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Section 12.3: Classical and Quantum-Mechanical Probabilities
n = ω =
check, then click the input values and play button to see the probability densities instead.
check, then click the input values and play button to see the classical probability distributions too.
Please wait for the animation to completely load.
In this Section we compare the energy eigenfunctions of the harmonic oscillator in position space to those in momentum space and then compare the resulting probability densities to their classical counterpart probability distributions.
Momentum-space energy eigenfunctions can be obtained by calculating the Fourier transform of the position-space energy eigenfunction (see Section 8.5). However, in the case of the harmonic oscillator, it is easier to consider the time-independent Schrödinger equation in momentum space:
[p2/2m − (mω2ħ2/2) d2/dp2] φ(p) = Eφ(p). (12.16)
In momentum space, the operator p represents the momentum and the operator ( d/dp) represents the position operator, x. Compare Eq.(12.6) to Eq.(12.16). What do you notice? It turns out that the two equations are in the same form, which can be seen if you make the substitution that p = mωx or x = p/mω. Therefore, the solutions to the two differential equations are the same, apart from a scaling factor. From Eq. (12.11) and Eq. (12.16), we have that3
φn(p) = Bn Hnp) exp(−η2p2/2) , (12.17)
where η = (mωħ)-1/2 = β/mω. The normalization constant becomes:
Bn = (2nn!(mωħπ)1/2)−1/2 , (12.18)
In the animations, you can change n and ω, and see the resulting changes in the position-space and momentum-space energy eigenfunctions. We have used 2m = ħ = 1 and initially ω = 2. Can you guess why we have chosen this particular value for ω? Using the first check box, you can view the probability densities in position and momentum space. In the animation, you can also check the box that superimposes the classical probability distributions (in pink) on the quantum-mechanical probability densities. Note the symmetry about x = 0 that the classical position-space and momentum-space probability distributions exhibit.
3If we had Fourier transformed the position-space energy eigenfunctions instead, we would have found the same result as Eq.(12.6), but multiplied by a phase exp(inπ/4) where n is the particular state's quantum number. This adds an overall phase to the momentum-space wave function and, as such, is not important.
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01389bc9ecce3642 | Quantum field theory
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"Relativistic quantum field theory" redirects here. For other uses, see Relativity.
In theoretical physics, quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. A QFT treats particles as excited states of an underlying physical field, so these are called field quanta.
For example, quantum electrodynamics (QED) has one electron field and one photon field; quantum chromodynamics (QCD) has one field for each type of quark; and, in condensed matter, there is an atomic displacement field that gives rise to phonon particles. Edward Witten describes QFT as "by far" the most difficult theory in modern physics.[1]
In QFT, quantum mechanical interactions between particles are described by interaction terms between the corresponding underlying fields. QFT interaction terms are similar in spirit to those between charges with electric and magnetic fields in Maxwell's equations. However, unlike the classical fields of Maxwell's theory, fields in QFT generally exist in quantum superpositions of states and are subject to the laws of quantum mechanics.
Quantum mechanical systems have a fixed number of particles, with each particle having a finite number of degrees of freedom. In contrast, the excited states of a QFT can represent any number of particles. This makes quantum field theories especially useful for describing systems where the particle count/number may change over time, a crucial feature of relativistic dynamics.
Because the fields are continuous quantities over space, there exist excited states with arbitrarily large numbers of particles in them, providing QFT systems with an effectively infinite number of degrees of freedom. Infinite degrees of freedom can easily lead to divergences of calculated quantities (i.e., the quantities become infinite). Techniques such as renormalization of QFT parameters or discretization of spacetime, as in lattice QCD, are often used to avoid such infinities so as to yield physically meaningful results.
Most theories in standard particle physics are formulated as relativistic quantum field theories, such as QED, QCD, and the Standard Model. QED, the quantum field-theoretic description of the electromagnetic field, approximately reproduces Maxwell's theory of electrodynamics in the low-energy limit, with small non-linear corrections to the Maxwell equations required due to virtual electron–positron pairs.
In the perturbative approach to quantum field theory, the full field interaction terms are approximated as a perturbative expansion in the number of particles involved. Each term in the expansion can be thought of as forces between particles being mediated by other particles. In QED, the electromagnetic force between two electrons is caused by an exchange of photons. Similarly, intermediate vector bosons mediate the weak force and gluons mediate the strong force in QCD. The notion of a force-mediating particle comes from perturbation theory, and does not make sense in the context of non-perturbative approaches to QFT, such as with bound states.
The gravitational field and the electromagnetic field are the only two fundamental fields in nature that have infinite range and a corresponding classical low-energy limit, which greatly diminishes and hides their "particle-like" excitations. Albert Einstein in 1905, attributed "particle-like" and discrete exchanges of momenta and energy, characteristic of "field quanta", to the electromagnetic field. Originally, his principal motivation was to explain the thermodynamics of radiation. Although the photoelectric effect and Compton scattering strongly suggest the existence of the photon, it is now understood that they can be explained without invoking a quantum electromagnetic field; therefore, a more definitive proof of the quantum nature of radiation is now taken up into modern quantum optics as in the antibunching effect.[2]
There is currently no complete quantum theory of the remaining fundamental force, gravity. Many of the proposed theories to describe gravity as a QFT postulate the existence of a graviton particle that mediates the gravitational force. Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will behave like Einstein's general theory of relativity in the low-energy limit. Quantum field theory of the fundamental forces itself has been postulated to be the low-energy effective field theory limit of a more fundamental theory such as superstring theory.
The early development of the field involved Dirac, Fock, Pauli, Heisenberg and Bogolyubov. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.
Gauge theory[edit]
Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed by the 1970s through the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.
Grand synthesis[edit]
Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics, which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s, which led to the seminal reformulation of quantum field theory by Kenneth G. Wilson in 1975.
Classical and quantum fields[edit]
A classical field is a function defined over some region of space and time.[3] Two physical phenomena which are described by classical fields are Newtonian gravitation, described by Newtonian gravitational field g(x, t), and classical electromagnetism, described by the electric and magnetic fields E(x, t) and B(x, t). Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom.[3]
Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—photons—rather than continuous fields. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a quantum field.
It is not immediately clear how to write down such a quantum field, since quantum mechanics has a structure very unlike a field theory. In its most general formulation, quantum mechanics is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically observable quantities and the state space represents the possible states of the system under study.[4] For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators \hat{x} and \hat{p}. Field theory, in contrast, treats x as a way to index the field rather than as an operator.[5]
There are two common ways of developing a quantum field: the path integral formalism and canonical quantization.[6] The latter of these is pursued in this article.
Lagrangian formalism[edit]
Quantum field theory frequently makes use of the Lagrangian formalism from classical field theory. This formalism is analogous to the Lagrangian formalism used in classical mechanics to solve for the motion of a particle under the influence of a field. In classical field theory, one writes down a Lagrangian density, \mathcal{L}, involving a field, φ(x,t), and possibly its first derivatives (∂φ/∂t and ∇φ), and then applies a field-theoretic form of the Euler–Lagrange equation. Writing coordinates (t, x) = (x0, x1, x2, x3) = xμ, this form of the Euler–Lagrange equation is[3]
\frac{\partial}{\partial x^\mu} \left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial x^\mu)}\right] - \frac{\partial\mathcal{L}}{\partial\phi} = 0,
where a sum over μ is performed according to the rules of Einstein notation.
By solving this equation, one arrives at the "equations of motion" of the field.[3] For example, if one begins with the Lagrangian density
\mathcal{L}(\phi,\nabla\phi) = -\rho(t,\mathbf{x})\,\phi(t,\mathbf{x}) - \frac{1}{8\pi G}|\nabla\phi|^2,
and then applies the Euler–Lagrange equation, one obtains the equation of motion
4\pi G \rho(t,\mathbf{x}) = \nabla^2 \phi.
This equation is Newton's law of universal gravitation, expressed in differential form in terms of the gravitational potential φ(t, x) and the mass density ρ(t, x). Despite the nomenclature, the "field" under study is the gravitational potential, φ, rather than the gravitational field, g. Similarly, when classical field theory is used to study electromagnetism, the "field" of interest is the electromagnetic four-potential (V/c, A), rather than the electric and magnetic fields E and B.
Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields. These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.
Single- and many-particle quantum mechanics[edit]
In quantum mechanics, a particle (such as an electron or proton) is described by a complex wavefunction, ψ(x, t), whose time-evolution is governed by the Schrödinger equation:
-\frac{{\hbar}^2}{2m}\frac{{\partial}^2}{\partial x^2}\psi(x,t) + V(x)\psi(x,t) = i \hbar \frac{\partial}{\partial t} \psi(x,t).
Here m is the particle's mass and V(x) is the applied potential. Physical information about the behavior of the particle is extracted from the wavefunction by constructing expected values for various quantities; for example, the expected value of the particle's position is given by integrating ψ*(x) x ψ(x) over all space, and the expected value of the particle's momentum is found by integrating ψ*(x)dψ/dx. The quantity ψ*(x)ψ(x) is itself in the Copenhagen interpretation of quantum mechanics interpreted as a probability density function. This treatment of quantum mechanics, where a particle's wavefunction evolves against a classical background potential V(x), is sometimes called first quantization.
This description of quantum mechanics can be extended to describe the behavior of multiple particles, so long as the number and the type of particles remain fixed. The particles are described by a wavefunction ψ(x1, x2, …, xN, t), which is governed by an extended version of the Schrödinger equation.
Often one is interested in the case where N particles are all of the same type (for example, the 18 electrons orbiting a neutral argon nucleus). As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. This is achieved by using a Slater determinant as the wavefunction of a fermionic system (and a Slater permanent for a bosonic system), which is equivalent to an element of the symmetric or antisymmetric subspace of a tensor product.
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_{p\in S_N} |\phi_{p(1)}\rang \otimes \cdots \otimes |\phi_{p(N)} \rang,
where |\phi_i\rang 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. \sqrt{\frac{\prod_j N_j!}{N!}} is a normalizing factor.
There are several shortcomings to the above description of quantum mechanics, which are addressed by quantum field theory. First, it is unclear how to extend quantum mechanics to include the effects of special relativity.[7] Attempted replacements for the Schrödinger equation, such as the Klein–Gordon equation or the Dirac equation, have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions not having a well-defined probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept. The second shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation;[8] this is crucial for describing phenomena such as pair production, which result from the conversion between mass and energy according to the relativistic relation E = mc2.
Second quantization[edit]
Main article: Second quantization
In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics.
Several other approaches exist, such as the Feynman path integral,[9] which uses a Lagrangian formulation. For an overview of some of these approaches, see the article on quantization.
For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states which are possible in the system by |\phi_1\rang, |\phi_2\rang, |\phi_3\rang, and so on. For example, the 3-particle state with one particle in state |\phi_1\rang and two in state |\phi_2\rang is
\frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang
|\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang
|\phi_2\rang |\phi_1\rang \right].
The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states |\phi_1\rang, |\phi_2\rang, etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as
|1, 2, 0, 0, 0, \dots \rangle.
An N-particle state belongs to a space of states describing systems of N particles. The next step is to combine the individual N-particle state spaces into an extended state space, known as Fock space, which can describe systems of any number of particles. This is composed of the state space of a system with no particles (the so-called vacuum state, written as |0\rang), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. States describing a definite number of particles are known as Fock states: a general element of Fock space will be a linear combination of Fock states. There is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.
At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number j indicating which of the single-particle states |\phi_1\rang, |\phi_2\rang,\dots,|\phi_j\rang,\dots it refers to:
| N_1, N_2, N_3, \dots, N_j, \dots \rang .
The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to ladder operators in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator a_2 and creation operator a_2^\dagger are easily defined in the occupation number representation as having the following effects:
a_2 | N_1, N_2, N_3, \dots \rang = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \dots \rang,
a_2^\dagger | N_1, N_2, N_3, \dots \rang = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \dots \rang.
It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation
\left[a_i , a_j \right] = 0 \quad,\quad
\left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad
\left[a_i , a_j^\dagger \right] = \delta_{ij},
where \delta 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.
Applying an annihilation operator a_k followed by its corresponding creation operator a_k^\dagger returns the number N_k of particles in the kth single-particle eigenstate:
a_k^\dagger\,a_k|\dots, N_k, \dots \rangle=N_k| \dots, N_k, \dots \rangle.
The combination of operators a_k^\dagger a_k is known as the number operator for the kth eigenstate.
The Hamiltonian operator of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, for a field of free (non-interacting) bosons, the total energy of the field is found by summing the energies of the bosons in each energy eigenstate. If the kth single-particle energy eigenstate has energy E_k and there are N_k bosons in this state, then the total energy of these bosons is E_k N_k. The energy in the entire field is then a sum over k:
E_\mathrm{tot} = \sum_k E_k N_k
This can be turned into the Hamiltonian operator of the field by replacing N_k with the corresponding number operator, a_k^\dagger a_k. This yields
H = \sum_k E_k \, a^\dagger_k \,a_k.
It turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers Ni can only take on the value 0 or 1. The fermionic annihilation operators c and creation operators c^\dagger are defined by their actions on a Fock state thus
c_j | N_1, N_2, \dots, N_j = 0, \dots \rangle = 0
c_j | N_1, N_2, \dots, N_j = 1, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 0, \dots \rangle
c_j^\dagger | N_1, N_2, \dots, N_j = 0, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 1, \dots \rangle
c_j^\dagger | N_1, N_2, \dots, N_j = 1, \dots \rangle = 0.
These obey an anticommutation relation:
\left\{c_i , c_j \right\} = 0 \quad,\quad
\left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad
\left\{c_i , c_j^\dagger \right\} = \delta_{ij}.
One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.
Field operators[edit]
We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states. However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position.
To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator \phi(\mathbf{r}) is
\phi(\mathbf{r}) \ \stackrel{\mathrm{def}}{=}\ \sum_{j} e^{i\mathbf{k}_j\cdot \mathbf{r}} a_{j}.
The bosonic field operators obey the commutation relation
\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad
\left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad
\left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})
where \delta(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
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 a quantum-mechanical amplitude for finding a particle in some position. 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 (in the non-relativistic limit and for negligible self-interactions) is
H = - \frac{\hbar^2}{2m} \int d^3\!r \ \phi^\dagger(\mathbf{r}) \nabla^2 \phi(\mathbf{r}) + \frac{1}{2}\int\!d^3\!r \int\!d^3\!r' \; \phi^\dagger(\mathbf{r}) \phi^\dagger(\mathbf{r}') 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 \phi 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.
Once the Hamiltonian operator is obtained as part of the canonical quantization process, the time dependence of the state is described with the Schrödinger equation, just as with other quantum theories. Alternatively, the Heisenberg picture can be used where the time dependence is in the operators rather than in the states.
Unification of fields and particles[edit]
The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon.[10] In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. For free (non-interacting) quantum fields, the quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.
Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field), so long as one can treat interactions as "perturbations" of free fields. There are still unsolved problems relating to the more general case of interacting fields that may or may not be adequately described by perturbation theory. For more on this topic, see Haag's theorem.
Physical meaning of particle indistinguishability[edit]
The second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.
Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.
Particle conservation and non-conservation[edit]
During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles (N), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations N is an important and perfectly well-defined quantity, e.g. if we are describing a gas of atoms sealed in a box. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator \hat{N}, which measures the total number of particles present. As with any quantum mechanical observable, \hat{N} is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the N-particle subspace of the total Fock space, and the situation could equally well be described by ordinary N-particle quantum mechanics. (Strictly speaking, this is only true in the noninteracting case or in the low energy density limit of renormalized quantum field theories)
For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator a_k is immediately put back by the creation operator a_k^\dagger.
On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of \hat{N}, which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of N. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as
H_I = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k,
where a_k^\dagger and a_k denotes the bosonic creation and annihilation operators, c_k^\dagger and c_k denotes the fermionic creation and annihilation operators, and V_q is a parameter that describes the strength of the interaction. This "interaction term" 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 type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. The interaction between electrons and photons is treated in a similar way, but 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 typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are particularly 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. In addition, the concept of a coherent state (used to model the laser and the BCS ground state) refers to a state with an ill-defined particle number but a well-defined phase.
Axiomatic approaches[edit]
The preceding description of quantum field theory follows the spirit in which most physicists approach the subject. However, it is not mathematically rigorous. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.
The first class of axioms, first proposed during the 1950s, include the Wightman, Osterwalder–Schrader, and Haag–Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.
During the 1980s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. However, most of the physically relevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang–Mills theory—is linked to this issue.
Associated phenomena[edit]
In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.
Main article: Renormalization
Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances that each give a finite contribution which results in a divergent series.
Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field that it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf with help from Furry, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.
The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, comes from recognizing that all the infinities in the interactions of photons and electrons can be isolated into redefining a finite number of quantities in the equations by replacing them with the observed values: specifically the electron's mass and charge: this is called renormalization. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by 't Hooft and Veltman, is a symmetry-preserving cutoff for perturbation theory (this process is called regularization). There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.
On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit.
The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories. A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The Standard Model of particle physics is perturbatively renormalizable, and so are its component theories (quantum electrodynamics/electroweak theory and quantum chromodynamics). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined.
The renormalization group describes how renormalizable theories emerge as the long distance low-energy effective field theory for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.
Haag's theorem[edit]
See also: Haag's theorem
From a mathematically rigorous perspective, there exists no interaction picture in a Lorentz-covariant quantum field theory. This implies that the perturbative approach of Feynman diagrams in QFT is not strictly justified, despite producing vastly precise predictions validated by experiment. This is called Haag's theorem, but most particle physicists relying on QFT largely shrug it off.
Gauge freedom[edit]
A gauge theory is a theory that 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. Consequently, 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 the shift may be different at every point in space-time. 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 gauge of variables is termed gauge transformation.
In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics.
The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non-unitary and again inconsistent (see optical theorem).
In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis).
All the fundamental interactions in nature are described by gauge theories. These are:
Multivalued gauge transformations[edit]
The gauge transformations which leave the theory invariant involve, by definition, only single-valued gauge functions \Lambda(x_i) which satisfy the Schwarz integrability criterion
\partial_{x_i x_j} \Lambda = \partial_{x_jx_i} \Lambda.
An interesting extension of gauge transformations arises if the gauge functions \Lambda(x_i) are allowed to be multivalued functions which violate the integrability criterion. These are capable of changing the physical field strengths and are therefore no proper symmetry transformations. Nevertheless, the transformed field equations describe correctly the physical laws in the presence of the newly generated field strengths. See the textbook by H. Kleinert cited below for the applications to phenomena in physics.
Main article: Supersymmetry
Supersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa. It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory.
The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider. The Higgs particle has been detected at the LHC, and no such superparticles have been discovered.
See also[edit]
1. ^ "Beautiful Minds, Vol. 20: Ed Witten". la Repubblica. 2010. Retrieved 22 June 2012. See here.
2. ^ J. J. Thorn et al. (2004). Observing the quantum behavior of light in an undergraduate laboratory. . J. J. Thorn, M. S. Neel, V. W. Donato, G. S. Bergreen, R. E. Davies, and M. Beck. American Association of Physics Teachers, 2004.DOI: 10.1119/1.1737397.
3. ^ a b c d David Tong, Lectures on Quantum Field Theory, chapter 1.
4. ^ Srednicki, Mark. Quantum Field Theory (1st ed.). p. 19.
5. ^ Srednicki, Mark. Quantum Field Theory (1st ed.). pp. 25–6.
6. ^ Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 61.
7. ^ David Tong, Lectures on Quantum Field Theory, Introduction.
8. ^ Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 3.
9. ^ Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0-19-851997-4. Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.
10. ^ Newton, T.D.; Wigner, E.P. (1949). "Localized states for elementary systems". Reviews of Modern Physics 21 (3): 400–406. Bibcode:1949RvMP...21..400N. doi:10.1103/RevModPhys.21.400.
Further reading[edit]
General readers
Introductory texts
Advanced texts
External links[edit] |
bcd16b50465345be | Take the 2-minute tour ×
Question. Let $(S(t))_{t \ge 0}$ be a continuous semigroup of linear operators on some Banach space $X$. Might there exist $f, g\in X$ and $0<t_0<t_1$ such that \begin{equation}S(t_0)f=S(t_1)g\end{equation} but \begin{equation} S(t_0-\varepsilon_0)f\ne S(t_1-\varepsilon_1)g \end{equation} for all $0<\varepsilon_0\le t_0$ and $0<\varepsilon_1\le t_1$ (in particular, $f\ne g$)?
Pictorially, I am wondering if the following configuration is possible:
enter image description here
Of course we know that, when the evolution is given by a group, this is not possible: orbits either coincide or are disjoint. This is the case of autonomous ODE systems or of the Schrödinger equation. But here we have a semi group, such as the heat one, which only goes forward in time, not backwards. So the only obvious thing that we can say is that, as soon as they touch, orbits merge into one. But in principle I don't see why they should coincide in the past.
Added: After some searches, I have found that for the special case of the heat equation, the answer is negative. This is commonly referred to as backward uniqueness property. Here is a simplified version, which takes into account classical solutions on bounded domains:
Theorem (Taken from Evans's book on PDE, 2nd ed., pag.64) Let $U\subset \mathbb{R}^n$ be an open and bounded domain. Suppose $u, \bar{u}$ are classical solutions of \begin{equation} \begin{cases} u_t=\Delta u & \text{in }U\times(0, T) \\ u=0 &\text{on }\partial U \times [0, T] \end{cases} \end{equation} If at time $T$ we have \begin{equation} u(x, T)=\bar{u}(x, T),\quad \forall x\in U, \end{equation} then $u\equiv \bar{u}$ on the whole parabolic cylinder $U\times (0, T]$.
The property holds in much more general functional settings, as I read here (look for the keyword backward uniqueness).
All of this leaves the general question open. Is the backward uniqueness property true for all continuous linear semigroups? I guess that the answer should be negative, otherwise this would not be regarded as a special feature of the heat equation. However, I cannot find an explicit example.
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2 Answers 2
up vote 2 down vote accepted
I believe this is a counterexample:
Consider the semigroup on $L_2[0,\infty)$ given by $$ S(t) f(x) = f(x+t)$$ Let $f = I_{[0,1)}$, and $g = 2I_{[0,2)}$. Then $S(1)f = S(2)g = 0$, but $S(1-\epsilon_1)f \ne S(2-\epsilon_2)g$.
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I think that it is a good question. To simplify, let us study the equation $$ u_t=-\Lambda u, $$ where $\Lambda=\sqrt{-\Delta}$. This equation is locally well-posed (both, forward and backward in time) for analytic initial data in a complex strip around the real axis. Then, at some time $T>0$, two solutions with a different initial data (at time $t=0$) can not coincide. Otherwise the backward problem posed at time $T$ would be ill-posed.
I think that for the heat equation the situation is similar.
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This example is a bit too complicated for me. Could you do something more basic? It's fine to discuss the heat equation only. – Giuseppe Negro May 24 '13 at 13:31
The point of this second equaton is that I know how to solve it forward and backward (because is a first order equation both in space and time you can apply a Cauchy-Kowalevsky Theorem). For the heat equation all this is much more difficult. You have to solve backward a heat equation and you know that your initial data is an entire function. if this problem is well-posed the solution can not touch themselves. – guacho May 24 '13 at 13:50
I agree that it is more difficult. We are lucky that someone already did this for us. See the update to the question. – Giuseppe Negro May 27 '13 at 19:17
Your Answer
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a970f70654965668 | Understanding energy bands in solids
1. For an electron in a periodic potential the Schrödinger equation has solutions for which there are large gaps in the energy. This is used to explain properties relating to the electric conduction in solids.
In my book the formation of energy bands is explained using the Bragg diffraction in a crystal with reciprocal lattice vectors G. As far as I can understand the idea is that we consider the electrons as free electrons, i.e. waves in the solid, which then bounce off the potenial walls formed by the different nuclei in the crystal. The bragg diffraction is:
k' = k + G
which are fulfilled by wavevectors on the boundary of the Brillouin zone.
I guess some of that makes sense. But what about the k-vectors which do no lie near the zone boundaries. My teacher told me nothing happens to these electron waves. WHY is that? Given the nature of the model we should also expect these to be reflected at the potential walls - whether or not they fulfill the Bragg condition.
Maybe I am wrong in assuming that we see the electrons as free waves, which scatter of the periodic potential?
2. jcsd
3. What do you mean by: "Given the nature of the model" ?
The electrons only scatter if their wavenumber fulfills the Bragg-condition. In that case, the electron wave is diffracted, we have constructive interference and the electron is scattered.
Now, if the Bragg-condition is not fulfilled, then we always have destructive interference. There will be always a diffracted sub-wave and a second sub-wave that is in opposite phase so that we get destructive interference.
But I do not understand well what you mean by:"reflected at the potential walls"..
4. Isnt it the idea that we view the electrons as free electrons moving in free space but with periodic potential peaks, where the waves are reflected and transmitted. If there is destructive interference for electron waves in the region of k space, where the bragg condition is not fulfilled, why is it that nothing happens to the energy of the solutions in this area? My teacher said these solutions are just what you would expect if the electron where indeed completely free to move.
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7fd51ff038bf0557 | LIVE VIDEO: Live Newscast Watch
VIDEO: 12-year-old math prodigy Jacob Barnett teaches Calculus 2
12:10 PM, Mar 29, 2011 | comments
As a child, Jake Barnett showed signs of autism, but now he is considered a savant, a genius in math skills who at a young age is tutoring much older students on campus.
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Indianapolis (Indy Star) -- When Jacob Barnett first learned about the Schrödinger equation for quantum mechanics, he could hardly contain himself.
For three straight days, his little brain buzzed with mathematical functions.
From within his 12-year-old, mildly autistic mind, there gradually flowed long strings of pluses, minuses, funky letters and upside-down triangles -- a tapestry of complicated symbols that few can understand.
He grabbed his pencil and filled every sheet of paper before grabbing a marker and filling up a dry erase board that hangs in his bedroom. With a single-minded obsession, he kept on, eventually marking up every window in the home.
Strange, say some.
Genius, say others.
But entirely normal for Jacob, a child prodigy who used to crunch his cereal while calculating the volume of the cereal box in his head.
"Whenever I try talking about math with anyone in my family," he said, "they just stare blankly."
So do many of his older classmates at Indiana University-Purdue University Indianapolis, who marvel at seeing this scrawny little kid in the front row of the calculus-based physics class he has been taking this semester.
"When I first walked in and saw him, I thought, 'Oh my God, I'm going to school with Doogie Howser,' " said Wanda Anderson, a biochemistry major, referring to a television show that featured a 16-year-old boy-genius physician.
Elementary school couldn't keep Jacob interested. And courses at IUPUI have only served to awaken a sleeping giant.
Just a few weeks shy of his 13th birthday, Jake, as he's often called, is starting to move beyond the level of what his professors can teach.
In fact, his work is so strong and his ideas so original that he's being courted by a top-notch East Coast research center. IUPUI is interested in him moving from the classroom into a funded researcher's position.
"We have told him that after this semester . . . enough of the book work. You are here to do some science," said IUPUI physics Professor John Ross, who vows to help find some grant funding to support Jake and his work.
"If we can get all of those creative juices in a certain direction, we might be able to see some really amazing stuff down the road."
Watch as Jake teaches us Calculus 2:
"My fear was that he would never be in our world"
Teenage college student?
Developer of his own original theory on quantum physics?
Paid researcher at 13?
This is not what Jake's parents expected from a child whose first few years were spent in silence.
"Oh my gosh, when he was 2, my fear was that he would never be in our world at all," said Kristine Barnett, 36, Jake's mother.
"He would not talk to anyone. He would not even look at us."
Child psychologists assessed Jake at the time and diagnosed behavioral characteristics of a borderline autistic child. He was impaired, they said, and had a lack of "spontaneous seeking to share enjoyment," difficulty showing emotion and interacting with others.
Diagnosis: mildly autistic.
"My biggest fear," his mom said last week, with tears welling up in her eyes, "was that he had lost the ability to say, 'I love you' to us."
By age 3, Jake was the focus of a more intense evaluation from a team of psychologists, therapists and a diagnostic teacher.
Their report indicated that while Jake continued to struggle with social activities and physical development, he was showing signs of academic skills that were above his age level.
Diagnosis: Asperger's syndrome, a somewhat milder condition related to autism.
After hearing this, Jake's parents decided to pay closer attention to the things their first-born son was doing -- rather than the things he was not.
For example, Jake often recited the alphabet -- forward and then backward. He used Q-tips to create vivid geometrical shapes on the living room floor. He solved 5,000-piece puzzles (rather quickly). And he once soaked in a state road map and ended up memorizing every highway and license plate prefix.
And perhaps most amazingly, he could recite the mathematical constant pi out to 70 digits.
"I'm at 98 now," Jake said, interrupting his mom during an interview.
