconductors including CdS and CdSe [1-6]. However, large-scale CBD deposition of CdS films raises considerable environmental concerns due to utilization of highly volatile and toxic ammonia. On the other hand, the volatility of ammonia modifies pH of the reacting solution during the deposition process, causing irreproducibility of thin film properties for the material obtained in different batches [1,3].
We manufacture CdS, CdSe, and PbS films using the CBD process to minimize the production cost and energy consumption. Ammonia-free CBD process was used to avoid negative environmental impact (see [7] reporting an example of CBD-made solar cell with structure glass/ITO/CdS/PbS/conductive graphite with quantum efficiency of 29% and energy efficiency of 1.6% ). All these materials have the melting temperatures above 1,000°C, remaining stable during the deposition process. It is also known that PbS is very promising for solar cell applications, confirmed by the recent discovery of multiple exciton generation in their nanocrystals [8].
Chemical bath-deposited films [9] have a particular structure. As a rule, at initial deposition stages, small (3 to 5 nm) nanocrystals are formed. They exhibit strong quantum confinement leading to large blueshift of the fundamental absorption edge. Historically, blueshift was first discovered namely in CBD-made CdSe films [9,10]. At later stages, the crystallite size becomes larger so that the corresponding blueshift decreases. Another feature characteristic to the process is a considerable porosity [3,9] inherent to the growth mechanism, which takes place ion by ion or cluster by cluster depending on the conditions or solution used (see also [11,12]). The degree of porosity decreases for larger deposition time because the film becomes denser. At the initial stage, the porosity can be up to 70% [9], and at final stages, it will be only about 5% to 10% .
In this paper, we present the experimental results for the investigation of porosity effects for relatively large deposition times upon the optical characteristics of CBD-made semiconductor materials such as CdS, CdSe, and PbS. We show that the nanoporosity can blueshift the absorption edge, leading to the variation observed for material with pronounced nanocrystallinity. For theoretical study of nanopores in a semiconductor, we use mirror boundary conditions to solve the Schrödinger equation, which were successfully applied to nanostructures of different geometries [13-15]. We show that the same treatment of pores allows to achieve a good correlation between theorical and experimental data.
The authors successfully developed ammonia-free CBD technology for polycrystalline CdS, CdSe, and PbS films, described in detail elsewhere [4-7,11,12]. We characterize the obtained structures by composition, microstructure (including average grain size), and morphology using X-ray diffraction, SEM, and EDS measurements. Optical properties were investigated with UV–vis and FTIR spectrometers. All experimental methods are described in the aforementioned references, together with the detailed results of this complex material study. Here, we would like to discuss optical phenomena characteristic to the entire group of semiconductor film studied, skipping the technological details that are given in [4-7,11,12].
Results and discussion
For CBD-made materials obtained after long deposition time (which resulted into dense films with crystallite size of about 20 nm), we observed a blueshift of the fundamental absorption edge relative to the bulk material data [16] in all cases with the following shift values: 0.06 eV for CdS [7], 0.15 eV for CdSe [6] (see also Figure 1), and 0.1 to 0.4 eV for different samples of PbS (Figure 2). This effect was accompanied by reduction of refractive index n (in comparison with bulk crystal data, see Figure 3 for CdSe and Figure 4 for PbS). This reduction is larger for samples obtained with small deposition times, but it is always present in the films discussed here. We connect both effects with pronounced porosity of the films obtained by CBD method. In particular, the blueshift in the dense CBD films is attributed to the quantum confinement in pores.
thumbnailFigure 1 . Transmission spectrum of 0.5-μm thick CdSe film.
thumbnailFigure 2 . Diagram used to determine bandgap of PbS CBD sample with growth time of 3 h. The value of D corresponds to optical density.
thumbnailFigure 3 . Refractive index of CdSe. Squares indicate the data for the bulk material adapted from [14], and circles correspond to CBD film.
thumbnailFigure 4 . Optical constantsn, kof PbS CBD films with different deposition times.
Figure 1 presents the transmission spectrum of 0.5-μm-thick CdSe films (deposition time of 4 h) displaying a clear interference pattern, characterized with transmission maxima at 2dn = and minima at 2dn = (N − 1/2) λ. Here, λ is the wavelength, d is the film thickness, and N is an integer defining the order of interference pattern. With these expressions, we calculated the spectrum of refractive index (Figure 3, circles). The squares in the same figure present the data for the bulk material [17] displaying a considerable drop of refractive index for the film in comparison with bulk material.
Figure 2 presents the diagram for PbS allowing to determinate the bandgap via direct interband transitions observed for all the materials studied by plotting the squared product of optical density and photon energy as a function of the latter. The similar diagrams for CdS and CdSe were given in [6,7].
The case of PbS requires more attention. Figure 5 presents the dependence of the crystallite size upon the deposition time. Figure 4 shows the spectra for optical constants (refractive index n and extinction coefficient k) measured for four PbS films deposited with growth time ranging from 1 to 4 h; in the latter case, the result was a 100-nm-thick film. It is clear that for larger deposition time, the film becomes denser so that refraction index and extinction coefficients increase. Their spectral behavior follows qualitatively the corresponding curves of the bulk material, but the values are essentially lower, even when deposited film has a considerable thickness. For example, the refractive index for film is 4 at most for the wavelength 450 nm, whereas for the bulk material, the corresponding value is 4.3. As for extinction coefficient k, the maximum of 2.75 is achieved at the wavelength of 350 nm, with the corresponding bulk value of 3.37.
thumbnailFigure 5 . Dependence of the grain size of PbS CBD samples on growth time. The line is given as eye guide only.
We assume that the pores in a dense CBD film correspond to the spaces between crystallites' boundaries. Therefore, in cubic crystals, the pores most probably will be of prismatic shape, defined by plane boundaries of the individual grains. These prismatic pores most probably will have a length (height) equal to the grain size, with quadratic or rectangular triangle cross-section. As pores and crystallites are considered to be of equal height, the question of a volume fraction of pores reduces to two dimensions by being equal to the ratio of pore cross-sectional area to the total cross-section of the film, assuming that in the average there will be one pore per one crystallite. The dimensions of the pore will define the blueshift observed, which can be seen from the following theoretical consideration.
Electron confined in pores: quantum mechanical approach
It was proposed (see [13-15]) to treat semiconductor quantum dots (QDs) as ‘mirror-wall boxes’ confining the particle, resulting in mirror boundary conditions for analytical solution of the Schrödinger equation in the framework of the effective mass approximation. The basic assumption is that a particle (an electron or a hole) is specularly reflected by a QD boundary, which sets the boundary conditions as equivalence of particle's Ψ-function in an arbitrary point r inside the semiconductor (Ψr) with wave function in the image point im (Ψim). It must be mentioned that Ψ-function in real and image points can be equated by its absolute values since the physical meaning is connected with |Ψ|2, so that mirror boundary conditions can have even and odd forms (Ψr = Ψim in the former case, and Ψr = −Ψim in the latter). The ‘odd’ case is equivalent to the impenetrable boundary conditions and strong confinement because Ψ-function vanishes at the boundary. The milder case of even mirror boundary conditions represents weak confinement and occurs when a particle is allowed to have tunnel probability inside the boundary.
It is evident that our basic assumption is favorable for effective mass approximation as it increases the length of effective path for a particle in a semiconductor material. Besides, in high symmetry case, the assumption of mirror boundary conditions forms a periodic structure filling the space. We have shown [15] that the use of even mirror boundary conditions gives the same solution as Born-von Karman boundary conditions applied to a periodic structure. The treatment performed in [13-15] of the QDs with different shapes (rectangular prism, sphere and square base pyramid) yielded the energy spectra that have a good agreement with the published experimental data achievable without any adjustable parameters.
Let us consider an inverted system: a pore formed by a void surrounded by a semiconductor material. The reflection accompanied with a partial tunneling into QD boundary (for the case of even mirror boundary conditions) can be described as equivalence of Ψ-function values in a real point in the vicinity of the boundary and a reflection point in a mirror boundary. Hence, the solution of the Schrödinger equation for a pore within semiconductor material will be the same as that for a QD of equal geometry with an equal expression for the particle's energy spectrum.
Table 1 summarizes the expressions for energy spectra obtained for QDs of several basic shapes with application of even mirror boundary conditions. All spectra have the same character, with a quadratic dependence on quantum numbers (all integers or odd numbers for a particular case of spherical QD [15]) and an inverse quadratic dependence on QD's dimensions. Besides, the position of energy levels has an inverse dependence on the effective mass [18,19].
Table 1. Energy spectra of different QDs
Comparison with the experiment
In the following discussion, we take into account that typical pores in CBD materials have a characteristic size a of several nanometers [3,9], being much smaller than the Bohr radius αB for an exciton, α/2 < < αB, which is especially important for the case of exciton formation under the action of a light beam incident on semiconductor. The energy difference defines the blueshift of absorption edge. In all semiconductors studied, the value of αB exceeds 15 nm according to the expression below:
a B = 4 π 2 ϵ ϵ 0 μ e 2 w i t h r e d u c e d m a s s μ = m e m h m e + m h (1)
Here, me,h is the electron/hole effective mass, ϵ is the dielectric constant of the material, and ϵ0 is a permittivity constant. Following the argumentation given in [18,19], we see that one can directly apply the expressions for energy spectra because the separation between the quantum levels proportional to ħ2/ma2 is large compared to the Coulomb interaction between the carriers which is proportional to e2/ϵϵ0α. Therefore, Coulomb interaction can be neglected, and the energy levels could be found from quantum confinement effect alone. Accordingly, we shall calculate the emission/absorption photon energy for transitions corresponding to the exciton ground state, which is given by n = 0 for spherical QD and n = 1 for other geometries. From Table 1, it follows that the lowest energy value can be obtained for a spherical QD, whereas for a prism with quadratic section, the energy value is twice larger. For all other geometries, the energy has the latter order of magnitude. For the estimation of porosity effects, we will use the expression for a prismatic QD with a square base, assuming that the fundamental absorption edge corresponds to generation of an exciton with ground state energy:
ω min = E g + h 2 4 μ a 2 (2)
with the semiconductor bandgap Eg.
In the case of CdSe (exciton reduced mass of 0.1 m0) using the expression (2) and the band edge shift ħωminEg = 0.15 eV (1.88 − 1.73), we calculate the pore size of 7 nm. For the average crystallite dimension of 22 nm, the pore fraction, thus, would be (7/22)2 ≈ 10% , which is twice as big as the relative reduction of refractive index found (Figure 3).
To explain the edge shift observed in CdS (exciton reduced mass 0.134 m0[16]), one obtains the pore size of 8 nm. Here, the crystallite size is 20.1 nm, making the total pore fraction of approximately 12% . The observed reduction of refractive index changes from 2.5 for the bulk material [7,16] to 2.3 for 600-nm-thick film, yielding the pore fraction of 9% that is close to our predictions.
The reduced mass for PbS is 0.0425 m0[16], and the observed edge shift is 0.4 eV, yielding the average pore size of 6.5 nm. Having the crystallite size of 20 nm, it will give the pore fraction of 10% (observed reduction of refractive index in [7] was 8% , and from Figure 4 we obtain the value of 7.5% ). We see that in all cases, the volumetric percentage of pores calculated using the blueshift values renders the correct order that is verified from the refractive index reduction. However, the latter value is always smaller that may mean that the pores' height is about 30% to 40% less than that of the grains.
It should be noted that in cases of PbS, due to high value of dielectric constant (17) and small exciton reduced mass, the Bohr radius for an exciton (21 nm) appears to be the same order of magnitude as the grain size. It means that the quantum confinement effect can be observed even without taking into account the porosity of the material. This effect was studied experimentally in [20] for PbS spherical quantum dots. It was found that in PbS quantum dots with diameter of 3.5 nm, the blue band edge shift of 1.05 eV is observed. Taking into account that the blueshift due to quantum confinement is reversely proportional to the square of the dot's diameter, we find that the shift caused by the crystallite size of 20 nm will be equal to 0.03 eV, which is about 10 times smaller than the observed values. We also note that the smaller crystallite size observed in our experiments at early stages of CBD process (variation from 8 to 18 nm, see Figure 5) does not allow to explain the experimentally observed blueshift. Thus, we conclude the mandatory accounting of nanopores, which offers improved agreement between theoretical and experimental data.
We report on ammonia-free CBD method that provides cheap, efficient, and environmentally harmless production of CdS, CdSe, and PbS films. Material porosity inherent to CBD technique can be used to fine-tune the material bandgap towards the required values, paving promising ways for solar cell applications. The theoretical description of porosity based on the solution of the Schrödinger equation with even mirror boundary conditions provides a good correlation of theoretical and experimental data.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YVV suggested the treatment of pores as inverted quantum dots. PPH realized the theoretical description and drafted the manuscript. JHB conducted the experiments on CdS and PbSe. HEEP made the experiments on CdSe. RRB adjusted the chemical part of CBD method and helped in drafting the manuscript. PV performed modeling of a porous semiconductor. CP realized the experiments with PbS. JGH supervised all the study. All authors read and approved the final manuscript.
The authors are grateful to Editor Prof. Andres Cantarero for the support and encouragement in the revision of the manuscript. PV and CP wish to thank CONACYT for their scholarships.
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0fd3d71054feddd0 | I’ve just uploaded to the arXiv my paper “An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation“. This paper is another technical component of my “heatwave project“, which aims to establish the global regularity conjecture for energy-critical wave maps into hyperbolic space. I have been in the process of writing the final paper of that project, in which I will show that the only way singularities can form is if a special type of solution, known as an “almost periodic blowup solution”, exists. However, I recently discovered that the existing function space estimates that I was relying on for the large energy perturbation theory were not quite adequate, and in particular I needed a certain “inverse theorem” for a standard bilinear estimate which was not quite in the literature. The purpose of this paper is to establish that inverse theorem, which may also have some application to other nonlinear wave equations.
To explain the inverse theorem, let me first discuss the bilinear estimate that it inverts. Define a wave to be a solution to the free wave equation -\phi_{tt} + \Delta \phi = 0. If the wave has a finite amount of energy, then one expects the wave to disperse as time goes to infinity; this is captured by the Strichartz estimates, which establish various spacetime L^p bounds on such waves in terms of the energy (or related quantities, such as Sobolev norms of the initial data). These estimates are fundamental to the local and global theory of nonlinear wave equations, as they can be used to control the effect of the nonlinearity.
In some cases (especially in low dimensions and/or low regularities, and with equations whose nonlinear terms contain derivatives), Strichartz estimates are too weak to control nonlinearities; roughly speaking, this is because waves decay too slowly in low dimensions. (For instance, one-dimensional waves \phi(t,x) = f(x+t)+g(x-t) do not decay at all.) However, it has been understood for some time that if the nonlinearity has a special null structure, which roughly means that it consists only of interactions between transverse waves rather than parallel waves, then there is more decay that one can exploit. For instance, while one-dimensional waves do not decay in time, the product between a left-propagating wave f(x+t) and a right-propagating wave g(x-t) does decay in time. In particular, if f and g are bounded in L^2({\Bbb R}), then this product is bounded in spacetime L^2_{t,x}({\Bbb R}), thanks to the Fubini-Tonelli theorem.
There is a similar “bilinear L^2” estimate for products of transverse waves in higher dimensions. This estimate is the basic building block for the bilinear X^{s,b} estimates and their variants as developed by Bourgain, Klainerman-Machedon, Kenig-Ponce-Vega, Tataru, and others, and which are the tool of choice for establishing local and global control on nonlinear wave equations, particularly at low dimensions and at critical regularities. In particular, these estimates (or more precisely, a complicated variant of these estimates in sophisticated function spaces, due to Tataru and myself), are used in the theory of the energy-critical wave map equation. [These bilinear (and trilinear) estimates are not, by themselves, enough to handle this equation; one also needs an additional gauge fixing procedure before the equation is sufficiently close to linear in behaviour that these estimates become effective. But I do not wish to discuss the (significant) gauge fixing issue here.]
To cut a (very) long story short, these estimates, when combined with a suitable perturbative theory, allow one to control energy-critical wave maps as long as the energy is small. However, the whole point of the “heatwave” project is to control the non-perturbative setting when the energy is large (but finite), and one wants to control the solution for long periods of time.
In my previous “heatwave” paper, in which I established large data local well-posedness for this equation, I finessed this issue by localising time to very short intervals, which made certain spacetime norms small enough for the perturbation theory to apply. This sufficed for the local well-posedness theory, but is not good enough for the global perturbative theory, because the number of very short intervals needed to cover the entire time axis becomes unbounded. For that, one needs the ability to make certain norms or estimates “small” by only chopping up time into a bounded number of intervals. I refer to this property as divisibility (I used to refer to it, somewhat incorrectly, as fungibility).
In the case of semilinear wave (or Schrödinger equations), in which Strichartz estimates are already sufficient to obtain a satisfactory perturbative theory, divisibility is well-understood, and boils down to the following simple observation: if a function \phi: {\Bbb R} \times {\Bbb R}^n \to {\Bbb C} obeys a global spacetime integrability bound such as
\int_{\Bbb R} \int_{\Bbb R}^n |\phi(t,x)|^p\ dx dt \leq M
for some finite exponent p and some finite bound M, then one can partition {\Bbb R} into intervals I on which
\int_I \int_{\Bbb R}^n |\phi(t,x)|^p\ dx dt \leq \varepsilon
for some \varepsilon > 0 at one’s disposal to select. Indeed the number of such intervals is bounded by M/\varepsilon, and the intervals can be selected by a simple “greedy algorithm” argument. This divisibility property of L^p-type spacetime norms allows one to easily generalise the small-data perturbation theory to the large-data setting, and is relied upon heavily in the modern theory of the critical nonlinear wave and Schrödinger equations; see for instance this survey of Killip and Visan.
Unfortunately, the function spaces used in wave maps are not easily divisible in this manner (very roughly speaking, this is because the function space norms contain too many L^\infty_t type norms within them). So one cannot rely purely on refining the function space; one must also work on refining the bilinear (and trilinear) estimates on these spaces. The standard way to do this is to strengthen the L^p exponents in these estimates, and for the basic bilinear L^2 estimate this has indeed been done (in work of Wolff and myself). This suffices for “equal-frequency” interactions, in which one is multiplying two transverse waves of the same frequency, but turns out to be inadequate for “imbalanced-frequency” interactions, when one is multiplying a low-frequency wave by a high-frequency transverse wave. For this, I rely instead on establishing an inverse theorem for the estimate.
Generally speaking, whenever one is faced with an estimate, e.g. a linear estimate
\| Tf \|_Y \leq C \|f\|_X,
one can pose the inverse problem of trying to classify the functions f for which the estimate is tight in the sense that
\| Tf \|_Y \geq \delta \|f\|_X
for some \delta > 0 which is not too small. Such inverse theorems are a current area of study in additive combinatorics, and have recently begun making an appearance in PDE as well. For instance:
• Young’s inequality \|f*g\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^r} or the Hausdorff-Young inequality \|\hat f\|_{L^{p'}} \leq \|f\|_{L^p}, is only tight (for non-endpoint p,q,r) when f, g are concentrated on balls, arithmetic progressions, or Bohr sets (this is a consequence of several basic theorems in additive combinatorics, including Freiman’s theorem and the Balog-Szemeredi-Gowers theorem);
• The trivial inequality \|f\|_{U^k} \leq \|f\|_{L^\infty} for the Gowers uniformity norms is only expected to be tight when f correlates with a highly algebraic object, such as a polynomial phase or nilsequence (this is the inverse conjecture for the Gowers norm, which is partially proven so far);
• The Sobolev embedding \| f \|_{L^q} \leq C \|f\|_{W^{s,r}} is only tight when f is concentrated on a unit ball (for non-endpoint estimates) or a ball of arbitrary radius (for endpoint estimates);
• Strichartz estimates are only tight when f is concentrated on a ball (for non-endpoint estimates) or a tube (for endpoint estimates).
Inverse theorems for such estimates as Sobolev inequalities and Strichartz estimates are also closely related to the theory of concentration compactness and profile decompositions; see this previous blog post of mine for a discussion.
I can now state informally, the main result of this paper:
Theorem 1 (informal statement). A bilinear L^2 estimate between two waves of different frequency is only tight when the waves are concentrated on a small number of light rays. Outside of these rays, the L^2 norm is small.
This leads to a corollary which will be used in my final heatwave paper:
Corollary 2 (informal statement). Any large-energy wave \phi can have its time axis subdivided into a bounded number of intervals, such that on each interval the bilinear estimates for that wave (when interacted against any high-frequency transverse wave) behave “as if” \phi was small-energy rather than large energy.
The method of proof relies on a paper of mine from several years ago on bilinear L^p estimates for the wave equation, which in turn is based on a celebrated paper of Wolff. Roughly speaking, the idea is to use wave packet decompositions and the combinatorics of light rays to isolate the regions of spacetime where the waves are concentrating, cover these regions by tubular neighbourhoods of light rays, then remove the light rays to reduce the energy (or mass) of the solution and iterate. The wave packet analysis is moderately complicated, but fortunately I can use a proposition on this topic from my paper as a black box, leaving only the other components of the argument to write out in detail. |
df8cf4fdfaf695aa | Transferring in 7 seconds to the latest research in Biocosmology
PDF versions:
1. Prebiotic Epoch: Cosmic Symmetry-breaking and Molecular Evolution (pdf)
2. Evolutionary Epoch: Complexity, Chaos and Complementarity (pdf)
3. Consummating Cosmology: Quantum Cosmology and the Hard Problem of the Conscious Brain (pdf)
Critical research developments
1. Chemist Shows How RNA Can Be the Starting Point for Life May 14, 2009
2. First Cells, Proton-Pumping and Undersea Rock Pores (research article pdf password "model") Oct 19, 2009
Fig 1 (a) Divergence of the four forces from a single superforce. (b) Non-zero vacuum polarization at minimum energy is the cause of electro-weak symmetry-breaking. (c) Cosmic abundances of the bioelements. (d) Global t-RNA structure and (e) protein lysozyme with substrate (purple). Primary, secondary, tertiary and global structures and conformation changes in biomolecules are the result of a fractal hierarchy of strong covalent and ionic and weaker chemical interactions - H-bonding, hydrophilic, polar and ven der Waal's interactions, arising from the unresolved non-linear nature of chemical bonding (figures from King, except (e) Watson et. al.).
Biocosmology: Cosmic Symmetry-Breaking and Molecular Evolution
We now explore the structural relationship between cosmological symmetry-breaking and the form of molecular evolution leading to biological systems on Earth. It thus forms an alternative to historical hypotheses in which the form of biogenesis is believed to be the product of a linked sequence of specific conditions, bridged by stochastic selection processes.
The rich diversity of structure in molecular systems is made possible by the profound asymmetries between the nuclear forces and electromagnetism. Although molecular dynamics is founded on electromagnetic orbitals, the diversity of the elements and their asymmetric charge structure, with electrons captured by a spectrum of positively charged nuclei, is made possible through the divergence of symmetry of the four fundamental forces. The non-linear electromagnetic charge interactions of these asymmetric structures is responsible for both chemical bonding and the hierarchy of weak bonding interactions which result in the non-periodic secondary and tertiary structures of proteins and nucleic acids. It also provides the basis for a bifurcation theory which could give biogenesis the same generality that nucleogenesis has.
Differentiation and Inflation: The Microscopic and Cosmic Scales
Force Differentiation: The strong (nuclear binding) and weak (neutron decay) forces, electromagnetism and gravity are believed to have emerged from a single superforce shortly after the big bang, fig1(a). The strong force is believed to be a secondary effect of the colour force in much the same way that molecular bonding is a secondary consequence of the formation of atoms. The weak force has become short range because it is mediated by massive particles, which are believed to gain an extra degree of freedom by assimilating a Higg's boson (Georgi 1981, t'Hooft 1980, Veltman 1986). The symmetry between the Z and W particles of the weak force and the massless photon of elecromagnetism is thus broken by the lower energy of the polarized configuration, fig 1(b). Even heavier particles are believed to separate the strong force from these two. Force reconvergence occurs at the unification temperature fig 1(c). The strong force mesons gain mass from a different mechanism, being the energies of the bound states of the colour force, whose gluons are massless, but confined. The separation of gravity from the other forces is more fundamental because it involves the structure of space-time and may be described by a higher-dimensional superstring force in which particles become excited loops or strings in a higher dimensional space-time which is compactified into our 4-dimensional form (Green 1985, 1986, Goldman et.al. 1988, Freedman & van Nieuwenhuizen 1985).
Cosmic Inflation: A universe in a symmetrical state, but below its unification temperature is in an unstable high-energy false vacuum. The energy of the Higg's field causes inflation, in which the universe has net gravitational repulsion and expands exponentially, smoothing irregularities to fractal structures on the scale of galaxies (Guth & Steinhardt 1984). The breakdown of the false vacuum then releases a stream of high-energy particles as latent heat, to form the hot expanding universe under attractive gravitation. The gravitational potential energy thus gained equals that of the energetic particles, making the generation of the universe possible from a quantum fluctuation. However variations in the cosmic background radiation are consistent with a big-bang smoothed by inflation (Smoot 1992).
Interactive Dynamics:
The interaction between the resulting wave-particles also results in distinct effects on the microscopic and cosmic scales, namely galaxy and star formation and genesis of nuclei, chemical elements, and finally molecules, in which the non-linear nature of chemical bonding becomes fully expressed in complex tertiary structures. These interactions are modified indirectly by the nuclear forces which contribute asymmetries, spin-effects, weak decay and the nuclear energy of stars.
Particle Interaction-1: Nucleosynthesis as a Cosmological Dynamical System. The nucleosynthesis pathway generates over 100 atomic nuclei from the already composite proton and neutron. Parity between protons and neutrons is slightly broken via weak decay, fig 1(e) to balance between the lowest nuclear quantum states being filled and increasing electromagnetic repulsion. The process is exothermic and moderated by the catalytic action of several of the isotopes of lighter elements such as carbon and oxygen. The cosmic abundance of the elements fig1(d) reflects the binding energies of the nuclei and stable a-particle-like shells (Moeller et. al. 1984). The nucleosynthesis pathway has a cosmologically-general form despite having some variation in individual star systems.
Particle Interaction-2: Moleculosynthesis. The Culminating Dynamic Although, by comparison with the energies of cosmic creation or even astronomical bodies, the structures of biomolecules seem much too fragile to be a cosmological feature, symmetry-breaking of the forces leads inevitably to molecular structures as a hierarchical culmination of the interactive phase. Quarks are bound by gluons into composite particles such as the proton p+ and neutron n. These interact by the strong force via the nucleosynthesis pathway to form the elementary nuclei. Subsequently the weaker electromagnetic force interacts, also in two phases, firstly by the formation of atoms around nuclei and then by secondary interaction to form molecules. The latter phase occurs in a sequence of stages through successive strong and weak bonding interactions, producing the complex tertiary structures of biomolecules, fig 1(f,g).
The Cosmic Interaction Sequence: The Pathway to the Planetary Biosphere Galaxy formation is followed by the generation of the chemical nuclei in the supernova explosion of a short-lived hot star. In the second phase these elements are drawn into a lower energy long-lived sun-like star, the lighter [bio]elements, occurring in high cosmic abundance as a result of nucelosynthesis dynamics, fig1(d), becoming concentrated on mid-range planets. The final re-entry of the forces is thus represented by stellar photon irradiation of molecular systems, under gravitational stabilization on a planetary surface.
A Brief Survey of Non-linear Orbital Theory
The fact that the laws of chemistry were discovered sooner and were relatively easier to explore than the conditions underlying the unification of electromagnetism with the nuclear forces has resulted in an anomalous historical perspective which has helped to obscure some of the most interesting and complex manifestations of chemistry as a final interactive consequence of cosmological quantum symmetry-breaking. The increasing nuclear charge permits an unparalleled richness and complexity of quantum bonding structures in which electron-electron repulsions, spin-obit coupling, and other effects perturb the periodicity of orbital properties and lead to the development of higher-order molecular structures.
Although quanta obey linear wave amplitude superposition, chemistry inherits non-linearity in the form of the attractive and repulsive charge interactions between orbital systems. Such non-linear interaction, combined with Pauli exclusion, is responsible for the diversity of chemical interaction from the covalent bond to the secondary and tertiary effects manifest in the complex structures of proteins and nucleic acids.
The source of this non-linear interaction is the foundation of all chemical bonding, the electric potential. Although the state vector of a quantum-mechanical system comprises a linear combination of eigenfunctions, the electrostatic charge of the electron causes orbital interaction to have non-linear energetics.
The atomic and molecular structure of molecular orbitals illustrate the geometrical complexity that arises from the asymmetrical charge distributions of atomic orbitals. If the nuclear force did not provide up to 100 stable nuclei and the electromagnetic force were not also asymmetrically distributed between the nucleus and the electrons this complexity would be impossible. Top left: the radial densitywaves of 1s 2s 3s orbitals. Top Right: the 1+3+5 pattern of s, p, and d orbitals showing their geometry. This explains the periodic properties of the table of elements. Bottom left: s and p orbitals can form hybrids by super-position in elements such as carbon, nitrogen, and oxygen. Botton centre: two types of molecular orbitals are illustrated in which s and p orbitals are combined to create a chemical bond. Bottom right: the linear, planar, and tetrahedral arrangement of the hybrid orbitals. Pi-orbitals are also capable of forming delocalized molecular orbitals which span a whole molecule. These and the tenrahedral sp3-hybrids form the backbone of biomolecular bonding.
In the Schrödinger equation for the hydrogen atom, where ,
the potential function results in charge attraction and a negative energy. The electric potential provides the principal non-linear basis for subsequent bonding phenomena because it results in an inverse square law force and non-linear attraction-repulsion dynamics in four-dimensional space-time. Further effects such as spin-orbit coupling add complicating terms to the Hamiltonian.
The the underlying linearity of wave superposition is illustrated in the formation of linear combinations of s & p wave functions to form the four sp3 hybrid orbitals. The treatment of more complex atoms is generally simplified by approximation by perturbation theory or the self-consistent field method, in which a hydrogen-like orbital is based on purely radial repulsion factors for the inner electrons. Variation theory succinctly illustrates the interactive non-linearity of bond formation. The total energy is represented by the resonance integral of the Hamiltonian composed with the wave function, divided by the normalizing overlap integral S.
In the case of the one-electron Hydrogen molecule ion, with Saa= Sbb normalized to 1, we have 2 solutions, as indicated:
Quantum matrix methods are generally simplified to take account of only one aspect of molecular interaction and involve extensive approximations such as the independent particle approximation and Hükel theory (Brown 1972). The non-linear interactions of electron repulsions and spin-orbit coupling in the global context of molecular tertiary structure require complex computer techniques for example to predict the 3-D structure of protein molecules. These are only beginning to simulate the folding of complex molecules, again requiring approximation techniques.
The capacity of orbitals, including unoccupied orbitals, to cause successive perturbations of bonding energetics results in an interaction succession from strong covalent and ionic bond types [200-800 kj/mole] through to their residual effects in the variety of weaker H-bonding, polar, hydrophobic, and van der Waals interactions [4-40 kj/mole] merging into the average kinetic energies at biological temperatures [2.6 kj/mole at 25oC], (Watson et. al. 1988). These are responsible for secondary structures such as the a-helix of proteins and base-pairing and stacking of nucleic acids, and result in the tertiary effects central to enzyme action, whose energetics are determined by global interactions in complex molecules.
The cooperative reactivity of the active site of hexokinase demonstrates how, even after resolving the covalent and successive weaker bonding effects, and the local interactions of individual sides chains, and the larger fractal structures arising from weak bonds forming secondayry and tertiary protein structure, the entire enzyme is still capable of marked global conformation changes of a highly energetic nature. Chemical forces are thus fractal, leading right up to the globally fractal tissue structures we see in organismic biology, from the lungs to the brain. This is confirmed in the fractal dynamics of key cell structures (Watson et. al.).
Beloushov-Zhabotinskii type reaction giving rise to three-dimensional scroll waves (CK).
2.2 Fractal and Chaotic Dynamics and Structure in Molecular Systems.
Most minerals adopt periodic crystal geometries. Although some anomalies are disordered, many such as those superconducting perovskites have higher-order geometrical regularity. By contrast, the irregularties in polymers such as polypeptides and RNA is critical are establishing the richness of their tertiary structures, and their bio-activity. Variable sequence polymers with significant tertiary structure are non-periodic because the unlimited variety of monomeric primary sequences induce irregular secondary and tertiary structures. These irregularities are central to biochemistry because they result in powerful catalysts which can alter the reaction dynamics because of the generation of local activating sites globally potentiated through intermolecular weak-bonding associations. They also permit allosteric regulation. Despite being genetically coded, such molecules form fractal structures both in stereochemical terms and in terms of their relaxation dynamics.
Prigogine's theory of non-equilibrium thermodynamics, in which maximum entropy is replaced by a more general critical point of entropy production, which in an open system may not be a maximum. The associated oscillating chemical systems such as the Beloushov-Zhabotinskii reaction have demonstrated the capacity of chemical systems to enter into non-linear concentration dynamics, including limit cycle bifurcations. Period-doubling bifurcations and chaotic concentration dynamics have also been observed . Similar dynamics occur in electrochemical membrane excitation. The living cell is a non-equilibrium open thermodynamic system whose boundary, the memerane, exchanges material with the outside world. This makes it possible for life to be a negentropic system within a universe where entropy is increasing. The photosynthetic conversion of light to chemical energy and structural growth in our great forests is a prime example.
By contrast, viruses do not form a thermodynamic system as such, but rather a system of pure information. The first emergence of polynucleotides may similarly have been associated with the acrual of such information by a more direct negentropic route, phase transition.
Fig 2: (a) Symmetry-breaking model of selection of bioelements, as an interference interaction between H and CNO, followed by secondary ionic, covalent and catalytic interactions. (b) Boiling points of hydrides illustrate the optimality of H2O as polar H-bonding and structural medium for biological structure (CK). 3-D periodic table Sci Am. Sep 98
Biogenesis as a Central Synthetic Pathway
One of the central ideas of the cosmological biogenesis model is that the molecular interactions forming the pathways to the origins of life as we know it are not just an accidental set of chemical reactions out of a great variety of ad-hoc initial conditions, but that they represent a fundamental biforcation arising ultimately from cosmological symmetry-breaking of the four forces. The non-linear properties of electron orbitals cause the periodic table to have a critical sequence of bifurcations relating to the fundamental interactions.
Traditionally chemists have become so wedded to the idea of atoms and molecules as simply the "building blocks of the universe", as Isaac Asimov once put it that they cannot comprehend how they might interact as a quantum dynamical system. The fact that chemical bonding is possible between a large variety of atoms in some form or other leads to the loss of an understanding for how the non-linear electronic interactions gave rise to chemical bonding in the first place. It also leads to a mechanistic view of biogenesis, in which there is no underlying dynamical theory, but simply a search for the underlying special or initial conditions which caused the first self-replicating reaction to get going. The aim is thus either to set up a laboratory reaction by placing extreme order on the system, to elucidate this reaction pathway, or an attempt to use random processes and probabilistc arguments to model the likelihood that some collection of replicating molecules might accidentally come together. This has marred prebiotic research and profoundly slowed its advances.
Two illustrations hightlight this conceptual barrier. There is a 40 year time span between Miller and Urey's first spark experiments elucidating pathways from simple precursors to the purine nucleic acid bases, and the modification of this synthesis which led to good yields of the pyrimidines. Likewise there has been two decades of research in attempting to polymerize ribonucleotides, littered with failures due to oversimiplification of RNA interactions and mechanistic variants such as peptide-nucleic acids, before the ribonucleotide evolution techniques of Szostak and finally simple relationship between polymerizing ribonucleotides and montmorrilonite clays became obvious.
Atomic radii and electronegativity scales indicate the optimal bonding energetics of CNO and particularly the strong polarity of O. A clear differentiation can also be seen between the ionic potentials of Na and K.
The cosmological biogenesis theory asserts the following three points:
1. All molecular interaction is highly non-linear, and forms an unresolved fractal interactive milieu which permits not only the cascade of weaking bonding and global interactions characterizing protein enzymes and nucleic acids but also on a larger scale the tissue structures of whole organisms. This means that, while nature can be crystalline, it can also display emergent properties on larger scales which are very difficult to predict from an examination of the components "the whole is more than the sum of the parts". The non-linear perspective realizes emergence within an in-principle reductionist viewpoint because the underlying principles are quantum chemistry, but the consequences are emergent fractal interaction. This situation is clearly illustrated in the great difficulty of fully accurate modelling of the electronic dynamics of even simple atoms because they are many-body problems, and by the complexity of the protein-folding problem (Sci Am. Jan 91 see also Shape is All NS 24 Oct 98 42).
2. The entire molecular environment is non-linear in a way which is capable of exploring its phase space in the manner of a chaotic dynamical system. This means that planetary, terrestrial and molecular systems display sufficient chaos to generate all the varieties of structural interaction possible. These non-linearities make the natural environment a quantum equivalent of a Mandelbrot set in which a potentially infinite variety of dynamics are possible. The overwhelming majority of chemical experiments into the origins of life (with the notable exception of the original spark experiments) attempt to defeat this process by introducing simple overweaning conditions of order to force simple clear-cut products out of the system.
3. Underlying this rich chaotic interaction is a universal bifurcation pathway which is a direct consequence of the form of cosmological symmetry-breaking of the four quantum forces. While there may be more than one way that molecular replication could occur in chemistry, the RNA-based form of life is nevertheless a central bifurcation product of the interaction between the fundamental forces and by no means a mere accident of unlikely circumstances.
The unique and key nature of H-bonding in biogenesis is illustrated in the
structure of the alpha helix of proteins and the pairing of nucleotide bases (Watson et. al.).
Principal Symmetry-splitting : The Covalent Interaction of H with C, N, O.
Quantum interference interaction between the two-electron 1s orbital and the eight-electron 2sp3 hybrid. The resulting three dimensional covalent bonds give C, N and O optimal capacity to form diverse polymeric structures in association with H. Symmetry is split, because the 1s has only one binding electron state, while the 2sp3 has a series from with differing energies and varied occupancy, as the nuclear charge increases. The 1s orbital is unique in the generation of the hydrogen bond through the capacity of the bare proton to interact with a lone pair orbital. The CNO group all possess the same tetrahedral sp3 bonding geometry and form a graded sequence in electronegativity, with one and two lone pairs appearing successively in N and O.
Polymeric condensation of unstable high-energy multiple-bonded forms. Some of the strongest covalent bonds are the multiple-bonds such as CC , CN, and > C = O. These can be generated by applying any one of several high-energy sources such as u.v. light, high temperatures (900oC), or spark discharge. Because of the higher energy of the resulting pi-orbitals, these bonds possess a specific type of structural instability in which one or two pi-bonds can open to form polymeric structures, particularly when bound to H and alkyl groups, as under reducing conditions. Most of the prebiotic molecular complexity generated by such energy sources can be derived from mutual polymerizations of HCCH, HCN, and H2C = O, including purines, pyrimidines, key sugar types, amino acids, porphyrins etc. They form a core pathway from high energy stability to structurally unstable polymerization, which we will examine in the next section.
Radio-telescope data demonstrates clouds of HCN and H2CO spanning the region in the Orion nebula where several new stars are forming. All of A, U, G, and C have been detected in carbonaceous chondrite meteorites, which also contain membrane-forming products. HCN and HCHO polymerizations also lead to membranous microcellular structures. Although the presence of CO2 as a principal atmospheric gas on the early earth could have reduced the quantities of such reduced molecules, HCN could have been produced as a transient in the early atmosphere leading to heterocyclic products. A variety of microenvironments would still have had access to reducing conditions.
The formation of conjugated double and single bonds in these reactions results is the appearance of delocalized pi-orbitals. Such orbitals in heterocyclic (N, C) rings with conjugated resonance configurations also enable lone pair n > p* and also p > p* transitions, resulting in increased photon absorption. These effects in combination play a key role in many biological processes including photosynthesis, electron transport and bioluminescence.
The dynamical subtlety over simple crystalline form is demonstrated by the exterme variety of snowflake form which displays both fractal dendrite growth from the vapour solid interface, which varies with external conditions of temperature and humidity and a global coherence of geometry which develops partly from quantum interactions including H2O molecules'walking' across the crystal surface.
Secondary Splitting between C, N, and O : Electronegativity Bifurcation.
In addition to varying covalent valencies, lone pairs etc., the 8-electron 2sp3 hybrid generates a sequence of elements with increasing electronegativity, arising from the increasing nuclear charge. This results in a variety of secondary effects in addition to the oxidation-reduction parameter, from the polarity bifurcation into aqueous and hydrophobic phases to the complementation of CO2 and NH3 as organic acid and base.
The tetrahedral H-bonding of H2O molecules in ice. In liquid water, 80% of the molecules are in such ordered relationships at any given time. The structures of DNA and proteins derive their energetics from their interaction with water. The stability of polynucleotides is a combination of hydrophobic base-stacking and polar interaction with the phosphate-sugar backbone. Soluable proteins (myoglobin is illustrated) derive their stability and energetics through their interactions with induced water structures. Generally these form micelle structures which have a hydrophobic interior surrounded by polar amino acids (see the illustration of lysozyme). All polymerization pathways of polysaccharides, polynucleotides and polypeptides share a common feature - dehydration is the basis of polymerization, making water the common factor in the negentropic assimilation of order (Watson et. al.).
Optimality of H2O: Polarity, Phase and Acid-base bifurcations. Ionic and Hydrogen bonding.
Outside metals such as mercury, water has one of the highest specific heats. This is a reflection of the large number of conformational degress of freedom it contains. It is also capable of a very unusual number of interactions, ionic, polar, H-bonding, acid-base and the polarity bifurcation into hydrophilic (water-loving) and hydrophobic (oily) phases in biological molecules and structures such as the lipid membrane, which is a sandwich of oily and watery moieties.
(a) Isolation of non-polar molecules into the hydrophobic phase is facilitated by water-bonding clathrate structures whose impressed order is reduced and entropy increased, by reducing these strictures to one common envelope. (b) polypeptides and (c) polynucleotides both form by elimination of H2O, accompanied by ATP energy in the latter as illustrated in the left of the two monomers.
Dehydration is the common currency of polymerization, beginning with the mineral pyrophosphate linkage of ATP. The central biopolymers, polynucleotides, polypeptides and polysacharides are uniformly linked by the removal of a molecule of water, dehydration in the aqueous medium. Furthermore the three-dimensional structures of the nucleic acid double-helix, globular enzymes, membranes and ion channels are all made structurally and energetically possible only through the interactions of these molecules with water and the induced H2O structures that form around them in solution. Both nucleic acids and proteins consist of a balance of hydrophilic and hydrophobic interactions which in the former give hydrophobic base-stacking within a polar back-bone and with enzymes a non-polar micelle surrounded by hydrophilic groups.
Differential electronegativity results in several coincident bifurcations associated with water structure. A symmetry-breaking occurs between the relatively non-polar CH bond and the increasingly polar NH and OH. This results in phase bifurcation of the aqueous medium into polar and non-polar phases in association with low-entropy water bonding structures induced around non-polar molecules. This is directly responsible for the development a variety of structures from the membrane in the context of lipid molecules, to the globular enzyme form and base-stacking of nucleic acids.
The optimal nature of water as a hydride is illustrated in boiling points. By comparison with ammonia H3N, water H2O has balanced doning and accepting H-bonds and a stronger polarity. Such polar properties are also clearly optimal over H2S, alcohols etc.
The discovery by the ISO Infra-red Space Observatory, of widespread incidence of water around stars, planets and throughout the universe where stars are forming has led increasing weight tothe cosmological status of water as a pre-cursor to life. - AP Apr 98
Water provides several other secondary bifurcations besides polarity. The dissociation of H2O into H+ and OH- lays the foundation for the acid-base bifurcation, while ionic solubility generates anion-cation. H-bonding structures are also pivotal in determining the form of polymers including the alpha helix, base pairing and solubility of molecules such as sugars. Many properties of proteins and nucleic acids, are derived from water bonding structures in which a mix of H-bonding and phase bifurcation effects occur. The large diversity of quantum modes in water is exemplified by its high specific heat contrasting with that of proteins (Cochran 1971). Polymerization of nucleotides, amino-acids and sugars all involve dehydration elimination of H2O, giving water a central role in polymer formation.
P and S as Low-energy Covalent Modifiers - the delicate role of Silicon.
The second-row covalent elements are sub-optimal in their mutual covalent interactions and their interaction with H. Their size is more compatible with interaction with O, forming e.g. SiO32-, PO43- & SO42- ions including crystalline minerals. The silicones are notable for their O content by comparison with hydrocarbons. However in the context of the primary H-CNO interaction, two new optimal properties are introduced.
PO43- is unique in its capacity to form a series of dehydration polymers, both in the form of pyro- and poly-phosphates, and in interaction with other molecules such as sugars. The energy of phosphorylation falls neatly into the weak bond range (30-60 kj/mole) making it suitable for conformational changes. The universality of dehydration as a polymerization mechanism in polynucleotides, polypeptides, polysaccharides and lipids, the involvement of phosphate in ATP energetics, RNA and membrane structure, and the fact that the dehydration mechanism easily recycles, unlike the organic condensing agents, give phosphate uniqueness and optimality as a dehydrating salt.
The function of S in biosystems highlights a second optimality. The lowered energy of oxidation transitions in S particularly S-S ´ S-H , by comparison with first row elements, gives S a unique role both in terms of tertiary bonding and low energy respiration and photosynthesis pathways.
It has recently been discovered that oligoribonucleotides will polymerize effectively on silicate clay surfaces, where the positive ions of atoms such as Al make polar interactions with the phosphate backbones of RNA, stabilizing the molecules and making further polymerization possible in an ordered geometry. This consitiutes a major breakthrough in the modelling of life's origins and demonstrates the sensitivity of the biogenic pathway to the subtle differences of electronegativity of the second-row covalent elements phosphorus and silicon.
Ionic Bifurcation.
The cations bifurcate in two phases : monovalent-divalent, and series (Na-K, Mg-Ca). Although ions such as K+ and Na+ are chemically very similar, their radii of hydration differ significantly enough to result in a bifurcation between their properties in relation to water structures and the membrane. Smaller Na+ and H3O+ require water structures to resolve their more intense electric fields. Larger K+ is soluble with less hydration, making it smaller in solution and more permeable to the membrane (King 1978) . Ca2+ and Mg2+ have a similar divergence, Ca2+ having stronger chelating properties. This causes a crossed bifurcation between the two series in which K+ and Mg2+ are intracellular, Mg2+ having a pivotal role in RNA tranesterifications. Cl- remains the central anion along with organic groups. These bifurcations are the basis of membrane excitability and the maintenance of concentration gradients in the intracellular medium which distinguish the living medium from the environment at large.
Transition Element Catalysis
These add d-orbital effects, forming a catalytic group. Almost all of the transition elements e.g. Mn, Fe, Co, Cu, Zn are essential biological trace elements (Frieden 1972), promote prebiotic syntheses (Kobayashi and Ponnamperuma 1985) and are optimal in their catalytic ligand-forming capacity and valency transitions. Zn2+ for example, by coupling to the PO43- backbone, catalyses RNA polymerization in prebiotic syntheses and occurs both in polymerases and DNA binding proteins. Both the Fe2+-Fe3+ transition, and spin-orbit coupling conversion of electrons into the triplet-state in Fe-S complexes occur in electron and oxygen transport (McGlynn et. al. 1964). Other metal atoms such as Mo, Mn have similar optimal functions, e.g. in N2 fixation.
Fig 3 : (a) The perturbing effect of the neutral weak force results in violation of chiral symmetry in electron orbits. Without perturbation (i) the orbits are non-chiral, but the action of Zo results in a perturbing chiral rotation. (b) Autocatalytic symmetry-breaking causes random chiral bifurcation (i). Weak perturbation results in only one chiral form (iii) (King).
Chirality bifurcation.
Although the electromagnetic force has chiral symmetry, the electron also interacts via the neutral weak force when close to the nucleus. This causes a perturbation to the electronic orbit causing it to become selectively chiral, fig 3(a) (Bouchiat & Pottier 1984, Hegstrom & Kondputi 1990). In a polymeric system with competing D and L variants, in which there is negative feedback between the two chiral forms of polymerization, making the system unstable, the chiral weak force provides a symmetry-breaking perturbation. In a simulation, fig 3(bi) high [S][T] causes autocatalytic bifurcation of system (ii), resulting in random symmetry-breaking into products D or L. Chiral weak perturbation (iii) results in one form only. The selection of D-nucleotides could have resulted in L-amino acids by a stereochemical association (Lacey et. al. 1988).
Inner Circles New Scientist 8 Aug 98 11 reports on findings that there is a 17% net circular polarization in light in gas clouds in the Orion nebula where new stars are forming. Although this was infra-red light James Hough says it should also apply to the ultra-violet light. This would explain the excess of L-amino-acids found in the Murchison meteorite, suggesting a cosmic rather than accidental origin for the handedness of biological molecules on earth.
Tertiary Interaction of Mineral Interface.
Both silicates such as clays and volcanic magmas have been the subject of intensive interest as catalytic or information organizing adjuncts to prebiotic evolution. Clays have been proposed as a primitive genetic system and both include adsorbent and catalytic sites. The mineral interface involves crucial processes of selective adsorption, chromatographic migration, and fractional concentration, which may be essential to explain how rich concentrations of nucleotide monomers could have occurred over geologic time scales.
More recently a fundamental interaction between RNA and clays has been elucidated wich appears to be central in enabling oligo-ribonucleotides to polymerize in an ordered way while bound to the positively charged metal groups in montmorrilonite clays, bridging the gap between small random ribo-oligomers and RNA molecules of a length capable of self-replication.
Key polymerizations
Key polymerizations such as those of HCN and HCHO are proposed to generate a series of generic bifurcation structures through combined autocatalytic and quantum bond effects, which include major components of the metabloism including nucleotides, polypeptides and key membrane components. These will be examined in the next section
The astronomical perspective
The presence of HCN and HCHO clouds in the Orion nebula (left) attests to the ubiquitous nature of the energetic monomeric precursors of biomolecules, in particular RNA. CO clouds (right) occupy a protosolar disc 20 times the size of the orbit of Pluto (King, Scientific American).
The occurrence of the key precursors of biomeolecules are not in any way confined to Earth of the specific conditions of Earth. Much of the organic material on earth is believed to have peppered down from comets and carbonaceous meteorites especially earlier in the evolution of the solar systems when less of the original material from the proto-solar gas and dust cloud had been swept away by collision. Protosolar gas clouds in the Orion nebula are known to contain precisely HCN and HCHO as shown above. Certain parts of the universe give off an infra-red signal not unlike that of carbohydrates. Interstellar dust grains are also known to contain organic molecules. In fact the occurrence of organic molecules is essentially ubiquiitous to all second generation sun-like stars containing a mix of elements of nucleosynthesis formed from the material of previous supernovas.
Indeed their presence is so commonplace and the incidence of life on Earth is so early that the possibility that arose previously to the formation of Earth cannot entirely be ruled out. Cosmological biogenesis is however ideally suited to the conditions actually occurring on Earth with plentiful water, a temperature just above the liquefaction of water, a good supply of organic molecules and a steady mild solar input.
Terrestrial planets Venus to Uranus: It is clear that the extreme variety of our own planetary system and the like variety of the many galilean satellites spells out a very important feature of astronomical law. In four-dimensional space-time gravitational forces are inverse square law. In addition to the natural graduation of temerature noted above in the proposolar disc, prviding for life as we know it on the outer edge of the inner region where water is liquid, the chaotic variation of conditions produced by such non-linear force fields generates a situation of extreme variety in the universe, much like the Mandelbrot set does. While this may seem to make the other planets of our system a little alien, it does guarantee the universe will explore its entire phase space of possibilities virtually guaranteeing it will leave no stone (planet) unturned in its exploration of the possibilities that life will emerge (Hubble public gallery).
Just as one can consider the non-linearities of the electromagnetic force in developing the fractal dynamics of molecules, one can also appreciate the significance of non-linearities in gravitation in forming the rich diversity of planets and satellites we see in our own solar system. Other stars now seem to be quite richly endowed with planets, but these again show very marked variation. Such marked variationis characteristic of non-linear dynamics, which serves to accentuate existing differences for example in temperature and composition between the planets to cause unique effects, such as the highly acidic, electrified runaway greenhouse atmosphere of venus.
Far from considering these extreme variations as reducing the likelihood of finding life on other planets, what it demonstrates is that on an astronomical scale as well as the microscopic, the universe behaves very much like a Mandelbrot set in establishing dynamics of uniqueness and diversity which explores the dynamical space of possibilities.
On to Biocosmology Part 2: Central Polymerization Pathways? |
46a406f3aadcfa2b | Record Waves in the North Atlantic
In the last day or two, there have been numerous press reports of a 19 meter (62.3 ft) wave, recorded by an automated buoy in the North Atlantic between Iceland and the UK off the Outer Hebrides. This is a new record for a wave recorded by a buoy.
What does this really mean? Rogue waves are often larger than 19 meters. The first scientifically reported rogue, the Draupner wave, which struck a drilling platform of the same name on New Year’s Day in 1995, was recorded to be 25.6 metres (84 ft) high. So why is a 19 meter wave such a big deal?
The best answer may be that there are different sorts of waves and they are measured differently. Automated buoys generally cannot accurately measure rogue waves. They are designed to measure the overall seastate. What makes the 19 meter wave so remarkable is that it was not a single wave but an average of a series of waves.
Technically, the World Meteorological Organization expert committee which reviewed the event called it “the highest significant wave height as measured by a buoy”. The WMO World Weather Climate Extremes Archive explains that “significant wave height” is “in essence, what an observer would have seen if he/she averaged over 15-20 waves passing by the buoy.”
So, an average wave height of 19 meters of a series of waves is rather remarkable. Or was remarkable. The WMO announcement about the new record significant wave height was released yesterday. The waves themselves were recorded in February 2013.
But what about rogue waves? They are something quite different. Often called “freak” waves, they do not fit the standard wave models and are generally explained using quantum physics, specifically the nonlinear Schrödinger equation. If you are familiar with Schrödinger’s cat, a rogue might be called Schrödinger’s wave.
Rogue waves are often defined as waves whose height is more than twice the significant wave height. If conditions were right for the formation of a rogue wave in a sea with a 19 meter significant wave height, that would an impressive wave indeed.
Thanks to Alaric Bond for contributing to this post.
2 Responses to Record Waves in the North Atlantic
1. Willy says:
That is quite the story lag. Recorded in 2013 and posted in 2016?
2. Cormac McGrane says:
Interesting, a similar post that has been doing the rounds on Facebook here in Ireland and on Coast Monkey.
I came across this one – – about a wave recorded off the Irish coast by Buoy M4, 44NM off the Donegal Coast, so well into the N Atlantic. It was an individual wave height as opposed to the scary average in your post but at 23.4m it was a whopper.
That NW corner of Ireland is apparently world famous in the surfing community for monster waves, both inshore and offshore. |
9fdb4b01017cc324 |
2Physics Quote:
-- Kathrin Altwegg and the ROSINA Team
Sunday, April 19, 2015
Percolation in Laser Filamentation
Wahb Ettoumi
Author: Wahb Ettoumi
Affiliation: GAP-Biophotonics, University of Geneva, Switzerland.
Other coauthors of the PRL paper: Jérôme Kasparian (left) and Jean-Pierre Wolf.
The discovery of laser filamentation can be attributed to M. Hercher [1], who observed damage tracks along the laser path in crystals. Later, the filamentation phenomenon was shown for a laser propagating in air (For a review, see Ref.[2]). For the first time, the optical power at hand could allow one to witness a new type of light propagation based on the Kerr effect, a non-linear phenomenon which acts as a focusing lens and overcomes the beam natural diffraction. As a consequence, the propagation medium is ionized, and produces a plasma filament of tens of microns wide, which can be sustained over meters in air.
The beam collapse is eventually stopped by this newly created plasma, which acts as a defocusing lens, and counter-balances the Kerr effect. This subtile equilibrium is broken when the energy losses along the propagation cause the Kerr effect to be negligible again, and the beam finally diffracts.
Image 1
For powers largely exceeding the critical power needed for the observation of a single filament, the initial beam inhomogeneities seed the emergence of many single filaments, as if many small beamlets were each undergoing filamentation. In 2010, an experimental campaign in Dresden [3] was aimed at characterizing the number of filaments with respect to the initial power (Image 1). However, we only noticed until recently the similarity between the laser burns obtained there on photographic paper and the numerical simulations of systems relevant to the statistical physics community. More particularly, we decided to probe the resemblance of the experimental recordings with percolation patterns.
Initially, the laser beam exhibits a noisy profile, but with rather small fluctuations around an average fluence. As the laser propagates, the Kerr effect drives the light to concentrate more and more around the peaks of the highest amplitude, leading to the clustering of light into islands of different sizes, each one potentially holding one or multiple filaments.
Image 2
At larger distances, typically of several meters in usual experimental setups, the energy flux towards the inner cores of the multiple filaments causes the fluence islands to shrink in size, destroying the previously well held light clusters into smaller, disconnected parts (Image 2). At higher distances, the losses due to the medium's absorption eventually wipe out the smallest clusters.
Because of the lack of experimental data, we turned to the numerical simulation of the non-linear Schrödinger equation, well-known for its remarkable agreement with real filamentation experiments. We showed [4] that the precise way light clusters depending to each other is a phase transition: we measured a set of seven critical exponents governing the pattern dynamics at the vicinity of the transition between a fully connected state and a non-connected one. The similarity with the percolation universality class is striking, but the clusters' size distribution in the laser case exhibits a finite cut-off physically associated to fluence islands withholding a single filament (their area is approx. 2 mm2).
An interesting issue subsists, however. The finite-size scaling techniques we used are intrinsically equilibrium methods, so that we implicitely assumed that each slice during the laser propagation could be treated as a statistical equilibrium of a given system. But the laser obviously evolves in time, and is not trapped into a quasi-stationary state, nor a fluctuating equilibrium. A hand waving argument can be drawn by saying that the evolution is quasi-static, but a correct theoretical argument remains to be found.
[1] M. Hercher, "Laser-induced damage in transparent media". Journal of Optical Society of America, 54, 563 (1964).
[2] A. Couairon, A. Mysyrowicz, "Femtosecond filamentation in transparent media". Physics Report, 441, 47-189 (2007). Abstract.
[3] S. Henin, Y. Petit, J. Kasparian, J.-P. Wolf, A. Jochmann, S. D. Kraft, S. Bock, U. Schramm, R. Sauerbrey, W. M. Nakaema, K. Stelmaszczyk, P. Rohwetter, L. Wöste, C.-L. Soulez, S. Mauger, L. Bergé, S. Skupin, "Saturation of the filament density of ultrashort intense laser pulses in air". Applied Physics B, 100, 77 (2010). Abstract.
[4] W. Ettoumi, J. Kasparian, J.-P. Wolf, "Laser Filamentation as a New Phase Transition Universality Class". Physical Review Letters, 114, 063903 (2015). Abstract.
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At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are contour integrals and Laplace transform, but of course this should be a "real" complex analysis course which I could later refer to in honours courses. I am now confident (after this discussion, especially after Gauss complaints given in Keith's comment) that the name "complex" is quite discouraging to average students.
Why do we need to study numbers which do not belong to the real world?
Of course, we all know that the thesis is wrong and I have in mind some examples where the use of complex variable functions simplify solving considerably (I give two below). The drawback of all them is assuming already some knowledge from students.
So I would be really happy to learn elementary examples which may convince students in usefulness of complex numbers and functions in complex variable. As this question runs in the community wiki mode, I would be glad to see one example per answer.
Thank you in advance!
Here comes the two promised example. The 2nd one was reminded by several answers and comments about relations with trigonometric functions (but also by notification "The bounty on your question Trigonometry related to Rogers--Ramanujan identities expires within three days"; it seems to be harder than I expect).
Example 1. Find the Fourier expansion of the (unbounded) periodic function $$ f(x)=\ln\Bigl|\sin\frac x2\Bigr|. $$
Solution. The function $f(x)$ is periodic with period $2\pi$ and has poles at the points $2\pi k$, $k\in\mathbb Z$.
Consider the function on the interval $x\in[\varepsilon,2\pi-\varepsilon]$. The series $$ \sum_{n=1}^\infty\frac{z^n}n, \qquad z=e^{ix}, $$ converges for all values $x$ from the interval. Since $$ \Bigl|\sin\frac x2\Bigr|=\sqrt{\frac{1-\cos x}2} $$ and $\operatorname{Re}\ln w=\ln|w|$, where we choose $w=\frac12(1-z)$, we deduce that $$ \operatorname{Re}\Bigl(\ln\frac{1-z}2\Bigr)=\ln\sqrt{\frac{1-\cos x}2} =\ln\Bigl|\sin\frac x2\Bigr|. $$ Thus, $$ \ln\Bigl|\sin\frac x2\Bigr| =-\ln2-\operatorname{Re}\sum_{n=1}^\infty\frac{z^n}n =-\ln2-\sum_{n=1}^\infty\frac{\cos nx}n. $$ As $\varepsilon>0$ can be taken arbitrarily small, the result remains valid for all $x\ne2\pi k$.
Example 2. Let $p$ be an odd prime number. For an integer $a$ relatively prime to $p$, the Legendre symbol $\bigl(\frac ap\bigr)$ is $+1$ or $-1$ depending on whether the congruence $x^2\equiv a\pmod{p}$ is solvable or not. One of elementary consequences of (elementary) Fermat's little theorem is $$ \biggl(\frac ap\biggr)\equiv a^{(p-1)/2}\pmod p. \qquad\qquad\qquad {(*)} $$ Show that $$ \biggl(\frac2p\biggr)=(-1)^{(p^2-1)/8}. $$
Solution. In the ring $\mathbb Z+\mathbb Zi=\Bbb Z[i]$, the binomial formula implies $$ (1+i)^p\equiv1+i^p\pmod p. $$ On the other hand, $$ (1+i)^p =\bigl(\sqrt2e^{\pi i/4}\bigr)^p =2^{p/2}\biggl(\cos\frac{\pi p}4+i\sin\frac{\pi p}4\biggr) $$ and $$ 1+i^p =1+(e^{\pi i/2})^p =1+\cos\frac{\pi p}2+i\sin\frac{\pi p}2 =1+i\sin\frac{\pi p}2. $$ Comparing the real parts implies that $$ 2^{p/2}\cos\frac{\pi p}4\equiv1\pmod p, $$ hence from $\sqrt2\cos(\pi p/4)\in\{\pm1\}$ we conclude that $$ 2^{(p-1)/2}\equiv\sqrt2\cos\frac{\pi p}4\pmod p. $$ It remains to apply ($*$): $$ \biggl(\frac2p\biggr) \equiv2^{(p-1)/2} \equiv\sqrt2\cos\frac{\pi p}4 =\begin{cases} 1 & \text{if } p\equiv\pm1\pmod8, \cr -1 & \text{if } p\equiv\pm3\pmod8, \end{cases} $$ which is exactly the required formula.
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Maybe an option is to have them understand that real numbers also do not belong to the real world, that all sort of numbers are simply abstractions. – Mariano Suárez-Alvarez Jul 1 '10 at 14:50
Probably your electrical engineering students understand better than you do that complex numbers (in polar form) are used to represent amplitude and frequency in their area of study. – Gerald Edgar Jul 1 '10 at 15:36
Not an answer, but some suggestions: try reading the beginning of Needham's Visual Complex Analysis ( and the end of Levi's The Mathematical Mechanic (…). – Qiaochu Yuan Jul 1 '10 at 17:05
Your example has a hidden assumption that a student actually admits the importance of calculating F.S. of $\ln\left|\sin{x\over 2}\right|$, which I find dubious. The examples with an oscillator's ODE is more convincing, IMO. – Paul Yuryev Jul 2 '10 at 3:02
@Mariano, Gerald and Qiaochu: Thanks for the ideas! Visual Complex Analysis sounds indeed great, and I'll follow Levi's book as soon as I reach the uni library. @Paul: I give the example (which I personally like) and explain that I do not consider it elementary enough for the students. It's a matter of taste! I've never used Fourier series in my own research but it doesn't imply that I doubt of their importance. We all (including students) have different criteria for measuring such things. – Wadim Zudilin Jul 2 '10 at 5:06
32 Answers 32
The nicest elementary illustration I know of the relevance of complex numbers to calculus is its link to radius of convergence, which student learn how to compute by various tests, but more mechanically than conceptually. The series for $1/(1-x)$, $\log(1+x)$, and $\sqrt{1+x}$ have radius of convergence 1 and we can see why: there's a problem at one of the endpoints of the interval of convergence (the function blows up or it's not differentiable). However, the function $1/(1+x^2)$ is nice and smooth on the whole real line with no apparent problems, but its radius of convergence at the origin is 1. From the viewpoint of real analysis this is strange: why does the series stop converging? Well, if you look at distance 1 in the complex plane...
More generally, you can tell them that for any rational function $p(x)/q(x)$, in reduced form, the radius of convergence of this function at a number $a$ (on the real line) is precisely the distance from $a$ to the nearest zero of the denominator, even if that nearest zero is not real. In other words, to really understand the radius of convergence in a general sense you have to work over the complex numbers. (Yes, there are subtle distinctions between smoothness and analyticity which are relevant here, but you don't have to discuss that to get across the idea.)
Similarly, the function $x/(e^x-1)$ is smooth but has a finite radius of convergence $2\pi$ (not sure if you can make this numerically apparent). Again, on the real line the reason for this is not visible, but in the complex plane there is a good explanation.
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Thanks, Keith! That's a nice point which I always mention for real analysis students as well. The structure of singularities of a linear differential equation (under some mild conditions) fully determines the convergence of the series solving the DE. The generating series for Bernoulli numbers does not produce sufficiently good approximations to $2\pi$, but it's just beautiful by itself. – Wadim Zudilin Jul 2 '10 at 5:14
You can solve the differential equation y''+y=0 using complex numbers. Just write $$(\partial^2 + 1) y = (\partial +i)(\partial -i) y$$ and you are now dealing with two order one differential equations that are easily solved $$(\partial +i) z =0,\qquad (\partial -i)y =z$$ The multivariate case is a bit harder and uses quaternions or Clifford algebras. This was done by Dirac for the Schrodinger equation ($-\Delta \psi = i\partial_t \psi$), and that led him to the prediction of the existence of antiparticles (and to the Nobel prize).
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Students usually find the connection of trigonometric identities like $\sin(a+b)=\sin a\cos b+\cos a\sin b$ to multiplication of complex numbers striking.
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Not sure about the students, but I do. :-) – Wadim Zudilin Jul 1 '10 at 12:21
This is an excellent suggestion. I can never remember these identities off the top of my head. Whenever I need one of them, the simplest way (faster than googling) is to read them off from $(a+ib)(c+id)=(ac-bd) + i(ad+bc)$. – alex Jul 1 '10 at 20:35
When I first started teaching calculus in the US, I was surprised that many students didn't remember addition formulas for trig functions. As the years went by, it's gotten worse: now the whole idea of using an identity like that to solve a problem is alien to them, e.g. even if they may look it up doing the homework, they "get stuck" on the problem and "don't get it". What is there to blame: calculators? standard tests that neglect it? teachers who never understood it themselves? Anyway, it's a very bad omen. – Victor Protsak Jul 2 '10 at 1:43
@Victor: It can be worse... When I taught Calc I at U of Toronto to engineering students, I was approached by some students who claimed they had heard words "sine" and "cosine" but were not quite sure what they meant. – Yuri Bakhtin Jul 2 '10 at 8:51
From "Birds and Frogs" by Freeman Dyson [Notices of Amer. Math. Soc. 56 (2009) 212--223]:
One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926. Schrödinger was a bird who started from the idea of unifying mechanics with optics. A hundred years earlier, Hamilton had unified classical mechanics with ray optics, using the same mathematics to describe optical rays and classical particle trajectories. Schrödinger’s idea was to extend this unification to wave optics and wave mechanics. Wave optics already existed, but wave mechanics did not. Schrödinger had to invent wave mechanics to complete the unification. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrödinger’s idea seemed to be going nowhere. But then came the surprise. Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrödinger as well as to everybody else. According to Schrödinger, his fourteen-year-old girl friend Itha Junger said to him at the time, "Hey, you never even thought when you began that so much sensible stuff would come out of it." All through the nineteenth century, mathematicians from Abel to Riemann and Weierstrass had been creating a magnificent theory of functions of complex variables. They had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.
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Here are two simple uses of complex numbers that I use to try to convince students that complex numbers are "cool" and worth learning.
1. (Number Theory) Use complex numbers to derive Brahmagupta's identity expressing $(a^2+b^2)(c^2+d^2)$ as the sum of two squares, for integers $a,b,c,d$.
2. (Euclidean geometry) Use complex numbers to explain Ptolemy's Theorem. For a cyclic quadrilateral with vertices $A,B,C,D$ we have $$\overline{AC}\cdot \overline{BD}=\overline{AB}\cdot \overline{CD} +\overline{BC}\cdot \overline{AD}$$
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And even more amazingly, one can completely solve the diophantine equation $x^2+y^2=z^n$ for any $n$ as follows: $$x+yi=(a+bi)^n, \ z=a^2+b^2.$$ I learned this from a popular math book while in elementary school, many years before studying calculus. – Victor Protsak Jul 2 '10 at 1:21
If the students have had a first course in differential equations, tell them to solve the system
$$x'(t) = -y(t)$$ $$y'(t) = x(t).$$
This is the equation of motion for a particle whose velocity vector is always perpendicular to its displacement. Explain why this is the same thing as
$$(x(t) + iy(t))' = i(x(t) + iy(t))$$
hence that, with the right initial conditions, the solution is
$$x(t) + iy(t) = e^{it}.$$
On the other hand, a particle whose velocity vector is always perpendicular to its displacement travels in a circle. Hence, again with the right initial conditions, $x(t) = \cos t, y(t) = \sin t$. (At this point you might reiterate that complex numbers are real $2 \times 2$ matrices, assuming they have seen this method for solving systems of differential equations.)
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One of my favourite elementary applications of complex analysis is the evaluation of infinite sums of the form $$\sum_{n\geq 0} \frac{p(n)}{q(n)}$$ where $p,q$ are polynomials and $\deg q > 1 + \deg p$, by using residues.
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One cannot over-emphasize that passing to complex numbers often permits a great simplification by linearizing what would otherwise be more complex nonlinear phenomena. One example familiar to any calculus student is the fact that integration of rational functions is much simpler over $\mathbb C$ (vs. $\mathbb R$) since partial fraction decompositions involve at most linear (vs quadratic) polynomials in the denominator. Similarly one reduces higher-order constant coefficient differential and difference equations to linear (first-order) equations by factoring the linear operators over $\mathbb C$. More generally one might argue that such simplification by linearization was at the heart of the development of abstract algebra. Namely, Dedekind, by abstracting out the essential linear structures (ideals and modules) in number theory, greatly simplified the prior nonlinear theory based on quadratic forms. This enabled him to exploit to the hilt the power of linear algebra. Examples abound of the revolutionary power that this brought to number theory and algebra - e.g. for one little-known gem see my recent post explaining how Dedekind's notion of conductor ideal beautifully encapsulates the essence of elementary irrationality proofs of n'th roots.
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If you really want to "demystify" complex numbers, I'd suggest teaching what complex multiplication looks like with the following picture, as opposed to a matrix representation:
If you want to visualize the product "z w", start with '0' and 'w' in the complex plane, then make a new complex plane where '0' sits above '0' and '1' sits above 'w'. If you look for 'z' up above, you see that 'z' sits above something you name 'z w'. You could teach this picture for just the real numbers or integers first -- the idea of using the rest of the points of the plane to do the same thing is a natural extension.
You can use this picture to visually "demystify" a lot of things:
• Why is a negative times a negative a positive? --- I know some people who lost hope in understanding math as soon as they were told this fact
• i^2 = -1
• (zw)t = z(wt) --- I think this is a better explanation than a matrix representation as to why the product is associative
• |zw| = |z| |w|
• (z + w)v = zv + wv
• The Pythagorean Theorem: draw (1-it)(1+it) = 1 + t^2 etc.
One thing that's not so easy to see this way is the commutativity (for good reasons).
After everyone has a grasp on how complex multiplication looks, you can get into the differential equation: $\frac{dz}{dt} = i z , z(0) = 1$ which Qiaochu noted travels counterclockwise in a unit circle at unit speed. You can use it to give a good definition for sine and cosine -- in particular, you get to define $\pi$ as the smallest positive solution to $e^{i \pi} = -1$. It's then physically obvious (as long as you understand the multiplication) that $e^{i(x+y)} = e^{ix} e^{iy}$, and your students get to actually understand all those hard/impossible to remember facts about trig functions (like angle addition and derivatives) that they were forced to memorize earlier in their lives. It may also be fun to discuss how the picture for $(1 + \frac{z}{n})^n$ turns into a picture of that differential equation in the "compound interest" limit as $n \to \infty$; doing so provides a bridge to power series, and gives an opportunity to understand the basic properties of the real exponential function more intuitively as well.
But this stuff is less demystifying complex numbers and more... demystifying other stuff using complex numbers.
Here's a link to some Feynman lectures on Quantum Electrodynamics (somehow prepared for a general audience) if you really need some flat out real-world complex numbers
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Several motivating physical applications are listed on wikipedia
You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q). Here is a nice quote by Yuri Manin picked from here
On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics. (Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 )
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Thanks for the tip! I'll better not cite Yuri Ivanovich to my electrical engineers; this will hardly encourage them to do complex analysis. :-) – Wadim Zudilin Jul 1 '10 at 13:24
• If they have a suitable background in linear algebra, I would not omit the interpretation of complex numbers in terms of conformal matrices of order 2 (with nonnegative determinant), translating all operations on complex numbers (sum, product, conjugate, modulus, inverse) in the context of matrices: with special emphasis on their multiplicative action on the plane (in particular, "real" gives "homotety" and "modulus 1" gives "rotation").
• The complex exponential, defined initially as limit of $(1+z/n)^n$, should be a good application of the above geometrical ideas. In particular, for $z=it$, one can give a nice interpretation of the (too often covered with mystery) equation $e^{i\pi}=-1$ in terms of the length of the curve $e^{it}$ (defined as classical total variation).
• A brief discussion on (scalar) linear ordinary differential equations of order 2, with constant coefficients, also provides a good motivation (and with some historical truth).
• Related to the preceding point, and especially because they are from engineering, it should be worth recalling all the useful complex formalism used in Electricity.
• Not on the side of "real world" interpretation, but rather on the side of "useful abstraction" a brief account of the history of the third degree algebraic equation, with the embarrassing "casus impossibilis" (three real solutions, and the solution formula gives none, if taken in terms of "real" radicals!) should be very instructive. Here is also the source of such terms as "imaginary".
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@Wadim: The $(1+z/n)^n$ definition of the exponential is exactly what you get by applying Euler's method to the defining diff Eq of the exponential function, if you travel along the straight line from 0 to z in the domain, and use n equal partitions. – Steven Gubkin Aug 27 '12 at 13:24
They're useful just for doing ordinary geometry when programming.
A common pattern I have seen in a great many computer programs is to start with a bunch of numbers that are really ratios of distances. Theses numbers get converted to angles with inverse trig functions. Then some simple functions are applied to the angles and the trig functions are used on the results.
Trig and inverse trig functions are expensive to compute on a computer. In high performance code you want to eliminate them if possible. Quite often, for the above case, you can eliminate the trig functions. For example $\cos(2\cos^{-1} x) = 2x^2-1$ (for $x$ in a suitable range) but the version on the right runs much faster.
The catch is remembering all those trig formulae. It'd be nice to make the compiler do all the work. A solution is to use complex numbers. Instead of storing $\theta$ we store $(\cos\theta,\sin\theta)$. We can add angles by using complex multiplication, multiply angles by integers and rational numbers using powers and roots and so on. As long as you don't actually need the numerical value of the angle in radians you need never use trig functions. Obviously there comes a point where the work of doing operations on complex numbers may outweigh the saving of avoiding trig. But often in real code the complex number route is faster.
(Of course it's analogous to using quaternions for rotations in 3D. I guess it's somewhat in the spirit of rational trigonometry except I think it's easier to work with complex numbers.)
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How about how the holomorphicity of a function $f(z)=x+yi$ relates to, e.g., the curl of the vector $(x,y)\in\mathbb{R}^2$? This relates nicely to why we can solve problems in two dimensional electromagnetism (or 3d with the right symmetries) very nicely using "conformal methods." It would be very easy to start a course with something like this to motivate complex analytic methods.
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Thanks, Jeremy! I'll definitely do the search, the magic word "the method of conformal mapping" is really important here. – Wadim Zudilin Jul 1 '10 at 11:45
I think most older Russian textbooks on complex analysis (e.g. Lavrentiev and Shabat or Markushevich) had examples from 2D hydrodynamics (Euler-D'Alambert equations $\iff$ Cauchy-Riemann equations). Also, of course, the Zhukovsky function and airwing profile. They serve more as applications of theory than motivations, since nontrivial mathematical work is required to get there. – Victor Protsak Jul 2 '10 at 2:04
I never took a precalculus class because every identity I've ever needed involving sines and cosines I could derive by evaluating a complex exponential in two different ways. Perhaps you could tell them that if they ever forget a trig identity, they can rederive it using this method?
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In answer to
"Why do we need to study numbers which do not belong to the real world?"
you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe.
I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check. (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes. After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.)
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Perhaps you are referring to Feynman's book QED? – S. Carnahan Jul 2 '10 at 4:41
Tristan Needham's book Visual Complex Analysis is full of these sorts of gems. One of my favorites is the proof using complex numbers that if you put squares on the sides of a quadralateral, the lines connecting opposite centers will be perpendicular and of the same length. After proving this with complex numbers, he outlines a proof without them that is much longer.
The relevant pages are on Google books:
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This is not exactly an answer to the question, but it is the simplest thing I know to help students appreciate complex numbers. (I got the idea somewhere else, but I forgot exactly where.)
It's something even much younger students can appreciate. Recall that on the real number line, multiplying a number by -1 "flips" it, that is, it rotates the point 180 degrees about the origin. Introduce the imaginary number line (perpendicular to the real number line) then introduce multiplication by i as a rotation by 90 degrees. I think most students would appreciate operations on complex numbers if they visualize them as movements of points on the complex plane.
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I now remember where I got the idea:… – Joel Reyes Noche Feb 3 '11 at 13:48
Try this: compare the problems of finding the points equidistant in the plane from (-1, 0) and (1, 0), which is easy, with finding the points at twice the distance from (-1, 0) that they are from (1, 0). The idea that "real" concepts are the only ones of use in the "real world" is of course a fallacy. I suppose it is more than a century since electrical engineers admitted that complex numbers are useful.
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I do see an undeniable benefit. If you are later asked about it in $\mathbb{R}^3$ then you use vectors and dot product. The historical way would have been to use quaternions; indeed, this is how the notion of dot product crystallized in the work of Gibbs, and more relevantly for your EE students, Oliver Heaviside. – Victor Protsak Jul 2 '10 at 1:26
Maybe artificial, but a nice example (I think) demonstrating analytic continuation (NOT just the usual $\mathrm{Re}(e^{i \theta})$ method!) I don't know any reasonable way of doing this by real methods.
As a fun exercise, calculate $$ I(\omega) = \int_0^\infty e^{-x} \cos (\omega x) \frac{dx}{\sqrt{x}}, \qquad \omega \in \mathbb{R} $$ from the real part of $F(1+i \omega)$, where $$ F(k) = \int_0^\infty e^{-kx} \frac{dx}{\sqrt{x}}, \qquad \mathrm{Re}(k)>0 $$ (which is easily obtained for $k>0$ by a real substitution) and using analytic continuation to justify the same formula with $k=1+i \omega$.
You need care with square roots, branch cuts, etc.; but this can be avoided by considering $F(k)^2$, $I(\omega)^2$.
Of course all the standard integrals provide endless fun examples! (But the books don't have many requiring genuine analytic continuation like this!)
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I rather suspect analytic continuation is a conceptual step above what the class in question coud cope with... – Yemon Choi Jul 8 '10 at 1:22
Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression formula, you can expand f(x)=1-x^2+... . This series converges for |x|<1 but diverges for all other x. Why this is so? The function looks nice and smooth everywhere on the real line.
This example is taken from the Introduction of the textbook by B. V. Shabat.
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From the perspective of complex analysis, the theory of Fourier series has a very natural explanation. I take it that the students had seen Fourier series first, of course. I had mentioned this elsewhere too. I hope the students also know about Taylor theorem and Taylor series. Then one could talk also of the Laurent series in concrete terms, and argue that the Fourier series is studied most naturally in this setting.
First, instead of cos and sin, define the Fourier series using complex exponential. Then, let $f(z)$ be a complex analytic function in the complex plane, with period $1$.
Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i \infty$ corresponds to $q = 0$. The Fourier expansion of $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$.
Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis.
I am am electrical engineer; I have an idea what they all study; so I can safely override any objections that this won't be accessible to electrical engineers. Moreover, the above will reduce their surprise later in their studies when they study signal processing and wavelet analysis.
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This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. Most are probably already familiar with its content, but it might be useful to state it again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late.
If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$
It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers.
If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for.
Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.
Reference: Dummit & Foote: Abstract Algebra, 13.1
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This answer is an expansion of the answer of Yuri Bakhtin.
Here is a kind of mime show.
Silently write the formulas for $\cos(2x)$ and $\sin(2x)$ lined up on the board, something like this: $$\cos(2x) = \cos^2(x) \hphantom{+ 2 \cos(x) \sin(x)} - \sin^2(x) $$ $$\sin(2x) = \hphantom{\cos^2(x)} + 2 \cos(x) \sin(x) \hphantom{- \sin^2(x)} $$
Do the same for the formulas for $\cos(3x)$ and $\sin(3x)$, and however far you want to go: $$\cos(3x) = \cos^3(x) \hphantom{+ 3 \cos^2(x) \sin(x)} - 3 \cos(x) \sin^2(x) \hphantom{- \sin^3(x)} $$ $$\sin(3x) = \hphantom{\cos^3(x)} + 3 \cos^2(x) \sin(x) \hphantom{- 3 \cos(x) \sin^2(x)} - \sin^3(x) $$
Maybe then let out a loud noise like "hmmmmmmmmm... I recognize those numbers..."
Then, on a parallel board, write out Pascal's triangle, and parallel to that write the application of Pascal's triangle to the binomial expansions $(x+y)^n$. Make some more puzzling sounds regarding those pesky plus and minus signs.
Then maybe it's time to actually say something: "Eureka! We can tie this all together by use of an imaginary number $i = \sqrt{-1}$". Then write out the binomial expansion of $$(\cos(x) + i\,\sin(x))^n $$ break it into its real and imaginary parts, and demonstrate equality with $$\cos(nx) + i\, \sin(nx). $$
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I always like to use complex dynamics to illustrate that complex numbers are "real" (i.e., they are not just a useful abstract concept, but in fact something that very much exist, and closing our eyes to them would leave us not only devoid of useful tools, but also of a deeper understanding of phenomena involving real numbers.) Of course I am a complex dynamicist so I am particularly partial to this approach!
Start with the study of the logistic map $x\mapsto \lambda x(1-x)$ as a dynamical system (easy to motivate e.g. as a simple model of population dynamics). Do some experiments that illustrate some of the behaviour in this family (using e.g. web diagrams and the Feigenbaum diagram), such as:
• The period-doubling bifurcation
• The appearance of periodic points of various periods
• The occurrence of "period windows" everywhere in the Feigenbaum diagram.
Then let x and lambda be complex, and investigate the structure both in the dynamical and parameter plane, observing
• The occurence of beautiful and very "natural"-looking objects in the form of Julia sets and the (double) Mandelbrot set;
• The explanation of period-doubling as the collision of a real fixed point with a complex point of period 2, and the transition points occuring as points of tangency between interior components of the Mandelbrot set;
• Period windows corresponding to little copies of the Mandelbrot set.
Finally, mention that density of period windows in the Feigenbaum diagram - a purely real result, established only in the mid-1990s - could never have been achieved without complex methods.
There are two downsides to this approach: * It requires a certain investment of time; even if done on a superficial level (as I sometimes do in popular maths lectures for an interested general audience) it requires the better part of a lecture * It is likely to appeal more to those that are mathematically minded than engineers who could be more impressed by useful tools for calculations such as those mentioned elsewhere on this thread.
However, I personally think there are few demonstrations of the "reality" of the complex numbers that are more striking. In fact, I have sometimes toyed with the idea of writing an introductory text on complex numbers which uses this as a primary motivation.
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Having been through the relevant mathematical mill, I subsequently engaged with Geometric Algebra (a Clifford Algebra interpreted strictly geometrically).
Once I understood that the square of a unit bivector is -1 and then how rotors worked, all my (conceptual) difficulties evaporated.
I have never had a reason to use (pure) complex numbers since and I suspect that most engineering/physics/computing types would avoid them if they were able.
Likely you have the above group mixed together with pure mathematicians that feel quite at home with the non-physical aspects of complex numbers and wouldn't dream of asking such an impertinent question:-)
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I don't think you can answer this in a single class. The best answer I can come up with is to show how complicated calculus problems can be solved easily using complex analysis.
As an example, I bet most of your students hated solving the problem $\int e^{-x}cos(x) dx.$ Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $\int e^{-x}cos(x) dx=\Re \int e^{-x(1-i)}dx.$ They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world.
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An interesting example of usage of complex numbers can be found in (Michael Eastwood, Roger Penrose, Drawing with Complex Numbers).
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Is it too abstract to motivate complex numbers in terms of the equations we can solve depending on whether we choose to work in ${\mathbb N, \mathbb Z, \mathbb Q, \mathbb R, \mathbb C}$? The famous "John and Betty" ( takes such an approach.
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As an example to demonstrate the usefulness of complex analysis in mechanics (which may seem counterintuitive to engineering students, since mechanics is introduced on the reals), one may consider the simple problem of the one dimensional harmonic oscillator, whose Hamiltonian equations of motion are diagonalized in the complex representation, equivalently one needs to integrate a single (holomorphic) first order ODE instead of a single second order or two first order ODEs.
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Motivating complex analysis
The physics aspect of motivation should be the strongest for engineering students. No complex numbers, no quantum mechanics, no solid state physics, no lasers, no electrical or electronic engineering (starting with impedance), no radio, TV, acoustics, no good simple way of understanding of the mechanical analogues of RLC circuits, resonance, etc., etc.
Then the "mystery" of it all. Complex numbers as the consequence of roots, square, cubic, etc., unfolding until one gets the complex plane, radii of convergence, poles of stability, all everyday engineering. Then the romance of it all, the "self secret knowledge", discovered over hundreds of years, a new language which even helps our thinking in general. Then the wider view of say Smale/Hirsch on higher dimensional differential equations, chaos etc. They should see the point pretty quickly. This is a narrow door, almost accidentally discovered, through which we see and understand entire new realms, which have become our best current, albeit imperfect,descriptions of how to understand and manipulate a kind of "inner essence of what is" for practical human ends, i.e. engineering. (True, a little over the top, but then pedagogical and motivational).
For them to say that they just want to learn a few computational tricks is a little like a student saying, "don't teach me about fire, just about lighting matches". It's up to them I suppose, but they will always be limited.
There might be some computer software engineer who needs a little more, but then I suppose there is also modern combinatorics. :-)
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Rae §2.1, B&J §3.1, B&M §5.1 2.1 An equation for the matter waves: the time-dependent Schrődinger equation*** Classical wave equation (in one dimension):
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1 Rae §2.1, B&J §3.1, B&M §5.1 2.1 An equation for the matter waves: the time-dependent Schrődinger equation*** Classical wave equation (in one dimension): e.g. Transverse waves on a string: x Can we use this to describe the matter waves in free space? 2222 Quantum Physics
2 An equation for the matter waves (2)
Seem to need an equation that involves the first derivative in time, but the second derivative in space (for matter waves in free space) 2222 Quantum Physics
3 An equation for the matter waves (3)
For particle with potential energy V(x,t), need to modify the relationship between energy and momentum: Total energy = kinetic energy + potential energy Suggests corresponding modification to Schrődinger equation: Time-dependent Schrődinger equation 2222 Quantum Physics Schrődinger
4 The Schrődinger equation: notes
This was a plausibility argument, not a derivation. We believe the Schrődinger equation to be true not because of this argument, but because its predictions agree with experiment. There are limits to its validity. In this form it applies to A single particle, that is Non-relativistic (i.e. has non-zero rest mass and velocity very much below c) The Schrődinger equation is a partial differential equation in x and t (like classical wave equation) The Schrődinger equation contains the complex number i. Therefore its solutions are essentially complex (unlike classical waves, where the use of complex numbers is just a mathematical convenience) 2222 Quantum Physics
5 The Hamiltonian operator
Can think of the RHS of the Schrődinger equation as a differential operator that represents the energy of the particle. This operator is called the Hamiltonian of the particle, and usually given the symbol Kinetic energy operator Potential energy operator Hence there is an alternative (shorthand) form for time-dependent Schrődinger equation: 2222 Quantum Physics
6 2.2 The significance of the wave function***
Rae §2.1, B&J §2.2, B&M §5.2 2.2 The significance of the wave function*** Ψ is a complex quantity, so what can be its significance for the results of real physical measurements on a system? Remember photons: number of photons per unit volume is proportional to the electromagnetic energy per unit volume, hence to square of electromagnetic field strength. Postulate (Born interpretation): probability of finding particle in a small length δx at position x and time t is equal to Note: |Ψ(x,t)|2 is real, so probability is also real, as required. δx |Ψ|2 Total probability of finding particle between positions a and b is x Born 2222 Quantum Physics a b
7 Example Suppose that at some instant of time a particle’s wavefunction is What is: (a) The probability of finding the particle between x=0.5 and x=0.5001? (b) The probability per unit length of finding the particle at x=0.6? (c) The probability of finding the particle between x=0.0 and x=0.5? 2222 Quantum Physics
8 Normalization Total probability for particle to be anywhere should be one (at any time): Normalization condition Suppose we have a solution to the Schrődinger equation that is not normalized, Then we can Calculate the normalization integral Re-scale the wave function as (This works because any solution to the S.E., multiplied by a constant, remains a solution, because the equation is linear and homogeneous) Alternatively: solution to Schrödinger equation contains an arbitrary constant, which can be fixed by imposing the condition (2.7) 2222 Quantum Physics
9 Normalizing a wavefunction - example
2222 Quantum Physics
10 2.3 Boundary conditions for the wavefunction
Rae §2.3, B&J §3.1 2.3 Boundary conditions for the wavefunction The wavefunction must: 1. Be a continuous and single-valued function of both x and t (in order that the probability density be uniquely defined) 2. Have a continuous first derivative (unless the potential goes to infinity) 3. Have a finite normalization integral. 2222 Quantum Physics
11 2.4 Time-independent Schrődinger equation***
Rae §2.2, B&J §3.5, B&M §5.3 2.4 Time-independent Schrődinger equation*** Suppose potential V(x,t) (and hence force on particle) is independent of time t: RHS involves only variation of Ψ with x (i.e. Hamiltonian operator does not depend on t) LHS involves only variation of Ψ with t Look for a solution in which the time and space dependence of Ψ are separated: Substitute: 2222 Quantum Physics
12 Time-independent Schrődinger equation (contd)
Solving the time equation: The space equation becomes: or Time-independent Schrődinger equation 2222 Quantum Physics
13 Notes In one space dimension, the time-independent Schrődinger equation is an ordinary differential equation (not a partial differential equation) The sign of i in the time evolution is determined by the choice of the sign of i in the time-dependent Schrődinger equation The time-independent Schrődinger equation can be thought of as an eigenvalue equation for the Hamiltonian operator: Operator × function = number × function (Compare Matrix × vector = number × vector) [See 2246] We will consistently use uppercase Ψ(x,t) for the full wavefunction (time-dependent Schrődinger equation), and lowercase ψ(x) for the spatial part of the wavefunction when time and space have been separated (time-independent Schrődinger equation) Probability distribution of particle is now independent of time (“stationary state”): For a stationary state we can use either ψ(x) or Ψ(x,t) to compute probabilities; we will get the same result. 2222 Quantum Physics
14 2.6 SE in three dimensions Rae §3.1, B&J §3.1, B&M §5.1
To apply the Schrődinger equation in the real (three-dimensional) world we keep the same basic structure: BUT Wavefunction and potential energy are now functions of three spatial coordinates: Kinetic energy now involves three components of momentum Interpretation of wavefunction: 2222 Quantum Physics
15 Puzzle The requirement that a plane wave
plus the energy-momentum relationship for free-non-relativistic particles led us to the free-particle Schrődinger equation. Can you use a similar argument to suggest an equation for free relativistic particles, with energy-momentum relationship: 2222 Quantum Physics
16 3.1 A Free Particle Free particle: experiences no forces so potential energy independent of position (take as zero) Linear ODE with constant coefficients so try Time-independent Schrődinger equation: General solution: Combine with time dependence to get full wave function: 2222 Quantum Physics
17 Notes Plane wave is a solution (just as well, since our plausibility argument for the Schrődinger equation was based on this being so) Note signs: Sign of time term (-iωt) is fixed by sign adopted in time-dependent Schrődinger Equation Sign of position term (±ikx) depends on propagation direction of wave There is no restriction on the allowed energies, so there is a continuum of states 2222 Quantum Physics
18 3.2 Infinite Square Well Rae §2.4, B&J §4.5, B&M §5.4 V(x)
Consider a particle confined to a finite length –a<x<a by an infinitely high potential barrier x No solution in barrier region (particle would have infinite potential energy). -a a In well region: Boundary conditions: Continuity of ψ at x=a: Note discontinuity in dψ/dx allowable, since potential is infinite Continuity of ψ at x=-a: 2222 Quantum Physics
19 Infinite square well (2)
Add and subtract these conditions: Even solution: ψ(x)=ψ(-x) Odd solution: ψ(x)=-ψ(-x) Energy: 2222 Quantum Physics
20 Infinite well – normalization and notes
Notes on the solution: Energy quantized to particular values (characteristic of bound-state problems in quantum mechanics, where a particle is localized in a finite region of space. Potential is even under reflection; stationary state wavefunctions may be even or odd (we say they have even or odd parity) Compare notation in 1B23 and in books: 1B23: well extended from x=0 to x=b Rae and B&J: well extends from x=-a to x=+a (as here) B&M: well extends from x=-a/2 to x=+a/2 (with corresponding differences in wavefunction) 2222 Quantum Physics
21 The infinite well and the Uncertainty Principle
Position uncertainty in well: Momentum uncertainty in lowest state from classical argument (agrees with fully quantum mechanical result, as we will see in §4) Compare with Uncertainty Principle: Ground state close to minimum uncertanty 2222 Quantum Physics
22 3.3 Finite square well Rae §2.4, B&J §4.6 V(x) x -a a V0 I II III
Now make the potential well more realistic by making the barriers a finite height V0 Region I: Region II: Region III: 2222 Quantum Physics
23 Finite square well (2) Match value and derivative of wavefunction at region boundaries: Match ψ: Match dψ/dx: Add and subtract: 2222 Quantum Physics
24 Finite square well (3) Divide equations:
Must be satisfied simultaneously: Cannot be solved algebraically. Convenient form for graphical solution: 2222 Quantum Physics
25 Graphical solution for finite well
k0=3, a=1 2222 Quantum Physics
26 Notes Penetration of particle into “forbidden” region where V>E (particle cannot exist here classically) Number of bound states depends on depth of potential well, but there is always at least one (even) state Potential is even function, wavefunctions may be even or odd (we say they have even or odd parity) Limit as V0→∞: 2222 Quantum Physics
27 Example: the quantum well
Quantum well is a “sandwich” made of two different semiconductors in which the energy of the electrons is different, and whose atomic spacings are so similar that they can be grown together without an appreciable density of defects: Material A (e.g. AlGaAs) Material B (e.g. GaAs) Electron potential energy Position Now used in many electronic devices (some transistors, diodes, solid-state lasers) Kroemer Esaki 2222 Quantum Physics
28 3.4 Particle Flux Rae §9.1; B&M §5.2, B&J §3.2
In order to analyse problems involving scattering of free particles, need to understand normalization of free-particle plane-wave solutions. Conclude that if we try to normalize so that will get A=0. This problem is related to Uncertainty Principle: Position completely undefined; single particle can be anywhere from -∞ to ∞, so probability of finding it in any finite region is zero Momentum is completely defined 2222 Quantum Physics
29 Particle Flux (2) x a b More generally: what is rate of change of probability that a particle exists in some region (say, between x=a and x=b)? Use time-dependent Schrődinger equation: 2222 Quantum Physics
30 Particle Flux (3) Integrate by parts: Flux entering at x=a -
Flux leaving at x=b Interpretation: x a b Note: a wavefunction that is real carries no current 2222 Quantum Physics Note: for a stationary state can use either ψ(x) or Ψ(x,t)
31 Particle Flux (4) Sanity check: apply to free-particle plane wave.
Makes sense: # particles passing x per unit time = # particles per unit length × velocity Wavefunction describes a “beam” of particles. 2222 Quantum Physics
32 3.5 Potential Step Rae §9.1; B&J §4.3 V(x)
Consider a potential which rises suddenly at x=0: Case 1 V0 x Boundary condition: particles only incident from left Case 1: E<V0 (below step) x<0 x>0 2222 Quantum Physics
33 Potential Step (2) Continuity of ψ at x=0:
Solve for reflection and transmission: 2222 Quantum Physics
34 Transmission and reflection coefficients
2222 Quantum Physics
35 Potential Step (3) Case 2: E>V0 (above step)
Solution for x>0 is now Matching conditions: Transmission and reflection coefficients: 2222 Quantum Physics
36 Summary of transmission through potential step
Notes: Some penetration of particles into forbidden region even for energies below step height (case 1, E<V0); No transmitted particle flux, 100% reflection (case 1, E<V0); Reflection probability does not fall to zero for energies above barrier (case 2, E>V0). Contrast classical expectations: 100% reflection for E<V0, with no penetration into barrier; 100% transmission for E>V0 2222 Quantum Physics
37 3.6 Rectangular Potential Barrier
Rae §2.5; B&J §4.4; B&M §5.9 3.6 Rectangular Potential Barrier V(x) III I II Now consider a potential barrier of finite thickness: V0 x a Boundary condition: particles only incident from left Region I: Region II: Region III: 2222 Quantum Physics
38 Rectangular Barrier (2)
Match value and derivative of wavefunction at region boundaries: Match ψ: Match dψ/dx: Eliminate wavefunction in central region: 2222 Quantum Physics
39 Rectangular Barrier (3)
Transmission and reflection coefficients: For very thick or high barrier: Non-zero transmission (“tunnelling”) through classically forbidden barrier region: 2222 Quantum Physics
40 Examples of tunnelling
Tunnelling occurs in many situations in physics and astronomy: 1. Nuclear fusion (in stars and fusion reactors) 2. Alpha-decay Distance x of α-particle from nucleus V Initial α-particle energy V Coulomb interaction (repulsive) Incident particles Internuclear distance x Strong nuclear force (attractive) V Distance x of electron from surface Work function W Material 3. Field emission of electrons from surfaces (e.g. in plasma displays) Vacuum 2222 Quantum Physics
41 3.7 Simple Harmonic Oscillator
Rae §2.6; B&M §5.5; B&J §4.7 3.7 Simple Harmonic Oscillator Mass m Example: particle on a spring, Hooke’s law restoring force with spring constant k: x Time-independent Schrődinger equation: Problem: still a linear differential equation but coefficients are not constant. Simplify: change variable to 2222 Quantum Physics
42 Simple Harmonic Oscillator (2)
Asymptotic solution in the limit of very large y: Check: Equation for H: 2222 Quantum Physics
43 Simple Harmonic Oscillator (3)
Must solve this ODE by the power-series method (Frobenius method); this is done as an example in 2246. We find: The series for H(y) must terminate in order to obtain a normalisable solution Can make this happen after n terms for either even or odd terms in series (but not both) by choosing Hn is known as the nth Hermite polynomial. Label resulting functions of H by the values of n that we choose. 2222 Quantum Physics
44 The Hermite polynomials
For reference, first few Hermite polynomials are: NOTE: Hn contains yn as the highest power. Each H is either an odd or an even function, according to whether n is even or odd. 2222 Quantum Physics
45 Simple Harmonic Oscillator (4)
Transforming back to the original variable x, the wavefunction becomes: Probability per unit length of finding the particle is: Compare classical result: probability of finding particle in a length δx is proportional to the time δt spent in that region: For a classical particle with total energy E, velocity is given by 2222 Quantum Physics
46 Notes “Zero-point energy”: “Quanta” of energy: Even and odd solutions
Applies to any simple harmonic oscillator, including Molecular vibrations Vibrations in a solid (hence phonons) Electromagnetic field modes (hence photons), even though this field does not obey exactly the same Schrődinger equation You will do another, more elegant, solution method (no series or Hermite polynomials!) next year For high-energy states, probability density peaks at classical turning points (correspondence principle) 2222 Quantum Physics
47 4 Postulates of QM This section puts quantum mechanics onto a more formal mathematical footing by specifying those postulates of the theory which cannot be derived from classical physics. Main ingredients: The wave function (to represent the state of the system); Hermitian operators (to represent observable quantities); A recipe for identifying the operator associated with a given observable; A description of the measurement process, and for predicting the distribution of outcomes of a measurement; A prescription for evolving the wavefunction in time (the time-dependent Schrődinger equation) 2222 Quantum Physics
48 4.1 The wave function Postulate 4.1: There exists a wavefunction Ψ that is a continuous, square-integrable, single-valued function of the coordinates of all the particles and of time, and from which all possible predictions about the physical properties of the system can be obtained. Examples of the meaning of “The coordinates of all the particles”: For a single particle moving in one dimension: For a single particle moving in three dimensions: For two particles moving in three dimensions: The modulus squared of Ψ for any value of the coordinates is the probability density (per unit length, or volume) that the system is found with that particular coordinate value (Born interpretation). 2222 Quantum Physics
49 4.2 Observables and operators
Postulate 4.2.1: to each observable quantity is associated a linear, Hermitian operator (LHO). An operator is linear if and only if Examples: which of the operators defined by the following equations are linear? Note: the operators involved may or may not be differential operators (i.e. may or may not involve differentiating the wavefunction). 2222 Quantum Physics
50 Hermitian operators An operator O is Hermitian if and only if:
for all functions f,g vanishing at infinity. Compare the definition of a Hermitian matrix M: Analogous if we identify a matrix element with an integral: (see 3226 course for more detail…) 2222 Quantum Physics
51 Hermitian operators: examples
2222 Quantum Physics
52 Eigenvectors and eigenfunctions
Postulate 4.2.2: the eigenvalues of the operator represent the possible results of carrying out a measurement of the corresponding quantity. Definition of an eigenvalue for a general linear operator: Compare definition of an eigenvalue of a matrix: Example: the time-independent Schrődinger equation: 2222 Quantum Physics
53 Important fact: The eigenvalues of a Hermitian operator are real (like the eigenvalues of a Hermitian matrix). Proof: Postulate 4.2.3: immediately after making a measurement, the wavefunction is identical to an eigenfunction of the operator corresponding to the eigenvalue just obtained as the measurement result. Ensures that we get the same result if we immediately re-measure the same quantity. 2222 Quantum Physics
54 4.3 Identifying the operators
Postulate 4.3: the operators representing the position and momentum of a particle are (one dimension) (three dimensions) Other operators may be obtained from the corresponding classical quantities by making these replacements. Examples: The Hamiltonian (representing the total energy as a function of the coordinates and momenta) Angular momentum: 2222 Quantum Physics
55 Eigenfunctions of momentum
The momentum operator is Hermitian, as required: Its eigenfunctions are plane waves: 2222 Quantum Physics
56 Orthogonality of eigenfunctions
The eigenfunctions of a Hermitian operator belonging to different eigenvalues are orthogonal. If then Proof: 2222 Quantum Physics
57 Orthonormality of eigenfunctions
What if two eigenfunctions have the same eigenvalue? (In this case the eigenvalue is said to be degenerate.) Any linear combination of these eigenfunctions is also an eigenfunction with the same eigenvalue: So we are free to choose as the eigenfunctions two linear combinations that are orthogonal. If the eigenfunctions are all orthogonal and normalized, they are said to be orthonormal. 2222 Quantum Physics
58 Orthonormality of eigenfunctions: example
Consider the solutions of the time-independent Schrődinger equation (energy eigenfunctions) for an infinite square well: We chose the constants so that normalization is correct: 2222 Quantum Physics
59 Complete sets of functions
The eigenfunctions φn of a Hermitian operator form a complete set, meaning that any other function satisfying the same boundary conditions can be expanded as If the eigenfunctions are chosen to be orthonormal, the coefficients an can be determined as follows: We will see the significance of such expansions when we come to look at the measurement process. 2222 Quantum Physics
60 Normalization and expansions in complete sets
The condition for normalizing the wavefunction is now If the eigenfunctions φn are orthonormal, this becomes Natural interpretation: the probability of finding the system in the state φn(x) (as opposed to any of the other eigenfunctions) is 2222 Quantum Physics
61 Expansion in complete sets: example
2222 Quantum Physics
62 4.4 Eigenfunctions and measurement
Postulate 4.4: suppose a measurement of the quantity Q is made, and that the (normalized) wavefunction can be expanded in terms of the (normalized) eigenfunctions φn of the corresponding operator as Then the probability of obtaining the corresponding eigenvalue qn as the measurement result is Corollary: if a system is definitely in eigenstate φn, the result measuring Q is definitely the corresponding eigenvalue qn. What is the meaning of these “probabilities” in discussing the properties of a single system? Still a matter for debate, but usual interpretation is that the probability of a particular result determines the frequency of occurrence of that result in measurements on an ensemble of similar systems. 2222 Quantum Physics
63 Commutators In general operators do not commute: that is to say, the order in which we allow operators to act on functions matters: For example, for position and momentum operators: We define the commutator as the difference between the two orderings: Two operators commute only if their commutator is zero. So, for position and momentum: 2222 Quantum Physics
64 Compatible operators Two observables are compatible if their operators share the same eigenfunctions (but not necessarily the same eigenvalues). Consequence: two compatible observables can have precisely-defined values simultaneously. Measure observable R, definitely obtain result rm (the corresponding eigenvalue of R) Measure observable Q, obtain result qm (an eigenvalue of Q) Re-measure Q, definitely obtain result qm once again Wavefunction of system is corresponding eigenfunction φm Wavefunction of system is still corresponding eigenfunction φm Compatible operators commute with one another: Expansion in terms of joint eigenfunctions of both operators Can also show the converse: any two commuting operators are compatible. 2222 Quantum Physics
65 Example: measurement of position
2222 Quantum Physics
66 Example: measurement of position (2)
2222 Quantum Physics
67 Expectation values The average (mean) value of measurements of the quantity Q is therefore the sum of the possible measurement results times the corresponding probabilities: We can also write this as: 2222 Quantum Physics
68 4.5 Evolution of the system
Postulate 4.5: Between measurements (i.e. when it is not disturbed by external influences) the wave-function evolves with time according to the time-dependent Schrődinger equation. Hamiltonian operator. This is a linear, homogeneous differential equation, so the linear combination of any two solutions is also a solution: the superposition principle. 2222 Quantum Physics
69 Calculating time dependence using expansion in energy eigenfunctions
Suppose the Hamiltonian is time-independent. In that case we know that solutions of the time-dependent Schrődinger equation exist in the form: where the wavefunctions ψ(x) and the energy E correspond to one solution of the time-independent Schrődinger equation: We know that all the functions ψn together form a complete set, so we can expand Hence we can find the complete time dependence (superposition principle): 2222 Quantum Physics
70 Time-dependent behaviour: example
Suppose the state of a particle in an infinite square well at time t=0 is a `superposition’ of the n=1 and n=2 states Wave function at a subsequent time t Probability density 2222 Quantum Physics
71 Rate of change of expectation value
Consider the rate of change of the expectation value of a quantity Q: 2222 Quantum Physics
72 Example 1: Conservation of probability
Rate of change of total probability that the particle may be found at any point: Total probability is the “expectation value” of the operator 1. Total probability conserved (related to existence of a well defined probability flux – see §3.4) 2222 Quantum Physics
73 Example 2: Conservation of energy
Consider the rate of change of the mean energy: Even although the energy of a system may be uncertain (in the sense that measurements of the energy made on many copies of the system may be give different results) the average energy is always conserved with time. 2222 Quantum Physics
74 5.1 Angular momentum operators
Reading: Rae Chapter 5; B&J§§6.1,6.3; B&M§§ 5.1 Angular momentum operators Angular momentum is a very important quantity in three-dimensional problems involving a central force (one that is always directed towards or away from a central point). In that case it is classically a conserved quantity: Central point r F The origin of r is the same central point towards/away from which the force is directed. We can write down a quantum-mechanical operator for it by applying our usual rules: Individual components: 2222 Quantum Physics
75 5.2 Commutation relations***
The different components of angular momentum do not commute with one another. By similar arguments get the cyclic permutations: 2222 Quantum Physics
76 Commutation relations (2)
The different components of L do not commute with one another, but they do commute with the (squared) magnitude of the angular momentum vector: Note a useful formula: Important consequence: we cannot find simultaneous eigenfunctions of all three components. But we can find simultaneous eigenfunctions of one component (conventionally the z component) and L2 2222 Quantum Physics
77 5.3 Angular momentum in spherical polar coordinates
On this slide, hats refer to unit vectors, not operators. Spherical polar coordinates are the natural coordinate system in which to describe angular momentum. In these coordinates, z θ y r (see 2246) φ So the full (vector) angular momentum operator can be written x To find z-component, note that unit vector k in z-direction satisfies 2222 Quantum Physics
78 L2 in spherical polar coordinates
On this slide, hats refer to unit vectors, not operators. Depends only on angular behaviour of wavefunction. Closely related to angular part of Laplacian (see 2246 and Section 6). 2222 Quantum Physics
79 5.4 Eigenvalues and eigenfunctions
Look for simultaneous eigenfunctions of L2 and one component of L (conventional to choose Lz) Eigenvalues and eigenfunctions of Lz: Physical boundary condition: wave-function must be single-valued Quantization of angular momentum about z-axis (compare Bohr model) 2222 Quantum Physics
80 Eigenvalues and eigenfunctions (2)
Now look for eigenfunctions of L2, in the form (ensures solutions remain eigenfunctions of Lz, as we want) Eigenvalue condition becomes 2222 Quantum Physics
81 The Legendre equation Make the substitution
This is exactly the Legendre equation, solved in 2246 using the Frobenius method. 2222 Quantum Physics
82 Legendre polynomials and associated Legendre functions
In order for solutions to exist that remain finite at μ=±1 (i.e. at θ=0 and θ=π) we require that the eigenvalue satisfies (like SHO, where we found restrictions on energy eigenvalue in order to produce normalizable solutions) The finite solutions are then the associated Legendre functions, which can be written in terms of the Legendre polynomials: where m is an integer constrained to lie between –l and +l. Legendre polynomials: 2222 Quantum Physics
83 Spherical harmonics The full eigenfunctions can also be written as spherical harmonics: Because they are eigenfunctions of Hermitian operators with different eigenvalues, they are automatically orthogonal when integrated over all angles (i.e. over the surface of the unit sphere). The constants C are conventionally defined so the spherical harmonics obey the following important normalization condition: First few examples (see also 2246): 2222 Quantum Physics
84 Shapes of the spherical harmonics
z x y Imaginary Real To read plots: distance from origin corresponds to magnitude (modulus) of plotted quantity; colour corresponds to phase (argument). 2222 Quantum Physics (Images from
85 Shapes of spherical harmonics (2)
86 5.5 The vector model for angular momentum***
To summarize: l is known as the principal angular momentum quantum number: determines the magnitude of the angular momentum m is known as the magnetic quantum number: determines the component of angular momentum along a chosen axis (the z-axis) These states do not correspond to well-defined values of Lx and Ly, since these operators do not commute with Lz. Semiclassical picture: each solution corresponds to a cone of angular momentum vectors, all with the same magnitude and the same z-component. 2222 Quantum Physics
87 The vector model (2) Lz Example: l=2 Ly L
Magnitude of angular momentum is Component of angular momentum in z direction can be Lx 2222 Quantum Physics
88 6.1 The three-dimensional square well
Reading: Rae §3.2, B&J §7.4; B&M §5.11 6.1 The three-dimensional square well z Consider a particle which is free to move in three dimensions everywhere within a cubic box, which extends from –a to +a in each direction. The particle is prevented from leaving the box by infinitely high potential barriers. y x Time-independent Schrödinger equation within the box is free-particle like: V(x) -a a Separation of variables: take x, or y, or z with boundary conditions 2222 Quantum Physics
89 Three-dimensional square well (2)
Substitute in Schrödinger equation: Divide by XYZ: Three effective one-dimensional Schrödinge equations. 2222 Quantum Physics
90 Three-dimensional square well (3)
Wavefunctions and energy eigenvalues known from solution to one-dimensional square well (see §3.2). Total energy is 2222 Quantum Physics
91 6.2 The Hamiltonian for a hydrogenic atom***
Reading: Rae §§ , B&M Chapter 7, B&J §7.2 and §7.5 6.2 The Hamiltonian for a hydrogenic atom*** -e For a hydrogenic atom or ion having nuclear charge +Ze and a single electron, the Hamiltonian is r Note spherical symmetry – potential depends only on r +Ze Note: for greater accuracy we should use the reduced mass corresponding to the relative motion of the electron and the nucleus (since nucleus does not remain precisely fixed – see 1B2x): The natural coordinate system to use is spherical polar coordinates. In this case the Laplacian operator becomes (see 2246): This means that the angular momentum about any axis, and also the total angular momentum, are conserved quantities: they commute with the Hamiltonian, and can have well-defined values in the energy eigenfunctions of the system. 2222 Quantum Physics
92 6.3 Separating the variables
Write the time-independent Schrődinger equation as: Now look for solutions in the form Substituting into the Schrődinger equation: 2222 Quantum Physics
93 The angular equation We recognise that the angular equation is simply the eigenvalue condition for the total angular momentum operator L2: This means we already know the corresponding eigenvalues and eigenfunctions (see §5): Note: all this would work for any spherically-symmetric potential V(r), not just for the Coulomb potential. 2222 Quantum Physics
94 6.4 Solving the radial equation
Now the radial part of the Schrődinger equation becomes: Note that this depends on l, but not on m: it therefore involves the magnitude of the angular momentum, but not its orientation. Define a new unknown function χ by: 2222 Quantum Physics
95 The effective potential
This corresponds to one-dimensional motion with the effective potential V(r) First term: Second term: r 2222 Quantum Physics
96 Atomic units*** Atomic units: there are a lot of physical constants in these expressions. It makes atomic problems much more straightforward to adopt a system of units in which as many as possible of these constants are one. In atomic units we set: In this unit system, the radial equation becomes 2222 Quantum Physics
97 Solution near the nucleus (small r)
For small values of r the second derivative and centrifugal terms dominate over the others. Try a solution to the differential equation in this limit as We want a solution such that R(r) remains finite as r→0, so take 2222 Quantum Physics
98 Asymptotic solution (large r)
Now consider the radial equation at very large distances from the nucleus, when both terms in the effective potential can be neglected. We are looking for bound states of the atom, where the electron does not have enough energy to escape to infinity: Inspired by this, let us rewrite the solution in terms of yet another unknown function, F(r): 2222 Quantum Physics
99 Differential equation for F
Can obtain a corresponding differential equation for F: This equation is solved in 2246, using the Frobenius (power-series) method. The indicial equation gives 2222 Quantum Physics
100 Properties of the series solution
If the full series found in 2246 is allowed to continue up to an arbitrarily large number of terms, the overall solution behaves like (not normalizable) Hence the series must terminate after a finite number of terms. This happens only if So the energy is Note that once we have chosen n, the energy is independent of both m (a feature of all spherically symmetric systems, and hence of all atoms) and l (a special feature of the Coulomb potential, and hence just of hydrogenic atoms). n is known as the principal quantum number. It defines the “shell structure” of the atom. 2222 Quantum Physics
101 6.5 The hydrogen energy spectrum and wavefunctions***
Each solution of the time-independent Schrődinger equation is defined by the three quantum numbers n,l,m For each value of n=1,2,… we have a definite energy: For each value of n, we can have n possible values of the total angular momentum quantum number l: l=0,1,2,…,n-1 -1 l=0 l=1 l=2 l=3 For each value of l and n we can have 2l+1 values of the magnetic quantum number m: Traditional nomenclature: l=0: s states (from “sharp” spectral lines) l=1: p states (“principal”) l=2: d states (“diffuse”) l=3: f states (“fine”) …and so on alphabetically (g,h,i… etc) The total number of states (statistical weight) associated with a given energy En is therefore 2222 Quantum Physics
102 The radial wavefunctions
Radial wavefunctions Rnl depend on principal quantum number n and angular momentum quantum number l (but not on m) Full wavefunctions are: Normalization chosen so that: Note: Probability of finding electron between radius r and r+dr is Only s states (l=0) are finite at the origin. Radial functions have (n-l-1) zeros. 2222 Quantum Physics
103 Comparison with Bohr model***
Quantum mechanics Angular momentum (about any axis) shown to be quantized in units of Planck’s constant: Angular momentum (about any axis) assumed to be quantized in units of Planck’s constant: Electron otherwise moves according to classical mechanics and has a single well-defined orbit with radius Electron wavefunction spread over all radii. Can show that the quantum mechanical expectation value of the quantity 1/r satisfies Energy quantized and determined solely by angular momentum: Energy quantized, but is determined solely by principal quantum number, not by angular momentum: 2222 Quantum Physics
104 6.6 The remaining approximations
This is still not an exact treatment of a real H atom, because we have made several approximations. We have neglected the motion of the nucleus. To fix this we would need to replace me by the reduced mass μ (see slide 1). We have used a non-relativistic treatment of the electron and in particular have neglected its spin (see §7). Including these effects gives rise to “fine structure” (from the interaction of the electron’s orbital motion with its spin), and “hyperfine structure” (from the interaction of the electron’s spin with the spin of the nucleus) We have neglected the fact that the electromagnetic field acting between the nucleus and the electron is itself a quantum object. This leads to “quantum electrodynamic” corrections, and in particular to a small “Lamb shift” of the energy levels. 2222 Quantum Physics
105 7.1 Atoms in magnetic fields
Reading: Rae Chapter 6; B&J §6.8, B&M Chapter 8 (all go further than 2B22) 7.1 Atoms in magnetic fields Interaction of classically orbiting electron with magnetic field: Orbit behaves like a current loop: μ r v In the presence of a magnetic field B, classical interaction energy is: Corresponding quantum mechanical expression (to a good approximation) involves the angular momentum operator: 2222 Quantum Physics
106 Splitting of atomic energy levels
Suppose field is in the z direction. The Hamiltonian operator is We chose energy eigenfunctions of the original atom that are eigenfunctions of Lz so these same states are also eigenfunctions of the new H. 2222 Quantum Physics
107 Splitting of atomic energy levels (2)
(2l+1) states with same energy: m=-l,…+l (Hence the name “magnetic quantum number” for m.) Predictions: should always get an odd number of levels. An s state (such as the ground state of hydrogen, n=1, l=0, m=0) should not be split. 2222 Quantum Physics
108 7.2 The Stern-Gerlach experiment***
Produce a beam of atoms with a single electron in an s state (e.g. hydrogen, sodium) Study deflection of atoms in inhomogeneous magnetic field. Force on atoms is N Results show two groups of atoms, deflected in opposite directions, with magnetic moments S Consistent neither with classical physics (which would predict a continuous distribution of μ) nor with our quantum mechanics so far (which always predicts an odd number of groups, and just one for an s state). 2222 Quantum Physics Gerlach
109 7.3 The concept of spin*** Try to understand these results by analogy with what we know about the ordinary (“orbital”) angular momentum: must be due to some additional source of angular momentum that does not require motion of the electron. Known as “spin”. Introduce new operators to represent spin, assumed to have same commutation relations as ordinary angular momentum: Corresponding eigenfunctions and eigenvalues: (will see in Y3 that these equations can be derived directly from the commutation relations) 2222 Quantum Physics Goudsmit Uhlenbeck Pauli
110 Spin quantum numbers for an electron
From the Stern-Gerlach experiment, we know that electron spin along a given axis has two possible values. So, choose But we also know from Stern-Gerlach that magnetic moments associated with the two possibilities are Spin angular momentum is twice as “effective” at producing magnetic moment as orbital angular momentum. So, have General interaction with magnetic field: 2222 Quantum Physics
111 A complete set of quantum numbers
Hence the complete set of quantum numbers for the electron in the H atom is: n,l,m,s,ms. Corresponding to a full wavefunction Note that the spin functions χ do not depend on the electron coordinates r,θ,φ; they represent a purely internal degree of freedom. H atom in magnetic field, with spin included: 2222 Quantum Physics
112 7.4 Combining different angular momenta
So, an electron in an atom has two sources of angular momentum: Orbital angular momentum (arising from its motion through the atom) Spin angular momentum (an internal property of its own). To think about the total angular momentum produced by combining the two, use the vector model once again: L S Lz |L-S| Vector addition between orbital angular momentum L (of magnitude L) and spin S (of magnitude S): produces a resulting angular momentum vector J: quantum mechanics says its magnitude lies somewhere between |L-S| and L+S.(in integer steps). S Ly L For a single electron, corresponding `total angular momentum’ quantum numbers are Lx Determines length of resultant angular momentum vector L+S 2222 Quantum Physics Determines orientation
113 Example: the 1s and 2p states of hydrogen
The 1s state: The 2p state: 2222 Quantum Physics
114 Combining angular momenta (2)
The same rules apply to combining other angular momenta, from whatever source. For example for two electrons in an excited state of He atom, one in 1s state and one in 2p state (defines what is called the 1s2p configuration in atomic spectroscopy): First construct combined orbital angular momentum L of both electrons: Then construct combined spin S of both electrons: Hence there are two possible terms (combinations of L and S): …and four levels (possible ways of combining L and S to get different total angular momentum quantum numbers) 2222 Quantum Physics
115 Term notation Spectroscopists use a special notation to describe terms and levels: The first (upper) symbol is a number giving the number of spin states corresponding to the total spin S of the electrons The second (main) symbol is a letter encoding the total orbital angular momentum L of the electrons: S denotes L=0 P denotes L=1 D denotes L=2 (and so on); The final (lower) symbol gives the total angular momentum J obtained from combining the two. Example: terms and levels from previous page would be: 2222 Quantum Physics
116 7.5 Wavepackets and the Uncertainty Principle revisited (belongs in §4 – non-examinable)
Can think of the Uncertainty Principle as arising from the structure of wavepackets. Consider a normalized wavefunction for a particle located somewhere near (but not exactly at) position x0 Probability density: Can also write this as a Fourier transform (see 2246): x (expansion in eigenstates of momentum) k 2222 Quantum Physics
117 Fourier transform of a Gaussian
2222 Quantum Physics
118 Wavepackets and Uncertainty Principle (2)
Mean-squared uncertainty in postion Mean momentum: Mean-squared uncertainty in momentum: In fact, can show that this form of wavepacket (“Gaussian wavepacket”) minimizes the product of Δx and Δp, so: 2222 Quantum Physics
119 Wavepackets and Uncertainty Principle (3)
Summary Three ways of thinking of Uncertainty principle: Arising from the physics of the interaction of different types of measurement apparatus with the system (e.g. in the gamma-ray microscope); Arising from the properties of Fourier transforms (narrower wavepackets need a wider range of wavenumbers in their Fourier transforms); Arising from the fact that x and p are not compatible quantities (do not commute), so they cannot simultaneously have precisely defined values. General result (see third year, or Rae §4.5): 2222 Quantum Physics
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String field theory (in which string theory undergoes "second quantization") seems to reside in the backwaters of discussions of string theory. What does second quantization mean in the context of a theory that doesn't have Feynman diagrams with point-like vertices? What would a creation or annihilation operator do in the context of string field theory? Does string field theory provide a context in which (a) certain quantities are more easily calculated or (b) certain concepts become less opaque?
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Dear Andrew, despite Moshe's expectations, I fully agree with him, but let me say it differently.
In QFT, we're talking about "first quantization" - this is not yet a quantum field theory but either a classical field theory or quantum mechanics for 1 particle. Those two have different interpretations - but a similar description. When it is "second-quantized", we arrive to QFT.
Feynman diagrams in QFT may be derived from "sums over histories" of quantum fields in spacetime; for example, the vertices come from the interaction terms in the Lagrangian, and the propagators arise from Wick contractions of quantum fields. This is the "second-quantized" interpretation of the Feynman diagrams.
There is also a first quantized interpretation. You may literally think that the propagators are amplitudes for an individual particle to get from $x$ to $y$, and the vertices allow you to split or merge particles. You may think in terms of particles instead of fields. In QFT, this is an awkward approach because most particles have spins and it's confusing to write a 1-particle Schrödinger equation for a relativistic spin-one photon, for example.
However, in string theory, spin is derived and the first-quantized interpretation is very natural. So the cylindrical world sheet describes the history of a closed string much like a world line describes the history of a particle. And it's enough to change the topology of the world sheet to get the interactions as well. So in string theory, one may produce the amplitudes "directly" from the first-quantized approach because the changed topology of the world sheet, which we sum over, knows all about multi-particle states and their interactions, too. We say that the interactions are already determined by the behavior of a single string.
Needless to say, like any Feynman diagrams, these sums over topologies are just perturbative in their reach.
Now, you may also write down string theory as a string field theory, in terms of quantized string fields in spacetime. Somewhat non-trivially, an appropriate interaction term - that "knows" about the merging and splitting of strings - may be constructed in terms of a "star-product" (a generalization of noncommutative geometry). In this way, string theory becomes formally equivalent to a quantum field theory with infinitely many fields in spacetime - for every possible internal vibration of the string, there is one string field in spacetime.
It used to be believed that this formalism would tell us much more than the perturbative expansions because, for example, lattice QCD in principle can be used to define the theory completely, beyond perturbative expansions. However, this belief has been showed largely untrue. At least so far.
It's been shown that string field theory indeed offers an equivalent way to calculate all the amplitudes of perturbative string theory - especially for bosonic strings with external open strings (closed strings are possible, and surely appear as internal resonances, but they are awkward to include directly as external states; superstrings are probably possible but require a substantially heavier formalism).
Also, string field theory has been very useful to explicitly verify various conjectures about the tachyon potential in bosonic string theory (or, equivalently, about the fate of unstable D-branes which emerge as classical solutions in string field theory). These investigations, started by Ashoke Sen, led to some nice mathematical identities that had to work - because string theory works in all legitimate descriptions - but that were still surprising from a mathematical viewpoint. But all the physical insights confirmed by string field theory had already been known from more direct calculations in string theory.
So because string field theory is widely believed not to tell us anything really new about physics, only a dozen of string theorists in the world dedicate most of their time to string field theory.
Moshe is surely no exception in thinking that it is not too important to work on SFT. Still, it is conceivable that sometime in the future, a more universal definition of string theory will be a refinement of string field theory we know today. However, it's also possible that this will never occur because it's not true: string field theory seems too tightly connected with a particular spacetime and with particular objects (strings) while we know that the true string theory finds it much easier to switch to another spacetime and other objects by dualities.
Cheers LM
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Thanks, Lubos. So it sounds like anything that can be calculated in string field theory can also be calculated in string theory, and more directly to boot. Fair statement? – Andrew Wallace Jan 19 '11 at 12:28
This is a matter of opinion, all I can state here is my opinion, and I’d fully expect other string theorists will have very different ones.
So, second quantization in field theory is a way to proceed from one particle QM to free field theory whose quantization will in turn give many particles of the type you started with. The point is that this is a way to introduce free (or by generalization, weakly interacting) field theory. It is a process which is perturbative in nature and does not provide us with non-perturbaive information. Nevertheless it is useful, for example it is easier to discuss things like the vacuum structure, off-shell observables, and some other issues which are completely obscured in a first quantized formulation.
Now, in string theory, the original formalism is first quantized, you discuss one string, or a fixed number of strings. It is natural to try to second quantize the theory, but in my view you'd be learning about perturbative string theory only in this process (as a sociological fact, all the non-perturbative knowledge we have about string theory does not connect very naturally to SFT). The formalism does have the advantage of having off-shell observables, like an ordinary quantum field theory, but it is not clear that in theory of gravity (which does not have local observables) this is necessarily a good thing.
This was all true for closed string field theory. There is a bit more motivation and results for open string field theory. Especially, there are some successes in exploring the space of (open string) vacua in this formalism (by discussing the so-called tachyon condensation and brane decay processes).
In this context I can elaborate a bit on what it means to second quantize the theory. You have creation operators that create an open string (in a particular state). You can add interactions (turns out you only need cubic ones) and create Feynman diagrams whose propagators and vertices are extended. Turns out that the set of Feynman diagrams you get is precisely the set of world-sheets you can build in open string theory. When you build loop amplitude, you can see that you have closed strings propagating as well, in intermediate states.
There is hope and some indications that quantization of open SFT will lead in the future to non-perturbative formulation of string theory (including the closed strings), but the state of the art is not very close to this goal, I think.
So, in my view the answer to both your questions is sadly negative, by and large, with some exceptions, but many really smart people continue working on the subject, and I may well be wrong in my expectations. I'd be happy if one of these smart people who has a more positive take will give their own answer here.
(I realize there may be obscure points in my reply, since I don't know your background. If you ask I'll try to explain things better).
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Thanks, Moshe. What is an example of an "off-shell observable" in string theory? – Andrew Wallace Jan 19 '11 at 12:25
An example of and off-shell observable is a correlation function, whose external states (in momentum space) are not necessarily on-shell (meaning they don't necessarily satisfy $p^2=m^2$. An example of on-shell quantity is the S-matrix. In string theory the former do not exist, only the latter do. In SFT correlation functions would exist as well - however in gravity such things are not expected to exist because they are not diffeomorphism invariant. – user566 Jan 19 '11 at 14:42
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c785385c1d219d6e | Antimatter Drives
"Any sufficiently advanced technology is indistinguishable from magic," - Arthur C. Clarke
Antimatter is one of the most promising fuels for future space travel. Mainly because it is the most efficient fuel possible. The following discussion covers how antimatter was predicted before it was even found, how it is created, and how it can be used for Starship propulsion. If you would like a "less-technical" explanation of what antimatter is. Please read my essay, entitled What is Antimatter? Otherwise, continue reading below:
The Schrödinger equation of Quantum Mechanics is a highly useful equation. It only works, however, for slow moving particles (in the same same sense that Newton's laws of motion only work for slow moving objects). As it turns out, Einstein's theory of special relativity can be applied to Quantum Mechanics as well. A Cambridge physicist named Paul Dirac combined special relativity and quantum mechanics to create a relativistic wave equation for an electron. This joining solved one mystery: previously, electrons were observed to be able to have two different energies while in what appeared to be the same quantum state. In fact, however, the electrons were not in the same state. Dirac's equation showed clearly that electrons (actually all particles) have an additional intrinsic property. This property is known as "spin" because it adds angular momentum to the particle (however, it is considered unlikely that the electron is actually spinning in the conventional sense of the word). Dirac showed that this spin is actually a relativistic effect, which is why it was not predicted by the non-relativistic Schrödinger equation. Furthermore, this "spin" could take on only two values for the electron: -1/2 (called "down") and +1/2 (called "up").
Dirac's equation, though, does not simply predict the existence of spin up and spin down particles, it actually predicts four different kinds of electrons: spin up with positive energy, spin down with positive energy, spin up with negative energy, and spin down with negative energy. A negative energy electron would speed up as it lost energy, until it was traveling at the speed of light with an energy of negative infinity. Since particles can jump to lower energy states by emitting photons, an electron in the lowest positive energy state could emit a photon with energy equal to 2mc^2, and jump to the highest negative energy state! The electron should, then, continue on its merry way, emitting photons and dropping ever farther into negative energy, speeding up as it did so. Dirac arrived at the startling conclusion that the world would end in 10 billionths of a second!
Obviously, a tremendous amount of experimental data indicated that the world did, indeed, exist. Therefore, Dirac theorized that the universe was already filled with densely packed electrons (called the "Dirac Sea"). These electrons were all negative energy electrons, but because the universe was uniformly filled with them, they could not directly be observed. Each of these negative energy electrons had the following properties: negative mass, negative energy, negative charge. Because electrons are a kind of particle known as a fermion, they obey the Pauli exclusion principle that states that no two fermions can exist in the same place at the same time in the same quantum state. Because of the Dirac sea, the normal, positive energy electrons could not turn into negative energy electrons, because that quantum state was already filled.
So, what if an extremely high energy photon (such as a gamma ray), with energy equal to 2mc^2, promoted a negative energy electron into a positive energy electron (effectively popping it out of the Dirac Sea). Well, that electron would now exist as a normal, positive energy electron. However, since there was a "hole" in the Dirac sea, that hole would behave as particle as well. Since the hole is the absence of negative energy electron, the hole would have exactly the opposite properties of a negative energy electron. It would have positive mass, positive energy, and positive charge! These particles would be anti-electrons (or positrons). Furthermore, if a normal electron ever encountered a positron (which was actually a hole in the Dirac sea), the electron would emit 2mc^2 energy, and fall into place in the Dirac sea.
A graphical depiction of the Dirac sea is shown in figure 5, the hole in the negative energy states represents the existance of a positron, while the blue ball in the positive energy states represents the existence of an electron:
Figure 5: Dirac sea, showing both an electron and a positron.
In 1932 the existence of antimatter (as predicted by Dirac's theory) was verified when Carl Anderson at CalTech discovered some positrons being produced in cosmic ray induced events. Since then, not only has the existence of antimatter been verified again and again, but small amounts of it are actually produced on a daily basis. In fact, the existance of anti-bosons has been discovered (bozons are dissimilar from fermions, in that they do not obey the Pauli exclusion principle). Since bozons do not obey the Pauli exclusion principle, Dirac's sea doesn't work for them. Independently, Richard Feynman and Ernst Carl Stückelberg discovered a way to solve the negative-energy problem, without using a Dirac sea. Instead, the Stückelberg-Feynman (S-F) theory utilizes the time symmetry of the Dirac equation. S-F states that negative energy electrons always travel backwards in time. To us humans, who only perceive time in one direction, a negative energy electron traveling backwards in time looks exactly as if it were a positive energy positron traveling forward in time. Again, antimatter-matter annihilation is possible (just as it was in the Dirac sea theory). A positive energy electron may be traveling merrily along, forward through time, then suddenly emit a super energetic photon (energy equal to 2mc^2), and turn into a negative energy electron traveling backward in time. To us, it would appear as if an electron and a positron came together and disappeared, emitting a gamma ray while doing so. To many, this theory seems a bit far fetched. However, for fermions it yields all the same results as the Dirac sea, plus it has the added benefit of working for bozons as well as fermions. Since anti-bozons do exist, one must conclude that the S-F theory is somehow closer to the truth than the Dirac Sea.
No matter which theory of antimatter one accepts, the existence of antimatter is undeniable. Currently, both FNAL (in the U.S.) and CERN (in Switzerland) produce small amounts of antimatter every day (for a grand total of 1 to 10 nanograms a day). At FNAL (Also known as Fermilab), normal protons are accelerated to extremely high velocities by a particle accelerator. The protons are then allowed to strike a target, made of nickel, tungsten, or copper. Since the protons have so much energy (from the acceleration process), a plethora of various particles are produced by the collision. Some of these particles are anti-protons. At FNAL, the antiprotons are almost immediately used again in scientific experiments, but at CERN some of them are actually stored. Figure 6 shows the general method currently used to produce antimatter (not just antiprotons are produced by this process, but they are the only ones that are captured):
Figure 6: Antimatter production
The normal protons (blue) are accelerated, then smashed into a metal target. From the spew of particles, a few of the antiprotons that just happen to have the right speed and ejection angle are captured. These antiprotons are then decelerated and stored in a storage unit.
Storing antimatter is a particularly difficult problem because it can't be allowed to touch normal matter (for obvious reasons). Although antiparticles can interact normally with normal particles of a different type (for example, collisions with electrons can be used to slow down fast moving antiprotons), this is not really useful in the trapping problem. The reason being that you can not build a container out of pure electrons, or pure protons, etc... Therefore, magnetic and electric fields must be used to trap the antimatter.
The collision of one antiproton with one proton produces their entire combined mass in energy, as given by the equation E=mc^2. That means that one gram of antimatter is capable of producing the total energy equivalent to twenty-three Space Shuttle External Fuel Tanks. Of course, this is useless for powering Earth-based systems, because the amount of energy required to make the antimatter in the first place is enormously higher. However, it offers a perfect fuel for long range spacecraft. The reason for this is that, traditionally, if a spacecraft needs to go a very long distance, it needs to go very fast. Otherwise, its crew would die of boredom long before it reached its destination. In order to go very fast, the spacecraft must carry a lot of fuel. The fuel, in turn, adds weight, which means the spacecraft will need an even greater increase in fuel to go faster! Obviously, before long, the fuel requirements are quite preventative to space travel. Antimatter, however, has such a high energy-density (energy yield per mass of fuel), that it provides a way for a spacecraft to actually reach a significant fraction of the speed of light. Current ideas for "Beamed Core" pure antimatter/matter reaction drives would allow speeds of up to 120 million meters per second. That's 2/5ths of the speed of light! The antimatter requirements for this drive are way beyond our current production rate, but someday we will most likely travel to nearby stars using just such an antimatter drive.
There are a large number of engineering problems associated with using antimatter as a fuel for starships. However, the problems are just that, engineering... there is no theoretical reason to disallow this huge energy source from being used to propel starships. The obstacles that must still be overcome are mainly in the production and storage of large (on the microgram scale) quantities of antimatter. However, two spacecraft designs are currently in the works that would utilize a hybrid antimatter/nuclear drive to allow manned exploration of our own solar system. Certainly, we shouldn't complain about not being able to reach other star systems until we've fully explored our own!
The current drive/spacecraft concepts, and their relative ranges and fuel requirements are listed below:
Drive Concept: Spacecraft: Max Antimatter Required: Max Range Recommended:
Antimatter catalyzed
micro fission/fusion
ICAN-II 1 microgram Pluto (intrasystem)
Antimatter initiated
micro fusion
AIM Star
10 milligrams Oort cloud (10,000 AU)
slow (50 years)
Plasma Core,
pure antimatter/matter
Theoretical 10 kilograms Oort cloud (10,000 AU)
not as slow as AIM
Beamed Core,
pure antimatter/matter
Theoretical 1,000 megagrams Interstellar
Obviously the extreamly high antimatter requirements for the "pure" drives are quite prohibitive, which is why there are no current designs for such drives. However the hybrid drives seem likely possibilities for near-term use.
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Next Topic: Antimatter: Fission/Fusion Drive |
61fc9f853357871e |
Pseudo-differential calculus in anisotropic Gelfand-Shilov setting
Ahmed Abdeljawad Dipartimento di Matematica, Università di Torino, Italy Marco Cappiello Dipartimento di Matematica, Università di Torino, Italy and Joachim Toft Department of Mathematics, Linnæus University, Växjö, Sweden
We study some classes of pseudo-differential operators with symbols admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type . Moreover, we deduce algebraic and certain invariance properties of these classes.
0. Introduction
Gelfand-Shilov spaces of type have been introduced in the book [16] as an alternative functional setting to the Schwartz space of smooth and rapidly decreasing functions for Fourier analysis and for the study of partial differential equations. Namely, fixed , the space can be defined as the space of all functions satisfying an estimate of the form
for some constant , or the equivalent condition
for some constants . For , represents a natural global counterpart of the Gevrey class but, in addition, the condition (0.2) encodes a precise description of the behavior at infinity of . Together with one can also consider the space , which has been defined in [25] by requiring (0.1) (respectively (0.2)) to hold for every (respectively for every ). The duals of and and further generalizations of these spaces have been then introduced in the spirit of Komatsu theory of ultradistributions, see [14, 25].
After their appearance, Gelfand-Shilov spaces have been recognized as a natural functional setting for pseudo-differential and Fourier integral operators, due to their nice behavior under Fourier transformation, and applied in the study of several classes of partial differential equations, see e. g. [1, 3, 4, 5, 6, 7, 8].
According to the condition on the decay at infinity of the elements of and , we can define on these spaces pseudo-differential operators with symbols admitting an exponential growth at infinity. These operators are commonly known as operators of infinite order and they have been studied in [2] in the analytic class and in [34, 12, 24] in the Gevrey spaces where the symbol has an exponential growth only with respect to and applied to the Cauchy problem for hyperbolic and Schrödinger equations in Gevrey classes, see [12, 13, 23, 15]. Parallel results have been obtained in Gelfand-Shilov spaces for symbols admitting exponential growth both in and , see [3, 4, 11, 7, 8, 27].
We stress that the above results concern the non-quasi-analytic isotropic case In [10] we considered the more general case , which is interesting in particular in connection with Shubin-type pseudo-differential operators, cf. [5, 9]. Although the extension of the complete calculus developed in [3, 4] in this case is out of reach due to the lack of compactly supported functions in and , nevertheless some interesting results can be achieved also in this case by using different tools than the usual micro-local techniques, namely a method based on the use of modulation spaces and of the short time Fourier transform.
In the present paper, we further generalize the results of [10] to the case when and may be different from each other. Thus the symbols we consider may have different rates of exponential growth and anisotropic Gevrey-type regularity in and . More precisely, the symbols should obey conditions of the form
for suitable restrictions on the constants (cf. (0.2)). We prove that if , and (0.3) holds true for every , then the pseudo-differential operator is continuous on and on . If instead , and (0.3) holds true for every , then we prove that is continuous on and on (cf. Theorems 3.7 and 3.13). We also prove that pseudo-differential operators with symbols satisfying such conditions form algebras (cf. Theorems 3.16 and 3.17). Finally we show that our span of pseudo-differential operators is invariant under the choice of representation (cf. Theorem 3.6).
An important ingredient in the analysis which is used to reach these properties concerns characterizations of symbols above in terms of suitable estimates of their short-time Fourier transforms. Such characterizations are deduced in Section 2.
The paper is organized as follows. In Section 1, after recalling some basic properties of the spaces and , we introduce several general symbol classes. In Section 2 we characterize these symbols in terms of the behavior of their short time Fourier transform. In Section 3 we deduce continuity on and , composition and invariance properties for pseudo-differential operators in our classes.
1. Preliminaries
In this section we recall some basic facts, especially concerning Gelfand-Shilov spaces, the short-time Fourier transform and pseudo-differential operators.
We let be the Schwartz space of rapidly decreasing functions on together with their derivatives, and by the corresponding dual space of tempered distributions. Moreover will denote the vector space of real matrices.
1.1. Gelfand-Shilov spaces
We start by recalling some facts about Gelfand-Shilov spaces. Let be fixed. Then is the Banach space of all such that
endowed with the norm (1.1).
The Gelfand-Shilov spaces and are defined as the inductive and projective limits respectively of . This implies that
and that the topology for is the strongest possible one such that the inclusion map from to is continuous, for every choice of . The space is a Fréchet space with seminorms , . Moreover, , if and only if and , and , if and only if .
The spaces and can be characterized also in terms of the exponential decay of their elements, namely (respectively ) if and only if
for some (respectively for every ). Moreover we recall that for the elements of admit entire extensions to satisfying suitable exponential bounds, cf. [16] for details.
The Gelfand-Shilov distribution spaces and are the projective and inductive limit respectively of . This means that
We remark that in [26] it is proved that is the dual of , and is the dual of (also in topological sense).
For every we have
for every . If , then the last two inclusions in (1.3) are dense, and if in addition , then the first inclusion in (1.3) is dense.
From these properties it follows that when , and if in addition , then .
The Gelfand-Shilov spaces possess several convenient mapping properties. For example they are invariant under translations, dilations, and to some extent tensor products and (partial) Fourier transformations.
The Fourier transform is the linear and continuous map on , given by the formula
when . Here denotes the usual scalar product on . The Fourier transform extends uniquely to homeomorphisms from to , and from to . Furthermore, it restricts to homeomorphisms from to , and from to .
Some considerations later on involve a broader family of Gelfand-Shilov spaces. More precisely, for , , the Gelfand-Shilov spaces and consist of all functions such that
for some respective for every . The topologies, and the duals
respectively, and their topologies are defined in analogous ways as for the spaces and above.
The following proposition explains mapping properties of partial Fourier transforms on Gelfand-Shilov spaces, and follows by similar arguments as in analogous situations in [16]. The proof is therefore omitted. Here, and are the partial Fourier transforms of with respect to and , respectively.
Proposition 1.1.
Let , . Then the following is true:
1. the mappings and on restrict to homeomorphisms
2. the mappings and on are uniquely extendable to homeomorphisms
The same holds true if the -spaces and their duals are replaced by corresponding -spaces and their duals.
The next two results follow from [14]. The proofs are therefore omitted.
Proposition 1.2.
Let , . Then the following conditions are equivalent.
1. ();
2. for some (for every ) it holds
We notice that if for some , then and are equal to the trivial space . Likewise, if for some , then .
1.2. The short time Fourier transform and Gelfand-Shilov spaces
We recall here some basic facts about the short-time Fourier transform and weights.
Let be fixed. Then the short-time Fourier transform of is given by
Here is the unique extension of the -form on to a continuous sesqui-linear form on . In the case , for some , then is given by
The following characterizations of the , and their duals follow by similar arguments as in the proofs of Propositions 2.1 and 2.2 in [31]. The details are left for the reader.
Proposition 1.3.
Let be such that , , and . Also let and let be a Gelfand-Shilov distribution on . Then the following is true:
1. , if and only if
holds for some ;
2. if in addition , then if and only if
holds for every .
A proof of Proposition 1.3 can be found in e. g. [20] (cf. [20, Theorem 2.7]). The corresponding result for Gelfand-Shilov distributions is the following improvement of [30, Theorem 2.5].
Proposition 1.4.
1. , if and only if
holds for every ;
2. if in addition , then , if and only if
holds for some .
A function on is called a weight or weight function, if are positive everywhere. It is often assumed that is -moderate for some positive function on . This means that
If is even and satisfies (1.9) with , then is called submultiplicative.
For any , let () be the set of all weights on such that
for some (for every ). In similar ways, if , then () consists of all submultiplicative weight functions on such that
for some (for every ). In particular, if (), then
for some (for every ).
1.3. Pseudo-differential operators
Let and be fixed, and let . Then the pseudo-differential operator with symbol is the continuous operator on , defined by the formula
We set when , and is the identity matrix, and notice that this definition agrees with the Shubin type pseudo-differential operators (cf. e. g. [29]).
If instead , then is defined to be the continuous operator from to with the kernel in , given by
It is easily seen that the latter definition agrees with (1.11) when .
If , then is equal to the Weyl operator for . If instead , then the standard (Kohn-Nirenberg) representation is obtained.
1.4. Symbol classes
Next we introduce function spaces related to symbol classes of the pseudo-differential operators. These functions should obey various conditions of the form
for functions on the phase space . For this reason we consider semi-norms of the form
indexed by ,
Definition 1.5.
Let , and be positive constants, let be a weight on , and let
1. The set consists of all such that in (1.14) is finite. The set consists of all such that is finite for every , and the topology is the projective limit topology of with respect to ;
2. The sets and are given by
and their topologies are the inductive respective the projective topologies of with respect to .
Furthermore we have the following classes.
Definition 1.6.
For , , and and , let
where the supremum is taken over all and .
1. consists of all such that is finite for some ;
2. consists of all such that for some , is finite for every ;
3. consists of all such that for some , is finite for every ;
4. consists of all such that is finite for every .
In order to define suitable topologies of the spaces in Definition 1.6, let be the set of such that is finite. Then is a Banach space, and the sets in Definition 1.6 are given by
and we equip these spaces by suitable mixed inductive and projective limit topologies of .
In Appendix A we show some further continuity results of the symbol classes in Definition 1.6.
2. The short-time Fourier transform and regularity
In this section we deduce equivalences between conditions on the short-time Fourier transforms of functions or distributions and estimates on derivatives.
In what follows we let be defined as
In the sequel we shall frequently use the well known inequality |
e70b2d30779b9a7b | Quantum theory of observation
The quantum theory of observation consists in studying the processes of observation with the tools of quantum physics. Both the observed system and the observer system (the measuring apparatus) are considered as quantum systems. The measurement process is determined by their interaction and is described by a unitary evolution operator.
The computed quantum presence (wave function) of an initially very localized particle. Click to animate
This theoretical approach was initiated by John von Neumann (1932). It differs from the usual interpretations of quantum mechanics (Niels Bohr, Copenhagen interpretation) which require that the measuring apparatus be considered as a classical system, which does not obey quantum physics. This requirement is not justified because quantum laws are universal. They apply to all material, microscopic and macroscopic, systems. This universality is a direct consequence of the principles: if two quantum systems are combined, they together form a new quantum system (cf. 2.1, third principle of quantum physics). Therefore the number of components does not change the quantum nature of a system.
The quantum theory of observation invites us to give up the postulate of the wave function collapse, because it is not necessary to explain the correlations between successive observations, and because it contradicts the Schrödinger equation. Thus conceived quantum theory of observation is another name for Everett's theory, also called the many-worlds interpretation, the theory of the universal wave function, or the "relative state" formulation of quantum mechanics, because by applying the Schrödinger equation to observation processes, we obtain solutions that represent the multiple destinies of observers and their relative worlds.
1. Quantum theory for beginners
2. Fundamental concepts and principles
3. Examples of measurements
4. Entanglement
5. General theory of quantum measurement
6. The forest of destinies
7. The appearance of relative classical worlds in the quantum Universe
About this book
The first chapter offers an introduction, intended for a reader who approaches quantum physics for the first time. It presents the great quantum principle, the principle of the existence of superposition of states, and begins to show how it can be understood.
All the quantum principles are stated and explained in the second chapter, and we deduce from them the first consequences: the existence of multiple destinies, the incomplete discernability of states and the incompatibility of measurements.
The next chapter applies the quantum theory of observation to a few simple examples (the Mach-Zehnder interferometer, the CNOT and SWAP gates).
Chapter 4 is the most important of the book because quantum entanglement is fundamental to explain the reality of observations. From the definition of the relativity of states (Everett), it shows that the postulate of the reduction of the wave function is not necessary, because the reduction of the state vector by observation is an appearance which results from the real entanglement between the observer system and the observed system. We then deduce many consequences: the impossibility of seeing non-localized macroscopic states (but we can still observe them), the quantum explanation of intersubjectivity, the observation of correlations in an entangled pair, co-presence without possible encounter and entanglement of space-time, the non-cloning theorem, the possibility of ideal measurements of entangled states and why it does not allow to observe our other destinies, why entangled pairs do not permit to communicate, decoherence through entanglement and why it explains at the same time Feynman's rules, the posterior reconstruction of interference patterns and the fragility of non-localized macroscopic states, and finally, the possibility, and the reality, of experiments of the "Schrödinger's cat" type. We can conclude that the existence theorem of multiple destinies is empirically verifiable. An observer can not observe her other destinies, but a second observer can in principle observe them, with experiments of the "Schrödinger's cat" type.
In Schrödinger's imagined experiment, the paradoxical state is produced, but the experiment is not designed to verify by observation that it was actually produced, because it is destroyed by opening the box. A slightly modified experiment, however, makes it possible to observe that a state similar to is actually produced. We can therefore in principle verify that two destinies of an observer system are simultaneously real. But this conclusion is limited to reversible observation processes. As the processes of life are irreversible, the simultaneous existence of the multiple destinies of a living being can not be observed.
Quantum theory of observation has so far been exposed for ideal measurements. Chapter 5 shows that it can be generalized for all observation systems, and that the results obtained for the ideal measurements (multiple destinies, Born rule ...) remain valid. It also shows that decoherence by the environment is sufficient to explain the selection of the pointer states of measuring instruments.
The multiple destinies of an observer form a tree. Chapter 6 applies the theory to a universe that contains many observers and obtains as a solution a forest of multiple destinies, a tree for each observer. Each branch is a destiny. All branches of forest trees can become entangled when observers meet or communicate. But some branches can never meet. The destinies they represent are inexorably separated. This book calls them incomposable destinies.
To speak of the growth of a forest of destinies is only one way of describing the solutions of the Schrödinger equation when applied to systems of ideal observers. It is a question of describing mathematical solutions which result from the simple assumptions which have been made. It is not a delusional imagination but a calculation of the consequences of mathematical principles.
The chapter ends by showing that we must distinguish between multiple destinies and Feynman's paths, that the parallelism of quantum computation is different from the parallelism of destinies.
The last chapter shows that quantum physics even explains the classical appearances of matter. The quantum evolution of the Universe can not be identified with a classical destiny, but it is sufficient to determine the growth of a forest of destinies of observers and their relative worlds. We thus explain the classical appearances of relative worlds without postulating that the Universe itself must have this appearance. The classical appearances of observers emerge from a quantum evolution that describes a forest of multiple destinies.
It is sometimes wrongly believed that the explanation of quantum principles (cf 2.1) requires advanced mathematics. The great concepts of quantum physics, superposition (1.1) and incomplete discernability (2.6) of states, incompatibility of measurements (2.7), entanglement of parts (4.1), relativity of states (4.3), decoherence by entanglement (4.17), selection of pointer states (5.4) and incomposability of destinies (6.4) ... can all be explained with minimal mathematical formalism. It suffices to know complex numbers (1.4) and to know how to add vectors in finite dimensional spaces. The applications of quantum physics often require advanced mathematical techniques, but not the explanation of the principles. This applies to all sciences. The principles are what we have to understand when we start studying. They are the main tools that enable us to progress. It is therefore normal and natural that they can be explained without exceeding a fairly basic level.
A philosophical introduction: quantum theory of multiple destinies, from my handbook of epistemology.
Co-presence without possible encounter and the incomposability of destinies are published in this book for the first time. These are discoveries for educational purposes, which is why it is natural to publish them in an educational library. They explain how a single space-time can accommodate worlds relative to the multiple destinies of observers, and why these destinies can coexist without meeting each other.
Who is this book addressed to ? Primarily to students who have already had a first course in quantum physics (for example, the first chapters of Feynman 1966, Cohen-Tannoudji, Diu & Laloë 1973, Griffiths 2004). More generally, to any interested reader who is not too frightened by the expressions Hilbert space or unitary operator.
Pedagogical objectives: At the end of the book, the reader will have the main elements to study the research work on the quantum theory of observation. They can also prepare for research on quantum computation and information (Nielsen and Chuang 2010).
Detailed contents
1. Quantum physics for beginners
1. The great principle : the existence of quantum superpositions
2. Wave-particle duality
3. The polarization of light
4. What is a complex number ?
5. Why is quantum reality represented by complex numbers ?
6. Scalar product and unitary operators
7. Tensor product and entanglement
8. Quantum bricks of the Universe: the qubits
2. Fundamental concepts and principles
1. The principles of quantum physics
2. Ideal measurements
3. The existence theorem of multiple destinies
4. The destruction of information by observation
5. The Born Rule
6. Can we observe quantum states?
7. Orthogonality and incomplete discernability of quantum states
8. The incompatibility of quantum measurements
9. Uncertainty and density operators
3. Examples of measurements
1. Observation of quantum superpositions with the Mach-Zehnder interferometer
2. An ideal measurement: the CNOT gate
3. A non-ideal measurement: the SWAP gate
4. Experimental realization of quantum gates
4. Entanglement
1. Definition
2. Interaction, entanglement and disentanglement
3. Everett relative states
4. The collapse of the state vector through observation is a disentanglement.
5. Apparent disentanglement results from real entanglement between the observed system and the observer.
6. Dirac's error
7. Can we see non-localized macroscopic states?
8. The quantum explanation of intersubjectivity
9. Einstein, Bell, Aspect and the reality of quantum entanglement
10. Co-presence without a possible encounter
11. Entangled space-time
12. Action, reaction and no cloning
13. The ideal measurement of entangled states
14. Why does not the measurement of entangled states enable us to observe other destinies?
15. Reduced density operators
16. Relative density operators
17. Why do not entangled pairs enable us to communicate?
18. Decoherence through entanglement
19. The Feynman Rules
20. The a posteriori reconstitution of interference patterns
21. The fragility of non-localized macroscopic states
22. Experiments of the "Schrödinger's cat" type
23. Is the existence theorem of multiple destinies empirically verifiable?
5. General theory of quantum measurement
1. Measurement operators
2. Observables and projectors
3. Uncertainty about the state of the detector and measurement superoperators
4. The selection of pointer states and environmental pressure
5. The pointer states of microscopic probes
6. A double constraint for the design of observation instruments
6. The forest of destinies
1. The arborescence of the destinies of an ideal observer
2. Absolute destiny of the observer and relative destiny of its environment
3. The probabilities of destinies
4. The incomposability of destinies
5. The growth of a forest of destinies
6. Virtual quantum destinies and Feynman paths
7. The parallelism of quantum computation and the multiplicity of virtual pasts
8. Can we have many pasts if we forget them?
7. The appearance of relative classical worlds in the quantum Universe
1. Are not classical appearances proofs that quantum physics is incomplete?
2. Space and mass
3. The quantum evolution of the Universe determines the classical destinies of the relative worlds |
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Schrodinger Equation and The Wave Function
Schrodinger Equation
In this quantum mechanics lecture you will learn the Schrödinger equation (1d and 3d, time-independent and time-dependent) within 45 minutes.
Content of the video
1. [00:10] What is a partial second-order DEQ?
2. [01:08] Classical Mechanics vs. Quantum Mechanics
3. [04:38] Applications
4. [05:24] Derivation of the time-independent Schrödinger equation (1d)
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9.3: Solving the H₂⁺ System Exactly (Optional)
• Page ID
• The hydrogen molecule ion \(\ce{H_{2}^{+}}\) is the only molecule for which we can solve the electronic Schrödinger equation exactly (and only within the Born Oppenheimer Approximation). Note that it has just one electron! In fact, there are no multi-electron molecules we can solve exactly. Thus from \(H_2\) on up to more complicated molecules, we only have approximate solutions for the allowed electronic energies and wavefunctions. Before we discuss these, however, let us examine the exact solutions for \(\ce{H_{2}^{+}}\) starting with a brief outline of how the exact solution is carried out.
Figure \(\PageIndex{1}\): Definitions of distances in the \(\ce{H2+}\) molecular ion. (CC BY-NC; Ümit Kaya)
Figure \(\PageIndex{1}\) shows the geometry of the \(\ce{H_{2}^{+}}\) molecule ion and the coordinate system we will use. The two protons are labeled A and B, and the distances from each proton to the one electron are \(r_A\) and \(r_B\), respectively. Let \(R\) be the distance between the two protons (this is the only nuclear degree of freedom that is important, and the electronic wavefunction will depend parametrically only on \(R\)). The coordinate system is chosen so that the protons lie on the z-axis one at a distance \(R/2\) above the \(xy\) plane and one a distance \(-R/2\) below the \(xy\) plane. The classical energy of the electron is
\[\frac{p^2}{2m_e}-\frac{e^2}{4\pi \epsilon_0}\left ( \frac{1}{r_A}+\frac{1}{r_B} \right )=E\]
The nuclear-nuclear term \(V_{nn}(R)=e^2 /4\pi \epsilon_0 R\) is a constant, and we can define the potential energy relative to this quantity.
The energy is not a simple function of energies for the \(x\), \(y\), and \(z\) directions, so we try another coordinate system to see if we can simplify the problem. In fact, this problem has a natural cylindrical symmetry (analogous to the spherical symmetry of the hydrogen atom) about the z-axis. Thus, we try cylindrical coordinates. In cylindrical coordinates the distance of the electron from the z-axis is denoted \(\rho\), the angle \(\phi\) is the azimuthal angle, as in spherical coordinates, and the last coordinate is just the Cartesian \(z\) coordinate. Thus,
• \(x=\rho \cos\phi\)
• \(y=\rho \sin\phi \)
• \(z=z\)
Using right triangles, the distance \(r_A\) and \(r_B\) can be shown to be
\[r_A =\sqrt{\rho^2 +(R/2-z)^2}\;\;\;\; r_B =\sqrt{\rho^2 +(R/2 +z)^2}\]
The classical energy becomes
\[\frac{p_{\rho}^{2}}{2m_e}+\frac{p_{\phi}^{2}}{2m_e \rho^2}+\frac{p_{z}^{2}}{2m_e}-\frac{e^2}{4\pi \epsilon_0}\left ( \frac{1}{\sqrt{\rho^2 +(R/2-z)^2}}+\frac{1}{\sqrt{\rho^2 +(R/2+z)^2}} \right ) =E\]
First, we note that the potential energy does not depend on \(\phi\), and the classical energy can be written as a sum \(E=\varepsilon_{\rho ,z}+\varepsilon_\phi\) of a \(\rho\) and \(z\) dependent term and an angular term. Moreover, angular momentum is conserved as it is in the hydrogen atom. However, in this case, only one component of the angular momentum is actually conserved, and this is the z-component, since the motion is completely symmetric about the z-axis. Thus, we have only certain allowed values of \(L_z\), which are \(m\bar{h}\), as in the hydrogen atom, where \(m=0,\pm 1,\pm 2,...\). The electronic wavefunction (now dropping the "(elec)" label, since it is understood that we are discussing only the electronic wavefunction), can be written as a product
\[\psi (\rho ,\phi ,z)=G(\rho ,z)y(\phi)\]
and \(y(\phi )\) is given by
\[y_m (\phi )=\frac{1}{\sqrt{2\pi}}e^{im\phi}\]
which satisfies the required boundary condition \(y_m (0)=y_m (2\pi )\). Unfortunately, what is left in \(\rho\) and \(z\) is still not that simple. But if we make one more change of coordinates, the problem simplifies. We introduce two new coordinates \(\mu\) and \(\nu\) defined by
\[\mu =\frac{r_A +r_B}{R}\;\;\;\; \nu =\frac{r_A -r_B}{R}\]
Note that when \(\nu =0\), the electron is in the \(xy\) plane. Thus, \(\nu \) is analogous to \(z\) in that it varies most as the electron moves along the z-axis. The presence or absence of a node in the \(xy\) plane will be an important indicator of wavefunctions that lead to a chemical bond in the molecule or not. The coordinate \(\mu \), on the other hand, is minimum when the electron is on the z-axis and grows as the distance of the electron from the z-axis increases. Thus, \(\mu \) is analogous to \(\rho\). The advantage of these coordinates is that the wavefunction turns out to be a simple product
\[\psi (\mu ,\nu ,\phi ,R)=M(\mu )N(\nu )y(\phi )\]
which greatly simplifies the problem and allows the exact solution.
The mathematical structure of the exact solutions is complex and nontransparent, so we will only look at these graphically, where we can gain considerably insight. First, we note that the quantum number \(m\) largely determines how the solutions appear. First, let us introduce the nomenclature for designating the orbitals (solutions of the Schrödinger equation, wavefunctions) of the system
• If \(m=0\), the orbitals are called \(\sigma\) orbitals, analogous to the \(s\) orbitals in hydrogen.
• If \(m=1\), the orbitals are called \(\pi\) orbitals, analogous to the \(p\) orbitals in hydrogen.
• If \(m=2\), the orbitals are called \(\delta\) orbitals, analogous to the \(d\) orbitals in hydrogen.
• If \(m=3\), the orbitals are called \(\phi\) orbitals, analogous to the \(f\) orbitals in hydrogen.
These orbitals are known as molecular orbitals because the describe the electronic wavefunctions of an entire molecule.
Designations of Molecular Orbitals
There are four designators that we use to express each molecular orbital:
1. A Greek letter, \(\sigma\), \(\pi\), \(\delta\), \(\phi\), .... depending on the quantum number \(m\).
2. A subscript qualifier g or u depending on how an orbital \(\psi\) behaves with respect to a spatial reflection or parity operation \(r\rightarrow -r\). If \[\psi (-r)=\psi (r)\] then \(\psi (r)\) is an even function of \(r\), so we use the \("g"\) designator, where \(g\) stands for the German word gerade, meaning "even". If \[\psi (-r)=-\psi (r)\] then \(\psi (r)\) is an odd function of \(r\), and we use the \("u"\) designator, where \(u\) stands for the German word ungerade, meaning "odd".
3. An integer \(n\) in front of the Greek letter to designate the energy level. This is analogous to the integer we use in atomic orbitals \((1s, 2s, 2p,...)\).
4. An asterisk or no asterisk depending on the presence or absence of nodes between the nuclei. If there is significant amplitude between the nuclei, then the orbital favors a chemical bond, and the orbital is called a bonding orbital. If there is a node between the nuclei, the orbital does not favor bonding, and the orbital is called an antibonding orbital.
So, the first few orbitals, in order of increasing energy are:
\[1\sigma_g ,1\sigma_{u}^{*},2\sigma_g ,2\sigma_{u}^{*}, 1\pi_u ,3\sigma_g ,1\pi_{g}^{*},3\sigma_u\]
These results are near quantitatively described by the Linear Combination of Atomic Orbitals approximation discussed in the last Section.
Contributors and Attributions |
8a3476864155f0ba |
Resonator-zero-qubit architecture for superconducting qubits
Andrei Galiautdinov, Alexander N. Korotkov, and John M. Martinis Department of Electrical Engineering, University of California, Riverside, California 92521, USA
Department of Physics, University of California, Santa Barbara, California 93106, USA
January 2, 2021
We analyze the performance of the Resonator/zero-Qubit (RezQu) architecture in which the qubits are complemented with memory resonators and coupled via a resonator bus. Separating the stored information from the rest of the processing circuit by at least two coupling steps and the zero qubit state results in a significant increase of the ON/OFF ratio and the reduction of the idling error. Assuming no decoherence, we calculate such idling error, as well as the errors for the MOVE operation and tunneling measurement, and show that the RezQu architecture can provide high fidelity performance required for medium-scale quantum information processing.
03.67.Lx, 85.25.-j
I Introduction
Superconducting circuits with Josephson junctions are steadily gaining attention as promising candidates for the realization of a quantum computer CLARKE2008 . Over the last several years, significant progress has been made in preparing, controlling, and measuring the macroscopic quantum states of such circuits Yamamoto03 ; Plantenberg07 ; Clarke-06 ; Steffen06 ; DiCarlo09 ; DiCarlo10 ; Mariantoni-Science ; Simmonds-10 ; Oliver-11 ; Esteve-09 ; Siddiqi-11 . However, the two major roadblocks – scalability and decoherence – still remain, impeding the development of a workable prototype. The Resonator/zero-Qubit (RezQu) protocol presented here aims to address these limitations at a low-level (hardware cell) architecture DiVincenzo .
A RezQu device consists of a set of superconducting qubits (e.g., phase qubits Martinis-QIP ), each of which is capacitively coupled to its own memory resonator and also capacitively coupled to a common resonator bus, as shown in Fig. 1 WANG2011 ; Mariantoni-NatPhys ; Mariantoni-Science ; WILHELM_NOON_2010 . The bus is used for coupling operations between qubits, while the memory resonators are used for information storage when the logic qubits are idling. With coupling capacitors being fixed and relatively small, qubit coupling is adjusted by varying the qubit frequency, which is brought in and out of resonance with the two resonators. For a one-qubit operation, quantum information is moved from the memory to the qubit, where a microwave pulse is applied. A natural two-qubit operation is the controlled- gate, for which one qubit state is moved to the bus, while the other qubit frequency is tuned close to resonance with the bus for a precise duration Strauch03 ; Haack10 ; DiCarlo10 ; Yamamoto10 .
Most importantly, the information stored in resonators is separated from the rest of the processing circuit by the known qubit state and at least two coupling steps, thus reducing crosstalk error during idling. Also, the problem of spectral crowding is essentially eliminated because the two-step resonance between empty qubits is not harmful, while the four-step coupling between memory resonators is negligible. Therefore the resonator frequencies, which are set by fabrication, can be close to each other, decreasing sensitivity to phase errors in the clock. Thus the RezQu architecture essentially solves the inherent ON/OFF ratio problem of the fixed capacitive coupling without using a more complicated scheme of a tunable coupling Clarke-06 ; Nakamura-07-tunable ; BIALCZAK2010 . As an additional benefit, information storage in resonators increases coherence time compared to storage in the qubits. We note that the idea of using resonators to couple qubits has been suggested by many authors Zhu03 ; Blais03 ; Zhou04 ; Blais04 ; Cleland04 ; Blais07 ; Koch07 ; MAJER2007 ; Girvin08 ; NORI11 . The use of resonators as quantum memories has also been previously proposed Pritchett05 ; Silanpaa07 ; Girvin08 ; Johnson10 . However, putting the two ideas together in a single architecture results in the new qualitative advantages, which have not been discussed.
Schematic diagram of the RezQu architecture:
Figure 1: Schematic diagram of the RezQu architecture: – memory resonators, – qubits, – bus. We assume frequencies 7 GHz for the memories, 6 GHz for the bus, and qubit frequencies are varied between these values.
In this paper we briefly consider the relation between the logical and the physical qubit states and then analyze several basic operations in the RezQu architecture. In particular, for a truncated three-component memory-qubit-bus RezQu device we focus on the idling error, information transfer (MOVE) between the qubit and its memory, and the tunneling measurement. The analysis of the controlled- gate will be presented elsewhere JoydipGhosh11 . For simplicity decoherence is neglected.
Ii Logical vs physical qubits
We begin by recalling an important difference between logical and actual, physical qubits. The difference stems from the fact that in the language of quantum circuit diagrams the idling qubits are always presumed to be stationary NIELSEN&CHUANG , while superconducting qubits evolve with a 6-7 GHz frequency even in idling, which leads to accumulation of relative phases. Also, the always-on coupling leads to fast small-amplitude oscillations of the “bare”-state populations during the off-resonant idling. A natural way to avoid the latter problem is to define logical multi-qubit states to be the (dressed) eigenstates of the whole system (see e.g., Blais04 ; Blais07 ; Koch07 ; Boissonneault09 ; DiCarlo09 ; Pinto10 for discussion of the dressed states). Then the only evolution in idling is the phase accumulation for each logic state. However, there are logic states for qubits, and using “local clocks” (rotating frames) would require an exponential overhead to calibrate the phases. The present-day experiments with two or three qubits often use this unscalable way, but it will not work for . A scalable way is to choose only rotating frames, which correspond to one-qubit logical states, and treat the frequencies of multi-qubit states only approximately, as sums of the corresponding single-qubit frequencies. The use of such non-exact rotating frames for multi-qubit states leads to what we call an idling error, which is analyzed in the next section.
Notice that it is sufficient to establish a correspondence between logical and physical states only at some time moments between the gates. Moreover, this correspondence may be different at different moments. At those moments, the bus is empty, the system components ( qubits, memories, and the bus) are well-detuned from their neighbors, and for each logical qubit it is unambiguously known whether the corresponding quantum information is located in the memory or in the qubit. Thus the eigenstates corresponding to the logical states are well-defined and the physical-to-logical correspondence is naturally established by projecting onto the computational eigenstates, while occupations of other eigenstates should be regarded as an error. In the simplest modular construction of an algorithm we should not attempt to correct the error of a given gate by the subsequent gates. Then for the overall error we only need to characterize the errors of individual gates, as well as the idling error.
One may think that defining logical states via the eigenstates of the whole system may present a technical problem in an algorithm design. However, this is not really a problem for the following reasons. First, we need the conversion into the basis of eigenstates only at the start and end of a quantum gate, while the design of a gate is modular and can be done using any convenient basis. Second, in practice, we can truncate the system to calculate the eigenstates approximately, making sure that the error due to truncation is sufficiently small. Similar truncation with a limited error is needed in practical gate design.
As mentioned above, the physical-to-logical correspondence rule can be different at every point between the gates. For the correspondence based on eigenstates we are free to choose single-excitation phases arbitrarily. In spite of this freedom, for definiteness, it makes sense to relate all the single-excitation phases to a particular fixed time moment in an algorithm. Then a shift of the gate start time leads to easily calculable phase shifts, which accumulate with the frequencies equal to the change of single-excitation frequencies before and after the gate. Such shift is useful for the adjustment of relative single-excitation phases Hofheinz . Another way of the single-excitation phase adjustment is by using “qubit frequency excursions” Mariantoni-Science . The ease of these adjustments significantly simplifies design of quantum gates, because we essentially should not wary about the single-excitation phases.
Notice that initial generation of high-fidelity single-excitation eigenstates is much easier experimentally than generation of the bare states. This is because a typical duration of the qubit excitation pulses is significantly longer than inverse detuning between the qubit and resonators. We have checked this advantage of using eigenstates versus bare states numerically in a simple model with typical parameters Mariantoni-Science of a RezQu device (the error decrease is about two orders of magnitude). Similarly, the standard one-qubit operations essentially operate with the eigenstates rather than with the bare states.
Iii Idling error
Before discussing the idling error in the RezQu architecture, let us consider a simpler case of two directly coupled qubits. Then in idling the wavefunction evolves as , where we denote the logical (eigen)states with an overline, their corresponding (eigen)energies with , and amplitudes at with . However, for the desired evolution the last term should be replaced with ; then only two rotating frames (clocks) with frequencies and are needed. We see that the phase difference accumulates with the frequency , and therefore the idling error due to qubit coupling accumulates over a time as
where we assumed . (The frequency is defined in the same way as in Ref. Pinto10 for a two-qubit interaction.) The error is state-dependent, but in this paper we will always consider error estimates for the worst-case scenario.
In the RezQu architecture, the main contribution to the idling error comes from interaction between a memory resonator, in which quantum information is stored, and the bus, which is constantly used for quantum gates between other qubits. By analogy with the above case, for the truncated memory-qubit-bus () system the idling error can be estimated as
where the eigenenergies correspond to the logical eigenstates , , , , and in our notation the sequence of symbols represents the states of the memory resonator, the qubit, and the bus. Notice that the qubit here is always in state , and is essentially the difference between the effective frequencies of the memory resonator in the presence and absence of the bus excitation.
To find we use the rotating wave approximation (RWA); then the dynamics of the system is described by the Hamiltonian (we use )
where the qubit frequency may vary in time, while the qubit anharmonicity is assumed to be constant,
are the qubit lowering and raising operators, , are the memory and the bus frequencies (which are presumed to be fixed), , , , are the creation/annihilation operators for the memory and the bus photons, and , are the memory-qubit and qubit-bus coupling constants. The last term in Eq. (III) describes the direct (electrostatic) memory-bus coupling; replacing a qubit in Fig. 1 with a lumped tank circuit it is found PRYADKO&KOROTKOV_iSWAP to be
It is typically smaller than the effective memory-bus coupling via the virtual excitation of the qubit because the detunings and between the elements are much smaller than their frequencies; because of that, we often neglect . From the physical model it is easy to show that and are proportional to and therefore change when the qubit frequency is varied; however for simplicity we will assume constant and .
Neglecting , in fourth order we find (see Appendix),
which is very close to the exact value found by direct diagonalization of the Hamiltonian (Fig. 2), and the effect of is of a higher order and therefore very small (see Appendix and Fig. 2). Notice that because in a linear system , and nonlinearity comes from the qubit. Equation (6) shows that an optimal choice of the qubit “parked” frequency is , midway between the memory and the bus frequencies; then the idling error in this order goes to zero (this happens because the contribution of in becomes zero – see Appendix). Notice that in the RezQu architecture the frequencies of the memory resonators are assumed to be relatively close to each other (forming a “memory band” of frequencies). Then the optimal “parked” frequencies of the qubits are also close to each other. This is not a problem when all qubits are in state ; however, when a qubit is excited this may lead to a significant resonant coupling with another qubit via the bus. To avoid this “spectral crowding” effect, it is useful to reserve two additional frequencies, situated sufficiently far from the “parked” frequencies, at which a pair of qubits may undergo local rotations (simultaneous rotations of two qubits are often useful before and after two-qubit gates).
(Color online) The frequency
Figure 2: (Color online) The frequency for a truncated memory-qubit-bus system [the idling error is ] for two values of the coupling: MHz (blue lines) and MHz (red lines). The solid lines show the results of exact diagonalization of the RWA Hamiltonain (III) with and without . The effect of is not visible (smaller than the line thickness). The blue and red dashed lines show the analytical result (6).
The idling error (2) scales quadratically with time. This is because we use a definition for which not the error itself but its square root corresponds to a metric, and therefore for a composition of quantum gates in the worst-case scenario we should sum square roots of the errors NIELSEN&CHUANG . For the same reason, the worst-case idling error scales quadratically, , with the number of qubits in a RezQu device. In principle, an average idling error may scale linearly with and time (for that we would need to define the memory “clock” frequency using an average occupation of the bus); however, here we use only the worst-case analysis.
It is convenient to replace the time-dependence in the idling error estimate by the dependence on the number of operations in an algorithm. (The corresponding quadratic dependence on can also be interpreted as the worst-case-scenario error for a composition of quantum operations.) Assuming that each operation crudely takes time (this estimate comes from MOVE operations discussed later and also from controlled- gate) and neglecting the second factor in Eq. (6) (i.e. assuming non-optimal “parked” qubit frequencies), we obtain the following estimate for the worst-case idling error:
where and are typical detunings at idling. Using for an estimate MHz, MHz, and MHz, we obtain .
To demonstrate the advantage of the RezQu architecture, Eq. (8) may be compared with the corresponding result for the conventional bus-based architecture (without additional memories). Then the idling error is due to -interaction between the qubit and the bus: the frequency of an idling qubit is affected by the bus occupation due to logic operations between other qubits. In this case , that gives
Assuming a typical ratio between the coupling and detuning, we have a reduction in the idling error in the RezQu architecture by at least even before considering that can be zeroed in this order. Using Eq. (9) we see that a conventional architecture allows only a very modest number of qubits and operations before the idling error becomes significant. In principle, the problem can be solved by a constantly running dynamical decoupling (which would be quite nontrivial in a multi-qubit device). The RezQu idea eliminates the need for such dynamical decoupling.
All our estimates so far were for the idling error due to the memory-bus interaction. Now let us discuss errors due to the four-step memory-memory interaction in the RezQu architecture. The -interaction between the memory and another (th) memory can be calculated Pinto10 as , where additional subscript indicates parameters for the th section of the device. The -interaction does not produce a phase error accumulating in idling, but leads to the error every time the information is retrieved from memory, where is the typical spacing between memory frequencies. Assuming similar sections of the RezQu device with and , we obtain an error estimate per operation. Since the worst-case scaling with the number of operations is always , we obtain the worst-case estimate
This error is smaller than the idling error (8) if . For we find a very small error , which means that there is essentially no spectral crowding problem for memories. Notice that for a conventional bus-based architecture the error estimate (10) is replaced by and presents a difficult scaling problem due to the spectral crowding.
Besides the -interaction between the two memories, there is also the -interaction. Using the same approximate derivation as in Appendix [see Eq. (36)] and assuming , we find an estimate . Then using (the scaling is because each pair brings a contribution), with , we obtain the worst-case estimate
This error is smaller than the memory-bus idling error (8) if , which is always the case in practice.
Iv MOVE operation
Any logic gate in the RezQu architecture requires moving quantum information from one system element (memory, qubit, bus) to another. Therefore the MOVE operation is the most frequent one. It is important to mention that the one-way MOVE operation PRYADKO&KOROTKOV_iSWAP is easier to design than the SWAP (or SWAP) operation because we are not interested in the fidelity of the reverse transfer and can also assume zero occupation of the neighboring element. For example, for a perfect qubitmemory MOVE (MOVE) operation in the truncated system we search for a unitary, which transforms (notice that in RWA always), but we are not interested in what happens to the initial states and . Moreover, we can allow for an arbitrary phase, , because this phase can be compensated either by shifting the operation start time within one period of the initial memory-qubit detuning Hofheinz or by “qubit frequency excursion” with proper integral Mariantoni-Science . Therefore, we need to satisfy only two (complex) equations to design the unitary for this MOVE,
Experimentally the qubitmemory MOVE is done Hofheinz ; WANG2011 ; Mariantoni-NatPhys ; Mariantoni-Science by tuning the qubit in resonance (with some overshoot) with the memory resonator approximately for a duration . Equation (12) means that any reasonable shape of tune/detune pulse with four adjustable parameters can be used for a perfect MOVE operation in the truncated system. Actually, as will be discussed later, the use of only two adjustable parameters is sufficient to obtain an exponentially small error in the quasi-adiabatic regime. Such two-parameter construction is most convenient for practical purposes, but formally it is imperfect (non-zero error). So we will first discuss the perfect (zero error) four-parameter construction.
We have designed the qubitmemory MOVE pulses for the truncated device both analytically (in first order) and numerically. The initial and final frequencies of the qubit are allowed to be different. In the analytical design we do calculations in the bare basis, , but define the co-moving frame as
In this representation the only interesting initial state of the qubitmemory MOVE is (in first order)
where and are the detunings. The desired (target) final state at time is , i.e.
Notice that even though the phase is arbitrary, the relative phase between and is fixed by the absence of the relative phase between and . We see that the MOVE operation should eliminate the initial “tail” on the bus (this needs two real parameters in the pulse design) and transfer most of the excitation to the memory with correct magnitude and relative phase of (two more real parameters).
Similarly to the experimental pulse design Hofheinz ; WANG2011 ; Mariantoni-NatPhys ; Mariantoni-Science , we assume that the shape of pulse consists of a front ramp, rear ramp, and a flat part in between them (Fig. 3 illustrates a piecewise-linear construction of the pulse). As will be shown below, using two parameters for the front ramp shape we can ensure elimination of the “tail” ; we can choose a rather arbitrary rear ramp, and using two parameters for the flat part (its frequency overshoot and duration) we can provide proper .
(Color online) Illustration of a
piecewise-linear tune/detune pulse shape (qubit frequency as a
function of time) for the MOVE operation qubit
Figure 3: (Color online) Illustration of a piecewise-linear tune/detune pulse shape (qubit frequency as a function of time) for the MOVE operation qubitmemory in a three-component system. The front ramp consists of two straight segments. The solid blue line shows the result of a four-parameter numerical optimization, in which the slope of the first straight segment, the qubit frequency at the end of the first straight segment, the duration of the flat part and its overshoot have been optimized. This gives up to machine accuracy. The red dashed line shows analytical design based on Eqs. (17) and (IV); in this case, . System parameters: GHz, GHz, GHz, GHz, MHz, the slopes of the second front segment and of the final ramp have been fixed at 500 MHz/ns.
Let us start with the “tail” . As follows from the Schrödinger equation with the Hamiltonian (III),
Let us denote the end of the front ramp by and the start of the rear ramp by (see Fig. 4). For in Eq. (16) we can replace with because the qubit occupation cannot change much during a short ramp. For we can use integration by parts using , with changing approximately from 1 to 0. Finally, there is a negligible (second-order in ) contribution to the integral for because is already small (first order in ). Thus for the desired pulse shape, in first order we obtain
As we see, required elimination of the “tail” on the bus gives two equations (real and imaginary parts) for the front ramp shape. This can be done by using practically any shape with two adjustable parameters. Notice that condition (17) essentially means that in order to have correct (zero) “tail” on the bus at final time , this tail at time should be the same as the tail for the co-moving eigenstate of the qubit-bus system.
Now let us design the flat part of the pulse, which should give us the proper ratio from Eq. (IV). After designing the front ramp we know and at the start of the flat part : in first order, and
Similarly, for an arbitrarily chosen rear ramp shape we know desired and at the end of the flat part : in first order, and
During the flat part of the pulse we can use the two-level approximation with coupling , and essentially connect the two points on the Bloch sphere corresponding to Eqs. (19) and (20) by a “Rabi” pulse. These points are close to the North and South poles, so the pulse is close to the ideal -pulse; we assume a small constant overshoot with (Fig. 4), and duration with , where . Then using the leading-order relation for an almost perfect -pulse,
we obtain the needed pulse parameters and as
and also find the resulting phase .
We have checked numerically the analytical pulse design given by Eqs. (17) and (IV). For example, for a piecewise-linear pulse whose front ramp consists of two straight segments (Fig. 4), the error
for the analytically designed pulses is found to be below for typical parameters with MHz. As expected, the numerical four-parameter optimization of such pulse shape gives zero error (up to machine accuracy), and the shape of this perfect pulse is close to the analytically-designed shape (see Fig. 4).
(Color online) Implementation of the
Figure 4: (Color online) Implementation of the qubitmemory MOVE operation in a three-component system using a pulse with error-function-shaped ramps (sum of two time-shifted error functions for the front ramp). Four parameters of the pulse shape (see upper panel) are optimized: the time shift and amplitude ratio for the front-ramp error functions, the duration of the middle part of the pulse, and the overshoot magnitude. The error functions are produced by integrating Gaussians with standard deviation ns; the beginning and end of the pulse are at 3 from the nearest error-function centers (shown by vertical lines in the upper panel). The middle and lower panels show time-dependence of the level populations in the bare-state basis and comoving eigenbasis. The MOVE error (23) is zero up to machine accuracy.
Experimental pulses for the MOVE operation WANG2011 ; Mariantoni-NatPhys ; Mariantoni-Science are produced by a Gaussian filter and therefore have the error-function-shape ramps. We can use the same design idea for such pulses: shaping the front ramp using two parameters (see Fig. 4) takes care of the“tail” on the bus, for the rear ramp we use any convenient shape, and for the middle part we vary the overshoot frequency and duration to ensure proper population transfer between the qubit and the memory (for such pulses it is natural to define the duration to be between the inflection points of the error-function shapes). We have checked numerically that these four parameters are sufficient to achieve zero error (perfect transfer fidelity ) in the truncated three-element system.
As a further simplification of the MOVE pulse design, let us optimize only two middle-part parameters (overshoot and duration) and do not optimize the front ramp shape. In this case we cannot ensure the proper “tail” ; however, it is small by itself, and therefore the error is not large. Moreover, for sufficiently slow pulses the “tail” is almost correct automatically because of the adiabatic theorem. It is important to notice that the bus is well-detuned, , and then the adiabaticity condition is , which is well-satisfied even by rather fast pulses. To estimate the corresponding error, we consider the two-level system bus-memory during the front ramp and write the differential equation for the variable , which describes deviation from the co-moving eigenstate. Assuming and , we obtain . The “tail” error at the end of the front ramp is (notice that ), and it does not change significantly during the rest of the pulse. If this is the major contribution to the MOVE error, then
where is defined in Eq. (18). Numerical optimization of only the middle part of the pulse (overshoot and duration) confirms that Eq. (24) is a good approximation for the MOVE error in this case. Notice that for an error-function ramp obtained by integrating a Gaussian with the standard deviation (time-width) , the error (24) decreases exponentially with (we assume sufficiently long ramp time ) and is typically quite small. For example, for changing from 0.5 GHz to 1 GHz and MHz, the error is below for ns (for ns if MHz).
So far we have only considered the MOVE qubitmemory. The MOVE in the opposite direction memoryqubit can be designed by using the time-reversed pulse shape. Perfect MOVE still requires optimizing four parameters (overshoot and duration of the middle part and the two parameters for the rear ramp), while using only the two parameters for the middle part is sufficient for a high-fidelity MOVE.
In designing the MOVEs between the qubit and the memory we assumed no quantum information on the bus. The presence of an excitation on the bus makes the previously designed perfect MOVE imperfect. We checked numerically that the corresponding error for typical parameters is about , i.e. quite small. Moreover, a typical RezQu algorithm never needs a MOVE between a qubit and memory with occupied bus, so the unaccounted MOVE errors due to truncation are even much smaller. The analysis in this paper assumes the RWA Hamiltonian (III), which neglects terms and , which change the number of excitations. We have checked numerically that addition of these terms into the Hamiltonian leads to negligibly small changes of the system dynamics during the MOVE operations.
Designing MOVEs between the qubit and the bus is similar to designing MOVEs between the qubit and memory, if we consider the truncated three-element system. However, in reality the situation is more complicated because the bus is coupled with other qubits. Our four-parameter argument in this case does not work, and designing a perfect single-excitation MOVE would require parameters (for a truncated system with qubits, one memory, and the bus), which is impractical. However, the occupation of additional qubits is essentially the effect of the “tails” (if the discussed below problem of level crossing is avoided). Therefore, the desired “tails” can be obtained automatically by using sufficiently adiabatic ramps in the same way as discussed above (for a MOVE qubitbus the front ramp will be important for the “tails” from both sides, i.e. on the memory and other qubits). In analyzing the dynamics of the “tails” at other qubits, it is useful to think it terms of eigenstates of a truncated system, which includes the bus and other qubits (while excluding the qubit involved in the MOVE). Then the “tail” error is the occupation of the eigenstates, mainly localized on other qubits. Since the frequencies of the bus and other qubits do not change with time, for the error calculation it is still possible to use Eq. (24), in which is replaced with (for the “tail” on th qubit), is replaced with , and is replaced with , where the subscript labels additional qubit. This gives the estimate
of the error due to the “tail” on th qubit for the qubitbus MOVE, in which integration is within the front ramp, , and . The formula for the busqubit MOVE is similar, but the integration should be within the rear ramp. The error (25) should be summed over additional qubits (index ) and therefore can be significantly larger than in our calculations for a truncated system; however, the error increase is partially compensated by smaller effective coupling . Crudely, we expect errors below for smooth ramps of few-nanosecond duration and . We emphasize that this simple solution of the “tail” problem is possible only when we use eigenstates to represent the logical states.
Another problem which we did not encounter in the analysis of the truncated system is the level crossing with other (empty) qubits during the MOVE operation. A simple estimate of the corresponding error is the following. Effective resonant coupling between the moving qubit and another (th qubit) via the bus is , where and are two qubit-bus couplings and the detuning is the same for both qubits at the moment of level crossing. Then using the Landau-Zener formula we can estimate the error (population of the other qubit after crossing) as , where is the rate of the qubit frequency change at the crossing. Our numerical calculations show that this estimate works well, though up to a factor of about 2 [when curvature of at the point of crossing is significant]. Using this estimate for MHz, MHz, and MHz/ns, we obtain a quite significant error of about . A possible way to compensate this error is by using interference of the Landau-Zener transitions L-Z-compensation on the qubit return transition. Another solution of the problem is to park empty qubits outside the frequency range between the bus and memories (above 7 GHz in our example). This would make impossible to cancel the idling error of Eq. (6) by using the “midway parking”, but the idling error is still small even without this cancellation [see the estimate below Eq. (8)]. Besides the qubit-qubit level crossings, there are also level crossings between a moving qubit and other memories. This is a higher-order (weaker) process because of three steps between the qubit and memory. The effective coupling with th memory is then , and the level crossing error estimate is .
In this paper we do not analyze two-qubit gates. Our preliminary numerical simulation of the controlled- gate has shown possibility of a high-fidelity gate design (with the error of about , mainly due to level crossing). However, we have not studied this gate in detail. A detailed analysis of two-qubit gates in the RezQu architecture will be presented elsewhere JoydipGhosh11 .
V Tunneling measurement
Finally, let us discuss whether or not using the eigenstates as the logical states presents a problem for measurement. Naively, one may think about a projective measurement of an individual qubit; in this case the logic state “1” would be erroneously measured as “0” with probability of about because the eigenstate spreads to the neighboring memory and bus. This would be a very significant error, and the bare-state representation of logical states would be advantageous. However, this is not actually the case because any realistic measurement is not instantaneous (not projective). In fact, if a measurement takes longer than , then the eigenbasis is better than the bare basis.
As a particular example let us analyze tunneling measurement of a phase qubit Martinis-QIP (we expect a similar result for the qubit measurement in the circuit-quantum-electrodynamics (cQED) setup Blais04 ; Girvin08 ). The bare states and of a phase qubit correspond to the two lowest energy states in a quantum well, and the measurement is performed by lowering the barrier separating the well from essentially a continuum of states Martinis-QIP . Then the state tunnels into the continuum with a significant rate , while the tunneling rate for the state is negligible. The event of tunneling is registered by a detector “click” (the detector is a SQUID, which senses the change of magnetic flux produced by the tunneling). In the ideal case after waiting for a time the measurement error is negligibly small (in real experiments the ratio of the two tunneling rates is only , which produces a few-per-cent error; however, we neglect this error because in principle it can be decreased by transferring the state population to a higher level before the tunneling, and also because here we are focusing on the effect of “tails” in the neighboring elements).
In presence of the memory and resonator coupled to the qubit, the logic state “0” still cannot be misidentified, because the tunneling is impossible without an excitation. However, the logic state “1” can be misidentified as “0”, when sometimes the expected tunneling does not happen (because part of the excitation is located in the memory and resonator). Let us find probability of this error. For simplicity we consider a two-component model in which a phase qubit is coupled to its memory resonator only, and restrict the state space to the single-excitation subspace of this system. Then the tunneling process can be described by the non-Hermitian Hamiltonian (e.g., GOTTFRIED_ParticlePhysicsIandII )
and the error in measuring the logic state “1” (identifying it as “0” after measurement for time ) is its survival probability
where the initial state is normalized, . Our goal is to compare this error for the cases when the initial state is the bare state or the eigenstate (in this notation the qubit state is shown at the second place, and is the eigenstate before the measurement, i.e. when ).
The solution of the time-dependent Schrödinger equation, , is given by the linear combination, . Here the eigenstate notation with the tilde sign reminds of a non-zero , the constants depend on the initial conditions, and are the complex eigenenergies, which include the corresponding decay rates and of the eigenstates located mainly on the memory and the qubit. Diagonalizing the Hamiltonian (26) and assuming weak coupling, , , we find
(Color online) Time dependence of the
squared amplitudes
Figure 5: (Color online) Time dependence of the squared amplitudes and of the state , decaying in the process of tunneling measurement. Blue curves correspond to the system initially prepared in the bare state , while for red curves the initial state is the eigenstate . For the measurement error (27) is mainly the residual occupation of the memory resonator (solid curves). For the depicted system parameters at ns, i.e. the error of the eigenstate measurement is 40 times smaller than the error of the bare state measurement.
For measurement during a sufficiently long time , only the -term in survives, and correspondingly the error (27) is , where . Thus we obtain
for the measurement errors starting either with the eigenstate or with the bare state. Even though both errors decrease with the measurement time as , the rate is small [see Eq. (28)], so for a realistically long measurement we can use .
Equation (29) shows that from the measurement point of view it is advantageous to use the eigenstates to represent the logical states rather than the bare states if . For a typical value GHz this requires ns, which is always the case.
Figure 4 shows the state dynamics during the tunneling measurement in the bare basis , starting either with the eigenstate or with the bare state . The oscillations correspond to the beating frequency GHz. We see that similarly to the above-analyzed dynamics in the eigenbasis, becomes exponentially small after , while essentially saturates (decaying with a much smaller rate ). Then for the assumed tunneling rate ns the ratio of errors (with the subscript denoting the initial state) saturates at approximately the value 0.025 given by Eq. (29).
We emphasize that even though we have only considered the tunneling measurement, the result (29) for the measurement error is expected to remain crudely valid for most of realistic (i.e. “weak”) measurements with a time scale . In particular, for the cQED setup we expect that the role of is played (up to a factor) by the ensemble dephasing rate due to measurement.
Vi Conclusion
In summary, we have discussed the main ideas of the RezQu architecture and analyzed several error mechanisms, excluding analysis of two-qubit gates. The main advantage of the RezQu architecture is the strong ( times) reduction of the idling error compared to the conventional bus-based architecture, and also an effective solution to most of the problems related to spectral crowding. In the absence of decoherence this makes possible a simple scaling of a RezQu device to qubits without the need for dynamical decoupling. For further scaling the next architectural level of communication between the RezQu devices seems to be needed.
We have shown that instead of using bare states it is much better to use eigenstates to represent logical states. In this case there is essentially no dynamics in idling (except for the phase errors), which greatly simplifies a modular construction of a quantum algorithm. The logical encoding by eigenstates is also advantageous for the single-qubit state generation and measurement. We have presented a simple design for the MOVE operation, which is the most frequent operation in the RezQu architecture. We have shown that a four-parameter optimization is sufficient for designing a perfect MOVE in a truncated three-component system. Moreover, optimization of only two experimentally-obvious parameters is sufficient for high-fidelity MOVEs (with errors less than ). While we have not analyzed two-qubit gates, we expect that their design with similar high fidelity is also possible. Overall, we believe that the RezQu architecture offers a very significant advantage compared to the previously proposed architectures for superconducting qubits, and we believe that this is the practical way to progress towards a medium-scale quantum computing device.
This work was supported by IARPA under ARO grant W911NF-10-1-0334. The authors thank Michael Geller, Farid Khalili, Matteo Mariantoni, Leonid Pryadko, and Frank Wilhelm for useful discussions.
Appendix A Derivation of
In this Appendix we derive Eq. (6) for in the truncated system. We assume that the couplings and are of the same order, , and do calculations in fourth order in .
The RWA Hamiltonian (III) leads to the formation of three subspaces, which do not interact with each other: the ground state (with zero energy, ), the single-excitation subspace , and the two-excitation subspace . It is rather easy to find eigenenergies in the single-excitation subspace; neglecting the direct coupling , in fourth order in we obtain
To find , we write the eigenstate as a superposition of all elements of the two-excitation subspace,
with unimportant normalization. Then the Schrödinger equation (again neglecting ) gives six equations:
From the first three of them we obtain
which gives in fourth order in if we use second-order in the denominators (which is obtained from the same equation using zeroth-order ) and second-order amplitudes , , . These amplitudes can be found from the last three equations (A) using the first-order values , :
Finally, substituting Eq. (35) into Eq. (A), and using Eqs. (30), (31), we obtain Eq. (6) for in fourth order.
The above was the formal derivation of Eq. (6). Let us also obtain it approximately. Since in a linear system (excitations do not interact with each other), a non-zero value can come only from the qubit nonlinearity . Assuming small , we can use the first-order perturbation theory in to find the energy shift of due to contribution from (occupation of the qubit second level),
To find we start with the first-order (in ) eigenstate |
7961c73f5b5b6be9 | Tuesday, 31 March 2009
More Geekfreak / The Schrödinger Equation
Schrödinger Equation
In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics.
In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only atomic and subatomic systems, electrons and atoms, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger, who discovered it in 1926.
Schrödinger's equation can be mathematically transformed into Heisenberg's matrix mechanics, and into Feynman's path integral formulation. The Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem which is not as severe in Heisenberg's formulation and completely absent in the path integral.
(From Wikipedia: http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)
It's things like these that just make me wish I was better at the mathematical part of science. I really frickin' wish I knew exactly what all this is about. It's akin to a wormhole in my head leading to a whole new paradigm of knowledge that I could have but don't because I lack the electromagnetic field and space-time requirements for it.
Quantum physics - The dreams stuff is made of.
Audio Candy:
Avenged Sevenfold - Dear God
angie said...
hello you are such a geek but i like (:
Jose said... |
6051676f92597b45 | The old never gave in to the new. It became superfluous.
Anticipation – A Spooky Computation
Conference on Computing Anticipatory Systems (CASYS 99), Liege, Belgium, August 8-11, 1999
Mihai Nadin
Program in Computational Design
University of Wuppertal
Computer Science, Center for the Study of Language and Information
201 Cordura Hall
Stanford University
Robert Rosen, in memoriam
As the subject of anticipation claims its legitimate place in current scientific and technological inquiry, researchers from various disciplines (e.g., computation, artificial intelligence, biology, logic, art theory) make headway in a territory of unusual aspects of knowledge and epistemology. Under the heading anticipation, we encounter subjects such as preventive caching, robotics, advanced research in biology (defining the living) and medicine (especially genetically transmitted disease), along with fascinating studies in art (music, in particular). These make up a broad variety of fundamental and applied research focused on a controversial concept. Inspired by none other than Einstein–he referred to spooky actions at distance, i.e., what became known as quantum non-locality–the title of the paper is meant to submit my hypothesis that such processes are related to quantum non-locality. The second goal of this paper is to offer a cognitive framework–based on my early work on mind processes (1988)–within which the variety of anticipatory horizons invoked today finds a grounding that is both scientifically relevant and epistemologically coherent. The third goal of this paper is to identify the broad conceptual categories under which we can identify progress made so far and possible directions to follow. The fourth and final goal is to submit a co-relation view of anticipation and to integrate the inclusive recursion in a logic of relations that handles co-relations.
Keywords: auto-suggestive memory, co-relation, non-locality, quantum semiotics, self-constitution, interactive computation
1 Introduction
Anticipation could become the new frontier in science. Trends, scientific fashions, and priority funding programs succeed one another rapidly in a society that experiences a dynamics of change reflected in ever shorter cycles of discovery, production, and consumption. Frontiers mark stark discontinuities that ascertain fundamentally new knowledge horizons. Einstein stated, “No problem can be solved from the same consciousness that created it. We must learn to see the world anew.” It is in this respect that I find it extremely important to begin by putting the entire effort into a broad perspective.
2 The Philosophic Foundation of Anticipation is Not Trivial
Philosophical considerations cannot be avoided (provided that they are not pursued as a means in themselves). Robert Rosen (1985) quoted David Hawkins, “Philosophy may be ignored but not escaped.” Rosen, whose work deserves to be integrated in current scientific dialog more than was been the case until his untimely death, understood this thought very well.
Anticipation bears a heavy burden of interpretations. As initial attempts (Rosen, 1985; Nadin, 1988; Dubois, 1992) to recover the concept and to give it a scientific foundation prove, the task is difficult. We face here the dominant deterministic view inspired by a model of the universe in which a net distinction between cause and effect can be made. We also face a reductionist understanding of the world, which claims that physics is paradigmatic for everything else. Moreover, we are captive to an understanding of time and space that corresponds to the mathematical descriptions of the physical world: Time is uniquely defined along the arrow from past to future; space is homogeneous. Finally, we are given to the hope that science leads to laws on whose basis we may make accurate predictions. Once we accept these laws, anticipation can at best be accepted as one of these predictions, but not as a scientific endeavor on its own terms.
A clear image of the difficulties in establishing this foundation results from revisiting Rosen’s work on anticipatory systems, above all his fundamental work, Life Itself (1991). Indeed, his rigorous argumentation, based on solid mathematical work and on a grounding in biology second to none among his peers, makes sense only against the background of the philosophic considerations set forth in his writings. It might not matter to a programmer whether Aristotle’s causa finalis (final cause) can be ascertained or justified, or deemed as passé and unacceptable. A programmer’s philosophy does not directly affect lines of code; neither do disputes among those partial to a certain world view. What is affected is the general perspective, i.e., the understanding of a program’s meaning. If the program displays characteristics of anticipation, the philosophic grounding might affect the realization that within a given condition–such as embodied in a machine–the simulation of anticipatory features should not be construed as anticipation per se.
The philosophic foundation is also a prerequisite for defining how far the field can be extended without ending up in a different cognitive realm. Regarding this aspect, it is better to let those trying to expand the inquiry of anticipation–let me mention again Dubois (since 1996) and the notions of incursion and hyperincursion, Holmberg (since 1997) and space aspects–express themselves on the matter. Van de Vijver (1997), among few others (cf. CASYS 98 and the contributions listed in the Program for CASYS 99) has already attempted to shed light on what seems philosophically pertinent to the subject. She is right in stating that the global/local relation more adequately pertains to anticipation than does the pair particular/universal. The practical implications of this observation have not yet been defined.
From my own perspective–based on pragmatics, which means grounding in the practical experience through which humans become what they are–anticipation corresponds to a characteristic of live beings as they attain the condition at which they constitutes their own nature. At this level, predictive models of themselves become possible, and progressively necessary. The thematization of anticipation, which as far as we know is a human being’s expression of self-awareness and connectedness, is only one aspect of this stage in the unfolding of our species. According to the premise of this perspective, pragmatics–expressed in what we do and how and why we do what we do–is where our understanding of anticipation originates. This is also where it returns, in the form of optimizing our actions, including those of defining what these actions should be, what sequence they follow, and how we evaluate them. All these are projections against a future towards which each of us is moving, all tainted by some form of finality (telos), or at least by its less disputed relative called intentionality. The generic why of our existence is embedded in this intentionality. The source of this finality are the others, those we interact with either in cooperating or in competing, or in a sense of belonging, which over time allowed for the constitution of the identity called humanness. Gordon Pask (1980), the almost legendary cybernetician, called such an entity a cognitive system.
2.1 Self-Entailment and Anticipation
In a dialog on entailment (cf.–a fundamental concept in Rosen’s explanation of anticipation–a line originating with François Jacob was dropped: “Theories come and go, the frog stays.” (Incidentally, Jacob is the author of The Logic of Life, Princeton University Press, 1993). This brings us back to a question formulated above: Does it matter to a programmer (the reader may substitute his/her profession for the word programmer) that anticipation is based on the self-entailment characteristic of the living? Or that evolution is the source of entailment? If we compare the various types of computation acknowledged since people started building computers and writing software programs, we find that during the syntactically driven initial phases, such considerations actually could not affect the pragmatics of programming. Only relatively recently has a rudimentary semantic dimension been added to computation. In the final analysis, it does not matter which microelectronics, computer architecture, programming languages, operating systems, networks, or communication protocols are used. For all practical purposes, what matters is that between the world and the computation pertinent to some aspects of this world, the relations are still extremely limited. If a programmer is not just in the business of writing lines of code for a specific application that might improve through a syntactically supported emulation of anticipatory characteristics–think about macros that save typing time by “guessing” which word or expression a user started to type in and “filling in” the letters or words–then it matters that there is something like self-entailment. It matters, too, that the notion of self-entailment supports more adequate explanations of biological processes than any other concept of the physical sciences. On a semantic level, the awareness of self-entailment (through self-associative memory) leads to better solutions in speech and handwriting recognition.
However, once the pragmatic level is reached–we are still far from this–understanding the philosophic implications of the nature and condition of anticipation becomes crucial. The reason is that it is not at all clear that characteristics of the living–self-repair, metabolism, and anticipation–can be effectively embodied in machines. This is why the notion of frontier science was mentioned in the Introduction. The frontier is that of conceiving and implementing life-like systems. Whether Rosen’s (M, R)-model, defined by metabolism and repair, or others, such as those advanced in neural networks, evolutionary computation, and ALife, will qualify as necessary and sufficient for making anticipation possible outside the realm of the living remains to be seen. I (Nadin, 1988, 1991) argue for computers with a variable configuration based on anticipatory procedures. This model is inspired by the dynamics of the constitution and interaction of minds, but does not suggest an imitation of such processes. The issue is not, however, reducible to the means (digital computation, algorithmic, non-algorithmic, or heterogenous processing, signal processing, quantum computation, etc.), but to the encompassing goal.
2.2 Specializations
To nobody’s surprise, anticipation, in some form or another, is part of the research program of logic, cognitive science, computer science, robotics, networking, molecular biology, genetics, medicine, art and design, nanotechnology, the mathematics of dynamic systems, and what has become known as ALife, i.e., the field of inquiry into artificial life. Anticipation involves semiotic notions, as it involves a deep understanding of complexity, or, better yet, of an improved understanding of complexity that integrates quantitative and qualitative aspects.
It is not at all clear that full-fledged anticipation, in the form of machine-supported anticipatory functioning, is a goal within the reach of the species through whose cognitive characteristics it came into being and who became aware of it. Machines, or computations, for those who focus on the various data processing machines, able to anticipate earthquakes, hurricanes, aesthetic satisfaction, disease, financial market performance, lottery drawings, military actions, scientific breakthroughs, social unrest, irrational human behavior, etc., could well claim total control of our universe of existence. Indeed, to correctly anticipate is to be in control. This rather simplistic image of machines or computations able to anticipate cannot be disregarded or relegated to science fiction. Cloning is here to stay; so are many techniques embodying the once disreputed causa finalis. A philosophic foundation of anticipation has to entertain the many questions and aspects that pertain to the basic assertion according to which anticipation reflects part of our cognitive make-up, moreover, constitutes its foundation. Even if Kuhn’s model of scientific paradigm change had not been abused to the extent of its trivialization, I would avoid the suggestion that anticipation is a new paradigm. Rather, as a frontier in science, it transcends its many specializations as it establishes the requirement for a different way of thinking, a fundamentally different epistemological foundation.
3 Pro-Action vs. Re-Action
Now that the epistemological requirement of a different way of thinking has been brought up, I would like to revisit work done during the years when the very subject of anticipation seemed not to exist (except in the title of Rosen’s book). My claim in 1988 (on the occasion of a lecture presented at Ohio State University) was that anticipation lies at the foundation of the entire cognitive activity of the human being. Moreover, through anticipation, we humans gain insight into what keeps our world together as a coherent whole whose future states stand in correlation to the present state as minds grasp it. Minds exist only in relation to other minds; they are instantiations of co-relations. This is also the main thesis of this paper.
For over 300 years–since Descartes’ major elaborations (1637, 1644) and Newton’s Principia (1687)–science has advanced in understanding what for all practical purposes came to be known as the reactive modality. Causality is experienced in the reactive model of the universe, to the detriment of any pro-active manifestations of phenomena not reducible to the cause-and-effect chain or describable in the vocabulary of determinism. It is important to understand that what is at issue here is not some silly semantic game, but rather a pragmatic horizon: Are human actions (through which individuals and groups identify themselves, i.e., self-constitute, Nadin 1997) in reaction to something assumed as given, or are human actions in anticipation of something that can be described as a goal, ideal, or value? But even in this formulation (in which the vocabulary is as far as it can be from the vitalistic notions to which Descartes, Newton, and many others reacted), the suspicion of teleological dynamics–is there a given goal or direction, a final vector?–is not erased. Despite progress made in the last 30 years in understanding dynamic systems, it is still difficult to accept the connection between goal and self-organization, between ideal, or value, and emergent properties.
3.1 Minds Are Anticipations
The mind is in anticipation of events, that is, ahead of them–this was my main thesis over ten years ago. Advanced research (Libet 1985, 1989) on the so-called “readiness potential” supported this statement. In recent years, work on the “wet brain” as well as work supported by MR-based visualization technologies have fully confirmed this understanding. Having entered the difficult dialog on the nature of cognitive processes from a perspective that no longer accepted the exclusive premise of representation –another heritage from Descartes–I had to examine how processes of self-constitution eventually result in shared knowledge without the assumption of a homunculus. What seemed inexplicable from a perspective of classical or relativist physics–a vast amount of actions that seemed instantaneous, in the absence of a better explanation for their connectedness–was coming into focus as constitutive of the human mind. Anticipatory cognitive and motoric scripts, from which in a given context one or another is instantiated, were advanced at that time as a possible description for how, from among many pro-active possible courses of action, one would be realized. Today I would call those possible scripts models and insist that a coherent description of the functioning of the mind is based on the assumption that there are many such models. Additionally, I would add that learning, in its many realizations, is to be understood as an important form of stimulating the generation of models, and of stimulating a competitive relation among them. [Von Foerster (1999) entertains a motto on his e-mail address that is an encapsulation of what I just described: “Act always as to increase the number of choices.”] In a subtle way, defense mechanisms–from blinking to reflexes of all types–belong to this family. Anticipatory nausea and vomiting (whether on a ship or related to chemotherapy) is another example. The phantom limb phenomenon (sensation in the area of an amputated limb) is mirrored by pain or discomfort before something could have actually caused them. There is a descriptive instance in Lewis Carroll’s Through the Looking Glass. Before accidentally pricking her finger, the White Queen cries: “I haven’t pricked it yet, but I soon shall.” She lives life in reverse, which is what anticipation ultimately affords–provided that the interpretation process is triggered and made part of the self-constitutive pragmatics.
3.1.1 Anticipation is Distributed
As recently as this year, results in the study of the anticipation of moving stimuli by the retina (Berry, et al 1999) made it clear that anticipation is distributed. The research proved that anticipation of moving stimuli begins in the retina. It is no longer that we expect the visual cortex to do some heavy extrapolation of trajectory (this was the predominant model until recently) but that we know that retinal processing is pro-active. Even if pro-activity is not equally distributed along all sensory channels–some are slower in anticipating than others, not the least because sound travels at a slower speed than light does, for example–it defines a characteristic of human perception and sheds new light on motoric activity.
3.1.2 Knowledge as Construction
But there is also Kelly’s (1995) constructivist position, which must be acknowledged by researchers in the psychological foundation of anticipation. The adequacy of our constructs is, in his view, their predictive utility. Coherence is gained as we improve our capacity to anticipate events. Knowledge is constructed; validated anticipations enhance cognitive confidence and make further constructs possible. In Kelly’s terms, human anticipation originates in the psychological realm (the mind) and reflects the intention to make possible a correspondence between a future experience and certain of our anticipations (Kelly, 1955; Mancuso & Adams-Weber, 1982). Since states of mind somehow represent states of the world, adequacy of anticipations remains a matter of the test of experience. The basic function of all our representations, as the “fundamental postulate” ascertains, is anticipation (a temporal projection). Alternative courses of action in respect to their anticipated consequences represent the pragmatic dimension of this view.
Observed phenomena and their descriptions are not independent of the assumptions we make. This applies to the perceptual control theory, as it applies to Kelly’s perspective and to any other theory. Moreover, assumptions facilitate or hinder new observations. For those who adopted the view according to which a future state cannot affect a present state, anticipation makes no sense, regardless of whether one points to the subject in various religious schemes, in biology, or in the quantum realm. The situation is not unlike that of Euclidean geometry vs. non-Euclidean geometries. To see the world anew is not an easy task!
Anticipation of moving stimuli, to get back to the discovery mentioned above, is recorded in the form of spike trains of many ganglion cells in the retina. It follows from known mechanisms of retinal processing; in particular, the contrast-gain control mechanism suggests that there will be limits to what kinds of stimuli can be anticipated. Researchers report that variations of speed, for instance, are important; variations of direction are not. Furthermore, since space-based anticipation and time-based anticipation have a different metric, it remains to be seen whether a dominance of one mode over the other is established. As we know, in many cases the meeting between a visual map (projection of the retina to the tectum) and an auditory map takes place in a process called binding. How the two maps are eventually aligned is far from being a matter of semantics (or terminology, if you wish). Synchronization mechanisms, of a nature we cannot yet define, play an important role here. Obviously, this is not control of imagination, even if those pushing such terms feel more forceful in the de facto rejection of anticipation.
Arguing from a formal system to existence is quite different from the reverse argumentation (from existence to formalism). Arguing from computation can take place only within the confines of this particular experience: the more constrained a mechanism, the more programmable it is (as Rosen pointed out, 1991, p. 238). Albeit, reaction is indeed programmable, even if at times it is not a trivial task. Pro-active characteristics make for quite a different task. The most impressive success stories so far are in the area of modeling and simulation. To give only one example: Chances are that your laptop (or any other device you use) will one day fall. The future state–stress, strain, depending upon the height, angle, weight, material, etc.–and the current state are in a relation that most frequently does not interest the user of such a portable device. It used to be that physical models were built and subjected to tests (this applies, for instance, to cars as well as to photo cameras). We can model, and thus to a certain point anticipate, the effects of various possible crashes through simulations based on finite-element analysis. That anticipation itself, in its full meaning, is different in nature from such simulations passes without too much comment. The kind of model we need in order to generate anticipations is a question to which we shall return.
3.2 A Rapidly Expanding Area of Inquiry
An exhaustive analysis of the database of the contributions to fundamental and applied research of anticipation reveals that this covers a wide area of inquiry. In many cases, those involved are not even aware of the anticipatory theme. They see the trees, but not yet the forest. More telling is the fact that the major current directions of scientific research allow for, or even require, an anticipatory angle. The simulation mentioned above does not anticipate the fall of the laptop; rather, it visualizes–conveniently for the benefit of designers, engineers, production managers, etc.–what could happen if this possibility were realized. From this possibilistic viewpoint, we infer to necessary characteristics of the product, corresponding to its use (how much force can be exercised on the keyboard, screen, mouse, etc.?) or to its accidental fall. That is, we design in anticipation of such possibilities. Or we should! I would like to mention other examples, without the claim of even being close to a complete list.
3.2.1 An Example from Genetics
But more than Rosen, whose work belongs rather to the meta-level, it was genetics that recovered the terminology of heredity. Having done so, it established a framework of implicit anticipations grounded in the genetic program. Of exceptional importance are the resulting medical alternatives to the “fix-it” syndrome of healthcare practiced as a “car repair” (including the new obsession with spare parts and artificial surrogates). Genetic medicine, as slow in coming as it is, is fundamentally geared towards the active recognition of anticipatory traits, instead of pursuing the reactive model based on physical determinism. Although there is not yet a remedy to Huntington’s disease, myotonic dystrophy, schizophrenia, Alzheimer’s disease, or Parkinson’s disease, medical researchers are making progress in the direction of better understanding how the future (the eventual state of diagnosed disease) co-relates to a present state (the unfolding of the individual in time). In the language of medicine, anticipation describes the tendency of such hereditary diseases to become symptomatic at a younger age, and sometimes to become more severe with each new generation.
We now have two parallel paths of anticipation: one is that of the disorder itself, i.e., the observed object; the other, that of observation. The elaborations within second-order cybernetics (von Foerster, 1976) on the relation between these paths (the classical subject-object problem) make any further comment superfluous. The convergence of the two paths, in what became known as eigen behavior (or eigen value), is of interest to those actively seeking to transcend the identification of genetic defects through the genetic design of a cure. After all, a cure can be conceived as a repair mechanism, related to the process of anticipation.
3.2.2 Art, Simulacrum, Fabrication
That art (healing was also seen as a special type of art not so long ago), in all its manifestations, including the arts of writing (poetry, fiction, drama), theatrical performance, and design–driven by purpose (telos) and in anticipation of what it makes possible–incorporates anticipatory features might be accepted as a metaphor. But once one becomes familiar with what it means to draw, paint, compose, design, write, sing, or perform (with or without devices), anticipation can be seen as the act through which the future (of the work) defines the current condition of the individual in the process of his or her self-constitution as an artist. What is interesting in both medicine and art is that the imitation can result only in a category of artifacts to be called simulacrum. In other words, the mimesis approach (for example, biomimesis as an attempt to produce organisms, i.e., replicate life from the inanimate; aesthetic mimesis, replicating art by starting with a mechanism such as the one embodied in a computer program) remains a simulacrum. Between simulacra and what was intended (organisms, and, respectively, art) there remains the distance between the authentic and the imitation, human art and machine art. They are, nevertheless, justified in more than one aspect: They can be used for many applications, and they deserve to be valued as products of high competence and extreme performance. But no one could or should ignore that the pragmatics of fabrication, characteristic of machines, and the pragmatics of human self-constitution within a dynamic involving anticipation are fundamentally different.
3.2.3 Learning (Human and Machined-Based)
Learning–to mention yet another example–is by its nature an anticipatory activity: The future associates with learning expectations and a sui generis reward mechanism. These are very often disassociated from the context in which learning takes place. That this is fundamentally different from generating predictive models and stimulating competition among them might not be totally clear to the proponents of the so-called computational learning theory (COLT), or to a number of researchers of learning–all from reputable fields of scientific inquiry but captive to the action-reaction model dominant in education. It is probably only fair to remark in this vein that teaching and learning experiences within the machine-based model of current education are not different from those mimicked in some computational form. Computer-based training, a very limited experience focused on a well defined body of information, can provide a cost-efficient alternative to a variety of training programs. What it cannot do is to stimulate and trigger anticipatory characteristics because, by design, it is not supposed to override the action-reaction cycle.
3.2.4 Reward
Alternatively, one can see promise in the formalism of neural networks. For instance, anticipation of reward or punishment was observed in functional neuroanatomy research (cf. Knutson, 1998). Activation of circuitry (to use the current descriptive language of brain activity) running from the medial dorsal thalamus through the anterior cingulate and mesial prefrontal cortex was co-related not to motor response but to personality variations. Accordingly, it is quite tempting to look at such mechanisms and to try to introduce reward anticipation in neural networks procedures as a method of increasing the performance of artificially mimicked decision-making. Homan (1997) reports on neural networks that “can anticipate rewards before they occur, and use these expectations to make decisions.” The focus of this type of research is to emulate biological processes, in particular the dopamine-based rewarding mechanism that lies behind a variety of goal-oriented mechanisms. Dynamic programming supports a similar objective. It focuses on states; their dynamic reassessment is propagated through the neural network in ways considered similar to those mapped in the successful enlisting of brain capabilities. Training, as a form of conditioning based on anticipation, is probably complementary to what one would call instinct-based (or natural) action.
3.2.5 Motion Planning
Animation and robot motion planning, as distant from each other as they appear to some of us, share the goal of providing path planning, that is, to find a collision-free path between an initial position (the robot’s arm or the arm of an animated character) and a goal position. It is clear that the future state influences the current state and that those planning the motion actually coordinate the relation between the two states. In predictive programs, anticipation is pursued as an evaluation procedure among many possibilities, as in economics or in the social sciences. The focus changes from movement (and planning) to dynamics and probability. A large number of applications, such as pro-active error detection in networks, hard-disk arm movement in anticipation of future requests, traffic control, strategic games (including military confrontation), and risk management prompted interest in the many varieties under which anticipatory characteristics can be identified.
3.3 Aspects of Anticipation
At this point, where understanding the difference between anticipation as a natural entailment process and embodying anticipatory features in machine-like artifacts meet, it is quite useful to mention that expectation, prediction, and planning–to which others add forecasting and guessing–are not fully equivalent to anticipation, but aspects of it. Let us also make note of the fact that we are not pursuing distinctions on the semantic level, but on the pragmatic–the only level at which it makes sense to approach the subject.
3.3.1 Expectation, Prediction, Forecast
The practical experience through which humans constitute themselves in expectation of something–rain (when atmospheric conditions are conducive), meeting someone, closing a transaction, etc.–has to be understood as a process of unfolding possibilities, not as an active search within a field of potential events. Expectation involves waiting; it is a rather passive state, too, experienced in connection with something at least probable. Predictions are practical experiences of inferences (weak or strong, arbitrary or motivated, clear-cut or fuzzy, explicit or implicit, etc.) along the physical timeline from past to the future. Checking the barometer and noticing pain in an arthritic knee are very different experiences; so are the outcomes: imperative prediction or tentative, ambiguous foretelling. To predict is to connect what is of the nature of a datum (information received as cues, indices, causal identifiers, and the like) experienced once or more frequently, and the unfolding of a similar experience, assumed to lead to a related result. It should be noted here that the deterministic perspective implies that causality affords us predictive power. Based on the deterministic model, many predictive endeavors of impressive performance are succesfully carried out (in the form of astronomical tables, geomagnetic data, and calculations on which the entire space program relies). Under certain circumstances (such as devising economic policies, participating in financial markets, or mining data for political purposes), predictions can form a pragmatic context that embodies the prediction. In other words, a self-referential loop is put in place.
Not fundamentally different are forecasts, although the etymology points to a different pragmatics, i.e., one that involves randomness. What pragmatically distinguishes these from predictions is the focus on specific future events (weather forecasting is the best known pragmatic example, that is, the self-constitution of the forecaster through an analytic activity of data acquisition, processing, and interpretation, whose output takes very precise forms corresponding to the intended communication process). These events are subject to a dynamics for which the immediate deterministic descriptions no longer suffice. Whether economic, meteorological, geophysical (regarding earthquakes, in particular), such forecasts are subject to an interplay of initial conditions, internal and external dynamics, linearity, and nonlinearity (to name only a few factors) that is still beyond our capacity to grasp, moreover to express in some efficient computational form. Although forecasts involve a predictive dimension, the two differ in scope and in the specific method. A computer program for predicting weather could process historic data (weather patterns over a long period of time). Its purpose is global prediction (for a season, a year, a decade, etc.). A forecasting algorithm, if at all possible, would be rather local and specific: Tomorrow at 11:30 am. Dynamic systems theory tells us how much more difficult forecasting is in comparison with prediction.
Our expectations, predictions, and forecasts co-constitute our pragmatics. That is, they participate in making the world of our actions. There is formative power in each of them. Although expecting, predicting, and forecasting good weather will not bring the sun out, they can lead to better chances for a political candidate in an election. Indeed, we need to distinguish between categories of events to which these forms of anticipation apply. Some are beyond our current efforts to shape events and will probably remain so; others belong to the realm of human interaction. Recursion would easily describe the self-referential nature of some particular anticipations: expected outcome = f(expectation). That such cases basically belong to the category of indeterminate problems is more suspected than acknowledged. Mutually reinforcing expectations, predictions, and forecasts are the result of more than one hypothesis and their comparative (not necessarily explicit) evaluation. This model can be relatively efficiently implemented in genetic computations.
3.3.2 Plans, Design, Management
Plans are the expression of well or less well defined goals associated with means necessary and sufficient to achieve them. They are conceived in a practical experience taking place under the expectation of reaching an acceptable, optimal, or high ratio between effort and result. Planning is an active pursuit within which expectations are encoded, predictions are made, and forecasts of all kind (e.g., price of raw materials and energy sources, weather conditions, individual and collective patterns of behavior, etc.) are considered. Design and architecture as pragmatic endeavors with clearly defined goals (i.e., to conceive of everything that qualifies as shelter and supports life and work in a “sheltered” society: housing, workplace, various institutions, leisure, etc.) are particular practical experiences that involve planning, but extend well beyond it, at least in the anticipatory aesthetic dimension. Every design is the expression of a possible future state–a new chip, a communication protocol, clothing, books, transportation means, medicine, political systems or events, erotic stimuli, meals–that affects the current state–of individuals, groups, society, etc.–through constitution of perceived and acknowledged needs, expectations, and desires. The dynamics of change embodied in design anticipations is normally higher than that of all other known human practical experiences.
Policy, management, and prevention (to name a few additional aspects or dimensions of anticipation) involve giving advance thought, looking forward, directing towards something that as a goal influences our actions in reaching it. All these characteristics are part of the dictionary definitions of anticipation. The various words (such as those just referred to) involved in the scientific discourse on anticipation, i.e., its various meanings, pertain to its many aspects; but they are not equivalent.
3.4 Resilience
It is probably useful to interrupt this account of the many ways through which anticipation penetrates the scientific agenda and to invoke a distinction that, in the beginning, defies our acquired understanding of anticipation, at least along the distinctions made above. In a deceptively light presentation, Postrel (1997) suggests a counterdistinction: resilience vs. anticipation. If the subject were only what distinguishes Silicon Valley from the Boston area, both known as regions of technical innovation and fast economic growth, the two elements invoked–predictable weather patterns, and earthquakes, anything but predictable–we would not have to bother. However, her article presents the political theory of a proficient political scholar, Wildawski (1988), focused on meeting the challenge of risk through anticipation, understood as planning that aspires to perfect foresight, or through resilience, a dynamic response based on providing adjustments. The definitions are quite telling: “Anticipation is a mode of control by a central mind; efforts are made to predict and prevent potential dangers before damage is done. . . . Resilience is the capacity to cope with unanticipated dangers after they have become manifest, learning to bounce back.” Not surprising is the inference that “anticipation seeks to preserve stability: the less fluctuation, the better. Resilience accommodates variability. . . .”
We seem to have here a reverse view of all that has been presented so far: Anticipation means to see the world as predictable. But it also qualifies anticipation as being quite inappropriate within dynamic systems, that is, exactly where anticipation makes a difference! Rapid changes, especially unexpected turns of events, seem the congenial weakness of anticipation in this model. (Those critical of the evolution theory refer to punctuated equilibrium, i.e., fast change for which evolution theory has yet to produce a convincing account.) Hubristic central planning and over-caution can undermine anticipation. This view of anticipation would also imply that it cannot be properly pursued within open systems or within transitory processes–again, where we could most benefit from it. Resilience depends on spontaneity, serendipity, on the unforeseeable. Wildavsky expressed this in rather sweeping statements: “. . . not only markets rely on spontaneity; science and democracy do as well. . . .” Computations of risk are, of course, also part of the subject of anticipation.
3.5 Synchronization
Yet another element of this methodological overview (far from being complete) is synchronization. It can serve here as a terminological cue, or, to recall Rosen (1991), co-temporality or simultaneity would do. In the canonical description of anticipation–the current state of the system is defined by a future state–one aspect of time, sequentiality or precedence (one instant precedes the other) takes over. Yet in the universe of simultaneous events, we encounter anticipation, not only as it refers to space aspects, but as it takes the form of synchronization mechanisms. Whether in genetic mechanisms, in musical perceptions (where temporality is definitory), or in the perception of the world (I have already mentioned above the way in which the visual and the auditory “map” are brought in sync, the so-called binding problem, i.e., integration of sensory information arriving on different channels), to name just a few, the coordination mechanism is the final guarantor of the system’s coherent functioning. As a synchronization mechanism, anticipation means to “know” (the quotation marks are used to identify a way of speaking) when relatively unrelated, or even related, events have to be integrated in order to make sense. It is therefore helpful to consider this particular kind of anticipation as the result of the work of a “conductor” (or switch, for those technically inclined) eliciting the various sound streams originating from independent sources, each operating within its own confines, to merge in a synchronized concert. Cognitively, this means to ensure that what is synchronous in the world is ultimately perceived as such, although information arrives asynchronously in the brain.
Synchronization, as opposed to precedence, is not tolerant of error. Precedence is less restrictive: The cold temperatures that might affect the viability (survival) of a deciduous tree, and the cycle of days and night affected by the cycle of seasons allow for a range. This is why leaves fall over a relatively long time, depending upon tree kinds and configurations (lone trees, groves, forests, etc.). So we learn that not only is there a variety of soft-defined forms of anticipation (weather prediction, even after data collection, processing, and interpretation have made spectacular advances, is as soft as soft gets), but also that there are high precision mechanisms that deserve to be accounted for if we expect to understand, and moreover make use of, anticipatory technologies.
3.6 Some Working Hypotheses
3.6.1 Rosen’s Model
Rosen distinguishes the difference between the dynamics of the coupled given object system S and the model M; that is, the difference between real time in S and the modeling time of M (faster than that of S) is indicative of anticipation. True, time in this particular description ceases to be an objective dimension of the world, since we can produce quite a variety of related and unrelated time sequences. He also remarks that the requirement of M to be a perfect model is almost never fulfilled. Therefore, the behavior of such a coupled system can only be qualified as quasi-anticipatory (in which E represents effectors through which action is triggered by M within S); cf. Fig. 1.
Fig. 1 Rosen’s model
As aspects of this functioning, Rosen names, rather ambiguously, planning, management, and policies. Essential here are the parametrization of M and S and the choice of the model. The standard definition, quoted again and again, is that an anticipatory system “contains a predictive model of itself and/or of its environment, which allows it to change state at an instant in accord with the model’s predictions pertaining to a later instant” (Rosen 1985, p. 339). The definition is not only contradictory–as Dubois (1997) noticed–but also circular–anticipation as a result of a weaker form of anticipation (prediction) exercised through a model.
Much more interesting are Rosen’s examples: “If I am walking in the woods and I see a bear appear on the path ahead of me, I will immediately tend to vacate the premises”; the “wired-in” winterizing behavior of deciduous trees; the biosynthetic pathway with a forward activation. Each sheds light on the distinction between processes that seem vaguely correlated: background information (what could happen if the encounter with the bear took place, based on what has already happened to others); the cycle of day and night and the related pattern of lower temperatures as days get shorter with the onset of autumn; the pathway for the forward activation and the viability of the cell itself. What is not at all clear is how less than obvious weak correlations end up as powerful anticipation links: heading away from the bear (“I change my present course of action, in accordance with my model’s prediction,” 1985, p. 7) usually eliminates the danger; loss of leaves saves the tree from freezing; forward activation, as an adaptive process, increases the viability of the cell. We have a “temporal spanning,” as Rosen calls it. In his example of senescence (“an almost ubiquitous property of organisms,” “a generalized maladaptation without any localizable failure in specific subsystems,” 1985, p. 402), it becomes even more clear that the time factor is of essence in the biological realm.
3.6.2 Inclusive Recursion (the Dubois Path)
Dubois (1997, p. 4) is correct in pointing out that this approach is reminiscent of classical control theory. He submits a formal language of inclusive (or implicit) recursion, more precisely, of self-referential systems, in which the value of a variable at a later time (t+1) explicitly contains a predictive model of itself (p. 6):
x(t+1) = f[x(t), x(t+1), p), p] (1a)
In this expression, x is the state variable of the system, t stands for time (present, t–1 is the past, t+1 is the future), and p is a control parameter. Dubois starts from recursion within dynamical discrete systems, where the future state of a system depends exclusively on its present and past
x(t+1) = f[… x(t–1), x(t), x(t+1), p] (1b)
He further defines incursion, i.e., an inclusive or implicit recursion, as
x(t+1) = f[… x(t–2), x(t–1), x(t), x(t+1), …, p] (2)
and exemplifies its simplest case as a self-referential system (cf. 1a and 1b). The embedded nature of such a system (it contains a model of itself) explains some of its characteristics, in particular the fact that it is purpose (i.e., finality, or telos) driven.
Having provided a mathematical description, Dubois further reasons from the formalism submitted to the mechanism of anticipation:
The dynamic of the system is represented by
D S/D t = [S(t+D t) – S(t)]/ D t = F[S(t), M(t+D t)] (3)
That of the predictive model is:
D M/D t = [M(t+D t) – M(t) = G[M(t)] (4)
In order to avoid the contradiction in Rosen’s model, Dubois suggests that
D M/D t = [M(t+D t) – M(t)]/ D t = F[S(t), M(t+D t)] (5)
Obviously, what he ascertains is that there is no difference between the system S and the anticipatory model, the result being
D S/D t = [S(t+D t) – S(t)]/ D t = F[S(t), S(t+D t)] (6)
which is, according to his definition, an incursive system.
That Rosen and Dubois take very different positions is clear. In Rosen’s view, since the “heart of recursion is the conversion of the present to the future” (1991, p. 78), and anticipation is an arrow pointing in the opposite direction, recursions could not capture the nature of anticipatory processes. Dubois, in producing a different type of recursion, in which the future affects the dynamics, partially contradicts Rosen’s view.
Incursion (inclusive or implicit recursion) and hyperincursion (an incursion with multiple solutions) describe a particular kind of predictive behavior, according to Dubois. Building upon McCulloch and Pitts (1943) formal neuron and taking von Neumann’s suggestion that a hybrid digital-analog neuron configuration could explain brain dynamics, Dubois (1990, 1992) submitted a fractal model of neural systems and furthered a non-linear threshold logic (with Resconi, 1993). The incursive map
x(t) = 1 – abs(1–2x(t+1)) (7)
where “abs” means “the absolute value” and in which the iterated x(t) is a function of its iterate at a future time t+1, can subsequently be transformed into a hyper-recursive map:
1 – 2x(t+1) = ± (1–x(t)) (8)
so that
x(t+1) = [1 ± x(t)–1]/2 (9)
It is clear that once an initial condition x(0) is defined, successive iterated values x(t+1), for t=0,1,2,…T, produce two iterations corresponding to the ± sign. In order to avoid the increase of the number of iterated values, i.e., in order to define a single trajectory, a control function u(T–k) is introduced. The resulting hyperincursive process is expressed through
x(t+1) = [1 + (1–2u(t+1))(x(t)–1]/2 = x(t)/2 + u(t+1) – x(t) · u(t+1)(10)
It turns out that this equation describes the von Neumann hybrid version through the x(t) as a floating point variable and the control function u(t) as a digital variable, accepting 0 and 1 as values, so that the sign + or – result from
Sg = 2u(t) – 1, for t=1, 2,…T (11)
It is tempting to see this hybrid neuron as a building block of a functional entity endowed with anticipatory properties. Let me add here that Dubois has continued his work in the direction of producing formal descriptions for neural net applications, memory research, and brain modeling (1998). His work is convincing, but, again, it takes a different direction from the work pursued by Rosen, if we correctly understand Rosen’s warning (1991) concerning the non-fractionability of the (M, R)-system, i.e., its intrinsic relational character. Nevertheless, Dubois’ results will be seen by many as another suggestion that the hybrid analog/digital computation better reflects the complexity of the living and thus might support effective information processing for applications in which the living is not reduced to the physical.
3.6.3 Space-Based Computation
Cellular automata, as discrete space-time models, constitute yet another way of modeling anticipation as a space-based computation. More details can be found in the work of Holmberg (1997), who introduces the concept of spatial automata and correctly positions this approach, as well as some basic considerations on the nature of anticipation in technological applications, within systems theory. Not surprisingly, the community of researchers of anticipation is generating further working hypotheses (Julià 1998; Sommer, 1998, addressing intentionality and learnability, respectively). It is very difficult to keep a record of all of these contributions, and even more difficult to comment on works in their incipient phase. Applications of fundamental theoretical anticipatory models are also being submitted in increasing numbers. Dubois himself suggested quite a number of applications, including robotics and neural machines. My focus is on variable configuration computers (regardless of the nature of computation). Obviously, those and similar attempts (many in the program of the CASYS conferences) are quite different from training in various sports, sports performance (think about anticipation in fencing!), political action, the functioning of the judicial system, the dissemination of writing rules for achieving suspense, the automatic generation of jokes (Barker, 1996), the building of economic models, and so on.
3.6.4 Dynamic Competing Models
Without attempting to submit a full-fledged alternative to either Rosen’s or Dubois’ anticipation descriptions, I will only mention once more that my own work speaks in favor of a changing set of models and of a procedure for maintaining competition among them.
Fig. 2 Changing models and competition among models
Since a diagram is a formalism of sorts, not unlike a mathematical or logical expression, I also reason from it to the dynamics of the system. The diagram ascertains that anticipation implies awareness, and thus processes of interpretation—hence semiotic processes. Mathematical or logical descriptions do not explicitly address awareness, but rather build upon it as a given. Some scientists subsequently commit the error of assuming that because awareness is not explicitly encoded in the formulae, it plays no role whatsoever in the system described. As we shall see in the discussion of the non-local nature of anticipation, quantum experiments suggest that in the absence of the observer, our descriptions of the universe make no sense.
3.6.5 Variability and Computation
To make things even more challenging, there are instances in which anticipation, resulting from the dynamics of natural evolution, is subject to variability, i.e., change. In every game situation, anticipations are at work in a competitive environment. Chess players, not unlike “black-box” traders on the financial or stock markets, as well as professional gamblers, could provide a huge amount of testimony regarding “anticipation as a moving target.” In my model of an anticipation mechanism based on a changing number of models and on stimulating competition among them, games can serve as a source of information in the validation process. The mathematics of game theory, not unlike the mathematics of ALife formal descriptions applied to trading mechanisms or to flocking behavior, is in many respects pertinent to questions of anticipation.
What is not explicitly provided through the ever expanding list of application examples is the broad perspective. Indeed, when the performing musician of a well known musical score seeks an expression that deviates from the expected sound (without being unfaithful to the composer), we have anticipation at work: not necessarily as a result of an understanding of its many implications, rather as a spontaneously developed means of expression. Many similar anticipation-based characteristics are recognizable in the practical human experience of self-constitution in competitive situations, in survival instances (some action performed ahead of the destructive instant), in the interpretation of various types of symptoms. After all, the immune system is one of the most impressive examples of the (M,R) models that Rosen describes. It is in anticipation of an infinity of potential possible factors that affect the organism during its unfolding from inception to death. The metabolism component and the repair component, although different, are themselves co-related. From the perspective opened by the subject of anticipation, it is implausible that a cure for a deficient immune system will be found in any place other than its repair function.
In contradistinction, as we shall see, when one searches for information on the World-Wide Web, there is anticipation involved in the mechanism of pre-fetching information that eventually gives the user the feeling of interactivity, even though what technology makes possible is a simulacrum. The question to be asked, but not necessarily answered in this paper, is: To what extent does becoming aware of anticipation, or living in a particular anticipation (of a concert, of a joke, or of an inherited disease), affect our practical experiences of self-constitution, regardless of whether we build a technology inspired by it or only use the technology, or to what extent are such experiences part of the technology? Friedrich Dü rrenmatt, the Swiss writer, once remarked (1962, in a play entitled The Physician Sits), “A machine only becomes useful when it has grown independent of the knowledge that led to its discovery.” This statement will follow us as we get closer to the association between anticipation and computation. It suggests that if we are able to endow machines with anticipatory characteristics (prediction, expectancy, planning, etc.), chances are that our relation to such machines will eventually become more natural. This might change our relation to anticipation altogether, either by further honing natural anticipation capabilities or by effecting their extinction.
The broader picture that results from the examination of what actually defines the field of inquiry identifiable as anticipation–in living systems and in machines–is at best contradictory. To be candid, it is also disconcerting, especially in view of the many so-called anticipation-based claims. But this should not be a discouraging factor. Rather, it should make the need for foundational work even more obvious. One or two books, many disparate articles in various journals, plus the Proceedings of the Computing Anticipatory Systems (CASYS) conferences do not yet constitute a sufficient grounding. It is with this understanding in mind that I have undertaken this preliminary overview (which will eventually become my second book on the subject of anticipation).
Since the time my book (1991) was published, and even more after its posting on the World-Wide Web, I have faced colleagues who were rather confused. They wanted to know what, in my opinion, anticipation is; but they were not willing to commit themselves to the subject. It impressed them; but it also made them feel uneasy because the solid foundation of determinism, upon which their reputations were built, and from which they operate, seemed to be put in question. In addition, funding agencies have trouble locating anticipation in their cubbyholes, and even more in providing peer reviews from people willing to jump over their shadow and entertain the idea that their own views, deeply rooted in the paradigm of physics and machines, deserve to be challenged. My research at Stanford University–which constituted the basis for this report–provided a stimulating academic environment, but not many possible research partners. Students in my classes turned out to be far more receptive to the idea of anticipation than my colleagues. The summary given in this section stands as a testimony to progress, but no more than that, unless it is integrated in the articulation of research hypotheses and models for future development.
4 Minds, Knowledge, Computation–a Borgesian Horizon
The anticipatory nature of the mind–and by this I mean the processes of mind constitution as well as mind interaction–together with the understanding of anticipation as a distributed characteristic of the human being, represents an epistemological and cognitive premise. Let us put these ascertainments in the broader perspective of knowledge–the ultimate goal of our inquiry (knowledge at work included, of course). Niels Bohr (1934), well ahead of the illustrious founders of second-order cybernetics or of today’s constructivist model of science, risked a rather scandalous sentence: “It is wrong to think that the task of physics is to find out how nature is.” He went on to claim that “Physics concerns what we can say about nature.” In this vein, we can say that Rosen and others have proven that anticipation is a characteristic of natural processes. We can also take this description and try to make it the blueprint of various applications (some of which were reported above).
4.1 Computation and Prolepsis
Computation is the dominant aspect of the Weltanschauung today. It is not only a representation, but also the mechanism for processing representations (for which reason I call the computer a semiotic engine). The attempt to reduce everything there is to computation is not new. Science might be rigorous, but it is also inherently opportunistic. That is, those constituting themselves as scientists, (i.e., defining themselves in pragmatic endeavors labeled as science) are human beings living in the reality of a generic conflict between goals and means. Having said this, well aware that Feyerabend (1975) et al articulated this thought even more obliquely, I have to add that anticipation as computation is, from an epistemological perspective, probably more appropriate to our understanding of the concept than what various pre-computation disciplines had to say or to speculate about anticipation.
Between Epicurus’ (cf. 1933) term prolepsis–rule, or standard of judgment (the second criterion for truth)–and the variety of analytical interpretations leading to the current infatuation with anticipation, there is a succession of epistemological viewpoints. It is not that background knowledge–”the idea of an object previously acquired through sensations” to which Epicurus referred as a necessary condition for understanding–changed its condition from a criterion of truth to a computational entity. After all, computer systems used in speech recognition or in vision involve a proleptic component. (The machine is trained to recognize something identified as such.) Rather, the pragmatic framework changed, and accordingly we constitute ourselves as researchers of the world in which we live by means of computation rather than by means used in Epicurus’ physics and corresponding theory of knowledge (the canon, as it is known). What I want to say is that computation and the subsequent attempt to see anticipation as computation are but another description of the world and, particularly in the latter case, of our attempts to form an effective body of knowledge about it. In his discussion of prolepsis, in Critique of Pure Reason, Kant (1781) saw it within his description of the world, that is, in the form of “something that can be known a priori.” In Kant’s view, only the “property of possessing a degree” is subject to anticipation. Indeed, in computation we can attach certain weights to various data before the data are actually input. These weights will affect the result and, in many cases, the art; that is, the appropriateness of specifying weights influences predictions and forecasts. But no one would infer à rebours that Kant saw the world as a computation, or that knowledge was the result of a computational process.
4.2 Evolutionary Computation
The substratum of basic principles on which a theory of anticipation relies (Epicurus, Kant, Rosen, etc.) affects the theory itself, and thus its possible technological implementations. It has not actually been convincingly demonstrated that we can compute anticipation. What has been accomplished, again and again, is the embodiment of anticipatory characteristics, such as prediction, expectation, management, planning, etc., in computer programs. What has also been carried out is the implementation of control mechanisms, and, bringing us closer to our subject, the modeling of selection mechanisms in the now well known genetic computing models inspired by the guiding Darwinian concept. Evolutionary computation might well end up displaying anticipatory characteristics if we take the time and the knowledge needed to apply ourselves to the task. It will not be a spontaneous birth, rather a designed and carefully executed computation. Entailment might prove the critical element, as Rosen’s work seems to indicate.
4.2.1 Co-Relation vs. Computation
Once a modeling relation is established between a natural system and a formal one, we can start inferring from the formal system to the natural. Let me mention that here we are in the territory of views that often contradict each other. (For instance, Daniel Dubois and myself are still in dialog over some of the examples to follow.) Neural networks or models of ALife, such as the simulation of collections of concurrently interacting agents, qualify as candidates for such an exercise. However, almost no effort has been made to elucidate the functioning of the causal arrow from the future to the present. In winter, temperatures will fall below the freezing point; leaves fall from deciduous trees in anticipation, but the trigger comes from a different process, i.e., the diminishing length of daylight, which stands in no direct causal relation to the phenomenon mentioned yet again. This is a co-relation of processes, not a computation, or at least not a Turing machine-based computation.
The migration of birds is another example; yet others are the immune system, the sleep mechanism, the blinking mechanism, and the behavior of Pfiesteria (the single-cell microorganisms that produce deadly toxins in anticipation of the fish they will eventually kill). But if we want to stick to computation, which is a description different from the one pursued until now, we land in a domain of parallel processes, not very sophisticated, probably even less sophisticated than the level of a UNIX operating system, but of a much higher order of magnitude. We are in what was described as a big numbers-based reality. If we could control the process “shorter days,” we could eventually graph the inter-relation among the various components at work leading to the shedding of leaves during autumn, or to the sophisticated patterns of behavior of birds preparing for migration.
4.3 Large Numbers and Simple Processes
In respect to brain activity, things are definitely more complicated, but they also fall in the realm of incredibly large numbers applying to rather simple entities and processes. The ongoing CAM-Brain Project (Hugo de Garis, 1994) is supposed to result in an artificial brain of one billion neurons (compare this to the 100 to 120 billion neurons of a wet brain implemented) on Field Programmable Gate Arrays. These digital circuits can be reconfigured as the tasks at hand might require. The notion of reconfiguration elicits our understanding of anticipation. Still, it remains to be seen whether the artificial brain will actually drive a robot or only simulate the robot’s functioning, as it also remains to be seen whether evolutionary patterns will support vision, hearing, their binding, coordinated movements, and, farther down the line, decision-making. The mind in anticipation of events (as I defined mind) is a lead. If we could parametrize the cognitive process and control the various channels, we could in principle learn more about how neuroactivity precedes moving one’s hand by 800 milliseconds, and what the consequences of this forecast for human anticipation abilities are. These are all possible experiments, after each of which we will end up not only with more data (the blessing and curse of our age!), but also necessarily with the desire to gain a better understanding of what these data mean.
If Rosen’s hypothesis that anticipation is what distinguishes the biological realm (life) from the physical world, it remains to be seen whether we can do more than to compute only particular aspects of it–prediction, expectation, planning, etc.–outside the living. Pseudo-anticipation is already part of our practical experience: satellite launches, virtual surgery, pre-fetching data in order to optimize networks are but three examples of effective pseudo-anticipation. If we could create life, we could study how anticipation emerges as one of its irreducible, or only as one of its specific, properties. Short of this, ALife is involved in the simulation of lifelike processes. Rosen, in defining complexity as not simulatable, comes close to Feynman’s (1982) hope that one can best study physics by actually conducting the calculations of the world of physics on the physical entities to be studied. One can call this epistemological horizon Borgesian, knowing that an ideal Borgesian map was none other than the territory mapped.
At this point, we need to arrive at a deeper understanding of what we want to do. Regardless of the metaphor, the epistemological foundation does not change. The knowing subject is already shaped by the implicit anticipatory dimension of mind interaction; in other words, the answer to the question meant to increase our knowledge is anticipated. Computation is as adequate a metaphor as we can have today, provided that we do not expect the metaphor to automatically generate the answers to our many questions. Regardless, the question concerning anticipation in the living and in the non-living is far from being settled, even after we might agree on a computational model or expand to something else, such as co-relation, which could either transcend computation or expand it beyond Turing’s universal machine.
5 Revisiting Non-Locality
I took it upon myself to approach these matters well aware that I am advancing in mined territory. Comparisons notwithstanding, such was the situation faced by the proponents of quantum theory. To nobody’s surprise, Einstein took quantum mechanics, as developed by Heisenberg, Schrödinger, Dirac, et al, under scrutiny, and, well before the theory was even really established, raised objections to it, as well as to Bohr’s interpretation. From these objections (the complete list is known as the EPR Paper, 1935, for Einstein, Podolski, and Rosen), one in particular seems connected to the subject of anticipation. Einstein had a major problem with the property of non-locality–the correlations among separated parts of a quantum system across space and time. He defined such correlations as “spooky actions at distance” (“spukhafte Fernwirkungen”), remarking that they have to take place at speeds faster than that of light in order to make various parts of the quantum system match. In simple terms, this spooky action at distance refers to the links that can develop between two or more photons, electrons, or atoms, even if they are remotely placed in the world. One example often mentioned is the decay of a pion (a subatomic particle). The resulting electron and positron move in opposite directions. Regardless how far apart they are, they remain connected. We notice the connection only when we measure some of their properties (well aware of the influence measurement has), their spin, for example. Since the initial pion had no spin, the electron and the positron will have opposite sense spins, so that the net spin is conserved at zero. So, at distance, if the spin of the electron is clockwise, the spin of the positron is counter-clockwise.
It would be out of place to enter here into the details of the discussion and the ensuing developments. Let me mention only that in support of the EPR document, Bohm (1951) tried, through his notion of a local hidden variable, to find a way for the correlations to be established at a speed lower than that of light. He wanted to save causality within quantum predications. Bohm’s attempt recalls what the community of researchers is trying to accomplish in approaching aspects of anticipation (such as prediction, expectation, forecast, etc.) with the idea that they cover the entire subject. Bell (1964, 1966) produced a theorem demonstrating that certain experimental tests could distinguish the predictions of quantum mechanics from those of any local hidden variable theory. (Incidentally, physicist Henry P. Stapp, 1991 characterized Bell’s theorem as “the greatest discovery of all science.”) Again, this recalls by analogy Rosen’s position, according to which anticipation is what (among other things) distinguishes the living from the rest of the world. It states that we can clearly discern a particular aspect of anticipation provided in some formal description or in some computer implementation from one that is natural. I mention these two episodes from a history still unfolding in order to explain that what we say in respect to nature–as Bohr defined the goal of physics–will be ultimately subjected to the test of our practical experiences. Einstein has been proven wrong in respect to his understanding of non-locality through many experiments that baffle our common sense, but his theory of relativity still stands. Spooky actions at distance are a very intuitive description of how someone educated in the spirit of physical determinism and thinking within this spirit understands how the future impacts the present, or how anticipation computes backwards from the future to the present. He, like many others, preached the need for learning “to see the world anew,” but was unable to position himself in a different consciousness than the one embodied in his theory.
As I worked on this text (more precisely, after reworking a draft dated July 22, 1999), Daniel Dubois graciously drew my attention to a number of his research accomplishments pertinent to the connection between anticipation and non-locality. Indeed, over the last seven years, he has applied his mathematical formalism to quite a number of computational aspects of anticipation. Consequently, he was able to establish, by means of incursion and hyperincursion, that the computation pertinent to the membrane neural potential (used as a model of a brain) “gives rise to non-locality effects” (Dubois, 1999). His argument is in line with von Neumann’s analogy between the computer and the brain.
But we are not yet beyond a first analogy (or reference). Non-locality is, in the last analysis, distance independent. Furthermore, non-locality is not a limited characteristic of the universe, but a global rule. In the words of Gribbin (1998), non-locality “cuts into the idea of the separateness of things.” If the “no-signaling” criterion (energy or information travel no faster than the speed of light) protects the “chain of cause and effect,” (effects can never happen before their causes), non-locality ensures the coherence of the universe. Reconciliation between non-locality and causality might therefore be suggestive for our understanding of anticipation. In such a case, the co-relation among elements involved in anticipation can be seen as a computation, but one different in nature from a digital computer, i.e., in a Turing machine. It follows from here that anticipation understood as co-relation–a notion we will soon focus on–must be a computation different in type than that embodied in a Turing machine.
5.1 Quantum Semiotics, Link Theory, Co-Relation
Let me preface this section ascertaining that anticipation is a particular form of non-locality, which is quite different from saying that there is non-locality in anticipation. (This is what actually distinguishes my thesis from the results of Dubois.) More precisely, its object is co-relations (over space and time) resulting from entanglements characteristic of the living, and eventually extending beyond the living, as in the quantum universe. These co-relations correspond to the integrated character of the world, moreover, of the universe. Our descriptions ascertain this character and are ultimately an active constituent of this universe. We introduce in this statement a semiotic notion of special significance to the quantum realm: Sign systems not only represent, but also constitute our universe. As with qubits (information units in the quantum universe), we can refer to qusigns as particular semiotic entities through which our descriptions and interpretations of quantum phenomena are made possible.
5.1.1 The Semiotic Engine
As a semiotic engine (Nadin, 1998), a digital computer processes a variety of possible descriptions of ourselves and of the universe of our existence. These descriptions can be indexical (marks left by the entity described), iconic (based on resemblance), or symbolic (established through convention). Anticipatory computation is based on the notion that every sign is in anticipation of its interpretation. Signs are not constituted at the object level, but in an open-ended infinite sign process (semiosis). In sign processes, the arrow of time can run in both directions: from the past through the present to the future, or the other way around, from the future to the present. Signs carry the future (intentions, desires, needs, ideals, etc., all of a nature different from what is given, i.e., all in the range of a final cause) into the present and thus allow us to derive a coherent image of the universe. Actually, not unlike the solution given in the Schrödinger equation, a semiosis is constituted in both directions: from the past into the future, and from the future into the present, and forward into the past. The interpretant (i.e., infinite process of sign interpretation) is probably what the standard Copenhagen Interpretation of quantum mechanics considered in defining the so-called “intelligent observer.” The two directions of semiosis are in co-relation. In the first case, we constitute understandings based on previous semiotic processes. In the second, we actually make up the world as we constitute ourselves as part of it. This means that the notion of sign has to reflect the two arrows. In other words, the Peircean sign definition (i.e., arrow from object to representamen to interpretant) has to be “reworded”:
Fig. 3 Qusign definition
The language of the diagram allows for such a “rewording” much better than so-called natural language: The interpretant as a sign refers to something else anticipated in and through the sign. (Peirce’s original definition of sign is, “something which stands to somebody in some respect or capacity,” 2.228.) Qusigns are thus the unity between the analytical and the synthetic dimension of the sign; their “spin” (to borrow from the description of qubits) can well describe the particular pragmatics through which their meaning is constituted.
5.1.2 Knowing in Advance
The 1930 Copenhagen Interpretation of quantum mechanics (developed primarily by Bohr and Heisenberg) should make us aware of the fact that observation (as in the examples advanced by Rosen, et al), measurement (as in the evaluation of learning performance of neural networks), and descriptions (such as those telling us how a certain software with anticipatory features works) are more pertinent to our understanding of what we observe, measure, or describe than to understanding the phenomena from which they derive. To measure is to describe the dynamics of what we measure. The coherence we gain is that of our own knowledge, where dynamics resides as a description. Albeit, the anticipation chain takes the path of something that smacks of backward causality, which the established scientific community excluded for a long time and still has difficulty in understanding. Quantum particle “tunneling”–a phenomenon related to quantum uncertainty and to wave-particle duality–might explain our own existence on the planet, but we still don’t know what it means (as Feynman repeatedly stated it, verbally and in writing, 1965). Quite a number of experiments (cf. Raymond Chiao, University of California-Berkeley; Paul Kwiat, University of Innsbruck; Aephraim Steinberg, US National Institute of Standards and Technology, Maryland, among others) ended up confirming that “the way in which a photon starting out on its journey behaves” in different experimental set-ups suggests that anticipation is at work in the quantum realm. They behave (cf. Gribbin, 1999) as if they “knew in advance what kind of experiment they were about to go through.” In view of these experiments, Rosen would have a hard time trying to argue that anticipation is a property exclusive of the living. Moreover, we find in such examples the justification for quantum semiotics: “The behavior of the photons at the beam-splitter is changed by how we are looking at them, even when we have not yet made up our minds about how we are going to look at them. The computer-controlled pseudo-random layout of the device used in the experiment is anticipated by the photon,” (Gribbin and Chimsky, 1996). In other words, it is an interpretant process.
I should mention here that within the relatively young field of mathematical research called link theory, a framework that generalizes the notion of causality is established in a way that removes its unidirectionality (cf. Etter, 1999). The relational aspect of this theory makes it a very good candidate for a closer look at anticipation, in particular, at what I call co-relations.
5.1.3 Coupling Strength
In various fields of human inquiry, the clear-cut distinction between past, present, and future is simply breaking down. No matter how deep and broad grudges against a reductionist physical model (such as Newton’s) are, Newtonian dynamics is reversible in time, and so is quantum mechanics. The goal of producing a “unified” description of the universe can be justified in more than one way, but regardless of the perspective, coupling strength is what interests us, that is, what “holds” the “universe” together. This applies to the coherence of the human mind, as it applies to monocellular organisms or to the cosmos at large. It might be that anticipation, in a manner yet unknown to us, plays a role in the coupling of the many parts of the universe and of everything else that appears as coherent to us. Galileian and Newtonian mechanics advanced answers, which were subsequently reformulated and expressed in a more comprehensive way in the theory of relativity (special and general), and afterwards in quantum theories (quantum mechanics, quantum field theory, quantum gravity). In the mechanical universe, to anticipate could mean to pre-compute the trajectory of the moving entity seen as constitutive of broad physical reality. But the causal chain is so tight that the fundamental equation allows only for the existence of recursions (from the present to the future), which we can represent by stacks and compute relatively easily. The past is closed; the future, however, is open, since we can define ad infinitum the coordinates of the changing position of a moving entity. No guesswork: Everything is determined, at least up to a certain level of complexity. Relativity does not do away with the openness of the future, but makes it more difficult to grasp. Within black holes, inherent in the relativistic description but not reducible to them, time is cyclic. In Einstein’s curved space-time, a circular “time-line” (Etter’s pun) is no more surprising than a “circle around a cylinder in ordinary space.” This, however, leads to a cognitive problem: how to accommodate a cycle with openness. Anticipation related to this description of time is quite different from that which might be associated with a physical-mechanical description.
5.2 Possible and Probable
Quantum theories, as we have suggested, pose even more difficult questions in regard to non-locality, and thus to entanglement. In this new cognitive territory, things get even more difficult to comprehend. Determinism, which means that something is (1) or is not (0) caused by something else, gives way to a probabilistic and/or possibilistic distribution: Something is caused probably (i.e., to a certain degree expressed in terms of probability, that is, statistic distribution) by something else. Or it is caused possibly (in Zadeh’s sense, 1977), which is a determination different from probability (although not totally unrelated), by something else. Probabilistic influences can be represented through a transition matrix. Given the relation between two entities A and B and their respective states, we can define a Markov chain, i.e., a transition matrix whose ijth entry is the probability of i given j. Such a chain tells us how influences are strung together (chained) and can serve as a predictive mechanism, thus covering some subset of what we call anticipation.
Recently, weather satellite observations of the density of green vegetation in Africa (an indication of rainfall) were connected through such processes to the danger of an outbreak of Rift Valley Fever, in which Linthicum (1999) devised a metrics based on climate indicators for a forecasting procedure. The “black boxes” chained in such processes have a single input and a single output representing the complete state variable of the system as it changes over time. Climate and health (the risk of malaria, Hanta virus, cholera) are related in more than one way (Epstein, 1999). These examples are less probabilistic than possibilistic. If we pursue possibilities, that is, infer from a determined set of what is possible, a different form of prediction can eventually be achieved. Abductive inferences belong to this category and are characteristic of functional diagnosis procedures. Here we have an example of semiotics at work, i.e., abductions on symptoms, not really far from what Epicurus meant by prolepsis.
5.2.1 Linked Incursions
For the aspects of anticipation that belong to a non-deterministic realm, we can further try to link descriptions of the form
y = f(x) or z = g(w) (12a, b)
Indeed, if we substitute y for w, our descriptions become
y = f(x) and z = g(y), that is, z = g(f(x)) (13a, b, c)
The result is a functional relation of the composed functions. Without going into the details of Etter’s theory, let me suggest that it can serve as an efficient method for encoding a variety of relations (not only in the case of the identity of two variables). If in the functional description we substitute not the variables (w with y, as shown in the example given above) but the relation between them, we reach a different level of relational encoding that can better support modeling. I even suggest that recursions, incursions, and hyperincursions can be defined for co-related events. For example:’
x(ti+1) = f[x(ti), x (ti +1), p] (14)
y(tj+1) = g[y (tj), y(tj+1), r] (15)
in which time in the two systems is obviously not the same (ti ¹ tj). A co-relation of time can be established, as can a co-relation among the states x(ti) and y(tj) or the two systems, through the intermediary of a third system acting as the “conductor,” or coordinator, z(ti, tj, tk), i.e., dependent upon both the time in each system and its own time metrics. To elaborate on the mathematics of linked incursions goes beyond the intentions of this paper. Let us not forget that we are pursuing an analysis of the particular ways in which anticipation takes place in the successive unified descriptions of the universe produced so far.
5.2.2 Alternative Computations
In the quantum perspective of a double identity–particle and wave–trajectory is the superposition of every possible location that a moving entity could conceivably occupy. This is where recursivity, in the classic sense, breaks down. I suspect that Dubois was motivated to look beyond recursivity for improved mathematical tools, to what he calls incursion and hyperincursion, for this particular reason. But I also suspect that linked incursions and hyperincursions will eventually afford more results in dealing with various aspects of anticipation and non-locality.
In respect to the explicit statement, prompted by quantum mechanics non-locality, that anticipation could be a form of computation different from that described by a Turing machine, it is only in the nature of the argument to say that a full-fledged anticipation, not just some anticipatory characteristics (prediction, planning, forecasting, etc.) is probably inherent in quantum computation. Rosen recognized early on (1972) that quantum descriptions were a promising path, although among his publications (even more manuscripts belong to his legacy, cf. 1999) there are no further leads in this direction. Efforts to transcend digital computing through quantum computation are significant in many ways. From the perspective of anticipation, I think Feynman’s concept comes closer to what we are after: understanding the quantum dynamics not by using a digital computer (as in the tradition of reductionist thinking), but by making use of the elements involved in quantum interactions. As the situation is loosely described: Nature does this calculation all the time! The same thing can be said about protein folding, a typical anticipatory process–a small increase in energy (warming up) drives the folding process back, only in order to have it repeated as the energy decreases. This process might also well qualify as an anticipatory computation, with a particular scope, not reducible to digital computation. (As a matter of fact, protein folding exceeds the complexity of digital computation.) It is an efficient procedure, this much we know; but about how it takes place we know as little as about anticipation itself.
5.2.3 Anticipation as Co-Relation (Or: Co-relation as Anticipation?)
Having advanced the notion of anticipation as a co-relation, I would like to point to instances of co-relation that are characteristic of experiences of practical human self-constitution in fields other than the much researched control theory of mechanisms, economic modeling, medicine, networking, and genetic computing. There is, as Peat (undated) once remarked, a strong concern with “a non-local representation of space” in art and literature. The integration of many viewpoints (perspectives) of the same event illustrates the thought. Reconstruction (in the perception of art and literature) means the realization of a future state (describable as understanding or as coordination of the aesthetic intent with the aesthetic interpretation) in the current state of the dynamic system represented by the work of art or of writing, and by its many interpreters (open-ended process). In Descartes’ and Newton’s traditions, space and time are local: a taming of artistic expression took place. Peat claims that the “tableau,” i.e., the painting, becomes a snapshot in which “motion and change is frozen in a single instant of time. This is a form of objectivity which the concert, the novel, and the diarist express.” With the advent of relativity and quantum physics, many perspectives are overlaid. As Peat puts it, “In our century, painting has returned to the non-local order.” This holds true for writing (think about Joyce), as well as it does for the dynamic arts (performance, film, video, multimedia). Complementary elements, entangled throughout the unifying body of the work or of its re-presentation, are brought into coherence by co-relations within non-locality-based interactions. Peat goes on to show that communication “cries our for a non-local” description: source and receiver cannot be treated as separable entities. (They are linked, as he poetically describes the process, “by a weak beam of coherent light.”) Meaning—which “cannot be associated exclusively with either participant” (n.b., in communication)—could be “said to be ‘non-local’.”
6 The Relational Path to Co-Relations
That computation, in one of its very many current forms or in a combination of such forms (such as hybrid algorithmic-nonalgorithmic computations), can embody and serve as a test for hypotheses about anticipation should not surprise. Neither should the use of computation imply the understanding that anticipation is ultimately a computation, that it is the only form, or the appropriate form, through which we can implement anticipation-based notions. It is an exciting but dangerous path: If everything is described as a computation—no matter how different computation forms can be—then nothing is a computation, because we lose any distinguishing reference. Epistemologically, this is a dead end. Furthermore, it has not yet been established whether information processing is a prerequisite of anticipation or only one means among many for describing it. While we could, in principle, embody anticipatory features in computer programs, we might miss a broad variety of anticipation characteristics. For instance, progress was made in describing the behavior of flocks (cf. The Swarm Simluation System at the Santa Fe Institute). But bird migration goes far beyond the modeled behavioral interrelationships. Trigger information differentials, group interaction, learning, orientation, etc. are far more sophisticated than what has been modeled so far. The immune system is yet another example of a complexity level that by far exceeds everything we can imagine within the computational model. Be all this as it may, our current challenge is to express co-relations, which appear as predefined or emerging relations in a dynamic system, by means of information processing in some computational form, or by means of describing natural entanglements. If we could reach these goals, we would effect a change in quality–from a functional to a relational model. Here are some suggestions for this approach.
6.1 Function and Relation
Relations between two or among several entities can be quite complicated. A solid relational foundation requires the understanding of what distinguishes relation from function. For all practical purposes, functions (also called mappings) can be linear or non-linear. (Of course, further distinctions are also important: They can be many or single-valued, real or complex-valued, etc.) Relations, however, cover a broader spectrum. A relation of dependence (or independence) can be immediate or intermediated. It can involve hierarchical aspects (as to what affects the relation more within a polyvalent connection), as well as order or randomness. Relations, not unlike functions, can be one-to-one, one-to-many, many-to-one, many-to-many. We can define a negation of a relation, a double negation, inverse relation, etc. A full logic of relations has not been developed, as far as I know. Rudimentary aspects are, however, part of what after Peirce (1870, 1883) and Schröder (The Circle of Operation of Logical Calculus, 1877) became known as a logic of relations. Russell and Whitehead (Principia Mathematica, 1910) made further clarifications.
Let us assume a simple case: xRy, in which x stands in relation to y (son of, higher than, warmer than, premise of, etc.). If we consider various aspects of the world and describe them as relationally connected, we can wind up with statements such as xR1y, zR2w, etc. In this form, it is not clear that Ri exhausts all the relations between the related entities; neither is it clear to what extent we can establish further relations between two relations Ri and Rj and thus eventually infer from their interrelationship new relations among entities that did not have an apparent relation in the first place. In a wide sense, a relation is an n-ary (n=1, 2, 3….) “connection”; a binary relation is a particular case and means that the relation xRy is true or false for a pair x,y in the Cartesian product XxY. As opposed to functions, for which we have relatively good mathematical descriptions, relations are more difficult to encode, but richer in their encodings. Their classification (e.g., inverse relation, reflexive, symmetric, transitive, equivalent, etc.) is important insofar it leads to higher orders (e.g., a reflexive and transitive relation is called a pre-ordering, while an ordering is a reflexive, transitive, and antisymmetric relation).
6.1.1 N-ary Relations
If we revisit some of the examples of anticipation produced so far in the literature–Rosen’s deciduous trees, Peat’s communication as a non-local unifying process, Linthicum’s and Epstein’s metrics of weather data and disease patterns, the cognitive implications of the many competing models from which one is eventually instantiated in an action, or the hyperincursion mechanism developed by Dubois (to name but a few)–it becomes obvious that we have chains of n-ary relations: xRin y (in which Rin is a specific Ri n-ary relation); that is, in a given situation, several relations are possible, and from all those possible, some are more probable than others.
To anticipate means to establish which co-relations, i.e., which relations among relations are possible, and from those, which are most probable. Anticipation is a process. It takes place within a system and we interpret it as being part of the dynamics of the system. Observed from outside the system–deciduous trees lose their leaves, birds migrate, tennis players anticipate the served ball–anticipation appears as goal-driven (teleologic). In particular, coherence is preserved through anticipation; or a different coherence among the variables of a situation is introduced (such as playing chess, or predicting market behavior). Pragmatically, this results in choices driven by possibilities, which appear as embodied in future states. The tennis ball is served and has to be returned in a well defined area–and this is an important constraint, an almost necessary condition for the game ever to take place! At a speed of over 100 miles per hour, the served ball is not returned through a reaction-based hit, but as a result of an anticipated course of action, one from among many continuously generated well ahead of the serve or as it progresses. If the serving area is increased by only 10%, chances for anticipation are reduced in a proportion that changes the game from one of resemblance and order to a chaotic, incoherent action that makes no competitive sense. The competition among the various models (all possibilities, but along a probability distribution corresponding to the particular style of the serving player) allows for a successful return, itself subject to various models and competition among them. The whole game can be seen as an unfolding chain of co-relations, i.e., a computation controlled by a range of acceptable parameters. The immune system works in a fundamentally similar fashion. Co-relations corresponding to a wide variety of acceptable parameters are pursued on a continuous basis. Acclimatization, i.e., the way humans adapt to changes in seasons, is but a preservation of the coherence of our individual and collective existence under the influence of anticipated changes in temperature, humidity, day-night cycle, and a number of other parameters, some of which we are not even aware.
6.1.2 Instantiated Co-Relations
But having given the example of an unfolding sequence does not place us in the domain of non-locality. For this we need to distinguish between the diachronic and synchronic axes. A strictly deterministic explanation will always place the anticipated in the sequence of cause-and-effect/action-reaction. The tennis ball is served, days are getting shorter, a virus causes an infection–all seen as causes. In the anticipatory view, the ball is actually not yet served as the sequence of models, from among which one will become the return, started being generated. The anticipation leading to the fall of leaves is the result of a co-relation involving more than one parameter. What appears as a reaction of the immune system is actually also a co-relation involving the metabolism and self-repair function. On the one hand, we have an unfolding over time; on the other, a synchronic relation that appears as an infinitely fast process. In reality we have a co-relation, an intertwining of many relations among a huge number of variables of which we are only marginally, if at all, aware.
Assuming that we have a good description of the n-ary relations R1n, R2n,… Rin, moreover that we can even “relate” relations of a different order (n=3 vs. n=4, for instance), and express this relation in a co-relation, it becomes clear that co-relations are descriptive of higher order relations. For example, two binary relations are identical when their converses are identical. In any sequences of the form xRiy, zRjw, uRkv, etc. we are trying to identify what the relation is among the various relations Ri, Rj, Rk, etc., represented by Ri Ra Rj, Rj Rb Rk, etc. The co-relations, Ra , Rb , Rg (e.g., son of and daughter of correspond to progeny, but among the co-relations, we will find similarity or distinction, among other things) can apply to the subsets of all Ri (i=1,… n) sharing a certain distinctive characteristic (such as similarity). We can further define referents (Ref) and relata (Rel), as well as a relation between referents or relata denoted as Sg (Sagitta, i.e., arrow). By no accident, the arrow can graphically suggest a dynamics from the present to the future (prediction), or the other way around, from the future to the present (anticipation). After Peirce, Tarski (1941) produced an axiomatized theory of relation that, not unlike Boolean logic, could serve as a basis for effective computations of relations and co-relations. It is quite possible that the computation of co-relation could be built around the formalism of quantum computing. In this case, we would operate on the value of the entanglement, not on the state of a particle. It is a task that invites further work. Last but not least, we invite the thought of considering relations among incursions and hyperincursions as a means of testing their descriptive power even more deeply.
6.2 Making Use of the Co-Relation Model
Having advanced this model of anticipation as a form of computation, based on the dynamic generalization of models and on competition among them, and encoded in a formalism that captures co-relations (thus the spirit of non-locality), I would like to present some examples speaking in favor of an understanding of anticipation that occasionally comes close to what I have proposed above. These are not direct applications of the theory I have advanced so far, rather they are suggestive of its possible directions, if not of its meaning.
6.2.1 Anticipatory Document Caching
Incidentally, anticipatory document caching with the purpose of reducing latency on Web transactions is introduced in a language reminiscent of Einstein’s observation, “Everyone talks about the speed of light but nobody ever does anything about it.” The reason for the provocative introduction is obvious: interactive HTML (i.e., text transmission through the Web) requires at least T-1 connection speeds (i.e., 1.5M bps). Once images are used, the requirement increases to T-3 lines (45M bps). Cross-country interactive screen images push the limit to 155M bps. Places such as the major cities on the West Coast of the USA (San Francisco, Los Angeles) are at least 85 milliseconds away from cities on the East Coast (Boston, New York). Interactivity under the limitations of the speed of light–assuming that we can send data at such speed and on the shortest path–is an illusion. In view of this practical observation, those involved in the design of networks, of communication protocols, of client-server access and the like are faced with the task of reducing the time between access request and delivery. Among the methods used are the utilization of inter-request bandwidth (transfer of unrequested files when no other use is made), proactive requests (preloading a client or intermediate cache with anticipated requests), optimization of topology (checking where files will be best used, combining identical requests and responses over shared links).
What Touch et al (1992, 1996, 1998) accomplished is an effective procedure for providing co-relations. Evidently, they realize that such correlations cannot rely on a second channel through which requests would travel faster than the information itself. Accordingly, they initiate processes in fact independent of the communication between the client and the remote server. Such processes facilitate an anticipatory behavior based on predictive cues corresponding to the searched information. They also define where in a network of such optimization servers should be placed. I insist upon this mechanism of implementation not only because of its significance for the networked community, but primarily in view of the understanding that anticipatory computation is one of producing meaningful co-relations. The entanglement between the search process and pre-fetching data is stricto sensu a pseudo-anticipation. But so are all other implementations known to date. These are all models of possible actions, and it is quite practical to think of generating even more models as the user gets involved in a certain transaction.
6.2.2 Software Design
The same idea was implemented by high-end 3D modeling software (e.g., UNIGRAPHICS), under the guidance of a better understanding of what designers can and would do at a certain juncture in visualizing their projects. The use of computation resources within such programs makes for the necessity to anticipate what is possible and to almost preclude functions and utilities that make no sense at a certain point. This is realized through a STRIM function. Instead of allowing the program to react to any and all possible courses of action, some functions are disabled. Henceforth, the functions essential to the task can take advantage of all available resources. (This is what STRIM makes possible.) It is by all practical means a pro-active concept based on realizing the co-relations within the various components of the program.
6.2.3 Agents Coordination
Another aspect of co-relation is coordination. It can be ascertained that cooperative activities can take place only if a minimum of anticipation–in one or several of the forms discussed so far–is provided. This applies to every form of cooperation we can think of: commerce, work on an assembly line (where anticipation is built in through planning and control mechanisms), the pragmatics of erecting a building, the performing arts, sports.
Coordination is a particular embodiment of anticipation. It can be expressed, for instance, in requirements of synchronization defined to ensure that from a set of possibilities the optimum is actually pursued. Thus, in a given situation, from a broad choice of what is possible, what is optimal is accomplished. The goal is to maximize the probability of successful cooperation. This is achieved by implementing anticipatory characteristics. I would like to mention here as an example the Robo Cup world champion, designed and implemented by Manuela Veleso, Peter Stone, and Michael Bowling (of Carnegie Mellon University). This is an autonomous agent collaboration with the purpose of achieving precise goals (in this case, winning a soccer game between robotic teams) in a competitive environment. Stated succinctly in the words of the authors, “Anticipation was one of the major differences between our team and the other teams,” (1998). Let us focus on this aspect and briefly describe the solution. What was accomplished in this implementation is a model of an unfolding soccer game. But instead of the limited action-reaction description, the authors endowed the “players” (i.e., agents) with the ability to maximize their contributions through anticipatory movements corresponding to increasing the team’s chance to execute successful passes leading to scoring. It is a relational approach: Agents are placed in co-relation (“taking into account the position of the other robots–both teammates and adversaries”) and in respect to the current and possible future positions of the ball. It is evidently a multi-objective description, that is, a dynamic set of models, with what the authors call “repulsion and attraction points.” The anticipation algorithm (SPAR, Strategic Positioning with Attraction and Repulsion) contains weighted single-objective decisions. Correctly assuming that transitions among states (i.e., choices among the various models) for each of the cooperating agents takes time (computing cost, in a broader sense), the authors implement the anticipatory feature in the form of selection procedures. The goal is to increase (ideally, to find the maximum) the probability of future collaboration as the game unfolds. The agents are given a degree of flexibility that results in adjustment supposed to enhance the probability of individual actions useful to the team. Additionally, an algorithm was designed in order to allow the “players” (team agents) to position themselves in anticipation of possible collaboration needs among teammates. Individual action and team collaboration are coordinated in anticipation (i.e., predictive form) of the actions of the opponents. At times, though, the anticipatory focus degrades to reactive moves.
Less successful in the competition, but inspired by Rosen’s definition, the team of the University of Caen (France) defined the following program: “Anticipation allows the consideration of global phenomena that cannot be treated through a local reactive approach. The anticipation of the actions of the adversary or of its teammates, the anticipation of the change of the other teamplayers’ roles, the anticipation of the ball’s movements, and the anticipation of conflicts among teammates are some of the forms of anticipation that our system tries to account for,” (Stinckwich, Girault, 1999).
6.2.4 Auto-Associative Memories
Along the same line of thought, it is worth mentioning that in the area of cognitive sciences, neural architectures involving auto-associative memories are used in attempts to implement anticipatory characteristics. Such memories reproduce input patterns as output. In other words, they mimic the fact that we remember what we memorize, which in essence we can describe through recursive or, better yet, incursive functions. The association of patterns of memorized information with themselves is powerful because, in remembering, we provide ourselves part of what we are looking for; that is, we anticipate. The context is supportive of anticipation because it supports the human experience of constituting co-relations. We can apply this to computer memory. Instead of memory-gobbling procedures, which hike the cost of computation and affect its effectiveness, auto-associative memory suggests that we can better handle fewer units, even if these are of a bigger size. Jeff Hawkins (1999), who sees “intelligence as an ability … to make successful predictions about its input,” i.e., as an internal measure of sensory prediction, not as a measure of behavior (still an AI obsession) applied his pattern classifier to handprinted-character recognition. The Palm Pilot�™ might sooner than we think profit from the anticipatory thought that went into its successful handwriting recognition program that Hawkins authored.
6.3 Interactivity
Such and similar examples are computational expressions of the many aspects of anticipation. Their interactive nature draws our attention towards the very telling distinction between algorithmic and interaction computation. In algorithmic computation, we basically start with a description (called algorithm) of what it takes to accomplish a certain task. The computer–a Turing machine–executes a single thread operation (the van Neumann paradigm of computation) on data appropriately formatted according to syntactic constraints. As such, the process of computation is disconnected from the outside world. Accordingly, there is no room for anticipation, which always results from interaction. In the interactive model, the outside world drives the process: Agents react to other agents; robots operate in a dynamic environment and need to be endowed with anticipatory traits. Searches over networks, not unlike airline ticket purchasing and other interactive tasks, are driven by those who randomly or systematically pursue a goal (find something or let something surprise you).
As Peter Wegner (1996), one of the proponents of interactive computation expresses it, “Algorithms are ‘sales contracts’ that deliver an output in exchange for an input. A marriage contract specifies behavior for all contingencies of interaction (‘in sickness and health’) over the lifetime of the object (’till death do us part’).” The important suggestion here is that we can conceive of object-based computation in which object operations (two or more) share a hidden state.
Fig. 4 Interactive computation: the shared state
None of the operations (or processes) are algorithmic, since they do not control the shared state, but participate in an interaction through the shared state. They are also subject to external interaction. What is of exceptional importance here is that the response of each operation to messages from outside depends on the shared state accessed through non-local variables of operations. The non-locality made possible here corresponds to the nature of anticipation. Interactive systems are inherently incomplete, thus decidable in Gödel’s sense (i.e., not subject to Gödelian strictures in respect to their consistency). Interactivity requires that the computation remain connected to the practical experiences of human self-constitution, i.e., that we overcome the limitations of syntactically limited processing, or even of semantic referencing, and reach the pragmatic level. Processes in this kind of computation are multi-threaded, open-ended, and subject to predictive or not predictive interactions. The Turing machine could not describe them; and implementation in anticipatory computing machines per se is probably still far away.
This brings up, somehow by association, the question of whether the category of artifacts called programs are anticipatory by design or by their condition. The question is pertinent not only to computers, since in the language of modern genetics, programming (as the encoding of DNA, for example) plays an important role. It is, however, obvious that silicon hardware (as one possible embodiment of computers) and DNA are quite different, not only in view of their make-up, but more in view of their condition. If birds are “programmed” for their migratory behavior, then these “programs” are based on entailment schemes of extreme complexity. The same applies even more to the immune system.
6.3.1 Virtual Reality
A special category of interactive computation is represented by virtual reality implementations, all intrinsically pseudo-anticipatory environments of multi-sensorial condition. In the virtual domain, a given set of co-relations can be established or pursued. Entanglement is part of the broader design. Various processes are triggered in a confined space-and-time, i.e., in a subset of the world. Non-locality is a generic metaphor in the virtual realm made possible by the integration of the human subject. Sure, as we advance towards molecular, biological, and genetic computation–where the distinction between real and virtual is less than clear-cut–we reach new levels of pragmatic integration. Evolutionary computation will probably be driven by the inherent anticipatory characteristic of the living. As designs of computation processes at the chromosome level are advanced, a foundation is laid for computation that involves and facilitates self-awareness. Interaction at this level goes deeper than interaction embodied in the examples mentioned above; that is, at this level, mind-interaction-like mechanisms are possible, and thus true anticipation (not just the pseudo type) emerges as a structural property.
We are used to the representation of anticipatory processes through models that have a higher speed than the systems modeled: A rocket launch is anticipated in the simulation that “runs” ahead of the real time of the launch. The program anticipates, i.e., searches for all kinds of correlations–the proper functioning of a very complex system consisting of various elements tightly integrated in the whole. We have here, not unlike the case of data pre-fetching, or of integration through search in a space of possibilities, or of auto-associative memory, a mechanism for ensuring that co-relations are maintained above and beyond the deterministic one-directional temporal chain. The more interesting bi-directional chain is not even imaginable in such applications.
The spookiness of anticipatory computation is not only reducible to the speed of interactions that worried Einstein. It also involves a bi-directional time arrow. The account given in this paper, which simultaneously occasioned the advancement of my own model, identifies the many perspectives of the possible frontier in science represented by the subject of anticipation.
7. Conclusion
In order to ascertain anticipatory computation as an effective method, working models that display anticipatory characteristics need to be realized. The examples given herein can be seen as the specs for such possible models. Work in alternative computing models is illustrative of what can be done and of the return expected. Co-relations, difficult to deal with once we part from the world of first-order objects, are another promising avenue, as are possibilistic-based computations. Finally, if quantum effects prove to take place also in a world of large scale, anticipation, as entanglement (i.e., co-relation), might turn out to be the binding substratum of our universe of existence.
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85d01a06c3402b8e | A Tensor Form of the Order Parameters
Fluctuating Stripes in Strongly Correlated Electron Systems and the Nematic-Smectic Quantum Phase Transition
We discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy “fluctuating stripes”. We study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively, for both a model with full rotational symmetry and for a system with an underlying lattice, which has a discrete point group symmetry. We find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. On the other hand, the quasiparticles in the smectic phase exhibit Fermi liquid behavior. We also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.
71.10.Hf, 71.45.Lr, 71.10.Ay
I Introduction
The discovery of the high temperature superconductors in the quasi-two-dimensional copper-oxide materials in the late 1980s, and of novel correlated phases in other complex oxides, has brought to the forefront the problem of the physics of strongly correlated electron systems. To this date the understanding of the behavior of these systems remains one of the main open and challenging problems in condensed matter physics. The central conundrum in this field is the fact that these strongly coupled electron systems are best regarded as doped Mott insulators for which both the band theory of metals and the Landau theory of the Fermi liquid (FL) fail.
One characteristic feature of the physics of doped Mott insulators is their inherent tendency to electronic phase separation, frustrated by the effects of Coulomb interactions.Emery and Kivelson (1993); Kivelson and Emery (1994) The ground states resulting from these competing tendencies typically break the translation invariance and/or the point group symmetry of the underlying lattice. From a symmetry point of view, the ground states of doped Mott insulators are charge-ordered phases, which share many similarities with classical liquid crystals, and should be regarded as electronic liquid crystal phases.Kivelson et al. (1998) However, unlike classical liquid crystals, electronic liquid crystals are strongly quantum mechanical states whose transport properties range from insulating to metallic and even superconducting. In contrast with classical liquid crystals, whose ordered phases represent the spontaneous breaking of the continuous translation and rotational symmetry of spacede Gennes and Prost (1993); Chaikin and Lubensky (1998), the electronic liquid crystal phases of strongly correlated systems are sensitive to the effects of the underlying lattice and the symmetry breaking patterns involve the point and space groups, as well as to disorder. More complex ordered states, involving simultaneously charge and spin degrees of freedom, may also arise.Wu et al. (2007)
The sequence of quantum phase transitions described above, electron crystal smectic (stripe) nematic isotropic fluid, representing the progressive restoration of symmetry, is natural from a strong correlation perspective. Indeed, the electron crystal state(s) are naturally insulating (much as in the case of a Wigner crystal), the smectic or stripe phases are either anisotropic metals or superconductors, and the charged isotropic fluids are either metallic or superconducting. While the isotropic metallic phase is essentially a FL (albeit with strongly renormalized parameters), the nematic and smectic metallic phases have a strong tendency to show non-FL character. Indeed, much of the theoretical description of the stripe or smectic phases is usually based on a quasi-one-dimensional analysis, which makes explicit use of this strong correlation physics. Such approaches give a good description of this state deep inside this phase and at energies high compared to a “dimensional crossover” scale below which the state is fully two-dimensional (and strongly anisotropic)Emery et al. (1997); Carlson et al. (2000); Emery et al. (2000); Vishwanath and Carpentier (2001); Granath et al. (2001); Arrigoni et al. (2004); Carlson et al. (2004). Stripe phases (insulating, metallic, and superconducting) have been found in mean-field studies of generalized two-dimensional Hubbard and t-J modelsZaanen and Gunnarsson (1989); Machida (1989); Kato et al. (1990); Poilblanc and Rice (1989); Schulz (1990); Vojta and Sachdev (1999); Vojta et al. (2000); Park and Sachdev (2001); Sachdev (2003); Lorenzana and Seibold (2002); Anisimov et al. (2004); Himeda et al. (2002); Raczkowski et al. (2007).
The same pattern of quantum phase transitions can also be considered in reverse order, with a weak coupling perspective, as a sequence of symmetry breaking phase transitions beginning from the isotropic metal: FL electron nematic electron smectic insulating electron crystal. In this case, one begins with a uniform isotropic metal, well described at low energies by the Landau theory of the FL, with well-defined quasiparticles and a Fermi surface (FS), and considers possible instabilities of the isotropic fluid into a nematic (or hexatic and other such states), as well as phase transitions into various possible charge-density-wave (CDW) phases. The unidirectional CDW-ordered states are the weak coupling analog of the smectic (or stripe) phases, and have the same order parameters as they break the same symmetries. The main difference between a CDW and a smectic resides in the fact that while the CDW arises as a weak coupling (infinitesimal) instability of a FL in which parts of the FS are gappedMcMillan (1975) (which requires the existence of a FS with sharp quasiparticles), the stripe phases do not require such description. While a CDW phase at high energies is essentially a FL, the high-energy regime of a stripe phase is a quasi-one-dimensional Luttinger liquid.Carlson et al. (2000); Emery et al. (2000) A direct quantum phase transition from a FL to a CDW phase is, naturally, possible and this quantum phase transition has been studied in some detail,Altshuler et al. (1995); Chubukov et al. (2005) as well as to a metallic spin-density wave (SDW)Vekhter and Chubukov (2004); v. Löhneysen et al. (2007).
The weak coupling description of an electron nematic phase uses a Pomeranchuk instability of a Fermi liquid statePomeranchuk (1958). Oganesyan, Kivelson, and FradkinOganesyan et al. (2001) showed that the nematic quantum phase transition is a quadrupolar instability of the FS, and gave a characterization of the properties of the nematic Fermi fluid in a continuum model. An electron nematic quantum phase transition has also been found in lattice modelsHalboth and Metzner (2000); Metzner et al. (2003); Dell’Anna and Metzner (2006), which show, however, a strong tendency to exhibit a first-order quantum phase transitionKee et al. (2003); Khavkine et al. (2004); Yamase et al. (2005). Pomeranchuk instabilities in the Landau theory of the FL have also shown the existence of an electron nematic transitionNilsson and Castro Neto (2005); Wölfle and Rosch (2007). Perturbative renormalization group analysis of the stability of the FL in Hubbard-type modelsHonerkamp et al. (2002), as well as high-temperature expansionsPryadko et al. (2004), has also shown that in such models there is a strong tendency to a nematic state. An electron nematic state was shown to be the exact ground state in the strong coupling limit of the Emery model of the copper oxides at low hole dopingKivelson et al. (2004).
The upshot of the work on the electron nematic quantum phase transition is that, at the QCP (if the transition is continuous) and in the nematic phase (in the continuum) the electron quasiparticle essentially no longer exists as an asymptotically stable state at low energies, except along symmetry determined directions in the ordered phase. A full solution of this QCP by bosonization methods has confirmed these results, which were gleaned from mean-field theory, and have also provided strong evidence for local quantum criticality at this QCP Lawler et al. (2006); Lawler and Fradkin (2007).
In this paper we will be interested in the quantum phase transition from an electron nematic phase to a charge stripe phase, a unidirectional CDW. For simplicity we will not consider here the spin channel, which plays an important role in many systems. We will only consider the simpler case of unidirectional order. Extensions to the more general case of multidirectional order are straightforward. Here we develop a quantum-mechanical version of the nematic-smectic transition in a metallic system. This is a quantum-mechanical version of the McMillan-deGennes theory for the quantum phase transition from a metallic nematic phase to a metallic smectic (or CDW) phase. The construction of such a generalization of the McMillan-deGennes theory is the main purpose of this paper.
As it is discussed in detail in subsequent sections, here we will follow the “weak-coupling” sequence of quantum phase transitions described above, beginning with the transition from a FL to an electron nematic, and from the latter to a stripe or unidirectional CDW state. The main advantages of this approach are that it allows to address the fate of the electronic quasiparticles and non-Fermi liquid behaviors as the correlations that give rise to these electronic liquid crystal phases develop, as well as to study the quantum critical behavior following the standard Hertz-Millis approach Hertz (1976); Millis (1993); Sachdev (1999). However, the main disadvantage is that this approach does not do justice to the physics of strong correlation. For this reason, in spite of the important insights that are gained through this line of analysis, this approach cannot explain the physics of the “strange metal” regime observed in the “normal state” of high superconductors where non-Fermi liquid effects are widely reported. To do that would require studying this problem as a sequence of quantum melting transitions. An important first step in this direction has been made by Cvetkovic and coworkers Cvetkovic et al. (2006, 2008); Cvetkovic (2007) who have studied a purely bosonic model of such quantum melting. The inclusion of fermionic degrees of freedom in this strong coupling approach is an interesting but challenging open problem.
We have both conceptual and phenomenological motivations for considering this problem. At the conceptual level the main question is to develop a theory of the quantum critical behavior at the electron nematic-smectic phase transition, and of the low-energy physics of both phases near quantum criticality. Although the static properties are the same as in the classical theory (as required by symmetry) the quantum dynamics changes the physics substantially. Thus, physical properties, which determine the transport properties and the fermion spectral function, cannot be gleaned from the classical problem. Provided that the quantum phase transition is continuous or, at most weakly first order, the low-energy fluctuations in one phase (say the nematic metal) must reflect the character of the nearby ordered stripe phase. In other words, under these assumptions, as the quantum phase transition is approached the metallic nematic phase behaves as a state with “fluctuating stripes”. The ample experimental evidence in high temperature superconductors for “fluctuating stripe order” should be interpreted instead as evidence of a nematic phase proximate to a quantum phase transition to a stripe (or smectic)-ordered stateKivelson et al. (2003).
Ii Summary of Results
In this work we follow a phenomenological approach to study the quantum phase transition between an electronic nematic state and electronic smectic state. We postulate the existence of both an electron nematic and a smectic phases with a possible direct phase transition between them. This physics will be represented by an effective field theory involving the nematic and CDW order parameters. The static part of the effective action of the order-parameter theory has the same form as in the classical theory of the nematic-smectic transition, the McMillan-deGennes theory. We will assume that aside from the effects of the coupling to the fermionic quasiparticles, this effective field theory is analytic in the order parameters and their derivatives as this dependence is determined by local physics. As shown below, this assumption implies a dynamical quantum critical exponent .
The fermionic quasiparticles couple to the nematic and smectic (CDW) order parameters in their natural symmetry-dictated way. The fermions will be assumed to be a normal FL, with well-defined quasiparticles and a FS. Thus, we will not attempt to explain why the phase transition exists, which requires a microscopic theory, but rather describe its character. One of our most important results is that this theory gives a description of a phase with fluctuating stripe (smectic) order, of much interest in current experiments. The effective theory that we consider also allows for a possible direct transition between the normal and isotropic FL state and a CDW phase, without going through an intermediate nematic phase, as in the direct transition between a FL and a CDW state, discussed by Altshuler, Ioffe, and MillisAltshuler et al. (1995). Thus, the theory we present here actually describes the behavior of a FL in the vicinity of a possible bicritical point which, as we shall see, is not directly accessible.
Nematic Smectic Mode at the Electronic Nematic-Smectic QCP Smectic
inflection continuous discrete
incommensurate commensurate point rotational rotational
symmetry symmetry
Anisotropic Scaling
Gaussian Fixed Point Stable Stable Unstable / First Order Stable Stable Stable Stable
or or
Table 1: Summary of results. See the text for a detailed explanation.
The main results of our theory are summarized in Table 1. In Sec. III we discuss the current experimental status of electronic liquid crystal phases in a number of different materials. In Sec. IV we set up the order parameter theory for the electronic liquid crystal phases based on symmetry and analyticity. The static part of this phenomenological theory is (as it should be) similar to its classical counterpart, but we add proper dynamics to describe the quantum fluctuations. We next couple the order parameter theory to the fermionic quasiparticles, in Sec. V. The coupling between the fermionic quasiparticles and the order parameters is completely determined by symmetry. This is a standard approach to study quantum phase transitions in metallic systems Sachdev (1999). It is a consistent scheme for the study of the quantum phase transition provided the effective dimension is close to 4 (here is the dimensionality of space). Several different non-analytic dependences on the order parameters in the effective action appear as a consequence of their coupling to the fermions. We show that these nonanalytic dynamical terms dominate over the dynamics prescribed phenomenologically. Hence, the dynamics of fermionic liquid crystal phase is very different from that of the simple phenomenological theory. We present a detailed analysis of the behavior of the dynamical susceptibilities in both phases and at the QCP.
The nematic-smectic QCP is studied in Sec. VI. In classical liquid crystals, the Goldstone mode of the nematic phase plays a very important role at the nematic-smectic transition. There, this relevant coupling drives the transition weakly first order through a fluctuation-induced first order transitionHalperin et al. (1974). However, in the case of the electronic liquid crystals, we find that the coupling between the nematic Goldstone mode and the smectic field is actually irrelevant at the electronic nematic-smectic QCP. Therefore, these two modes can be treated separately, as they are weakly coupled to each other. Several different nematic-smectic critical theories are studied, depending on the relation between the magnitude of the ordering wave vector of the CDW, , and the Fermi wave vector, . For (Fig. 2(a)), we find that the critical smectic field has a dynamic critical exponent , which will result in a contribution to the heat capacity. This is a correction to the conventional linear behavior of Fermi liquids. These quantum fluctuations lead to the existence of four points on the FS where the assumptions of FL theory are violated (Fig. 2(a)). At these points the imaginary part of the fermion self-energy correction . For (Fig. 2(b)), the system exhibits anisotropic scaling: , and for the incommensurate CDW, while , and for the commensurate case. Besides, a non-analytic term, where is the smectic order parameter, is generated in the action of the low-energy effective theory. This non-analytic term is relevant under the renormalization group (RG) for the incommensurate case, suggesting a weak, fluctuation-induced, first-order transition. This coupling is irrelevant in the commensurate case. Here we also find two points on the FS (Fig. 2(b)), where the system has marginal FL behavior, with a quasiparticle scattering rate , and a low temperature correction to the heat capacity , which is subleading. We also consider the special case of a CDW caused by a nearly nested FS, for which we find that the low-temperature heat capacity correction , which is also subleading, and the fermions form a FL, with . We also calculated the dynamic CDW susceptibility for both cases. The case will not be discussed here. In the presence of a lattice this case is quite trivial (see Sec. VI) while for it to occur in a continuum system, where it is non-trivial, requires unphysical assumptions.
The smectic phase is discussed in Sec. VII. In the smectic phase the anisotropic scaling associated with the Goldstone fluctuations are , . We find that the low-temperature heat capacity correction , which is also subleading. The quasiparticle scattering rate in this case is for much of the FS while at the two special points where the Fermi velocity is parallel to the ordering wave vector. Thus, in this case fermions behave as a FL. We also calculated both the longitudinal and transverse dynamic CDW susceptibilities in the smectic phase.
Lattice effects are also discussed. For the case of an incommensurate smectic phase, we show that there is an unpinned smectic phase close to the nematic-smectic critical point. In this phase, the smectic Goldstone mode has a dynamic critical exponent and the system is a FL with at most of the FS and at some special point on the FS described below. Due to the unpinned smectic ordering, the system receives a correction to the low-temperature heat capacity , and we also computed the dynamic transverse CDW susceptibility. Deep into the smectic phase, an incommensurate CDW may be pinned down by lattice distortion. As expected, the fermions in a pinned smectic are in a conventional FL state.
In Sec. VIII we present a brief discussion of the role of thermal fluctuations for these phases and of the classical-to-quantum crossovers. We conclude with a summary of our main results and a discussion of open questions in Sec. IX. Details of the calculations are presented in several appendices. In Appendix A we discuss the tensor structure of the order parameters. In Appendix B we present details of the nematic-smectic QCP for the case , while the nonanalytic terms induced for the case are presented in Appendix C. In Appendix D we present details of the calculation of the spectrum of Goldstone modes in the smectic phase. In Appendix E we summarize the random phase approximation (RPA) calculation of the fermion self-energy at the nematic-smectic QCP and in the smectic phase.