And then, a week later, he was up to 200 digits after the decimal point -- forward and backward.
At 3, his head was in the stars
The Barnetts decided it was time to follow Jake's lead, adopting a method that some parents of children with autism use -- floor-time therapy -- to help foster developmental growth. They let their children focus intently on subjects they like, rather than trying to conform them to "normal" things.
For Jake, that meant astronomy. As a 3-year-old, he loved looking at a book about stars, over and over again.
So off they went on a tour of the Holcomb Observatory and Planetarium at Butler University.
Kristine Barnett will never forget the day.
"We were in the crowd, just sitting, listening to this guy ask the crowd if anyone knew why the moons going around Mars were potato-shaped and not round," she recalls. "Jacob raised his hand and said, 'Excuse me, but what are the sizes of the moons around Mars?' "
The lecturer answered, and "Jacob looked at him and said the gravity of the planet . . . is so large that (the moon's) gravity would not be able to pull it into a round shape."
"That entire building . . . everyone was just looking at him, like, 'Who is this 3-year-old?' "
After that, the Barnetts began to feed Jake's hunger for knowledge, through more books and more visits to the planetarium. By the time he was 8, he got permission to sit in on an advanced astronomy class at IUPUI.
Meanwhile, his math skills were reaching astronomical levels.
By the time he was in fifth grade, Jake had become bored with elementary math. He was a student, first at Carey Ridge Elementary School and then at Westfield Intermediate School, an experience he now says he enjoyed for a while.
"The first couple of years were great, but then eventually the math started being, like, OK, we've been discussing this for a while, and it really isn't that hard," Jake said. "Can I move on to calculus now? Can I move on to algebra now?"
The boredom did not go unnoticed at home. Jake was coming home from school quiet, huddling in a safe space in the house and starting to show signs of withdrawing.
"I was really afraid we were going to lose him back into the world he was in when he was 2," his mom said.
Frank Lawlis, a Texas-based psychologist who serves as a testing supervisor for the American Mensa organization -- a society for geniuses -- said it would not have been unusual for a child with symptoms of autism to regress backward after a brief time of growth.
"One of the aspects of autism is that these kids' brains grow at an accelerated rate and then, generally speaking, there is kind of a reversal that happens," said Lawlis, who last year wrote "The Autism Answer," a book for parents of children with autism.
"The theory is that the brain reaches a certain capacity, can't grow, becomes inflamed, and then a reversal effect occurs. It's just a theory, but it's very common."
That did not happen to Jake, thanks in part to a third psychological evaluation done nearly two years ago. It showed that this fifth-grader was not regressing but was simply bored and needed to be stimulated -- in a very big way.
As in dropping out of school.
"Indeed, it would not be in Jacob's best interest to force him to complete academic work that he has already mastered," clinical neurophysiologist Carl S. Hale, Merrillville, said in a report provided by the Barnetts.
The Barnetts were blown away. They knew Jake was smart, but doctorate-level smart?
"I flunked math," Kristine said with a laugh. "I know this did not come from me."
Off to college, where he tutors classmates
Encouraged by this new assessment, the Barnetts made the tough decision to pull Jake out of Westfield Washington Schools and enroll him in IUPUI's early college entrance program that caters to gifted and talented kids -- although typically they are advanced high school students, not 12-year-old whiz kids.
As he prepared for the more rigorous work of a college class, Jake decided he ought to make sure he could master all high school-level math that would be required in college.
"In one two-week period, he sat on our front porch and learned all of his high school math," Kristine said. "He tested out of algebra 1 and 2, geometry, trigonometry and calculus."
At this point, Jake's math IQ -- which has been measured at 170 (top of the Wechsler Intelligence Scale for Children) -- could not get any higher.
"You could tell right off the bat, his performance has been outstanding," said Ross, who, at age 46 with a Ph.D. from Boston University, has never seen a kid as smart as Jake.
"When he asks a question, he is always two steps ahead of the lecture," Ross said. "Everyone in the class gets quiet. Poor kid. . . . He sits right in the front row, and they all just look at him.
"He will come to see me during office hours and ask even more detailed questions. And you can tell he's been thinking these things through."
Jake is driven by Mom or Dad from his home in Hamilton County to IUPUI's campus, where he attends classes a few days each week. In between classes, he spends time at the Honors College lounge, where he has become a go-to guy for much older classmates needing tutoring.
"A lot of people come to him for help when they don't understand a physics problem," said Anderson, his class partner. "People come up to him all the time and say, 'Hey Jake, can you help me?' "
"A lot of people think a genius is hard to talk to, but Jake explains things that would still be over their head."
His professor has noticed.
"Is he a genius? Well, yeah," Ross said. "Kids his age would normally have problems adding fractions, and he is helping out some of his fellow students."
If Jake stays on track, Ross could see him working someday at a government lab or an observatory. Maybe he'll be a professor or a highly respected researcher.
"He can do anything he wants."
A normal boy, except for the numbers
Despite this new experience, his parents insist that Jake remain close with his friends in Westfield. Social activity is important, they know.
For Jake, life is not all centered on math and astrophysics.
He also likes playing video games. ("Guitar Hero" and "Halo: Reach" are his current favorites.) He plays basketball with friends, has a girlfriend and recently attended his first dance.
He likes music -- classical, which he plays by memory on a piano, but he also plays some contemporary songs he hears on the radio. He loves sci-fi movies and the Disney Channel. He watches documentaries on the History Channel.
A normal kid.
But then, late at night, when the TV is off, the homework is done and everyone in the house is sleeping, the numbers start to percolate again.
They percolate so much that he has trouble sleeping. His parents got so worried a few years ago that they took him for medical tests, but no malady was diagnosed. He just can't fall asleep easily.
"A lot keeps me awake," Jake said. "I scare people."
The numbers that keep him from snoozing are the same that led him to develop his own theory of physics -- an original work that proposed a "new expanded theory of relativity" and takes what Einstein developed even further.
His mom, still not sure whether her son was truly a genius at work or a kid at play, decided to send a video of Jake explaining his theory to the prestigious Institute for Advanced Study near Princeton University, one of the world's leading centers for theoretical research and intellectual inquiry.
That's where astrophysics Professor Scott Tremaine does his work. Tremaine is one of the world's leading scientists and is an expert in the evolution of planetary systems, comets, black holes, galaxies -- all the stuff Jake really likes.
In a letter to the Barnetts, Tremaine confirmed the brilliance.
"I'm impressed by his interest in physics and the amount that he has learned so far," Tremaine wrote in an email, provided by the family. "The theory that he's working on involves several of the toughest problems in astrophysics and theoretical physics.
He then encouraged Jake to spend as much time as possible to learn more and to further develop his theory.
Contacted by The Indianapolis Star, Tremaine confirmed the exchange of notes.
"I have seen a YouTube video in which Jake describes his theory, and I have spoken with his mother and corresponded with both her and Jake by email," Tremaine said. "I hope that Jake continues his interest in physics and mathematics."
Thinking big is what he does
Meanwhile, Jake is moving on to his next challenge: proving that the big-bang theory, the event some think led to the formation of the universe, is, well, wrong.
He explains.
"There are two different types of when stars end. When the little stars die, it's just like a small poof. They just turn into a planetary nebula. But the big ones, above 1.4 solar masses, blow up in one giant explosion, a supernova," Jake said. "What it does, is, in larger stars there is a larger mass, and it can fuse higher elements because it's more dense."
OK . . . trying to follow you.
"So you get all the elements, all the different materials, from those bigger stars. The little stars, they just make hydrogen and helium, and when they blow up, all the carbon that remains in them is just in the white dwarf; it never really comes off.
"So, um, in the big-bang theory, what they do is, there is this big explosion and there is all this temperature going off and the temperature decreases really rapidly because it's really big. The other day I calculated, they have this period where they suppose the hydrogen and helium were created, and, um, I don't care about the hydrogen and helium, but I thought, wouldn't there have to be some sort of carbon?"
He could go on and on.
And he did.
"Otherwise, the carbon would have to be coming out of the stars and hence the Earth, made mostly of carbon, we wouldn't be here. So I calculated, the time it would take to create 2 percent of the carbon in the universe, it would actually have to be several micro-seconds. Or a couple of nano-seconds, or something like that. An extremely small period of time. Like faster than a snap. That isn't gonna happen."
"Because of that," he continued, "that means that the world would have never been created because none of the carbon would have been given 7 billion years to fuse together. We'd have to be 21 billion years old . . . and that would just screw everything up."
So, we had to ask.
If not the big bang, then how did the universe come about?
"I'm still working on that," he said. "I have an idea, but . . . I'm still working out the details."
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c16720e0dd17ad73 | The Official String Theory Web Site:--> Basics -->Particles and relativity (basic / advanced)
The story so far... particles and relativity
The sense of achievement and closure for theoretical physics that came with the brilliant success of the classical field theory of electromagnetism was short lived. The new technology invented out of the mathematical unification of electricity with magnetism produced copious data about the nature of matter and light that snapped all of the mathematical threads that physicists had just succeeded in tying down.
And after this new data was unraveled and understood and explained using mathematics, the unified worldview of classical theoretical physics became split into two very different views of the universe -- the particle view and the geometric view.
Particles and waves
The first sign of trouble was when J.J. Thomson discovered the electron in 1897. Experimentalists began to see data that suggested a model of the atom with negatively charged particles orbiting around a positively charged core. But according to Maxwell's equations, such a system should be physically unstable. Classical field theory was unable to explain or describe the emerging data on atomic structure.
Another big mystery that came out of Maxwell's equations was the thermal behavior of light. Hot objects, like a hot coal, glow by emitting light and that light is observed to consist of a distribution of waves of different frequencies. But physicists who tried to explain the observed distribution of frequencies using light waves as described by Maxwell's equations met with continued failure.
Then as the new 20th century was beginning, a young German physicist, in an "act of despair" over the gaps in the understanding of thermal radiation, made a guess called the Quantum Hypothesis, which explained the observed thermal spectrum of light as coming from a collection of identical discrete quanta of energy. His formula worked, but he didn't know why.
This was the beginning of the idea known as particle-wave duality, and the field of quantum mechanics.
Einstein used Planck's idea to explain the newly-observed photoelectric effect. Einstein proposed that light was emitted or absorbed by an excited electron in discrete quanta called photons whose energy was proportional to the frequency of the light according to the relation
Planck formula,
where h is a number called Planck's constant, determined by measurement to be 6.6 x 10-34 joule seconds.
If a light wave could behave like a particle, then could a particle behave like a wave of some kind? In 1923, French aristocrat Louis de Broglie put forward the idea that an electron traveling with some momentum p could act like a continuous wave with wavelength l according to the relation
deBroglie wavelength
When the dust was settled, the new quantum theory described a given physical system not in terms of the path of a particle or the strength of a field, but as the probability amplitude for a given system to be in a given quantum state. This probability amplitude is the square of a function called the wave function Y(x,t), which is a solution to the Schrodinger equation
Schrodinger equation
Solutions to Schödinger equation for more then one identical particle have an interesting symmetry. For example, let's consider a two particle system and exchange the two particles. The wave function will obey the relation
Exchange symmetry
In the plus case, the two particles are what we call bosons. Two bosons can occupy the same quantum state at the same time.
In the minus case, the two particles are what we call fermions. Two fermions cannot occupy the same quantum state at the same time. This effect is called Pauli repulsion, and Pauli repulsion explains the structure of the periodic table of elements and the stability of atoms, and hence of all matter.
Relativity and geometry
The radical new idea of the quantum physics of atoms and light marked one direction of departure from the comforting sureness of 19th century classical field theory. The other big surprise of the 20th century came with the astounding observation in an experiment by Michelson and Morley that the speed of light was independent of the motion of the observer.
Now normally one would think that is a person were capable of throwing a javelin at 5 miles per hour while standing still, that same person, when running across the ground at 10 miles per hour, would be capable of making the javelin travel across the ground at a speed of 15 miles per hour.
But according to the data from the Michelson-Morley experiment, if one uses a laser instead of a javelin, then whether the person is sanding still or running 60 miles per hour or in a rocket traveling near the speed of light -- the light from the laser still travels the same speed!
This was an astounding result! How could it be explained using physics? Einstein came up with a powerful, simple theory, called the Special Theory of Relativity. Einstein used the geometric notion of a metric. The most familiar metric is just the Pythagorean Rule, which in three space dimensions in differential form looks like
Pythagorean rule
This formula has the special property that it is invariant under rotations. In other words, the length of a straight line does not change when you rotate the line in space. In the Special Theory of Relativity the idea of a metric is extended to include time, with a very crucial minus sign:
Minkowski metric
Like the space metric, the spacetime is invariant under rotations in space. But now there is a new twist -- the spacetime metric is also invariant under a kind of rotation of space and time called a Lorentz transformation, and this transformation tells us how different observers who are moving with some constant velocity relative to one another see the world.
And under a Lorentz transformation, the speed of light always stays the same, which is consistent with the shocking Michelson-Morley experiment.
Einstein's next target of revision was Newton's Universal Law of Gravitation. In Newton's formula the gravitational force F12 between two planets of masses m1 and m2 as depending on the inverse square of the distance r12 between the planets
Newton's law of gravity
GN is called Newton's constant and is measured to be 6.7x10-8 cm3 /(gm sec2).
Newton's Law was extremely successful at explaining the observed motions of the planets around the Sun, and of the moon around the Earth, and easily extendible through the techniques of classical field theory to continuous systems.
However, there was no hint in Newton's theory as to how a gravitational field would change in time, especially not in a manner that was consistent with the new understanding in Special Relativity that nothing can travel faster than the speed of light.
Einstein took a very bold step, and reached out to some radical new mathematics called non-Euclidean geometry, where the Pythagorean rule is generalized to include metrics with coefficients that depend on the spacetime coordinates in the form
Metric tensor
where repeated indices imply a sum over all space and time directions in the chosen coordinate system. Einstein extended the idea of Lorentz invariance to general coordinate invariance, proposing that the values of physical observables should be independent of a choice of coordinate system used to chart points in spacetime. He called this new theory the General Theory of Relativity.
In Einstein's new theory, spacetime can have curvature, like the surface of a beach ball has curvature, compared to the flat top of a table, which doesn't. The curvature is a function of the metric gab and its first and second derivatives. In the Einstein equation
Einstein equation
the spacetime curvature (represented by Rmn and R) is determined by the total energy and momentum Tmn of the "stuff" in the spacetime like the planets, stars, radiation, interstellar dust and gas, black holes, etc.
The Einstein equation is not strictly a departure from classical field theory, and the Einstein equation can be derived as the solution to Euler-Lagrange equations that represent the stationary point, or extremum, of the action
Einstein action
Two views of the world
Using quantum mechanics, the typical questions that can be answered concern the types of quantum states and allowed transitions in a system that features one or more particles that has some type of potential energy represented by the potential V(x). A typical method of working is to take some given V(x) and use the Schrödinger equation find the wave function, the energies of the quantum states of the system, and the allowed transitions between those states.
In general relativity, things are very different. One performs calculations that compute the evolution and structure of an entire universe at a time. A typical way of working is to propose some particular collection of energy and matter in the universe,to provide the Tmn. Given a particular Tmn, the Einstein equation turns into a system of second order nonlinear differential equations whose solutions give us the metric of spacetime, gmn, which holds all the information about the structure and evolution of a universe with that given Tmn.
Given the difference in the fundamental questions and methodologies used in quantum mechanics and in general relativity, it seems hardy surprising that uniting quantum physics with gravity, for a theory of quantum gravity, would prove to be a very tough challenge.
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electron-positron annihilation and creation
The particle view of nature is a description that works exceedingly well to describe three of the four observed forces of nature
Black Holes
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The geometric view of nature works very well for describing gravity at astronomical distance scales
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This article is about the general notion of determinism in philosophy. For other uses, see Determinism (disambiguation).
Determinism is the philosophical position that for every event, including human action, there exist conditions that could cause no other event. "There are many determinisms, depending upon what pre-conditions are considered to be determinative of an event."[1] Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. Some forms of determinism can be empirically tested with ideas from physics and the philosophy of physics. The opposite of determinism is some kind of indeterminism (otherwise called nondeterminism). Determinism is often contrasted with free will.[2]
Determinism often is taken to mean causal determinism, which in physics is known as cause-and-effect. It is the concept that events within a given paradigm are bound by causality in such a way that any state (of an object or event) is completely determined by prior states. This meaning can be distinguished from other varieties of determinism mentioned below.
Other debates often concern the scope of determined systems, with some maintaining that the entire universe is a single determinate system and others identifying other more limited determinate systems (or multiverse). Numerous historical debates involve many philosophical positions and varieties of determinism. They include debates concerning determinism and free will, technically denoted as compatibilistic (allowing the two to coexist) and incompatibilistic (denying their coexistence is a possibility).
Determinism should not be confused with self-determination of human actions by reasons, motives, and desires. Determinism rarely requires that perfect prediction be practically possible.
Below appear some of the more common viewpoints meant by, or confused with "determinism".
Many philosophical theories of determinism frame themselves with the idea that reality follows a sort of predetermined path
• Causal determinism is "the idea that every event is necessitated by antecedent events and conditions together with the laws of nature".[3] However, causal determinism is a broad enough term to consider that "one's deliberations, choices, and actions will often be necessary links in the causal chain that brings something about. In other words, even though our deliberations, choices, and actions are themselves determined like everything else, it is still the case, according to causal determinism, that the occurrence or existence of yet other things depends upon our deliberating, choosing and acting in a certain way".[4] Causal determinism proposes that there is an unbroken chain of prior occurrences stretching back to the origin of the universe. The relation between events may not be specified, nor the origin of that universe. Causal determinists believe that there is nothing uncaused or self-caused. Historical determinism (a sort of path dependence) can also be synonymous with causal determinism. - Causal determinism has also been considered more generally as the idea that everything that happens or exists is caused by antecedent conditions.[5] In the case of nomological determinism, these conditions are considered events also, implying that the future is determined completely by preceding events—a combination of prior states of the universe and the laws of nature.[3] Yet they can also be considered metaphysical of origin (such as in the case of theological determinism).[4]
• Nomological determinism is the most common form of causal determinism. It is the notion that the past and the present dictate the future entirely and necessarily by rigid natural laws, that every occurrence results inevitably from prior events. Quantum mechanics and various interpretations thereof pose a serious challenge to this view. Nomological determinism is sometimes illustrated by the thought experiment of Laplace's demon.[6] Nomological determinism is sometimes called 'scientific' determinism, although that is a misnomer. Physical determinism is generally used synonymously with nomological determinism (its opposite being physical indeterminism).
• Necessitarianism is very related to the causal determinism described above. It is a metaphysical principle that denies all mere possibility; there is exactly one way for the world to be. Leucippus claimed there were no uncaused events, and that everything occurs for a reason and by necessity.[7]
• Predeterminism is the idea that all events are determined in advance.[8][9] The concept of predeterminism is often argued by invoking causal determinism, implying that there is an unbroken chain of prior occurrences stretching back to the origin of the universe. In the case of predeterminism, this chain of events has been pre-established, and human actions cannot interfere with the outcomes of this pre-established chain. Predeterminism can be used to mean such pre-established causal determinism, in which case it is categorised as a specific type of determinism.[8][10] It can also be used interchangeably with causal determinism—in the context of its capacity to determine future events.[8][11] Despite this, predeterminism is often considered as independent of causal determinism.[12][13] The term predeterminism is also frequently used in the context of biology and hereditary, in which case it represents a form of biological determinism.[14]
• Fatalism is normally distinguished from "determinism".[15] Fatalism is the idea that everything is fated to happen, so that humans have no control over their future. Fate has arbitrary power, and need not follow any causal or otherwise deterministic laws.[5] Types of Fatalism include hard theological determinism and the idea of predestination, where there is a God who determines all that humans will do. This may be accomplished either by knowing their actions in advance, via some form of omniscience[16] or by decreeing their actions in advance.[17]
• Theological determinism is a form of determinism which states that all events that happen are pre-ordained, or predestined to happen, by a monotheistic deity, or that they are destined to occur given its omniscience. Two forms of theological determinism exist, here referenced as strong and weak theological determinism.[18] The first one, strong theological determinism, is based on the concept of a creator deity dictating all events in history: "everything that happens has been predestined to happen by an omniscient, omnipotent divinity".[19] The second form, weak theological determinism, is based on the concept of divine foreknowledge—"because God's omniscience is perfect, what God knows about the future will inevitably happen, which means, consequently, that the future is already fixed".[20] There exist slight variations on the above categorisation. Some claim that theological determinism requires predestination of all events and outcomes by the divinity (i.e. they do not classify the weaker version as 'theological determinism' unless libertarian free will is assumed to be denied as a consequence), or that the weaker version does not constitute 'theological determinism' at all.[21] With respect to free will, "theological determinism is the thesis that God exists and has infallible knowledge of all true propositions including propositions about our future actions", more minimal criteria designed to encapsulate all forms of theological determinism.[22] Theological determinism can also be seen as a form of causal determinism, in which the antecedent conditions are the nature and will of God.[4]
• Logical determinism or Determinateness is the notion that all propositions, whether about the past, present, or future, are either true or false. Note that one can support Causal Determinism without necessarily supporting Logical Determinism and vice versa (depending on one's views on the nature of time, but also randomness). The problem of free will is especially salient now with Logical Determinism: how can choices be free, given that propositions about the future already have a truth value in the present (i.e. it is already determined as either true or false)? This is referred to as the problem of future contingents.
Adequate determinism focuses on the fact that, even without a full understanding of microscopic physics, we can predict the distribution of 1000 coin tosses
• Often synonymous with Logical Determinism are the ideas behind Spatio-temporal Determinism or Eternalism: the view of special relativity. J. J. C. Smart, a proponent of this view, uses the term "tenselessness" to describe the simultaneous existence of past, present, and future. In physics, the "block universe" of Hermann Minkowski and Albert Einstein assumes that time is a fourth dimension (like the three spatial dimensions). In other words, all the other parts of time are real, like the city blocks up and down a street, although the order in which they appear depends on the driver (see Rietdijk–Putnam argument).
• Adequate determinism is the idea that quantum indeterminacy can be ignored for most macroscopic events. This is because of quantum decoherence. Random quantum events "average out" in the limit of large numbers of particles (where the laws of quantum mechanics asymptotically approach the laws of classical mechanics).[23] Stephen Hawking explains a similar idea: he says that the microscopic world of quantum mechanics is one of determined probabilities. That is, quantum effects rarely alter the predictions of classical mechanics, which are quite accurate (albeit still not perfectly certain) at larger scales.[24] Something as large as an animal cell, then, would be "adequately determined" (even in light of quantum indeterminacy).
Philosophical connections[edit]
With nature/nurture controversy[edit]
Nature and nurture interact in humans. A scientist looking at a sculpture after some time does not ask whether we are seeing the effects of the starting materials or of environmental influences.
Although some of the above forms of determinism concern human behaviors and cognition, others frame themselves as an answer to the debate on nature and nurture. They will suggest that one factor will entirely determine behavior. As scientific understanding has grown, however, the strongest versions of these theories have been widely rejected as a single-cause fallacy.[25]
In other words, the modern deterministic theories attempt to explain how the interaction of both nature and nurture is entirely predictable. The concept of heritability has been helpful in making this distinction.
Biological determinism, sometimes called genetic determinism, is the idea that each of human behaviors, beliefs, and desires are fixed by human genetic nature.
Behaviorism involves the idea that all behavior can be traced to specific causes—either environmental or reflexive. John B. Watson and B. F. Skinner developed this nurture-focused determinism.
Cultural determinism or social determinism is the nurture-focused theory that the culture in which we are raised determines who we are.
Environmental determinism, also known as climatic or geographical determinism, proposes that the physical environment, rather than social conditions, determines culture. Supporters of environmental determinism often[quantify] also support Behavioral determinism. Key proponents of this notion have included Ellen Churchill Semple, Ellsworth Huntington, Thomas Griffith Taylor and possibly Jared Diamond, although his status as an environmental determinist is debated.[26]
With particular factors[edit]
A technological determinist might suggest that technology like the mobile phone is the greatest factor shaping human civilization.
Other 'deterministic' theories actually seek only to highlight the importance of a particular factor in predicting the future. These theories often use the factor as a sort of guide or constraint on the future. They need not suppose that complete knowledge of that one factor would allow us to make perfect predictions.
Psychological determinism can mean that humans must act according to reason, but it can also be synonymous with some sort of Psychological egoism. The latter is the view that humans will always act according to their perceived best interest.
Linguistic determinism claims that our language determines (at least limits) the things we can think and say and thus know. The Sapir–Whorf hypothesis argues that individuals experience the world based on the grammatical structures they habitually use.
Economic determinism is the theory which attributes primacy to the economic structure over politics in the development of human history. It is associated with the dialectical materialism of Karl Marx.
Technological determinism is a reductionist theory that presumes that a society's technology drives the development of its social structure and cultural values.
With free will[edit]
Main article: Free will
A table showing the different positions related to free will and determinism
Philosophers have debated both the truth of determinism, and the truth of free will. This creates the four possible positions in the figure. Compatibilism refers to the view that free will is, in some sense, compatible with determinism. The three incompatibilist positions, on the other hand, deny this possibility. The hard incompatibilists hold that both determinism and free will do not exist, the libertarianists that determinism does not hold, and free will might exist, and the hard determinists that determinism does hold and free will does not exist.
The standard argument against free will, according to philosopher J. J. C. Smart focuses on the implications of determinism for 'free will'.[27] However, he suggests free will is denied whether determinism is true or not. On one hand, if determinism is true, all our actions are predicted and we are assumed not to be free; on the other hand, if determinism is false, our actions are presumed to be random and as such we do not seem free because we had no part in controlling what happened.
In his book, The Moral Landscape, author and neuroscientist Sam Harris also argues against incompatibilist free will. He offers one thought experiment where a mad scientist represents determinism. In Harris' example, the mad scientist uses a machine to control all the desires, and thus all the behavior, of a particular human. Harris believes that it is no longer as tempting, in this case, to say the victim has "free will". Harris says nothing changes if the machine controls desires at random - the victim still seems to lack free will. Harris then argues that we are also the victims of such unpredictable desires (but due to the unconscious machinations of our brain, rather than those of a mad scientist). Based on this introspection, he writes "This discloses the real mystery of free will: if our experience is compatible with its utter absence, how can we say that we see any evidence for it in the first place?"[28] adding that "Whether they are predictable or not, we do not cause our causes."[29] That is, he believes there is compelling evidence of absence of free will. Harris' viewpoint implicitly assumes a philosophy of materialism, that is, that mental events are reducible to neurological occurrences.
Some research (founded by the John Templeton Foundation) suggested that reducing a person's belief in free will is dangerous, making them less helpful and more aggressive.[30] This could occur because the individual's sense of self-efficacy suffers.
With the "soul"[edit]
A number of positions can be delineated:
1. Immaterial souls are all that exist (Idealism).
2. Immaterial souls exist and exert a non-deterministic causal influence on bodies. (Traditional free-will, interactionist dualism).[31][32]
3. Immaterial souls exist, but are part of deterministic framework.
4. Immaterial souls exist, but exert no causal influence, free or determined (epiphenomenalism, occasionalism)
5. Immaterial souls do not exist — there is no mind-body dichotomy, and there is a Materialistic explanation for intuitions to the contrary.
With ethics and morality[edit]
Another topic of debate is the implication that Determinism has on morality. Hard determinism (a belief in determinism, and not free will) is particularly criticized for seeming to make traditional moral judgments impossible. Some philosophers, however, find this an acceptable conclusion.
Philosopher and incompatibilist Peter van Inwagen introduces this thesis as such:
Argument that Free Will is Required for Moral Judgments
1. The moral judgment that you shouldn't have done X implies that you should have done something else instead
2. That you should have done something else instead implies that there was something else for you to do
3. That there was something else for you to do implies that you could have done something else
4. That you could have done something else implies that you have free will
5. If you don't have free will to have done other than X we cannot make the moral judgment that you shouldn't have done X.[33]
However, a compatibilist might have an issue with Inwagen's process because one can not change the past like his arguments center around. A compatibilist who centers around plans for the future might posit:
1. The moral judgment that you shouldn't have done X implies that you can do something else instead
2. That you can do something else instead implies that there is something else for you to do
3. That there is something else for you to do implies that you can do something else
4. That you can do something else implies that you have free will for planning future recourse
5. If you have free will to do other than X we can make the moral judgment that you should do other than X, and punishing you as a responsible party for having done X that you know you should not have done can help you remember to not do X in the future.
Some of the main philosophers who have dealt with this issue are Marcus Aurelius, Omar Khayyám, Thomas Hobbes, Baruch Spinoza, Gottfried Leibniz, David Hume, Baron d'Holbach (Paul Heinrich Dietrich), Pierre-Simon Laplace, Arthur Schopenhauer, William James, Friedrich Nietzsche, Albert Einstein, Niels Bohr, Ralph Waldo Emerson and, more recently, John Searle, Ted Honderich, and Daniel Dennett.