Iii Experimental Status of Electronic Liquid Crystal Phases
During the past decade or so experimental evidence has been mounting of the existence of electronic liquid crystal phases in a variety of strongly correlated (as well as not as strongly correlated) electronic systems. We will be particularly interested in the experiments in the copper oxide high temperature superconductors, in the ruthenate materials (notably SrRuO), and in two-dimensional electron gases (2DEG) in large magnetic fields. However, as we will discuss below, our results are also relevant to more conventional CDW systems such as the quasi-two-dimensional dichalcogenides.
iii.1 High temperature superconductors
In addition to high temperature superconductivity, the copper oxide materials display a strong tendency to have charge-ordered states, such as stripes. The relation between charge ordered statesKivelson and Fradkin (2007), as well as other proposed ordered statesChakravarty et al. (2001); Varma (2005), and the mechanism(s) of high temperature superconductivity is a subject of intense current research. It is not, however, the focus of this paper.
Stripe phases have been extensively investigated in high temperature superconductors and detailed and recent reviews are available on this subjectKivelson et al. (2003); Tranquada (2007). Stripe phases in high temperature superconductors have unidirectional order in both spin and charge (although not always) and it is typically incommensurate. In general the detected stripe order (by low-energy inelastic neutron scattering) in LaSrCuO, LaBaCuO and YBaCuO (see Refs.Kivelson et al. (2003) and Tranquada (2007) and references therein) is not static but “fluctuating”. As emphasized in Ref.Kivelson et al. (2003), “fluctuating order” means that there is no true long range unidirectional order. Instead, the system is in a (quantum) disordered phase, very close to a quantum phase transition to such an ordered phase, with very low-energy fluctuations that reveal the character of the proximate ordered state. On the other hand, in LaBaCuO near (and in LaNdSrCuO also near ), the order detected by elastic neutron scatteringTranquada et al. (2004), and resonant x-ray scattering in LaBaCuO Abbamonte et al. (2005) also near , becomes true long-range static order.
In the case of LaSrCuO, away from , and particularly on the more underdoped side, the in-plane resistivity has a considerable temperature-dependent anisotropyAndo et al. (2002), which has been interpreted as an indication of electronic nematic order. From these experiments it has been suggested that this phase be identified as an electron nematicAndo et al. (2002). The same series of experiments also showed that very underdoped YBaCuO is an electron nematic as well. The most striking evidence for electronic nematic order in high temperature superconductors are the recent neutron scattering experiments in YBaCuO at Hinkov et al. (2008). In particular, the temperature-dependent anisotropy of the inelastic neutron scattering in YBaCuO shows that there is a critical temperature for nematic order (with ) where the inelastic neutron peaks also become incommensurate. Similar effects were reported by the same groupHinkov et al. (2006) at higher doping levels () who observed that the nematic signal was decreasing in strength suggesting the existence of a nematic-isotropic quantum phase transition closer to optimal doping. Fluctuating stripe order in underdoped YBaCuO has been detected earlier on in inelastic neutron scattering experiments Mook et al. (2000); Stock et al. (2004) which, in hindsight, can be reinterpreted as evidence for nematic order. However, as doping increases the strength of the temperature-independent anisotropic background, due to the increased orthorhombicity of the crystal, also increases thus making this phase transition difficult to observe.
Recent inelastic neutron scattering experiments have found similar effects in LaSrCuO materials where fluctuating stripes where in fact first discoveredTranquada et al. (1995). Matsuda et al Matsuda et al. (2008) have given qualitatively similar evidence for nematic order in underdoped LaSrCuO () which was known to have “fluctuating diagonal stripes”. In the same doping range it has also been found by resonant x-ray scattering experiments that 5% Zn doping stabilizes a static diagonal stripe-ordered state with a very long persistence length which sets in at quite high temperaturesRusydi et al. (2007).
These recent results strongly suggest that the experiments that had previously identified the high temperature superconductors as having “fluctuating stripe order” (both inside and outside the superconducting phase) were most likely detecting an electronic nematic phase, quite close to a state with long-range stripe (smectic) order. In all cases the background anisotropy (due to the orthorhombic distortion of the crystal structure) acts as a symmetry breaking field that couples linearly to the nematic order, thus rounding the putative thermodynamic transition to a state with spontaneously broken point group symmetry. These effects are much more apparent at low doping where the crystal orthorhombicity is significantly weaker.
The nature of the fluctuating spin order changes substantially as a function of doping: in the very underdoped systems there is no spin gap while inside much of the superconducting dome there is a finite spin gap. In fact in LaBaCuO at there is strong evidence for a complex stripe-ordered state which combines charge, spin and superconducting orderLi et al. (2007); Berg et al. (2007). These experiments have also established that static long-range stripe charge and spin orders do not have the same critical temperature, with static charge order having a higher .
An important caveat to our analysis is that in doped systems there is always quenched disorder, and has different degrees of short range “organization” in different high temperature superconductors. Since disorder also couples linearly to the charge order parameters it ultimately also rounds the transitions and renders the system to a glassy state (as noted in Refs.Kivelson et al. (1998, 2003)). Such effects are evident in scanning tunneling microscopy (STM) experiments in BiSrCaCuO which revealed that the high-energy (local) behavior of the high temperature superconductors has charge order and it is glassyHowald et al. (2003); Kivelson et al. (2003); Hanaguri et al. (2004); Kohsaka et al. (2007); Vershinin et al. (2004).
Finally, we note that in the recently discovered iron pnictides based family of high temperature superconductors, such as La (OF)FeAs Kamihara et al. (2008); Mu et al. (2008), a unidirectional spin-density wave has been found. It has been suggestedFang et al. (2008) that the undoped system LaOFeAs may have a high-temperature nematic phase and that quantum phase transitions also occur as a function of fluorine dopingXu et al. (2008). This suggests that many of the ideas and results that we present here may be relevant to these still poorly understood materials.
iii.2 Other complex oxides
The existence of stripe-ordered phases is well established in other complex oxide materials, particularly the manganites and the nickelates. In general, these materials tend to be “less quantum mechanical” than the cuprates in that they are typically insulating (although with interesting magnetic properties) and the observed charge-ordered phases are very robust. These materials typically have larger electron-phonon interactions and electronic correlations are comparatively less dominant in their physics. For this reason they tend to be “more classical” and less prone to quantum phase transitions. However, at least at the classical level, many of the issues we discussed above, such as the role of phase separation and Coulomb interactions, also play a key roleDagotto et al. (2001). The thermal melting of a stripe state to a nematic has been seen in the manganite material BiCaMnORübhausen et al. (2000).
iii.3 Ruthenates
Recent magneto-transport experiments in the quasi-two-dimensional bilayer ruthenate SrRuO by the St. Andrews groupBorzi et al. (2007) have given strong evidence of a strong temperature-dependent in-plane transport anisotropy in these materials at low temperatures mK and for a window of perpendicular magnetic fields around Tesla. These experiments provide strong evidence that the system is in an electronic nematic phase in that range of magnetic fieldsBorzi et al. (2007); Fradkin et al. (2007). The electronic nematic phase appears to have preempted a metamagnetic QCP in the same range of magnetic fieldsGrigera et al. (2001); Millis et al. (2002); Perry et al. (2004); Green et al. (2005). This suggests that proximity to phase separation may be a possible microscopic mechanism to trigger such quantum phase transitions, consistent with recent ideas on the role of Coulomb-frustrated phase separation in 2EDGsJamei et al. (2005); Lorenzana et al. (2002).
iii.4 2DEGs in large magnetic fields
To this date, the best documented electron nematic state is the anisotropic compressible state observed in 2DEGs in large magnetic fields near the middle of a Landau level, with Landau index Lilly et al. (1999a, b); Du et al. (1999); Pan et al. (1999). In ultrahigh-mobility samples of a 2DEG in AlAs-GaAs heterostructures, transport experiments in the second Landau level (and above) near the center of the Landau level show a pronounced anisotropy of the longitudinal resistance rising sharply below mK, with an anisotropy that increases by orders of magnitude as the temperature is lowered. These experiments were originally interpreted as evidence for a quantum Hall smectic (stripe) phase Koulakov et al. (1996); Moessner and Chalker (1996); Fradkin and Kivelson (1999); MacDonald and M. P. A. Fisher (2000); Barci et al. (2002). Further experimentsCooper et al. (2001, 2002, 2003) did not show any evidence of pinning of this putative unidirectional CDW as the curves are strictly linear at low bias and no broadband noise was detected. In contrast, extremely sharp threshold electric fields and broadband noise in transport was observed in a nearby reentrant integer quantum Hall phase, suggesting a crystallized electronic state. These facts, together with a detailed analysis of the experimental data, suggested that the compressible state is in an electron nematic phaseFradkin and Kivelson (1999); Fradkin et al. (2000); Wexler and Dorsey (2001); Radzihovsky and Dorsey (2002); Doan and Manousakis (2007), which is better understood as a quantum melted stripe phase.
iii.5 Conventional CDW materials
CDWs have been extensively studied since the mid-seventies and there are extensive reviews on their propertiesGrüner (1988, 1994). From the symmetry point of view there is no difference between a CDW and a stripe (or electron smectic). The CDW states are usually observed in systems which are not particularly strongly correlated, such as the quasi-one-dimensional and quasi-two-dimensional dichalcogenides, and the more recently studied tritellurides. These CDW states are reasonably well described as FLs which undergo a CDW transition, commensurate or incommensurate, triggered by a nesting condition of the FSMcMillan (1975, 1976). As a result, a part or all of the FS is gapped in which case the CDW may or may not retain metallic properties. Instead, in a strongly correlated stripe state, which has the same symmetry breaking pattern, at high energy has Luttinger liquid behaviorKivelson et al. (1998); Emery et al. (2000); Carlson et al. (2004).
What will interest us here is that conventional quasi-2D dichalcogenides, the also quasi-2D tritellurides and other similar CDW systems can quantum melt as a function of pressure in TiSeSnow et al. (2003), or by chemical intercalation as in CuTiSeMorosan et al. (2006); Barath et al. (2008) and NbTaSDai et al. (1991). Thus, CDW phases in chalcogenides can serve as a weak-coupling version of the problem of quantum melting of a quantum smectic. Interestingly, there is strong experimental evidence that both TiSeSnow et al. (2003) and NbTaSDai et al. (1991) do not melt directly to an isotropic Fermi fluid but go instead through an intermediate phase, possibly hexatic. (CuTiSe is known to become superconductingMorosan et al. (2006).) Whether or not the intermediate phases are anisotropic is not known as no transport data is available in the relevant regime.
The case of the CDWs in tritellurides is more directly relevant to the theory we present in this paper. Tritellurides are quasi-2D materials which for a broad range of temperatures exhibit a unidirectional CDW (i.e. an electronic smectic phase) and whose anisotropic behavior appears to be primarily of electronic originBrouet et al. (2004); Laverock et al. (2005); Sacchetti et al. (2006, 2007); Fang et al. (2007). However, the quantum melting of this phase has not been observed yet. Theoretical studies have also suggested that it may be possible to have a quantum phase transition to a state with more than one CDW in these materialsYao et al. (2006).
Iv Order-Parameter Theory
In this section we will construct, using phenomenological arguments, an effective order parameter theory that will describe both the electron nematic and the electron smectic (or unidirectional CDW) phases. Although by symmetry the order parameter theory must be very similar to the ones used in classical liquid crystal phases, we will go through the construction of the phenomenological theory in some detail for several reasons. In 2D the rotation group is Abelian which allows for a significant simplification of the formulas by using a complex order parameter for the nematic phase, instead of a tensor expressions commonly used for 3D classical liquid crystals. Proper dynamical terms now need to be included to describe the quantum fluctuations at zero-temperature. Besides, in order to provide a clear relation between this paper and earlier studies of the CDW state of fermions, we would like to discuss also the relation between the smectic phase and the CDW state.
iv.1 The normal-electronic nematic transition
The nematic order parameter in 2D is a representation of the rotational group Oganesyan et al. (2001). It is defined as a symmetric traceless tensor of rank two.
The 2D rotational group is isomorphic to . Hence, we define instead the complex order-parameter field
where and are the space and time coordinates. We will use this complex order parameter field in this paper to take the advantage of the Abelian nature of .
The conjugate field is . Under a global rotation by an angle , the fields and transform, respectively, as and . Hence, and carry the angular momentum quantum numbers and , respectively.
This complex order parameter can be generalized easily to other angular momentum channels , but not to higher dimensions , since it relies heavily on the special property of the 2D rotational group . In higher dimensions, the rotational group will no longer be Abelian, so one will need to use the tensor formula as in the classical liquid crystal theories. In Appendix A, formulas using the complex order parameter are translated into the conventional tensor form for comparison.
The order-parameter field we just defined is invariant under spatial-inversion and time-reversal
In even spatial dimensions, including 2D in which our system lives, a chiral transformation is different from a space inversion. To change the chirality in 2D, we can reverse the direction and keep the direction unchanged. Under this chiral transformation, the nematic field will be changed into the conjugate field
Here, is the chiral transformation operator.
The effective action must preserve the symmetries of the system, both continuous, as the translational and rotational symmetries, and discrete, as the time reversal, spatial inversion and chiral symmetries. With the assumption of analyticity, the action must be
Here the dynamical term is quadratic in time derivatives. This is because the term linear in time derivatives is not allowed by the chiral symmetry. It is the imaginary part of . It corresponds to a pseudoscalar, and is not allowed.
In 2D, cubic terms in the nematic field are not allowed. Hence, if , the normal-nematic transition is second order, instead of a first-order transition as in the 3D case de Gennes and Prost (1993); Chaikin and Lubensky (1998). For and , the rotational invariant ground state will be stable. When becomes negative, will develop an expectation value with module , which breaks the rotation symmetry. The residual rotational symmetry would be . The argument of determines the direction of the nematic order parameter.
The action of Eq. (LABEL:eq:N_action) has an internal symmetry associated with the phase of the complex field , which is not physical. By symmetry, terms of the form
are allowed Oganesyan et al. (2001); chi (). This kind of terms are irrelevant at the QCP and in the isotropic phase, which leads to the existence of an “emergent” internal symmetry at quantum criticality. But it will be important in the nematic phase, as it makes the two Frank constants to attain different values. (This effect is formally analogous to the role of spin-orbit interactions in the Schrödinger equation: in their absence spin is an internal degree of freedom.) This emergent symmetry of the normal phase and at the critical point is very important for the classical normal-nematic transition, especially in the study about the fluctuation effects Priest and Lubensky (1976); Korzhenevskii and Shalaev (1979); Nelson and Toner (1981).
iv.2 The electronic nematic phase
In the nematic phase, the rotational symmetry is broken. Hence, we expect the fluctuations of the amplitude of the nematic order parameter, , to correspond to a massive mode with an energy gap of (), and the fluctuations of the phase, , constitute the gapless Goldstone mode. Without loss of generality, throughout this paper, we assume that is real and positive. This state corresponds to a nematic order in the main axis direction. In this state, the action of is
where and are the two Frank constants. This action is only valid for small nematic fluctuations. It cannot be used to study topological defects of the nematic phase, known as disclinationsde Gennes and Prost (1993). The field has dynamic critical exponent . This makes the effective dimension of this system , which is above the lower critical dimension of the theory , and nematic order will not be destroyed by fluctuations.
iv.3 CDW multi-critical point
The smectic order is a unidirectional CDW, described by a single complex order parameter field. If we assume analyticity, the effective low-energy theory of the bosonic field can be determined as:
where is the term in the quadratic order of , and are the cubic and quartic terms respectively.
The term in the momentum space is,
The function has the physical meaning of the inverse of the CDW susceptibility. If we assume the ordering wave-vector of the CDW is (with its magnitude), will have the form
where is the energy gap of the CDW excitations Brazovskii (1975), and is a positive constant.
When the energy gap decreases to zero, all the density wave modes with will become soft and critical when . This is very different from an ordinary - or -theory, where we only need to consider one mode (or two modes for a complex field) at small momentum. Here, we need to consider all the modes with the wave vector whose magnitude is close to . In other words, the point is not a critical point but a multi-critical point with an infinite number of critical modes. Even if a lattice background is present, the point may still be a multi-critical point of critical modes, if the lattice has a -fold rotational symmetry for . For a multi-critical point, higher-order terms become important. Without a detailed knowledge of these higher-order terms, it is not possible to determine whether the transition is first or second order, or how many CDWs will be formed in the ordered phase.
Brazovskii Brazovskii (1975) studied the classical version of this problem, considering only the isotropic interactions. Chubukov and co-workers Chubukov et al. (2005) studied the quantum problem in a fermionic system in the high-density regime where the cubic and quartic terms of can be ignored.
In general, depending on the non-Gaussian terms, the ordered phase may have only one or several CDWs Brazovskii (1975). For a rotational invariant system, it is often assumed that CDWs form a triangular lattice to minimize the breaking of the rotational symmetry, as the 2D Wigner crystal state Wigner (1934). For systems with a strong lattice potential, the system is often assumed to become an electron crystal state which preserves the point group rotational symmetry of the background lattice, e.g. the rare-earth tritelluridesYao et al. (2006).
For isotropic systems, outside the nematic phase, the term in Eq. (9) favors that three CDWs form by a first-order transition Brazovskii (1975). However, inside a nematic phase, as we will show below, the nematic order parameter, which is coupled to , favors only one CDW and will compete with the term. For a continuous quantum phase transition, will be a subleading perturbation compared to , at least close enough to the transition. Hence, the smectic phase, a unidirectional CDW, will be energetically favorable. On the other hand, in the case of a first-order transition, depending on microscopic details either the smectic phase or the state with three CDWs would be preferred. We represent these different possibilities in the schematic phase diagram shown in Fig. 1.
On a square lattice, due to the point group symmetry of the lattice, the electron crystal phase usually consists of CDWs perpendicular to each other. The phase transition between this phase and the FL may be second order due to the absence of the cubic term which, in contrast to isotropic systems, is prohibited by momentum conservation. We have confirmed this structure of the phase diagram in a microscopic mean-field calculation. However, at the multi-critical point where both the CDW modes and the nematic mode are critical (the point in Fig. 1), the coupling between CDWs and the nematic order parameter (Eq. (12)) is relevant. This suggests a fluctuation driven first-order transition near the multi-critical point. Hence, this multi-critical point is essentially unreachable.
Figure 1: (online color) Schematic phase diagram at as a function of and defined in Eqs.(LABEL:eq:N_action) and (11). The cross point of the two dash lines is the multi-critical point . The red thick lines stand for first order phase boundaries. Other phase boundaries may be first or second order. More complex electron crystal phases are possible, for example, an anisotropic electron crystal phase where more than one CDWs and nematic coexist, but they are beyond the discussion of this paper.
In this paper, we study the nematic-smectic phase transition and the smectic phase using a weak coupling approach by perturbing about a FL state. This approach is consistent provided the nematic phase is narrow enough in coupling constant space so that the nematic-smectic transition is not too far from the FL phase.
It is useful to compare to the classical version of this problem. The theory of classical (thermal) melting in two dimensions, the Kosterlitz-Thouless-Halperin-Nelson-Young theoryNelson and Halperin (1979); Young (1979) (see Ref.Chaikin and Lubensky (1998)), is a theory of a phase transition driven by the proliferation of topological defects: a dislocation unbinding transition in the case of melting of a 2D Wigner crystal (a triangular lattice) into a hexatic phase, and disclination unbinding transition in the hexatic-isotropic phase transition. (The case of the square lattice was discussed only recently in Ref.DelMaestro and Sachdev (2005)). The reason for the success of the classical theory of melting in two dimensions is that, as in all Kosterlitz-Thouless phase transitionsKosterlitz and Thouless (1973); Chaikin and Lubensky (1998), at finite temperatures the classical ordered state with a spontaneously broken continuous symmetry is not possible in two dimensions. Instead, there is a line (or region) of classical critical behavior with exactly marginal operators. The defect-unbinding phase transition appears as an irrelevant operator becoming marginally relevant.
In the case of the quantum phase transitions in two dimensions that we are interested in, there are no such exact marginal operators available at zero temperature, and hence, no lines of fixed points available. Thus, the quantum phase transition is not triggered by a defect-unbinding operator becoming marginal, but instead by making the coupling constant of an irrelevant operator large (as in standard continuous phase transitions, classical or quantum). Instead, the quantum phase transition is closer to Landau-type (or, rather, Hertz-Millis like) description in that it is governed (as we will see) by a quantum-mechanical analog of the celebrated McMillan-deGennes theory for a nematic-smectic phase transition in classical liquid crystals in three dimensionsde Gennes and Prost (1993); Chaikin and Lubensky (1998). The approach that we will pursue here does not contain much of the physics of strong correlations as it begins with a state with well-defined fermionic quasiparticles. It also does not treat correctly the tendency of strongly correlated systems to exhibit inhomogeneous states and phase separation. The only way to account for this physics correctly is to use the opposite approach, a strong coupling theory of quantum melting of the crystal and stripe phases, as advocated in Ref.Kivelson et al. (1998). So far, this theory only treats the physics deep inside a stripe phases, and the theory of their quantum melting to a nematic phase does not yet exist. Thus, although from a strong-coupling perspective it would be highly desirable to have such a defect unbinding theory of this quantum phase transition (such a description does exist for an insulating systemZaanen et al. (2004) but its extension to a metallic state is not available and it is highly non-trivial), we will pursue instead a Hertz-Millis approach Hertz (1976); Millis (1993); Sachdev (1999) to this quantum phase transition.
iv.4 The electronic nematic-smectic transition
Nematic order will remove the degeneracy of CDW modes in different directions and select one CDW. As a result, the Brazovskii CDW multi-critical point becomes just a critical point. For simplicity, we assume that the nematic order parameter is small enough so that a Landau-type expansion still makes sense, which is equivalent to assuming that the system is still “close enough” to the nematic-isotropic QCP. However, as we will show later, the critical theory we get using these assumptions has the only form allowed by symmetry, assuming analyticity.
By symmetry, the coupling between the CDW and the nematic field is
whose tensor form is shown in Appendix A. Here, is the polar angle of . This term is irrelevant in the isotropic phase, but in the nematic phase, where gets the expectation value ; this term will be of the same order as , which was defined in Eq. (10), and hence it becomes important.
Inside the nematic phase the amplitude fluctuations of the nematic order parameter are gapped while the orientational fluctuations, the nematic Goldstone modes, are gapless, at least strictly in the absence of a lattice and other orientational symmetry breaking couplings. Thus, deep enough in the nematic phase it is possible to integrate out the gapped nematic amplitude fluctuations and derive an effective theory involving the gapless nematic Goldstone mode. However, as the nematic-smectic phase transition is approached, the gap of the fluctuations of the smectic order parameter will get smaller and will approach zero at the QCP. Thus, in this regime, the nematic phase has low-energy “fluctuating stripes”. This regime is the analog of that in conventional liquid crystals where the McMillan-deGennes classical theory applies . We will now see how this theory arises in the quantum case.
The leading term in of Eq. (12) will be
This term will stabilize the density wave in either or direction and destabilize the other, depending on the sign of . As a result, the nematic order will select a special direction along which only one CDW will form. Past this phase transition the system will be in a smectic state, a unidirectional CDW. For simplicity, we assume , which selects in the direction.
Only the density fluctuations close to matter for the low-energy theory. We define a complex field , describing the density fluctuations around as
where is small. The real part of measures the density fluctuations.
Under a spatial inversion, will become its conjugate field . Hence, the term is not allowed in the Lagrangian, and the dynamical term for is at least quadratic in time derivatives.
The cubic term of the field in the isotropic-CDW transition vanishes in the nematic-smectic transition, due to momentum conservation. By expanding Eq. (12) around , and , we obtain
Here is the action of the nematic Goldstone mode defined in Eq. (LABEL:eq:nematic_goldstone).
The action of Eq. (LABEL:eq:ns_action) is just a 2D version McMillan-de Gennes theory of the nematic-smectic transition in the classical liquid crystals but with quantum dynamics. The constants in Eq.(LABEL:eq:ns_action) are
Here is the energy gap of field, which mainly comes from the CDW gap defined in Eq. (11). The correction term comes from the nematic ordering. The term comes from the interactions between CDWs and it gets a correction from the amplitude fluctuations of the nematic order, which has been integrated out. The nematic Goldstone field couples to the CDW field as a gauge field with a “charge” . Here the two in the denominator comes from the fact that the nematic order parameter has an angular momentum . This gauge-like coupling is required by the rotational symmetry since, under spatial rotation by a small angle , the fields transform as and (for the angular momentum channel ). In fact, with the symmetry constrain and the assumption of analyticity, the action we show in Eq. (LABEL:eq:ns_action) is the only allowed form for the effective low-energy theory, provided the topological excitations of are ignored Renn and Lubensky (1988). Therefore, although we only keep linear terms of in our calculations above, which is valid close to the normal-nematic critical point, the action in Eq. (LABEL:eq:ns_action) will have the same form even deep inside the nematic phase.
The theory with the effective action given in Eq.(LABEL:eq:ns_action) has a critical field and gapless Goldstone boson . A naive mean-field theory would suggest that this is a continuous phase transition. In the case of the theory of classical liquid crystals, where the same naive argument also holds, Halperin, Lubensky and Ma Halperin et al. (1974) used the expansion to show that there is a run-away behavior in the renormalization group flows, similar to that of superconducting transition coupled to a fluctuating electromagnetic field. They concluded that in both cases the transition is probably weakly first order, a fluctuation-induced first-order transition. In other terms, in the classical theory the coupling of the smectic to the nematic Goldstone mode (which has the same form as a coupling to a gauge field) is relevant. To ascertain what happens in the case of the metallic nematic-smectic QCP we will also need to take into account the effects of the fermionic degrees of freedom. We will see that the fermionic fluctuations change the critical behavior in an essential way.
iv.5 The electronic smectic phase: a unidirectional CDW
In the smectic phase, the amplitude fluctuations of the order parameter, , are gapped but the phase fluctuations, , are gapless, as required by the Ward-identity. This happens in systems for which lattice effects can be neglected, and hence are described formally in a continuum, or if the smectic order is sufficiently incommensurate. Therefore, upon integrating out the gapped amplitude fluctuations , the effective low-energy theory of the Goldstone mode becomes
When we are close to the nematic-smectic critical point, the coefficients of this effective action are
with being the expectation value of the CDW order parameter. The vanishing of the stiffness term Peierls (1935); Landau (1937) is required by the Ward identity of rotational invariance. Thus, an underlying lattice, which will break the continuous rotational symmetry down to its discrete point group, will lead to a non-vanishing stiffness. Nevertheless, in many cases and particularly away from situations in which the FS is strongly nested, the breaking of rotational invariance can be parametrically small enough that at low temperatures its effects to a first approximation can be neglected and treated perturbatively afterward.
A simple scaling analysis of the effective action of Eq. (17) shows that, at the tree level, the scaling dimensions of space and time , and , are , and , respectively. Although the time direction and the direction scales in the same way, the and directions now scale differently. This is a typical phenomenon for anisotropic states. Now the effective dimensions of this quantum theory is . Hence, our theory is above its (upper) critical dimension. So the higher-order interactions of the smectic Goldstone mode will be irrelevant, if we don’t consider topological defects. The fact that we are above the critical dimension also tells us that the quantum fluctuations in the quantum smectic phase will not destroy the long-range order.
This scaling is very different from the classical smectic phase of 3D, where and , if the modulation is on the direction. This classical theory is at its lower critical dimension, and long-range order is destroyed by fluctuations Peierls (1935); Landau (1937), resulting in a power-law quasi-long-range order. This system has a line of critical points, so the higher order terms of the action that need to be considered were found to lead to logarithmic corrections to the power-law behavior Grinstein and Pelcovits (1981).
The above analysis implies that our quantum problem is above the lower critical dimension. Therefore all these effects of the 3D classical smectic phase will not be present in the 2D quantum case. The scaling behavior of a 2D quantum system is similar to the columnar state of the classical liquid crystals, instead of that of classical smectics. The classical columnar state has two density waves so that it is a solid in two directions but a liquid in the third direction. The Goldstone fluctuations of this state scale as and de Gennes and Prost (1993), which is the same as in the present case, if we consider the time direction in our problem as the direction. The difference between the classical columnar state and the 2D quantum smectic state is that in the 3D columnar state, the Goldstone mode is a planar vector but in the present case it is a scalar.
V Coupling the Order Parameter Theory to Fermions
Figure 2: (online color) The FS of the nematic phase (a and b) and the reconstructed FS in the smectic phase (c and d). (a) and (c) are for at the QCP and in the smectic phase respectively, while (b) and (d) are , also at the QCP and in the smectic phase respectively. In (a) and (b), the black dots marked the non-FL points on the FS caused by the smectic mode fluctuations at the nematic-smectic QCP. The relevance of the points in (c) is explained in Sec. VII. In (c) we have show the case of to be comparable to so as to keep the FS reconstruction simple. Here we show the effective Brillouin Zone with an open orbit and a closed pocket. The reconstructed FS of case (d) is partially gapped and the FS has an open orbit.
We will now proceed to couple the phenomenological theory of the nematic and smectic phases to a system of a priori well-defined fermionic quasiparticles described by the Landau theory of the FL. In a fermionic liquid crystal state, the bosonic order-parameter fields, defined above, will couple to the fermions.
Let us define and to be the fermion creation and annihilation operators of a FL. We will assume that the FL has a well-defined FS, which for simplicity we will assume is circular. (For lattice systems the FS will have the symmetry of the point group of the lattice.) The Fermi wave vector is . The Fermi velocity is set to so that the energy and momentum have the same units. Consistent with the assumptions of the Landau theory of the FLBaym and Pethick (1991) the effective Hamiltonian of the fermionic quasiparticles will be taken to be that of a free Fermi system, with a well-defined FS, and a set of quasiparticle interactions parametrized by the Landau parameters. These interactions are irrelevant in the low-energy limit of the FL but play an important role in the physics of electronic liquid crystal phases Oganesyan et al. (2001). In any case in our discussion it will be unnecessary to include the Landau parameters explicitly since their effects will already be taken into account through the coupling to the liquid crystal order parameters.
By symmetry, the nematic order-parameter field, , couples to the fermion density quadrupole Oganesyan et al. (2001)
In 2D, since the rotational group is , the density quadrupole can be defined in terms of a two-component real director field (i.e. a headless vector) or, in terms of complex field
Same as the nematic order parameter, is also invariant under rotations by .
The coupling between and is
Here is a coupling constant. Again, the chiral symmetry of the system requires that the effective action depends only on the real part of , and that there is no dependence on the imaginary part, since it is a pseudo scalar. The tensor form of this coupling is shown in Appendix A. In what follows we choose the sign of to be negative, so that a positive expectation value of the nematic order parameter means a FS stretched along the direction and compressed in the direction, as shown in Figs. 2(a) and (b).
The sign of alone is not important. What matters is the relative sign between and the coupling constant defined in Eq. (12). Under a redefinition of becoming , both and change sign. If , prefers the direction in which the FS is stretched, but when , it prefers the direction where the FS is compressed. In general, the sign of is determined by microscopic details of the system to which this model may apply.
If , very close to a nesting condition the curvature of the FS controls the CDW instability as it controls how singular the charge susceptibility is near the nesting wave vector. In this case one finds that it leads to the condition , when connects two points on the Fermi surface where the curvature is smallest, as shown in Fig.2(b). In general, far from a nesting condition, the curvature of the FS alone is not the dominant factor, and the sign of may be positive or negative, depending on the microscopic details.
The smectic order-parameter field should be coupled to the CDW of the fermions. The CDW operator of the fermions, close to the ordering wave vector , is
where . The smectic order-parameter field couples to this fermion density wave as
Integrating out the bosons, attractive four-fermion interactions are generated of the form
Hence, the order-parameter fields can be regarded as Hubbard-Stratonovich fields used to decouple four-fermion interactions. In this picture, the couplings between the order-parameter fields and fermions are measuring the strength of the attractive four-fermion term.
Gapless fermions will introduce nonanalytic terms to the low-energy effective theory of the nematics and smectics. For the case of the nematic order parameter, it was shown by Oganesyan and co-workers Oganesyan et al. (2001) that the fermions generate nonanalytic Landau damping termsHertz (1976); Millis (1993), so the theory of the isotropic-nematic metallic QCP becomes
The nematic susceptibility at this FL-nematic QCP isOganesyan et al. (2001)
The phase mode of the nematic order-parameter field in the nematic phase, the nematic Goldstone mode, has an effective action of the form
where is the expectation value of the nematic order parameter and is the angle between and the main axis direction of the nematic ordering. The stiffnesses and (the Frank constants) are given in Ref.Oganesyan et al. (2001). With this action, it follows that the transverse nematic susceptibility in the electron nematic phase isOganesyan et al. (2001)
For the case of a nematic order parameter aligned along the -axis, the angular factor becomes .
For the case of a charged smectic, a unidirectional CDW, a similar effect will be observed. Besides, if connects to points on the FS which have just the opposite Fermi velocity as shown in Fig. 2(b), the discontinuity leads to another type of nonanalytic terms as will be shown in Sec. VI.2.
Vi the Nematic-Smectic Metallic Quantum Critical Point
In this section, we study the metallic nematic-smectic QCP. The two cases shown in Figs. 2(a) , and (b) , are studied separately.
Deep in the nematic phase, the amplitude fluctuations of the nematic order parameter are gapped, and the low-energy fluctuations are due to the nematic Goldstone mode, , whose action is given in Eq. (LABEL:eq:quantum_nematic_goldstone). However, as the nematic-smectic QCP is approached (from the nematic side) the fluctuations of the smectic order parameter become progressively softer and, provided the quantum phase transition is continuous, become gapless at this QCP. In this scenario, the nematic phase looks like a “fluctuating stripe” phase qualitatively similar to the phenomenology of the cuprate superconductors, as discussed in Sec. III.1.
The case will not be discussed here. The reason is that since now the CDW fluctuations with cannot decay into particle-hole pairs, in this case fermions only renormalize the coefficients of the smectic effective action, while the nematic fluctuations will still be Landau damped. For isotropic systems and for , the CDW (Lindhard) susceptibility in general decreases faster than linear as , where is a constant. This implies that a CDW with is unlikely to be realized as it would require an anomalously attractive interaction at a large . However, for a lattice system the phase fluctuations of the nematic mode get gapped by lattice anisotropies and in this case the fermions only yield the trivial effect of renormalizing the coefficients of the effective action at the CDW transition.
For , the leading contribution to the effective action of the order parameter field, resulting from integrating out the fermions, has the form
Here is the CDW susceptibility of the fermions, given by the fermion loop integral (bubble)
where is the Fermi-Dirac distribution function.
The static part of the fermion CDW susceptibility depends on the details of the dispersion relation from way above the FS to the bottom of the band. However, since is analytic for , the static part, , will not change the analytic structure of Eq. (LABEL:eq:ns_action), but just renormalize the coefficients, in particular the critical value of the coupling constant. The important contribution comes from the dynamical part, . The singular contributions to this integral are dominated by the behavior of the integrand around the four points on the FS, which are connected by the ordering wave vector , as marked with black dots on Fig. 2(a). If we expand the dispersion relation of the fermions around these four points, , to leading order we get a Landau damping contribution
which is linear in . The formula above can be checked by taking the limit of or . In these two regimes, the fermion loop integral can be computed by RPA without expanding the dispersion relations around the four points. After setting , for , one finds with being the density of states and for , , which can be reached by expanding Eq. (37). Both of them agree with the general formula given above.
The term linear in in the effective action for the smectic field of Eq. (31), which is due to the contributions of the fermions, dominates over the “naive” dynamical term proportional to of the phenomenological theory. We can thus write an effective action for the electron nematic-smectic quantum phase transition of the form
where , and . The point and is the nematic-smectic critical point. With the nonanalytic dynamical term, the dynamic critical exponent of the field becomes , instead of as it would generally be in the absence of fermions (or, if the fermions were gapped as in the case of an insulator).
The nematic Goldstone mode has a dynamic critical exponent Oganesyan et al. (2001), larger than the exponent for the smectic fluctuations. Thus, the Goldstone mode of the nematic order parameter and the smectic fluctuate on very different energy scales, with being the low-energy mode. If we only focus on the asymptotic low-energy theory, we should integrate out the high-energy mode . This process will lead to an effective theory of . In turn, the low-energy mode will mediate interactions of the field . However, we will show by a scaling argument that in the case of the quantum metallic system the coupling between the smectic field and the nematic Goldstone mode is irrelevant.
The action of Eq.(LABEL:eq:quantum_mcmillan) is invariant under a rescaling parametrized by a factor
where , , , , and are the stiffness in Eq. (LABEL:eq:quantum_mcmillan).
When , at the tree level and in the long-wavelength regime, both the gauge-like “coupling constant” and scale to infinite, but the ratio scales to as a function of . This implies that the gauge-like coupling is irrelevant. Quantum fluctuations may change the tree-level scaling behavior as we include loop corrections. However, for large enough or small enough , the irrelevancy of the gauge-like coupling will not be changed. As a byproduct, we notice that and scale to zero in the long-wavelength regime, which means that these two terms are irrelevant also. However, we should keep in mind that these operators are actually dangerous irrelevant, in the sense that is necessary to find the proper equal-time correlation function for and is necessary for stability in the ordered phase, and they are only irrelevant at this QCP.
Notice that at the QCP there are two critical modes: the amplitude of the CDW order parameter, which has , and the transverse (Goldstone) mode of the nematic phase, which has (and it is clearly dominant at low enough energies). Thus, we also need to check the scaling behavior of and for the high-energy mode. Under this rescaling,
At the critical point where , it can be seen that also scales to as in the long-wavelength limit, which means, for the mode, the gauge-like coupling is still irrelevant.
These conclusions are confirmed by one-loop perturbation theory calculations, presented in Appendix B, where we show that integrating out (or ) does not change the action of (or ). This is one of our main results.
Figure 3: (color online) The spectral density of the smectic susceptibility at the nematic-smectic QCP, , as a function of and for . The spectral density is singular near the origin (lower right corner) and decays monotonically away from there. Here we show contour plots at constant spectral density with values from up to . The red line, , marks the peak of the spectral density as a function of momentum parallel to the nematic orientation. The inset is the energy dependence of at a fixed small momentum (along the dashed vertical line).
In conclusion, there are two essentially decoupled soft modes at the nematic-smectic QCP. The nematic Goldstone mode is governed by the same action as in the nematic phase, Eq. (LABEL:eq:quantum_nematic_goldstone). Since as the nematic-smectic QCP is approached from the nematic side, the nematic Goldstone mode and the smectic order parameters effectively decouple; the effective action for the smectic field in this limit reduces to
which implies that the dynamic smectic susceptibility is |
2a65ff957006269a | Conical Quantum Dot
Application ID: 723
Quantum dots are nano- or microscale devices created by confining free electrons in a 3D semiconducting matrix. Those tiny islands or droplets of confined “free electrons” (those with no potential energy) present many interesting electronic properties. They are of potential importance for applications in quantum computing, biological labeling, or lasers, to name only a few.
Quantum dots can have many geometries including cylindrical, conical, or pyramidal. This model studies the electronic states of a conical InAs quantum dot grown on a GaAs substrate. To compute the electronic states taken on by the quantum dot/wetting layer assembly embedded in the GaAs surrounding matrix, the 1-band Schrödinger equation is solved. The four lowest electronic energy levels with corresponding eigenwave functions for those four states are solved in this model using the Coefficient form in COMSOL Multiphysics.
The model was based upon the paper: R. Melnik and M. Willatzen, “Band structure of conical quantum dots with wetting layers,” Nanotechnology, vol. 15, 2004, pp. 1-8.
COMSOL Multiphysics® |
d7e8b5574a329297 | Skip to main content
Chemistry LibreTexts
4.4: The Time-Dependent Schrödinger Equation
There are two "flavors" of Schrödinger equations: the time-dependent and the time-independent versions. While the time-dependent Schrödinger equation predicts that wavefunctions can form standing waves (called stationary states), that if classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by the time-independent Schrödinger equation (used only when the Hamiltonian is not explicitly time dependent). However, it should be noted that the solutions to the time-independent Schrödinger equation still have time dependencies.
Time-Dependent Wavefunctions
Recall that the time-independent Schrödinger equation
\[\hat{H}\psi (x)=E\psi (x) \label{4.4.1}\]
yields the allowed energies and corresponding amplitude (wave) functions. But it does not tell us how the system evolves in time. It would seem that something is missing, since, after all, classical mechanics tells us how the positions and velocities of a classical system evolve in time. The time dependence is given by solving Newton's second law
\[m\dfrac{d^2 x}{dt^2}=F(x) \label{4.4.2}\]
But where is \(t\) in quantum mechanics? First of all, what is it that must evolve in time? The answer is that the wavefunction (and associated probability density) must evolve. Suppose, therefore, that we prepare a system at \(t=0\) according to a particular probability density \(p(x,0)\) related to an amplitude \(\Psi (x,0)\) by
\[p(x,0) =|\Psi (x,0)|^2 \label{4.4.3}\]
How will this initial amplitude \(\Psi (x,0)\) look at time \(t\) later? Note, by the way, that \(\Psi (x,0)\) does not necessarily need to be one of the eigenstates \(\psi_n (x)\). To address this, we refer to the time-dependent Schrödinger equation that tells us how \(\Psi (x,t)\) will evolve starting from the initial condition \(\Psi (x,0)\):
\[\hat{H}\Psi (x,t)=i\hbar\dfrac{d}{dt}\Psi (x,t) \label{4.4.4}\]
It is important to know how it works physically and when it is sufficient to work with the time-independent version of the Schrödinger equation \(\hat{H}\psi (x)=E\psi (x)\)
Postulate 5
The time dependence of wavefunctions is governed by the Time-Dependent Schrödinger Equation (Equation \(\ref{4.4.1}\)).
Suppose that we are lucky enough to choose
\[\Psi (x,0) =\psi_n (x), \;\;\;\; p(x,0) =|\psi_n (x)|^2 \label{4.4.5}\]
We will show that
\[\Psi (x,t) =\psi_n (x)e^{-iE_n t/\hbar} \label{4.4.5A}\]
From the time-dependent Schrödinger equation
\[\begin{align*}\dfrac{d\Psi}{dt} &= \psi_n (x) \left ( \dfrac{-iE_n}{\hbar} \right ) e^{-iE_n t/\hbar} \\ i\hbar \dfrac{d\Psi }{dt} &= E_n \psi_n (x) e^{-iE_n t/\hbar} \end{align*} \label{4.4.6}\]
\[\hat{H}\Psi (x,t)=e^{-iE_n t/\hbar}\hat{H}\psi_n (x)=e^{-iE_n t/\hbar}E_n \psi_n (x) \label{4.4.7}\]
Hence \(\psi_n (x) exp(-iE_n t/\hbar)\) satisfies the equation.
Nonstationary States
Consider the probability density \(p(x,t)=|\Psi (x,t)|^2\):
\[\begin{align*}p(x,t) &= \left [ \psi_n (x) e^{iE_n t/\hbar} \right ] \left [ \psi_n (x)e^{-iE_n t/\hbar} \right ] \\ &= \psi_{n}^{2}(x)e^{iE_n t/\hbar}e^{-iE_n t/\hbar}\\ &= |\psi_n (x)|^2 =p(x,0)\end{align*} \label{4.4.8}\]
the probability does not change in time and for this reason, \(\psi_n (x)\) is called a stationary state. In such a state, the energy remains fixed at the well-defined value \(E_n\)Suppose, however, that we had chosen \(\Psi (x,0)\) to be some arbitrary combination of the two lowest energy states:
\[\Psi (x,0) =a\psi_1 (x)+b\psi_2 (x) \label{4.4.9}\]
for example
\[\Psi (x,0) =\dfrac{1}{\sqrt{2}}[\psi_1 (x)+\psi_2 (x)] \label{4.4.10}\]
as in the previous example. Then, the probability density at time \(t\)
\[p(x,t) = |\Psi (x,t)|^2 \neq p(x,0) \label{4.4.11}\]
For such a mixture to be possible, there must be sufficient energy in the system that there is some probability of measuring the particle to be in its excited state.
Finally, suppose we start with a state \(\Psi (x,0)=(1/\sqrt{2})[\psi_1 (x)+\psi_2 (x)]\), and we let this state evolve in time. At any point in time, the state \(\Psi (x,t)\) will be some mixture of \(\psi_1 (x)\) and \(\psi_2 (x)\), and this mixture changes with time. Now, at some specific instance in time \(t\), we measure the energy and obtain a value \(E_1\). What is the state of the system just after the measurement is made? Once we make the measurement, then we know with 100% certainty that the energy is \(E_1\). From the above discussion, there is only one possibility for the state of the system, and that has to be the wavefunction \(\psi_1 (x)\), since in this state we know with 100% certainty that the energy is \(E_1\). Hence, just after the measurement, the state must be \(\psi_1 (x)\), which means that because of the measurement, any further dependence on \(\psi_2 (x)\) drops out, and for all time thereafter, there is no dependence on \(\psi_2 (x)\). Consequently, any subsequent measurement of the energy would yield the value \(E_1\) with 100% certainty. This discontinuous change in the quantum state of the system as a result of the measurement is known as the collapse of the wavefunction. The idea that the evolution of a system can change as a result of a measurement is one of the topics that is currently debated among quantum theorists.
The fact that measuring a quantum system changes its time evolution means that the experimenter is now completely coupled to the quantum system. In classical mechanics, this coupling does not exist. A classical system will evolve according to Newton's laws of motion independent of whether or not we observe it. This is not true for quantum systems. The very act of observing the system changes how it evolves in time. Put another way, by simply observing a system, we change it! |
9d1060c9da760243 | Ditching the Indeterminate Cat: Griffith Scientists Suggest Testability of their Many Interacting Worlds Theory
Author: Belinda Ongaro
Institution: University of Alberta
It may be a bit mind-bending to consider the possibility of parallel worlds interacting, let alone existing, but a recent theory proposed by Griffith University researchers suggests that there may be more to their model than science fiction. Professor Howard Wiseman and researcher Michael Hall, PhD, from Griffith University, along with Dirk-Andre Deckert, PhD, of the University of California, recently reported that their radical “Many Interacting-Worlds”(MIW) theory is not only plausible, but also potentially testable.
MIW theory postulates an abundance of parallel worlds that coexist and interact with one another, including our own. On the other hand, the prevailing Many-Worlds Interpretation (MWI) envisions a continuous propagation of worlds that never interact, with “each universe branch[ing] into a bunch of new universes every time a quantum measurement is made,” according to Wiseman. Clearly, the prospect of coming in contact with parallel worlds holds a little more intrigue.
The MIW approach posits that the parallel universes have always existed and that they experience interaction with other worlds. The “interaction” they refer to is one of mutual repulsion between similar worlds, comparable to how the common poles of a magnet will repel each other, which consequently renders them more different.
The research team describe the “interacting” aspect of their theory by means of analogy. If world A and world B are represented by independent gas entities they will experience the presence of each other much the same as they would experience the presence of particles of their own respective entities. The interaction is negligible unless all of the A gas particles are in close proximity to their B counterparts, at which point their interaction becomes quantifiable. The associated particles hold similar locations in both worlds; therefore, when they are in close proximity they experience repulsion. The whole point of parallel worlds is that they are unique, so this force prevents the configuration of any two parallel worlds from becoming too similar.
Furthermore, the Griffith researchers propose that worlds comprising our universe are equally real; there is no “true” world that serves as a reference point outside of our own subjective frame of reference — a common and natural assumption. Egocentric people often need to be reminded of the fact that they are not the center of the universe; coincidentally, our universe may not be the center of all universes either.
As a result, MIW approach differs from preceding models on the basis of wave function; the solution to the Schrödinger equation, which describes how a physical system’s quantum state changes with time. In the approach taken by the Griffith scientists, probabilities of different states being realized only arise because we don’t know which world we occupy, so wave function is essentially irrelevant. How could all possibilities “collapse” or settle on existing in a single state if the observer’s perspective is arbitrary? Dead or alive, Schrödinger’s cat gets to sit this one out.
"Once you have six or seven quantum particles interacting, the Schrödinger equation is far too hard to solve, even approximately," Hall says. "With our theory, we just have to worry about where the particles are in each world, and calculate that inter-world force between them."
Conceptualization aside, the findings of this model, where a large but finite quantity of worlds are considered, may someday give rise to real-world applications in technology, although the “how” and “what” are still a little fuzzy. Being in its early stages, their model has yet to explain quantum entanglement – the idea that particles are linked in terms of their properties although they may be located a distance away from one another.
“By no means have we answered all the questions that such a shift entails,” says Wiseman.
Looking ahead, Wiseman hopes to delve deeper into the nature of world interaction, looking at the forces and conditions behind the theoretical phenomenon. The team anticipates that the deviations from quantum theory will be subtle but valuable – that is if they can successfully identify means of testing their theory. Perhaps in one parallel universe they already have.
This News Brief was produced under the guidance of Science Writing Mentor Brian Clark Howard.
1. Griffith Scientists Propose Existence and Interaction of Parallel Worlds
2. Quantum Phenomena Modeled by Interactions Between Many Classical Worlds
3. A Quantum World Arising from Many Ordinary Ones |
2e559cf2ea4403bb | Take the tour ×
What is the Madelung transformation and how is it used?
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I am interested in an answer but perhaps it would be nice to give a little motivation why you are interested in this? – Marek Nov 26 '10 at 11:00
Indeed this is intended to be a Google-magnet; I know something about it because of my superfluid helium work. – mbq Nov 26 '10 at 11:26
What do you mean by "Google-magnet" here? Are you looking for an answer or not? If so, you should address the concerns of Marek. – j.c. Nov 27 '10 at 2:33
Well, this question is hit #8 on Google for "Madelung transformation," I guess that counts for something ;-) But I agree with Marek that expanding on your motivation would make this a better question. – David Z Nov 27 '10 at 4:19
Your comments looks like I'm asking about building a nuclear bomb ;-) . I think this question is quite straightforward. I can and probably I'll answer it myself if no one answers it within a week, yet I hope this won't be the case and I'll learn something interesting from those answer(s). The fact it may attract few more people here is just an additional gain. – mbq Nov 27 '10 at 18:03
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I googled "Madelung Transformation" in German and found the following answer in an article about the theory of super conductivity.
The Schrödinger equation is an axiom of quantum physics that is difficult to interpret i.a. because of its usage of complex numbers: $i\hbar \frac{\partial}{\partial t}\Psi = \mathcal{H} \Psi$.
In the Kopenhagen interpretation $\Psi\Psi^*$ is known to describe the probability distribution of a particle. Erwin Madelung tried (already in 1926) to understand the nature of the Schrödinger equation from a different point of view. He substituted $\Psi = ae^{i\phi}$ and split the Schrödinger equation in its real part and its imaginary part (first and second Madelung equation).
Because $\Psi\Psi^* = a^2$, the second Madelung equation (see above reference) becomes a continuity equation of the probability density $a^2$. This equation can thus be interpreted in a more physical way than the Schrödinger equation and that's the point. Similar arguments hold for the first equation but these arguments are more involved (see above reference).
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This is extremely interesting - does the paper contain the expressions for the Madelung equations? – Sklivvz Nov 27 '10 at 21:07
Isn't this essentially the path followed in the Bohmian approach towards quantum mechanics? ... To answer my own question, the wikipedia page on Bohm's theory says: "Around this time Erwin Madelung[29] also developed a hydrodynamic version of Schrödinger's equation which is incorrectly considered as a basis for the density current derivation of the de Broglie–Bohm theory. The Madelung equations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohm mechanics[30] and are the basis of the hydrodynamic interpretation of quantum mechanics." – user346 Nov 27 '10 at 23:45
For a recent paper which compares the Bohmian and Madelung approaches see this. Also contains all the math ! Cheers. – user346 Nov 27 '10 at 23:58
First, Madelung equations. Second, this is not the transformation that is relevant to quantum fluid dynamics (at which mbq was hinting in the first comment under the question). The interesting one comes from nonlinear Schroedinger equation and is apparently somehow also related to KdV equation. Just google for papers. Still, I'll won't add this as an answer because I don't quite understand the stuff (I just looked at papers for a little while). And for the same reason I think Gerard's answer is pretty unsatisfactory. – Marek Nov 28 '10 at 15:01
@Marek It is related, you just do the same trick on Gross-Pitaevskii equation. Related paper: springerlink.com/content/0v2l261ru255n721 – mbq Nov 29 '10 at 12:30
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2b38482e566b060f | Eigenvalues and eigenvectors
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In this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping, and since its length is unchanged its eigenvalue is 1.
An eigenvector of a square matrix A is a non-zero vector v that, when the matrix is multiplied by v, yields a constant multiple of v, the multiplier being commonly denoted by \lambda. That is:
A v = \lambda v
(Because this equation uses post-multiplication by v, it describes a right eigenvector.)
The number \lambda is called the eigenvalue of A corresponding to v.[1]
In analytic geometry, for example, a three-element vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector v is an arrow whose direction is either preserved or exactly reversed after multiplication by A. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.
In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors \mathbb{R}^n is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function f(x) = a^x is an eigenfunction of the derivative operator " {}' ", with eigenvalue \lambda = \ln a, since its derivative is f'(x) = (\ln a)a^x = \lambda f(x).
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.[2] Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix A is the set of all eigenvectors with the same eigenvalue, together with the zero vector.[1] An eigenbasis for A is any basis for the set of all vectors that consists of linearly independent eigenvectors of A. Not every matrix has an eigenbasis, but every symmetric matrix does.
The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen- is adopted from the German word eigen for "self-" or "unique to", "peculiar to", or "belonging to."
Eigenvectors and eigenvalues of a real matrix[edit]
In many contexts, a vector can be assumed to be a list of real numbers (called elements), written vertically with brackets around the entire list, such as the vectors u and v below. Two vectors are said to be scalar multiples of each other (also called parallel or collinear) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a scaling factor, or a scalar). For example, the vectors
u = \begin{bmatrix}1\\3\\4\end{bmatrix}\quad\quad\quad and \quad\quad\quad v = \begin{bmatrix}-20\\-60\\-80\end{bmatrix}
are scalar multiples of each other, because each element of v is −20 times the corresponding element of u.
A vector with three elements, like u or v above, may represent a point in three-dimensional space, relative to some Cartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition "u is parallel to v" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line.
If we multiply any square matrix A with n rows and n columns by such a vector v, the result will be another vector w = A v , also with n rows and one column. That is,
\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad\quad is mapped to
\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \;=\;
\begin{bmatrix} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\
A_{2,1} & A_{2,2} & \ldots & A_{2,n} \\
A_{n,1} & A_{n,2} & \ldots & A_{n,n} \\
where, for each index i,
w_i = A_{i,1} v_1 + A_{i,2} v_2 + \cdots + A_{i,n} v_n = \sum_{j = 1}^{n} A_{i,j} v_j
In general, if v is not all zeros, the vectors v and A v will not be parallel. When they are parallel (that is, when there is some real number \lambda such that A v = \lambda v) we say that v is an eigenvector of A. In that case, the scale factor \lambda is said to be the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix A may change both the direction and the magnitude of an arrow v in three-dimensional space. However, if v is an eigenvector of A with eigenvalue \lambda, the operation may only change its length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if |\lambda| > 1, remain the same if |\lambda| = 1, and decrease it if |\lambda|< 1. Moreover, the direction will be precisely the same if \lambda > 0, and flipped if \lambda < 0. If \lambda = 0, then the length of the arrow becomes zero.
An example[edit]
The transformation matrix \bigl[ \begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix} \bigr] preserves the direction of vectors parallel to \bigl[ \begin{smallmatrix} 1 \\ 1 \end{smallmatrix} \bigr] (in blue) and \bigl[ \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr] (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. See also: An extended version, showing all four quadrants.
For the transformation matrix
A = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix},
the vector
v = \begin{bmatrix} 4 \\ -4 \end{bmatrix}
is an eigenvector with eigenvalue 2. Indeed,
A v = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 4 + 1 \cdot (-4) \\ 1 \cdot 4 + 3 \cdot (-4) \end{bmatrix}
= \begin{bmatrix} 8 \\ -8 \end{bmatrix} = 2 \cdot \begin{bmatrix} 4 \\ -4 \end{bmatrix}.
On the other hand the vector
is not an eigenvector, since
\begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 0 + 1 \cdot 1 \\ 1 \cdot 0 + 3 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix},
and this vector is not a multiple of the original vector v.
Another example[edit]
For the matrix
A= \begin{bmatrix} 1 & 2 & 0\\0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix},
we have
A \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1\\0\\0 \end{bmatrix},\quad\quad
A \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 0\\0\\3 \end{bmatrix} = 3 \cdot \begin{bmatrix} 0\\0\\1 \end{bmatrix}.\quad\quad
Therefore, the vectors [1,0,0]^\mathsf{T} and [0,0,1]^\mathsf{T} are eigenvectors of A corresponding to the eigenvalues 1 and 3 respectively. (Here the symbol {}^\mathsf{T} indicates matrix transposition, in this case turning the row vectors into column vectors.)
Trivial cases[edit]
The identity matrix I (whose general element I_{i j} is 1 if i = j, and 0 otherwise) maps every vector to itself. Therefore, every vector is an eigenvector of I, with eigenvalue 1.
More generally, if A is a diagonal matrix (with A_{i j} = 0 whenever i \neq j), and v is a vector parallel to axis i (that is, v_i \neq 0, and v_j = 0 if j \neq i), then A v = \lambda v where \lambda = A_{i i}. That is, the eigenvalues of a diagonal matrix are the elements of its main diagonal. This is trivially the case of any 1 ×1 matrix.
General definition[edit]
The concept of eigenvectors and eigenvalues extends naturally to abstract linear transformations on abstract vector spaces. Namely, let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V. We say that a non-zero vector v of V is an eigenvector of T if (and only if) there is a scalar \lambda in K such that
T(v)=\lambda v.
This equation is called the eigenvalue equation for T, and the scalar \lambda is the eigenvalue of T corresponding to the eigenvector v. Note that T(v) means the result of applying the operator T to the vector v, while \lambda v means the product of the scalar \lambda by v.[3]
The matrix-specific definition is a special case of this abstract definition. Namely, the vector space V is the set of all column vectors of a certain size n×1, and T is the linear transformation that consists in multiplying a vector by the given n\times n matrix A.
Some authors allow v to be the zero vector in the definition of eigenvector.[4] This is reasonable as long as we define eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of T as a scalar \lambda in K such that there is a nonzero vector v in V with T(v) = \lambda v . We then define an eigenvector to be a vector v in V such that there is an eigenvalue \lambda in K with T(v) = \lambda v . This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.
Eigenspace and spectrum[edit]
If v is an eigenvector of T, with eigenvalue \lambda, then any scalar multiple \alpha v of v with nonzero \alpha is also an eigenvector with eigenvalue \lambda, since T(\alpha v) = \alpha T(v) = \alpha(\lambda v) = \lambda(\alpha v). Moreover, if u and v are eigenvectors with the same eigenvalue \lambda, then u+v is also an eigenvector with the same eigenvalue \lambda. Therefore, the set of all eigenvectors with the same eigenvalue \lambda, together with the zero vector, is a linear subspace of V, called the eigenspace of T associated to \lambda.[5][6] If that subspace has dimension 1, it is sometimes called an eigenline.[7]
The geometric multiplicity \gamma_T(\lambda) of an eigenvalue \lambda is the dimension of the eigenspace associated to \lambda, i.e. number of linearly independent eigenvectors with that eigenvalue.
The eigenspaces of T always form a direct sum (and as a consequence any family of eigenvectors for different eigenvalues is always linearly independent). Therefore the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the space on which T operates, and in particular there cannot be more than n distinct eigenvalues.[8]
The set of eigenvalues of T is sometimes called the spectrum of T.
An eigenbasis for a linear operator T that operates on a vector space V is a basis for V that consists entirely of eigenvectors of T (possibly with different eigenvalues). Such a basis exists precisely if the direct sum of the eigenspaces equals the whole space, in which case one can take the union of bases chosen in each of the eigenspaces as eigenbasis. The matrix of T in a given basis is diagonal precisely when that basis is an eigenbasis for T, and for this reason T is called diagonalizable if it admits an eigenbasis.
Generalizations to infinite-dimensional spaces[edit]
The definition of eigenvalue of a linear transformation T remains valid even if the underlying space V is an infinite dimensional Hilbert or Banach space. Namely, a scalar \lambda is an eigenvalue if and only if there is some nonzero vector v such that T(v) = \lambda v.
A widely used class of linear operators acting on infinite dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator in on the space \mathbf{C^\infty} of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
D f = \lambda f
The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator d/dt, an eigenfunction is a function that, when differentiated, yields a constant times the original function. If \lambda is zero, the generic solution is a constant function. If \lambda is non-zero, the solution is an exponential function
Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoretical fields. For instance, the exponential functions are eigenfunctions of any shift invariant linear operator. This fact is the basis of powerful Fourier transform methods for solving all sorts of problems.
Spectral theory[edit]
If \lambda is an eigenvalue of T, then the operator T-\lambda I is not one-to-one, and therefore its inverse (T-\lambda I)^{-1} is not defined. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional ones. In general, the operator T - \lambda I may not have an inverse, even if \lambda is not an eigenvalue.
For this reason, in functional analysis one defines the spectrum of a linear operator T as the set of all scalars \lambda for which the operator T-\lambda I has no bounded inverse. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
Associative algebras and representation theory[edit]
More algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize the algebraic object that is acting on the space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
A closer analog of eigenvalues is given by the representation-theoretical concept of weight, with the analogs of eigenvectors and eigenspaces being weight vectors and weight spaces.
Eigenvalues and eigenvectors of matrices[edit]
Characteristic polynomial[edit]
The eigenvalue equation for a matrix A is
A v - \lambda v = 0,
which is equivalent to
(A-\lambda I)v = 0,
where I is the n\times n identity matrix. It is a fundamental result of linear algebra that an equation M v = 0 has a non-zero solution v if, and only if, the determinant \det(M) of the matrix M is zero. It follows that the eigenvalues of A are precisely the real numbers \lambda that satisfy the equation
\det(A-\lambda I) = 0
The left-hand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial function of the variable \lambda. The degree of this polynomial is n, the order of the matrix. Its coefficients depend on the entries of A, except that its term of degree n is always (-1)^n\lambda^n. This polynomial is called the characteristic polynomial of A; and the above equation is called the characteristic equation (or, less often, the secular equation) of A.
For example, let A be the matrix
A =
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
The characteristic polynomial of A is
\det (A-\lambda I) \;=\; \det \left(\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
\end{bmatrix} - \lambda
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\right) \;=\;
\det \begin{bmatrix}
2 - \lambda & 0 & 0 \\
0 & 3 - \lambda & 4 \\
0 & 4 & 9 - \lambda
which is
(2 - \lambda) \bigl[ (3 - \lambda) (9 - \lambda) - 16 \bigr] = -\lambda^3 + 14\lambda^2 - 35\lambda + 22
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of A, corresponding to the eigenvectors [1,0,0]', [0,2,-1]', and [0,1,2]' (or any non-zero multiples thereof).
In the real domain[edit]
Since the eigenvalues are roots of the characteristic polynomial, an n\times n matrix has at most n eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than n real roots, or no real roots at all.
For example, consider the cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Its characteristic polynomial is 1 - \lambda^3 which has one real root \lambda_1 = 1. Any vector with three equal non-zero elements is an eigenvector for this eigenvalue. For example,
A \begin{bmatrix} 5\\5\\5 \end{bmatrix} =
1 \cdot \begin{bmatrix} 5\\5\\5 \end{bmatrix}
In the complex domain[edit]
The fundamental theorem of algebra implies that the characteristic polynomial of an n\times n matrix A, being a polynomial of degree n, has exactly n complex roots. More precisely, it can be factored into the product of n linear terms,
\det(A-\lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda)\cdots(\lambda_n - \lambda)
where each \lambda_i is a complex number. The numbers \lambda_1, \lambda_2, ... \lambda_n, (which may not be all distinct) are roots of the polynomial, and are precisely the eigenvalues of A.
Even if the entries of A are all real numbers, the eigenvalues may still have non-zero imaginary parts (and the elements of the corresponding eigenvectors will therefore also have non-zero imaginary parts). Also, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers, or all are integers. However, if the entries of A are algebraic numbers (which include the rationals), the eigenvalues will be (complex) algebraic numbers too.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugate values, namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. If the degree is odd, then by the intermediate value theorem at least one of the roots will be real. Therefore, any real matrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no real eigenvalues.
In the example of the 3×3 cyclic permutation matrix A, above, the characteristic polynomial 1 - \lambda^3 has two additional non-real roots, namely
\lambda_2 = -1/2 + \mathbf{i}\sqrt{3}/2\quad\quad and \quad\quad\lambda_3 = \lambda_2^* = -1/2 - \mathbf{i}\sqrt{3}/2,
where \mathbf{i}= \sqrt{-1} is the imaginary unit. Note that \lambda_2\lambda_3 = 1, \lambda_2^2 = \lambda_3, and \lambda_3^2 = \lambda_2. Then
A \begin{bmatrix} 1 \\ \lambda_2 \\ \lambda_3 \end{bmatrix} =
\begin{bmatrix} \lambda_2\\ \lambda_3 \\1 \end{bmatrix} =
\lambda_2 \cdot \begin{bmatrix} 1\\ \lambda_2 \\ \lambda_3 \end{bmatrix}
A \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix} =
\begin{bmatrix} \lambda_3 \\ \lambda_2 \\ 1 \end{bmatrix} =
\lambda_3 \cdot \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix}
Therefore, the vectors [1,\lambda_2,\lambda_3]' and [1,\lambda_3,\lambda_2]' are eigenvectors of A, with eigenvalues \lambda_2, and \lambda_3, respectively.
Algebraic multiplicities[edit]
Let \lambda_i be an eigenvalue of an n\times n matrix A. The algebraic multiplicity \mu_A(\lambda_i) of \lambda_i is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (\lambda - \lambda_i)^k divides evenly that polynomial.
Like the geometric multiplicity \gamma_A(\lambda_i), the algebraic multiplicity is an integer between 1 and n; and the sum \boldsymbol{\mu}_A of \mu_A(\lambda_i) over all distinct eigenvalues also cannot exceed n. If complex eigenvalues are considered, \boldsymbol{\mu}_A is exactly n.
It can be proved that the geometric multiplicity \gamma_A(\lambda_i) of an eigenvalue never exceeds its algebraic multiplicity \mu_A(\lambda_i). Therefore, \boldsymbol{\gamma}_A is at most \boldsymbol{\mu}_A.
If \gamma_A(\lambda_i) = \mu_A(\lambda_i), then \lambda_i is said to be a semisimple eigenvalue.
For the matrix: A= \begin{bmatrix}
2 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 1 & 3 & 0 \\
0 & 0 & 1 & 3
the characteristic polynomial of A is \det (A-\lambda I) \;=\;
\det \begin{bmatrix}
2- \lambda & 0 & 0 & 0 \\
1 & 2- \lambda & 0 & 0 \\
0 & 1 & 3- \lambda & 0 \\
0 & 0 & 1 & 3- \lambda
\end{bmatrix}= (2 - \lambda)^2 (3 - \lambda)^2 ,
being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by the vector [0,1,-1,1], and is therefore 1 dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by [0,0,0,1]. Hence, the total algebraic multiplicity of A, denoted \mu_A, is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity \gamma_A is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.
Diagonalization and eigendecomposition[edit]
If the sum \boldsymbol{\gamma}_A of the geometric multiplicities of all eigenvalues is exactly n, then A has a set of n linearly independent eigenvectors. Let Q be a square matrix whose columns are those eigenvectors, in any order. Then we will have A Q = Q\Lambda , where \Lambda is the diagonal matrix such that \Lambda_{i i} is the eigenvalue associated to column i of Q. Since the columns of Q are linearly independent, the matrix Q is invertible. Premultiplying both sides by Q^{-1} we get Q^{-1}A Q = \Lambda. By definition, therefore, the matrix A is diagonalizable.
Conversely, if A is diagonalizable, let Q be a non-singular square matrix such that Q^{-1} A Q is some diagonal matrix D. Multiplying both sides on the left by Q we get A Q = Q D . Therefore each column of Q must be an eigenvector of A, whose eigenvalue is the corresponding element on the diagonal of D. Since the columns of Q must be linearly independent, it follows that \boldsymbol{\gamma}_A = n. Thus \boldsymbol{\gamma}_A is equal to n if and only if A is diagonalizable.
If A is diagonalizable, the space of all n-element vectors can be decomposed into the direct sum of the eigenspaces of A. This decomposition is called the eigendecomposition of A, and it is the preserved under change of coordinates.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and that of diagonal matrix to a Jordan form matrix. Over an algebraically closed field, any matrix A has a Jordan form and therefore admits a basis of generalized eigenvectors, and a decomposition into generalized eigenspaces
Further properties[edit]
Let A be an arbitrary n\times n matrix of complex numbers with eigenvalues \lambda_1, \lambda_2, ... \lambda_n. (Here it is understood that an eigenvalue with algebraic multiplicity \mu occurs \mu times in this list.) Then
\operatorname{tr}(A) = \sum_{i=1}^n A_{i i} = \sum_{i=1}^n \lambda_i = \lambda_1+ \lambda_2 +\cdots+ \lambda_n.
• The determinant of A is the product of all eigenvalues:
\operatorname{det}(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n.
• The eigenvalues of the kth power of A, i.e. the eigenvalues of A^k, for any positive integer k, are \lambda_1^k,\lambda_2^k,\dots,\lambda_n^k
• The matrix A is invertible if and only if all the eigenvalues \lambda_i are nonzero.
• If A is invertible, then the eigenvalues of A^{-1} are 1/\lambda_1,1/\lambda_2,\dots,1/\lambda_n
• If A is equal to its conjugate transpose A^* (in other words, if A is Hermitian), then every eigenvalue is real. The same is true of any a symmetric real matrix. If A is also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positive respectively.
• Every eigenvalue of a unitary matrix has absolute value |\lambda|=1.
Left and right eigenvectors[edit]
The use of matrices with a single column (rather than a single row) to represent vectors is traditional in many disciplines. For that reason, the word "eigenvector" almost always means a right eigenvector, namely a column vector that must be placed to the right of the matrix A in the defining equation
A v = \lambda v.
There may be also single-row vectors that are unchanged when they occur on the left side of a product with a square matrix A; that is, which satisfy the equation
u A = \lambda u
Any such row vector u is called a left eigenvector of A.
The left eigenvectors of A are transposes of the right eigenvectors of the transposed matrix A^\mathsf{T}, since their defining equation is equivalent to
A^\mathsf{T} u^\mathsf{T} = \lambda u^\mathsf{T}
It follows that, if A is Hermitian, its left and right eigenvectors are complex conjugates. In particular if A is a real symmetric matrix, they are the same except for transposition.
Computing the eigenvalues[edit]
The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.
It turns out that any polynomial with degree n is the characteristic polynomial of some companion matrix of order n. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. [9] Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.[citation needed] For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.[9]
Computing the eigenvectors[edit]
A = \begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}
we can find its eigenvectors by solving the equation A v = 6 v, that is
\begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = 6 \cdot \begin{bmatrix}x\\y\end{bmatrix}
This matrix equation is equivalent to two linear equations
\left\{\begin{matrix} 4x + {\ }y &{}= 6x\\6x + 3y &{}=6 y\end{matrix}\right.
\quad\quad\quad that is
\left\{\begin{matrix} -2x+ {\ }y &{}=0\\+6x-3y &{}=0\end{matrix}\right.
Both equations reduce to the single linear equation y=2x. Therefore, any vector of the form [a,2a]', for any non-zero real number a, is an eigenvector of A with eigenvalue \lambda = 6.
The matrix A above has another eigenvalue \lambda=1. A similar calculation shows that the corresponding eigenvectors are the non-zero solutions of 3x+y=0, that is, any vector of the form [b,-3b]', for any non-zero real number b.
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.[14] Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.[15]
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.[16] He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.[17]
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis[18] and Vera Kublanovskaya[19] in 1961.[20]
Eigenvalues of geometric transformations[edit]
scaling unequal scaling rotation horizontal shear hyperbolic rotation
illustration Equal scaling (homothety) Vertical shrink () and horizontal stretch () of a unit square. Rotation by 50 degrees
Horizontal shear mapping
matrix \begin{bmatrix}k & 0\\0 & k\end{bmatrix}
\begin{bmatrix}k_1 & 0\\0 & k_2\end{bmatrix}
\begin{bmatrix}c & -s \\ s & c\end{bmatrix}
\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}
\begin{bmatrix} c & s \\ s & c \end{bmatrix}
c=\cosh \varphi
s=\sinh \varphi
\ (\lambda - k)^2 (\lambda - k_1)(\lambda - k_2) \lambda^2 - 2c\lambda + 1 \ (\lambda - 1)^2 \lambda^2 - 2c\lambda + 1
eigenvalues \lambda_i \lambda_1 = \lambda_2 = k \lambda_1 = k_1
\lambda_2 = k_2
\lambda_1 = e^{\mathbf{i}\theta}=c+s\mathbf{i}
\lambda_2 = e^{-\mathbf{i}\theta}=c-s\mathbf{i}
\lambda_1 = \lambda_2 = 1 \lambda_1 = e^\varphi
\lambda_2 = e^{-\varphi},
algebraic multipl.
\mu_1 = 2 \mu_1 = 1
\mu_2 = 1
\mu_1 = 1
\mu_2 = 1
\mu_1 = 2 \mu_1 = 1
\mu_2 = 1
geometric multipl.
\gamma_i = \gamma(\lambda_i)
\gamma_1 = 2 \gamma_1 = 1
\gamma_2 = 1
\gamma_1 = 1
\gamma_2 = 1
\gamma_1 = 1 \gamma_1 = 1
\gamma_2 = 1
eigenvectors All non-zero vectors u_1 = \begin{bmatrix}1\\0\end{bmatrix}
u_2 = \begin{bmatrix}0\\1\end{bmatrix}
u_1 = \begin{bmatrix}{\ }1\\-\mathbf{i}\end{bmatrix}
u_2 = \begin{bmatrix}{\ }1\\ +\mathbf{i}\end{bmatrix}
u_1 = \begin{bmatrix}1\\0\end{bmatrix} u_1 = \begin{bmatrix}{\ }1\\{\ }1\end{bmatrix}
u_2 = \begin{bmatrix}{\ }1\\-1\end{bmatrix}.
Note that the characteristic equation for a rotation is a quadratic equation with discriminant D = -4(\sin\theta)^2, which is a negative number whenever \theta is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, \cos\theta \pm \mathbf{i}\sin\theta; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
Schrödinger equation[edit]
H\psi_E = E\psi_E \,
where H, the Hamiltonian, is a second-order differential operator and \psi_E, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for \psi_E within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which \psi_E and H can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
Bra-ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by |\Psi_E\rangle. In this notation, the Schrödinger equation is:
H|\Psi_E\rangle = E|\Psi_E\rangle
where |\Psi_E\rangle is an eigenstate of H. It is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable). As in the matrix case, in the equation above H|\Psi_E\rangle is understood to be the vector obtained by application of the transformation H to |\Psi_E\rangle.
Molecular orbitals[edit]
Geology and glaciology[edit]
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered v_1, v_2, v_3 by their eigenvalues E_1 \geq E_2 \geq E_3;[24] v_1 then is the primary orientation/dip of clast, v_2 is the secondary and v_3 is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of E_1, E_2, and E_3 are dictated by the nature of the sediment's fabric. If E_1 = E_2 = E_3, the fabric is said to be isotropic. If E_1 = E_2 > E_3, the fabric is said to be planar. If E_1 > E_2 > E_3, the fabric is said to be linear.[25]
Principal components analysis[edit]
PCA of the multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878,0.478) direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
Vibration analysis[edit]
1st lateral bending (See vibration for more types of vibration)
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by
m\ddot x + kx = 0
m\ddot x = -k x
that is, acceleration is proportional to position (i.e., we expect x to be sinusoidal in time).
In n dimensions, m becomes a mass matrix and k a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
-k x = \omega^2 m x
where \omega^2 is the eigenvalue and \omega is the angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of k alone. Furthermore, damped vibration, governed by
m\ddot x + c \dot x + kx = 0
leads to what is called a so-called quadratic eigenvalue problem,
(\omega^2 m + \omega c + k)x = 0.
Eigenfaces as examples of eigenvectors
Tensor of moment of inertia[edit]
Stress tensor[edit]
Eigenvalues of a graph[edit]
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix (see also Discrete Laplace operator), which is either T - A (sometimes called the combinatorial Laplacian) or I - T^{-1/2}A T^{-1/2} (sometimes called the normalized Laplacian), where T is a diagonal matrix with T_{i i} equal to the degree of vertex v_i, and in T^{-1/2}, the ith diagonal entry is \sqrt{\operatorname{deg}(v_i)}. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest or kth smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
Basic reproduction number[edit]
See Basic reproduction number
The basic reproduction number (R_0) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then R_0 is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, t_G, from one person becoming infected to the next person becoming infected. In a heterogenous population, the next generation matrix defines how many people in the population will become infected after time t_G has passed. R_0 is then the largest eigenvalue of the next generation matrix.[27][28]
See also[edit]
1. ^ a b Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.
2. ^ William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), Numerical Recipes: The Art of Scientific Computing, Chapter 11: Eigensystems., pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688
5. ^ Shilov 1977, p. 109
6. ^ Lemma for the eigenspace
7. ^ Schaum's Easy Outline of Linear Algebra, p. 111
10. ^ See Hawkins 1975, §2
13. ^ See Kline 1972, p. 673
14. ^ See Kline 1972, pp. 715–716
15. ^ See Kline 1972, pp. 706–707
16. ^ See Kline 1972, p. 1063
17. ^ See Aldrich 2006
18. ^ Francis, J. G. F. (1961), "The QR Transformation, I (part 1)", The Computer Journal 4 (3): 265–271, doi:10.1093/comjnl/4.3.265 and Francis, J. G. F. (1962), "The QR Transformation, II (part 2)", The Computer Journal 4 (4): 332–345, doi:10.1093/comjnl/4.4.332
19. ^ Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics 3: 637–657 . Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 1 (4), 1961: 555–570
21. ^ Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method", Earth Surface Processes and Landforms 25 (13): 1473–1477, doi:10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
22. ^ Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis", Journal of Geology 66 (2): 114–150, doi:10.1086/626490
23. ^ Knox-Robinson, C; Gardoll, Stephen J (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system", Computers & Geosciences 24 (3): 243, doi:10.1016/S0098-3004(97)00122-2
24. ^ Stereo32 software
26. ^ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), Online paper in PDF format, National Technical University of Athens
27. ^ Diekmann O, Heesterbeek JAP, Metz JAJ (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology 28 (4): 365–382, doi:10.1007/BF00178324, PMID 2117040
28. ^ Odo Diekmann and J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
• Korn, Granino A.; Korn, Theresa M. (2000), "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review", New York: McGraw-Hill (1152 p., Dover Publications, 2 Revised edition), Bibcode:1968mhse.book.....K, ISBN 0-486-41147-8 .
• Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series (2nd ed.), New York, NY: McGraw-Hill Companies, ISBN 0-07-038007-4 .
• Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs, NJ 07632: Prentice Hall, ISBN 0-13-537102-3 .
• Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, MA, ISBN 0-9614088-5-5 .
• Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA, ISBN 0-03-010567-6 .
• Bowen, Ray M.; Wang, Chao-Cheng (1980), Linear and multilinear algebra, Plenum Press, New York, NY, ISBN 0-306-37508-7 .
• Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd ed.), Addison-Wesley Publishing Company, ISBN 0-201-83999-7 (international edition) Check |isbn= value (help) .
• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix computations (3rd Edition), Johns Hopkins University Press, Baltimore, MD, ISBN 978-0-8018-5414-9 .
• Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4 .
• Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press, ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback) Check |isbn= value (help) .
• Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of Computational and Applied Mathematics 123: 35–65, doi:10.1016/S0377-0427(00)00413-1 .
• Akivis, Max A.; Vladislav V. Goldberg (1969), Tensor calculus, Russian, Science Publishers, Moscow .
• Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York, NY: Springer Science + Business Media, LLC, ISBN 978-0-387-72828-5 .
• Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: Dover Publications, ISBN 0-486-63518-X .
• Halmos, Paul R. (1987), Finite-dimensional vector spaces (8th ed.), New York, NY: Springer-Verlag, ISBN 0-387-90093-4 .
• Pigolkina, T. S. and Shulman, V. S., Eigenvalue (in Russian), In:Vinogradov, I. M. (Ed.), Mathematical Encyclopedia, Vol. 5, Soviet Encyclopedia, Moscow, 1977.
• Greub, Werner H. (1975), Linear Algebra (4th Edition), Springer-Verlag, New York, NY, ISBN 0-387-90110-8 .
• Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), ISBN 0-387-90992-3.
• Sharipov, Ruslan A. (1996), Course of Linear Algebra and Multidimensional Geometry: the textbook, arXiv:math/0405323, ISBN 5-7477-0099-5 .
External links[edit]
Online calculators
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23b8dfbcdae771f7 | Quantum graph
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In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected by bonds (or edges) with a differential or pseudo-differential operator acting on functions defined on the bonds. Such systems were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They arise in a variety of mathematical contexts, e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.[1]
Metric graphs[edit]
A metric graph embedded in the plane with three open edges. The dashed line denotes the metric distance between two points x and y.
A metric graph is a graph consisting of a set V of vertices and a set E of edges where each edge e=(v_1,v_2)\in E has been associated with an interval [0,L_e] so that x_e is the coordinate on the interval, the vertex v_1 corresponds to x_e=0 and v_2 to x_e=L_e or vice versa. The choice of which vertex lies at zero is arbitrary with the alternative corresponding to a change of coordinate on the edge. The graph has a natural metric: for two points x,y on the graph, \rho(x,y) is the shortest distance between them where distance is measured along the edges of the graph. It is possible to embed a metric graph in the plane, {\mathbb R}^2, or space, {\mathbb R}^3, if the lengths are chosen appropriately.
Open graphs: in the combinatorial graph model edges always join pairs of vertices however in a quantum graph one may also consider semi-infinite edges. These are edges associated with the interval [0,\infty) attached to a single vertex at x_e=0. A graph with one or more such open edges is referred to as an open graph.
Quantum graphs[edit]
Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function f on a metric graph is defined as the |E|-tuple of functions f_e(x_e) on the intervals. The Hilbert space of the graph is \bigoplus_{e\in E} L^2([0,L_e]) where the inner product of two functions is
\langle f,g \rangle = \sum_{e\in E} \int_{0}^{L_e} f_e^{*}(x_e)g_e(x_e) \, dx_e,
L_e may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. The operator on an edge is -\frac{\textrm{d}^2}{\textrm{d} x_e^2} where x_e is the coordinate on the edge. To make the operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space H^2 of functions on the edges of the graph and specifying matching conditions at the vertices.
The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary conditions, f_e(0)=f_e(L_e)=0 for every edge. An eigenfunction on a finite edge may be written as
f_e(x_e) = \sin \left( \frac{n \pi x_e}{L_e} \right)
for integer n. If the graph is closed with no infinite edges and the lengths of the edges of the graph are rationally independent then an eigenfunction is supported on a single graph edge and the eigenvalues are \frac{n^2\pi^2}{L_e^2}. The Dirichlet conditions don't allow interaction between the intervals so the spectrum is the same as that of the set of disconnected edges.
More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or natural matching conditions. A function f in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,
\sum_{e\sim v} f'(v) = 0 \ ,
where f'(v)=f'(0) if the vertex v is at x=0 and f'(v)=-f'(L_e) if v is at x=L_e.
The properties of other operators on metric graphs have also been studied.
• These include the more general class of Schrŏdinger operators,
\left( i \frac{\textrm{d}}{\textrm{d} x_e} + A_e(x_e) \right)^2 + V_e(x_e) \ ,
where A_e is a "magnetic vector potential" on the edge and V_e is a scalar potential.
• Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector valued functions that describe the quantum mechanics of particles with an intrinsic angular momentum of one half such as the electron.
• The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of photonic crystals.
All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by Kuchment, see,[2] which automatically yields an operator in variational form.
Let v be a vertex with d edges emanating from it. For simplicity we choose the coordinates on the edges so that v lies at x_e=0 for each edge meeting at v. For a function f on the graph let
\mathbf{f}=(f_{e_1}(0),f_{e_2}(0),\dots,f_{e_{d}}(0))^T , \qquad \mathbf{f}'=(f'_{e_1}(0),f'_{e_2}(0),\dots,f'_{e_{d}}(0))^T.
Matching conditions at v can be specified by a pair of matrices A and B through the linear equation,
A \mathbf{f} +B \mathbf{f}'=\mathbf{0}.
The matching conditions define a self-adjoint operator if (A, B) has the maximal rank d and AB^{*}=BA^{*}.
The spectrum of the Laplace operator on a finite graph can be conveniently described using a scattering matrix approach introduced by Kottos and Smilansky .[3] The eigenvalue problem on an edge is,
-\frac{d^2}{dx_e^2} f_e(x_e)=k^2 f_e(x_e).\,
So a solution on the edge can be written as a linear combination of plane waves.
f_e(x_e) = c_e \textrm{e}^{i k x_e} + \hat{c}_e \textrm{e}^{-i k x_e}.\,
where in a time-dependent Schrödinger equation c is the coefficient of the outgoing plane wave at 0 and \hat{c} coefficient of the incoming plane wave at 0. The matching conditions at v define a scattering matrix
S(k)=-(A+i kB)^{-1}(A-ikB).\,
The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at v, \mathbf{c}=S(k)\hat{\mathbf{c}}. For self-adjoint matching conditions S is unitary. An element of \sigma_{(uv)(vw)} of S is a complex transition amplitude from a directed edge (uv) to the edge (vw) which in general depends on k. However, for a large class of matching conditions the S-matrix is independent of k. With Neumann matching conditions for example
1& -1 & 0 & 0 & \dots \\
0 & 1 & -1 & 0 & \dots \\
& & \ddots & \ddots & \\
0& \dots & 0 & 1 & -1 \\
0 &\dots & 0 & 0& 0 \\
0& 0 & \dots & 0 \\
\vdots & \vdots & & \vdots \\
0& 0 & \dots & 0 \\
1 &1 & \dots & 1 \\
\end{array} \right).
Substituting in the equation for S produces k-independent transition amplitudes
where \delta_{uw} is the Kronecker delta function that is one if u=w and zero otherwise. From the transition amplitudes we may define a 2|E|\times 2|E| matrix
U_{(uv)(lm)}(k)= \delta_{vl} \sigma_{(uv)(vm)}(k) \textrm{e}^{i kL_{(uv)}}.\,
U is called the bond scattering matrix and can be thought of as a quantum evolution operator on the graph. It is unitary and acts on the vector of 2|E| plane-wave coefficients for the graph where c_{(uv)} is the coefficient of the plane wave traveling from u to v. The phase \textrm{e}^{i kL_{(uv)}} is the phase acquired by the plane wave when propagating from vertex u to vertex v.
Quantization condition: An eigenfunction on the graph can be defined through its associated 2|E| plane-wave coefficients. As the eigenfunction is stationary under the quantum evolution a quantization condition for the graph can be written using the evolution operator.
Eigenvalues k_j occur at values of k where the matrix U(k) has an eigenvalue one. We will order the spectrum with 0\leqslant k_0 \leqslant k_1 \leqslant \dots.
The first trace formula for a graph was derived by Roth (1983). In 1997 Kottos and Smilansky used the quantization condition above to obtain the following trace formula for the Laplace operator on a graph when the transition amplitudes are independent of k. The trace formula links the spectrum with periodic orbits on the graph.
d(k):=\sum_{j=0}^{\infty} \delta(k-k_j)=\frac{L}{\pi}+\frac{1}{\pi}
\sum_p \frac{L_p}{r_p} A_p \cos(kL_p).
d(k) is called the density of states. The right hand side of the trace formula is made up of two terms, the Weyl term \frac{L}{\pi} is the mean separation of eigenvalues and the oscillating part is a sum over all periodic orbits p=(e_1,e_2,\dots,e_n) on the graph. L_p=\sum_{e\in p} L_e is the length of the orbit and L=\sum_{e\in E}L_e is the total length of the graph. For an orbit generated by repeating a shorter primitive orbit, r_p counts the number of repartitions. A_p=\sigma_{e_1 e_2} \sigma_{e_2 e_3} \dots \sigma_{e_n e_1} is the product of the transition amplitudes at the vertices of the graph around the orbit.
Naphthalene molecule
Quantum graphs were first employed in the 1930s to model the spectrum of free electrons in organic molecules like Naphthalene, see figure. As a first approximation the atoms are taken to be vertices while the σ-electrons form bonds that fix a frame in the shape of the molecule on which the free electrons are confined.
A similar problem appears when considering quantum waveguides. These are mesoscopic systems - systems built with a width on the scale of nanometers. A quantum waveguide can be thought of as a fattened graph where the edges are thin tubes. The spectrum of the Laplace operator on this domain converges to the spectrum of the Laplace operator on the graph under certain conditions. Understanding mesoscopic systems plays an important role in the field of nanotechnology.
In 1997 [citation needed]Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic Markov chain where the probability of scattering from edge e to edge f is given by the absolute value of the quantum transition amplitude squared, |\sigma_{ef}|^2. For almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic.
Quantum graphs embedded in two or three dimensions appear in the study of photonic crystals [citation needed]. In two dimensions a simple model of a photonic crystal consists of polygonal cells of a dense dielectric with narrow interfaces between the cells filled with air. Studying dielectric modes that stay mostly in the dielectric gives rise to a pseudo-differential operator on the graph that follows the narrow interfaces.
Periodic quantum graphs like the lattice in {\mathbb R}^2 are common models of periodic systems and quantum graphs have been applied to the study the phenomena of Anderson localization where localized states occur at the edge of spectral bands in the presence of disorder.
See also[edit]
1. ^ M. Freedman, L. Lovász & A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs, J. Amer. Math. Soc. 20, 37-51 (2007); MR2257396
2. ^ P. Kuchment, Quantum graphs I. Some basic structures, Waves in Random Media 14, S107-S128 (2004)
3. ^ S. Gnutzman & U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics, Adv. Phys. 55 527-625 (2006) |
4613f06e4ac39582 | Take the tour ×
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. There is a "smallest" one (Friedrichs) and a largest one (Krein), and all others are in some sense in between. Considering the corresponding Schrödinger equations, to each of these extensions there is a (completely different) unitary group solving it. My question is: what is the physical meaning of these extensions? How do you distinguish between the different unitary groups? Is there one which is physically "relevant"? Why is the Friedrichs extension chosen so often?
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I am asking this question as a mathematician trying to understand the meaning and motivation of the objects I am working with. – András Bátkai Sep 15 '11 at 19:49
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up vote 23 down vote accepted
The differential operator itself (defined on some domain) encodes local information about the dynamics of the quantum system . Its self-adjoint extensions depend precisely on choices of boundary conditions of the states that the operator acts on, hence on global information about the kinematics of the physical system.
This is even true fully abstractly, mathematically: in a precise sense the self-adjoint extensions of symmetric operators (under mild conditions) are classified by choices of boundary data.
More information on this is collected here
See the references on applications in physics there for examples of choices of boundary conditions in physics and how they lead to self-adjoint extensions of symmetric Hamiltonians. And see the article by Wei-Jiang there for the fully general notion of boundary conditions.
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A typical interpretation of the self-adjoint extensions for the free hamiltonian in a line segment is that you get a four parametric family of possible boundary conditions, to preserve unitarity. Some of them just "bounce" the wave, some others "teletransport" it from one wall to the other. So it is also traditional to imagine this segment as a circle where you have removed a point, and then you are in the mood of studying "point interactions" or generalisations of dirac-delta potentials. The topic resurfaces from time to time, but surely some old references can be digged starting from M. Carreau. Four-parameter point-interaction in 1d quantum systems. Journal of Physics A, 26:427, 1993. In some works, I quote also Seba and Polonyi.
Sometimes the extensions are linked to the question of the domain of definition for the operator and then to the existence of anomalies. Here Phys.Rev.D34: 674-677, 1986, "Anomalies in conservation laws in the Hamiltonian formalism", revisited by the same autor, J G Esteve, later in Phys.Rev.D66:125013,2002 ( http://arxiv.org/abs/hep-th/0207164 ). These topics have been live for years in the university of Zaragoza; some related material, perhaps more about boundary conditions than about extensions, is http://arxiv.org/abs/0704.1084, http://arxiv.org/abs/quant-ph/0609023, http://arxiv.org/abs/0712.4353
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I hadn't been aware of the reference by Esteve. I have added it to the references of the nLab entry ncatlab.org/nlab/show/quantum+anomaly (many more references are currently still missing there, of course). – Urs Schreiber Sep 16 '11 at 11:14
@Urs Schreiber Thanks for the add. The topic was common folklore in Zaragoza in the nineties and it was not infrequent in PhD theses, but I think that its main role was motivational, either aiming towards other topics, or used as a guide when exploring some other concept. For instance, it was very valuable to me in order to navigate Albeverio et al, who had got into a confusing notation/naming for some self adjoint extensions classifying these "1D point interactions". – user135 Sep 16 '11 at 11:35
Thank, I like both answers very much. The references are great. Unfortunately, I have to choose an answer to accept... – András Bátkai Sep 16 '11 at 19:23
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9bff94a10aa065c5 | Take the tour ×
I have no background in physics. This isn't for homework, just for interest.
In quantum physics, it's described that a particle can act as both a particle and a wave.
Quoted from HowStuffWorks "Wave-Particle Duality"
Today, physicists accept the dual nature of light. In this modern view, they define light as a collection of one or more photons propagating through space as electromagnetic waves. This definition, which combines light's wave and particle nature, makes it possible to rethink Thomas Young's double-slit experiment in this way: Light travels away from a source as an electromagnetic wave. When it encounters the slits, it passes through and divides into two wave fronts. These wave fronts overlap and approach the screen. At the moment of impact, however, the entire wave field disappears and a photon appears. Quantum physicists often describe this by saying the spread-out wave "collapses" into a small point.
I have trouble visualizing a particle transforming into a wave and vice-versa. The quote says that light travels away from a source as an electromagnetic wave. What does that even look like? How can I visualize "a wave"? Is that supposed to look like some thin wall of advancing light? And then, the quote says, at the moment of impact, the wave disappears and a photon appears. So, a ball of light appears? Something that resembles a sphere? How does a sphere become something like an ocean wave? What does that look like?
My (completely uneducated) guess is, by a particle becoming a wave, does that mean that this expansive wave is filled with tons of ghost copies of itself, like the one electron exists everywhere in this expansive area of the wave, and then when it hits the wall, that property suddenly disappears and you're left with just one particle. So, this "wave", is really tons of identical copies of the same photon in the shape and form and with the same properties of, a wave? My guess comes from reading about how shooting just one photon still passes through two slits in the double-slit experiment. So the photon actually duplicated itself?
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Possible duplicate: physics.stackexchange.com/q/33333/2451 – Qmechanic Nov 6 '12 at 22:13
The wave nature is what evolves with time and the particle nature is what is observed. This link may help: upload.wikimedia.org/wikipedia/commons/e/e7/… – DIMension10 Jun 18 at 9:41
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What we observe in nature exists in several scales. From the distances of stars and galaxies and clusters of galaxies to the sizes of atoms and elementary particles.
Now we have to define "observe".
Observing in human size scale means what our ears hear, what our eyes see, what our hands feel, our nose smells , our mouth tastes. That was the first classification and the level of "proxy", i.e. intermediate between fact and our understanding and classification, which is biological. (the term proxy is widely used in climate researches)
A second level of observing comes when we use proxies, like meters, thermometers, telescopes and microscopes etc. which register on our biological proxies and we accumulate knowledge. At this level we can overcome the limits of the human scale and find and study the enormous scales of the galaxies and the tiny scales of the bacteria and microbes. A level of microns and milimeters. We observe waves in liquids with such size wavelengths
Visible light is of the order of Angstroms, 10^-10 meters. As science progressed the idea of light being corpuscles ( Newton) became overcome by the observation of interference phenomena which definitely said "waves".
Then came the quantum revolution, the photoelectric effect (Particle), the double slit experiments( wave) that showed light had aspects of a corpuscle and aspects of a wave. We our now in a final level of use of proxy, called mathematics
The wave particle duality was understood in the theory of quantum mechanics. In this theory depending on the observation a particle will either react as a "particle" i.e. have a momentum and location defined , or as a wave, i.e. have a frequency/wavelength and geometry defining its presence BUT, and it is a huge but, this wavelength is not in the matter/energy itself that is defining the particle , but in the probability of finding that particle in a specific (x,y,z,t) location. If there is no experiment looking for the particle at specific locations its form is unknown and bounded by the Heisenberg Uncertainty Principle.
What is described with words in the last paragraph is rigorously set out in mathematical equations and it is not possible to understand really what is going on if one does not acquire the mathematical tools, as a native on a primitive island could not understand airplanes. Mathematics is the ultimate proxy for understanding quantum phenomena.
Now light is special in the sense that collectively it displays the wave properties macroscopically, and the specialness comes from the Maxwell Equations which work as well in both systems, the classical and the quantum mechanical, but this also needs mathematics to be comprehended.
So a visualization is misleading in the sense that the mathematical wave function coming from the quantum mechanical equations is like a "statistical" tool whose square gives us the probability of observing the particle at (x,y,z,t). Suppose that I have a statistical probability function for you, that you may be in New York on 17/10/2012 and probabilities spread all over the east coast of the US. Does that mean that you are nowhere? does that mean that you are everywhere? Equally with the photons and the elementary particles. It is just a mathematical probability coming out of the inherent quantum mechanical nature of the cosmos.
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Thanks for the really detailed post and the concept of proxies. Surely mathematics isn't "the final" proxy to observe and learn? There may be others? – Jason Oct 17 '12 at 7:38
Mathematics itself has several levels used in physics, and those are continually expanding as research progresses. The question becomes esoteric to mathematics, imo. – anna v Oct 17 '12 at 10:35
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Any source that is content to describe scientific theories in terms of black magic is worse than useless.
How can I visualize "a wave"?
You can visualize it as y=sin(x). The wave's strength oscillates both over time (if you stand in one place and watch it pass), and space (if you "freeze" it in time). Light is more complex than more familiar waves (eg, waves in water), in that it's made up of oscillating electrical and magnetic waves.
So the photon actually duplicated itself?
No, it hasn't duplicated itself, just spread itself out so that it can pass through two slits simultaneously (just like a wave in water would do). The "collapsing" occurs due to the quantization of light, which is evident when the light gets absorbed by matter.
Realize that trying to visualize is a very limited vehicle for understanding quantum-scale physics. Since such tiny scales are out the domain of our ordinary sense perception, all we have available is hypothesis based on experiments. So, to understand a theory is to understand the paradoxes and experiments that gave rise to it; in the case of the wave-particle duality, that would be (among others) the double slit experiment, as you mentioned. On the quantization of light, a good class of experiments to ponder is those of emission/absorption spectra of elements.
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Visualization is difficult since, for example, the 'waves' that we're talking about a probability amplitude density waves. (In fact, we (teachers) should probably discourage initiates into the subject from trying to do this outright.)
One thing that has always helped when I describe this to folks is something I got from Paul Tipler's undergrad book (!) a long time ago when I was a teaching assistant. He makes a very useful distinction: when an electron (as a canonical example of a quantum 'particle') propagates it behaves in a wavelike fashion; when it exchanges energy with other systems, it does so discretely, like a particle.
In this sense then the 'duality' of quantum mechanics is less paradoxical and perhaps less seemingly contradictory. Electrons behave as waves and particles but never 'at the same time.'
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Particles in quantum mechanics are always particles and act as particles. E.g. an electron or a photon are always defined as particles according to the Standard Model and Wigner representation. E.g. an electron is defined as a particle with mass $m_e$, spin 1/2 and charge e and always behaves as a particle, never as a wave. As emphasized in CERN website "everything in the Universe is found to be made from twelve basic building blocks called fundamental particles".
There is not need for a wave-particle duality in modern interpretations of quantum mechanics (in fact quantum mechanics can be formulated without wavefunctions) and the wave-particle duality term is often considered a "myth", Klein prefers the term "misnomer". The historical roots of the wave-particle duality myth are explained in Ballentine's celebrated textbook Quantum Mechanics: A Modern Mechanics Development:
As stated by Klein:
"The miraculous "wave-particle duality" continues to flourish in popular texts and elementary text books. However, the rate of appearance of this term in scientific works has been decreasing in recent years."
Look Akira Tonomura’s video clip .wmv .mpeg for a beautiful demonstration of the appearance of statistical wave pattern on a double-slit interference experiment when a large number of independent single particles (electrons) impact the detector.
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In advanced formulations of QM, wavefunctions are substituted by kets, density matrices, Wigner distributions... Have you heard of some ket-particle, matrix-particle, or distribution-particle duality? No, because there is none. Moreover, wavefunctions are not waves, are functions. – juanrga Oct 21 '12 at 11:07
I would be interested in the reasons for the -1 votes because i can see nothing wrong here. – ungerade Oct 22 '12 at 12:27
Thank you! It was -4 some days ago. I would like to know two things: (i) why do they appeal to a hypothetical wave-particle duality, when $\Psi$ is not a physical wave but a mathematical function? and (ii) what term would they use in those advanced formulations of QM, where there is no wavefunction $\Psi$? E.g. in the Wigner-Moyal formulation of QM the state of the system is given by the Wigner distribution W and the evolution equation is not the Schrödinger equation but the Moyal equation $\dot{W} = \{H, W\}_{M}$. – juanrga Oct 22 '12 at 14:59
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87f9dbc59fbd26c2 | Project info for QuantuMagic
Share This Created 15 May 2000 at 21:22 UTC by Aspuru.
QuantuMagiC is a FORTRAN program based upon the quantum Monte Carlo (QMC) method for solving the electronic, non-relativistic, clamped-nuclei Schrödinger equation. The current version performs variational Monte Carlo (VMC) and fixed-node diffusion Monte Carlo (DMC) computations of the energy and other properties of atoms and molecules. Version 7.7 can perform both all-electron and effective-core potential calculations. Before using this program I suggest you read as many of the following as possible: 1. P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, Jr., ``Diffusion Monte Carlo for Atoms and Molecules,'' J. Chem. Phys. 77, 5593-5603 (1982). 2. W. A. Lester, Jr. and B. L. Hammond, ``Fixed Node Quantum Monte Carlo for Molecules,'' Annu. Rev. Phys. Chem. 41, 283-311 (1990). 3. B. H. Wells, ``Green's function Monte Carlo,'' in Methods in Computational Chemistry 1, 311-50 (1987). 4. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Vol. 1: Basics. Wiley. 5. B. L. Hammond, W. A. Lester, Jr., P. J. Reynolds, Monte Carlo Methods in ab initio Electronic Structure Theory. World Scientific Press, 1994 (ISBN 981-02-0322-5).
License: Modified BSD / Elsevier-Like
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Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground-state energy $E_0$ via the Schrödinger equation.
I'm completely stuck on this one. I've set up the time-independent Schrödinger equation, but it can't be solved without using special functions. I don't see how I can calculate the energy without solving the schrodinger equation. Any hints?
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Have you found the general shape of the curve? Have you found the minimum? Have you deduced the shape of the curve in the neighborhood of the minimum? How do you proceed? – dmckee Mar 21 '12 at 3:04
This is exactly the point. As the particle is large, it is localized. The localization will be around the minimum of the potential. Could you write up a solution and post it, answering your own question? – yohBS Mar 21 '12 at 7:27
It is really extremely confusing if you change the question completely so that previous comments suddenly don't make any sense at all. – Lagerbaer Mar 21 '12 at 17:59
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The energy spectrum of the problem(v3) with potential
$$\Phi(x)~=~\frac{A}{\cos x},\qquad\qquad x\in\mathbb{R}_{+}, $$
is unbounded from below, i.e., there is no ground state.
This can e.g. be seen using semiclassical methods a la this answer. Semiclassically, the reason is:
1. because the potential $\Phi$ has infinitely many periods, and
2. because the classically accessible length within one period is non-zero for any (potential) energy-level $V$.
The total accessible length $\ell(V)$ is therefore infinite for any (potential) energy-level $V$, no matter how negative $V$ is. In other words, the accessible region of phase space is always bigger than Planck constant $h$, and we can hence fit a semiclassical state, for any energy-level $E$, no matter how negative $E$ is.
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fac9e26f4620e34f | Simulating Hamiltonian evolution with quantum computers
In 1982, Richard Feynman proposed the concept of a quantum computer as a means of simulating physical systems that evolve according to the Schrödinger equation. I will explain various quantum algorithms that have been proposed for this simulation problem, including my recent work (jointly with Dominic Berry and Rolando Somma) that significantly improves the running time as a function of the precision of the output data.
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689abb9b60754181 | This is a good article. Click here for more information.
From Wikipedia, the free encyclopedia
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"Exponent" redirects here. For other uses, see Exponent (disambiguation).
Graphs of y = bx for various bases b: base 10 (green), base e (red), base 2 (blue), and base 1/2 (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:
b^n = \underbrace{b \times \cdots \times b}_n
In that case, bn is called the n-th power of b, or b raised to the power n.
The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the square of b (b2) or b squared; the exponent 3 (or 3rd power) is called the cube of b (b3) or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b.
When n is a negative integer and b is not zero, bn is naturally defined as 1/bn, preserving the property bn × bm = bn + m.
Exponentiation for integer exponents can be defined for a wide variety of algebraic structures, including matrices.
Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.
Calculation results
Addition (+)
\scriptstyle\left.\begin{matrix}\scriptstyle\text{summand}+\text{summand}\\\scriptstyle\text{augend}+\text{addend}\end{matrix}\right\}= \scriptstyle\text{sum}
Subtraction (−)
\scriptstyle\text{minuend}-\text{subtrahend}= \scriptstyle\text{difference}
Multiplication (×)
\scriptstyle\left.\begin{matrix}\scriptstyle\text{multiplicand}\times\text{multiplicand}\\\scriptstyle\text{multiplicand}\times\text{multiplier}\end{matrix}\right\}= \scriptstyle\text{product}
Division (÷)
\scriptstyle\left.\begin{matrix}\scriptstyle\frac{\scriptstyle\text{dividend}}{\scriptstyle\text{divisor}}\\\scriptstyle\frac{\scriptstyle\text{numerator}}{\scriptstyle\text{denominator}}\end{matrix}\right\}= \scriptstyle\text{quotient}
Modulation (mod)
\scriptstyle\text{dividend}\bmod\text{divisor}= \scriptstyle\text{remainder}
\scriptstyle\text{base}^\text{exponent}= \scriptstyle\text{power}
nth root (√)
\scriptstyle\sqrt[\text{degree}]{\scriptstyle\text{radicand}}= \scriptstyle\text{root}
Logarithm (log)
\scriptstyle\log_\text{base}(\text{antilogarithm})= \scriptstyle\text{logarithm}
History of the notation[edit]
The term power was used by the Greek mathematician Euclid for the square of a line.[1] Archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10.[2] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kab for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.[3]
In the late 16th century, Jost Bürgi used Roman Numerals for exponents.[4]
Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.[5]
Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word "exponent" was coined in 1544 by Michael Stifel.[6] Samuel Jeake introduced the term indices in 1696.[1] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).[7] Biquadrate has been used to refer to the fourth power as well.
Some mathematicians (e.g., Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d.
Another historical synonym, involution,[8] is now rare and should not be confused with its more common meaning.
The expression b2 = bb is called the square of b because the area of a square with side-length b is b2. It is pronounced "b squared".
The expression b3 = bbb is called the cube of b because the volume of a cube with side-length b is b3. It is pronounced "b cubed".
The exponent says how many copies of the base are multiplied together. For example, 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3, 3 raised to the fifth power, or 3 to the power of 5.
The word "raised" is usually omitted, and very often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five". The exponentiation bn can be read as b raised to the n-th power, or b raised to the power of n, or b raised by the exponent of n, or most briefly as b to the n.
Exponentiation may be generalized from integer exponents to more general types of numbers.
Integer exponents[edit]
The exponentiation operation with integer exponents requires only elementary algebra.
Positive integer exponents[edit]
Formally, powers with positive integer exponents may be defined by the initial condition[9]
b^1 = b
and the recurrence relation
b^{n+1} = b^n \cdot b
From the associativity of multiplication, it follows that for any positive integers m and n,
b^{m+n} = b^m \cdot b^n
Zero exponent[edit]
Any nonzero number raised by the exponent 0 is 1;[10] one interpretation of such a power is as an empty product. The case of 00 is discussed below.
Negative exponents[edit]
The following identity holds for an arbitrary integer n and nonzero b:
b^{-n} = 1/b^n
Raising 0 by a negative exponent is left undefined.
The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.
For non-zero b and positive n, the recurrence relation from the previous subsection can be rewritten as
b^{n} = {b^{n+1}}/{b}, \quad n \ge 1 .
By defining this relation as valid for all integer n and nonzero b, it follows that
b^0 &= {b^{1}}/{b} = 1 \\
b^{-1} &= {b^{0}}/{b} = {1}/{b}
and more generally for any nonzero b and any nonnegative integer n,
b^{-n} = {1}/{b^n} .
This is then readily shown to be true for every integer n.
Combinatorial interpretation[edit]
For nonnegative integers n and m, the power nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet).
05 = │ { } │ = 0 There is no 5-tuple from the empty set.
14 = │ { (1,1,1,1) } │ = 1 There is one 4-tuple from a one-element set.
23 = │ { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } │ = 8 There are eight 3-tuples from a two-element set.
32 = │ { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } │ = 9 There are nine 2-tuples from a three-element set.
41 = │ { (1), (2), (3), (4) } │ = 4 There are four 1-tuples from a four-element set.
50 = │ { () } │ = 1 There is exactly one 0-tuple.
Identities and properties[edit]
The following identities hold for all integer exponents, provided that the base is non-zero:
b^{m + n} &= b^m \cdot b^n \\
(b^m)^n &= b^{m\cdot n} \\
(b \cdot c)^n &= b^n \cdot c^n
Exponentiation is not commutative. This contrasts with addition and multiplication, which are. For example, 2 + 3 = 3 + 2 = 5 and 2 ⋅ 3 = 3 ⋅ 2 = 6, but 23 = 8, whereas 32 = 9.
Exponentiation is not associative either. Addition and multiplication are. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 ⋅ 3) ⋅ 4 = 2 ⋅ (3 ⋅ 4) = 24, but 23 to the 4 is 84 or 4096, whereas 2 to the 34 is 281 or 2417851639229258349412352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:
b^{p^q} = b^{(p^q)} \ne (b^p)^q = b^{(p \cdot q)} = b^{p \cdot q} .
Note that some computer programs (notably Microsoft Office Excel) associate to the left instead, i.e. a^b^c is evaluated as (a^b)^c.
Particular bases[edit]
Powers of ten[edit]
In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 103 = 1000 and 10−4 = 0.0001.
Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299792458 m/s (the speed of light in vacuum, in metres per second) can be written as 2.99792458×108 m/s and then approximated as 2.998×108 m/s.
SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 103 = 1000, so a kilometre is 1000 m.
Powers of two[edit]
The positive powers of 2 are important in computer science because there are 2n possible values for an n-bit binary register.
Powers of 2 are important in set theory since a set with n members has a power set, or set of all subsets of the original set, with 2n members.
The negative powers of 2 are commonly used, and the first two have special names: half, and quarter.
In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written as 1000 in binary.
Powers of one[edit]
The powers of one are all one: 1n = 1.
Powers of zero[edit]
If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.
If the exponent is negative, the power of zero (0n, where n < 0) is undefined, because division by zero is implied.
If the exponent is zero, some authors define 00 = 1, whereas others leave it undefined, as discussed below under § Zero to the power of zero.
Powers of minus one[edit]
If n is an even integer, then (−1)n = 1.
If n is an odd integer, then (−1)n = −1.
Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see § Powers of complex numbers.
Large exponents[edit]
The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:
bn → ∞ as n → ∞ when b > 1
This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".
Powers of a number with absolute value less than one tend to zero:
bn → 0 as n → ∞ when |b| < 1
Any power of one is always one:
bn = 1 for all n if b = 1
If the number b varies tending to 1 as the exponent tends to infinity then the limit is not necessarily one of those above. A particularly important case is
(1 + 1/n)ne as n → ∞
See § The exponential function below.
Other limits, in particular of those that take on an indeterminate form, are described in § Limits of powers below.
Rational exponents[edit]
Main article: nth root
From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.
An nth root of a number b is a number x such that xn = b.
If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal nth root of b. It is denoted nb, where is the radical symbol; alternatively, the principal root may be written b1/n. For example: 41/2 = 2, 81/3 = 2.
The fact that x=b^{1/n} solves x^n=b follows from noting that
x^n = \underbrace{ b^\frac{1}{n} \times b^\frac{1}{n} \times \cdots \times b^\frac{1}{n} }_n = b^{\left( \frac{1}{n} + \frac{1}{n} + \cdots + \frac{1}{n} \right)} = b^\frac{n}{n} = b^1 = b.
If n is even, then xn = b has two real solutions if b is positive, which are the positive and negative nth roots (the positive one being denoted b^{1/n}). If b is negative, the equation has no solution in real numbers for even n.
If n is odd, then xn = b has one real solution. The solution is positive if b is positive and negative if b is negative.
The principal root of a positive real number b with a rational exponent u/v in lowest terms satisfies
b^\frac{u}{v} = \left(b^u\right)^\frac{1}{v} = \sqrt[v]{b^u}
where u is an integer and v is a positive integer.
Rational powers u/v, where u/v is in lowest terms, are positive if u is even (and hence v is odd) (because then bu is positive), and negative for negative b if u and v are odd (because then bu is negative). There are two roots, one of each sign, if b is positive and v is even (as exemplified by the case in which u = 1 and v = 2, whereby a positive b has two square roots); in this case the principal root is defined to be the positive one.
Thus we have (−27)1/3 = −3 and (−27)2/3 = 9. The number 4 has two 3/2th roots, namely 8 and −8; however, by convention 43/2 denotes the principal root, which is 8. Since there is no real number x such that x2 = −1, the definition of bu/v when b is negative and v is even must use the imaginary unit i, as described more fully in the section § Powers of complex numbers.
Care needs to be taken when applying the power identities with negative nth roots. For instance, −27 = (−27)((2/3)⋅(3/2)) = ((−27)2/3)3/2 = 93/2 = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section § Failure of power and logarithm identities.
Real exponents[edit]
The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity
(b^r)^s = b^{r\cdot s}
cannot be extended consistently to cases where b is a negative real number (see § Real exponents with negative bases). The failure of this identity is the basis for the problems with complex number powers detailed under § Failure of power and logarithm identities.
Exponentiation to real powers of positive real numbers can be defined either by extending the rational powers to reals by continuity, or more usually as given in § Powers via logarithms below.
Limits of rational exponents[edit]
Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule[11]
b^x = \lim_{r (\in\mathbb Q)\to x} b^r\quad(b \in\mathbb R^+,\,x\in\mathbb R)
where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used, this involves showing that for any desired accuracy of the result bx one can choose a sufficiently small interval around x so all the rational powers in the interval are within the desired accuracy.
For example, if x = π, the nonterminating decimal representation π = 3.14159... can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers
[b^3,b^4], [b^{3.1},b^{3.2}], [b^{3.14},b^{3.15}], [b^{3.141},b^{3.142}], [b^{3.1415},b^{3.1416}], [b^{3.14159},b^{3.14160}], ...
The bounded intervals converge to a unique real number, denoted by b^\pi. This technique can be used to obtain any irrational power of a positive real number b. The function fb(x) = bx is thus defined for any real number x.
The exponential function[edit]
Main article: Exponential function
The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents.
As a consequence, the notation ex usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by:
\exp(x) = \lim_{n \rightarrow \infty} \left(1+\frac x n \right)^n
Among other properties, exp satisfies the exponential identity:
\exp(x+y) = \exp(x) \cdot \exp(y)
The exponential function is defined for all integer, fractional, real, and complex values of x. In fact, the matrix exponential is well-defined for square matrices (in which case the exponential identity only holds when x and y commute), and is useful for solving systems of linear differential equations.
Since exp(1) is equal to e and exp(x) satisfies the exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of ex for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ex definitions in the previous section for all real x by continuity.
Powers via logarithms[edit]
The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for b > 0, and satisfies
b = e^{\ln b}
If bx is to preserve the logarithm and exponent rules, then one must have
b^x = (e^{\ln b})^x = e^{x \cdot\ln b}
for each real number x.
This can be used as an alternative definition of the real number power bx and agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.
Real exponents with negative bases[edit]
Powers of a positive real number are always positive real numbers. The solution of x2 = 4, however, can be either 2 or −2. The principal value of 41/2 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved.
Neither the logarithm method nor the rational exponent method can be used to define br as a real number for a negative real number b and an arbitrary real number r. Indeed, er is positive for every real number r, so ln(b) is not defined as a real number for b ≤ 0.
The rational exponent method cannot be used for negative values of b because it relies on continuity. The function f(r) = br has a unique continuous extension[11] from the rational numbers to the real numbers for each b > 0. But when b < 0, the function f is not even continuous on the set of rational numbers r for which it is defined.
For example, consider b = −1. The nth root of −1 is −1 for every odd natural number n. So if n is an odd positive integer, (−1)(m/n) = −1 if m is odd, and (−1)(m/n) = 1 if m is even. Thus the set of rational numbers q for which (−1)q = 1 is dense in the rational numbers, as is the set of q for which (−1)q = −1. This means that the function (−1)q is not continuous at any rational number q where it is defined.
On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b.
Irrational exponents[edit]
If a is a positive algebraic number, and b is a rational number, it has been shown above that ab is algebraic. This remains true even if one accepts any algebraic number for a, with the only difference that ab may take several values (see below), all algebraic. Gelfond–Schneider theorem provides some information on the nature of ab when b is irrational (that is not rational). It states:
If a is an algebraic number different of 0 and 1, and b an irrational algebraic number, then all the values of ab are transcendental numbers (that is, not algebraic).
Complex exponents with positive real bases[edit]
Imaginary exponents with base e[edit]
Main article: Exponential function
The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus e is the limit of (1 + /N)N. In this animation N takes values increasing from 1 to 100. The computation of (1 + /N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + /N)N. It can be seen that as N gets larger (1 + /N)N approaches a limit of −1. Therefore, e = −1, which is known as Euler's identity.
A complex number is an expression of the form z=x+iy, where x and y are real numbers, and i is the so-called imaginary unit, a number that satisfies the rule i^2=-1). A complex number can be visualized as a point in the (x,y) plane. The polar coordinates of a point in the (x,y) plane consist of a non-negative real number r and angle θ such that x = r cos θ and y = r sin θ. So
x+iy=r(\cos\theta + i\sin\theta).
The product of two complex numbers z1 = x1 + iy1, z2 = x2 + iy2 is obtained by expanding out the product of the binomials and simplifying using the rule i^2=-1:
z_1z_2=(x_1+iy_1)(x_2+iy_2) = (x_1x_2-y_1y_2) + i(x_1y_2+x_2y_1).
As a consequence of the angle sum formulas of trigonometry, if z1 and z2 have polar coordinates (r1, θ1), (r2, θ2), then their product z1z2 has polar coordinates equal to (r1r2, θ1 + θ2).
Consider the right triangle in the complex plane which has 0, 1, 1 + ix/n as vertices. For large values of n, the triangle is almost a circular sector with a radius of 1 and a small central angle equal to x/n radians. 1 + ix/n may then be approximated by the number with polar coordinates (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)n approaches (1, x/n)n = (1n, nx/n) = (1, x), the point on the unit circle whose angle from the positive real axis is x radians. The cartesian coordinates of this point are (cos x, sin x). So e ix = cos x + isin x; this is Euler's formula, connecting algebra to trigonometry by means of complex numbers.
The solutions to the equation ez = 1 are the integer multiples of 2πi:
\{ z : e^z = 1 \} = \{ 2k\pi i : k \in \mathbb{Z} \}
More generally, if ev = w, then every solution to ez = w can be obtained by adding an integer multiple of 2πi to v:
\{ z : e^z = w \} = \{ v + 2k\pi i : k \in \mathbb{Z} \}
Thus the complex exponential function is a periodic function with period 2πi.
More simply: e = −1; ex + iy = ex(cos y + i sin y).
Trigonometric functions[edit]
Main article: Euler's formula
It follows from Euler's formula stated above that the trigonometric functions cosine and sine are
\cos(z) = \frac{e^{iz} + e^{-iz}}{2}; \qquad \sin(z) = \frac{e^{iz} - e^{-iz}}{2i}
Before the invention of complex numbers, cosine and sine were defined geometrically. The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula
e^{i(x+y)}=e^{ix}\cdot e^{iy}
Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.
Complex exponents with base e[edit]
The power z = ex + iy can be computed as exeiy. The real factor ex is the absolute value of z and the complex factor eiy identifies the direction of z.
Complex exponents with positive real bases[edit]
If b is a positive real number, and z is any complex number, the power bz is defined as ez ⋅ ln(b), where x = ln(b) is the unique real solution to the equation ex = b. So the same method working for real exponents also works for complex exponents.
For example:
2i = e i⋅ln(2) = cos(ln(2)) + i⋅sin(ln(2)) ≈ 0.76924 + 0.63896i
ei ≈ 0.54030 + 0.84147i
10i ≈ −0.66820 + 0.74398i
(e)i ≈ 535.49i ≈ 1
The identity (b^z)^u=b^{zu} is not generally valid for complex powers. A simple counterexample is given by:
(e^{2\pi i})^i=1^i=1\neq e^{-2\pi}=e^{2\pi i\cdot i}
The identity is, however, valid when z is a real number, and also when u is an integer.
Powers of complex numbers[edit]
Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then in equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4.
Complex powers of positive reals are defined via ex as in section Complex exponents with positive real bases above. These are continuous functions.
Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory.
The rational power of a complex number must be the solution to an algebraic equation. Therefore, it always has a finite number of possible values. For example, w = z1/2 must be a solution to the equation w2 = z. But if w is a solution, then so is −w, because (−1)2 = 1. A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers.
Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.
Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as for the corresponding real numbers.
Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However, in the common case of a positive real number the principal value is the same.
The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.
Complex exponents with complex bases[edit]
For complex numbers w and z with w ≠ 0, the notation wz is ambiguous in the same sense that log w is.
To obtain a value of wz, first choose a logarithm of w; call it log w. Such a choice may be the principal value Log w (the default, if no other specification is given), or perhaps a value given by some other branch of log w fixed in advance. Then, using the complex exponential function one defines
w^z = e^{z \log w}
because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of log w is used.
If z is an integer, then the value of wz is independent of the choice of log w, and it agrees with the earlier definition of exponentiation with an integer exponent.
If z is a rational number m/n in lowest terms with z > 0, then the countably infinitely many choices of log w yield only n different values for wz; these values are the n complex solutions s to the equation sn = wm.
If z is an irrational number, then the countably infinitely many choices of log w lead to infinitely many distinct values for wz.
The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below.
A similar construction is employed in quaternions.
Complex roots of unity[edit]
Main article: Root of unity
The three 3rd roots of 1
A complex number w such that wn = 1 for a positive integer n is an nth root of unity. Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1.
If wn = 1 but wk ≠ 1 for all natural numbers k such that 0 < k < n, then w is called a primitive nth root of unity. The negative unit −1 is the only primitive square root of unity. The imaginary unit i is one of the two primitive 4th roots of unity; the other one is −i.
The number e2πi/n is the primitive nth root of unity with the smallest positive argument. (It is sometimes called the principal nth root of unity, although this terminology is not universal and should not be confused with the principal value of n1, which is 1.[12])
The other nth roots of unity are given by
\left( e^{ \frac{2}{n} \pi i } \right) ^k = e^{ \frac{2}{n} \pi i k }
for 2 ≤ kn.
Roots of arbitrary complex numbers[edit]
Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power wq in the important special case where q = 1/n and n is a positive integer. These are the nth roots of w; they are solutions of the equation zn = w. As with real roots, a second root is also called a square root and a third root is also called a cube root.
It is conventional in mathematics to define w1/n as the principal value of the root. If w is a positive real number, it is also conventional to select a positive real number as the principal value of the root w1/n. For general complex numbers, the nth root with the smallest argument is often selected as the principal value of the nth root operation, as with principal values of roots of unity.
The set of nth roots of a complex number w is obtained by multiplying the principal value w1/n by each of the nth roots of unity. For example, the fourth roots of 16 are 2, −2, 2i, and −2i, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, i, and −i.
Computing complex powers[edit]
It is often easier to compute complex powers by writing the number to be exponentiated in polar form. Every complex number z can be written in the polar form
z = re^{i\theta} = e^{\log(r) + i\theta}
where r is a nonnegative real number and θ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. That is, r is the "radius" r2 = u2 + v2 and θ is the "angle" θ = atan2(v, u). The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A branch cut can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.
In order to compute the complex power wz, write w in polar form:
w = r e^{i\theta}
\log(w) = \log(r) + i \theta
and thus
w^z = e^{z \log(w)} = e^{z(\log(r) + i\theta)}
If z is decomposed as c + di, then the formula for wz can be written more explicitly as
\left( r^c e^{-d\theta} \right) e^{i (d \log(r) + c\theta)} = \left( r^c e^{-d\theta} \right) \left[ \cos(d \log(r) + c\theta) + i \sin(d \log(r) + c\theta) \right]
This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).
The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute ii, write i in polar and Cartesian forms:
i &= 1 \cdot e^{\frac{1}{2} i \pi} \\
i &= 0 + 1i
Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields:
i^i = \left( 1^0 e^{-\frac{1}{2}\pi} \right) e^{i \left[1 \cdot \log(1) + 0 \cdot \frac{1}{2}\pi \right]} = e^{-\frac{1}{2}\pi} \approx 0.2079
Similarly, to find (−2)3 + 4i, compute the polar form of −2,
-2 = 2e^{i \pi}
and use the formula above to compute
(-2)^{3 + 4i} = \left( 2^3 e^{-4\pi} \right) e^{i[4\log(2) + 3\pi]} \approx (2.602 - 1.006 i) \cdot 10^{-5}
The value of a complex power depends on the branch used. For example, if the polar form i = 1e5πi/2 is used to compute ii, the power is found to be e−5π/2; the principal value of ii, computed above, is e−π/2. The set of all possible values for ii is given by:[13]
i &= 1 \cdot e^{\frac{1}{2} i\pi + i 2 \pi k} \big| k \isin \mathbb{Z} \\
i^i &= e^{i \left(\frac{1}{2} i\pi + i 2 \pi k\right)} \\
&= e^{-\left(\frac{1}{2} \pi + 2 \pi k\right)}
So there is an infinity of values which are possible candidates for the value of ii, one for each integer k. All of them have a zero imaginary part so one can say ii has an infinity of valid real values.
Failure of power and logarithm identities[edit]
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:
• The identity log(bx) = x ⋅ log b holds whenever b is a positive real number and x is a real number. But for the principal branch of the complex logarithm one has
i\pi = \log(-1) = \log\left[(-i)^2\right] \neq 2\log(-i) = 2\left(-\frac{i\pi}{2}\right) = -i\pi
Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
\log(w^z) \equiv z \cdot \log(w) \pmod{2 \pi i}
This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) contain those of z ⋅ log w as a subset. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:
\left\{\log(w^z)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n + 2 \pi i m \right\} \\
\left\{z \cdot \log(w)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n \right\}
• The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x is a real number. But a calculation using principal branches shows that
1 = (-1\times -1)^\frac{1}{2} \not = (-1)^\frac{1}{2}(-1)^\frac{1}{2} = -1
i = (-1)^\frac{1}{2} = \left (\frac{1}{-1}\right )^\frac{1}{2} \not = \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i
On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.
If exponentiation is considered as a multivalued function then the possible values of (−1×−1)1/2 are {1, −1}. The identity holds but saying {1} = {(−1×−1)1/2} is wrong.
• The identity (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:[14]
For any integer n, we have:
1. e^{1 + 2 \pi i n} = e^{1} e^{2 \pi i n} = e \cdot 1 = e
2. \left( e^{1+2\pi i n} \right)^{1 + 2 \pi i n} = e
3. e^{1 + 4 \pi i n - 4 \pi^{2} n^{2}} = e
4. e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e
5. e^{-4 \pi^2 n^2} = 1
but this is false when the integer n is nonzero.
There are a number of problems in the reasoning:
The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.
From the multi-valued point of view, the first error occurs even sooner. Implicit in the first line is that e is a real number, whereas the result of e1+2πin is a complex number better represented as e+0i. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have. \scriptstyle (e^z)^w \;\ne\; e^{z w}, but rather \scriptstyle (e^z)^w \;=\; e^{(z \,+\, 2\pi i n) w} multivalued over integers n.
Exponentiation can be defined in any monoid.[15] A monoid is an algebraic structure consisting of a set X together with a rule for composition ("multiplication") satisfying an associative law and a multiplicative identity, denoted by 1. Exponentiation is defined inductively by:
• x^0=1 for all x\in X
• x^{n+1}=x^nx for all x\in X and non-negative integers n
Monoids include many structures of importance in mathematics, including groups and rings (under multiplication), with more specific examples of the latter being matrix rings and fields.
Matrices and linear operators[edit]
If A is a square matrix, then the product of A with itself n times is called the matrix power. Also A^0 is defined to be the identity matrix,[16] and if A is invertible, then A^{-n}=(A^{-1})^n.
Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system.[17] This is the standard interpretation of a Markov chain, for example. Then A^2x is the state of the system after two time steps, and so forth: A^nx is the state of the system after n time steps. The matrix power A^n is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.
Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, d/dx, which is a linear operator acting on functions f(x) to give a new function (d/dx)f(x)=f'(x). The n-th power of the differentiation operator is the n-th derivative:
\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).
These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups.[18] Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.
Finite fields[edit]
A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties. The real numbers, for example, form a field, as do the complex numbers and rational numbers. Unlike these familiar examples of fields, which are all infinite sets, some fields have only finitely many elements. The simplest example is the field with two elements F_2=\{0,1\} with addition defined by 0+1=1+0=1 and 0+0=1+1=0, and multiplication 0\cdot 0=1\cdot 0 = 0\cdot 1=0 and 1\cdot 1=1.
Exponentiation in finite fields has applications in public key cryptography. For example, the Diffie–Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive.
Any finite field F has the property that there is a unique prime number p such that px=0 for all x in F; that is, x added to itself p times is zero. For example, in F_2, the prime number p = 2 has this property. This prime number is called the characteristic of the field. Suppose that F is a field of characteristic p, and consider the function f(x) = x^p that raises each element of F to the power p. This is called the Frobenius automorphism of F. It is an automorphism of the field because of the Freshman's dream identity (x+y)^p = x^p+y^p. The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield.
In abstract algebra[edit]
Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.
Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by
x^1 &= x \\
x^n &= x^{n-1}x \quad\hbox{for }n>1
One has the following properties
(x^i x^j) x^k &= x^i (x^j x^k) \quad\text{(power-associative property)} \\
x^{m+n} &= x^m x^n \\
(x^m)^n &= x^{mn}
If the operation has a two-sided identity element 1, then x0 is defined to be equal to 1 for any x.
x1 &= 1x = x \quad\text{(two-sided identity)} \\
x^0 &= 1
\end{align}[citation needed]
If the operation also has two-sided inverses and is associative, then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents.
x x^{-1} &= x^{-1} x = 1 \quad\text{(two-sided inverse)} \\
(x y) z &= x (y z) \quad\text{(associative)} \\
x^{-n} &= \left(x^{-1}\right)^n \\
x^{m-n} &= x^m x^{-n}
If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:
(xy)^n = x^n y^n
If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.
When there are several power-associative binary operations defined on a set, any of which might be iterated, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, xn is x ∗ ... ∗ x, while x#n is x # ... # x, whatever the operations ∗ and # might be.
Superscript notation is also used, especially in group theory, to indicate conjugation. That is, gh = h−1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.
Over sets[edit]
Main article: Cartesian product
If n is a natural number and A is an arbitrary set, the expression An is often used to denote the set of ordered n-tuples of elements of A. This is equivalent to letting An denote the set of functions from the set {0, 1, 2, ..., n−1} to the set A; the n-tuple (a0, a1, a2, ..., an−1) represents the function that sends i to ai.
For an infinite cardinal number κ and a set A, the notation Aκ is also used to denote the set of all functions from a set of size κ to A. This is sometimes written κA to distinguish it from cardinal exponentiation, defined below.
This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of
\bigoplus_{i \in \mathbb{N}} V_{i}
where each Vi is a vector space.
Then if Vi = V for each i, the resulting direct sum can be written in exponential notation as VN, or simply VN with the understanding that the direct sum is the default. We can again replace the set N with a cardinal number n to get Vn, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and n to be some natural number, we get the vector space that is most commonly studied in linear algebra, the real vector space Rn.
If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n-tuple, which can be represented by a function on a set of appropriate cardinality, SN becomes simply the set of all functions from N to S in this case:
S^N \equiv \{ f\colon N \to S \}
This fits in with the exponentiation of cardinal numbers, in the sense that |SN| = |S||N|, where |X| is the cardinality of X. When "2" is defined as {0, 1}, we have |2X| = 2|X|, where 2X, usually denoted by P(X), is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for xY and 0 for xY.
In category theory[edit]
In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 00 is isomorphic to any terminal object 1.
Of cardinal and ordinal numbers[edit]
In set theory, there are exponential operations for cardinal and ordinal numbers.
If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.[19] If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3-tuples of elements from a 2-element set has cardinality 8 = 23. In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero).
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
Repeated exponentiation[edit]
Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7625597484987 (= 327 = 333 = 33) respectively.
Zero to the power of zero[edit]
Discrete exponents[edit]
There are many widely used formulas having terms involving natural-number exponents that require 00 to be evaluated to 1. For example, regarding b0 as an empty product assigns it the value 1, even when b = 0. Alternatively, the combinatorial interpretation of b0 is the number of empty tuples of elements from a set with b elements. There is exactly one empty tuple, even if b = 0. Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.[19]
Polynomials and power series[edit]
Likewise, when working with polynomials, it is often necessary to assign 0^0 the value 1. A polynomial is an expression of the form a_0x^0+\cdots+a_nx^n where x is an indeterminate, and the coefficients a_n are real numbers (or, more generally, elements of some ring). The set of all real polynomials in x is denoted by \mathbb R[x]. Polynomials are added termwise, and multiplied by the applying the usual rules for exponents in the indeterminate x (see Cauchy product). With these algebraic rules for manipulation, polynomials form a polynomial ring. The polynomial x^0 is the identity element of the polynomial ring, meaning that it is the (unique) element such that the product of x^0 with any polynomial p(x) is just p(x).[20] Polynomials can be evaluated by specializing the indeterminate x to be a real number. More precisely, for any given real number x_0 there is a unique unital ring homomorphism \operatorname{ev}_{x_0}:\mathbb R[x]\to\mathbb R such that \operatorname{ev}_{x_0}(x^1)=x_0.[21] This is called the evaluation homomorphism. Because it is a unital homomorphism, we have \operatorname{ev}_{x_0}(x^0) = 1. That is, x^0=1 for all specializations of x to a real number (including zero).
This perspective is significant for many polynomial identities appearing in combinatorics. For example, the binomial theorem (1 + x)^n = \sum_{k = 0}^n \binom{n}{k} x^k is not valid for x = 0 unless 00 = 1.[22] Similarly, rings of power series require x^0=1 to be true for all specializations of x. Thus identities like \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n and e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} are only true as functional identities (including at x = 0) if 00 = 1.
In differential calculus, the power rule \frac{d}{dx} x^n = nx^{n-1} is not valid for n = 1 at x = 0 unless 00 = 1.
Continuous exponents[edit]
Plot of z = xy. The red curves (with z constant) yield different limits as (x, y) approaches (0, 0). The green curves (of finite constant slope, y = ax) all yield a limit of 1.
Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.[23] In fact, when f(t) and g(t) are real-valued functions both approaching 0 (as t approaches a real number or ±∞), with f(t) > 0, the function f(t)g(t) need not approach 1; depending on f and g, the limit of f(t)g(t) can be any nonnegative real number or +∞, or it can diverge. For example, the functions below are of the form f(t)g(t) with f(t), g(t) → 0 as t → 0+, but the limits are different:
\lim_{t \to 0^+} {t}^{t} = 1, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^t = 0, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^{-t} = +\infty, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t}}\right)^{at} = e^{-a}.
Thus, the two-variable function xy, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on any set containing (0, 0), no matter how one chooses to define 00.[24] However, under certain conditions, such as when f and g are both analytic functions and f is positive on the open interval (0, b) for some positive b, the limit approaching from the right is always 1.[25][26][27]
Complex exponents[edit]
In the complex domain, the function zw may be defined for nonzero z by choosing a branch of log z and defining zw as ew log z. This does not define 0w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.[28][29][30]
History of differing points of view[edit]
The debate over the definition of 0^0 has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0^0 = 1, until in 1821 Cauchy[31] listed 0^0 along with expressions like \frac{0}{0} in a table of indeterminate forms. In the 1830s Libri[32][33] published an unconvincing argument for 0^0 = 1, and Möbius[34] sided with him, erroneously claiming that \scriptstyle \lim_{t \to 0^+} f(t)^{g(t)} \;=\; 1 whenever \scriptstyle \lim_{t \to 0^+} f(t) \;=\; \lim_{t \to 0^+} g(t) \;=\; 0. A commentator who signed his name simply as "S" provided the counterexample of \scriptstyle (e^{-1/t})^t, and this quieted the debate for some time. More historical details can be found in Knuth (1992).[35]
More recent authors interpret the situation above in different ways:
• Some argue that the best value for 0^0 depends on context, and hence that defining it once and for all is problematic.[36] According to Benson (1999), "The choice whether to define 0^0 is based on convenience, not on correctness. If we refrain from defining 0^0 then certain assertions become unnecessarily awkward. The consensus is to use the definition 0^0=1, although there are textbooks that refrain from defining 0^0."[37]
• Others argue that 0^0 should be defined as 1. Knuth (1992) contends strongly that 0^0 "has to be 1", drawing a distinction between the value 0^0, which should equal 1 as advocated by Libri, and the limiting form 0^0 (an abbreviation for a limit of \scriptstyle f(x)^{g(x)} where \scriptstyle f(x), g(x) \to 0), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."[35]
Treatment on computers[edit]
IEEE floating point standard[edit]
The IEEE 754-2008 floating point standard is used in the design of most floating point libraries. It recommends a number of functions for computing a power:[38]
• pow treats 00 as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for pown, otherwise the result is as for powr (except for some exceptional cases).
• pown treats 00 as 1. The power must be an exact integer. The value is defined for negative bases; e.g., pown(−3,5) is −243.
• powr treats 00 as NaN (Not-a-Number – undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by epower×log(base).
Programming languages[edit]
Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 00 as 1. The later C[39] and C++ standards describe this as the normative behaviour. The Java standard[40] mandates this behavior. The .NET Framework method System.Math.Pow also treats 00 as 1.[41]
Mathematics software[edit]
• Sage simplifies b0 to 1, even if no constraints are placed on b.[42] It takes 00 to be 1, but does not simplify 0x for other x.
• Maple distinguishes between integers 0, 1, ... and the corresponding floats 0.0, 1.0, ... (usually denoted 0., 1., ...). If x does not evaluates to a number, then x0 and x0.0 are respectively evaluated to 1 (integer) and 1.0 (float); on the other hand, 0x is evaluated to the integer 0, while 0.0x is evaluated as 0.x. If both the base and the exponent are zero (or are evaluated to zero), the result is Float(undefined) if the exponent is the float 0.0; with an integer as exponent, the evaluation of 00 results in the integer 1, while that of 0.0 results in the float 1.0.
• Macsyma also simplifies b0 to 1 even if no constraints are placed on b, but issues an error for 00. For x>0, it simplifies 0x to 0.[citation needed]
• Mathematica and Wolfram Alpha simplify b0 into 1, even if no constraints are placed on b.[43] While Mathematica does not simplify 0x, Wolfram Alpha returns two results, 0 for x > 0, and "indeterminate" for real x.[44] Both Mathematica and Wolfram Alpha take 00 to be "(indeterminate)".[45]
• Matlab, Python, Magma, GAP, singular, PARI/GP and the Google and iPhone calculators evaluate 00 as 1.
Limits of powers[edit]
The section § Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.
More precisely, consider the function f(x, y) = xy defined on D = {(x, y) ∈ R2 : x > 0}. Then D can be viewed as a subset of R2 (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.
In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).[46] Accordingly, this allows one to define the powers xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.
Under this definition by continuity, we obtain:
• x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞.
• x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1.
• 0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞.
• 0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 0.
These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.
On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.
Efficient computation with integer exponents[edit]
The simplest method of computing bn requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, note that 100 = 64 + 32 + 4. Compute the following in order:
1. 22 = 4
2. (22)2 = 24 = 16
3. (24)2 = 28 = 256
4. (28)2 = 216 = 65,536
5. (216)2 = 232 = 4,294,967,296
6. (232)2 = 264 = 18,446,744,073,709,551,616
7. 264 232 24 = 2100 = 1,267,650,600,228,229,401,496,703,205,376
This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).
In general, the number of multiplication operations required to compute bn can be reduced to Θ(log n) by using exponentiation by squaring or (more generally) addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bn is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.[47]
Exponential notation for function names[edit]
Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus f 3(x) may mean f(f(f(x))); in particular, f −1(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root f 1/2(x).
However, for historical reasons, a special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin2x is just a shorthand way to write (sin x)2 without using parentheses, whereas sin−1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation; for example, 1/(sin x) = (sin x)−1 = csc x. A similar convention applies to logarithms, where log2x usually means (log x)2, not log log x.
In programming languages[edit]
The superscript notation xy is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. Many programming languages have alternate ways of expressing exponentiation that do not use superscripts:
Many programming languages lack syntactic support for exponentiation, but provide library functions.
In Bash, C, C++, C#, Go, Java, JavaScript, Perl, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection. In OCaml and Standard ML, it represents string concatenation.
List of whole-number exponentials[edit]
n n2 n3 n4 n5 n6 n7 n8 n9 n10
2 4 8 16 32 64 128 256 512 1,024
3 9 27 81 243 729 2,187 6,561 19,683 59,049
4 16 64 256 1,024 4,096 16,384 65,536 262,144 1,048,576
5 25 125 625 3,125 15,625 78,125 390,625 1,953,125 9,765,625
6 36 216 1,296 7,776 46,656 279,936 1,679,616 10,077,696 60,466,176
7 49 343 2,401 16,807 117,649 823,543 5,764,801 40,353,607 282,475,249
8 64 512 4,096 32,768 262,144 2,097,152 16,777,216 134,217,728 1,073,741,824
9 81 729 6,561 59,049 531,441 4,782,969 43,046,721 387,420,489 3,486,784,401
10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000
See also[edit]
1. ^ a b O'Connor, John J.; Robertson, Edmund F., "Etymology of some common mathematical terms", MacTutor History of Mathematics archive, University of St Andrews .
2. ^ For further analysis see The Sand Reckoner.
3. ^ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews .
4. ^ Cajori, Florian (2007). A History of Mathematical Notations; Vol I. Cosimo Classics. Pg 344 ISBN 1602066841
6. ^ See:
• Earliest Known Uses of Some of the Words of Mathematics
• Michael Stifel, Arithmetica integra (Nuremberg ("Norimberga"), (Germany): Johannes Petreius, 1544), Liber III (Book 3), Caput III (Chapter 3): De Algorithmo numerorum Cossicorum. (On algorithms of algebra.), page 236. Stifel was trying to conveniently represent the terms of geometric progressions. He devised a cumbersome notation for doing that. On page 236, he presented the notation for the first eight terms of a geometric progression (using 1 as a base) and then he wrote: "Quemadmodum autem hic vides, quemlibet terminum progressionis cossicæ, suum habere exponentem in suo ordine (ut 1ze habet 1. 1ʓ habet 2 &c.) sic quilibet numerus cossicus, servat exponentem suæ denominationis implicite, qui ei serviat & utilis sit, potissimus in multiplicatione & divisione, ut paulo inferius dicam." (However, you see how each term of the progression has its exponent in its order (as 1ze has a 1, 1ʓ has a 2, etc.), so each number is implicitly subject to the exponent of its denomination, which [in turn] is subject to it and is useful mainly in multiplication and division, as I will mention just below.) [Note: Most of Stifel's cumbersome symbols were taken from Christoff Rudolff, who in turn took them from Leonardo Fibonacci's Liber Abaci (1202), where they served as shorthand symbols for the Latin words res/radix (x), census/zensus (x2), and cubus (x3).]
7. ^ Quinion, Michael. "Zenzizenzizenzic - the eighth power of a number". World Wide Words. Retrieved 2010-03-19.
8. ^ This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary [1]. The most recent usage in this sense cited by the OED is from 1806.
9. ^ Hodge, Jonathan K.; Schlicker, Steven; Sundstorm, Ted (2014). Abstract Algebra: an inquiry based approach. CRC Press. p. 94. ISBN 978-1-4665-6706-1.
10. ^ Achatz, Thomas (2005). Technical Shop Mathematics (3rd ed.). Industrial Press. p. 101. ISBN 0-8311-3086-5.
11. ^ a b Denlinger, Charles G. (2011). Elements of Real Analysis. Jones and Bartlett. pp. 278–283. ISBN 978-0-7637-7947-4.
12. ^ This definition of a principal root of unity can be found in:
13. ^ Complex number to a complex power may be real at Cut The Knot gives some references to ii
14. ^ Steiner J, Clausen T, Abel NH (1827). "Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen" [Problems and propositions, the former to solve, the later to prove]. Journal für die reine und angewandte Mathematik 2: 286–287.
15. ^ Nicolas Bourbaki (1970). Algèbre. Springer. , I.2
16. ^ Chapter 1, Elementary Linear Algebra, 8E, Howard Anton
17. ^ Strang, Gilbert (1988), Linear algebra and its applications (3rd ed.), Brooks-Cole , Chapter 5.
18. ^ E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975.
19. ^ a b N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
20. ^ Nicolas Bourbaki (1970). Algèbre. Springer. , §III.2 No. 9: "L'unique monôme de degré 0 est l'élément unité de A[(X_i)_{i\in I}]; on l'identifie souvent à l'élément unité 1 de A".
21. ^ Nicolas Bourbaki (1970). Algèbre. Springer. , §IV.1 No. 3.
22. ^ "Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant".Ronald Graham, Donald Knuth, and Oren Patashnik (1989-01-05). "Binomial coefficients". Concrete Mathematics (1st ed.). Addison Wesley Longman Publishing Co. p. 162. ISBN 0-201-14236-8.
23. ^ Malik, S. C.; Savita Arora (1992). Mathematical Analysis. New York: Wiley. p. 223. ISBN 978-81-224-0323-7. In general the limit of φ(x)/ψ(x) when x = a in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞ 0 × ∞, ∞ − ∞, 00, 1 and ∞0.
24. ^ L. J. Paige (March 1954). "A note on indeterminate forms". American Mathematical Monthly 61 (3): 189–190. doi:10.2307/2307224. JSTOR 2307224.
25. ^ sci.math FAQ: What is 0^0?
26. ^ Rotando, Louis M.; Korn, Henry (1977). "The Indeterminate Form 00". Mathematics Magazine (Mathematical Association of America) 50 (1): 41–42. doi:10.2307/2689754. JSTOR 2689754.
27. ^ Lipkin, Leonard J. (2003). "On the Indeterminate Form 00". The College Mathematics Journal (Mathematical Association of America) 34 (1): 55–56. doi:10.2307/3595845. JSTOR 3595845.
28. ^ "Since log(0) does not exist, 0z is undefined. For Re(z) > 0, we define it arbitrarily as 0." George F. Carrier, Max Krook and Carl E. Pearson, Functions of a Complex Variable: Theory and Technique , 2005, p. 15
29. ^ "For z = 0, w ≠ 0, we define 0w = 0, while 00 is not defined." Mario Gonzalez, Classical Complex Analysis, Chapman & Hall, 1991, p. 56.
30. ^ "... Let's start at x = 0. Here xx is undefined." Mark D. Meyerson, The xx Spindle, Mathematics Magazine 69, no. 3 (June 1996), 198-206.
31. ^ Augustin-Louis Cauchy, Cours d'Analyse de l'École Royale Polytechnique (1821). In his Oeuvres Complètes, series 2, volume 3.
32. ^ Guillaume Libri, Note sur les valeurs de la fonction 00x, Journal für die reine und angewandte Mathematik 6 (1830), 67–72.
33. ^ Guillaume Libri, Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik 10 (1833), 303–316.
34. ^ A. F. Möbius (1834). "Beweis der Gleichung 00 = 1, nach J. F. Pfaff" [Proof of the equation 00 = 1, according to J. F. Pfaff]. Journal für die reine und angewandte Mathematik 12: 134–136.
35. ^ a b Donald E. Knuth, Two notes on notation, Amer. Math. Monthly 99 no. 5 (May 1992), 403–422 (arXiv:math/9205211 [math.HO]).
36. ^ Examples include Edwards and Penny (1994). Calculus, 4th ed, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. Addison-Wesley, p. 32.
37. ^ Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 978-0-19-511721-9
38. ^ Handbook of Floating-Point Arithmetic. Birkhäuser Boston. 2009. p. 216. ISBN 978-0-8176-4704-9.
39. ^ John Benito (April 2003). "Rationale for International Standard—Programming Languages—C" (PDF). Revision 5.10. p. 182.
40. ^ "Math (Java Platform SE 8) pow". Oracle.
41. ^ ".NET Framework Class Library Math.Pow Method". Microsoft.
42. ^ "Sage worksheet calculating x^0". Jason Grout.
43. ^ "Wolfram Alpha calculates b^0". Wolfram Alpha LLC, accessed April 25, 2015.
44. ^ "Wolfram Alpha calculates 0^x". Wolfram Alpha LLC, accessed April 25, 2015.
45. ^ "Wolfram Alpha calculates 0^0". Wolfram Alpha LLC, accessed April 25, 2015.
46. ^ N. Bourbaki, Topologie générale, V.4.2.
47. ^ Gordon, D. M. (1998). "A Survey of Fast Exponentiation Methods". Journal of Algorithms 27: 129–146. doi:10.1006/jagm.1997.0913.
External links[edit] |
75e4a7e8a4ecfbfc | Cubical atom
From Wikipedia, the free encyclopedia
Jump to: navigation, search
The cubical atom was an early atomic model in which electrons were positioned at the eight corners of a cube in a non-polar atom or molecule. This theory was developed in 1902 by Gilbert N. Lewis and published in 1916 in the article "The Atom and the Molecule" and used to account for the phenomenon of valency.[1] Lewis's theory was based on Abegg's rule. It was further developed in 1919 by Irving Langmuir as the cubical octet atom.[2] The figure below shows structural representations for elements of the second row of the periodic table.
Cubical atom 1.svg
Although the cubical model of the atom was soon abandoned in favor of the quantum mechanical model based on the Schrödinger equation, and is therefore now principally of historical interest, it represented an important step towards the understanding of the chemical bond. The 1916 article by Lewis also introduced the concept of the electron pair in the covalent bond, the octet rule, and the now-called Lewis structure.
Bonding in the cubical atom model[edit]
Single covalent bonds are formed when two atoms share an edge, as in structure C below. This results in the sharing of two electrons. Ionic bonds are formed by the transfer of an electron from one cube to another, without sharing an edge (A). An intermediate state B where only one corner is shared was also postulated by Lewis.
Cubical atom 2.svg
Double bonds are formed by sharing a face between two cubic atoms. This results in sharing four electrons:
Cubical atom 3.svg
Triple bonds could not be accounted for by the cubical atom model, because there is no way of having two cubes share three parallel edges. Lewis suggested that the electron pairs in atomic bonds have a special attraction, which result in a tetrahedral structure, as in the figure below (the new location of the electrons is represented by the dotted circles in the middle of the thick edges). This allows the formation of a single bond by sharing a corner, a double bond by sharing an edge, and a triple bond by sharing a face. It also accounts for the free rotation around single bonds and for the tetrahedral geometry of methane.
Cubical atom 4.svg
See also[edit]
1. ^ Lewis, Gilbert N. (1916-04-01). "The Atom and the Molecule.". Journal of the American Chemical Society 38 (4): 762–785. doi:10.1021/ja02261a002. See images of original article
2. ^ Langmuir, Irving (1919-06-01). "The Arrangement of Electrons in Atoms and Molecules.". Journal of the American Chemical Society 41 (6): 868–934. doi:10.1021/ja02227a002. |
115784ba82fcc799 | From New World Encyclopedia
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1 (none)hydrogenhelium
Name, Symbol, Number hydrogen, H, 1
Chemical series nonmetals
Group, Period, Block 1, 1, s
Appearance colorless
Atomic mass 1.00794(7) g/mol
Electron configuration 1s1
Electrons per shell 1
Physical properties
Phase gas
Density (0 °C, 101.325 kPa)
0.08988 g/L
Melting point 14.01 K
(−259.14 °C, −434.45 °F)
Boiling point 20.28 K
(−252.87 °C, −423.17 °F)
Triple point 13.8033 K, 7.042 kPa
Critical point 32.97 K, 1.293 MPa
Heat of fusion (H2) 0.117 kJ/mol
Heat of vaporization (H2) 0.904 kJ/mol
Heat capacity (25 °C) (H2)
28.836 J/(mol·K)
Vapor pressure
P/Pa 1 10 100 1 k 10 k 100 k
at T/K 15 20
Atomic properties
Crystal structure hexagonal
Oxidation states 1, −1
(amphoteric oxide)
Electronegativity 2.20 (Pauling scale)
Ionization energies 1st: 1312.0 kJ/mol
Atomic radius 25 pm
Atomic radius (calc.) 53 pm (Bohr radius)
Covalent radius 37 pm
Van der Waals radius 120 pm
Thermal conductivity (300 K) 180.5 mW/(m·K)
CAS registry number 1333-74-0 (H2)
Notable isotopes
Main article: Isotopes of hydrogen
iso NA half-life DM DE (MeV) DP
1H 99.985% H is stable with 0 neutrons
2H 0.0115% H is stable with 1 neutron
3H trace 12.32 years β 0.019 3He
Hydrogen (chemical symbol H, atomic number 1) is the lightest chemical element and the most abundant of all elements, constituting roughly 75 percent of the elemental mass of the universe.[1] Stars in the main sequence are mainly composed of hydrogen in its plasma state.
In the Earth's natural environment, free (uncombined) hydrogen is relatively rare. At standard temperature and pressure, it takes the form of a colorless, odorless, tasteless, highly flammable gas made up of diatomic molecules (H2). On the other hand, the element is widely distributed in combination with other elements, and many of its compounds are vital for living systems. Its most familiar compound is water (H2O).
Elemental hydrogen is industrially produced from hydrocarbons such as methane, after which most elemental hydrogen is used "captively" (meaning locally, at the production site). The largest markets are about equally divided between fossil fuel upgrading (such as hydrocracking) and ammonia production (mostly for the fertilizer market).
The most common naturally occurring isotope of hydrogen, known as protium, has a single proton and no neutrons. In ionic compounds, it can take on either a positive charge (becoming a cation, H+, which is a proton) or a negative charge (becoming an anion, H, called a hydride). It plays a particularly important role in acid-base chemistry, in which many reactions involve the exchange of protons between soluble molecules. As the only neutral atom for which the Schrödinger equation can be solved analytically, study of the energetics and bonding of the hydrogen atom has played a key role in the development of quantum mechanics.
The term hydrogen (Latin: 'hydrogenium') can be traced to a combination of the ancient Greek words hydor, meaning "water," and genes, meaning "forming." This refers to the observation that when hydrogen burns, it produces water.
Natural occurrence
Hydrogen is the most abundant element in the universe, making up 75 percent of normal matter by mass and over 90 percent by number of atoms.[2] This element is found in great abundance in stars and gas giant planets. Molecular clouds of H2 are associated with star formation. Hydrogen plays a vital role in powering stars through proton-proton reaction nuclear fusion.
Under ordinary conditions on Earth, elemental hydrogen exists as the diatomic gas, H2 (for data see table). However, hydrogen gas is very rare in the Earth's atmosphere (1 part per million by volume) because of its light weight, which enables it to escape Earth's gravity more easily than heavier gases. Although H atoms and H2 molecules are abundant in interstellar space, they are difficult to generate, concentrate and purify on Earth. Still, hydrogen is the third most abundant element on the Earth's surface.[3] Most of the Earth's hydrogen is in the form of chemical compounds such as hydrocarbons and water.[4] Hydrogen gas is produced by some bacteria and algae and is a natural component of flatus. Methane is a hydrogen source of increasing importance.
Discovery of H2
Role in history of quantum theory
The hydrogen atom
Electron energy levels
The ground state energy level of the electron in a hydrogen atom is 13.6 eV, which is equivalent to an ultraviolet photon of roughly 92 nanometers.
Hydrogen has three naturally occurring isotopes, denoted 1H, 2H, and 3H. Other, highly unstable nuclei (4H to 7H) have been synthesized in the laboratory but not observed in nature.[7][8]
• 1H is the most common hydrogen isotope with an abundance of more than 99.98 percent. Because the nucleus of this isotope consists of only a single proton, it is given the descriptive but rarely used formal name protium.
• 2H, the other stable hydrogen isotope, is known as deuterium and contains one proton and one neutron in its nucleus. Deuterium comprises 0.0026–0.0184 percent (by mole-fraction or atom-fraction) of hydrogen samples on Earth, with the lower number tending to be found in samples of hydrogen gas and the higher enrichments (0.015 percent or 150 parts per million) typical of ocean water. Deuterium is not radioactive, and does not represent a significant toxicity hazard. Water enriched in molecules that include deuterium instead of normal hydrogen is called heavy water. Deuterium and its compounds are used as a non-radioactive label in chemical experiments and in solvents for 1H-NMR spectroscopy. Heavy water is used as a neutron moderator and coolant for nuclear reactors. Deuterium is also a potential fuel for commercial nuclear fusion.
Hydrogen is the only element that has different names for its isotopes in common use today (During the early study of radioactivity, various heavy radioactive isotopes were given names, but such names are no longer used. The symbols D and T (instead of 2H and 3H) are sometimes used for deuterium and tritium, but the corresponding symbol P is already in use for phosphorus and thus is not available for protium. IUPAC states that while this use is common, it is not preferred.
Elemental molecular forms
First tracks observed in liquid hydrogen bubble chamber at the Bevatron
There are two different types of diatomic hydrogen molecules that differ by the relative spin of their nuclei.[9] In the orthohydrogen form, the spins of the two protons are parallel and form a triplet state; in the parahydrogen form the spins are antiparallel and form a singlet. At standard temperature and pressure, hydrogen gas contains about 25 percent of the para form and 75 percent of the ortho form, also known as the "normal form."[10] The equilibrium ratio of orthohydrogen to parahydrogen depends on temperature, but since the ortho form is an excited state and has a higher energy than the para form, it is unstable and cannot be purified. At very low temperatures, the equilibrium state is composed almost exclusively of the para form. The physical properties of pure parahydrogen differ slightly from those of the normal form.[11] The ortho/para distinction also occurs in other hydrogen-containing molecules or functional groups, such as water and methylene.
Hydrogen is the lightest element in the periodic table, with an atomic mass of 1.00794 g/mol. For lack of a better place, it is generally shown at the top of group 1 (former group 1A). It is, however, a nonmetal, whereas the other members of group 1 are alkali metals.
Hydrogen can combust rapidly in air. It burned rapidly in the Hindenburg airship disaster May 6, 1937.
Hydrogen gas is highly flammable and will burn at concentrations as low as four percent H2 in air. The combustion reaction may be written as follows:
The reaction generates a large amount of heat. The enthalpy of combustion is – 286 kJ/mol.
When mixed with oxygen across a wide range of proportions, hydrogen explodes upon ignition. Pure hydrogen-oxygen flames are nearly invisible to the naked eye, as illustrated by the faintness of flame from the main space shuttle engines (as opposed to the easily visible flames from the shuttle boosters). Thus it is difficult to visually detect if a hydrogen leak is burning.
The Hindenburg airship flames seen in the adjacent picture are hydrogen flames colored with material from the covering skin of the zeppelin which contained carbon and pyrophoric aluminum powder, as well as other combustible materials.[18] Regardless of the cause of this fire, this was clearly primarily a hydrogen fire since skin of the airship alone would have taken many hours to burn.[19] Another characteristic of hydrogen fires is that the flames tend to ascend rapidly with the gas in air, as illustrated by the Hindenburg flames, causing less damage than hydrocarbon fires. For example, two-thirds of the Hindenburg passengers survived the hydrogen fire, and many of the deaths that occurred were from falling or from gasoline burns.[20]
Reaction with halogens
Covalent and organic compounds
Hydrogen forms a vast array of compounds with carbon. Because of their general association with living things, these compounds came to be called organic compounds; the study of their properties is known as organic chemistry and their study in the context of living organisms is known as biochemistry. By some definitions, "organic" compounds are only required to contain carbon, but most of them also contain hydrogen, and the carbon-hydrogen bond is responsible for many of their chemical characteristics.
Compounds of hydrogen are often called hydrides, a term that is used fairly loosely. To chemists, the term "hydride" usually implies that the H atom has acquired a negative or anionic character, denoted H. The existence of the hydride anion, suggested by G. N. Lewis in 1916 for group I and II salt-like hydrides, was demonstrated by Moers in 1920 with the electrolysis of molten lithium hydride (LiH), that produced a stoichiometric quantity of hydrogen at the anode.[21] For hydrides other than group I and II metals, the term is quite misleading, considering the low electronegativity of hydrogen. An exception in group II hydrides is BeH2, which is polymeric. In lithium aluminum hydride, the AlH4 anion carries hydridic centers firmly attached to the Al(III). Although hydrides can be formed with almost all main-group elements, the number and combination of possible compounds varies widely; for example, there are over one hundred binary borane hydrides known, but only one binary aluminum hydride.[22] Binary indium hydride has not yet been identified, although larger complexes exist.[23]
"Protons" and acids
H2 is produced in chemistry and biology laboratories, often as a byproduct of other reactions; in industry for the hydrogenation of unsaturated substrates; and in nature as a means of expelling reducing equivalents in biochemical reactions.
Laboratory syntheses
Zn + 2 H+ → Zn2+ + H2
Aluminum produces H2 upon treatment with an acid or a base:
The electrolysis of water is a simple method of producing hydrogen, although the resulting hydrogen necessarily has less energy content than was required to produce it. A low-voltage current is run through the water, and gaseous oxygen forms at the anode while gaseous hydrogen forms at the cathode. Typically the cathode is made from platinum or another inert metal when producing hydrogen for storage. If, however, the gas is to be burnt on site, oxygen is desirable to assist the combustion, and so both electrodes would be made from inert metals (iron, for instance, would oxidize, and thus decrease the amount of oxygen given off). The theoretical maximum efficiency (electricity used vs. energetic value of hydrogen produced) is between 80 and 94 percent.[26]
In 2007 it was discovered that an alloy of aluminum and gallium in pellet form added to water could be used to generate hydrogen.[27] The process creates also creates alumina, but the expensive gallium, which prevents the formation of an oxide skin on the pellets, can be reused. This potentially has important implications for a hydrogen economy, since hydrogen can be produced on-site and does not need to be transported.
Industrial syntheses
Hydrogen can be prepared in several different ways but the economically most important processes involve removal of hydrogen from hydrocarbons. Commercial bulk hydrogen is usually produced by the steam reforming of natural gas.[28] At high temperatures (700–1100 °C; 1,300–2,000 °F), steam (water vapor) reacts with methane to yield carbon monoxide and H2.
CH4 + H2OCO + 3 H2
CH4 → C + 2 H2
Consequently, steam reforming typically employs an excess of H2O.
CO + H2OCO2 + H2
Other important methods for H2 production include partial oxidation of hydrocarbons:
CH4 + 0.5 O2CO + 2 H2
and the coal reaction, which can serve as a prelude to the shift reaction above:[28]
C + H2OCO + H2
Hydrogen is sometimes produced and consumed in the same industrial process, without being separated. In the Haber process for the production of ammonia (the world's fifth-most produced industrial compound), hydrogen is generated from natural gas.
Biological syntheses
H2 is a product of some types of anaerobic metabolism and is produced by several microorganisms, usually via reactions catalyzed by iron- or nickel-containing enzymes called hydrogenases. These enzymes catalyze the reversible redox reaction between H2 and its component two protons and two electrons. Evolution of hydrogen gas occurs in the transfer of reducing equivalents produced during pyruvate fermentation to water.[29]
The triple point temperature of equilibrium hydrogen is a defining fixed point on the International Temperature Scale of 1990 (ITS-90).
Hydrogen as an energy carrier
See also
1. Hydrogen in the Universe, NASA. Retrieved December 27, 2007.
2. Steve Gagnon, It’s Elemental: Hydrogen, Jefferson Lab. Retrieved December 27, 2007.
3. Basic Research Needs for the Hydrogen Economy, Argonne National Laboratory, U.S. Department of Energy, Office of Science Laboratory. Retrieved December 27, 2007.
4. 4.0 4.1 4.2 G. L. Miessler and D. A. Tarr, Inorganic Chemistry, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2004, ISBN 0130354716).
5. Webelements – Hydrogen Historical Information. Retrieved December 27, 2007.
6. R. Berman, A. H. Cooke and R. W. Hill, “Cryogenics,” Ann. Rev. Phys. Chem. 7 (1956): 1–20.
7. Y. B. Gurov, D. V. Aleshkin, M. N. Berh, S. V. Lapushkin, et al., Spectroscopy of superheavy hydrogen isotopes in stopped-pion absorption by nuclei, Physics of Atomic Nuclei 68 (3) (2004): 491–497.
8. A. A. Korsheninnikov, et al., Experimental evidence for the existence of 7H and for a specific structure of 8He, Phys. Rev. Lett. 90 (2003)” 082501.
9. Hydrogen (H2) Applications and Uses, Universal Industrial Gases, Inc. Retrieved December 27, 2007.
10. V. I. Tikhonov and A. A. Volkov, Separation of water into its ortho and para isomers, Science 296 (5577) (2002) :2363.
11. CH. 6 - NASA Glenn Research Center Glenn Safety Manual: Hydrogen, Document GRC-MQSA.001, March 2006. Retrieved December 27, 2007.
12. Y. Y. Milenko, R. M. Sibileva and M. A. Strzhemechny, Natural ortho-para conversion rate in liquid and gaseous hydrogen, J. Low. Temp. Phys. 107 (1-2) (1997): 77–92.
13. R. E. Svadlenak and A. B. Scott, The conversion of ortho-to parahydrogen on iron oxide-zinc oxide catalysts, J. Am. Chem. Soc. 79(20) (1957): 5385–5388.
14. H3+ Resource Center. Universities of Illinois and Chicago. Retrieved December 27, 2007.
15. T. Takeshita, W. E. Wallace and R. S. Craig, Hydrogen solubility in 1:5 compounds between yttrium or thorium and nickel or cobalt, Inorg. Chem. 13 (9) (1974): 2282.
16. R. Kirchheim, T. Mutschele and W. Kieninger, Hydrogen in amorphous and nanocrystalline metals, Mater. Sci. Eng. 99 (1988): 457–462.
17. R. Kirchheim, Hydrogen solubility and diffusivity in defective and amorphous metals, Prog. Mater. Sci. 32 (4) (1988): 262–325.
18. A. Bain and W. D. Van Vorst, The Hindenburg tragedy revisited: the fatal flaw exposed, International Journal of Hydrogen Energy 24 (5) (1999): 399–403.
19. John Dziadecki, Hindenburg Hydrogen Fire. Retrieved December 27, 2007.
20. The Hindenburg Disaster, Swiss Hydrogen Association. Retrieved December 27, 2007.
21. K. Moers,. 2. Z. Anorg. Allgem. Chem. 113 (1920): 191.
22. A. J. Downs and C. R. Pulham, The hydrides of aluminium, gallium, indium, and thallium: a re-evaluation, Chem. Soc. Rev. 23 (1994): 175–183.
23. D. E. Hibbs, C. Jones and N. A. Smithies, A remarkably stable indium trihydride complex: Synthesis and characterization of [InH3{P(C6H11)3}], Chem. Commum. (1999): 185–186.
24. M. Okumura, L. I. Yeh, J. D. Myers and Y. T. Lee, Infrared spectra of the solvated hydronium ion: Vibrational predissociation spectroscopy of mass-selected H3O+•H2On•H2m (1990).
25. A. Carrington and I. R. McNab, The infrared predissociation spectrum of triatomic hydrogen cation (H3+), Accounts of Chemical Research 22 (1989): 218–222.
26. Bellona Report on Hydrogen. Retrieved December 27, 2007.
27. “New process generates hydrogen from aluminum alloy to run engines, fuel cells,” Physorg.com (May 16, 2007). Retrieved December 27, 2007.
28. 28.0 28.1 28.2 D. W. Oxtoby, H. P. Gillis and N. H. Nachtrieb, Principles of Modern Chemistry, 5th ed. (Belmont, CA: Thomson Brooks/Cole, 2002, ISBN 0030353734).
29. R. Cammack, M. Frey and R. Robson, Hydrogen as a Fuel: Learning from Nature (London: Taylor & Francis, 2001).
30. O. Kruse, J. Rupprecht, K. P. Bader, S. Thomas-Hal, P. M. Schenk, G. Finazzi and B. Hankamer, Improved photobiological H2 production in engineered green algal cells, J. Biol. Chem. 280 (40) (2005): 34170–34177.
31. H.O. Smith and Q. Xu, Hydrogen from Water in a Novel Recombinant Oxygen-Tolerant Cyanobacteria System, United States Department of Energy FY2005 Progress Report, IV.E.6. Retrieved December 27, 2007.
32. Hydrogen, Los Alamos National Laboratory. Retrieved December 27, 2007.
33. Joseph Romm, The Hype About Hydrogen: Fact and Fiction in the Race to Save the Climate (New York: Island Press, 2004, ISBN 1559637048).
• General Electric. 1989. Chart of the Nuclides. General Electric Company. Retrieved December 27, 2007.
• Ferreira-Aparicio, P., M. J. Benito, and J. L. Sanz. 2005. New Trends in Reforming Technologies: from Hydrogen Industrial Plants to Multifuel Microreformers. Catalysis Reviews 47: 491–588.
• Krebs, Robert E. 1998. The History and Use of Our Earth's Chemical Elements: A Reference Guide. Westport, CT: Greenwood Press. ISBN 0313301239
• Newton, David E. 1994. The Chemical Elements. New York: Franklin Watts. ISBN 0531125017
• Rigden, John S. 2002. Hydrogen: The Essential Element. Cambridge, MA: Harvard University Press. ISBN 0531125017
• Romm, Joseph J. 2004. The Hype about Hydrogen, Fact and Fiction in the Race to Save the Climate. Washington, D.C.: Island Press. ISBN 155963703X
• Stwertka, Albert. 2002. A Guide to the Elements. New York: Oxford University Press. ISBN 0195150279
External links
All links retrieved December 27, 2007.
E numbers
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064c89ff84c12f05 | Mathematical formulation of quantum mechanics
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The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely: as spectral values (point spectrum plus absolute continuous plus singular continuous spectrum) of linear operators in Hilbert space.[1]
These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observable which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables.
Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus, and increasing in complexity up to differential geometry and partial differential equations. Probability theory was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, which was formulated entirely on the classical phase space.
History of the formalism[edit]
The "old quantum theory" and the need for new mathematics[edit]
In the 1890s, Planck was able to derive the blackbody spectrum which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Planck's constant in his honor.
In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.
light at the right frequency.
All of these developments were phenomenological and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.
In 1923 de Broglie proposed that wave-particle duality applied not only to photons but to electrons and every other physical system.
The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.
The "new quantum theory"[edit]
Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra. Later in the same year, Schrödinger created his wave mechanics. Schrödinger's formalism was considered easier to understand, visualize and calculate with as it led to differential equations, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.
Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrödinger's wave function can be seen to be closely related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.
To be more precise, already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics–– the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.
Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in all kinds of generalizations of the field.
The first complete mathematical formulation of this approach, known as the Dirac–von Neumann axioms, is generally credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.
Later developments[edit]
The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case.
On a different front, von Neumann originally dispatched quantum measurement with his infamous postulate on the collapse of the wavefunction, raising a host of philosophical problems. Over the intervening 70 years, the problem of measurement became an active research area and itself spawned some new formulations of quantum mechanics.
A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.
Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics.
Mathematical structure of quantum mechanics[edit]
A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
Postulates of quantum mechanics[edit]
The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms.
• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product . Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.
• The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J.M. Jauch, Foundations of quantum mechanics, section 11-7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (supersymmetry is another matter entirely).
• Physical observables are represented by Hermitian matrices on H.
The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector |ψH is
\langle\psi\mid A\mid\psi\rangle
By spectral theory, we can associate a probability measure to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.
More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator ρ normalized to be of trace 1. The expected value of A in the state ρ is
If ρψ is the orthogonal projector onto the one-dimensional subspace of H spanned by |ψ, then
\operatorname{tr}(A\rho_\psi)=\left\langle\psi\mid A\mid\psi\right\rangle
Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.
One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.
Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.
Pictures of dynamics[edit]
The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If |ψ(t) denotes the state of the system at any one time t, the following Schrödinger equation holds:
Schrödinger equation (general)
i\hbar\frac{d}{d t}\left|\psi(t)\right\rangle=H\left|\psi(t)\right\rangle
where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and ħ is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U(t): HH such that
for all times s, t. The existence of a self-adjoint Hamiltonian H such that
U(t)=e^{-(i/\hbar)t H}
is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H does not depend on time and that the perturbation starts at t0 = 0; otherwise one must use the Dyson series, formally written as
U(t)=\mathcal{T}\left[\exp\left(-\frac{i}{\hbar} \int_{t_0}^t \,{\rm d}t'\, H(t')\right)\right]\,,
where {\mathcal{T}} is Dyson's time-ordering symbol.
(This symbol permutes a product of noncommuting operators of the form
B_1(t_1)\cdot B_2(t_2)\cdot\dots \cdot B_n(t_n)
into the uniquely determined re-ordered expression
B_{i_1}(t_{i_1})\cdot B_{i_2}(t_{i_2})\cdot\dots \cdot B_{i_n}(t_{i_n}) with t_{i_1}\ge t_{i_2}\ge\dots\ge t_{i_n}\,.
The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the utmost l.h.s. .)
• The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:
\left|\psi\right\rangle = \left|\psi(0)\right\rangle
A(t) = U(-t)AU(t). \quad
It is then easily checked that the expected values of all observables are the same in both pictures
\langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle
and that the time-dependent Heisenberg operators satisfy
Heisenberg picture (general)
\frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\frac{\partial A(t)}{\partial t},
which is true for time-dependent A = A(t). Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.
• The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:
Dirac picture
i\hbar\frac{d }{dt}\left|\psi(t)\right\rangle ={H}_{\rm int}(t) \left|\psi(t)\right\rangle
i\hbar{d \over d t}A(t) = [A(t),H_{0}].
The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. H = H0 + V, in the interaction picture it does, at least, if V does not commute with H0, since
H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_0}\,V\,e^{{(-i/\hbar})tH_0}.
So the above-mentioned Dyson-series has to be used anyhow.
The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.
Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry (for instance, angular or linear momentum).
The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerations of quantization, the deformation extension from classical to quantum mechanics.
The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.
Time as an operator[edit]
The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H − E, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H − E (this requires the use of a rigged Hilbert space and a renormalization of the norm).
This is related to the quantization of constrained systems and quantization of gauge theories. It is also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).
In addition to their other properties, all particles possess a quantity called spin, an intrinsic angular momentum. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position r and time t as continuous variables, ψ = ψ(r, t), for spin wavefunctions the spin is an additional discrete variable: ψ = ψ(r, t, σ), where σ takes the values;
\sigma = -S \hbar , -(S-1) \hbar , \dots, 0, \dots ,+(S-1) \hbar ,+S \hbar \,.
That is, the state of a single particle with spin S is represented by a (2S + 1)-component spinor of complex-valued wave functions.
Two classes of particles with very different behaviour are bosons which have integer spin (S = 0, 1, 2...), and fermions possessing half-integer spin (S = 123252, ...).
Pauli's principle[edit]
The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have
Pauli principle
\psi (\dots, \,\mathbf r_i,\sigma_i, \, \dots, \,\mathbf r_j,\sigma_j, \,\dots) = (-1)^{2S}\cdot \psi ( \dots, \,\mathbf r_j,\sigma_j, \, \dots, \mathbf r_i,\sigma_i,\, \dots)
i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor (−1)2S which is +1 for bosons, but (−1) for fermions. Electrons are fermions with S = 1/2; quanta of light are bosons with S = 1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d = 2 can one construct entities where (−1)2S is replaced by an arbitrary complex number with magnitude 1, called anyons.
Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of the two properties.
The problem of measurement[edit]
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.[3] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain):
• Let A have spectral resolution
A = \int \lambda \, d \operatorname{E}_A(\lambda),
where EA is the resolution of the identity (also called projection-valued measure) associated to A. Then the probability of the measurement outcome lying in an interval B of R is |EA(Bψ|2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure
\langle \psi \mid \operatorname{E}_A \psi \rangle.
• If the measured value is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state EA(B)ψ. If the measured value does not lie in B, replace B by its complement for the above state.
For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λi, with corresponding eigenvectors ψi. The projection-valued measure associated with A, EA, is then
\operatorname{E}_A (B) = | \psi_i\rangle \langle \psi_i|,
where B is a Borel set containing only the single eigenvalue λi. If the system is prepared in state
| \psi \rangle \,
Then the probability of a measurement returning the value λi can be calculated by integrating the spectral measure
over Bi. This gives trivially
\langle \psi| \psi_i\rangle \langle \psi_i \mid \psi \rangle = | \langle \psi \mid \psi_i\rangle | ^2.
The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.
A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections
| \psi_i\rangle \langle \psi_i| \,
by a finite set of positive operators
F_i F_i^* \,
whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ1 ... λn} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λi. Instead of collapsing to the (unnormalized) state
| \psi_i\rangle \langle \psi_i |\psi\rangle \,
after the measurement, the system now will be in the state
F_i |\psi\rangle. \,
Since the Fi Fi* operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.
The same formulation applies to general mixed states.
In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.
In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).
The relative state interpretation[edit]
An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum mechanics.
List of mathematical tools[edit]
Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.
The main tools include:
• J. von Neumann, Mathematical Foundations of Quantum Mechanics (1932), Princeton University Press, 1955. Reprinted in paperback form.
• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
• A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin 1964 (Reprinted by Princeton University Press)
• R. Jost, The General Theory of Quantized Fields, American Mathematical Society, 1965.
• J.M. Jauch, Foundations of quantum mechanics, Addison-Wesley Publ. Cy., Reading, Mass., 1968.
• M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
• T.S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, Clarendon Press, Oxford and Oxford University Press, New York, 1978.
• D. Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, 42 (1979),pp. 1–70.
• E. Prugovecki, Quantum Mechanics in Hilbert Space, Dover, 1981.
• S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995.
• N. Weaver, "Mathematical Quantization", Chapman & Hall/CRC 2001.
• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators,, American Mathematical Society, 2009.
• V. Moretti, "Spectral Theory and Quantum Mechanics; With an Introduction to the Algebraic Formulation", Springer, 2013.
• B.C. Hall, "Quantum Theory for Mathematicians," Springer, 2013.
1. ^ Frederick W. Byron, Robert W. Fuller; Mathematics of classical and quantum physics; Courier Dover Publications, 1992.
2. ^ Dirac, P. A. M. (1925). "The Fundamental Equations of Quantum Mechanics". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 109 (752): 642. doi:10.1098/rspa.1925.0150.
3. ^ G. Greenstein and A. Zajonc |
74157ff0382f09a2 | Soliton (optics)
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In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
• spatial solitons: the nonlinear effect can balance the diffraction. The electromagnetic field can change the refractive index of the medium while propagating, thus creating a structure similar to a graded-index fiber. If the field is also a propagating mode of the guide it has created, then it will remain confined and it will propagate without changing its shape
• temporal solitons: if the electromagnetic field is already spatially confined, it is possible to send pulses that will not change their shape because the nonlinear effects will balance the dispersion. Those solitons were discovered first and they are often simply referred as "solitons" in optics.
Spatial solitons[edit]
how a lens works
In order to understand how a spatial soliton can exist, we have to make some considerations about a simple convex lens. As shown in the picture on the right, an optical field approaches the lens and then it is focused. The effect of the lens is to introduce a non-uniform phase change that causes focusing. This phase change is a function of the space and can be represented with \varphi (x), whose shape is approximately represented in the picture.
The phase change can be expressed as the product of the phase constant and the width of the path the field has covered. We can write it as:
\varphi (x) = k_0 n L(x)
where L(x) is the width of the lens, changing in each point with a shape that is the same of \varphi (x) because k_0 and n are constants. In other words, in order to get a focusing effect we just have to introduce a phase change of such a shape, but we are not obliged to change the width. If we leave the width L fixed in each point, but we change the value of the refractive index n(x) we will get exactly the same effect, but with a completely different approach.
That's the way graded-index fibers work: the change in the refractive index introduces a focusing effect that can balance the natural diffraction of the field. If the two effects balance each other perfectly, then we have a confined field propagating within the fiber.
Spatial solitons are based on the same principle: the Kerr effect introduces a Self-phase modulation that changes the refractive index according to the intensity:
\varphi (x) = k_0 n (x) L = k_0 L [n + n_2 I(x)]
if I(x) has a shape similar to the one shown in the figure, then we have created the phase behavior we wanted and the field will show a self-focusing effect. In other words, the field creates a fiber-like guiding structure while propagating. If the field creates a fiber and it is the mode of such a fiber at the same time, it means that the focusing nonlinear and diffractive linear effects are perfectly balanced and the field will propagate forever without changing its shape (as long as the medium does not change and if we can neglect losses, obviously). In order to have a self-focusing effect, we must have a positive n_2, otherwise we will get the opposite effect and we will not notice any nonlinear behavior.
The optical waveguide the soliton creates while propagating is not only a mathematical model, but it actually exists and can be used to guide other waves at different frequencies. This way it is possible to let light interact with light at different frequencies (this is impossible in linear media)
An electric field is propagating in a medium showing optical Kerr effect, so the refractive index is given by:
n(I) = n + n_2 I
we remember that the relationship between intensity and electric field is (in the complex representation):
I = \frac{|E|^2}{2 \eta}
where \eta = \eta_0 / n and \eta_0 is the impedance of free space, given by:
\eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} \approx 377 \Omega
The field is propagating in the z direction with a phase constant k_0 n. About now, we will ignore any dependence on the y axis, assuming that it is infinite in that direction. Then the field can be expressed as:
E(x,z,t) = A_m a(x,z) e^{i(k_0 n z - \omega t)}
where A_m is the maximum amplitude of the field and a(x,z) is a dimensionless normalized function (so that its maximum value is 1) that represents the shape of the electric field among the x axis. In general it depends on z because fields change their shape while propagating. Now we have to solve Helmholtz equation:
\nabla^2 E + k_0^2 n^2 (I) E = 0
where it was pointed out clearly that the refractive index (thus the phase constant) depend on intensity. If we replace the expression of the electric field in the equation, assuming that the envelope a(x,z) changes slowly while propagating, i.e.
\left| \frac{\partial^2 a(x,z)}{\partial z^2} \right| \ll \left|k_0 \frac{\partial a(x,z)}{\partial z} \right|
the equation becomes:
\frac{\partial^2 a}{\partial x^2} + i 2 k_0 n \frac{\partial a}{\partial z} + k_0^2 [n^2 (I) - n^2] a = 0
Let us introduce an approximation that is valid because the nonlinear effects are always much smaller than the linear ones:
[n^2 (I) - n^2] = [n (I) - n] [n (I) + n] = n_2 I (2 n + n_2 I) \approx 2 n n_2 I
now we express the intensity in terms of the electric field:
[n^2 (I) - n^2] \approx 2 n n_2 \frac{|A_m|^2 |a(x,z)|^2 }{2 \eta_0 / n} = n^2 n_2 \frac{|A_m|^2 |a(x,z)|^2 }{\eta_0}
the equation becomes:
\frac{1}{2 k_0 n} \frac{\partial^2 a}{\partial x^2} + i \frac{\partial a}{\partial z} + \frac{k_0 n n_2 |A_m|^2}{2 \eta_0} |a|^2 a = 0
We will now assume n_2 > 0 so that the nonlinear effect will cause self focusing. In order to make this evident, we will write in the equation n_2 = |n_2| Let us now define some parameters and replace them in the equation:
• \xi = \frac{x}{X_0}, so we can express the dependence on the x axis with a dimensionless parameter; X_0 is a length, whose physical meaning will be clearer later.
• L_d = X_0^2 k_0 n, after the electric field has propagated across z for this length, the linear effects of diffraction can not be neglected anymore.
• \zeta = \frac{z}{L_d}, for studying the z-dependence with a dimensionless variable.
• L_{nl} = \frac{2 \eta_0}{k_0 n | n_2 | \cdot |A_m|^2}, after the electric field has propagated across z for this length, the nonlinear effects can not be neglected anymore. This parameter depends upon the intensity of the electric field, that's typical for nonlinear parameters.
• N^2 = \frac{L_d}{L_{nl}}
The equation becomes:
\frac{1}{2} \frac{\partial^2 a}{\partial \xi^2} + i\frac{\partial a}{\partial \zeta} + N^2 |a|^2 a = 0
this is a common equation known as nonlinear Schrödinger equation. From this form, we can understand the physical meaning of the parameter N:
• if N \ll 1, then we can neglect the nonlinear part of the equation. It means L_d \ll L_{nl}, then the field will be affected by the linear effect (diffraction) much earlier than the nonlinear effect, it will just diffract without any nonlinear behavior.
• if N \gg 1, then the nonlinear effect will be more evident than diffraction and, because of self phase modulation, the field will tend to focus.
• if N \approx 1, then the two effects balance each other and we have to solve the equation.
For N=1 the solution of the equation is simple and it is the fundamental soliton:
a(\xi,\zeta) = \operatorname{sech} (\xi) e^{i \zeta /2}
where sech is the hyperbolic secant. It still depends on z, but only in phase, so the shape of the field will not change during propagation.
For N=2 it is still possible to express the solution in a closed form, but it has a more complicated form:
a(\xi,\zeta) = \frac{4[\cosh (3 \xi) + 3 e^{4 i \zeta} \cosh (\xi)] e^{i \zeta / 2}}{\cosh (4 \xi) + 4 \cosh (2 \xi) + 3 \cos (4 \zeta)}
It does change its shape during propagation, but it is a periodic function of z with period \zeta = \pi / 2.
Soliton's shape while propagating with N=1, it does not change its shape
Soliton's shape while propagating with N=2, it changes its shape periodically
For soliton solutions, N must be an integer and it is said to be the order or the soliton. For higher values of N, there are no closed form expressions, but the solitons exist and they are all periodic with different periods. Their shape can easily be expressed only immediately after generation:
a(\xi,\zeta = 0) = N \operatorname{sech} (\xi)
on the right there is the plot of the second order soliton: at the beginning it has a shape of a sech, then the maximum amplitude increases and then comes back to the sech shape. Since high intensity is necessary to generate solitons, if the field increases its intensity even further the medium could be damaged.
The condition to be solved if we want to generate a fundamental soliton is obtained expressing N in terms of all the known parameters and then putting N=1:
1 = N = \frac{L_d}{L_{nl}} = \frac{X_0^2 k_0^2 n^2 |n_2| |A_m|^2}{2 \eta_0}
that, in terms of maximum intensity value becomes:
I_{max} = \frac{|A_m|^2}{2 \eta_0 / n} = \frac{1}{X_0^2 k_0^2 n |n_2|}
in most of the cases, the two variables that can be changed are the maximum intensity I_\max and the pulse width X_0.
Generation of Spatial Solitons[edit]
The first experiment on spatial optical solitons was reported in 1974 by Ashkin and Bjorkholm[1] in a cell filled with sodium vapor. It was more than another ten years before this field was revisited in experiments at Limoges University[2] in liquid carbon disulphide. After these experiments spatial solitons have been demonstrated in glass, semiconductors[3] and polymers. During the last decade several experiments have been reported on solitons in nematic liquid crystals,[4] also referred to as nematicons.
Temporal solitons[edit]
The main problem that limits transmission bit rate in optical fibers is group velocity dispersion. It is because generated impulses have a non-zero bandwidth and the medium they are propagating through has a refractive index that depends on frequency (or wavelength). This effect is represented by the group delay dispersion parameter D; using it, it is possible to calculate exactly how much the pulse will widen:
\Delta \tau \approx D L \, \Delta \lambda
where L is the length of the fiber and \Delta \lambda is the bandwidth in terms of wavelength. The approach in modern communication systems is to balance such a dispersion with other fibers having D with different signs in different parts of the fiber: this way the pulses keep on broadening and shrinking while propagating. With temporal solitons it is possible to remove such a problem completely.
linear and nonlinear effects on Gaussian pulses
Consider the picture on the right. On the left there is a standard Gaussian pulse, that's the envelope of the field oscillating at a defined frequency. We assume that the frequency remains perfectly constant during the pulse.
Now we let this pulse propagate through a fiber with D > 0, it will be affected by group velocity dispersion. For this sign of D, the dispersion is anomalous, so that the higher frequency components will propagate a little bit faster than the lower frequencies, thus arriving before at the end of the fiber. The overall signal we get is a wider chirped pulse, shown in the upper right of the picture.
effect of self-phase modulation on frequency
Now let us assume we have a medium that shows only nonlinear Kerr effect but its refractive index does not depend on frequency: such a medium does not exist, but it's worth considering it to understand the different effects.
The phase of the field is given by:
\varphi (t) = \omega_0 t - k z = \omega_0 t - k_0 z [n + n_2 I(t)]
the frequency (according to its definition) is given by:
\omega (t) = \frac{\partial \varphi (t)}{\partial t} = \omega_0 - k_0 z n_2 \frac{\partial I(t) }{\partial t}
this situation is represented in the picture on the left. At the beginning of the pulse the frequency is lower, at the end it's higher. After the propagation through our ideal medium, we will get a chirped pulse with no broadening because we have neglected dispersion.
Coming back to the first picture, we see that the two effects introduce a change in frequency in two different opposite directions. It is possible to make a pulse so that the two effects will balance each other. Considering higher frequencies, linear dispersion will tend to let them propagate faster, while nonlinear Kerr effect will slow them down. The overall effect will be that the pulse does not change while propagating: such pulses are called temporal solitons.
History of temporal solitons[edit]
In 1973, Akira Hasegawa and Fred Tappert of AT&T Bell Labs were the first to suggest that solitons could exist in optical fibers, due to a balance between self-phase modulation and anomalous dispersion.[5] [6] Also in 1973 Robin Bullough made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton-based transmission system to increase performance of optical telecommunications.
Solitons in a fiber optic system are described by the Manakov equations.
In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton, in an optical fiber.
In 1988, Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named for the Indian scientist Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fiber.
In 1991, a Bell Labs research team transmitted solitons error-free at 2.5 gigabits over more than 14,000 kilometers, using erbium optical fiber amplifiers (spliced-in segments of optical fiber containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses.
In 1998, Thierry Georges and his team at France Télécom R&D Center, combining optical solitons of different wavelengths (wavelength division multiplexing), demonstrated a data transmission of 1 terabit per second (1,000,000,000,000 units of information per second).
An electric field is propagating in a medium showing optical Kerr effect through a guiding structure (such as an optical fiber) that limits the power on the xy plane. If the field is propagating towards z with a phase constant \beta_0, then it can be expressed in the following form:
E(\mathbf{r},t) = A_m a(t,z) f(x,y) e^{i(\beta_0 z - \omega_0 t)}
where A_m is the maximum amplitude of the field, a(t,z) is the envelope that shapes the impulse in the time domain; in general it depends on z because the impulse can change its shape while propagating; f(x,y) represents the shape of the field on the xy plane, and it does not change during propagation because we have assumed the field is guided. Both a and f are normalized dimensionless functions whose maximum value is 1, so that A_m really represents the field amplitude.
Since in the medium there is a dispersion we can not neglect, the relationship between the electric field and its polarization is given by a convolution integral. Anyway, using a representation in the Fourier domain, we can replace the convolution with a simple product, thus using standard relationships that are valid in simpler media. We Fourier-transform the electric field using the following definition:
\tilde{E} (\mathbf{r},\omega - \omega_0) = \int_{-\infty}^{\infty} E (\mathbf{r}, t ) e^{-i (\omega - \omega_0)t} dt
using this definition, a derivative in the time domain corresponds to a product in the Fourier domain:
\frac{\partial}{\partial t} E \Longleftrightarrow -i (\omega - \omega_0) \tilde{E}
the complete expression of the field in the frequency domain is:
\tilde{E} (\mathbf{r},\omega - \omega_0) = A_m \tilde{a} (\omega - \omega_0 , z) f(x,y) e^{i \beta_0 z}
Now we can solve Helmholtz equation in the frequency domain:
\nabla^2 \tilde{E} + n^2 (\omega) k_0^2 \tilde{E} = 0
we decide to express the phase constant with the following notation:
n(\omega) k_0 = \beta (\omega) = \underbrace{\beta_0}_{\mbox{linear non dispersive}} + \underbrace{\beta_l (\omega)}_{\mbox{linear dispersive}} + \underbrace{\beta_{nl}}_{\mbox{non linear}} = \beta_0 + \Delta \beta (\omega)
where we assume that \Delta \beta (the sum of the linear dispersive component and the non linear part) is a small perturbation, i.e. |\beta_0| \gg |\Delta \beta (\omega)|. The phase constant can have any complicated behavior, but we can represent it with a Taylor series centered on \omega_0:
\beta (\omega) \approx \beta_0 + (\omega - \omega_0) \beta_1 + \frac{(\omega - \omega_0)^2}{2} \beta_2 + \beta_{nl}
where, as known:
\beta_u = \left. \frac{d^u \beta (\omega)}{d \omega^u} \right|_{\omega = \omega_0}
we put the expression of the electric field in the equation and make some calculations. If we assume the slowly varying envelope approximation:
\left| \frac{\partial^2 \tilde{a}}{\partial z^2} \right| \ll \left| \beta_0 \frac{\partial \tilde{a}}{\partial z} \right|
we get:
2 i \beta_0 \frac{\partial \tilde{a}}{\partial z} + [\beta^2 (\omega) - \beta_0^2] \tilde{a} = 0
we are ignoring the behavior in the xy plane, because it is already known and given by f(x,y). We make a small approximation, as we did for the spatial soliton:
\beta^2 (\omega) - \beta_0^2 = [ \beta (\omega) - \beta_0 ] [ \beta (\omega) + \beta_0 ] =
[ \beta_0 + \Delta \beta (\omega) - \beta_0 ] [2 \beta_0 + \Delta \beta (\omega) ] \approx 2 \beta_0 \Delta \beta (\omega)
replacing this in the equation we get simply:
i \frac{\partial \tilde{a}}{\partial z} + \Delta \beta (\omega) \tilde{a} = 0
now we want to come back in the time domain. Expressing the products by derivatives we get the duality:
\Delta \beta (\omega) \Longleftrightarrow i \beta_1 \frac{\partial}{\partial t} - \frac{\beta_2}{2} \frac{\partial^2}{\partial t^2} + \beta_{nl}
we can write the non linear component in terms of the amplitude of the field:
\beta_{nl} = k_0 n_2 I = k_0 n_2 \frac{|E|^2}{2 \eta_0 / n} = k_0 n_2 n \frac{|A_m|^2}{2 \eta_0} |a|^2
for duality with the spatial soliton, we define:
L_{nl} = \frac{2 \eta_0}{k_0 n n_2 |A_m|^2}
and this symbol has the same meaning of the previous case, even if the context is different. The equation becomes:
i \frac{\partial a}{\partial z} + i \beta_1 \frac{\partial a}{\partial t} - \frac{\beta_2}{2} \frac{\partial^2 a}{\partial t^2} + \frac{1}{L_{nl}} |a|^2 a = 0
We know that the impulse is propagating among the z axis with a group velocity given by v_g = 1/\beta_1, so we are not interested in it because we just want to know how the pulse changes its shape while propagating. We decide to study the impulse shape, i.e. the envelope function a(.) using a reference that is moving with the field at the same velocity. Thus we make the substitution
T = t-\beta_1 z
and the equation becomes:
i \frac{\partial a}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 a}{\partial T^2} + \frac{1}{L_{nl}} |a|^2 a = 0
we assume the medium where the field is propagating to show anomalous dispersion, i.e. \beta_2 < 0 or in term of the group delay dispersion parameter D=\frac{- 2 \pi c}{\lambda^2} \beta_2 > 0 . We make this more evident replacing in the equation \beta_2 = - |\beta_2|. Let us define now the following parameters (the duality with the previous case is evident):
L_d = \frac{T_0^2}{|\beta_2|}; \qquad
\tau=\frac{T}{T_0}; \qquad
\zeta = \frac{z}{L_d} ; \qquad
replacing those in the equation we get:
\frac{1}{2} \frac{\partial^2 a}{\partial \tau^2} + i\frac{\partial a}{\partial \zeta} + N^2 |a|^2 a = 0
that is exactly the same equation we have obtained in the previous case. The first order soliton is given by:
a(\tau,\zeta) = \operatorname{sech} (\tau) e^{i \zeta /2}
the same considerations we have made are valid in this case. The condition N=1 becomes a condition on the amplitude of the electric field:
|A_m|^2 = \frac{2 \eta_0 |\beta_2|}{T_0^2 n_2 k_0 n}
or, in terms of intensity:
I_{max} = \frac{|A_m|^2}{2 \eta_0 / n} = \frac{|\beta_2|}{T_0^2 n_2 k_0}
or we can express it in terms of power if we introduce an effective area A_\mathit{eff} defined so that P = I A_\mathit{eff}:
P = \frac{|\beta_2| A_\mathit{eff}}{T_0^2 n_2 k_0}
Stability of solitons[edit]
We have described what optical solitons are and, using mathematics, we have seen that, if we want to create them, we have to create a field with a particular shape (just sech for the first order) with a particular power related to the duration of the impulse. But what if we are a bit wrong in creating such impulses? Adding small perturbations to the equations and solving them numerically, it is possible to show that mono-dimensional solitons are stable. They are often referred as (1 + 1) D solitons, meaning that they are limited in one dimension (x or t, as we have seen) and propagate in another one (z).
If we create such a soliton using slightly wrong power or shape, then it will adjust itself until it reaches the standard sech shape with the right power. Unfortunately this is achieved at the expense of some power loss, that can cause problems because it can generate another non-soliton field propagating together with the field we want. Mono-dimensional solitons are very stable: for example, if 0.5 < N < 1.5 we will generate a first order soliton anyway; if N is greater we'll generate a higher order soliton, but the focusing it does while propagating may cause high power peaks damaging the media.
The only way to create a (1 + 1) D spatial soliton is to limit the field on the y axis using a dielectric slab, then limiting the field on x using the soliton.
On the other hand, (2 + 1) D spatial solitons are unstable, so any small perturbation (due to noise, for example) can cause the soliton to diffract as a field in a linear medium or to collapse, thus damaging the material. It is possible to create stable (2 + 1) D spatial solitons using saturating nonlinear media, where the Kerr relationship n(I) = n + n_2 I is valid until it reaches a maximum value. Working close to this saturation level makes it possible to create a stable soliton in a three dimensional space.
If we consider the propagation of shorter (temporal) light pulses or over a longer distance, we need to consider higher-order corrections and therefore the pulse carrier envelope is governed by the higher-order nonlinear Schrödinger equation (HONSE) for which there are some specialized (analytical) soliton solutions.[7]
Effect of power losses[edit]
As we have seen, in order to create a soliton it is necessary to have the right power when it is generated. If there are no losses in the medium, then we know that the soliton will keep on propagating forever without changing shape (1st order) or changing its shape periodically (higher orders). Unfortunately any medium introduces losses, so the actual behavior of power will be in the form:
P(z) = P_0 e^{- \alpha z}
this is a serious problem for temporal solitons propagating in fibers for several kilometers. Let us consider what happens for the temporal soliton, generalization to the spatial ones is immediate. We have proved that the relationship between power P_0 and impulse length T_0 is:
if the power changes, the only thing that can change in the second part of the relationship is T_0. if we add losses to the power and solve the relationship in terms of T_0 we get:
T(z) = T_0 e^{\frac{\alpha}{2}z}
the width of the impulse grows exponentially to balance the losses! this relationship is true as long as the soliton exists, i.e. until this perturbation is small, so it must be \alpha z \ll 1 otherwise we can not use the equations for solitons and we have to study standard linear dispersion. If we want to create a transmission system using optical fibers and solitons, we have to add optical amplifiers in order to limit the loss of power.
Generation of soliton pulse[edit]
Experiments has been carried out to analyze the effect of high frequency (20 MHz-1 GHz) external magnetic field induced nonlinear Kerr effect on Single mode optical fiber of considerable length (50-100m) to compensate group velocity dispersion (GVD) and subsequent evolution of soliton pulse ( peak energy, narrow, secant hyperbolic pulse).[8] Generation of soliton pulse in fiber is an obvious conclusion as self phase modulation due to high energy of pulse offset GVD, whereas the evolution length is 2000 km. (the laser wavelength chosen greater than 1.3 micrometers). Moreover peak soliton pulse is of period 1-3ps so that it is safely accommodated in the optical bandwidth. Once soliton pulse is generated it is least dispersed over thousands of kilometers length of fiber limiting the number of repeater stations.
Dark solitons[edit]
In the analysis of both types of solitons we have assumed particular conditions about the medium:
• in spatial solitons, n_2 > 0, that means the self-phase modulation causes self-focusing
• in temporal solitons, \beta_2 < 0 or D > 0 , anomalous dispersion
Is it possible to obtain solitons if those conditions are not verified? if we assume n_2 < 0 or \beta_2 > 0, we get the following differential equation (it has the same form in both cases, we will use only the notation of the temporal soliton):
This equation has soliton-like solutions. For the first order (N=1):
a(\tau,\zeta) = \tanh (\tau) e^{i \zeta}.\
power of a dark soliton
The plot of |a(\tau, \zeta)|^2 is shown in the picture on the right. For higher order solitons ( N > 1 ) we can use the following closed form expression:
a(\tau,\zeta = 0) = N \tanh (\tau).\
It is a soliton, in the sense that it propagates without changing its shape, but it is not made by a normal pulse; rather, it is a lack of energy in a continuous time beam. The intensity is constant, but for a short time during which it jumps to zero and back again, thus generating a "dark pulse"'. Those solitons can actually be generated introducing short dark pulses in much longer standard pulses. Dark solitons are more difficult to handle than standard solitons, but they have shown to be more stable and robust to losses.
See also[edit]
1. ^ J.E. Bjorkholm, A. Ashkin (1974). "cw Self-Focusing and Self-Trapping of Light in Sodium Vapor". Phys. Rev. Lett. 32 (4): 129. Bibcode:1974PhRvL..32..129B. doi:10.1103/PhysRevLett.32.129.
2. ^ A. Barthelemy, S. Maneuf and C. Froehly (1985). "Propagation soliton et auto-confinement de faisceaux laser par non linearité optique de kerr". Opt. Comm. 55 (3): 201. Bibcode:1985OptCo..55..201B. doi:10.1016/0030-4018(85)90047-1.
3. ^ J.S. Aitchison et al. (1992). "Observation of spatial solitons in AlGaAs waveguides". Electron. Lett. 28 (20): 1879. doi:10.1049/el:19921203.
4. ^ J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, M. Haelterman (2004). "Simulations and Experiments on Self-focusing Conditions in Nematic Liquid-crystal Planar Cells". Opt. Express 12 (6): 1011–1018. Bibcode:2004OExpr..12.1011B. doi:10.1364/OPEX.12.001011. PMID 19474916. [1][2]
5. ^ ""Solitons in Telecommunications" in the book _Nonlinear Science_ (Chapter 3)".
6. ^ ""Making Waves: Solitons and Their Optical Applications" from SIAM News, Volume 31, Number 2".
7. ^ M. Gedalin, T.C. Scott, and Y.B. Band, "Optical Solitons in the Higher Order Nonlinear Schrödinger Equation", Phys. Rev. Lett. 78: 448-451 (1997) [3][4].
8. ^ S.Chakraborty, "Report of soliton pulse generation within 50 m length of SM fiber by high frequency induced nonlinear intelligent feedback method" , Proceedings, IEEE National Conference on Applications of Intelligent System, Sonepat, India, pp.91–94, 2008, ISBN 978-81-906531-0-7.[verification needed]
• Saleh, B. E. A.; Teich, M. C. (1991). Fundamentals of Photonics. New York: John Wiley & sons, inc. ISBN 0-471-83965-5
• Agrawal, Govind P. (1995). Nonlinear fiber optics (2nd ed.). San Diego (California): Academic Press. ISBN 0-12-045142-5
External links[edit] |
f44231001861ea34 | Take the tour ×
Can anybody give a list of the most fundamental assumptions of quantum mechanics in plain english ?
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what do you mean with assumptions ? – Stefano Borini Nov 10 '10 at 0:39
Just like we have a dimensionless point particle, the constant linear flow of time, and an immovable space fabric having straight lines in the classical Newtonian mechanics ? – mumtaz Nov 10 '10 at 0:45
Sadly, quantum mechanics is not written in plain English. – Mark Eichenlaub Nov 10 '10 at 1:57
as a quantum chemist would say, yes, it exists, but its projection in the english space has lower dimension, thus has a larger error due to the variational theorem. – Stefano Borini Nov 10 '10 at 8:39
Note that quantum mechanics is often taught in an axiomatic way, but it does not come from axioms: it comes from observations of how the world really works. – dmckee Apr 27 '12 at 17:16
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6 Answers
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Uncertain Principles: 7 essential elements of QM
Sorry but I can't be made plainer than that in English. The link has been provided by @user26143 , the copy by @DeerHunter .
Posted on: January 20, 2010 11:13 AM, by Chad Orzel ( who happens to be a USER here, on Physics.SE).
1) Particles are waves, and vice versa. Quantum physics tells us that every object in the universe has both particle-like and wave-like properties. It's not that everything is really waves, and just sometimes looks like particles, or that everything is made of particles that sometimes fool us into thinking they're waves. Every object in the universe is a new kind of object-- call it a "quantum particle" that has some characteristics of both particles and waves, but isn't really either.
Quantum particles behave like particles, in that they are discrete and (in principle) countable. Matter and energy come in discrete chunks, and whether you're trying to locate an atom or detect a photon of light, you will find it in one place, and one place only.
Quantum particles also behave like waves, in that they show effects like diffraction and interference. If you send a beam of electrons or a beam of photons through a narrow slit, they will spread out on the far side. If you send the beam at two closely spaced slits, they will produce a pattern of alternating bright and dark spots on the far side of the slits, as if they were water waves passing through both slits at once and interfering on the other side. This is true even though each individual particle is detected at a single location, as a particle.
3) Probability is all we ever know. When physicists use quantum mechanics to predict the results of an experiment, the only thing they can predict is the probability of detecting each of the possible outcomes. Given an experiment in which an electron will end up in one of two places, we can say that there is a 17% probability of finding it at point A and an 83% probability of finding it at point B, but we can never say for sure that a single given electron will definitely end up at A or definitely end up at B. No matter how careful we are to prepare each electron in exactly the same way, we can never say for definitiviely what the outcome of the experiment will be. Each new electron is a completely new experiment, and the final outcome is random.
4) Measurement determines reality. Until the moment that the exact state of a quantum particle is measured, that state is indeterminate, and in fact can be thought of as spread out over all the possible outcomes. After a measurement is made, the state of the particle is absolutely determined, and all subsequent measurements on that particle will return produce exactly the same outcome.
This seems impossible to believe-- it's the problem that inspired Erwin Schrödinger's (in)famous thought experiment regarding a cat that is both alive and dead-- but it is worth reiterating that this is absolutely confirmed by experiment. The double-slit experiment mentioned above can be thought of as confirmation of this indeterminacy-- until it is finally measured at a single position on the far side of the slits, an electron exists in a superposition of both possible paths. The interference pattern observed when many electrons are recorded one after another is a direct consequence of the superposition of multiple states.
The Quantum Zeno Effect is another example of the effects of quantum measurement: making repeated measurements of a quantum system can prevent it from changing its state. Between measurements, the system exists in a superposition of two possible states, with the probability of one increasing and the other decreasing. Each measurements puts the system back into a single definite state, and the evolution has to start over.
The effects of measurement can be interpreted in a number of different ways-- as the physical "collapse" of a wavefunction, as the splitting of the universe into many parallel worlds, etc.-- but the end result is the same in all of them. A quantum particle can and will occupy multiple states right up until the instant that it is measured; after the measurement it is in one and only one state.
5) Quantum correlations are non-local. One of the strangest and most important consequences of quantum mechanics is the idea of "entanglement." When two quantum particles interact in the right way, their states will depend on one another, no matter how far apart they are. You can hold one particle in Princeton and send the other to Paris, and measure them simultaneously, and the outcome of the measurement in Princeton will absolutely and unequivocally determine the outcome of the measurement in Paris, and vice versa.
The correlation between these states cannot possibly be described by any local theory, in which the particles have definite states. These states are indeterminate until the instant that one is measured, at which time the states of both are absolutely determined, no matter how far apart they are. This has been experimentally confirmed dozens of times over the last thirty years or so, with light and even atoms, and every new experiment has absolutely agreed with the quantum prediction.
It must be noted that this does not provide a means of sending signals faster than light-- a measurement in Paris will determine the state of a particle in Princeton, but the outcome of each measurement is completely random. There is no way to manipulate the Parisian particle to produce a specifc result in Princeton. The correlation between measurements will only be apparent after the fact, when the two sets of results are compared, and that process has to take place at speeds slower than that of light.
6) Everything not forbidden is mandatory. A quantum particle moving from point A to point B will take absolutely every possible path from A to B, at the same time. This includes paths that involve highly improbable events like electron-positron pairs appearing out of nowhere, and disappearing again. The full theory of quantum electro-dynamics (QED) involves contributions from every possible process, even the ridiculously unlikely ones.
It's worth emphasizing that this is not some speculative mumbo-jumbo with no real applicability. A QED prediction of the interaction between an electron and a magnetic field correctly describes the interaction to 14 decimal places. As weird as the idea seems, it is one of the best-tested theories in the history of science.
7) Quantum physics is not magic. [...] As strange as quantum physics is [...] it does not suspend all the rules of common sense. The bedrock principles of physics are still intact: energy is still conserved, entropy still increases, nothing can move faster than the speed of light. You cannot exploit quantum effects to build a perpetual motion machine, or to create telepathy or clairvoyance.
Quantum mechanics has lots of features that defy our classical intuition-- indeterminate states, probabilitistic measurements, non-local effects-- but it is still subject to the most important rule at all: If something sounds too good to be true, it probably is. Anybody trying to peddle a perpetual motion machine or a mystic cure using quantum buzzwords is deluded at best, or a scam artist at worst.
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Especially well told in point #5 is that the nonlocal effect is seen only when the measurements are brought together in one place. That detail is too often overlooked by writers trying to convey quantum mechanics to non-physicists. – DarenW Aug 1 '11 at 7:25
I cannot agree with Orzel, mainly because he is adding interpretation to the principles, which muddies the waters too much to be an acceptable answer. – joseph f. johnson Jan 14 '12 at 16:59
this is not a very clear list of underlying assumptions really.. this is a (popularized) list of interpretations and results of the current QM teachings. – Bjorn Wesen Oct 21 at 9:34
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In a rather concise manner, Shankar describes four postulates of nonrelativistic quantum mechanics.
I. The state of the particle is represented by a vector $|\Psi(t)\rangle$ in a Hilbert space.
II. The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $X$ and $P$ with the following matrix elements in the eigenbasis of $X$
$$\langle x|X|x'\rangle = x \delta(x-x')$$
$$\langle x|P|x' \rangle = -i\hbar \delta^{'}(x-x')$$
The operators corresponding to dependent variable $\omega(x,p)$ are given Hermitian operators
$$\Omega(X,P)=\omega(x\rightarrow X,p \rightarrow P)$$
III. If the particle is in a state $|\Psi\rangle$, measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $P(\omega)\propto |\langle \omega|\Psi \rangle|^{2}$. The state of the system will change from $|\Psi \rangle$ to $|\omega \rangle$ as a result of the measurement.
IV. The state vector $|\Psi(t) \rangle$ obeys the Schrödinger equation
$$i\hbar \frac{d}{dt}|\Psi(t)\rangle=H|\Psi(t)\rangle$$
where $H(X,P)=\mathscr{H}(x\rightarrow X, p\rightarrow P)$ is the quantum Hamiltonian operator and $\mathscr{H}$ is the Hamiltonian for the corresponding classical problem.
After that, Shankar discusses the postulates and the differences between quantum mechanics and classical mechanics, hopefully in plain english. Probably you want to take a look at that book: Principles of Quantum Mechanics
There are, of course, other sets of equivalent postulates.
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+1, I was thinking of the same thing, I just couldn't remember the list of assumptions in detail. – David Z Nov 10 '10 at 2:26
Ah, the first mention of Shankar I've seen! – Mark C Nov 10 '10 at 6:04
Thanks for the conciseness and reference to Shankar's book. I will definitely have a go at it. – mumtaz Nov 10 '10 at 8:27
This is hardly in plain english ;) – Stefano Borini Nov 10 '10 at 8:28
@Stefano ...yeah its not , thats why i up-voted it but could not mark it as the answer ;) – mumtaz Nov 10 '10 at 8:43
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Here are some ways of looking at it for an absolute beginner. It is only a start. It is not the way you will finally look at it, but there is nothing actually incorrect here.
First, plain English just won't do, you are going to need some math. Do you know anything about complex numbers? If not, learn a little more before you begin.
Second, in quantum mechanics, we don't work with probabilities, you know, numbers between 0 and 1 that represent the likelihood of something happening. Instead we work with the square root of probabilities, called "probability amplitudes", which are complex numbers. When we are done, we convert these back to probabilities. This is a fundamental change; in a way the complex number amplitudes are just as important as the probabilities. The logic is therefore different from what you are used to.
Third, in quantum mechanics, we don't imagine that we know all about objects in between measurements, rather we know that we don't. Therefore we only focus on what we can actually measure, and possible results of a particular measurement. This is because some of the assumed imagined properties of objects in between measurements contradict one another, but we can get a theory where we can always work with the measurements themselves in a consistent way. Any measurement (meter reading, etc.) or property in between measurements has only a probability amplitude of yielding a given result, and the property only co-arises with the measurement itself.
Fourth, if we have several mutually exclusive possibilities, the probability amplitude of either one or the other happening is just the sum of the probability amplitudes of each happening. Note that this does not mean that the probabilities themselves add, the way we are used to classically. This is new and different.
If we add up the probabilities (not the amplitudes this time but their "squares") of all possible mutually exclusive measurements, gotten by "squaring" the probability amplitudes of each mutually exclusive possibilities, we always get 1 as usual. This is called unitarity.
This is not the whole picture, but it should help somewhat. As a reference, try looking at three videos of some lectures that Feynman once gave at Cornell. That should help more.
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thanks for the link to vids – mumtaz Nov 10 '10 at 8:52
The probability amplitudes are not quite « square roots » of the probabilities, they differ from the square root by a phase factor which is important. Perhaps a parenthetical remark should be put into the third paragraph to this effect. – joseph f. johnson Jan 14 '12 at 17:07
@josephf.johnson: One can declare them to be signed square roots if one implements "operator i" and doubles the size of the Hilbert space. This is a clearer point of view for those who do not like complex numbers sneaking into physics without a physical argument. – Ron Maimon Apr 27 '12 at 18:22
The suggestion you sketch here will indeed work, but doesn't change the issue I brought up, of the phase factors. Relative phase factors are physical, whether you use the formalism of complex numbers or use an alternative real space with double the dimension (this procedure is called base change). – joseph f. johnson Apr 28 '12 at 15:36
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I don't really understand exactly what you mean with assumptions, but I guess you ask about the physical nature and behavior of the quantistic world. I am going to be very inaccurate in the following writing, but I need some space to get things understood. What you are going to read is a harsh simplification.
The first assumption is the fact that particles at the quantistic level don't have a position, like real-life objects, they don't even have a speed. Well, they sort of have it, but in reality they are "fuzzy". Fuzzy in the sense that you cannot really assign a position to them, or a speed.
To be fair, you perfectly know a real-world case scenario where you have to give up the concepts of speed and position for an entity. Take a guitar, and pluck a string. Where is the "vibration" ? Can you point at one specific point in the string where the "vibration" is, or how fast this vibration is traveling. I guess you would say that the vibration is "on the string" or "in the space between one end of the string, and the other hand". However, there are parts of the string that are vibrating more, and parts that are not vibrating at all, so you could rightfully say that a larger part of the "vibration" is in the center of the string, a smaller part is on the lateral parts close to the ends, and no vibration is at the very ends. In some sense, you will experience higher "presence" of the vibration in the center.
With electrons, it's exactly the same. Think "electron" as "vibration" in the previous example. Soon, you will realize that you cannot really claim anything about the position of an electron, for the fact that has wave nature. You are forced to see the electron not as a charged ball rolling here and there, but as a diffuse entity totally similar to the concept of "vibration". As a result, you cannot say anything about the position of an electron, but you can claim that there are zones in the space where the electron has "higher presence" and zones that have "smaller presence". These zones are probabilistic in nature, and this probability is directly obtained by a mathematical description called the wavefunction.
The wavefunction mainly depends on the external potential, meaning that the presence of charges, such as protons and other electrons, will affect the perceived potential and will have an effect on the probability distribution in space of an electron.
A second assumption is the mapping between physical quantities (such as energy, or momentum) and quantistic operators. Take for example the momentum of an everyday object: it is given by its mass times its speed. In the quantistic world, you don't really have the concept of position, hence you don't have the concept of speed, hence you have to reinterpret the whole thing in terms of probability (rememeber the string ?), which means that what you have about a quantistic object is its wavefunction, and you have to extract the information about the momentum from this wavefunction. How do you do it ?
Well, there's a dogma in quantum mechanics, that if you apply a magic "operator" to the wavefunction, it gives you the quantity you want. There are (simple) rules to generate these operators, and you can apply them to a wavefunction to query the momentum, or the total energy, the kinetic energy and so on.
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so the first thing we notice down there is a clear shift from a deterministic to a probabilistic notion of existence ? – mumtaz Nov 10 '10 at 0:57
I would not say "probabilistic existence". It's not like if you say "the electron has, say, 40% probability of being here" does not mean that it spends its time 40% here and 60% somewhere else. I think the plucked string is the most appropriate real-world example. It's not like in the middle of the string there's all the vibration 40 % of the time, and 60% is somewhere else. The vibration is just 40 % there and 60 % on the rest of the string. – Stefano Borini Nov 10 '10 at 1:05
so we dont have real dimensionless point particles then. These are extended in space , right ? ...but wait are we talking about the same space that we have an intuitive notion of from our empirical experience or is the space down there also funky ? – mumtaz Nov 10 '10 at 1:21
@mumtaz:The electron is technically handled as a dimensionless point, but it's irrelevant. What you work on is the coordinate of an electron, but what you have is a coordinate of space where an electron's wavefunction has a given value. when you say wavefunction $\psi(x)$, your $x$ is a dimensionless point, but it's not really like you have a infinitesimal particle stuck in point $x$. – Stefano Borini Nov 10 '10 at 8:36
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Dirac seems to have thought that the most fundamental assumption of Quantum Mechanics is the Principle of Superposition I read somewhere, perhaps an obituary, that he began every year his lecture course on QM by taking a piece of chalk, placing it on the table, and saying, now I paraphrase:
One possible state of the piece of chalk is that it is here. Another possible state is that it is over there (pointing to another table). Now according to Quantum Mechanics, there is another possible state of the chalk in which it is partly here, and partly there.
In his book he explains more about what this « partly » means: it means that the properties of a piece of chalk that is partly in state 1 (here) and partly in state 2 (there), are in between the properties of state 1 and state 2.
EDIT: I find I have compiled the basic explanations given by Dirac in the first edition of his book. Unfortunately, it is relativistic, but I am not going to re-type it all.
« we must consider the photons as being controlled by waves, in some way which cannot be understood from the point of view of ordinary mechanics. This intimate connexion between waves and particles is of very great generality in the new quantum mechanics. ...
« The waves and particles should be regarded as two abstractions which are useful for describing the same physical reality. One must not picture this reality as containing both the waves and particles together and try to construct a mechanism, acting according to classical laws, which shall correctly describe their connexion and account for the motion of the particles.
« Corresponding to the case of the photon, which we say is in a given state of polarizationn when it has been passed through suitable polarizing apparatus, we say that any atomic system is in a given state when it has been prepared in a given way, which may be repeated arbitrarily at will. The method of preparation may then be taken as the specification of the state. The state of a system in the general case then includes any information that may be known about its position in space from the way in which it was prepared, as well as any information about its internal condition.
« We must now imagine the states of any system to be related in such a way that whenever the system is definitely in one state, we can equally well consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannnot be conceived on classical ideas.
« When a state is formed by the superposition of two other states, it will have properties that are in a certain way intermediate between those of the two original states and that approach more or less closely to those of either of them according to the greater or less `weight' attached to this state in the superposition process.
« We must regard the state of a system as referring to its condition throughout an indefinite period of time and not to its condition at a particular time, ... A system, when once prepared in a given state, remains in that state so long as it remains undisturbed.....It is sometimes purely a matter of convenience whether we are to regard a system as being disturbed by a certain outside influence, so that its state gets changed, or whether we are to regard the outside influence as forming a part of and coming in the definition of the system, so that with the inclusion of the effects of this influence it is still merely running through its course in one particular state. There are, however, two cases when we are in general obliged to consider the disturbance as causing a change in state of the system, namely, when the disturbance is an observation and when it consists in preparing the system so as to be in a given state.
« With the new space-time meaning of a state we need a corresponding space-time meaning of an observation. [, Dirac has not actually mentioned anything about `observation' up to this point...]
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In plain English: Quantum physics is the realization that probabilities are Pythagorean quantities that evolve linearly.
That's it. All the rest is detail.
See slides 5-7 in Scott Aaronson's presentation: here.
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9f82dbf9ba7f8878 | De Broglie–Bohm theory
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The de Broglie–Bohm theory is explicitly nonlocal: The velocity of any one particle depends on the value of the guiding equation, which depends on the whole configuration of the universe. Because the known laws of physics are all local, and because nonlocal interactions combined with relativity lead to causal paradoxes, many physicists find this unacceptable.
This theory is deterministic. Most (but not all) variants of this theory that support special relativity require a preferred frame. Variants which include spin and curved spaces are known. It can be modified to include quantum field theory. Bell's theorem was inspired by Bell's discovery of the work of David Bohm and his subsequent wondering if the obvious nonlocality of the theory could be eliminated.
De Broglie–Bohm theory is based on the following postulates:
• There is a configuration q of the universe, described by coordinates q^k, which is an element of the configuration space Q. The configuration space is different for different versions of pilot wave theory. For example, this may be the space of positions \mathbf{Q}_k of N particles, or, in case of field theory, the space of field configurations \phi(x). The configuration evolves (for spin=0) according to the guiding equation
m_k\frac{d q^k}{dt} (t) = \hbar \nabla_k \operatorname{Im} \ln \psi(q,t) = \hbar \operatorname{Im}\left(\frac{\nabla_k \psi}{\psi} \right) (q, t) = \frac{m_k \bold{j}_k}{\psi^*\psi} = \mathrm{Re}\left ( \frac{\bold{\hat{P}}_k\Psi}{\Psi} \right ) .
Where \bold{j} is the probability current or probability flux and \bold{\hat{P}} is the momentum operator. Here, \psi(q,t) is the standard complex-valued wavefunction known from quantum theory, which evolves according to Schrödinger's equation
• The configuration is distributed according to |\psi(q,t)|^2 at some moment of time t, and this consequently holds for all times. Such a state is named quantum equilibrium. With quantum equilibrium, this theory agrees with the results of standard quantum mechanics.
Two-slit experiment[edit]
The Bohmian trajectories for an electron going through the two-slit experiment. A similar pattern was also extrapolated from weak measurements of single photons.[2]
In de Broglie–Bohm theory, the wavefunction travels through both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wave function interferes with itself and guides the particles in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.
The Theory[edit]
The ontology[edit]
While the ontology of classical mechanics is part of the ontology of de Broglie–Bohm theory, the dynamics are very different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the velocities of the particles are given by the wavefunction, which exists in a 3N-dimensional configuration space, where N corresponds to the number of particles in the system;[3] Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction.[4] Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie-Bohm theory, not localized at the position of the particle.[5][6]
The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrodinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles".[7] P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory".[8] It should be noted however that Holland has later called this a merely apparent lack of back reaction, due to the incompleteness of the description.[9]
Extensions to this theory include spin and more complicated configuration spaces.
We use variations of \mathbf{Q} for particle positions while \psi represents the complex-valued wavefunction on configuration space.
Guiding equation[edit]
For a spinless single particle moving in \mathbb{R}^3, the particle's velocity is given
Schrödinger's equation[edit]
For many particles, the equation is the same except that \psi and V are now on configuration space, \mathbb{R}^{3N}.
This is the same wavefunction of conventional quantum mechanics.
Relation to the Born Rule[edit]
In Bohm's original papers [Bohm 1952], he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by |\psi|^2. And that distribution is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies |\psi|^2.
For a given experiment, we can postulate this as being true and verify experimentally that it does indeed hold true, as it does. But, as argued in Dürr et al.,[10] one needs to argue that this distribution for subsystems is typical. They argue that |\psi|^2 by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for initial conditions of the positions of the particles. They then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e., |\psi|^2) for measurement outcomes. In summary, in a universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical.
It is in that qualified sense that the Born rule is, for the de Broglie–Bohm theory, a theorem rather than (as in ordinary quantum theory) an additional postulate.
It can also be shown that a distribution of particles that is not distributed according to the Born rule (that is, a distribution 'out of quantum equilibrium') and evolving under the de Broglie-Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as |\psi|^2. See, for example Ref. .[11] A video of the electron density in a 2D box evolving under this process is available here.
The conditional wave function of a subsystem[edit]
In the formulation of the De Broglie–Bohm theory, there is only a wave function for the entire universe (which always evolves by the Schrödinger equation). However, once the theory is formulated, it is convenient to introduce a notion of wave function also for subsystems of the universe. Let us write the wave function of the universe as \psi(t,q^{\mathrm I},q^{\mathrm{II}}), where q^{\mathrm I} denotes the configuration variables associated to some subsystem (I) of the universe and q^{\mathrm{II}} denotes the remaining configuration variables. Denote, respectively, by Q^{\mathrm I}(t) and by Q^{\mathrm{II}}(t) the actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The conditional wave function of subsystem (I) is defined by:
It follows immediately from the fact that Q(t)=(Q^{\mathrm I}(t),Q^{\mathrm{II}}(t)) satisfies the guiding equation that also the configuration Q^{\mathrm I}(t) satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wave function \psi replaced with the conditional wave function \psi^{\mathrm I}. Also, the fact that Q(t) is random with probability density given by the square modulus of \psi(t,\cdot) implies that the conditional probability density of Q^{\mathrm I}(t) given Q^{\mathrm{II}}(t) is given by the square modulus of the (normalized) conditional wave function \psi^{\mathrm I}(t,\cdot) (in the terminology of Dürr et al.[12] this fact is called the fundamental conditional probability formula).
then the conditional wave function of subsystem (I) is (up to an irrelevant scalar factor) equal to \psi^{\mathrm I} (this is what Standard Quantum Theory would regard as the wave function of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction term between subsystems (I) and (II) then \psi^{\mathrm I} does satisfy a Schrödinger equation. More generally, assume that the universal wave function \psi can be written in the form:
where \phi solves Schrödinger equation and \phi(t,q^{\mathrm I},Q^{\mathrm{II}}(t))=0 for all t and q^{\mathrm I}. Then, again, the conditional wave function of subsystem (I) is (up to an irrelevant scalar factor) equal to \psi^{\mathrm I} and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), \psi^{\mathrm I} satisfies a Schrödinger equation.
To incorporate spin, the wavefunction becomes complex-vector valued. The value space is called spin space; for a spin-½ particle, spin space can be taken to be \mathbb{C}^2. The guiding equation is modified by taking inner products in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a Pauli spin term.
i\hbar\frac{\partial}{\partial t}\psi = \left(-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}D_k^2 + V - \sum_{k=1}^{N} \mu_k \mathbf{S}_{k}/{S}_{k} \cdot \mathbf{B}(\mathbf{q}_k) \right) \psi
where \mu_k is the magnetic moment of the kth particle, \mathbf{S}_{k} is the appropriate spin operator acting in the kth particle's spin space, {S}_{k} is spin of the particle ({S}_{k} = 1/2 for electron),
\mathbf{B} and \mathbf{A} are, respectively, the magnetic field and the vector potential in \mathbb{R}^{3} (all other functions are fully on configuration space), e_k is the charge of the kth particle, and (\cdot,\cdot) is the inner product in spin space \mathbb{C}^d,
That is, its spin space is a 12 dimensional space.
Curved space[edit]
Quantum field theory[edit]
In Dürr et al.,[14][15] the authors describe an extension of de Broglie–Bohm theory for handling creation and annihilation operators, which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles. But under a stochastic process, particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space.
Exploiting nonlocality[edit]
There has been work in developing relativistic versions of de Broglie–Bohm theory. See Bohm and Hiley: The Undivided Universe, and [3], [4], and references therein. Another approach is given in the work of Dürr et al.[18] in which they use Bohm-Dirac models and a Lorentz-invariant foliation of space-time.
Initially, it had been considered impossible to set out a description of photon trajectories in the de Broglie–Bohm theory in view of the difficulties of describing bosons relativistically.[19] In 1996, Partha Ghose had presented a relativistic quantum mechanical description of spin-0 and spin-1 bosons starting from the Duffin–Kemmer–Petiau equation, setting out Bohmian trajectories for massive bosons and for massless bosons (and therefore photons).[19] In 2001, Jean-Pierre Vigier emphasized the importance of deriving a well-defined description of light in terms of particle trajectories in the framework of either the Bohmian mechanics or the Nelson stochastic mechanics.[20] The same year, Ghose worked out Bohmian photon trajectories for specific cases.[21] Subsequent weak measurement experiments yielded trajectories which coincide with the predicted trajectories.[22][23]
Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions.[24] He has developed a generalized relativistic-invariant probabilistic interpretation of quantum theory,[16][25][26] in which |\psi|^2 is no longer a probability density in space, but a probability density in space-time. He uses this generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theory without introducing a preferred foliation of space-time. His work also covers the extension of the Bohmian interpretation to a quantization of fields and strings.[27]
The basis for agreement with standard quantum mechanics is that the particles are distributed according to |\psi|^2. This is a statement of observer ignorance, but it can be proven[10] that for a universe governed by this theory, this will typically be the case. There is apparent collapse of the wave function governing subsystems of the universe, but there is no collapse of the universal wavefunction.
Measuring spin and polarization[edit]
Measurements, the quantum formalism, and observer independence[edit]
Collapse of the wavefunction[edit]
Operators as observables[edit]
There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to |\psi|^2 and no contradiction to experimental results is possible to detect.
Hidden variables[edit]
Bohm and Hiley have stated that they found their own choice of terms of an "interpretation in terms of hidden variables" to be too restrictive. In particular, a particle is not actually hidden but rather "is what is most directly manifested in an observation", even if position and momentum of a particle cannot be observed with arbitrary precision.[33] Put in simpler words, the particles postulated by the de Broglie–Bohm theory are anything but "hidden" variables: they are what the objects we see in everyday experience are made of; it is the wavefunction itself which is "hidden" in the sense of being invisible and not-directly-observable.
Heisenberg's uncertainty principle[edit]
The Heisenberg uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of \Delta x, and the momentum with an accuracy of \Delta p, then \Delta x\Delta p\gtrsim h. If we make further measurements in order to get more information, we disturb the system and change the trajectory into a new one depending on the measurement setup; therefore, the measurement results are still subject to Heisenberg's uncertainty relation.
Quantum entanglement, Einstein-Podolsky-Rosen paradox, Bell's theorem, and nonlocality[edit]
De Broglie–Bohm theory highlighted the issue of nonlocality: it inspired John Stewart Bell to prove his now-famous theorem,[34] which in turn led to the Bell test experiments.
In the Einstein–Podolsky–Rosen paradox, the authors describe a thought-experiment one could perform on a pair of particles that have interacted, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.[35]
Classical limit[edit]
Quantum trajectory method[edit]
Occam's razor criticism[edit]
Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that \psi is a real field then the associated particle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.[40]
In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the luminiferous ether, which was found to be unnecessary in special relativity. This argument of Everett's is sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of Occam's razor.[41]
It is usually overlooked that Bohm's theory contains the same "many worlds" of dynamically separate branches as the Everett interpretation (now regarded as "empty" wave components), since it is based on precisely the same . . . global wave function . . .[42]
David Deutsch has expressed the same point more "acerbically":[39]
pilot-wave theories are parallel-universe theories in a state of chronic denial.[43]
According to Brown & Wallace[39] the de Broglie-Bohm particles play no role in the solution of the measurement problem. These authors claim[39] that the "result assumption" (see above) is inconsistent with the view that there is no measurement problem in the predictable outcome (i.e. single-outcome) case. These authors also claim[39] that a standard tacit assumption of the de Broglie-Bohm theory (that an observer becomes aware of configurations of particles of ordinary objects by means of correlations between such configurations and the configuration of the particles in the observer's brain) is unreasonable. This conclusion has been challenged by Valentini[44] who argues that the entirety of such objections arises from a failure to interpret de Broglie-Bohm theory on its own terms.
De Broglie–Bohm theory has been derived many times and in many ways. Below are six derivations all of which are very different and lead to different ways of understanding and extending this theory.
Notice that this derivation does not use Schrödinger's equation.
• A fourth derivation was given by Dürr et al.[10] In their derivation, they derive the velocity field by demanding the appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, once the wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.
2. The wave is described mathematically by a solution \psi to Schrödinger's wave equation;
The fourth postulate is subsidiary yet consistent with the first three:
4. The probability \rho (\mathbf{x}(t)) to find the particle in the differential volume d^3 x at time t equals |\psi(\mathbf{x}(t))|^2.
Pilot-wave theory[edit]
Dr. de Broglie presented his pilot wave theory at the 1927 Solvay Conference,[47] after close collaboration with Schrödinger, who developed his wave equation for de Broglie's theory. At the end of the presentation, Wolfgang Pauli pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the case of inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particular technique could not be generalized for Pauli's purpose, although the audience might have been lost in the technical details and de Broglie's mild mannerism left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless in 1932 due to both the Copenhagen school's more successful P.R. efforts and his own inability to understand quantum decoherence. Also in 1932, John von Neumann published a paper,[48] that was widely (and erroneously[49]) believed to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie's theory for the next two decades.
Around this time Erwin Madelung also developed a hydrodynamic version of Schrödinger's equation which is incorrectly considered as a basis for the density current derivation of the de Broglie–Bohm theory.[52] The Madelung equations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohm mechanics[53] and are the basis of the hydrodynamic interpretation of quantum mechanics.
Peter R. Holland has pointed out that, earlier in 1927, Einstein had actually submitted a preprint with a similar proposal but, not convinced, had withdrawn it before publication.[54] According to Holland, failure to appreciate key points of the de Broglie–Bohm theory has led to confusion, the key point being "that the trajectories of a many-body quantum system are correlated not because the particles exert a direct force on one another (à la Coulomb) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them."[55] This entity is the quantum potential.
This stage applies to multiple particles, and is deterministic.
Bohm's paper was largely ignored or panned by other physicists. Albert Einstein did not consider it a satisfactory answer to the quantum nonlocality question, calling it "too cheap".,[56] and Werner Heisenberg considered it a "superfluous 'ideological superstructure' ".[57] Wolfgang Pauli, who had been unconvinced by de Broglie in 1927, conceded to Bohm as follows:
I just received your long letter of 20th November, and I also have studied more thoroughly the details of your paper. I do not see any longer the possibility of any logical contradiction as long as your results agree completely with those of the usual wave mechanics and as long as no means is given to measure the values of your hidden parameters both in the measuring apparatus and in the observe [sic] system. As far as the whole matter stands now, your ‘extra wave-mechanical predictions’ are still a check, which cannot be cashed.[58]
When Bohm's theory was presented at the Institute for Advanced Study in Princeton, many of the objections were ad hominem, focusing on Bohm's sympathy with communists as exemplified by his refusal to give testimony to the House Un-American Activities Committee.[citation needed]
Eventually John Bell began to defend the theory. In "Speakable and Unspeakable in Quantum Mechanics" [Bell 1987], several of the papers refer to hidden variables theories (which include Bohm's).
Bohmian mechanics[edit]
Causal interpretation and ontological interpretation[edit]
An in-depth analysis of possible interpretations of Bohm's model of 1952 was given in 1996 by philosopher of science Arthur Fine.[59]
See also[edit]
3. ^ David Bohm (1957). Causality and Chance in Modern Physics. Routledge & Kegan Paul and D. Van Nostrand. ISBN 0-8122-1002-6. , p. 117.
4. ^ D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory, p. 37.
5. ^ H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", Foundations of Physics, 1995, Volume 25, Number 2, pp. 329-347.
6. ^ J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", Foundations of Physics, March 1999, Volume 29, Issue 3, pp 333-348.
8. ^ Peter R. Holland: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, Cambridge (first published 25 June 1993), ISBN 0-521-35404-8 hardback, ISBN 0-521-48543-6 paperback, transferred to digital printing 2004, Chapter I. section (7) "There is no reciprocal action of the particle on the wave", p. 26
19. ^ a b Partha Ghose: Relativistic quantum mechanics of spin-0 and spin-1 bosons, Foundations of Physics, vol. 26, no. 11, pp. 1441-1455, 1996, doi:10.1007/BF02272366
20. ^ Nicola Cufaro Petroni, Jean-Pierre Vigier: Remarks on Observed Superluminal Light Propagation, Foundations of Physics Letters, vol. 14, no. 4, pp. 395-400, doi:10.1023/A:1012321402475, therein: section 3. Conclusions, page 399
22. ^ Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg: Observing the trajectories of a single photon using weak measurement, 19th Australian Institute of Physics (AIP) Congress, 2010 [1]
24. ^ Hrvoje Nikolić: Relativistic Quantum Mechanics and the Bohmian Interpretation, Foundations of Physics Letters, vol. 18, no. 6, November 2005, pp. 549-561, doi:10.1007/s10702-005-1128-1
28. ^ a b Bell, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0521334950.
35. ^ Einstein; Podolsky; Rosen (1935). "Can Quantum Mechanical Description of Physical Reality Be Considered Complete?". Phys. Rev. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
36. ^ Bell, page 115
37. ^ Maudlin, T. (1994). Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics. Cambridge, MA: Blackwell. ISBN 0631186093.
41. ^ Craig Callender, "The Redundancy Argument Against Bohmian Mechanics"
42. ^ Daniel Dennett (2000). With a little help from my friends. In D. Ross, A. Brook, and D. Thompson (Eds.), Dennett's Philosophy: a comprehensive assessment. MIT Press/Bradford, ISBN 0-262-68117-X.
45. ^ P. Holland, "Hamiltonian Theory of Wave and Particle in Quantum Mechanics I, II", Nuovo Cimento B 116, 1043, 1143 (2001) online
48. ^ von Neumann J. 1932 Mathematische Grundlagen der Quantenmechanik
52. ^ Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Zeit. f. Phys. 40 (3–4): 322–326. Bibcode:1927ZPhy...40..322M. doi:10.1007/BF01400372.
54. ^ Peter Holland: What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics?, Foundations of Physics (2004), vol. 35, no. 2, p. 177–196, doi:10.1007/s10701-004-1940-7, arXiv: quant-ph/0401017, p. 1
56. ^ (Letter of 12 May 1952 from Einstein to Max Born, in The Born–Einstein Letters, Macmillan, 1971, p. 192.
57. ^ Werner Heisenberg, Physics and Philosophy (1958), p. 133.
58. ^ Pauli to Bohm, 3 December 1951, in Wolfgang Pauli, Scientific Correspondence, Vol IV – Part I, [ed. by Karl von Meyenn], (Berlin, 1996), pp. 436-441.
59. ^ A. Fine: On the interpretation of Bohmian mechanics, in: J. T. Cushing, A. Fine, S. Goldstein (Eds.): Bohmian mechanics and quantum theory: an appraisal, Springer, 1996, pp. 231−250
Further reading[edit]
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Quantum mechanics
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, or quantum theory) is a fundamental branch of physics which deals with physical phenomena at nanoscopic scales where the action is on the order of the Planck constant. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. Quantum mechanics provides a substantially useful framework for many features of the modern periodic table of elements including the behavior of atoms during chemical bonding and has played a significant role in the development of many modern technologies.
In advanced topics of quantum mechanics, some of these behaviors are macroscopic (see macroscopic quantum phenomena) and emerge at only extreme (i.e., very low or very high) energies or temperatures (such as in the use of superconducting magnets). In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects.
The mathematical formulations of quantum mechanics are abstract. A mathematical function, the wavefunction, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve bra–ket notation which requires an understanding of complex numbers and linear functionals. The wavefunction formulation treats the particle as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance. Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics. For instance, in a quantum mechanical model the lowest energy state of a system, the ground state, is non-zero as opposed to a more "traditional" ground state with zero kinetic energy (all particles at rest). Instead of a traditional static, unchanging zero energy state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler.
The earliest versions of quantum mechanics were formulated in the first decade of the 20th century. About this time, the atomic theory and the corpuscular theory of light (as updated by Einstein)[1] first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Max Born and Pascual Jordan, (matrix mechanics); Louis de Broglie and Erwin Schrödinger (wave mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles). Moreover, the Copenhagen interpretation of Niels Bohr became widely accepted. By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann[2] with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer. Quantum mechanics has since permeated throughout many aspects of 20th-century physics and other disciplines including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Much 19th-century physics has been re-evaluated as the "classical limit" of quantum mechanics and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories.
The name quantum mechanics derives from the observation that some physical quantities can change only in discrete amounts (Latin quanta), and not in a continuous (cf. analog) way.
• History 1
• Mathematical formulations 2
• Mathematically equivalent formulations of quantum mechanics 3
• Interactions with other scientific theories 4
• Quantum mechanics and classical physics 4.1
• Relativity and quantum mechanics 4.2
• Attempts at a unified field theory 4.3
• Philosophical implications 5
• Applications 6
• Examples 7
• See also 8
• Notes 9
• References 10
• Further reading 11
• External links 12
Scientific inquiry into the wave nature of light began in the 17th and 18th centuries when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.[3] In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he later described in a paper entitled "On the nature of light and colours". This experiment played a major role in the general acceptance of the wave theory of light.
In 1838, with the discovery of cathode rays by Michael Faraday, these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.[4] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or "energy elements") precisely matched the observed patterns of black-body radiation.
In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation,[5] known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it was valid only at high frequencies, and underestimated the radiance at low frequencies. Later, Max Planck corrected this model using Boltzmann statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics.
Among the first to study quantum phenomena in nature were Arthur Compton, C.V. Raman, and Pieter Zeeman, each of whom has a quantum effect named after him. Robert A. Millikan studied the Photoelectric effect experimentally and Albert Einstein developed a theory for it. At the same time Niels Bohr developed his theory of the atomic structure which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld.[6] This phase is known as Old quantum theory.
According to Planck, each energy element, E, is proportional to its frequency, ν:
E = h \nu\
Max Planck is considered the father of the Quantum Theory
where h is Planck's constant. Planck (cautiously) insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.[7] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[8] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect in which shining light on certain materials can eject electrons from the material.
The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others. In the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the "Old Quantum Theory". Out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.
The other exemplar that led to quantum mechanics was the study of electromagnetic waves, such as visible and ultraviolet light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or "quanta", Albert Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon) with a discrete quantum of energy that was dependent on its frequency.[9] Einstein was able to use the photon theory of light to explain the photoelectric effect for which he won the 1921 Nobel Prize in Physics. This led to a theory of unity between subatomic particles and electromagnetic waves in which particles and waves are neither simply particle nor wave but have certain properties of each. This originated the concept of wave–particle duality.
While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain recently investigated
• PHYS 201: Fundamentals of Physics II by Ramamurti Shankar, Open Yale Course
• Lectures on Quantum Mechanics by Leonard Susskind
• Everything you wanted to know about the quantum world — archive of articles from New Scientist.
• Quantum Physics Research from Science Daily
• Overbye, Dennis (December 27, 2005). "Quantum Trickery: Testing Einstein's Strangest Theory". The New York Times. Retrieved April 12, 2010.
• Audio: Astronomy Cast Quantum Mechanics — June 2009. Fraser Cain interviews Pamela L. Gay.
• Many-worlds or relative-state interpretation.
• Measurement in Quantum mechanics.
• Quantum Physics Database - Fundamentals and Historical Background of Quantum Theory.
• Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics).
• MIT OpenCourseWare: Chemistry.
• MIT OpenCourseWare: Physics. See 8.04
• Stanford Continuing Education PHY 25: Quantum Mechanics by Leonard Susskind, see course description Fall 2007
• 5½ Examples in Quantum Mechanics
• Imperial College Quantum Mechanics Course.
• Spark Notes - Quantum Physics.
• Quantum Physics Online : interactive introduction to quantum mechanics (RS applets).
• Experiments to the foundations of quantum physics with single photons.
• AQME : Advancing Quantum Mechanics for Engineers — by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on nanohub
• Quantum Mechanics by Martin Plenio
• Quantum Mechanics by Richard Fitzpatrick
• Quantum TransportOnline course on
Course material
• 3D animations, applications and research for basic quantum effects (animations also available in (Université paris Sud))
• Quantum Cook Book by R. Shankar, Open Yale PHYS 201 material (4pp)
• The Modern Revolution in Physics - an online textbook.
• J. O'Connor and E. F. Robertson: A history of quantum mechanics.
• Introduction to Quantum Theory at Quantiki.
• Quantum Physics Made Relatively Simple: three video lectures by Hans Bethe
• H is for h-bar.
• Quantum Mechanics Books Collection: Collection of free books
External links
• Bernstein, Jeremy (2009). Quantum Leaps. Cambridge, Massachusetts: Belknap Press of Harvard University Press.
• Eisberg, Robert;
• Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc.
• Shankar, R. (1994). Principles of Quantum Mechanics. Springer.
• Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press.
Further reading
More technical:
• The beginning chapters make up a very clear and comprehensible introduction.
• Hugh Everett, 1957, "Relative State Formulation of Quantum Mechanics", Reviews of Modern Physics 29: 454-62.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. A standard undergraduate text.
• Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications.
1. ^ Ben-Menahem, Ari (2009). Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1. Springer. p. 3678. , Extract of page 3678
2. ^ van Hove, Leon (1958). "Von Neumann's contributions to quantum mechanics" (PDF). Bulletin of the American Mathematical Society 64: Part2:95–99.
5. ^ Kragh, Helge (2002). Quantum Generations: A History of Physics in the Twentieth Century. Princeton University Press. p. 58. , Extract of page 58
6. ^ E Arunan (2010). "Peter Debye". Resonance (journal) (Indian Academy of Sciences) 15 (12).
7. ^
9. ^ Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" [On a heuristic point of view concerning the production and transformation of light]. Reprinted in The collected papers of Albert Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also Einstein's early work on the quantum hypothesis, ibid. pp. 134-148.
10. ^ "Quantum interference of large organic molecules". Retrieved April 20, 2013.
11. ^ "Quantum - Definition and More from the Free Merriam-Webster Dictionary". Retrieved 2012-08-18.
12. ^
13. ^ Compare the list of conferences presented here
14. ^ at the Wayback Machine (archived October 26, 2009)
16. ^ D. Hilbert Lectures on Quantum Theory, 1915–1927
19. ^ Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics Symmetries, Second edition. Springer-Verlag. p. 52. , Chapter 1, p. 52
20. ^ "Heisenberg - Quantum Mechanics, 1925–1927: The Uncertainty Relations". Retrieved 2012-08-18.
21. ^ a b Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second edition. Jones and Bartlett Publishers, Inc. p. 215. , Chapter 8, p. 215
22. ^ "[Abstract] Visualization of Uncertain Particle Movement". Retrieved 2012-08-18.
23. ^ Hirshleifer, Jack (2001). The Dark Side of the Force: Economic Foundations of Conflict Theory. Campbridge University Press. p. 265. , Chapter , p.
24. ^
28. ^ Michael Trott. "Time-Evolution of a Wavepacket in a Square Well — Wolfram Demonstrations Project". Retrieved 2010-10-15.
30. ^ Mathews, Piravonu Mathews; Venkatesan, K. (1976). A Textbook of Quantum Mechanics. Tata McGraw-Hill. p. 36. , Chapter 2, p. 36
32. ^ [1]
33. ^ Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124-8 and 285-6.
34. ^
36. ^ Carl M. Bender, Daniel W. Hook, Karta Kooner (2009-12-31). "Complex Elliptic Pendulum". arXiv:1001.0131 [hep-th].
41. ^ "Between classical and quantum�" (PDF). Retrieved 2012-08-19.
43. ^ "Atomic Properties". Retrieved 2012-08-18.
44. ^
45. ^ "There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4. — V. B. Berestetskii, E. M. Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell (translators). Relativistic Quantum Theory 4, part I. Course of Theoretical Physics (Landau and Lifshitz) ISBN 0-08-016025-5
46. ^ Stephen Hawking; Gödel and the end of physics
47. ^ Excerpt from an article by Roger Penrose
48. ^ "Life on the lattice: The most accurate theory we have". 2005-06-03. Retrieved 2010-10-15.
54. ^ The Transactional Interpretation of Quantum Mechanics by John Cramer. Reviews of Modern Physics 58, 647-688, July (1986)
55. ^ See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14-11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp. 8-6), and lasers (vol III, pp. 9-13).
56. ^ Introduction to Quantum Mechanics with Applications to Chemistry - Linus Pauling, E. Bright Wilson. 1985-03-01.
57. ^ Anderson, Mark (2009-01-13). "Is Quantum Mechanics Controlling Your Thoughts? | Subatomic Particles". DISCOVER Magazine. Retrieved 2012-08-18.
58. ^ "Quantum mechanics boosts photosynthesis". Retrieved 2010-10-23.
59. ^ Davies, P. C. W.; Betts, David S. (1984). Quantum Mechanics, Second edition. Chapman and Hall. p. 79. , Chapter 6, p. 79
60. ^ Baofu, Peter (2007-12-31). The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos.
61. ^ Derivation of particle in a box,
See also
This is another example illustrating the quantization of energy for bound states.
and the corresponding energy levels are
H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)
where Hn are the Hermite polynomials,
This problem can either be treated by directly solving the Schrödinger, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by
Harmonic oscillator
The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wavefunction is not pinned to zero at the walls of the well. Instead, the wavefunction must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.
Finite potential well
E = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}.
k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.
\psi(L) = 0 = C\sin kL.\!
and D = 0. At x = L,
\psi(0) = 0 = C\sin 0 + D\cos 0 = D\!
\psi(x) = C \sin kx + D \cos kx.\!
or, from Euler's formula,
\psi(x) = A e^{ikx} + B e ^{-ikx} \qquad\qquad E = \frac{\hbar^2 k^2}{2m}
with state \psi in this case having energy E coincident with the kinetic energy of the particle.
\frac{1}{2m} \hat{p}_x^2 = E,
the previous equation is evocative of the classic kinetic energy analogue,
\hat{p}_x = -i\hbar\frac{d}{dx}
With the differential operator defined by
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and infinite potential energy everywhere outside that region. For the one-dimensional case in the x direction, the time-independent Schrödinger equation may be written[61]
1-dimensional potential energy box (or infinite potential well)
Particle in a box
Rectangular potential barrier
k_1=\sqrt{2m E/\hbar^2}, and
k_2=\sqrt{2m (E-V_0)/\hbar^2}
where the wave vectors are related to the energy via
\psi_1(x)= \frac{1}{\sqrt{k_1}} \left(A_\rightarrow e^{i k_1 x} + A_\leftarrow e^{-ik_1x}\right)\quad x<0
\psi_2(x)= \frac{1}{\sqrt{k_2}} \left(B_\rightarrow e^{i k_2 x} + B_\leftarrow e^{-ik_2x}\right)\quad x>0
V(x)= \begin{cases} 0, & x < 0, \\ V_0, & x \ge 0. \end{cases}
The potential in this case is given by:
Step potential
3D confined electron wave functions for each eigenstate in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot, the wave functions are mixed due to confinement symmetry.
For example, consider a free particle. In quantum mechanics, there is wave–particle duality, so the properties of the particle can be described as the properties of a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over space as a wave function. The position and momentum of the particle are observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be measured with complete precision. However, one can measure the position (alone) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large (a Dirac delta) at a particular position x, and zero everywhere else. If one performs a position measurement on such a wavefunction, the resultant x will be obtained with 100% probability (i.e., with full certainty, or complete precision). This is called an eigenstate of position—or, stated in mathematical terms, a generalized position eigenstate (eigendistribution). If the particle is in an eigenstate of position, then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown.[59] In an eigenstate of momentum having a plane wave form, it can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate.[60]
Free particle
. Since classical formulas are much simpler and easier to compute than quantum formulas, classical approximations are used and preferred when the system is large enough to render the effects of quantum mechanics insignificant.quantum numbers can often provide good approximations to results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large classical physics Even so, [58] Quantum theory also provides accurate descriptions for many previously unexplained phenomena, such as
Quantum tunneling is vital to the operation of many devices. Even in the simple light switch, the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide without quantum tunneling. Flash memory chips found in USB drives use quantum tunneling to erase their memory cells.
Quantum mechanics is also critically important for understanding how individual atoms combine covalently to form molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry. Quantum mechanics can also provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others and the magnitudes of the energies involved.[56] Furthermore, most of the calculations performed in modern computational chemistry rely on quantum mechanics.
Quantum mechanics has had enormous[55] success in explaining many of the features of our universe. Quantum mechanics is often the only tool available that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism).
Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. Einstein held that there should be a local hidden variable theory underlying quantum mechanics and, consequently, that the present theory was incomplete. He produced a series of objections to quantum theory, the most famous of which has become known as the Einstein–Podolsky–Rosen paradox. John Bell showed that this "EPR" paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thereby demonstrating that the physical world cannot be described by any local realistic theory.[52] The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view.
The Copenhagen interpretation - due largely to the Danish theoretical physicist Niels Bohr - remains the quantum mechanical formalism that is currently most widely accepted amongst physicists, some 75 years after its enunciation. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered a final renunciation of the classical idea of "causality." It is also believed therein that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the conjugate nature of evidence obtained under different experimental situations.
Philosophical implications
The quest to unify the fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most accurately tested physical theory in competition with general relativity,[47][48] (blog) has been successfully merged with the weak nuclear force into the electroweak force and work is currently being done to merge the electroweak and strong force into the electrostrong force. Current predictions state that at around 1014 GeV the three aforementioned forces are fused into a single unified field,[49] Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However — and while special relativity is parsimoniously incorporated into quantum electrodynamics — the expanded general relativity, currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. One of those searching for a coherent TOE is Edward Witten, a theoretical physicist who formulated the M-theory, which is an attempt at describing the supersymmetrical based string theory. M-theory posits that our apparent 4-dimensional spacetime is, in reality, actually an 11-dimensional spacetime containing 10 spatial dimensions and 1 time dimension, although 7 of the spatial dimensions are - at lower energies - completely "compactified" (or infinitely curved) and not readily amenable to measurement or probing.
Attempts at a unified field theory
Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th and 21st century physics. Many prominent physicists, including Stephen Hawking, have labored for many years in the attempt to discover a theory underlying everything. This TOE would combine not only the different models of subatomic physics, but also derive the four fundamental forces of nature - the strong force, electromagnetism, the weak force, and gravity - from a single force or phenomenon. While Stephen Hawking was initially a believer in the Theory of Everything, after considering Gödel's Incompleteness Theorem, he has concluded that one is not obtainable, and has stated so publicly in his lecture "Gödel and the End of Physics" (2002).[46]
The Einstein–Podolsky–Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner - although the two particles can be an arbitrary distance apart. However, this effect does not violate causality, since no transfer of information happens. Quantum entanglement forms the basis of quantum cryptography, which is used in high-security commercial applications in banking and government.
According to the paper of J. Bell and the Copenhagen interpretation—the common interpretation of quantum mechanics by physicists since 1927 - and contrary to Einstein's ideas, quantum mechanics was not, at the same time a "realistic" theory and a "local" theory.
Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept many of the more "philosophical consequences and interpretations" of quantum mechanics, such as the lack of deterministic causality. He is famously quoted as saying, in response to this aspect, "My God does not play with dice". He also had difficulty with the assertion that a single subatomic particle can occupy numerous areas of space at one time. However, he was also the first to notice some of the apparently exotic consequences of entanglement, and used them to formulate the Einstein–Podolsky–Rosen paradox in the hope of showing that quantum mechanics had unacceptable implications if taken as a complete description of physical reality. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that - although Einstein was correct in identifying seemingly paradoxical implications of quantum mechanical nonlocality - these implications could be experimentally tested. Alain Aspect's initial experiments in 1982, and many subsequent experiments since, have definitively verified quantum entanglement.
Relativity and quantum mechanics
Quantum coherence is an essential difference between classical and quantum theories as illustrated by the Einstein–Podolsky–Rosen (EPR) paradox — an attempt to disprove quantum mechanics by an appeal to local realism.[40] Quantum interference involves adding together probability amplitudes, whereas classical "waves" infer that there is an adding together of intensities. For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal phenomena characteristic of quantum systems.[41] Quantum coherence is not typically evident at macroscopic scales, though an exception to this rule may occur at extremely low temperatures (i.e. approaching absolute zero) at which quantum behavior may manifest itself macroscopically.[42] This is in accordance with the following observations:
Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.[37] According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles).[38] The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large quantum numbers.[39] However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.
Quantum mechanics and classical physics
List of unsolved problems in physics
In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "superposition of states" and "wavefunction collapse", give rise to the reality we perceive?
The rules of quantum mechanics are fundamental. They assert that the state space of a system is a Hilbert space and that observables of that system are Hermitian operators acting on that space—although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or—equivalently—larger quantum numbers, i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an established classical model of a particular system, then attempt to guess the underlying quantum model that would give rise to the classical model in the correspondence limit.
Interactions with other scientific theories
There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics - matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).[32]
Mathematically equivalent formulations of quantum mechanics
The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the "wave-like" behavior of quantum states. As it turns out, analytic solutions of the Schrödinger equation are available for only a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives. Even the helium atom—which contains just one more electron than does the hydrogen atom—has defied all attempts at a fully analytic treatment.
During a measurement, on the other hand, the change of the initial wavefunction into another, later wavefunction is not deterministic, it is unpredictable (i.e., random). A time-evolution simulation can be seen here.[28][29]
The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the time evolution. The time evolution of wave functions is deterministic in the sense that - given a wavefunction at an initial time - it makes a definite prediction of what the wavefunction will be at any later time.[27]
In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates). Usually, a system will not be in an eigenstate of the observable (particle) we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as wavefunction collapse, a controversial and much-debated process[25] that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0 (neither an eigenstate of position nor of momentum). When one measures the position of the particle, it is impossible to predict with certainty the result.[21] It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.[26]
Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a probability distribution; that is, it describes the probability of obtaining the possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.[22][23] Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as eigenstates of the observable ("eigen" can be translated from German as meaning "inherent" or "characteristic").[24]
According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate. The possible results of a measurement are the eigenvalues of the operator representing the observable—which explains the choice of Hermitian operators, for which all the eigenvalues are real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.
In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac,[15] David Hilbert,[16] John von Neumann,[17] and Hermann Weyl[18] the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors"). Formally, these reside in a complex separable Hilbert space—variously called the "state space" or the "associated Hilbert space" of the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.
Mathematical formulations
Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the spectra of light emitted by different isotopes of the same element, as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.
Quantum mechanics is essential to understanding the behavior of systems at atomic length scales and smaller. If the physical nature of an atom was solely described by classical mechanics electrons would not "orbit" the nucleus since orbiting electrons emit radiation (due to circular motion) and would eventually collide with the nucleus due to this loss of energy. This framework was unable to explain the stability of atoms. Instead, electrons remain in an uncertain, non-deterministic, "smeared", probabilistic, wave–particle orbital about the nucleus, defying the traditional assumptions of classical mechanics and electromagnetism.[14]
The word quantum derives from the Latin, meaning "how great" or "how much".[11] In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and sub-atomic systems which is today called quantum mechanics. It underlies the mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.[12] Some fundamental aspects of the theory are still actively studied.[13]
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b15e210268621461 | Neutrons Optical Devices
September 22, 2015 9:27 am Published by
Optics deals with waves and their transformation by objects, i.e., scattering. Accordingly, since quantum mechanics describes massive particles as traveling wavesneutron optics deals with scattering of the neutrons’ de Broglie (or matter) wave. The source of this phenomenon may be of solid nature – more precisely the fermi pseudo potential of atoms, or represented by magnetic fields which effect the neutron’s wave function likewise.
Direct Current (DC) Spin Rotators
rrrrrrrrrrrrrrrrrrNext generation (3D-printed) Spin Rotators
Radio Frequency (RF) Spin Flippers
Supermirror Polarizer
Neutrons in magnetic fields: Larmor precession
The motion of a free propagating neutron, interacting with magnetic is described by a nonrelativistic Schrödinger equation also referred to as Pauli equation, given by , where and are the mass (kg) and the magnetic moment (, with J/T) of the neutron, respectively and and contains the Pauli matrices . A solution is found by the two dimensional spinor wave function of the neutron, which is denoted as , with spatial wave function . The state vector for the spin eigenstates denoted as and is given by , introducing polar angle and azimuthal angle , which can be represented on a Bloch sphere as shown below (left). In the field of general two-level systems (qubits) the term Bloch sphere is conventionally used, whereas the term Poincaré sphere is more common for representation of light polarization.
The neutron couples via its permanent magnetic dipole moment to magnetic fields, which is described by the Hamiltonian , where magnetic fields of stationary and/or time dependent origin are utilized for arbitrary spinor rotations in neutron optics. When a neutron enters a stationary magnetic field region (non-adiabatically), the motion of its polarization vector, defined as the expectation value of the Pauli spin matrices is described by the Bloch-equation, exhibiting Larmor precession: , where is a gyromagnetic ratio given by . This is the equation of motion of a classical magnetic dipole in a magnetic field, which shows the precession of the polarization vector about the magnetic field with the Larmor frequency , which is depicted above (right). The Larmor precession angle (rotation angle), solely depends on the strength of the applied magnetic field and the propagation within the field and is given by , where and are the length of the magnetic-field region traversed by the neutrons and the neutron velocity, respectively (see here for a detailed derivation of Larmor precession).
Larmor precession is often utilized in co called Direct Current (DC) spin-rotators, or spin-flippers (if the spinor rotation angle is set to 180 deg), which is depicted below. The illustrated field configuration assures a highly non-adiabatic transit required for Larmor precession. In practice a second coils (-direction), perpendicular to the original coil (-direction) is necessary to compensate the field component of the guide field (-direction).
Using the formalism of quantum mechanics, a spin rotation through an angle about an axis pointing in direction is described by the unitary transformation operator , which can be written as (see here for a detailed derivation of the unitary transformation ). Note that for a stationary magnetic field the total energy of a neutron is indeed a conserved quantity since . Thus, neither the momentum nor the potential (Zeeman magnetic energy) is a conserved quantity due to and , which is depicted below (left).
In a purely time dependent magnetic field the total energy of a neutron is not a conserved quantity: . Energy can be exchanged with the magnetic field via photon interaction. However the momentum is conserved , since due to the fact that is purely time dependent. Therefore the change in the total energy must origin from the potential energy . A diagram of the kinetic, potential and total energy is shown above (right) 1.
RF_all copy
An oscillating RF field and a static magnetic field—a configuration used in nuclear magnetic resonance (NMR)—is also capable of spin flipping. An oscillating RF field can be viewed as two counter-rotating fields. In the frame of one of the rotating components, the other is rotating at double-frequency and can be neglected (rotating-wave approximation). The static field component of magnitude is fully suppressed in the case of frequency resonance, i.e. for the oscillation frequency . If, in addition, the amplitude-resonance condition — determining the amplitude of the rotating field — is fulfilled, a spin flip occurs. A consequence of the rotating-wave approximation is the so-called Bloch–Siegert shift, which gives rise to a correction term for the frequency resonance now reading . The above-explained combination of static and time-dependent magnetic fields is exploited in Radio Frequency (RF) flippers, as depicted in below (see here for detailed calculations).
Next generation (3D-printed) Spin Rotators
A recent projects originates in a cooperation with the group of Dieter Süss (Physics of functional materials) from the faculty of physics, university of vienna where they operate a Christian Doppler lab. This particular group demonstrated recently that an end-user 3D printer can be used to print polymer-bonded rare-earth magnets with a complex shape. A Fusing Deposition Modeling (FDM) 3D printer with a maximum temperature of 260°C, and a nozzle diameter of 0.4 mm is used to print magnetic structures with layer heights between 0.05 and 0.3 mm. The polymer-bonded magnetic compound consists of PA11 and magnetically isotropic NdFeB powder MQP-S-11-9. This source material is compounded and extruded into suitable filaments in the desired ratio of 85 wt.% MQP-S-11-9 powder. As example of an optimized permanent magnetic system a Larmor-spin rotator for a polarized neutron interferometer setup was 3D printed.
3D printes Magnets
Polarizer/Analyzer: Multi-Layer Supermirrors
For understanding the mode of operation of the applied neutron polariser we have to recap some basic concepts of of neutron optic such as the refraction index. The time dependent Schrödinger equation , with yields . From the strong (nuclear) interaction of the neutron we have the Fermi Pseudopotenital denoted as superm, with the coherent scattering length an the atom number density ( denotes the position of each scattering center). From the magnetic interaction the contribution is given by , with , where is the magnetic moment of the neutron. So for materials containing Fe, Ni, or Co we have for the index of refraction . Since can become complex, which accounts for absorption or incoherent scattering. In general we have < 1 so the potential is repulsive. For neutrons vacuum () is an optically denser medium compared to most elements ! So neutron are totally reflected, if . With and we get and a critical angle , which is depicted on the left side. For example Ni has a critical angle of 0.1°/Å. The polarizer and analyzer (spin filter) consist of multilayer structure of two media and having different coherent scattering length . For an incident angle at every single boundary layer there will occur a transmitted an a reflected sub beam. If the thickness of the layers is chosen in such way that the partial waves of the reflected sub beams have a opticalScreenshot 2017-04-07 16.26.06 path difference of constructive interference will be observed. If the thickness of the layers varies only slightly, from layer to layer, there will be an appropriate “lattice constant” for a diversity of wavelengths. If alternating a magnetic and a non-magnetic medium is utilized not only the nuclear scattering length, but also the magnetic scattering length has to be considered . This can be used for beam polarization, since the sign of the magnetic scattering length depends of the orientation of the spin towards the magnetization of the medium. If a combination is chosen such that the sum of the nuclear scattering length and the magnetic scattering length for one spin component (for instance ) equals the scattering length of the non-magnetic substance, then this spin component will not be reflected, since there is no difference in the refractive index of the two layers for this spin component. However, the other spin component () will be (partly) reflected. The transmitted spin component ( ) is absorbed after the last layer. An arrangement as discussed here is referred to as supermirror, often used as polarizer or analyzer 2,3. A multilayer composed of layers whose thicknesses are varied gradually layer by layer reflects neutrons with a wide wavelength range. The thicknesses of a layer thereby vary between 5 and 35 nm and the number of layers can be several thousand. A Supermirrors are usually characterised by its critical angle as which can go up to .
1. F. Mezei, Physica B (Amsterdam) 151, 74 (1988).
2. F. Mezei, Commun. Phys. 1, 81 (1976).
3. F. Mezei and P. A. Dagleis Commun. Phys. 2, 41 (1977). |
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Statistical mechanics
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Statistical mechanics
Statistical mechanics is a branch of theoretical physics and chemistry (and mathematical physics) that studies, using probability theory, the average behaviour of a mechanical system where the state of the system is uncertain.[1][2][3][note 1]
The classical view of the universe was that its fundamental laws are mechanical in nature, and that all physical systems are therefore governed by mechanical laws at a microscopic level. These laws are precise equations of motion that map any given initial state to a corresponding future state at a later time. There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor easy) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics is a collection of mathematical tools that are used to fill this disconnection between the laws of mechanics and the practical experience of incomplete knowledge.
A common use of statistical mechanics is in explaining the thermodynamic behaviour of large systems. Microscopic mechanical laws do not contain concepts such as temperature, heat, or entropy, however, statistical mechanics shows how these concepts arise from the natural uncertainty that arises about the state of a system when that system is prepared in practice. The benefit of using statistical mechanics is that it provides exact methods to connect thermodynamic quantities (such as heat capacity) to microscopic behaviour, whereas in classical thermodynamics the only available option would be to just measure and tabulate such quantities for various materials. Statistical mechanics also makes it possible to extend the laws of thermodynamics to cases which are not considered in classical thermodynamics, for example microscopic systems and other mechanical systems with few degrees of freedom.[1] This branch of statistical mechanics which treats and extends classical thermodynamics is known as statistical thermodynamics or equilibrium statistical mechanics.
Statistical mechanics also finds use outside equilibrium. An important subbranch known as non-equilibrium statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions, or flows of particles and heat. Unlike with equilibrium, there is no exact formalism that applies to non-equilibrium statistical mechanics in general and so this branch of statistical mechanics remains an active area of theoretical research.
• Principles: mechanics and ensembles 1
• Statistical thermodynamics 2
• Fundamental postulate 2.1
• Three thermodynamic ensembles 2.2
• Statistical fluctuations and the macroscopic limit 2.2.1
• Illustrative example (a gas) 2.3
• Calculation methods 2.4
• Exact 2.4.1
• Monte Carlo 2.4.2
• Other 2.4.3
• Non-equilibrium statistical mechanics 3
• Stochastic methods 3.1
• Near-equilibrium methods 3.2
• Hybrid methods 3.3
• Applications outside thermodynamics 4
• History 5
• See also 6
• Notes 7
• References 8
• External links 9
Principles: mechanics and ensembles
In physics there are two types of mechanics usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two ingredients:
1. The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).
2. An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)
Using these two ingredients, the state at any other time, past or future, can in principle be calculated.
Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,[note 2] and can be compactly summarized as a density matrix.
As is usual for probabilities, the ensemble can be interpreted in different ways:[1]
• an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or
• the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (empirical probability), in the limit of an infinite number of trials.
These two meanings are equivalent for many purposes, and will be used interchangeably in this article.
However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.
One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to that state.[note 3] The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.
Statistical thermodynamics
The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to explain the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.
As an example, one might ask what is it about a thermodynamic system of NH3 molecules that determines the free energy characteristic of that compound? Classical thermodynamics does not provide the answer. If, for example, we were given spectroscopic data, of this body of gas molecules, such as bond length, bond angle, bond rotation, and flexibility of the bonds in NH3 we should see that the free energy could not be other than it is. To prove this true, we need to bridge the gap between the microscopic realm of atoms and molecules and the macroscopic realm of classical thermodynamics. Statistical mechanics demonstrates how the thermodynamic parameters of a system, such as temperature and pressure, are related to microscopic behaviours of such constituent atoms and molecules.[4]
Although we may understand a system generically, in general we lack information about the state of a specific instance of that system. For this reason the notion of statistical ensemble (a probability distribution over possible states) is necessary. Furthermore, in order to reflect that the material is in a thermodynamic equilibrium, it is necessary to introduce a corresponding statistical mechanical definition of equilibrium. The analogue of thermodynamic equilibrium in statistical thermodynamics is the ensemble property of statistical equilibrium, described in the previous section. An additional assumption in statistical thermodynamics is that the system is isolated (no varying external forces are acting on the system), so that its total energy does not vary over time. A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).[1]
Fundamental postulate
There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.[1] An additional postulate is necessary to motivate why the ensemble for a given system should have one form or another.
A common approach found in many textbooks is to take the equal a priori probability postulate.[2] This postulate states that
For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.
The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:
• Ergodic hypothesis: An ergodic state is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.
• Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation.
• Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest Gibbs entropy (information entropy).[5]
Other fundamental postulates for statistical mechanics have also been proposed.[6]
In any case, the reason for establishing the microcanonical ensemble is mainly axiomatic.[6] The microcanonical ensemble itself is mathematically awkward to use for real calculations, and even very simple finite systems can only be solved approximately. However, it is possible to use the microcanonical ensemble to construct a hypothetical infinite thermodynamic reservoir that has an exactly defined notion of temperature and chemical potential. Once this reservoir has been established, it can be used to justify exactly the canonical ensemble or grand canonical ensemble (see below) for any other system by considering the contact of this system with the reservoir.[1] These other ensembles are those actually used in practical statistical mechanics calculations as they are mathematically simpler and also correspond to a much more realistic situation (energy not known exactly).[2]
Three thermodynamic ensembles
There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.[1] These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.
• The microcanonical ensemble describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
• The canonical ensemble describes a system of fixed composition that is in thermal equilibrium[note 4] with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
• The grand canonical ensemble describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
Thermodynamic ensembles[1]
Microcanonical Canonical Grand canonical
Fixed variables
N, V, E
N, V, T
μ, V, T
Microscopic features
Macroscopic function
Statistical fluctuations and the macroscopic limit
The thermodynamic ensembles' most significant difference is that they either admit uncertainty in the variables of energy or particle number, or that those variables are fixed to particular values. While this difference can be observed in some cases, for macroscopic systems the thermodynamic ensembles are usually observationally equivalent.
The limit of large systems in statistical mechanics is known as the thermodynamic limit. In the thermodynamic limit the microcanonical, canonical, and grand canonical ensembles tend to give identical predictions about thermodynamic characteristics. This means that one can specify either total energy or temperature and arrive at the same result; likewise one can specify either total particle number or chemical potential. Given these considerations, the best ensemble to choose for the calculation of the properties of a macroscopic system is usually just the ensemble which allows the result to be derived most easily.[7]
Important cases where the thermodynamic ensembles do not give identical results include:
• Systems at a phase transition.
• Systems with long-range interactions.
• Microscopic systems.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.[2]
Illustrative example (a gas)
The above concepts can be illustrated for the specific case of one liter of ammonia gas at standard conditions. (Note that statistical thermodynamics is not restricted to the study of macroscopic gases, and the example of a gas is given here to illustrate concepts. Statistical mechanics and statistical thermodynamics apply to all mechanical systems (including microscopic systems) and to all phases of matter: liquids, solids, plasmas, gases, nuclear matter, quark matter.)
A simple way to prepare one litre sample of ammonia in a standard condition is to take a very large reservoir of ammonia at those standard conditions, and connect it to a previously evacuated one-litre container. After ammonia gas has entered the container and the container has been given time to reach thermodynamic equilibrium with the reservoir, the container is then sealed and isolated. In thermodynamics, this is a repeatable process resulting in a very well defined sample of gas with a precise description. We now consider the corresponding precise description in statistical thermodynamics.
Although this process is well defined and repeatable in a macroscopic sense, we have no information about the exact locations and velocities of each and every molecule in the container of gas. Moreover, we do not even know exactly how many molecules are in the container; even supposing we knew exactly the average density of the ammonia gas in general, we do not know how many molecules of the gas happened to be inside our container at the moment when we sealed it. The sample is in equilibrium and is in equilibrium with the reservoir: we could reconnect it to the reservoir for some time, and then re-seal it, and our knowledge about the state of the gas would not change. In this case, our knowledge about the state of the gas is precisely described by the grand canonical ensemble. Provided we have an accurate microscopic model of the ammonia gas, we could in principle compute all thermodynamic properties of this sample of gas by using the distribution provided by the grand canonical ensemble.
Hypothetically, we could use an extremely sensitive weight scale to measure exactly the mass of the container before and after introducing the ammonia gas, so that we can exactly know the number of ammonia molecules. After we make this measurement, then our knowledge about the gas would correspond to the canonical ensemble. Finally, suppose by some hypothetical apparatus we can measure exactly the number of molecules and also measure exactly the total energy of the system. Supposing furthermore that this apparatus gives us no further information about the molecules' positions and velocities, our knowledge about the system would correspond to the microcanonical ensemble.
Even after making such measurements, however, our expectations about the behaviour of the gas do not change appreciably. This is because the gas sample is macroscopic and approximates very well the thermodynamic limit, so the different ensembles behave similarly. This can be demonstrated by considering how small the actual fluctuations would be. Suppose that we knew the number density of ammonia gas was exactly 3.04×1022 molecules per liter inside the reservoir of ammonia gas used to fill the one-litre container. In describing the container with the grand canonical ensemble, then, the average number of molecules would be \langle N\rangle = 3.04\times 10^{22} and the uncertainty (standard deviation) in the number of molecules would be \sigma_N = \sqrt{\langle N \rangle} \approx 2\times 10^{11} (assuming Poisson distribution), which is relatively very small compared to the total number of molecules. Upon measuring the particle number (thus arriving at a canonical ensemble) we should find very nearly 3.04×1022 molecules. For example the probability of finding more than 3.040001×1022 or less than 3.039999×1022 molecules would be about 1 in 103000000000.[note 5]
Calculation methods
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
There are some cases which allow exact solutions.
Monte Carlo
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
Non-equilibrium statistical mechanics
There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underling mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to add additional ingredients besides probability and reversible mechanics.
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.
Stochastic methods
One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.
• Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem). The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas. A quantum technique related in theme is the random phase approximation.
• BBGKY hierarchy: In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.
• Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions): A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.
Near-equilibrium methods
Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.[3]:664
This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.
A few of the theoretical tools used to make this connection include:
Hybrid methods
An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.[9][10]
Applications outside thermodynamics
The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:
In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.[11] Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell’s paper and was so inspired by it that he spent much of his life developing the subject further.
Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.[12] Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.
The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884.[13][note 6] "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched.[14] Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.[1] Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.[2]
See also
Fundamentals of Statistical Mechanics
1. ^ The term statistical mechanics is sometimes used to refer to only statistical thermodynamics. This article takes the broader view. By some definitions, statistical physics is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.
2. ^ The probabilities in quantum statistical mechanics should not be confused with quantum superposition. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.
3. ^ Statistical equilibrium should not be confused with mechanical equilibrium. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.
4. ^ The transitive thermal equilibrium (as in, "X is thermal equilibrium with Y") used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.
5. ^ This is so unlikely as to be practically impossible. The statistical physicist Émile Borel noted that, compared to the improbabilities found in statistical mechanics, it would be more likely that monkeys typing randomly on a typewriter would happen to reproduce the books of the world. See infinite monkey theorem.
2. ^ a b c d e f
3. ^ a b c d
4. ^ Nash, Leonard K. (1974). Elements of Statistical Thermodynamics, 2nd Ed. Dover Publications, Inc.
5. ^
6. ^ a b c J. Uffink, "Compendium of the foundations of classical statistical physics." (2006)
7. ^ Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. (p.227)
9. ^ Altshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. (1982). "Effects of electron-electron collisions with small energy transfers on quantum localisation". Journal of Physics C: Solid State Physics 15 (36): 7367.
10. ^ Aleiner, I.; Blanter, Y. (2002). "Inelastic scattering time for conductance fluctuations". Physical Review B 65 (11).
12. ^ Ebeling, Werner; Sokolov, Igor M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. World Scientific Publishing Co. Pte. Ltd. pp. 3–12. (section 1.2)
13. ^ J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II (1906), pp. 16.
14. ^ Mayants, Lazar (1984). The enigma of probability and physics. Springer. p. 174.
External links
• Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.
• Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials. SklogWiki is particularly orientated towards liquids and soft condensed matter.
• Statistical Thermodynamics - Historical Timeline
• Thermodynamics and Statistical Mechanics by Richard Fitzpatrick
• Lecture Notes in Statistical Mechanics and Mesoscopics by Doron Cohen
• Videos of lecture series in statistical mechanics on YouTube taught by Leonard Susskind.
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18ae6730014fff12 | 【University of Birmingham】Parabolic and hyperbolic Liouville equations-大连理工大学数学科学学院（新）
【University of Birmingham】Parabolic and hyperbolic Liouville equations
2019年10月18日 08:44 点击：[]
报告题目：Parabolic and hyperbolic Liouville equations
报告人：Dr. Yuzhao WANG (University of Birmingham)
内联系人：王文栋 联系电话：84708351-8139
: We will talk about some stochastic partial differential equations (SPDEs), which arise naturally in the context of Liouville quantum gravity and are proposed to preserve the Liouville measure which has been constructed recently in the work by David-Kupiainen-Rhodes-Vargas. We construct global solutions to these equations and then show the invariance of the Liouville measure under the resulting dynamics. As a by product, we also answer an open problem proposed by Sun-Tzvetkov recently.
Dr Wang is a Lecture at the University of Birmingham, School of Mathematics. His research interests lie in partial differential equations, harmonic analysis, and stochastic analysis. In particular, the study of nonlinear dispersive PDEs such as nonlinear Schrödinger equations, nonlinear wave equations, and the KdV equation by using techniques from PDEs, Harmonic Analysis, and Probability theory. Partial papers are pubilished in "Adv. Math", "SIAM J. Math. Anal.", "J. Funct. Anal.", "Comm. Partial Differential Equations.","Nonlinearity", "Proc. Lond. Math. Soc.", etc.
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Chemistry LibreTexts
8.4: Magnetic Properties and the Zeeman Effect
• Page ID
• Magnetism results from the circular motion of charged particles. This property is demonstrated on a macroscopic scale by making an electromagnet from a coil of wire and a battery. Electrons moving through the coil produce a magnetic field (Figure \(\PageIndex{1}\)), which can be thought of as originating from a magnetic dipole or a bar magnet.
Figure \(\PageIndex{1}\): Faraday’s apparatus for demonstrating that a magnetic field can produce a current. A change in the field produced by the top coil induces an emf and, hence, a current in the bottom coil. When the switch is opened and closed, the galvanometer registers currents in opposite directions. No current flows through the galvanometer when the switch remains closed or open. Image used with permission (CC BT 3.0; OpenStax).
Magnetism results from the circular motion of charged particles.
Electrons in atoms also are moving charges with angular momentum so they too produce a magnetic dipole, which is why some materials are magnetic. A magnetic dipole interacts with an applied magnetic field, and the energy of this interaction is given by the scalar product of the magnetic dipole moment, and the magnetic field, \(\vec{B}\).
\[E_B = - \vec{\mu} _m \cdot \vec{B} \label {8.4.1}\]
Magnets are acted on by forces and torques when placed within an external applied magnetic field (Figure \(\PageIndex{2}\)). In a uniform external field, a magnet experiences no net force, but a net torque. The torque tries to align the magnetic moment (\(\vec{\mu} _m\) of the magnet with the external field \(\vec{B}\). The magnetic moment of a magnet points from its south pole to its north pole.
Figure \(\PageIndex{2}\): A magnet will feel a force to realign in an external field, i.e. go from a higher energy to a lower energy. The energy of this system is determined by Equation \(ref{8.4.1}\) and classical can vary since the angle between \(\vec{\mu} _m\)and \(\vec{B}\) can vary continuously from 0 (low energy) to 180° (high energy).
In a non-uniform magnetic field a current loop, and therefore a magnet, experiences a net force, which tries to pull an aligned dipole into regions where the magnitude of the magnetic field is larger and push an anti-aligned dipole into regions where magnitude the magnetic field is smaller.
Quantum Effects
As expected, the quantum picture is different. Pieter Zeeman was one of the first to observe the splittings of spectral lines in a magnetic field caused by this interaction. Consequently such splittings are known as the Zeeman effect. Let’s now use our current knowledge to predict what the Zeeman effect for the 2p to 1s transition in hydrogen would look like, and then compare this prediction with a more complete theory. To understand the Zeeman effect, which uses a magnetic field to remove the degeneracy of different angular momentum states, we need to examine how an electron in a hydrogen atom interacts with an external magnetic field, \(\vec{B}\). Since magnetism results from the circular motion of charged particles, we should look for a relationship between the angular momentum \(\vec{L}\) and the magnetic dipole moment \(\vec{\mu} _m\).
The relationship between the magnetic dipole moment \(\vec{\mu} _m\) (also referred to simply as the magnetic moment) and the angular momentum \(\vec{L}\) of a particle with mass m and charge \(q\) is given by
\[ \vec{\mu} _m = \dfrac {q}{2m} \vec{L} \label {8.4.2}\]
For an electron, this equation becomes
\[ \vec{\mu} _m = - \dfrac {e}{2m_e} \vec{L} \label {8.4.3}\]
where the specific charge and mass of the electron have been substituted for \(q\) and \(m\). The magnetic moment for the electron is a vector pointing in the direction opposite to \(\vec{L}\), both of which classically are perpendicular to the plane of the rotational motion.
Exercise \(\PageIndex{1}\)
Will an electron in the ground state of hydrogen have a magnetic moment? Why or why not?
The relationship between the angular momentum of a particle and its magnetic moment is commonly expressed as a ratio, called the gyromagnetic ratio, \(\gamma\). Gyro is Greek for turn so gyromagnetic simply relates turning (angular momentum) to magnetism. Now you also know why the Greek sandwiches made with meat cut from a spit turning over a fire are called gyros.
\[ \gamma = \dfrac {\mu _m}{L} = \dfrac {q}{2m} \label {8.4.4}\]
In the specific case of an electron,
\[ \gamma _e = - \dfrac {e}{2m_e} \label {8.4.5}\]
Exercise \(\PageIndex{2}\)
Calculate the magnitude of the gyromagnetic ratio for an electron.
To determine the energy of a hydrogen atom in a magnetic field we need to include the operator form of the hydrogen atom Hamiltonian. The Hamiltonian always consists of all the energy terms that are relevant to the problem at hand.
\[ \hat {H} = \hat {H} ^0 + \hat {H} _m \label {8.4.6}\]
where \(\hat {H} ^0\) is the Hamiltonian operator in the absence of the field and \(\hat {H} _m\) is written using the operator forms of Equations \(\ref{8.4.1}\) and \(\ref{8.4.3}\)),
\[ \hat {H}_m = - \hat {\mu} \cdot \vec{B} = \dfrac {e}{2m_e} \hat {L} \cdot B \label {8.4.7}\]
The scalar product
\[ \hat {L} \cdot \vec{B} = \hat {L}_x B_x + \hat {L}_y B_y + \hat {L}_z B_z \label {8.4.8}\]
simplifies if the z-axis is defined as the direction of the external field because then \(B_x\) and \(B_y\) are automatically 0, and Equation \ref{8.4.6} becomes
\[ \hat {H} = \hat {H}^0 + \dfrac {eB_z}{2m_e} \hat {L} _z \label {8.4.9}\]
where \(B_z\) is the magnitude of the magnetic field, which is along the z-axis.
We now can ask, “What is the effect of a magnetic field on the energy of the hydrogen atom orbitals?” To answer this question, we will not solve the Schrödinger equation again; we simply calculate the expectation value of the energy, \(\left \langle E \right \rangle \), using the existing hydrogen atom wavefunctions and the new Hamiltonian operator.
\[ \left \langle E \right \rangle = \left \langle \hat {H}^0 \right \rangle + \dfrac {eB_z}{2m_e} \left \langle \hat {L} _z \right \rangle \label {8.4.10}\]
\[\left \langle \hat {H}^0 \right \rangle = \int \psi ^*_{n,l,m_l} \hat {H}^0 \psi _{n,l,m_l} d \tau = E_n \label {8.4.11}\]
\[\left \langle \hat {L}_z \right \rangle = \int \psi ^*_{n,l,m_l} \hat {L}_z \psi _{n,l,m_l} d \tau = m_l \hbar \label {8.4.12}\]
Exercise \(\PageIndex{3}\)
Show that the expectation value \(\left \langle \hat {L}_z \right \rangle = m_l \hbar\).
The expectation value approach provides an exact result in this case because the hydrogen atom wavefunctions are eigenfunctions of both \(\hat {H} ^0\) and \(\hat {L}_z\). If the wavefunctions were not eigenfunctions of the operator associated with the magnetic field, then this approach would provide a first-order estimate of the energy. First and higher order estimates of the energy are part of a general approach to developing approximate solutions to the Schrödinger equation. This approach, called perturbation theory, is discussed in the next chapter.
The expectation value calculated for the total energy in this case is the sum of the energy in the absence of the field, \(E_n\), plus the Zeeman energy, \(\dfrac {e \hbar B_z m_l}{2m_e}\)
\[\left \langle E \right \rangle = E_n + \dfrac {e \hbar B_z m_l}{2m_e} = E_n + \mu _B B_z m_l \label {8.4.13}\]
The factor
\[ \dfrac {e \hbar}{2m_e} = - \gamma _e \hbar = \mu _B \label {8.4.14}\]
defines the constant \(\mu _B\), called the Bohr magneton, which is taken to be the fundamental magnetic moment. It has units of \(9.2732 \times 10^{-21}\) erg/Gauss or \(9.2732 \times 10^{-24}\) Joule/Tesla. This factor will help you to relate magnetic fields, measured in Gauss or Tesla, to energies, measured in ergs or Joules, for any particle with a charge and mass the same as an electron.
Equation \(\ref{8.4.13}\) shows that the \(m_l\) quantum number degeneracy of the hydrogen atom is removed by the magnetic field. For example, the three states \(\Psi _{211}\) , \(\Psi _{21-1}\), and \(\Psi _{210}\), which are degenerate in zero field, have different energies in a magnetic field, as shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\): The Zeeman effect. Emission when an electron switches from a 2p orbital to a 1s orbital occurs at only one energy in the absence of a magnetic field, but can occur at three different energies in the presence of a magnetic field.
The \(m_l = 0\) state, for which the component of angular momentum and hence also the magnetic moment in the external field direction is zero, experiences no interaction with the magnetic field. The \(m_l = +1\) state, for which the angular momentum in the z-direction is +ħ and the magnetic moment is in the opposite direction, against the field, experiences a raising of energy in the presence of a field. Maintaining the magnetic dipole against the external field direction is like holding a small bar magnet with its poles aligned exactly opposite to the poles of a large magnet (Figure \(\PageIndex{5}\). It is a higher energy situation than when the magnetic moments are aligned with each other.
Figure \(\PageIndex{4}\): The effect of an external magnetic field (B) on the energy of a magnetic dipole (L) oriented a) with and b) against the applied magnetic field.
Exercise \(\PageIndex{4}\)
Carry out the steps going from Equation \(\ref{8.4.10}\) to Equation \(\ref{8.4.13}\).
Exercise \(\PageIndex{5}\)
Consider the effect of changing the magnetic field on the magnitude of the Zeeman splitting. Sketch a diagram where the magnetic field strength is on the x-axis and the energy of the three 2p orbitals is on the y-axis to show the trend in splitting magnitudes with increasing magnetic field. Be quantitative, calculate and plot the exact numerical values using a software package of your choice.
Exercise \(\PageIndex{6}\)
Based on your calculations in Exercise \(\PageIndex{2}\) sketch a luminescence spectrum for the hydrogen atom in the n = 2 level in a magnetic field of 1 Tesla. Provide the numerical value for each of the transition energies. Use cm-1 or electron volts for the energy units. |
9341f914a5530aff | RAZ Glossary
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This is a list of brief explanations and definitions for terms that Eliezer Yudkowsky uses in the book Rationality: From AI to Zombies, an edited version of the Sequences.
The glossary is a community effort, and you're welcome to improve on the entries here, or add new ones. See the Talk page for some ideas for unwritten entries.
• a priori. A sentence that is reasonable to believe even in the absence of any experiential evidence (outside of the evidence needed to understand the sentence). A priori claims are in some way introspectively self-evident, or justifiable using only abstract reasoning. For example, pure mathematics is often claimed to be a priori, while scientific knowledge is claimed to be a posteriori, or dependent on (sensory) experience. These two terms shouldn’t be confused with prior and posterior probabilities.
• ad hominem. A verbal attack on the person making an argument, where a direct criticism of the argument is possible and would be more relevant. The term is reserved for cases where talking about the person amounts to changing the topic. If your character is the topic from the outset (e.g., during a job interview), then it isn't an ad hominem fallacy to cite evidence showing that you're a lousy worker.
• affective death spiral. A halo effect that perpetuates and exacerbates itself over time.
• AI-Box Experiment. A demonstration by Yudkowsky that people tend to overestimate how hard it is to manipulate people, and therefore underestimate the risk of building an Unfriendly AI that can only interact with its environment by verbally communicating with its programmers. One participant role-plays an AI, while another role-plays a human whose job it is interact with the AI without voluntarily releasing the AI from its “box”. Yudkowsky and a few other people who have role-played the AI have succeeded in getting the human supervisor to agree to release them, which suggests that a superhuman intelligence would have an even easier time escaping.
• algorithm. A specific procedure for computing some function. A mathematical object consisting of a finite, well-defined sequence of steps that concludes with some output determined by its initial input. Multiple physical systems can simultaneously instantiate the same algorithm.
• alien god. One of Yudkowsky's pet names for natural selection.
• ambiguity aversion. Preferring small certain gains over much larger uncertain gains.
• amplitude. A quantity in a configuration space, represented by a complex number. Amplitudes are physical, not abstract or formal. The complex number’s modulus squared (i.e., its absolute value multiplied by itself) yields the Born probabilities, but the reason for this is unknown.
• anchoring. The cognitive bias of relying excessively on initial information after receiving relevant new information.
• anthropomorphism. The tendency to assign human qualities to non-human phenomena.
• ASCII. The American Standard Code for Information Exchange. A very simple system for encoding 128 ordinary English letters, numbers, and punctuation.
• beisutsukai. Japanese for "Bayes user." A fictional order of high-level rationalists, also known as the Bayesian Conspiracy.
• Berkeleian idealism. The belief, espoused by George Berkeley, that things only exist in various minds (including the mind of God).
• bias. (a) A cognitive bias. In Rationality: From AI to Zombies, this will be the default meaning. (b) A statistical bias. (c) An inductive bias. (d) Colloquially: prejudice or unfairness.
• black box. Any process whose inner workings are mysterious or poorly understood.
• blind god. One of Yudkowsky's pet names for natural selection.
• bucket. See “pebble and bucket.”
• comparative advantage. An ability to produce something at a lower cost than some other actor could. This is not the same as having an absolute advantage over someone: you may be a better cook than someone across-the-board, but that person will still have a comparative advantage over you at cooking some dishes. This is because your cooking skills make your time more valuable; the worse cook may have a comparative advantage at baking bread, for example, since it doesn’t cost them much to spend a lot of time on baking, whereas you could be spending that time creating a large number of high-quality dishes. Baking bread is more costly for the good cook than for the bad cook because the good cook is paying a larger opportunity cost, i.e., is giving up more valuable opportunities to be doing other things.
• conjunction. A compound sentence asserting two or more distinct things, such as "A and B" or "A even though B." The conjunction fallacy is the tendency to count some conjunctions as more probable than their components even though they can’t be more probable (and are almost always less probable).
• decision theory. (a) The mathematical study of correct decision-making in general, abstracted from an agent's particular beliefs, goals, or capabilities. (b) A well-defined general-purpose procedure for arriving at decisions, e.g., causal decision theory.
• econblog. Economics blog.
• edge. See “graph.”
• entanglement. (a) Causal correlation between two things. (b) In quantum physics, the mutual dependence of two particles' states upon one another. Entanglement in sense (b) occurs when a quantum amplitude distribution cannot be factorized.
• entropy. (a) In thermodynamics, the number of different ways a physical state may be produced (its Boltzmann entropy). E.g., a slightly shuffled deck has lower entropy than a fully shuffled one, because there are many more configurations a fully shuffled deck is likely to end up in. (b) In information theory, the expected value of the information contained in a message (its Shannon entropy). That is, a random variable’s Shannon entropy is how many bits of information one would be missing (on average) if one did not know the variable’s value. Boltzmann entropy and Shannon entropy have turned out to be equivalent; that is, a system’s thermodynamic disorder corresponds to the number of bits needed to fully characterize it.
• epistemic. Concerning knowledge.
• eutopia. Yudkowsky’s term for a utopia that’s actually nice to live in, as opposed to one that’s unpleasant or unfeasible.
• evolution. (a) In biology, change in a population’s heritable features. (b) In other fields, change of any sort.
• expected utility. A measure of how much an agent’s goals will tend to be satisfied by some decision, given uncertainty about the decision’s outcome. Accepting a 5% chance of winning a million dollars will usually leave you poorer than accepting a 100% chance of winning one dollar; nine times out of ten, the certain one-dollar gamble has higher actual utility. All the same, we say that the 10% shot at a million dollars is better (assuming dollars have utility for you) because it has higher expected utility in all cases: $1M multiplied by probability 0.05 > $1 multiplied by probability 1.
• fitness. See “inclusive fitness.”
• formalism. A specific way of logically or mathematically representing something.
• function. A relation between inputs and outputs such that every input has exactly one output. A mapping between two sets in which every element in the first set is assigned a single specific element from the second.
• graph. In graph theory, a mathematical object consisting of simple atomic objects (vertices, or nodes) connected by lines (edges) or arrows (arcs).
• halting oracle. An abstract agent that is stipulated to be able to reliably answer questions that no algorithm can reliably answer. Though it is provably impossible for finite rule-following systems (e.g., Turing machines) to answer certain questions (e.g., the halting problem), it can still be mathematically useful to consider the logical implications of scenarios in which we could access answers to those questions.
• happy death spiral. See “affective death spiral.”
• hat tip. A grateful acknowledgment of someone who brought information to one's attention.
• hedonic. Concerning pleasure.
• heuristic. An imperfect method for achieving some goal. A useful approximation. Cognitive heuristics are innate, humanly universal brain heuristics.
• idiot god. One of Yudkowsky's pet names for natural selection.
• iff. If, and only if.
• inclusive fitness. The degree to which a gene causes more copies of itself to exist in the next generation. Inclusive fitness is the property propagated by natural selection. Unlike individual fitness, which is a specific organism’s tendency to promote more copies of its genes, inclusive fitness is held by the genes themselves. Inclusive fitness can sometimes be increased at the expense of the individual organism’s overall fitness.
• instrumental value. A goal that is only pursued in order to further some other goal.
• Iterated Prisoner’s Dilemma. A series of Prisoner’s Dilemmas between the same two players. Because players can punish each other for defecting on previous rounds, they will usually more reason to cooperate than in the one-shot Prisoner’s Dilemma.
• Lamarckism. The 19th-century pre-Darwinian hypothesis that populations evolve via the hereditary transmission of the traits practiced and cultivated by the previous generation.
• Machine Intelligence Research Institute. A small non-profit organization that works on mathematical research related to Friendly AI. Yudkowsky co-founded MIRI in 2000, and is the senior researcher there.
• magisterium. Stephen Gould’s term for a domain where some community or field has authority. Gould claimed that science and religion were separate and non-overlapping magisteria. On his view, religion has authority to answer questions of “ultimate meaning and moral value” (but not empirical fact) and science has authority to answer questions of empirical fact (but not meaning or value).
• map and territory. A metaphor for the relationship between beliefs (or other mental states) and the real-world things they purport to refer to.
• materialism. The belief that all mental phenomena can in principle be reduced to physical phenomena.
• Maxwell’s Demon. A hypothetical agent that knows the location and speed of individual molecules in a gas. James Maxwell used this demon in a thought experiment to show that such knowledge could decrease a physical system’s entropy, “in contradiction to the second law of thermodynamics.” The demon’s ability to identify faster molecules allows it to gather them together and extract useful work from them. Leó Szilárd later pointed out that if the demon itself were considered part of the thermodynamic system, then the entropy of the whole would not decrease. The decrease in entropy of the gas would require an increase in the demon’s entropy. Szilárd used this insight to simplify Maxwell’s scenario into a hypothetical engine that extracts work from a single gas particle. Using one bit of information about the particle (e.g., whether it’s in the top half of a box or the bottom half), a Szilárd engine can generate log2(kT) joules of energy, where T is the system’s temperature and k is Boltzmann’s constant.