Mecca Chiesa notes that the probabilistic or selectionistic determinism of B.F. Skinner comprised a wholly separate conception of determinism that was not mechanistic at all. Mechanistic determinism assumes that every event has an unbroken chain of prior occurrences, but a selectionistic or probabilistic model does not.[34][35]
Eastern tradition[edit]
The idea that the entire universe is a deterministic system has been articulated in both Eastern and non-Eastern religion, philosophy, and literature.
In I Ching and Philosophical Taoism, the ebb and flow of favorable and unfavorable conditions suggests the path of least resistance is effortless (see wu wei).
In the philosophical schools of India, the concept of precise and continual effect of laws of Karma on the existence of all sentient beings is analogous to western deterministic concept. Karma is the concept of "action" or "deed" in Indian religions. It is understood as that which causes the entire cycle of cause and effect (i.e., the cycle called saṃsāra) originating in ancient India and treated in Hindu, Jain, Sikh and Buddhist philosophies. Karma is considered predetermined and deterministic in the universe, and in combination with the decisions (free will) of living beings, accumulates to determine futuristic situations that the living being encounters. See Karma in Hinduism.[citation needed]
Western tradition[edit]
In the West, some elements of determinism seem to have been expressed by the Presocratics Heraclitus[36] and Leucippus.[37] The first full-fledged notion of determinism appears to originate with the Stoics, as part of their theory of universal causal determinism.[38] The resulting philosophical debates, which involved the confluence of elements of Aristotelian Ethics with Stoic psychology, led in the 1st-3rd centuries CE in the works of Alexander of Aphrodisias to the first recorded Western debate over determinism and freedom,[39] an issue that is known in theology as the paradox of free will. The writings of Epictetus as well as Middle Platonist and early Christian thought were instrumental in this development.[40] The Jewish philosopher Moses Maimonides said of the deterministic implications of an omniscient god:[41] "Does God know or does He not know that a certain individual will be good or bad? If thou sayest 'He knows', then it necessarily follows that [that] man is compelled to act as God knew beforehand he would act, otherwise God's knowledge would be imperfect.…"[42]
Determinism in the West is often associated with Newtonian physics, which depicts the physical matter of the universe as operating according to a set of fixed, knowable laws. The "billiard ball" hypothesis, a product of Newtonian physics, argues that once the initial conditions of the universe have been established, the rest of the history of the universe follows inevitably. If it were actually possible to have complete knowledge of physical matter and all of the laws governing that matter at any one time, then it would be theoretically possible to compute the time and place of every event that will ever occur (Laplace's demon). In this sense, the basic particles of the universe operate in the same fashion as the rolling balls on a billiard table, moving and striking each other in predictable ways to produce predictable results.
Whether or not it is all-encompassing in so doing, Newtonian mechanics deals only with caused events, e.g.: If an object begins in a known position and is hit dead on by an object with some known velocity, then it will be pushed straight toward another predictable point. If it goes somewhere else, the Newtonians argue, one must question one's measurements of the original position of the object, the exact direction of the striking object, gravitational or other fields that were inadvertently ignored, etc. Then, they maintain, repeated experiments and improvements in accuracy will always bring one's observations closer to the theoretically predicted results. When dealing with situations on an ordinary human scale, Newtonian physics has been so enormously successful that it has no competition. But it fails spectacularly as velocities become some substantial fraction of the speed of light and when interactions at the atomic scale are studied. Before the discovery of quantum effects and other challenges to Newtonian physics, "uncertainty" was always a term that applied to the accuracy of human knowledge about causes and effects, and not to the causes and effects themselves.
Newtonian mechanics as well as any following physical theories are results of observations and experiments, and so they describe "how it all works" within a tolerance. However, old western scientists believed if there are any logical connections found between an observed cause and effect, there must be also some absolute natural laws behind. Belief in perfect natural laws driving everything, instead of just describing what we should expect, led to searching for a set of universal simple laws that rule the world. This movement significantly encouraged deterministic views in western philosophy,[43] as well as the related theological views of Classical Pantheism.
Modern scientific perspective[edit]
Generative processes[edit]
Main article: Emergence
Although it was once thought by scientists that any indeterminism in quantum mechanics occurred at too small a scale to influence biological or neurological systems, there is indication that nervous systems are influenced by quantum indeterminism due to chaos theory. It is unclear what implications this has for free will given various possible reactions to the standard problem in the first place.[44] Not all biologists grant determinism: Christof Koch argues against it, and in favour of libertarian free will, by making arguments based on generative processes (emergence).[45] Other proponents of emergentist or generative philosophy, cognitive sciences and evolutionary psychology, argue that determinism is true.[46][47][48][49] They suggest instead that an illusion of free will is experienced due to the generation of infinite behaviour from the interaction of finite-deterministic set of rules and parameters. Thus the unpredictability of the emerging behaviour from deterministic processes leads to a perception of free will, even though free will as an ontological entity does not exist.[46][47][48][49] Certain experiments looking at the neuroscience of free will can be said to support this possibility.[citation needed]
In Conway's Game of Life, the interaction of just four simple rules creates patterns that seem somehow "alive".
As an illustration, the strategy board-games chess and Go have rigorous rules in which no information (such as cards' face-values) is hidden from either player and no random events (such as dice-rolling) happen within the game. Yet, chess and especially Go with its extremely simple deterministic rules, can still have an extremely large number of unpredictable moves. When chess is simplified to 7 or fewer pieces, however, there are endgame tables available which dictate which moves to play to achieve a perfect game. The implication of this is that given a less complex environment (with the original 32 pieces reduced to 7 or fewer pieces), a perfectly predictable game of chess is possible to achieve. In this scenario, the winning player would be able to announce a checkmate happening in at most a given number of moves assuming a perfect defense by the losing player, or less moves if the defending player chooses sub-optimal moves as the game progresses into its inevitable, predicted conclusion. By this analogy, it is suggested, the experience of free will emerges from the interaction of finite rules and deterministic parameters that generate nearly infinite and practically unpredictable behaviourial responses. In theory, if all these events could be accounted for, and there were a known way to evaluate these events, the seemingly unpredictable behaviour would become predictable.[46][47][48][49] Another hands-on example of generative processes is John Horton Conway's playable Game of Life.[50] Nassim Taleb is wary of such models, and coined the term "ludic fallacy".
Mathematical models[edit]
Many mathematical models of physical systems are deterministic. This is true of most models involving differential equations (notably, those measuring rate of change over time). Mathematical models that are not deterministic because they involve randomness are called stochastic. Because of sensitive dependence on initial conditions, some deterministic models may appear to behave non-deterministically; in such cases, a deterministic interpretation of the model may not be useful due to numerical instability and a finite amount of precision in measurement. Such considerations can motivate the consideration of a stochastic model even though the underlying system is governed by deterministic equations.[51][52][53]
Quantum mechanics and classical physics[edit]
Day-to-day physics[edit]
Further information: Macroscopic quantum phenomena
Since the beginning of the 20th century, quantum mechanics—the physics of the extremely small—has revealed previously concealed aspects of events. Before that, Newtonian physics—the physics of everyday life—dominated. Taken in isolation (rather than as an approximation to quantum mechanics), Newtonian physics depicts a universe in which objects move in perfectly determined ways. At the scale where humans exist and interact with the universe, Newtonian mechanics remain useful, and make relatively accurate predictions (e.g. calculating the trajectory of a bullet). But whereas in theory, absolute knowledge of the forces accelerating a bullet would produce an absolutely accurate prediction of its path, modern quantum mechanics casts reasonable doubt on this main thesis of determinism.
Relevant is the fact that certainty is never absolute in practice (and not just because of David Hume's problem of induction). The equations of Newtonian mechanics can exhibit sensitive dependence on initial conditions. This is an example of the butterfly effect, which is one of the subjects of chaos theory. The idea is that something even as small as a butterfly could cause a chain reaction leading to a hurricane years later. Consequently, even a very small error in knowledge of initial conditions can result in arbitrarily large deviations from predicted behavior. Chaos theory thus explains why it may be practically impossible to predict real life, whether determinism is true or false. On the other hand, the issue may not be so much about human abilities to predict or attain certainty as much as it is the nature of reality itself. For that, a closer, scientific look at nature is necessary.
Quantum realm[edit]
Quantum physics works differently in many ways from Newtonian physics. Physicist Aaron D. O'Connell explains that understanding our universe, at such small scales as atoms, requires a different logic than day-to-day life does. O'Connell does not deny that it is all interconnected: the scale of human existence ultimately does emerge from the quantum scale. O'Connell argues that we must simply use different models and constructs when dealing with the quantum world.[54] Quantum mechanics is the product of a careful application of the scientific method, logic and empiricism. The Heisenberg uncertainty principle is frequently confused with the observer effect. The uncertainty principle actually describes how precisely we may measure the position and momentum of a particle at the same time — if we increase the accuracy in measuring one quantity, we are forced to lose accuracy in measuring the other. "These uncertainty relations give us that measure of freedom from the limitations of classical concepts which is necessary for a consistent description of atomic processes."[55]
Although it is not possible to predict the trajectory of any one particle, they all obey determined probabilities which do permit some prediction.
This is where statistical mechanics come into play, and where physicists begin to require rather unintuitive mental models: A particle's path simply cannot be exactly specified in its full quantum description. "Path" is a classical, practical attribute in our every day life, but one which quantum particles do not meaningfully possess. The probabilities discovered in quantum mechanics do nevertheless arise from measurement (of the perceived path of the particle). As Stephen Hawking explains, the result is not traditional determinism, but rather determined probabilities.[56] In some cases, a quantum particle may indeed trace an exact path, and the probability of finding the particles in that path is one (certain).[clarification needed] In fact, as far as prediction goes, the quantum development is at least as predictable as the classical motion, but the key is that it describes wave functions that cannot be easily expressed in ordinary language. As far as the thesis of determinism is concerned, these probabilities, at least, are quite determined. These findings from quantum mechanics have found many applications, and allow us to build transistors and lasers. Put another way: personal computers, Blu-ray players and the internet all work because humankind discovered the determined probabilities of the quantum world.[57] None of that should be taken to imply that other aspects of quantum mechanics are not still up for debate.
On the topic of predictable probabilities, the double-slit experiments are a popular example. Photons are fired one-by-one through a double-slit apparatus at a distant screen. Curiously, they do not arrive at any single point, nor even the two points lined up with the slits (the way you might expect of bullets fired by a fixed gun at a distant target). Instead, the light arrives in varying concentrations at widely separated points, and the distribution of its collisions with the target can be calculated reliably. In that sense the behavior of light in this apparatus is deterministic, but there is no way to predict where in the resulting interference pattern any individual photon will make its contribution (although, there may be ways to use weak measurement to acquire more information without violating the Uncertainty principle).
Some (including Albert Einstein) argue that our inability to predict any more than probabilities is simply due to ignorance.[58] The idea is that, beyond the conditions and laws we can observe or deduce, there are also hidden factors or "hidden variables" that determine absolutely in which order photons reach the detector screen. They argue that the course of the universe is absolutely determined, but that humans are screened from knowledge of the determinative factors. So, they say, it only appears that things proceed in a merely probabilistically determinative way. In actuality, they proceed in an absolutely deterministic way. These matters continue to be subject to some dispute. A critical finding was that quantum mechanics can make statistical predictions which would be violated if local hidden variables really existed. There have been a number of experiments to verify such predictions, and so far they do not appear to be violated. This would suggest there are no hidden variables, although many physicists believe better experiments are needed to conclusively settle the issue (see also Bell test experiments). Furthermore, it is possible to augment quantum mechanics with non-local hidden variables to achieve a deterministic theory that is in agreement with experiment. An example is the Bohm interpretation of quantum mechanics. This debate is relevant because it is easy to imagine specific situations in which the arrival of an electron at a screen at a certain point and time would trigger one event, whereas its arrival at another point would trigger an entirely different event (e.g. see Schrödinger's cat - a thought experiment used as part of a deeper debate).
Thus, quantum physics casts reasonable doubt on the traditional determinism of classical, Newtonian physics in so far as reality does not seem to be absolutely determined. This was the subject of the famous Bohr–Einstein debates between Einstein and Niels Bohr and there is still no consensus.[59][60]
Adequate determinism (see Varieties, above) is the reason that Stephen Hawking calls Libertarian free will "just an illusion".[56] Compatibilistic free will (which is deterministic) may be the only kind of "free will" that can exist. However, Daniel Dennett, in his book Elbow Room, says that this means we have the only kind of free will "worth wanting". For even more discussion, see Free will.
Other matters of quantum determinism[edit]
Chaotic radioactivity is the next explanatory challenge for physicists supporting determinism
All uranium found on earth is thought to have been synthesized during a supernova explosion that occurred roughly 5 billion years ago. Even before the laws of quantum mechanics were developed to their present level, the radioactivity of such elements has posed a challenge to determinism due to its unpredictability. One gram of uranium-238, a commonly occurring radioactive substance, contains some 2.5 x 1021 atoms. Each of these atoms are identical and indistinguishable according to all tests known to modern science. Yet about 12600 times a second, one of the atoms in that gram will decay, giving off an alpha particle. The challenge for determinism is to explain why and when decay occurs, since it does not seem to depend on external stimulus. Indeed, no extant theory of physics makes testable predictions of exactly when any given atom will decay. At best scientists can discover determined probabilities in the form of the element's half life.
The time dependent Schrödinger equation gives the first time derivative of the quantum state. That is, it explicitly and uniquely predicts the development of the wave function with time.
i\hbar\frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2}+V(x)\psi
So if the wave function itself is reality (rather than probability of classical coordinates), quantum mechanics can be said to be deterministic.
According to some,[citation needed][who?] quantum mechanics is more strongly ordered than Classical Mechanics, because while Classical Mechanics is chaotic, quantum mechanics is not. For example, the classical problem of three bodies under a force such as gravity is not integrable, while the quantum mechanical three body problem is tractable and integrable, using the Faddeev Equations.[clarification needed] This does not mean that quantum mechanics describes the world as more deterministic, unless one already considers the wave function to be the true reality. Even so, this does not get rid of the probabilities, because we can't do anything without using classical descriptions, but it assigns the probabilities to the classical approximation, rather than to the quantum reality.
Asserting that quantum mechanics is deterministic by treating the wave function itself as reality implies a single wave function for the entire universe, starting at the origin of the universe. Such a "wave function of everything" would carry the probabilities of not just the world we know, but every other possible world that could have evolved. For example, large voids in the distributions of galaxies are believed by many cosmologists to have originated in quantum fluctuations during the big bang. (See cosmic inflation, primordial fluctuations and large-scale structure of the cosmos.)
See also[edit]
2. ^ For example, see Richard Langdon Franklin (1968). Freewill and determinism: a study of rival conceptions of man. Routledge & K. Paul.
3. ^ a b Hoefer, Carl (Apr 1, 2008). "Causal Determinism". In Edward N. Zalta, ed. The Stanford Encyclopedia of Philosophy (Winter 2009 edition).
4. ^ a b c Eshleman, Andrew (Nov 18, 2009). "Moral Responsibility". In Edward N. Zalta, ed. The Stanford Encyclopedia of Philosophy (Winter 2009 ed.).
5. ^ a b Arguments for Incompatibilism (Stanford Encyclopedia of Philosophy)
6. ^ Laplace posited that an omniscient observer knowing with infinite precision all the positions and velocities of every particle in the universe could predict the future entirely. For a discussion, see Robert C. Solomon, Kathleen M. Higgins (2009). "Free will and determinism". The Big Questions: A Short Introduction to Philosophy (8th ed.). Cengage Learning. p. 232. ISBN 0495595152. Another view of determinism is discussed by Ernest Nagel (1999). "§V: Alternative descriptions of physical state". The Structure of Science: Problems in the Logic of Scientific Explanation (2nd ed.). Hackett. pp. 285–292. ISBN 0915144719. "a theory is deterministic if, and only if, given its state variables for some initial period, the theory logically determines a unique set of values for those variables for any other period."
7. ^ Leucippus, Fragment 569 - from Fr. 2 Actius I, 25, 4
8. ^ a b c McKewan, Jaclyn (2009). "Predeterminism". In H. James Birx". Encyclopedia of Time: Science, Philosophy, Theology, & Culture. SAGE Publications, Inc. pp. 1035–1036. doi:10.4135/9781412963961.n191. Retrieved 20 December 2012.
10. ^ "Some Varieties of Free Will and Determinism". Philosophy 302: Ethics. 09.10.09. Retrieved 19 December 2012. "Predeterminism: the philosophical and theological view that combines God with determinism. On this doctrine events throughout eternity have been foreordained by some supernatural power in a causal sequence."
11. ^ See for example Hooft, G. (2001). "How does god play dice? (Pre-)determinism at the Planck scale". arXiv preprint hep-th/0104219. "Predeterminism is here defined by the assumption that the experimenter's 'free will' in deciding what to measure (such as his choice to measure the x- or the y-component of an electron's spin), is in fact limited by deterministic laws, hence not free at all" , and Sukumar, CV (1996). "A new paradigm for science and architecture". City (Taylor & Francis) 1 (1-2): 181–183. doi:10.1080/13604819608900044. "Quantum Theory provided a beautiful description of the behaviour of isolated atoms and nuclei and small aggregates of elementary particles. Modern science recognized that predisposition rather than predeterminism is what is widely prevalent in nature."
12. ^ Borst, C. (1992). "Leibniz and the compatibilist account of free will". Studia leibnitiana (JSTOR): 49–58. "Leibniz presents a clear case of a philosopher who does not think that predeterminism requires universal causal determinism"
13. ^ Far Western Philosophy of Education Society (1971). Proceedings of the Annual Meeting of the Far Western Philosophy of Education Society. Far Western Philosophy of Education Society. p. 12. Retrieved 20 December 2012. ""Determinism" is, in essence, the position which holds that all behavior is caused by prior behavior. "Predeterminism" is the position which holds that all behavior is caused by conditions which predate behavior altogether (such impersonal boundaries as "the human conditions", instincts, the will of God, inherent knowledge, fate, and such)."
14. ^ "Predeterminism". Merriam-Webster Dictionary. Merriam-Webster, Incorporated. Retrieved 20 December 2012. See for example Ormond, A.T. (1894). "Freedom and psycho-genesis". Psychological Review (Macmillan & Company) 1 (3): 217. doi:10.1037/h0065249. "The problem of predeterminism is one that involves the factors of heredity and environment, and the point to be debated here is the relation of the present self that chooses to these predetermining agencies" , and Garris, M.D. and others (1992). "A Platform for Evolving Genetic Automata for Text Segmentation (GNATS)". Science of Artificial Neural Networks (Citeseer) 1710: 714–724. doi:10.1117/12.140132. "However, predeterminism is not completely avoided. If the codes within the genotype are not designed properly, then the organisms being evolved will be fundamentally handicapped."
15. ^ SEP, Causal Determinism
16. ^ Fischer, John Martin (1989) God, Foreknowledge and Freedom. Stanford, California: Stanford University Press. ISBN 1-55786-857-3
17. ^ Watt, Montgomery (1948) Free-Will and Predestination in Early Islam. London:Luzac & Co.
18. ^ Anne Lockyer Jordan; Anne Lockyer Jordan Neil Lockyer Edwin Tate; Neil Lockyer; Edwin Tate (25 June 2004). Philosophy of Religion for A Level OCR Edition. Nelson Thornes. p. 211. ISBN 978-0-7487-8078-5. Retrieved 22 December 2012.
19. ^ A. Pabl Iannone (2001). "determinism". Dictionary of World Philosophy. Taylor & Francis. p. 194. ISBN 978-0-415-17995-9. Retrieved 22 December 2012. "theological determinism, or the doctrine of predestination: the view that everything which happens has been predestined to happen by an omniscient, omnipotent divinity. A weaker version holds that, though not predestined to happen, everything that happens has been eternally known by virtue of the divine foreknowledge of an omniscient divinity. If this divinity is also omnipotent, as in the case of the Judeo-Christian religions, this weaker version is hard to distinguish from the previous one because, though able to prevent what happens and knowing that it is going to happen, God lets it happen. To this, advocates of free will reply that God permits it to happen in order to make room for the free will of humans."
20. ^ Wentzel Van Huyssteen (2003). "theological determinism". Encyclopedia of science and religion 1. Macmillan Reference. p. 217. ISBN 978-0-02-865705-9. Retrieved 22 December 2012. "Theological determinism constitutes a fifth kind of determinism. There are two types of theological determinism, both compatible with scientific and metaphysical determinism. In the first, God determines everything that happens, either in one all-determining single act at the initial creation of the universe or through continuous divine interactions with the world. Either way, the consequence is that everything that happens becomes God's action, and determinism is closely linked to divine action and God's omnipotence. According to the second type of theological determinism, God has perfect knowledge of everything in the universe because God is omniscient. And, as some say, because God is outside of time, God has the capacity of knowing past, present, and future in one instance. This means that God knows what will happen in the future. And because God's omniscience is perfect, what God knows about the future will inevitably happen, which means, consequently, that the future is already fixed."
21. ^ Raymond J. VanArragon (21 October 2010). Key Terms in Philosophy of Religion. Continuum International Publishing Group. p. 21. ISBN 978-1-4411-3867-5. Retrieved 22 December 2012. "Theological determinism, on the other hand, claims that all events are determined by God. On this view, God decree that everything will go thus-and-so and ensure that everything goes that way, so that ultimately God is the cause of everything that happens and everything that happens is part of God's plan. We might think of God here as the all-powerful movie director who writes script and causes everything to go accord with it. We should note, as an aside, that there is some debate over what would be sufficient for theological determinism to be true. Some people claim that God's merely knowing what will happen determines that it will, while others believe that God must not only know but must also cause those events to occur in order for their occurrence to be determined."
23. ^ The Information Philosopher website, "Adequate Determinism", from the site: "We are happy to agree with scientists and philosophers who feel that quantum effects are for the most part negligible in the macroscopic world. We particularly agree that they are negligible when considering the causally determined will and the causally determined actions set in motion by decisions of that will."
25. ^ de Melo-Martín I (2005). "Firing up the nature/nurture controversy: bioethics and genetic determinism". J Med Ethics 31 (9): 526–30. doi:10.1136/jme.2004.008417. PMC 1734214. PMID 16131554.
26. ^ Andrew, Sluyter. "Neo-Environmental Determinism, Intellectual Damage Control, and Nature/Society Science". Antipode 4 (35).
27. ^ J. J. C. Smart, "Free-Will, Praise and Blame,"Mind, July 1961, p.293-4.
28. ^ Sam Harris, The Moral Landscape (2010), pg.216, note102
29. ^ Sam Harris, The Moral Landscape (2010), pg.217, note109
30. ^ Baumeister RF, Masicampo EJ, Dewall CN. (2009). Prosocial benefits of feeling free: disbelief in free will increases aggression and reduces helpfulness. Pers Soc Psychol Bull. 35(2):260-8. PMID 19141628 doi:10.1177/0146167208327217
31. ^ By 'soul' in the context of (1) is meant an autonomous immaterial agent that has the power to control the body but not to be controlled by the body (this theory of determinism thus conceives of conscious agents in dualistic terms). Therefore the soul stands to the activities of the individual agent's body as does the creator of the universe to the universe. The creator of the universe put in motion a deterministic system of material entities that would, if left to themselves, carry out the chain of events determined by ordinary causation. But the creator also provided for souls that could exert a causal force analogous to the primordial causal force and alter outcomes in the physical universe via the acts of their bodies. Thus, it emerges that no events in the physical universe are uncaused. Some are caused entirely by the original creative act and the way it plays itself out through time, and some are caused by the acts of created souls. But those created souls were not created by means of physical processes involving ordinary causation. They are another order of being entirely, gifted with the power to modify the original creation. However, determinism is not necessarily limited to matter; it can encompass energy as well. The question of how these immaterial entities can act upon material entities is deeply involved in what is generally known as the mind-body problem. It is a significant problem which philosophers have not reached agreement about
32. ^ Free Will (Stanford Encyclopedia of Philosophy)
33. ^ van Inwagen, Peter (2009). The Powers of Rational Beings: Freedom of the Will. Oxford.
34. ^ Chiesa, Mecca (2004) Radical Behaviorism: The Philosophy & The Science.
35. ^ Ringen, J. D. (1993). Adaptation, teleology, and selection by consequences. Journal of Applied Behavior Analysis. 60,3–15. [1]
36. ^ Stobaeus Eclogae I 5 (Heraclitus)
37. ^ Stobaeus Eclogae I 4 (Leucippus)
38. ^ Susanne Bobzien Determinism and Freedom in Stoic Philosophy (Oxford 1998) chapter 1.
39. ^ Susanne Bobzien The Inadvertent Conception and Late Birth of the Free-Will Problem (Phronesis 43, 1998).
40. ^ Michael Frede A Free Will: Origins of the Notion in Ancient Thought (Berkeley 2011).
41. ^ Though Moses Maimonides was not arguing against the existence of God, but rather for the incompatibility between the full exercise by God of his omniscience and genuine human free will, his argument is considered by some as affected by Modal Fallacy. See, in particular, the article by Prof. Norman Swartz for Internet Encyclopedia of Philosophy, Foreknowledge and Free Will and specifically Section 6: The Modal Fallacy
42. ^ The Eight Chapters of Maimonides on Ethics (Semonah Perakhim), edited, annotated, and translated with an Introduction by Joseph I. Gorfinkle, pp. 99–100. (New York: AMS Press), 1966.
43. ^ Swartz, Norman (2003) The Concept of Physical Law / Chapter 10: Free Will and Determinism (
44. ^ Lewis, E.R.; MacGregor, R.J. (2006). "On Indeterminism, Chaos, and Small Number Particle Systems in the Brain". Journal of Integrative Neuroscience 5 (2): 223–247. doi:10.1142/S0219635206001112.
45. ^ Koch, Christof (September 2009). "Free Will, Physics, Biology and the Brain". In Murphy, Nancy; Ellis, George; O'Connor, Timothy. Downward Causation and the Neurobiology of Free Will. New York, USA: Springer. ISBN 978-3-642-03204-2.
46. ^ a b c Kenrick, D. T., Li, N. P., & Butner, J. (2003) "Dynamical evolutionary psychology: Individual decision rules and emergent social norms", Psychological Review 110: 3–28.
47. ^ a b c Nowak A., Vallacher R.R., Tesser A., Borkowski W., (2000) "Society of Self: The emergence of collective properties in self-structure", Psychological Review 107.
48. ^ a b c Epstein J.M. and Axtell R. (1996) Growing Artificial Societies - Social Science from the Bottom. Cambridge MA, MIT Press.
49. ^ a b c Epstein J.M. (1999) Agent Based Models and Generative Social Science. Complexity, IV (5)
50. ^ John Conway's Game of Life
51. ^ Werndl, Charlotte (2009). Are Deterministic Descriptions and Indeterministic Descriptions Observationally Equivalent?. Studies in History and Philosophy of Modern Physics 40, 232-242.
52. ^ Werndl, Charlotte (2009). Deterministic Versus Indeterministic Descriptions: Not That Different After All?. In: A. Hieke and H. Leitgeb (eds), Reduction, Abstraction, Analysis, Proceedings of the 31st International Ludwig Wittgenstein-Symposium. Ontos, 63-78.
53. ^ J. Glimm, D. Sharp, Stochastic Differential Equations: Selected Applications in Continuum Physics, in: R.A. Carmona, B. Rozovskii (ed.) Stochastic Partial Differential Equations: Six Perspectives, American Mathematical Society (October 1998) (ISBN 0-8218-0806-0).
54. ^ "Struggling with quantum logic: Q&A with Aaron O'Connell
55. ^ Heisenberg, Werner (1930 (1949)). Physikalische Prinzipien der Quantentheorie [Physical Principles of Quantum Theory]. Leipzig: Hirzel/University of Chicago Press. p. 4.
57. ^ Scientific American, "What is Quantum Mechanics Good For?"
58. ^ Albert Einstein insisted that, "I am convinced God does not play dice" in a private letter to Max Born, 4 December 1926, Albert Einstein Archives reel 8, item 180
• Daniel Dennett (2003) Freedom Evolves. Viking Penguin.
• John Earman (2007) "Aspects of Determinism in Modern Physics" in Butterfield, J., and Earman, J., eds., Philosophy of Physics, Part B. North Holland: 1369-1434.
• George Ellis (2005) "Physics and the Real World", Physics Today.
• Epstein J.M. (1999) "Agent Based Models and Generative Social Science", Complexity IV (5).
• -------- and Axtell R. (1996) Growing Artificial Societies — Social Science from the Bottom. MIT Press.
• Albert Messiah, Quantum Mechanics, English translation by G. M. Temmer of Mécanique Quantique, 1966, John Wiley and Sons, vol. I, chapter IV, section III.
• Ernest Nagel (March 3, 1960). "Determinism in history". Philosophy and Phenomenological Research, (International Phenomenological Society) 20 (8): 291–317. doi:10.2307/2105051. (Online version found here)
• John T Roberts (2006). "Determinism". In Sahotra Sarkar, Jessica Pfeifer, eds. The Philosophy of Science: A-M. Taylor & Francis. pp. 197 ff. ISBN 0415977096.
• Schimbera, Jürgen / Schimbera, Peter (2010) (in German), Determination des Indeterminierten. Kritische Anmerkungen zur Determinismus- und Freiheitskontroverse, Hamburg: Verlag Dr. Kovac, ISBN 978-3-8300-5099-5
• Reinhold Zippelius, Das Problem der Willensfreiheit, in: Rechtsphilosophie, § 25, 6th ed. 2011, C.H. Beck, Munich, ISBN 978-3-406-61191-9
External links[edit] |
e90cc763740f50aa | Friday, October 29, 2010
Maybe I/m Against Humans.
(A repsponse to Miguel Guhlin and too many other well-meaning writers.).
Being neither rich nor powerful, I’m unqualified to comment on ‘empowering’ vs. ‘domesticating’ education…wait, I did redesign the world’s most complex (and powerful) sensor-processor-effector system…at age 23. What the heck, my one cent:
We are ‘creativity’-ing ourselves down the path of the Roman Empire. We are a nation where it’s not important to walk from your touchdown to thank your blockers and focus ahead; it’s how creative a dance you do in the end zone. (Yes, those athletes are conditioned, but which part do the children see every week?). Now it’s to be unimportant to master math and logic, as long as we “create stuff”, no matter how distracting that might be.
Like the Roman citizens who grew bored with engineering and democracy and military art, turning instead to circuses and outdoing each other in bad poetry, we ‘create’ 500 bland television programs per hour, 24/365.
We build 14,000 ‘apps’ on top of Twitter alone. 250,000 for the iPhone. Every minute we upload another 24 hours of video to YouTube.
What do we know of the world? Our place in it? How many readers here know the fundamental difference between Shia and Sunni? Can describe the Iraqi and Afghan borders? Know the difference between a battalion and a brigade? Can guess the percentage of a school’s budget spent on personnel? Know—really know—why Washington was considered the “Father of our Country’? Understand why we don’t use much of the oil lying under our feet?
We need people who are productive and dependable. Especially when they are young and still learning what it is to be an adult, let alone lead adults. We need people who can care for the elderly and do repetitive research on sickle-cell anemia. We need people who will plant the seed each spring and gather the harvest each fall to feed a malnourished world.
We are, by the way, not as poor or unpowerful as you might think. Barack Obama is slave to his staff, cabinet, guards, and politicos. We have evenings and weekends free, can learn whatever we like, volunteer if we like to build parks, sing, deliver meals, guide youth groups, gather in spiritual need, organize a festival, build a business, golf, run.
There is, true, slavery in having a family at eighteen or twenty when you have no skill or education. And there’s the rub because you will not have time to read to your children, speak with them, sing to them. And they too will not learn, will head to slavery. Unless great teachers intervene.
Great teachers don’t teach you to be dangerous. All those dangerous people—they’re the ones keeping the sub-par teachers in place, distracting funds and resources from those in need, muddling the debate, spreading false economics, electing status-quo leaders. The useful idiots gathering at G-20 meetings to protest…well, to protest something, they have no idea what.
Great teachers don’t teacher you to be creative because no one is creative standing alone. All build on the shoulders of giants. It’s getting harder to learn everything the giants have given us. To be truly powerful you must master accounting and capital asset modeling and something of proteomics. Of foreign policy, but also of the difficulties of leading and sustaining a platoon in the field. Of statistics,…and of their limits. Of all the little things it takes to build something in your community.
One ought have time to have gathered intelligence like Sadaam Hussein’s offer of $25,000 to the family of every suicide bomber in Palestine. That 1 in 1200 teachers is delicensed, compared to more like 1 in 100 doctors or lawyers. That churches built most of our universities and hospitals. That the local auto-body shop is funding many of the local scholarships and public activities.
Things all learned over time, while being productive and dependable. While learning mental discipline, logical thought, patient disinterested analytical rigor.
Quadratic formulae, Schrödinger equations, and enantiomeric tranformations are hardly passionless, obsolete areas of study. They are the stuff of stars, of philosophy, of digital and analog empowerment.
Washington, by the way (with von Steuben) transformed a creative, individualistic, and un-dangerous army into a productive, dependable one which could throw off a Despotic King.
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fc071aad921b0247 | söndag 31 juli 2011
The Sky Dragon Strikes Back
Andrew Skolnick has mounted a ferocious attack on the Slayers of the Sky Dragon on Judy Curry's blog as a large set of comments (out of 2000) on the blog post Slaying a Greenhouse Dragon.
The attack is supported by a Youtube clip entitled Needling the Deniers aimed at disproving my new derivation of Planck's law of blackbody radiation showing that the basic postulate of CO2 alarmism of backradiation is fiction.
The clip shows that a needle can be heated in a microwave oven, which is known to everybody with some experience of a such a device. Skolnick thus demonstrates that low frequency waves (microwaves) can heat an absorber to higher temperature than the blackbody temperature corresponding to the frequency.
Does this mean that a blackbody can heat another blackbody of higher temperature, that a cold atmosphere can radiatively heat a warmer Earth surface? Of course not!
But what about the microwave oven then? Isn't this a counter-example? No, it is not because the amplitude of the microwave radiation is much larger than that of blackbody radiation of the corresponding temperature. The heating in a microwave oven is thus not blackbody heating; it is amplifed blackbody heating, and therefore the microwave heating of a needle is not a counter-example to my proof that blackbody backradiation from cold to warm is fiction.
But it is good that Skolnick brings this issue to the table, which allows one more head of the Sky Dragon to be eliminated. Thank you Andrew!
fredag 29 juli 2011
Mathematical Secret of Flight 6: Wikipedia Cover Up
onsdag 27 juli 2011
Mathematical Secret of Flight 5: Bird Wing
The thesis by Heather Falconsong Howard studies techniques for generating photo-realistic and fantasy digital bird and avian creatures in film, TV and games, based on an analysis of the design of real birds wings.
Particular attention is given to little covert feathers covering the space between groups of main fetahers, which also seem to act like little wing flaps delaying separation.
This is indicated in the above picture from the thesis which represents the classical Prandtl scenario of separation based on 2d recirculation to stagnation.
Our new analysis of separation and generation of lift opens to a different understanding of the action of birds wings. In particular we expect to find a connection between the separation pattern of our new analysis with point-stagnation and streamwise vorticity, and the arrangement of feathers of a bird wing including covert feathers and a periodic wavy trailing edge. We will report on our findings in upcoming posts...
The design of birds' wings thus suggest that the smooth surface and sharp straight trailing edge of a standard airplane wing may not be optimal. A further indication in this direction is given by the slotted wing tips of gliding hawks and the slotted jet flaps of Skywalk paragliders, to which we will also return...
måndag 25 juli 2011
Backradiation in Stefan-Boltzmann's Law: Folklore or Science?
Stefan-Boltzmann's Law can be formulated in the following two algebraically equivalent, but physically different forms:
1. E = sigma Te^4 - sigma Ta^4, (photon particle model: difference of two-way gross flows)
2. E = sigma (Te^4 - Ta^4) ~ 4 sigma Te^3 (Te - Ta), (wave model: net one-way flow)
where E is the intensity of the heat energy transferred from a blackbody (emitter) of temperature Te to a blackbody (absorber) of temperature Ta smaller than Te, and sigma is a constant.
Version 1 is the basis of CO2 alarmism based on "backradiation" of sigma Ta^4 from absorber to emitter, as transfer of heat energy from cold to warm.
In Slaying the Sky Dragon and Mathematical Physics of Blackbody Radiation I present a derivation of Version 2 based on a principle of finite precision computation in a wave model, without backradiation. And without backradiation CO2 alarmism crumbles.
The original version by Stefan and Boltzmann is formulated with Ta = 0 as Version 0. without backradiation (in which case 1. and 2. look identical), as an integrated version of Planck's law based on a statistical particle model.
Which is the correct formulation? Version o, 1 or 2? Particle statistics or waves? Let's list some answers from the web supposedly reflecting scientific sources:
The list can be made much longer, but we dont find any support of 1. and backradiation. And without backradiation CO2 alarmism crumbles.
The following questions present themselves:
• Why is 1. found only in the CO2 alarmism of IPCC, and not elsewhere?
• Is 1. a free invention which lacks original scientific source?
• Is 1. a form of hyper-reality for which the original is missing?
• Is 1. a form of folklore known by everybody to be true, yet without any individual scientist claiming to have demonstrated the statement.
• Is 1. an expression of "scientific consensus" for which no original scientific source is required?
What do you think? Is CO2 alarmism based on backradiation, folklore or real science?
fredag 22 juli 2011
The Emitter-Absorber Relation of Radiation
There is a lot of confusion concerning the physics behind Plank's radiation law and its integrated form of Stefan-Boltzmann's law in the following two algebraically equivalent but physically different forms:
Version 1 reflects two-way energy transfer by two way photon particles emitted by both emitter and absorber into a void (of zero Kelvin), and can be seen as an a hoc version cooked up from Planck's original law of one blackbody emitting into a void (of zero Kelvin).
Version 1 reflects simple physics of particles with the two bodies like two very young children playing side by side without interaction both spitting out photons in two directions into a void (of zero Kelvin).
Version 2 reflects more complex physics with the two bodies playing together, talking to each other by a two-way wave equation, but with one-way net transfer of heat: The effect of the finite precision computation is a high frequency cut-off depending on temperature limiting the ability of the absorber to re-emit only frequencies below cut-off, with frequencies above cut-off being absorbed and turned into heat.
Version 2 is like two educated people talking and listening to each other, with the emitter being the smarter and the frequencies above the cut-off of the dumber being absorbed by the dumber and then transformed into heat (frustration).
Which version is better? The trivial 1 or the educated 2? Is there an intimate relation between emitter and absorber into a system relation, where emission from one body is directly connected to absorption of another? Is the play between adults more interesting that than between babies?
Are these questions above your cut-off frequency and will only lead to heated frustration?
tisdag 19 juli 2011
Answer to Question by Roy Spencer
which can be turned around into:
• E = sigma Te^4 - sigma Ta^4
would be transferring sigma Te^4 to the atmosphere.
Does this answer your question Roy?
lördag 16 juli 2011
Monstrosity of Quantum Mechanics 7: Basic Postulates
In what sense are the basic postulates of quantum mechanics not Harry Potter fantasy?
Lubos Motl makes in The Unbreakable Postulates of Quantum Mechanics a heroic effort to justify quantum mechanics almost 100 years after its formulation, starting with:
The mission is to convince skeptics about the truths of the following basic postulates:
1. The set of possibilities in which a physical system may be found is described by a linear Hilbert space (more precisely by the rays in this space) equipped with an inner product.
2. Complex (nonzero) linear combinations of allowed states are allowed states, too.
3. A physical system composed out of N separated (or fully independent) subsystems has the Hilbert space equal to the tensor product of the Hilbert space describing the individual subsystems.
4. Physical quantities, also referred to as "observables" in the fancy quantum mechanical context, are encoded in Hermitean (linear) operators acting on the Hilbert space.
5. In particular, the evolution in time is generated by the operator known as the Hamiltonian.
6. The exponentials of its imaginary multiples are the operators that evolve the system over a finite interval and these operators are unitary; similarly, other symmetry transformations are given by other unitary (or anti-unitary, if the time reversal is included) operators.
7. The expectation values of the quantity "A" are given by the inner product ; if "A" is replaced by the projection operator "P", this expectation value expresses the probability that the condition connected with "P" will be satisfied once the system is measured.
The motivations for 1 - 7 presented by Lubos tell us something essential about the solidity of quantum mechanics. Let see how Lubos motivates 1 - 3:
1. Why do we know that there is a Hilbert space? If a physical theory has a content, it must be able to manipulate with the information. We insert some information that we know - and it spits out another piece of information that we didn't know but that is predicted, or postdicted, by the theory. So there must exist some states; which state was realized in Nature, is realized in Nature, or will be realized in Nature, is the way to phrase all the information we have or we want to have about the world or its pieces. That was true even in classical physics: different states of a physical system were given by points in the phase space (spanned by the positions and their canonical momenta).
2. The new thing about quantum mechanics is that the complex linear superpositions of two allowed states are also allowed states. How do we know that? Well, we may actually design procedures that create such combined states in practice.
3. Now, there are other postulates and universal rules of quantum mechanics. For example, the composite systems are described by tensor products of Hilbert spaces. It's not hard to see why: if the dimensions of Hilbert spaces H1, H2 are equal to d1, d2, there are clearly d1 basis vectors of H1 and d2 basis vectors of H2. These basic vectors parameterize some linearly independent (i.e. fully mutually exclusive) possibilities. The set of linearly independent possibilities for the composite system obviously has to be the Cartesian product of the two sets for the separate subsystems. And the "linear envelope" of this Cartesian product - the new basis - is the tensor product of the original spaces. Its dimension - its number of basis vectors - is equal to d1.d2 as expected. This conclusion is pretty much inevitable, by basic logic.
When you read this as a mathematician you understand that the motivation is weak, formal and touches triviality elevated to deep insight into the true inner mechanisms of the microscopic world. The Hilbert space assumption essentially reflects that the Schrödinger equation is linear. But why physics on atomic scales is linear allowing superposition, is not motivated. This appears as an ad hoc assumption which could be made by one who has recently fallen in love with linear Hilbert space theory and has been so overwhelmed by emotion that rational thinking has disappeared.
The argument that "we may actually design procedures that create such combined states (superposed) in practice" sounds hollow, knowing that this principle of quantum computing has shown to be very difficult to demonstrate.
Atomic physics concerns the interaction of elementary particles by certain forces and thus can be thought as N-body problems. But an N-body problem is not linear, and so it requires a lot of fantasy to believe that the N-body problem of quantum mechanics through some miracle decides to show up as linear.
without being able to find any reasonable one.
tisdag 12 juli 2011
Why Prandtl Was Wrong 4
Lift and drag of a NACA0012 wing in computation by Unicorn and experiment.
We have asked if it is possible to check if drag and lift of a body moving through a fluid originate from a thin boundary layer which separates from the body surface into the fluid, as is the mantra of Ludwig Prandtl, the father of modern fluid mechanics, formulated in an 8 page note in 1904.
To check in experiment is cumbersome because the viscosity of a real fluid is never exactly zero and thus it can be argued that no real fluid can satisfy a slip boundary condition with zero skin friction without any boundary layer.
But to check in computation is perfectly possible: just set the skin friction to zero in a Navier-Stokes code, that is use a slip boundary condition and see what happpens. Will drag and lift develop in accordance with observation in solutions of the Navier-Stokes equations with slip
without boundary layers?
Yes! Computations without boundary layer give correct drag and lift!
The conclusion is inevitable:
• Prandtl was wrong: Drag and lift do not originate from boundary layers.
• Prandtl's scenario of fluid separation is incorrect.
• The mantra of modern fluid mechanics is incorrect.
For further details see the new article Analysis of Separation in Turbulent Incompressible Flow which exhibits a scenario of fluid separation which is fundamentally different from that of Prandtl and which is supported by mathematical analysis, computation and observation.
måndag 11 juli 2011
Blackbody Radiation as a Generic Emergent Phenomenon
A body of temperature T emits radiation with an intensity E similar to that of an ideal blackbody given by Planck's law (Rayleigh-Jeans law with cut-off):
• E = gamma T f^2,
• with a high frequency cut-off proportional to T,
where f is the frequency and gamma is a constant.
The radiation spectrum thus only depends on the temperature and not of the material of the body. This indicates that blackbody radiation is a generic emergent phenomenon resulting from collective atomic vibrations and not from individual atoms or molecules which have different line spectra.
This idea is explored in the upcoming book Mathematical Physics of Blackbody Radiation (and in one of my chapters in Slaying the Sky Dragon) by an analysis of resonance in a wave equation with radiation.
In this model blackbody radiation can be thought of as the sound of a piano with all keys being struck at the same time with the same force: a complex chord which sounds the same for all pianos. An generic emergent phenomenon which cannot be understood by looking at just one key.
It can also be thought of as the complex sound of a big gong with a big range of frequencies. There is an interesting experiment showing that a gong can be made to sound by a short laser pulse kicking the gong atoms into an emergent collective vibrational motion producing sound waves hitting your ear.
söndag 10 juli 2011
Large Boundary Layer Collider: Why Prandtl Was Wrong 3
Part of the Large Boundary Layer Collider at the European Spallation Source in Lund, Sweden.
According to Ludwig Prandtl, named the father of modern fluid mechanics, both drag and lift of a body moving through air or water originate for a thin boundary layer.
This is the fundamental postulate of modern fluid mechanics formulated in 1904, but it is now being questioned. Is modern fluid mechanics based on a postulate which is does not correspond to physical reality?
The answer may be given by the European Spallation Source (ESS) in Lund, Sweden: The world's biggest proton accelerator (see picture).
The idea is to eliminate the boundary layer by bombarding it with high energy protons, and once the boundary layer has been removed completely this way, drag and lift will be measured. If drag and lift remain the same under removal of the boundary layer, then drag and lift do not originate from any boundary layer, and modern fluid mechanics is based on incorrect physics.
But ESS will not be ready to use before 2020, and thus it is natural to ask if there is some other quicker and cheaper way of eliminating a boundary layer? Yes, there is. But what is it?
Follow the thrilling uncovering of one of modern physics most well kept secrets...
PS An alternative to ESS would be to use liquid helium with next to zero viscosity, but to reach a sufficiently large Reynolds number, the dimension of the experiment needs to be 10 times bigger than that of the Large Hadron Collider and thus is out of reach, for the moment at least.
But as UN global warming alarmism is now fading away maybe this experiment could become the next big initiative by the UN backed by EU. DS
lördag 9 juli 2011
Why Prandtl Was Wrong 2
One way of eliminating a butterfly.
Question and Answer 1:
Question and Answer 2:
• How can one prove that a boundary layer is not the origin of drag and lift of a body?
• Eliminate the boundary layer and notice drag and lift without boundary layer.
But how to eliminate a butterfly and how to eliminate a boundary layer? Follow the thrilling
continuation of this story...
fredag 8 juli 2011
Why Prandtl Was Wrong 1
Prandtl initating modern fluid mecahnics in 1904: A very satisfactory explanation of the physical process in the boundary layer between a fluid and a solid body could be obtained by the hypothesis of an adhesion of the fluid to the walls, that is, by the hypothesis of a zero relative velocity between fluid and wall (no-slip).
Ludwig Prandtl is named the father of modern fluid mechanics because he discovered the boundary layer of a slightly viscous fluid flowing around a solid body, like air flowing around a moving car or airplane, as a thin layer where the fluid velocity rapidly changes from the free flow velocity away from the body to that of the body surface as an expression of a no-slip boundary condition.
Prandtl claimed that the that turbulent flow in the aft of a body results from separation of turbulent boundary layer away from the body surface into the free flow.
This has become the mantra of modern fluid mechanics: The truth of slightly viscous fluid flow is to be found in thin boundary layers. Both drag and lift of a body moving through a fluid are effects of a no-slip boundary condition creating a thin boundary layer.
In a sequence of posts we shall show that Prandtl was wrong: Drag and lift do not originate from a thin no-slip boundary layer.
But how can one show that Prandtl was wrong? Something to reflect upon a rainy summer day.
Hint 1: Suppose you observe the same drag and lift with the boundary layers eliminated. Can you then be sure that drag and lift do not originate from boundary layers? Yes, you probably say. But how to "eliminate" the boundary layers?
onsdag 6 juli 2011
The Secret of Separation
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3400a41545d056d7 | Next Contents Previous
4.3.2. Magnetic fields from Higgs field equilibration
In the previous section we have seen that, concerning the generation of magnetic fields, the QCDPT and the EWPT share several common aspects. However, there is one important aspect which makes the EWPT much more interesting than the QCDPT. In fact, at the electroweak scale the electromagnetic field is directly influenced by the dynamics of the Higgs field which drives the EWPT.
To start with we remind that, as a consequence of the Weinberg-Salam theory, before the EWPT is not even possible to define the electromagnetic field, and that this operation remains highly non-trivial until the transition is completed. In a sense, we can say that the electromagnetic field was "born" during the EWPT. The main problem in the definition of the electromagnetic field at the weak scale is the breaking of the translational invariance: the Higgs field module and its SU(2) and UY(1) phases take different values in different positions. This is either a consequence of the presence of thermal fluctuations, which close to Tc are locally able to break/restore the SU(2) × UY(1) symmetry or of the presence of large stable domains, or bubbles, where the broken symmetry has settled.
The first generalized definition of the electromagnetic field in the presence of a non-trivial Higgs background was given by t'Hooft [149] in the seminal paper where he introduced magnetic monopoles in a SO(3) Georgi-Glashow model. t'Hooft definition is the following
Equation 4.21 (4.21)
In the above G aµnu ident partialW aµ - partialW anu, where
Equation 4.22 (4.22)
(taua are the Pauli matrices) is a unit isovector which defines the "direction" of the Higgs field in the SO(3) isospace (which coincides with SU(2)) and (Dµphihat)a = partialµphihata + g epsilonabc Wµbphihatc, where Wµb are the gauge fields components in the adjoint representation. The nice features of the definition (4.21) are that it is gauge-invariant and it reduces to the standard definition of the electromagnetic field tensor if a gauge rotation can be performed so to have phihata = - deltaa3 (unitary gauge). In some models, like that considered by t'Hooft, a topological obstruction may prevent this operation to be possible everywhere. In this case singular points (monopoles) or lines (strings) where phi a = 0 appear which become the source of magnetic fields. t'Hooft result provides an existence proof of magnetic fields produced by non-trivial vacuum configurations.
The Weinberg-Salam theory, which is based on the SU(2) × UY(1) group representation, does not predict topologically stable field configurations. We will see, however, that vacuum non-topological configurations possibly produced during the EWPT can still be the source of magnetic fields.
A possible generalization of the definition (4.21) for the Weinberg-Salam model was given by Vachaspati [106]. It is
Equation 4.23 (4.23)
Dµ = partialµ - i[(g) / 2] taua Wµa -i[(g') / 2] Yµ.
This expression was used by Vachaspati to argue that magnetic fields should have been produced during the EWPT. Synthetically, Vachaspati argument is the following. It is known that well below the EWPT critical temperature Tc the minimum energy state of the Universe corresponds to a spatially homogeneous vacuum in which Phi is covariantly constant, i.e. Dnu Phi = Dµ phihata = 0. However, during the EWPT, and immediately after it, thermal fluctuations give rise to a finite correlation length xi ~ (eTc)-1. Therefore, there are spatial variations both in the Higgs field module |phi| and in its SU(2) and U(1)Y phases which take random values in uncorrelated regions (15). It was noted by Davidson [150] that gradients in the radial part of the Higgs field cannot contribute to the production of magnetic fields as this component is electrically neutral. While this consideration is certainly correct, it does not imply the failure of Vachaspati argument. In fact, the role played by the spatial variations of the SU(2) and U(1)Y "phases" of the the Higgs field cannot be disregarded. It is worthwhile to observe that gradients of these phases are not a mere gauge artifact as they correspond to a nonvanishing kinetic term in the Lagrangian. Of course one can always rotate Higgs fields phases into gauge boson degrees of freedom (see below) but this operation does not change Femµnu which is a gauge-invariant quantity. The contribution to the electromagnetic field produced by gradients of phihata can be readily determined by writing the Maxwell equations in the presence of an inhomogeneous Higgs background [151]
Equation 4.24 (4.24)
Even neglecting the second term on the righthand side of Eq. (4.24), which depends on the definition of Femµnu in a Higgs inhomogeneous background (see below), it is evident that a nonvanishing contribution to the electric 4-current arises from the covariant derivative of phihata. The physical meaning of this contribution may look more clear to the reader if we write Eq. (4.24) in the unitary gauge
Equation 4.25 (4.25)
Not surprisingly, we see that the electric currents produced by Higgs field equilibration after the EWPT are nothing but W boson currents.
Since, on dimensional grounds, Dnu Phi ~ v / xi where v is the Higgs field vacuum expectation value, Vachaspati concluded that magnetic fields (electric fields were supposed to be screened by the plasma) should have been produced at the EWPT with strength
Equation 4.26 (4.26)
Of course these fields live on a very small scale of the order of xi and in order to determine fields on a larger scale Vachaspati claimed that a suitable average has to be performed (see return on this issue below in this section).
Before discussing averages, however, let us try to understand better the nature of the magnetic fields which may have been produced by the Vachaspati mechanism. We notice that Vachaspati's derivation does not seem to invoke any out-of-equilibrium process and indeed the reader may wonder what is the role played by the phase transition in the magnetogenesis. Moreover, magnetic fields are produced anyway on a scale (eT)-1 by thermal fluctuations of the gauge fields so that it is unclear what is the difference between magnetic fields produced by the Higgs fields equilibration and these more conventional fields. In our opinion, although Vachaspati's argument is basically correct its formulation was probably oversimplified. Indeed, several works showed that in order to reach a complete understanding of this physical effect a more careful study of the dynamics of the phase transition is called for. We shall now review these works starting from the case of a first order phase transition.
The case of a first order EWPT
Before discussing the SU(2) × U(1) case we cannot overlook some important work which was previously done about phase equilibration during bubble collision in the framework of more simple models. In the context of a U(1) Abelian gauge symmetry, Kibble and Vilenkin [152] showed that the process of phase equilibration during bubble collisions give rise to relevant physical effects. The main tool developed by Kibble and Vilenkin to investigate this kind of processes is the, so-called, gauge-invariant phase difference defined by
Equation 4.27 (4.27)
where theta is the U(1) Higgs field phase and Dµ theta ident partialµ theta + e Aµ is the phase covariant derivative. A and B are points taken in the bubble interiors and k = 1,2,3. Deltatheta obeys the Klein-Gordon equation
Equation 4.28 (4.28)
where m = ev is the gauge boson mass. Kibble and Vilenkin assumed that during the collision the radial mode of the Higgs field is strongly damped so that it rapidly settles to its expectation value v everywhere. One can choose a frame of reference in which the bubbles are nucleated simultaneously with centers at (t, x, y, z) = (0,0,0, ± Rc). In this frame, the bubbles have equal initial radius Ri = R0. Their first collision occurs at (tc, 0, 0, 0) when their radii are Rc and tc = sqrt[Rc2 - R02]. Given the symmetry of the problem about the axis joining the nucleation centers (z-axis), the most natural gauge is the axial gauge. In this gauge
Equation 4.29 (4.29)
where alpha = 0, 1, 2 and tau2 = t2 - x2 - y2 . The condition thetaa(tau, 0) = 0 fixes the gauge completely. At the point of first contact z = 0, tau = tc the Higgs field phase was assumed to change from theta0 to -theta0 going from a bubble into the other. This constitutes the initial condition of the problem. The following evolution of theta is determined by the Maxwell equation
Equation 4.30 (4.30)
and the Klein-Gordon equation which splits into
Equation 4.31 (4.31)
Equation 4.32 (4.32)
The solution of the linearized equations (4.31) and (4.32) for tau > tc then becomes
Equation 4.33 (4.33)
Equation 4.34 (4.34)
where omega2 = k2+m2. The gauge-invariant phase difference is deduced by the asymptotic behavior at z rightarrow ± infty
Equation 4.35 (4.35)
Thus, phase equilibration occurs with a time scale tc determined by the bubble size, with superimposed oscillations with frequency given by the gauge-field mass. As we see from Eq. (4.34) phase oscillations come together with oscillations of the gauge field. It follows from Eq. (4.30) that these oscillations give rise to an "electric" current. This current will source an "electromagnetic" field strength Fµnu (16). Because of the symmetry of the problem the only nonvanishing component of Fµnu is
Equation 4.36 (4.36)
Therefore, we have an azimuthal magnetic field Bvarphi = F zrho = rhopartialz a and a longitudinal electric field Ez = F0z = -tpartialz a = -(t / rho) Bvarphi(tau, z), where we have used cylindrical coordinates (rho, phi). We see that phase equilibration during bubble collision indeed produces some real physical effects.
Kibble and Vilenkin did also consider the role of electric dissipation. They showed that a finite value of the electric conductivity sigma gives rise to a damping in the "electric" current which turns into a damping for the phase equilibration. They found
Equation 4.37 (4.37)
for small values of sigma, and
Equation 4.38 (4.38)
in the opposite case. The dissipation time scale is typically much smaller than the time which is required for two colliding bubble to merge completely. Therefore the gauge-invariant phase difference settles rapidly to zero in the overlapping region of the two bubbles and in its neighborhood. It is interesting to compute the line integral of Dk theta over the path ABCD represented in the Fig. 4.1.