• Maxwell’s equations. In classical physics, a set of differential equations that model the behavior of electromagnetic fields.
• meme. Richard Dawkins’s term for a thought that can be spread through social networks.
• meta level. A domain that is more abstract or derivative than some domain it depends on, the "object level." A conversation can be said to operate on a meta level, for example, when it switches from discussing a set of simple or concrete objects to discussing higher-order or indirect features of those objects.
• metaethics. A theory about what it means for ethical statements to be correct, or the study of such theories. Whereas applied ethics speaks to questions like "Is murder wrong?" and "How can we reduce the number of murders?", metaethics speaks to questions like "What does it mean for something to be wrong?" and "How can we generally distinguish right from wrong?"
• minimax. A decision rule for turn-based zero-sum two-player games, where one picks moves that minimize one's opponent’s chance of winning when their moves maximize their chance of winning. This rule is intended to perform well even in worst-case scenarios where one’s opponent makes excellent decisions.
• Minimum Message Length Principle. A formalization of Occam’s Razor that judges the probability of a hypothesis based on how long it would take to communicate the hypothesis plus the available data. Simpler hypotheses are favored, as are hypotheses that can be used to concisely encode the data.
• MIRI. See “Machine Intelligence Research Institute.”
• money pump. A person who is irrationally willing to accept sequences of trades that add up to an expected loss.
• monotonic logic. A logic that will always continue to assert something as true if it ever asserted it as true. For example, if “2+2=4” is proved, then in a monotonic logic no subsequent operation can make it impossible to derive that theorem again in the future. In contrast, non-monotonic logics can “forget” past conclusions and lose the ability to derive them.
• monotonicity. In mathematics, the property, loosely speaking, of always moving in the same direction (when one moves at all). If I have a preference ordering over outcomes, a monotonic change to my preferences may increase or decrease how much I care about various outcomes, but it won’t change the order -- if I started off liking cake more than cookies, I’ll end up liking cake more than cookies, though any number of other changes may have taken place. Alternatively, a monotonic function can flip all of my preferences. The only option ruled out is for the function to sometimes flip the ordering and sometimes preserve the ordering. A non-monotonic function, then, is one that at least once take an x<y input and outputs x>y, and at least once takes an x>y and outputs x<y.
• Moore’s Law. The observation that technological progress has enabled engineers to double the number of transistors they can fit on an integrated circuit approximately every two years from the 1960s to the 2010s. Other exponential improvements in computing technology (some of which have also been called “Moore’s Law”) may continue to operate after the end of the original Moore’s Law. The most important of these is the doubling of available computations per dollar. The futurist Ray Kurzweil has argued that the latter exponential trend will continue for many decades, and that this trend will determine rates of AI progress.
• motivated cognition. Reasoning and perception that is driven by some goal or emotion of the reasoner that is at odds with accuracy. Examples of this include non-evidence-based inclinations to reject a claim (motivated skepticism), to believe a claim (motivated credulity), to continue evaluating an issue (motivated continuation), or to stop evaluating an issue (motivated stopping).
• Murphy’s law. The saying “Anything that can go wrong will go wrong.”
• mutual information. For two variables, the amount that knowing about one variable tells you about the other's value. If two variables have zero mutual information, then they are independent; knowing the value of one does nothing to reduce uncertainty about the other.
• nanotechnology. Technologies based on the fine-grained control of matter on a scale of molecules, or smaller. If known physical law (or the machinery inside biological cells) is any guide, it should be possible in the future to design nanotechnological devices that are much faster and more powerful than any extant machine.
• Nash equilibrium. A situation in which no individual would benefit by changing their own strategy, assuming the other players retain their strategies. Agents often converge on Nash equilibria in the real world, even when they would be much better off if multiple agents simultaneously switched strategies. For example, mutual defection is the only Nash equilibrium in the standard one-shot Prisoner’s Dilemma (i.e., it is the only option such that neither player could benefit by changing strategies while the other player’s strategy is held constant), even though it is not Pareto-optimal (i.e., each player would be better off if the group behaved differently).
• natural selection. The process by which heritable biological traits change in frequency due to their effect on how much their bearers reproduce.
• negentropy. Negative entropy. A useful concept because it allows one to think of thermodynamic regularity as a limited resource one can possess and make use of, rather than as a mere absence of entropy.
• Neutral Point of View. A policy used by the online encyclopedia Wikipedia to instruct users on how they should edit the site’s contents. Following this policy means reporting on the different positions in controversies, while refraining from weighing in on which position is correct.
• Newcomb’s Problem. A central problem in decision theory. Imagine an agent that understands psychology well enough to predict your decisions in advance, and decides to either fill two boxes with money, or fill one box, based on their prediction. They put $1,000 in a transparent box no matter what, and they then put $1 million in an opaque box if (and only if) they predicted that you’d only take the opaque box. The predictor tells you about this, and then leaves. Which do you pick? If you take both boxes, you get only the $1000, because the predictor foresaw your choice and didn’t fill the opaque box. On the other hand, if you only take the opaque box, you come away with $1M. So it seems like you should take only the opaque box. However, many people object to this strategy on the grounds that you can’t causally control what the predictor did in the past; the predictor has already made their decision at the time when you make yours, and regardless of whether or not they placed the $1M in the opaque box, you’ll be throwing away a free $1000 if you choose not to take it. This view that we should take both boxes is prescribed by causal decision theory, which (for much the same reason) prescribes defecting in Prisoner’s Dilemmas (even if you’re playing against a perfect atom-by-atom copy of yourself).
• nonmonotonic logic. See “monotonic logic.”
• normality. (a) What’s commonplace. (b) What’s expected, prosaic, and unsurprising. Categorizing things as “normal” or weird” can cause one to conflate these two definitions, as though something must be inherently extraordinary or unusual just because one finds it surprising or difficult to predict. This is an example of confusing a feature of mental maps with a feature of the territory.
• normalization. Adjusting values to meet some common standard or constraint, often by adding or multiplying a set of values by a constant. E.g., adjusting the probabilities of hypotheses to sum to 1 again after eliminating some hypotheses. If the only three possibilities are A, B, and C, each with probability 1/3, then evidence that ruled out C (and didn’t affect the relative probability of A and B) would leave us with A at 1/3 and B at 1/3. These values must be adjusted (normalized) to make the space of hypotheses sum to 1, so A and B change to probability 1/2 each.
• normativity. A generalization of morality to include other desirable behaviors and outcomes. If it would be prudent and healthy and otherwise a good idea or me to go jogging, then there is a sense in which I should go jogging, even if it I’m not morally obliged to do so. Prescriptions about what one ought to do are normative, even when the kind of ‘ought’ involved isn’t moral or interpersonal.
• NP-complete. The hardest class of decision problems within the class NP, where NP consists of the problems that an ideal computer (specifically, a deterministic Turing machine) could efficiently verify correct answers to. The difficulty of NP-complete problems is such that if an algorithm were discovered to efficiently solve even one NP-complete problem, that algorithm would allow one to efficiently solve every NP problem. Many computer scientists hypothesize that this is impossible, a conjecture called “P ≠ NP.”
• null-op. A null operation; an action that does nothing in particular.
• object level. A base-case domain, especially one that is relatively concrete -- e.g., the topic of a conversation, or the target of an action. One might call one’s belief that murder is wrong "object-level" to contrast it with a meta-level belief about moral beliefs, or about the reason murder is wrong, or about something else that pertains to murder in a relatively abstract and indirect way.
• objective. (a) Remaining real or true regardless of what one’s opinions or other mental states are. (b) Conforming to generally applicable moral or epistemic norms (e.g., fairness or truth) rather than to one’s biases or idiosyncrasies. (c) Perceived or acted on by an agent. (d) A goal.
• Objectivism. A philosophy and social movement invented by Ayn Rand, known for promoting self-interest and laissez-faire capitalism as “rational.”
• Occam’s Razor. The principle that, all else being equal, a simpler claim is more probable than a relatively complicated one. Formalizations of Occam’s Razor include Solomonoff induction and the Minimum Message Length Principle.
• odds ratio. A way of representing how likely two events are relative to each other. E.g., if I have no information about which day of the week it is, the odds are 1:6 that it’s Sunday. If x:y is the odds ratio, the probability of x is x / (x + y); so the prior probability that it’s Sunday is 1/7. Likewise, if p is my probability and I want to convert it into an odds ratio, I can just write p : (1 - p). For a percent probability, this becomes p : (100 - p). If my probability of winning a race is 40%, my odds are 40:60, which can also be written 2:3. Odds ratios a are useful formalism because they are easy to update. If I notice that the mall is closing early, and that’s twice as likely to happen on a Sunday as it is on a non-Sunday (a likelihood ratio of 2:1), I can simply multiply the left and right sides of my prior it’s Sunday (1:6) by the evidence’s likelihood ratio (2:1) to arrive at a correct posterior probability of 2:6, or 1:3. This means that if I guess it’s Sunday, I should expect to be right 1/4 of the time -- 1 time for every 3 times I’m wrong. This is usually faster to calculate than Bayes’s rule for real-numbered probabilities.
• OLPC. See “One Laptop Per Child.”
• Omega. A hypothetical arbitrarily powerful agent used in various thought experiments.
• One Laptop Per Child. A program to distribute cheap laptops to poor children.
• ontology. An account of the things that exist, especially one that focuses on their most basic and general similarities. Things are “ontologically distinct” if they are of two fundamentally different kinds.
• OpenCog. An open-source AGI project based in large part on work by Ben Goertzel. MIRI provided seed funding to OpenCog in 2008, but subsequently redirected its research efforts elsewhere.
• opportunity cost. The value lost from choosing not to acquire something valuable. If I choose not to make an investment that would have earned me $10, I don’t literally lose $10 -- if I had $100 at the outset, I’ll still have $100 at the end, not $90. Still, I pay an opportunity cost of $10 for missing a chance to gain something I want. I lose $10 relative to the $110 I could have had. Opportunity costs can result from making a bad decision, but they also occur when you make a good decision that involves sacrificing the benefits of inferior options for the different benefits of a superior option. Many forms of human irrationality involve assigning too little importance to opportunity costs.
• optimization process. Yudkowsky’s term for an agent or agent-like phenomenon that produce surprisingly specific (e.g., rare or complex) physical structures. A generalization of the idea of efficiency and effectiveness, or “intelligence.” The formation of water molecules and planets isn’t “surprisingly specific,” in this context, because it follows in a relatively simple and direct way from garden-variety particle physics. For similar reasons, the existence of rivers does not seem to call for a particularly high-level or unusual explanation. On the other hand, the existence of trees seems too complicated for us to usefully explain it without appealing to an optimization process such as evolution. Likewise, the arrangement of wood into a well-designed dam seems too complicated to usefully explain without appealing to an optimization process such as a human, or a beaver.
• oracle. See “halting oracle.”
• orthogonality. A generalization of the property of being at a right angle to something. Perpendicularity, as it applies to lines in higher-dimensional spaces. If two variables are orthogonal, then knowing the value of one doesn’t tell you the value of the other.
• Overcoming Bias. The blog where Yudkowsky originally wrote most of the content of Rationality: From AI to Zombies. It can be found at [www.overcomingbias.com], where it now functions as the personal blog of Yudkowsky’s co-blogger, Robin Hanson. Most of Yudkowsky’s writing is now hosted on the community blog Less Wrong.
• P ≠ NP. A widely believed conjecture in computational complexity theory. NP is the class of mathematically specifiable questions with input parameters (e.g., “can a number list A be partitioned into two number lists B and C whose numbers sum to the same value?”) such that one could always in principle efficiently confirm that a correct solution to some instance of the problem (e.g., “the list {3,2,7,3,5} splits up into the lists {3,2,5} and {7,3}, and the latter two lists sum to the same number”) is in fact correct. More precisely, NP is the class of decision problems that a deterministic Turing machine could verify answers to in a polynomial amount of computing time. P is the class of decision problems that one could always in principle efficiently solve -- e.g., given {3,2,7,3,5} or any other list, quickly come up with a correct answer (like “{3,2,5} and {7,3}”) should one exist. Since all P problems are also NP problems, for P to not equal NP would mean that some NP problems are not P problems; i.e., some problems cannot be efficiently solved even though solutions to them, if discovered, could be efficiently verified.
• Pareto optimum. A situation in which no one can be made better off without making at least one person worse off.
• pebble and bucket. An example of a system for mapping reality, analogous to memory or belief. One picks some variable in the world, and places pebbles in the bucket when the variable’s value (or one’s evidence for its value) changes. The point of this illustrative example is that the mechanism is very simple, yet achieves many of the same goals as properties that see heated philosophical debate, such as perception, truth, knowledge, meaning, and reference.
• phase space. A mathematical representation of physical systems in which each axis of the space is a degree of freedom (a property of the system that must be specified independently) and each point is a possible state.
• phlogiston. A substance hypothesized in the 17th entity to explain phenomena such as fire and rust. Combustible objects were thought by late alchemists and early chemists to contain phlogiston, which evaporated during combustion.
• photon. An elementary particle of light.
• physicalism. See “materialism.”
• Planck units. Natural units, such as the Planck length and the Planck time, representing the smallest physically significant quantized phenomena.
• positive bias. Bias toward noticing what a theory predicts you’ll see instead of noticing what a theory predicts you won’t see.
• possible world. A way the world could have been. One can say “there is a possible world in which Hitler won World War II” in place of “Hitler could have won World War II,” making it easier to contrast the features of multiple hypothetical or counterfactual scenarios. Not to be confused with the worlds of the many-worlds interpretation of quantum physics or Max Tegmark's Mathematical Universe Hypothesis, which are claimed (by their proponents) to be actual.
• posterior probability. An agent’s beliefs after acquiring evidence. Contrasted with its prior beliefs, or priors.
• prior probability. An agent’s information -- beliefs, expectations, etc. -- before acquiring some evidence. The agent’s beliefs after processing the evidence are its posterior probability.
• Prisoner’s Dilemma. A game in which each player can choose to either "cooperate" or "defect" with the other. The best outcome for each player is to defect while the other cooperates; and the worst outcome is to cooperate while the other defects. Mutual cooperation is second-best, and mutual defection is second-worst. On conventional analyses, this means that defection is always the correct move; it improves your reward if the other player independently cooperates, and it lessens your loss if the other player independently defects. This leads to the pessimistic conclusion that many real-world conflicts that resemble Prisoner’s Dilemmas will inevitably end in mutual defection even though both players would be better off if they could find a way to force themselves to mutually cooperate. A minority of game theorists argue that mutual cooperation is possible even when the players cannot coordinate, provided that the players are both rational and both know that they are both rational. This is because two rational players in symmetric situations should pick the same option; so each player knows that the other player will cooperate if they cooperate, and will defect if they defect.
• probability. A number representing how likely a statement is to be true. Bayesians favor using the mathematics of probability to describe and prescribe subjective states of belief, whereas frequentists generally favor restricting probability to objective frequencies of events.
• probability theory. The branch of mathematics concerned with defining statistical truths and quantifying uncertainty.
• problem of induction. In philosophy, the question of how we can justifiably assert that the future will resemble the past (scientific induction) without relying on evidence that presupposes that very fact.
• proposition. Something that is either true or false. Commands, requests, questions, cheers, and excessively vague or ambiguous assertions are not propositions in this strict sense. Some philosophers identify propositions with sets of possible worlds -- that is, they think of propositions like “snow is white” not as particular patterns of ink in books, but rather as the thing held in common by all logically consistent scenarios featuring white snow. This is one way of abstracting away from how sentences are worded, what language they are in, etc., and merely discussing what makes the sentences true or false. (In mathematics, the word “proposition” has separately been used to refer to theorems -- e.g., “Euclid’s First Proposition.”)
• quantum mechanics. The branch of physics that studies subatomic phenomena and their nonclassical implications for larger structures; also, the mathematical formalisms used by physicists to predict such phenomena. Although the predictive value of such formalisms is extraordinarily well-established experimentally, physicists continue to debate how to incorporate gravitation into quantum mechanics, whether there are more fundamental patterns underlying quantum phenomena, and why the formalisms require a “Born rule” to relate the deterministic evolution of the wavefunction under Schrödinger’s equation to observed experimental outcomes. Related to the last question is a controversy in philosophy of physics over the physical significance of quantum-mechanical concepts like “wavefunction,” e.g., whether this mathematical structure in some sense exists objectively, or whether it is merely a convenience for calculation.
• quark. An elementary particle of matter.
• quine. A program that outputs its own source code.
• rationalist. A person interested in rationality, especially one who is attempting to use new insights from psychology and the formal sciences to become more rational.
• rationality. The property of employing useful cognitive procedures. Making systematically good decisions (instrumental rationality) based on systematically accurate beliefs (epistemic rationality).
• recursion. A sequence of similar actions that each build on the result of the previous action.
• reductio ad absurdum. Refuting a claim by showing that it entails a claim that is more obviously false.
• reduction. An explanation of a phenomenon in terms of its origin or parts, especially one that allows you to redescribe the phenomenon without appeal to your previous conception of it.
• reductionism. (a) The practice of scientifically reducing complex phenomena to simpler underpinnings. (b) The belief that such reductions are generally possible.
• representativeness heuristic. A cognitive heuristic where one judges the probability of an event based on how well it matches some mental prototype or stereotype.
• Ricardo’s Law of Comparative Advantage. See “comparative advantage.”
• satori. In Zen Buddhism, a non-verbal, pre-conceptual apprehension of the ultimate nature of reality.
• Schrödinger equation. A fairly simple partial differential equation that defines how quantum wavefunctions evolve over time. This equation is deterministic; it is not known why the Born rule, which converts the wavefunction into an experimental prediction, is probabilistic, though there have been many attempts to make headway on that question.
• scope insensitivity. A cognitive bias where large changes in an important value have little or no effect on one's behavior.
• screening off. Making something informationally irrelevant. A piece of evidence A screens off a piece of evidence B from a hypothesis C if, once you know about A, learning about B doesn’t affect the probability of C.
• search tree. A graph with a root node that branches into child nodes, which can then either terminate or branch once more. The tree data structure is used to locate values; in chess, for example, each node can represent a move, which branches into the other player’s possible responses, and searching the tree is intended to locate winning sequences of moves.
• self-anchoring. Anchoring to oneself. Treating one’s own qualities as the default, and only weakly updating toward viewing others as different when given evidence of differences.
• separate magisteria. See “magisterium.”
• sequences. Yudkowsky’s name for short series of thematically linked blog posts or essays.
• set theory. The study of relationships between abstract collections of objects, with a focus on collections of other collections. A branch of mathematical logic frequently used as a foundation for other mathematical fields.
• Shannon entropy. See “entropy.”
• Shannon mutual information. See “mutual information.”
• Simulation Hypothesis. The hypothesis that the world as we know it is a computer program designed by some powerful intelligence. An idea popularized in the movie The Matrix, and discussed more seriously by the philosopher Nick Bostrom.
• Singularity. One of several claims about a radical future increase in technological advancement. Kurzweil’s “accelerating change” singularity claims that there is a general, unavoidable tendency for technology to improve faster and faster. Vinge’s “event horizon” singularity claims that intelligences will develop that are too advanced for humans to model. Yudkowsky’s “intelligence explosion” singularity claims that self-improving AI will improve its own ability to self-improve, thereby rapidly achieving superintelligence. These claims are often confused with one another.
• Singularity Summit. An annual conference held by MIRI from 2006 to 2012. Purchased by Singularity University in 2013.
• skyhook. An attempted explanation of a complex phenomenon in terms of a deeply mysterious or miraculous phenomenon -- often one of even greater complexity.
• Solomonoff induction. An attempted definition of optimal (albeit computationally unfeasible) reasoning. A combination of Bayesian updating with a simplicity prior that assigns less probability to percept-generating programs the longer they are.
• stack trace. A retrospective step-by-step report on a program's behavior, intended to reveal the source of an error.
• strawman. An indefensible claim that is wrongly attributed to someone whose actual position is more plausible.
• subjective. (a) Conscious, experiential. (b) Dependent on the particular distinguishing features (e.g., mental states) of agents. (c) Playing favorites, disregarding others’ knowledge or preferences, or otherwise violating some norm as a result of personal biases. Importantly, something can be subjective in sense (a) or (b) without being subjective in sense (c); e.g., one’s ice cream preferences and childhood memories are “subjective” in a perfectly healthy sense.
• subjectivism. See “Berkeleian idealism.”
• superintelligence. An agent much smarter (more intellectually resourceful, rational, etc.) than present-day humans. This can be a purely hypothetical agent (e.g., Omega or Laplace’s demon), or it can be a predicted future technology (e.g., Friendly or Unfriendly AI).
• System 1. The brain’s fast, automatic, emotional, and intuitive judgments.
• System 2. The brain’s slow, deliberative, reflective, and intellectual judgments.
• Szilárd engine. See “Maxwell’s Demon.”
• Taboo. A game by Hasbro where you try to get teammates to guess what word you have in mind while avoiding conventional ways of communicating it. Yudkowsky uses this as an analogy for the rationalist skill of linking words to the concrete evidence you use to decide when to apply them. Ideally, one should be know what one is saying well enough to paraphrase the message in several different ways, and to replace abstract generalizations with concrete observations.
• Tegmark world. A mathematical structure resembling our universe, in Max Tegmark’s Mathematical Universe Hypothesis. Tegmark argues that our universe is mathematical in nature, and that it is contained in a vast ensemble in which all possible computable structures exist.
• terminal value. A goal that is pursued for its own sake, and not just to further some other goal.
• territory. See “map and territory.”
• theorem. A statement that has been mathematically or logically proven.
• Tit for Tat. A strategy in which one cooperates on the first round of an Iterated Prisoner’s Dilemma, then on each subsequent rounds mirrors what the opponent did the previous round.
• Traditional Rationality. Yudkowsky’s term for the scientific norms and conventions espoused by thinkers like Richard Feynman, Carl Sagan, and Charles Peirce. Yudkowsky contrasts this with the ideas of rationality in contemporary mathematics and cognitive science.
• truth-value. A proposition’s truth or falsity. True statements and false statements have truth-values, but questions, imperatives, strings of gibberish, etc. do not. “Value” is meant here in a mathematical sense, not a moral one.
• Turing-computability. The ability to be executed, at least in principle, by a simple process following a finite set of rules. "In principle" here means that a Turing machine could perform the computation, though we may lack the time or computing power to build a real-world machine that does the same. Turing-computable functions cannot be computed by all Turing machines, but they can be computed by some. In particular, they can be computed by all universal Turing machines.
• Turing machine. An abstract machine that follows rules for manipulating symbols on an arbitrarily long tape. It is impossible to build a true Turing machine with finite resources, but such machines are a very useful mathematical fiction for distilling the basic idea of computation.
• Type-A materialism. David Chalmers’s term for the view that the world is purely physical, and that there is no need to try to explain the relationship between the physical facts and the facts of first-person conscious experience. Type-A materialists deny that there is even an apparent mystery about why philosophical zombies seem conceivable. Other varieties of materialist accept that this is a mystery, but expect it to be solved eventually, or deny that the lack of a solution undermines physicalism.
• Unfriendly AI. A hypothetical smarter-than-human artificial intelligence that causes a global catastrophe by pursuing a goal without regard for humanity’s well-being. Yudkowsky predicts that superintelligent AI will be “Unfriendly” by default, unless a special effort goes into researching how to give AI stable, known, humane goals. Unfriendliness doesn’t imply malice, anger, or other human characteristics; a completely impersonal optimization process can be “Unfriendly” even if its only goal is to make paperclips. This is because even a goal as innocent as ‘maximize the expected number of paperclips’ could motivate an AI to treat humans as competitors for physical resources, or as threats to the AI’s aspirations.
• universal Turing machine. A Turing machine that can compute all Turing-computable functions. If something can be done by any Turing machine, then it can be done by every universal Turing machine. A system that can in principle do anything a Turing machine could is called “Turing-complete."
• updating. Revising one’s beliefs in light of new evidence. If the updating is epistemically rational -- that is, if it follows the follows the rules of probability theory -- then it counts as Bayesian inference.
• utilitarianism. An ethical theory asserting that one should act in whichever causes the most benefit to people, minus how much harm results. Standard utilitarianism argues that acts can be justified even if they are morally counter-intuitive and harmful, provided that the benefit outweighs the harm.
• utility. The amount some outcome satisfies a set of goals, as defined by a utility function.
• utility function. A function that ranks outcomes by how well they satisfy some set of goals.
• utility maximizer. An agent that always picks actions with better outcomes over ones with worse outcomes (relative to its utility function). An expected utility maximizer is more realistic, given that real-world agents must deal with ignorance and uncertainty: it picks the actions that are likeliest to maximize its utility, given the available evidence. An expected utility maximizer’s decisions would sometimes be suboptimal in hindsight, or from an omniscient perspective; but they won’t be foreseeably inferior to any alternative decision, given the agent’s available evidence. Humans can sometimes be usefully modeled as expected utility maximizers with a consistent utility function, but this is at best an approximation, since humans are not perfectly rational.
• utilon. Yudkowsky’s name for a unit of utility, i.e., something that satisfies a goal. The term is deliberately vague, to permit discussion of desired and desirable things without relying on imperfect proxies such as monetary value and self-reported happiness.
• vertex. See “graph.”
• wavefunction. A complex-valued function used in quantum mechanics to explain and predict the wave-like behavior of physical systems at small scales. Realists about the wavefunction treat it as a good characterization of the way the world really is, more fundamental than earlier (e.g., atomic) models. Anti-realists disagree, although they grant that the wavefunction is a useful tool by virtue of its mathematical relationship to observed properties of particles (the Born rule).
• winning. Yudkowsky’s term for getting what you want. The result of instrumental rationality.
• wu wei. “Non-action.” The concept, in Daoism, of effortlessly achieving one’s goals by ceasing to strive and struggle to reach them.
• XML. Extensible Markup Language, a system for annotating texts with tags that can be read both by a human and by a machine.
• ZF. The Zermelo–Fraenkel axioms, an attempt to ground standard mathematics in set theory. ZFC (the Zermelo–Fraenkel axioms supplemented with the Axiom of Choice) is the most popular axiomatic set theory.
• zombie. In philosophy, a perfect atom-by-atom replica of a human that lacks a human’s subjective awareness. Zombies behave exactly like humans, but they lack consciousness. Some philosophers argue that the idea of zombies is coherent -- that zombies, although not real, are at least logically possible. They conclude from this that facts about first-person consciousness are logically independent of physical facts, that our world breaks down into both physical and nonphysical components. Most philosophers reject the idea that zombies are logically possible, though the topic continues to be actively debated. |
fd5a42a1a3e819be | Saturday, February 9, 2008
Quantum field theory (QFT) provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed.
Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.
Origin of theory
Quantum mechanics in general deals with operators acting upon a (separable) Hilbert space. For a single nonrelativistic particle, the fundamental operators are its position and momentum,
hat{mathbf{x}}(t) and hat{mathbf{p}}(t).
These operators are time dependent in the Heisenberg picture, but we may also choose to work in the Schrödinger picture or (in the context of perturbation theory) the interaction picture.
Quantum field theory is a special case of quantum mechanics in which the fundamental operators are an operator-valued field
A single scalar field describes a spinless particle. More fields are necessary for more types of particles, or for particles with spin. For example, particles with spin are usually described by higher order tensor or spinor-valued (or matrix-valued) tensor fields which in turn can be reinterpreted as a possibly large set of scalar fields with appropriate transformation rules as one changes the system of coordinates used.
In quantum field theory, the energy is given by the Hamiltonian operator, which can be constructed from the quantum fields; it is the generator of infinitesimal time translations. (Being able to construct the generator of infinitesimal time translations out of quantum fields means many unphysical theories are ruled out, which is a good thing.)In order for the theory to be sensible, the Hamiltonian must be bounded from below. The lowest energy eigenstate (which may or may not be degenerate) is called the vacuum in particle physics and the ground state in condensed matter physics (QFT appears in the continuum limit of condensed matter systems).
Quantum field theory corrects several limitations of ordinary quantum mechanics. The time-dependent Schrödinger equation, in its most commonly encountered form, is
left[ frac{|mathbf{p}|^2}{2m} + V(mathbf{r}) right><br /> |psi(t)rang = i hbar frac{partial}{partial t} |psi(t)rang, where |psirang denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V.
Quantum field theory
QFT corrections to quantum mechanics
As described in the article on identical particles, quantum-mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as
|phi_1 cdots phi_N rang = sqrt{frac{prod_j N_j!}{N!}} sum_{pin S_N} |phi_{p(1)}rang cdots |phi_{p(N)} rang,
where |phi_irang are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro's number, approximately 10.
Large numbers of particles.
It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.
Schrödinger equation and special relativity
Quantum field theory Quantizing a classical field theory
Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is:
1. Each normal mode oscillation of the field is interpreted as a particle with frequency f.
2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles.
The energy associated with the mode of excitation is therefore E = (n+1/2)hbaromega which directly follows from the energy eigenvalues of a one dimensional harmonic oscillator in quantum mechanics. With some thought, one may similarly associate momenta and position of particles with observables of the field.
Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves.
Two caveats should be made before proceeding further:
The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.
Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.
Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system. Canonical quantization
Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |phi_1rang, |phi_2rang, |phi_3rang, and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state |phi_1rang and two in state|phi_2rang. The normal way of writing the wavefunction is
frac{1}{sqrt{3}} left[ |phi_1rang |phi_2rang<br /> |phi_2rang + |phi_2rang |phi_1rang |phi_2rang + |phi_2rang<br /> |phi_2rang |phi_1rang right>. In second quantized form, we write this as
|1, 2, 0, 0, 0, cdots rangle,
which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."
Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a2 and creation operator a_2^dagger have the following effects:
a_2 | N_1, N_2, N_3, cdots rangle = sqrt{N_2} mid N_1, (N_2 - 1), N_3, cdots rangle,
a_2^dagger | N_1, N_2, N_3, cdots rangle = sqrt{N_2 + 1} mid N_1, (N_2 + 1), N_3, cdots rangle.
We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way many have written them.
The bosonic creation and annihilation operators obey the commutation relation
<br /> left[a_i , a_j right> = 0 quad,quad<br /> left[a_i^dagger , a_j^dagger right] = 0 quad,quad<br /> left[a_i , a_j^dagger right] = delta_{ij},<br /> where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.
The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is
H = sum_k E_k , a^dagger_k ,a_k,
where Ek is the energy of the k-th single-particle energy eigenstate. Note that
a_k^dagger,a_k|cdots, N_k, cdots rangle=N_k| cdots, N_k, cdots rangle.
Canonical quantization for bosons
It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers Ni can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators c^dagger by
c_j | N_1, N_2, cdots, N_j = 0, cdots rangle = 0
c_j | N_1, N_2, cdots, N_j = 1, cdots rangle = (-1)^{(N_1 + cdots + N_{j-1})} | N_1, N_2, cdots, N_j = 0, cdots rangle
c_j^dagger | N_1, N_2, cdots, N_j = 0, cdots rangle = (-1)^{(N_1 + cdots + N_{j-1})} | N_1, N_2, cdots, N_j = 1, cdots rangle
c_j^dagger | N_1, N_2, cdots, N_j = 1, cdots rangle = 0
The fermionic creation and annihilation operators obey an anticommutation relation,
<br /> left{c_i , c_j right} = 0 quad,quad<br /> left{c_i^dagger , c_j^dagger right} = 0 quad,quad<br /> left{c_i , c_j^dagger right} = delta_{ij}<br />
Canonical quantization for fermions
When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator ak destroys a particle during an infinitesimal time step, the creation operator a_k^dagger to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.
On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by c_k^dagger and ck, we could add a "potential energy" term to our Hamiltonian such as:
V = sum_{k,q} V_q (a_q + a_{-q}^dagger) c_{k+q}^dagger c_k
This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k + q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.
Significance of creation and annihilation operators
We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.
phi(mathbf{r}) stackrel{mathrm{def}}{=} sum_{j} e^{imathbf{k}_jcdot mathbf{r}} a_{j}
The bosonic field operators obey the commutation relation
<br /> left[phi(mathbf{r}) , phi(mathbf{r'}) right> = 0 quad,quad<br /> left[phi^dagger(mathbf{r}) , phi^dagger(mathbf{r'}) right] = 0 quad,quad<br /> left[phi(mathbf{r}) , phi^dagger(mathbf{r'}) right] = delta^3(mathbf{r} - mathbf{r'})<br /> where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say
H = - frac{hbar^2}{2m} sum_i nabla_i^2 + sum_{i < j} U(|mathbf{r}_i - mathbf{r}_j|)
where the indices i and j run over all particles, then the field theory Hamiltonian is
H = - frac{hbar^2}{2m} int d^3!r ; phi(mathbf{r})^dagger nabla^2 phi(mathbf{r}) + int!d^3!r int!d^3!r' ; phi(mathbf{r})^dagger phi(mathbf{r}')^dagger U(|mathbf{r} - mathbf{r}'|) phi(mathbf{r'}) phi(mathbf{r})
This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.
Field operators
So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetism, so a quantum field theory must be formulated right from the start.
The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as
left[ q_i , p_j right> =i delta_{ij} For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential phi(mathbf{x}) and the vector potential mathbf{A}(mathbf{x}) at every point mathbf{x}. This is an uncountable set of variables, because mathbf{x} is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory. Only by accident electrons were not regarded as de Broglie waves and photons governed by geometrical optics were not the dominant theory when QFT was developed.
Quantization of classical fields
Path integral methods
The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics. Constructive quantum field theory is the construction of theories which satisfy one of these sets of axioms. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.
In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millennium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist.
The axiomatic approach
Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle's electric field. The energy density grows to infinity as one gets close to the charge.
A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture
|psi(t)rangle = e^{iH_It} |psi(0)rangle = left[1+iH_It-frac12 H_I^2t^2 -frac i{3!}H_I^3t^3 + frac1{4!}H_I^4t^4 + cdotsright> |psi(0)rangle. Taking the overlap with the initial state, one retains the even powers of HI. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave function renormalization and mass renormalization. Similar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory.
A gauge theory is a theory which admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical, so the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that in every point in space-time the shift is different. This is a local symmetry. However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In quantum electrodynamics this gauge field is the electromagnetic field. The change of local change of variables is termed gauge transformation.
Quantum electrodynamics, whose gauge transformation is a local change of phase, so that the gauge group is U(1). The gauge boson is the photon.
Quantum chromodynamics, whose gauge group is SU(3). The gauge bosons are eight gluons.
The electroweak Theory, whose gauge group is U(1)times SU(2) (a direct product of U(1) and SU(2)).
Gravity, whose classical theory is general relativity, admits the equivalence principle which is a form of gauge symmetry. Gauge theories
List of quantum field theories
Feynman path integral
Quantum chromodynamics
Quantum electrodynamics
Schwinger-Dyson equation
Relationship between string theory and quantum field theory
Abraham-Lorentz force
Photon polarization
Theoretical and experimental justification for the Schrödinger equation
Invariance mechanics Notes
Wilczek, Frank ; Quantum Field Theory, Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., Nobel laureate 2003. Full text available at : hep-th/9803075
Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X]. Introduction to relativistic Q.F.T. for particle physics.
Zee, Anthony ; Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6].
Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2]
Weinberg, Steven ; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979.
Loudon, Rodney ; The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0-19-851155-8]
Greiner, Walter and Müller, Berndt (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.
Paul H. Frampton , Gauge Field Theories, Frontiers in Physics, Addison-Wesley (1986), Second Edition, Wiley (2000).
Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5. |
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This work is licensed under a Creative Commons License.
The Elephant and the Event Horizon
26 October 2006
Exclusive from New Scientist Print Edition.
What happens when you throw an elephant into a black hole? It sounds like a bad joke, but it's a question that has been weighing heavily on Leonard Susskind's mind. Susskind, a physicist at Stanford University in California, has been trying to save that elephant for decades. He has finally found a way to do it, but the consequences shake the foundations of what we thought we knew about space and time. If his calculations are correct, the elephant must be in more than one place at the same time.
In everyday life, of course, locality is a given. You're over there, I'm over here; neither of us is anywhere else. Even in Einstein's theory of relativity, where distances and timescales can change depending on an observer's reference frame, an object's location in space-time is precisely defined. What Susskind is saying, however, is that locality in this classical sense is a myth. Nothing is what, or rather, where it seems.
This is more than just a mind-bending curiosity. It tells us something new about the fundamental workings of the universe. Strange as it may sound, the fate of an elephant in a black hole has deep implications for a "theory of everything" called quantum gravity, which strives to unify quantum mechanics and general relativity, the twin pillars of modern physics. Because of their enormous gravity and other unique properties, black holes have been fertile ground for researchers developing these ideas.
It all began in the mid-1970s, when Stephen Hawking of the University of Cambridge showed theoretically that black holes are not truly black, but emit radiation. In fact they evaporate very slowly, disappearing over many billions of years. This "Hawking radiation" comes from quantum phenomena taking place just outside the event horizon, the gravitational point of no return. But, Hawking asked, if a black hole eventually disappears, what happens to all the stuff inside? It can either leak back into the universe along with the radiation, which would seem to require travelling faster than light to escape the black hole's gravitational death grip, or it can simply blink out of existence.
Trouble is, the laws of physics don't allow either possibility. "We've been forced into a profound paradox that comes from the fact that every conceivable outcome we can imagine from black hole evaporation contradicts some important aspect of physics," says Steve Giddings, a theorist at the University of California, Santa Barbara.
blackhole imageResearchers call this the black hole information paradox. It comes about because losing information about the quantum state of an object falling into a black hole is prohibited, yet any scenario that allows information to escape also seems in violation. Physicists often talk about information rather than matter because information is thought to be more fundamental.
In quantum mechanics, the information that describes the state of a particle can't slip through the cracks of the equations. If it could, it would be a mathematical nightmare. The Schrödinger equation, which describes the evolution of a quantum system in time, would be meaningless because any semblance of continuity from past to future would be shattered and predictions rendered absurd. "All of physics as we know it is conditioned on the fact that information is conserved, even if it's badly scrambled," Susskind says.
For three decades, however, Hawking was convinced that information was destroyed in black hole evaporation. He argued that the radiation was random and could not contain the information that originally fell in. In 1997, he and Kip Thorne, a physicist at the California Institute of Technology in Pasadena, made a bet with John Preskill, also at Caltech, that information loss was real. At stake was an encyclopedia - from which they agreed information could readily be retrieved. All was quiet until July 2004, when Hawking unexpectedly showed up at a conference in Dublin, Ireland, claiming that he had been wrong all along. Black holes do not destroy information after all, he said. He presented Preskill with an encyclopedia of baseball.
What inspired Hawking to change his mind? It was the work of a young theorist named Juan Maldacena of the Institute for Advanced Study in Princeton, New Jersey. Maldacena may not be a household name, but he contributed what some consider to be the most ground-breaking piece of theoretical physics in the last decade. He did it using string theory, the most popular approach to understanding quantum gravity.
In 1997, Maldacena developed a type of string theory in a universe with five large dimensions of space and a contorted space-time geometry. He showed that this theory, which includes gravity, is equivalent to an ordinary quantum field theory, without gravity, living on the four-dimensional boundary of that universe. Everything happening on the boundary is equivalent to everything happening inside: ordinary particles interacting on the surface correspond precisely to strings interacting on the interior.
This is remarkable because the two worlds look so different, yet their information content is identical. The higher-dimensional strings can be thought of as a "holographic" projection of the quantum particles on the surface, similar to the way a laser creates a 3D hologram from the information contained on a 2D surface. Even though Maldacena's universe was very different from ours, the elegance of the theory suggested that our universe might be something of a grand illusion - an enormous cosmic hologram (New Scientist, 27 April 2002, p 22).
The holographic idea had been proposed previously by Susskind, one of the inventors of string theory, and by Gerard't Hooft of the University of Utrecht in the Netherlands. Each had used the fact that the entropy of a black hole, a measure of its information content, was proportional to its surface area rather than its volume. But Maldacena showed explicitly how a holographic universe could work and, crucially, why information could not be lost in a black hole.
According to his theory, a black hole, like everything else, has an alter ego living on the boundary of the universe. Black hole evaporation, it turns out, corresponds to quantum particles interacting on this boundary. Since no information loss can occur in a swarm of ordinary quantum particles, there can be no mysterious information loss in a black hole either. "The boundary theory respects the rules of quantum mechanics," says Maldacena. "It keeps track of all the information."
Of course, our universe still looks nothing like the one in Maldacena's theory. The results are so striking, though, that physicists have been willing to accept the idea, at least for now. "The opposition, including Hawking, had to give up," says Susskind. "It was so mathematically precise that for most practical purposes all theoretical physicists came to the conclusion that the holographic principle and the conservation of information would have to be true."
All well and good, but a serious problem remains: if the information isn't lost in a black hole, where is it? Researchers speculate that it is encoded in the black hole radiation (see "Black hole computers"). "The idea is that Hawking radiation is not random but contains subtle information on the matter that fell in," says Maldacena.
Susskind takes it a step further. Since the holographic principle leaves no room for information loss, he argues, no observer should ever see information disappear. That leads to a remarkable thought experiment.
Which brings us back to the elephant. Let's say Alice is watching a black hole from a safe distance, and she sees an elephant foolishly headed straight into gravity's grip. As she continues to watch, she will see it get closer and closer to the event horizon, slowing down because of the time-stretching effects of gravity in general relativity. However, she will never see it cross the horizon. Instead she sees it stop just short, where sadly Dumbo is thermalised by Hawking radiation and reduced to a pile of ashes streaming back out. From Alice's point of view, the elephant's information is contained in those ashes.
Inside or out?
There is a twist to the story. Little did Alice realise that her friend Bob was riding on the elephant's back as it plunged toward the black hole. When Bob crosses the event horizon, though, he doesn't even notice, thanks to relativity. The horizon is not a brick wall in space. It is simply the point beyond which an observer outside the black hole can't see light escaping. To Bob, who is in free fall, it looks like any other place in the universe; even the pull of gravity won't be noticeable for perhaps millions of years. Eventually as he nears the singularity, where the curvature of space-time runs amok, gravity will overpower Bob, and he and his elephant will be torn apart. Until then, he too sees information conserved.
Neither story is pretty, but which one is right? According to Alice, the elephant never crossed the horizon; she watched it approach the black hole and merge with the Hawking radiation. According to Bob, the elephant went through and floated along happily for eons until it turned into spaghetti. The laws of physics demand that both stories be true, yet they contradict one another. So where is the elephant, inside or out?
The answer Susskind has come up with is - you guessed it - both. The elephant is both inside and outside the black hole; the answer depends on who you ask. "What we've discovered is that you cannot speak of what is behind the horizon and what is in front of the horizon," Susskind says. "Quantum mechanics always involves replacing 'and' with 'or'. Light is waves or light is particles, depending on the experiment you do. An electron has a position or it has a momentum, depending on what you measure. The same is happening with black holes. Either we describe the stuff that fell into the horizon in terms of things behind the horizon, or we describe it in terms of the Hawking radiation that comes out."
Wait a minute, you might think. Maybe there are two copies of the information. Maybe when the elephant hits the horizon, a copy is made, and one version comes out as radiation while the other travels into the black hole. However, a fundamental law called the no-cloning theorem precludes that possibility. If you could duplicate information, you could circumvent the uncertainty principle, something nature forbids. As Susskind puts it, "There cannot be a quantum Xerox machine." So the same elephant must be in two places at once: alive inside the horizon and dead in a heap of radiating ashes outside.
The implications are unsettling, to say the least. Sure, quantum mechanics tells us that an object's location can't always be pinpointed. But that applies to things like electrons, not elephants, and it usually spans tiny distances, not light years. It is the large scale that makes this so surprising, Susskind says. In principle, if the black hole is big enough, the two versions of the same elephant could be separated by billions of light years. "People always thought quantum ambiguity was a small-scale phenomenon," he adds. "We're learning that the more quantum gravity becomes important, the more huge-scale ambiguity comes into play."
All this amounts to the fact that an object's location in space-time is no longer indisputable. Susskind calls this "a new form of relativity". Einstein took factors that were thought to be invariable - an object's length and the passage of time - and showed that they were relative to the motion of an observer. The location of an object in space or in time could only be defined with respect to an observer, but its location in space-time was certain. Now that notion has been shattered, says Susskind, and an object's location in space-time depends on an observer's state of motion with respect to a horizon.
What's more, this new type of "non-locality" is not just for black holes. It occurs anywhere a boundary separates regions of the universe that can't communicate with each other. Such horizons are more common than you might think. Anything that accelerates - the Earth, the solar system, the Milky Way - creates a horizon. Even if you're out running, there are regions of space-time from which light would never reach you if you kept speeding up. Those inaccessible regions are beyond your horizon.
As researchers forge ahead in their quest to unify quantum mechanics and gravity, non-locality may help point the way. For instance, quantum gravity should obey the holographic principle. That means there might be redundant information and fewer important dimensions of space-time in the theory. "This has to be part of the understanding of quantum gravity," Giddings says. "It's likely that this black hole information paradox will lead to a revolution at least as profound as the advent of quantum mechanics."
This paradox will lead to a revolution as profound as the birth of quantum mechanics
Black hole computers
According to Leonard Susskind of Stanford University, however, it makes no sense to talk about the location of information independent of an observer. To an outside observer, information never falls into the black hole in the first place. Instead, it is heated and radiated back out before ever crossing the horizon. The quantum computer model, he says, relies on the old notion of locality. "The location of a bit becomes ambiguous and observer-dependent when gravity becomes important," he says. So the idea of a black hole computer remains controversial.
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4f7a43f0f5a47d2f | A wave function or wavefunction is a probability amplitude? in quantum mechanics describing the quantum state of a particle or system of particles. Typically, it is a function of space or momentum or rotation and possibly of time that returns the probability amplitude? of a position or momentum for a subatomic particle. Mathematically, it is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
The common application is as a property of particles relating to their wave-particle duality, where it is denoted ψ(position,time) and where | ψ | 2 is equal to the chance of finding the subject at a certain time and position. For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.
A simple wave function is that for a particle in a box. Another simple example is a free particle (or a particle in a large box), whose wave function is a sinusoid where, in the spirit of the uncertainty principle?, the momentum is known but the position is not known.
The modern usage of the term wave function refers to a complex vector or function, i.e. an element in a complex Hilbert space?. Typically, a wave function is either:
a complex vector with finitely many components,
a complex vector with infinitely many components,
a complex function of one or more real variables (a continuously indexed complex vector).
In all cases, the wave function provides a complete description of the associated physical system. An element of a vector space can be expressed in different bases; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however the wave function itself is not dependent on the basis chosen. In this respect they are like spatial vectors in ordinary space because choosing a new set of cartesian axes by rotation of the coordinate frame does not alter the vector itself, only the representation of the vector with respect to the coordinate frame. A basis in quantum mechanics is analogous to the coordinate frame in that choosing a new basis does not alter the wavefunction, only its representation, which is expressed as the values of the components above. (underline added)
Because the probabilities that the system is in each possible state should add up to 1, the norm of the wave function must be 1. wikipedia - wavefunction (external link)
See Also
12.11 - Eighteen Attributes or Dimensions
Angular Momentum coupling
Born Oppenheimer approximation
magnetic moment
Quantum coupling
Renner-Teller Effect
Rotational-vibrational coupling
rovibronic coupling
Russell WaveFunction
Russell Wavefunction Equation
spin-orbit coupling
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aba10005cc1dc128 | Review of the fundamental theories behind small angle X-ray scattering, molecular dynamics simulations, and relevant integrated application
Lauren Boldon*, Fallon Laliberte and Li Liu*
Department of Mechanical Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA
Received: 6 August 2014; Revised: 24 November 2014; Accepted: 18 January 2015; Published: 25 February 2015
In this paper, the fundamental concepts and equations necessary for performing small angle X-ray scattering (SAXS) experiments, molecular dynamics (MD) simulations, and MD-SAXS analyses were reviewed. Furthermore, several key biological and non-biological applications for SAXS, MD, and MD-SAXS are presented in this review; however, this article does not cover all possible applications. SAXS is an experimental technique used for the analysis of a wide variety of biological and non-biological structures. SAXS utilizes spherical averaging to produce one- or two-dimensional intensity profiles, from which structural data may be extracted. MD simulation is a computer simulation technique that is used to model complex biological and non-biological systems at the atomic level. MD simulations apply classical Newtonian mechanics’ equations of motion to perform force calculations and to predict the theoretical physical properties of the system. This review presents several applications that highlight the ability of both SAXS and MD to study protein folding and function in addition to non-biological applications, such as the study of mechanical, electrical, and structural properties of non-biological nanoparticles. Lastly, the potential benefits of combining SAXS and MD simulations for the study of both biological and non-biological systems are demonstrated through the presentation of several examples that combine the two techniques.
Keywords: small angle X-ray scattering; molecular dynamics; protein folding; nanoparticles; MD-SAXS; atomistic simulation; ab initio; radius of gyration; pair distribution function; Newtonian equations of motion
Fallon Laliberte is a Ph.D. student studying under Dr. Li (Emily) Liu at Rensselaer Polytechnic Institute. Her interests are the use of small angle X-ray scattering (SAXS) and small angle neutron scattering (SANS) techniques to study the effects of radiation damage on materials, in particular semiconductors. Ms. Laliberte earned her M.Sc. in physics from the University of Rhode Island in 2013 and her B.A. in physics from the College of the Holy Cross in Worcester, MA in 2011.
Lauren Boldon is a Ph.D. student studying under Dr. Li (Emily) Liu at Rensselaer Polytechnic Institute (RPI) with a Department of Energy Nuclear Energy University Program Graduate Fellowship. She graduated with her B.S. and M.Eng in Nuclear Science and Engineering from RPI in 2012. Lauren is a member of the Diversity, Women’s Affairs and Outreach Committee and was Vice-Chair of the Student Advisory Council for the Mechanical, Aerospace, and Nuclear Engineering Department. Her research areas include small angle X-ray scattering and future energy systems incorporating small modular reactors.
Li (Emily) Liu, Associate Professor, Nuclear Engineering and Engineering Physics Program, Rensselaer Polytechnic Institute
As a Physicist and Nuclear Engineer by training, Prof. Liu’s research is focused on solving high-impact problems associated with energy and the environment through fundamental investigations into the structure-function relationships of materials. For this purpose, she is developing a variety of experimental and computational tools based on neutron, X-ray, and light scattering as well as molecular dynamics (MD) simulations. More importantly, her work focuses on direct nanoscale experimental validation of simulation results as well as the integration of simulation, experiments, and theories at various length scales.
Nano Reviews 2015. © 2015 Lauren Boldon et al. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), permitting all non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Citation: Nano Reviews 2015, 6: 25661 - http://dx.doi.org/10.3402/nano.v6.25661
The theory of small angle X-ray scattering (SAXS), molecular dynamics (MD) simulation, and the combination of the two techniques is reviewed along with potential applications in this article. SAXS is an experimental technique that has become very popular in the biological community with many substantial advancements made over the last few decades, particularly in data collection and analysis (1, 2). SAXS provides detailed structural analysis and physical information for a variety of 1–100 nm and beyond particle systems by characterizing average particle sizes and shapes (310). In a SAXS experiment, the sample is exposed to X-rays of a specific wavelength, which scatter elastically between 0 and 5 degrees to produce a spatially averaged intensity distribution (3, 4, 11, 12). In situ, static, and dynamical experiments may be performed to analyze samples with solid, liquid, and even gaseous components with crystalline, partially ordered, or randomly oriented structures (3). Data such as the pore size, specific inner surface, surface to volume ratio, solution structure factor, and lattice type and dimensions may be determined depending on the structure being analyzed (3).
Common methods of SAXS spatial or spherical averaging include the Debye formula, multipole expansion, numerical or spherical quadrature, Monte-Carlo sampling, Cubature formula, or Zernike polynomials, utilizing atomic, grid, and/or course grained models of the structure (11, 1325). The final intensity distribution provides significant information regarding the size, shape, and general structure of the sample through direct calculations of the radius of gyration, volume of correlation, and the Porod invariant in the Guinier, Fourier, and Porod regions, respectively (3, 12, 2628). The volume, mass, and pair distribution distances are just a few properties that may then be determined; the pair distribution, for example, is determined from an indirect Fourier transformation. Three-dimensional models, such as the ab initio model discussed in later sections, may be approximated to represent the most accurate fit to the SAXS profile (3, 11, 2933).
The spatial averaging required for a SAXS experiment results in low-resolution imaging, as it reduces structural information down to one or two dimensions dependent on the experiment. As a result, the extraction of three-dimensional structural information may be difficult (3, 2932, 34, 35). Other high-resolution techniques, such as X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, or even electron microscopy (EM), have limitations concerning the analysis of complex or dynamical configurations (11, 36). Despite the low resolution of SAXS imaging, it may be utilized to determine the structure of protein or macromolecular assemblies and to model the kinetics of the system over time (11, 3739). Furthermore, its application to polymers, nanoparticles, proteins, and so on with different organizational properties and physical states makes it very useful for samples that are otherwise difficult to examine experimentally. High-resolution techniques such as X-ray crystallography, NMR spectroscopy, and EM have limitations concerning the analysis of the complex conformational changes.
Developing higher resolution SAXS profiles requires supplemental geometric information, which may be derived from MD simulations. MD is a computer simulation technique, which models complex systems at the atomic level. An MD program simulates the motion of atoms by dividing the trajectory of the atoms into states and recording the velocity and position of each atom over time (40). The acting forces and the displacement of the particles are calculated for each time step to determine the new position and state of the particles in the system (40). To model systems of particles, MD simulations employ classical Newtonian mechanics to determine the forces acting on the system, which in turn provide information on the kinetic and thermodynamic properties of the system (40). The force field calculations provide information on the various features of the system at a particular time (41, 42) using the defined position, momentum, charge, bond information, and the potential energy functions (41, 42). Since most systems that are examined in an MD simulation are complex (i.e. more than one particle), it is necessary to calculate the potential functions and resulting forces for both the non-bonded atoms and bonded atoms comprising the system. From each of these potentials, a respective force is derived for the particle at every time step taken throughout the MD simulation.
Due to the complexity of both biological and non-biological systems, MD simulations are becoming increasingly popular for their power to predict and verify experimental results. They provide an opportunity to study the physical characteristics of systems that are not easily examined in the laboratory (43). For example, there is active research aimed toward enhancing the MD algorithms, so they may simulate protein folding and unfolding (4448). In addition to biological applications, MD simulations have been used to study the physical characteristics of non-biological nanoparticles (49). MD simulation is advantageous in the fields of biology, chemistry, physics, and engineering due to its ability to provide information on system dynamics at the atomic scale, yet has been sparingly utilized for non-biological applications.
Coupling SAXS and MD simulation, or MD-SAXS, holds tremendous potential, especially in the structural and mechanical analysis of complex particles like proteins and macromolecules with a multitude of conformational changes. SAXS techniques detail the folding patterns of these structures, while MD simulations model the movement between states. MD-SAXS may hold the key to furthering the study of nanoparticles and helping to overcome informational losses in SAXS experimentation due to spherical averaging and determining the theoretical profile, while retaining the inherent experimental flexibility of structures and states. Furthermore, advances in SAXS, such as time-resolved kinetic studies and the potential for ‘super-resolution’ will only further enhance the capabilities of MD-SAXS (50, 51). This review article will provide theory on SAXS, MD simulation, and MD-SAXS methods and illustrate several examples of each to demonstrate how the uses of MD-SAXS may be expanded to non-biological applications.
SAXS theory and methods
Calculating the theoretical intensity profile
In a SAXS experiment, the intensity is expressed as a function of the scattering vector q resulting from a photon of wavelength λ scattering off the sample at an angle 2θ (Equation 1) (3). For a macromolecule in solution, the intensity distribution of the macromolecule must be obtained by subtracting the buffer or solution profile from the total profile (3, 11).
The following parameters will directly alter the experimental intensity distribution of the system: particle size, volume, contrast (electron density), sample to detector distance, resolution, and beam collimation (3, 28). For calculating the theoretical intensity profile, all of these factors combined with the particle shape will result in greatly different profiles. To better detail the process, spherical and cylindrical particle profile equations are described followed by a flow chart. The electron contrast is calculated from the difference in the mean electron densities for the system and medium (12, 28).
The amplitude of the scattering intensity A(q) for a particle is a function of the electron radial density ρ(r) over particle volume Vp (Equations 23) (12, 28). For non-spherical shapes, there are amplitude components in each direction, such as the radial (Equation 4) and longitudinal directions for the cylindrical particle (Equations 5), where L is the length of the cylinder, R is the radius of the sphere or radial cylinder cross section, a is the cosine of the angle formed between the scattering vector and the z axis, and J1 is the first-order Bessel function (4, 10, 12, 28, 52, 53). For homogeneous spherical or cylindrical particles, the electron radial density ρ(r) is equal to unity (28, 52, 54).
The atomic form factor P(q) represents the particle shape or the interference pattern and is directly related to the size of the particle (1). P(q) is a function of the amplitude A(q) and the particle volume Vp (Equation 6) (10, 12, 50). For a sphere, the equation is straight forward, but for a cylinder, it becomes more complex (Equation 7) (12, 52, 54).
The theoretical intensity for a particle is a function of the form factor, electron contrast, and particle volume (Equation 8) (12). The total intensity profile is then the sum of each particle intensity for monodisperse particles (Equation 10) or a weighted, normalized summation for polydisperse particles (Equations 9 and 10) (12).
The overall process of calculating the intensity profile for comparison is outlined in a flow chart (Fig. 1). The equations may directly be utilized for spherical or disk/cylinder shaped particles.
Fig 1
Fig. 1. SAXS flow chart.
Comparing the theoretical and experimental intensity profiles
For an accurate fit between the experimental profile Iexp(qi) and the computed theoretical profile I(qi), the square root of the chi-squared term X must be minimized over the total number of data points or particles M (Equation 12) (11, 15):
where c is a scaling factor and σ(qi) is the experimental error. Different methods of calculating the theoretical intensity distribution will result in different statistical fits. For macromolecules in solution, several methods of spherical averaging with distinct representations – atomic, grid, or course grained – have been developed (5, 19, 20). As previously mentioned, variations on the Debye formula are commonly utilized (13, 14, 16, 22). Other methods, such as multipole expansion, may be used to speed up the calculation time (15). A balance must be met between the computational time and accuracy required for a specific application, resulting in many additional methods: Monte-Carlo sampling, numerical quadrature, cubature, and Zernike polynomial expansion (17, 18, 21, 23, 24).
The excluded volume and treatment of the solvent and hydration layer are significant for information on the shape of the particle in the SAXS profile calculation and are also noted within the brief description of each computational method (11). The excluded volume may be assumed to have an electron density equal to that of the bulk solvent (15, 55, 56); utilize explicit water molecules (21); or it may be adjusted based on the fit to the experimental data (15, 22, 23). Similarly, the hydration layer will also effect the profile and may be represented by explicit water molecules (19, 21, 57); solvent density maps (23, 58); or an implicit layer (15, 17, 22). Putnam (37) reviewed methods of constructing a theoretical SAXS profile, providing additional information regarding the equations presented in this paper and significant information on assessing different spherical averaging methods for macromolecular structures. Rambo and Tainer provide a more extensive review of the SAXS theories, including analytical and computational methods as they have developed since the early 1900s (51).
Extracting structural data from the SAXS experimental profile
The SAXS profile has three distinct regions from which information may be extracted: Guinier, Fourier, and Porod (Fig. 2).
Fig 2
Fig. 2. Regions of SAXS profile and data that may be extracted from each (3).
In the Guinier region, the experimental radius of gyration Rg may be approximated by fitting a line to the natural log of the intensity as a function of the square of the scattering vector q2 (3, 12, 26, 28). The radius of gyration will be greatly affected by aggregation of particles, polydispersity, and improper subtraction of the background or buffer.
In the Fourier region, the pair distribution function may be determined by an indirect Fourier transformation of the experimental form factor, providing significant information regarding the particle shape P(q) (Equation 13). It refers to the distribution of electrons averaged over a radius r. ρ(r) curves are used to determine the general particle shape, provided all the particles in the sample are of similar shape (3, 12, 26, 28, 59).
In the Porod region, the Porod invariant Q (Equation 14) may be determined, providing surface information such as the surface to volume ratio (Equation 15) and specific surface estimation for compact particles. The log of the form factor is proportional to q−4 for particles with uniform density (3, 4, 27, 28).
A Porod plot of q4I(q) vs q may provide other useful information such as the Porod volume and molecular weight for compact particles at high q values where asymptotic behavior dominates (3, 28). The Kratky plot of q2I(q) vs q is another useful tool particularly for protein analysis and determining whether disordered states are present in the sample or whether a protein is unfolded (28).
Ab initio shape determination
In small angle scattering, the three-dimensional shape may be reconstructed from the scattering patterns determined by sample geometries (3, 60). Ab initio shapes are primarily developed by two models, the envelope function and the bead modeling method. In the first method, the envelope function is used to represent the shape, r=F(w), where r and w are spherical coordinates of the particle, which must be completely contained within the envelope. Spherical harmonics are used to provide details of the envelope, where L is the maximum harmonic order (Equation 16). The envelope model provides a unique solution of the shape. However, it has limitations in accuracy if the shape is overly complicated (29, 33, 60, 61).
In the bead modeling method, Monte-Carlo methods are used to best fit the scattering profile as determined by one of the previous methods described, such as the adapted Debye formula, where f(q) is the form factor for each bead and rij is the distance between beads. (Equation 17) (13, 60). Densely packed beads or dummy atoms must fill a sphere of radius Dmax as determined by the scattering profile. The radius r of these beads must be less than Dmax and the particle must fit entirely within this sphere. The particle shape is represented by string X of M beads, such that when Xi=1, the bead belongs to the particle, and if Xi=0, it belongs to the solvent (31, 60, 62, 63).
Methods of combining simple harmonics and bead modeling have also been performed in order to speed up the computational time required (60).
SAXS applications
SAXS has been fundamental in the study of a plethora of materials and structures both biological and non-biological from complex and large proteins in physiological conditions to crystalline and semi-crystalline materials containing large lattice structures. It may be applied to multi-phase systems as well as porous materials. Furthermore, reaction kinetics or time-resolved SAXS allow for the study of conformational changes. Progress in time-resolved SAXS has improved tremendously with many new techniques and advanced software. Kirby and Cowieson detail the current state of time-resolved SAXS as it applies to the study of dynamic biomolecules (50). Stawski and Benning studies in situ and time-resolved SAXS as it applies to precipitation reactions that are difficult to analyze with other techniques (64). Sinko et al. (65) studied the use of SAXS to analyze porous nanoparticles, explicating how the general theory calculations would differ. Doniach and Lipfert provide background information on how SAXS is useful for understanding ligand binding, the folding of structures like RNA, and even in analyzing intermediate folding states (66).
Protein and macromolecular characterization
Understanding key macromolecular and protein functions within the body holds tremendous promise for future medical advancements. With the number of possible protein functions and conformations, being able to predict when and why a protein is in a particular state or how it reacts to environmental changes may hold clues to inducing or inhibiting undesired reactions. Furthermore the ability to analyze the behavior of macromolecules, which make up DNA, RNA, and even cell networks, is significant in understanding biological functions. Petoukhov and Svergun detailed modern techniques of obtaining low-resolution SAXS images, such as methods to characterize macromolecules in solution, identify the conditions in which the macromolecule is close to native conformation, and assess the oligomeric state and quaternary structure (67).
An unfolded protein is randomly oriented with an unpredictable configuration. Once stimulated, the protein will fold into its native conformation, which serves an important biological purpose. Several processes, such as dehydration and polypeptide collapse, occur to facilitate the folding. Barnase has two distinct folding phases, a burst intermediate phase and hidden intermediate phase, which are dependent on the surrounding conditions. Konuma et al. (68) studied the coupling of these processes and the time-resolved compaction that occurs during folding. Prior investigation into the barnase phases detailed how the burst intermediate phase is the denatured state of the protein when no denaturing substance is present. Kinetic SAXS was used to provide structural information for this phase. The Guinier approximation predicted the radius of gyration over time as barnase folding occurred. The intensity distribution as a function of vector space and the pair distribution function was also determined over time, starting from the unfolded state until the native state was reached. The results obtained align with the proposed folding mechanisms, made viewable with time-resolved SAXS (68).
SAXS has been utilized to study the temperature and pH dependence of three forms of hemoglobin Glossoscolex paulistus (HBGp) in order to determine what conditions result in denaturation and/or aggregation and whether these changes are reversible. The hemoglobin protein consists of heme and non-heme structures, linkers, monomers, and trimers (69). The three oxidation states analyzed by SAXS were the oxy, cyanomet, and met HbGp states. The resulting intensity and pair distribution information was utilized to determine critical temperatures and pH values of the hydrodynamic radius for the three oxidation states at various concentrations at which dissociation and denaturation occur (69). The radius of gyration, intensity, and maximum diameter were all determined for the three oxidation states and demonstrate when the hemoglobin protein structure will change. Oxy-HbGp, for example, will retain its native conformation at a pH of 5.0 for temperatures in the range 20–60°C (69). However, at a pH of 7.0, increasing the temperature above 50°Ccauses an increase in the radius of gyration and the maximum diameter, represented by protein enlargement during denaturation and aggregation. This information on changes to hemoglobin was made clear with SAXS analysis (69). Identifying how factors such as pH and temperature affect the human body and what specific results will occur to particular systems, such as the hemoglobin levels in the blood, provides greater insight into very specific biological functions. Macromolecular states are significant in the stability and action of cell networks. Multiple imaging techniques are required to develop models of the many macromolecular configurations. A process for identifying the flexibility of macromolecules in solution using SAXS profiles in combination with NMR and MX was presented, in which it is possible to differentiate between rigid and flexible particle conformations (Fig. 3) (70). A review of this process was applied to several macromolecule structures, including an Mre11-Rad50 complex, for which it was determined that flexibility increases when ATP is not present (70). Once again, it is clear that understanding the conditions in which macromolecular properties change will undoubtedly enhance one’s ability to induce or inhibit these changes by affecting specific conditions.
Fig 3
Fig. 3. Flow chart for the method of validating flexibility of macromolecules with SAXS profiles (a) and rigid body modeling (b). DNA Ligase III experimental SAXS profiles (black and blue) were compared to the theoretical profiles for crystal (red) and dynamic models (green) (70).
Nucleic acids may be characterized by coupling SAXS and NMR. Burke et al. (71) studied all-atom models for U2/U6 ribonucleic acids in this manner. SAXS provides significant information on the shape of particles in solution and thus proves useful in the low-resolution characterization of nucleic acids. RNA has a highly regular structure with many electrons, making SAXS an excellent method of analysis. Burke et al. (71) detailed how NMR (or SAXS) profiles may be used to refine the results obtained from all-atom modeling. The ability to analyze such significant macromolecular structures will prove useful in future studies of both RNA and DNA which are key to the functioning of all cells in the body.
Surface structural properties and organization
Many materials such as composites may be analyzed with SAXS to determine how surface characteristics are influenced. TiO2/CNT photocatalytic composites, for example, have been analyzed by SAXS to assess the charge transfer effects on surface roughness and micropore structure (72). These composites, created from hydration/dehydration methods, may theoretically be utilized to extract pollutants from contaminated water and air (72). The homogeneity of these composites, with different size TiO2 nanoparticles strengthened by single-wall carbon nanotube (CNT) supports, was analyzed with SAXS to view differences in surface roughness and pore structure that might explain prior imaging results, in which all samples except one displayed low removal efficiencies. SAXS demonstrated that this one sample contained large micropores of approximately 0.8–0.9 nm, which was indistinguishable by other methods (72). The additional absorption by these micropores resulted in discrepancies in the interfacial interactions and increased the photocatalytic efficiencies. Multiple imaging techniques, including SAXS, demonstrated that the method of hydration/dehydration is unable to provide TiO2/CNT composites with the required parameters, due to the distinct phases present in each of the materials. These phases must be combined in a way that promotes synergetic effects from the transfer of charges at the interface (72).
SAXS has often been coupled with a variety of techniques to obtain a more complete picture of organizational structure of a material. Yao Jin et al. (73) developed a method of combining SAXS with Ultrasound-coupled filtration cells to determine changes in the colloidal organization of concentrated Laponite layers. Laponite particles have concentrated layers that were analyzed and viewed for the first time with in situ time-resolved SAXS-Ultrasound method, demonstrating that the ultrasonic process reduces the concentrated layer when applied during the ultrafiltration process (73). A linear relationship between the volume fractions of Laponite dispersions and the scattering intensity was derived from the SAXS results (73). This type of study demonstrates the usefulness of SAXS and the possibility of coupling it with many other distinct methods to provide enhanced structural analyses.
Mechanical properties such as strain dependence on temperature may also be assessed through SAXS. Polyethylene samples have been tested with in situ SAXS to study the effects of temperature and microstructure on strain (74). As strain increases, the polyethylene structure contains both crystalline and amorphous regions. Four different polyethylene samples, PEA, PEB, PEC, and PED, were isothermally crystallized and annealed. The SAXS profiles were then determined and corrected with a Lorentz correction to study the polar and equatorial lamellae where the azimuthal angles are 0° and 90° (74). The strain distribution was analyzed and demonstrated heterogeneity, which could only be explained by interactions between the crystalline and amorphous regions in the polar and equatorial directions (74). In materials, the effects of stresses are significant. A method of analyzing changes between crystalline and amorphous structures and their interfaces in polyethylene will help determine how the plastic is affected when under applied stress. Improved behavior under stress through future material development, while maintaining other necessary properties, may be possible once the structural changes are fully understood. This will only help facilitate the development of improved materials in the future.
MD simulation methods
The following sections are meant to serve as a brief overview of MD theory and provide an example of an MD algorithm. The review by Bernardi et al. (75) addresess sampling methods and techniques for MD simulations of biological systems. Methods to apply MD simulations to small peptides and biological macromolecules are reviewed by Doshi et al. (76). The use of short time trajectory fragments to enhance sampling of kinetics is discussed in the review by Elber et al. (77). These three review articles may be referred to for further information regarding methods, techniques, and theory of how one may apply MD simulations to biological systems.
MD simulation theory: atomic force field
The atomic force field describes the system under study as a collection of atoms, which are held together by interatomic forces. To determine the interatomic force one must know the potential energy of the N atoms comprising the system as a function of their respective position (43). The position of each of the N atoms is given by the vector (Equation 18).
One of the first challenges for MD simulation is to determine a realistic potential energy function that will describe the system. To create an accurate potential energy function, the interactions between both bonded and non-bonded atoms must be accounted for. For non-bonded atoms, the two main interactions are the van der Waals force and the electrostatic attraction or repulsion. To describe the van der Waals force, the Lennard-Jones potential is commonly used, while Coulomb’s law is used to describe the electrostatic interaction. Thus, the potential energy function for non-bonded atoms in a biological system will take the form of where the value of εij and σij may be varied depending upon the environment of the system (43). This intermolecular potential energy function implicitly describes the geometric shapes of individual molecules or more specifically, their electron clouds. Therefore, when a potential energy function is defined, molecular behavior, such as rigidity or flexibility, the number of interaction sites per molecule, and so on are also determined (52). Note that the definition for non-bonded, bonded, and total potential energy is typical force field equations that are commonly used to describe biological systems. The terms included in the definition for the potential energy function may vary depending on the particular system (Equation 19).
For bonded atoms, the bond length potential, bond angle potential, and torsional potential must be determined. The additive potential of these three interactions is described by (Equation 20) (43). The first term describes the energies of deformation in the bond length, li, and bond angle, θi, from the equilibrium position, li0 and θi0, respectively (43). The third term describes the energy of deformation resulting from rotations around the chemical bond, where n describes the periodicity of energy terms for rotation (43).
The atomic force field model considers the potentials for bonded and non-bonded atoms to be additive, thus the force field that is used to describe a biological system will take the form of (Equation 21) (4149, 78). Once the potential energy of the system is known, the interatomic force is found by taking the gradient of the potential energy with respect to the atomic displacements (Equation 22).
MD algorithm
In an MD simulation, the forces acting on the system evolves with time. Thus, one can write the position vector as a function of time, where the position of the ith particle is given by (Equation 23).
Since MD simulations utilize classical Newtonian mechanics equations to study the time evolution of the system, one can write the force acting on a particular particle in the system at time t with mass mi as (43). Note that is simply Newton’s second law of motion (Equation 24).
Integrating the force (Equation 26) yields the atomic momenta, while integrating a second time yields the atomic positions (79). Repeatedly integrating the force several 1,000 times produces individual atomic trajectories from which time averages may be computed for macroscopic properties (Equation 25). At equilibrium, the time average does not depend on the initial time t0. The time average represents both static properties, such as thermodynamics, and dynamic properties such as transport coefficients (79).
The term ‘particle’ typically refers to a particular atom; however ‘particle’ could correspond to a larger entity. To perform the MD simulation, one must know all of the forces acting on each of the particles at every time step throughout the simulation, in addition to the initial positions and velocities of all particles. This creates what is called the ‘many body problem’. The many body problem states that the quantum Schrödinger equation for any atom other than hydrogen or the classical equations of motion for a system of more than two point masses may only be solved approximately. Therefore, exact solutions are unavailable for more complex systems such as proteins or nanoparticles suspended in a solution. However, the inclusion of parameters from quantum mechanical calculations may better approximate the quantum mechanical result. Furthermore, due to the many body problem, the classical equations of motion are discretized and solved numerically. The position and velocity vectors describe the evolution of the system in phase space as they are propagated for a finite interval by a numerical integrator. One common numerical integrator used in MD simulations is the Verlet algorithm, which can be derived from the Taylor expansion of the position (Equation 26) (80, 81).
The position and forces acting on each of the particles in the system are continuously updated in the neighbor list. Once the simulation is complete, the desired physical quantities are calculated and the results of the simulation are displayed.
Hard sphere MD simulation algorithm example
There are two types of MD algorithms: soft body algorithms and hard body algorithms. For soft bodies, the acting forces are continuous functions of the distances between molecules (79). For hard bodies, like the classic case of two billiard balls, the discontinuity in the force extends to the intermolecular potential. This section will provide an example of a hard body algorithm. Hard spheres with a diameter of σ interact via a potential energy function U(r) (Equation 27) (79).
The potential function describes a situation in which the spheres exert a force on one another only when they collide. In between collisions the spheres travel along straight lines at constant velocities (via Newton’s laws). Therefore, the simulation algorithm computes the times of the collisions (82). The algorithm calculations are algebraic since the collisions between the two spheres are assumed to be purely elastic. Thus, collisions do not affect the conservation of linear momentum or kinetic energy. The conservation of linear momentum and kinetic energy allow for the MD simulation algorithm to calculate collisions times (82).
The hard sphere algorithm is divided into three main tasks: initialization, equilibration, and production of the result. First, it is important to note that MD programs use a system of units in which the dimensional quantities are unitless. The fundamental dimensions are mass, length, energy, and time (82). In the case of a hard sphere with a diameter of σ, the number density ρ is defined by (82):
where N and V are the number and hard spheres and the volume of the primary cube enclosing the system, respectively. The packing fraction is then defined by (82):
The total energy of the system and the kinetic energy kT are made unitless in the following way (82):
MD simulations can be used to study a variety of motions and structures. Table 1 (83) below displays the times scales and length over which certain biological processes occur. The following flow chart outlines the process for each of the main tasks (initialization, equilibration, production of result) for an MD simulation (Fig. 4).
Fig 4
Fig. 4. Illustrates the overall process of an MD simulation for hard sphere collisions (82).
Table 1. Time scale and length over which molecular motions are calculated in a typical MD simulation program (83).
Motion Length scale Time
Local motions,
i.e. atomic fluctuations, side-chain motions, loop motions
5 Angstrom 10−15−10−1 sec
Rigid body motions,
i.e. helix motions, domain motion, subunit motions
10 Angstrom 10−9−1 sec
Large scale motions,
i.e. helix-coil transitions, dissociation/association, folding/unfolding
>5 Angstrom 10−7−104 sec
MD simulation applications
MD simulations allow one to accurately model and simulate a variety of biological and non-biological nanoscale to microscale systems and behavior. The results from MD simulation can be used for its predictive power in experimental work, or can be used to serve as a comparison to experimental results. Currently, MD simulations are being widely utilized to study protein folding and unfolding. One of the greatest challenges for MD simulation concerning the study of protein folding/unfolding to/from the natural state is the time scale and the statistical error. Future research in this area will therefore be aimed at improving the algorithms to better suit the time scale of the protein processes, thereby reducing the statistical error of the calculation. In addition to furthering our understanding of protein characteristics and behavior, future work will focus of the application of MD simulation to study complex nanoscale systems, such as colloidal solutions with nanoparticles. MD simulation has demonstrated its ability to accurately model and predict mechanical, structural, and electrical properties of nanoparticles. In sum, MD simulation has proved its ability to serve as both a powerful predictive tool and method for studying complex systems that are not easily examined in an experimental system.
Protein characterization
One biological application of MD simulations is to study protein folding and unfolding. Proteins fold during a time scale ranging from microseconds to seconds (84). Complications arise due to the length of time required for the folding process. For example, if one were to use the atom-based model for potential energy and solve the time-discretized Newtonian equation of motion for folding from the denatured to native state, the simulation would require years for a protein composing of 100 residues. The experimental transition to this folded state, however, takes only 1 millisecond (84). During an MD simulation, some large proteins will not fold to their native states due to systematic error and reduced stability in the native state (84). Thus, the greatest challenges for MD simulations concerning protein folding are the time scale and statistical error.
On the other hand, protein unfolding is a much simpler process, which can be simulated on shorter time scales of 1–100 nanoseconds (84). Understanding protein folding at the atomic level of detail is important, as protein folding is a complicated reaction that cannot be easily studied from experimentation. However, experimental outcomes are significant when for validation or comparison with simulation results. Experimental results may also be used to improve the force equations in a simulation by providing more realistic parameters.
Ferrara and Caflisch utilized MD simulation to study the reversible folding and free energy surface of two designed 20-residue sequences called beta3s (TWIQNGSTKWYQNGSTKIYT) and DPG (Ace-VFITSDPGKTYTEVDPG-Orn- KILQ-NH). Both of these sequences have a three-stranded antiparallel β-sheet topology (44, 84). The beta3s have been examined previously by Nuclear Overhauser enhancement spectroscopy and chemical shift data, both of which showed that at 10°C, beta3s populates a single structured form-the three-stranded antiparallel β-sheet conformation with turns at Gly6-Ser7 and Gly14-Ser15 (44, 85).
Chemical shift data have shown that the designed amino acid sequence DPG adopts the three-stranded β-sheet conformation at 24°C in aqueous solution (86). Ferrara and Caflisch performed MD simulations at 300 K and found that both peptides met most of the Nuclear Overhauser enhancement spectroscopy distance restraints (84). Furthermore, they determined that the average effective energy and free energy landscape were similar for both peptides at 360 K despite sequence dissimilarity (84). The average effective energy for the peptides showed a downhill profile at and above the melting temperature (330 K); hence, Ferrara and Caflisch determined that the free energy barriers resulted from entropic losses due to the formation of a β-hairpin (84). They concluded that the free energy surface of the β-sheet peptides is different from a helical peptide, as the helical peptides’ folding free energy barrier was much closer to the fully unfolded state than that of the β-sheet peptides (84). In summary, the topology of the peptide determines the free energy surface of the folding mechanism (84). This study demonstrates how MD simulation can be utilized to determine important physical characteristics of biological samples, such as the free energy barrier.
At physiological temperatures, small molecules can exhibit Arrhenius temperature dependence, meaning that they have a faster folding rate at high temperatures (84). However, at even higher temperatures, the folding rate can deviate and become non-Arrhenius (84). Ferrara and Apostolakis investigated the kinetics of folding for an α-helical structure and for a β-hairpin with MD simulations (87). They ran 862 simulations over a time period of four µs for the peptides (87). The negative activation enthalpy at high temperatures was an important feature of folding for both the α-helical and a β-hairpin structures (84). The folding rate increases with temperature, reaching a maximum and then decreases (87).
The observations made by Ferrara and Apostolakis are in agreement with experimental data, thereby validating that at high temperatures, non-Arrhenius behavior may arise despite the interactions being temperature independent (84). This might be explained by the temperature dependence on configuration space, meaning that at high temperatures, a larger portion of configuration space is accessible, resulting in a reduction in the folding rate (84). This study illustrates how environmental conditions, such as temperature can affect the rate of biological processes (i.e. folding/unfolding). It further demonstrates how MD simulation can strengthen experimental results through theoretical verification.
In addition to studying free energy surfaces and temperature dependence of folding rates, MD simulations have been valuable in studying the function of particular proteins such as the tumor suppressor protein, p53. Upon detection of DNA damage, p53 can either cause cell cycle arrest, allowing for DNA repair, or induce apoptosis. Therefore, the study of the function of p53 is important because this cell regulator protein is often inhibited in many forms of cancer. There is interest in drug development concerning how one can restore the p53 function. MD simulations play an important role because they allow for the study of structural changes and interactions at an atomic scale (87). To study the kinetics of p53, MD simulation packages like AMBER, CHARMM, and GROMACS can be used (87). The binding free energy from these programs may be utilized to study the binding affinities of certain p53 inhibitors, such as MDM2 (87). For example, one paper studied the role of electrostatic interactions concerning the formation of the p53-MDM2 complex by changing the distance of separation between p53 and MDM2 (85). It was found that electrostatic interactions dominate the p53-MDM2 complex formation at long distances while van der Waals interactions of three residues in p53 (Phe19, Trp23, and Leu26) control the formation of p53-MDM2 at short range (88).
The main challenges that must be overcome for MD simulations of protein folding are building accurate force fields, providing sufficient sampling, and providing robust data analysis (Fig. 5) (89). Concerning the force field models, more work is needed to ensure that the force fields are capable of folding proteins into their native state in agreement with experimental results (89). Sufficient sampling is required to overcome force field inaccuracies. Since MD simulations must integrate Newton’s equations of motion with femtosecond time steps, a typical folding simulation requires approximately 1012 time steps to reach a millisecond time scale (89). Furthermore, the average system is composed of many atoms (typically ≥105 atoms), thereby requiring significant computing power to maintain a neighbor list, which will track the forces acting on and positions of each of the atoms in the system (89). Today, millisecond simulations are now possible due to improvements in computer hardware, software, and sampling techniques (89).
Fig 5
Fig. 5. Protein folding simulations conducted using unbiased, all-atom MD in empirical force fields reported in literature. Some folding times for the same protein differ, due to mutations. For lambda with a (*), the longest timescale seen in that simulation, which was not the folding time, occurred on the order of 10 ms (89).
Another challenge for MD simulation is providing the physical knowledge of the system. Data analysis can be very complex, and depending upon the analysis method used, may result in different conclusions. Currently, there are two analysis techniques for protein folding, reaction coordinate, and associated transition states methods and Markov State Models (MSM) (Fig. 6) (89). For the reaction coordinate method, a single coordinate which is capable of describing the process from unfolded to folded is used, and then a kinetics model is built using that coordinate (86, 90, 9193). The benefit of this method is that it reduces information to a single coordinate and finds one or more transition state ensembles that have equal probability of folding or unfolding (87).
Fig 6
Fig. 6. Two methods of data analysis, the MSM (top) and reaction coordinate (bottom), shown for the same system (ACBP). The MSM represents folding as interconversion between structurally similar states, and can be illustrated as flow through a network. Reaction coordinates attempt to depict folding as progress along a single degree of freedom, such as the committers (pfold, shown). The MSM picture is more detailed, can capture parallel paths, has tunable resolution, and connects naturally to experiment – all advantages over the coordinate-based approach (89).
MSM represents folding as first-order kinetics between a set of discrete states, making the data analysis simpler by discarding dynamics below a defined ‘lag time’ (89). By changing the defined lag time, one can achieve different resolutions (89). Furthermore, MSM can track parallel paths of folding and is generally closer to the experimental results (89). The problem is that the results obtained using the reaction coordinate method and MSM do not always agree. Thus, there is a need to develop a general data analysis method for MD simulations of protein folding and unfolding.
Structural, mechanical, and electrical properties
Anisotropic conductive adhesives (ACAs) are of interest for microelectronic devices, specifically for producing ultra-thin liquid-crystal displays. ACAs in liquid-crystal displays are subjected to compressive stresses during manufacturing and operation; thus, it is important to understand how these mechanical stresses affect their performance (94). There has been experimental research performed on the mechanical response of polymers ranging in size from 2.6 to 25.1 µm (86). He et al. (93) observed that decreasing particle diameters resulted in increasing stiffness of the polymer material. This size effect is very important for the production of electronics.
Zhao et al. (86) utilized a coarse-grained MD model to verify the observed size dependence for the particles. A coarse-grained MD simulation was utilized because they can be simulated with less time than an all-atom model and because of the availability of coarse-grained potentials for surface tension (86). The coarse-grained model was performed in LAMMPS, an MD simulation source code, for linear polyethylene. As a result, they obtained an entangled molecular model (full atomic), which was then converted to a coarse-grained model where each bead represented three monomer units of polyethylene (86).
Five different polymer particles with different diameters from 5 to 40 nm were constructed (86). For comparison, a bulk coarse-grained model of linear polyethylene was developed using the same potential function (86). Compression loadings were simulated for each of the constructed particles to determine the influence of particle size on mechanical response. Using coarse-grained MD simulations, the nominal stress and strain curves were obtained. The continuum model represents a particle subjected to compressive loading between the two plates, as evaluated by finite element analysis and was included because the size effect does not occur; therefore, this model served as a control reference for comparison other models (86). In addition to studying the loading behavior of different sized particles, MD simulations were also performed for compression unloading of these particles.
From the compression loading and unloading simulations, it was determined that there is a size effect in polymer particles (86). In addition to studying loading and unloading properties of the particles, Zhao et al. (86) examined the surface energy vs. ratio of surface area to volume. Amorphous polymers have a mass density, which is higher at the surface than in the bulk of the material. This higher surface density can be explained by surface tension whereby polymer molecules at the surface are pulled by attractive forces toward the bulk, resulting in surface densification (86). The thickness of this dense region on the surface is constant; thus, for decreasing particle sizes, the volume fraction of the material will increase (86). This property gives smaller polymer particles stiffer mechanical responses to load. It was found that there is a linear relationship between the surface energy and the relative surface area (86). The increases in surface energy and particle stiffness result from the increase in the mass density of the material at the surface (86). This example illustrates how MD simulation can be utilized to study particular characteristics of a system that are difficult to examine in a laboratory setting.
In another study, Torres-Vega et al. (95) determined the threshold for the behavior of copper nanoparticles by studying their structural and electronic properties using MD simulations. Depending upon the size of the copper nanoparticles, they contained between 13 and 8,217 atoms (95). Metallic nanoparticles are of interest because their physical and chemical properties are strongly dependent on their size, structure, composition, and defects (95). MD simulations are crucial for the study of nanoparticles, because experiments can be very difficult to perform and the MD simulations allow one to study different properties at the atomic scale. Torres-Vega et al. (95) determined the threshold above which copper nanoparticles behave like their solid counterpart, in addition to the structural and electronic properties of small and large copper nanoparticles.
The MD simulations were performed using a force field derived from the Johnson potential for copper, which is based on the embedded atom method (95). The Johnson potential has been previously used to study the binding energy and atomic structure of copper nanoparticles and nanowires (95). Torres-Vega defined the total energy of the system Etotal as the summation of F(ρi), the embedded energy of atom i, and ϕij(rij), the two-body potential between atoms i and j (Equation 31) (95). The electron density for an atom due to all other atoms ρi is defined as the summation of f(rij), the electron density at atom i due to atom j as a function of distance rij between the two atoms (Equation 32) (95).
The Newtonian equations of motion were integrated using a fifth order predictor − corrector algorithm with a time step of 2 femtoseconds (95). For fewer than 2,000 atoms, Torres-Vega et al. (95) observed that the potential energy decreased drastically with increases in the size of the nanoparticle. However, when there were greater than 2,000 atoms comprising the copper nanoparticles, the potential energy stabilized around −3.4 eV/atom, which is approximately the cohesive energy for bulk copper, −3.49 eV/atom (95).
For structural analysis, Torres-Vega et al. (95) noted that nanoparticles with fewer than 2,000 atoms presented a spherical external form while nanoparticles with greater than 600 atoms but less than 2,000 atoms presented an irregular form. The different structures for the copper nanoparticles based upon the number of atoms occurred because the system seeks a structure that will result in a minimum energy. The cohesive energy of bulk copper was described by the power law fit function, U, where U0=−3.52eV/atom, α=2.31 eV/atom and α=−1/3 (Equation 33) (85, 95). Torres-Vega et al. (95) found that when the number of atoms composing the nanoparticle approached infinity, the value of U0 approached the value for bulk copper.
For electronic analysis the influence of size on the electronic density of states (EDOS) was analyzed utilizing a recursion method to calculate the EDOS. The results demonstrated that for small nanoparticles (those with less than 1,000 atoms), the total EDOS presents fluctuations which are caused by large surface to volume ratios; large nanoparticles (greater than 2,000 atoms), on the other hand, present negligible fluctuations (95). To compare the total EDOS for a nanoparticle of size N, the total DOS of crystal line copper model with half a million fcc structured a parameter ΔT was defined (Equation 34) (95):
where is the total EDOS of a nanoparticle of size N (Equation 35) (95).
Furthermore, the local DOS at position is defined by (Equation 36) (95):
where is the Green function and is a positive infinitesimal (85). Using Equation 23 for ΔT, the total EDOS of the copper nanoparticles verses the size was plotted to illustrate the relationship between EDOS and ΔT.
In summary, using MD simulations and the Johnson potential for copper, Torres-Vega et al. (95) studied the influence of nanoparticle size on the structural and electrical properties and determined the threshold for copper nanoparticle behavior. Small nanoparticles (fewer than 1,000 atoms) exhibit a strong dependence on the surface and have an icosahedral shape, while nanoparticles greater than 3.5 nm (more than 2,000 atoms) exhibit a spherical shape (95). The threshold size predicted by MD simulation was close to the experimentally determined size of 3.8 nm. Finally, for nanoparticles with greater than 2,000 atoms present, an electronic character similar to the corresponding macroscopic counterpart was observed.
Integration of SAXS and MD theory
General process overview for biomolecules
The following process outlines the general steps required to perform a restrained MD simulation where the SAXS profile provides constraints for protein analysis. A requirement for this process is some knowledge of the structure. For the case outlined by Kojima et al. (36) a crystalline structure was used. SAXS theoretical profiles were determined for incrementing atomic coordinates which were then used in the regular MD process. Additionally, in restrained MD, an extra step is added in which the potential energy is minimized by including a constrained energy value.
Kojima et al. (36) detailed the Restrained MD-SAXS process for proteins in solution using an AMBER united-atom force field model (95). The first stage is to calculate the theoretical intensity profile Icalc(q) based on the scattering amplitudes in both a vacuum Av(q) and in the excluded volume Ao(q) with an average bulk density ρo (Equations 37) (34, 96). The average bulk density for this case was calculated with the modified cube model (97). The vacuum and excluded volume intensity profiles Iv(q) and Ic(q) are calculated in the same manner as a typical SAXS profile (Equations 38 and 39) (11, 12, 36). It should be noted that all calculated intensities and gradients are averaged over all potential configurations (36).
The combined intensity profile Ivc(q) is calculated from the vacuum and excluded volume scattering amplitudes and their conjugates (Equation 40).
In order to solve for the potential energy, the gradients of each of the scattering intensity profiles with respect to the atomic coordinate must be determined (Equations 4145) (36). It should be noted that Kojima et al. (36) set the gradient of the excluded volume intensity profile to zero for calculation purposes.
Based on the specific system, the values for the scaling factor k and the relative weight w(qi) will vary. Kojima et al. (36) used specific formulas for these two variables to help reduce the variation between their observed and theoretical profiles (Equations 46 and 47). The normalization factor NA is also shown (Equation 58) (36).
Finally, the constraint potential energy and total potential energy may be calculated (Equations 49 and 50) (36, 98, 99). The normal potential energy is derived from the AMBER united-atom force field (96).
The weighting factor wconst is chosen to balance the potential energy contributions from the constrained and normal potential energies per the energy minimization (36). The potential energy is then used in the regular MD process. A general method of applying these equations and feeding them into the MD simulation is shown in a flow chart (Fig. 7).
Fig 7
Fig. 7. Integrated SAXS and MD flow chart.
MD-SAXS method for proteins
The MD-SAXS method detailed is for protein structural studies, in which the MD simulation trajectories are used to simulate the SAXS data. Typically, continuum solvent models are applied to do so; however, MD-SAXS utilizes explicit water molecule hydrating, where MD simulation is performed for both the pure solvent and the solution. Then the pure solvent intensity profile is subtracted from the solution intensity profile to calculate the excess intensity I(q) where U represents the solution and V represents the pure solvent. (Equation 51) (100).
In this integrative MD-SAXS method, two regions for the solvation and bulk layer are created in order to facilitate the necessary calculations. Region O must enclose the entire solvation layer and part of the bulk, so that the experimental and theoretical SAXS profiles will be similar. Region b contains the remaining bulk of the sample. Any particle shape may be analyzed as long as it is fully contained in region O (Fig. 8) (100). The SAXS results are low resolution. Thus, all-atom MD simulation is performed to extract additional information.
Fig 8
Fig. 8. Spherical (a), rectangular (b), and cylindrical (c) representations of regions o and b for MD-SAXS protein calculations (100).
To obtain the solution and pure solvent intensities, the form factor for the pure solvent Pv(q) and electron radial densities, ρo and of the bulk in solution and the pure solvent with respect to position r, must be determined over region O (Equation 52) (100). The brackets represent configurational averages over the possible protein ensembles that may exist in either the solution or pure solvent. The bracket for Ωq represents the orientational average. The tilde over a symbol represents an instantaneous value. Multipole expansion may be used for faster calculation of the intensity (100).
In MD simulations of spherical proteins, the spherical simulation box must include the entire protein and few water molecules, so the water molecules do not increase the computation time. If the shape is not spherical, then the box will contain more water molecules and the computational time will increase. This would happen, for instance, if the protein had an elongated shape (100).
SAXS-MD applications
Protein characterization
The integration of SAXS and MD simulation has already been used in several applications to analyze the more complex structures that are not amenable to the other conventional imaging methods (11, 101). Oroguchi et al. (101) studied intrinsically disordered proteins (IDPs) as well as multi-domain proteins, developing an MD-SAXS method to obtain low-resolution protein complexes in solution using SAXS and to analyze the dynamics of the proteins using MD simulation. The MD simulation helped overcome the loss of information that results in SAXS spatial averaging. The MD-SAXS method was utilized to characterize the structure of Endonuclease EcoO1091. The MD structural ensemble was validated with the SAXS profile (101). This work along with other advancements in structural analysis methods will help further the study of proteins.
SAXS combined with MD simulation is being used to study the structure of proteins in solution. To do this at an atomic resolution, it is necessary to develop a SAXS profile including scattering due to the hydration layer. Calculations for the hydration layer may be performed with MD simulations using either uniform density layers or explicit water molecules. Changes in ionic strength and the resulting changes in the SAXS profile were studied by performing MD simulations. Lysozymes present in hen egg whites were modeled over varying concentrations of NaCl (102). The SAXS profile demonstrated large fluctuations when the concentration of NaCl was varied from 0 to 100 mM and did not converge to the MD simulation predicted profile. Thus, the ionic concentrations and strength will affect the ability of SAXS and MD to be coupled and for the MD-SAXS method to produce high-resolution profiles (102). Once again, it is clear that MD-SAXS holds the potential to unlock new information on protein structures in solution and the effects on protein properties; however, there are limitations in the ability to provide the necessary input information in the MD simulation so the results converge with the SAXS experimental data.
Stabilizing membrane proteins, such as G-protein coupled receptors (GPCRs), are pivotal in the study of membrane protein structure. Midtgaad et al. (103) utilized SAXS and Small Angle Neutron Scattering to study self-assembling peptides, which combine with the membrane phospholipids to yield nanodiscs over time and increase the membrane protein stability (103). The solution structure of ApoA1 peptides in phospholipids before and after assembly into a disc structure was viewed in detail. Coarse-grained MD simulation of the change to disc shape and resulting dynamics to stabilize the membrane show clear aggregation of these peptides (103). Cell membrane protein behavior and stabilization methods are crucial to cell function and should be studied further.
These three examples represent the potential uses of combining distinct types of analysis – both simulation and experimental practices – and hold future promise for better understanding complex protein processes.
Structural and mechanical properties and conformational changes
There are a variety of properties such as charge that will impact electrical, mechanical, and surface characteristics. Often these characteristics can be analyzed with SAXS and MD coupling. For example, charge imbalance in archaeal lipid bilayers in cell membranes has been studied. Charge imbalance in cell membranes results in high transmembrane voltages, which affects the structure of archaeal lipid bilayers (104). MD simulations for DPPC bilayers mixed with archaeal lipids were performed to investigate these changes based on different charge imbalances and the resulting effects on electrical and pore properties. The electron density profiles, or pair distribution profiles, were compared to SAXS profiles for validation (104). The ability to accurately analyze the behavior of cell membranes under different conditions will greatly enhance our understanding of cells.
Dendrimers are another structure with complex conformations that has been effectively studied with SAXS and MD simulation. These highly branched structures have potential applications in biological and industrial fields and have been difficult to analyze in the past due to their many possible conformations. MD simulation to determine structural properties, such as the radius of gyration, monomer density distribution, molecular surface area and volume, and spatial array of branch points, was performed to analyze the PETIM dendrimer (generations G2–G4) in water at each branch point (105). SAXS imaging was also performed as validation for the MD results. It was determined that the radius of gyration follows the scaling law for lower generations and Rg~N28 for higher generations, where N is the number of monomers present in the dendrimer (105). The MD and SAXS monomer density distribution also showed that back folding was present and the dendrimer had flexible repeat units as expected. The PETIM structure combined with previous PAMAM studies indicate that dendrimers tend toward flexible folding non-spherical conformations (105). MD and SAXS demonstrate how complex and highly folded structures with many conformations may be accurately modeled, providing significant structural information. The behavior of dendrimers in different environments is crucial to many distinct fields and holds untold potential. The physical characteristics of cycloisomaltooligosaccharides (CIs) in solution were analyzed by three methods: viscosity measurements, SAXS, and MD simulation. SAXS was performed over multiple concentrations to determine whether the radius of gyration varied with concentration and how the degree of polymerization affected the shapes of CIs. The radius of gyration for CI was not directly affected by concentration. Cls with different degrees of polymerization resulted in similar SAXS profiles. Thus, there shapes are likely similar, but the radius of gyration will vary as a result of changes in the degree of polymerization. The experimental SAXS profile was compared to theoretical flexible Gaussian ring and rigid ring models. MD simulation was performed to find a ring model that better fit the SAXS data (106). The combination of MD and SAXS to determine a ring model fit demonstrates how simulation and experimental data may be incorporated in an effective manner. It holds tremendous promise in future applications for both biological and non-biological systems.
SAXS studies have been performed for a wide array of materials and structures, providing extremely useful low-resolution one or two-dimensional profiles meant to describe three-dimensional structures. Moving toward higher resolution profiles requires the use of MD simulation to extract relevant three-dimensional geometrical information. This coupled MD-SAXS method combines the advantages of both SAXS and MD simulation and provides a tremendous opportunity for advancement in the materials field for both biological and non-biological applications. Capitalizing on the ease of use and flexibility of sample type for SAXS and the ability of MD simulations to capture significant information on the motion of particles will allow for significant improvement on structural analysis and conformational changes of larger particles. For now, it is clear that MD-SAXS provides a solution to viewing complex protein and macromolecular changes, with SAXS allowing one to view folding and unfolding patterns and MD to track the movement. Furthermore, MD-SAXS may be the key to revolutionizing nanoparticle studies, as SAXS is well suited for imaging nanoscale materials, and MD is suited for tracking the trajectories of complex nanoparticle components, such as branch points in dendrimers. In the future, the use of MD-SAXS should be expanded to include the analysis of non-biological structures, which is rarely done with MD simulation alone.
This material is based upon work supported under a Department of Energy Nuclear Energy University Programs Graduate Fellowship. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the view of the Department of Energy Office of Nuclear Energy.
Conflict of interest and funding
There is no conflict of interest in the present study for any of the authors.
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*Li Liu
110 8th St, JEC 5046
Troy, NY 12180
Email: liue@rpi.edu
*Lauren Boldon
110 8th St, JEC 5046
Troy, NY 12180
Email: boldol@rpi.edu
About The Authors
Lauren Boldon
Rensselaer Polytechnic Institute
United States
PhD Student, Nuclear Engineering and Science
Department: Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Fallon Laliberte
Rensselaer Polytechnic Institute
United States
MSc. Physics, 2013, University of Rhode Island
PhD Student, Nuclear Engineering and Science
Department: Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Li Liu
Rensselaer Polytechnic Institute
United States
Associate Professor
Nuclear Engineering and Engineering Physics Program
Department of Mechanical, Aerospace, and Nuclear Engineering
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2838f5c6269917ce | relative dispersion
Coherent state
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926 while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent state, arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of the particle in a quadratic potential well. In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories they were introduced by the work of Roy J. Glauber in 1963. Here the coherent state of a field describes an oscillating field, the closest quantum state to a classical sinusoidal wave such as a continuous laser wave.
Coherent states in quantum optics
In quantum mechanics a coherent state is a specific kind of quantum state, applicable to the quantum harmonic oscillator, the electromagnetic field, etc. that describe a maximal kind of coherence and a classical kind of behavior. Erwin Schrödinger derived it as a minimum uncertainty Gaussian wavepacket in 1926 while searching for solutions of the Schrödinger equation that satisfy the correspondence principle . It is a minimum uncertainty state, with the single free parameter chosen to make the relative dispersion (standard deviation divided by the mean) equal for position and momentum, each being equally small at high energy. Further, while the expectation value of the Heisenberg equations of motion are zero for all energy eigenstates of the system, in a coherent state the expectation values of the equations of motion are precisely the classical equations of motion, and have small dispersion at high energy. (High energy is guaranteed when mean oscillatory amplitude and momentum have small classical values.) The quantum linear harmonic oscillator and hence, the coherent state, arise in the quantum theory of a wide range of physical systems. They are found in the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories.
While minimum uncertainty Gaussian wave-packets were well-known, they did not attract much attention until Roy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field . Glauber was prompted to do this to provide a description of the Hanbury-Brown & Twiss experiment that generated very wide baseline (hundreds or thousands of miles) interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence. (For more, see Quantum mechanical description.)
Quantum mechanical definition
Mathematically, the coherent state |alpharangle is defined to be the 'right' eigenstate of the annihilation operator hat a . Formally, this reads:
Since hat a is not hermitian, alpha is complex, and can be represented as
alpha = |alpha|e^{itheta}~~~,
where ~theta~ is a real number. Here ~|alpha|~ and ~theta~ are called the amplitude and phase of the state.
Physically, this formula means that a coherent state is left unchanged by the detection (or annihilation) of a particle. The eigenstate of the annihilation operator has a Poissonian number distribution (as shown below). A Poisson distribution is a necessary and sufficient condition that all detections are statistically independent. Compare this to a single-particle state (|1rangle Fock state): once one particle is detected, there is zero probability of detecting another.
The derivation of this will make use of dimensionless quadratures, X and P. These quadratures are related to the position and momentum of the mass in a spring and mass oscillator:
mathbf{P}=sqrt{frac{1}{2hbar momega }} mathbf{p}text{,}quad mathbf{X}=sqrt{frac{momega }{2hbar }} mathbf{x}text{,}quad quad text{where }omega equiv sqrt{k/m} ,
For an optical field,
~E_{rm R} = left(frac{hbaromega}{2epsilon_0 V} right)^{1/2} cos(theta) X~ and ~E_{rm I} = left(frac{hbaromega}{2epsilon_0 V}right)^{1/2} sin(theta) X~
With these quadratures, the Hamiltonian of either system becomes
mathbf{H}=hbar omega left(mathbf{P}^{2}+mathbf{X}^{2} right)text{, with }left[mathbf{X},mathbf{P} right]equiv mathbf{XP}-mathbf{PX}=i/2
Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally distributed in both X and P quadratures satisfies the equation
left(mathbf{P}- right),|alpha rangle =ileft(mathbf{X}- right),|alpha rangle text{,}quad text{or}quad left(mathbf{P}-imathbf{X} right),left| alpha rightrangle = leftlangle mathbf{P}-imathbf{X} rightrangle ,left| alpha rightrangle .
It is an eigenstate of the operator (P - iX). (If the uncertainty is not balanced between X and P, the state is now called a squeezed coherent state.)
Schrödinger found minimum uncertainty states for the linear harmonic oscillator to be the eigenstates of (X - iP), and using the notation for multi-photon states, Glauber found the state of complete coherence to all orders in the electromagnetic field to be the right eigenstate of the annihilation operator -- formally, in a mathematical sense, the same state. The name "coherent state" took hold after Glauber's work.
The coherent state's location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the same phase θ and amplitude (or the same complex electric field value for an electromagnetic wave). As shown in Figure 2, the uncertainty, equally spread in all directions, is represented by a disk with diameter 1/2. As the phase increases the coherent state circles the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space.
Since the uncertainty (and hence measurement noise) stays constant at 1/2 as the amplitude of the oscillation increases, the state behaves more and more like a sinusoidal wave, as shown in Figure 1. And, since the vacuum state |0rangle is just the coherent state with alpha=0, all coherent states have the same uncertainty as the vacuum. Therefore one can interpret the quantum noise of a coherent state as being due to the vacuum fluctuations.
(It should be noted that the notation ~|alpharangle~ does not refer to a Fock state. For example, at alpha=1 , one should not mistake ~|1rangle~ as a single-photon Fock state -- it represents a Poisson distribution of fixed number states with a mean photon number of unity.)
The formal solution of the eigenvalue equation is the vacuum state displaced to a location alpha in phase space, i.e., is the displacement operator D(alpha) operating on the vacuum:
|alpharangle=e^{alpha hat a^dagger - alpha^*hat a}|0rangle = D(alpha)|0rangle
where |nrangle are energy (number) eigenvectors of the Hamiltonian. This is a Poissonian distribution. The probability of detecting ~n~ photons is:
P(n)=e^{-langle n rangle}frac{langle n rangle^n}{n!}
Similarly, the average photon number in a coherent state is ~langle n rangle =langle hat a^dagger hat a rangle =|alpha|^2~ and the variance is ~(Delta n)^2={rm Var}left(hat a^dagger hat aright)= |alpha|^2~.
In the limit of large α these detection statistics are equivalent to that of a classical stable wave for all (large) values of ~alpha~. These results apply to detection results at a single detector and thus relate to first order coherence (see degree of coherence). However, for measurements correlating detections at multiple detectors, higher-order coherence is involved (e.g., intensity correlations, second order coherence, at two detectors). Glauber's definition of quantum coherence involves nth-order correlation functions (n-th order coherence) for all n. The perfect coherent state has all n-orders of correlation equal to 1 (coherent). It is perfectly coherent to all orders.
At ~alpha gg 1~, from Figure 5, simple geometry gives Deltatheta|alpha|=frac{1}{2} From this we can see that there is a tradeoff between number uncertainty and phase uncertainty Deltatheta~Delta n=1/2, which sometimes can be interpreted as the number-phase uncertainty relation. This is not a formal uncertainty relation: there is no uniquely defined phase operator in quantum mechanics
Mathematical characteristics
The coherent state does not display all the nice mathematical features of a Fock state; for instance two different coherent states are not orthogonal:
so that if the oscillator is in the quantum state |α> it is also with nonzero probability in the other quantum state |β> (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis in which one can diagonally decompose any state. This is the premise for the Sudarshan-Glauber P representation. This closure relation can be expressed by the resolution of the identity:
frac{1}{pi} int |alpharanglelanglealpha| d^2alpha = 1.
Another difficulty is that a has no eigenket (and a has no eigenbra). The following formal equality is the closest substitute and turns out to be very useful for technical computations:
a^{dagger}|alpharangle=left({partialoverpartialalpha}+{alpha^*over 2}right)|alpharangle
The last state is known as Agarwal state denoted as |alpha,1rangle. Agarwal states for order {n} can be expressed as |alpha,nrangle=hat{a}^{dagger~n}|alpharangle
Coherent states of Bose–Einstein condensates
• A Bose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. In a thermodynamic system, the ground state becomes macroscopically occupied below a critical temperature — roughly when the thermal de Broglie wavelength is longer than the interatomic spacing. Superfluidity in liquid helium-4 is believed to be associated with the Einstein-Bose condensation in the ideal gas. But, 4He has strong interactions, and the liquid structure factor (a 2nd-order statistic) plays an important role. The use of a coherent state to represent the superfluid component of 4He provided a good estimate of the condensate / non-condensate fractions in superfluidity , consistent with results of slow neutron scattering. Most of the special superfluid properties follow directly from the use of a coherent state to represent the superfluid component — that acts as a macroscopically occupied single-body state with well-defined amplitude and phase over the entire volume. (The superfluid component of 4He goes from zero at the transition temperature to 100% at absolute zero. But the condensate fraction is less than 20% at absolute zero temperature.)
• Early in the study of superfluidity, Penrose and Onsager proposed a metric ("order parameter") for superfluidity. It was represented by a macroscopic factored component (a macroscopic eigenvalue) in the first-order reduced density matrix. Later, C. N. Yang proposed a more generalized measure of macroscopic quantum coherence, called "Off-Diagonal Long-Range Order" (ODLRO), that included fermion as well as boson systems. ODLRO exists whenever there is a macroscopically large factored component (eigenvalue) in a reduced density matrix of any order. Superfluidity corresponds to a large factored component in the first-order reduced density matrix. (And, all higher order reduced density matrices behaved similarly.) Superconductivity involves a large factored component in the 2nd-order ("Cooper electron-pair") reduced density matrix.
Coherent electron states in superconductivity
• In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.
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The Internet of Things Is the Next Digital Evolution—What Will It Mean? | Amazing Science |
But respondents also expressed concerns about algorithms.
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Australian stingless honeybee (Tetragonula carbonaria) makes spiral nests
Australian stingless honeybee (Tetragonula carbonaria) makes spiral nests | Amazing Science |
This tiny stingless bee (3 - 5 mm) is the only native species, in Sydney, that lives in a social colony and makes honey. The queen bee lays all the eggs, and the thousands of sterile female worker bees and male drone bees work harmoniously in building, maintaining and sustaining the colony and its hive. The bees’ hive is usually nested inside hollow trees, branches and logs. The nest is constructed from cerumen, a dark brown mixture of wax (secreted by the workers) and resins (collected by the foragers from ‘bleeding’ trees). Pollen, nectar and honey are stored in clusters of small pots around the edges of the nest. In the centre, the queen lays the colony’s future eggs, in hexagonal cells within a horizontal spiraling brood comb. Native Bee enthusiasts have developed techniques for keeping these unique bees in specially constructed hive boxes. Colonies can then be propagated by splitting a single hive in two. This will help to conserve and re-establish colonies of stingless bees back into urbanized areas.
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Optical shock wave: Scientists imaged light going faster than itself
Optical shock wave: Scientists imaged light going faster than itself | Amazing Science |
A team of researchers at Washington University in St. Louis has taken images of a laser pulse generating an optical Mach cone: the equivalent of a sonic boom, but for light. To make an optical Mach cone, a pulse of light would need to be traveling faster than the waves it’s emitting can propagate forward. But the researchers were able to peel apart the properties of a laser beam, interacting separately with velocity, wavelength, and frequency. They directed the beam through a layered confection of silicone panels, aluminum oxide powder, and dry ice. The source of the light waves was moving faster than the waves themselves as they passed through the layers, leaving behind the optical Mach cone.
To capture the cone itself, the researchers set up CCD cameras next to the cone-generating apparatus. One of the cameras was a streak camera, which exploits the motion of charged particles to create a spatial “pulse profile” that characterizes the light waveform in 3-space over time. Using the streak camera and the CCDs, the researchers captured a 2D sequence of images from three perspectives in a single take. They then spliced the images back together like a CAT scan to make a 3D model of the cone.
Lead author Jinyang Liang hopes that these developments can be pressed into use not just in physics, but in neuroscience. Their imaging setup can capture 100 billion frames a second. With that kind of temporal resolution, researchers could capture neurons firing in real time. Optical shockwaves, as it turns out, are pretty easy to photograph if you have the right kind of camera setup. This jet is producing a Mach wave at its nose because the air molecules can’t get out of the way fast enough for the plane to make its way neatly through them — not unlike the physical wave that happens at the bow of a fast-moving ship as it plows through the water. It was photographed with a method called Schlieren photography, which takes high-speed images and then compares the backgrounds, to see where the wave of distortion traveled through the frame.
And optical shockwaves aren’t constrained to places with an atmosphere, either. They also happen at a much, much larger scale. When a body in space gets moving fast enough, it can produce a phenomenon called a bow shock. Kappa Casseiopeiae is an enormous rogue star traveling at 2.5 million miles an hour, which is a quarter of a percent the speed of light. It’s so big, and moving so fast, that its bow shock is twelve light-years across and stands four light-years apart from the star itself.
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Spermbot catches single sperm cell, moves it to egg cell and delivers it
Spermbot catches single sperm cell, moves it to egg cell and delivers it | Amazing Science |
It is hard to believe: A tiny spiral - a micro robot - catches a single sperm, moves it directly to an egg cell and delivers it right there.
The company Nanoscribe is producing the most accurate 3D-printers in the world to print structures 250 times finer than a human hair. Researchers use them to print tiny robots, which may one day move inside the body. So far, this spermbot only functions in a petri dish and with bovine sperm. But maybe one day, it could help women who wish to get pregnant, Oliver Schmidt, Professor at the Leibniz Institute for Solid State and Materials Research (IFW) Dresden told DW. "With some men, the sperm are not moving, but still healthy. We would like to propel them artificially to be able to reach their final destination," he said. But the physicist admitted there is still a long journey ahead before the technique becomes a medical application.
Right now, the main challenge for using such micro robots inside a human body is imaging: "In a petri dish we can do all our experiments with high resolution microscopy," Schmidt said. "But when we operate deeply inside the tissue, the resolution fades."
Even the most modern computer tomographs, which are used to display a cross-section of a human body, are not strong enough to help guide the micro robot to its target. One would need real time imaging to observe the robot, he added.
The researchers from Dresden control their spiral with a magnetic field that rotates outside around the experiment. "It can not just be a permanent magnetic field. But on the other hand, the field does not need to be very strong." Certainly, for the human body it would not cause any harm, Schmidt stresses.
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Fun with the Koch snowflake
Fun with the Koch snowflake | Amazing Science |
The Koch snowflake is a remarkable tile which only tiles the plane if tiles of two (or more) different sizes are used. It is constructed by a recursive process. Begin with an equilateral triangle with side 1 and to each side attach in the middle an equilateral triangle with side 1/3. Each side of the new tile has side 1/3 also. Attach to the middle of each side an equilateral triangle of side 1/9 and so on. Alternatively, start with a regular hexagon of side 1 and cut equilateral triangles of side 1/3 from the middle of each side, and continue by cutting triangles of side 1/9 from the middle of each side, and so on. These constructions converge and for the area between two constructions after each iterations is reduced by a factor 4/9 each time. Congruent copies of the Koch snowflake do not tile the plane.
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China’s central bank has begun cautiously testing a digital currency
China’s central bank has begun cautiously testing a digital currency | Amazing Science |
Speeches and research papers from officials at the People's Bank of China show that the bank’s strategy is to introduce the digital currency alongside China’s renminbi. But there is currently no timetable for this, and the bank seems to be proceeding cautiously. Nonetheless the test is a significant step. It shows that China is seriously exploring the technical, logistical, and economic challenges involved in deploying digital money, something that could ultimately have broad implications for its economy and for the global financial system.
Private digital currencies, also known as cryptocurrencies, have shot to prominence in recent years following a wave of excitement, investment, and speculation focused on Bitcoin, a distributed, cryptographically secured form of money invented by an anonymous individual or group in 2008 (see “What Bitcoin Is, and Why It Matters”). Bitcoin’s distributed ledger of transactions, known as a blockchain, makes it possible for it to operate without any central authority.
Via Ben van Lier
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Need to Fix a Heart Attack? Try Photosynthesis Provided by Cyanobacteria
Need to Fix a Heart Attack? Try Photosynthesis Provided by Cyanobacteria | Amazing Science |
Injecting plant-like creatures like cyanobacteria into a rat's heart can jumpstart the recovery process, study finds.
Coronary artery disease is one of the most common causes of death and disability, afflicting more than 15 million Americans. Although pharmacological advances and revascularization techniques have decreased mortality, many survivors will eventually succumb to heart failure secondary to the residual microvascular perfusion deficit that remains after revascularization. A group of scientists now present a novel system that rescues the myocardium from acute ischemia, using photosynthesis through intramyocardial delivery of the cyanobacterium Synechococcus elongatus.
Via LilyGiraud
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One billion suns: World's brightest laser sparks new behavior in light
One billion suns: World's brightest laser sparks new behavior in light | Amazing Science |
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Near instantaneous evolution discovered in bacteria
Near instantaneous evolution discovered in bacteria | Amazing Science |
How fast does evolution occur? In certain bacteria, it can occur almost instantaneously, a University at Buffalo molecular biologist has discovered.
The discovery was made almost by accident, O'Brian said. The bacteria Bradyrhizobium japonicum was placed in a medium along with a synthetic compound to extract all the iron. O'Brian expected the bacteria to lie dormant having been deprived of the iron needed to multiply. But to his surprise, the bacteria started multiplying.
"We had the DNA of the bacteria sequenced on campus, and we discovered they had mutated and were using the new compound to take iron in to grow," he said. "It suggests that a single mutation can do that. So we tried it again with a natural iron-binding compound, and it did it again." The speed of the genetic mutations—17 days—was astounding.
"The machinery to take up iron is pretty complicated, so we would have thought many mutations would have been required for it to be taken up," he said. The evolution of the bacteria does not mean it is developing into some other type of creature. Evolution can also change existing species "to allow them to survive," O'Brian said.
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Artificial iris could let cameras react to light like the eye does
Artificial iris could let cameras react to light like the eye does | Amazing Science |
While the pupil may be the opening in the eye that lets light through to the retina, the iris is the tissue that opens and closes to determine the size of the pupil. Although mechanical irises are already a standard feature in cameras, scientists from Finland and Poland have recently created an autonomous artificial iris that’s much more similar to those found in the eye – it may even eventually be able to replace damaged or defective ones.
The contact lens-like device was created by researchers from Finland’s Tampere University of Technology, along with Poland’s University of Warsaw and Wrocław Medical University. It’s made from a polymer (a liquid crystal elastomer) that expands when exposed to light, then shrinks back when the light is lessened. This causes an opening in the middle to get smaller or larger, depending on the light levels – in this way, it works very much like a natural iris. Unlike automatic irises in cameras, it requires no power source or external light detection system.
With an “eye” towards one being able to use it as an optical implant, the scientists are now adapting it to work in an aqueous environment. They’re also working at increasing its sensitivity, so that its opening and closing are triggered by smaller changes in the amount of incoming light.
The research is being led by Tampere’s Prof. Arri Priimägi, and was recently described in a paper published in the journal Advanced Materials.
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Thousands of mouse genes could help decipher human disease providing important disease models
Thousands of mouse genes could help decipher human disease providing important disease models | Amazing Science |
Thousands of mouse genes (around 15% of the mouse genome) are now available from the IMPC - an important milestone for genotype-to-phenotype research.
Researchers at the European Bioinformatics Institute (EMBL-EBI) and their collaborators in the International Mouse Phenotyping Consortium (IMPC) have fully characterised thousands of mouse genes for the first time. Published in Nature Genetics, the results offer hundreds of new disease models and reveal previously unknown gene functions. The 3328 genes described in this publication by the IMPC represent approximately 15% of the mouse genome.
Via Integrated DNA Technologies
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Scientists illuminate structures vital to virus replication
Scientists illuminate structures vital to virus replication | Amazing Science |
“The challenge is a bit like being a car mechanic and not being able to see the engine or how it’s put together in detail,” says Paul Ahlquist, director of virology at the Morgridge Institute and professor of oncology and molecular virology at the University of Wisconsin–Madison. “This work is our first look at the engine.”
The research, published June 27 in the journal eLife, uses pioneering cryo-electron tomography to reveal the complex viral replication process in vivid detail, opening up new avenues to potentially disrupt, dismantle or redirect viral machinery.
One of several goals in the Ahlquist Lab is to understand genome replication for positive strand RNA viruses, the largest genetic class of viruses that includes many human pathogens such as the Zika, Dengue, SARS, and Chikungunya viruses. The group studies processes in a strategic way, focusing not on the fine details of a single important virus, but on large principles that apply to the whole class.
Via Integrated DNA Technologies
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By 2021 Most Internet Will Not Be For Humans. It Will Exceed Three Zettabytes
By 2021 Most Internet Will Not Be For Humans. It Will Exceed Three Zettabytes | Amazing Science |
Five years from now, there will be more machines talking to one another than people using smartphones, tablets and laptops, according to Cisco's annual internet forecast. Machine-to-machine communication, also called M2M, will soar to 51 percent of internet usage, with humans picking up the rest of the slack. The machines in question? Devices in your smart home, hospitals and offices. They'll account for more than half of 27.1 billion devices and connections, Cisco projects.
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Moonshine Master Cheng Toys With String Theory and K3 Surfaces
Moonshine Master Cheng Toys With String Theory and K3 Surfaces | Amazing Science |
The physicist-mathematician Dr Miranda Cheng is working to harness a mysterious connection between string theory, algebra and number theory. She happened to have read a book about the “monstrous moonshine,” a mathematical structure of truly monstrous dimensions that unfolded out of a similar bit of numerology: In the late 1970s, the mathematician John McKay noticed that 196,884, the first important coefficient of an object called the j-function, was the sum of one and 196,883, the first two dimensions in which a giant collection of symmetries called the monster group could be represented. By 1992, researchers had traced this farfetched (hence “moonshine”) correspondence to its unlikely source: string theory of all places – a candidate for the fundamental theory of physics that casts elementary particles as tiny oscillating strings. The j-function describes the strings’ oscillations in a particular string theory model, and the monster group captures the symmetries of the space-time fabric that these strings inhabit.
By the time of the Eyjafjallajökull’s eruption in 2010, “this was ancient stuff,” Cheng said — a mathematical volcano that, as far as physicists were concerned, had gone dormant. The string theory model underlying monstrous moonshine was nothing like the particles or space-time geometry of the real world. But Cheng sensed that a new moonshine, if it was one, might be different. It involved K3 surfaces — the geometric objects that she and many other string theorists study as possible toy models of real space-time.
By the time she flew home from Paris, Cheng had uncovered more evidence that the new moonshine existed. She and collaborators John Duncan and Jeff Harvey gradually teased out evidence of not one but 23 new moonshines: mathematical structures that connect symmetry groups on the one hand and fundamental objects in number theory called mock modular forms (a class that includes the j-function) on the other. The existence of these 23 moonshines, posited in their Umbral Moonshine Conjecture in 2012, was proved by Duncan and coworkers late last year.
Meanwhile, Cheng is 37, and is on the hot trail of the K3 string theory underlying the 23 moonshines — a particular version of the theory in which space-time has the geometry of a K3 surface. She and other string theorists hope to be able to use the mathematical ideas of umbral moonshine to study the properties of the K3 model in detail. This in turn could be a powerful means for understanding the physics of the real world where it can’t be probed directly — such as inside black holes. An assistant professor at the University of Amsterdam on leave from France’s National Center for Scientific Research, Cheng spoke with Quanta Magazine about the mysteries of moonshines, her hopes for string theory, and her improbable path from punk-rock high school dropout to a researcher who explores some of the most abstruse ideas in math and physics. An edited and condensed version of the conversation follows.
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Final Kepler Report Includes 219 New Potential Exoplanets
Final Kepler Report Includes 219 New Potential Exoplanets | Amazing Science |
Earth is the only planet we know of right now that supports life, but considering the scale of the universe it seems like there could be others. The first step in finding these Earth-like worlds is to detect planets orbiting nearby stars. That’s what NASA’s Kepler space telescope has been doing these past eight years. The project has had its ups and downs, but the final planetary catalog unveiled by astronomers during a recent meeting at the Ames Research Center signifies the final chapter of Kepler. Its grand total: 4,034 objects.
These exoplanets were detected by Kepler using what’s known as the transit method. The telescope watches a patch of the sky, recording dips in luminance that could indicate a planet passing between its host star and us. By monitoring over time, astronomers can determine the size, mass, and orbit of such a planet. This requires the distant solar system to be at just the right angle, and Kepler can only see a small segment of the sky. Still, we knew of only 300 probably exoplanets when Kepler launched. Now, there are thousands.
However, the Kepler satellite hit a snag four years ago when two of its four reaction wheels failed, leaving it unable to maintain orientation. The mission seemed doomed, but NASA worked out a clever solution in 2013 using the solar wind to stabilize the spacecraft in certain parts of its orbit. This K2 search program has been underway ever since, and it slated to come to an end on September 30th, 2017.
The list includes many objects that are confirmed planets, but everything on the list is at least 90 percent certain to be an exoplanet. To date, more than 1,200 exoplanets have been confirmed using Kepler data. The latest update to Kepler’s survey of the sky includes 219 new planetary candidates. Ten of them are in the habitable zone of their stars, meaning they (or their moons) could support life.
Kepler’s discoveries with new planets indicated in yellow on the above graph. One analysis presented alongside the new data sheds light on the way smaller planets form. The most common “small” planets come in two sizes. There are rocky worlds about 1.5-times the diameter of Earth, known as super-Earths. Then, the commonality of planets drops off until you get to “mini-Neptunes” at about two times Earth’s diameter. All planets seem to begin with roughly the same amount of solid material in the core, then gas adheres in large quantities to create a gas giant. Alternatively, a small envelope of gas sticks and you get a planet like Earth.
The data acquired by Kepler will instrumental in the search for life.
NASA plans to deploy a satellite in the 2030s that could capture images of these planets. In the meantime, the Webb Space Telescope might be able to image some of these planets with its 6.5-meter mirror. It launches in October 2018.
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Physicists Have Made the Impossible Quantum Hologram with a Single Photon
Physicists Have Made the Impossible Quantum Hologram with a Single Photon | Amazing Science |
A hologram with a single particle of light!
Scientists at the Faculty of Physics, University of Warsaw, have created the first ever hologram of a single light particle. The spectacular experiment, reported in the prestigious journal Nature Photonics, was conducted by Dr. Radoslaw Chrapkiewicz and Michal Jachura under the supervision of Dr. Wojciech Wasilewski and Prof. Konrad Banaszek. Their successful registering of the hologram of a single photon heralds a new era in holography: quantum holography, which promises to offer a whole new perspective on quantum phenomena. "We performed a relatively simple experiment to measure and view something incredibly difficult to observe: the shape of wavefronts of a single photon," says Dr. Chrapkiewicz.
In standard photography, individual points of an image register light intensity only. In classical holography, the interference phenomenon also registers the phase of the light waves (it is the phase which carries information about the depth of the image). When a hologram is created, a well-described, undisturbed light wave (reference wave) is superimposed with another wave of the same wavelength but reflected from a three-dimensional object (the peaks and troughs of the two waves are shifted to varying degrees at different points of the image). This results in interference and the phase differences between the two waves create a complex pattern of lines. Such a hologram is then illuminated with a beam of reference light to recreate the spatial structure of wavefronts of the light reflected from the object, and as such its 3D shape.
One might think that a similar mechanism would be observed when the number of photons creating the two waves were reduced to a minimum, that is to a single reference photon and a single photon reflected by the object. And yet you'd be wrong! The phase of individual photons continues to fluctuate, which makes classical interference with other photons impossible. Since the Warsaw physicists were facing a seemingly impossible task, they attempted to tackle the issue differently: rather than using classical interference of electromagnetic waves, they tried to register quantum interference in which the wave functions of photons interact.
Wave function is a fundamental concept in quantum mechanics and the core of its most important equation: the Schrödinger equation. In the hands of a skilled physicist, the function could be compared to putty in the hands of a sculptor: when expertly shaped, it can be used to 'mould' a model of a quantum particle system. Physicists are always trying to learn about the wave function of a particle in a given system, since the square of its modulus represents the distribution of the probability of finding the particle in a particular state, which is highly useful.
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Metafluorophores: Extremely colorful, incredibly bright and highly multiplexed
Metafluorophores: Extremely colorful, incredibly bright and highly multiplexed | Amazing Science |
Biomedical researchers are understanding the functions of molecules within the body’s cells in ever greater detail by increasing the resolution of their microscopes. However, what’s lagging behind is their ability to simultaneously visualize the many different molecules that mediate complex molecular processes in a single snap-shot.
These fluorescence images show a matrix representing 124 distinct metafluorophores, that are generated by combining three fluorescent dyes with varying intensity levels. In the future, the metafluorophore’s unique and identifiable color patterns can be used to analyze the molecular components of complex samples.
“We use DNA nanostructures as molecular pegboards: by functionalizing specific component strands at defined positions of the DNA nanostructure with one of three different fluorescent dyes, we achieve a broad spectrum of up to 124 fluorescent signals with unique color compositions and intensities,” said Peng Yin, who is a Core Faculty member at the Wyss Institute and Professor of Systems Biology at Harvard Medical School. “Our study provides a framework that allows researchers to construct a large collection of metafluorophores with digitally programmable optical properties that they can use to visualize multiple targets in the samples they are interested in.”
“We developed a triggered version of our metafluorophore that dynamically self-assembles from small component strands that take on their prescribed shape only when they bind their target,” said Ralf Jungmann, Ph.D., who is faculty at the LMU Munich and the Max Planck Institute of Biochemistry and co-conducted the study together with Yin. “These in-situ assembled metafluorophores can not only be introduced into complex samples with similar combinatorial possibilities as the prefabricated ones to visualize DNA, but they could also be leveraged to label antibodies as widely used detection reagents for proteins and other biomolecules.”
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A day lasting 80,000 Earth years? Possible on a strange exoplanet!
A day lasting 80,000 Earth years? Possible on a strange exoplanet! | Amazing Science |
So it’s a good moment to note how good we have it here on Earth. There are longer days in our solar system, but none are quite so pleasant. If “day” refers to the time it takes for a planet to rotate exactly once on its axis (a sidereal day), then the Venusian day is the longest, lasting two hundred and forty-three Earth days. That’s even longer, by nineteen Earth days, than a Venusian year, which is the time it takes the planet to orbit the sun. If, instead, “day” refers to the period between sunrise and sunset (a solar day), Neptune’s is the longest: the gas giant orbits the sun on its side, such that one pole or the other receives daylight for forty-two years non-stop.
Farther out in the universe, the days are longer still. Since 1995, some thirty-five hundred extrasolar planets have been discovered, but scientists only gained the ability to measure their spin rates in 2014. A great many of the known ones, though, orbit very close to their host stars and are probably tidally locked, with one side of the planet perpetually facing the star, just as our moon always presents the same face to Earth. “This leads to an infinitely long day, since if you are on the night side, you will never see the sun,” Konstantin Batygin, an astrophysicist at Caltech, explains. Last January, Batygin and the astronomer Mike Brown, also at Caltech, announced the possible existence of a ninth planet in the solar system, a relictual ice giant so distant that it orbits our sun once every twelve thousand to twenty thousand years. Last August, scientists discovered Proxima b, an exoplanet just 4.3 light-years away, which is about as close to us as any extrasolar planet will ever come. It, too, is probably tidally locked, its day eternal. But, even being so near, Proxima b would take us eighty thousand years (some thirty million days) to reach—a very long day’s journey into day.
Summer is a separate matter. A planet’s seasons are shaped by two factors: the eccentricity of its orbit—whether it’s closer to the sun at some times of the year than at others—and the tilt of its axis. Earth’s orbit is essentially circular, so the effect on our climate is negligible. But the planet itself leans twenty-three degrees to the side; as we orbit, there comes a day when the North Pole is maximally tilted toward the sun and the Northern Hemisphere sees more daylight than it will all year. That’s today, the summer solstice. (Below the equator, it’s the winter solstice, of course, and in six months our situations will reverse.) If we weren’t off-kilter, we’d have no summer nor any seasons at all. Every day would be as long as every other, and changes in the weather would be driven more by the local geography—latitude, elevation, that mountain range to the west that keeps the rain from falling—than by shifts in the jet stream, or the massive blooms of Pacific plankton in the winter that fuel El Niño, or the decline in sunlight that triggers autumn leaves to change color. Mercury, Venus, and Jupiter, standing all but upright, are seasonless. Sad.
Perhaps the weirdest summer of all unfolds on HD 131399Ab, an extrasolar gas giant that was discovered last July by Daniel Apai, an astronomer at the University of Arizona, and his colleagues. The planet belongs to a system with three stars but orbits only one of them, the biggest, which is eighty per cent larger than our sun. The other two stars orbit each other and, together, like a spinning dumbbell, orbit the big one. The view from HD 131399Ab would be spectacular if not for the ferocious winds, the lack of solid ground, and a steady rain of liquid iron. For much of the year, which lasts five hundred and fifty Earth years, the three stars appear close together in the sky, giving the planet “a familiar night side and day side, with a unique triple sunset and sunrise each day,” Kevin Wagner, one of the discoverers, remarked at the time. But as HD 131399Ab progresses in its orbit and the stars drift apart, a day arrives when the setting of one coincides with the rising of the other, and a period of near-constant daylight begins—a solstice of sorts, the start of a summer that will last about a hundred and forty Earth years.
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Can Human Mortality Really Be Hacked?
Can Human Mortality Really Be Hacked? | Amazing Science |
Backed by the digital fortunes of Silicon Valley, biotech companies are brazenly setting out to “cure” aging. Can this really be done?
Via LilyGiraud
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Groundbreaking discovery confirms the existence of two orbiting supermassive black holes
Groundbreaking discovery confirms the existence of two orbiting supermassive black holes | Amazing Science |
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Water exists as two different liquids
Water exists as two different liquids | Amazing Science |
"It is very exciting to be able to use X-rays to determine the relative positions between the molecules at different times", says Fivos Perakis, postdoc at Stockholm University with a background in ultrafast optical spectroscopy. "We have in particular been able to follow the transformation of the sample at low temperatures between the two phases and demonstrated that there is diffusion as is typical for liquids".
"I have studied amorphous ices for a long time with the goal to determine whether they can be considered a glassy state representing a frozen liquid", says Katrin Amann-Winkel, researcher in Chemical Physics at Stockholm University. "It is a dream come true to follow in such detail how a glassy state of water transforms into a viscous liquid which almost immediately transforms to a different, even more viscous, liquid of much lower density".
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Mice Provide Insight Into Genetics of Autism Spectrum Disorders
Mice Provide Insight Into Genetics of Autism Spectrum Disorders | Amazing Science |
Via Mariaschnee
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One in five 'healthy' adults seems to carry disease-related genetic mutations
One in five 'healthy' adults seems to carry disease-related genetic mutations | Amazing Science |
Some doctors dream of diagnosing diseases—or at least predicting disease risk—with a simple DNA scan. But others have said the practice, which could soon be the foundation of preventative medicine, isn’t worth the economic or emotional cost. Now, a new pair of studies puts numbers to the debate, and one is the first ever randomized clinical trial evaluating whole genome sequencing in healthy people. Together, they suggest that sequencing the genomes of otherwise healthy adults can for about one in five people turn up risk markers for rare diseases or genetic mutations associated with cancers.
What that means for those people and any health care system considering genome screening remains uncertain, but some watching for these studies welcomed the results nonetheless. “It's terrific that we are studying implementation of this new technology rather than ringing our hands and fretting about it without evidence,” says Barbara Biesecker, a social and behavioral researcher at the National Human Genome Research Institute in Bethesda, Maryland.
The first genome screening study looked at 100 healthy adults who initially reported their family history to their own primary care physician. Then half were randomly assigned to undergo an additional full genomic workup, which cost about $5000 each and examined some 5 million subtle DNA sequence changes, known as single-nucleotide variants, across 4600 genes—such genome screening goes far beyond that currently recommended by the American College of Medical Genetics and Genomics (ACMG), which suggests informing people of results for just 59 genes known or strongly expected to cause disease.
Via Integrated DNA Technologies
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Discovery of a new mechanism for bacterial division
Discovery of a new mechanism for bacterial division | Amazing Science |
Most rod-shaped bacteria divide by splitting into two around the middle after their DNA has replicated safely and segregated to opposite ends of the cell. This seemingly simple process actually demands tight and precise coordination, which is achieved through two biological systems: nucleoid occlusion, which protects the cell's genetic material from dividing until it replicates and segregates, and the "minicell" system, which localizes the site of division around the middle of the cell, where a dividing wall will form to split it in two.
But some pathogenic bacteria, e.g. Mycobacterium tuberculosis, don't use these mechanisms. EPFL scientists have now combined optical and atomic force microscopy to track division in such bacteria for the first time and have discovered that they use instead an undulating "wave-pattern" along their length to mark future sites of division. The findings are published in Nature Microbiology.
The work was carried out jointly by the labs of John McKinney and Georg Fantner at EPFL. The scientists wanted to understand how bacteria that do not have the genes for nucleoid occlusion and the minicell system "decide" where and when to divide. This is important, as many pathogenic bacteria fall into this category, and knowing how they divide can open up new ways to fight them.
The researchers focused on Mycobacterium smegmatis, a non-pathogenic relative of M. tuberculosis. Neither of these bacteria uses the two "conventional" biological systems for coordinating division, meaning that a non-conventional approach was needed for studying them.
The researchers combined two types of microscopy to track the life cycle of the bacteria. The first technique was optical microscopy, which uses fluorescent labels for "seeing" various biological structures and biomolecules. The second technique was atomic force microscopy, which provides extremely high-resolution images of structures on the cell surface by "feeling" the surface with a tiny mechanical probe, much like a blind person can form a three-dimensional mental image of an object by passing their hands over its surface.
"This experiment constitutes the longest continuous atomic force microscopy experiment ever performed on growing cells," says Georg Fantner, while John McKinney adds: "It illustrates the power of new technologies not only to analyze the things we already knew about with greater resolution, but also to discover new things that we hadn't anticipated."
Via Mariaschnee
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Cancer Immunotherapy - Where Are We Today?
Cancer Immunotherapy - Where Are We Today? | Amazing Science |
The immune system is naturally equipped to protect us against cancer. Cytotoxic T lymphocytes—otherwise known as killer T cells—are especially effective at targeting tumors. However, cancers sometimes figure out how to outsmart the immune system and protect themselves. Immunotherapy aims to reverse that situation.
This review is highlighting a wide scope of immune-based approaches that are already improving outcomes for patients. Many of these treatments work by either directly or indirectly enhancing the activity of T cells.
Much of what we know about the immune system and its relationship to cancer was discovered by scientists affiliated with the Cancer Research Institute (CRI), which since 1953 has served as the world’s leading (and for several decades only) nonprofit organization dedicated exclusively to transforming cancer patient care by advancing immunotherapy and the science behind it.
It’s clear now that immunotherapy can provide long-term benefits to sizable subsets of patients with diverse types of cancer, and new clinical breakthroughs are happening all the time. Thus far in 2017 there have been eight immunotherapy approvals, with two immunotherapies (durvalumab and avelumab) gaining approval for the first time. These clinical breakthroughs were made possible in part by decades of discoveries made by Cancer Research Institute (CRI) scientists.
One of the most important figures who helped advance immunotherapy (and the science that supports it) to this point was Dr. Lloyd J. Old, who actually worked with Dr. William B. Coley’s daughter, Helen, at CRI. Now known as the “Father of Modern Tumor Immunology,” Dr. Old directed CRI’s scientific and medical efforts for 40 years (1971-2011), during which time he made major discoveries about the immune system and cancer, and helped establish the scientific foundation upon which today’s immunotherapies were developed.
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1ac667063443c247 | March 15, 2012
This is a follow up to the design electron article from yesterday
Manoharan lab covers their own work here
The work could lead to new materials and devices.
Nature - Designer Dirac fermions and topological phases in molecular graphene
Phantom Fields
A version of molecular graphene in which the electrons respond as if they're experiencing a very high magnetic field (red areas) when none is actually present. Scientists from Stanford and SLAC National Accelerator Laboratory calculated the positions where carbon atoms in graphene should be to make its electrons believe they were being exposed to a magnetic field of 60 Tesla, more than 30 percent higher than the strongest continuous magnetic field ever achieved on Earth. (A 1 Tesla magnetic field is about 20,000 times stronger than the Earth's.) The researchers then used a scanning tunneling microscope to place carbon monoxide molecules (black circles) at precisely those positions. The electrons responded by behaving exactly as expected — as if they were exposed to a real field, but no magnetic field was turned on in the laboratory. Image credit: Hari Manoharan / Stanford University.
Schrödinger Meets Dirac
Visualization depicting the transformation of an electron moving under the influence of the non-relativistic Schrödinger equation (upper planar quantum waves) into an electron moving under the prescription of the relativistic Dirac equation (lower honeycomb quantum waves). The light blue line shows a quasiclassical path of one such electron as it enters the molecular graphene lattice made of carbon monoxide molecules (black/red atoms) positioned individually by an STM tip (comprised of iridium atoms, dark blue). The path shows that the electron becomes trapped in synthetic chemical bonds that bind it to a honeycomb lattice and allow it to quantum mechanically tunnel between neighboring honeycomb sites, just like graphene. The underlying electron density in a honeycomb pattern (lower part of image, yellow-orange) is the quantum superposition formed from all such electron paths as they transmute into a new tunable species of massless Dirac fermions. Image credit: Hari Manoharan / Stanford University.
Designer Electrons
This graphic shows the effect that a specific pattern of carbon monoxide molecules (black/red) has on free-flowing electrons (orange/yellow) atop a copper surface. Ordinarily the electrons behave as simple plane waves (background). But the electrons are repelled by the carbon monoxide molecules, placed here in a hexagonal pattern. This forces the electrons into a honeycomb shape (foreground) mimicking the electronic structure of graphene, a pure form of carbon that has been widely heralded for its potential in future electronics. The molecules are precisely positioned with the tip of a scanning tunneling microscope (dark blue). Image credit: Hari Manoharan / Stanford University.
Molecular Graphene PNP Junction Device
Stretching or shrinking the bond lengths in molecular graphene corresponds to changing the concentrations of Dirac electrons present. This image shows three regions of alternating lattice spacing sandwiched together. The two regions on the ends contain Dirac "hole" particles (p-type regions), while the region in the center contains Dirac "electron" particles (n-type region). A p-n-p structure like this is of interest in graphene transistor applications. Image credit: Hari Manoharan / Stanford University.
The observation of massless Dirac fermions in monolayer graphene has generated a new area of science and technology seeking to harness charge carriers that behave relativistically within solid-state materials. Both massless and massive Dirac fermions have been studied and proposed in a growing class of Dirac materials that includes bilayer graphene, surface states of topological insulators and iron-based high-temperature superconductors. Because the accessibility of this physics is predicated on the synthesis of new materials, the quest for Dirac quasi-particles has expanded to artificial systems such as lattices comprising ultracold atoms. Here we report the emergence of Dirac fermions in a fully tunable condensed-matter system—molecular graphene—assembled by atomic manipulation of carbon monoxide molecules over a conventional two-dimensional electron system at a copper surface5. Using low-temperature scanning tunnelling microscopy and spectroscopy, we embed the symmetries underlying the two-dimensional Dirac equation into electron lattices, and then visualize and shape the resulting ground states. These experiments show the existence within the system of linearly dispersing, massless quasi-particles accompanied by a density of states characteristic of graphene. We then tune the quantum tunnelling between lattice sites locally to adjust the phase accrual of propagating electrons. Spatial texturing of lattice distortions produces atomically sharp p–n and p–n–p junction devices with two-dimensional control of Dirac fermion density and the power to endow Dirac particles with mass. Moreover, we apply scalar and vector potentials locally and globally to engender topologically distinct ground states and, ultimately, embedded gauge fields wherein Dirac electrons react to ‘pseudo’ electric and magnetic fields present in their reference frame but absent from the laboratory frame. We demonstrate that Landau levels created by these gauge fields can be taken to the relativistic magnetic quantum limit, which has so far been inaccessible in natural graphene. Molecular graphene provides a versatile means of synthesizing exotic topological electronic phases in condensed matter using tailored nanostructures.
14 pages of supplemental material
Molecular graphene assembly
Molecular graphene assembly. A movie shows the nanoscale assembly sequence of an electronic honeycomb lattice by manipulating individual CO molecules on the Cu(111) two-dimensional electron surface state with the STM tip. The video comprises 52 topographs (30 × 30 nm2, bias voltage V = 10 mV, tunnel current I = 1 nA) acquired during the construction phase and between manipulation steps.
Tunable Pseudomagnetic Field
Molecular Manipulation
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0d850ccd3c8c0974 | Dirac equation
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In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles, for which parity is a symmetry, such as electrons and quarks, and is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics.
It accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved, and actually predated its experimental discovery. It also provided a theoretical justification for the introduction of several-component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represent one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him.[2] In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.
Mathematical formulation[edit]
The Dirac equation in the form originally proposed by Dirac is:[3]
Dirac equation (original)
\left(\beta mc^2 + c(\sum_{n \mathop =1}^{3}\alpha_n p_n)\right) \psi (x,t) = i \hbar \frac{\partial\psi(x,t) }{\partial t}
where ψ = ψ(x, t) is the wave function for the electron of rest mass m with spacetime coordinates x, t. The p1, p2, p3 are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, c is the speed of light, and ħ is the Planck constant divided by 2π. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.
Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have bearing on the problem of atomic spectra. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.
The new elements in this equation are the 4 × 4 matrices αk and β, and the four-component wave function ψ. There are four components in ψ because evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion).
The 4 × 4 matrices αk and β are all Hermitian and have squares equal to the identity matrix:
and they all mutually anticommute (if i and j are distinct):
\alpha_i\alpha_j + \alpha_j\alpha_i = 0
\alpha_i\beta + \beta\alpha_i = 0
The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices, and the form of the wave function, have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th century work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre (Theory of Linear Extensions). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.
Making the Schrödinger equation relativistic[edit]
The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:
which says that the length of this four-vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,
\left(-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2\right)\phi = \frac{m^2c^2}{\hbar^2}\phi
with the wave function ϕ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. The space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression
and this density is convected according to the probability current vector
with the conservation of probability current and density following from the continuity equation:
which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression
The continuity equation is as before. Everything is compatible with relativity now, but we see immediately that the expression for the density is no longer positive definite – the initial values of both ψ and tψ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.
Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson). Historically, Schrödinger himself arrived at this equation before the one that bears his name, but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the charge density, which can be positive or negative, and not the probability density.
Dirac's coup[edit]
Dirac thus thought to try an equation that was first order in both space and time. One could, for example, formally take the relativistic expression for the energy
E = c\sqrt{p^2 + m^2c^2}\,,
replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.
As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:
\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right).
On multiplying out the right side we see that, in order to get all the cross-terms such as xy to vanish, we must assume
AB + BA = 0, \;\ldots
A^2 = B^2 = \ldots = 1.\,
Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B, C and D are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.
Given the factorization in terms of these matrices, one can now write down immediately an equation
\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t\right)\psi = \kappa\psi
with κ to be determined. Applying again the matrix operator on both side yields
\left(\nabla^2 - \frac{1}{c^2}\partial_t^2\right)\psi = \kappa^2\psi.
On taking κ = mc/ħ we find that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is
\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t - \frac{mc}{\hbar}\right)\psi = 0.
A = i\beta \alpha_1, B = i\beta \alpha_2, C = i\beta \alpha_3, D = \beta \, ,
we get the Dirac equation as written above.
Covariant form and relativistic invariance[edit]
To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:
\gamma^0 = \beta \,
\gamma^k = \gamma^0 \alpha^k. \,
and the equation takes the form
Dirac equation
i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0
where there is an implied summation over the values of the twice-repeated index μ = 0, 1, 2, 3. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is
\gamma^0 = \left(\begin{array}{cccc} I_2 & 0 \\ 0 & -I_2 \end{array}\right),
\gamma^1 = \left(\begin{array}{cccc} 0 & \sigma_x \\ -\sigma_x & 0 \end{array}\right),
\gamma^2 = \left(\begin{array}{cccc} 0 & \sigma_y \\ -\sigma_y & 0 \end{array}\right),
\gamma^3 = \left(\begin{array}{cccc} 0 & \sigma_z \\ -\sigma_z & 0 \end{array}\right).
The complete system is summarized using the Minkowski metric on spacetime in the form
\{\gamma^\mu,\gamma^\nu\} = 2 \eta^{\mu\nu} \,
where the bracket expression
\{a, b\} = ab + ba
denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature (+ − − −). The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.
The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:
P_\mathrm{op}\psi = mc\psi. \,
Using {\partial\!\!\!\big /} (pronounced: "d-slash"[4]) in Feynman slash notation, which includes the gamma matrices as well as a summation over the spinor components in the derivative itself, the Dirac equation becomes:
i \hbar {\partial\!\!\!\big /} \psi - m c \psi = 0
In practice, physicists often use units of measure such that ħ = c = 1, known as natural units. The equation then takes the simple form
Dirac equation (natural units)
(i{\partial\!\!\!\big /} - m) \psi = 0\,
A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:
\gamma^{\mu\prime} = S^{-1} \gamma^\mu S.
If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary;
\gamma^{\mu\prime} = U^\dagger \gamma^\mu U.
The transformation U is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator γμμ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form
( iU^\dagger \gamma^\mu U\partial_\mu^\prime - m)\psi(x^\prime,t^\prime) = 0
U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi(x^\prime,t^\prime) = 0.
If we now define the transformed spinor
\psi^\prime = U\psi
then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:
(i\gamma^\mu\partial_\mu^\prime - m)\psi^\prime(x^\prime,t^\prime) = 0.
Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function (see below). The representation shown here is known as the standard representation – in it, the wave function's upper two components go over into Pauli's 2-spinor wave function in the limit of low energies and small velocities in comparison to light.