Figure 4.1
Figure 4.1. Two colliding bubbles. It is showed the closed path along which the displacement of the gauge invariant phase difference is computed. (From Ref. [152]).
From the previous considerations it follows that DeltathetaAB = 0, DeltathetaAD = DeltathetaBC = 0 and DeltathetaDC = 2theta0. It is understood that in order for the integral to be meaningful, the vacuum expectation value of the Higgs field has to remain nonzero in the collision region and around it, so that the phase theta remains well defined and interpolates smoothly between its values inside the bubbles. Under these hypothesis we have
Equation 4.39 (4.39)
The physical meaning of this quantity is recognizable at a glance in the unitary gauge, in which each Deltatheta is given by a line integral of the vector potential A. We see that the gauge-invariant phase difference computed along the loop is nothing but the magnetic flux trough the loop itself
Equation 4.40 (4.40)
In other words, phase equilibration give rise to a ring of magnetic flux near the circle on which bubble walls intersect. If the initial phase difference between the two bubbles is 2pi, the total flux trapped in the ring is exactly one flux quantum, 2pi / e.
Kibble and Vilenkin did also consider the case in which three bubbles collide. They argued that in this case the formation of a string, in which interior symmetry is restored, is possible. Whether or not this happens is determined by the net phase variation along a closed path going through the three bubbles. The string forms if this quantity is larger than 2pi. According to Kibble and Vilenkin strings cannot be produced by two bubble collisions because, for energetic reasons, the system will tend to choose the shorter of the two paths between the bubble phases so that a phase displacement geq 2pi can never be obtained. This argument, which was first used by Kibble [153] for the study of defect formation, is often called the "geodesic rule".
The work of Kibble and Vilenkin was reconsidered by Copeland and Saffin [154] and more recently by Copeland, Saffin and Törnkvist [155] who showed that during bubble collision the dynamics of the radial mode of the Higgs field cannot really be disregarded. In fact, violent fluctuations in the modulus of the Higgs field take place and cause symmetries to be restored locally, allowing the phase to "slip" by an integer multiple of 2pi violating the geodesic rule. Therefore strings, which carry a magnetic flux, can be produced also by the collision of only two bubbles. Saffin and Copeland [156] went a step further by considering phase equilibration in the SU(2) × U(1) case, namely the electroweak case. They showed that for some particular initial conditions the SU(2) × U(1) Lagrangian is equivalent to a U(1) Lagrangian so that part of Kibble and Vilenkin [152] considerations can be applied. The violation of the geodesic rule allows the formation of vortex configurations of the gauge fields. Saffin and Copeland argued that these configurations are related to the Nielsen-Olesen vortices [157]. Indeed, it is know that such a kind of non-perturbative solutions are allowed by the Weinberg-Salam model [158] (for a comprehensive review on electroweak strings see Ref. [159]). Although electroweak string are not topologically stable, numerical simulations performed in Ref. [156] show that in presence of small perturbations the vortices survives on times comparable to the time required for bubble to merge completely.
The generation of magnetic fields in the SU(2) × U(1)Y case was not considered in the work by Saffin and Copeland. This issue was the subject of a following paper by Grasso and Riotto [151]. The authors of Ref. [151] studied the dynamics of the gauge fields starting from the following initial Higgs field configuration
Equation 4.41 (4.41)
which represents the superposition of the Higgs fields of two bubbles which are separated by a distance b. In the above na is a unit vector in the SU(2) isospace and taua are the Pauli matrices. The phases and the orientation of the Higgs field were chosen to be uniform across any single bubble. It was assumed that Eq. (4.41) holds until the two bubble collide (t = 0). Since na taua is the only Lie-algebra direction which is involved before the collision, one can write the initial Higgs field configuration in the form [156]
Equation 4.42 (4.42)
In order to disentangle the peculiar role played by the Higgs field phases, the initial gauge fields Wµa and their derivatives were assumed to be zero at t = 0. This condition is of course gauge dependent and should be interpreted as a gauge choice. It is convenient to write the equation of motion for the gauge fields in the adjoint representation. For the SU(2) gauge fields we have
Equation 4.43 (4.43)
where the isovector phihata has been defined in Eq. (4.22). Under the assumptions mentioned in the above, at t = 0, this equation reads
Equation 4.44 (4.44)
In general, the unit isovector phihata can be decomposed into
Equation 4.45 (4.45)
where phihatT0 ident - (0, 0, 1). It is straightforward to verify that in the unitary gauge, phihat reduces to phihat0. The relevant point in Eq. (4.42) is that the versor [^(n)], about which it is performed the SU(2) gauge rotation, does not depend on the space coordinates. Therefore, without loosing generality, we have the freedom to choose nhat to be everywhere perpendicular to phihat0. In other words, phihat can be everywhere obtained by rotating phihat0 by an angle theta in the plane identified by nhat and phihat0. Formally, phihat = costheta phihat0 + sintheta nhat × phihat0, which clearly describes a simple U(1) transformation. In fact, since it is evident that the condition nhat perp phihat0 also implies nhat perp phihat, the equation of motion (4.44) becomes
Equation 4.46 (4.46)
As expected, we see that only the gauge field component along the direction nhat, namely Aµ = na Waµ, that has some initial dynamics which is created by a nonvanishing gradient of the phase between the two domains. When we generalize this result to the full SU(2) × U(1)Y gauge structure, an extra generator, namely the hypercharge, comes-in. Therefore in this case is not any more possible to choose an arbitrary direction for the unit vector nhat since different orientations of the unit vector nhat with respect to phihat0 correspond to different physical situations. We can still consider the case in which nhat is parallel to phihat0 but we should take in mind that this is not the only possibility. In this case we have
Equation 4.47 (4.47)
Equation 4.48 (4.48)
where g and g' are respectively the SU(2) and U(1)Y gauge coupling constants. It is noticeable that in this case the charged gauge fields are not excited by the phase gradients at the time when bubble first collide. We can combine Eqs. (4.47) and (4.48) to obtain the equation of motion for the Z-boson field
Equation 4.49 (4.49)
This equation tells us that a gradient in the phases of the Higgs field gives rise to a nontrivial dynamics of the Z-field with an effective gauge coupling constant sqrt[g2 + g'2]. We see that the equilibration of the phase (theta + varphi) can be now treated in analogy to the U(1) toy model studied by Kibble and Vilenkin [152], the role of the U(1) "electromagnetic" field being now played by the Z-field. However, differently from Ref. [152] the authors of Ref. [151] left the Higgs field modulus free to change in space. Therefore, the equation of motion of rho(x) has to be added to (4.49). Assuming the charged gauge field does not evolve significantly, the complete set of equations of motion that we can write at finite, though small, time after the bubbles first contact, is
Equation 4.50 (4.50)
where dµ = partialµ + i[g / (2costhetaW)] Zµ, eta is the vacuum expectation value of Phi and lambda is the quartic coupling. Note that, in analogy with [152], a gauge invariant phase difference can be introduced by making use of the covariant derivative dµ. Equations (4.50) are the Nielsen-Olesen equations of motion [157]. Their solution describes a Z-vortex where rho = 0 at its core [160]. The geometry of the problem implies that the vortex is closed, forming a ring which axis coincide with the conjunction of bubble centers. This result provides further support to the possibility that electroweak strings are produced during the EWPT.
In principle, in order to determine the magnetic field produced during the process that we illustrated in the above, we need a gauge-invariant definition of the electromagnetic field strength in the presence of the non-trivial Higgs background. We know however that such definition is not unique [161]. For example, the authors of Ref. [151] used the definition given in Eq. (4.23) to find that the electric current is
Equation 4.51 (4.51)
whereas other authors [162], using the definition
Equation 4.52 (4.52)
found no electric current, hence no magnetic field, at all. We have to observe, however, that the choice between these, as others, gauge invariant definitions is more a matter of taste than physics. Different definitions just give the same name to different combinations of the gauge fields. The important requirement which acceptable definitions of the electromagnetic field have to fulfill is that they have to reproduce the standard definition in the broken phase with a uniform Higgs background. This requirement is fulfilled by both the definitions used in the Refs. [151] and [162]. In our opinion, it is not really meaningful to ask what is the electromagnetic field inside, or very close to, the electroweak strings. The physically relevant question is what are the electromagnetic relics of the electroweak strings once the EWPT is concluded.
One important point to take in mind is that electroweak strings are not topologically stable (see [159] and references therein) and that, for the physical value of the Weinberg angle, they rapidly decay after their formation. Depending on the nature of the decay process two scenarios are possible. According to Vachaspati [163] long strings should decay in short segments of length ~ mW-1. Since the Z-string carry a flux of Z-magnetic flux in its interior
Equation 4.53 (4.53)
and the Z gauge field is a linear superposition of the W3 and Y fields then, when the string terminates, the Y flux cannot terminate because it is a U(1) gauge field and the Y magnetic field is divergenceless. Therefore some field must continue even beyond the end of the string. This has to be the massless field of the theory, that is, the electromagnetic field. In some sense, a finite segment of Z-string terminates on magnetic monopoles [158]. The magnetic flux emanating from a monopole is:
Equation 4.54 (4.54)
This flux may remain frozen-in the surrounding plasma and become a seed for cosmological magnetic fields.
Another possibility is that Z-strings decay by the formation of a W-condensate in their cores. In fact, it was showed by Perkins [164] that while electroweak symmetry restoration in the core of the string reduces mW, the magnetic field via its coupling to the anomalous magnetic moment of the W-field, causes, for eB > mW2, the formation of a condensate of the W-fields. Such a process is based on the Ambj orn-Olesen instability which will be discussed in some detail in the Chap.5 of this review. As noted in [151] the presence of an inhomogeneous W-condensate produced by string decay gives rise to electric currents which may sustain magnetic fields even after the Z string has disappeared. The formation of a W-condensate by strong magnetic fields at the EWPT time, was also considered by Olesen [165].
We can now wonder what is the predicted strength of the magnetic fields at the end of the EWPT. An attempt to answer to this question has been done by Ahonen and Enqvist [166] (see also Ref. [167]) where the formation of ring-like magnetic fields in collisions of bubbles of broken phase in an Abelian Higgs model were inspected. Under the assumption that magnetic fields are generated by a process that resembles the Kibble and Vilenkin [152] mechanism, it was concluded that a magnetic field of the order of B appeq 2 × 1020 G with a coherence length of about 102 GeV-1 may be originated. Assuming turbulent enhancement the authors of Ref. [166] of the field by inverse cascade [51], a root-mean-square value of the magnetic field Brms appeq 10-21 G on a comoving scale of 10 Mpc might be present today. Although our previous considerations give some partial support to the scenario advocated in [166] we have to stress, however, that only in some restricted cases it is possible to reduce the dynamics of the system to the dynamics of a simple U(1) Abelian group. Furthermore, once Z-vortices are formed the non-Abelian nature of the electroweak theory shows due to the back-reaction of the magnetic field on the charged gauge bosons and it is not evident that the same numerical values obtained in [166] will be obtained in the case of the EWPT.
However the most serious problem with the kind of scenario discussed in this section comes form the fact that, within the framework of the standard model, a first order EWPT seems to be incompatible with the Higgs mass experimental lower limit [143]. Although some parameter choice of the minimal supersymmetric standard model (MSSM) may still allow a first order transition [144], which may give rise to magnetic fields in a way similar to that discussed in the above, we think it is worthwhile to keep an open mind and consider what may happen in the case of a second order transition or even in the case of a cross over.
The case of a second order EWPT
As we discussed in the first part of this section, magnetic fields generation by Higgs field equilibration share several common aspects with the formation of topological defects in the early Universe. This analogy holds, and it is even more evident, in the case of a second order transition. The theory of defect formation during a second order phase transition was developed in a seminal paper by Kibble [153]. We shortly review some relevant aspects of the Kibble mechanism. We start from the Universe being in the unbroken phase of a given symmetry group G. As the Universe cools and approach the critical temperature Tc protodomains are formed by thermal fluctuations where the vacuum is in one of the degenerate, classically equivalent, broken symmetry vacuum states. Let M be the manifold of the broken symmetry degenerate vacua. The protodomains size is determined by the Higgs field correlation function. Protodomains become stable to thermal fluctuations when their free energy becomes larger than the temperature. The temperature at which this happens is usually named Ginsburg temperature TG. Below TG stable domains are formed which, in the case of a topologically nontrivial manifold M, give rise to defect production. Rather, if M is topologically trivial, phase equilibration will continue until the Higgs field is uniform everywhere. This is the case of the Weinberg-Salam model, as well as of its minimal supersymmetrical extension.
Higgs phase equilibration, which occurs when stable domains merge, gives rise to magnetic fields in a way similar to that described by Vachaspati [106] (see the beginning of this section). One should keep in mind, however, that as a matter of principle, the domain size, which determine the Higgs field gradient, is different from the correlation length at the critical temperature [151]. At the time when stable domains form, their size is given by the correlation length in the broken phase at the Ginsburg temperature. This temperature was computed, in the case of the EWPT, by the authors of Ref. [151] by comparing the expansion rate of the Universe with the nucleation rate per unit volume of sub-critical bubbles of symmetric phase (with size equal to the correlation length in the broken phase) given by
Equation 4.55 (4.55)
where lb is the correlation length in the broken phase. S3ub is the high temperature limit of the Euclidean action (see e.g. Ref. [168]). It was shown that for the EWPT the Ginsburg temperature is very close to the critical temperature, TG = Tc within a few percent. The corresponding size of a broken phase domain is determined by the correlation length in the broken phase at T = TG
Equation 4.56 (4.56)
where V(phi, T) is the effective Higgs potential. ell(TG)2b is weakly dependent on MH, ellb(TG) appeq 11/ TG for MH = 100 GeV and ellb(TG) appeq 10 / TG for MH = 200 GeV. Using this result and Eq. (4.23) the authors of Ref. [151] estimated the magnetic field strength at the end of the EWPT to be of order of
Equation 4.57 (4.57)
on a length scale ellb(TG).
Although it was showed by Martin and Davis [169] that magnetic fields produced on such a scale may be stable against thermal fluctuations, it is clear that magnetic fields of phenomenological interest live on scales much larger than ellb(TG). Therefore, some kind of average is required. We are ready to return to the discussion of the Vachaspati mechanism for magnetic field generation [106]. Let us suppose we are interested in the magnetic field on a scale L = N ell. Vachaspati argued that, since the Higgs field is uncorrelated on scales larger than ell, its gradient executes a random walk as we move along a line crossing N domains. Therefore, the average of the gradient Dµ Phi over this path should scale as sqrtN. Since the magnetic field is proportional to the product of two covariant derivatives, see Eq. (4.23), Vachaspati concluded that it scales as 1 / N. This conclusion, however, overlooks the difference between <Dµ Phi> <Dµ Phi;> and <Dµ Phi Dµ Phi>. This point was noticed by Enqvist and Olesen [107] (see also Ref. [109]) who produced a different estimate for the average magnetic field, <B>rms, L ident B(L) ~ Bell / sqrtN. Neglecting the possible role of the magnetic helicity (see the next section) and of possible related effects, e.g. inverse cascade, and using Eq. (4.57), the line-averaged field today on a scale L ~ 1 Mpc (N ~ 1025) is found to be of the order B0(1 Mpc) ~ 10-21 G.
Another important point of this kind of scenario (for the reasons which will become clear in the next section) is that it naturally gives rise to a nonvanishing vorticity. This point can be understood by the analogy with the process which lead to the formation of superfluid circulation in a Bose-Einstein fluid which is rapidly taken below the critical point by a pressure quench [170]. Consider a circular closed path through the superfluid of length C = 2piR. This path will cross N appeq C / ell domains, where ell is the characteristic size of a single domain. Assuming that the phase theta of the condensate wave function is uncorrelated in each of the N domains (random-walk hypothesis) the typical mismatch of theta is given by:
Equation 4.58 (4.58)
where nabla theta is the phase gradient across two adjacent domains and ds is the line element along the circumference. It is well known (see e.g. [171]) that from the Schrödinger equation it follows that the velocity of a superfluid is given by the gradient of the phase trough the relation vs = (hbar / m) nabla theta, therefore (4.58) implies
Equation 4.59 (4.59)
It was argued by Zurek [170] that this phenomenon can effectively simulate the formation of defects in the early Universe. As we discussed in the previous section, although the standard model does not allows topological defects, embedded defects, namely electroweak strings, may be produced through a similar mechanism. Indeed a close analogy was showed to exist [172] between the EWPT and the 3He superfluid transition where formation of vortices is experimentally observed. This hypothesis received further support by some recent lattice simulations which showed evidence for the formation of a cluster of Z-strings just above the cross-over temperature [173] in the case of a 3D SU(2) Higgs model. Electroweak strings should lead to the generation of magnetic fields in the same way we discussed in the case of a first order EWPT. Unfortunately, to estimate the strength of the magnetic field produced by this mechanism requires the knowledge of the string density and net helicity which, so far, are rather unknown quantities.
15 Vachaspati [106] did also consider Higgs field gradients produced by the presence of the cosmological horizon. However, since the Hubble radius at the EWPT is of the order of 1 cm whereas xi ~ (eTc)-1 ~ 10-16 cm, it is easy to realize that magnetic fields possibly produced by the presence of the cosmological horizon are phenomenologically irrelevant. Back.
16 It is understood that since the toy model considered by Kibble e Vilenkin is not SU(2) × U(1)Y, Fµnu is not the physical electromagnetic field strength. Back.
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6e97a1bc651ca737 | Take the 2-minute tour ×
When finding the discrete energy states of a operator I have been taught to use the time-independent Schrodinger equation which restates the definition of eigenvalues and eigenvectors. What I don’t understand is why the eigenvalues are the energy states, is there firstly a mathematical reason and secondly a physical reason?
Does this arise from Hamiltonian or Lagrangian mechanics which I am not familiar with?
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Sorry I mean eigenvalues of the operator not wave funciton – Josh Apr 6 '11 at 17:00
Keep in mind that we are only dealing with Hermetian operators, because their eigenvalues are real, and hence correspond to positive definite probabilities. – Matt Calhoun Apr 8 '11 at 16:44
7 Answers 7
up vote 4 down vote accepted
As has been remarked by others and explained clearly, and mathematically, the eigenvalues are important because a) they allow you to solve the time-dependent equation, i.e., solve for the evolution of the system and b) a state which belongs to the eigenvalue $E$, i.e., as we say, a state which is an eigenstate with eigenvalue $E$, has an expectation value of the energy operator which is easy to see has to be $E$ itself. But those explanations are advanced and rely on the maths. And they do not explain why $E$ should be considered 'an energy level'. At some risk, I will try to answer your question more physically.
What is the physical reason why the energy states of a system, e.g., an atom, are the eigenvalues of the operator $H$ that appears in the time-independent Schroedinger equation? Well, first, note that it's absolutely the same $H$ that appears in the time-dependent Schrodinger equation, $$H\cdot \psi = -i{\partial \psi \over \partial t}$$ which controls the rate of change of $\psi$.
The answer doesn't come from the classical Hamiltonian or Lagrangian mechanics, but from the then-new quantum properties of Nature. A non-classical feature of QM is that some states are stationary, which means they do not change in time. E.g., the electron in a Bohr orbit is actually not moving, not orbiting at all, and this solves the classical paradoxes about the atom (why the rotating charge doesn't radiate its energy away and fall into the centre).
The first key point is that an eigenstate is a stationary state: what is the explanation for this? well, Schroedinger's time dependent equation clearly says that, up to a constant of proportionality, the time-rate of change of any state $\psi$ is found by applying the operator $H$ (the Hamiltonian: we do not yet know it is also the energy operator) to it: the new vector or function $H\cdot\psi$ is the change in $\psi$ per unit time. Obviously if this is zero, $\psi$ does not change (this was the only classical possibility). But also if $H\cdot\psi$ is even a non-zero multiple of $\psi$, call it $E\psi$, then $\psi$ plus this rate of change is still a multiple of $\psi$, so as time goes on, $\psi$ changes in a trivial fashion: just to another multiple of itself. In QM, a multiple of the wave function represents the same quantum state, so we see the quantum state does not change.
Now the next key point is that a state with a definite energy value must be stationary. Why? In QM, it is not automatic that a system has a definite value of a physical quantity, but if it does, that means its measurement always leads to the same answer, so there is no uncertainty. So if there is no uncertainty in the energy, by Heisenberg's uncertainty principle there must be infinite uncertainty in something else, whatever is 'conjugate' to energy. And that is time. You cannot tell the time using this system, which implies it is not changing. So it is stationary. (remember, we are not assuming that $H$ is also the energy operator and we are not assuming the formula for expectations).
Thus being an eigenstate of $H$ implies $\psi$ is stationary. And having a definite energy value implies it is stationary. Being physicists, we now conclude that being an eigenstate implies it has a definite energy value, which answers your question, and these are the 'energy levels' of a system such as an atom: a system, even an atom, might not possess a definite energy, but if it doesn't, it won't be stationary, and being microscopic, the time-scale in which it will evolve will be so rapid we are unlikely to be able to observe its energy, or even care (since it won't be relevant to molecules or chemistry). So, 'most' atoms for which we can actually measure their energy must be stationary: this is 'why' the definite values of energy which a stationary state can possess are called the 'energy levels' of the system, and historically were discovered first, before Schroedingers equation. From a human perspective, most atoms that we care about spend most of their time that matters to us in an approximately stationary state.
In case you are wondering why time is the conjugate to energy, whereas Heisenberg's original analysis of his uncertainty principle showed that position was conjugate to momentum, we rely on relativity: time is just another coordinate of space-time, and so is analogous to position. And in relativistic mechanics, momentum in a spatial direction is analogous to energy (or mass, same thing). In the standard relativistic equation $$p^2-m^2=E^2,$$ we see that momentum ($p$) and mass $m$ are symmetric (except for the negative sign) with each other. So since momentum is conjugate to position, $m$ or energy must be conjugate to time. For this reason, Bohr was able to extend Heisenberg's analysis, of the uncertainty relations between measurements of position and measurements of momentum, to show the same relations between energy and time.
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Both eigenvalues and eigenstates belong to some operator. In your case, this is the Hamiltonian operator $\hat H$. It's fundamental because of many reasons. First is that it is indeed an operator that represents energy in a sense that possible energy levels are encoded in its spectrum (i.e. a set of eigenvalues). The second important reason is that it is the operator that can be found in Schrodinger equation $i \hbar \partial_t \left | \psi(t) \right > = \hat H \left | \psi(t) \right >$. This equation can then be solved by writing $\left | \psi(t) \right >$ as superposition of eigenstates of $\hat H$: $\left | \psi(t) \right > = \sum_n c_n(t) \left | \psi_n \right >$. If we can find these states, we are done as $c_n(t) = \exp({-iE_n t \over \hbar}) c_n(t=0)$ solves the equation (and it also shows the importance of these eigenstates because they are preserved by time evolution).
So this means the problem of time evolution in quantum mechanics can be reduced to the problem of finding the eigenvalues and eigenstates of $\hat H$, the equation for that being $\hat H \left | \psi_n \right> = E_n \left| \psi_n \right>$.
Note: the above assumes that $\hat H$ is time-independent. If it's not (as is the case in basically all practical applications, using perturbation theory) then we use different techniques, e.g. of path integration, or various scattering formulas.
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You seem to be confusing two things, namely the eigenstates of an operator and Schrödinger's equation. A priori, these two have nothing to do with each other.
In Quantum Mechanics, measurable quantities are represented by (hermitian) operators on a Hilbert space. For instance there is an operator $P$ corresponding to the momentum. In general, when measuring the momentum of a state $|\psi \rangle$, the result will not be deterministic. However, the average over several measurements will be equal to the expection value
$$ \langle \psi | P | \psi \rangle $$
However, when $|\psi\rangle$ is an eigenvector of the operator, $P|\psi\rangle = \lambda |\psi\rangle$, then the measurement will always be the same value $\lambda$.
In particular, there is an operator corresponding to the total energy, the Hamiltionian $H$. The form of this operator can be obtained from classical physics if you replace momentum and location by their corresponding operators. For instance, the Hamiltonian of an electron in an electric potential $V$ is
$$ H = \frac1{2m} P^2 + eV(X) .$$
Thus, the expectation value for the energy of a state $|\psi\rangle$ is $\langle \psi|H|\psi\rangle$.
Now, the Hamiltionian is a very interesting operator because it features prominently in the equation of motion, the Schrödinger equation.
$$ i\hbar \partial_t |\psi(t)\rangle = H |\psi(t)\rangle .$$
What does this have to do with the eigenvalues of the Hamiltionian? A priori nothing, but the point is that knowing the eigenvectors and -values of $H$ allow you to solve this equation. Namely, if you have an eigenvator $|\psi_n\rangle$, then you have
$$ i\hbar \partial_t |\psi_n(t)\rangle = H |\psi_n(t)\rangle = E_n|\psi(t)\rangle$$
which can be solved to
$$ |\psi_n(t)\rangle = e^{-\frac{i}{\hbar} E_nt} |\psi(0)\rangle $$
To summarize, the eigenvalues of an operator tells you something about what happens when you perform measurements, but in addition, the eigenvalues of the energy operator help you solve the equations of motion.
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The reason why it is the eigenvalues of the Hamiltonian and not some other operator that will give you the energy states is that in classical Mechanics, the Hamiltonian function is just the energy of your system, expressed as a function of position $x$ and momentum $p$. As a simple example, the Hamiltonian for a harmonic oscillator is $$H(x,p) = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ Note that this really is just the sum of kinetic and potential energy, so we could write $$H(x,p) = E$$.
To get to quantum mechanics, one now performs what is called canonical quantization. There is no mathematically rigorous reason why this will give you a correct quantum mechanics. Since quantum isn't classical, we cannot really expect to find a seamless and watertight derivation of the former from the latter. To my knowledge, this approach has, however, always given correct results.
So, in canonical quantization, what one does is to replace the variables of the Hamiltonian, i.e., $x$ and $p$, with their operator versions, $\hat x$ and $\hat p$. Now we cannot simply write $H(x,p) = E$ anymore, since the energy is a scalar, but the Hamiltonian $H$ is now an operator. Operators are functions that take a wavefunction and modify it in some way and give you a new wavefunction. Now, another postulate of quantum mechanics is that you get the expectation value of an operator $\hat A$ in a given state $\Psi$ by calculation the integral $$ \int dx \Psi^*(x) \hat A \Psi(x) $$ Hence, we get the expectation value of the energy by calculating $$ \int dx \Psi^*(x) H \Psi(x)$$ Obviously, if $H\Psi(x) = E\Psi(x)$, then the expectation value yields $E$, and it is not hard to show that for such an eigenstate, the variance of $E$ will be $0$, i.e. every measurement of the energy in state $\Psi$ will yield the same value $E$.
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A transformation from one set to another can be regarded as a matrix if we define a particular basis. Likewise, an operator can be thought of as a matrix. What is the matrix equation that relates eigenvalues and eigenvectors? You are solving an eigenvalue problem when you are solving the time independent Schrodinger equation.
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The basic experimental fact the inventors of QM had to deal with was the uncertainty principle. The mathematics behind this principle has two major parts, one involving linear algebra and another involving Fourier analysis.
In other words, the operator algebra of QM is necessary in order to have a theory which obeys the uncertainty principle, and if you want to know why this is true, you have to study the mathematics.
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I think you should specify you answer. – Self-Made Man Nov 14 '13 at 14:29
The physics of this is the DeBroglie relation for particles, which relates the energy to the frequency of some wave. The energy of a photon is the frequency of the emitted electromagnetic wave.
When a quantum mechanical atom is weakly interacting with the photon field, and goes from a state with frequency f to a state with frequency f', it can only emit photons with frequency f-f'. The reason is that the transition process is only resonant with waves of frequency equal to the beat frequency \Delta f= f-f'. The atomic relative phases during the transition process recurs with time $1\over \Delta f$, and for any outgoing wave whose frequency does not match this, the process will be cancelling at long times, and no wave will be emitted.
This means that atomic transitions from f to f' are accompanied by a loss of energy of $h\Delta f$, so that one must identify the frequency with the energy in general quantum systems. The Schrodinger waves of definite frequency are the solutions of the time independent problem, since when
$$i{d\over dt} \psi = H \psi $$
and $H\psi = E\psi$, that is, if $\psi$ is an eigenvector of H, then $\psi(t) = e^{-iEt} \psi(0)$, so the time dependence of the wave has a definite frequency. I am giving a physical argument here, because the notion that energy is frequency is engrained into the foundation of quantum mechanics, and it is hard to argue that it is true using a formalism built upon this as a foundation.