The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation – they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as γμγν represent oriented surface elements, and so on. With this in mind, we can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is
V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta.
For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of g, where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus
V = i \gamma^0\gamma^1\gamma^2\gamma^3.\
This matrix is given the special symbol γ5, owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the standard representation it is
\gamma_5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}.
This matrix will also be found to anticommute with the other four Dirac matrices:
\gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0
It takes a leading role when questions of parity arise, because the volume element as a directed magnitude changes sign under a spacetime reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime .
Conservation of probability current[edit]
By defining the adjoint spinor
where ψ is the conjugate transpose of ψ, and noticing that
(\gamma^\mu)^\dagger\gamma^0 = \gamma^0\gamma^\mu \,,
we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by γ0, the adjoint equation:
\bar{\psi}(-i\gamma^\mu\partial_\mu - m) = 0 \,
where μ is understood to act to the left. Multiplying the Dirac equation by ψ from the left, and the adjoint equation by ψ from the right, and subtracting, produces the law of conservation of the Dirac current:
\partial_\mu \left( \bar{\psi}\gamma^\mu\psi \right) = 0.
Now we see the great advantage of the first-order equation over the one Schrödinger had tried – this is the conserved current density required by relativistic invariance, only now its 4th component is positive definite and thus suitable for the role of a probability density:
J^0 = \bar{\psi}\gamma^0\psi = \psi^\dagger\psi.
Because the probability density now appears as the fourth component of a relativistic vector, and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.
See Dirac spinor for details of solutions to the Dirac equation. Note that since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected – see the hole theory section below for more details.
Comparison with the Pauli theory[edit]
See also: Pauli equation
The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with Lz = −1, 0, +1. The conclusion is that silver atoms have net intrinsic angular momentum of 12. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, where as the Minkowski four-vector A_\mu can be defined as A_\mu = (\phi, c\mathbf{A}).)
H = \frac{1}{2m}\left( \boldsymbol{\sigma}\cdot\left(\mathbf{p} - e \mathbf{A}\right)\right)^2 + e\phi.
Here A and φ represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units:
H = \frac{1}{2m}\left(\mathbf{p} - e \mathbf{A}\right)^2 + e\phi - \frac{e\hbar}{2m}\boldsymbol{\sigma}\cdot \mathbf{B}.
This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology— Dirac now had a theoretical argument that implied that spin was somehow the consequence of the marriage of quantum mechanics to relativity. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units)
(i\gamma^\mu(\partial_\mu + i\frac{e}{c}A_\mu) - m) \psi = 0\,
A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:
\begin{pmatrix} (mc^2 - E + e \phi) & c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \\ -c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) & \left(mc^2 + E - e \phi\right) \end{pmatrix} \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
(E - e\phi) \psi_+ - c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_- = mc^2 \psi_+
-(E - e\phi) \psi_- + c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_+ = mc^2 \psi_-
Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum going over to the classical value,
E - e\phi \approx mc^2
and so the second equation may be written
\psi_- \approx \frac{1}{2mc} \boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right) \psi_+
which is of order v/c - thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement
(E - mc^2) \psi_+ = \frac{1}{2m} \left[\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e \mathbf{A}\right)\right]^2 \psi_+ + e\phi \psi_+
The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.
It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an irreducible whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.
Comparison with the Weyl theory[edit]
In the limit m → 0, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-1/2 particles.[5]
Dirac Lagrangian[edit]
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:
\mathcal{L}=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^{2}\overline{\psi}\psi
If one varies this with respect to ψ one gets the Adjoint Dirac equation. Meanwhile if one varies this with respect to ψ one gets the Dirac equation.
Physical interpretation[edit]
The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions.[citation needed]
Identification of observables[edit]
The critical physical question in a quantum theory is—what are the physically observable quantities defined by the theory? According to general principles, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be
H = \gamma^0 \left[mc^2 + c \gamma^k \left(p_k-\frac{q}{c}A_k\right) \right] + qA^0.
where, as always, there is an implied summation over the twice-repeated index k = 1, 2, 3. This looks promising, because we see by inspection the rest energy of the particle and, in case A = 0, the energy of a charge placed in an electric potential qA0. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is
H = c\sqrt{\left(p - \frac{q}{c}A\right)^2 + m^2c^2} + qA^0.
Thus the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behaviour implied by the Dirac equation amounts to a misidentification of these observables.
Hole theory[edit]
The negative E solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of photons. Real electrons obviously[clarification needed] do not behave in this way.
To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.
If an electron is forbidden from simultaneously occupying positive-energy and negative-energy eigenstates, then the feature known as Zitterbewegung, which arises from the interference of positive-energy and negative-energy states, would have to be considered to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the preceding paragraph. Recent results have been published in Nature [R. Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)] in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment impacts the hole interpretation if one infers that the physics-laboratory experiment is not merely a check on the mathematical correctness of a Dirac-equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach.
Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy, since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons has to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.
In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.
In quantum field theory[edit]
See also: Fermionic field
In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.
Other formulations[edit]
The Dirac equation can be formulated in a number of other ways.
As a differential equation in one real component[edit]
Generically (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.[6]
Curved spacetime[edit]
This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.
The algebra of physical space[edit]
This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.
See also[edit]
The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was inaugurated on November 13, 1995.[7]
1. ^ P.W. Atkins (1974). Quanta: A handbook of concepts. Oxford University Press. p. 52. ISBN 0-19-855493-1.
2. ^ T.Hey, P.Walters (2009). The New Quantum Universe. Cambridge University Press. p. 228. ISBN 978-0-521-56457-1.
3. ^ Dirac, P.A.M. (1982) [1958]. Principles of Quantum Mechanics. International Series of Monographs on Physics (4th ed.). Oxford University Press. p. 255. ISBN 978-0-19-852011-5.
4. ^ see for example Brian Pendleton: Quantum Theory 2012/2013, section 4.3 The Dirac Equation
5. ^ Tommy Ohlsson (22 September 2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. p. 86. ISBN 978-1-139-50432-4.
6. ^ Akhmeteli, Andrey (2011). "One real function instead of the Dirac spinor function" (PDF). Journal of Mathematical Physics 52 (8): 082303. arXiv:1008.4828. Bibcode:2011JMP....52h2303A. doi:10.1063/1.3624336.
7. ^ Gisela Dirac-Wahrenburg. "Paul Dirac". Dirac.ch. Retrieved 2013-07-12.
Selected papers[edit]
• Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum.
• Bjorken, J D & Drell, S. Relativistic Quantum mechanics.
• Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer.
• Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
• Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.
External links[edit] |
5b9364ffed3c3c29 | Ab Initio Molecular Orbital Theory
To make a quantum mechanical model of the electronic structure of a molecule, we must solve the Schrödinger equation.
Solving this equation is a very difficult problem and cannot be done without making approximations. We have covered some of these approximations in the Semiempirical MO Theory handout. In this handout we focus on ab initio methods of solving the equation, in which no integrals are neglected in the course of the calculation.
The Born-Oppenheimer Approximation
The first approximation is known as the Born-Oppenheimer approximation, in which we take the positions of the nuclei to be fixed so that the internuclear distances are constant. Because nuclei are very heavy in comparison with electrons, to a good approximation we can think of the electrons moving in the field of fixed nuclei. We first choose a geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy.
The Independent Electron Approximation
When more than one electron is present, the Schrödinger equation is impossible to solve because of the interelectron terms in the Hamiltonian. Consider, for instance, the Hamiltonian for the hydrogen molecule in the Born-Oppenheimer approximation.
The first two terms are due to the kinetic energy of the electrons. The last six terms express the potential energy of the system of four particles. The potential energy term due to the repulsion of the electrons makes the Schrödinger equation impossible to solve.
To produce a solvable Schrödinger equation we assume that the Hamiltonian is a sum of one-electron functions, fi, with an approximate potential energy that takes the average interaction of the electrons into account. This leads to a set of one-electron equations, called the Hartree-Fock equations, where is a one-electron wavefunction.
The total wavefunction that is a solution to the total Schrödinger equation, , is approximated as the product of the solutions to the one-electron equations.
This product must be adjusted to satisfy the Pauli Exclusion principle, but we won't get into that here. If you are familiar with determinants, it involves writing the wavefunction as a determinant.
The Hartree-Fock Self-Consistent Field (SCF) Approximation
The question remains about the approximate potential energy in the one-electron functions that take the average interaction of the electrons into account. What is the form of the functions fi in the Hartree-Fock equations? The most common way of handling this is to define
where vi is an average potential energy due to the interaction of one electron with all the other electrons and nuclei in the molecule. The average potential depends on the orbitals, , of the other electrons, which means we must solve the Hartree-Fock equations iteratively.
The iterative solution of the Hartree-Fock equation is as follows.
1. Guess reasonable one-electron orbitals (wavefunctions), , and calculate the average potential energies, vi.
2. Using the variation principle, solve the Hartree-Fock equations,
to give new one-electron orbitals, . Use these new orbitals to calculate new and improved average potential energies, vi. Because the solution of the Hartree-Fock equations depends on the variation principle, the Hartree-Fock energy should be higher than the true energy.
3. Repeat the second step until the one-electron orbitals and potential energies don't change (are self-consistent).
Restricted Hartree-Fock Calculations
To take the Pauli Principle into account, we must include electron spin in our wavefunctions. The orbitals that are calculated by the Hartree-Fock method actually are spin orbitals that are a product of a spatial wavefunction and a spin function.
In a spin orbital, is the spatial wavefunction describing the probability of finding the electron in space and or are spin wavefunctions.
For a closed shell system, in which all of the electrons are paired, during the solution of the self-consistent field equations, we can restrict the solution so that the spatial wavefunctions for paired electrons are the same. This is called a restricted Hartree-Fock (RHF) calculation and generally is used for molecules in which all the electrons are paired. When the spin functions are removed, we are left with a set of spatial orbitals, each occupied by two electrons.
An example would be the restricted Hartree-Fock solution to the Schrödinger equation for the hydrogen molecule, H2. This would lead to two spatial orbitals, one occupied by the pair of electrons and one unoccupied. The orbitals holding electrons are called occupied orbitals and the unoccupied orbitals are called virtual orbitals.
Unrestricted Hartree-Fock Calculations
For open shell systems that contain unpaired electrons, the assumption made in the restricted Hartree-Fock method obviously won't work. There is more than one way of handling this type of problem. One way is to not constrain pairs of electrons to occupy the same spatial orbital - the unrestricted Hartree-Fock (UHF) method. In this method there are two sets of spatial orbitals - those with spin up () electrons and those with spin down
() electrons. This leads to two sets of orbitals as pictured at the right and to a lower energy than if the restricted method were used.
Basis Sets
For molecular calculations, the Hartree-Fock SCF equations
still cannot be solved without one further approximation. To solve the equations, each SCF orbital,, is written as a linear combination of atomic orbitals. For instance, for the H2 molecule, the simplest approximation is to write each spatial SCF orbital as a combination of 1s atomic orbitals, each centered on one of the protons.
This reduces the problem to solving for the coefficients, c1 and c2, since the atomic orbitals do not change.
The set of atomic orbitals that is chosen to represent the SCF orbitals is called a basis set. The {1sA, 1sB} basis set shown above is a minimal basis set - the smallest set of orbitals possible that describe an SCF orbital. Usually, the quality of a basis set depends on its size. For instance, a larger basis set, such as {1sA, 1sB, 2sA, 2sB}would do a better job approximating the SCF orbital than {1sA, 1sB}.
For many-electron atoms, we don't know the actual mathematical functions for the atomic orbitals, so substitutes are used - usually either Slater-type orbitals (STO) or Gaussian-type orbitals (GTO). We won't concern ourselves with the exact form of STO and GTO. Suffice it to say that they are chosen to behave mathematically like the actual atomic orbitals: s-type, p-type, d-type, and f-type, for instance. A few commonly used basis sets are listed below. The symbol of the basis set is given in the left column and the characteristics of the basis set in the center. At the right is the basis set that would be used to represent methane. For instance, the STO-3G basis set for methane would be {1sH, 1sH, 1sH, 1sH, 1sC, 2sC, 2pxC, 2pyC, 2pzC}.
Basis Sets1
Basis Set Example (CH4)
A minimal basis set (although not the smallest possible) using three GTOs to approximate each STO. This basis set should only be used for qualitative results on very large systems
Each H: 1s
C: 1s, 2s, 2px, 2py, 2pz
Inner shell basis functions made of three GTOs. Valence s- and p-orbitals each represented by two basis functions (one made of two GTOs, the other of a single GTO). Use for very large molecules for which 6-31G is too expensive.
Each H: 1s, 1s'
C: 1s, 2s, 2px, 2py, 2pz, 2s', 2px', 2py', 2pz'
Inner shell basis functions made of six GTOs. Valence s- and p- orbitals each represented by two basis functions (one made of three GTOs, the other of a single GTO). Adds six d-type basis functions to non-hydrogen atoms. This is a popular basis set that often is used for medium and large systems.
Each H: 1s, 2s
C: 1s, 2s, 2px, 2py, 2pz, 2s', 2px', 2py', 2pz', 3dx2, 3dy2, 3dz2, 3dxy, 3dxz, 3dyz
Like 6-31G(d) except p-type functions also are added for hydrogen atoms. Use when hydrogens are of interest and for final, accurate energy calculations.
Each H: 1s, 2s, 2px, 2py, 2pz
Generally, the larger the basis set the more accurate the calculation (within limits) and the more computer time that is required. As an example, consider the calculation of the bond length of H-F using different basis sets, as shown below. 1
Basis Set
Bond Length (Å)
| Error (Å) |
You might notice that although the large basis set, 6-311++G(d,p), predicts the correct answer to within 0.001 Å, several others are correct to within 0.01 Å (well within the criteria of chemical accuracy). Although a larger basis set usually gives better results, you often have diminishing returns as you choose larger sets. A point may be reached beyond which the additional computer time is not worth it.
Post-SCF Calculations
Even with a very large basis set calculation, Hartree-Fock results are not exact because they rely on the independent electron approximation. Hartree-Fock SCF Theory is a good base-level theory that is reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states2. Electrons are not independent, though. We say that they are correlated with each other and that the Hartree-Fock method neglects electron correlation. This means that Hartree-Fock calculations do not do a good job modeling the energetics of reactions or bond dissociation. There are several ways of correcting SCF results to take electron correlation into account.
One method of taking electron correlation into account is Møller-Plesset many-body perturbation theory, which is used after a RHF or UHF calculation has been made. It is assumed that the relationship between the exact and Hartree-Fock Hamiltonians is expressed by an additional term, H(1), so that H = fi + H(1). Calculations based on this assumption lead to corrections that can improve SCF results. Various levels of perturbation theory can be applied to the problem. They are called MP2, MP3, MP4, etc. MP2 calculations are not time-consuming and usually give quite accurate geometries and about one-half of the correlation energy. Because perturbation theory is not based on the variation principle, the energy predicted by MP calculations can fall below the actual energy.
Another important method of correcting for the correlation energy is configuration interaction (CI). Conceptually we can think of CI calculations as using the variation principle to combine various SCF excited states with the SCF ground state, which lowers its energy. We won't use CI calculations in our exercises at this level.
SCF Molecular Orbitals
When calculating molecular orbitals, you should remember that molecular orbitals are not real physical quantities. Orbitals are a mathematical convenience that help us think about bonding and reactivity, but they are not physical observables. In fact, several different sets of molecular orbitals can lead to the same energy. Nevertheless, they are quite useful. We will use ethylene as an example to illustrate MO concepts.
The basis functions in SCF molecular orbitals are like atomic orbitals. A RHF/6-31G(d) calculation on ethylene uses 38 basis functions (15 for each carbon and 2 for each hydrogen). Since the molecular orbital wavefunction is expanded in terms of the all the basis functions,
it might seem that constructing a picture of the orbital would be difficult. Luckily, most of the coefficients are zero, so the molecular orbitals are easy to picture. Consider, for instance, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of ethylene.
The HOMO is a bonding-orbital.
The LUMO is an antibonding -orbital.
Scaling Vibrational Frequencies
In the last part of the job output from a frequency calculation you will find the predicted vibrational frequencies (cm-1) of the normal modes of the molecule. Also supplied are the predicted intensities of the IR and Raman bands corresponding to these normal modes.
1 2 3
B1 B2 A1
Frequencies -- 1335.5948 1383.4094 1679.4157
4 5 6
A1 A1 B2
Frequencies -- 2027.8231 3160.8817 3232.9970
Computational results usually have systematic errors. In the case of Hartree-Fock level calculations, for instance, it is known that calculated frequency values are almost always too high by 10% - 12%. To compensate for this systematic error, it is usual to multiply frequencies predicted at the HF/6-31G(d) level by an empirical factor of 0.893. Similarly, frequencies calculated at the MP2/6-31G(d) level are scaled by 0.943. 1
The predicted frequencies after applying the 0.893 scale factor are listed below.
1 2 3
B1 B2 A1
Scaled Frequencies -- 1193 1235 1450
4 5 6
A1 A1 B2
Scaled Frequencies -- 1811 2822 2887
1J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p 102.
2J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1996, p 115.
Physical Chemistry Consortium |
9af3914f59a99920 | Dirac sea
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Dirac sea for a massive particle. • particles, • antiparticles
The equation relating energy, mass and momentum in special relativity is:
In the special case of a particle at rest (i.e. p = 0), the above equation reduces to E^2=m^2c^4, which is usually quoted as the familiar E=mc^2. However, this is a simplification because, while x \cdot x=x^2, we can also see that (-x) \cdot (-x)=x^2. Therefore, the correct equation to use to relate energy and mass in the Hamiltonian of the Dirac equation is:
Here the negative solution was used to predict the existence of antimatter, discovered by Carl Anderson as the positron. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum field theory, but it also implies that the number of particles cannot be conserved.[1]
The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, an extension of the Schrödinger equation that is consistent with special relativity, that Dirac had formulated in 1928. Although the equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for each quantum state possessing a positive energy E, there is a corresponding state with energy -E. This is not a big difficulty when an isolated electron is considered, because its energy is conserved and negative-energy electrons may be left out. However, difficulties arise when effects of the electromagnetic field are considered, because a positive-energy electron would be able to shed energy by continuously emitting photons, a process that could continue without limit as the electron descends into lower and lower energy states. Real electrons clearly do not behave in this way.
Dirac's solution to this was to turn to the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom. Dirac hypothesized that what we think of as the "vacuum" is actually the state in which all the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy.
Dirac also pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively charged particle. Initially, Dirac identified this hole as a proton. However, Robert Oppenheimer pointed out that an electron and its hole would be able to annihilate each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atoms would not exist.[2] Hermann Weyl also noted that a hole should act as though it has the same mass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932 when the positron was discovered by Carl Anderson, with all the physical properties predicted for the Dirac hole.
Inelegance of Dirac sea[edit]
Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite positive electric charge filling all of space. In order to make any sense out of this, one must assume that the "bare vacuum" must have an infinite negative charge density which is exactly cancelled by the Dirac sea. Since the absolute energy density is unobservable—the cosmological constant aside—the infinite energy density of the vacuum does not represent a problem. Only changes in the energy density are observable. Landis also notes that Pauli exclusion does not definitively mean that a filled Dirac sea cannot accept more electrons, since, as Hilbert elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have a chiral anomaly and a gauge instanton.
The development of quantum field theory (QFT) in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture is much more convincing, especially since it recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular the problem of the vacuum possessing infinite energy.
Modern interpretation[edit]
The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformation, an identification between the creation and annihilation operators of two different free field theories. In the modern interpretation, the field operator for a Dirac spinor is a sum of creation operators and annihilation operators, in a schematic notation:
\psi(x) = \sum a^\dagger(k) e^{ikx} + a(k)e^{-ikx}
An operator with negative frequency lowers the energy of any state by an amount proportional to the frequency, while operators with positive frequency raise the energy of any state.
In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For a Fermionic field, the creation operator \scriptstyle a^\dagger (k) gives zero when the state with momentum k is already filled, while the annihilation operator \scriptstyle a(k) gives zero when the state with momentum k is empty.
But then it is possible to reinterpret the annihilation operator as a creation operator for a negative energy particle. It still lowers the energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states. Instead of being states with no antiparticle, these are states that are already filled with a negative energy particle.
The price is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number. The number operator for a Fermi field[3] is:
N = a^\dagger a = 1 - a a^\dagger
which means that if one replaces N by 1-N for negative energy states, there is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum is Lorentz invariant, but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation.
Revival in the theory of causal fermion systems[edit]
Dirac's original concept of a sea of particles was revived in the theory of causal fermion systems, a recent proposal for a unified physical theory. In this approach, the problems of the infinite vacuum energy and infinite charge density of the Dirac sea disappear because these divergences drop out of the physical equations formulated via the causal action principle.[4] These equations do not require a preexisting space-time, making it possible to realize the concept that space-time and all structures therein arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea.
Popular culture[edit]
• There is a song called Sea of Dirac by the band The Superconducting Supercolliders.
See also[edit]
1. ^ Alvarez-Gaume, Luis; Vazquez-Mozo, Miguel A. (2005). "Introductory Lectures on Quantum Field Theory". CERN Yellow Report CERN-, pp. - 1 (96): 2010–001. arXiv:hep-th/0510040. Bibcode:2005hep.th...10040A.
2. ^ "Dirac, P A M - 1931 - Quantized Singularities In The Electromagnetic Fields". Scribd.com. 1931-05-29. Retrieved 2013-02-18.
3. ^ Sattler, Klaus D. (2010). Handbook of Nanophysics: Principles and Methods. CRC Press. pp. 10–4. ISBN 978-1-4200-7540-3. Retrieved 2011-10-24.
4. ^ F. Finster, "A formulation of quantum field theory realizing a sea of interacting Dirac particles," arXiv:0911.2102 [hep-th], Lett. Math. Phys. 97 (2011), no. 2, 165–183, Online.
5. ^ Cocoro Books, Martin Foster. Neon Genesis Evangelion: The Unofficial Guide. DH Publishing Inc. p. 34. ISBN 9780974596143.
External links[edit] |
99abd277d3fa663c | Michael Mann: Will the USA Join the Green Energy Revolution or Fall Behind?
This is Michael Mann, Distinguished University Professor of Atmospheric Sciences and Geosciences, Penn State.
CREDIT Patrick Mansell, Penn State
Guest essay by Eric Worrall
The part I don’t get – why is it important to greens not to “fall behind”?
Climate Change Impacts ‘No Longer Subtle,’ Scientist Says
Here & Now’s Jeremy Hobson talks with Michael Mann (@MichaelEMann), distinguished professor of atmospheric science and director of the Earth System Science Center at Penn State University.
“What we’re seeing right now across the Northern Hemisphere is extreme weather in the form of unprecedented heat waves, droughts, floods, wildfires. In isolation, it might seem like any one of these things could be dismissed as an anomaly, but it’s the interconnectedness of all these events and their extreme nature that tells us that we are now seeing the face of climate change. The impacts of climate change are no longer subtle.
“That’s a huge lost opportunity when the media does not connect those dots for the people, because this is the face of climate change, and we have to understand it’s not just about polar bears up in the Arctic. It’s about extreme damaging weather events that we’re experiencing now in real time.”
“So the world is moving on, and the question now is, simply, is the United States going to join the rest of the world in what is the great economic revolution of this century, the green energy revolution, or are we going to fall behind the rest of the world? That’s the decision that we have to make, and if we don’t like the path that we’re on right now with the Trump administration and Republican leadership in Congress, we’ve got a midterm election in less than 100 days, where we can speak out and say, ‘We want a different path. We want to join the rest of the world, rather than be the last holdout towards progress.’ ”
Read more: http://www.wbur.org/hereandnow/2018/08/02/summer-weather-climate-change
Why is it so important to greens for the USA not to “fall behind” the alleged renewable energy revolution?
Back in 2014, an engineering team working for Google discovered to their horror that there is no economically viable path to 100% renewable energy. No matter what they tried, the cost of building all the green energy infrastructure which would be required to get anywhere close to 100% renewable energy was an insurmountable barrier.
Spending money on green energy R&D – I don’t have a problem with that. But spending money on green energy infrastructure is a high risk investment in an unready technology which currently has no chance of delivering. Even Google’s engineers couldn’t crack the problem.
My question – instead of risking national economic ruin by trying to jump to the front of the pack, why not continue with the status quo? Encourage US entrepreneurs servicing the technology needs of green states and other countries to solve the problems, without risking the national economy.
That way countries or states which are enthusiastic about solar and wind take the risks, while states which are less enthusiastic about green energy provide an economic safety net in case the green energy revolution doesn’t work out.
165 thoughts on “Michael Mann: Will the USA Join the Green Energy Revolution or Fall Behind?
• Climate alarmist seem to fail to understand the concept of forward and backwards. I think they have lost their sense of direction, and I suspect they never had one which accounts for their willingness to grab at anything that drifts by.
Perhaps I missed the announcement where we had determined returning to dreadful past lifestyles (nasty, brutish and short) was progress.
• They’ve also missed the direction of causality: “climate change” can’t CAUSE anything; rather, a persistent, 30-year pattern of changed weather is the DEFINITION of climate change.
One mayas well say “wet sidewalks cause rain.”
• Try to explain that to Michael (The Lemming) Mann
Sage advice from my Grandpa
“If all your friends were jumping off a cliff, would you do it?”
• That reminds me of an old joke a friend of mine once told me:
A couple of guys where taking a drive through town at night. They came to a red light and the driver flew through it. The passenger was quite upset: “What the heck are you doing you just ran through a red light”. The driver said, relax, it’s no big deal, my brother does it all the time”.
They come up to the next red light and same thing happens. Passenger again says: “you did it again, that was a red light”. The driver responded “relax, my brother does it all the time”.
Next they come to a green light and the driver slams on the breaks bringing the car to a screeching halt. The passenger says “why’d you do that, the light is green?” to which the driver replied “My brother might be coming the other way.”
• beng135, sometimes one must call a spade a spade. If something is stupid, pretending otherwise won’t change the facts.
And BTW, calling someone childish & immature, that indicates you lost the argument – by your own standards.
• Perhaps this is a reasonable case for 0bama’s policy of “leading from behind”, so that we can see what tomfoolery masquerading as STEM does to cause havoc to other nations and avoid it ourselves.
1. US will join the green energy revolution when
1. it’s cost effective,
2. won’t raise energy prices to the consumer and
3. isn’t subsidized by the Federal Government (which we pay for)
I for one don’t want to pay $0.30 to $0.40/KwHr like the EU.
2. I’d present Dr Mann with my dissenting views, only he blocked me long ago. I don’t even know why he did so.
• Some people on social media make use of apps that pre-block large numbers of people for simply liking or viewing the “wrong” person elsewhere.
• Could be the reason. I do recall asking him a question on his TL (respectfully, something like “Sir, I don’t understand, can you enlighten me as to why..)) that may have prompted him to block me because I am not ‘of the body’. It’s why I deleted my ‘facebook’ account after they assigned me a ‘political persuasion’ I had no control over, didn’t agree with, and couldn’t ‘opt out’ of. I also switched from Google search to ‘DuckDuckGo’ (not perfect, but better. If Twitter goes forward with their current plans to ‘reduce ‘hate speech’, etc, I will regretfully abandon that as well. The age of dominance by ‘Big Corp’ is upon us. I don’t know that government is capable of (or even wants to) stopping them anymore.
3. Oddly, the Green renewable snake-oil sellers seem to avoid talking about reliability in energy delivery when discussing renewable sources for electricity. /s
Reliability is the fatal flaw in the Green’s push for renewable power. Because lacking reliability, means either 1) people, businesses, and industry will either endure black-outs, and/or 2) endure much higher electricity costs as the grid operators duplicate power delivery with fossil fuel generators to mitigate #1. And duplicating means most of the claimed CO2 emissions savings is lost when entire systems level view is taken. The Google engineers took into account the total life-cycle CO2 fossil fuels used costs of renewable sources: the mining more copper and lithium for generators and batteries, the wind turbine production, the transport, installation. They can’t change that. They can only avoid discussing it.
We can’t let them get away with that.
• Somewhere along the way they seem to have lost contact with the fact that the laws of physics prohibit us having perpetual motion devices, perpetual motion processes or perpetual motion systems.
• The reason is they don’t give a fig about sufficient, reliable power. They ‘give a fig’ about what control over our energy brings them – total control over our lives.
• Reliability is a secondary issue to me. The requirement for storage effectively doubles the cost of ALL Green power. Buying a truck to drag your dead horse around in.
• Elon Musk will build a battery capable of powering the entire North American grid for a month, to protect us against fickle winds and sunshine. The battery will be the size of Rhode Island and it will contain all the rare earths, lithium and cobalt that the world can produce for the next 150 years.
4. “a huge lost opportunity when the media does not connect those dots for the people”
*because ‘the people’ are all too damn stupid to connect them themselves*
I think the ‘media’ has been trying (and failing) to ‘connect the dots’ for quite some time, but, ‘the people’ seem to have an innate ‘BS’ meter about stuff like this.
• Theorise.
Produce your propostion.
Give other scientists your data in order to produce the same results and prove your proposition.
In answer to your question Mann, by refusing to release his data, is not a scientist.
• Thomas: You didn’t use a “?”, but if it were a question, my answer is this- he is at most a very flawed scientist. Even if he is the Einstein of Atmospherics, does that translate to expertise in energy production? Don’t get me wrong, alarmists like to say “if your not a peer-reviewed climate scientist, your facts don’t count.” Mann can have an opinion, but how does he account for Google’s failure? I didn’t read the full article, but I’m sure Jeremy Hobson really grilled him on it. A journalist, right? Bet he challenged him, “are you an authority on electric grids and the like, Dr. Mann?” Right?
5. If – repeat IF – we finally see cost-effective renewable energy generation and storage on an industrial scale, it will be most rapidly adopted by those economies that are prosperous…… not those which have impoverished themselves chasing green dreams.
6. It’s pretty standard politics to assert that other nations are doing something better and we have to act now to catch up, or risk suffering dire consequences. It’s normal to hear it from politicians, so when you hear it from an alleged scientist you know that this person probably has an agenda and is mixing politics with their day job, probably to the detriment of their science.
I’ll also bet he didn’t mention the shockingly high, and rising, electricity prices enjoyed by the unlucky citizens of these other ‘green’ nations.
• “Mr. President, we cannot afford a mine shaft gap!!!”
— Gen. Turgisson, “Dr. Strangelove (or How I Stopped Worrying and Learned to Love the Bomb”)
• Canada just scaled back it’s carbon taxes ( which haven’t even come into effect yet) due to international competitive issues ( read re-election issues for Trudeau). Very possibly a preliminary step to dropping them altogether.
Multiple provinces are taking the Feds to court to stop the taxes and the political opposition to Trudeau’s fatuous, posturing nonsense is growing.
People might vote stupidly but they’re also kinda fickle.
7. Not sure what Dr Mann is referring to as ‘behind’. Since every nation that has adopted the idea of the ‘Green Revolution’ has actually increased it’s CO2 from power generation (for various reasons) while the US has decreased it’s CO2 from power generation, perhaps Dr Mann wants the US to not ‘fall behind; and increase our CO2?
8. If it were such an exciting field of opportunity there is nothing stopping Michael Mann or anyone else risking their own capital on non-government-sponsored so-called green energy developments.
Genuinely exciting technological developments don’t need government mandates, in fact quite the opposite, fortunes are made ‘getting in on the ground floor’.
9. The world’s #1 snake oil salesman! The only left out was unprecedented sameness! This guy should go work on a ranch. I’ve never seen a human being shovel so much s#!t!
One of the very few people in the world who I find disgusting.
10. As soon as that fraud Michael Mann mentioned the polar bears he lost all credibility.
He wouldn’t make a scientist’s arse hole.
11. If we need any more proof that there is not going to be enough electricity:
Smart meters will allow energy firms to introduce “surge pricing”, one of Britain’s biggest gas and electric providers has admitted for the first time.
Were these the dots you’re connecting Mr Mann?
From here:
Meanwhile Brexit rumbles on towards a total trainwreck:
A drugs giant is amassing a 14-WEEK medicine stockpile as fears grow of a “nightmare” no deal Brexit. Pharmaceutical giant Sanofi is a major supplier of insulin.
From The Mirror somewhere.
Insulin eh?
Any pre-diabetics out there had better start eating and stocking up on the coconut oil
Wonder how the continental inter-connectors will fare in this mess – even less elektrickery for your average punter?
I’m sure the creators of Smart Meters have that one worked out, probably involving truckloads of £20 notes and Drax power station
• Those smart meters rely on getting a signal to the outside world. Quite a lot of them can’t get a signal from the cupboard under the stairs. Just saying.
12. The United States of America is leading the world out of the mann-made dark ages of climate change fraud and into a new age of low cost, abundant, 24/7/365 reliable energy.
To the rest of the world: “Follow the Leader!”
13. We should offer to send Mann to Zimbabwe or some other such beacon of enlightenment where he can don his golden mantle of Eco Saviour and be fed peeled grapes while he looks down his nose at us common, ignoramuses. (Ignorami?)
14. “The impacts of climate change are no longer subtle.”
My, my, so until now they were, in Mann’s view, subtle? In stark contrast to the maistream alarmist propaganda, that would be.
It is Mann who is falling behind, he should read more on this site and get updated about the fate of renewable energy projects and policies worldwide. But that would be an intrusion of reality upon illusion, and possibly dangerous.
• Yes, I took it (actually my wife did for me) and hereby release it to the public domain. Have at it. Location is at the intersection of hwy 46 and 50 about 6 miles East of Wagner, SD. This is looking East from the stop sign on hwy 50. If I had panned 90 deg left to the North there is a large wind farm a few miles North-North-West busily chopping up raptors and annoying the locals.
15. Why isn’t there a big discussion about nuclear energy. If climate change is such a disaster in the making why aren’t the alarmists talking about the only real option of less carbon and that is nuclear power. In their minds is it better to destroy the planet than use nuclear power? I don’t get it.
• To his credit, James Hansen does call for lots more nuclear power, but he seems to be in a small minority among global-warmers and he got called the D-word by Oreskes for the sin.
His problem is that they were only too happy to recruit the most rabid of environmentalists and other assorted loud mouths as climate-change shock troops. There really are some allies who are just not worth the trouble. These same people have been brought up on a diet of undiluted Jane Fonda propaganda, and would rather see the end of human civilization than countenance building up a large fleet of nuclear reactors.
• Mikey is “skeptical” about nuclear energy. When he mentions it he claims there is room for a diversity of opinions on the subject. Reading some of his other stuff it looks like he wants complete devolution of energy production away from big business to local level – but for some reason small modular nuclear reactors doesn’t fit his vision of self sufficient communities.
• Nuclear energy has this thing called “radiation” which may be a bigger bug-a-boo to environmentalists than CO2.
16. Why doesn’t the rest of the world go green and the US stay on fossil fuels? We could then charge the world for helping to fertilize their crops.
17. Unfortunately, I had my car radio on while driving to the site of my afternoon run today. I turned it on in the middle of this interview and didn’t know who the interviewee was though it was obvious it was the voice of a propagandizer and proselytizer of the CAGW religion.
I couldn’t help but wonder if it was Hansen or Gavin or another of the usual nutters.
Eventually the transgressor was identified as none other than the infamous Michael “Piltdown” Mann.
WBUR is, of course, located in Boston. One of WBUR’s largest funders is Jeremy Grantham. You may rest assured that the station’s attitude and programming in respect of CAGW is bought and paid for.
18. If the inventor of the hokey Shtick wants the media to “connect those dots”, why doesn’t he help them by providing the data that he must have somewhere showing all those events he lists are in actual fact anomalies or unprecedented or a deviation from normal weather. I could list all of the demented psychopathic eco wing-nuts in published climate journals and claim they were signs of the decay of the human race but it would mean nothing without statistics.
By the way what sort of PhD thesis made this Mann an expert on connecting the dots. Last time I did that I was in grade three.
19. Looking at that photo….What an arrogant, smug, pompous ass.
O.K., now to answer the question, “Why are they afraid to fall behind?”
A. Because somehow, they think that their campfire histrionics and 1KiloWatt solar installs are going to save our nation from some global catastrophe. The truth, as most of us on this thread know, is that it is adherence to their maxims and mandates that will doom our futures and those of our children and grandchildren. They just will not admit how wrong they are and how pitifully lacking in capacity their foolish schemes are and how they will condemn the world to a darker and less prosperous future that they, being dead, will not have to suffer….either the effects or the blame.
20. Leave the SunWorshipers and WindWonks behind. Why on continue on research into the cheapest and move advanced energy source — advanced Thorium reactors. Whilst they tilt at windmills and whine,.. new natural nuclear may give us the edge for a millennium more. Hanson might even become the PR man for the Energy Department in a hilarious twist of technical advanced. Back to the future – nuclear powered cars (probably fuel cells some day as storage in H2,..) {must be careful,.. China is doing some good basic research and may surprise all — just without the panels}
• Sparky says: “cheapest and move advanced energy source — advanced Thorium reactors.” However there is not a single Thorium reactor in commercial operation, so we have no idea if it is the “cheapest”
• I was suggesting R&D for advanced 3rd Ten Nuclear. Reduced and competitive Cost is a result of research and development and the implementation of commercial architectures viable to be competitive. Nuclear’s proven cost competitiveness has been shown for years (unless you conflate excessive regulatory burdens and false assessments of ‘storage’ of waste.). Thorium is in early stages of development but with much promise of reduced costs, risk mitigation and eventual broad commercial success.
• I was suggesting R&D for …
GOVERNMENT money? Why? Let Apple take a stab at it for a change …
21. I don’t have any problem with green energy, but unfortunately, it is not up to the task. The reason is fundamental. It’s not a lack of research, it is the low energy density and non dispatchability of green energy that is the root problem.
22. We are in danger of falling behind the leading edge of power production and thus being unable to compete in the global market place, but it’s not in the way the left and the main stream media think. The future is high efficiency coal, not green energy:
• The future is high efficiency coal,
Can we revisit this prediction, in, oh, say just three years? I think one Dr. Randell L. Mills will have operating field units of his inaptly named “SunCell” working at that time …
23. If he’s soooooooo interested in reducing CO2, why does he focus so much on “green” energy instead of nuclear? His politics, of course.
Well, even Whackopedia contradicts the “unprecedented heat waves” claim:
also, floods:
As for droughts:
and, concerning wildfires, again, even Whackopedia contradicts the “unprecedented” claim by Dr. Doom (I mean, “Mann”):
Even COLLECTIVELY, all these things CAN be dismissed as an anomaly. None of them, either standing alone or in view of all the others, is an anomaly. Their extreme nature today is no more extreme than their extreme nature yesterday or yesteryear or yestercentury.
What IS extreme is the NUMBER of people now observing these weather events, the NUMBER of news media reporting all this, the NUMBER of people now witnessing, talking about, hyping, misassociating, conflating, and confusing all this to create a popular lie.
Furthermore, attributing all this to “climate change”, in and of itself, would not be unusual, even if the “unprecedented” claim WERE true (which it is NOT). But what Dr. Doom (I mean, “Mann”) subtly tries to sneak into favor is the hidden premise that “climate change” means ONLY changes to climate caused by humans, and he trusts that this insidious hijacking of the general word meaning has successfully overtaken the minds of the masses, but this writer here (Robert K) is NOT one of those masses who has fallen for this verbal hijacking.
So, Dr. Doom (I mean, “Mann”), I call total bullshit on your ego-inflated, uneducated claims that support your completely ignorant faith in “Green energy” replacing the current fossil-fuel-energized civilization.
• extreme weather in the form of unprecedented heat waves, droughts, floods, wildfires.
Can Dr Doom (oops! Mann) guarantee that there will be no more extreme weather events when fossil fuels are no longer used and we are all using electricity in intermittent stages?
When he puts his nuts on the block to demonstrate his commitment to his beliefs, then I might believe he is serious. The first heat wave, drought or flood after his green energy revolution is in place causes forfeit of his family jewels.
25. What Green Energy Revolution is Mann talking about when the world’s leading polluters like China, India, and Russia are continuing to build coal plants and/or aren’t showing any great enthusiasm for cutting emissions until they find it convenient. And when all forecasts show fossil fuel use continuing to rise until 2040 at least, who’s really making such a great shift toward renewable energy besides countries with small populations and areas? Mann had better take a closer look at the facts.
26. So right off the bat he is speaking with forked tongue.
He is wrong on heatwaves, floods, fires, storms and droughts.
None of those weather related events are unprecedented in size, degree, or scope.
When will ethical scientists prove they are reputable and stand up to his less than accurate claims?
27. Many continue to believe that this all has something to do with climate. It does not, it is about power, control and destroying capitalism and freedom but especially the USA.
People like Mann love the accolades and fame they have gendered. It gives their sad little lives meaning, more than they ever dreamed. Sadly there are those seeking power that use just such people to get what they want. Mann and his ilk would be the first in re-education camps. Just ask yourself why have the socialists, social justice crowd, Antifa, radical feminists, animal rights folks, Islamists, etc, etc allied themselves with the CAGW orthodoxy, priests and priestesses?
28. Who in their right mind would want to pay forty and fifty something cents a KWH for unreliable electricity that most people couldn’t even afford. Like they do in Europe and Australia and in Ontario, where Doug Ford has started to throw it all out.
29. Is “the great economic revolution” going to be upward or downward? As for extreme weather and unprecedented heat waves, how many thousands of years back do his usable information and well written observations go? The way I look at it, going back for only about 233 years is talking about weather and not climate. As for wild fires, how much “credit” is he giving to human caused excessive density of trees and lower story vegetation and accumulation of debris from stopping natural fire activity and other forms of mismanagement?
30. Mann 2018:
Lenin 1918 (paraphrased):
“…join the comrads of the World in what is the Great Economic Revolution of this century, the Red Communist Economic Revolution, or are we going to fall behind the rest of the World…”.
31. My question. If Michael Mann is a scientist, why doesn’t he just stick to doing science?
32. When is his court case with Mark Steyn going to start? I am waiting to see this goose squirm as he is publicly humiliated in like fashion to the way he has denigrated others.
33. China has a 2025 plan to dominate world trade while at the same time doubling coal electrical production. So Mann is saying he is smarter than all the planners and engineers of China put together as they are going to get behind.
34. Mann wrote
Because science cant do it. If science could connect the dots, then media could report that with confidence.
35. Because if they “fall behind” they don’t make a dime. Remember, it’s money that’s at the heart of it all. The redistribution of wealth. Everything that is hyped, made to look like a climate or ecological crisis, and breed climate hysteria is all about taking money out of all of your pocket and putting it into theirs. They’re greedy and they want their cut of all of our money.
36. The part I don’t get – why is it important to greens not to “fall behind”? – The reason they don’t want to fall behind is because it hits them in the wallet. They want to redistribute wealth from your wallet to theirs. That’s why they have this fake crisis mentality. They want their cut of our money to line their pockets and pay for their agenda. Fortunately, we have the ability to stop them in their tracks. Trump has kept his promises and pulled us out out Climate Change agreements and executive ordered the reversal of much of Obama’s environmentalist agenda.
37. They are getting desperate. They have lost every argument, and now only have ‘be part of the first world’ left.
38. Well now, As a Professional Engineering practitioner, there is a BIG disconnect between the Practitioners on the ground in everyday life and the “professors” in their ivory towers shouting along the corridors, across the vennels to their fellow elitists ( politico, etc). ……. could go on , but I have a job to do and a wage to earn…
I mean, …. that we have to Listen to the continuous drabble from them and the Media with their cohorts – any story this past while, seems to include a reason for failure is attributed to climate change. …
• Yeah, Green Energy Revolution as a movement is so passe. Come on Mann get caught up with the times! The term green energy is so Obama era. Its time and their fantasies and failed policies has come and gone.
39. What is the carbon footprint of building all this “green” infrastructure? What is the carbon footprint of a wind farm compared to the annual carbon emission savings, and what is the “payback period?” Given that we have only five years to blah blah blah, I hope it’s not more than five years.
(Besides, how much wind do those things actually create? They look like they use up a lot of electricity. )
40. The “all in” renoobles countries are trying to back out while they still have an economy. Trump saved the world but I hope he has a Kevlar suit on. There are sane sober believers in the AGW alarm stuff but it is an attractive type of issue for psychologically impaired types, angry haters and the like. I dont thing the come-down is going to be pretty.
41. why is it important to greens not to “fall behind”?
Because they’re so far behind they think they’re ahead.
42. I find it interesting this perspective that a nation or people would be “falling behind” when failing to adopt the green approach to energy, when observations are that adopting green policies actually are set-backs in advancing a culture and society.
43. They have to press right now. They have to try to imply that we need to catch up and that we need to act fast…while they have the chance.
We’re currently undergoing a string of inclement weather (I seriously doubt the “unprecedented” part) that they have to exploit while it’s still in the forefront of people’s minds.
They sure can’t do it when this dies down and we have a run of “run of the mill” weather.
Actually, that’s pretty much the summer we’ve had in my area…pretty much exactly what we expect from summer in this area.
I wonder how many other areas are like that too, but by the time the media gets done reporting on it, anyone that’s NOT from there gets the impression that it’s been an extreme, “unprecedented” summer.
California’s got a bad wildfire going on. Um…doesn’t that happen pretty much every year?
“New heat records in death valley and Morrocco!!!!1!!11!!!!1111” The desert is hot. Duh.
We’ve had a heat wave. We’ve had heat waves for as long as I’ve been alive. We’ve also had blizzards, deep freezes and really mild weather too at various times. None of that, in my limited experience, is the harbinger of the end of the world…it’s called “weather”.
Anyway…Mann and company have to try to exploit the weather to push their agenda while they can because they know as well as we do that it won’t last forever and it’s a lot harder to push their agenda when it’s nice outside.
44. Oh, Mann means the green energy revolution like in the Netherlands where they put out hundreds of electric generator windmills only to take them down because they were costing more to operate than the power they were supplying. Until there is a battery technology breakthrough, renewable energy will never be feasible for more than a fraction of the world’s energy needs.
• Even with batteries, they fail. It has to do with energy density and consistent performance (the battery bank required to cover a week of no wind would probably dwarf a city). I guess we could change the laws of physics. They vote on that, right?
• Until there is a battery technology breakthrough
This will be quite literally swamped by the (inaptly named, I think, because it has nothing to do with the sun as in solar) “SunCell”.
• Until there is a battery technology breakthrough
Battery tech is mature — has been. Incremental improvements like lithium batteries can occur, but that doesn’t change the fact of very-low energy density.
• To some of us in the UK this is reminiscent of the history of liquid crystal displays . The materials were first explored in Germany and in the UK ( by Gray at Univ of Hull) and the research on application to displays was promoted by the UK Govt through , eg the Royal Radar Establishment in Malvern and the work of Prof Cyril Hilsum .
However it is Japan , S Korea and Taiwan and the companies there that reaped the major commercial benefit through their massive industrial investment from the 1980s.
45. Woou8ldn’t it be better if this “disgrace to his profession” actually limited his disgrace to his profession?
• Leaders in science fields are a “disgrace to (their) profession.”
We get it. You don’t like scientists who are published and peer reviewed, especially when their work is confirmed by many studies in the next few decades.
• Ally, I suggest you read “”A Disgrace To The Profession” The World’s Scientists, In Their Own Words, On Michael E Mann, His Hockey Stick And Their Damage To Science Volume I”. You might learn just what his peer think about this “leader” you so blindly defend.
• No, he’s doing fine. Director of the Earth System Science Center at Pennsylvania State University and several books (well written too, not sure if he had a ghost writer.)
I think the sites that peddle non-science are hurting.
46. There’s an old saw that seems to apply, if I could just remember it correctly… Something like, “At first I was proud of the General, as a great leader, for running out in front of his troops, until I realized they were in full retreat.”
Does anyone else remember such a quote? Who said it? Unless I’m the one that made it up, in which case I take full credit.
What I’m trying to say is, there’s all kinds of leadership, not all of it desirable.
• I’ve seen leadership defined as “Figure out which direction people are moving, then run that way.”
47. “Spending money on green energy R&D – I don’t have a problem with that.”
I do. There is no upside in it. Windmills are thousand year old technology. Batteries 220 years. Solar thermal ancient. Solar cells are only at 110.But the point is made. Pouring money into “renewable” R&D is just welfare for white people. A waste of money and time.
48. 1. Virtue-signaling “Ahh others are ahead op us in the virtue race! Unacceptable!”
2. Attempt to appeal to naive competitiveness “Americans love to be first. Tell them they are falling behind.”
49. I can not for the life of me figure out the motivation of Michael Mann. As a climate scientist he must be aware of and studied all the data, yet he makes these statements which he must know are false. He must know that amid all this hysteria over heatwaves, the northern hemisphere is only 0.2° C above the average, and that the northern Atlantic SST has dropped by over 0.5° C. He must also know that parts of the southern hemisphere are suffering abnormally low winter temperatures. Maybe it was getting away with the deception of the hockey stick chart that went to his head.
• I can not for the life of me figure out the motivation of Michael Mann.
Hubris overtakes rationality; the desire for fame and a ‘name’ outstrips all other motivating factors, including the exercise of discretion.
• “the northern hemisphere is only 0.2° C above the average”
That makes no sense. Scientists would indicate what average they are talking about. You seem to have picked an arbitrary average.
And picking specific parts of the globe is interesting, but are duly noted in the global average.
50. Idiots all. The “SunCell” courtesy of Dr. Mills is going to eat a lot of “lunches” in the energy field …
• I add this for ristvan’s edification: “From the time of its inception, quantum mechanics (QM) has been controversial because its foundations are in conflict with physical laws and are internally inconsistent. Interpretations of quantum mechanics such as hidden variables, multiple worlds, consistency rules, and spontaneous collapse have been put forward in an attempt to base the theory in reality. Unfortunately many theoreticians ignore the requirement that the wave function must be real and physical in order for it to be considered a valid description of reality.
These issues and other such flawed philosophies and interpretations of experiments that arise from quantum mechanics are discussed in the Retrospect section and Ref. [8, 10, 12]. Reanalysis of old experiments and many new experiments including electrons in superfluid helium and data confirming the existence of hydrinos challenge the Schrödinger equation predictions.
Many noted physicists rejected quantum mechanics, even those whose work undermined classical laws. Feynman attempted to use first principles including Maxwell’s Equations to discover new physics to replace quantum mechanics [34] and Einstein searched to the end. “Einstein […] insisted […] that a more detailed, wholly deterministic theory must underlie the vagaries of quantum mechanics [35].” He believed scientists were misinterpreting the data.
From: https://brilliantlightpower.com/wp-content/uploads/theory/GUT-CP-2016-Ed-Volume1-Web-121517.pdf
51. Michael Mann apparently lacks the engineering/science skill to adequately understand what is wrong with depending on solar and wind for energy.
The fallacy of renewables is revealed with simple arithmetic.
5 mW wind turbine, avg output 1/3 nameplate, 20 yr life, electricity @ wholesale 3 cents per kwh produces $8.8E6.
Installed cost @ $1.7E6/mW = $8.5E6.
Add the cost of energy storage facility and energy loss during storage/retrieval, or standby CCGT for low wind periods. Add the cost of land lease, maintenance, administration.
Solar voltaic and solar thermal are even worse with special concern for disposal and/or recycling at end-of-life (about 15 yr for PV).
The dollar relation is a proxy for energy relation. Bottom line, the energy consumed to design, manufacture, install, maintain and administer renewables exceeds the energy they produce in their lifetime.
Without the energy provided by other sources renewables could not exist.
52. Newsflash, “revolutions” are created, funded and promoted by Governments in power. Revolutions are usually the result of Tyrannical Governments wasting peoples money to the point of destroying an economy so the people can do nothing but rebel. Think of Venezuela and Iran.
Also, Revolutions usually aren’t based upon lies.
Isolating the Impact of CO2 on Atmospheric Temperatures; Conclusion is CO2 has No Measurable Impact
53. China has cut back on solar projects. And, California utilities have paid Arizona to take excess green energy. Leadership isn’t all its cracked up to be. LOL
Comments are closed. |
e536fb2b2273d9cb | Saturday, May 20, 2017
Cosmo : supremely relaxing fishing video
The Seychelles are an angler’s paradise – if you can actually get to them.
Follow the crew of the Alphonse Fishing Co. as they wade the flats of the Cosmoledo Atoll, hoping for a shot at Giant Trevally.
see the story
Cosmoledo island with the GeoGarage platform
Friday, May 19, 2017
Terrifying 20m-tall 'rogue waves' are actually real
The Wave painting by Ivan Aivazovsky
From BBC by Nic Fleming
For centuries sailors told stories of enormous waves tens of metres tall.
They were dismissed as tall tales, but in fact they are alarmingly common
TEN-storey high, near-vertical walls of frothing water.
Smashed portholes and flooded cabins on the upper decks.
Thirty-metre behemoths that rise up from nowhere to throw ships about like corks, only to slip back beneath the depths moments later.
Evocative descriptions of abnormally large "rogue waves" that appear out of the blue have been shared among sailors for centuries.
With little or no hard evidence, and the size of the waves often growing with each telling, there is little surprise that scientists long dismissed them as tall tales.
Until around half a century ago, this scepticism chimed with the scientific evidence.
According to scientists' best understanding of how waves are generated, a 30m wave might be expected once every 30,000 years.
Rogue waves could safely be classified alongside mermaids and sea monsters.
However, we now know that they are no maritime myths.
A wave is a disturbance that moves energy between two points.
The most familiar waves occur in water, but there are plenty of other kinds, such as radio waves that travel invisibly through the air.
Although a wave rolling across the Atlantic is not the same as a radio wave, they both work according to the same principles, and the same equations can be used to describe them.
A rogue wave is one that is at least twice the "significant wave height", which refers to the average of the third highest waves in a given period of time.
According to satellite-based measurements, rogue waves do not only exist, they are relatively frequent.
The sceptics had got their sums wrong, and what was once folklore is now fact.
This led scientists to altogether more difficult questions.
Given that they exist, what causes rogue waves?
More importantly for people who work at sea, can they be predicted?
Until the 1990s, scientists' ideas about how waves form at sea were heavily influenced by the work of British mathematician and oceanographer Michael Selwyn Longuet-Higgins. In work published from the 1950s onwards, he stated that, when two or more waves collide, they can combine to create a larger wave through a process called "constructive interference".
According to the principle of "linear superposition", the height of the new wave should simply be the total of the heights of the original waves.
A rogue wave can only form if enough waves come together at the same point according to this view.
However, during the 1960s evidence emerged that things might not be so simple.
The key player was mathematician and physicist Thomas Brooke Benjamin, who studied the dynamics of waves in a long tank of shallow water at the University of Cambridge.
With his student Jim Feir, Benjamin noticed that while waves might start out with constant frequencies and wavelengths, they would change unexpectedly shortly after being generated.
Those with longer wavelengths were catching those with shorter ones.
This meant that a lot of the energy ended up being concentrated in large, short-lived waves.
At first Benjamin and Feir assumed there was a problem with their equipment.
However, the same thing happened when they repeated the experiments in a larger tank at the UK National Physical Laboratory near London.
What's more, other scientists got the same results.
For many years, most scientists believed that this "Benjamin-Feir instability" only occurred in laboratory-generated waves travelling in the same direction: a rather artificial situation.
However, this assumption became increasingly untenable in the face of real-life evidence.
At 3am on 12 December 1978, a German cargo ship called The München sent out a mayday message from the mid-Atlantic.
Despite extensive rescue efforts, she vanished never to be found, with the loss of 27 lives.
A lifeboat was recovered.
Despite having been stowed 66ft (20m) above the water line and showing no signs of having been purposefully lowered, the lifeboat seemed to have been hit by an extreme force.
The significant wave height was 35.4ft (10.8m).
When scientists from the European Union's MAXWAVE project analysed 30,000 satellite images covering a three-week period during 2003, they found 10 waves around the globe had reached 25 metres or more.
"Satellite measurements have shown there are many more rogue waves in the oceans than linear theory predicts," says Amin Chabchoub of Aalto University in Finland.
"There must be another mechanism involved."
In the last 20 years or so, researchers like Chabchoub have sought to explain why rogue waves are so much more common than they ought to be.
Instead of being linear, as Longuet-Higgins had argued, they propose that rogue waves are an example of a non-linear system.
A non-linear equation is one in which a change in output is not proportional to the change in input.
If waves interact in a non-linear way, it might not be possible to calculate the height of a new wave by adding the originals together.
Instead, one wave in a group might grow rapidly at the expense of others.
When physicists want to study how microscopic systems like atoms behave over time, they often use a mathematical tool called the Schrödinger equation.
It turns out that certain non-linear version of the Schrödinger equation can be used to help explain rogue wave formation.
The basic idea is that, when waves become unstable, they can grow quickly by "stealing" energy from each other.
Researchers have shown that the non-linear Schrödinger equation can explain how statistical models of ocean waves can suddenly grow to extreme heights, through this focusing of energy.
In a 2016 study, Chabchoub applied the same models to more realistic, irregular sea-state data, and found rogue waves could still develop.
"We are now able to generate realistic rogue waves in the laboratory environment, in conditions which are similar to those in the oceans," says Chabchoub.
"Having the design criteria of offshore platforms and ships being based on linear theory is no good if a non-linear system can generate rogue waves they can't cope with."
Still, not everyone is convinced that Chabchoub has found the explanation.
"Chabchoub was examining isolated waves, without allowing for interference with other waves," says optical physicist Günter Steinmeyer of the Max Born Institute in Berlin.
"It's hard to see how such interference can be avoided in real-world oceans."
Instead, Steinmeyer and his colleague Simon Birkholz looked at real-world data from different types of rogue waves.
They looked at wave heights just before the 1995 rogue at the Draupner oil platform, as well as unusually bright flashes in laser beams shot into fibre optic cables, and laser beams that suddenly intensified as they exited a container of gas.
Their aim was to find out whether these rogue waves were at all predictable.
The pair divided their data into short segments of time, and looked for correlations between nearby segments.
In other words, they tried to predict what might happen in one period of time by looking at what happened in the periods immediately before.
They then compared the strengths of these correlations with those they obtained when they randomly shuffled the segments.
The results, which they published in 2015, came as a surprise to Steinmeyer and Birkholz.
It turned out, contrary to their expectations, that the three systems were not equally predictable.
They found oceanic rogue waves were predictable to some degree: the correlations were stronger in the real-life time sequence than in the shuffled ones.
There was also predictability in the anomalies observed in the laser beams in gas, but at a different level, and none in the fibre optic cables.
However, the predictability they found will be little comfort to ship captains who find themselves nervously eyeing the horizon as the winds pick up.
"In principle, it is possible to predict an ocean rogue wave, but our estimate of the reliable forecast time needed is some tens of seconds, perhaps a minute at most," says Steinmeyer.
"Given that two waves in a severe North Sea storm could be separated by 10 seconds, to those who say they can build a useful device collecting data from just one point on a ship or oil platform, I'd say it's already been invented.
It's called a window."
However, others believe we could foresee rogue waves a little further ahead.
The complexity of waves at sea is the result of the winds that create them.
While ocean waves are chaotic in origin, they often organise themselves into packs or groups that stay together.
In 2015 Themis Sapsis and Will Cousins of MIT in Cambridge, Massachusetts, used mathematical models to show how energy can be passed between waves within the same group, potentially leading to the formation of rogue waves.
The following year, they used data from ocean buoys and mathematical modelling to generate an algorithm capable of identifying wave groups likely to form rogues.
Most other attempts to predict rogue waves have attempted to model all the waves in a body of water and how they interact.
This is an extremely complex and slow process, requiring immense computational power.
Instead, Sapsis and Cousins found they could accurately predict the focusing of energy that can cause rogues, using only the measurements of the distance from the first to last waves in a group, and the height of the tallest wave in the pack.
"Instead of looking at individual waves and trying to solve their dynamics, we can use groups of waves and work out which ones will undergo instabilities," says Sapsis.
He thinks his approach could allow for much better predictions.
If the algorithm was combined with data from LIDAR scanning technology, Sapsis says, it could give ships and oil platforms 2-3 minutes of warning before a rogue wave formed.
Others believe the emphasis on waves' ability to catch other waves and steal their energy – which is technically called "modulation instability" – has been a red herring.
"These modulation instability mechanisms have only been tested in laboratory wave tanks in which you focus the energy in one direction," says Francesco Fedele of Georgia Tech in Atlanta.
"There is no such thing as a uni-directional stormy sea.
In real-life, oceans' energy can spread laterally in a broad range of directions."
In a 2016 study, Fedele and his colleagues argued that more straightforward linear explanations can account for rogue waves after all.
They used historic weather forecast data to simulate the spread of energy and ocean surface heights in the run up to the Draupner, Andrea and Killard rogue waves, which struck respectively in 1995, 2007 and 2014.
Their models matched the measurements, but only when they factored in the irregular shapes of ocean waves.
Because of the pull of gravity, real waves have rounded troughs and sharp peaks – unlike the perfectly smooth wave shapes used in many models.
Once this was factored in, interfering waves could gain an extra 15-20% in height, Fedele found.
"When you account for the lack of symmetry between crest and trough, and add it to constructive interference, there is an enhancement of the crest amplitudes that allows you to predict the occurrence observed in the ocean," says Fedele.
What's more, previous estimates of the chances of simple linear interference generating rogue waves only looked at single points in time and space, when in fact ships and oil rigs occupy large areas and are in the water for long periods.
This point was highlighted in a 2016 report from the US National Transportation Safety Board, written by a group overseen by Fedele, into the sinking of an American cargo ship, the SS El Faro, on 1 October 2015, in which 33 people died.
"If you account for the space-time effect properly, then the probability of encountering a rogue wave is larger," Fedele says.
Also in 2016, Steinmeyer proposed that linear interference can explain how often rogue waves are likely to form.
As an alternative approach to the problem, he developed a way to calculate the complexity of ocean surface dynamics at a given location, which he calls the "effective" number of waves.
"Predicting an individual rogue wave event might be hopeless or non-practical, because it requires too much data and computing power.
But what if we could do a forecast in the meteorological sense?" says Steinmeyer.
"Perhaps there are particular weather conditions that we can foresee that are more prone to rogue wave emergence."
Steinmeyer's group found that rogue waves are more likely when low pressure leads to converging winds; when waves heading in different directions cross each other; when the wind changes direction over a wide range; and when certain coastal shapes and subsea topographies push waves together.
They concluded that rogue waves could only occur when these and other factors combined to produce an effective number of waves of 10 or more.
Steinmeyer also downplays the idea that anything other than simple interference is required for rogue wave formation, and agrees that wave shape plays a role.
However, he disagrees with Fedele's view that sharp peaks can have a significant impact on wave height.
"Non-linearities have a role, but it's a minor one," he says.
"Their main role is that ocean waves are not perfect sine waves, but have more spikey crests and depressed troughs.
However, what we calculated for the Draupner wave is that the effect of non-linearities on wave height was in the order of a few tens of centimetres."
In fact, Steinmeyer thinks that Longuet-Higgins had it pretty much right 60 years ago, when he emphasised basic linear interference as the driver of large waves, rogue or otherwise.
But not everyone agrees.
In fact, the argument over exactly why rogue waves form seems set to rumble on for some time.
Part of the issue is that several kinds of scientists are studying them – experimentalists and theoreticians, specialists in optical waves and fluid dynamics – and they have not as yet done a good job of integrating their different approaches.
There is no sign that a consensus is developing.
But it is an important question to solve, because we will only be able to predict these deadly waves when we understand them.
For anyone sitting on an isolated oil rig or ship, watching the swell of the waves under a stormy sky, those few minutes of warning could prove crucial.
Links :
Thursday, May 18, 2017
North Sea wind power hub: A giant wind farm to power all of north Europe
North Sea Infrastructure
The future development of a North Sea energy system up to approx. 2050 will require a rollout, coordinated at European level, of interlinked offshore interconnectors, i.e. a so-called interconnection hub, combined with large-scale wind power.
Any surplus wind power could be converted into other forms of energy, or stored.
Situating this interconnection hub on a modularly constructed island in a relatively shallow part of the North Sea would result in significant cost savings.
These are the starting points for a proposed efficient, affordable and reliable energy system on the North Sea, which will contribute to European objectives being met.
This vision does not preclude the option of providing renewably generated power from the wind farms to nearby oil and gas platforms to reduce Europe's CO2 emissions.
From Ars Technica by William Steel
While the forecast for offshore wind farms of the future is for ever-larger projects featuring ever-larger wind turbines, an unprecedented plan from electricity grid operators in the Netherlands, Germany, and Denmark aims to rewrite the rulebook on offshore wind development.
A proposed North Sea power link island, as conceived by TenneT
with a map of the North Sea, with the location of the Dogger Bank
and the possible interconnectors highlighted
The proposal is relatively straight-forward: build an artificial island in the middle of the North Sea to serve as a cost-saving base of operations for thousands of wind turbines, while at the same time doubling up as a hub that connects the electricity grids of countries bordering the North Sea, including the UK.
In time, more islands may be built too; daisy chained via underwater cables to create a super-sized array of wind farms tapping some of best wind resources in the world.
“Don’t be mistaken, this is really a very large, very ambitious project—there’s nothing like it anywhere in the world. We’re taking offshore wind to the next level,” Jeroen Brouwers, spokesperson for the organisation that first proposed the plan, Dutch-German transmission system operator (TSO) TenneT, tells Ars Technica.
“As we see it, each island could facilitate approximately 30 gigawatts (GW) of offshore wind energy; but the concept is modular, so we could establish multiple interconnected islands, potentially supporting up to 70 to 100GW.”
The London Array
To add some context to those figures, consider that the world’s largest offshore wind farm in operation today, the London Array, has a max capacity of 630MW (0.63GW), and that all the wind turbines installed in European waters to date amount to a little over 12.6GW.
The Danish TSO Energinet says 70GW could supply power for some 80 million Europeans.
Undoubtedly ambitious, the North Sea Wind Power Hub—as the project is titled—is nevertheless being taken seriously by key stakeholders.
The project was centre of attention at the seminal North Seas Energy Forum held in Brussels at the end of March.
There, the consortium behind the project (Dutch-German TSO TenneT, alongside the Danish TSO Energinet) took the opportunity to sign a memorandum of understanding (MoU) that will drive the project forward over the coming decades.
Dagmara Koska, a member of the cabinet of the EU vice-president in charge of the Energy Union (Maroš Šefčovič), tells Ars Technica: “We’re incredibly supportive of the project and welcome the MoU. The agreement demonstrates commitment to a very exciting prospect; one that stands to create a lot of synergies to benefit the growth of renewables energy in northern Europe.”
On the intentions of the Wind Power Hub, Koska says: “From our perspective, the project fully reflects the spirit of the North Seas Energy Cooperation—the political agreement signed last yearto facilitate deployment of offshore renewable energy alongside interconnection capacity across the region. As Maroš Šefčovič said at the signing, it’s an ingenious solution.”
The London Array wind farm is the largest in operation with 175 wind turbines generating enough power for close to half a million UK homes annually.
A paradigm shift
The North Sea Wind Power Hub represents a fundamentally new approach to the development of offshore wind; one that tackles multiple challenges faced by the wind industry head on and capitalises on economies of scale in a bid to deliver access to the wind resources of the North Sea at reduced costs.
Something of a case of necessity being the mother of invention, Brouwers explains that the Wind Power Hub concept is a response to a looming problem faced by the wind industry: ”At the moment, offshore wind is focused on sites relatively close to shore where development costs are lower. The problem is that there’s not space for the 150GW of offshore wind power that the EU has called for. There are other industrial and economic interests in those near-shore regions—fishing, shipping lanes, military areas and so on.
"This pushes things farther out to sea, but the costs can rapidly rise as you move to deeper waters. The solution? Create near-shore costs, or even lower, out at sea.”
Construction of offshore wind farms is a highly complex logistical and engineering operation
So how would the Wind Power Hub deliver on this objective?
Well, the wind farms envisioned by the project wouldn’t be dissimilar from those we see today, but their proximity and connection to artificial "power link islands" represents a substantial departure from the conventional model for offshore wind.
“The idea is that islands as large as six square kilometres would feature a harbour, a small airstrip, transmission infrastructure, and all equipment necessary to maintain the surrounding wind farms, alongside accommodation and workshops for staff,” Brouwers says.
London Array construction
These novel features would open up a lot of possibilities for wind power developers and operators.
With a base of operations out at sea—complemented with storage of components, assembly lines, and other logistical assets—the installation of wind turbines would be more convenient, efficient, and ultimately cheaper than is achieved by today’s methods which rely on specialised ships journeying out from ports.
Savings on installation would be coupled with reduced expenditure over the twenty-year lifetime of wind turbines, too.
Operations and maintenance of offshore wind turbines—a crucial, albeit expensive, affair that stands to be transformed with a base of operations located out at sea.
Onshore wind farms require a lot of support.
But in harsh marine environments, that need is paramount.
Operations and maintenance, or O&M, is key to ensuring turbines avoid downtime and remain productive. By convention (and presently also by necessity) offshore O&M run out of ports; it's logistically complex and pricey, easily representing some 20% of a wind turbine's levellised cost of energy (LCOE), and increasing with distance from shore.
O&M is a permanent fixture on the wind industry’s list of areas within which it aims to lower expenditure, and highlighted as such by the International Renewable Energy Agency, which reports: “It is clear that reducing O&M costs for offshore wind farms remains a key challenge and one that will help improve the economics of offshore wind.”
“In contrast to what we see today,” says Brouwers, “operating from an island on the doorstep of the wind farms would be a game-changer in terms of reducing costs and simplification of O&M activities.”
Subsea DC cables would not only export power from the wind farms, but will serve as interconnectors between countries bordering the North Sea.
High Voltage Direct Current
Alongside savings on installation and reductions on O&M, a third major cost saving feature of the Wind Power Hub concerns grid connections—the electrical infrastructure that links wind farms with electricity grids.
Typically, grid connection is a significant cost component in offshore wind, representing between 15 to 30% of the capital costs for an offshore wind farm, with costs creeping higher the farther from shore you go.
Like O&M, grid connection is a cost component that holds potential for improvement.
With the Wind Power Hub, instead of alternate current (AC) cables taking electricity from a wind farm to grids onshore—the typical arrangement we see today—the output of multiple wind farms would be directed to a power link island.
There, electricity would be aggregated, conditioned for transmission, and then dispatched to onshore grids of the North Sea countries.
It’s a setup that would reduce the amount of export cables running to individual wind farms, and enable cost-effective use of high-voltage direct current (DC) transmission that boasts the added benefit of reduced losses compared to AC transmission.
International electricity interconnections are the set of lines and substations that allow the exchange of energy between neighbouring countries and generate a number of advantages in connected countries.
North Sea Super Grid: The key to sustainable energy in Europe
As significant as the North Sea Wind Power Hub would be terms of clean energy production and cost reduction of offshore wind power, the broader proposition for the concept goes beyond island-building and supporting wind farms.
It would provide a solution to one of the central challenges in transitioning to a sustainable future.
As Brouwers says: “When we talk about the transition towards 100% sustainable energy production, it’s simply not possible from a national point of view. We need to consider things on a European level, and we need the infrastructure to transport the renewable electricity to where it is needed.”
The inherent difficulty with renewable energy is its intermittency: power generation relies on variable resources like the Sun and wind that we cannot control.
It’s an immutable characteristic of renewables, and one that creates problems for grids trying to balance supply and demand, and ensure efficient use of generated electricity.
At least part of the solution is interconnectors—cables that function as long distance energy conduits across and between electricity grids.
Interconnectors allow for electricity generated in one region to be transmitted to another, and allow countries to import and export electricity.
The UK, for example, has interconnectors with France (2GW), the Netherlands (1GW), Northern Ireland (500MW), and the Republic of Ireland (500MW).
“Without interconnectors we’re not able to balance supply and demand and that’s crucial for the energy transition. It’s absolutely key,” explains the EU Energy Union’s Koska.
“We have cables between some North Sea countries already, but considering the amount of renewables coming online in the region, it’s not enough if we are to optimise use of resources available.”
The imperative and current efforts to establish a European super grid are part of another story for another day, but the significance of interconnectors is neatly outlined in the YouTube video above from the Spanish TSO Red Eléctrica.
In this matter of interconnectors and energy distribution, the Wind Power Hub would serve an extraordinarily valuable purpose; one Koska describes as “a clear response to needs of the European grid, and the goals set by the European Union that would contribute to a crucial part of the energy transition.”
As noted earlier, undersea cables would transmit electricity from islands to countries bordering the North Sea, but the same DC cables would also function as interconnectors between those nations.
Something similar is already under development in the Baltic Sea, where the Combined Grid Solution will connect Danish and German electrical grids via the Kriegers Flak wind farm.
The Wind Power Hub applies a similar logic, albeit connecting via islands and not wind farms, and on a much grander scale.
The Netherlands, Denmark, Germany, the UK, Norway and Belgium are all potential players in this new North Sea grid.
Construction of Mischief island by China has resulted in some 1,379 acres of land.
Specialized ships involved in the construction process can be seen in this image.
The dark lines seen connected to ships are floating pipes that pump sediment to be deposited.
photo : CSIS Asia Maritime Transparency Initiative /Digital Globe
Building islands
Construction of islands is nothing new.
Prominent examples of the practice come from China and Dubai.
Although motivated by radically different intentions (in the former instance, to establish a military presence in waters of the South China Sea; in the latter, to support luxurious hotels and residences) both nations have demonstrated the validity of creating artificial islands to varying specifications.
In the simplest of terms, island-building involves dumping a huge amount of rock and sediment on the seabed until an island emerges.
In reality, a little more finesse and a significant amount of engineering skill goes into the process.
Acumen here means that islands may be built to survive waves, storms, and erosion, as well ensure that the newly minted land can physically support whatever is destined to be built on the island.
Expertise will be especially critical for islands of the North Sea Wind Hub where the northerly climate and rough waters of the North Sea offer up considerable challenges.
Still, with the Netherlands party to the project, there will be no shortage of world-class engineers on hand to deliver solutions.
The Dutch have a long history in land reclamation and have been at the helm of some of the most prominent examples of island building around the world, including those of Dubai.
A European wind power infographic produced by WindEurope in 2016.
The task ahead
The North Sea Wind Power Hub is a vast, multinational project that won't just pop up overnight. Brouwers notes that the consortium imagines a first island could be realised by 2035.
Project literature frames the project as one providing a vision for joint European collaboration out to 2050.
“It’s a long-term project, but it’s important to begin now and that the industry knows what on the horizon,” says Brouwers.
For its part, numerous bodies within the European wind industry have acknowledged and expressed optimism about the project.
Andrew Ho, senior offshore wind analyst of the wind power trade association Wind Europe, tells Ars Technica: “Setting out a long term ambition for offshore wind provides a great signal to the wind sector. It’s not governments that are behind the target yet, it’s TSOs laying out the vision—but it’s still important to know that they see a big role for offshore wind in the future of European energy.
"The reality is we need a lot more clean energy if we’re going to decarbonise and really commit to the actions of COP21. For that, we need the technologies that can deliver vast amounts of clean power with relatively stable output—and that’s what offshore wind gets you.The wind industry would certainly be ready to deliver the volume of offshore wind envisioned by the Wind Power Hub.”
Ho emphasised that the wind industry’s activities over the forthcoming decade will lay the groundwork for the Wind Power Hub's success: “The project would give us a pathway from 2030 to 2050, but we’re missing policy targets for 2023 to 2030. To explore the project’s full potential we need to support development through the next decade to ensure we’re fully cost competitive with other sources of energy in the period leading up to 2030.”
As the industry works towards reducing costs, the consortium will busy itself with more practical matters. Brouwers explains: “The next steps involve feasibility studies. We’re also underway in collaborating with environmental groups about the construction of the islands and in talks with infrastructure companies beyond the energy sector, of the sort that would provide critical insight on the project. There’s certainly a lot of work ahead of us.”
The North Sea Wind Power Hub is an unquestionably mammoth project.
But in so being it aptly reflects the enormity of challenges we face in tackling climate change.
Many would contend that we already have the technologies necessary for transitioning to a sustainable energy system.
The Wind Power Hub project reminds us that boldly pursuing the extraordinary, and resolving to commit to collaborative solutions, are traits that will serve us well in application of those technologies.
Links :
Wednesday, May 17, 2017
How an uninhabited island got the world’s highest density of trash
with the highest density of plastic debris reported anywhere on the planet.
From National Geographic by Laura Parker
Henderson Island lies in the South Pacific, halfway between New Zealand and Chile.
No one lives there.
It is about as far away from anywhere and anyone on Earth.
Yet, on Henderson’s white sandy beaches, you can find articles from Russia, the United States, Europe, South America, Japan, and China.
All of it is trash, most of it plastic.
It bobbed across global seas until it was swept into the South Pacific gyre, a circular ocean current that functions like a conveyor belt, collecting plastic trash and depositing it onto tiny Henderson’s shore at a rate of about 3,500 pieces a day.
One researcher claims that a hermit crab that has made its home in a blue Avon cosmetics pot is a 'common sight' on the island.
The plastic is very old and toxic, and is damaging to much of the island's diverse wildlife
Jennifer Lavers, co-author of a new study of this 38-million-piece accumulation, told the Associated Press she found the quantity “truly alarming.”
Much of the trash consists of fishing nets and floats, water bottles, helmets, and large, rectangular pieces.
Two-thirds of it was invisible at first because it was buried about four inches (10 cm) deep on the beach.
“Although alarming, these values underestimate the true amount of debris, because items buried 10 cm below the surface and particles less than 2 mm and debris along cliff areas and rocky coastlines could not be sampled,” Lavers and a colleague wrote in their study, published Tuesday in the scientific journal, Proceedings of the National Academy of Sciences.
The accumulation is even more disturbing when considering that Henderson is also a United Nations World Heritage site and one of the world’s biggest marine reserves.
The UNESCO website describes Henderson as “a gem” and “one of the world’s best remaining examples of a coral atoll,” that is “practically untouched by human presence.”
Henderson Island, with the GeoGarage platform
a coral atoll in the south Pacific, is just 14.5 square miles (37.5 square km), and the nearest cities are some 3,000 miles (4,800 km) away
Henderson is one of the four-island Pitcairn Group, a cluster of small islands whose namesake is famed as the home to the descendants of the HMS Bounty’s mutineers.
Pitcairn’s population, which has dwindled to 42 people, uses Henderson as an idyllic get-away from the day-to-day life on Pitcairn.
But aside from the neighboring Pitcairners, the occasional scientist or boatload of tourists making the two-day sail from the Gambier Islands, Henderson supports only four kinds of land birds, ten kinds of plants, and a large colony of seabirds.
Lavers, a scientist at Australia’s University of Tasmania, and her co-author, Alexander Bond, a conservation biologist, arrived on Henderson in 2015 for a three-month stay.
They measured the density of debris and collected nearly 55,000 pieces of trash, of which about 100 could be traced back to their country of origin.
The duo’s analysis concluded that nearly 18 tons of plastic had piled up on the island—giving Henderson the highest density of plastic debris recorded anywhere in the world—at least so far.
Henderson Island has the highest density of plastic debris in the world, with 3,570 new pieces of litter washing up on its beaches every day.
Jenna Jambeck, a University of Georgia environmental engineering professor, who was one of the first scientists to quantify ocean trash on a global scale, was not surprised that Lavers and Bond discovered plastic in such abundance on Henderson.
Jambeck’s 2015 study concluded that 8 million tons of trash flow into the ocean every year, enough to fill five grocery store shopping bags for every foot of coastline on Earth.
“One of the most striking moments to me while working in the field was when I was in the Canary Islands, watching microplastic being brought onto the shore with each wave,” she says.
“There was an overwhelming moment of ‘what are we doing?’ It’s like the ocean is spitting this plastic back at us. So I understand when you’re there on the beach on Henderson, it’s shocking to see.”
The Henderson research ranks with earlier discoveries of microplastics in places so remote, such as embedded in the deep ocean floor or in Arctic sea ice, that finding plastic in such abundance touched a nerve.
“People are always surprised to find trash in what’s supposed to be an uninhabited paradise island. It does not fit our mental paradigms, and this might be the reason why it continues to be shocking,” says Enric Sala,a marine scientist who led a National Geographic Pristine Seas expedition to the Pitcairn Islands, including Henderson, in 2012.
“There are no remote islands anymore. We have turned the ocean into a plastic soup.”
Links :
Tuesday, May 16, 2017
The incredible 'x-ray' map of the world's oceans that reveals the damage mankind has done to them
Darker colors, which can be seen in the East China and the North Seas, for example, show just where the ocean has been hit hardest.
Source: NGM Maps, 'Spatial and Temporal Changes in Cumulative Human Impacts on the World's Ocean,' Ben S. Halpern and others, Nature Communications; UNEP-WCMC World Database on Protected Areas (2016
From DailyMail by Cheyenne MacDonald
• Study used satellite images and modelling software, to compare cumulative impact in 2008 and 2013
• Over this span of time, researchers found that nearly two-thirds of the ocean shows increased impact
• These impacts stem from fishing, shipping, or climate change – and some areas are experiencing all three
The stunning map comes from the April 2017 issue of National Geographic magazine, based on data from a recent study published to Nature Communications, and the World Database on Protected Areas.
Darker colors, which can be seen in East China and the North Seas, for example, show just where the ocean has been hit hardest.
‘The ocean is crowded with human uses,’ the authors explain in the paper.
‘As human populations continue to grow and migrate to the coasts, demand for ocean space and resources is expanding, increasing the individual and cumulative pressures from a range of human activities.
‘Marine species and habitats have long experienced detrimental impacts from human stressors, and these stressors are generally increasing globally.’
Using satellite images and modelling software, the researchers calculated the cumulative impact of 19 different types of human-caused stress on the ocean, comparing the effects seen in 2008 with those occurring five years later.
The map above reveals the cumulative human impact to marine ecosystems as of 2013, based on 19 anthropogenic stressors.
Shades of red indicate higher impact scores, while blue shows lower scores
This revealed that nearly two-thirds (66 percent) of the ocean, and more than three-quarters (77 percent) of coastal areas experienced increased human impact, which the researchers note are ‘driven mostly by climate change pressures.’
‘A lot of the ocean is getting worse, and climate change in particular is driving a lot of those changes,’ lead author Ben Halpern told National Geographic.
While the Southern Ocean was found to be subjected to a ‘patchy mix’ of increases and decreases, the researchers found that other areas, especially the French territorial holdings in the Indian Ocean, Tanzania, and the Seychelles, saw major increases.
Just 13 percent of the ocean saw a decrease in human impact over the years included in the study.
These regions were concentrated in the Northeast and Central pacific, along with the Eastern Atlantic, according to the researchers.
In a comprehensive study analyzing changes over a five-year period, researchers found that nearly two-thirds of the ocean shows increased impact.’The graphic shows (a) the difference from 2013 to 2008, with shades of red indicating an increase, while blue shows decrease.
It also reveals (b) the 'extreme combinations of cumulative impact and impact trend'
Links :
Monday, May 15, 2017
Netherlands NLHO layer update in the GeoGarage platform
1 new inset added
see GeoGarage news
Changes to Traffic Separation Scheme TSS to be implemented on 1st June, 2017
Prelimary notice of changes of the shipping routing Southern North Sea (Belgium and Netherlands).
changes in charts (see NTMs Berichten aan Zeevarenden week 17 / 15 / 09)
New Zealand Linz layer update in the GeoGarage platform
7 nautical raster charts updated
China revises mapping law to bolster claims over South China Sea land, Taiwan
From JapanTimes
China’s National People’s Congress Standing Committee, a top law-making body, passed a revised version of China’s surveying and mapping law intended to safeguard the security of China’s geographic information, lawmakers told reporters in Beijing.
Hefty new penalties were attached to “intimidate” foreigners who carry out surveying work without permission.
President Xi Jinping has overseen a raft of new legislature in the name of safeguarding China’s national security by upgrading and adding to already broad laws governing state secrets and security.
Laws include placing management of foreign nongovernmental organizations under the Security Ministry and a cybersecurity law requiring that businesses store important business data in China, among others.
Overseas critics say that these laws give the state extensive powers to shut foreign companies out of sectors deemed “critical” or to crack down on dissent at home.
The revision to the mapping law aims to raise understanding of China’s national territory education and promotion among the Chinese people, He Shaoren, head spokesman for the NPC Standing Committee, said, according to the official China News Service.
When asked about maps that “incorrectly draw the countries boundaries” by labeling Taiwan a country or not recognizing China’s claims in the South China Sea, He said, “These problems objectively damage the completeness of our national territory.”
China claims almost all the South China Sea and regards neighboring self-ruled Taiwan as a breakaway province.
The rise of technology companies which use their own mapping technology to underpin ride-hailing and bike-sharing services made the need for revision pressing, the official Xinhua News Agency said Tuesday.
Foreign individuals or groups who break the law could be fined up to 1 million yuan ($145,000), an amount chosen to “intimidate,” according to Yue Zhongming, deputy head of the NPC Standing Committee’s legislation planning body.
According to MoT, China cleared the wreckage of stranded fishing boat on Scarborough Shoal to ensure the security of navigation.
China’s Southeast Asian neighbors are hoping to finalize a code of conduct in the South China Sea, but those working out the terms remain unconvinced of Beijing’s sincerity.
Signing China up to a legally binding and enforceable code for the strategic waterway has long been a goal for claimant members of the Association of Southeast Asian Nations.
The South China Sea Dispute – An Update, Lecture Delivered on April 23, 2015 at a forum sponsored by the Bureau of Treasury and the Asian Institute of Journalism and Communications at the Ayuntamiento de Manila.
But the DOC was not stuck to, especially by China, which has built seven islands in the Spratly archipelago.It is now capable of deploying combat planes on three reclaimed reefs, where radars and surface-to-air missile systems have also been installed, according to the Asia Maritime Transparency Initiative think tank.
Beijing insists its activities are for defense purposes in its waters. Malaysia, Taiwan, Brunei, Vietnam and the Philippines, however, all claim some or all of the resource-rich waterway and its myriad of shoals, reefs and islands.
There will be no mention of the Hague ruling in an ASEAN leaders’ statement at a summit in Manila on Saturday, nor will there be any reference to concerns about island-building or militarization that appeared in last year’s text, according to excerpts of a draft.
The map’s most valuable and relevant feature is found on the upper left section where a cluster of land mass called “Bajo de Masinloc” and “Panacot” – now known as Panatag or Scarborough Shoal – located west of the Luzon coastline
(see YouTube : An ancient map is reinforcing Manila's arbitration victory against China on the disputed South China Sea.)
Duterte said Thursday that he sees no need to gather support from his neighbors about the July 2016 landmark decision.
His predecessor, Benigno Aquino III, brought the territorial disputes to the Permanent Court of Arbitration in The Hague in 2013 amid China’s aggressive assertion of its claims in the South China Sea by seizing control of Scarborough Shoal located less than about 300 km (200 miles) from the Philippines’ Luzon island, and harassment of Philippine energy surveillance groups near the Reed Bank, among others.
While the arbitration case was heard, China completed a number of reclamation projects on some of the disputed features and fortified them with structures, including those military in nature.
China did not participate in the arbitration hearing, and does not honor the award, insisting it only seeks to settle the matter bilaterally with the Philippines.
He rejected the view that China can be pressed by way of international opinion, saying, “You are just dreaming.”
The Philippines, meanwhile, has completed an 18-day scientific survey in the South China Sea to assess the condition of coral reefs and draw a nautical map of disputed areas.
Two survey ships, including an advanced research vessel acquired from the United States, conducted surveys around Scarborough Shoal and on three islands, including Thitu, in the Spratly group, National Security Adviser Hermogenes Esperon said Thursday.
“This purely scientific and environmental undertaking was pursued in line with Philippine responsibilities under the U.N. Convention of the Law of the Sea to protect the marine biodiversity and ensure the safety of navigation within the Philippines’ EEZ,” Esperon said in a statement.
He gave no details of the findings from the reef assessments and nautical mapping of the area, which was carried out between April 7 to 25.
Links :
Sunday, May 14, 2017
Rock and Roll in the Roaring Forties - Dagmar Aaen of Arved Fuchs expeditions
Dagmar Aaen on her "Ocean Change" Expeditions by Arved Fuchs.
Here on the way from Ushuaia Argentina to Piriapolis in Uruguay.
Footage by Arved Fuchs, Felix Hellmann and Heimir Harðarson.
The Dagmar Aaen was built as a fishing cutter in 1931 in the Danish city of Esbjerg at the N. P. Jensen shipyard and was given the registration number E 510.
The hull was built out of six cm oak planks and oak frames.
The space between the single frames is sometimes so small, that a fist can hardly fit between them.
Because of this and due to the addition of extra waterproof bulkheads, the hull was given a remarkably high strength.
The ship was often used in the Greenland region because of its solid built and its choice building materials.
Journeys through ice-fields and months of overwintering in frozen fjords and bays meant daily routine to a ship of this type.
The famous Greenland explorer Knut Rasmussen chose just such a ship for one of his expeditions in the Arctic regions.
The Dagmar Aaen was employed for the fishing industry until 1977.
Niels Bach purchased her in 1988 together with the Peters shipyard in Wewelsfleth Germany and the Skibs & Bædebyggeri shipyard, owned by Christian Jonsson in Egernsund Denmark, built her into expedition ship with ice reinforcements.
Since this time there have been many repairs and changes done at the shipyard, in order to adapt the ship to the different conditions on each expedition. |
dbb170ae13d20dbe | Monday, December 18, 2017
The Osmosis Of Jazz
Style meets substance on the road ahead and merges with the traffic. Curaçao is blue in the red house of logarithms. Chintz plus fungus equals llama. And there, I said it. Everybody gets a shot to be deputy. Me, I’m the deputy of whimsicality. The pollen is random but the momentum is real. Cognition is mostly ants going off in all directions.
Is it difficult to change? Yes, it is. Extremely. But it can be done. Cosmetic is a Greek word. So is cosmos. There is a universe in your cologne, revelations in lather.
My life has been an odyssey, erratic turns, novelties, joys, the willingness to experience incongruities, thriftless fugues, slippery latitudes, irritations like small edible fruit that get you drunk and turn you mad with memory.
Rubber Soul, age 18, San José, California. Streams of consciousness nourish the flame at the tip of a candle.
But why say ‘tip’? Where else would a flame be? Fire can be mesmerizing.
Fifty-two years later I go outside to see what asshole is throwing cherry bombs in the parking lot. Kids in the park. Can’t see them. I just shout into the darkness.
Down below, at the south end of the lake, where all the high-tech companies are moving in, I see tall building cranes festooned with blue Christmas lights. There is salvation in stars. But if you don’t have stars, there’s always the lights of the city.
I worry about the lack of birds this winter. I know what you’re thinking. You’re thinking birds migrate. But not as much as you think. And yes, I, too believe that paradise can sometimes be found in a capsule. But will it last? That’s the question.
For example, there’s a hardware store down on 15th West with a big display of doorknobs in the window. I find that fascinating. I imagine Marcel Duchamp standing there gazing at all those knobs, choosing to mount one as a readymade. He’s dead now, of course, but maybe I could do it for him. So imagine that. Imagine this paragraph full of doorknobs. Now reach out your hand and turn one. Does it turn? Does a door open? Good.
My life has been a shipwreck at times, the rumble of a big barn door, the lowing of cattle, that mournful sound, those smells of shit and straw, and later the whistle of a kettle on a wood-burning stove. I go in and sit down and listen to the furniture. Surrealism sparkles like pearls of irrational beauty.
Things to do in Martinique: breathe the air, smell the many fragrances, ride a horse, watch sunlight pass through a glass full of Chablis, a group of gynecologists peering into a hole in the ground. I see sensuality as a flowering of being. An openness that comes over you and shakes your senses loose as you sit and absorb the atmosphere, no division between you and the external world, voices lifted in hymn, the pullulation of words seeking life and fulfillment in the eyes of an attentive reader.
The ego is propped up by wealth. There’s a certain brilliance in the conception of money. But you can’t trust it. Money cannot be trusted. It’s too ethereal, too volatile. It’s like the slosh of sauce, the piquancy of spice, a man jerked out of a stupor in time to see a train go by where he was standing just a minute ago counting the money in his wallet.
Once, there was a snowman arrested for loitering. His lawyer came for a visit, but the snowman couldn’t be found. There was just a puddle on the floor.
Do snowmen have lawyers? Sure they do. Lawyers made of snow.
Let’s drip.
…sings Irma Thompson: it’s raining so hard, looks like it’s going to rain all night.
Singing is different from thinking. Singing is infused with feeling. Thinking is a hungry mind trying to relieve its own inflammation.
By thinking. Ain’t that a gas?
Thinking transpires in the act by which the thinking subject differentiates itself from its thought. A fiddle is a violin, after all, it’s just played a little differently.
The first time I saw the ocean I couldn’t take it all in at once. Nothing is ever so near to us as the personal, physical feeling of our own being letting the world in. It’s like gazing at an ascension of angels on a cloth of stains. The females have organs on the dorsal webs of their arms. Everything feels like a nebular holiday of junkyard secrets. Birds in dizzying formations. The actuality of twigs. The tortured constancy of lava. The play of light and shadow in a Lisbon bistro. The jubilant brightness of morning in the Valley of the Moon.
Here’s a coupon for ointment, the delicacy of prepositions. We’re all trapped in an illusion of choice, each of us a personality churning in animal tissues. I feel like an ordained fool, an isthmus of unsatisfied consequence condensed into a diving board. Here I go, leaping into space.
Obscurity works best as a meringue of equivocation, a web of abstract commitments. What I want is an augmentation of choice. Not destiny. Who needs that? Destiny is for mythologies. Byzantine monks seeking the ascetic life. Princess Syringe and her system of doors. I want something more geometric, more like the glories of distillation, the colors of the athanor, the feeling that something is about to happen, something real, something exciting, something like photosynthesis or lingerie.
Petroglyphs in the Draa River Valley of Morocco.
The osmosis of jazz.
Sunday, December 10, 2017
We must be careful not to punish the whim or wham the whim with whatnot. The whim within, the whim without, the whim whom folly molds in wobbly wonder. The whim of whims, which is a worldly whim, and is whimful with whimfallity. The inscrutability of the whim is notably willy-nilly. The whim whims to whim itself. The boil of the whim loiters in ham. The whimsical whim has goose whimples. To ogle a whim is to whim oneself into whimsiness. Heideggerian whims hold Being as it moves toward the shore in ripples of time. Whipples of Rhyme that rim the whim in lime. The whim, the great whim, the whim of whims, is whittles and wheels. The wink of the whim is tender. The lion of whims is wholesome and wide. The whale is awash in whim. The whim is full of mirth and mirth is a mirror of life. The whim protects the mileage of the old. The solutions of whims merge on the play of isms. The philosophy of the whim is puzzling but suggests a superstructure of moose antlers. The shortcake whim is a bolt in the door of time. The whim that is wisdom is a wiliness of whims. The guava whim, the jerk whim, the hallelujah hallway haphazard whim. Synthetic whims do not work. They decline into checkers. The true whim is an outcropping in polite society, intrinsically fluid, thermodynamically preposterous. Ladies and gentlemen, we stand at the end of empire, cradling whims in our thoughts, holding to them dearly, as newly ordained codicils to a rip tide of fools.
Thursday, December 7, 2017
Thermodynamics As A Community Of Nouns
Antiques guarantee the gravity of the blowtorch. The welder lifts his helmet and nods to the apparitions dancing on the walls. The zoom lens moves in on stilts. Our tour begins its journey of unicorns and broccoli. Clouds scudding through the sky inform us of feelings yet to be felt, celebrations yet to be celebrated, funerals prophesied in the guts of frogs, tendrils of sinister cloud hanging down from the heavens like twisting anacondas of hell’s colorful aristocracy.
Is life a simulacrum of somewhere finer and better, or is this it, is this the sleep from which we must awaken? Puddles return the sky to itself after all the water that has fallen out of it. Isn't that what writing is? What words are? A refund, a redemption, items from a lost and found, sad, enigmatic objects with stories to be told?
You can masturbate almost anywhere. But try to be discreet. Sometimes all you need is a sack in the hand and a destination in mind to survive the hazards of impulse. If you manage to keep your pants up, the world will reveal the magic of espionage. Let us tromp through the world like God’s spies, quiet, unassuming blokes boiling with paradigms and saints, temperamental philosophers painting despair on the good soft linen of our redundancies.
My gaze sometimes turns to the mouthwash on the counter and stays there, lost in that beautiful blue of the liquid, cool and divine.
Who was the first human to say ‘water’? And what was their word? Their word for water. In Norwegian vann. German wasser. Zulu amanzi. Welsh dŵr. Vietnamese nước.
Tôi muốn đi bơi.
Or, as Hegel put it, Die Externalisierung des Willens als subjektiver oder moralischer Wille ist Handlung.
This plywood is nascent. That is to say, the spice is in the rack, and the senses are aroused. The revolution snaps into place and everything begins to look seaworthy. The embryo of a novel crawls into its pages and begins to evolve. Characters develop, ideas are floated, a cake is baked, pleasantries are exchanged. The world crackles as it turns in space. Virtues are decided. The novel ends with a symposium on perception: is it true that we all see things differently?
And no.
Night glitters in its empire. The horses jingle in their bells. The concept of property decays in its archaisms. What is it to own something? Is it simply to exclude others from the use or enjoyment of something, or is there an actual bond, a eucalyptus hardened against the vagaries of the sidewalk? I am silver in my reflections, but platinum at my wedding. Audacity talks a good game but in the end it all comes down to pineapple. Yellow winds bronze the face of history. Pharmaceutical concerns are packed in cotton. The cows are built with kettledrums. Fog rolls in. The light turns red. We hear a faint music in the background. I lean forward to kiss you.
Still here? Still reading? Thank you.
All it takes is a puff or two to blow the little hairs off of the computer screen.
Nothing is really empty. Not even nothingness is empty. This is what makes Mallarmé so unpredictable. Lightning riddles the conjurations of his words. Galaxies hurl through the room proposing an end to pain. I find a wilderness in my skull teeming with resurrection when I shave. Why resurrection? What is not brought back when we most think it dead? Gone and buried? Nothing dies. Energy can neither be created or destroyed. It just assumes different forms. It stumbles into a flint and becomes a spark. It merges with traffic and becomes a horn. It flings itself into moonlight and becomes a trout. It is expressed in numbers. It becomes calculus. It becomes chalk on a blackboard. Nipples and ripples and wildly expressed panaceas.
Splendor, glory, magnificence and softball.
Hardball is different. Hardballs are stitched by hand and have a round cushioned cork center. I mention this because embroidery only enters the picture later, when there is time for discussion, and no one needs to be goaded or tilted in order to talk. There is a loud whack and the ball bounces to left field where it is caught by a pterodactyl and carried to the end of this sentence and dropped.
I pick it up and hear a giant monotony walking around inside of it. Cork. Or Corky, if you prefer. Consider the sport healed at last. A line drive to first will simply be a luminous stream of consciousness that might be talked about later, when it’s quiet and the crowds have gone home. There is a cure for the clarinet as well. But it must be taken in abstract form or there is a tendency to smear the air with drums.
Monday, December 4, 2017
Slow Henry
I like the light bulbs in the bathroom. The little bulbs at the top rim of the mirror. Where it begins in the morning. My face. That person standing in the brightness wandering how it all began, where did it all go, why is there something rather than nothing? How much longer before the artic ice disappears? Before we all disappear?
A lot of us hope that doesn’t happen. But you can’t stand on hope.
Hope is nonsense. I don’t like hope. It sets you up for disappointment. It swarms with delusion.
I offer, as an alternative, dispensation. I can’t give it to you. I don’t have that kind of authority. Not even in a place like this, which isn’t a place so much as a process. But I leave it here at your doorstep as a suggestion. A proposal. An invitation.
There are times, I think, when thinking makes things emerge, all that energy in the brain, whatever one chooses to call it, does sometimes produce a helpful image, a furnace, an athanor, a Slow Henry, as the alchemists called it. There are experiences and translations of those experiences. Distillations, sublimations, compounds. One can make of the world a loom of golden parables. A bonfire. A surf. A thunderous pounding of water on a sandy beach in Tabatinga.
Lumber, at the very least. Planks of pine and oak in a drafty building.
I think I’m a carpenter who builds things with ink, and the next thing you know, I’ve created a birdhouse of words, a wordhouse.
Ok, maybe that’s not just a good example. It’s a nice wordhouse, as wordhouses go. Why abuse it with rumination? The glow of a hinge in the hodge-podge of the ponderous shines forth to inform the senses of phenomena begging description and definition.
Is why.
Am I negligent? I try not to be. I try to be careful. I try to notice things. I try to notice what I’m doing. Even though, much of the time, I don’t know what I’m doing.
Sometimes I see Herculean colors fissioning in the pretext of a sunflower and amalgamate it into a heliotrope.
Drugs can be adjectives. Adjectives can be excursions. Excursions can occur on water. Water can be random. Water loves being random. Though I think it’s a mistake to arbitrarily attribute self-awareness to water. Water is water. Sloppy. Like me. Who is 60% water.
If mistakes were money I’d be a millionaire. This is why I believe singing belongs in an elevator. My singing. Which is strange and full of experience. You can’t boycott experience. Experience just happens. I was born to be a comma.
The lobster has a weird body. But it’s not the fault of the lobster. It is the responsibility of the lobster to be a lobster, to eat what a lobster needs to eat to continue being a lobster, take some time out to reproduce, make more lobsters, bring more lobsters into the world, in whatever manner lobsters have devised for themselves to reproduce. And what makes the body of the lobster weird to me? These are simply my perceptions. I’m sure that my body is weird to the lobster. If (as one might assume) the lobster has any sense of what might be an anomaly, an anatomical eccentricity, then certainly the lobster will perceive the human body as extraordinary. Skin, for example, might seem strange to a lobster, adorned as it is in a carapace equipped with claws and antennae. It’s hard to think what a lobster thinks. Meanwhile, I listen to the Rolling Stones sing “Blue Turns To Gray,” which has little to do with lobsters and everything to do with feeling troubled, feeling uneasy, feeling unsatisfied.
I’m tangled up in gray. I squeeze the morning sun. The Beast hands me a shaker of salt. The horizon splits the day from night. I feel eloquent as a speed bump.
I belong to a strange group of people called poets. Imagine being immersed in an activity with no commercial potential. Abstraction feeds on reverie. I keep feeding abstraction. Abstraction plays comparisons into prospect. The whipped cream articulates the rhythms of our conversation.
Money is always hypothetical. Surround yourself with healthy advantages.
My species has not been successful.
A book is written each time someone reads it. There is redemption in the present. The only cure for summer is more summer. I’m soaked in phenomenology. Who knew that everything in the world was so delicately interrelated? Let’s go searching for mushrooms in Iceland. Interactions heal the poverty of power.
Here are some artifacts of the 17th century: an embroidered shoe, Constance Hopkin’s beaver hat, a lobed Delft dish with a swan.
The frenetic taste of conflict keeps words churning in my brain. I see Buffalo Bill filling an SUV with gas. The pregnant charm of a drugstore. The spectral dots of Dagwood.
As much as I ingest the world, I exhibit the world. I like swimming in swimming pools. Rivers freak me out a little. It’s hard to carry a generation in your voice. The kiss of wealth decomposes rapidly. What you want to do is get reborn. Look what happens when you stay alive this long. A broken escalator is just another set of steps. Use them carefully. Each step is important.
A man gets into a red Mazda and it coughs into action, electricity careening through the wires.
The hammer is immersed in its purpose. All the electrical cords get tangled up here in the eternally humid Northwest. My fingers respect the feeling of aluminum. Don’t panic if the immaterial materializes. Celebrate the fact of your existence. The drapery redeems the view. An embryonic telecast bubbles on my lap. Pathos is a giant sip of universe.
Friday, December 1, 2017
Here I Am
Everyone knows how life happens. It’s over in a flash. Meanwhile, there’s soup and mythology. Light gleaming on the Seine as it roams through Paris. Bombs and machine guns everywhere the U.S. claims empire. A woman in Rome bending over to pick up a beach ball. It’s 7:27 p.m. November 14th and I’m sitting on a bed with a tuxedo cat reading Persian Pony by Michael McClure, “THE SOFT NEW SOUL / with its capsule of masks / tender and quivering / ascends into matter / and here I am.” Fingers, fingernails, laptop, breath. A presence to myself until the absence I try to imagine occurs, and I cease to occur, hopefully before the arctic ice melts, and tens of millions of tons of methane are released into the already stressed and out-of-balance atmosphere. You can’t stop extinction. You can’t stop habitat loss. But you can focus on the present. The slippery, elusive present. Now it’s here, and now it’s gone. Here again, gone again.
And so some words walk around trying to be a pineapple. Let’s let them. Welcome to smart investing. Welcome to the play of the concertina. Opinions shaved in the rain. Indigo octopi.
Curls, corkscrews, swirls, convolutions. Nothing in life is linear. It’s waves and oscillations, embellishments and sleep. It’s the weight of a dream, the murmur of wind in the trees. Cool water in a Peruvian jungle. A scratched Parisian angel. The mercurial spur of gossip, broken rain crumpled into gold. Theorems in serums. Sandstone arch in August heat.
We live in a world of flux. We are flux. Everything is soaked in phenomenology. Some say it’s the singer not the song. I say it’s elves riding on the backs of swans. Running over tree roots to avoid puddles. The opinions of a lotus. The force of subtlety in a drug taking effect. Truffles in the Dordogne. The thunder of giants punching eternity with improvisations of water.
Neon chrysanthemums. My bare feet resting on the blue sheep of a white blanket.
Eager fingers on a limestone ledge.
Puff on the seeds to be born into myth. The tongue is soaked in redemption. The stones of Iceland aren’t there to glitter in idleness they’re a punctuation of convergences, druid moons and Viking purgatories. Lug the pilgrim to the call of the lake. The singer of the song is unknown, but the song itself is exempt from agriculture, and paddles like a swan across a pond of belief.
Belief is a diversion. Agriculture was a mistake. Let us convene instead with the spirits.
Remember the spirits?
The spirits of water, the spirits of lingering, the spirits of sustenance and fever. Rosie and The Originals. Angel Baby.
I have a silver buckle and a hat of chaotic mahogany. Streams of consciousness percolate through the roots. A musician buys diamonds for his guitar. I talk about the problems of aging and mortality with a friend while a foreign melody gets dressed in a person with leprosy. I don’t feel like ironing today.
I’m the Rembrandt of butter. Regret is a drawer in my skull. If I see the weirdness of wax drooling down the stick of a candle I want to paint it. There’s a momentary pause in time that sometimes reveals itself as a pale morning sun. It’s that moment of stillness right after the waitress has cleared and wiped the table and no one has been asked if they want more coffee yet.
Pains have personalities. Some of them emerge in music and some of them enter into Being like 150 pounds of pressure in a tire designed for 125 pounds of pressure. I like the ones that float in the air like astronauts looking down at Planet Earth weeping. The ones I don’t like churn in the brain unendingly with no resolution. They’re like the entanglement of vines in a blackberry bush, the insane repetitions of traffic around the Arc de Triomphe.
Rumination is a dead end. Time suspended in a cuckoo clock. A stuffed wildcat with its mouth open. Mickey Rourke gazing into a tank of rumble fish.
Think of chiaroscuro as an old man scrounging for change. The soul of white is black. Defining anything is a delicate process.
I’ve always loved the effects of darkness and light in Rembrandt’s paintings. Among my favorites is The Philosopher in Meditation. An old man sits by a window through which a golden light diffuses its warmth. To the immediate right is a spiral staircase. And to the right of the staircase an old woman bends over to tend a fire in an open hearth. The philosopher is very calm, hands folded, head tilted slightly forward, as if with a weight of thought, or immersed in reverie. All around is darkness. It’s the darkness that makes the light so voluminous and alive.
How does one get to the essence of something? We all want to see the interior of things. Interiority is a constant fascination. Everyone feels deceived on some level. Everyone seeks quiddity. The vital truth of a thing. A chair, a table, a person, a cat.
Though perhaps not its essence so much as its whatness. Its presence as a thing in itself.
Time walks around in my head dropping memories. Some of them are long and delicate, and some of them are abrupt and brutal. A few are dopy. A lot of them are thematic. There is one in which I am crowned King of England and introduced to the dining room staff. I take long steps of introversion in a royal chamber of books and ledgers. Liquids bubble in tubes and flasks. I create a new velocity for the indecisions of purple. Malachite and jasper sparkle around my neck. A jet flies over a Fed Ex Office. I keep trying to write my way out of this world. Autonomy is a prompt solution. I use it carefully. But even that is a mistake. The train is a hymn of steel feeding on its own reverie.
Throw another log into the fire. The poem is ample that never loses it clarity. But you’re not going to solve any riddles that way. What you need is a salute to nothingness, the superfluity of leaves blowing around in the wind. I can offer you a place to sprawl and dream. Do you feel the sting of a needle? Don’t worry, it's just a spark from the foundry of apples.
Sunday, November 19, 2017
The Emperor Of Macaroni
Speed is aromatic when it becomes lightning. Who are you? Ribbon is one solution. Moccasins are another. Density is magnificent with mermaids. Think of this as a phenomenology of reaching and reading and reaching for something to read. Of pianos and cockpits. Syncopation and garlic. Wax and honey, which are lieutenants of bric-a-brac, and dare to matter in a world of geeks and grossly inflated salaries. Even though, when you think about it, the sponge is every bit as brilliant as a whale, and a crisis such as this can loosen our frosting. I think it's wonderful that things exist. That the nose is naturally Zen and that one’s chains are imaginary. Break them. Drop them. It’s wonderful that magnesium can be a waitress and that the color gray can fall into the hands of a dwarf and televise the chlorophyll of a milkweed. That lips have their own brand of chivalry. That success can mean so many different things to so many different people. This hour will dissolve within the limits of another hour and various sensations will hatch out of that and become words in a sentence. Drop everything and run into the sky. Pasta is sensual because the streets are full of wasps, not because hope is cruel, and it takes courage to foster a load of despair. Hope is a delegation from a future that doesn’t exist. Don’t go there.
Wednesday, November 15, 2017
Nothingness Wins Again
Enigmatic reverie package that comes from sparkling spray. Gotta buy me some fire and a bear, some foliage and a bell, a pair of pants and a nice green pump and a warehouse of suede. Anything that models the cylinders of a pretty nickname. I will be the meaning I want, lavender buttons and bliss. Antenna gum that stretches all the dust of life and a cricket singing from a high edge of facts about brackets, which are delegations of pearl. Wild abstraction with a reason to open a bean.
Oil and turpentine and a shade to undertake the lassitude of ash.
If space is within space, then spatiality must have something to do with recompense. I will call it a dollar. Which expresses the camel hiding among my nerves. The desert, the wind, the dunes, the drift of detachment. And this is happening with a claw and a negligence of rocks. Acute crumbling of a cabbage sorbet. Bienvenue au Palais Idéal.
This is my electric yellow pin. I am in the east licking the power that is nature. I smell sweet from my locomotive stomach but I don’t really care about the friendliness of furniture unless it starts talking like sparrows, which reminds me of Hamlet, and the sweet beginnings of stars, and then I cry the long thin tears of supplication and collapse to the floor and become a chair.
What is causation? Does anyone really know?
I have been talking about what is ready-to-hand. But what about assemblage? The sandwich on the counter at the diner? What about jaws, and brightness, and indigestion?
Equipment, too, has its place, or it just lies around collecting dust. The nosegay doesn’t appear at random. It is there in accordance with its involvements.
Allegiances are further complicated by disagreements over what events, facts, and these other creatures are. Some seem precise, like the praying mantis, whereas others are whiskered, and whistle like steam. How is it possible for one mind to know another? Is there a phenomenology that cooks like rice but is better than caviar?
I believe that there is gold in the cave and that it doesn’t harm the glory of being a little lost among the shadows when they bring a little reflection to the glitter of its veins. Listen to the bullish scrap woman who does ironing on the sidewalk of a rose. The thorn clock piloting the edge of a wave at the monastery. These are reasonable and sipped. Indications assembled to accommodate the decipherment of cause.
As for causation, let’s explain it with quarks. Binoculars and breakfast. Causation is the cause of cause. The cause of giants lifting the ocean into rain. The cause of hope, which is appalling in its constipation. The cause of the cashew, which is expensive, and the cause of the peach, which is lips. A hammer causes itself by hammering.
Exclaims nothingness, which is now a nail in a two-by-four of an insect cycling around an apple.
Sunday, November 12, 2017
Out Of Control
I enjoy the sensations of things, doorknobs, laundry warm from the dryer, spider legs scampering over my palm, water when I’m thirsty, symphony strings, Buddy Guy doing some straight up insane things on his guitar, the weight of a book in my hands.
Did you know that horses are able to identify emotion in human facial expressions? I can’t even do that. What I can do is reveal or conceal an emotion depending on circumstances.
There are landscapes I could never describe. Not with paint, not with words, not with echoes or inclines or swamps. The whole is always going to be greater than the sum of its parts. This is especially true of landscapes, fjords, inlets, lakes, clouds, late afternoon light on a Tuscany hill.
I like the feeling of the word ‘seethe’ as it seethes through my teeth. As this from Shakespeare’s Timon of Athens, “go, suck the subtle blood ‘o the grape till the high fever seethe your blood to froth.” Or this, from Pencillings, by N.P. Willis, “Cold meat, seethed, Italian fashion, in nauseous oil.”
Do you see? Each word is a history, a palimpsest, a landscape. Cold meat seethed in nauseous oil. The workings of wine in the blood, turning it to froth, delirium and groping. Daydreaming. Musing on the grain of the wood of an old dark bar. Big arguments with the hands waving. Voices raised in speech, or singing, or the flutter of syllables on the ear in a foreign country, where the weight of what is being said is hidden among its vowels.
The word ‘landscape’ comes from Old Saxon ‘landscepi.’ Old Norse ‘landscap.’ The word was later introduced as a technical term by painters, a picture representing natural inland scenery. Or as I like to call it: the language of earth as it is spoken by wind and rock.
The loose dirt of the Palouse is called ‘loess.’ It’s soft and fine and nourishes the soft white wheat of the Palouse, which goes into the making of pastries, apple strudel and cinnamon rolls.
Since consciousness seems to be localized within my head, I always have the feeling of being in an airplane, in which case the landscape I’m looking down at is generally a carpet, if I’m barefoot in our apartment, or the sidewalk, one of many sidewalks, here in Seattle or in Paris or Minneapolis, which is a little like Paris, in that it has a river running through the city, about the same size as the Seine, but called the Mississippi, and is legendary, and full of catfish.
I remember standing on the Pont Neuf in the winter of 2015 looking down at the Seine, which looked wild and turbulent, weirdly green in color, heavy with French dirt, French landscape, paysage as they call it.
My eyes fill with the light of a thousand bright yellow leaves stuck to the sidewalk at the top of Highland Drive. The temperature is 45 degrees and is invigorating and moist. The sky is gray. It’s mid-November and Seattle’s skyline gleams below. I feel good, but can’t shake the sadness caused by hearing Guy McPherson’s grim predictions. McPherson was a professor of ecology and evolutionary biology at the University of Arizona until he left his position to live on an off-grid homestead in southern New Mexico. He has since moved to Belize and put his property in New Mexico up for sale. He is best known for his talks on imminent mass extinction due to the accumulation of greenhouse gases in earth’s atmosphere, a situation he deems long out of our control. He states a paradox: if all industrial production stopped this minute and no more pollution entered the atmosphere, the heating of the planet would be accelerated since the pollutants in the atmosphere act as a filter, diffusing the sun’s heat.
McPherson delivers his talks in a calm, measured, eminently rational voice. He supports his claims with compelling facts. He has a warm presence and emphasizes the importance of enjoying life to its fullest, living in the present moment, seeking excellence in a culture of mediocrity and continuing to floss one’s teeth. He tries to put a redemptive spin on our imminent doom by urging us to do what we love, disburden ourselves from the encumbering shackles of false hope and the oppressive tyranny of jobs and money and live to the fullest while we still can. But it doesn’t work. Extinction sounds horrible. The death he describes sounds awful: when heat and humidity rise to a certain level, we behave drunkenly, because our organs are boiling.
Other climate scientists, such as Michael Tobis at the University of Wisconsin, say McPherson’s claims are incompetent and grossly misleading. I don’t know what to think. I tend to think Tobis is correct and McPherson is wrong. I want Tobis to be correct and McPherson to be wrong: way wrong. I’m not a big fan of human beings, they’ve been responsible for a great deal of ruin and savagery and pain, but I don’t want to see humanity go extinct, any more than I want to see other species go extinct. I mean, didn’t the dinosaurs do better? They managed to stick around for 165 million years. Think of it: big old walking Walmarts of bone and flesh. And what about dinosaur farts? I don’t get it. Is it all this cortical activity that’s gotten us humans into so much trouble in such a short amount of time?
It would be so much nicer if I could just reject McPherson’s claims wholesale and get on with my life. But I can’t, not quite. I can’t shake the sadness nor the truthfulness implicit in McPherson’s words that easily. It will take more than Tobis’s rigorous mathematics to do it. The wildfires and hurricanes and droughts this last summer were horrendous. Clearly, something very, very wrong is occurring to our planet. And it’s just the one planet; there aren’t any more available when this one is finally, irreparably lost.
Flash drought destroyed half the wheat crops this year.
But enough of that.
Why is it that the things over which I have the least amount of control are the things hardest to let go of?
I think the answer is right there in the question: no control.
Most of the time, the only thing I truly have control over is how to respond to things. And even there I have to separate instinct from intellect.
I have no control over the maniacs using leaf blowers in the rain when everything is sopping wet and stuck to the ground, or the jerks whose leviathan SUVs and four-by-fours won’t fit in their driveways and stick out over the sidewalk blocking everyone’s way, or the ongoing looting of the American population by their “elected” officials, and their cronies, the banks.
Making money out of thin air. “Don’t think money does everything or you are going to end up doing everything for money,” said Voltaire. Amen to that.
Thursday, November 9, 2017
The Elegance Of Leaving
Why are ghosts always represented as bed-sheets? Death is nothing. Nothing without England and its historical debris. Nothing without a fugitive understanding of life’s most basic courtesies. Sleep and nuance. Umbrellas and cows.
The invention of thirst comes to us dressed as a mythical taffeta in the tenacity of an ant. Red feathers on a white table. Certitude. Incertitude. The philosophy of yourself.
An X-ray and the light behind the X-ray. Bones. Prisms. Gregorian chant.
Nothing beats the elegance of leaving a job. A party. A bad marriage. An excruciating eulogy. A firm decision. An endless war. An ideology gone sour.
Death is nothing. The fragrance of a casket is unaffected by its mystery. It is sometimes sudden, sometimes long and inquiring.
Death is nothing but hoes in a row in a pink garage. Brian Jones smiling at Howlin’ Wolf. Letters thrashing around in a sentence. Nothingness is underrated. So is the shine on the shell of a crab. Eyebrows are incidental, like molasses and papier collé.
The poet is a nomad with nowhere to go. The United States has become an open-air prison with an extortionate hellcare system. I’m old enough to remember streetcars. So when I say that the poet has nowhere to go, I mean nothingness articulate as a gravel driveway. I mean clumsy indications of death walking through the eye of a needle. I mean camel. I mean rich man. I mean crinkly old dollar and words in a process of waves moving up and down a cobra neck-tie.
Poetry is an engine of ice, helter-skelter at a Cincinnati gas station. Caress the spine of a dragon. I will tell you what it’s like to eat lobster on a private jet. I will tell you how to articulate the gravel of a driveway without using nails or nutmeg. I once corresponded with a cringe. Which I later pumped to the surface of my skin and showed it around town like a tattoo of shadows boiling in the midnight of a woman’s fingernail. I’m sympathetic to most vibrations, but I’m mostly favorable to the forehead when it’s lit up by a crown of electricity. It’s a good look. I agree to nothing but what goes on in my fingers. Golden oarlocks on a red boat. Think of it as symbolism, something out of the late 19th century. A huge barroom metaphor that answers the demands of reason with a tiger’s head and a snake between its teeth. I feel the exclamation of stalagmites in my guts. Opinions slam the door on discussion. If you have an opinion nail it to the wall and shoot it with a .38 caliber toad.
Do you like cream in your gridlock? Feathers are marvels of engineering. Can I offer another version of myself that explains these things? Some people like to punch the air when they dance. But I’m not going to pretend I’m Mick Jagger. You don’t know who I am. Who am I? I am you. I am us. I am her. I am him. I am everyone. But mostly I’m a guy looking for a way out of here.
Gravity is a cure for science. But nothing cures a heartache like the bone black in a painting by Rembrandt. No amount of logic can explain a clam. But I can tell you what a sparkle looks like in the eye of a monkey. Watch it dilate. The mind dilates. Did you know? Yes. And I’m hooked on polyphony. A crinkly old dollar. Zen mosquitos on a hairy arm. Bend the milk into asphalt. The forklift lifts a pallet of formaldehyde and so concludes: death is nothing. What is the source of this emotion? Flames thundering out of the bottom of a rocket. The lure of Titan. Buffalo on the plains in 1752.
Tuesday, November 7, 2017
Cartoon Noises In A Kitchen Sink
Cartoon noises in a kitchen sink. Metal crabs tap-dancing on a China plate. Water running. Two pieces of meat stuck to a spoon. What shall we do with this loaf of elevator? Give it a little baptism. The biology of a feeling, which is soon felt going chromosomal, like a rattlesnake chandelier, or a hymn to the speed bump. Everything in life sooner or later gets to feeling reptilian, or naked, the way a fork throws itself into space.
Words are sticks of meaning soaked in pain.
I stood on the stepladder trying to open a little plastic sack with two little screws in it for the ceiling light mount. It opened of a sudden and the screws went flying. That’s how it always is. Just listen to Gregorio Allegri. Or the murmur of doctors focusing on a bone.
Breakfast explains nothing. I can hear the rustle of rain. It’s early November. I can see a discarded bikini in the Hall of Mirrors. The pulse of a sawhorse wrapped in cloth. How many pounds are in the ghost of a hammer? I agree with my spine. The Renaissance is mostly about music. Science came later, blistered and stubborn, like language. Except language isn’t very scientific. It’s more like swans perched on the top of a barn. You can smell it as it gropes for a coat, or enters the parlor goofy as a traffic cone and sits down on a concertina. Oops.
Concentration is the essence of the concertina. The dreams of a halibut are different. The dreams of a halibut resemble the furniture of winter. The reason is obvious as cocoa.
I wouldn’t characterize myself as jaunty. I fuss over the issue of subjectivity much of the time, but it leads nowhere testimonial. Nothing like an elephant, whose subjectivity is intellectual, and drinks experience from a waterhole of stillness and quiet, poised as a mosquito on a policeman’s arm.
What does it mean to be ambitious? I’m not pleased by the taste of oysters. Never have been. I see a mockingbird on a barbed wire fence and think about the many unseen gears of the escalator. Let your eyes carry this sentence to the end of itself. When you arrive at the end, you will find an abyss. You will see ice and snow. Pain floating in the eyes of a stranger. And that stranger is you.
Or not. Maybe it’s just another bend in the river, random and wide and full of reflection.
What do we mean when we speak of a music as “heavy metal?”
Consciousness is a rag of emotion, the crackle of feeling in a ball of thought. Stars in a jug of white lightning, the many doors to perception.
Did you forget to fall in love today? I didn’t. I just now fell in love with a Dutch apple pie. Oats are easily made, but the many subtleties of sleep are not so easily described. I would like to further explore the idea of Sam Elliott’s mustache. Has it been a boost to his career? Probably. Is it eloquent? Yes. Like a popped balloon, or a star hanging from a thread of music. Crystals sparkling in the arctic night.
My plan throughout life has been to evade too much planning. Stepladders make me angry. They never fold back up right. If I see puddles in a row I think of vertebrae. I think about singing in Montana. Belonging to a choir. I watch the cat as she rolls on the floor, exposing a white fur belly.
Can we bring some words into this sentence that usurp their own progression, that swirl back on themselves and duplicate the invasion of an eggplant? Sure. Why not. I don’t want to get too fancy. Let’s keep things simple and enjoy a sip of universe. It’s calm tonight and my needs are congenial to the employment of various prepositions. Sometimes it takes a powerful drug to walk through a wall. And sometimes all you need is a few prepositions and a warped sense of oligarchy. A jug of conflict and a jar of argument. The heart is an armchair for feelings. So sit back, and let yourself float. The ugliness of time is remedied by oak. And the swans on the barn are quiet as Sam Elliott brushing his hair.
Saturday, November 4, 2017
Close Shave
A careless hue expresses dyeing. A beach bonfire crumbles in the lung kite. I go frisking past scratching my right leg. The wind is melting in an ebony crate. I’m floating in the weather of a bee making words come together.
Mechanical rain a dragon in my head. Neon money for a ruptured chocolate. I see a solitary radical ball that stuns the value of grease. The expansibility of shoe ash excites the senses like a swamp that jumps into an old New England spoon and begins varnishing oats. The lush spring of a streaming friend powers a tug of antique sugar as it journeys across space and time and so begins another rag with which to solace the groan of coupons burdened with impersonating raspberries in the butter Marie Laurencin spreads across this particular slice of bread.
Of course, when I say particular, I really mean bulky and round. You shouldn’t have to think of this as surrealism. It's more like undressing a landscape of sage and smelling the sexuality of noon. Surrealism is for banquets and airports. This is more like lunch with a Q-tip. Anarchic chairs pondered in wild benediction. Fingers on an open G tuning.
It’s almost irritating the way shaving lather keeps coming out of the can when I am sure it must be empty. But let’s face it. Facial hair is intrinsic to the dominion of ivory. It’s not like heresy, not entirely, despite some obvious resemblances. A beard must be worn as a portable device for heroic deeds.
Sometimes sitting in the garage chattering to the shelves about mutiny is the closest I can come to unbending the fizz of lacrosse.
This is where flirtatious 35-year old Charlotte (Laura Prepon) stumbles into the poem, explaining that she has a thing for older men, along with the poetry of Edna St. Vincent Millay.
I tell her she has the wrong poem and open the door to let her out.
My allegorical knee has a carpenter’s scratch. I win everything by throwing chocolate at a bureau drawer and selling pineapples to a hoe. I attempt to do the same thing to an authoritarian tattoo. How cannot it not know what it is? Who doesn’t like hats? The mission fails miserably and I console myself with ichthyology. I can always try to sputter a few opinions later when the meaning of being reawakens. I’m not going to argue with a menu built around augury.
Holes pause for an eon in a Mediterranean hamburger and the world gets sliced into turf. Nebulous and soft, I sift an obscure hill of dormant tinsel and thereby welcome butter, which is good to me, and simple like sleep. Later, when the proximities loom, luminous insects display their emotions in elevator eyebrows and an aromatic silverware creates a craze for openly indiscriminate music.
Which is the best kind of music. It dreams it’s a cupboard with a canine tooth and plates crashed together and is the sage way to the salt beard. I am bitter about frozen agitation. I like the hint of flexibility on the street, the pendulum of tomorrow mingled with loops of iron like the crashing of words in a foundry. Anything else is just structure, a profession brought up on the hind legs of a uterus.
Wednesday, November 1, 2017
The Paint With No Name
It’s not infrequent for something very small to get on my nerves. Such was the case in our bathroom. I was taking a shower one day when I noticed the paint in the upper corner had blistered and flaked just a little and that there were a few mold stains along the ceiling where it met the wall. It wasn’t a big deal. I tried to dismiss it but I couldn’t. Once something gets on my nerves, it grinds down and stays there. It would have to be painted.
And why had that corner flaked and blistered? What was going on there? I worried about a leak. Were there any pipes in that spot? I hoped not. I got a small stepladder out of the hallway closet of our building and got up there and poked softly and felt around with my fingers for moisture or gumminess. It didn’t feel like anything was leaking. Maybe it had been the old showerhead that I replaced, a huge bulbous thing with little nodes and holes all over it that sprayed water everywhere. I hope that’s the explanation. So, no plumber (knock on wood) would be required, but it definitely would need to be painted. How was I going to find a match for this paint? It was an off-white with a soupçon of yellow. I didn’t have a name for it. It had been fifteen years at least since I had painted the bathroom. It was probably called something like eggshell white or water lily blonde or coronation champagne. Who knows. I had not been prudent and kept the can. Or written it down.
Finding a match turned out to be embarrassingly simple, albeit a tad pricey. After I scraped some paint away in the corner I was able to collect the flakes that had fallen into the tub in an envelope, which I took to a paint store on Stone Way called Daly’s. A pleasant young man at the counter explained that I could get a mix, but the smallest they could sell me for doing that would be a quart. A quart would be way too much, but if they could find a good match, it would be worth the price, which was around thirty bucks. I picked the paint up a few days later. The match was perfect.
I asked the clerk (this time a young woman) what the name of the paint was. She said it didn’t have a name. It had a number. I looked at the number. It was big, a formidable number. The tint had been calculated with such perfection that it had entered the realm of science, astronomy and quantum mechanics. This wasn’t just paint, it was a Schrödinger equation.
I got the paint home and got everything ready to paint, newspaper and stepladder in the tub, can of paint on the floor also on a sheet of newspaper. I had a critical decision to make. Should I find a small container that I can hold in my hand, or should I dip the brush in the paint and hold it so that the paint does not drip on the tub or floor?
I decided on the latter. It was riskier, but simpler. If I was careful, I would make less of a mess than if I tried to pour the paint into another container. I donned a pair of surgical gloves, opened the can with a screwdriver, and dipped the paintbrush into the smooth white surface of the paint. There is something very sensual about paint. The gooeyness, the viscosity, the weight of the brush, the richness of color in liquid form, slowly turning the brush in my hand while the excess paint drooled back into the can, then slowly and gracefully raising the brush while positioning myself simultaneously on the stepladder in the tub, all these actions performed with great concentration were a form of meditation, an immersion in a medium of sumptuous stickiness.
Mistakes were made. Mistakes are inevitable. I forgot about our cat, Athena. Athena came wondering in and was naturally curious about what I was doing. She’s fascinated by the shower to begin with. She loves to get her front paws on the edge of the tub after I shower and gaze with great fascination at whatever it was that just took place. She licks herself. She doesn’t see us licking ourselves, but we do get into a shiny place and make water fall on us. In her world, that’s phenomenal.
Roberta had been outside raking leaves. When she came in I hollered to her to remove Athena from the bathroom as I had paint on my hands. Unfortunately, I forgot to warn her about the wet paint on the corner of the wall by the door. She scraped past and got paint on her fleecy blue bed jacket. I told her the paint was not water soluble. She would have to toss the jacket. I would buy her a new one.
And, inevitably, I spotted a few places that I missed, went to reopen the can, got paint on my hands and realized that I’d forgotten to put surgical gloves on. Getting the paint off with rubbing alcohol and soap was the most difficult part of the job.
Whether it was the tension of doing the job or the smell and fumes of the paint in an unventilated room I don’t know, but I got a terrible headache later in the evening. My brain felt like it had swelled in overall size by about an inch and was pressing against my skull which was beginning to crack. If headaches - like hurricanes - had names, I would name this one Vercingetorix after the Celtic warrior king who proved to be such a headache for Julius Caesar during the Gallic Wars. It was tough and stubborn and shaggy and unruly. Celtic to the core. A mean headache. The kind of headache that brings down empires. I could name it that, or I could name it Jon Brower Minnoch, the heaviest man in medical history, who weighed over 1,400 pounds when he was admitted to Seattle’s University Hospital in March, 1978. Two beds were lashed together and it took thirteen people to roll him over for linen changes.
I took some ibuprofen, and the headache dissipated some minutes later. That’s always such a good feeling. It’s as if Jon Brower Minnoch lost 1,211 pounds and strolled out of the hospital at a trim 189 with a smile on his face.
The next day I removed the painter’s tape from the upper wall by the ceiling where I’d painted. It was riddled with paint, which hadn’t yet dried. I tossed the tape, got out the rubbing alcohol, and went to work on my hands again. Paint has a genius for getting and going everywhere. There was even some paint under the tip of my thumbnail. I solved that with a pair of clippers.
I was happy with the result. The bathroom looks great. That yellow tint, the indefinable hue that put the off in off-white, that required calculations as formidable as those assembled at the Jet Propulsion Laboratory in Pasadena or the Large Hadron Collider in Geneva, Switzerland, brightens things up, makes it seem like a fun place to be. Showering and shaving and brushing my teeth and other ceremonies performed to maintain my hygiene are not activities I generally choose to celebrate, or characterize as fun (I would choose very different words), but it’s nice to perform them in a space that’s been augmented by a nameless color of paint, a paint whose hue is so specific in its charm that it eludes the syllables of the mortal realm and hovers somewhere between transcendence and dream.
Monday, October 23, 2017
Each Moment Is A Universe
There is sometimes a good clean feeling of being alive and wet in the rain. Doesn’t matter what age, but if it happens late in life, so much the better. I didn’t think much about being alive in my youth, I was simply alive, simply living, trying to resist some things and experimenting with others, trying to get a sense of what’s good, what’s bad, what’s exciting and stupid, and what’s just stupid. But as an older person, not just older but old, an old person, I think about being alive. Because one, I’m still alive, still doing it, still living and breathing and eating and sleeping and all that good stuff. But two, I’m stuck with all those decisions I made in my youth, and three, there’s not much in the way of destiny at my age.
Destiny is about the future. What happens in old age, what is important in old age, is to stay focused on the immediate, to experience the immediate, squeeze the immediate, hug the immediate, all the while trying to get used to the idea of one’s life coming to an end. The ephemerality of life, its ultimate temporariness, is more acutely felt as we age, and it is both a great sadness and a great liberation. We are brief custodians of a life energy running through us. Our reality is something far greater than the life we encapsulate in blood and bone for X number of years. Not to put to morbid a spin on it, but that’s what’s real at my age. The immediate, the imminent, the actual. The universe at large, of which I am a part, a temporary manifestation of hair and skin, ideas and fingers, daydreams and DNA. “Here feel we not the penalty of Adam,” observes Duke Senior in Shakespeare’s As You Like It, “the season’s difference,
………as the icy fang
And churlish chiding of the winter’s wind,
Which when it bites and blows upon my body
Even till I shrink with cold, I smile and say
‘This is no flattery; these are counsellors
That feelingly persuade me what I am.’
So going out in the afternoon of a day in late October when the air is honing its knife and getting ready for the real cutting cold of December and January and it’s raining and gloomy and gray and there are still people up on Bigelow hunting down chestnut burrs is a luxury of sorts. I can still move, still run, still get wet. The immediate and actual are large and multiple and keenly felt. Each moment is a universe, reads the title of a book by Sōtō Zen roshi Dainin Katagiri.
It’s hard to appreciate just how vast the universe is. I can’t. Can’t do it. Can’t wrap my head around it. For one thing, it’s infinite. I can’t wrap my head around infinity. I know what it is, it’s boundlessness. Infinity is forever. It’s beyond time, beyond space, beyond Google. It’s beyond my ability to imagine what forever is. I’m a drop in the infinity bucket. Drops are easy. I can imagine myself as a drop. I’ve seen drops. I’ve seen them on the windshield of our car and I’ve seen them run down the windows of our apartment. But the space outside of the bucket and the space outside of the space surrounding the bucket surpasses the limits of my bucket.
My brain - the human brain - weighs approximately three pounds. Planet Earth weighs about 1,000 trillion metric pounds. I can’t squeeze a 1,000 trillion metric ton planet into a three-pound brain. I can, however, form an image of Planet Earth which will fit nicely into my brain. It’s round, it’s pretty, it’s blue and white, it’s clearly defined against the black of infinite space. That part is easy. Thank you, language.
Some things I can picture, some things I can’t. I can picture Wyoming. I can picture a helicopter flying over Wyoming. I can picture a helicopter hovering over a herd of wild mustangs up north in the Pryor Mountains of Montana but I can’t picture myself floating forever into space. I can picture myself floating, I can even picture space, but I can’t picture endlessness. Is there anyone who can? What did George Clooney feel like in Gravity when he let go of the parachute strap holding both he and Sandra Bullock to the remains of the International Space Station and went floating to his death as he utters his last words to Bullock about the beauty of the sunrise on the Ganges. I don’t mean Clooney, of course, but the fictitious character he was embodying, veteran astronaut Matt Kowalski. Suppose it was real, an actual catastrophe, and these events actually occurred: you let go of a strap and go floating for eternity in space. Your air supply will soon be depleted and you will die what I hope would be a peaceful death. How long might your body go traveling through space? Would it go into orbit? Would internal bacteria survive long enough to eat the flesh and leave a skeleton in the suit? Would it soon by hit by a rock? Torn to pieces by debris? You see what happens: the mind begins adding details, tossing them into this fiction and avoiding the central issue: infinity. The horror of eternity.
Endlessness isn’t an image endlessness is endless abstraction. It’s a philosophical concept whose appearance might take the form of infinitesimal calculus, a Taylor series, the mathematics of continuous change. The mind needs limits to form definitions, contours, meanings. Meanings require shapes, purpose, infinity signs. Endlessness has no meaning because it exceeds all boundary and zone and the ghosts of departed quantities. The river never reaches the ocean. The ocean never ceases heaving itself onto land and receding back into the infinite undulation that is the living manifestation of its being. Water is being. It’s why it has waves. It’s why it splashes and swirls. And it’s everywhere. Water is everywhere. It’s in me. It’s in us. It’s all above us, below us, and all around us. It’s in bugs and wolves and scorpions and centipedes. We’re all carriers of water carrying water from one form of water to another, boiling it, pouring it, drinking it. Floating on it, swimming in it, squirting it. The transformations of water are endless. The movement of its ripples on a pond parallel the words in a sentence that remain separate in sound and movement but are a coherence of moving pattern that results in meaning and emotion.
Last night at a reading I heard a writer refer to a Japanese scientist named Masaru Emoto who has discovered that molecules of water are affected by our thoughts, words, and feelings. Water exhibits properties of molecular coherence, and is the main carrier of all the electric signals our bodies generate. Beethoven’s pastoral symphonies, played between two bottles of water, produced beautiful and well-formed crystals. Mozart’s 40th symphony, a graceful prayer to beauty, “created crystals that were delicate and elegant.” “And the crystals formed by Chopin’s Étude in E, Op 10, No. 3, surprised us with their lovely detail.” I’m assuming that the better the crystal the better the signal produced, leading to a happier, more profound sense of well-being.
But what about heavy metal? What about the rages and hammering rhymes of rap? The big brass sounds of John Philip Sousa’s military marches?
What about polka? What does water do under the influence of polka? Does it Hoop-Dee-Doo? Do the crystals form licorice sticks and peanuts?
What if I sing in the shower? Does the water pelting my body alter its crystals in accordance with “Knock, Knock, Kockin’ on Heaven’s Door?” I just hear it as it gurgles down the drain. It is I who feel changed when I leave the shower. Water always has a soothing effect on my body. It’s like music heard by my skin. I feel like I stepped out of Mozart dripping symphonies of water. I dry myself with an étude and get dressed in a bisbigliando.
The best possible place to get wet is in the comedy of your own lilypond.
Infinity hurts the head. It tastes like totems on the Kwakiutl shore. Are mind and body one? This should not be a question. This should be leaves glossed with rain. A name in the mud written with a stick. Trek to the store for butter under a black umbrella. This is the mind in the body in the rain of a soggy day. And this is a piece of infinity discarded by time and secreted by hope.
The slop of water the honeycombs of bees. Halibuts are angels of circumstance. Schools of smelt in the emerald calm of the sound. It’s there, infinity. You know it, you can feel it, it’s what gives life this particular taste, feeling. Because we appear here, we are brought here, through conduits of fluid, and it’s by fluid we go, turn to vapor and cloud, if that, and so what, who wants to hang around forever? |
b7c2cee4a4cb2bd9 | Resume Reading — Haunted by His Brother, He Revolutionized Physics
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Haunted by His Brother, He Revolutionized Physics
To John Archibald Wheeler, the race to explain time was personal.
T he postcard contained only two words: “Hurry up.”John Archibald Wheeler, a 33-year-old physicist, was in Hanford, Wash., working…By Amanda Gefter
he postcard contained only two words: “Hurry up.”
John Archibald Wheeler, a 33-year-old physicist, was in Hanford, Wash., working on the nuclear reactor that was feeding plutonium to Los Alamos, when he received the postcard from his younger brother, Joe. It was late summer, 1944. Joe was fighting on the front lines of World War II in Italy. He had a good idea what his older brother was up to. He knew that five years earlier, Wheeler had sat down with Danish scientist Niels Bohr and worked out the physics of nuclear fission, showing that unstable isotopes of elements like uranium or soon-to-be-discovered plutonium would, when bombarded with neutrons, split down the seams, releasing unimaginable stores of atomic energy. Enough to flatten a city. Enough to end a war.
After the postcard’s arrival, Wheeler worked as quickly as he could, and the Manhattan Project completed its construction of the atomic bomb the following summer. Over the Jornada del Muerto Desert in New Mexico, physicists detonated the first nuclear explosion in human history, turning 1,000 feet of desert sand to glass. J. Robert Oppenheimer, the project’s director, watched from the safety of a base camp 10 miles away and silently quoted Hindu scripture from the Bhagavad Gita: “Now I am become Death, the destroyer of worlds.” In Hanford, Wheeler was thinking something different: I hope I’m not too late. He didn’t know that on a hillside near Florence, lying in a foxhole, Joe was already dead.
When Wheeler learned the news, he was devastated. He blamed himself. “One cannot escape the conclusion that an atomic bomb program started a year earlier and concluded a year sooner would have spared 15 million lives, my brother Joe’s among them,” he wrote in his memoir. “I could—probably—have influenced the decision makers if I had tried.”
Outsmarting the CERNageddon
It’s a sunny summer day in Geneva, Switzerland. The birds are singing as lovers canoodle near the Jet d’Eau. Somewhere, someone is listening to techno music a little too loudly.Nearby, 570 feet belowground, the Large Hadron Collider, or LHC, hums...READ MORE
Time. As a physicist, Wheeler had always been curious to untangle the nature of that mysterious dimension. But now, in the wake of Joe’s death, it was personal.
Wheeler would spend the rest of his life struggling against time. His journals, which he always kept at hand (and which today are stashed, unpublished, in the archives of the American Philosophical Society Library in Philadelphia), reveal a stunning portrait of an obsessed thinker, ever-aware of his looming mortality, caught in a race against time to answer not a question, but the question: “How come existence?”
“Of all obstacles to a thoroughly penetrating account of existence, none looms up more dismayingly than ‘time,’” Wheeler wrote. “Explain time? Not without explaining existence. Explain existence? Not without explaining time.”
As the years raced on, Wheeler’s journal entries about time grew more frequent and urgent, their lines shakier. In one entry, he quoted the Danish scientist and poet Piet Hein:
“I’d like to know
what this whole show
is all about
before it’s out.”
Before his curtain came down, Wheeler changed our understanding of time more radically than any thinker before him or since—a change driven by the memory of his brother, a revolution fueled by regret.
In 1905, six years before Wheeler was born, Einstein formulated his theory of special relativity. He discovered that time does not flow at a steady pace everywhere for everyone; instead, it’s relative to the motion of an observer. The faster you go, the slower time goes. If you could go as fast as light, you’d see time come to a halt and disappear.
But in the years following Einstein’s discovery, the formulation of quantum mechanics led physicists to the opposite conclusion about time. Quantum systems are described by mathematical waves called wavefunctions, which encode the probabilities for finding the system in any given state upon measurement. But the wavefunction isn’t static. It changes. It evolves in time. Time, in other words, is defined outside the quantum system, an external clock that ticks away second after absolute second, in direct defiance of Einstein.
That’s where things stood—the two theories in a stalemate, the nature of time up in the air—when Wheeler first came onto the physics scene in the 1930s. As he settled into an academic career at Princeton University, Wheeler was soft-spoken and impossibly polite, donning neatly pressed suits and ties. But behind his conservative demeanor lay a fearlessly radical mind. Raised by a family of librarians, Wheeler was a voracious reader. As he struggled with thorny problems in general relativity and quantum mechanics, he consulted not only Einstein and Bohr but the novels of Henry James and the poetry of Spanish writer Antonio Machado. He lugged a thesaurus in his suitcase when he travelled.
Wheeler’s first inkling that time wasn’t quite what it seemed came one night in the spring of 1940 at Princeton. He was thinking about positrons. Positrons are the antiparticle alter egos of electrons: same mass, same spin, opposite charge. But why should such alter egos exist at all? When the idea struck, Wheeler called his student Richard Feynman and announced, “They are all the same particle!”
Imagine there’s only one lone electron in the whole universe, Wheeler said, winding its way through space and time, tracing paths so convoluted that this single particle takes on the illusion of countless particles, including positrons. A positron, Wheeler declared, is just an electron moving backwards in time. (A good-natured Feynman, in his acceptance speech for the 1965 Nobel Prize in Physics, said he stole that idea from Wheeler.)
The puzzle of existence: “I am not ‘I’ unless I continue to hammer at that nut,” wrote John Archibald Wheeler.Corbis Images
After working on the Manhattan Project in the 1940s, Wheeler was eager to get back to Princeton and theoretical physics. Yet his return was delayed. In 1950, still haunted by his failure to act quickly enough to save his brother, he joined physicist Edward Teller in Los Alamos to build a weapon even deadlier than the atomic bomb—the hydrogen bomb. On November 1, 1952, Wheeler was on board the S.S. Curtis, about 35 miles from the island of Elugelab in the Pacific. He watched the U.S. detonate an H-bomb with 700 times the energy of the bomb that destroyed Hiroshima. When the test was over, so was the island of Elugelab.
With his work at Los Alamos complete, Wheeler “fell in love with general relativity and gravitation.” Back at Princeton, just down the street from Einstein’s home, he stood at a chalkboard and gave the first course ever taught on the subject. General relativity described how mass could warp spacetime into strange geometries that we call gravity. Wheeler wanted to know just how strange those geometries could get. As he pushed the theory to its limits, he became fascinated by an object that seemed to turn time on its head. It was called an Einstein-Rosen bridge, and it was a kind of tunnel that carves out a cosmic shortcut, connecting distant points in spacetime so that by entering one end and emerging from the other, one could travel faster than light or backward in time. Wheeler, who loved language, knew that one could breathe life into obscure convolutions of mathematics by giving them names; in 1957, he gave this warped bit of reality a name: wormhole.
As he pushed further through spacetime, he came upon another gravitational anomaly, a place where mass is so densely packed that gravity grows infinitely strong and spacetime infinitely mangled. This, too, he gave a name: black hole. It was a place where “time” lost all meaning, as if it never existed in the first place. “Every black hole brings an end to time,” Wheeler wrote.
As he pushed further through spacetime, Wheeler came upon another gravitational anomaly. This, too, he gave a name: black hole.
n the 1960s, as the Vietnam War tore the fabric of American culture, Wheeler struggled to mend a rift in physics between general relativity and quantum mechanics—a rift called time. One day in 1965, while waiting out a layover in North Carolina, Wheeler asked his colleague Bryce DeWitt to keep him company for a few hours at the airport. In the terminal, Wheeler and DeWitt wrote down an equation for a wavefunction, which Wheeler called the Einstein-Schrödinger equation, and which everyone else later called the Wheeler-DeWitt equation. (DeWitt eventually called it “that damned equation.”)
Instead of a wavefunction describing some system of particles moving around in a lab, Wheeler and DeWitt’s wavefunction described the whole universe. The only problem was where to put the clock. They couldn’t put it outside the universe, because the universe, by definition, has no outside. So while their equation successfully combined the best of both relativity and quantum theory, it also described a universe that couldn’t evolve—a frozen universe, stuck in a single, eternal instant.
Wheeler’s work on wormholes had already shown him that, like electrons and positrons, we too might be capable of bending and breaking time’s arrow. Meanwhile his work on the physics of black holes had led him to suspect that time, deep down, does not exist. Now, at the Raleigh International Airport, that damned equation left Wheeler with a nagging hunch that time couldn’t be a fundamental ingredient of reality. It had to be, as Einstein said, a stubbornly persistent illusion, a result of the fact that we are stuck inside a universe that only has an inside.
Wheeler was convinced the central clue to the puzzle of existence—and in turn of time—was quantum measurement. He saw that the profound strangeness of quantum theory lies in the fact that when an observer makes a measurement, he doesn’t measure something that already exists in the world. Instead, his measurement somehow brings that very thing into existence—a bizarre fact that no one in his right mind would have bought, except that it had been proven again and again with a mind-melting experiment known as the double-slit. It was an experiment that Wheeler could not get out of his head.
In the experiment, single photons are shot from a laser at a screen with two tiny parallel slits, then land on a photographic plate on the other side, where they leave a dot of light. Each photon has a 50/50 chance of passing through either slit, so after many rounds of this, you’d expect to see two big blobs of light on the plate, one showing the pile of photons that passed through slit A and the other showing the pile that passed through slit B. You don’t. Instead you see a series of black and white stripes—an interference pattern. “Watching this actual experiment in progress makes vivid the quantum behavior,” Wheeler wrote. “Simple though it is in concept, it strikingly brings out the mind-bending strangeness of quantum theory.”
As impossible as it sounds, the interference pattern can only mean one thing: each photon went through both slits simultaneously. As the photon heads toward the screen, it is described by a quantum wavefunction. At the screen, the wavefunction splits in two. The two versions of the same photon travel through each slit, and when they emerge on the other side, their wavefunctions recombine—only now they are partially out of phase. Where the waves align, the light is amplified, producing stripes of bright light on the plate. Where they are out of sync, the light cancels itself out, leaving stripes of darkness.
Things get even stranger, however, when you try to catch the photons passing through the slits. Place a detector at each slit and run the experiment again, photon after photon. Dot by dot, a pattern begins to emerge. It’s not the stripes. There are two big blobs on the plate, one opposite each slit. Each photon took only one path at a time. As if it knows it’s being watched.
Photons, of course, don’t know anything. But by choosing which property of a system to measure, we determine the state of the system. If we don’t ask which path the photon takes, it takes both. Our asking creates the path.
Could the same idea be scaled up, Wheeler wondered. Could our asking about the origin of existence, about the Big Bang and 13.8 billion years of cosmic history, could that create the universe? “Quantum principle as tiny tip of giant iceberg, as umbilicus of the world,” Wheeler scrawled in his journal on June 27, 1974. “Past present and future tied more intimately than one realizes.”