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be3f8154951f5f94 | GPGPU with WebGL: solving Laplace’s equation
This is the first post in what will hopefully be a series of posts exploring how to use WebGL to do GPGPU (General-purpose computing on graphics processing units). In this installment we will solve a partial differential equation using WebGL, the Laplace’s equation more specifically.
Discretizing the Laplace’s equation
The Laplace’s equation, \nabla^2 \phi = 0, is one of the most ubiquitous partial differential equations in physics. It appears in lot of areas, including electrostatics, heat conduction and fluid flow.
To get a numerical solution of a differential equation, the first step is to replace the continuous domain by a lattice and the differential operators with their discrete versions. In our case, we just have to replace the Laplacian by its discrete version:
\displaystyle \nabla^2 \phi(x) = 0 \rightarrow \frac{1}{h^2}\left(\phi_{i-1\,j} + \phi_{i+1\,j} + \phi_{i\,j-1} + \phi_{i\,j+1} - 4\phi_{i\,j}\right) = 0,
where h is the grid size.
If we apply this equation at all internal points of the lattice (the external points must retain fixed values if we use Dirichlet boundary conditions) we get a big system of linear equations whose solution will give a numerical approximation to a solution of the Laplace’s equation. Of the various methods to solve big linear systems, the Jacobi relaxation method seems the best fit to shaders, because it applies the same expression at every lattice point and doesn’t have dependencies between computations. Applying this method to our linear system, we get the following expression for the iteration:
\displaystyle \phi_{i\,j}^{(k+1)} = \frac{1}{4}\left(\phi_{i-1\,j}^{(k)} + \phi_{i+1\,j}^{(k)} + \phi_{i\,j-1}^{(k)} + \phi_{i\,j+1}^{(k)}\right),
where k is a step index.
Solving the discretized problem using WebGL shaders
If we use a texture to represent the domain and a fragment shader to do the Jacobi relaxation steps, the shader will follow this general pseudocode:
1. Check if this fragment is a boundary point. If it’s one, return the previous value of this point.
2. Get the four nearest neighbors’ values.
3. Return the average of their values.
To flesh out this pseudocode, we need to define a specific representation for the discretized domain. Taking into account that the currently available WebGL versions don’t support floating point textures, we can use 32 bits RGBA fragments and do the following mapping:
R: Higher byte of \phi.
G: Lower byte of \phi.
B: Unused.
A: 1 if it’s a boundary value, 0 otherwise.
Most of the code is straightforward, but doing the multiprecision arithmetic is tricky, as the quantities we are working with behave as floating point numbers in the shaders but are stored as integers. More specifically, the color numbers in the normal range, [0.0, 1.0], are multiplied by 255 and rounded to the nearest byte value when stored at the target texture.
My first idea was to start by reconstructing the floating point numbers for each input value, do the required operations with the floating numbers and convert the floating point numbers to color components that can be reliably stored (without losing precision). This gives us the following pseudocode for the iteration shader:
// wc is the color to the "west", ec is the color to the "east", ...
float w_val = wc.r + wc.g / 255.0;
float e_val = ec.r + ec.g / 255.0;
// ...
float val = (w_val + e_val + n_val + s_val) / 4.0;
float hi = val - mod(val, 1.0 / 255.0);
float lo = (val - hi) * 255.0;
fragmentColor = vec4(hi, lo, 0.0, 0.0);
The reason why we multiply by 255 in place of 256 is that we need val_lo to keep track of the part of val that will be lost when we store it as a color component. As each byte value of a discrete color component will be associated with a range of size 1/255 in its continuous counterpart, we need to use the “low byte” to store the position of the continuous component within that range.
Simplifying the code to avoid redundant operations, we get:
float val = (wc.r + ec.r + nc.r + sc.r) / 4.0 +
(wc.g + ec.g + nc.g + sc.g) / (4.0 * 255.0);
float lo = (val - hi) * 255.0;
The result of running the full code, implemented in GLSL, is:
Solving the Laplace's equation using a 32x32 grid. Click the picture to see the live solving process (if your browser supports WebGL).
As can be seen, it has quite low resolution but converges fast. But if we just crank up the number of points, the convergence gets slower:
Incompletely converged solution in a 512x512 grid. Click the picture to see a live version.
How can we reconcile these approaches?
The basic idea behind multigrid methods is to apply the relaxation method on a hierarchy of increasingly finer discretizations of the problem, using in each step the coarse solution obtained in the previous grid as the “starting guess”. In this mode, the long wavelength parts of the solution (those that converge slowly in the finer grids) are obtained in the first coarse iterations, and the last iterations just add the finer parts of the solution (those that converge relatively easily in the finer grids).
The implementation is quite straightforward, giving us fast convergence and high resolution at the same time:
Multigrid solution using grids from 8x8 to 512x512. Click the picture to see the live version.
It’s quite viable to use WebGL to do at least basic GPGPU tasks, though it is, in a certain sense, a step backward in time, as there is no CUDA, floating point textures or any feature that helps when working with non-graphic problems: you are on your own. But with the growing presence of WebGL support in modern browsers, it’s an interesting way of partially accessing the enormous computational power present in modern video cards from any JS application, without requiring the installation of a native application.
In the next posts we will explore other kinds of problem-solving where WebGL can provide a great performance boost.
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5 thoughts on “GPGPU with WebGL: solving Laplace’s equation
1. Evgeny says:
Very nice application. There are floating point textures in the nightly Chrome (for about 2 months)
There is “The Energy2D Simulator” open source Java based project
with very nice turbulent flows (3-5 applets). They used implicit scheme and relaxation. You could move in this directions too :)
• mchouza says:
You can see a more complex example of the same techniques in this (not very accurate and still unfinished) simulation of the two slits experiment with the Schrödinger equation:
In my next posts I will probably transition to floating point textures for this kind of simulations, as working with the combination of integer textures and floating point values in the shaders is quite painful :-D
Thanks for your comment and your very interesting website!
2. […] This is very cool indeed — GPGPU with WebGL: solving Laplace’s equation […]
3. […] In a previous post we solved Laplace’s Equation using WebGL. We will see how to implement the Lattice Boltzmann algorithm using WebGL shaders in the next post, but this post has a preview of the solution: Click on the image to go to the demo. New obstacles can be created by dragging the mouse over the simulation area. […]
4. […] method is introduced with WebGL demos in this blog. Demidov wrote something about Multigrid recently. Real-Time Gradient-Domain Painting is an […]
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c544c02bc00d3fa0 | Quantum Mechanics I: Wave Functions
Wave functions: The wave function for a particle contains all of the information about that particle. If the particle moves in one dimension in the presence of a potential energy function U(x), the wave function \Psi(x,t) obeys the one-dimensional Schrödinger equation: -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}+U(x)\Psi(x,t)=i\hbar\frac{\partial\Psi(x,t)}{\partial t}. (For a free particle on which no forces act, U(x)=0.) The quantity |\Psi(x,t)|^2, called the probability distribution function, determines the relative probability of finding a particle near a given position at a given time. If the particle is in a state of definite energy, called a stationary state, \Psi(x,t) is a product of a function \psi(x) that depends on only spatial coordinates and a function e^{-iEt/\hbar} that depends on only time: \Psi(x,t)=\psi(x)e^{iEt/\hbar}. For a stationary state, the probability distribution function is independent of time.
A spatial stationary-state wave function \psi(x) for a particle that moves in one dimension in the presence of a potential-energy function U(x) satisfies the time-independent Schrödinger equation: -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+U(x)\psi(x)=E\psi(x). More complex wave functions can be constructed by super-imposing stationary-state wave functions. These can represent particles that are localized in a certain region, thus representing both particle and wave aspects.
Particles Behaving as Waves
De Broglie waves and electron diffraction: Electrons and other particles have wave properties. A particle’s wavelength depends on its momentum in the same way as for photons: \lambda=\frac hp=\frac h{mv}, E=hf. A non-relativistic electron accelerated from rest through a potential difference V_{ba} has a wavelength \lambda=\frac hp=\frac h{\sqrt{2meV_{ba}}}. Electron microscopes use the very small wavelengths of fast-moving electrons to make images with resolution thousands of times finer than is possible with visible light.
The nuclear atom: The Rutherford scattering experiments show that most of an atom’s mass and all of its positive charge are concentrated in a tiny, dense nucleus at the center of the atom.
Atomic line spectra and energy levels: The energies of atoms are quantized: They can have only certain definite values, called energy levels. When an atom makes a transition from an energy level E_i to a lower level E_f, it emits a photon of energy E_i-E_f: hf=\frac{hc}{\lambda}=E_i-E_f. The same photon can be absorbed by an atom in the lower energy level, which excites the atom to the upper level.
The Bohr model: In the Bohr model of the hydrogen atom, the permitted values of angular momentum are integral multiples of h/2\pi: L_n=mv_nr_n=n\frac{h}{2\pi}, (n=1,2,3,\ldots). The integer multiplier n is called the principal quantum number for the level. The orbital radii are proportional to n^2: r_n=\epsilon_0\frac{n^2h^2}{\pi me^2}=n^2a_0, v_n=\frac{1}{\epsilon_0}\frac{e^2}{2nh}. The energy levels of the hydrogen atoms are given by E_n=-\frac{hcR}{n^2}=-\frac{13.60\,\mathrm{eV}}{n^2}, (n=1,2,3,\ldots), where R is the Rydberg constant.
The laser: The laser operates on the principle of stimulated emission, by which many photons with identical wavelength and phase are emitted. Laser operation requires a nonequilibrium condition called population inversion, in which more atoms are in a higher-energy state than are in a lower-energy state.
Blackbody radiation: The total radiated intensity (average power radiated per area) from a blackbody surface is proportional to the fourth power of the absolute temperature T: I=\sigma T^4 (Stefan-Boltzmann law). The quantity \sigma=5.67\times 10^{-8}\,\mathrm{W/m^2\cdot K^4} is called the Stefan-Boltzmann constant. The wavelength \lambda_m at which a blackbody radiates most strongly is inversely proportional to T: \lambda_mT=2.90\times 10^{-3}\,\mathrm{m\cdot K} (Wien displacement law). The Planck radiation law gives the spectral emittance I(\lambda) (intensity per wavelength interval in blackbody radiation): I(\lambda)=\frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)}.
The Heisenberg uncertainty principle for particles: The same uncertainty considerations that apply to photons also apply to particles such as electrons. The uncertainty \Delta E in the energy of a state that is occupied for a time \Delta t is given by equation \Delta t\Delta E\geq\hbar/2.
Photons: Light Waves behaving as Particles
Photons: Electromagnetic radiation behaves as both waves and particles. The energy in an electromagnetic wave is carried in units called photons. The energy E of one photon is proportional to the wave frequency f and inversely proportional to the wavelength \lambda, and is proportional to a universal quantity h called Planck’s constant: E=hf=\frac{hc}{\lambda}. The momentum of a photon has magnitude E/c: p=\frac Ec=\frac{hf}c=\frac h{\lambda}.
The photo-electric effect: In the photo-electric effect, a surface can eject an electron by absorbing a photon whose energy hf is greater than or equal to the work function \phi of the material. The stopping potential V_0 is the voltage required to stop a current of ejected electrons from reaching an anode: eV_0=hf-\phi.
Photon production, photon scattering, and pair production: X rays can be produced when electrons accelerated to high kinetic energy across a potential increase V_{AC} strike a target. The photon model explains why the maximum frequency and minimum wavelength produced are given by the equation: eV_{AC}=hf_{\max}=\frac{hc}{\lambda_{\min}} (bremsstrahlung). In Compton scattering a photon transfers some of its energy and momentum to an electron with which it collides. For free electrons (mass m), the wavelengths of incident and scattered photons are related to the photon scattering angle \phi: \lambda'-\lambda=\frac{h}{mc}(1-\cos\phi) (Compton scattering). In pair production a photon of sufficient energy can disappear and be replaced by electron-positron pair. In the inverse process, an electron and positron can annihilate and be replaced by a pair of photons.
The Heisenberg uncertainty principle: It is impossible to determine both a photon’s position and its momentum at the same time to arbitrarily high precision. The precision of such measurements for the x-components is limited by the Heisenberg uncertainty principle, \Delta x\Delta p_x\geq\hbar/2; there are corresponding relationships for the y– and z-components. The uncertainty \Delta E in the energy of a state that is occupied for a time \Delta t is given by equation \Delta t\Delta E\geq\hbar/2. In these expressions, \hbar=h/2\pi.
Invariance of physical laws, simultaneity: All of the fundamental laws of physics have the same form in all inertial frames of reference. The speed of light in vacuum is the same in all inertial frames and is independent of the motion of the source. Simultaneity is not an absolute concept; events that are simultaneous in one frame are not necessarily simultaneous in a second frame moving relative to the first.
Time dilation: If two events occur at the same space point in a particular frame of reference, the time interval \Delta t_0 between the events as measured in that frame is called a proper time interval. If this frame moves with constant velocity u relative to a second frame, the time interval \Delta t between the events as observed in the second frame is longer than \Delta t_0: \Delta t=\frac{\Delta t_0}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma\Delta t_0, \gamma=\frac1{\sqrt{1-u^2/c^2}}.
Length contraction: If two points are at rest in a particular frame of reference, the distance l_0 between the points as measured in that frame is called a proper length. If this frame moves with constant velocity u relative to a second frame and the distances are measured parallel to the motion, the distance l between the points as measured in the second frame is shorter than l_0. l=l_0\sqrt{1-\frac{u^2}{c^2}}=\frac{l_0}{\gamma}.
The Lorentz transformation: The Lorentz coordinate transformations relate the coordinates and time of an event in an inertial frame S to the coordinates and time of the same event as observed in a second inertial frame S' moving at velocity u relative to the first. For one-dimensional motion, a particle’s velocities v_x in S and v_x' in S' are related by the Lorentz velocity transformation. x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}=\gamma(x-ut), y'=y, z'=z, t'=\frac{t-ux/c^2}{\sqrt{1-u^2/c^2}}=\gamma(t-ux/c^2), v_x'=\frac{v_x-u}{1-uv_x/c^2}, v_x=\frac{v_x'+u}{1+uv_x'/c^2}.
The Doppler effect for electromagnetic waves: The Doppler effect is the frequency shift in light from a source due to the relative motion of source and observer. For a source moving toward the observer with speed u, the received frequency f in terms of the emitted frequency f_0 is f=\sqrt{\frac{c+u}{c-u}}f_0.
Relativistic momentum and energy: For a particle of rest mass m moving with velocity \vec{v}, the relativistic momentum is \vec{p}=\frac{m\vec{v}}{\sqrt{1-v^2/c^2}}=\gamma m\vec{v}, the relativistic kinetic energy is K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2=(\gamma-1)mc^2. The total energy E is the sum of the kinetic energy and the rest energy mc^2: E=K+mc^2=\frac{mc^2}{\sqrt{1-v^2/c^2}}=\gamma mc^2. Also, E^2=(mc^2)^2+(pc)^2.
Interference and coherent sources: Monochromatic light is light with a single frequency. Coherence is a definite, unchanging phase relationship between two waves. The overlap of waves from two coherent sources of monochromatic light forms an interference pattern. The principle of superposition states that the total wave disturbance at any point is the sum of the disturbances from the separate waves.
Two-source interference of light: When two sources are in phase, constructive interference occurs where the difference in path length from the two sources is zero or an integer number of wavelengths; destructive interference occurs where path difference is a half-integer number of wavelengths. If two sources separated by a distance d are both very far from a point P, and the line from the sources make an angle \theta with the line perpendicular to the line of the sources, then the condition for constructive interference at P is d\sin\theta=m\lambda, (m=0,\pm 1,\pm 2,\ldots). The condition for destructive interference is d\sin\theta=(m+\frac12)\lambda, (m=0,\pm1,\pm2,\ldots). When \theta is very small, the position y_m of the mth bright fringe on a screen located at distance R from the sources is: y_m=R\frac{m\lambda}d, (m=0,\pm1,\pm2,\ldots).
Intensity in interference pattern: When two sinusoidal waves with equal amplitude E and phase difference \phi are superimposed, the resultant amplitude E_P and intensity I are as follows: E_P=2E|\cos\frac{\phi}2|, I=I_0\cos^2\frac{\phi}2. If the two sources emit in phase, the phase difference \phi at a point P (located a distance r_1 from source 1 and a distance r_2 from source 2) is directly proportional to the path difference r_2-r_1: \phi=\frac{2\pi}{\lambda}(r_2-r_1)=k(r_2-r_1).
Interference in thin films: When light is reflected from both sides of a thin film of thickness t and no phase shift occurs at either surface, constructive interference between the reflected waves occurs when 2t is equal to an integral number of wavelengths. If a half-cycle phase shift occurs at one surface, this is the condition for destructive interference. A half-cycle phase shift occurs during reflection whenever the index of refraction in the second material is greater than in the first.
Michelson interferometer: The Michelson interferometer uses a monochromatic light source and can be used for high-precision measurements of wavelengths. Its original purpose was to detect motion of the earth relative to a hypothetical ether, the supposed medium for electromagnetic waves. The ether has never been detected, and the concept has been abandoned; the speed of light is the same relative to all observers. This is part of the foundation of the special theory of relativity. |
ceea68c15b1bb13c | De Broglie–Bohm theory
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The de Broglie–Bohm theory, also known as the pilot-wave theory, Bohmian mechanics, the Bohm or Bohm's interpretation, and the causal interpretation, is an interpretation of quantum theory. In addition to a wavefunction on the space of all possible configurations, it also postulates an actual configuration that exists even when unobserved. The evolution over time of the configuration (that is, of the positions of all particles or the configuration of all fields) is defined by the wave function via a guiding equation. The evolution of the wave function over time is given by Schrödinger's equation. The theory is named after Louis de Broglie (1892–1987), and David Bohm (1917–1992).
The theory is deterministic[1] and explicitly nonlocal: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of the system given by its wavefunction; the latter depends on the boundary conditions of the system, which in principle may be the entire universe.
The theory was historically developed by de Broglie in the 1920s, who in 1927 was persuaded to abandon it in favour of the then-mainstream Copenhagen interpretation. David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory in 1952. Bohm's suggestions were not widely received then, partly due to reasons unrelated to their content, connected to Bohm's youthful communist affiliations.[2] De Broglie–Bohm theory was widely deemed unacceptable by mainstream theorists, mostly because of its explicit non-locality. Bell's theorem (1964) 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. Since the 1990s, there has been renewed interest in formulating extensions to de Broglie–Bohm theory, attempting to reconcile it with special relativity and quantum field theory, besides other features such as spin or curved spatial geometries. [3]
The Stanford Encyclopedia of Philosophy article on Quantum decoherence (Guido Bacciagaluppi, 2012) groups "approaches to quantum mechanics" into five groups, of which "pilot-wave theories" are one (the others being the Copenhagen interpretation, objective collapse theories, many-world interpretations and modal interpretations).
There are several equivalent mathematical formulations of the theory and it is known by a number of different names. The de Broglie wave has a macroscopic analogy termed Faraday wave.[4]
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
This already completes the specification of the theory for any quantum theory with Hamilton operator of type H=\sum \frac{1}{2m_i}\hat{p}_i^2 + V(\hat{q}).
• 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.
Notably, even if this latter relation is frequently presented as an axiom of the theory, in Bohm's original papers of 1952 it was presented as derivable from statistical-mechanical arguments. This argument was further supported by the work of Bohm in 1953 and was substantiated by Vigier and Bohm's paper of 1954 in which they introduced stochastic fluid fluctuations that drive a process of asymptotic relaxation from quantum non-equilibrium to quantum equilibrium (ρ → |ψ|2).[5]
Double-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.[6]
The double-slit experiment is an illustration of wave-particle duality. In it, a beam of particles (such as electrons) travels through a barrier which has two slits. If one puts a detector screen on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen).
If we modify this experiment so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. We can also arrange to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When we do that, the interference pattern disappears.
The Copenhagen interpretation states that the particles are not localised in space until they are detected, so that, if there is not any detector on the slits, there is no information about which slit the particle has passed through. If one slit has a detector on it, then the wavefunction collapses due to that detection.
In de Broglie–Bohm theory, the wavefunction is defined at 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. In Bohm's 1952 papers he used the wavefunction to construct a quantum potential that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wave function interferes with itself and guides the particles via the quantum potential 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.
To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space.
The theory[edit]
The ontology[edit]
The ontology of de Broglie-Bohm theory consists of a configuration q(t)\in Q of the universe and a pilot wave \psi(q,t)\in\mathbb{C}. The configuration space Q can be chosen differently, as in classical mechanics and standard quantum mechanics.
Thus, the ontology of pilot wave theory contains as the trajectory q(t)\in Q we know from classical mechanics, as the wave function \psi(q,t)\in\mathbb{C} of quantum theory. So, at every moment of time there exists not only a wave function, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrodinger equation). The correspondence to our experiences is made by the identification of the configuration of our brain with some part of the configuration of the whole universe q(t)\in Q, as in classical mechanics.
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;[7] 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 via the quatum potential.[8] 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.[9][10]
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".[11] 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".[12] 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.[13]
In what follows below, we will give the setup for one particle moving in \mathbb{R}^3 followed by the setup for N particles moving in 3 dimensions. In the first instance, configuration space and real space are the same while in the second, real space is still \mathbb{R}^3, but configuration space becomes \mathbb{R}^{3N}. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space which is how particles are entangled with each other in this theory.
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
\frac{d \mathbf{Q}}{dt} (t) = \frac{\hbar}{m} \operatorname{Im} \left(\frac{\nabla \psi}{\psi} \right) (\mathbf{Q}, t).
For many particles, we label them as \mathbf{Q}_k for the kth particle and their velocities are given by
\frac{d \mathbf{Q}_k}{dt} (t) = \frac{\hbar}{m_k} \operatorname{Im} \left(\frac{\nabla_k \psi}{\psi} \right) (\mathbf{Q}_1, \mathbf{Q}_2, \ldots, \mathbf{Q}_N, t).
The main fact to notice is that this velocity field depends on the actual positions of all of the N particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe.
Schrödinger's equation[edit]
The one particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on \mathbb{R}^3. The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function V on \mathbb{R}^3:
For many particles, the equation is the same except that \psi and V are now on configuration space, \mathbb{R}^{3N}.
i\hbar\frac{\partial}{\partial t}\psi=-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}\nabla_k^2\psi + V\psi
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.,[14] 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.
The situation is thus analogous to the situation in classical statistical physics. A low entropy initial condition will, with overwhelmingly high probability, evolve into a higher entropy state: behavior consistent with the second law of thermodynamics is typical. There are, of course, anomalous initial conditions which would give rise to violations of the second law. However, in the absence of some very detailed evidence supporting the actual realization of one of those special initial conditions, it would be quite unreasonable to expect anything but the actually observed uniform increase of entropy. Similarly, in the de Broglie–Bohm theory, there are anomalous initial conditions which would produce measurement statistics in violation of the Born rule (i.e., in conflict with the predictions of standard quantum theory). But the typicality theorem shows that, in the absence of some specific reason to believe that one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect.
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. .[15] 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). It should however be noted that the "universe" is simply the system limited by the same boundary conditions used to solve the Schrodinger 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:
\psi^{\mathrm I}(t,q^{\mathrm I})=\psi(t,q^{\mathrm I},Q^{\mathrm{II}}(t)). \,
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.[16] this fact is called the fundamental conditional probability formula).
Unlike the universal wave function, the conditional wave function of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does. For instance, if the universal wave function factors as:
\psi(t,q^{\mathrm I},q^{\mathrm{II}})=\psi^{\mathrm I}(t,q^{\mathrm I})\psi^{\mathrm{II}}(t,q^{\mathrm{II}}) \,
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:
\psi(t,q^{\mathrm I},q^{\mathrm{II}})=\psi^{\mathrm I}(t,q^{\mathrm I})\psi^{\mathrm{II}}(t,q^{\mathrm{II}})+\phi(t,q^{\mathrm I},q^{\mathrm{II}}), \,
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.
The fact that the conditional wave function of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of Standard Quantum Theory emerges from the Bohmian formalism when one considers conditional wave functions of subsystems.
Pilot wave theory is explicitly nonlocal, which is in ostensible conflict with special relativity. Various extensions of "Bohm-like" mechanics exist that attempt to resolve this problem. Bohm himself in 1953 presented an extension of the theory satisfying Dirac equation for a single particle. However, this was not extensible to the many-particles case because it used an absolute time.[17] A renewed interest in constructing Lorentz-invariant extensions of Bohmian theory arose in the 1990s; see Bohm and Hiley: The Undivided Universe, and [2], [3], 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.
Thus, Dürr et al. (1999) showed that it is possible to formally restore Lorentz invariance for the Bohm-Dirac theory by introducing additional structure. This approach still requires a foliation of space-time. While this is in conflict with the standard interpretation of relativity, the preferred foliation, if unobservable, does not lead to any empirical conflicts with relativity.
The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles. On the other hand, in the theory of relativity the concept of instantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal 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]
Chris Dewdney and G. Horton have proposed a relativistically covariant, wave-functional formulation of Bohm's quantum field theory[24][25] and has extended it to a form that allows the inclusion of gravity.[26]
Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions.[27] He has developed a generalized relativistic-invariant probabilistic interpretation of quantum theory,[28][29][30] 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.[31]
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.
\frac{d \mathbf{Q}_k}{dt} (t) &= \frac{\hbar}{m_k} Im \left(\frac{(\psi,D_k \psi)}{(\psi,\psi)} \right) (\mathbf{Q}_1, \mathbf{Q}_2, \ldots, \mathbf{Q}_N, t) \\
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),
D_k = \nabla_k-\frac{ie_k}{c\hbar}\mathbf{A}(\mathbf{q}_k),
\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,
(\phi,\psi) = \sum_{s=1}^d \phi_s^* \psi_s.
For an example of a spin space, a system consisting of two spin 1/2 particle and one spin 1 particle has a wavefunctions of the form
\psi: \mathbb{R}^{9}\times \mathbb{R} \to \mathbb{C}^{2}\otimes \mathbb{C}^{2} \otimes \mathbb{C}^{3}.
That is, its spin space is a 12 dimensional space.
Quantum field theory[edit]
In Dürr et al.,[32][33] 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.
Hrvoje Nikolić[28] introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, according to which particle trajectories are continuous, but particle detectors behave as if particles have been created or destroyed even when a true creation or destruction of particles does not take place.
Curved space[edit]
To extend de Broglie–Bohm theory to curved space (Riemannian manifolds in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as gradients and Laplacians. Thus, we use equations that have the same form as above. Topological and boundary conditions may apply in supplementing the evolution of Schrödinger's equation.
For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.[34]
Exploiting nonlocality[edit]
Antony Valentini[35] has extended the de Broglie–Bohm theory to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but it has the virtue that it makes the parallel universes of the chaotic inflation theory observable in principle.
Unlike de Broglie–Bohm theory, Valentini's theory has the wavefunction evolution also depend on the ontological variables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heat death". The resulting theory becomes nonlinear and non-unitary.
Below are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimental results agree with all of the standard predictions of quantum mechanics in so far as the latter has predictions. However, while standard quantum mechanics is limited to discussing the results of 'measurements', de Broglie–Bohm theory is a theory which governs the dynamics of a system without the intervention of outside observers (p. 117 in Bell[36]).
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[14] 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]
According to ordinary quantum theory, it is not possible to measure the spin or polarization of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way. For an ensemble of particles, if we expect the particles to be aligned, the results are all 1. If we expect them to be aligned oppositely, the results are all -1. For other alignments, we expect some results to be 1 and some to be -1 with a probability that depends on the expected alignment. For a full explanation of this, see the Stern-Gerlach Experiment.
In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible[37] to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin up while in the other setup it registers as spin down. Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle—instead spin is, so to speak, in the wave function of the particle in relation to the particular device being used to measure the spin. This is an illustration of what is sometimes referred to as contextuality, and is related to naive realism about operators.[38]
Measurements, the quantum formalism, and observer independence[edit]
De Broglie–Bohm theory gives the same results as quantum mechanics. It treats the wavefunction as a fundamental object in the theory as the wavefunction describes how the particles move. This means that no experiment can distinguish between the two theories. This section outlines the ideas as to how the standard quantum formalism arises out of quantum mechanics. References include Bohm's original 1952 paper and Dürr et al.[14]
Collapse of the wavefunction[edit]
De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation. Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial |\psi|^2 distribution for the particles in the system (see the section on the conditional wave function of a subsystem for details).
It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, the conditional wavefunction of the system evolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction of the system appears to be in a superposition of distinct states. But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results.
Collapse of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed by Schrödinger's equation and the particles' evolutions are governed by the guiding equation. Collapse only occurs in a phenomenological way for systems that seem to follow their own Schrödinger's equation. As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include and this will affect when "collapse" occurs.
Operators as observables[edit]
In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem.[39] A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction.
In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.
There have also been claims that experiments reject the Bohm trajectories [4] in favor of the standard QM lines. But as shown in [5] and [6], such experiments cited above only disprove a misinterpretation of the de Broglie–Bohm theory, not the theory itself.
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.
Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie–Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al.[40] for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators.
Hidden variables[edit]
De Broglie–Bohm theory is often referred to as a "hidden variable" theory. Bohm used this description in his original papers on the subject, writing, "From the point of view of the usual interpretation, these additional elements or parameters [permitting a detailed causal and continuous description of all processes] could be called 'hidden' variables." Bohm and Hiley later stated that they found Bohm's choice of the term "hidden variables" to be too restrictive. In particular, they argued that a particle is not actually hidden but rather "is what is most directly manifested in an observation [though] its properties cannot be observed with arbitrary precision (within the limits set by uncertainty principle)."[41] However, others nevertheless treat the term "hidden variable" as a suitable description.[42]
Generalized particle trajectories can be extrapolated from numerous weak measurements on an ensemble of equally prepared systems, and such trajectories coincide with the de Broglie–Bohm trajectories, and thus may seem to be evidence of the existence of the otherwise "hidden variables". However, the results of the weak measurements are also consistent with many other interpretations, that do not include such trajectories.
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.
In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Each particle has a well-defined trajectory. Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum). It is the lack of knowledge of the particle's trajectory that accounts for the uncertainty relation. What one can know about a particle at any given time is described by the wavefunction. Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived (in the epistemic sense mentioned above), on the de Broglie–Bohm theory.
To put the statement differently, the particles' positions are only known statistically. As in classical mechanics, successive observations of the particles' positions refine the experimenter's knowledge of the particles' initial conditions. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with the normal use of the Schrödinger equation.
For the derivation of the uncertainty relation, see Heisenberg uncertainty principle, noting that it describes it from the viewpoint of the Copenhagen interpretation.
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,[43] 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.[44]
Decades later John Bell proved Bell's theorem (see p. 14 in Bell[36]), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanics must either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce unique results (see counterfactual definiteness and many-worlds interpretation). In particular, Bell proved that any local theory with unique results must make empirical predictions satisfying a statistical constraint called "Bell's inequality".
Alain Aspect performed a series of Bell test experiments that test Bell's inequality using an EPR-type setup. Aspect's results show experimentally that Bell's inequality is in fact violated—meaning that the relevant quantum mechanical predictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles are separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent nonlocality of the effect.
The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments as ordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejected based on this; Bell's attitude was: "It is a merit of the de Broglie–Bohm version to bring this [nonlocality] out so explicitly that it cannot be ignored."[45]
The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. An analysis of exactly what kind of nonlocality is present and how it is compatible with relativity can be found in Maudlin.[46] Note that in Bell's work, and in more detail in Maudlin's work, it is shown that the nonlocality does not allow for signaling at speeds faster than light.
Classical limit[edit]
Bohm's formulation of de Broglie–Bohm theory in terms of a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952. Modern methods of decoherence are relevant to an analysis of this limit. See Allori et al.[47] for steps towards a rigorous analysis.
Quantum trajectory method[edit]
Work by Robert E. Wyatt in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion. At each time-step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. (QuickTime movies of this for H+H2 reactive scattering can be found on the Wyatt group web-site at UT Austin.) This approach has been adapted, extended, and used by a number of researchers in the Chemical Physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A recent (2007) issue of the Journal of Physical Chemistry A was dedicated to Prof. Wyatt and his work on "Computational Bohmian Dynamics".
Eric R. Bittner's group at the University of Houston has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat-capacity of small clusters Nen for n~100.
There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effects lead to the case where R^{-1}\nabla^2R\rightarrow\infty. This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged.
These methods, as does Bohm's Hamilton-Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account.
Occam's razor criticism[edit]
Both Hugh Everett III and Bohm treated the wavefunction as a physically real field. Everett's many-worlds interpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations. When we see the particle detectors flash or hear the click of a Geiger counter then Everett's theory interprets this as our wavefunction responding to changes in the detector's wavefunction, which is responding in turn to the passage of another wavefunction (which we think of as a "particle", but is actually just another wave-packet).[48] No particle (in the Bohm sense of having a defined position and velocity) exists, according to that theory. For this reason Everett sometimes referred to his own many-worlds approach as the "pure wave theory". Talking of Bohm's 1952 approach, Everett says:
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.[50]
Many authors have expressed critical views of the de Broglie-Bohm theory, by comparing it to Everett's many worlds approach. Many (but not all) proponents of the de Broglie-Bohm theory (such as Bohm and Bell) interpret the universal wave function as physically real. According to some supporters of Everett's theory, if the (never collapsing) wave function is taken to be physically real, then it is natural to interpret the theory as having the same many worlds as Everett's theory. In the Everettian view the role of the Bohm particle is to act as a "pointer", tagging, or selecting, just one branch of the universal wavefunction (the assumption that this branch indicates which wave packet determines the observed result of a given experiment is called the "result assumption"[48]); the other branches are designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.[48] H. Dieter Zeh comments on these "empty" branches:
David Deutsch has expressed the same point more "acerbically":[48]
According to Brown & Wallace[48] the de Broglie-Bohm particles play no role in the solution of the measurement problem. These authors claim[48] 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[48] 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[53] 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.
The guiding equation can be derived in a similar fashion. We assume a plane wave: \psi(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}. Notice that i\mathbf{k}= \nabla\psi /\psi. Assuming that \mathbf{p} = m \mathbf{v} for the particle's actual velocity, we have that \mathbf{v}= \frac{\hbar}{m} Im \left(\frac{\nabla\psi}{\psi}\right) . Thus, we have the guiding equation.
Notice that this derivation does not use Schrödinger's equation.
• Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites. It is this method which generalizes to many possible alternative theories. The starting point is the continuity equation -\frac{\partial \rho}{\partial t} = \nabla \cdot (\rho v^{\psi}) for the density \rho=|\psi|^2. This equation describes a probability flow along a current. We take the velocity field associated with this current as the velocity field whose integral curves yield the motion of the particle.
• A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the continuity equation from above and the Hamilton–Jacobi equation. This is the method used by Bohm in 1952. The decomposition and equations are as follows:
Decomposition: \psi(\mathbf{x},t) = R(\mathbf{x},t)e^{i S(\mathbf{x},t) / \hbar}. Note R^2(\mathbf{x},t) corresponds to the probability density \rho (\mathbf{x},t) = |\psi (\mathbf{x},t)|^2.
Continuity Equation: -\frac{\partial \rho(\mathbf{x},t)}{\partial t} = \nabla \cdot \left(\rho (\mathbf{x},t)\frac{\nabla S(\mathbf{x},t)}{m}\right)
Hamilton–Jacobi Equation: \frac{\partial S(\mathbf{x},t)}{\partial t} = -\left[ V + \frac{1}{2m}(\nabla S(\mathbf{x},t))^2 -\frac{\hbar ^2}{2m} \frac{\nabla ^2R(\mathbf{x},t)}{R(\mathbf{x},t)} \right].
The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential V-\frac{\hbar ^2}{2m} \frac{\nabla ^2 R}{R} and velocity field \frac{\nabla S}{m}. The potential V is the classical potential that appears in Schrödinger's equation and the other term involving R is the quantum potential, terminology introduced by Bohm.
This leads to viewing the quantum theory as particles moving under the classical force modified by a quantum force. However, unlike standard Newtonian mechanics, the initial velocity field is already specified by \frac{\nabla S}{m} which is a symptom of this being a first-order theory, not a second-order theory.
• A fourth derivation was given by Dürr et al.[14] 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.
• A fifth derivation, given by Dürr et al.[32] is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator H, the equation to satisfy for all functions f (with associated multiplication operator \hat{f}) is
(v(f))(q) = \mathrm{Re} \frac{(\psi, \frac{i}{\hbar} [H,\hat f] \psi)}{(\psi,\psi)}(q) where (v,w) is the local Hermitian inner product on the value space of the wavefunction.
This formulation allows for stochastic theories such as the creation and annihilation of particles.
• A further derivation has been given by Peter R. Holland, on which he bases the entire work presented in his quantum physics textbook The Quantum Theory of Motion, a main reference book on the de Broglie–Bohm theory. It is based on three basic postulates and an additional fourth postulate that links the wave function to measurement probabilities:[55]
1. A physical system consists in a spatiotemporally propagating wave and a point particle guided by it;
2. The wave is described mathematically by a solution \psi to Schrödinger's wave equation;
3. The particle motion is described by a solution to \mathbf{\dot x}(t) = [\nabla S (\mathbf{x}(t),t))]/m in dependence on initial condition \mathbf{x}(t=0), with S the phase of \psi.
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.
De Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference.
Pilot-wave theory[edit]
Dr. de Broglie presented his pilot wave theory at the 1927 Solvay Conference,[56] 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 manner left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless because he was "discouraged by criticisms which [it] roused."[57] De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as no one understood quantum decoherence at the time. An analysis of de Broglie's presentation is given in Bacciagaluppi et al.[58][59] Also, in 1932 John von Neumann published a paper,[60] that was widely (and erroneously, as shown by Jeffrey Bub[61]) believed to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie's theory for the next two decades.
In 1926, Erwin Madelung had 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.[62] The Madelung equations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohm mechanics[63] and are the basis of the stochastic 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.[64] 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."[65] This entity is the quantum potential.
After publishing a popular textbook on Quantum Mechanics which adhered entirely to the Copenhagen orthodoxy, Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. The result was 'A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It was an independent origination of the pilot wave theory, and extended it to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers that there should be disturbances to this, in the way Brownian motion disturbs Newtonian mechanics). This stage is known as the de Broglie–Bohm Theory in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory of Motion' [Holland 1993].
This stage applies to multiple particles, and is deterministic.
The de Broglie–Bohm theory is an example of a hidden variables theory. Bohm originally hoped that hidden variables could provide a local, causal, objective description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as Schrödinger's cat, the measurement problem and the collapse of the wavefunction. However, Bell's theorem complicates this hope, as it demonstrates that there can be no local hidden variable theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is causal but not local.
Bohm's paper was largely ignored or panned by other physicists. Albert Einstein, who had suggested that Bohm search for a realist alternative to the prevailing Copenhagen approach, did not consider Bohm's interpretation to be a satisfactory answer to the quantum nonlocality question, calling it "too cheap",[66] while Werner Heisenberg considered it a "superfluous 'ideological superstructure' ".[67] 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.[68]
He subsequently described Bohm's theory as "artificial metaphysics".[69]
According to physicist Max Dresden, 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.[2]
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]
This term is used to describe the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the quantum equilibrium hypothesis that the probability follows the Born rule. The term "Bohmian mechanics" is also often used to include most of the further extensions past the spin-less version of Bohm. While de Broglie–Bohm theory has Lagrangians and Hamilton-Jacobi equations as a primary focus and backdrop, with the icon of the quantum potential, Bohmian mechanics considers the continuity equation as primary and has the guiding equation as its icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-less particles. The papers of Dürr et al. popularized the term.
All of non-relativistic quantum mechanics can be fully accounted for in this theory.
Causal interpretation and ontological interpretation[edit]
Bohm developed his original ideas, calling them the Causal Interpretation. Later he felt that causal sounded too much like deterministic and preferred to call his theory the Ontological Interpretation. The main reference is 'The Undivided Universe' [Bohm, Hiley 1993].
This stage covers work by Bohm and in collaboration with Jean-Pierre Vigier and Basil Hiley. Bohm is clear that this theory is non-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not, strictly speaking, a formulation of the de Broglie–Bohm theory. However, it deserves mention here because the term "Bohm Interpretation" is ambiguous between this theory and the de Broglie–Bohm theory.
An in-depth analysis of possible interpretations of Bohm's model of 1952 was given in 1996 by philosopher of science Arthur Fine.[70]
See also[edit]
1. ^ Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden Variables' I". Physical Review 85: 166–179. Bibcode:1952PhRv...85..166B. doi:10.1103/PhysRev.85.166. ("In contrast to the usual interpretation, this alternative interpretation permits us to conceive of each individual system as being in a precisely definable state, whose changes with time are determined by definite laws, analogous to (but not identical with) the classical equations of motion. Quantum-mechanical probabilities are regarded (like their counterparts in classical statistical mechanics) as only a practical necessity and not as an inherent lack of complete determination in the properties of matter at the quantum level.")
2. ^ a b F. David Peat, Infinite Potential: The Life and Times of David Bohm (1997), p. 133. James T. Cushing, Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (1994) discusses "the hegemony of the Copenhagen interpretation of quantum mechanics" over theories like Bohmian mechanics as an example of how the acceptance of scientific theories may be guided by social aspects.
3. ^ David Bohm and Basil J. Hiley, The Undivided Universe - An Ontological Interpretation of Quantum Theory appreared after Bohm's death, in 1993; reviewed by Sheldon Goldstein in Physics Today (1994). J. Cushing, A. Fine, S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory - An Appraisal (1996).
4. ^ John W. M. Bush: Quantum mechanics writ large -
5. ^ Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in: Antony Valentini; Hans Westman (8 January 2005). "Dynamical origin of quantum probabilities". Proc. R. Soc. A 461 (2053): 253–272. arXiv:quant-ph/0403034. Bibcode:2005RSPSA.461..253V. doi:10.1098/rspa.2004.1394. p. 254
6. ^ - Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer
7. ^ David Bohm (1957). Causality and Chance in Modern Physics. Routledge & Kegan Paul and D. Van Nostrand. ISBN 0-8122-1002-6. , p. 117.
8. ^ D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory, p. 37.
9. ^ 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.
10. ^ 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.
12. ^ 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
13. ^ * P. Holland: Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction, Nuovo Cimento B 116, 2001, pp. 1143-1172, full text preprint p. 31)
14. ^ a b c d Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Origin of Absolute Uncertainty", Journal of Statistical Physics 67: 843–907, 1992.
15. ^ Towler, M.D., Russell, N.J., Valentini A., pbs.,"Timescales for dynamical relaxation to the Born rule" quant-ph/11031589
16. ^ Quantum Equilibrium and the Origin of Absolute Uncertainty, D. Dürr, S. Goldstein and N. Zanghì, Journal of Statistical Physics 67, 843-907 (1992),
17. ^ Oliver Passon, What you always wanted to know about Bohmian mechanics but were afraid to ask, Invited talk at the spring meeting of the Deutsche Physikalische Gesellschaft, Dortmund, 2006, arXiv:quant-ph/0611032, p. 13.
18. ^ Dürr, D., Goldstein, S., Münch-Berndl, K., and Zanghì, N., 1999, "Hypersurface Bohm-Dirac Models", Phys. Rev. A 60: 2729–2736.
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
21. ^ Partha Ghose, A.S. Majumdar, S. Guhab, J. Sau: Bohmian trajectories for photons, Physics Letters A 290 (2001), pp. 205–213, 10 November 2001
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]
23. ^ Sacha Kocsis, Boris Braverman, Sylvain Ravets, Martin J. Stevens, Richard P. Mirin, L. Krister Shalm, Aephraim M. Steinberg: Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer, Science, vol. 332 no. 6034 pp. :1170-1173, 3 June 2011, doi:10.1126/science.1202218 (abstract)
24. ^ Chris Dewdney, George Horton (2002): Relativistically invariant extension of the de Broglie Bohm theory of quantum mechanics, Journal of Physics A: Mathematical and General, 35 (47). pp. 10117-10127. doi:10.1088/0305-4470/35/47/311
25. ^ Chris Dewdney, George Horton (2004): A relativistically covariant version of Bohm's quantum field theory for the scalar field, Journal of Physics A: Mathematical and General, 37 (49). pp. 11935-11943. doi:10.1088/0305-4470/37/49/011
26. ^ Chris Dewdney, George Horton (2010): A relativistic hidden-variable interpretation for the massive vector field based on energy-momentum flows, Foundations of Physics, 40 (6). pp. 658-678. doi:10.1007/s10701-010-9456-9
27. ^ 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 Nikolic, H. 2010 "QFT as pilot-wave theory of particle creation and destruction", Int. J. Mod. Phys. A 25, 1477 (2010)
29. ^ Hrvoje Nikolić: Time in relativistic and nonrelativistic quantum mechanics, arXiv:0811/0811.1905v2 (submitted 12 November 2008 (v1), revised 12 Jan 2009)
30. ^ Hrvoje Nikolić: Making nonlocal reality compatible with relativity, arXiv:1002.3226v2 [quant-ph] (submitted on 17 Feb 2010, version of 31 May 2010)
31. ^ Hrvoje Nikolić: Bohmian mechanics in relativistic quantum mechanics, quantum field theory and string theory, 2007 J. Phys.: Conf. Ser. 67 012035
32. ^ a b Dürr, D., Goldstein, S., Tumulka, R., and Zanghì, N., 2004, "Bohmian Mechanics and Quantum Field Theory", Phys. Rev. Lett. 93: 090402:1–4.
33. ^ Dürr, D., Tumulka, R., and Zanghì, N., J. Phys. A: Math. Gen. 38, R1–R43 (2005), quant-ph/0407116
34. ^ Dürr, D., Goldstein, S., Taylor, J., Tumulka, R., and Zanghì, N., J. "Quantum Mechanics in Multiply-Connected Spaces", Phys. A: Math. Theor. 40, 2997–3031 (2007)
35. ^ Valentini, A., 1991, "Signal-Locality, Uncertainty and the Subquantum H-Theorem. II," Physics Letters A 158: 1–8.
36. ^ a b Bell, John S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0521334950.
37. ^ Albert, D. Z., 1992, Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press
38. ^ Daumer, M., Dürr, D., Goldstein, S., and Zanghì, N., 1997, "Naive Realism About Operators", Erkenntnis 45: 379–397.
39. ^ Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory" Journal of Statistical Physics 116, 959–1055 (2004)
40. ^ Hyman, Ross et al Bohmian mechanics with discrete operators, J. Phys. A: Math. Gen. 37 L547–L558, 2004
43. ^ Bell J. S. (1964). "On the Einstein Podolsky Rosen Paradox" (PDF). Physics 1: 195.
44. ^ 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.
45. ^ Bell, page 115
46. ^ Maudlin, T. (1994). Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics. Cambridge, MA: Blackwell. ISBN 0631186093.
47. ^ Allori, V., Dürr, D., Goldstein, S., and Zanghì, N., 2002, "Seven Steps Towards the Classical World", Journal of Optics B 4: 482–488.
48. ^ a b c d e f g Brown, Harvey R; Wallace, David (2005). "Solving the measurement problem: de Broglie-Bohm loses out to Everett" (PDF). Foundations of Physics 35: 517–540. arXiv:quant-ph/0403094. Bibcode:2005FoPh...35..517B. doi:10.1007/s10701-004-2009-3. Abstract: "The quantum theory of de Broglie and Bohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural home in the Everett interpretation."
49. ^ See section VI of Everett's thesis:Theory of the Universal Wavefunction, pp 3-140 of Bryce Seligman DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 0-691-08131-X
50. ^ Craig Callender: "The Redundancy Argument Against Bohmian Mechanics"
51. ^ 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.
52. ^ David Deutsch, Comment on Lockwood. British Journal for the Philosophy of Science 47, 222228, 1996
53. ^ Valentini A., "De Broglie-Bohm pilot wave theory: many worlds in denial?" 'Many Worlds? Everett, Quantum Theory, and Reality', eds. S. Saunders et al. (Oxford University Press, 2010), pp. 476--509
54. ^ P. Holland, "Hamiltonian Theory of Wave and Particle in Quantum Mechanics I, II", Nuovo Cimento B 116, 1043, 1143 (2001) online
55. ^ Peter R. Holland: The quantum theory of motion, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), ISBN 0-521-48543-6, p. 66 ff.
56. ^ Solvay Conference, 1928, Electrons et Photons: Rapports et Descussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 October 1927 sous les auspices de l'Institut International Physique Solvay
57. ^ Louis be Broglie, in the foreword to David Bohm's Causality and Chance in Modern Physics (1957). p. x.
58. ^ Bacciagaluppi, G., and Valentini, A., Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference
59. ^ See the brief summary by Towler, M., "Pilot wave theory, Bohmian metaphysics, and the foundations of quantum mechanics"
60. ^ von Neumann J. 1932 Mathematische Grundlagen der Quantenmechanik
62. ^ Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Zeit. f. Phys. 40 (3–4): 322–326. Bibcode:1927ZPhy...40..322M. doi:10.1007/BF01400372.
63. ^ Tsekov, Roumen (2012). "Bohmian Mechanics versus Madelung Quantum Hydrodynamics". doi:10.13140/RG.2.1.3663.8245.
64. ^ 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
66. ^ (Letter of 12 May 1952 from Einstein to Max Born, in The Born–Einstein Letters, Macmillan, 1971, p. 192.
67. ^ Werner Heisenberg, Physics and Philosophy (1958), p. 133.
68. ^ 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.
69. ^ Pauli, W. (1953). "Remarques sur le probleme des parametres caches dans la mecanique quantique et sur la theorie de l’onde pilote." In A. George (Ed.), Louis de Broglie—physicien et penseur (pp. 33–42). Paris: Editions Albin Michel.
70. ^ 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
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Foundations for Guided-Wave Optics
Foundations for Guided-Wave Optics
Chin-Lin Chen
ISBN: 978-0-470-04221-2
Sep 2006
425 pages
A classroom-tested introduction to integrated and fiber optics
This text offers an in-depth treatment of integrated and fiber optics, providing graduate students, engineers, and scientists with a solid foundation of the principles, capabilities, uses, and limitations of guided-wave optic devices and systems. In addition to the transmission properties of dielectric waveguides and optical fibers, this book covers the principles of directional couplers, guided-wave gratings, arrayed-waveguide gratings, and fiber optic polarization components.
The material is fully classroom-tested and carefully structured to help readers grasp concepts quickly and apply their knowledge to solving problems. Following an overview, including important nomenclature and notations, the text investigates three major topics:
• Integrated optics
• Fiber optics
• Pulse evolution and broadening in optical waveguides
Each chapter starts with basic principles and gradually builds to more advanced concepts and applications. Compelling reasons for including each topic are given, detailed explanations of each concept are provided, and steps for each derivation are carefully set forth. Readers learn how to solve complex problems using physical concepts and simplified mathematics.
Illustrations throughout the text aid in understanding key concepts, while problems at the end of each chapter test the readers' grasp of the material.
The author has designed the text for upper-level undergraduates, graduate students in physics and electrical and computer engineering, and scientists. Each chapter is self-contained, enabling instructors to choose a subset of topics to match their particular course needs. Researchers and practitioners can also use the text as a self-study guide to gain a better understanding of photonic and fiber optic devices and systems.
1. Brief review of Electromagnetics and Guided Waves.
1.1 Introduction.
1.2 Maxwell's equations.
1.3 Uniform plane waves in isotropic media.
1.4 State of polarization.
1.5 Reflection and refraction by a planar boundary between two dielectric media.
1.5.1. Perpendicular polarization. Reflection and refraction. Total internal reflection.
1.5.2. Parallel polarization. Reflection and refraction. Total internal reflection.
1.6 Guided waves.
1.6.1 TE modes.
1.6.2 TM modes.
1.6.3 Waveguides with constant index regions.
List of Figures.
2. Step-index Thin-film Waveguides.
2.1 Introduction.
2.2 Dispersion of step-index thin-film waveguides.
2.2.1 TE modes.
2.2.2 TM modes.
2.3 Generalized parameters.
2.3.1 a, b, c, d and V.
2.3.2 bV diagram.
2.3.3 Cutoff thickness and cutoff frequencies.
2.3.4 Number of guided modes.
2.3.5 Birefringence in thin-film waveguides.
2.4 Fields of step-index thin-film waveguides.
2.4.1 TE modes.
2.4.2 TM modes.
2.5 Cover and substrate modes.
2.6 Time-average power and confinement factors.
2.6.1 Time-average power transported by TE modes.
2.6.2 Confinement factor of TE modes.
2.6.3 Time-average power transported by TM modes.
2.7 Phase and group velocities.
List of figures.
3. Graded-index Thin-film waveguides.
3.1 Introduction.
3.2 TE modes guided by linearly graded dielectric waveguides.
3.3 Exponentially graded dielectric waveguides.
3.3.1 TE modes.
3.3.2 TM modes.
3.4 WKB method.
3.4.1 Auxiliary function.
3.4.2 Fields in the R Zone.
3.4.3 Fields in the L Zone.
3.4.4 Fields in the transition zone.
3.4.5 The constants.
3.4.6 The dispersion relation.
3.4.7 An example.
3.5 Hocker and Burns’ numerical method.
3.5.1 TE modes.
3.5.2 TM modes.
3.6 Step-index thin-film waveguides vs. graded-index dielectric waveguides.
List of figures.
4. Propagation Loss in Thin-film Waveguides.
4.1 Introduction.
4.2 Complex relative dielectric constant and complex refractive index.
4.3 Propagation loss in step-index waveguides.
4.3.1 Waveguides having weakly absorbing materials.
4.3.2 Metal-clad waveguides.
4.4 Attenuation in thick waveguides with step-index profiles.
4.5 Loss in TM0 mode.
4.6 Metal-clad waveguides with graded index profiles.
List of Figures.
5. Three-dimensional Waveguides with Rectangular Boundaries.
5.1 Fields and modes guided by rectangular waveguides.
5.2 Orders of magnitude of fields.
5.2.1 modes.
5.2.2 modes.
5.3 Marcatili's method.
5.3.1 modes. Expressions for Hx. Boundary conditions along horizontal boundaries, y = ±h/2, |x| Boundary conditions along vertical boundaries, x = ±w/2, |y| Transverse wave vector K,sub>x. Transverse wave vector Ky. Approximate dispersion relation.
5.3.2 modes.
5.3.3 Discussions.
5.3.4 Generalized guide index.
5.4 Effective index method.
5.4.1 A pseudo waveguide.
5.4.2 An alternate pseudo waveguide.
5.4.3 Generalized guide index.
5.5 Comparison of methods.
List of figures.
6. Optical directional couplers and their applications.
6.1 Introduction.
6.2 Qualitative description of the operation of directional couplers.
6.3 Marcatili’s improved coupled mode equations.
6.3.1 Fields of isolated waveguides.
6.3.2 Normal mode fields of the composite waveguide.
6.3.3 Marcatili’s relation.
6.3.4 Approximate normal mode fields.
6.3.5 Improved coupled mode equations.
6.3.6 Coupled mode equation in an equivalent form.
6.3.7 Coupled mode equation in an alternate form.
6.4 Directional couplers with uniform cross section and constant spacing.
6.4.1 Transfer matrix.
6.4.2 Essential characteristics of couplers with K1 = K2 = K.
6.4.3 3 dB directional couplers.
6.4.4 Directional couplers as electrically controlled optical switches.
6.4.5. Switching diagram.
6.5 Switched δβ directional couplers.
6.6 Optical directional couplers filters.
6.6.1 Directional coupler filters with identical waveguides and uniform spacing.
6.6.2 Directional coupler filters with non-identical waveguides and uniform spacing.
6.6.3 Tapered directional coupler filters.
6.7 Intensity modulators based on directional couplers.
6.7.1 Electrooptic properties of lithium niobate.
6.7.2 Dielectric waveguide with an electrooptic layer.
6.7.3 Directional coupler modulator built on a Z-cut LiNbO3 plate.
6.8 Normal mode theory of directional couplers with two waveguides.
6.9 Normal mode theory of directional couplers with three or more waveguides.
List of Figures.
7. Guided-wave Gratings.
7.1 Introduction.
7.1.1 Types of guided-wave gratings. Static gratings. Programmable gratings. Moving grating.
7.1.2 Applications of guided-wave gratings.
7.1.3. Two methods for analyzing guided-wave grating problems.
7.2 Perturbation theory.
7.2.1 Waveguide perturbation.
7.2.2 Fields of perturbed waveguide.
7.2.3 Coupled mode equations and coupling coefficients.
7.2.4 Co-directional coupling.
7.2.5 Contra-directional coupling.
7.3 Coupling coefficient of a rectangular grating-an example.
7.4 Graphical representation of grating equation.
7.5 Grating reflectors.
7.5.1 Coupled mode equations.
7.5.2 Filter response of grating reflectors.
7.5.3 Bandwidth of grating reflectors.
7.6 Distributed feedback lasers.
7.6.1 Coupled mode equations with optical gain.
7.6.2 Boundary conditions and symmetric condition.
7.6.3 Eigen value equations.
7.6.4 Mode patterns.
7.6.5 Oscillation frequency and threshold gain.
List of Figures.