In his journal, Wheeler drew a picture of a capital-U for “universe,” with a giant eye perched atop the left-hand peak, staring across the letter’s abyss to the tip of the right-hand side: the origin of time. As you follow the swoop of the U from right to left, time marches forward and the universe grows. Stars form and then die, spewing their carbon ashes into the emptiness of space. In a corner of the sky, some carbon lands on a rocky planet, merges into some primordial goo, grows, evolves until … an eye! The universe has created an observer and now, in an act of quantum measurement, the observer looks back and creates the universe. Wheeler scribbled a caption beneath the drawing: “The universe as a self-excited system.”
The problem with the picture, Wheeler knew, was that it conflicted with our most basic understanding of time. It was one thing for electrons to zip backward through time, or for wormholes to skirt time’s arrow. It was something else entirely to talk about creation and causation. The past flows to the present and then the present turns around and causes the past?
Have to come through to a resolution of these issues, whatever the cost,” Wheeler wrote in his journal. “Nowhere more than here can I try to live up to my responsibilities to mankind living and dead, to [his wife] Janette and my children and grandchildren; to the child that might have been but was not; to Joe…” He glued into the journal a newspaper clipping from The Daily Telegraph. The headline read: “Days are Getting Shorter.”
In 1979, Wheeler gave a lecture at the University of Maryland in which he proposed a bold new thought experiment, one that would become the most dramatic application of his ideas about time: the delayed choice.
Wheeler had realized that it would be possible to arrange the usual double slit experiment in such a way that the observer can decide whether he wants to see stripes or blobs—that is, he can create a bit of reality—after the photon has already passed through the screen. At the last possible second, he can choose to remove the photographic plate, revealing two small telescopes: one pointed at the left slit, the other at the right. The telescopes can tell which slit the photon has passed through. But if the observer leaves the plate in place, the interference pattern forms. The observer’s delayed choice determines whether the photon has taken one path or two after it has presumably already done one or the other.
For Wheeler, this wasn’t a mere curiosity. This was a clue to the universe’s existence. It was the mechanism he needed to get his U-drawing to work, a bending of the rules of time that might allow the universe—one that was born in a Big Bang 13.8 billion years ago—to be created right now. By us.
To see the point, Wheeler said, just take the delayed choice experiment and scale it up. Imagine light traveling toward Earth from a quasar a billion light years away. A massive galaxy sits between the quasar and the Earth, diverting the light’s path with its gravitational field like a lens. The light bends around the galaxy, skirting either left or right with equal probability and, for the sake of the thought experiment, arrives on Earth a single photon at a time. Again we are faced with a similar choice: We can center a photographic plate at the light’s arrival spot, where an interference pattern will gradually emerge, or we can point our telescope to the left or right of the galaxy to see which path the light took. Our choice determines which of two mutually exclusive histories the photon lived. We determine its route (or routes) start to finish, right now—despite the fact that it began its journey a billion years ago.
Listening intently in the audience was a physicist named Carroll Alley. Alley had known Wheeler in Princeton, where he had studied under the physicist Robert Henry Dicke, whose research group had come up with the idea of putting mirrors on the moon.
Dicke and his team were interested in studying general relativity by looking at subtle gravitational interactions between the moon and the Earth, which would require exquisitely accurate measurements of the distance to the moon as it swept along its orbit. They realized if they could put mirrors on the lunar surface, they could bounce lasers off of them and time how long it took the light to return. Alley became the principle investigator of the NASA project and got three mirrors on the moon; the first one was set down in 1969 by Neil Armstrong.
Now, as Alley listened to Wheeler speak, it dawned on him that he might be able to use the same techniques he had used for measuring laser light bouncing off the moon to realize Wheeler’s vision in the lab. The light signals returning from the mirrors on the moon had been so weak that Alley and his team had developed sophisticated ways to measure single photons, which was exactly what Wheeler’s delayed choice setup required.
In 1984, Alley—along with Oleg Jakubowicz and William Wickes, both of whom had also been in the audience that day—finally got the experiment to run. It worked just as Wheeler had imagined: measurements made in the present can create the past. Time as we once knew it does not exist; past does not come indelibly before future. History, Wheeler discovered—the kind that brews guilt, the kind that lies dormant in foxholes—is never set in stone.
Later that year, he wrote, “How come existence? How come the quantum? Is death the penalty for raising such a question?”
Still, some fundamental insight eluded Wheeler. He knew that quantum measurement allowed observers in the present to create the past, the universe hoisting itself into existence by its bootstraps. But how did quantum measurement do it? And if time was not a primordial category, why was it so relentless? Wheeler’s journals became a postcard of their own, written again and again to himself. Hurry up. The puzzle of existence taunted him. “I am not ‘I’ unless I continue to hammer at that nut,” he wrote. “Stop and I become a shrunken old man. Continue and I have a gleam in my eye.”
In 1988, Wheeler’s health was wavering; he had already undergone cardiac surgery two years before. Now, his doctors gave him an expiration date. They told him he could expect to live for another three to five years. Under the threat of his own mortality, Wheeler grew despondent, worried that he would not solve the mystery of existence in time to even the score for what he saw his personal failure to save his brother. Under the heading “Apology,” he wrote in his journal, “It will take years of work to develop these ideas. I—76—don’t have them.”
Luckily, like scientists before them, the doctors had gotten the nature of time all wrong. The gleam in Wheeler’s eye continued to shine, and he hammered away at the mystery of quantum mechanics and the strange loops of time. “Behind the glory of the quantum—shame,” he wrote on June 11, 1999. “Why shame? Because we still don’t understand how come the quantum. Quantum as signal of self-created universe?” Later that year, he wrote, “How come existence? How come the quantum? Is death the penalty for raising such a question—”
Although Wheeler’s journals reveal a driven man on a lonely quest, his influence was widespread. In recent years, Stephen Hawking, along with his collaborator Thomas Hertog of the Institute for Theoretical Physics at the KU Leuven in Belgium, has been developing an approach known as top-down cosmology, a direct descendant of Wheeler’s delayed choice. Just as photons from a distant quasar take multiple paths simultaneously when no one’s looking, the universe, Hawking and Hertog argue, has multiple histories. And just as observers can make measurements that determine a photon’s history stretching back billions of years, the history of the universe only becomes reality when an observer makes a measurement. By applying the laws of quantum mechanics to the universe as a whole, Hawking carries the torch that Wheeler lit that day back at the North Carolina airport, and challenges every intuition we have about time in the process. The top-down approach “leads to a profoundly different view of cosmology,” Hawking wrote, “and the relation between cause and effect.” It’s exactly what Wheeler had been driving at when he drew the eye atop his self-creating universe.
In 2003, Wheeler was still chasing the meaning of existence. “I am as far as can be imagined from being able to speak so reasonably about ‘How come existence’!” he wrote in his journal. “Not much time left to find out!”
On April 13, 2008, in Hightstown, N.J., at the age of 96, John Archibald Wheeler finally lost his race against time. That stubbornly persistent illusion.
Amanda Gefter’s book Trespassing on Einstein’s Lawn—a memoir about physics, her father, the universe, John Wheeler, nothing and everything—is published this month by Bantam.
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2b9ce37999f5ada6 | Monday, 22 February 2016
Macro and Credit - The Monkey and banana problem
"There are three growth industries in Japan: funerals, insolvency and securitization." - Dominic Jones, Asset Finance International Nov., 1998
Looking at the evolution of markets and convolution in which central bankers have fallen, we reminded ourselves for our chosen title analogy of the "Monkey and banana problem", which is a famous toy problem in artificial intelligence, particularly in logic programming and planning. The problem goes as follows:
While there are many applications to this problem. One is as a toy problem for computer science, the other, we think is a "credit impulse" problem" for central bankers. The issue at hand is given financial conditions are globally tightening, as shown as well in the US in the latest publications of the Fed Senior Loan Officer Survey and the bloodbath in European bank shares, the problem is how on earth our central bankers or "monkeys" (yes it's after all the Chinese year of the fire monkey...) are going to avoid the contraction in loan growth? As put it bluntly by our friend Cyril Castelli from Rcube, the European credit channel is at risk as banks' share of credit transmission is much higher in EU than US, which of course is bound to create a negative feedback loop and could therefore stall much needed growth:
- source Rcube - @CyrilRcube
Another possible tongue in cheek purpose of our analogy and problem is to raise the question: Are central bankers intelligent? Of course they are, and most of them have to deal with complete lack of political support (or leadership). It seems we have reached the limits of what monetary policies can do in many instances.
Although both humans and monkeys have the ability to use mental maps to remember things like where to go to find shelter, or how to avoid danger, it seems to us that in recent years central bankers have lost the ability to avoid danger. While monkeys can also remember where to go to gather food and water, as well as how to communicate with each other, it seems to us as of late, that central bankers are losing their ability to communicate, not only with each other but, as well with markets, hence our chosen title. Could it be that monkeys have indeed superior abilities than central bankers given their ability not only to remember how to hunt and gather but to learn new things, as is the case with the monkey and the bananas? Despite the facts that the monkey may never have been in an identical situation, with the same artifacts at hand (printing press), a monkey is capable of concluding that it needs to make a ladder, position it below the bananas, and climb up to reach for them. It seems to us that despite the glaring evidence that the "wealth effect" is not translating into strong positive effects into the "real economy", yet it seems central bankers have decided to all embrace the Negative Interest Rate Policy aka NIRP as the new "banana". One would argue that, to some extent, central bankers have gone "bananas" but we ramble again...
In this week's conversation we will voice our concern relating to the heightened probability of a credit crunch in Europe thanks to banking woes and the unresolved Italian Nonperforming loans issue (NPLs). We will as well look at the credit markets from a historical bear market perspective and muse around the relief rally experienced so far.
• Macro and Credit - The risk of another credit crunch in Europe is real
• Why NIRP matters on the asset side of a bank balance sheet
• Credit spreads and FX movements - Why we are watching the Japanese yen
• Final chart: US corporate sector leverage approaching crisis peak
• Credit - The risk of another credit crunch in Europe is real
The fast deterioration in European bank stocks in conjunction with the rising and justified concerns relating to Italian NPLs, constitute a direct threat to the "credit impulse" needed to sustain growth in Europe we think. As we pointed out on numerous occasions, the ECB and the FED have taken different approaches in tackling their banking woes following the Great Financial Crisis (GFC). In various conversations we have been highlighting the growth differential between the US and Europe ("Shipping is a leading deflationary indicator"):
Exactly. The issue with Italian NPLs is that the tepid Italian growth of the last few years is in no way alleviating the bloated balance sheets of Italian banks which would help sustain credit growth for consumption purposes in Italy as evidently illustrated in a recent Société Générale chart we have used in our conversation the "Vasa ship":
And no earnings thanks to NIRP means now, no reduction in Italian NPLs which according to Euromoney's article entitled "Italy's bad bad bank" from February 2016 have now been bundled up into a new variety of CDOs" - source Macronomics, February 2016
- source Société Générale.
So, all in all, the ECB is going to have to find a way to shift these impaired assets onto its balance sheet, if it wants to swiftly and clearly deal with the worsening Italian situation. While some pundits would point out that the new "bail-in" resolutions in place since the 1st of January are sufficient to deal with such an issue, we do not share their optimism. This is a potential "political" problem of the first order, should the ECB decides to deal with this sizable problem à la Cyprus. Caveat emptor.
You could expect once more politicians and the ECB to somewhat twist the rule book in order to facilitate through this "securitization" process and an ECB take up of part of the capital structure (Senior tranches probably) of these new NPLs CDOs. A new LTRO at this point might once again alleviate funding issues for some but in no way alter the debilitating course of the credit profile of the Italian banks. On a side note we joked in our last conversation around these new NPLs CDOs being the new "Big Short":
But, when it comes to the "credit impulse" and its potential "impairment" in Europe thanks to bloated banks balance sheets, and equities bleeding, we read with interest Deutsche Bank's take in their note Focus Europe note from the 19th of February entitled "Moving down a gear":
"The balance between the growth drivers and detractors is being tipped towards the negative as the questioning of confidence in European banks threatens to result in a less beneficial credit impulse. In last week’s Focus Europe we presented a scenario analysis to demonstrate the sensitivity of euro area GDP growth to the provision of bank credit. We did this via the credit impulse relationship. Our earlier assumption of 1.6% real GDP growth in 2016 was consistent with 2% credit growth. If, on the other hand, banks issue no net new credit this year, domestic demand would fall, confidence deteriorate and financial markets tighten. With no reaction from the ECB, 2016 GDP growth would fall to about 0.5%
The recent fall in bank equities and rise in bank debt costs combined with increasing economic risks, the balance of probabilities suggests that lending standards will tighten relative to what we expected previously. Therefore, to some degree the provision of bank credit, and hence economic growth, will suffer. The revision we are announcing is an attempt to capture this effect. There are considerable uncertainties as to the scale of the problem, but we feel a modestly weaker lending impulse is now a more appropriate baseline.
Credit (-0.2pp). Our previous baseline forecast of 1.6% GDP growth was consistent with an acceleration in bank credit growth from broadly zero in 2015 to about 2% this year. The improvement in credit conditions in the last Bank Lending Survey implied a modest upside risk relative to forecasts. The last Bank Lending Survey was conducted in December and published in January. There were no indications at that point of concern about capital, liquidity or risk. However, as we said above, the balance of probabilities implies that lending standards will now tighten. We are conservatively allowing for a scenario in which the contribution to GDP from bank credit is now 0.2pp weaker than our previous baseline.
The ECB can help minimize the damage…
The onus is on the ECB to achieve two things at the next meeting on 10 March. First, to set an appropriately accommodative policy stance given the worsen outlook for both growth and inflation. Note, since December, our headline and core HICP inflation forecasts for 2016 have fallen from 0.9% and 1.3% to 0.2% and 1.1% respectively (see page 2 for updated country inflation forecasts).
Second, to set a policy stance that does not compound the pressures on a banking system that may be perceived as being more vulnerable.
We presented a detailed discussion of the ECB’s options in last week’s Focus Europe (pages 8-10)2. Suffice to say, the choice of policies will be affected by conditions in the banking system. Our expectation prior to this episode of banking stress in recent weeks was a 10bp deposit rate cut and a temporary acceleration in the pace of purchases. The bank stress implies that a further deposit rate cut may be unwise without a system of exemptions. A refi cut, for example targeted at TLTROs, would be more effective.
The bank stress also implies the ECB should offer some kind of supplementary liquidity tender. Excess liquidity is running at about EUR700bn, but if there was any sense of fragmentation re-emerging between strong and weak banks, it would be in the ECB’s interest to remove all doubt about bank access to liquidity. Finally, the justification for more QE has increased given the widening of credit and sovereign spreads. More QE would reduce the risk of a negative feedback loop between banks and sovereigns.
The ECB is conducting a technical review of the asset purchase programme (APP). This might result in some changes. In terms of broadening the eligible asset base, we suspect the ECB will remain in the sovereign/quasi-sovereign space for now. Corporate bonds are possible but not very impactful.
Purchasing unsecured bank debt might not be inconsistent with the Treaty but would be complex and politically controversial. Stresses would have to increase markedly to bring this option onto the table.
There is no relaxation of regulation, however. Both Mario Draghi, President of the ECB, and Daniele Nouy, head of the ECB Single Supervisory Mechanism (SSM), were consistent in their messages this week that (a) the new regulatory regime is resulting is a more stable and sustainable banking system - there is no sense that regulation is a net cost - and (b) “all else unchanged” there will be no significant additional capital requirements imposed on banks.
Benoit Coeure, ECB Executive Board Member, said that if bank profits are under pressure the onus would be on governments to implement structural reforms and growth-friendly fiscal policies. In short, no change in ECB message.
…but negative feedback loops cannot be ruled out either
Last week we showed how sensitive the economic cycle can be to the bank credit cycle. This negative dynamic can become self-reinforcing. One direction is via the private sector and another is through the public sector. A tightening of lending standards weakens demand, undermining growth and asset quality, triggering a second-order tightening in credit. At the same time, weaker demand undermines sovereign sustainability which can tightening bank funding cost and additionally contribute to second-order tightening.
Fiscal dynamics have deteriorated. The primary balance gap is the difference between the debt-stabilising primary balance and the primary balance. It captures the underlying dynamic of the public debt-to-GDP ratio. A negative gap means the public debt ratio is falling. Our previous forecast was for a primary balance gap of -0.6% of GDP in 2016, the first genuine decline in the public debt ratio since the start of the crisis. Following the growth and inflation revisions, the primary balance gap is expected to be positive again. In other words, the benefits of lower funding rates thanks to ECB QE are not enough to compensate for the loss of economic momentum. Moreover, if the scenario of zero net new bank credit were to materialize, 2016 could see the primary balance gap rise back to levels not seen since 2012. That would imply a euro area public debt-to-GDP ratio of about 98%." - source Deutsche Bank
Whereas indeed we are waiting to Le Chiffre aka Mario Draghi to come up with new tricks in March to alleviate the renewed pressure on European banks, where we disagree with Deutsche Bank is that providing a new TLROs would provide much needed support for funding in a situation where some banking players are seeing their cost of capital rise thanks to a flattening of their credit curve (in particular Deutsche Bank as per our previous conversation), this intervention would in no way remove the troubled growing impaired assets from the likes of Italian banks. It is one thing to deal with the flow (funding) and another entirely to deal with the stocks (impaired assets). Whereas securitization of the lot seems to be latest avenue taken, you need to find a buyer for the various tranches of these new NPLs CDOs. Also more QE. will not deal with the stocks of impaired assets unless these assets are purchased directly by the ECB.
When it comes to credit conditions in Europe, not only do we closely monitor the ECB lending surveys, we also monitor on a monthly basis the “Association Française des Trésoriers d’Entreprise” (French Corporate Treasurers Association) surveys. In their latest survey, while it is difficult to assess for now a clear trend in the deterioration of financial conditions for French corporate treasurers, it appears to us that NIRP has already been impacting the margin paid on credit facilities given a small minority of French corporate treasurers are indicating that, since December, there is an increasing trend in margin paid on the aforementioned credit facilities:
"Does the margin paid on your credit facilities has a tendency, to rise, fall or remains stable?"
- source AFTE
Going forward we will closely be monitoring these additional signals coming from French corporate treasurers to measure up the impact on the overall financial conditions as well as the impact NIRP has on the margins they are getting charged on their credit facilities. For now conditions for French corporate treasurers do not warrant caution on the microeconomic level.
While we have recently indicated our medium to long term discomfort with the current state of affaires akin to 2007 in terms of credit markets, the recent "relief rally" witnessed so far is for us a manifestation of the "overshoot" we discussed last week. Some welcome stabilization was warranted, yet we do feel that the credit cycle has turned and that you should be selling into strength and move towards more defensive position, higher into the rating quality spectrum that is and rise your cash levels.
When it comes to enticing banks to "lend" more, as far as our analogy is concerned we wonder how the "monkey" bankers are going to react if indeed additional NIRP is going to remove more "bananas".
This brings us to our second point, namely that the flatter the yield curve, the less effective NIRP is.
Whereas, Europe overall has been moving more into the NIRP phenomenon, with over $7 trillion worth of global government bonds now yielding “less” than zero percent, the FED in the US is weighting joining the NIRP club in 2016 apparently, NIRP being vaunted as the new "banana" tool in the box to stimulate the "monkeys".
The issue of course at hand is that NIRP does matter and particularly when it comes to the asset side of a bank balance sheet as put forward by Deutsche Bank in their note from the 22nd of February entitled "Three things the market taught me this year":
"Negative rates – much more complicated
Negative rates look powerful at face value. Bank profits can be protected by exempting excess liquidity while market rates are pushed down. The turmoil in Japan points to three considerations that mute this view.
First, the impact of negative rates on the asset side of banks’ balance sheet can matter much more than the charge on excess liquidity. Banks that own large amounts of fixed income assets relative to the size of their total balance sheet (and their excess liquidity) are hit the hardest as returns on these assets drop. Japan and the US stand out as economies where the cost to the banks is biggest, Switzerland and Sweden the least, while Europe is somewhere in between:
Second, super-flat yield curves reduce the impact of negative rates. When bonds don’t offer risk premia, a perfect Keynesian liquidity trap exists: fixed income is the same as cash, and negative rates instantaneously transmit to the entire yield curve. The portfolio rebalancing into riskier assets declines as the marginal holders of zero-yielding bonds are naturally risk-averse. Japan’s yield curve, the flattest in the world, failed to steepen when the BoJ cut rates earlier this month – all yields just shifted down. Sweden, the UK and Europe stand out as yield curves where there’s still more risk premium to be squeezed, Japan, Canada and Norway the least:
Third, sub-zero rates can send a negative forward-looking signal. Until the technological and institutional framework is designed to pass negative rates to depositors without triggering banknote withdrawal, there will eventually be a (negative) lower bound. As this is approached the signaling cost of easing being exhausted may be bigger than the benefit of lower rates. At the extreme cash and bonds turn into “Giffen goods”: the substitution effect of lower return is more than offset by expected lower future income. Lower rates then end up raising, rather than lowering the demand for bonds as the saving rate goes up. The limitations of additional BoJ easing in addition to changing Japanese hedging behaviour, are some of the factors, that have led us to revise our USD/JPY forecasts for this year. We now think 2015 marked the peak in USD/JPY for this cycle and forecast a move down to as low as 105 this year." - source Deutsche Bank
Exactly, this negative feedback-loop, doesn't stop the frenzy for bonds and the "over-allocation process. On the contrary, as the "yield frenzy" gather pace thanks to NIRP. This push yields lower and bond prices even higher. This is exactly what we discussed in our conversation "Le Chiffre" in October 2015:
- source Deutsche Bank (h/t Tracy Alloway).
Furthermore the significant repricing in European equities (where many pundits had been "overweight" at the beginning of the year) has led to a significant switch from equities to bonds as indicated by Bloomberg in their article from the 22nd of February entitled "They'd Rather Get Nothing in Bonds Than Buy Europe Stocks":
• "Estimates for Euro Stoxx 50 dividend yield at 4.3 percent
• The region's government debt is yielding 0.6 percent
The cash reward for owning European stocks is about seven times larger than for bonds. Investors are ditching the equities anyway.
Even with the Euro Stoxx 50 Index posting its biggest weekly rally since October, managers pulled $4.2 billion from European stock funds in the period ended Feb. 17, the most in more than a year, according to a Bank of America Corp. note citing EPFR Global. The withdrawals are coming even as corporate dividends exceed yields on fixed-income assets by the most ever:
Investors who leaped into stocks during a similar bond-stock valuation gap just four months ago aren’t eager to do it again: an autumn equity rally quickly evaporated come December. A Bank of America fund-manager survey this month showed cash allocations rose to a 14-year high and expectations for global growth are the worst since 2011.
If anything, the valuation discrepancy between stocks and bonds is likely to get wider, said Simon Wiersma of ING Groep NV.
“The gap between bond and dividend yields will continue expanding,” said Wiersma, an investment manager in Amsterdam. “Investors fear economic growth figures. We’re still looking for some confirmations for the economic growth outlook.”
Dividend estimates for sectors like energy and utilities may still be too high for 2016, Wiersma says. Electricite de France SA and Centrica Plc lowered their payouts last week, and Germany’s RWE AG suspended its for the first time in at least half a century. Traders are betting on cuts at oil producer Repsol SA, which offers Spain’s highest dividend yield.
With President Mario Draghi signaling in January that more European Central Bank stimulus may be on its way, traders have been flocking to the debt market. The average yield for securities on the Bloomberg Eurozone Sovereign Bond Index fell to about 0.6 percent, and more than $2.2 trillion -- or one-third of the bonds -- offer negative yields. Shorter-maturity debt for nations including Germany, France, Spain and Belgium have touched record, sub-zero levels this month." - source Bloomberg
In that instance, while the equity "banana" appears more enticing from a "yield" perspective, it seems that the "electric shock" inflicted to our investor "monkey" community has no doubt change their "psyche".
When it comes to our outlook and stance relating to the credit cycle we would like to point out again towards chapter 5 of Credit Crisis authored by Dr Jochen Felsenheimer and Philip Gisdakis where they highlight the work of Hyman Minsky's work on the equity-debt cycle and particularly in the light of the Energy sector upcoming bust:
"His cyclical theory of financial crises describes the fragility of financial markets as a function of the business cycle. In the aftermath of a recession, firms finance themselves in a very safe way. As the economy grows and expected profits rise, firms take on more speculative financing, anticipating profits and that loans can be repaid easily. Increased financing translates into rising investment triggering further growth of the economy, making lenders confident that they will receive a decent return on their investments. In such a boom period, lenders tend to abstain from guarantees of success, i.e; reflected in less covenants or in rising investments in low-quality companies. Even if lenders knowsthat the firms are not able to repay their debt, they believe these firms will refinance elsewhere as their expected profits rise. While this is still a positive scenario for equity markets, the economy has definitely taken on too much credit risk. Consequently the next stage of the cycle is characterized by rising defaults. This translates into tighter lending standards of banks. Here, the similarities to the subprime turmoil become obvious. Refinancing becomes impossible especially for lower-rated companies and more firms default. This is the beginning of a crisis in the real economy, while during the recession, firms start to turn to more conservative financing and the cycle closes again" - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
The issue with NIRP and the relationship between credit spreads and safe-haven yields is that while the traditional pattern being lower government bond yields and a flatter yield curve is accompanied by wider spreads in "risk-off" scenarios, given more and more pundits (such as Hedge Funds) have been playing the total return game, they have become less and less dependent on traditional risk-return optimized approach as they are less dependent on movements on the interest rate side. The consequence for this means that classical theories based on allocation become more and more challenged in a NIRP world because correlation patterns change in a crisis period particularly when correlations are becoming more and more positive (hence large standard deviations move).
But if indeed, the behavior of credit is affected in relation to safe-haven yields by changes in correlations, then you might rightly ask yourself about the relationship of credit spreads and FX movements given the focus as of late has been around the surge in the US dollar and the fall in oil prices in conjunction of the rise of the cost in capital since mid 2014.
In our next point we think that, from a credit perspective, once more you should focus your attention on the Japanese yen.
However, NIRP doesn't reduce the cost of capital. NIRP is a currency play. This is clearly the case in Japan and has been well described in Deutsche Bank's note from the 22nd of February entitled "Yen hedging cycle risks rapid reverse":
"A lot of negative things have been said about negative rates, not least in Japan. Negative rates do not work by reducing financing costs materially, providing a ‘price of money’ stimulus. If negative rates do support activity, they primarily work through the exchange rate, adding to the portfolio substitution into risky asset.
For Japan the biggest problem is that macro policies have been directing capital toward risky assets for the last 4+ years. There are inevitably diminishing returns to this strategy, not least because ‘value’ matters. Value matters when it comes to the underlying domestic and foreign ‘risky’ asset, and the value of the exchange rate. Specifically on the latter, the yen even after the recent appreciation is still close to 20% cheap in PPP terms.
It is also cheap on a FEER and BEER basis, helped by a terms of trade shock that is seen lifting the Current Account surplus to near 5% of GDP in 2016. Figure 2 shows that in the last year, Japan has had the most favorable terms of trade shock of any major economy.
While FDI can recycle up to half of the C/A surplus, the question is whether other BoP components, notably net portfolio flows, will do the rest of the recycling, and at what exchange rate. For much of 2013 – H1 2015, vehicles like the GPIF were used to recycle (a much smaller) C/A surplus, that even briefly went into deficit in 2014. By June 2015, the GPIF’s portfolio of riskier assets inclusive of domestic stocks (22.3%), international bonds (13.1%), and international stocks (22.3%) was well within the desired base/benchmark ranges (see Figure 3).
For the latest data available for Q3 2015, GPIF domestic and international equity holdings declined led by weaker equity prices and a stronger yen – price action that underscores the risky nature of these investments.
As the C/A surplus grows and the above ‘structural’ pension shift toward capital flows abroad diminishes, there is a danger that we have already entered the realm where yen strength becomes self-fulfilling, as many of the hedging activities that were associated with a weak yen in the first four years of Abeconomics go into reverse.
Prior to Abeconomics, hedging on Japan equity flows was limited. Since 2011 when BOJ policy encouraged yen weakness, foreign inflows into Japan equities typically included much higher currency hedge ratios, while fully hedged instruments became popular. Precise numbers are not available, but it is estimated that as much as a quarter of the stock of foreign holdings of Japan equities of Y183tr has a currency hedge – a hedge that quickly becomes much less attractive with a stronger yen.
In contrast, for Japan investments abroad, FX hedge ratios declined. This particularly relates to USD investments, where expectations of USD gains increased rapidly in the Abeconomics years, and FX hedges on USD investments dropped. At the end of Q3 2015, Japan had a total of Y770trillion non-Central Banks assets abroad, inclusive of Y418tr portfolio assets, of which Y153tr are equity and investment fund shares, and Y266tr are debt securities. Even if much of the investment abroad has only limited hedges, one observation is that a very small adjustment in hedge ratios can have a huge flow impact. A shift in the hedge ratio on foreign fixed income assets by 10% is roughly equivalent to a year’s C/A surplus. Secondly, to the extent that hedge ratios are very low, as is the case for, say, the GPIF, there are sizable potential losses for funds recently adding to foreign exposure, and an emerging disincentive to invest in the most risky assets abroad.
Of the large players that actively hedge FX exposure, the life insurance companies’ activities can most closely be tracked through quarterly statements, and the time series can provide a useful standard to benchmark recent activity. Life insurance companies as of Q3 2015 had some Y65tr in foreign securities. As per Figure 4, their currency hedge ratios on dollar-based investments are estimated to have dropped to ~46% by the end of Q3 2015, the lowest levels recorded since the Great Financial Crisis in 2008. The hedge ratio is down from a peak of 79% in September 2009.
Life insurance company hedge ratios have likely reached a cycle nadir at the end of 2015, as concerns about JPY appreciation start to rise.
Among the other largest participants that have foreign portfolios of comparable size to the Lifers, both Toshins (foreign securities of ~77tr) and particularly public pension funds ( ~ Y57tr foreign securities) have very low currency hedge ratios and are heavily exposed to currency risk. Japan investments abroad, so actively encouraged by policymakers, are slowly being shown to have a familiar ‘catch’ – interest parity! Nominal yields may be more attractive abroad, but the long-term currency risks are enormous, at least when placed in the context of a yen that is still significantly undervalued.
A crucial element of hedging activity is the expected exchange rate. Here three bigger macro forces are at play for the remainder of 2016: i) Japan/BOJ policy; ii) the Fed; and iii) China FX policy. Firstly on BOJ intervention, the market should not expect any official BOJ intervention barring extreme FX volatility. It would run counter to G20 rules and risk a serious rift with the US. On rates policy, adding significantly to NIRP looks increasingly unpalatable, with our Tokyo Economics team expecting only one more 10bp cut in Q3. The next set of actions will likely need to revolve around ‘qualitative QE’ and the buying of more risky assets, notably securitized products.
On the Fed, we expect USD/JPY to remain sensitive to Fed expectations, but not to the point where more Fed tightening is likely to lead to new USD/JPY highs. The yen has a history of doing well in 5 of the last 7 Fed tightening cycles, although it did weaken in the two big USD upswings." - source Deutsche Bank
As we posited in our conversation "Information cascade" back in March 2015, you should very carefully look at what the GPIF, and their friends are doing:
"Go with the flow:
One should closely watch Japan's GPIF (Government Pension Investment Fund) and its $1.26 trillion firepower. Key investor types such as insurance companies, pension funds and toshin companies have been significant net buyers of foreign assets." - source Macronomics, March 2015
We also added more recently in our conversation "The Ninth Wave" the following:
"So, moving on to why US high quality Investment Grade credit is a good defensive play? Because of attractiveness from a relative value perspective versus Europe and as well from a flow perspective. The implementation of NIRP by the Bank of Japan will induced more foreign bonds buying by the Japanese Government Pension Investment Fund (GPIF) as well as Mrs Watanabe (analogy for the retail investors) through their Toshin funds. These external source of flows will induce more "financial repression" on European government yield curves, pushing most likely in the first place German Bund and French OATs more towards negative territory à la Swiss yield curve, now negative up to the 10 year tenor.
When it comes to Mrs Watanabe, Toshin funds are significant players and you want to track what they are doing, particularly in regards to the so-called "Uridashi" funds. The Japanese levered "Uridashi" funds (also called "Double-Deckers") used to have the Brazilian Real as their preferred speculative currency. Created in 2009, these levered Japanese products now account for more than 15 percent of the world’s eighth-largest mutual-fund market and funds tied to the real accounted previously for 46 percent of double-decker funds in 2009 with close to a record 80% in 2010 and now down to only 22.8%.
As our global macro "reverse osmosis" theory has been playing out, so has been the allocation to the US dollar in selection-type Toshin
Because GPIF and other large Japanese pension funds as well as retail investors such as Mrs Watanabe are likely to increase their portfolios into foreign assets, you can expect them to keep shifting their portfolios into foreign assets, meaning more support for US Investment Grade credit, more negative yields in the European Government bonds space with renewed buying thanks to a weaker "USD/JPY" courtesy of NIRP." - source Macronomics, January 2016
So from a "flow" perspective and like any trained "monkey" looking to reach out for "bananas", at least the slippery type, whereas other "monkeys" are focusing on the US dollar and Oil related woes, we'd rather for now focus our attention onto the Japanese yen, and the allocation implications of a stronger yen. For us, like others, a PBOC devaluation move on the Yuan would send a deflationary impulse worldwide but, in terms of risk assets, it would have serious consequences on Japanese asset allocations and would lead to an acceleration in capital repatriation (this would mean liquidation of some existing positions rest assured) as indicated in Deutsche Bank's note:
"Even modest JPY gains against the USD should translate to a strong yen against all the other G10 currencies and EMG Asia FX, not least because of global macro risks elsewhere. Nothing is capable of lifting the yen trade weighted index more than a speed up in the Rmb’s depreciation rate, leading to knock-on devaluations in EM Asia. This risk alone should encourage higher Japan hedge ratios for investment abroad, inclusive of the stock of Japan FDI assets abroad. A risk-off China shock would tend to concentrate JPY gains against currencies of other G4, but initially would likely include additional yen strength against all currencies. It should also drive the Nikkei sharply lower.
The Nikkei and yen have a long and sometimes tortured history of moving in lock-step. A stronger yen has hurt the Nikkei for obvious reasons, but a weaker Nikkei also tends to lead to a repatriation of capital and a stronger yen. Interestingly, the current Nikkei levels are already consistent with a USD/JPY below Y105." - source Deutsche Bank
When it comes to the year of the Fire Monkey, the slippery banana type, no doubt could come from Japanese investors hurt by the violent appreciation of the Japanese yen, which has indeed been a significant "sucker punch" when it comes to the large standard deviation move experienced by the Japanese yen versus the US dollar. If Mrs Watanabe goes into "liquidation" mode, things could indeed become interesting to say the least.
When it comes to Minsky and the equity-credit cycle, whereas central banks can affect the amplitude and the duration of the cycle, in no way can they alter the character of the cycle. In our final chart, we once again indicate our 2007 feeling thanks to the rise in leverage, tightening financial conditions with the issuance markets closing down on the weaker players, which bode poorly from a risk-reward perspective.
• Final chart: US corporate sector leverage approaching crisis peak
Like many pundits, we have voiced our concerns on the increasing leverage thanks to buybacks financed by debt issuance and the lack of the use of proceeds for investment purposes. Our final chart comes from the same Deutsche Bank note from the 22nd of February entitled "Three things the market taught me this year" quoted previously and displays the US corporate sector leverage which is approaching crisis peak:
"US deleveraging – not that great
US consumer deleveraging stands out as one of the major achievements of the Yellen Fed. Yet the corporate picture looks much less impressive. Total amount
of US corporate debt has approached the highs seen in the financial crisis (chart 3).
Not only that but the bulk of the leverage has been directed towards corporate stock buybacks (chart 4), explaining how low investment but high borrowing have existed at the same time. Persistent volatility in the US credit market has highlighted vulnerabilities that weren’t a concern last year.
- source Deutsche Bank
While a respite is always welcome, when it comes to the rally seen recently, as far as the Monkey and banana problem is concerned as everyone is hoping from additional tricks from our "Generous gamblers" aka our central bankers, this rally might have some more room ahead but then again, it doesn't change our belief in the stage of the credit cycle and our focus on what's the Japanese yen will be doing.
"Life is full of banana skins. You slip, you carry on." - Daphne Guinness, British artist
Stay tuned!
Sunday, 14 February 2016
Macro and Credit - The disappearance of MS München
"Hope, the best comfort of our imperfect condition." - Edward Gibbon, English historian
While thinking about correlations in particular and risk in general, we reminded ourselves of one of our pet subject we have touched in different musings, namely the fascinating destructive effect of "Rogue waves". It is a subject we discussed in details, particularly in our post "Spain surpasses 90's perfect storm":
"We already touched on the subject of "Rogue Waves" in our conversation "the Italian Peregrine soliton", being an analytical solution to the nonlinear Schrödinger equation (which was proposed by Howell Peregrine in 1983), and being as well "an attractive hypothesis" to explain the formation of those waves which have a high amplitude and may appear from nowhere and disappear without a trace, the latest surge in Spanish Nonperforming loans to a record 10.51% and the unfortunate Sandy Hurricane have drawn us towards the analogy of the 1991 "Perfect Storm".
Generally rogues waves require longer time to form, as their growth rate has a power law rather than an exponential one. They also need special conditions to be created such as powerful hurricanes or in the case of Spain, tremendous deflationary forces at play when it comes to the very significant surge in nonperforming loans.", source Macronomics, October 2012
You might already asking yourselves why our title and where we are going with all this?
The MS München was a massive 261.4 m German LASH carrier of the Hapag-Lloyd line that sank with all hands for unknown reasons in a severe storm in December 1978. The most accepted theory is that one or more rogue waves hit the München and damaged her, so that she drifted for 33 hours with a list of 50 degrees without electricity or propulsion. The München departed the port of Bremerhaven on December 7, 1978, bound for Savannah, Georgia. This was her usual route, and she carried a cargo of steel products stored in 83 lighters and a crew of 28. She also carried a replacement nuclear reactor-vessel head for Combustion Engineering, Inc. This was her 62nd voyage, and took her across the North Atlantic, where a fierce storm had been raging since November. The München had been designed to cope with such conditions, and carried on with her voyage. The exceptional flotation capabilities of the LASH carriers meant that she was widely regarded as being practically unsinkable (like the Titanic...). That was of course until she encountered "non-linear phenomena such as solitons.
While a 12-meter wave in the usual "linear" model would have a breaking force of 6 metric tons per square metre (MT/m2), although modern ships are designed to tolerate a breaking wave of 15 MT/m2, a rogue wave can dwarf both of these figures with a breaking force of 100 MT/m2. Of course for such "freak" phenomenon to occur, you need no doubt special conditions, such as the conjunction of fast rising CDS spreads (high winds), global tightening financial conditions and NIRP (falling pressure towards 940 MB), as well as rising nonperforming loans and defaults (swell). So if you think having a 99% interval of confidence in the calibration of you VaR model will protect you againtst multiple "Rogue Waves", think again...
Of course the astute readers would have already fathomed between the lines that our reference to the giant ship MS München could be somewhat a veiled analogy to banking giant Deutsche Bank. It could well be...
But given our recent commentaries on the state of affairs in the credit space, we thought it would be the right time to reach again for a book collecting dust since 2008 entitled Credit Crisis authored by Dr Jochen Felsenheimer (which we quoted on numerous occasions on this very blog for good reasons) and Philip Gisdakis.
Before we go into the nitty gritty of our usual ramblings, it is important we think at this juncture to steer you towards chapter 5 entitled "The Anatomy of a Credit Crisis" and take a little detour worth our title analogy to "Rogue Waves" which sealed the fate of MS München. What is of particular interest to us, in similar fashion to the demise of the MS München is page 215 entitled "LTCM: The arbitrage saga" and the issue we have discussing extensively which is our great discomfort with rising positive correlations and large standard deviations move. This amounts to us as increasing rising instability and the potential for "Rogue Waves" to show up in earnest:
"LTCM's trading strategies generally showed no or almost very little correlation. In normal times or even in crises that are limited to a specific segment, LTCM benefited from this high degree of diversification. Nevertheless, the general flight to liquidity in 1998 caused a jump in global risk premiums, hitting the same direction. All (in normal times less-correlated) positions moved in the same direction. Finally, it is all about correlation! Rising correlations reduces the benefit from diversification, in the end hitting the fund's equity directly. This is similar with CDO investments (ie, mezzanine pieces in CDOs), which also suffer from a high (default) correlation between the underlying assets. Consequently, a major lesson of the LTCM crisis was that the underlying Covariance matrix used in Value-at-Risk (VaR) analysis is not static but changes over time." - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
You might probably understand by now from our recent sailing analogy (The Vasa ship) and wave analogy (The Ninth Wave) where we are heading: A financial crisis is more than brewing.
Moving back to the LTCM VaR reference, the Variance-Covariance Method assumes that returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation. Value-at-Risk (VaR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence.
LTCM and the VaR issue reminds us of a regular quote we have used, particularly in May 2015 in our conversation "Cushing's syndrome":
To that effect and in continuation to Martin Hutchinson's LTCM reference, we would like to repeat the quote used in the conversation "The Unbearable Lightness of Credit":
Today investors face the same "optimism bias" namely that they overstate their ability to exit.
So what is VaR really measuring these days?
This what we had to say about VaR in our May 2015 conversation "Cushing's syndrome" and ties up nicely to our world of rising positive correlations. Your VaR measure doesn't measure today your maximum loss, but could be only measuring your minimum loss on any given day. Check the recent large standard deviation moves dear readers such as the one on the Japanese yen and ask yourself if we are anymore in a VaR assumed "normal market" conditions:
"On a side note while enjoying a lunch with a quant fund manager friend of ours, we mused around the ineptness of VaR as a risk model. When interviewing fellow quants for a position within his fund, he has always asked the same question: What does VaR measures? He always get the same answer, namely that VaR measures the maximum loss at any point during the period. VaR is like liquidity, it is a backward-looking yardstick. It does not measure your maximum loss at any point during the period but, in today "positively correlated markets" we think it measures your "minimum loss" at any point during the period as it assumes "normal" markets. We are not in "normal" markets anymore rest assured." - source Macronomics, May 2015
Therefore this week's conversation we will look at what positive correlations entails for risk and diversification and also we will look at the difference cause of financial crisis and additional signs we are seriously heading into one like the MS München did back in 1978, like we did in 2008 and like we are most likely heading in 2016 with plenty of menacing "Rogue Waves" on the horizon. So fasten your seat belt for this long conversation, this one is to be left for posterity.
• Credit - The different types of credit crises and where do we stand
• A couple of illustrations of on-going nonlinear "Rogue Waves" in the financial world of today
• The overshooting phenomenon
• The Yuan Hedge Fund attack through the lense of the Nash Equilibrium Concept
Rising positive correlations, are rendering "balanced funds" unbalanced and as a consequence models such as VaR are becoming threatened by this sudden rise in non-linearity as it assumes normal markets. The rise in correlations is a direct threat to diversification, particularly as we move towards a NIRP world:
When it comes to the classification of credit crises and their potential area of origins both the authors for the book "Credit Crisis" shed a light on the subject:
• "Currency crisis: A speculative attack on the exchange rate of a currency which results in a sharp devaluation of the currency; or it forces monetary authorities to intervene in currency markets to defend the currency (eg. by sharply hiking interest rates).
• Foreign Debt Crisis: a situation where a country is not able to service its foreign debt.
• Banking crisis: Actual or potential bank runs. Banks start to suspend the internal convertibility of their liabilities or the government has to bail out the banks.
• Systemic Financial crisis: Severe disruptions of the financial system, including a malfunctioning of financial markets, with large adverse effect on the real economy. It may involves a currency crisis and also a banking crisis, although this is not necessarily true the other way around.
In many cases, a crisis is characterized by more than one type, meaning we often see a combination of at least two crises. These involve strong declines in asset values, accompanied by defaults, in the non-financials but also in the financials universe. The effectiveness of government support or even bailout measures combined with the robustness of the economy are the most important determinants of the economy's vulneability, and they therefore have a significant impact on the severity of the crisis. In addition, a crucial factor is obviously the amplitude of asset price inflation that preceded the crisis.
We classify a credit crisis as something between a banking crisis and a systematic financial crisis. A credit crisis affects the banking system or arises in the financial system; the huge importance of credit risk for the functioning of the financial system as a whole bears also a systematic component. The trigger event is often an exogenous shock, while the pre-credit crisis situation is characterized by excessive lending, excessive leverage, excessive risk taking, and lax lending standards. Such crises emerge in periods of very high expectations on economic development, which in turns boosts loan demand and leverage in the system. When an exogenous shock hits the market, it triggers an immediate repricing of the whole spectrum of credit-risky assets, increasing the funding costs of borrowers while causing an immense drop in the asset value of credit portfolios.
A so-called credit crunch scenario is the ugliest outcome of a credit crisis. It is characterized by a sharp reduction of lending activities by the banking sector. A credit crunch has a severe impact on the real economy, as the basic transmission mechanism of liquidity (from central banks over the banking sector to non-financial corporations) is distorted by the fact that banks do a liquidity squeeze, finally resulting in rising default rates. A credit crunch is a full-fledged credit crisis, which includes all major ingredients for a banking and a systemic crisis spilling over onto several parts of the financial market and onto the real economy. A credit crunch is probably the most costly type of financial crisis, also depending on the efficiency of regulatory bodies, the shape of the economy as a whole, and the health of the banking sector itself." - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
The exogenous shock started in earnest in mid-2014 which saw a conjunction of factors, a significant rise in the US dollar that triggered the fall in oil prices, the unabated rise in the cost of capital.
If we were to build another schematic of the current market environment, here what we think it should look like to name a few of the issues worth looking at:
- source Macronomics
So if you think diversification is a "solid defense" in a world of "positive correlations", think again, because here what the authors of "Credit Crisis" had to say about LTCM and tail events (Rogue Waves):
"Even if there are arbitrage opportunities in the sense that two positions that trade at different prices right now will definitely converge at a point in the future, there is a risk that the anomaly will become even bigger. However typically a high leverage is used for positions that have a skewed risk-return profile, or a high likelihood of a small profit but a very low risk of a large loss. This equals the risk-and-return profile of credit investments but also the risk that selling far-out-of-the-money puts on equities. In case of a tail event occurs, all risk parameters to manage the overall portfolio are probably worthless, as correlation patterns change dramatically during a crisis. That said, arbitrage trades are not under fire because the crisis has an impact on the long-term-risk-and-return profile of the position. However, a crisis might cause a short-term distortion of capital market leading to immense mark-to-market losses. If the capital adequacy is not strong enough to offset the mark-to-market losses, forced unwinding triggers significant losses in arbitrage portfolios. The same was true for many asset classes during the summer of 2007, when high-quality structures came under pressure, causing significant mark-to-market losses. Many of these structures did not bear default risk but a huge liquidity risk, and therefore many investors were forced to sell." source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
You probably understand by now why we have raised the "red flag" so many times on our fear in the rise of "positive correlations". They do scare us, because they entail, larger and larger standard deviation moves and potentially trigger "Rogue Waves" which can wipe out even the biggest and most reputable "Investment ships" à la MS München.
The big question is not if we are in a bubble again but if this "time it's different". It is not. It's worse, because you have all the four types of crisis evolving at the same time.
Here is what Chapter 5 of "Credit Crisis" is telling us about the causes of the bubble:
"A mainstream argument is that the cause of the bubbles is excessive monetary liquidity in the financial system. Central banks flood the market with liquidity to support economic growth, also triggering rising demand for risky assets, causing both good assets and bad assets to appreciate excessively beyond their fundamentally fair valuation. In the long run, this level is not sustainable, while the trigger of the burst of the bubble is again policy shifts of central banks. The bubble will burst when central banks enter a more restrictive monetary policy, removing excess liquidity and consequently causing investors to get rid of risky assets given the rise in borrowing costs on the back of higher interest rates.
This is the theory, but what about the practice? The resurfacing discussion about rate cuts in the United States and in the Euroland in mid-2005 was accompanied by expectations that inflation will remain subdued. Following this discussion, the impact of inflation on credit spreads returned to the spotlight. An additional topic regarding inflation worth mentioning is that if excess liquidity flows into assets rather than into consumer goods, this argues for low consumer price inflation but rising asset price inflation. In late 2000, the Fed and the European Central Banks (ECB) started down a monetary easing path, which was boosted by external shocks (9/11 and the Enron scandal), when central banks flooded the market with additional liquidity to avoid a credit crunch. Financial markets benefited in general from this excess liquidity, as reflected in the positive performance of almost all asset classes in 2004, 2005, and 2006, which argued for overall liquidity inflows but not for allocation shifts. It is not only excess liquidity held by investors and companies that underpins strong performing assets in general, but also the pro-cyclical nature of banking. In a low default rate environment, lending activities accelerate, which might contribute to an overheating of the economy accompanied by rising inflation. From a purely macroeconomic viewpoint, private households have two alternatives to allocate liquidity: consuming or saving. The former leads to rising price inflation, whereas the latter leads to asset price inflation." - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
Where we slightly differ from the author's take in terms of liquidity allocation is in the definition of "saving". The "Savings Glut" view of economists such as Ben Bernanke and Paul Krugman needs to be vigorously rebuked. This incorrect view which was put forward to attempt to explain the Great Financial Crisis (GFC) by the main culprits was challenged by economists at the Bank for International Settlements (BIS), particularly in one paper by Claudio Borio entitled "The financial cycle and macroeconomics: What have we learnt?".
"The core objection to this view is that it arguably conflates “financing” with “saving” –two notions that coincide only in non-monetary economies. Financing is a gross cash-flow concept, and denotes access to purchasing power in the form of an accepted settlement medium (money), including through borrowing. Saving, as defined in the national accounts, is simply income (output) not consumed. Expenditures require financing, not saving. The expression “wall of saving” is, in fact, misleading: saving is more like a “hole” in aggregate expenditures – the hole that makes room for investment to take place. … In fact, the link between saving and credit is very loose. For instance, we saw earlier that during financial booms the credit-to-GDP gap tends to rise substantially. This means that the net change in the credit stock exceeds income by a considerable margin, and hence saving by an even larger one, as saving is only a small portion of that income." - source BIS paper, December 2012
Their paper argues that it was unrestrained extensions of credit and the related creation of money that caused the problem which could have been avoided if interest rates had not been set too low for too long through a "wicksellian" approach dear to Charles Gave from Gavekal Research.
Borio claims that the problem was that bank regulators did nothing to control the credit booms in the financial sector, which they could have done. We know how that ended before.
But, guess what: We have the same problem today and suprise, it's worse.
Look at the issuance levels reached in recent years and the amount of cov-lite loans issued (again...). Look at mis-allocation of capital in the Energy sector and its CAPEX bubble.
Look at the $9 trillion debt issued by Emerging Markets Corporates.
We could go on and on.
Now the credit Fed induced credit bubble is bursting again. One only has to look at what is happening in credit markets (à la 2007). By the way Financial Conditions are tightening globally and the process has started in mid 2014. CCC companies are now shut out of primary markets and default rates will spike. Credit always lead equities...The "savings glut" theory of Ben Bernanke and the FED is hogwash:
"Asset price inflation in general, is not a phenomenon which is limited to one specific market but rather has a global impact. However, there are some specific developments in certain segments of the market, as specific segments are more vulnerable against overshooting than others. Therefore, a strong decline in asset prices effects on all risky asset classes due to the reduction of liquidity.
This is a very important finding, as it explains the mechanism behind a global crisis. Spillover effects are liquidity-driven and liquidity is a global phenomenon. Against the background of the ongoing integration of the financial markets, spillover effects are inescapable, even in the case there is no fundamental link between specific market segments. How can we explain decoupling between asset classes during financial crises? During the subprime turmoil in 2007, equity markets held up pretty well, although credit markets go hit hard." - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
As a reminder, a liquidity crisis always lead to a financial crisis. That simple, unfortunately.
This brings us to lead you towards some illustration of rising instability and worrying price action and the formation of "Rogue Waves" we have been witnessing as of late in many segments of the credit markets.
Rogue waves present considerable danger for several reasons: they are rare, unpredictable, may appear suddenly or without warning, and can impact with tremendous force. Looking at the meteoric rise in US High yield spreads in the Energy sector is an illustration we think about the destructive power of a High Yield "Rogue Wave":
- source Thomson Reuters Datastream (H/T Eric Burroughs on Twitter)
When it comes to the "short gamma" investor crowd and with Contingent Convertibles aka "CoCos" making the headlines, the velocity in the explosion of spreads has been staggering:
- graph source Barclays (H/T TraderStef on Twitter)
When it comes to the unfortunate truth about wider spreads, what the flattening of German banking giant Deutsche bank is telling you is that it's cost of capital is going up, this is what a flattening of credit curve is telling you:
Also the percentage of High Yield bonds trading at Distressed levels is at the highest level since 2009 according to S&P data:
2015: 20.1%*
2013: 11.2%
2011: 16.8%
2009: 23.2%
- source H/T - Lawrence McDonald - Twitter feed
In our book a flattening of the High Yield curve is a cause for concern as illustrated by the one year point move on the US CDS index CDX HY (High Yield) series 25:
- source CMA part of S&P Capital IQ
This is a sign that cost of capital is steadily going up. Also the basis being the difference between the index and the single names continues to be as wide as it was during the GFC. A basis going deeper into negative territory is a main sign of stress.
We have told you recently we have been tracking the price action in the Credit Markets and particularly in the CMBS space. What we are seeing is not good news to say the least and is a stark reminder of what we saw unfold back in 2007. On that subject we would like to highlight Bank of America Merrill Lynch's CMBS weekly note from the 12th of February entitled "The unfortunate truth about wider spreads":
"Key takeaways
• We anticipate that spread volatility, liquidity stress and credit tightening will persist. Look for wider conduit spreads.
• While CMBX.BBB- tranche prices fell sharply this week we think further downside exists, particularly in series 6&7.
As investors ponder the likelihood that economic growth may slow and that CRE prices may have risen too quickly (Chart 3), recent CMBX price action indicates that a growing number of investors may have begun to short it since it is a liquid, levered way to voice the opinion that CRE is considered to be a good proxy for the state of the economy.
In the past, this type of activity began by investors shorting tranches that were most highly levered to a deteriorating economy and could fall the most if fundamentals eroded. This includes the lower rated tranches of CMBX.6-8, which, as of last night’s close, have seen the prices for their respective BBB-minus and BB tranches fall by 13-17 points for CMBX.6 (Chart 4), 14-20 points for CMBX.7 (Chart 5) and 17-19 points for CMBX.8 (Chart 6) since the beginning of the year.
We agree that underwriting standards loosened over the past few years, which, all else equal, could imply loans in CMBX.8 have worse credit metrics compared to either the CMBX.6 or CMBX.7 series. Despite this, and although prices have already fallen considerably, for several reasons we think it makes sense to short the BBBminus tranche from either CMBX.6 or CMBX.7 instead of the CMBX.8. First, the dollar price of the BBB-minus tranche from CMBX.6 and CMBX.7 is materially higher that of CMBX.8 (Chart 7).
Additionally, although the CMBX.8 does have more loans with IO exposure than series 6 or 7 do, we think this becomes more meaningful when considering maturity defaults. By contrast, the earlier series not only have lower subordination attachment points at the BBB-minus tranche, but they also have more exposure to the retail sector, which could realize faster fundamental deterioration if the economy does contract." - source Bank of America Merrill Lynch
Now having read seen the movie "The Big Short" and also read the book and also recently read in Bloomberg about Hedge Fund pundits thinking about shorting Subprime Auto-Loans, as the next new "big kahuna" trade, we would like to make another suggestion. If you want to make it big, here is what we suggest à la "Big Short", given last week we mentioned that Italian NPLs have now been bundled up into a new variety of CDOs according to Euromoney's article entitled "Italy's bad bad bank" from February 2016 and that the Italian state guarantees the senior debt of such operations and thinks it is unlikely ever to have to honour the guarantee (as equity and subordinated debt tranches will take the first hit from any shortfall to the price the SPV paid for the loans), maybe you want to find someone stupid enough to sell you protection on the senior tranche of these "new CDOs". In essence, like in the "Big Short", if the whole of the capital structure falls apart, your wager might make a bigger return because of the assumed low probability of such a "tail risk" to ever materialize. and will be cheaper to implement in terms of negative carry than, placing a bet on the lower part of the capital structure. This is just a thought of course...
Moving back to the disintegration of the CMBS space, Bank of America Merrill Lynch made some additional interesting points on the fate of SEARS and CMBS:
"To this point, Sears’s management announced this week that revenues for the year ending January 31, 2016, decreased to about $25.1 billion (Chart 8) and that the company would accelerate the pace of store closings, sell assets and cut costs.
Why could CMBX.6 be more negatively impacted by the negative Sears news than some of the other CMBX series? Among the more recently issued CMBX series (6-9), CMBX.6 has the highest percentage of retail exposure. When we focus solely on CMBX.6 and CMBX.7, which have the highest percentage exposure to retail among the postcrisis series, we see that although the headline exposure to retail properties is similar, CMBX.6 has considerably more exposure to B/C quality malls than CMBX.7 does" - source Bank of America Merrill Lynch
Not really this is all part of what is known as the overshooting phenomenon.
• The overshooting phenomenon
The overshooting phenomenon is closely related to the bubble theory we have discussed earlier on through the comments of both authors of the book "Credit Crisis. The overshooting paper mentioned below in the book is of great interest as it was written by Rudi Dornbusch, a German economist who worked for most of his career in the United States, who also happened to have had Paul Krugman and Kenneth Rogoff as students:
"Closely linked to the bubble theory, Rudiger Dornbusch's famous overshooting paper set a milestone for explaining "irrational" exchange rate swings and shed some light on the mechanism behind currency crises. This paper is one of the most influential papers writtten in the field of international economics, while it marks the birth of modern international macroeconomics. Can we apply some of the ideas to credit markets? The major input from the Dornbusch model is not only to better understand exchange rate moves; it also provides a framework for policymakers. This allow us to review the policy actions we have seen during the subprime turmoil of 2007.
The background of the model is the transition from fix to flexible exchange rates, while changes in exchange rates did not simply follow the inflation differentials as previous theories suggest. On the contrary, they proved more volatile than most experts expected they would be. Dornsbusch explained this behavior of exchange rates with sticky prices and an instable monetary policy, showing that overshooting of exchange rates is not necessarily linked to irrational behavior of investors ("herding"). Volatility in FX markets is a necessary adjustment path towards a new equilibrium in the market as a response to exogenous shocks, as the price of adjustment in the domestic markets is too slow.
The basic idea behind the overshooting model is based on two major assumptions. First, the "uncovered interest parity" holds. Assuming that domestic and foreign bonds are perfect substitutes, while international capital is fully mobile (and capital markets are fully integrated), two bonds (a domestic and a foreign one) can only pay different interest rates if investors expect compensating movement in exchange rates. Moreover, the home country is small in world capital markets, which means that the foreign interest rate can be taken as exogenous. The model assumes "perfect foresight", which argues against traditional bubble theory. The second major equation in the model is the domestic demand for money. Higher interest rates trigger rising opportunity costs of holding money, and hence lower demand for money. In the contrary, an increase in output raises demand for money while demand for money is proportional to the price level.
In order to explain what overshooting means in this context, we have to introduce additional assumptions. First of all, domestic prices do not immediately follow any impulses from the monetary side, while they adjust only slower over time, which is a very realistic assumption. Moreover, output is assumed to be exogenous, while in the long run, a permanent rise in money supply causes a proportional rise in prices and in exchange rates. The exogenous shock to the system is now defined as unexpected permanent increase in money supply, while prices are sticky in the short term. And as also output is fixed, interest rates (on domestic bonds) have to fall to equilibrate the system. As interest-rate parity holds, interest rates can only fall if the domestic currency is expected to appreciate. As the assumption of the model is that in the long run rising money supply must be accompanied by a proportional depreciation in the exchange rate must be larger than the long term depreciation! That said the exchange rate must overshoot the long-term equilibrium level. The idea of sticky prices is in the current macroeconomic discussion fully accepted, as it is a necessary assumption to explain many real-world data.
This is exactly what we need to explain the link to the credit market. The basic assumption of the majority of buy-and-hold investors is that credit spreads are mean reverting. Ignoring default risk, spreads are moving around their fair value through the cycle. Overshooting is only a short-term phenomenon and it can be seen as a buying opportunity rather than the establishment of a lasting trend. This is true, but one should not forget that this is only true if we ignore default risk. This might be a calamitous assumption. Transferring this logic to the first subprime shock in 2007, it is exactly what happened as an initial reaction regarding structured credit investments. For example, investment banks booked structured credit investments in marked-to-model buckets (Level 3 accounting) to avoid mark-to-market losses.
A credit crisis can be the trigger point of overshooting in other markets. This is exactly what we have observed during the subprime turmoil of 2007.
This is a crucial point, especially from the perspective of monetary policy makers. Providing additional liquidity would mean that there will be further distortions. Healing a credit crunch at the cost of overshooting in other markets. Consequently liquidity injections can be understood as a final hope rather than the "silver bullet" in combating crises. In the context of the overshooting approach, liquidity injections could help to limit some direct effects from credit crises, but they will definitely trigger spillover effects onto other markets. In the end, the efficiency of liquidity injections by central banks depends on the benefit on the credit side compared to the cost in other markets. In any case, it proved not to be the appropriate instrument as a reaction to the subprime crisis in 2007" - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
On that subject we would like to highlight again Bank of America Merrill Lynch's CMBS weekly note from the 12th of February entitled "The unfortunate truth about wider spreads":
"As spreads widened over the past few weeks, a significant number of conversations we’ve had with investors have revolved around the concern that the recent spread widening may not represent a transient opportunity to add risk at wider levels, but instead could represent a new reality earmarked by tighter credit standards, lower liquidity and higher required returns for a given level of risk. While it may be easy to look at CRE fundamentals and dismiss the recent spread widening as being due to market technicals, it is important to realize that while that may be true today, if investors are pricing in what they expect could occur in the future, there may be some validity to the recent spread moves. As a case in point, given the recent new issue CMBS spread widening, breakeven whole loan spreads have widened substantially over the past two months (Chart 16).
Not only do wider whole loan breakeven spreads result in higher coupons to CMBS borrowers, which, effectively tightens credit standards, but it also can reduce the profitability of CMBS originators, which may cause some of them to exit the business. As a case in point, this week Redwood Trust, Inc. announced it is repositioning its commercial business to focus solely on investing activities and will discontinue commercial loan originations for CMBS distribution. Marty Hughes, the CEO of Redwood said:
"We have concluded that the challenging market conditions our CMBS conduit has faced over the past few quarters are worsening and are not likely to improve for the foreseeable future. The escalation in the risks to both source and distribute loans through CMBS, as well as the diminished economic opportunity for this activity, no longer make our commercial conduit activities an accretive use of capital."
If, as we wrote last week, CRE portfolio lenders also tighten credit standards, it stands to reason that some proportion of borrowers that would have previously been able to successfully refinance may no longer be able to do so. The upshot is that it appears that we have entered into a phase where it becomes increasingly possible that negative market technicals and less credit availability form a feedback loop that negatively affects CRE fundamentals.
To this point, although a continued influx of foreign capital into trophy assets in gateway markets can support CRE prices in certain locations, it won’t help CRE prices for properties located in many secondary or tertiary markets. If borrowers with “average” quality properties located away from gateway markets are faced with higher borrowing costs and more stringent underwriting standards, the result may be fewer available proceeds and wider cap rates." - source Bank of America Merrill Lynch
This is another sign that credit will no doubt overshoot to the wide side and that you will, rest assured see more spillover in other asset classes. Given credit leads equities, you can expect equities to trade "lower" for "longer" we think.
Furthermore, Janet Yellen's recent performance is confirming indeed the significant weakening of the Fed "put" as described in Bank of America Merrill Lynch's note:
"With Fed Chair Yellen’s Humphry Hawkins testimony, in which she stressed the notion that the Fed’s decision to raise rates is not on a predetermined course, the probability that the Fed would raise interest rates at its March 2016 plummeted as did the probability of rate hikes over the next year. During her testimony, however, the Fed Chair mentioned that the current global turmoil could cause the Fed to alter the timing of upcoming rate hikes, not abandon them.
As a result, risky asset prices broadly fell and a flight to quality ensued due to the uncertainty of the timing of future rate hikes, the notion that the Fed put may be further out of the money than was previously anticipated and the prospect that a growing policy divergence among global central banks could contribute to a U.S. recession. While delaying the next rate hike may be viewed positively in the sense that it could help keep risk free rates low, which would allow a greater number of borrowers to either refinance or acquire new properties, we think it is likely that many investors will view it as a canary in the coalmine that presages slower economic growth, more capital market volatility, wider credit spreads and lower asset prices.
Ultimately, the framework that has been put in place by regulators over the past few years effectively severely limits banks’ collective abilities to provide liquidity during periods of stress. As global economic concerns have increased, investors and dealers alike have become increasingly aware of the extremely limited amount of liquidityavailable, which has manifested through a surge in liquidity stress measures (Chart 21) and wider spreads across risky asset classes.
- source Bank of America Merrill Lynch
When it comes to rising risk, it certainly looks to us through the "credit lense" that indeed it certainly feels like 2007 and that once again we are heading towards a Great Financial Crisis version 2.0. For us, it's a given.
When it comes to the much talked about Kyle Bass significant "short yuan" case, we would like to offer our views through the lens of the Nash Equilibrium Concept in our next point.
Hyman Capital’s Kyle Bass has recently commented on the $34 trillion experiment and his significant currency play against the Chinese currency (a typical old school Soros type of play we think).
Indirectly, our HKD peg break idea which we discussed back in September t2015 our conversation "HKD thoughts - Strongest USD peg in the world...or most convex macro hedge?", we indicated that the continued buying pressure on the HKD had led the Hong-Kong Monetary Authority to continue to intervene to support its peg against the US dollar. At the time, we argued that the pressure to devalue the Hong-Kong Dollar was going to increase, particularly due to the loss of competitivity of Hong-Kong versus its peers and in particular Japan, which has seen many Chinese turning out in flocks in Japan thanks to the weaker Japanese Yen. This Yuan trade is of interest to us as we won the "best prediction" from Saxo Bank community in their latest Outrageous Predictions for 2016 with our call for a break in the HKD currency peg as per our September conversation and with the additional points made in our recent "Cinderella's golden carriage".
We also read with interest Saxo Bank's French economist Christopher Dembik's take on the Yuan in his post "The Chinese yuan countdown is on".
Overall, we think that if the Yuan goes, so could the Hong Dollar peg. Therefore we would like again to quote once again the two authors of the book "Credit Crisis" and their Nash Equilibrium reasoning in order to substantiate the probability of this bet paying off:
"Financial panic models are based on the idea of a principle-agent: There is a government which is willing to maintain the current exchange rate using its currency reserves. Investors or speculators are building expectations regarding the ability of the government to maintain the current exchange-rate level. An as answer to a speculative attack on the currency, the government will buy its own currency using its currency reserves. There are three possible outcomes in this situation. First, currency reserves are big enough to combat the speculative attack successfully, and the government is able to keep the current exchange rate. In this case there will be no attack as speculators are rational and able to anticipate the outcome. Second, the reserves of central banks are not large enough to successfully avert the speculative attack, even if only one speculator is starting the attack. Thus, the attack will occur and will be successful. The government has to adjust the exchange rate. Third, the attack will only be successful if speculators join forces and start to attack the currency simultaneously. In this case, there are two possible equilibriums, a "good one" and a "bad one". The good one means the government is able to defend the currency peg, while the bad one means that the speculators are able to force the government to adjust the exchange rate. In this simple approach, the amount of currency reserves is obviously the crucial parameter to determine the outcome, as a low reserve leads to a speculative attack while a high reserve prevents attacks. However, the case of medium reserves, in which a concerted action of speculators is needed is the most interesting case. In this case, there are two equilibriums (based on the concept of the Nash equilibrium): independent from the fundamental environment, both outcomes are possible. If both speculators believe in the success of the attack, and consequently both attack the currency, the government has to abandon the currency peg. The speculative attack would be self-fulfilling. If at least one speculator does not believe in the success, the attack (if there is one) will not be successful. Again, this outcome is also self-fulfilling. Both outcomes are equivalent in the sense of our basic equilibrium assumption (Nash). It also means that the success of an attack depends not only on the currency reserves of the government, but also on the assumption what the other speculator is doing. This is interesting idea behind this concept: A speculative attack can happen independent from the fundamental situation. In this framework, any policy actions which refer to fundamentals are not the appropriate tool to avoid a crisis. " - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
If indeed the amount of currency reserves is obviously the crucial parameter when it comes to assessing the pay off for the Yuan bet, we have to agree with Deutsche Bank recent House View note from the 9th of February 2016 entitled "Still deep in the woods" that problems in China remains unresolved:
"The absence of new news has helped divert attention away from China – but the underlying problem remains unresolved
• After surprise devaluation in early January, China has stopped being a source of new bad news
• Currency stable since, though authorities no longer taking cues from market close to set yuan level*
• Macro data soft as expected, pointing to a gradual deceleration not a sharp slowdown
• Underlying issue of an overvalued yuan remains unresolved, current policy unsustainable long-term
−At over 2x nominal GDP growth, credit growth remains too high
−FX intervention to counter capital outflows – at the expense of foreign reserves
- source Deutsche Bank
When it comes to the risk of a currency crisis breaking and the Yuan devaluation happening, as posited by the Nash Equilibrium Concept, it all depends on the willingness of the speculators rather than the fundamentals as the Yuan attacks could indeed become a self-fulfilling prophecy in the making.
This self-fulfilling process is as well a major feature of credit crises and a prominent feature of credit markets (CDS) as posited again in Chapter 5 of the book from Dr Jochen Felsenheimer and Philip Gisdakis:
"Self-fulfilling processes are a major characteristics of credit crises and we can learn a lot from the idea presented above. The self-fulfilling process of a credit crisis is that short-term overshooting might end up in a long-lasting credit crunch - assuming that spreads jump initially above the level that we would consider "fundamentally justified; for instance reflected in the current expected loss assumption. That said, the implied default rate is by far higher than the current one (e.g., the current forecast of the future default rate from rating agencies or from market participants in general). However the longer the spreads remains at an "overshooting level", the higher the risk that lower quality companies will encounter funding problems, as liquidity becomes more expensive for them. this can ultimately cause rising default rate at the beginning of the crisis; a majority of market participants refer to it as short-term overshooting. Self fulfilling processes are major threat in a credit crisis, as was also the case during the subprime meltdown. If investors think that higher default rates are justified, they can trigger rising default rates just by selling credit-risky assets and causing wider spreads. This is independent from what we could call the fundamentally justified level!
The other interesting point is that the assumption of concerted action is not necessary in credit markets to trigger a severe action. If we translate the role of the government (defending a currency peg) into credit markets, we can define a company facing some aggressive investors who can send the company into default. Buying protection on an issuer via Credit Default Swaps (CDS) leads to wider credit spreads of the company, which can be seen as an impulse for the self-fulfilling process described above. If some players are forced to hedge their exposure against a company by buying protection on the name, the same mechanism might be put to work." - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
As we highlighted above with the flattening of MS München and/or Deutsche Bank and the flattening of the CDX HY curve, the flattening trend means that the funding costs for many companies is rising across all maturities:
"Such a technically driven concerted action of many players, consequently can also cause an impulse for a crisis scenario, as in the case for currency markets in financial panic models" - source Credit Crises, published in 2008, authored by Dr Jochen Felsenheimer and Philip Gisdakis
So there you go, you probably understand by now the disappearance of MS München due to a conjunction of "Rogue Waves":
"The laws of probability, so true in general, so fallacious in particular." - Edward Gibbon, English historian
And this dear readers is the story of VaR in a world of rising "positive correlations" but we are ranting again...
Stay tuned!
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c3e2ae7d8c40b062 | Skip to main content
September 16, 2015
Speaker: Professor I. G. Kaplan (Materials Research Institute, UNAM, Mexico)
Discovery and the modern state of the Pauli Exclusion Principle. Can it be proved?
Abstract: In the introduction, we will present how Wolfgang Pauli came to the formulation of his exclusion principle, and the dramatic history of the discovery of the fundamental quantum-mechanical conception of spin. Then we will discuss the modern state of the Pauli Exclusion Principle (PEP). If all experimental data agree with the PEP and the best to day limit on the violation of the PEP is negligible1, its theoretical foundations are still absent.
PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is a so-called spin-statistics connection. As we will discuss, the reasons the spin-statistics connection exists are still unknown. On the other hand, according to the PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric; all other types of permutation symmetry are forbidden. However, the solutions of the Schrödinger equation may belong to any representation of the permutation group, including the multi-dimensional ones. It is demonstrated that the proofs of the PEP in some textbooks on quantum mechanics, including the famous course of Landau and Lifshitz, are incorrect. The indistinguishability principle is insensitive to the permutation symmetry of the wave function and cannot be used as a criterion for the verification of PEP.
The heuristic arguments have been given in favor that the existence in nature only one-dimensional permutation representations (symmetric and antisymmetric) is not accidental. As follows from the analysis of possible scenarios, the permission of multi-dimensional representations of the permutation group leads to contradictions with the concept of particle identity and their independence2. Thus, the prohibition of the degenerate permutation stated by the PEP follows from the general physical assumptions underlying quantum theory. |
5a48e25fc8c82c5d | If I want to do a very accurate simulation of a molecular system (e.g. 2 hydrogen atoms), then I'll want to use something like diffusion Monte Carlo to determine the energies of these atoms in different configurations.
For example, if I want to create a bonding potential like the harmonic
$$E = K(r - r_0)^2$$
or Morse potential
$$E = D[1-e^{-\alpha(r-r_0)}]^2$$
then I assume what I would do is perform DMC to find the energies for various separation distances, and then perform a regression to fit one of those equations.
However, I was recently thinking perhaps this isn't as accurate as it could be, because you're using quantum mechanics to find energies, and then using classical physics (Newton's laws of motion) to evolve the system.
A more accurate alternative, I assume, would be to evolve a system of two hydrogen atoms using the time-dependent Schrödinger equation and then somehow fit this to some easily computable model.
What I'm curious though is how the resulting simulations would differ? I don't know enough about QM to speculate on what kind of effects the classical system evolution would be missing, or exactly how much (numerically speaking), the two approaches would differ. Does anyone have any idea?
I suppose I could always try both methods and see, but that's going to take quite a bit of work...
migrated from physics.stackexchange.com Feb 15 '13 at 17:28
• $\begingroup$ I'm not an expert on the topic, but I know that Jim Mitroy's group at CDU (click through to their publications if interested) are doing a lot of high accuracy few-body (up to about 7 particles) QM calculations. I hear they set up a simulation then let it run on the supercomputer for a month before checkin on the answer. :) But that's to get a ridiculous number of sig. figs. You may be able to get away with much less. $\endgroup$ – Michael Brown Feb 15 '13 at 5:18
• $\begingroup$ Should mention that the computational complexity goes up exponentially in the number of particles, so if you just have two electrons + two nuclei then there may be cheaper options for you. $\endgroup$ – Michael Brown Feb 15 '13 at 5:20
• $\begingroup$ Not necessarily. What you are doing first is solving the electronic Schrödinger equation parametrically for each internuclear position. Once you have the potential you can use classical mechanics or quantum mechanics to treat the nuclear (molecular) Hamiltonian dynamics. If you solve this at the quantum level the approximation is in the treatment of the coupling of the electronic and nuclear degrees of freedom. But you can also include these couplings in your study. $\endgroup$ – perplexity Feb 15 '13 at 17:09
It sounds like you are interested in Born-Oppenheimer molecular dynamics, i.e. using classical equations of motion for the nuclei while using quantum mechanics for the electrons. This is a fairly common method implemented in a range of quantum chemistry packages.
A related approximate method is Car-Parinello molecular dynamics, which may be of use to you. This may be your best bet if your system of study is large.
(This is more like a comment but I don't have enough reputation to comment and is kind of long; however this may be useful for you if I am understanding your question).
If you are only interested in calculating a bonding like potential for a two-electron system such as two hydrogen atoms, that can be done exactly (within the Born-Oppenheimer approx. and for a given basis set, $i.e.$ set of orbitals) using standard quantum chemistry methods. The methods of full configuration interaction (http://en.wikipedia.org/wiki/Full_configuration_interaction), CISD (http://www.gaussian.com/g_tech/g_ur/k_cid.htm) or Coupled Cluster CCSD (http://www.gaussian.com/g_tech/g_ur/k_ccd.htm) would give all exact answers for two electron systems. You can calculate the energy at different bond lengths with these methods using standard quantum chemistry programs. You can then use a program such a gnuplot to fit the obtained point to your harmonic or Morse potential equation.
The methods that I am mentioning above are computationally very expensive, but for your small two electron system it is not a problem to do many calculations at different bond lengths with any modern computer.
Hope this helps, cause I am not sure if I understood perfectly your question. Your question talked about mixing QM with classical methods, but I am not mentioning about anything about the latter because you can get the "exact" quantum answer for your small system (and you said you wanted to do a very accurate calculation).
• $\begingroup$ Well, ultimately I would like to accurately describe C-C, C-Li, and Li-Li interactions but I think there's too many particles for a fully accurate simulation. $\endgroup$ – Nick Feb 15 '13 at 5:36
• $\begingroup$ Yes, you cannot do exact calculations for these larger systems (except perhaps Li-Li). I cannot help you much more because I am not very familiar with mixed QM methods. But I can warn you that it will be very hard to get very accurate results for C-C. Even using purely quantum mechanics, many sophisticated methods fail for this system so be very careful on what you do for C-C as you might even get unphysical answers. $\endgroup$ – Goku Feb 15 '13 at 5:41
• $\begingroup$ @Nick If you have access, you can look at this article jcp.aip.org/resource/1/jcpsa6/v118/i4/p1610_s1?isAuthorized=no. The authors compare dissociation curves of several quantum methods with the "exact" (FCI) solution. The make calculations for potential energy curves of molecules of about the size you are interested in (they show HF, BH and methane). $\endgroup$ – Goku Feb 15 '13 at 5:46
Your Answer
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fe94151c808b6ab5 | Revealing quantum chaos with machine learning
Revealing quantum chaos with machine learning
Understanding the properties of quantum matter is an outstanding challenge in science. In this work, we demonstrate how machine learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. By taking this method further, we show that machine learning techniques allow to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. Our results pave the way for exploring the power of machine learning tools for revealing exotic phenomena in complex quantum many-body systems.
Introduction. Significant attention to machine learning techniques is related to their applications in tasks of finding patterns in data, such as image recognition, speech analysis, computer vision, and many other domains LeCun2015 (). Quantum physics is well known to produce atypical patterns in data, which are in principle can be revealed using machine learning methods Biamonte2017 (). This idea has stimulated an intensive ongoing research of this subject. The scope so far includes identification of phases of quantum matter and detecting phase transitions Broecker2016 (); Melko2017 (); Schindler2017 (); Chng2017 (); Nieuwenburg2017 (); Ringel2018 (); Beach2018 (); Greitemann2018 (), as well as representing quantum states of many-body systems in regimes that are intractable for existing exact numerical approaches Troyer2017 (); Glasser2018 (); Lu2018 (). Another branch of research is related to the applications of machine learning tools to the analysis of experimental data Troyer2018 (); Zhang2018 (); Sriarunothai2018 (). Recently, a machine learning approach has been used for processing data from gas microscopes and evaluating predictions of competing theories that describe the doped Hubbard model without a bias towards one of the theories Knap2019 ().
Remarkable progress on building large-scale quantum simulators has opened fascinating prospects for the observation of novel quantum phases and exotic states Monroe2013 (); Rey2017 (); Lukin2017 (); Monroe2018 (). They also provide interesting insights to traditionally challenging problems in studies of complex quantum systems, such as investigation of quantum critical dynamics and quantum chaos Polkovnikov2016 (). Quantum systems with chaotic behaviour are of great interest in the view of a possibility to explore quantum scars in them Papic2018 (). Quantum many-body scars can be potentially compatible with long-lived states, which are of importance for quantum information processing. A standard criterion for the separation between regular and chaotic regimes is based on the nearest-neighbor energy level statistics Berry1977 (); Bohigas1984 (): Poisson and Wigner-Dyson distributions correspond to integrable and chaotic systems, respectively. However, the energy level statistics of highly excited states is not always directly accessible in experiments.
From the machine learning perspective, an interesting problem is to understand whereas it is possible to distinguish between regular and chaotic behavior, in the best-case scenario, based on experimentally accessible quantities such as data from projective measurements. Within this context, finding an appropriate criteria to identify a transition within machine learning tools is essential.
Figure 1: Neural network approach for identifying a transition between chaotic and regular states in quantum billiards and Heisenberg spin chains. The input data contains probability distribution in the configuration space, the two neuron activation functions are used for the identification of the two regimes.
Figure 2: Convolutional neuron network outputs for (a) the Sinai billiard, (b) the Bunimovich stadium, and (c) the Pascal’s limaçon as functions of the chaoticity parameter characterizing billiard’s boundary shape. The highlighted critical region corresponds to the regions of “uncertainty” in neuron network output activation curves. The analysis of the chaotic/regular transition for the Bunimovich stadium is the most challenging due to its extreme sensitivity to the variation of the chaoticity parameter (see Ref. Tao2008 ()).
In the present paperá we implement a neural network based algorithm to perform a classification between regular and chaotic states in single-particle and many-body systems. The input data contains the wavefunction amplitudes of excited states and the output is represented by two neurons corresponding to integrable and chaotic classes (Fig. 1). In the single-particle case, we consider a paradigmatically important models of quantum billiards, such as Sinai billiard, the Bunimovich stadium, and the Pascal’s billiard. We then apply a semisupervised “learning by confusion” scheme Nieuwenburg2017 () in order to detect the integrability/chaos transition and to evaluate a critical transition region. This approach is then extended in order to study the transition in Heisenberg XXZ spin-1/2 chain in the presence of additional interactions that break integrability, such as next-nearest neighbour spin-spin interaction and a coupling of a single spin to a local magnetic field or a magnetic impurity. In our work, regular/chaos transitions are identified with the classification accuracy up to . We show that our results based on the machine learning approach are in a good agreement with the analysis of level spacing distributions.
To address the problem of revealing the transition between regular and chaotic behaviour, we propose a learning approach based on the prior evaluation of the critical region and further detecting the critical point within its boundaries, performed by the confusion scheme Nieuwenburg2017 (). At the first stage, we train the network to distinguish states belonging to the extreme cases of regular () and chaotic () regimes, where is the chaoticity parameter. We then determine the critical domain where the neural network predicts a transition between the two regimes. At the second stage, we perform the ‘learning by confusion’ scheme and we refer the middle peak on W-like performance curves of the neural network as the transition point Nieuwenburg2017 ().
Quantum billiards. Quantum billiards are among the simplest models exhibiting quantum chaos. The problem of transition from regular to chaotic behaviour in quantum billiards have been intensively studied for decades Jain2017 (). The transition from integrability to chaos is controlled by the shape of the billiard boundary. Quantum billiards have been realized in various experimental setups including microwave cavities Sridhar1991 (), ultracold atoms Raizen2001 (), and graphene quantum dots Geim2008 (). Quantum scars Heller1993 (), which are regions with enhanced amplitude of the wavefunction in the vicinity of unstable classical periodic trajectories, is the hallmark of quantum chaos. Quantum scars are of a great interest in quantum billiards Heller1993 (); Tao2008 () and their analogs have recently been found in many-body systems Papic2018 ().
We consider three standard types of two-dimensional quantum billiards: Sinai billiard, Bunimovich stadium, and Pascal’s limaçon (Robnik) billiard. We define a dimensionless parameter of chaoticity for each billiard type, where is determined by the billiard shape. In Sinai billiard the chaoticity parameter is controlled by the ratio of the radius of the inner circle to the width/height of the external rectangle, so that . In the case of Bunimovich stadium the parameter is and in the of Pascal’s limaçon billiard shape is defined via the conformal map on the complex plane , where . At the limit of these billiards have regular shapes and therefore are integrable. Varying the parameter allows one to trace out a continuous transition from integrability to quantum chaos.
Figure 3: Universal W-like NN perormance curves in the “learning by confusion” scheme for the Sinai billiard (top panel) and the Pascal’s limaçon (bottom panel). The predicted transition point is highlighted. The estimated position of the transition point predicted from the KL divergence calculation [see Eq. (1)] based on the lowest 500 energy levels is shown with a red dot.
Figure 4: Unsupervised learning of regular and chaotic states in quantum billiards with variational autoencoder (VAE). Latent space representation for the wavefunctions in (a) Bunimovich stadium, (b) Sinai billiard, are coordinates in the latent space with dimension 2.
We use a supervised learning approach for revealing chaotic/regular transitions in quantum billiard models. We train a binary classifier based on convolutional neural network (CNN) using real space images of the probability density function (PDF) . The training dataset consist of randomly sampled snapshots of the PDF in fragments excluding the billiard’s boundary in the regions of interest (ROI). The datasets are prepared separately for the each billiard type. The wave functions are obtained from the numerical solution of the stationary Shrödinger equation (for details on the numerical solution of the Shrödinger equation and the dataset preparation, see Ref. SM ()). Since the information about the transition from the regular to chaotic regimes is mostly represented in the properties of highly excited states, we use wavefunctions with sufficiently large values of in our dataset.
The snapshots corresponding to we label as “regular” (class 1), and snapshots corresponding to a large value of we label as “chaotic” (class 2). The activations of the neurons in the last layer allow to classify between chaotic/regular snapshots in the test dataset with a high accuracy, see Fig. 2. The activation curves for each of the three billiard types (Sinai, Bunimovich, Pascal) for different values of in Fig. 2 demonstrate that the CNN algorithm is able to learn the difference between regular and chaotic wavefunctions and reveals existence of a transition region. The CNN confidence for the binary classification for away from the transition region. The transition region determined by the CNN is highlighted in red in Fig. 2. In Sinai and Bunimovich billiards the critical region detected by the CNN algorithm is . The critical region for the Pascal billiard is . The boundaries of the transition regions provided by the CNN classifier are in a good agreement with the ones obtained from the analysis of the energy levels spacing statistics SM ().
The transition region can be analyzed in more details within the “learning by confusion” scheme Nieuwenburg2017 () by performing a dynamical reassignment of the class labels with respect to a given value of . We present the NN “confusion curves” with a typical W-like shape for Sinai and Pascal billiards in Fig. 3. The central peak of the W-like CNN performance curve gives an estimate of the position of the critical point separating regular and chaotic regimes. We note that a precise definition of the transition point is somewhat ambiguous and depends on selected criteria, because all observables have a smooth dependence on the parameter . Therefore, in our approach we only estimate the location of a characteristic critical point , separating regular and chaotic regimes. The prior identification of the critical region is important in the “learning by confusion” scheme, since it allows to garantee the presence of the transition point inside of the selected range of . The estimated position of the critical point is in Sinai billiard and in Pascal limaçon billiard. We note that the analysis of the chaotic/regular transition for the Bunimovich stadium is challenging due to its extreme sensitivity to the variation of the chaoticity parameter (see Ref. Tao2008 ()).
Figure 5: Neural network classification accuracy between integrable and chaotic XXZ spin chains for spins. In order to independently pinpoint a location of the transition from integrability to chaos we present distribution of energy level spacings and the Poisson/Wigner-Dyson distributions (). Top panels: XXZ model with the next-nearest neighbor interactions; bottom panels: XXZ model in the presence of a local magnetic field (a magnetic impurity) at the central site of the spin chain.
One of the key features that allows us to perform machine learning of the regular-to-chaos transition is the difference in statistical properties of in the two regimes. While in the chaotic case the wavefunctions have Gaussian statistics, in regular case the probability distribution is non-universal and has a power-law singularity at small values of Beugeling2017 ().
The standard approach to identify a transition from an integrability to a quantum chaos is based on the comparison of the energy level spacing statistics with the Poisson distribution and the Wigner-Dyson distributions. In order to characterize a “degree of chaoticity” of the system it is convenient to introduce a single scalar quantity, a measure of chaos. One of the examples of a such measure is the average ratio of consecutive level spacings , where and Atas2013 (). In the present work we introduce a different measure based on the Kullback-Leibler (KL) divergence, defined as follows:
where is the level spacing distribution for a given value of , and is the Wigner-Dyson or Poisson distribution: , . Here is the unfolded nearest neighbour energy level spacing.
In the transition region between regular and chaotic regimes the energy spacings distribution is neither the Poisson nor the Wigner-Dyson. The KL distance between and () is the measure of integrability (chaoticity) of the system. There exists a point when is equidistant from both Poisson and Wigner-Dyson distributions in the KL metric, , which we refer as a “critical point”. As it is shown in Fig. 3, the critical points predicted by the confusion scheme and KL divergence curves are in a good agreement. It is important to note that the confusion scheme uses experimentally accessible quantities, whereas extracting the energy levels statistics from experimental data is hardly achievable in condensed matter and atomic simulator experiments.
An alternative approach to differentiate between regular and chaotic wavefunctions is to use unsupervised machine learning techniques, such as Variational Autoencoder (VAE). VAEs are generative NN models that are able to directly learn statistical distributions in raw data and can be efficiently used for solving clasterization problems Kingma2014 (); Sohn2015 (). VAE consists of encoding NN, latent space and decoding NN, Fig. 4a. During the training VAE “learns” to reproduce initial data by optimizing the weight in the encoder and decoder NN and parameters in the latent layer. Training VAE on the images with corresponding regular () and chaotic () cases and by taking samples from the latent space with the dimension 2 results in two clearly separated clusters representing regular and chaotic wavefunctions. In Figs. 4(b) and 4(c) we demonstrate latent space distributions for the cases of Bunimovich and Sinai billiards. The separation in the two clusters shows that VAE is able to learn the difference in the statistical properties of in regular and chaotic billiards. Similar approach was used for unsupervised learning of phase transitions Wetzel2017 (). Exploring a full potential of unsupervised machine learning methods for clasterizing quantum states is beyond the scope of the present work.
Quantum chaos in XXZ spin chains. While quantum billiards is an instructive example of a single particle quantum chaos, the most interesting and challenging problem is quantum chaos in many-body systems. Developing machine learning approaches to characterize/classify many-body states in chaotic and integrable regimes using only limited information from measurements is a non-trivial task. For example, such techniques can benefit the analysis of experimental data from quantum simulators. As a prototypical example of a quantum many-body integrable system we consider 1D Heisenberg XXZ spin chain, which is of great interest for realizing models of quantum magnetism using quantum simulators Bloch2017 (). Recent experimental advances have opened exciting prospects for exploiting a rich variety of tunable interactions in Rydberg atoms Browaeys2016 (); Lukin2016 (); Lukin2017 (); Browaeys2018 (); Browaeys2018-2 () and cold polar molecules Buchler2012 (); Ye2012 (); Rey2013 () for the engineering of spin Hamiltonians including the XXZ model.
The Hamiltonian of the Heisenberg XXZ model reads:
where is the number of spins, and are the Heisenberg exchange constants and are Pauli spin-1/2 operators. For simplicity we consider only antiferromagnetic XXZ model, . Hereafter we set . The XXZ model is integrable and exactly solvable by the Bethe ansatz Buchler2012 (), however it can be non-integrable in the presence of additional interactions.
Here we consider two types of perturbations that violate integrability of the XXZ model: (i) antiferromagnetic next-nearest neighbour spin-spin interaction (NNN), (ii) a local static magnetic field acting on a single spin (impurity). We parametrize the perturbation Hamiltonians in the following form:
We consider spin chains with an odd number of spins , so that in the case (ii) the local magnetic field is acting on the spin in the middle of the chain, i.e. . Hence, the Hamiltonian of the perturbed XXZ model reads:
We train a multilayer perceptron on the dataset containing the probabilities of the spin configurations in representation ( refers to basis states in -representation), e.g. , that are experimentally accessible data. The eigenfunctions are obtained by exact diagonalization of spin-chain Hamiltonian, here we consider system size . Similarly to the case of quantum billiards, we consider only highly excited states with corresponding to the levels lying in the middle of the energy spectrum, .
Further, in order to pindown transition in these systems we evaluate NN classification prediction for the test dataset as a function of , see Fig. 5. The transition region is highlighted with red. For XXZ + NNN and XXZ + impurity detected critical regions are and respectively, which turn out to be in agreement with level spacing distributions represented in Fig. 5. Within these critical regions we performed “learning by confusion”, that resulted in W-like NN performance curves, see Fig. 6, and detected transition points for XXZ + NNN and for XXZ + impurity. We note that we have a reasonable agreement with the results based on the KL divergence calculations.
Figure 6: Reconstructed universal W-like NN performance curves for (a) XXZ chain with NNN and (b) XXZ chain in the presence of an impurity, the predicted transition point is highlighted. The transition point predicted by the KL divergence calculation (1) for the energy spacing distribution is also presented.
Conclusion. In summary, we have shown the potential of classical supervised and unsupervised machine learning techniques for classification of regular/chaotic regimes in single-particle and many-body systems. For quantum billiards and XXZ spin chains we demonstrated that neural networks can serve as a binary classifier to distinguish between the two regimes with remarkably high accuracy. We revealed the integrability-chaos transition region purely based on machine learning techniques and located the transition point using “learning by confusion” approach. The extension of our work opens a new avenue to study chaotic and integrable regimes in quantum systems using experimentally accessible data in different many-body quantum systems including atomic simulators. Harnessing machine learning methods could open up exciting possibilities for studying exotic many-body phenomena with controlled quantum many-body systems, such as many-body localization Altshuler2006 (), many-body quantum scars Papic2018 (), and ergodic/non-ergodic phase transitions Shlyapnikov2018 () and near-critical properties of these systems.
Acknowledgements. We are grateful to M.B. Zvonarev and V.V. Vyborova for valuable suggestions. We thank G.V. Shlyapnikov, V.I. Yudson, and B.L. Altshuler for fruitful discussions and useful comments. The work was supported by the RFBR (Grant No. 18-37-00096).
Supplementary material
s0.1 Numerical solution of the Schrödinger equation for quantum billiards.
We solve a stationary Shrödinger equation describing a single particle in a quantum billiard with the Dirichlet boundary condition:
where is the wavefunction and is the energy of a particle in the billiard with the boundary ; is the two-dimensional Laplace operator. Hereafter we set the Plank’s constant and the mass to unity, . In order to solve Eq. (S1) for an arbitrary 2D billiard boundary shape we use Matlab PDE toolbox. The PDE solver is based on the finite element method with an adaptive triangular mesh for a given boundary geometry. In order to reduce computational complexity and to avoid additional complications due to degeneracies of eigenstates, we constrain the eigenfunctions to a specific symmetry (parity) sector. We remove degeneracies by considering the lowest symmetry segments of billiards. In the case of the Bunimovich stadium we consider a quarter of the billiard [see inset of Fig. 2(b) in the main text]. For the Sinai billiard we consider a boundary with the incommensurate ratio of vertical and horizontal dimensions of the external rectangle, (we denote in the main text). In the case of the Pascal limaçon billiard, the degeneracy is lifted when considering only the upper part of the billiard .
s0.2 Dataset preparation for quantum billiards
Wavefunctions obtained from numerical solution of the Schrödinger equation are converted into images of PDFs . From original images with pixels we randomly select a square fragments (region of interest) which exclude the billiard boundary, pixels. In order to reduce the size of the images we perform a coarse graining (downsampling) to images with dimensions . The dataset for each billiard type contains wavefunctions corresponding to highly energy states, . In order to increase amount of images in the dataset we perform an augmentation of the dataset by adding horizontal and vertical reflections, discrete rotations by angles and rotations by random angles from the uniform distribution . The total number of images in the resulting dataset for each billiard type and each value of is . The trial samples from the dataset for the Bunimovich billiard are shown in Fig. S1.
Figure S1: Sample images of in the dataset for Bunimovich billiard. Regular case () and chaotic case ().
The training dataset consists of labeled images from the class 1 (regular, ) and class 2 (chaotic, ). The value of we independently choose for each billiard type: Sinai - , Bunimovich - , Pascal - . In order to check that at the system is in the chaotic regime we compare the energy level spacing distribution with the Wigner-Dyson distribution. As long as the value of is much greater than the critical , , the NN activations curves remain practically unchanged (see Fig. 2 in the main text).
The training and test dataset are split in the proportion . The test set for each billiard type consists of images for several values of (including values of not present in the training dataset), evaluation of the NN output for the sample images from the test dataset for each value of results in the NN prediction curves presented in Fig. 2 in the main text.
s0.3 Convolutional neural network
The used CNN consists of two convolutional layers followed by a fully connected layer and a final softmax layer. The output from the second convolutional layer is the subject to dropout regularization and batch normalization. The cost function for the binary classifier is the cross-entropy. The neuron activation function is ReLU. The scheme of the CNN architecture is presented in Fig. S2.
Figure S2: CNN used for recognizing chaotic regimes in quantum billiards.
The weights in the CNN are optimized with the use of the Adam optimizer. The batch size is 60, the number of training epochs is of about , the learning rate is .
Energy level spacing statistics in quantum billiards
Figure S3: Left column: The CNN activation functions (Fig. 2 from the main text). The histograms show the energy level spacing distributions (lowest 500 energy levels). In order to compare NN prediction for the regular-to-chaos transition region we compare the energy level spacing distribution with the standard Poisson/GOE distributions.
s0.4 Unsupervised learning with VAE
We perform unsupervised learning of two classes (“regular” and “chaotic”) using a variational autoencoder (VAE). The unlabeled dataset was prepared in a similar way as for the supervised learning. Dataset consist of randomly sampled images of with the dimensions , number of samples in the training dataset for each billiard type is , number of testing samples is for each billiard type. VAE was trained and tested for the states with in Bunimovich and Sinai’s billiards, corresponds to the “regular” class, corresponds to the “chaotic” class. VAE consists of the encoder, decoder, sampler and the latent space of dimension 2 (latent space parameters and ) representing the two classes, “regular” and “chaotic”. The sampler generates random latent space variables with the mean and the dispersion . Encoder and decoder are represented by a fully connected NN with two hidden layers and neurons in each layer. The objective function is a sum of reconstruction loss (binary cross entropy) and KL divergence loss. VAE was trained over 50 epochs using Adam optimizer Kingma2014 (), learning rate is , batch size is samples.
Figure S4: Architecture of variational autoencoder (VAE) for unsupervised learning of regular-chaos transition in quantum billiards.
s0.5 Exact diagonalization of the Hamiltonian of XXZ model
We find eigenstates of Heisenberg XXZ model for an arbitrary value of perturbation parameter by the exact diagonalization method based on the Lancsoz algorithm Sandvik2011 (). We used Python implementation of QuSpin software package Weinberg2017 (). In order to avoid extensive computational costs, the size of Hamiltonian matrix was reduced by considering only the eigenstates in certain parity and magnetization sectors of the XXZ Heisenberg model. Specifically, we find eigenstates in the even parity sector and the lowest magnetization sector. The lowest magnetization sector corresponds to the states with (for odd spin chains), where and the number of up and down spins, respectively.
s0.6 Dataset preparation for Heisenberg XXZ chains
Dataset for Heisenberg XXZ chains consists of vectors of probability densities (PDs) corresponding to integrable and chaotic Hamiltonians. We take the wavefunction corresponding to a quantum state with the energy lying in the center of the spectrum. In order to prepare a diverse dataset for a given value of we randomly select from the uniform distribution . Since the XXZ model is integrable for any value of we build a dataset corresponding to a set of different Hamiltonians by varying . In the training set we include PDs for regular systems () and chaotic systems () and label the samples, accordingly. The test set contains PDs corresponding to a discrete set of lying in the interval . The training set contains 400 samples, testing set consists of 100 samples.
s0.7 Multi-layer perceptron
Figure S5: Multilayer perceptron used for investigation integrable/chaotic transitions in Heisenberg XXZ chains.
We used a standard multi-layer perceptron neural network that consists of an input layer with the size , which is equal to the size of vector with probability densities in the specified symmetry (parity and total magnetization) sector of the eigenstates; one hidden layer with neurons, and an output softmax layer. Neurons of the hidden layer receive input and a weight () and compute output , where . An output is computed with a sigmoid activation function . Further, each output with a corresponding weight () is passed to a neuron of an output layer, which finally results a scalar value between 0 and 1. The objective function is the binary cross-entropy. Neural network’s weights are optimized using Adam optimizer Kingma2014 () with the learning rate , batch size of samples, training epochs. The scheme of the neural network architecture is presented in Fig. S5.
1. preprint: APS/123-QED
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92e359364576d019 | Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A–B), and according to the Schrödinger equation of quantum mechanics (C–H). In A–B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.[1][2][3]
One-dimensional harmonic oscillatorEdit
Hamiltonian and energy eigenstatesEdit
Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. The horizontal axis shows the position x. Note: The graphs are not normalized, and the signs of some of the functions differ from those given in the text.
Corresponding probability densities.
The Hamiltonian of the particle is:
where m is the particle's mass, k is the force constant, is the angular frequency of the oscillator, is the position operator (given by x), and is the momentum operator (given by ). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.
One may write the time-independent Schrödinger equation,
where E denotes a to-be-determined real number that will specify a time-independent energy level, or eigenvalue, and the solution |ψ denotes that level's energy eigenstate.
One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function x|ψ⟩ = ψ(x), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,
The functions Hn are the physicists' Hermite polynomials,
The corresponding energy levels are
This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħω/2 above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle.
The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.
Ladder operator methodEdit
Probability densities |ψn(x)|2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.
The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a,
This leads to the useful representation of and ,
The operator a is not Hermitian, since itself and its adjoint a are not equal. The energy eigenstates |n, when operated on by these ladder operators, give
It is then evident that a, in essence, appends a single quantum of energy to the oscillator, while a removes a quantum. For this reason, they are sometimes referred to as "creation" and "annihilation" operators.
From the relations above, we can also define a number operator N, which has the following property:
The following commutators can be easily obtained by substituting the canonical commutation relation,
And the Hamilton operator can be expressed as
so the eigenstate of N is also the eigenstate of energy.
The commutation property yields
and similarly,
This means that a acts on |n to produce, up to a multiplicative constant, |n–1⟩, and a acts on |n to produce |n+1⟩. For this reason, a is called a annihilation operator ("lowering operator"), and a a creation operator ("raising operator"). The two operators together are called ladder operators. In quantum field theory, a and a are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, since
the smallest eigen-number is 0, and
In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that
Finally, by acting on |0⟩ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates
such that
which matches the energy spectrum given in the preceding section.
Arbitrary eigenstates can be expressed in terms of |0⟩,
Analytical questionsEdit
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation . In the position representation, this is the first-order differential equation
whose solution is easily found to be the Gaussian[4]
Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the energy eigenstates constructed by the ladder method form a complete orthonormal set of functions.[5]
Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by ,
so that , and so on.
Natural length and energy scalesEdit
The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization.
The result is that, if energy is measured in units of ħω and distance in units of ħ/(), then the Hamiltonian simplifies to
while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
where Hn(x) are the Hermite polynomials.
To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
For example, the fundamental solution (propagator) of H−i∂t, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel,[6][7]
where K(x,y;0) =δ(xy). The most general solution for a given initial configuration ψ(x,0) then is simply
Coherent statesEdit
The coherent states of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty σx σp = 2, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.
The coherent states are indexed by α ∈ ℂ and expressed in the |n⟩ basis as
Because and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: . The position space wave functions are
Highly excited statesEdit
Excited state with n=30, with the vertical lines indicating the turning points
When n is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy En can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.
The frequency of oscillation at x is proportional to the momentum p(x) of a classical particle of energy En and position x. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately
This is also given, asymptotically, by the integral
Phase space solutionsEdit
In the phase space formulation of quantum mechanics, solutions to the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution, which has the solution
and Ln are the Laguerre polynomials.
This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.
Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have
This claim can be verified using the Segal–Bargmann transform. Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply . At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.
N-dimensional harmonic oscillatorEdit
The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x1, ..., xN. Corresponding to each position coordinate is a momentum; we label these p1, ..., pN. The canonical commutation relations between these operators are
The Hamiltonian for this system is
As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x1, ..., xN would refer to the positions of each of the N particles. This is a convenient property of the potential, which allows the potential energy to be separated into terms depending on one coordinate each.
This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:
In the ladder operator method, we define N sets of ladder operators,
By an analogous procedure to the one-dimensional case, we can then show that each of the ai and ai operators lower and raise the energy by ℏω respectively. The Hamiltonian is
This Hamiltonian is invariant under the dynamic symmetry group U(N) (the unitary group in N dimensions), defined by
where is an element in the defining matrix representation of U(N).
The energy levels of the system are
As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional ground energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.
The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define n = n1 + n2 + n3. All states with the same n will have the same energy. For a given n, we choose a particular n1. Then n2 + n3 = n − n1. There are n − n1 + 1 possible pairs {n2n3}. n2 can take on the values 0 to n − n1, and for each n2 the value of n3 is fixed. The degree of degeneracy therefore is:
Formula for general N and n [gn being the dimension of the symmetric irreducible nth power representation of the unitary group U(N)]:
The special case N = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particles, or one particle in N dimensions (as dimensions are distinguishable). For the case of N bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer n using integers less than or equal to N.
This arises due to the constraint of putting N quanta into a state ket where and , which are the same constraints as in integer partition.
Example: 3D isotropic harmonic oscillatorEdit
Schrödinger 3D Spherical Harmonic orbital solutions in 2D Density plots with Mathematica(TM) generating source code snippet at the top
The Schrödinger equation of a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables; see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with the spherically symmetric potential
where μ is the mass of the problem. Because m will be used below for the magnetic quantum number, mass is indicated by μ, instead of m, as earlier in this article.
The solution reads[8]
is a normalization constant; ;
are generalized Laguerre polynomials; The order k of the polynomial is a non-negative integer;
is a spherical harmonic function;
ħ is the reduced Planck constant:
The energy eigenvalue is
The energy is usually described by the single quantum number
Because k is a non-negative integer, for every even n we have ℓ = 0, 2, ..., n − 2, n and for every odd n we have ℓ = 1, 3, ..., n − 2, n . The magnetic quantum number m is an integer satisfying −ℓ ≤ m ≤ ℓ, so for every n and ℓ there are 2 + 1 different quantum states, labeled by m . Thus, the degeneracy at level n is
where the sum starts from 0 or 1, according to whether n is even or odd. This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of SU(3),[9] the relevant degeneracy group.
Harmonic oscillators lattice: phononsEdit
We can extend the notion of a harmonic oscillator to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.
As in the previous section, we denote the positions of the masses by x1,x2,..., as measured from their equilibrium positions (i.e. xi = 0 if the particle i is at its equilibrium position). In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is
where m is the (assumed uniform) mass of each atom, and xi and pi are the position and momentum operators for the i th atom and the sum is made over the nearest neighbors (nn). However, it is customary to rewrite the Hamiltonian in terms of the normal modes of the wavevector rather than in terms of the particle coordinates so that one can work in the more convenient Fourier space.
We introduce, then, a set of N "normal coordinates" Qk, defined as the discrete Fourier transforms of the xs, and N "conjugate momenta" Π defined as the Fourier transforms of the ps,
The quantity kn will turn out to be the wave number of the phonon, i.e. 2π divided by the wavelength. It takes on quantized values, because the number of atoms is finite.
This preserves the desired commutation relations in either real space or wave vector space
From the general result
it is easy to show, through elementary trigonometry, that the potential energy term is
The Hamiltonian may be written in wave vector space as
Note that the couplings between the position variables have been transformed away; if the Qs and Πs were hermitian(which they are not), the transformed Hamiltonian would describe N uncoupled harmonic oscillators.
The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the (N + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
The harmonic oscillator eigenvalues or energy levels for the mode ωk are
If we ignore the zero-point energy then the levels are evenly spaced at
So an exact amount of energy ħω, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.
All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later.[10]
In the continuum limit, a→0, N→∞, while Na is held fixed. The canonical coordinates Qk devolve to the decoupled momentum modes of a scalar field, , whilst the location index i (not the displacement dynamical variable) becomes the parameter x argument of the scalar field, .
Molecular vibrationsEdit
• The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by
where is the reduced mass and and are the masses of the two atoms.[11]
• The Hooke's atom is a simple model of the helium atom using the quantum harmonic oscillator.
• Modelling phonons, as discussed above.
• A charge , with mass , in a uniform magnetic field , is an example of a one-dimensional quantum harmonic oscillator: the Landau quantization.
See alsoEdit
1. ^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-805326-0.
2. ^ Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison–Wesley. ISBN 978-0-8053-8714-8.
3. ^ Rashid, Muneer A. (2006). "Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian" (PDF-Microsoft PowerPoint). M.A. Rashid – Center for Advanced Mathematics and Physics. National Center for Physics. Retrieved 19 October 2010.
4. ^ The normalization constant is , and satisfies the normalization condition .
5. ^ See Theorem 11.4 in Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158
6. ^ Pauli, W. (2000), Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics). ISBN 978-0486414621 ; Section 44.
7. ^ Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158–164. online
8. ^ Albert Messiah, Quantum Mechanics, 1967, North-Holland, Ch XII, § 15, p
9. ^ Fradkin, D. M. "Three-dimensional isotropic harmonic oscillator and SU3." American Journal of Physics 33 (3) (1965) 207-211.
10. ^ Mahan, GD (1981). many particle physics. New York: springer. ISBN 978-0306463389.
11. ^ "Quantum Harmonic Oscillator". Hyperphysics. Retrieved 24 September 2009.
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Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky. A discrete parametrized surface theory in R3. International Mathematics Research Notices, 2017(14):4217-4258, 2017. [ bib | DOI | arXiv ]
Tim Hoffmann. On local deformations of planar quad-meshes. In Proceedings of the Third International Congress Conference on Mathematical Software, ICMS'10, pages 167-169, Berlin, Heidelberg, 2010. Springer-Verlag. [ bib | http ]
T. Hoffmann. Discrete Differential Geometry of Curves and Surfaces, volume 18 of MI Lecture Note Series. Faculty of Mathematics, Kyushu University, 2009. [ bib ]
Steffen Weißmann, Charles Gunn, Peter Brinkmann, Tim Hoffmann, and Ulrich Pinkall. jreality: a java library for real-time interactive 3d graphics and audio. In MM '09: Proceedings of the seventeen ACM international conference on Multimedia, pages 927-928, New York, NY, USA, 2009. ACM. [ bib | DOI ]
Tim Hoffmann. Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow. In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 95-115. Birhkäuser Basel, 2008. [ bib | arXiv ]
In this paper Bäcklund transformations for smooth and space discrete Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Hashimoto or smoke-ring flow. A doubly discrete Hashimoto flow is derived and it is shown that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.
W. K. Schief, A. I. Bobenko, and Hoffmann T. On the integrability of infinitesimal and finite deformations of polyhedral surfaces. In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 67-93. Birkhäuser Basel, 2008. [ bib ]
T. Hoffmann and M. Schmies. jreality, jtem, and oorange - a way to do math with computers. In ICMS, volume 4151 of Lecture Notes in Computer Science, pages 74-85. Springer, 2006. [ bib | http ]
A. Bobenko, T. Hoffmann, and B. Springborn. Minimal surfaces from circle patterns: Geometry from combinatorics. Ann. Math., 164(1):231-264, 2006. [ bib | arXiv ]
T. Hoffmann and N. Kutz. Discrete curves in CP1 and the Toda lattice. Stud. Appl. Math., 113(1):31-55, 2004. [ bib | arXiv ]
In this paper we investigate flows on discrete curves in 2, 1, , and 2. A novel interpretation of the one dimensional Toda lattice hierachy and reductions thereof as flows on discrete curves will be found.
A. I. Bobenko and T. Hoffmann. Hexagonal circle patterns and integrable systems. patterns with constant angles. Duke Math. J., 116(3), 2003. [ bib | arXiv ]
Hexagonal circle patterns with constant intersection angles are introduced and stud- ied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings zc and log z are constructed as special isomonodromic solutions. Circle patterns studied in the paper include Schramm's circle patterns with the combina- torics of the square grid as a special case.
A. I. Bobenko, T. Hoffmann, and Suris Yu. B. Hexagonal circle patterns and integrable systems. patterns with the multi-ratio property and lax equations on the regular triangular lattice. Int. Math. Res. Notices, 3:111-164, 2002. [ bib | arXiv ]
Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to −1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B ̈acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogs to the analytic functions zα and log z.
Tim Hoffmann. jDvi - a way to put interactive TeX on the web. In Multimedia Tools for Communicating Mathematics, pages 117-130. Springer-Verlag, Berlin, Heidelberg, 2002. [ bib ]
A. Bobenko and T. Hoffmann. Conformally symmetric circle packings. a generalization of Doyle spirals. J. Exp. Math., 10(1), 2001. [ bib | arXiv ]
T. Hoffmann, J. Kellendonk, N. Kutz, and N. Reshetikhin. Factorization dynamics and Coxeter-Toda lattices. Comm. Math. Phys., 212(2):297-321, 2000. [ bib | arXiv ]
T. Hoffmann. On the equivalence of the discrete nonlinear Schrödinger equation and the discrete isotropic Heisenberg magnet. Phys Lett. A, 265(1-2):62-67, 2000. [ bib ]
T. Hoffmann. Discrete curves and surfaces. PhD thesis, Technische Universität Berlin, 2000. [ bib | .pdf ]
T. Hoffmann. Discrete amsler surfaces and a discretePainlevé III equation. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 83-96. Oxford University Press, 1999. [ bib ]
T. Hoffmann. Discrete cmc surfaces and discrete holomorphic maps. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 97-112. Oxford University Press, 1999. [ bib ]
U. Hertrich-Jeromin, T. Hoffmann, and Pinkall U. A discrete version of the Darboux transform for isothermic surfaces. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 59-81. Oxford University Press, 1999. [ bib | arXiv ]
T. Hoffmann. Rauchringe in der mathematik/smoke rings in mathmatics. International forum man and architecture, 27/28, 1999. [ bib ]
T. Hoffmann. Discrete rotational cmc surfaces and the elliptic billiard. In H.-C. Hege and K. Polthier, editors, Mathematical Visualisation, pages 117-124. Springer, 1998. [ bib ]
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d583b3d6aa527e41 | Nov 212018
On this date in 1676, the Danish astronomer Ole Rømer published the first quantitative measurements of the speed of light. Until the early modern period, it was not known whether light travelled instantaneously or at a very fast finite speed. The first extant recorded examination of this subject was in ancient Greece. The ancient Greeks, Muslim scholars, and classical European scientists long debated this until Rømer provided the first calculation of the speed of light. Einstein’s Theory of Special Relativity concluded that the speed of light is constant regardless of one’s frame of reference. That is, if you are traveling towards a light source or away from it or stationary in relation to it, the light from the source comes at you at exactly the same speed. That is an astounding fact that most people fail to grasp. Today is also a milestone for Einstein and the speed of light which I posted on three years ago
Empedocles (c. 490–430 BC) was the first person to propose a theory of light, as far as we know, and he claimed that light has a finite speed. He maintained that light was something in motion, and therefore must take some time to travel. Aristotle argued, to the contrary, that “light is due to the presence of something, but it is not a movement.” Euclid and Ptolemy advanced Empedocles’ emission theory of vision, arguing that light is emitted from the eye, thus enabling sight. Based on that theory, Heron of Alexandria argued that the speed of light must be infinite because distant objects such as stars appear immediately upon opening the eyes.
Early Islamic philosophers initially agreed with the Aristotelian view that light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the Book of Optics, in which he presented a series of arguments dismissing the emission theory of vision in favor of the now accepted intromission theory, in which light moves from an object into the eye. This led Alhazen to propose that light must have a finite speed, and that the speed of light is variable, decreasing in denser bodies. He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from our senses. Also in the 11th century, Abū Rayhān al-Bīrūnī agreed that light has a finite speed, and observed that the speed of light is much faster than the speed of sound. In the 13th century, Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle. In the 1270s, the friar/natural philosopher Witelo considered the possibility of light traveling at infinite speed in vacuum, but slowing down in denser bodies.
In the early 17th century, Johannes Kepler believed that the speed of light was infinite, since empty space presents no obstacle to it. René Descartes argued that if the speed of light were to be finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. Since such misalignment had not been observed, Descartes concluded the speed of light was infinite. Descartes speculated that if the speed of light were found to be finite, his whole system of philosophy might be demolished. In Descartes’ derivation of Snell’s law (concerning the angle that light refracts when passing through media of different densities), he assumed that even though the speed of light was instantaneous, the denser the medium, the faster was light’s speed. Pierre de Fermat derived Snell’s law using the opposing assumption, the denser the medium the slower light traveled. Fermat also argued in support of a finite speed of light – and, of course, if you know your physics, Fermat was right and Descartes was wrong.
In 1629, Isaac Beeckman proposed an experiment in which a person observes the flash of a cannon reflecting off a mirror about one mile (1.6 km) away. In 1638, Galileo Galilei proposed an experiment, with an apparent claim to having performed it some years earlier, to measure the speed of light by observing the delay between uncovering a lantern and its perception some distance away. He was unable to distinguish whether light travel was instantaneous or not, but concluded that if it were not, it must nevertheless be extraordinarily rapid. In 1667, the Accademia del Cimento of Florence reported that it had performed Galileo’s experiment, with the lanterns separated by about one mile, but no delay was observed. The actual delay in this experiment would have been about 11 microseconds.
The first quantitative estimate of the speed of light was made in 1676 by Rømer. From the observation that the periods of Jupiter’s innermost moon Io appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth’s orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth’s orbit to obtain an estimate of speed of light of 220000 km/s, 26% lower than the actual value.
In his 1704 book Opticks, Isaac Newton reported Rømer’s calculations of the finite speed of light and gave a value of “seven or eight minutes” for the time taken for light to travel from the Sun to the Earth (the modern value is 8 minutes 19 seconds). Newton queried whether Rømer’s eclipse shadows were colored; hearing that they were not, he concluded the different colors traveled at the same speed. In 1729, James Bradley discovered stellar aberration. From this effect he determined that light must travel 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.
I’ll return to molecular gastronomy one more time for this physics post to be consistent, even though there’s an awful lot of spherical liquid things involved. It does get a tad tiresome after a while.
Jun 132018
Today is the birthday (1773) of Thomas Young FRS, an English polymath, called “The Last Man Who Knew Everything” by Andrew Robinson in his biography, subtitled, Thomas Young, the Anonymous Polymath Who Proved Newton Wrong, Explained How We See, Cured the Sick, and Deciphered the Rosetta Stone, Among Other Feats of Genius. Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony, and Egyptology. He was mentioned favorably by, among others, William Herschel, Hermann von Helmholtz, James Clerk Maxwell, and Albert Einstein. It’s also Maxwell’s birthday today, by the way:
Young was born in Milverton in Somerset, the eldest of 10 children in a Quaker family. By the age of 14 Young had learned Greek and Latin and was acquainted with French, Italian, Hebrew, German, Aramaic, Syriac, Samaritan, Arabic, Persian, Turkish and Amharic. He began to study medicine in London at St Bartholomew’s Hospital in 1792, moved to the University of Edinburgh Medical School in 1794, and a year later went to the University of Göttingen in Lower Saxony where he obtained the degree of doctor of medicine in 1796. In 1797 he entered Emmanuel College, Cambridge. In the same year he inherited the estate of his grand-uncle, Richard Brocklesby, which made him financially independent, and in 1799 he established himself as a physician at 48 Welbeck Street, London (now recorded with a blue plaque). Young published many of his first academic articles anonymously to protect his reputation as a physician.
In 1801, Young was appointed professor of natural philosophy (mainly physics) at the Royal Institution. In two years, he delivered 91 lectures. In 1802, he was appointed foreign secretary of the Royal Society, of which he had been elected a fellow in 1794. He resigned his professorship in 1803, fearing that its duties would interfere with his medical practice. His lectures were published in 1807 in the Course of Lectures on Natural Philosophy and contain a number of anticipations of later theories. In 1811, Young became physician to St George’s Hospital, and in 1814 he served on a committee appointed to consider the dangers involved in the general introduction of gas for lighting into London. In 1816 he was secretary of a commission charged with ascertaining the precise length of the seconds pendulum (the length of a pendulum whose period is exactly 2 seconds), and in 1818 he became secretary to the Board of Longitude and superintendent of the HM Nautical Almanac Office.
Young was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1822. A few years before his death he became interested in life insurance, and in 1827 he was chosen one of the eight foreign associates of the French Academy of Sciences. In 1828, he was elected a foreign member of the Royal Swedish Academy of Sciences. He died in London on 10th May 1829, and was buried in the cemetery of St. Giles Church in Farnborough, Kent, England. Westminster Abbey houses a white marble tablet in memory of Young bearing an extended epitaph by Hudson Gurney:
Sacred to the memory of Thomas Young, M.D., Fellow and Foreign Secretary of the Royal Society Member of the National Institute of France; a man alike eminent in almost every department of human learning. Patient of unintermitted labour, endowed with the faculty of intuitive perception, who, bringing an equal mastery to the most abstruse investigations of letters and of science, first established the undulatory theory of light, and first penetrated the obscurity which had veiled for ages the hieroglyphs of Egypt. Endeared to his friends by his domestic virtues, honoured by the World for his unrivalled acquirements, he died in the hopes of the Resurrection of the just. — Born at Milverton, in Somersetshire, 13 June 1773. Died in Park Square, London, 10 May 1829, in the 56th year of his age.
Young was highly regarded by his friends and colleagues. He was said never to impose his knowledge, but if asked was able to answer even the most difficult scientific question with ease. Although very learned he had a reputation for sometimes having difficulty in communicating his knowledge. It was said by one of his contemporaries that, “His words were not those in familiar use, and the arrangement of his ideas seldom the same as those he conversed with. He was therefore worse calculated than any man I ever knew for the communication of knowledge.” Young is quite well known by scholars in different fields but they usually know him only for his work in their specialties, not as a polymath.
I’ll just list briefly the areas where he made significant contributions – with a small synopsis.
Wave theory of light
In Young’s own judgment, of his many achievements the most important was to establish the wave theory of light. To do so, he had to overcome the view, expressed in the highly esteemed Isaac Newton’s Opticks, that light is a particle. Nevertheless, in the early-19th century Young put forth a number of theoretical reasons supporting the wave theory of light, and he developed two enduring demonstrations to support this viewpoint. With the ripple tank he demonstrated the idea of interference in the context of water waves. With his interference experiment (the now-classic double-slit experiment), he demonstrated interference in the context of light as a wave.
After publishing a paper on interference, he published a paper entitled “Experiments and Calculations Relative to Physical Optics” in 1804. Young describes an experiment in which he placed a narrow card (approximately 1/30th inch) in a beam of light from a single opening in a window and observed the fringes of color in the shadow and to the sides of the card. He observed that placing another card before or after the narrow strip so as to prevent light from the beam from striking one of its edges caused the fringes to disappear. This supported the contention that light is composed of waves. Young performed and analyzed a number of experiments, including interference of light from reflection off nearby pairs of micrometer grooves, from reflection off thin films of soap and oil, and from Newton’s rings. He also performed two important diffraction experiments using fibers and long narrow strips. In his Course of Lectures on Natural Philosophy and the Mechanical Arts (1807) he gives Grimaldi credit for first observing the fringes in the shadow of an object placed in a beam of light. Within ten years, much of Young’s work was reproduced and then extended by others.
Young’s modulus
Engineers all know Young’s modulus, which describes the elasticity of materials beyond the limits of Hook’s Law. Hook’s Law describes the direct, proportional correlation between the load on a spring, and the extension of the spring “provided the load is not too great.” The proviso is there because if the load is “too great” all bets are off. Young’s modulus takes care of that. Young described his findings in his Course of Lectures on Natural Philosophy and the Mechanical Arts. However, the first use of the concept of Young’s modulus in experiments was by Giordano Riccati in 1782, predating Young by 25 years. Furthermore, the idea can be traced to a paper by Leonhard Euler published in 1727, 80 years before Young’s 1807 paper on the subject. Nonetheless, Young’s application was the one generally adopted by engineers. Young’s Modulus allowed, for the first time, prediction of the strain in a component subject to a known stress (and vice versa). Prior to Young’s contribution, engineers were required to apply Hooke’s F = kx relationship to identify the deformation (x) of a body subject to a known load (F), where the constant (k) is a function of both the geometry and material under consideration. Finding k required physical testing for any new component, as the F = kx relationship is a function of both geometry and material. Young’s Modulus depends only on the material, not its geometry, thus allowing a revolution in engineering strategies.
Vision and color theory
Young has sometimes been called the founder of physiological optics. In 1793 he explained the mode in which the eye accommodates itself to vision at different distances as depending on change of the curvature of the crystalline lens; in 1801 he was the first to describe astigmatism; and in his lectures he presented the hypothesis, afterwards developed by Hermann von Helmholtz, (the Young–Helmholtz theory), that color perception depends on the presence in the retina of three kinds of nerve fibers. This foreshadowed the modern understanding of color vision, in particular the finding that the eye does indeed have three colour receptors which are sensitive to different wavelength ranges.
Young–Laplace equation
In 1804, Young developed the theory of capillary action based on the principle of surface tension. He also observed the constancy of the angle of contact of a liquid surface with a solid, and showed how to deduce the phenomenon of capillary action from these two principles. In 1805, Pierre-Simon Laplace, the French philosopher, discovered the significance of meniscus radii with respect to capillary action. In 1830, Carl Friedrich Gauss, the German mathematician, unified the work of these two scientists to derive the Young–Laplace equation, the formula that describes the capillary pressure difference sustained across the interface between two static fluids. Young’s equation describes the contact angle of a liquid drop on a plane solid surface as a function of the surface free energy, the interfacial free energy and the surface tension of the liquid. Young’s equation was developed further some 60 years later by Dupré to account for thermodynamic effects, and this is known as the Young–Dupré equation.
In physiology Young made an important contribution to haemodynamics in the Croonian lecture for 1808 on the “Functions of the Heart and Arteries,” where he derived a formula for the wave speed of the pulse and his medical writings included An Introduction to Medical Literature, including a System of Practical Nosology (1813) and A Practical and Historical Treatise on Consumptive Diseases (1815). Young devised a rule of thumb for determining a child’s drug dosage. Young’s Rule states that the child dosage is equal to the adult dosage multiplied by the child’s age in years, divided by the sum of 12 plus the child’s age.
In an appendix to his Göttingen dissertation (1796; “De corporis hvmani viribvs conservatricibvs. Dissertatio.”) there are four pages added proposing a universal phonetic alphabet (so as ‘not to leave these pages blank’ – Ne vacuae starent hae paginae, libuit e praelectione ante disputationem habenda tabellam literarum vniuersalem raptim describere”). It includes 16 “pure” vowel symbols, nasal vowels, various consonants, and examples of these, drawn primarily from French and English. In his Encyclopædia Britannica article “Languages”, Young compared the grammar and vocabulary of 400 languages. In a separate work in 1813, he introduced the term “Indo-European” languages, 165 years after the Dutch linguist Marcus Zuerius van Boxhorn proposed the grouping to which this term refers in 1647.
Egyptian hieroglyphs
Young made significant contributions in the decipherment of Egyptian hieroglyphs. He started his Egyptology work rather late, in 1813, when the work was already in progress among other researchers. He began by using an Egyptian demotic alphabet of 29 letters built up by Johan David Åkerblad in 1802 (14 turned out to be incorrect). Åkerblad was correct in stressing the importance of the demotic text in trying to read the inscriptions, but he wrongly believed that demotic was entirely alphabetic. By 1814 Young had completely translated the “enchorial” text of the Rosetta Stone (using a list with 86 demotic words), and then studied the hieroglyphic alphabet but initially failed to recognize that the demotic and hieroglyphic texts were paraphrases and not simple translations. There was considerable rivalry between Young and Jean-François Champollion while both were working on hieroglyphic decipherment. At first they briefly cooperated in their work, but later, from around 1815, a chill arose between them. For many years they kept details of their work away from each other. Some of Young’s conclusions appeared in the famous article “Egypt” he wrote for the 1818 edition of the Encyclopædia Britannica. When Champollion finally published a translation of the hieroglyphs and the key to the grammatical system in 1822, Young (and many others) praised his work. Nevertheless, a year later Young published an Account of the Recent Discoveries in Hieroglyphic Literature and Egyptian Antiquities, to have his own work recognized as the basis for Champollion’s system. Young had correctly found the sound value of six hieroglyphic signs, but had not deduced the grammar of the language. Young, himself, acknowledged that he was somewhat at a disadvantage because Champollion’s knowledge of the relevant languages, such as Coptic, was much greater. Several scholars have suggested that Young’s true contribution to Egyptology was his decipherment of the demotic script. He made the first major advances in this area. He also correctly identified demotic as being composed of both ideographic and phonetic signs.
Young developed two systems of tuning a piano so that it was well tempered (Wohltemperiert), that is, was tuned so as to be able to modulate between all major and minor scales without sounding obviously out of tune in any of them. Discussions of temperaments get really technical really quickly. Young’s first temperament was designed to sound best in the keys that were the commonest, and his second was a kind of inversion of the first. Unless you know the difference between BƄ and A#, and the differences that their major and minor thirds make in chords, this will not make any sense to you. It is a problem in the physics of acoustics, essentially.
Historians and critics vary enormously in their assessment of Young. Without question he was well versed in all the fields above – and more – and was able to expound on them critically (if not always clearly). How original his contributions were to the various fields, is the subject of ongoing debate. The idea than he was the last man to know everything, is obvious (and intentional) hyperbole. But it also highlights the fact that at the beginning of the 19th century it was still possible to gain expert knowledge in widely diverse fields. Furthermore, Young not only knew a lot of stuff, he was able to make contributions to diverse fields. Whether or not he was always entirely original is beside the point as far as I am concerned. We’re talking about a man who made contributions – recognized as significant by experts – in half a dozen specialties, that most of us do not even understand, let alone are capable of mastering.
As I have done quite a number of times with birthdays recently, I’ll celebrate Young with a recipe from his home region, Somerset. Somerset is well known for apples, cider, and dairying, and this recipe for Somerset chicken, which is traditional, combines all three.
Somerset Chicken
6 boneless chicken breasts, skin on
salt and freshly ground black pepper
75 gm/2½ oz butter
3 tbsp olive oil
2 onions, peeled and sliced
4 tbsp plain flour
2 tbsp wholegrain mustard
2 dessert apples, peeled, cored and sliced
110 gm/4 oz button mushrooms, sliced
250 ml/9 fl oz chicken stock
300 ml/10½ fl oz cider
1 tbsp finely chopped fresh sage
250 ml/9 fl oz double cream
300 gm/10½ oz cheddar cheese, grated
Preheat the oven to 200˚C/400˚F.
Season the chicken breasts with salt and freshly ground black pepper.
Heat a large skillet until smoking, then add half of the butter and oil. Fry the chicken breasts in batches, skin-side down first, for 5 minutes on each side, making sure they are golden-brown all over. Transfer the chicken breasts to a baking dish and keep warm.
Return the skillet to the heat and add the remaining butter and oil. Add the onions and cook for 4-5 minutes, or until softened but without taking on color. Stir in the flour and the mustard and cook for a further 1-2 minutes. Add the apples and mushrooms and cook for a further minute, then pour the chicken stock over ingredients.
Bring the skillet to the boil, add the cider and return to the boil. Cook for 1-2 minutes, then lower the heat, add the sage and stir in the cream. Simmer for a further 5-6 minutes, then season with salt and freshly ground black pepper to taste.
Pour the sauce over the chicken in the baking pan.
Preheat the broiler to high.
Sprinkle the cheddar cheese over the chicken and place under the broiler for 4-5 minutes, or until the cheese is melted, golden-brown and bubbling.
Serve with baked or boiled new potatoes.
Feb 182018
Today is the birthday (1838) of Ernst Waldfried Josef Wenzel Mach, Austrian physicist and philosopher. The ratio of an object’s speed to that of sound is named the Mach number in his honor. As a philosopher of science, he was a major influence on logical positivism and American pragmatism. Through his criticism of Newton’s theories of space and time, he foreshadowed Einstein’s theory of relativity.
Mach was born in Chrlice (German: Chirlitz) in Moravia (then in the Austrian empire, now part of Brno in the Czech Republic). His father, who had attended Charles University in Prague, acted as tutor to the noble Brethon family in Zlín in eastern Moravia. Up to the age of 14, Mach received his education at home from his parents. He then entered a Gymnasium in Kroměříž (German: Kremsier), where he studied for 3 years. In 1855 he became a student at the University of Vienna. There he studied physics and medical physiology, receiving his doctorate in physics in 1860 under Andreas von Ettingshausen with a thesis titled “Über elektrische Ladungen und Induktion”, and his habilitation the following year. His early work focused on the Doppler effect in optics and acoustics. In 1864 he took a job as Professor of Mathematics at the University of Graz, having turned down the position of a chair in surgery at the University of Salzburg to do so, and in 1866 he was appointed as Professor of Physics. During that period, Mach continued his work in psycho-physics and in sensory perception. In 1867, he took the chair of Experimental Physics at the Charles University, Prague, where he stayed for 28 years before returning to Vienna.
Mach’s main contribution to physics involved his description and photographs of spark shock-waves and then ballistic shock-waves. He described how when a bullet or shell moved faster than the speed of sound, it created a compression of air in front of it. Using schlieren photography, he and his son Ludwig were able to photograph the shadows of the invisible shock waves. During the early 1890s Ludwig was able to invent an interferometer which allowed for much clearer photographs. But Mach also made many contributions to psychology and physiology, including his anticipation of gestalt phenomena, his discovery of the oblique effect and of Mach bands, an inhibition-influenced type of visual illusion, and especially his discovery of a non-acoustic function of the inner ear which helps control human balance.
One of the best-known of Mach’s ideas is the so-called “Mach principle,” the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The idea is that local inertial frames are determined by the large-scale distribution of matter, as exemplified by this anecdote:
You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don’t move?
Mach’s principle says that this is not a coincidence—that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force. There are a number of rival formulations of the principle. It is often stated in vague ways, like “mass out there influences inertia here”. A very general statement of Mach’s principle is “local physical laws are determined by the large-scale structure of the universe.” This concept was a guiding factor in Einstein’s development of the general theory of relativity. Einstein realized that the overall distribution of matter would determine the metric tensor, which tells you which frame is rotationally stationary
Mach also became well known for his philosophy developed in close interplay with his science. Mach defended a type of phenomenalism recognizing only sensations as real. This position seemed incompatible with the view of atoms and molecules as external, mind-independent things. He famously declared, after an 1897 lecture by Ludwig Boltzmann at the Imperial Academy of Science in Vienna: “I don’t believe that atoms exist!” From about 1908 to 1911 Mach’s reluctance to acknowledge the reality of atoms was criticized by Max Planck as being incompatible with physics. Einstein’s 1905 demonstration that the statistical fluctuations of atoms allowed measurement of their existence without direct individuated sensory evidence marked a turning point in the acceptance of atomic theory. Some of Mach’s criticisms of Newton’s position on space and time influenced Einstein, but later Einstein realized that Mach was basically opposed to Newton’s philosophy and concluded that his physical criticism was not sound.
In 1898 Mach suffered from cardiac arrest and in 1901 retired from the University of Vienna and was appointed to the upper chamber of the Austrian parliament. On leaving Vienna in 1913 he moved to his son’s home in Vaterstetten, near Munich, where he continued writing and corresponding until his death in 1916, only one day after his 78th birthday.
Most of Mach’s initial studies in the field of experimental physics concentrated on the interference, diffraction, polarization and refraction of light in different media under external influences. From there followed important explorations in the field of supersonic fluid mechanics. Mach and physicist-photographer Peter Salcher presented their paper on this subject in 1887; it correctly describes the sound effects observed during the supersonic motion of a projectile. They deduced and experimentally confirmed the existence of a shock wave of conical shape, with the projectile at the apex. The ratio of the speed of a fluid to the local speed of sound vp/vs is now called the Mach number. It is a critical parameter in the description of high-speed fluid movement in aerodynamics and hydrodynamics.
From 1895 to 1901, Mach held a newly created chair for “the history and philosophy of the inductive sciences” at the University of Vienna. In his historico-philosophical studies, Mach developed a phenomenalistic philosophy of science which became influential in the 19th and 20th centuries. He originally saw scientific laws as summaries of experimental events, constructed for the purpose of making complex data comprehensible, but later emphasized mathematical functions as a more useful way to describe sensory appearances. Thus, scientific laws while somewhat idealized have more to do with describing sensations than with reality as it exists beyond sensations.
In accordance with empirio-critical philosophy, Mach opposed Ludwig Boltzmann and others who proposed an atomic theory of physics. Since one cannot observe things as small as atoms directly, and since no atomic model at the time was consistent, the atomic hypothesis seemed to Mach to be unwarranted, and perhaps not sufficiently “economical”. Mach had a direct influence on the Vienna Circle philosophers and the school of logical positivism in general.
According to Alexander Riegler, Ernst Mach’s work was a precursor to the influential perspective known as constructivism. Constructivism holds that all knowledge is constructed rather than received by the learner. He took an exceptionally non-dualist, phenomenological position. The founder of radical constructivism, von Glasersfeld, gave a nod to Mach as an ally.
In 1873, independently of each other Mach and the physiologist and physician Josef Breuer discovered how the sense of balance (i.e., the perception of the head’s imbalance) functions, tracing its management by information which the brain receives from the movement of a fluid in the semicircular canals of the inner ear. That the sense of balance depended on the three semicircular canals was discovered in 1870 by the physiologist Friedrich Goltz, but Goltz did not discover how the balance-sensing apparatus functioned. Mach devised a swivel chair to enable him to test his theories, and Floyd Ratliff has suggested that this experiment may have paved the way to Mach’s critique of a physical conception of absolute space and motion.
Mach’s home town of Brno is in Moravia which is now part of the Czech Republic, and much of the cuisine is common to the nation as a whole. But there are some distinctive dishes. Moravian chicken pie is one. It can be made as a simple two-crust pie, but is often made with a crumb topping as well, as in this recipe.
Moravian Chicken Pie
Pie Crust
2 cups all-purpose flour
1 tsp salt
3⁄4 cup shortening
6 -8 tbsp cold water
2 ½ cups chopped cooked chicken
salt and pepper
3 tbsp flour
1 cup chicken broth
1 -2 tbsp butter, cut in small pieces
Crumb Topping
¼ cup all-purpose flour
1 tbsp butter
For the pie crust: combine the flour and salt in a food processor. Add the shortening and pulse until the mixture is like coarse cornmeal. Gradually stir in cold water just until a dough forms. Divide the dough into two equal pieces. Cover and chill 30 minutes, or until ready to use.
Preheat the oven to 375˚F/190˚C degrees.
Roll out one piece of dough to cover the bottom and sides of a 9-inch pie plate and place in the plate. Roll out the second piece of dough for the top crust and set aside.
For the filling: combine all the ingredients in a bowl and season with salt and pepper to taste. Pour the ingredients into the pie crust and top with the second crust, moisten the edges, and crimp to seal.
For the crumb topping: pulse the butter and flour in a food processor until it is like coarse cornmeal. Sprinkle the topping over the top crust of pie. Cut a few slits in the top crust to allow steam to escape.
Bake the pie 45 minutes to 1 hour, until golden and bubbly.
Apr 282016
If a system is consistent, it cannot be complete.
The consistency of the axioms cannot be proven within the system.
Sauerkraut Soup
7 oz sauerkraut
6 cups light broth
2 potatoes, diced
2 tbs flour
1 onion, diced
4 slices bacon, finely diced
1 large sausage, sliced thinly
5-10 white mushrooms, diced
1 tsp caraway seeds
1-2 tsp sweet paprika
Nov 212015
On this date in 1905 Albert Einstein’s paper, “Does the Inertia of a Body Depend Upon Its Energy Content?” was published in the journal Annalen der Physik. This paper explored the relationship between energy and mass via Special Relativity, and, thus, led to the mass–energy equivalence formula E = mc² — arguably the most famous formula in the world. I would also argue that it is the most misunderstood formula in the world, although I notice in researching this post that a lot of physicists in trying to help non-physicists understand it, seriously misrepresent its implications.
The problem frequently in trying to explain physics to the mathematically and scientifically challenged is that scientists and science teachers fall back on analogies – often involving cats for some inscrutable reason. The problem, as I have stated many times in many places before, is that analogies can help, but they can also be misleading.
There is a second problem in that E = mc² does not represent the whole story. That’s the part about theory that non-scientists rarely get. Einstein’s theories of relativity, Darwin’s theory of natural selection, etc. are not complete, hence they are called “theories.” No one is seeking radically new alternatives (although some day they might); scientists are just trying to explain messy bits in the theories that cannot be explained now. That’s how Einstein came to unravel Newton. Newton was not totally wrong; it’s just that his “laws” of motion, for example, are incomplete – as stated by Newton they apply only to mass, force, acceleration, etc. as we encounter them in the everyday world. When physicists started looking at interstellar, and subatomic worlds at the turn of the 20th century, Newton’s physics did not work very well for them. That’s when Einstein came along and added bits to Newton to make his equations more encompassing.
Here’s a couple of provisos before I get into things more. First, for the non-mathematically inclined I am going to have to be simplistic and in doing so I will have to be a little misleading, or, you might say, downright wrong. The only way to understand physics deeply is to understand the underlying mathematics deeply (which, incidentally, I don’t, although I am better at it than most non-scientists). Second, my usual caveat, I don’t find physics per se very interesting. My son switched from being a physics major to an anthropology major a few years ago for precisely the same reason. Physics does very well in helping us build computers, cell phones, and what not, and I use them all the time. Thanks physics. It is useless when it comes to issues that I really care about such as the existence of God, how to mend a broken heart, and so forth. To be sure, philosophers and theologians can sometimes gain insight into problems they are working on by learning some physics, and vice versa. But the one realm cannot explain the other. Their methods and goals are radically different. When it comes to understanding the formation of the universe as we now know it, I’ll study physics; when it comes to understanding God, I’ll read the Bible and other spiritual texts. Both areas still have a long way to go.
The formula E = mc² is incomplete, but let’s stick with it for now. In the formula, E is energy, m is mass, and c is the velocity of light (here it is squared). Most people know that. Where they go drastically wrong is in thinking that mass = matter. That is false. Mass is mass, matter is matter, and energy is energy. E = mc² does not talk about matter directly, but about the relationship between energy and mass. It’s not about the conversion of matter into energy, as most people think the atom bomb or atomic energy are all about (as in the TIME cover photo). Einstein was not involved in the Manhattan project because he lacked the proper security clearance. But even if he had been, E = mc² has very little application in making a bomb. Atomic bombs and atomic energy concern releasing energy within the atom, not converting matter into energy as such. This is a bit of a semantic quibble, but at least you can get the general idea that matter is made up of particles and energy. Under certain conditions it is possible to set the energy free – and a little goes a long way.
In a nuclear bomb or energy plant, you are not converting the particles into energy; you are setting energy free that keeps the atoms together. It requires an incredible amount of energy to keep the particles of the atomic nucleus together, so, if you can tear them apart, you can release that energy. That’s why it’s called nuclear energy. You are not converting the particles into energy, you are simply setting it free. The particles remain as particles, just much less organized since nothing is holding them together.
So, then, what is it about E = mc²? It’s simply telling you that energy has mass, but it’s very, very small. Nonetheless, if you add energy to something, you increase its mass. For example, if you accelerate something it gains energy, therefore mass. At usual speeds the increase in mass is minute. But when you start approaching the speed of light you have to increase the energy input enormously, and in so doing, what you are pushing gets enormously massive. That fact is an important component of Special Relativity.
Explaining all of this will help you understand why I generally find physics dull. I’ll leave professional physics up to people who care about mathematical puzzles. I am not especially interested in how atomic bombs or cell phones work at a deep level. I’m much more interested in how and why people use (or don’t use) them, and for that kind of question physics is no help.
Physics burst on the scene a few years ago in the form of so-called “molecular” gastronomy. I’ve mentioned this fad before as a trend that I feel is more trickery than artistry – a way to amuse the eyes once in a while, but not much of an enhancement on classic cooking techniques. I expect it will vanish ere long. So . . . you can make spherical stuff, and foams, and “instant” frozen things. Big whoop. The equipment to do this is expensive, especially If you are only going to use it occasionally for a flashy dinner party. I will admit that I bought a rechargeable soda siphon once, about 40 years ago, which allowed me to make carbonated liquids – usually water. But it’s a whole lot cheaper to buy carbonated water than to have a machine. If I want something fizzy these days I’ll turn to chemistry and put a little sodium bicarbonate in an acidulated liquid. But I usually only do that when I have an upset stomach.
For the sake of completeness, though, here’s a video on making mock fried eggs with mango “yolks” and coconut milk “whites.” I expect they are delicious, but I’ll content myself with the video, and settle for mango balls in coconut milk for my next dessert.
Oct 072015
Today is the birthday (1885) of Niels Henrik David Bohr, a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research. Bohr developed the Bohr model of the atom, in which he proposed that energy levels of electrons are discrete and that the electrons revolve in stable orbits around the atomic nucleus but can jump from one energy level (or orbit) to another. Although the Bohr model has been supplanted by other models, its underlying principles remain valid. He also conceived the principle of complementarity: that items could be separately analyzed in terms of contradictory properties, like behaving as a wave or a stream of particles. The notion of complementarity dominated Bohr’s thinking in both science and philosophy.
I don’t want to delve too deeply into Bohr’s physics because I know I will lose a big chunk of my audience before I get started. But I will make one point before I fly off in different directions. Long-time readers know that I have a bee in my bonnet about certain superlatives – the BEST painting/painter or composer or mathematician or whatever. There are lots and lots of smart and talented people throughout history. If this were not so, this would be a very limited blog. My main limitation is that their birthdays are not spread evenly through the year, coupled with my intrinsic favoritism. In the latter case I am allowed because it is MY blog. I make the rules. What gets me wound up is the popular idea that the yardstick of hyper-genius is Einstein. He had a phenomenal mind – no question. He was, however, far from being the ONLY genius of the 20th century, yet his is the name that automatically comes to mind. I’ve always countered this prejudice when it comes up by mentioning Niels Bohr.
The first half of the 20th century was almost wallpapered with brilliant mathematicians and physicists, not to mention anthropologists, writers, philosophers, painters, and all the rest of it. Niels Bohr is one of them, but he is far from being a household name. Yet he helped usher in the age of quantum mechanics (with other brilliant minds), the dominant model of atomic physics to this day.
Bohr was born in Copenhagen, the second of three children of Christian Bohr, a professor of physiology at the University of Copenhagen, and Ellen Adler Bohr, who came from a wealthy Danish Jewish family prominent in banking and parliamentary circles. He had an elder sister, Jenny, and a younger brother Harald. Jenny became a teacher, while Harald became a mathematician and Olympic footballer who played for the Danish national team at the 1908 Summer Olympics in London. Niels was a passionate footballer as well, and the two brothers played several matches for the Copenhagen-based Akademisk Boldklub (Academic Football Club), with Niels as goalkeeper. My son and I love goalies.
In 1910, Bohr met Margrethe Nørlund, the sister of the mathematician Niels Erik Nørlund. Bohr resigned his membership in the Church of Denmark on 16 April 1912, and he and Margrethe were married in a civil ceremony at the town hall in Slagelse on 1 August. Their honeymoon was delayed, however, because Bohr had an insight into the nature of orbiting electrons within the atom that he felt could not wait. For reasons that are not clear to me, he was unable to sit and write the paper himself, so he dictated it to Margrethe. Maybe this was his idea of marital bliss?
Planetary models of atoms were fairly recent but not new. Bohr’s treatment was. The old planetary model could not explain why the negatively charged electron did not simply collapse into the positively charged nucleus. He advanced the theory of electrons travelling in nested orbits of different energies around the atom’s nucleus, with the chemical properties of each element being largely determined by the number of electrons in the outer orbits of its atoms. He introduced the idea that an electron could drop from a higher-energy orbit to a lower one, in the process emitting a quantum of discrete energy. This became a basis for what is now known as the old quantum theory. In 1922 Bohr received the Nobel Prize in physics for his work.
Bohr became convinced that light behaved like both waves and particles, and in 1927, experiments confirmed the de Broglie hypothesis that matter (like electrons) also behaved like waves. He conceived the philosophical principle of complementarity: that items could have apparently mutually exclusive properties, such as being a wave or a stream of particles, depending on the experimental framework. He felt that it was not fully understood by contemporary philosophers. Einstein never fully accepted quantum mechanics and complementarity. Einstein preferred the determinism of classical physics over the probabilistic new quantum physics to which he himself had contributed. Philosophical issues that arose from the novel aspects of quantum mechanics became widely celebrated subjects of discussion. Einstein and Bohr had good-natured arguments over such issues throughout their lives.
Bohr’s model of the atomic nucleus helped him explain the nature of nuclear fission which he published in a paper in 1939, “The Mechanism of Nuclear Fission,” along with John Wheeler. Thus the age of nuclear energy and the atom-bomb was born.
Bohr was aware of the possibility of using uranium-235 to construct an atomic bomb, referring to it in lectures in Britain and Denmark shortly before and after the war started, but he did not believe that it was technically feasible to extract a sufficient quantity of uranium-235 (fissionable material). In September 1941, Werner Heisenberg, who had become head of the German nuclear energy project, visited Bohr in Copenhagen. During this meeting the two men took a private moment outside, the content of which has caused much speculation, as both gave differing accounts. According to Heisenberg, he began to address nuclear energy, morality and the war, to which Bohr seems to have reacted by terminating the conversation abruptly while not giving Heisenberg hints about his own opinions.
In 1957, Heisenberg wrote to Robert Jungk, who was then working on the book Brighter than a Thousand Suns: A Personal History of the Atomic Scientists. Heisenberg explained that he had visited Copenhagen to communicate to Bohr the views of several German scientists, that production of a nuclear weapon was possible with great efforts, and this raised enormous responsibilities on the world’s scientists on both sides. When Bohr saw Jungk’s depiction in the Danish translation of the book, he drafted (but never sent) a letter to Heisenberg, stating that he never understood the purpose of Heisenberg’s visit, was shocked by Heisenberg’s opinion that Germany would win the war, and that atomic weapons could be decisive.
In September 1943, word reached Bohr and his brother Harald that the Nazis considered their family to be Jewish, since their mother, Ellen Adler Bohr, had been a Jew, and that they were therefore in danger of being arrested. The Danish resistance helped Bohr and his wife escape by sea to Sweden on 29 September. The next day, Bohr persuaded King Gustaf V of Sweden to make public Sweden’s willingness to provide asylum to Jewish refugees. On 2 October 1943, Swedish radio broadcast that Sweden was ready to offer asylum, and the mass rescue of the Danish Jews by their countrymen followed swiftly thereafter. Some historians claim that Bohr’s actions led directly to the mass rescue, while others say that, though Bohr did all that he could for his countrymen, his actions were not a decisive influence on the wider events. Eventually, over 7,000 Danish Jews escaped to Sweden.
When the news of Bohr’s escape reached Britain, Lord Cherwell sent a telegram to Bohr asking him to come to Britain. Bohr arrived in Scotland on 6 October in a de Havilland Mosquito operated by the British Overseas Airways Corporation (BOAC). The Mosquitos were unarmed high-speed bomber aircraft that had been converted to carry small, valuable cargoes or important passengers. By flying at high speed and high altitude, they could cross German-occupied Norway, and yet avoid German fighters. Bohr, equipped with parachute, flying suit and oxygen mask, spent the three-hour flight lying on a mattress in the aircraft’s bomb bay. During the flight, Bohr did not wear his flying helmet as it was too small, and consequently did not hear the pilot’s intercom instruction to turn on his oxygen supply when the aircraft climbed to high altitude to overfly Norway. He passed out from oxygen starvation and only revived when the aircraft descended to lower altitude over the North Sea.
On 8 December 1943, Bohr arrived in Washington, D.C., where he met with the director of the Manhattan Project, Brigadier General Leslie R. Groves, Jr,and went to Los Alamos in New Mexico, where the nuclear weapons were being designed. Bohr did not remain at Los Alamos, but paid a series of extended visits over the course of the next two years. Robert Oppenheimer credited Bohr with acting “as a scientific father figure to the younger men”, most notably Richard Feynman. Bohr is quoted as saying, “They didn’t need my help in making the atom bomb.”
Bohr recognized early that nuclear weapons would change international relations. In April 1944, he received a letter from Peter Kapitza, written some months before when Bohr was in Sweden, inviting him to come to the Soviet Union. The letter convinced Bohr that the Soviets were aware of the Anglo-American project, and would strive to catch up. He sent Kapitza a non-committal response, which he showed to the authorities in Britain before posting. Bohr met Churchill on 16 May 1944. Churchill disagreed with the idea of openness towards the Russians to the point that he wrote in a letter: “It seems to me Bohr ought to be confined or at any rate made to see that he is very near the edge of mortal crimes.”
Oppenheimer suggested that Bohr visit President Franklin D. Roosevelt to convince him that the Manhattan Project should be shared with the Soviets in the hope of speeding up its results. Bohr’s friend, Supreme Court Justice Felix Frankfurter, informed President Roosevelt about Bohr’s opinions, and a meeting between them took place on 26 August 1944. Roosevelt suggested that Bohr return to the United Kingdom to try to win British approval. When Churchill and Roosevelt met at Hyde Park on 19 September 1944, they rejected the idea of informing the world about the project, and the aide-mémoire of their conversation contained a rider that “enquiries should be made regarding the activities of Professor Bohr and steps taken to ensure that he is responsible for no leakage of information, particularly to the Russians”.
Bohr died of heart failure at his home in Carlsberg on 18 November 1962. He was cremated, and his ashes were buried in the family plot in the Assistens Cemetery in the Nørrebro section of Copenhagen, along with those of his parents, his brother Harald, and his son Christian. Years later, his wife’s ashes were also interred there.
I’ve mentioned traditional Danish cuisine several times before. It shares features with the other Sandinavian countries, and, like them as well as Britain, tends to be unfairly disdained by foreigners. Danish food does not involve a lot of herbs and spices, but it is noted for combinations of flavors and colorful presentation. Historically lunch was usually an open faced sandwich known as smørrebrød. Smørrebrød (originally smør og brød, meaning “butter and bread”) usually consists of a piece of buttered rye bread (rugbrød), a dense, dark brown bread. Pålæg (“put on”), the topping, which can be cold cuts, pieces of meat or fish, cheese or spreads. More elaborate, finely decorated, varieties have contributed to the international reputation of the smørrebrød. A slice or two of pålæg is placed on the buttered bread and decorated with the right accompaniments to create a tasty and visually appealing lunch or snack. Standards include:
Dyrlægens natmad (Veterinarian’s late night snack). On a piece of dark rye bread, a layer of liver pâté (leverpostej), topped with a slice of saltkød (salted beef) and a slice of sky (meat jelly). This is all decorated with raw onion rings and garden cress.
Røget ål med røræg Smoked eel on dark rye bread, topped with scrambled eggs, herbs and a slice of lemon.
Leverpostej Warm rough-chopped liverpaste served on dark rye bread, topped with bacon, and sauteed mushrooms. Additions can include lettuce and sliced pickled cucumber.
Roast beef, thinly sliced and served on dark rye bread, topped with a portion of remoulade, and decorated with a sprinkling of shredded horseradish and crispy fried onions.
Ribbensteg (roast pork), thinly sliced and served on dark rye bread, topped with red cabbage, and decorated with a slice of orange.
Rullepølse, (rolled stuffed pork) with a slice of meat jelly, onions, tomatoes and parsley.
Tartar, with salt and pepper, served on dark rye bread, topped with raw onion rings, grated horseradish and a raw egg yolk.
Røget laks. Slices of cold-smoked salmon on white bread, topped with shrimp and decorated with a slice of lemon and fresh dill.
Stjerneskud (lit. shooting star). On a base of buttered toast, two pieces of fish: a piece of steamed white fish on one half, a piece of fried, breaded plaice or rødspætte on the other half. On top is piled a mound of shrimp, which is then decorated with a dollop of mayonnaise, sliced cucumber, caviar or blackened lumpfish roe, and a lemon slice.
Here’s a small gallery to get you thinking:
bohr3 bohr2 bohr4 bohr1
Aug 122015
es1 es3
Today is the birthday (1887) of Erwin Rudolf Josef Alexander Schrödinger (1887 ), a Nobel Prize-winning Austrian physicist who developed a number of fundamental ideas in the field of quantum theory, which formed the basis of wave mechanics. He formulated the basic wave equation (stationary and time-dependent Schrödinger equation) and, more popularly, proposed an original interpretation of the physical meaning of the wave function which led to his famous thought experiment “Schrödinger’s Cat” which supposedly illustrates the absurdity of the Copenhagen interpretation of quantum mechanics.
For those who know (and care) about the implications of this thought experiment I have to say that I’ve never seen the point of it. The Copenhagen interpretation states that the wave function of certain subatomic particles exists in two (or more) simultaneously contradictory states until they are observed, at which point the function “collapses” or resolves to one or the other. Erwin Schrödinger’s thought experiment involved a closed box within which was a chamber containing a very small amount of radioactive material a particle of which within a fixed span of time might decay or not decay. The state of the particle would be measured by a Geiger counter and if it had decayed would trigger the release of cyanide gas. Also in the box was a cat. Schrödinger’s point was that it was absurd to imagine that until the box was opened by an “observer” the decay state of the particle was unknown and therefore that the cat was simultaneously alive and dead. Here’s the original:
Einstein had always been troubled by the idea that matter could simultaneously exist in two contradictory states and was so delighted by the thought experiment that he wrote:
You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gunpowder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.
I don’t know where he got the gunpowder from, but that’s not the only mistake that he made. I’ve wondered for years why they thought that the wave function had to be observed by a human for it to collapse. Why isn’t the cat an observer? Why not the Geiger counter? Anything in the macro world that interacts with the wave function is an observer. Oh dear, Erwin, you should have taken an anthropology class with me. I gather from recent reading that I am not the only person to have spotted the fallacy. Niels Bohr apparently made the same observation a long time ago. Oh well, it’s not surprising; he’s much smarter than I am.
The experiment as described is a purely theoretical one, and the machine proposed is not known to have been constructed. However, successful experiments involving similar principles, e.g. superpositions (that is matter in 2 states at the same time) of relatively large (by the standards of quantum physics) objects have been performed. These experiments do not show that a cat-sized object can be superposed (both alive and dead), but the known upper limit on “cat states” has been pushed upwards by them. In many cases the state is short-lived, even when cooled to near absolute zero.
1. A “cat state” has been achieved with photons.
2 A beryllium ion has been trapped in a superposed state.
3. An experiment involving a superconducting quantum interference device (“SQUID”) has been linked to the theme of the thought experiment: “The superposition state does not correspond to a billion electrons flowing one way and a billion others flowing the other way. Superconducting electrons move en masse. All the superconducting electrons in the SQUID flow both ways around the loop at once when they are in the Schrödinger’s cat state.”
4. A piezoelectric “tuning fork” has been constructed, which is both vibrating and still at the same time.
All this thinking makes me hungry but I don’t think I can do much with cyanide and a dead cat. So I am left with Schrödinger’s home of Vienna, which has already given me enough headaches. But . . . Vienna is well known for dishes made with a cheese called quark, which by silly coincidence is the name of an elementary sub-atomic particle. By an even sillier coincidence there are different types of quark particles which are referred to as “flavors.”
Quark the dairy product is made by warming soured milk until the desired degree of coagulation (denaturation, curdling) of milk proteins is met, and then strained. It can be classified as fresh acid-set cheese, though in some countries it is traditionally considered a distinct fermented milk product. Traditional quark is made without rennet, but in some modern dairies rennet is added. It is soft, white and unaged, and usually has no salt added.
Last time I gave a Viennese recipe I included a link for this video on how to make apple strudel:
Well , you can adapt it to make Viennese Topfenstrudel, using sweetened quark in place of apples. Problem solved. |
6f4866e9fd9b95e6 | Collision of Two Neutrons in the de Broglie–Bohm Approach
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Neutron scattering is a spectroscopic method of measuring the motions of magnetic particles. This can be used to probe a wide variety of different physical phenomena: interference effects, diffusional or hopping motions of atoms, rotational modes of molecules, acoustic modes and molecular vibrations, recoil in quantum fluids and magnetic quantum excitations [1].
In the causal interpretation of nonrelativistic quantum mechanics, a particle such as a neutron possesses a definite position and a momentum at all times. The possible trajectories are determined by the gradient of real phase function (or quantum potential) in the total wavefunction (pilot wave) [2]. We show a very simple model of a time-dependent collision of two neutrons, without spin or magnetic moment, neglecting the influence of gravitation and quark structure and with low momentum, displayed in a three-dimensional configuration space.
Two neutrons represented by two three-dimensional Gaussian wave profiles with different initial positions and momenta are used as projectiles. These interact in some regions, if the initial momenta are chosen appropriately [3]. In an elastic neutron scattering event, the momentum is transferred from one neutron to the another.
For a symmetric initial momentum distribution of the two waves (monochromatic beam), but with opposite signs in one direction, for example, in the direction, the model can be interpreted as a neutron interferometer (beam-splitter). After splitting by amplitude division (Mach–Zehnder type) via Bragg reflection, but with only one neutron present in the device at a time, the interference effect still appears. In this case, the neutron wave packet is split into two coherent waves (sub-beams) and the possible trajectories of one particle are affected by the wave that is not carrying the other particle (empty wave) [4].
In the graphics you see the wave density (if enabled), the initial momentum (large red arrows), the velocity vector field, the initial starting points of the eight trajectories (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and eight possible trajectories of the two neutrons.
Contributed by: Klaus von Bloh (January 2020)
Open content licensed under CC BY-NC-SA
The Schrödinger equation for the motion of a free particle is given by:
It is readily shown that the general normalized solution of the two sub-beams is of the form
where is the considered point of space and is the midpoint of the wave. The dispersion relationship holds with , here (atomic units). The solution as stated is defined in the infinite space . Obviously, the initial condition at determines (inverse Fourier transform). It follows
For the neutron beam, the is chosen by:
with initial density , , and with the wavenumber vector for the particle. The total wavefunction becomes
From the total wavefunction for , the equation for the phase function could be calculated, and therefore the components of the velocity. This is a standard procedure [5–7] in Bohmian mechanics, which will not be described here.
When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the starting trajectories is determined by the factor .
[1] Wikipedia. "Neutron Spectroscopy." (Jan 21, 2020)
[2] H. R. Brown, C. Dewdney and G. Horton, "Bohm Particles and Their Detection in the Light of Neutron Interferometry," Foundations of Physics, 25(2), 1995 pp. 329–347. doi:10.1007/BF02055211.
[3] C. Dewdney, "The Quantum Potential Approach to Neutron Interferometry Experiments," Physica B+C, 151(1–2), 1988 pp. 160–170. doi:10.1016/0378-4363(88)90161-1.
[4] D. M. Greenberger, "The Neutron Interferometer as a Device for Illustrating the Strange Behavior of Quantum Systems," Reviews of Modern Physics, 55(4), 1983 pp. 875-905. doi:10.1103/RevModPhys.55.875.
[5] "" (Jan 21, 2020)
[6] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jan 14, 2020)
[7] P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics, New York: Cambridge University Press, 1993.
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The Conscious Observer in Quantum Measurement
Can We Have Quantum Measurement Without an Observer?
Niels Bohr, Werner Heisenberg, John von Neumann, and Eugene Wigner
insisted that a measurement depends on the mind of a conscious observer.
Pascual Jordan, David Bohm, John Bell, and textbook authors like Landau and Lifshitz, Albert Messiah, and Kurt Gottfried denied this.
Hugh Everett invented automatic measuring equipment without a mind. John Bell asked whether the observer needs a Ph.D.?
Without Recorded Information, an Observation Is Not Possible.
Every interaction between quantum systems may lead to a collapse of the wave function. But most collapses are never recorded.
When a change of quantum state is recorded as information in a macroscopic apparatus, it becomes available for observation at a later time. Most measurements in quantum physics are made by complex, computer-controlled apparatus, whose data may wait for days, weeks, or even months to be completely analyzed and thought about by physicists.
Once Recorded, the Observer Can Then Look at the "Observable."
When the new information is observed (recorded in a human mind), it becomes an observation. This last stage has no effect on the measurement (it's after the fact!),
but it does increase human knowledge.
So When Does the Wave Function Collapse Without an Observer?
At some random time and place, before which ψ evolution is unitary, time-reversible, and deterministic. For example, when a radioactive nucleus decays, when a spontaneous photon is emitted, when an electron jumps between eigenstates of an atom, or when a collision between atoms changes their internal states.
If we could predict exactly when these events will happen, it would not be quantum mechanics.
The Heisenberg "Cut" or "Schnitt"
Werner Heisenberg described the collapse of the wave function as requiring a "cut" (Schnitt in German) somewhere along the transition from the microscopic quantum system through the "classical" apparatus to the observer and the observer's "knowledge" about the quantum system. He asked, "Where is the cut to be between the description by the wave function and the classical description?" He said it did not matter where this cut was placed, because the mathematics would produce the same experimental results wherever it was placed.
Like Niels Bohr, his goal was to describe quantum mechanical observations in the normal everyday language about a classically understandable measuring system. For Heisenberg, an observing system could be the human eye or a familiar photograph, because for the Bohr-Heisenberg "Copenhagen Interpretation" the final aim of physics is to describe experiments and their results like we describe the things and events in everyday life, i. e., by intuitive, common sense concepts of the space-time world and in the words we use for this classical space-time world.
The "cut" is frequently conflated with the "quantum to classical transition, the point at which the "classical" laws of physics, for example Newton's laws of motion, emerge from the quantum world.
There has been a lot of controversy and confusion about the location of this cut. Eugene Wigner placed it outside a room which includes the measuring apparatus and an observer A, and just before observer B makes a measurement of the physical state of the room, which is imagined to evolve deterministically according to John von Neumann's "process 2" according to the Schrödinger equation.
The case of Schrödinger's Cat is thought to present a similar paradoxical problem. Is the cat simultaneously (in a "superposition" of) dead and alive just before the observer learns which is the case? The simple answer is that live and dead are "possibilities," with calculable probabilities.
John von Neumann contributed a lot to this confusion in his discussion of subjective perceptions and "psycho-physical parallelism," which was encouraged by Neils Bohr. Bohr interpreted his "complementarity principle" as explaining the difference between subjectivity and objectivity (as well as several other dualisms). von Neumann wrote:
The difference between these two processes is a very fundamental one: aside from the different behaviors in regard to the principle of causality, they are also different in that the former is (thermodynamically) reversible, while the latter is not.
Let us now compare these circumstances with those which actually exist in nature or in its observation. First, it is inherently entirely correct that the measurement or the related process of the subjective perception is a new entity relative to the physical environment and is not reducible to the latter. Indeed, subjective perception leads us into the intellectual inner life of the individual, which is extra-observational by its very nature (since it must be taken for granted by any conceivable observation or experiment).
Nevertheless, it is a fundamental requirement of the scientific viewpoint -- the so-called principle of the psycho-physical parallelism -- that it must be possible so to describe the extra-physical process of the subjective perception as if it were in reality in the physical world -- i.e., to assign to its parts equivalent physical processes in the objective environment, in ordinary space. (Of course, in this correlating procedure there arises the frequent necessity of localizing some of these processes at points which lie within the portion of space occupied by our own bodies. But this does not alter the fact of their belonging to the "world about us," the objective environment referred to above.)
In a simple example, these concepts might be applied about as follows: We wish to measure a temperature. If we want, we can pursue this process numerically until we have the temperature of the environment of the mercury container of the thermometer, and then say: this temperature is measured by the thermometer. But we can carry the calculation further, and from the properties of the mercury, which can be explained in kinetic and molecular terms, we can calculate its heating, expansion, and the resultant length of the mercury column, and then say: this length is seen by the observer.
Going still further, and taking the light source into consideration, we could find out the reflection of the light quanta on the opaque mercury column, and the path of the remaining light quanta into the eye of the observer, their refraction in the eye lens, and the formation of an image on the retina, and then we would say: this image is registered by the retina of the observer.
And were our physiological knowledge more precise than it is today, we could go still further, tracing the chemical reactions which produce the impression of this image on the retina, in the optic nerve tract and in the brain, and then in the end say: these chemical changes of his brain cells are perceived by the observer. But in any case, no matter how far we calculate -- to the mercury vessel, to the scale of the thermometer, to the retina, or into the brain, at some time we must say: and this is perceived by the observer. That is, we must always divide the world into two parts, the one being the observed system, the other the observer. In the former, we can follow up all physical processes (in principle at least) arbitrarily precisely. In the latter, this is meaningless.
the Schnitt
The boundary between the two is arbitrary to a very large extent. In particular we saw in the four different possibilities in the example above, that the observer in this sense needs not to become identified with the body of the actual observer: In one instance in the above example, we included even the thermometer in it, while in another instance, even the eyes and optic nerve tract were not included. That this boundary can be pushed arbitrarily deeply into the interior of the body of the actual observer is the content of the principle of the psycho-physical parallelism -- but this does not change the fact that in each method of description the boundary must be put somewhere, if the method is not to proceed vacuously, i.e., if a comparison with experiment is to be possible. Indeed experience only makes statements of this type: an observer has made a certain (subjective) observation; and never any like this: a physical quantity has a certain value.
Now quantum mechanics describes the events which occur in the observed portions of the world, so long as they do not interact with the observing portion, with the aid of the process 2, but as soon as such an interaction occurs, i.e., a measurement, it requires the application of process 1. The dual form is therefore justified.* However, the danger lies in the fact that the principle of the psycho-physical parallelism is violated, so long as it is not shown that the boundary between the observed system and the observer can be displaced arbitrarily in the sense given above.
(The Mathematical Foundations of Quantum Mechanics, pp.418-21)
Information physics places the cut or boundary at the place and time of information creation. It is only after information is created that an observer could make an observation. Beforehand, there is no information to be observed.
Information creation occurs as a result of the interaction between the microscopic system and the measuring apparatus. It was a severe case of anthropomorphism to think it required the consciousness of an observer for the wave function to collapse.
The collapse of a wave function and information creation has been going on in the universe for billions of years before human consciousness emerged.
Fifty years after Heisenberg and von Neumann, John Bell drew a diagram of possible locations for the "cut" which he called his "shifty split." The Bell diagram identifies the correct moment when irreversible information enters the universe.
In the information physics solution to the problem of measurement, the timing and location of Bell's "shifty split" (the "cut" or "Schnitt" of Heisenberg and von Neumann) are identified with the interaction between quantum system and classical apparatus that leaves the apparatus in an irreversible stable state providing information to the observer.
As Bell may have seen, it is therefore not a "measurement" by a conscious observer that is needed to "collapse" wave functions. It is the irreversible interaction of the quantum system with another system, whether quantum or approximately classical. The interaction must be one that changes the information about the system. And that means a local entropy decrease (the recorded information) and overall entropy increase to make the information stable enough to be observed by an experimenter and therefore be a measurement.
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66d66bcdb9338fdc | Many-worlds versus discrete knowledge
[epistemic status: I’m a mathematical and philosophical expert but not a QM expert; conclusions are very much tentative]
There is tension between the following two claims:
• The fundamental nature of reality consists of the wave function whose evolution follows the Schrödinger equation.
• Some discrete facts are known.
(What is discrete knowledge? It is knowledge that some nontrivial proposition X is definitely true. The sort of knowledge a Bayesian may update on, and the sort of knowledge that logic applies to.)
The issue here is that facts are facts about something. If quantum mechanics has any epistemic basis, then at least some things are known, e.g. the words in a book on quantum mechanics, or the outcomes of QM experiments. The question is what this knowledge is about.
If the fundamental nature of reality is the wave function, then these facts must be facts about the wave function. But, this runs into problems.
Suppose the fact in question is “A photon passed through the measurement apparatus”. How does this translate to a fact about the wave function?
The wave function consists of a mapping from the configuration space (some subset of R^n) to complex numbers. Some configurations (R^n points) have a photon at a given location and some don’t. So the fact of a photon passing through the apparatus or not is a fact about configurations (or configuration-histories), not about wave functions over configurations.
Yes, some wave functions assign more amplitude to configurations in which the photon passes through the apparatus than others. Still, this does not allow discrete knowledge of the wave function to follow from discrete knowledge of measurements.
The Bohm interpretation, on the other hand, has an answer to this question. When we know a fact, we know a fact about the true configuration-history, which is an element of the theory.
In a sense, the Bohm interpretation states that indexical information about which world we are in is part of fundamental reality, unlike the many-worlds interpretation which states that fundamental reality contains no indexical information. (I have discussed the trouble of indexicals with respect to physicalism previously)
Including such indexical information as “part of reality” means that discrete knowledge is possible, as the discrete knowledge is knowledge of this indexical information.
For this reason, I significantly prefer the Bohm interpretation over the many-worlds interpretation, while acknowledging that there is a great deal of uncertainty here and that there may be a much better interpretation possible. Though my reservations about the many-worlds interpretation had led me to be ambivalent about the comparison between the many-worlds interpretation and the Copenhagen interpretation, I am not similarly ambivalent about Bohm versus many-worlds; I significantly prefer the Bohm interpretation to both many-worlds and to the Copenhagen interpretation.
Modeling naturalized decision problems in linear logic
The following is a model of a simple decision problem (namely, the 5 and 10 problem) in linear logic. Basic familiarity with linear logic is assumed (enough to know what it means to say linear logic is a resource logic), although knowing all the operators isn’t necessary.
The 5 and 10 problem is, simply, a choice between taking a 5 dollar bill and a 10 dollar bill, with the 10 dollar bill valued more highly.
While the problem itself is trivial, the main theoretical issue is in modeling counterfactuals. If you took the 10 dollar bill, what would have happened if you had taken the 5 dollar bill? If your source code is fixed, then there isn’t a logically coherent possible world where you took the 5 dollar bill.
I became interested in using linear logic to model decision problems due to noticing a structural similarity between linear logic and the real world, namely irreversibility. A vending machine may, in linear logic, be represented as a proposition “$1 → CandyBar”, encoding the fact that $1 may be exchanged for a candy bar, being consumed in the process. Since the $1 is consumed, the operation is irreversible. Additionally, there may be multiple options offered, e.g. “$1 → Gumball”, such that only one option may be taken. (Note that I am using “→” as notation for linear implication.)
This is a good fit for real-world decision problems, where e.g. taking the $10 bill precludes also taking the $5 bill. Modeling decision problems using linear logic may, then, yield insights regarding the sense in which counterfactuals do or don’t exist.
First try: just the decision problem
As a first try, let’s simply try to translate the logic of the 5 and 10 situation into linear logic. We assume logical atoms named “Start”, “End”, “$5”, and “$10”. Respectively, these represent: the state of being at the start of the problem, the state of being at the end of the problem, having $5, and having $10.
To represent that we have the option of taking either bill, we assume the following implications:
TakeFive : Start → End ⊗ $5
TakeTen : Start → End ⊗ $10
The “⊗” operator can be read as “and” in the sense of “I have a book and some cheese on the table”; it combines multiple resources into a single linear proposition.
So, the above implications state that it is possible, starting from the start state, to end up in the end state, yielding $5 if you took the five dollar bill, and $10 if you took the 10 dollar bill.
The agent’s goal is to prove “Start → End ⊗ $X”, for X as high as possible. Clearly, “TakeTen” is a solution for X = 10. Assuming the logic is consistent, no better proof is possible. By the Curry-Howard isomorphism, the proof represents a computational strategy for acting in the world, namely, taking the $10 bill.
Second try: source code determining action
The above analysis is utterly trivial. What makes the 5 and 10 problem nontrivial is naturalizing it, to the point where the agent is a causal entity similar to the environment. One way to model the agent being a causal entity is to assume that it has source code.
Let “M” be a Turing machine specification. Let “Ret(M, x)” represent the proposition that M returns x. Note that, if M never halts, then Ret(M, x) is not true for any x.
How do we model the fact that the agent’s action is produced by a computer program? What we would like to be able to assume is that the agent’s action is equal to the output of some machine M. To do this, we need to augment the TakeFive/TakeTen actions to yield additional data:
TakeFive : Start → End ⊗ $5 ⊗ ITookFive
TakeTen : Start → End ⊗ $10 ⊗ ITookTen
The ITookFive / ITookTen propositions are a kind of token assuring that the agent (“I”) took five or ten. (Both of these are interpreted as classical propositions, so they may be duplicated or deleted freely).
How do we relate these propositions to the source code, M? We will say that M must agree with whatever action the agent took:
MachineFive : ITookFive → Ret(M, “Five”)
MachineTen : ITookTen → Ret(M, “Ten”)
These operations yield, from the fact that “I” have taken five or ten, that the source code “M” eventually returns a string identical with this action. Thus, these encode the assumption that “my source code is M”, in the sense that my action always agrees with M’s.
Operationally speaking, after the agent has taken 5 or 10, the agent can be assured of the mathematical fact that M returns the same action. (This is relevant in more complex decision problems, such as twin prisoner’s dilemma, where the agent’s utility depends on mathematical facts about what values different machines return)
Importantly, the agent can’t use MachineFive/MachineTen to know what action M takes before actually taking the action. Otherwise, the agent could take the opposite of the action they know they will take, causing a logical inconsistency. The above construction would not work if the machine were only run for a finite number of steps before being forced to return an answer; that would lead to the agent being able to know what action it will take, by running M for that finite number of steps.
This model naturally handles cases where M never halts; if the agent never executes either TakeFive or TakeTen, then it can never execute either MachineFive or MachineTen, and so cannot be assured of Ret(M, x) for any x; indeed, if the agent never takes any action, then Ret(M, x) isn’t true for any x, as that would imply that the agent eventually takes action x.
Interpreting the counterfactuals
At this point, it’s worth discussing the sense in which counterfactuals do or do not exist. Let’s first discuss the simpler case, where there is no assumption about source code.
First, from the perspective of the logic itself, only one of TakeFive or TakeTen may be evaluated. There cannot be both a fact of the matter about what happens if the agent takes five, and a fact of the matter about what happens if the agent takes ten. This is because even defining both facts at once requires re-using the Start proposition.
So, from the perspective of the logic, there aren’t counterfactuals; only one operation is actually run, and what “would have happened” if the other operation were run is undefinable.
On the other hand, there is an important sense in which the proof system contains counterfactuals. In constructing a linear logic proof, different choices may be made. Given “Start” as an assumption, I may prove “End ⊗ $5” by executing TakeFive, or “End ⊗ $10” by executing TakeTen, but not both.
Proof systems are, in general, systems of rules for constructing proofs, which leave quite a lot of freedom in which proofs are constructed. By the Curry-Howard isomorphism, the freedom in how the proofs are constructed corresponds to freedom in how the agent behaves in the real world; using TakeFive in a proof has the effect, if executed, of actually (irreversibly) taking the $5 bill.
So, we can say, by reasoning about the proof system, that if TakeFive is run, then $5 will be yielded, and if TakeTen is run, then $10 will be yielded, and only one of these may be run.
The logic itself says there can’t be a fact of the matter about both what happens if 5 is taken and if 10 is taken. On the other hand, the proof system says that both proofs that get $5 by taking 5, and proofs that get $10 by taking 10, are possible.
How to interpret this difference? One way is by asserting that the logic is about the territory, while the proof system is about the map; so, counterfactuals are represented in the map, even though the map itself asserts that there is only a singular territory.
And, importantly, the map doesn’t represent the entire territory; it’s a proof system for reasoning about the territory, not the territory itself. The map may, thus, be “looser” than the territory, allowing more possibilities than could possibly be actually realized.
What prevents the map from drawing out logical implications to the point where it becomes clear that only one action may possibly be taken? Given the second-try setup, the agent simply cannot use the fact of their source code being M, until actually taking the action; thus, no amount of drawing implications can conclude anything about the relationship between M and the agent’s action. In addition to this, reasoning about M itself becomes harder the longer M runs, i.e. the longer the agent is waiting to make the decision; so, simply reasoning about the map, without taking actions, need not conclude anything about which action will be taken, leaving both possibilities live until one is selected.
This approach aligns significantly with the less-formal descriptions given of subjective implication decision theory and counterfactual nonrealism. Counterfactuals aren’t real in the sense that they are definable after having taken the relevant action; rather, an agent in a state of uncertainty about which action it will take may consider multiple possibilities as freely selectable, even if they are assured that their selection will be equal to the output of some computer program.
The linear logic formalization increases my confidence in this approach, by providing a very precise notion of the sense in which the counterfactuals do and don’t exist, which would be hard to make precise without similar formalism.
I am, at this point, less worried about the problems with counterfactual nonrealism (such as global accounting) than I was when I wrote the post, and more worried about the problems of policy-dependent source code (which requires the environment to be an ensemble of deterministic universes, rather than a single one), such that I have updated towards counterfactual nonrealism as a result of this analysis, although I am still not confident.
Overall, I find linear logic quite promising for modeling embedded decision problems from the perspective of an embedded agent, as it builds critical facts such as non-reversibility into the logic itself.
Appendix: spurious counterfactuals
The following describes the problem of spurious counterfactuals in relation to the model.
Assume the second-try setup. Suppose the agent becomes assured that Ret(M, “Five”); that is, that M returns the action “Five”. From this, it is provable that the agent may, given Start, attain the linear logic proposition 0, by taking action “Ten” and then running MachineTen to get Ret(M, “Ten”), which yields inconsistency with Ret(M, “Five”). From 0, anything follows, e.g. $1000000, by the principle of explosion.
If the agent is maximizing guaranteed utility, then they will take the $10 bill, to be assured of the highest utility possible. So, it cannot be the case that the agent can be correctly assured that they will take action five, as that would lead to them taking a different action.
If, on the other hand, the agent would have provably taken the $5 bill upon receiving the assurance (say, because they notice that taking the $10 bill could result in the worst possible utility), then there is a potential issue with this assurance being a self-fulfilling prophecy. But, if the agent is constructing proofs (plans for action) so as to maximize guaranteed utility, this will not occur.
This solution is essentially the same as the one given in the paper on UDT with a known search order.
Topological metaphysics: relating point-set topology and locale theory
The following is an informal exposition of some mathematical concepts from Topology via Logic, with special attention to philosophical implications. Those seeking more technical detail should simply read the book.
There are, roughly, two ways of doing topology:
• Point-set topology: Start with a set of points. Consider a topology as a set of subsets of these points which are “open”, where open sets must satisfy some laws.
• Locale theory: Start with a set of opens (similar to propositions), which are closed under some logical operators (especially and and or), and satisfy logical relations.
What laws are satisfied?
• For point-set topology: The empty set and the full set must both be open; finite intersections and infinite unions of opens must be open.
• For local theory: “True” and “false” must be opens; the opens must be closed under finite “and” and infinite “or”; and some logical equivalences must be satisfied, such that “and” and “or” work as expected.
Roughly, open sets and opens both correspond to verifiable propositions. If X and Y are both verifiable, then both “X or Y” and “X and Y” are verifiable; and, indeed, even countably infinite disjunctions of verifiable statements are verifiable, by exhibiting the particular statement in the disjunction that is verified as true.
What’s the philosophical interpretation of the difference between point-set topology and locale theory, then?
• Point-set topology corresponds to the theory of possible worlds. There is a “real state of affairs”, which can be partially known about. Open sets are “events” that are potentially observable (verifiable). Ontology comes before epistemology. Possible worlds are associated with classical logic and classical probability/utility theory.
• Locale theory corresponds to the theory of situation semantics. There are facts that are true in a particular situation, which have logical relations with each other. The first three lines of Wittgenstein’s Tracatus Logico-Philosophicus are: “The world is everything that is the case. / The world is the totality of facts, not of things. / The world is determined by the facts, and by these being all the facts.” Epistemology comes before ontology. Situation semantics is associated with intuitionist logic and Jeffrey-Bolker utility theory (recently discussed by Abram Demski).
Thus, they correspond to fairly different metaphysics. Can these different metaphysics be converted to each other?
• Converting from point-set topology to locale theory is easy. The opens are, simply, the open sets; their logical relations (and/or) are determined by set operations (intersection/union). They automatically satisfy the required laws.
• To convert from locale theory to point-set topology, construct possible worlds as sets of opens (which must be logically coherent, e.g. the set of opens can’t include “A and B” without including “A”), which are interpreted as the set of opens that are true of that possible world. The open sets of the topology correspond with the opens, as sets of possible words which contain the open.
From assumptions about possible worlds and possible observations of it, it is possible to derive a logic of observations; from assumptions about the logical relations of different propositions, it is possible to consider a set of possible worlds and interpretations of the propositions as world-properties.
Metaphysically, we can consider point-set topology as ontology-first, and locale theory as epistemology-first. Point-set topology starts with possible worlds, corresponding to Kantian noumena; locale theory starts with verifiable propositions, corresponding to Kantian phenomena.
While the interpretation of a given point-set topology as a locale is trivial, the interpretation of a locale theory as a point-set topology is less so. What this construction yields is a way of getting from observations to possible worlds. From the set of things that can be known (and knowable logical relations between these knowables), it is possible to conjecture a consistent set of possible worlds and ways those knowables relate to the possible worlds.
Of course, the true possible worlds may be finer-grained than these consistent set; however, it cannot be coarser-grained, or else the same possible world would result in different observations. No finer potentially-observable (verifiable or falsifiable) distinctions may be made between possible worlds than the ones yielded by this transformation; making finer distinctions risks positing unreferenceable entities in a self-defeating manner.
How much extra ontological reach does this transformation yield? If the locale has a countable basis, then the point-set topology may have an uncountable point-set (specifically, of the same cardinality as the reals). The continuous can, then, be constructed from the discrete, as the underlying continuous state of affairs that could generate any given possibly-infinite set of discrete observations.
In particular, the reals may be constructed from a locale based on open intervals whose beginning/end are rational numbers. That is: a real r may be represented as a set of (a, b) pairs where a and b are rational, and a < r < b. The locale whose basis is rational-delimited open intervals (whose elements are countable unions of such open intervals, and which specifies logical relationships between them, e.g. conjunction) yields the point-set topology of the reals. (Note that, although including all countable unions of basis elements would make the locale uncountable, it is possible to weaken the notion of locale to only require unions of recursively enumerable sets, which preserves countability)
If metaphysics may be defined as the general framework bridging between ontology and epistemology, then the conversions discussed provide a metaphysics: a way of relating that-which-could-be to that-which-can-be-known.
I think this relationship is quite interesting and clarifying. I find it useful in my own present philosophical project, in terms of relating subject-centered epistemology to possible centered worlds. Ontology can reach further than epistemology, and topology provides mathematical frameworks for modeling this.
That this construction yields continuous from discrete is an added bonus, which should be quite helpful in clarifying the relation between the mental and physical. Mental phenomena must be at least partially discrete for logical epistemology to be applicable; meanwhile, physical theories including Newtonian mechanics and standard quantum theory posit that physical reality is continuous, consisting of particle positions or a wave function. Thus, relating discrete epistemology to continuous ontology is directly relevant to philosophy of science and theory of mind.
Two Alternatives to Logical Counterfactuals
The following is a critique of the idea of logical counterfactuals. The idea of logical counterfactuals has appeared in previous agent foundations research (especially at MIRI): here, here. “Impossible possible worlds” have been considered elsewhere in the literature; see the SEP article for a summary.
I will start by motivating the problem, which also gives an account for what a logical counterfactual is meant to be.
Suppose you learn about physics and find that you are a robot. You learn that your source code is “A”. You also believe that you have free will; in particular, you may decide to take either action X or action Y. In fact, you take action X. Later, you simulate “A” and find, unsurprisingly, that when you give it the observations you saw up to deciding to take action X or Y, it outputs action X. However, you, at the time, had the sense that you could have taken action Y instead. You want to be consistent with your past self, so you want to, at this later time, believe that you could have taken action Y at the time. If you could have taken Y, then you do take Y in some possible world (which still satisfies the same laws of physics). In this possible world, it is the case that “A” returns Y upon being given those same observations. But, the output of “A” when given those observations is a fixed computation, so you now need to reason about a possible world that is logically incoherent, given your knowledge that “A” in fact returns X. This possible world is, then, a logical counterfactual: a “possible world” that is logically incoherent.
To summarize: a logical counterfactual is a notion of “what would have happened” had you taken a different action after seeing your source code, and in that “what would have happened”, the source code must output a different action than what you actually took; hence, this “what would have happened” world is logically incoherent.
It is easy to see that this idea of logical counterfactuals is unsatisfactory. For one, no good account of them has yet been given. For two, there is a sense in which no account could be given; reasoning about logically incoherent worlds can only be so extensive before running into logical contradiction.
To extensively refute the idea, it is necessary to provide an alternative account of the motivating problem(s) which dispenses with the idea. Even if logical counterfactuals are unsatisfactory, the motivating problem(s) remain.
I now present two alternative accounts: counterfactual nonrealism, and policy-dependent source code.
Counterfactual nonrealism
According to counterfactual nonrealism, there is no fact of the matter about what “would have happened” had a different action been taken. There is, simply, the sequence of actions you take, and the sequence of observations you get. At the time of taking an action, you are uncertain about what that action is; hence, from your perspective, there are multiple possibilities.
Given this uncertainty, you may consider material conditionals: if I take action X, will consequence Q necessarily follow? An action may be selected on the basis of these conditionals, such as by determining which action results in the highest guaranteed expected utility if that action is taken.
This is basically the approach taken in my post on subjective implication decision theory. It is also the approach taken by proof-based UDT.
The material conditionals are ephemeral, in that at a later time, the agent will know that they could only have taken a certain action (assuming they knew their source code before taking the action), due to having had longer to think by then; hence, all the original material conditionals will be vacuously true. The apparent nondeterminism is, then, only due to the epistemic limitation of the agent at the time of making the decision, a limitation not faced by a later version of the agent (or an outside agent) with more computation power.
This leads to a sort of relativism: what is undetermined from one perspective may be determined from another. This makes global accounting difficult: it’s hard for one agent to evaluate whether another agent’s action is any good, because the two agents have different epistemic states, resulting in different judgments on material conditionals.
A problem that comes up is that of “spurious counterfactuals” (analyzed in the linked paper on proof-based UDT). An agent may become sure of its own action before that action is taken. Upon being sure of that action, the agent will know the material implication that, if they take a different action, something terrible will happen (this material implication is vacuously true). Hence the agent may take the action they were sure they would take, making the original certainty self-fulfilling. (There are technical details with how the agent becomes certain having to do with Löb’s theorem).
The most natural decision theory resulting in this framework is timeless decision theory (rather than updateless decision theory). This is because the agent updates on what they know about the world so far, and considers the material implications of themselves taken a certain action; these implications include logical implications if the agent knows their source code. Note that timeless decision theory is dynamically inconsistent in the counterfactual mugging problem.
Policy-dependent source code
A second approach is to assert that one’s source code depends on one’s entire policy, rather than only one’s actions up to seeing one’s source code.
Formally, a policy is a function mapping an observation history to an action. It is distinct from source code, in that the source code specifies the implementation of the policy in some programming language, rather than itself being a policy function.
Logically, it is impossible for the same source code to generate two different policies. There is a fact of the matter about what action the source code outputs given an observation history (assuming the program halts). Hence there is no way for two different policies to be compatible with the same source code.
Let’s return to the robot thought experiment and re-analyze it in light of this. After the robot has seen that their source code is “A” and taken action X, the robot considers what would have happened if they had taken action Y instead. However, if they had taken action Y instead, then their policy would, trivially, have to be different from their actual policy, which takes action X. Hence, their source code would be different. Hence, they would not have seen that their source code is “A”.
Instead, if the agent were to take action Y upon seeing that their source code is “A”, their source code must be something else, perhaps “B”. Hence, which action the agent would have taken depends directly on their policy’s behavior upon seeing that the source code is “B”, and indirectly on the entire policy (as source code depends on policy).
We see, then, that the original thought experiment encodes a reasoning error. The later agent wants to ask what would have happened if they had taken a different action after knowing their source code; however, the agent neglects that such a policy change would have resulted in seeing different source code! Hence, there is no need to posit a logically incoherent possible world.
The reasoning error came about due to using a conventional, linear notion of interactive causality. Intuitively, what you see up to time t depends only on your actions before time t. However, policy-dependent source code breaks this condition. What source code you see that you have depends on your entire policy, not just what actions you took up to seeing your source code. Hence, reasoning under policy-dependent source code requires abandoning linear interactive causality.
The most natural decision theory resulting from this approach is updateless decision theory, rather that timeless decision theory, as it is the entire policy that the counterfactual is on.
Before very recently, my philosophical approach had been counterfactual nonrealism. However, I am now more compelled by policy-dependent source code, after having analyzed it. I believe this approach fixes the main problem of counterfactual nonrealism, namely relativism making global accounting difficult. It also fixes the inherent dynamic inconsistency problems that TDT has relative to UDT (which are related to the relativism).
I believe the re-analysis I have provided of the thought experiment motivating logical counterfactuals is sufficient to refute the original interpretation, and thus to de-motivate logical counterfactuals.
The main problem with policy-dependent source code is that, since it violates linear interactive causality, analysis is correspondingly more difficult. Hence, there is further work to be done in considering simplified environment classes where possible simplifying assumptions (including linear interactive causality) can be made. It is critical, though, that the linear interactive causality assumption not be used in analyzing cases of an agent learning their source code, as this results in logical incoherence.
What is metaphysical free will?
This is an attempt to explain metaphysical free will. This serves to explain metaphysics in general.
First: on the distinction between subject-properties and object-properties. The subject-object relation holds between some subject and some object. For example, a person might be a subject looking at a table, which is an object. Objects are, roughly, entities that could potentially be beheld by some subject.
Metaphysical free will is a property of subjects rather than objects. This will make more sense if I first contrast it with object-properties.
Objects can be defined by some properties: location, color, temperature, and so on. These properties yield testable predictions. Objects that are hot will be painful to touch, for example.
Object properties are best-defined when they are closely connected with testable predictions. The logical positivist program, though ultimately unsuccessful, is quite effective when applied to defining object properties. Similarly, the falsificationist program is successful in clarifying the meaning of a variety of scientific hypotheses in terms of predictions.
Intuitively, free will has to do with the ability of a someone to choose from one of multiple options. This implies a kind of unpredictability, at least from the perspective of the one making the choice.
Hence, there is a tension in considering free will as an object-property, in that object properties are about predictable relations, whereas free will is about choice. (Probabilistic randomness would not much help either, as e.g. taking an action with 50% probability does not match the intuitive notion of choice)
The most promising attempts to define free will as an object-property are within the physicalist school that includes Gary Drescher and Daniel Dennett. These define choice in terms of optimization: selection of the best action from a list of options, based upon anticipated consequences. This remains an object-property, because it yields a testable prediction: that the chosen action will be the one that is predicted to lead to the best consequences (and if the agent is well-informed, one that actually will). Drescher calls this “mechanical choice”.
I will now contrast object-properties (including mechanical choice) with subject-properties.
The distinction between subjects and objects is, to a significant extent, grammatical. Subjects do things, objects have things done to them. “I repaired the table with some glue.”
It is easy to detect notions of choice in ordinary language. “I could have gone to the store but I chose not to”; “you don’t have to do all that work”; “this software has so many options and capabilities“.
Functional definitions of objects are often defined in terms of the capabilities the subject has in using the object. For example, an axe can (roughly) be defined as an object that can be swung to hit another object and create a rift.
The desiderata of products, including software, are about usability. The desire is for an object that can be used in a number of ways.
Moral language, too, refers to capabilities. What one should do depends on what one can do; see Ought implies Can.
We could say, then, that this sort of subjunctive language is tied with orienting towards reality in a certain way. The orientation is, specifically, about noticing the capabilities that one’s self (and perhaps others) have, and communicating about these capabilities. I find that replacing the word “metaphysics” with the word “orientation” is often illuminating.
When this orientation is coupled with language, the language describes itself as between observation and action. That is: we talk as if we may take action on the basis of our speech. Thus, our language refers to, among other things, our capabilities, which are decision-relevant. This is in contrast to thinking of language as a side effect, or as an action in itself.
This could be studied in AI terms. An AI may be programmed to assume it has control of “its action”, and may have a model of what the consequences of various actions are, which correspond to its capabilities. From the AI’s perspective, it has a choice among multiple actions, hence in a sense “believing in metaphysical free will”. To program an AI to take effective actions, it isn’t sufficient for it to develop a model of what is; it must also develop a model of what could be made to happen. (The AI may, like a human, generate verbal reports of its capabilities, and select actions on the basis of these verbal reports)
Even relatively objective ways of orienting towards reality notice capabilities. I’ve already noted the phenomenon of functional definitions. If you look around, you will see many objects, and you will also likely notice affordances: ways these objects may be used. It may seem that these affordances inhere in the objects, although it would be more precise to say that affordances exist in the subject-object relationship rather than the object itself, as they depend on the subject.
Metaphysics isn’t directly an object of scientific study, but can be seen in the scientific process itself, in the way that one must comport one’s self towards reality to do science. This comportment includes tool usage, logic, testing, observation, recording, abstraction, theorizing, and so on. The language scientists use in the course of their scientific study, and their communication about the results, reveals this metaphysics.
(Yes, recordings of scientific practice may be subject to scientific study, but interpreting the raw data of the recordings as e.g. “testing” requires a theory bridging between the objective recorded data and whatever “testing” is, where “testing” is naively a type of intentional action)
Upon noticing choice in one’s metaphysics, one may choose to philosophize on it, to see if it holds up to consistency checks. If the metaphysics leads to inconsistencies, then it should be modified or discarded.
The most obvious possible source of inconsistency is in the relation between the metaphysical “I” and the physical body. If the “I” is identical with one’s own physical body, then metaphysical properties of the self, such as freedom of choice, must be physical properties, leading to the usual problems.
If, on the other hand, the “I” is not identical with one’s physical body, then it must be explained why the actions and observations of the “I” so much align with the actions of the body; the mind-body relation must be clarified.
Another issue is akrasia; sometimes it seems that the mind decides to take an action but the body does not move accordingly. Thus, free will may be quite partial, even if it exists.
I’ve written before about reconciliation between metaphysical free will and the predictions of physics. I believe this account is better than the others I have seen, although nowhere near complete.
It is worth contrasting the position of believing in metaphysical free will with its converse. For example, in the Bhagavad Gita, Krishna states that the wise do not identify with the doer:
All actions are performed by the gunas of prakriti. Deluded by identification with the ego, a person thinks, “I am the doer.” But the illumined man or woman understands the domain of the gunas and is not attached. Such people know that the gunas interact with each other; they do not claim to be the doer.
Bhagavad Gita, Easwaran translation, ch. 3, 27-28
In this case the textual “I” is dissociated from the “doer” which takes action. Instead, the “I” is more like a placeholder in a narrative created by natural mental processes (gunas), not an agent in itself. (The interpretation here is not entirely clear, as Krishna also gives commands to Arjuna)
This specific discussion of metaphysical free will generalizes to metaphysics in general. Metaphysics deals with the basic entities/concepts associated with reality, subjects, and objects. It is contrasted with physics, which deals with objects, generalizing from observable properties of them (and the space they exist in and so on) to lawful theories.
To summarize metaphysical free will:
• We talk in ways that imply that we and others have capabilities and make choices.
• This way of talking is possible and sufficiently-motivated because of the way we comport ourselves towards reality, noticing our capabilities.
• Effective AIs should similarly be expected to model their own capabilities as distinct from the present state of the world.
• It is difficult to coherently identify these capabilities we talk as if we have, with physical properties of our bodies.
• Therefore, it may be a reasonable (at least provisional) assumption that the capabilities we have are not physical properties of our bodies, and are metaphysical.
• The implications of this assumption can be philosophically investigated, to build out a more coherent account, or to find difficulties in doing so.
• There are ways of critiquing metaphysical free will. The assumption may lead to contradictions, with observations, well-supported scientific theories, and so on.
The absurdity of un-referenceable entities
Whereof one cannot speak, thereof one must be silent.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Some criticism of my post on physicalism is that it discusses reference, not the world. To quote one comment: “I consider references to be about agents, not about the world.” To quote another: “Remember, you have only established that indexicality is needed for reference, ie. semantic, not that it applies to entities in themselves” and also “you need to show that standpoints are ontologically fundamental, not just epistemically or semantically.” A post containing answers says: “However, everyone already kind of knows the we can’t definitely show the existence of any objective reality behind our observations and that we can only posit it.” (Note, I don’t mean to pick on these commentators, they’re expressing a very common idea)
These criticisms could be rephrased in this way:
“You have shown limits on what can be referenced. However, that in no way shows limits on the world itself. After all, there may be parts of the world that cannot be referenced.”
This sounds compelling at first: wouldn’t it be strange to think that properties of the world can be deduced from properties of human reference?
But, a slight amount of further reflection betrays the absurdity involved in asserting the possible existence of un-referenceable entities. “Un-referenceable entities” is, after all, a reference.
A statement such as “there exist things that cannot be referenced” is comically absurd, in that it refers to things in the course of denying their referenceability.
We may say, then, that it is not the case that there exist things that cannot be referenced. The assumption that this is the case leads to contradiction.
I believe this sort of absurdity is quite related to Kantian philosophy. Kant distinguished phenomena (appearances) from noumena (things-in-themselves), and asserted that through observation and understanding we can only understand phenomena, not noumena. Quoting Kant:
Appearances, to the extent that as objects they are thought in accordance with the unity of the categories, are called phaenomena. If, however, I suppose there to be things that are merely objects of the understanding and that, nevertheless, can be given to an intuition, although not to sensible intuition, then such things would be called noumena.
Critique of Pure Reason, Chapter III
Kant at least grants that noumena are given to some “intuition”, though not a sensible intuition. This is rather less ridiculous than asserting un-referenceability.
It is ironic that noumena-like entity being hypothesized in the present case (the physical world) would, by Kant’s criterion, be considered a scientific entity, a phenomenon.
Part of the absurdity in saying that the physical world may be un-referenceable is that it is at odds with the claim that physics is known through observation and experimentation. After all, un-referenceable observations and experimental results are of no use in science; they couldn’t made their way into theories. So the shadow of the world that can be known (and known about) by science is limited to the referenceable. The un-referenceable may, at best, be inferred (although, of course, this statement is absurd in refererring to the un-referenceable).
It’s easy to make fun of this idea of un-referenceable entities (infinitely more ghostly than ghosts), but it’s worth examining what is compelling about this (absurd) position, to see what, if anything, can be salvaged.
From a modern perspective, we can see things that a pre-modern perspective cannot conceptualize. For example, we know about gravitational lensing, quantum entanglement, Cesium, and so on. It seems that, from our perspective, these things-in-themselves did not appear in the pre-modern phenomenal world. While they had influence, they did not appear in a way clear enough for a concept to be developed.
We may believe it is, then, normative for the pre-moderns to accept, in humility, that there are things-in-themselves they lack the capacity to conceptualize. And we may, likewise, admit this of the modern perspective, in light of the likelihood of future scientific advances.
However, conceptualizability is not the same as referenceability. Things can be pointed to that don’t yet have clear concepts associated with them, such as the elusive phenomena seen in dreams.
In this case, pre-moderns may point to modern phenomena as “those things that will be phenomena in 500 years”. We can talk about those things our best theories don’t conceptualize that will be conceptualized later. And this is a kind of reference; it travels through space-time to access phenomena not immediately present.
This reference is vague, in that it doesn’t clearly define what things are modern phenomena, and also doesn’t allow one to know ahead of time what these phenomena are. But it’s finitely vague, in contrast to the infinite vagueness of “un-referenceable entities”. It’s at least possible to imagine accessing them, by e.g. becoming immortal and living until modern times.
A case that our current condition (e.g. modernity) cannot know about something can be translated into a reference: a reference to that which we cannot know on account of our conditions but could know under other imaginable conditions. Which is, indeed, unsurprising, given that any account of something outside our understanding existing, must refer to that thing outside our understanding.
My critique of an un-refererenceable physical world is quite similar to Nietzsche’s of Kant’s unknowable noumena. Nietzsche wrote:
The “thing-in-itself” nonsensical. If I remove all the relationships, all the “properties,” all the “activities” of a thing, the thing does not remain over; because thingness has only been invented by us owing to the requirements of logic, thus with the aim of defining, communication (to bind together the multiplicity of relationships, properties, activities).
Will to Power, sec. 558
I continue to be struck by the irony of the transition from physical phenomena to physical noumena. Kant’s positing of a realm of noumena was, perhaps, motivated by a kind of humility, a kind of respect for morality, an appeasement of theological elements in society, while still making a place for thinking-for-one’s-self, science, and so on, in a separate magisterium that can’t collide with the noumenal realm.
Any idea, whether it’s God, Physics, or Objectivity, can disconnect from the human cognitive faculty that relates ideas to the world of experience, and remain as a mere signifier, which persists as a form of unfalsifiable control. When Physics and Objectivity take on theological significance (as they do in modern times), a move analogous to Kant’s will place them in an un-falsifiable noumenal realm, with the phenomenal realm being the subjective and/or intersubjective. This is extremely ironic.
Puzzles for physicalists
The following is a list of puzzles that are hard to answer within a broadly-physicalist, objective paradigm. I believe critical agentialism can answer these better than competing frameworks; indeed, I developed it through contemplation on these puzzles, among others. This post will focus on the questions, though, rather than the answers. (Some of the answers can be found in the linked post)
In a sense what I have done is located “anomalies” relative to standard accounts, and concentrated more attention on these anomalies, attempting to produce a theory that explains them, without ruling out its ability to explain those things the standard account already explains well.
(This section would be philosophical plagiarism if I didn’t cite On the Origin of Objects.)
Indexicals are phrases whose interpretation depends on the speaker’s standpoint, such as “my phone” or “the dog over there”. It is often normal to treat indexicals as a kind of shorthand: “my phone” is shorthand for “the phone belonging to Jessica Taylor”, and “the dog over there” is shorthand for “the dog existing at coordinates 37.856570, -122.284176”. This expansion allows indexicals to be accounted for within an objective, standpoint-independent frame.
However, even these expanded references aren’t universally unique. In a very large universe, there may be a twin Earth which also has a dog at coordinates 37.856570, -122.284176. As computer scientists will find obvious, specifying spacial coordinates requires a number of bits logarithmic in the amount of space addressed. These globally unique identifiers get more and more unwieldy the more space is addressed.
Since we don’t expand out references enough to be sure they’re globally unique, our use of them couldn’t depend on such global uniqueness. An accounting of how we refer to things, therefore, cannot posit any causally-effective standpoint-independent frame that assigns semantics.
Indeed, the trouble of globally unique references can also be seen by studying physics itself. Physical causality is spacially local; a particle affects nearby particles, and there’s a speed-of-light limitation. For spacial references to be effective (e.g. to connect to observation and action), they have to themselves “move through” local space-and-time.
This is a bit like the problem of having a computer refer to itself. A computer may address computers by IP address. The IP address “” always refers to this computer. These references can be resolved even without an Internet connection. It would be totally unnecessary and unwieldy for a computer to refer to itself (e.g. for the purpose of accessing files) through a globally-unique IP address, resolved through Internet routing.
Studying enough examples like these (real and hypothetical) leads to the conclusion that indexicality (and more specifically, deixis) are fundamental, and that even spacial references that appear to be globally unique are resolved deictically.
How does this relate to physics? It means references to “the objective world” or “the physical world” must also be resolved indexically, from some standpoint. Paying attention to how these references are resolved is critical.
The experimental results you see are the ones in front of you. You can’t see experimental results that don’t, through spacio-temporal information flows, make it to you. Thus, references to the physical which go through discussing “the thing causing experimental predictions” or “the things experiments failed to falsify” are resolved in a standpoint-dependent way.
It could be argued that physical law is standpoint-independent, because it is, symmetrically, true at each point in space-time. However, this excludes virtual standpoints (e.g. existing in a computer simulation), and additionally, this only means the laws are standpoint-independent, not the contents of the world, the things described by the laws.
Pre-reduction references
(For previous work, see “Reductive Refrerence”.)
Indexicality by itself undermines view-from-nowhere mythology, but perhaps not physicalism itself. What presents a greater challenge for physicalism is the problem of pre-reduced references (which are themselves deictic).
Let’s go back to the twin Earth thought experiment. Suppose we are in pre-chemistry times. We still know about water. We know water through our interactions with it. Later, chemistry will find that water has a particular chemical formula.
In pre-chemistry times, it cannot be known whether the formula is H2O, XYZ, etc, and these formulae are barely symbolically meaningful. If we discover that water is H2O, we will, after-the-fact, define “water” to mean H2O; if we discover that water is XYZ, we will, after-the-fact, define “water” to mean XYZ.
Looking back, it’s clear that “water” has to be H2O, but this couldn’t have been clear at the time. Pre-chemistry, “water” doesn’t yet have a physical definition; a physical definition is assigned later, which rationalizes previous use of the word “water” into a physicalist paradigm.
A philosophical account of reductionism needs to be able to discuss how this happens. To do this, it needs to be able to discuss the ontological status of entities such as “water” (pre-chemistry) that do not yet have a physical definition. In this intermediate state, the philosophy is talking about two entities, pre-reduced entities and physics, and considering various bridgings between them. So the intermediate state needs to contain entities that are not yet conceptualized physically.
A possible physicalist objection is that, while it may be a provisional truth that water is definitionally the common drinkable liquid found in rivers and so on, it is ultimately true that water is H20, and so physicalism is ultimately true. (This is very similar to the two truths doctrine in Buddhism).
Now, expanding out this account needs to provide an account of the relation between provisional and ultimate truth. Even if such an account could be provided, it would appear that, in our current state, we must accept it as provisionally true that some mental entities (e.g. imagination) do not have physical definitions, since a good-enough account has not yet been provided. And we must have a philosophy that can grapple with this provisional state of affairs, and judge possible bridgings as fitting/unfitting.
Moreover, there has never been a time without provisional definition. So this idea of ultimate truth functions as a sort of utopia, which is either never achieved, or is only achieved after very great advances in philosophy, science, and so on. The journey is, then, more important than the destination, and to even approach the destination, we need an ontology that can describe and usably function within the journeying process; this ontology will contain provisional definitions.
The broader point here is that, even if we have the idea of “ultimate truth”, that idea isn’t meaningful (in terms of observations, actions, imaginations, etc) to a provisional perspective, unless somehow the provisional perspective can conceptualize the relation between itself and the ultimate truth. And, if the ultimate truth contains all provisional truths (as is true if forgetting is not epistemically normative), the ultimate truth needs to conceptualize this as well.
Epistemic status of physics
Consider the question: “Why should I believe in physics?”. The conventional answer is: “Because it predicts experimental results.” Someone who can observe these experimental results can, thus, have epistemic justification for belief in physics.
This justificatory chain implies that there are cognitive actors (such as persons or social processes) that can do experiments and see observations. These actors are therefore, in a sense, agents.
A physicalist philosophical paradigm should be able to account for epistemic justifications of physics, else fails to self-ratify. So the paradigm needs to account for observers (and perhaps specifically active observers), who are the ones having epistemic justification for belief in physics.
Believing in observers leads to the typical mind-body problems. Disbelieving in observers fails to self-ratify. (Whenever a physicalist says “an observation is X physical entity”, it can be asked why X counts as an observation of the sort that is epistemically compelling; the answer to this question must bridge the mental and the physical, e.g. by saying the brain is where epistemic cognition happens. And saying “you know your observations are the things processed in this brain region because of physics” is circular.)
What mind-body problems? There are plenty.
The anthropic principle states, roughly, that epistemic agents must believe that the universe contains epistemic agents. Else, they would believe themselves not to exist.
The language of physics, on its own, doesn’t have the machinery to say what an observer is. Hence, anthropics is a philosophical problem.
The standard way of thinking about anthropics (e.g. SSA/SIA) is to consider the universe from a view-from-nowhere, and then assume that “my” body is in some way sampled “randomly” from this viewed-from-nowhere universe, such that I proceed to get observations (e.g. visual) from this body.
This is already pretty wonky. Indexicality makes the view-from-nowhere problematic. And the idea that “I” am “randomly” placed into a body is a rather strange metaphysics (when and where does this event happen?).
But perhaps the most critical issue is that the physicalist anthropic paradigm assumes it’s possible to take a physical description of the universe (e.g. as an equation) and locate observers in it.
There are multiple ways of considering doing so, and perhaps the best is functionalism, which will be discussed later. However, I’ll note that a subjectivist paradigm can easily find at least one observer: I’m right here right now.
This requires some explaining. Say you’re lost in an amusement park. There are about two ways of thinking about this:
1. You don’t know where you are, but you know where the entrance is.
2. You don’t know where the entrance is, but you know where you are.
Relatively speaking, 1 is an “objective” (relatively standpoint-independent) answer, and 2 is a “subjective” (relatively standpoint-dependent) answer.
2 has the intuitive advantage that you can point to yourself, but not to the entrance. This is because pointing is deictic.
Even while being lost, you can still find your way around locally. You might know where the Ferris wheel is, or the food stand, or your backpack. And so you can make a local map, which has not been placed relative to the entrance. This map is usable despite its disconnection from a global reference frame.
Anthropics seems to be saying something similar to (1). The idea is that I, initially, don’t know “where I am” in the universe. But, the deictic critique applies to anthropics as it applies to the amusement park case. I know where I am, I’m right here. I know where the Earth is, it’s under me. And so on.
This way of locating (at least one) observer works independent of ability to pick out observers given a physical description of the universe. Rather than finding myself relative to physics, I find physics relative to me.
Of course, the subjectivist framework has its own problems, such as difficulty finding other observers. So there is a puzzle here.
Tool use and functionalism
Functionalism is perhaps the current best answer as to how to locate observers in physics. Before discussing functionalism, though, I’ll discuss tools.
What’s a hammer? It’s a thing you can swing to apply lots of force to something at once. Hammers can be made of many physical materials, such as stone, iron, or wood. It’s about the function, not the substance.
The definition I gave refers to a “you” who can swing the hammer. Who is the “you”? Well, that’s standpoint-dependent. Someone without arms can’t use a conventional hammer to apply lots of force. The definition relativizes to the potential user. (Yes, a person without arms may say conventional hammers are hammers due to social convention, but this social convention is there because conventional hammers work for most people, so it still relativizes to a population.)
Let’s talk about functionalism now. Functionalism is based on the idea of multiple realizability: that a mind can be implemented on many different substrates. A mind is defined by its functions rather than its substrate. This idea is very familiar to computer programmers, who can hide implementation details behind an interface, and don’t need to care about hardware architecture for the most part.
This brings us back to tools. The definition I gave of “hammer” is an interface: it says how it can be used (and what effects it should create upon being used).
What sort of functions does a mind have? Observation, prediction, planning, modeling, acting, and so on. Now, the million-dollar question: Who is (actually or potentially) using it for these functions?
There are about three different answers to this:
1. The mind itself. I use my mind for functions including planning and observation. It functions as a mind as long as I can use it this way.
2. Someone or something else. A corporation, a boss, a customer, the government. Someone or something who wants to use another mind for some purpose.
3. It’s objective. Things have functions or not independent of the standpoint.
I’ll note that 1 and 2 are both standpoint-dependent, thus subjectivist. They can’t be used to locate minds in physics; there would have to be some starting point, of having someone/something intending to use a mind for something.
3 is interesting. However, we now have a disanalogy from the hammer case, where we could identify some potential user. It’s also rather theological, in saying the world has an observer-independent telos. I find the theological implications of functionalism to be quite interesting and even inspiring, but that still doesn’t help physicalism, because physicalist ontology doesn’t contain standpoint-independent telos. We could, perhaps, say that physicalism plus theism yields objective functionalism. And this requires adding a component beyond the physical equation of the universe, if we wish to find observers in it.
Causality versus logic
Causality contains the idea that things “could” go one way or another. Else, causal claims reduce to claims about state; there wouldn’t be a difference between “if X, then Y” and “X causes Y”.
Pearlian causality makes this explicit; causal relations are defined in terms of interventions, which come from outside the causal network itself.
The ontology of physics itself is causal. It is asserted, not just that some state will definitely follow some previous state, but that there are dynamics that push previous states to new states, in a necessary way. (This is clear in the case of dynamical systems)
Indeed, since experiments may be thought of as interventions, it is entirely sensible that a physical theory that predicts the results of these interventions must be causal.
These “coulds” have a difficult status in relation to logic. Someone who already knows the initial state of a system can logically deduce its eventual state. To them, there is inevitability, and no logically possible alternative.
It appears that, while “could”s exist from the standpoint of an experimenter, they do not exist from the standpoint of someone capable of predicting the experimenter, such as Laplace’s demon.
This is not much of a problem if we’ve already accepted fundamental deixis and rejected the view-from-nowhere. But it is a problem for those who haven’t.
Trying to derive decision-theoretic causality from physical causality results in causal decision theory, which is known to have a number of bugs, due to its reliance on hypothetical extra-physical interventions.
An alternative is to try to develop a theory of “logical causality”, by which some logical facts (such as “the output of my decision process”, assuming you know your source code) can cause others. However, this is oxymoronic, because logic does not contain the affordance for intervention. Logic contains the affordance for constructing and checking proofs. It does not contain the affordance for causing 3+4 to equal 8. A sufficiently good reasoner can immediately see that “3+4=8” runs into contradiction; there is no way to construct a possible world in which 3+4=8.
Hence, it is hard to say that “coulds” exist in a standpoint-independent way. We may, then, accept standpoint-dependence of causation (as I do), or reject causation entirely.
My claim isn’t that physicalism is false, or that there don’t exist physicalist answers to these puzzles. My claim, rather, is that these puzzles are at least somewhat difficult, and that sufficient contemplation on them will destabilize many forms of physicalism. The current way I answer these puzzles is through a critical agential framework, but other ways of answering them are possible as well.
A conversation on theory of mind, subjectivity, and objectivity
I recently had a Twitter conversation with Roko Mijic. I believe it contains ideas that a wider philosophical/rationalist audience may find valuable, and so include here a transcript (quoted with permission).
Jessica: There are a number of “runs on top of” relations in physicalism:
• mind runs on top of body
• discrete runs on top of continuous
• choice runs on top of causality
My present philosophy inverts the metaphysical order: mind/discrete/choice is more basic.
This is less of a problem than it first appears, because mind/discrete/choice can conceptualize, hypothesize, and learn about body/continuous/causality, and believe in a “effectively runs on” relation between the two.
In contrast, starting from body/continuous/causality has trouble with getting to mind/discrete/choice as even being conceptualizable, hence tending towards eliminativism.
Roko: Eliminitivism has a good track record though.
Jessica: Nah, it can’t account for what an “observation” is so can’t really explain observations.
Roko: I don’t really see a problem here. It makes perfect sense within a reductionist or eliminativist paradigm for a robot to have some sensors and to sense its environment. You don’t need a soul, or god, or strong free will, or objective person-independent values for that.
Jessica: Subjective Occam’s razor (incl. Solomonoff induction) says I should adopt the explanation that best explains my observations. Eliminativism can’t really say what “my” means here. If it believed in “my observations” it would believe in consciousness.
It has to do some ontological reshuffling around what “observations” are that, I think, undermines the case for believing in physics in the first place, which is that it explains my observations.
Roko: It means the observations that are caused by sensors plugged into the hardware that your algorithm instance is running on.
Jessica: That means “my algorithm instance” exists. Sounds like a mental entity. Can’t really have those under eliminativism (but can under functionalism etc).
Roko: I don’t want to eliminate my mental instance from my philosophy, that would be kind of ridiculous.
Jessica: Well, yes, so eliminativism is false. I understand eliminativism to mean there is only physical, no mental. Believing mental runs on physical could be functionalism, property dualism, or some other non-eliminativist position.
Roko: I think it makes more sense to think of mental things as existing subjectively (i.e. if they belong to you) and physical things as existing objectively. I definitely think that dualism is making a mistake in thinking of objectively-existing mental things
Jessica: I don’t think this objective/subjective dichotomy works out. I haven’t seen a good positive case, and my understanding of deixis leads me to believe that references to the objective must be resolved subjectively. See also On the Origin of Objects.
Basically I don’t see how we can, in a principled way, have judgments like “X exists but only subjectively, not objectively”. It would appear that by saying “X exists” I am asserting that X is an existent object (i.e. I’m saying something objective).
See also Thomas Nagel’s The View From Nowhere. Spoiler alert: there isn’t a view from nowhere, it’s an untenable concept.
Roko: My sensation of the flavor of chocolate exists but only subjectively.
Jessica: We’re now talking about the sensation of the flavor of the chocolate though. Is this really that different from talking about “that car over there”? I don’t see how some entities can, in a principled way, be classified as objective and some as subjective.
Like, in talking about “X” I’m porting something in my mental world-representation into the discursive space. I don’t at all see how to classify some of these portings as objective and some as subjective.
See also writing on the difficulty of the fact/opinion distinction.
Roko: It’s not actually the flavor “of” the chocolate though. It’s the sensation of flavor that your brain generates for you only, in response to certain nerve stimuli.
It’s very easy actually. Subjectives are the things that you cannot possibly be mistaken about, the “I think therefore I am’s”.
No deceiving demon can fool you into thinking that you’re experiencing the taste of chocolate, the color purple, or an orgasm. No deceiving demon can fool you into thinking that you’re visualizing the number 4.
Jessica: I don’t think this is right. The thought follows the experience. There can be mistranslations along the way. This might seem like a pedantic point but we’re talking about linguistic subjective statements so it’s relevant.
Translating the subjective into words can introduce errors. It’s at least as hard as, say, adding small numbers. So your definition means “1+1=2” is also subjective.
Roko: I think that it’s reasonable to see small number math instances as subjectives. I can see 3 pens. I can conceive of 3 dots, that’s a subjective thing. It’s in the same class as seeing red or smelling a rose.
[continuing from the deceiving demon thread] These are the things that are inherently part of your instance or mind. The objective, on the other hand, is always somewhat uncertain and inferred. Things are out there and they send signals to you. But you are inferring their existence.
Jessica: Okay, I agree with this sort of mental/outside-mental distinction, and you can define subjective/objective to mean that. This certainly doesn’t bring in other connotations of the objective, such as view-from-nowhere or observer-independence; I can be wrong about indexicals too.
Roko: Well it happens to be a property of our world that when different people infer the shape of the objective (i.e. draw maps), they always converge. This is what being in a shared reality means.
I mean they always converge if they follow the right principles, e.g. complexity priors, and those same principles are the ones that allow us to successfully manipulate reality via actions. That’s what the objective world out there is.
Jessica: Two reasons they could converge:
1. Symmetry (this explains math)
2. Existence of same entities (e.g. landmarks)
I’m fine with calling 1 observer-independent. Problem: your view of, and references to, 2, depend on your standpoint. Because of deixis.
Obvious deictic references are things like “the car over there” or “the room I’m in”. It is non-obvious but, I think, true, that all physical references are deictic. Which makes sense because physical causality is deictic (locally causal and symmetric).
Even “the Great Wall of China” refers to the Great Wall of China on our Earth. It couldn’t refer to the one on the twin Earth. And the people on twin Earth have “the Great Wall of China” refer to the one on the twin Earth, not ours.
At the same time, maps created starting from different places can be patched together, in a collage. However, pasting these together requires taking into account the standpoint-dependence of the individual maps being pasted together.
And at no point does this pasting-together result in a view from nowhere. It might seem that way because it keeps getting bigger and more zoomed-out. But at each individual time it’s finite.
Roko: Yes this is all nice but I think the point where we get to hard questions is when we think about mental phenomena that I would classify as subjectives as being part of the objective reality.
This is the problem, or @reducesuffering worrying about whether plankton or insects “really do” have subjective experiences etc
Jessica: In my view “my observation” is an extremely deictic reference, to something maximally here-and-now, such that there isn’t any stabilization to do. Intermediate maps paste these extremely deictic maps together into less-deictic, but still deictic, maps. It never gets non-deictic.
It’s hard to pin down intersubjectively precisely because it’s so deictic. I can’t really port my here-and-now to your here-and-now without difficulty.
Subjective implication decision theory in critical agentialism
This is a follow-up to a previous post on critical agentialism, to explore the straightforward decision-theoretic consequences. I call this subjective implication decision theory, since the agent is looking at the logical implications of their decision according to their beliefs.
We already covered observable action-consequences. Since these are falsifiable, they have clear semantics in the ontology. So we will in general assume observable rewards, as in reinforcement learning, while leaving un-observable goals for later work.
Now let’s look at a sequence of decision theory problems. We will assume, as before, the existence of some agent that falsifiably believes itself to run on at least one computer, C.
5 and 10
Assume the agent is before a table containing a 5 dollar bill and a 10 dollar bill. The agent will decide which dollar bill to take. Thereafter, the agent will receive a reward signal: 5 if the 5 dollar bill is taken, and 10 if the 10 dollar bill is taken.
The agent may have the following beliefs about action-consequences: “If I take action 5, then I will get 5 reward. If I take action 10, then I will get 10 reward.” These beliefs follow directly from the problem description. Notably, the beliefs include beliefs about actions that might not actually be taken; it is enough that these actions are possible that their consequences are falsifiable.
Now, how do we translate these beliefs about action-consequences into decisions? The most straightforward way to do so is to select the policy that is believed to return the most reward. (This method is ambiguous under conditions of partial knowledge, though that is not a problem for 5 and 10).
This method (which I will call “subjective implication decision theory”) yields the action 10 in this case.
This is all extremely straightforward. We directly translated the problem description into a set of beliefs about action consequences. And these beliefs, along with the rule of subjective causal decision theory, yield an optimal action.
The difficulty of 5 and 10 comes when the problem is naturalized. The devil is in the details: how to naturalize the problem? The previous post examined a case of both external and internal physics, compatible with free will. There is no obvious obstacle to translating these physical beliefs to the 5 and 10 case: the dollar bills may be hypothesized to follow physical laws, as may the computer C.
Realistically, the agent should assume that the proximate cause of the selection of the dollar bill is not their action, but C’s action. Recall that the agent falsifiably believes it runs on C, in the sense that its observations/actions necessarily equal C’s.
Now, “I run on C” implies in particular: “If I select ‘pick up the 5 dollar bill’ at time t, then C does. If I select ‘pick up the 10 dollar bill’ at time t, then C does.” And the assumption that C controls the dollar bill implies: “If C selects ‘pick up the 5 dollar bill at time t‘, then the 5 dollar bill will be held at some time between t and t+k“, and also for the 10 dollar bill (for some k that is an upper bound of the time it takes for the dollar bill to be picked up). Together, these beliefs imply: “If I select ‘pick up the 5 dollar bill’ at time t, then the 5 dollar bill will be held at some time between t and t+k“, and likewise for the 10 dollar bill. At this point, the agent’s beliefs include ones quite similar to the ones in the non-naturalized case, and so subjective implication decision theory selects the 10 dollar bill.
Twin prisoner’s dilemma
Consider an agent that believes itself to run on computer C. It also believes there is another computer, C’, which has identical initial state and dynamics to C.
Each computer will output an action; the agent will receive 10 reward if C’ cooperates plus 1 reward if C defects (receiving 0 reward for defection).
As in 5 and 10, the agent believes: “If I cooperate, C cooperates. If I defect, C defects.” However, this does not specify the behavior of C’ as a function of the agent’s action.
It can be noted at this point that, because the agent believes C’ has identical initial state and dynamics to C, the agent believes (falsifiably) that C’ must output the same actions as C on each time step, as long as C and C’ receive idential observations. Since, in this setup, observations are assumed to be equal until C receives the reward (with C’ perhaps receiving a different reward), these beliefs imply: “If I cooperate, C’ cooperates. If I defect, C’ defects”.
In total we now have: “If I cooperate, C and C’ both cooperate. If I defect, C and C’ both defect”. Thus the agent believes itself to be straightforwardly choosing between a total reward of 10 for cooperation, and a total of 1 reward for defection. And so subjective implication decision theory cooperates.
Note that this comes apart from the conventional interpretation of CDT, which considers interventions on C’s action, rather than on “my action”. CDT’s hypothesized intervention updates C but not C’, as C and C’ are physically distinct.
Newcomb’s problem
This is very much similar to twin prisoner’s dilemma. The agent may falsifiably believe: “The Predictor filled box A with $1,000,000 if and only if I will choose only box A.” From here it is straightforward to derive that the agent believes: “If I choose to take only box A, then I will have $1,000,000. If I choose to take both boxes, then I will have $1,000.” Hence subjective implication decision theory selects only box A.
The usual dominance argument for selecting both boxes does not apply. The agent is not considering interventions on C’s action, but rather on “my action”, which is falsifiably predicted to be identical with C’s action.
Counterfactual mugging
In this problem, a Predictor flips a coin; if the coin is heads, the Predictor asks the agent for $10 (and the agent may or may not give it); if the coin is tails, the Predictor gives the agent $1,000,000 iff the Predictor predicts the agent would have given $10 in the heads case.
We run into a problem with translating this to a critical agential ontology. Since both branches don’t happen in the same world, it is not possible to state the Predictor’s accuracy as a falsifiable statement, as it relates two incompatible branches.
To avoid this problem, we will say that the Predictor predicts the agent’s behavior ahead of time, before flipping the coin. This prediction is not told to the agent in the heads case.
Now, the agent falsifiably believes the following:
• If the coin is heads, then the Predictor’s prediction is equal to my choice.
• If the coin is tails, then I get $1,000,000 if the Predictor’s prediction is that I’d give $10, otherwise $0.
• If the coin is heads, then I get $0 if I don’t give the predictor $10, and -$10 if I do give the predictor $10.
From the last point, it is possible to show that, after the agent observes heads, the agent believes they get $0 if they don’t give $10, and -$10 if they do give $10. So subjective implication decision theory doesn’t pay.
This may be present a dynamic inconsistency in that the agent’s decision does not agree with what they would previously have wished they would decide. Let us examine this.
In a case where the agent chooses their action before the coin flip, the agent believes that, if they will pay up, the Predictor will predict this, and likewise for not paying up. Therefore, the agent believes they will get $1,000,000 if they decide to pay up and then the coin comes up tails.
If the agent weights the heads/tails branches evenly, then the agent will decide to pay up. This presents a dynamic inconsistency.
My sense is that this inconsistency should be resolved by considering theories of identity other than closed individualism. That is, it seems possible that the abstraction of receiving an observation and taking on action on each time step, while having a linear lifetime, is not a good-enough fit for the counterfactual mugging problem to achieve dynamic consistency.
It seems that subjective implication decision theory agrees with timeless decision theory and evidential decision theory on the problems considered, while diverging from causal decision theory and functional decision theory.
I consider this a major advance, in that the ontology is more cleanly defined than the ontology of timeless decision theory, which considers interventions on logical facts. It is not at all clear what it means to “intervene on a logical fact”; the ontology of logic does not natively contain the affordance of intervention. The motivation for considering logical interventions was the belief that the agent is identical with some computation, such that its actions are logical facts. Critical agential ontology, on the other hand, does not say the agent is identical with any computation, but rather than the agent effectively runs on some computer (which implements some computation), while still being metaphysically distinct. Thus, we need not consider “logical counterfactuals” directly; rather, we consider subjective implications, and consider whether these subjective implications are consistent with the agent effectively running on some computer.
To handle cases such as counterfactual mugging in a dynamically consistent way (similar to functional decision theory), I believe that it will be necessary to consider agents outside the closed-individualist paradigm, in which one is assumed to have a linear lifetime with memory and observations/actions on each time step. However, I have not proceeded exploring in this direction yet.
[ED NOTE: After the time of writing I realized subjective implication decision theory, being very similar to proof-based UDT, has problems with spurious counterfactuals by default, but can similarly avoid these problems by “playing chicken with the universe”, i.e. taking some action it has proven it will not take.]
A critical agential account of free will, causation, and physics
This is an account of free choice in a physical universe. It is very much relevant to decision theory and philosophy of science. It is largely metaphysical, in terms of taking certain things to be basically real and examining what can be defined in terms of these things.
The starting point of this account is critical and agential. By agential, I mean that the ontology I am using is from the point of view of an agent: a perspective that can, at the very least, receive observations, have cognitions, and take actions. By critical, I mean that this ontology involves uncertain conjectures subject to criticism, such as criticism of being logically incoherent or incompatible with observations. This is very much in a similar spirit to critical rationalism.
Close attention will be paid to falsifiability and refutation, principally for ontological purposes, and secondarily for epistemic purposes. Falsification conditions specify the meanings of laws and entities relative to the perspective of some potentially falsifying agent. While the agent may believe in unfalsifiable entities, falsification conditions will serve to precisely pin down that which can be precisely pinned down.
I have only seen “agential” used in the philosophical literature in the context of agential realism, a view I do not understand well enough to comment on. I was tempted to use “subjective”; however, while subjects have observations, they do not necessarily have the ability to take actions. Thus I believe “agential” has a more concordant denotation.
You’ll note that my notion of “agent” already assumes one can take actions. Thus, a kind of free will is taken as metaphysically basic. This presupposition may cause problems later. However, I will try to show that, if careful attention is paid, the obvious problems (such as contradiction with determinism) can be avoided.
The perspective in this post can be seen as starting from agency, defining consequences in terms of agency, and defining physics in terms of consequences. In contrast, the most salient competing decision theory views (including framings of CDT, EDT, and FDT) define agency in terms of consequences (“expected utility maximization”), and consequences in terms of physics (“counterfactuals”). So I am rebasing the ontological stack, turning it upside-down. This is less absurd than it first appears, as will become clear.
(For simplicity, assume observations and actions are both symbols taken from some finite alphabet.)
Naive determinism
Let’s first, within a critical agential ontology, disprove some very basic forms of determinism.
Let A be some action. Consider the statement: “I will take action A”. An agent believing this statement may falsify it by taking any action B not equal to A. Therefore, this statement does not hold as a law. It may be falsified at will.
Let f() be some computable function returning an action. Consider the statement: “I will take action f()”. An agent believing this statement may falsify it by taking an action B not equal to f(). Note that, since the agent is assumed to be able to compute things, f() may be determined. So, indeed, this statement does not hold as a law, either.
This contradicts a certain strong formulation of naive determinism: the idea that one’s action is necessarily determined by some known, computable function.
But wait, what about physics? To evaluate what physical determinism even means, we need to translate physics into a critical agential ontology. However, before we turn to physics, we will first consider action-consequences, which are easier to reason about.
Consider the statement: “If I take action A, I will immediately there-after observe O.” This statement is falsifiable, which means that if it is false, there is some policy the agent can adopt that will falsify it. Specifically, the agent may adopt the policy of taking action A. If the agent will, in fact, not observe O after taking this action, then the agent will learn this, falsifying the statement. So the statement is falsifiable.
Finite conjunctions of falsifiable statements are themselves falsifiable. Therefore, the conjunction “If I take action A, I will immediately there-after observe O; if I take action B, I will immediately there-after observe P” is, likewise, falsifiable.
Thus, the agent may have falsifiable beliefs about observable consequences of actions. This is a possible starting point for decision theory: actions having consequences is already assumed in the ontology of VNM utility theory.
Falsification and causation
Now, the next step is to account for physics. Luckily, the falsificationist paradigm was designed around demarcating scientific hypotheses, such that it naturally describes physics.
Interestingly, falsificationism takes agency (in terms of observations, computation, and action) as more basic than physics. For a thing to be falsifiable, it must be able to be falsified by some agent, seeing some observation. And the word able implies freedom.
Let’s start with some basic Popperian logic. Let f be some testable function (say, connected to a computer terminal) taking in a natural number and returning a Boolean. Consider the hypothesis: “For all x, f(x) is true”. This statement is falsifiable: if it’s false, then there exists some action-sequence an agent can take (typing x into the terminal, one digit at a time) that will prove it to be false.
The given hypothesis is a kind of scientific law. It specifies a regularity in the environment.
Note that there is a “bridge condition” at play here. That bridge condition is that the function f is, indeed, connected to the terminal, such that the agent’s observations of f are trustworthy. In a sense, the bridge condition specifies what f is, from the agent’s perspective; it allows the agent to locate f as opposed to some other function.
Let us now consider causal hypotheses. We already considered action-consequences. Now let us extend this analysis to reasoning about causation between external entities.
Consider the hypothesis: “If the match is struck, then it will alight immediately”. This hypothesis is falsifiable by an agent who is able to strike the match. If the hypothesis is false, then the agent may refute it by choosing to strike the match and then seeing the result. However, an agent who is unable to strike the match cannot falsify it. (Of course, this assumes the agent may see whether the match is alight after striking it)
Thus, we are defining causality in terms of agency. The falsification conditions for a causal hypothesis refer to the agent’s abilities. This seems somewhat wonky at first, but it is quite similar to Pearlian casuality, which defines causation in terms of metaphysically-real interventions. This order of definition radically reframes the determinism vs. free will apparent paradox, by defining the conditions of determinism (causality) in terms of potential action.
External physics
Let us now continue, proceeding to more universal physics. Consider the law of gravity, according to which a dropped object will accelerate downward at a near-constant weight. How might we port this law into an agential ontology?
Here is the assumption about how the agent interacts with gravity. The agent will choose some natural number as the height of an object. Thereafter, the object will fall, while a camera will record the height of the object at each natural-number time expressed in milliseconds, to the nearest natural-number millimeter from the ground. The agent may observe a printout of the camera data afterwards.
Logically, constant gravity implies, and is implied by, a particular quadratic formula for the height of the object as a function of the object’s starting height and the amount of time that has passed. This formula implies the content of the printout, as a function of the chosen height. So, the agent may falsify constant gravity (in the observable domain) by choosing an object-height, placing an object at that height, letting it fall, and checking the printout, which will show the law of constant gravity to be false, if the law in fact does not hold for objects dropped at that height (to the observed level of precision).
Universal constant gravity is not similarly falsifiable by this agent, because this agent may only observe this given experimental setup. However, a domain-limited law, stating that the law of constant gravity holds for all possible object-heights in this setup, up to the camera’s precision, is falsifiable.
It may seem that I am being incredibly pedantic about what a physical law is and what the falsification conditions are; however, I believe this level of pedantry is necessary for critically examining the notion of physical determinism to a high-enough level of rigor to check interaction with free will.
Internal physics
We have, so far, considered the case of an agent falsifying a physical law that applies to an external object. To check interaction with free will, we must interpret physical law applied to the agent’s internals, on which the agent’s cognition is, perhaps, running in a manner similar to software.
Let’s consider the notion that the agent itself is “running on” some Turing machine. We will need to specify precisely what such “running on” means.
Let C be the computer that the agent is considering whether it is running on. C has, at each time, a tape-state, a Turing machine state, an input, and an output. The input is attached to a sensor (such as a camera), and the output is attached to an actuator (such as a motor).
For simplicity, let us say that the history of tapes, states, inputs, and outputs is saved, such that it can be queried at a later time.
We may consider the hypothesis that C, indeed, implements the correct dynamics for a given Turing machine specification. These dynamics imply a relation between future states and past states. An agent may falsify these dynamics by checking the history and seeing if the dynamics hold.
Note that, because some states or tapes may be unreachable, it is not possible to falsify the hypothesis that C implements correct dynamics starting from unreachable states. Rather, only behavior following from reachable states may be checked.
Now, let us think on an agent considering whether they “run on” this computer C. The agent may be assumed to be able to query the history of C, such that it may itself falsify the hypothesis that C implements Turing machine specification M, and other C-related hypotheses as well.
Now, we can already name some ways that “I run on C” may be falsified:
• Perhaps there is a policy I may adopt, and a time t, such that if I implement this policy, I will observe O at time t, but C will observe something other than O at time t.
• Perhaps there is a policy I may adopt, and a time t, such that if I implement this policy, I will take action A at time t, but C will take an action other than A at time t.
The agent may prove these falsification conditions by adopting a given policy until some time t, and then observing C’s observation/action at time t, compared to their own observation/action.
I do not argue that the converse of these conditions exhaust what it means that “I run on C”. However, they at least restrict the possibility space by a very large amount. For the falsification conditions given to not hold, the observations and behavior of C must be identical with the agent’s own observations and behavior, for all possible policies the agent may adopt.
I will name the hypothesis with the above falsification conditions: “I effectively run on C”. This conveys that these conditions may not be exhaustive, while still being quite specific, and relating to effects between the agent and the environment (observations and actions).
Note that the agent can hypothesize itself to effectively run on multiple computers! The conditions for effectively running on one computer do not contradict the conditions for effectively running on another computer. This naturally handles cases of identical physical instantiations of a single agent.
At this point, we have an account of an agent who:
• Believes they have observations and take free actions
• May falsifiably hypothesize physical law
• May falsifiably hypothesize that some computer implements a Turing machine specification
• May falsifiably hypothesize that they themselves effectively run on some computer
I have not yet shown that this account is consistent. There may be paradoxes. However, this at least represents the subject matter covered in a unified critical agential ontology.
Paradoxes sought and evaluated
Let us now seek out paradox. We showed before that the hypothesis “I take action f()” may be refuted at will, and therefore does not hold as a necessary law. We may suspect that “I effectively run on C” runs into similar problems.
Remember that, for the “I effectively run on C” hypothesis to be falsified, it must be falsified at some time, at which the agent’s observation/action comes apart from C’s. In the “I take action f()” case, we had the agent simulate f() in order to take an opposite action. However, C need not halt, so the agent cannot simulate C until halting. Instead, the agent may select some time t, and run C for t steps. But, by the time the agent has simulated C for t steps, the time is already past t, and so the agent may not contradict C’s behavior at time t, by taking an opposite action. Rather, the agent only knows what C does at time t at some time later than t, and only their behavior after this time may depend on this knowledge.
So, this paradox is avoided by the fact that the agent cannot contradict its own action before knowing it, but cannot know it before taking it.
We may also try to create a paradox by assuming an external super-fast computer runs a copy of C in parallel, and feeds this copy’s action on subjective time-step t into the original C’s observation before time t; this way, the agent may observe its action before it takes it. However, now the agent’s action is dependent on its observation, and so the external super-fast computer must decide which observation to feed into the parallel C. The external computer cannot know what C will do before producing this observation, and so this attempt at a paradox cannot stand without further elaboration.
We see, now, that if free will and determinism are compatible, it is due to limitations on the agent’s knowledge. The agent, knowing it runs on C, cannot thereby determine what action it takes at time t, until a later time. And the initial attempt to provide this knowledge externally fails.
Downward causation
Let us now consider a general criticism of functionalist views, which is that of downward causation: if a mental entity (such as observation or action) causes a physical entity, doesn’t that either mean that the mental entity is physical, or that physics is not causally closed?
Recall that we have defined causation in terms of the agent’s action possibilities. It is straightforwardly the case, then, that the agent’s action at time t causes changes in the environment.
But, what of the physical cause? Perhaps it is also the case that C’s action at time t causes changes in the environment. If so, there is a redundancy, in that the change in the environment is caused both by the agent’s action and by C’s action. We will examine this possible redundancy to find potential conflicts.
To consider ways that C’s action may change the environment, we must consider how the agent may intervene on C’s action. Let us say we are concerned with C’s action at time t. Then we may consider the agent at some time u < t taking an action that will cause C’s action at time t to be over-written. For example, the agent may consider programming an external circuit that will interact with C’s circuit (“its circuit”).
However, if the agent performs this intervention, then the agent’s action at time t has no influence on C’s action at time t. This is because C’s action is, necessarily, equal to the value chosen at time u. (Note that this lack of influence means that the agent does not effectively run on C, for the notion of “effectively run on” considered! However, the agent may be said to effectively run on C with one exception.)
So, there is no apparent way to set up a contradiction between these interventions. If the agent decides early (at time u) to determine C’s action at time t, then that decision causes C’s action at time t; if the agent does not do so, then the agent’s decision at time t causes C’s action at time t; and these are mutually exclusive. Hence, there is not an apparent problem with redundant causality.
It may be suspected that the agent I take to be real is epiphenomenal. Perhaps all may be explained in a physicalist ontology, with no need to posit that there exists an agent that has observations and takes actions. (This is a criticism levied at some views on consciousness; my notion of metaphysically-real observations is similar enough to consciousness that these criticisms are potentially applicable)
The question in regards to explanatory power is: what is being explained, in terms of what? My answer is: observations are being explained, in terms of hypotheses that may be falsified by action/observations.
An eliminativist perspective denies the agent’s observations, and thus fails to explain what ought to be explained, in my view. However, eliminativists will typically believe that “scientific observation” is possible, and seek to explain scientific observations.
A relevant point to make here is that the notion of scientific observation assumes there is some scientific process happening that has observations. Indeed, the scientific method includes actions, such as testing, which rely on the scientific process taking actions. Thus, scientific processes may be considered as agents in the sense I am using the term.
My view is that erasing the agency of both individual scientists, and of scientific processes, puts the ontological and epistemic status of physics on shaky ground. It is hard to say why one should believe in physics, except in terms of it explaining observations, including experimental observations that require taking actions. And it is hard to say what it means for a physical hypothesis to be true, with no reference to how the hypothesis connects with observation and action.
In any case, the specter of epiphenomenalism presents no immediate paradox, and I believe that it does not succeed as a criticism.
Comparison to Gary Drescher’s view
I will now compare my account to Gary Drescher’s view. I have found Drescher’s view to be both particularly systematic and compelling, and to be quite similar to the views of other relevant philosophers such as Daniel Dennett and Eliezer Yudkowsky. Therefore, I will compare and contrast my view with Drescher’s. This will dispel the illusion that I am not saying anything new.
Notably, Drescher makes a similar observation to mine on Pearl: “Pearl’s formalism models free will rather than mechanical choice.”
Quoting section 5.3 of Good and Real:
Why did it take that action? In pursuit of what goal was the action selected? Was that goal achieved? Would the goal have been achieved if the machine had taken this other action instead? The system includes the assertion that if the agent were to do X, then Y would (probably) occur; is that assertion true? The system does not include the assertion that if it were to do P, Q would probably occur; is that omitted assertion true? Would the system have taken some other action just now if it had included that assertion? Would it then have better achieved its goals?
Insofar as such questions are meaningful and answerable, the agent makes choices in at least the sense that the correctness of its actions with respect to its designated goals is analyzable. That is to say, there can be means-end connections between its actions and its goals: its taking an action for the sake of a goal can make sense. And this is so despite the fact that everything that will happen-including every action taken and every goal achieved or not-is inalterably determined once the system starts up. Accordingly, I propose to call such an agent a choice machine.
Drescher is defining conditions of choice and agency in terms of whether the decisions “make sense” with respect to some goal, in terms of means-end connections. This is a “outside” view of agency in contrast with my “inside” view. That is, it says some thing is an agent when its actions connect with some goal, and when the internal logic of that thing takes into account this connection.
This is in contrast to my view, which takes agency to be metaphysically basic, and defines physical outside views (and indeed, physics itself) in terms of agency.
My view would disagree with Drescher’s on the “inalterably determined” assertion. In an earlier chapter, Drescher describes a deterministic block-universe view. This view-from-nowhere implies that future states are determinable from past states. In contrast, the view I present here rejects views-from-nowhere, instead taking the view of some agent in the universe, from whose perspective the future course is not already determined (as already argued in examinations of paradox).
Note that these disagreements are principally about metaphysics and ontology, rather than scientific predictions. I am unlikely to predict the results of scientific experiments differently from Drescher on account of this view, but am likely to account for the scientific process, causation, choice, and so on in different language, and using a different base model.
Conclusion and further research
I believe the view I have presented to be superior to competing views on multiple fronts, most especially logical/philosophical systematic coherence. I do not make the full case for this in this post, but take the first step, of explicating the basic ontology and how it accounts for phenomena that are critically necessary to account for.
An obvious next step is to tackle decision theory. Both Bayesianism and VNM decision theory are quite concordant with critical agential ontology, in that they propose coherence conditions on agents, which can be taken as criticisms. Naturalistic decision theory involves reconciling choice with physics, and so a view that already includes both is a promising starting point.
Multi-agent systems are quite important as well. The view presented so far is near-solipsistic, in that there is a single agent who conceptualizes the world. It will need to be defined what it means for there to be “other” agents. Additionally, “aggregative” agents, such as organizations, are important to study, including in terms of what it means for a singular agent to participate in an aggregative agent. “Standardized” agents, such as hypothetical skeptical mathematicians or philosophers, are also worthy subjects of study; these standardized agents are relevant in reasoning about argumentation and common knowledge. Also, while the discussion so far has been in terms of closed individualism, alternative identity views such as empty individualism and open individualism are worth considering from a critical agential perspective.
Other areas of study include naturalized epistemology and philosophy of mathematics. The view so far is primarily ontological, secondarily epistemological. With the ontology in place, epistemology can be more readily explored.
I hope to explore the consequences of this metaphysics further, in multiple directions. Even if I ultimately abandon it, it will have been useful to develop a coherent view leading to an illuminating refutation. |
bc945fa36e93717e | I'm an undergrad studying mathematics right now. I have a very big interest in mathematical applications of physics, and I have been debating whether or not I should change my degree to a dual major (math/physics), because of this annoying little fear that only the math degree won't prepare me for a career in physics research later down the road, at the graduate level. However, the Mathematics is the degree I want (I believe there are more opportunities - I'd also enjoy going into a [bandwagon?] research field like Artificial Intelligence)..
Would a BS (and of course grad school) in Mathematics alone be enough to prepare me for a decent graduate school future, if I decided to pursue research in theoretical/mathematical physics? Or would it be better for me to tack on the dual major in physics? I would rather not stay in school an extra year and a half (it's already taking long enough after the A.S. in computer science!), but if it's necessary, I could do that.
I found another thread on ASE about this, with someone recommending that of course a mathematics major take physics courses - and I have.
My specific question is if it would behoove me to pursue a dual major as opposed to just taking extra physics courses (or a minor).
This is essentially the bare minimum I have to do for just the mathematics major.
These would be the physics courses I could tack on. The darker colors are required for the dual major, the lighter ones are electives.
The grad program I'm looking at, by the way, has a Master's program in Math that offers courses like Riemannian Geometry, Riemann surfaces, Group Theory, etc, which I think are useful in theoretical/mathematical physics at the graduate research level. Here's a picture
enter image description here
(Speaking from the perspective of a US grad student who did double major, but who has no direct experience working on a grad admissions committee.)
In general, the extra degree itself probably isn't much added value for what you want to do. You have your statement of purpose and the rest of your application to show what specific knowledge, skills, and interest you have. Grad schools will look at your transcript, seeing every course you took anyway. If you take enough physics courses, it shouldn't really matter what your diploma says. And by the time you finish grad school, your undergrad major will be even less important to people.
The question is how much is enough physics? I don't have any course descriptions to work with, but "Principles of Physics" sounds like a survey course for non-physicists, the kind of thing you won't learn anything useful from. If you want to do research in physics, you'll need a graduate-level understanding of physics, and for that you'll need plenty of undergraduate-level physics. You should have at least a core foundation of
• special relativity,
• classical mechanics (Lagrangians and Hamiltonians),
• electromagnetism,
• statistical mechanics, and
• quantum mechanics
by the time you finish undergrad. Now since you are planning to do primarily math for the PhD there might be some flexibility in that you can hold off on some of this until grad school, but note you will still need even more physics knowledge to do research. If you don't know any physics discovered in the last 100 years, there's no way you can discover anything new, and all the subjects I listed are at least 100 years old.
To be sure, some math is very useful for physics. But just knowing differential geometry doesn't mean you know general relativity, and particle physics is more than just the pure group theory you'll see in a math course.
On the topic of labs: I've met a fair number of student mathematicians-interested-in-physics who say labs aren't important for what they want to do. That might be true for what I would call a mathematical physicist, but I don't believe it for theoretical physics. Some exposure to a laboratory setting is important for knowing what physics really is. This is based on my own experiences, as well as advice I got from my (Nobel-prize-winning, theoretical physicist) undergrad adviser. The thing is, undergrad physics courses are unfortunately largely doable with blind symbol manipulation. Those who are good at it (and mathematicians usually are) often end up thinking physics is just easier, less abstract math. But physics is an empirical science, which math most certainly isn't. A (well-organized) lab will show you how much more there is to thinking like a physicist than solving the Schrödinger equation with yet another potential. And if it turns out thinking like a physicist isn't your cup of tea, it's better to find that out earlier.
In summary: Decide what classes you want to take, keeping in mind that you'll need a number of actual physics courses, taught by physicists, if you want to do physics proper (rather than, say, prove pure math theorems deemed important by those with more of a connection to physics). If that list of classes is a superset of the physics requirements, sure, do the extra major -- if nothing else it might impress an industry recruiter should your career go in that direction. If it isn't a superset, just take the physics courses of interest and skip the extra major.
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My answer is specific to the course lists you have linked to, and written from a US point of view.
I recommend taking
• Modern
• Electromagnetism (at least a whole year)
• Quantum (also whole year)
• Mechanics
• Some kind of statistical mechanics. 2311 may or may not be sufficient.
Electronics lab is for experimentalists. Since you say you want to do mathematical physics, I don't see any point in that (unless it causes you to change your goals). Take those electives that interest you.
Also, check the admissions requirements of graduate programs you are interested in.
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• So, you're saying that the full dual major program would be overkill? Also, instead of taking [undergrad] Quantum Mechanics, there's a course called "Quantum Theory of Two State Systems" (photon polarization states and fermion spin states). Would that be an ok substitute? – galois Dec 11 '14 at 8:53
• Without more information, I would not consider it an okay substitute. – Anonymous Physicist Dec 11 '14 at 16:29
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d4aa854c23839307 | Книга еврейской мудрости
...Подавляющее большинство человеческих существ получило столь скромную часть разума или сознания, что ее едва достаточно для того, чтобы стоило говорить об этом.
Зигмунд Фрейд
Richard Feynman - biography
Feynman was a keen popularizer of physics through both books and lectures, notably a 1959 talk on top-down nanotechnology called There's Plenty of Room at the Bottom and The Feynman Lectures on Physics. Feynman also became known through his semi-autobiographical books (Surely You're Joking, Mr. Feynman! and What Do You Care What Other People Think?) and books written about him, such as Tuva or Bust! Feynman also had a deep interest in biology, and was a friend of the geneticist and microbiologist Esther Lederberg, who developed replica plating and discovered bacteriophage lambda. They had several mutual physicist friends who, after beginning their careers in nuclear research, moved for moral reasons into genetics, among them Leó Szilárd, Guido Pontecorvo, and Aaron Novick.
Early life
Richard Phillips Feynman was born on May 11, 1918, in Far Rockaway, Queens, New York. His family originated from Russia and Poland; both of his parents were Jewish, but they were not devout. In fact, by his early youth, Feynman described himself as an "avowed atheist". Feynman (in common with the famous physicists Edward Teller and Albert Einstein) was a late talker; by his third birthday he had yet to utter a single word. The young Feynman was heavily influenced by his father, Melville, who encouraged him to ask questions to challenge orthodox thinking. From his mother, Lucille, he gained the sense of humor that he had throughout his life. As a child, he delighted in repairing radios and had a talent for engineering. His sister Joan also became a professional physicist.
In high school, his IQ was determined to be 125 (on an unspecified test, making this statement dubious): high, but "merely respectable" according to biographer Gleick. Feynman later scoffed at psychometric testing. By 15, he had learned differential and integral calculus. Before entering college, he was experimenting with and re-creating mathematical topics, such as the half-derivative, utilizing his own notation. In high school, he was developing the mathematical intuition behind his Taylor series of mathematical operators.
His habit of direct characterization sometimes rattled more conventional thinkers; for example, one of his questions when learning feline anatomy was "Do you have a map of the cat?" (referring to an anatomical chart).
Feynman attended Far Rockaway High School, a school that also produced fellow laureates Burton Richter and Baruch Samuel Blumberg. A member of the Arista Honor Society, in his last year in high school, Feynman won the New York University Math Championship; the large difference between his score and those of his closest competitors shocked the judges.
He applied to Columbia University, but was not accepted, because of the "Jewish quota" (a discriminatory practice of limiting the number of places available to students of Jewish background). Instead he attended the Massachusetts Institute of Technology, where he received a bachelor's degree in 1939, and in the same year was named a Putnam Fellow. While there, Feynman took every physics course offered, including a graduate course on theoretical physics while only in his second year. He obtained a perfect score on the graduate school entrance exams to Princeton University in mathematics and physics — an unprecedented feat — but did rather poorly on the history and English portions. Attendees at Feynman's first seminar included Albert Einstein, Wolfgang Pauli, and John von Neumann. He received a Ph.D. from Princeton in 1942; his thesis advisor was John Archibald Wheeler. Feynman's thesis applied the principle of stationary action to problems of quantum mechanics, laying the groundwork for the "path integral" approach and Feynman diagrams, and was entitled "The Principle of Least Action in Quantum Mechanics".
The Manhattan Project
At Princeton, the physicist Robert R. Wilson encouraged Feynman to participate in the Manhattan Project—the wartime U.S. Army project at Los Alamos developing the atomic bomb. Feynman said he was persuaded to join this effort to build it before Nazi Germany developed their own bomb.
He was assigned to Hans Bethe's theoretical division, and impressed Bethe enough to be made a group leader. He and Bethe developed the Bethe-Feynman formula for calculating the yield of a fission bomb, which built upon previous work by Robert Serber.
He immersed himself in work on the project, and was present at the Trinity bomb test. Feynman claimed to be the only person to see the explosion without the very dark glasses or welder's lenses provided, reasoning that it was safe to look through a truck windshield, as it would screen out the harmful ultraviolet radiation.
As a junior physicist, he was not central to the project. The greater part of his work was administering the computation group of human computers in the Theoretical division (one of his students there, John G. Kemeny, later went on to co-write the computer language BASIC). Later, with Nicholas Metropolis, he assisted in establishing the system for using IBM punched cards for computation. Feynman succeeded in solving one of the equations for the project that were posted on the blackboards. However, they did not "do the physics right" and Feynman's solution was not used.
Feynman's other work at Los Alamos included calculating neutron equations for the Los Alamos "Water Boiler", a small nuclear reactor, to measure how close an assembly of fissile material was to criticality. On completing this work he was transferred to the Oak Ridge facility, where he aided engineers in devising safety procedures for material storage so that criticality accidents (for example, due to sub-critical amounts of fissile material inadvertently stored in proximity on opposite sides of a wall) could be avoided. He also did theoretical work and calculations on the proposed uranium-hydride bomb, which later proved not to be feasible.
Due to the top secret nature of the work, Los Alamos was isolated. In Feynman's own words, "There wasn't anything to do there". Bored, he indulged his curiosity by learning to pick the combination locks on cabinets and desks used to secure papers. Feynman played many jokes on colleagues. In one case he found the combination to a locked filing cabinet by trying the numbers a physicist would use (it proved to be 27-18-28 after the base of natural logarithms, e = 2.71828...), and found that the three filing cabinets where a colleague kept a set of atomic bomb research notes all had the same combination. He left a series of notes as a prank, which initially spooked his colleague, Frederic de Hoffman, into thinking a spy or saboteur had gained access to atomic bomb secrets. On several occasions, Feynman drove to Albuquerque to see his ailing wife in a car borrowed from Klaus Fuchs, who was later discovered to be a real spy for the Soviets, transporting nuclear secrets in his car to Santa Fe.
On occasion, Feynman would find an isolated section of the mesa to drum in the style of American natives; "and maybe I would dance and chant, a little". These antics did not go unnoticed, and rumors spread about a mysterious Indian drummer called "Injun Joe". He also became a friend of laboratory head J. Robert Oppenheimer, who unsuccessfully tried to court him away from his other commitments after the war to work at the University of California, Berkeley.
Feynman alludes to his thoughts on the justification for getting involved in the Manhattan project in The Pleasure of Finding Things Out. As mentioned earlier, he felt the possibility of Nazi Germany developing the bomb before the Allies was a compelling reason to help with its development for the US. However, he goes on to say that it was an error on his part not to reconsider the situation when Germany was defeated. In the same publication, Feynman also talks about his worries in the atomic bomb age, feeling for some considerable time that there was a high risk that the bomb would be used again soon so that it was pointless to build for the future. Later he describes this period as a "depression."
Early academic career
Following the completion of his Ph.D. in 1942, Feynman held an appointment at the University of Wisconsin-Madison (UW) as an assistant professor of physics. The appointment was spent on leave for his involvement in the Manhattan project. In 1945, he received a letter from Dean Mark Ingraham of the College of Letters and Science requesting his return to UW to teach in the coming academic year. His appointment was not extended when he did not commit to return. In a talk given several years later at UW, Feynman quipped, "It's great to be back at the only University that ever had the good sense to fire me".
After the war, Feynman declined an offer from the Institute for Advanced Study in Princeton, New Jersey, despite the presence there of such distinguished faculty members as Albert Einstein, Kurt Gödel, and John von Neumann. Feynman followed Hans Bethe, instead, to Cornell University, where Feynman taught theoretical physics from 1945 to 1950. During a temporary depression following the destruction of Hiroshima by the bomb produced by the Manhattan Project, he focused on complex physics problems, not for utility, but for self-satisfaction. One of these was analyzing the physics of a twirling, nutating dish as it is moving through the air. His work during this period, which used equations of rotation to express various spinning speeds, soon proved important to his Nobel Prize-winning work. Yet because he felt burned out, and had turned his attention to less immediately practical but more entertaining problems, he felt surprised by the offers of professorships from renowned universities.
Despite yet another offer from the Institute for Advanced Study, which would have included teaching duties (which was not included in the Institute's initial offer, a factor in his rejection of it), Feynman opted for the California Institute of Technology (Caltech) — as he says in his book Surely You're Joking Mr. Feynman! — because a desire to live in a mild climate had firmly fixed itself in his mind while installing tire chains on his car in the middle of a snowstorm in Ithaca.
Feynman has been called the "Great Explainer". He gained a reputation for taking great care when giving explanations to his students and for making it a moral duty to make the topic accessible. His guiding principle was that if a topic could not be explained in a freshman lecture, it was not yet fully understood. Feynman gained great pleasure from coming up with such a "freshman-level" explanation, for example, of the connection between spin and statistics. What he said was that groups of particles with spin 1/2 "repel", whereas groups with integer spin "clump". This was a brilliantly simplified way of demonstrating how Fermi-Dirac statistics and Bose-Einstein statistics evolved as a consequence of studying how fermions and bosons behave under a rotation of 360°. This was also a question he pondered in his more advanced lectures and to which he demonstrated the solution in the 1986 Dirac memorial lecture. In the same lecture, he further explained that antiparticles must exist, for if particles only had positive energies, they would not be restricted to a so-called "light cone".
During one sabbatical year, he returned to Newton's Principia Mathematica to study it anew; what he learned from Newton, he passed along to his students, such as Newton's attempted explanation of diffraction.
Caltech years
Feynman did significant work while at Caltech, including research in: Quantum electrodynamics. The theory for which Feynman won his Nobel Prize is known for its accurate predictions. This theory was developed in the earlier years during Feynman's work at Cornell. He helped develop a functional integral formulation of quantum mechanics, in which every possible path from one state to the next is considered, the final path being a sum over the possibilities (also referred to as sum-over-paths or sum over histories).
Physics of the superfluidity of supercooled liquid helium, where helium seems to display a complete lack of viscosity when flowing. Applying the Schrödinger equation to the question showed that the superfluid was displaying quantum mechanical behavior observable on a macroscopic scale. This helped with the problem of superconductivity; however, the solution eluded Feynman. It was solved with the BCS theory of superconductivity, proposed by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer.
A model of weak decay, which showed that the current coupling in the process is a combination of vector and axial (an example of weak decay is the decay of a neutron into an electron, a proton, and an anti-neutrino). Although E. C. George Sudarshan and Robert Marshak developed the theory nearly simultaneously, Feynman's collaboration with Murray Gell-Mann was seen as seminal because the weak interaction was neatly described by the vector and axial currents. It thus combined the 1933 beta decay theory of Enrico Fermi with an explanation of parity violation. He also developed Feynman diagrams, a bookkeeping device which helps in conceptualizing and calculating interactions between particles in spacetime, notably the interactions between electrons and their antimatter counterparts, positrons. This device allowed him, and later others, to approach time reversibility and other fundamental processes. Feynman's mental picture for these diagrams started with the hard sphere approximation, and the interactions could be thought of as collisions at first. It was not until decades later that physicists thought of analyzing the nodes of the Feynman diagrams more closely. Feynman famously painted Feynman diagrams on the exterior of his van.
From his diagrams of a small number of particles interacting in spacetime, Feynman could then model all of physics in terms of those particles' spins and the range of coupling of the fundamental forces. Feynman attempted an explanation of the strong interactions governing nucleons scattering called the parton model. The parton model emerged as a complement to the quark model developed by his Caltech colleague Murray Gell-Mann. The relationship between the two models was murky; Gell-Mann referred to Feynman's partons derisively as "put-ons". In the mid 1960s, physicists believed that quarks were just a bookkeeping device for symmetry numbers, not real particles, as the statistics of the Omega-minus particle, if it were interpreted as three identical strange quarks bound together, seemed impossible if quarks were real. The Stanford linear accelerator deep inelastic scattering experiments of the late 1960s showed, analogously to Ernest Rutherford's experiment of scattering alpha particles on gold nuclei in 1911, that nucleons (protons and neutrons) contained point-like particles which scattered electrons. It was natural to identify these with quarks, but Feynman's parton model attempted to interpret the experimental data in a way which did not introduce additional hypotheses. For example, the data showed that some 45% of the energy momentum was carried by electrically neutral particles in the nucleon. These electrically neutral particles are now seen to be the gluons which carry the forces between the quarks and carry also the three-valued color quantum number which solves the Omega — problem. Feynman did not dispute the quark model; for example, when the fifth quark was discovered in 1977, Feynman immediately pointed out to his students that the discovery implied the existence of a sixth quark, which was duly discovered in the decade after his death.
After the success of quantum electrodynamics, Feynman turned to quantum gravity. By analogy with the photon, which has spin 1, he investigated the consequences of a free massless spin 2 field, and was able to derive the Einstein field equation of general relativity, but little more. However, the computational device that Feynman discovered then for gravity, "ghosts", which are "particles" in the interior of his diagrams which have the "wrong" connection between spin and statistics, have proved invaluable in explaining the quantum particle behavior of the Yang-Mills theories, for example QCD and the electro-weak theory.
In 1965, Feynman was appointed a foreign member of the Royal Society. At this time in the early 1960s, Feynman exhausted himself by working on multiple major projects at the same time, including a request, while at Caltech, to "spruce up" the teaching of undergraduates. After three years devoted to the task, he produced a series of lectures that eventually became the Feynman Lectures on Physics, one reason that Feynman is still regarded as one of the greatest teachers of physics. He wanted a picture of a drumhead sprinkled with powder to show the modes of vibration at the beginning of the book. Outraged by many rock and roll and drug connections that could be made from the image, the publishers changed the cover to plain red, though they included a picture of him playing drums in the foreword. Feynman later won the Oersted Medal for teaching, of which he seemed especially proud.
Partly as a way to bring publicity to progress in physics, Feynman offered $1000 prizes for two of his challenges in nanotechnology, claimed by William McLellan and Tom Newman, respectively. He was also one of the first scientists to conceive the possibility of quantum computers.
Many of his lectures and other miscellaneous talks were turned into books, including The Character of Physical Law and QED: The Strange Theory of Light and Matter. He gave lectures which his students annotated into books, such as Statistical Mechanics and Lectures on Gravity. The Feynman Lectures on Physics occupied two physicists, Robert B. Leighton and Matthew Sands as part-time co-authors for several years. Even though they were not adopted by most universities as textbooks, the books continue to be bestsellers because they provide a deep understanding of physics. As of 2005, The Feynman Lectures on Physics has sold over 1.5 million copies in English, an estimated 1 million copies in Russian, and an estimated half million copies in other languages.
In 1984-86, he developed a variational method for the approximate calculation of path integrals which has led to a powerful method of converting divergent perturbation expansions into convergent strong-coupling expansions (variational perturbation theory) and, as a consequence, to the most accurate determination of critical exponents measured in satellite experiments.
In the late 1980s, according to "Richard Feynman and the Connection Machine", Feynman played a crucial role in developing the first massively parallel computer, and in finding innovative uses for it in numerical computations, in building neural networks, as well as physical simulations using cellular automata (such as turbulent fluid flow), working with Stephen Wolfram at Caltech. His son Carl also played a role in the development of the original Connection Machine engineering; Feynman influencing the interconnects while his son worked on the software.
Feynman diagrams are now fundamental for string theory and M-theory, and have even been extended topologically. The world-lines of the diagrams have developed to become tubes to allow better modeling of more complicated objects such as strings and membranes. However, shortly before his death, Feynman criticized string theory in an interview: "I don't like that they're not calculating anything," he said. "I don't like that they don't check their ideas. I don't like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, 'Well, it still might be true.'" These words have since been much-quoted by opponents of the string-theoretic direction for particle physics.
Challenger disaster
Feynman played an important role on the Presidential Rogers Commission, which investigated the Challenger disaster. Feynman devoted the latter half of his book What Do You Care What Other People Think? to his experience on the Rogers Commission, straying from his usual convention of brief, light-hearted anecdotes to deliver an extended and sober narrative. Feynman's account reveals a disconnect between NASA's engineers and executives that was far more striking than he expected. His interviews of NASA's high-ranking managers revealed startling misunderstandings of elementary concepts. He concluded that the NASA management's space shuttle reliability estimate was fantastically unrealistic. He warned in his appendix to the commission's report, "For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled."
M8 Entertainment Inc. announced in May 2006 that a movie would be made about the disaster. Challenger, scheduled for a 2010 release, is to be directed by Philip Kaufman—whose 1983 film The Right Stuff chronicled the early history of the space program—and will focus on the role of Feynman in the ensuing investigation. David Strathairn will play Feynman.
Personal life
While researching for his Ph.D., Feynman married his first wife, Arline Greenbaum (often spelled Arlene). She was diagnosed with tuberculosis, but she and Feynman were careful, and he never contracted it. She succumbed to the disease in 1945. This portion of Feynman's life was portrayed in the 1996 film Infinity, which featured Feynman's daughter Michelle in a cameo role.
He was married a second time in June 1952, to Mary Louise Bell of Neodesha, Kansas; this marriage was brief and unsuccessful. He later married Gweneth Howarth from Ripponden, Yorkshire, who shared his enthusiasm for life and spirited adventure. Besides their home in Altadena, California, they had a beach house in Baja California, purchased with the prize money from Feynman's Nobel Prize, his one third share of $55,000. They remained married until Feynman's death. They had a son, Carl, in 1962, and adopted a daughter, Michelle, in 1968.
Feynman had a great deal of success teaching Carl, using discussions about ants and Martians as a device for gaining perspective on problems and issues; he was surprised to learn that the same teaching devices were not useful with Michelle.
Mathematics was a common interest for father and son; they both entered the computer field as consultants and were involved in advancing a new method of using multiple computers to solve complex problems—later known as parallel computing. The Jet Propulsion Laboratory retained Feynman as a computational consultant during critical missions. One co-worker characterized Feynman as akin to Don Quixote at his desk, rather than at a computer workstation, ready to do battle with the windmills. Feynman traveled a great deal, notably to Brazil, and near the end of his life schemed to visit the Russian land of Tuva, a dream that, because of Cold War bureaucratic problems, never became reality. The day after he died, a letter arrived for him from the Soviet government giving him authorization to travel to Tuva. During this period, he discovered that he had a form of cancer, but, thanks to surgery, he managed to hold it off. Out of his enthusiastic interest in reaching Tuva came the phrase "Tuva or Bust" (also the title of a book about his efforts to get there), which was tossed about frequently amongst his circle of friends in hope that they, one day, could see it firsthand. The documentary movie Genghis Blues mentions some of his attempts to communicate with Tuva, and chronicles the successful journey there by his friends.
Responding to Hubert Humphrey's congratulation for his Nobel Prize, Feynman admitted to a long admiration for the then vice president. In a letter to an MIT professor dated December 6, 1966, Feynman expressed interest in running for the governor of California.
Feynman took up drawing at one time and enjoyed some success under the pseudonym "Ofey", culminating in an exhibition dedicated to his work. He learned to play a metal percussion instrument (frigideira) in a samba style in Brazil, and participated in a samba school.
In addition, he had some degree of synesthesia for equations, explaining that the letters in certain mathematical functions appeared in color for him, even though invariably printed in standard black-and-white.
According to Genius, the James Gleick-authored biography, Feynman experimented with LSD during his professorship at Caltech. Somewhat embarrassed by his actions, Feynman largely sidestepped the issue when dictating his anecdotes; he mentions it in passing in the "O Americano, Outra Vez" section, while the "Altered States" chapter in Surely You're Joking, Mr. Feynman! describes only marijuana and ketamine experiences at John Lilly's famed sensory deprivation tanks, as a way of studying consciousness.
Feynman gave up alcohol when he began to show early signs of alcoholism, as he did not want to do anything that could damage his brain—the same reason given in "O Americano, Outra Vez" for his reluctance to experiment with LSD.
In Surely You're Joking, Mr. Feynman!, he gives advice on the best way to pick up a girl in a hostess bar. At Caltech, he used a nude/topless bar as an office away from his usual office, making sketches or writing physics equations on paper placemats. When the county officials tried to close the place, all visitors except Feynman refused to testify in favor of the bar, fearing that their families or patrons would learn about their visits. Only Feynman accepted, and in court, he affirmed that the bar was a public need, stating that craftsmen, technicians, engineers, common workers "and a physics professor" frequented the establishment. While the bar lost the court case, it was allowed to remain open as a similar case was pending appeal.
Feynman developed two rare forms of cancer, Liposarcoma and Waldenström's macroglo
Автор статьи: Zipora Galitski
Темы статьи: biography
Источник статьи: http://en.wikipedia.org/wiki/Richard_Feynman
В статье упоминаются люди: Ричард Фейнман
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f4733df212aac720 | Precision tests of General Relativity with Matter Waves
Michael A. Hohensee Department of Physics, 366 Le Conte Hall MS 7300, University of California, Berkeley, California 94720, USA Holger Müller Department of Physics, 366 Le Conte Hall MS 7300, University of California, Berkeley, California 94720, USA
February 21, 2021
We review the physics of atoms and clocks in weakly curved spacetime, and how each may be used to test the Einstein Equivalence Principle (EEP) in the context of the minimal Standard Model Extension (mSME). We find that conventional clocks and matter-wave interferometers are sensitive to the same kinds of EEP-violating physics. We show that the analogy between matter-waves and clocks remains true for systems beyond the semiclassical limit. We quantitatively compare the experimentally observable signals for EEP violation in matter-wave experiments. We find that comparisons of Li and Li are particularly sensitive to such anomalies. Tests involving unstable isotopes, for which matter-wave interferometers are well suited, may further improve the sensitivity of EEP tests.
I Introduction
The gravitational redshift is an important prediction of general relativity, was the first experimental signature considered by Einstein in 1911 Einstein1911 , and its experimental verification remains central to our confidence in the theory. Clock comparison tests Hafele:1972 ; Chou have reached accuracies of parts in Vessot , while experiments based on matter waves, in which a redshift anomaly would modify the Compton frequency of material particles, have reached an accuracy of parts in redshift ; Poli ; redshiftPRL . These experiments complement a wide array of other tests of the equivalence principle, including tests of the universality of free fall (UFF) and local Lorentz invariance Schlamminger:2008 ; datatables . We briefly review the physics of the gravitational redshift and the acceleration of free fall relevant to clocks, and moving test masses, both quantum and classical, in the limit of a weak, static gravitational potential. Using the mSME KosteleckyGravity ; KosteleckyTassonPRL ; KosteleckyTasson , we determine the phenomenological parameters for EEP violation that are constrained by redshift and UFF tests. Focusing on those terms of the mSME that are only observable by gravitational experiments, we find that using metastable nuclides in a matter-wave interferometer may offer improved sensitivity to such effects.
Ii Action and the Gravitational Redshift
If are the coordinates of a clock moving on a path parameterized by the affine parameter through space-time, and is the metric, the proper time experienced by a locally inertial clock as it moves a distance is MTW . If the metric differs between two points in spacetime, then two otherwise identical oscillators with the same proper frequency can appear to an observer to tick at different frequencies , since the relationship between the proper time and coordinate time is a function of position. The difference frequency for locally inertial clocks moving with nonrelativistic velocities and in a weak static gravitational potential becomes
The first term is the gravitational redshift, first measured by Pound and Rebka in 1960 PoundRebka . The second is the time dilation due to the clock’s motion, and can be subtracted if the trajectories are known. This equation is universal up to terms proportional to ; at and beyond, measurements of the instantaneous depend upon how the signals carrying the clocks’ frequencies propagate and where the comparison takes place.
It has recently been argued comment that matter wave experiments do not constitute tests of the gravitational redshift, but should rather be understood as probes of UFF. We note that similar arguments have been leveled at clock comparison tests in the past schiff , and that tests of UFF and the gravitational redshift are generally not independent of one another in any theory which conserves energy and momentum nordtvedt . In order to explain the analogy between clocks and matter waves, let us consider two clocks which are initially synchronized to have identical phase at and then transported to the same point along different paths, where they are compared. Then, the phase of clock relative to clock is , is given by
where and ; we have specialized to a homogenous gravitational field so that . If the clocks are freely falling, then their motion is an extremum of their respective actions, given by MTW
where, since we work in the non-relativistic limit, we take the clocks’ coordinate time to be , and thus use in the place of .
Gravity also acts upon the quantum phase of matter waves Colella ; Bonse ; Horne ; Werner ; Littrell . In Feynman’s path integral formulation FeynmanHibbs of quantum mechanics, the wavefunction for a particle with mass at is obtained from its value at according to
where indicates that the integral is taken over all paths. In the semiclassical limit, the matter-wavepacket follows the classical path of least action, and acquires a phase shift , with , where . We thus conclude that the relative phase accumulated by two identical matter-wavepackets that travel along separated paths is the same, up to a constant factor of , as that acquired by two conventional clocks which follow the same trajectory. Note that although this expression applies to the semiclassical limit, we need not work in this limit to interpret matter-wave experiments as redshift tests. In the appendix, we derive the non-relativistic Schrödinger equation from the path integral in the weak gravitational field limit.
For any pair of conventional clocks, (e.g. electronic oscillators referenced to a microwave, or optical transition) the phase difference accumulated over a given period of coordinate time is a small fraction of the total quantum phase they may accumulate. Since the phase of a matter-wave oscillates at the Compton frequency ( Hz), the intrinsic sensitivity of a matter-wave interferometer to variations in the proper time is between and times greater than that of a conventional clock. The greater precision in the phase readout and the greater separation available to optical and microwave clocks can bridge part, but not all of this divide.
Iii QED and the Gravitational Redshift
With the exception of the Hafele-Keating experiment Hafele:1972 , all redshift tests prior to the advent of matter-wave interferometry have used electromagnetic signals to compare the relative ticking rate of two clocks. Since a Mach-Zehnder matter-wave interferometer more closely resembles the Hafele-Keating experiment in that the relative clock rates are never encoded in the frequency of a photon, one might reasonably be concerned that matter-wave interferometers might be unable to observe anomalous redshift physics detectable by more conventional tests. In the absence of an anomalous gravitational coupling to spin (i.e. torsion), or wavelength-dependent gravitational coupling to light, and in the limit that the photon remains massless in the vacuum, this concern can be resolved by applying general covariance, i.e. our freedom to choose our coordinate system.
The properties of a curved spacetime metric and of a lone field propagating within that metric are never directly observable Mattingly:2005 ; KosteleckyTasson . Instead, we must infer these properties by comparing the effects of the metric on several different fields. General covariance affords us complete freedom to choose the coordinate chart upon which the metric tensor is defined, and the freedom to arbitrarily define the coordinates of the local Lorentz frame at a single point in spacetime. Any anomaly in the coupling of light to gravity can be expressed as a modification of the metric tensor . This modification can be formally eliminated from the electromagnetic sector of the theory by a redefinition of the local Lorentz frame, so that photons behave conventionally. This moves the anomalous QED couplings into the physics of all other particle fields. The existence of any photon-mass and spin-dependent anomalies, while not considered in detail here, has been strongly constrained by spectro-polarimetric studies of light emitted by distant gamma ray bursts Kostelecky:2006 . More recently, a broader class of wavelength-dependent QED anomalies has also been tightly bounded by astrophysical observations Kostelecky:2008a . While neither study explicitly considered anomalies arising from gravitational interactions, their results suggest that such effects, if they exist, are likely to be extremely small in any terrestrial experiment.
Iv Equivalence Principle Tests in the Standard Model Extension
In the non-relativistic limit, the motion and gravitational redshift experienced by a freely falling particle are determined by the same element of the metric tensor. It is therefore no surprise that tests of UFF and the gravitational redshift are not independent of one another. Indeed, the two must be linked in any energy-conserving theory nordtvedt . We will now explore this relationship in the context of the minimal gravitational standard model extension KosteleckyGravity ; KosteleckyTassonPRL ; KosteleckyTasson . The EEP requires that the laws of physics be the same in all local inertial frames, no matter where they are or how fast they are moving, and that gravity must act through the curvature of spacetime alone, affecting all particles in the same way MTW ; TEGP . Both clock comparison and matter-wave interferometer tests can be used to test the EEP, and their results can be used to quantitatively restrict the degree to which weak position- or velocity-dependent effects described by the mSME are consistent with the observed laws of physics. The mSME framework is formulated from the Lagrangian of the standard model by adding all Lorentz- and CPT violating (and thus EEP-violating) terms that can be formed from the standard model fields and Lorentz tensors ColladayKostelecky . Some of these terms, which can represent the vacuum expectation values of heretofore unknown fields, are only detectable via gravitationally-induced fluctuations in their mean values KosteleckyTasson . They can also contribute to the metric tensor via their effect on the stress-energy tensor. Since the effective particle Lagrangian that results is not an explicit function of space or time, the mSME conserves energy and momentum.
Most, but not all, coefficients in the gravitational mSME produce Lorentz violation that is measurable in flat space-time. We focus on an isotropic subset of the theory and thereby upon some of the comparatively weakly constrained flat-space-observable terms, and the dominant elements of other EEP-violating vectors that are hard to detect with non-gravitational tests. Up to , isotropic spin-independent EEP violation is governed by the six coefficients and , where the superscript may take the values , indicating that the new physics enters via the action of the electron, neutron, or proton fields, respectively. As the subscripts suggest, these respective coefficients are elements of a four-vector and a four-tensor: the other elements of which would give rise to spatially anisotropic anomalies. These coefficients generate measurable violations of EEP in two ways: First, they modify the effective value of for the electrons, neutrons, and protons which make up a clock or moving test particle. This channel is responsible for most of the signal in experiments which measure the total phase accumulated by a test particle’s wavefunction, or which compare the effective gravitational acceleration of different objects. It also contributes to the signal in conventional clock comparison tests by perturbing the motion of any test mass used to map the gravitational potential . These terms can also modify the rest-frame energy and energy levels of composite systems as a function of the gravitational potential, shifting the Compton and transition frequencies of a bound system in a species and state-dependent manner. This is the primary signal available to EEP tests using conventional clocks. These position-dependent binding energy shifts also produce correction to the motion of the freely falling composite particle. While this correction is small, it is important because it increases the difference between the linear combinations of mSME coefficients constrained by individual experiments. While the first mechanism is a simple function of the number of electrons, neutrons and protons in any given composite particle, estimates of the second mechanism for EEP-violation determined by the particle’s internal structure, discussed in more detail below.
Without loss of generality, we choose coordinates such that light propagates in the usual way through curved spacetime (see Sec. III). The Lorentz-violating properties of an object composed of of the neutrons, protons, and electrons can often be represented by effective coefficients
where we have neglected Lorentz-violating contributions from particles other than photons mediating the binding forces (e.g. W-bosons, -mesons, etc.). These Lorentz vectors and tensors are defined in one particular inertial reference frame. Although it is conventional to adopt a sun-centered celestial equatorial reference frame BaileyKostelecky when performing such analyses, the distinction between it and any Earth-centered frame is unimportant to a derivation of the effects of the isotropic subset of the minimal gravitational SME up to terms appearing at higher powers of , and will not be made here.
As derived in KosteleckyTasson , the effects of Lorentz symmetry violation on the motion of a test particle, up to Post-Newtonian order PNO(3), as defined by their suppression by no more than 3 powers of , are described by the particle action
where , and , for a non-rotating spherical source with gravitational potential . The vector, where the overbar indicates the value of in the absence of gravity, is typically unobservable in non-gravitational experiments, as it can be eliminated from the action by a global phase shift. If has a non-minimal coupling (parameterized here by ) to the gravitational potential, however, it does not drop out of the action under such a field redefinition, and produces observable effects. In general, is itself modified by the contributions of the pure gravity sector coefficients and any Lorentz-symmetry violating terms in the action for the gravitational source body. We consider only experiments performed in the Earth’s gravitational field, and thus neglect the effects of such modifications of as being common to all experiments. The isotropic subset and is of particular interest because the former can only be observed by gravitational tests, and are not yet individually constrained; while the , though measurable in non-gravitational experiments, are comparatively weakly constrained relative to the and terms.
The expansion of Eq. (6) up to PNO(2) terms, dropping the constant term associated with the rest particle mass, and redefining , yields
where is the relative velocity of the Earth and the test particle. Thus, at leading order, a combination of and coefficients rescale the particle’s gravitational mass (proportional to ) relative to its inertial mass (proportional to ).
V Experimental Observables
The gravitational acceleration of a test mass is obtained by finding the extremum of Eq. (7), so that
Thus the test mass moves in the gravitational potential as if it were actually in a rescaled potential . The terms can also give rise to position-dependent shifts in the binding energy of composite particles. Appearing at in the expansion of Eq. (6), terms proportional to produce an anomalous -dependent rescaling of a particle’s inertial mass. Though these terms are in most cases negligible for systems of non-relativistic, gravitationally bound particles, the internal velocities of the constituents of a composite particle held together by electromagnetic or nuclear forces are large enough to make the terms significant. To leading order, in gravitational fields that are weak compared to the non-gravitational binding forces, it has been shown KosteleckyTasson that the bound particles’ equations of motion are unchanged save for the substitution
causing the energy (as measured in its local frame) of a bound system of particles to vary as a function of the gravitational potential . For a clock referenced to a transition between different bound states of a system of particles, the substitution in Eq. (9) gives rise to an anomalous rescaling of its measured redshift by a factor of KosteleckyTasson ; redshiftPRL . The factor may be different for clocks referenced to different transitions, depending upon how the bound system’s energy levels scale with the constituent particles’ masses. The value of for a Bohr transition in hydrogen has been estimated to be KosteleckyTasson
since this energy is proportional to the reduced mass . Energy conservation and the principle of stationary action requires that this effect also contribute to the motion of the composite particle nordtvedt , since any increase in the energy of a given configuration of composite particle with increasing must be offset by an increase in the amount of work necessary to elevate it, implying that the effective for the composite system is also increased. Thus the fractional modification of the effective gravitational force acting on the hydrogen atom due to corrections of the Bohr energy would be . In general, this effect scales as , where is the ratio of the relevant binding energy to the bound particle’s rest mass. Even for atoms with higher , this is as small as . Note that the exact value of the binding energy correction to the motion depends upon the details of the bound system. Contributions from -dependent variations in the binding energy of the nucleus can be substantially larger, as the mass defect of many nucleons can represent between and of an atom’s overall mass. The form of depends on the details of the atomic nucleus, and is model dependent.
All EEP tests compare the action of gravity on one system to its effects on another. Relative, or null redshift tests compare the frequencies of two different clocks as they are moved about in the gravitational potential, and the precision to which they agree with one another constrains the difference . Tests involving the gravitationally-determined motion of two matter-wave clocks redshiftPRL or test masses constrain the difference , where is the binding energy of the test particle . Clock comparison tests in which the clocks’ motion is not determined by the gravitational potential (e.g., they are at rest, or on continuously monitored trajectories, as in Gravity Probe A Vessot or the proposed ACES mission ACES ) limit the difference , where the superscript “grav” denotes terms applicable to the gravimeter used to measure the potential used to calculate the expected value of the clock’s redshifted signal. In principle, the gravimeter could also be another clock. See redshiftPRL for a more detailed analysis relevant to some specific tests of EEP.
Vi Sensitivity to the coefficients
The coefficients of the gravitational mSME are of particular interest because they are difficult to observe in non-gravitational experiments ColladayKostelecky . In a flat spacetime, these terms can be eliminated from each particle’s Lagrangian by a global phase shift. This is not necessarily the case in a curved spacetime KosteleckyTasson ; KosteleckyTassonPRL ; gravitationally induced fluctuations (proportional to the potential and an arbitrary gravitational interaction constant ) in are observable.
Sensitivity to
Figure 1: Sensitivity to (vertical axis) and (horizontal axis) for different nuclear isotopes. Experiments that compare two nuclides that are widely separated on this plot have greater sensitivity than those that use neighboring nuclides. Gray points indicate stable isotopes, while blue, green, and orange points indicate isotopes with lifetimes of over 1 Gyr, 1 Myr - 1 Gyr, or 1 yr - 1 Myr, respectively. Red points indicate isotopes with lifetimes measured in hours. The sum and difference factors for Ti and SiO are defined for objects made with natural isotopic abundances. Not shown are the coefficients for H, H, H, or He. Nuclide data is taken from nubase .
These coefficients are readily found in any test sensitive to the of one or more test particles, as given by Eqs. (5) and (8). Since different materials are made of different numbers of neutrons, protons, and electrons, UFF or matter-wave tests involving different species can be used to set limits on the coefficients. Practical limitations, however, can make it difficult to set independent constraints on all three terms. Tests involving neutral particles, for example, are only sensitive to the sum . The fractional -dependent shift in the gravitational potential reduces to .
Placing constraints upon these two neutral-particle parameters is further complicated by the fact that the number of neutrons relative to the number of protons typically satisfies for stable nuclei. This often results in a significant suppression of the EEP-violating signal proportional to and when the effect of gravity on different systems is compared. It is therefore useful to consider which combination of atomic species might best be employed to obtain limits on the coefficients. Most experiments are primarily sensitive to the combination , with a small residual sensitivity to proportional to deviations from the trend in vs. . The numerical factors multiplying these sum () and difference () coefficients are plotted for nuclides with lifetimes greater than one hour nubase in Figure 1. Species which have been or may soon be used to test the EEP are explicitly indicated. Also plotted are the coefficients for natural abundance SiO, since many modern gravimeters employ falling corner cubes made largely out of glass, and bulk Ti metal, as the best modern UFF tests compare it with Be Schlamminger:2008 .
A UFF or matter-wave interferometer test which compares H with H or He would have the greatest intrinsic sensitivity to the coefficients. If we restrict ourselves to heavier nuclides with equal proton numbers, Li and Li are the clear favorites, with a suppression of only on the difference term , and on the sum . Comparisons between K and Rb rbvsk , are nearly as sensitive, with a suppression factor of on the difference and on the sum signals. Comparisons between different stable isotopes of the same element become less sensitive with increased atomic weight. A test comparing Li versus Cs or a Li versus any isotope of potassium would yield better sensitivity to the signal, with only a factor of suppression. The more recently analyzed Cs matter-wave redshift test redshift had a slightly greater sensitivity to .
Vii Conclusion
We have presented a quantitative analysis of the experimental signals for EEP violation in matter-wave interferometers in the context of the mSME, with a particular focus on anomalies that are difficult to constrain in non-gravitational experiments. We find that it is unnecessary to exchange photons to carry out definitive tests of the gravitational redshift, as anomalous physics in the electromagnetic sector is either well constrained, or transferrable to other sectors by a judicious choice of coordinate chart. We use the mSME to quantitatively determine the relative sensitivities of existing and proposed experimental tests of the EEP rbvsk , illustrated in Figure 1. This figure also reveals that tests employing one or more metastable nuclides can potentially offer greater sensitivity to these parameters than would otherwise be possible for stable isotopes with large . Matter-wave interferometers may be particularly well suited to carry out such tests, since the atomic source need not be isotopically pure, and particle decay on timescales longer than a single experimental shot (typically less than 10 s) will not affect the measured signal.
Appendix A Equivalence to the Schrödinger equation
From Sec. 2, it is clear that many effects in quantum mechanics are connected to the gravitational redshift and special relativistic time dilation. They can, therefore, be employed in testing general relativity. It is thus interesting to develop the above ideas into a more familiar form that is directly applicable to nonrelativistic quantum mechanics. Here, we will show that the interpretation of matter-wave interferometry as redshift tests is mathematically equivalent to the Schrödinger equation of an atom in a gravitational field. We shall follow the approach of Feynman Feynman1948 . This approach is, thus, not fundamentally new. However, there is pleasure in viewing familiar things from a new point of view. We start by using a post newtonian approximation
where is the usual 3-velocity. We replaced the parameter by the coordinate time , which is possible at . We now compute the path integral for propagation over an infinitesimal distance between and over an infinitesimal time interval , during which the integrand can be treated as constant. We denote . For an infinitesimal , so
where is a normalization factor, , and . We can expand in powers of :
where . Computing the Gaussian integrals Zee , we obtain
where is the determinant of and is the inverse matrix. We determine the normalization factor from the fact that must approach for and carry out the derivatives with respect to and inserting and . Working in post newtonian order 3, we can neglect and terms proportional to . This leads to a Schrödinger equation
where we have substituted . The 3-vector is defined by . We neglected a term proportional to and one proportional to . From the path integral approach, the usual commutation relations can also be derived Feynman1948 . This shows that quantum mechanics is a description of waves oscillating at the Compton frequency that explore all possible paths through curved spacetime.
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For everything else, email us at [email protected]. |
5b24bf968d238695 | Kategorier Priser Virksomhedsløsninger
Free Textbook
Atomistic Models
Concepts in Computational Chemistry
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Language: English
1. Introduction
1. What is molecular modeling?
2. Briefsummary
2. Molecular quantum mechanics
1. The Schrödinger equation
2. The molecular Hamiltonian
3. Some basic properties of the wavefunction
4. The Born-Oppenheimer approximation
5. Atomic orbitals
6. Molecular orbitals
7. The variational principle
8. Perturbation theory
9. First and second-order electric properties
10. The Hartree-Fock approximation
11. Basis set expansion
12. Electron correlation
13. Density functional theory
3. Force fields
1. Introduction to force fields
2. Force-field terms for covalent bonding
3. Intermolecular interactions
4. Intermolecular forces from quantum mechanics
About the Author
Per-Olof Åstrand
Per-Olof Åstrand was born in Sätila in western Sweden in 1965 and grew up in Tygelsjö just south of Malmö in the very southern part of Sweden. After a compulsory military service, he moved to Lund in 1985 to study at Lund University for a degree in chemical engineering which was completed in 1990. He then started on a Ph.D. degree in theoretical chemistry with Gunnar Karlström as thesis supervisor and Anders Wallqvist as cosupervisor. His thesis work was on the development of the polarizable force field named NEMO, and the Ph.D. thesis included also theoretical work on far infrared spectroscopy in collaboration with Anders Engdahl and Bengt Nelander as well as molecular dynamics simulations with Per Linse and Kurt V. Mikkelsen using the NEMO force field.
He moved to Denmark in 1995 for a postdoc with Kurt V. Mikkelsen, first one year at Aarhus University and then at the University of Copenhagen in 1996-97. In 1998, he moved to Risø National Laboratory just north of Roskilde outside Copenhagen, and in 2001-02 he was part time at the University of Copenhagen and at Risø. In this period he was also included in a collaboration with Kenneth Ruud and Trygve Helgaker at the University of Oslo.
In 2002, he was appointed as full professor in computational chemistry at the Norwegian University of Science and Technology (NTNU) in Trondheim, Norway and he thereby moved to his third Scandinavian country. He has been teaching molecular modeling and computational chemistry since 2002. In 2008, he established a new chemistry-oriented course for the new nanotechnology engineering program in statistical thermodynamics, which he has been teaching since then. He is also giving a biannual Ph.D. course in advanced molecular modeling with a focus on the theory of intermolecular forces, the connection between force fields and quantum mechanics as well as the connection between microscopic and macroscopic polarization.
Since his Ph.D. work, his research has covered a broad range of theoretical chemistry including applied quantum chemistry, vibrational motion, force-field development and molecular dynamics simulations. Many of the projects have been carried out in close collaboration with experimental groups, most noteably work on far infrared spectroscopy on bimolecular complexes (Anders Engdahl and Bengt Nelander, Lund), THz spectroscopy on liquid water (Cecilie Rønne and Søren R. Keiding, Aarhus), azobenzenes in optical storage materials (P.S. Ramanujam and Søren Hvilsted, Risø), and more recently work on heterogenous catalysis (Magnus Rønning and De Chen, NTNU) and electrically insulating properties of dielectric liquids (Lars Lundgaard, SINTEF Energy, and Mikael Unge, ABB Corporate Research). |
5890c89864de9bc1 | In Re “CopenHagen” and “COLLAPSE”
I was having an e-mail conversation the other day with a friend from olden days — another MIT student who made it out with a physics degree the same year I did — and that led me to set down some thoughts about history and terminology that may be useful to share here.
My primary claim is the following:
We should really expunge the term “the Copenhagen interpretation” from our vocabularies.
What Bohr thought was not what Heisenberg thought, nor was it what Pauli thought; there was no single unified “Copenhagen interpretation” worthy of the name. Indeed, the term does not enter the written literature until the 1950s, and that was mostly due to Heisenberg acting like he and Bohr were more in agreement back in the 1920s than they actually had been.
For Bohr, the “collapse of the wavefunction” (or the “reduction of the wave packet”, or whatever you wish to call it) was not a singular concept tacked on to the dynamics, but an essential part of what the quantum theory meant. He considered any description of an experiment as necessarily beginning and ending in “classical language”. So, for him, there was no problem with ending up with a measurement outcome that is just a classical fact: You introduce “classical information” when you specify the problem, so you end up with “classical information” as a result. “Collapse” is not a matter of the Hamiltonian changing stochastically or anything like that, as caricatures of Bohr would have it, but instead, it’s a question of what writing a Hamiltonian means. For example, suppose you are writing the Schrödinger equation for an electron in a potential well. The potential function $V(x)$ that you choose depends upon your experimental arrangement — the voltages you put on your capacitor plates, etc. In the Bohrian view, the description of how you arrange your laboratory apparatus is in “classical language”, or perhaps he’d say “ordinary language, suitably amended by the concepts of classical physics”. Getting a classical fact at your detector is just the necessary flipside of starting with a classical account of your source.
(Yes, Bohr was the kind of guy who would choose the yin-yang symbol as his coat of arms.)
To me, the clearest expression of all this from the man himself is a lecture titled “The causality problem in atomic physics”, given in Warsaw in 1938 and published in the proceedings, New Theories in Physics, the following year. This conference is notable for several reasons, among them the fact that Hans Kramers, speaking both for himself and on behalf of Heisenberg, suggested that quantum mechanics could break down at high energies. More than a decade after what we today consider the establishment of the quantum theory, the pioneers of it did not all trust it in their bones; we tend to forget that nowadays.
As to how Heisenberg disagreed with Bohr, and what all this has to do with decoherence, I refer to Camilleri and Schlosshauer.
Do I find the Bohrian position that I outlined above satisfactory? No, I do not. Perhaps the most important reason why, the reason that emotionally cuts the most deeply, is rather like the concern which Rudolf Haag raised while debating Bohr in the early 1950s:
I tried to argue that we did not understand the status of the superposition principle. Why are pure states described as [rays] in a complex linear space? Approximation or deep principle? Niels Bohr did not understand why I should worry about this. Aage Bohr tried to explain to his father that I hoped to get inspiration about the direction for the development of the theory by analyzing the existing formal structure. Niels Bohr retorted: “But this is very foolish. There is no inspiration besides the results of the experiments.” I guess he did not mean that so absolutely but he was just annoyed. […] Five years later I met Niels Bohr in Princeton at a dinner in the house of Eugene Wigner. When I drove him afterwards to his hotel I apologized for my precocious behaviour in Copenhagen. He just waved it away saying: “We all have our opinions.”
Why rays? Why complex linear space? I want to know too. |
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