8. Arrayed-waveguide Gratings.
8.1 Introduction.
8.2 Arrays of isotropic radiators.
8.3 Two examples.
8.3.1 Arrayed-waveguide gratings as dispersive components.
8.3.2 Arrayed-waveguide gratings as focusing components.
8.4 1x2 arrayed-waveguide grating multiplexers and demultiplexers.
8.4.1 Waveguide grating elements.
8.4.2 Output waveguides.
8.4.3 Spectral response.
8.5 NxN arrayed-waveguide grating multiplexers and demultiplexers.
8.6 Applications in WDM communications.
List of Figures.
9. Transmission characteristics of step-index optical fibers.
9.1. Introduction.
9.2. Fields and propagation characteristic of modes guided by step-index fibers.
9.2.1 Electromagnetic fields.
9.2.2 Characteristic equation.
9.2.3 Traditional mode designation and fields.
9.3. Linearly polarized modes guided by weakly guiding step-index fibers.
9.3.1 Basic properties of fields of weakly guiding fibers..
9.3.2 Fields and boundary conditions.
9.3.3 Characteristic equation and mode designation.
9.3.4 Fields of x-polarized LP0m modes.
9.3.5 Time-average power.
9.3.6 Single mode operation.
9.4. Phase velocity, group velocity and dispersion of linearly polarized modes.
9.4.1 Phase velocity and group velocity.
9.4.2 Dispersion. Intermodal dispersion. Intramodal dispersion. Zero dispersion wavelengths.
List of Figures.
10. Input and output characteristics of weakly guiding step-index fibers.
10.1 Radiation of LP modes.
10.1.1 Radiated fields in the Fraunhofer zone.
10.1.2 Radiation by a Gaussian aperture field.
10.1.3 Experimental determination of ka and V.
10.2 Excitation of LP modes.
10.2.1 Power coupled to LP mode .
10.2.2 Gaussian beam excitation.
List of Figures.
11. Birefringence in Single-mode Fibers.
11.1 Introduction.
11.2 Geometrical birefringence.
11.3 Birefringence due to build-in stress.
11.4 Birefringence due to externally applied mechanical stress.
11.4.1 Lateral stress.
11.4.2 Bending. Pure bending. Bending under tension.
11.4.3 Mechanical twisting.
11.5 Birefringence due to externally applied electric and magnetic fields.
11.5.1 Strong transverse electric fields.
11.5.2 Strong axis magnetic fields.
11.6 Jones matrices of birefringent fibers.
11.6.1 Linearly birefringent fibers with stationary birefringent axes.
11.6.2 Linearly birefringent fiber with a continuous rotating axis.
11.6.3 Circularly birefringent fibers.
11.6.4 Linearly and circularly birefringent fibers.
11.6.5 Fibers with linear and circular birefringence and axis rotation.
12. Manufactured fibers.
12.1 Introduction.
12.2 Power-law index fibers.
12.3 Key propagation and dispersion parameters of graded index fibers.
12.3.1 Generalized guide index b.
12.3.2 Normalized group delay.
12.3.3 Group delay and the confinement factor.
12.3.4 Normalized waveguide dispersion.
12.3.5 An example.
12.4 Radiation and excitation characteristics of graded index fibers.
12.4.1 Radiation.
12.4.2 Excitation by a linearly polarized Gaussian beam.
12.5 Mode field radius.
12.5.1 Marcuse?s mode field radius.
12.5.2 First Petermann?s mode field radius.
12.5.3 Second Petermann?s mode field radius.
12.5.4 Comparison of three mode field radii.
12.6 Mode field radius and key propagation and dispersion parameters.
List of Figures.
13. Propagation of pulses in single-mode fibers.
13.1 Introduction.
13.2 Dispersion and group velocity dispersion.
13.3 Fourier transform method.
13.4 Propagation of Gaussian pulses in fibers.
13.4.1 Effects of? the first order group dispersion.
13.4.2 Effects of the second order group dispersion.
13.5 Impulse response.
13.5.1 Approximate impulse response function with β" ignored.
13.5.2 Approximate impulse response function with β" ignored.
13.6 Propagation of rectangular pulses in fibers.
13.7 Envelope equation.
13.7.1 Monochromatic waves.
13.7.2 Envelop equation.
13.7.3 Pulse envelop in non-dispersive media.
13.7.4 Effect of the first order group velocity dispersion.
13.7.5 Effect of the second order group velocity dispersion.
13.8 Dispersion compensation.
List of Figures.
14. Optical Solitons in Optical Fibers.
14.1 Introduction.
14.2 Optical Kerr effect in isotropic media.
14.2.1 Electric susceptibility tensor.
14.2.2 Refractive index.
14.3 Nonlinear envelope equation.
14.3.1 Linear and third-order polarizations.
14.3.2 Nonlinear envelope equation for nonlinear media.
14.3.3 Self-phase modulation.
14.3.4 Nonlinear envelope equation for nonlinear fibers.
14.3.5 Nonlinear Schrödinger equation.
14.4 Qualitative description of solitons.
14.5 Fundamental solitons.
14.5.1 Canonical expression.
14.5.2 General expression.
14.5.3 Basic soliton parameters.
14.5.4 Basic soliton properties.
14.6 Higher-order solitons.
14.6.1 Second-order solitons.
14.6.2 Third-order solitons.
14.7 Generation of solitons.
14.7.1 Integer A.
14.7.2 Non-integer A.
14.8 Soliton units of time, distance and power.
14.9 Interaction of solitons.
List of Figures.
Appendix A: Brown Identity.
A.1 Wave equations for inhomogeneous media.
A.2 Brown identity.
A.3 Two special cases.
A.4 Effect of material dispersion.
Appendix B: Two-dimensional Divergence Theorem and Green’s Theorem.
Appendix C. Orthogonality and Orthonormality of Guided Modes.
C.1 Lorentz’ reciprocity.
C.2 Orthogonality of guided modes.
C.3 Orthonormality of guided modes.
Appendix D: Elasticity, Photoelasticity and Electrooptic Effects.
D1 Strain tensors.
D1.1 Strain tensors in one-dimensional objects.
D1.2 Strain tensors in two-dimensional objects.
D1.3 Strain tensors in three-dimensional objects.
D2 Stress tensors.
D3 Hook’s law in isotropic materials.
D4 Strain and stress tensors in abbreviated indices.
D5 Relative dielectric constant tensors and relative dielectric impermeability tensors.
D6 Photoelastic effect and photoelastic constant tensors.
D7 Index change in isotropic solids: an example.
D8 Linear electrooptic effects.
D9 Quadratic electrooptic effects.
List of Figures.
Appendix E: Effect of mechanical twisting on fiber birefringence.
E1. Relative dielectric constant tensor of a twisted medium.
E2. LP modes in weakly guiding, untwisted fibers.
E3. Eigen polarization modes in twisted fibers.
Appendix F: Derivation of (12.7), (12.8) and (12.9).
Appendix G: Two Hankel transform relations.
"…an interesting, well-balanced, useful book, addressing an increasing educational need for works on optical engineering and communications." (CHOICE, June 2007) |
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Next: 2.9 Summary Up: 2 Designing Parallel Algorithms Previous: 2.7 Case Study: Floorplan Optimization
2.8 Case Study: Computational Chemistry
Our third case study, like the first, is from computational science. It is an example of an application that accesses a distributed data structure in an asynchronous fashion and that is amenable to a functional decomposition.
2.8.1 Chemistry Background
Computational techniques are being used increasingly as an alternative to experiment in chemistry. In what is called ab initio quantum chemistry , computer programs are used to compute fundamental properties of atoms and molecules, such as bond strengths and reaction energies, from first principles, by solving various approximations to the Schrödinger equation that describes their basic structures. This approach allows the chemist to explore reaction pathways that would be hazardous or expensive to explore experimentally. One application for these techniques is in the investigation of biological processes. For example, Plate 6
shows a molecular model for the active site region in the enzyme malate dehydrogenase, a key enzyme in the conversion of glucose to the high-energy molecule ATP. This image is taken from a simulation of the transfer of a hydride anion from the substrate, malate, to a cofactor, nicotinamide adenine diphosphate. The two isosurfaces colored blue and brown represent lower and higher electron densities, respectively, calculated by using a combined quantum and classical mechanics methodology. The green, red, blue, and white balls are carbon, oxygen, nitrogen, and hydrogen atoms, respectively.
Fundamental to several methods used in quantum chemistry is the need to compute what is called the Fock matrix, a two-dimensional array representing the electronic structure of an atom or molecule. This matrix, which is represented here as F, has size N N and is formed by evaluating the following summation for each element:
where D is a two-dimensional array of size N N that is only read, not written, by this computation and the I represent integrals that are computed using elements i , j , k , and l of a read-only, one-dimensional array A with elements. An integral can be thought of as an approximation to the repulsive force between two electrons.
Because Equation 2.3 includes a double summation, apparently 2 integrals must be computed for each element of F, for a total of 2 integrals. However, in practice it is possible to exploit redundancy in the integrals and symmetry in F and reduce this number to a total of . When this is done, the algorithm can be reduced to the rather strange logic given as Algorithm 2.3. In principle, the calculation of each element of F requires access to all elements of D and A; furthermore, access patterns appear highly irregular. In this respect, the Fock matrix construction problem is representative of many numeric problems with irregular and nonlocal communication patterns.
For the molecular systems of interest to chemists, the problem size N may be in the range . Because the evaluation of an integral is a fairly expensive operation, involving operations, the construction of the Fock matrix may require operations. In addition, most methods require that a series of Fock matrices be constructed, each representing a more accurate approximation to a molecule's electronic structure. These considerations have motivated a considerable amount of work on both efficient parallel algorithms for Fock matrix construction and improved methods that require the computation of less than integrals.
2.8.2 Chemistry Algorithm Design
Because the Fock matrix problem is concerned primarily with the symmetric two-dimensional matrices F and D, an obvious partitioning strategy is to apply domain decomposition techniques to these matrices to create N(N+1)/2 tasks, each containing a single element from each matrix (, ) and responsible for the operations required to compute its . This yields N(N+1)/2 tasks, each with data and each responsible for computing 2 integrals, as specified in Equation 2.3.
This domain decomposition strategy is simple but suffers from a significant disadvantage: it cannot easily exploit redundancy and symmetry and, hence, performs eight times too many integral computations. Because an alternative algorithm based on functional decomposition techniques is significantly more efficient (it does not perform redundant computation and does not incur high communication costs), the domain decomposition algorithm is not considered further.
Figure 2.31: Functional decomposition of Fock matrix problem. This yields about data tasks, shown in the upper part of the figure, and computation tasks, shown in the lower part of the figure. Computation tasks send read and write requests to data tasks.
Quite a different parallel algorithm can be developed by focusing on the computation to be performed rather than on the data structures manipulated, in other words, by using a functional decomposition. When redundancy is considered, one naturally thinks of a computation as comprising a set of integrals (the integral procedure of Algorithm 2.3), each requiring six D elements and contributing to six F elements. Focusing on these computations, we define ``computation'' tasks, each responsible for one integral.
Having defined a functional decomposition, we next need to distribute data structures over tasks. However, we see no obvious criteria by which data elements might be associated with one computation task rather than another: each data element is accessed by many tasks. In effect, the F, D, and A arrays constitute large data structures that the computation tasks need to access in a distributed and asynchronous fashion. This situation suggests that the techniques described in Section 2.3.4 for asynchronous communication may be useful. Hence, for now we simply define two sets of ``data'' tasks that are responsible only for responding to requests to read and write data values. These tasks encapsulate elements of the two-dimensional arrays D and F (, ) and of the one-dimensional array A (), respectively. In all, our partition yields a total of approximately computation tasks and data tasks (Figure 2.31).
Communication and Agglomeration.
We have now defined computation tasks and data tasks. Each computation task must perform sixteen communications: six to obtain D matrix elements, four to obtain A matrix elements, and six to store F matrix elements. As the computational costs of different integrals can vary significantly, there does not appear to be any opportunity for organizing these communication operations into a regular structure, as is advocated in Section 2.3.2.
On many parallel computers, the cost of an integral will be comparable to the cost of a communication. Hence, communication requirements must be reduced by agglomeration. We describe two alternative strategies that can be used to achieve this goal. Their data requirements are illustrated in Figure 2.32.
Figure 2.32: Agglomeration strategies for Fock matrix construction with N=P=5 , for (a) the total replication algorithm and (b) the partial replication algorithm. In each case, the five tasks are shown with shading used to represent the portion of the symmetric D and F matrices allocated to each task. In (a), each matrix is replicated in each task. In (b), each task is given a single row and column; this corresponds to a factor of two replication.
1. Total replication. Communication costs can be cut dramatically by replicating the F and D matrices in each of P tasks, one per processor of a parallel computer. Each task is given responsibility for 1/P of the integrals. Computation can then proceed in each task without any communication. The only coordination required is a final summation to accumulate partial F matrices. This can be achieved using a parallel vector reduction algorithm described in Section 11.2.
The technique of replicating data structures on each processor of a parallel computer is commonly used in parallel computing to reduce software engineering costs. It allows an existing sequential code to be adapted quickly for parallel execution, since there is no need to modify data structures. The principal disadvantage of the technique is that it is nonscalable. Because total memory requirements scale with the number of tasks created, the largest problem that can be solved is determined not by the total amount of memory in a parallel computer, but by the amount available in a single processor. For example, on a 512-processor computer with 16 MB of memory per processor, an implementation of the quantum chemistry code DISCO that uses this strategy cannot solve problems with N>400 . In principle, it would be interesting to solve problems where N is 10 times larger.
Figure 2.33: Data requirements for integral clusters. Each task accesses three rows (and sometimes columns) of the D and F matrices.
2. Partial replication. An alternative approach is as follows. First, we agglomerate computation in what seems an obvious way, namely, by making the inner loop of the procedure fock_build in Algorithm 2.3 into a task. This yields computation tasks, each responsible for integrals. Next, we examine the communication requirements of each such task. We find that there is considerable locality in the data required by these clusters of integrals: each cluster accesses the i th, j th, and k th row (and sometimes column) of D and F (Figure 2.33). To exploit this locality, we agglomerate data to create N data tasks, each containing a row/column pair of the two-dimensional arrays D and F and all of the one-dimensional array A. In this scheme, each element of D and F is replicated once, and A is replicated N times, so total storage requirements are increased from an average of N to 3N per task. Because of this replication, each computation task now requires data from just three data tasks. Hence, the number of messages is reduced from to . The total volume communicated remains . Because the cost of communicating a word is typically much less than the cost of computing an integral, this is an efficient parallel algorithm.
The ``partial replication'' Fock matrix construction algorithm creates N data tasks and computation tasks. We use the notation (i j k) to identify the computation task responsible for computing the integrals I ; this task requires data from data tasks i , j , and k . To complete the parallel algorithm, we must define a mapping of data and computation tasks to processors.
We assume processors. Since each data task will receive roughly the same number of requests, we allocate one data task to each processor. This leaves the problem of mapping computation tasks. We can imagine a variety of approaches:
1. A simple mapping, in which task (i j k) is mapped to the same processor as data task i ; since each task communicates with data tasks i , j , and k , off-processor communication requirements are reduced by one third. A disadvantage of this strategy is that since both the number of integrals in a task and the amount of computation per integral can vary, different processors may be allocated different amounts of computation.
2. A probabilistic mapping, in which computation tasks are mapped to processors at random or using a cyclic strategy.
3. A task-scheduling algorithm to allocate tasks to idle processors. Since a problem can be represented by three integers ( i , j , k ) and multiple problems can easily be agglomerated into a single message, a simple centralized scheduler can be used. (Empirical studies suggest that a centralized scheduler performs well on up to a few hundred processors.)
4. Hybrid schemes in which, for example, tasks are allocated randomly to sets of processors, within which a manager/worker scheduler is used.
The best scheme will depend on performance requirements and on problem and machine characteristics.
2.8.3 Chemistry Summary
We have developed two alternative parallel algorithms for the Fock matrix construction problem.
1. The F and D matrices are replicated in each of N tasks. Integral computations are distributed among the tasks, and a summation algorithm is used to sum F matrix contributions accumulated in the different tasks. This algorithm is simple but nonscalable.
2. The F, D, and A matrices are partitioned among N tasks, with a small amount of replication. Integral computations are agglomerated into tasks, each containing integrals. These tasks are mapped to processors either statically or using a task-scheduling scheme.
This case study illustrates some of the tradeoffs that can arise in the design process. The first algorithm slashes communication and software engineering costs; however, it is not scalable. In contrast, the second algorithm has higher communication costs but is highly scalable: its memory requirements increase only with problem size, not the number of processors. To choose between the two algorithms, we need to quantify their parallel performance and then to determine the importance of scalability, by assessing application requirements and the characteristics of the target parallel computer.
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© Copyright 1995 by Ian Foster |
5c8bdcdd305e70bd | tisdag 26 september 2017
Update of Real Quantum Mechanics: Electron vs Kernel
I have made a discovery resolving an issue with poor correspondence between theory and observation in the new approach to quantum mechanics termed realQM presented here and here and here.
In the original setting of realQM the same set-up was used as in the standard version of quantum mechanics based on the Schrödinger equation as concerns the kernel assumed to act like a point source with no extension with a corresponding potential $-Z/r$ with $Z$ the kernel charge and $r$ the distance to the kernel, thus with a singularity at the kernel with $r=0$.
In this setting realQM gave a ground state energy for Helium (with two electrons meeting the kernel) of about -3.0 which was substantially lower than the observed -2.9034.
Something was thus wrong with realQM in this original form, and I could not figure out what. I have now understood that this mismatch comes from the kernel singularity which, like all singularities, introduces a dark horse into the model, which has to be handled properly to not lead astray. It is thus natural to give the kernel a positive radius and study the dependence of the ground state energy on the kernel radius.
The question of the boundary condition for the electron as it meets the kernel at a positive radius then comes up, something which is hidden if the radius is zero. Recalling that the boundary condition on the free boundary separating different electrons is a homogeneous Neumann condition, it is natural to try the same condition for the kernel, understanding that it requires the kernel to have positive radius.
An alternative is to use a Robin boundary condition of the form $\frac{\partial\phi}{\partial r}=-Z\phi$ for a positive radius. This is the effective condition at zero radius built into the Schrödinger equation with a point source kernel.
And indeed, both approaches seem to work (very similarly) as recorded in the above references.
More specifically, the kernel radius (which comes out to be small (of size 0.05 - 0.01 atomic units for kernel charge 2-10) can be used as a model parameter, which can be adjusted to give exact agreement with observations as a calibration of the realQM model for two electron ions, which can serve to build a model with more electrons in outer shells.
The model of realQM thus opens to inspection of the inner mechanics of an atom, including information on the effective radius of the kernel as seen by an electron in the innermost shell, something which is hidden to direct experimental observation.
We recall that standard quantum mechanics stdQM does not offer a physical model of the atom and thus with stdQM the inner mechanics of an atom is closed to human understanding, a defect made into a virtue in the Copenhagen interpretation of stdQM filling text books.
3 kommentarer:
1. The inner mechanics of an atom?
Do you mean the mechanics of the protons and neutrons in the kernel?
2. More the mechanics of the electrons with mutual interaction and with the kernel.
3. Would a physical model of the atom include geometrical distribution of the internal interactions within the atom structure. I have an idea about the internal energy interactions based on my model of earth, mars and venus heat flow, which use the shell theorem with both volume and surface area of the sphere to find exact solutions to surface temperature. Including thermodynamic work in the form of gravity. I think there might be a simnilarity between the atom and planets. They can maybe be treated as particles, both of them. You don´t seem fond of my comments though, since you don+t let them through. I wonder why?
I feel there is way to much censorship in these "scientific" blogs. People in the academic world just want to promote themselves everywhere. |
208ad9cf1ffacd7b | Monday, May 28, 2018
What do physicists mean when they say the laws of nature are beautiful?
Simplicity in photographic art.
“Monday Blues Chat”
By Erin Photography
In my upcoming book “Lost in Math: How Beauty Leads Physics Astray,” I explain what makes a theory beautiful in the eyes of a physicist and how beauty matters for their research. For this, I interviewed about a dozen theoretical physicists (full list here) and spoke to many more. I also read every book I could find on the topic, starting with Chandrasekhar’s “Truth and Beauty” to McAllister’s “Beauty and Revolution in Science” and Orrell’s “Truth or Beauty”.
Turns out theoretical physicists largely agree on what they mean by beauty, and it has the following three aspects:
A beautiful theory is simple, and it is simple if it can be derived from few assumptions. Currently common ways to increase simplicity in the foundations of physics is unifying different concepts or adding symmetries. To make a theory simpler, you can also remove axioms; this will eventually result in one or the other version of a multiverse.
Please note that the simplicity I am referring to here is absolute simplicity and has nothing to do with Occam’s razor, which merely tells you that from two theories that achieve the same you should pick the simpler one.
A beautiful theory is also natural, meaning it does not contain numbers without units that are either much larger or much smaller than 1. In physics-speak you’d say “dimensionless parameters are of order 1.” In high energy particle physics in particular, theorists use a relaxed version of naturalness called “technical naturalness” which says that small numbers are permitted if there is an explanation for their smallness. Symmetries, for example, can serve as such an explanation.
Note that in contrast to simplicity, naturalness is an assumption about the type of assumptions, not about the number of assumptions.
Elegance is the fuzziest aspect of beauty. It is often described as an element of surprise, the “aha-effect,” or the discovery of unexpected connections. One specific aspect of elegance is a theory’s resistance to change, often referred to as “rigidity” or (misleadingly, I think) as the ability of a theory to “explain itself.”
By no way do I mean to propose this as a definition of beauty; it is merely a summary of what physicists mean when they say a theory is beautiful. General relativity, string theory, grand unification, and supersymmetry score high on all three aspects of beauty. The standard model, modified gravity, or asymptotically safe gravity, not so much.
But while physicists largely agree on what they mean by beauty, in some cases they disagree on whether a theory fulfills the requirements. This is the case most prominently for quantum mechanics and the multiverse.
For quantum mechanics, the disagreement originates in the measurement axiom. On the one hand it’s a simple axiom. On the other hand, it covers up a mess, that being the problem of defining just what a measurement and a measurement apparatus are.
For the multiverse, the disagreement is over whether throwing out an assumption counts as a simplification if you have to add it again later because otherwise you cannot describe our observations.
If you want to know more about how arguments from beauty are used and abused in the foundations of physics, my book will be published on June 12th and then it’s all yours to peruse!
Matthew Rapaport said...
Thanks Dr. H. I'm looking forward to it. Of course physicists use the term in a technical way only tangentially connected to beauty as the term is commonly used and like the question "what is goodness?" or "what is truth?" Beauty in its common usage is a slippery concept. But as I've noted before, like the other two, beauty is a VALUE and its slippery nature stems from our very vague recognition of what constitutes those values
CapitalistImperialistPig said...
The epigraph to Chandrasekhar's Mathematical Theory of Black Holes has quotes from Heisenberg and Bacon on the subject. I don't have the volume at hand, but they go something like this:
Heisenberg: Beauty consists of the proper proportion of the parts to the whole, and to each other.
Bacon: There is no thing of excellent beauty which hath not some strangeness in the proportion.
neo said...
where does loop quantum gravity score on aspects of beauty?
Thomas said...
Every time I read something like "the laws of nature are beautiful?" someone want to sell a new book to the public :-(
Bill said...
A beautiful equation is also one that exhibits the fewest free parameters while explaining the most physics. That's why general relativity is beautiful while the Lagrangian of the Standard Model is ugly as hell. They both work, one by itself and the other by brute force, although I would never compare one with the other.
Looking forward to purchasing your book!
Uncle Al said...
Simple, natural, elegant: The answer exits a printer after a one-hour observation.
... 1) Baryogenesis is post-Big Bang excess matter over antimatter violating conservation laws via selective leakage.
... 2) Sakharov conditions. Vacuum is neither exactly mirror-symmetric nor exactly isotropic toward quarks then hadrons.
... 3) Einstein-Cartan-Kibble-Sciama spacetime torsion chiral dopant.
... 4) Milgrom acceleration and the cosmological constant emerge. Dark matter, SUSY, and M-theory wither.
... 5) Extreme opposite shoes embed within a vacuum left foot with measurably different energies.
... 6) Measure spacetime trace chiral anisotropy ̶ one hour in a microwave spectrometer, 40,000:1 signal to noise, using molecular lollipop enantiomers.
Enantiomeric balls ( short sticks (2-CN group for dipole moment) Divergent rotational spectra.
Enrico said...
I only recognize simplicity of the Occam's razor kind. Make only a few assumptions that can be tested empirically. Naturalness is just modern numerology of the ancient Pythagoreans. (They believed the square root of two is evil) Elegance is epistemology because no scientific theory explains itself. They explain observations. We can make theories that explain themselves or anything except the observable universe. Theoretical physicists are envious of mathematicians because their imaginations can fly without constraint from the physical world. It's flight-of-fantasy envy
Sabine Hossenfelder said...
Something's wrong with the comment feature at blogger - I'm not getting comment notifications, meaning I basically don't know if anyone submitted a comment until I check the website. If anyone has an idea what's the issue, please let me know. For the rest, may I kindly ask for your patience. I'll be traveling today but will look for a fix once back home.
Space Time said...
To me, as a mathematician, beauty in physics/science is something that is almost impossible to describe and define but easy to tell when you see it. It is also very subjective and different people may disagree. for instance general relativity is beautiful, modified gravity is not. The orthodox quantum mechanics is beautiful, Bohmian mechanics is not, and so on.
Uncle Al said...
arxiv:1712.07969 k. How can that be empirically furiously wrong?
XENON1T, 1300 kg active liquid xenon target of total 2000 kg. ZERO net output.
XENONnT, 5200 kg active liquid xenon target of total 7500 kg. 2019 launch.
Zero signal crashes physical theory. Simple, natural, elegant: The math is rigorous but empirically irrelevant. It's a curve fit. It's phlogiston.
Newton becomes special relativity given Maxwell. Lightspeed is unchanged from all reference frames. What rubbish! No, it's true. Baryogenesis happened. Conservation laws are inexact. What rubbish! Is it true?
"There is no reason to look. Physical theory cannot be fundamentally defective." Nothing predicts. Simple, natural, elegant: Look at the answer not at its guesses.
M. J. Glaeser said...
Dr Hossenfelder:
How do the following score in beauty,in your view?
The C*-algebraic version of LQG (see LOST theorem).
The spectral derivation of the standard model (see Connes's work).
Isham and Döring's topos-theoretic formulation of the Kochen-Specker Theorem.
Geroch's Einstein algebras
Lawrence Crowell said...
Beauty is not something we can easily codify. Much of this has to do with induction nature of proposing a grand theory. There is no deductive structure to proposing some set of physical axioms or postulates as the most economical and elegant foundation to the universe. Beauty is a relative of “quality” and as Pirsig wrote in Zen in the Art of Motorcycle Maintenance is not something that can be codified.
A good theory has a limited number of physical axioms that upon their introduction change how we think and that are then by deductive reasoning able to lead to a large set of expected results. Such theories are often considered to be beautiful, elegant and natural. Simplicity comes with the limited number of postulates, elegance is in the new structures proposed, and naturalness comes with the relative ease with which physical predictions occur.
One thing that can happen of course is a very beautiful and elegant theory can be wrong. Supersymmetry is an intertwine between the boson and fermion structures of quantum mechanics and the structure of spacetime. It also short circuits the Coleman-Mandula “no-go” theorem on obstructions to unification of internal symmetries of gauge fields with the external symmetry of gravitation. This is a big problem with Lisi's program that includes the SL(2,C) of gravitation. Supersymmetry is though a framework more than an exact description of nature, and one must “hang” a model of gauge bosons and fermions on it. This has been the case with minimally supersymmetric standard model (MSSM). Interestingly MSSM is on the verge of being falsified, and many thousands of papers on this topic, including those by luminaries as Gordon Kane, may be completely trashed. The question is whether this is a failure of supersymmetry or a particular model.
Maybe supersymmetry occurs in ways completely different from what has been thought. This is my thesis. I frankly welcome the prospect that MSSM is falsified; it clears the decks for small players like me and many others. Whether following beauty has been the downfall or not is not clear to me. Obviously people tried to work light mass supersymmetric partners with the standard model. The standard model is not considered to be the most elegant theory out there, but it sure works like a top for TeV scale physics. The MSSM was built because it was what seemed most reasonable at the time, and it had some level of beauty to it. It though appears to be headed for the trash heap.
sean s. said...
You may want to check your spam folder; sometimes a setting gets broken and many things go there. I've seen it happen to others.
Travel safely.
sean s.
The Universe said...
Beautiful mathematics is absolutely no substitute for understanding. Dirac was the epitome of that. He had absolutely no understanding of the electron, but didn't care, and he even ignored the likes of Gustav Mie and Charles Galton Darwin. In fact, seeing as his 1962 paper an extensible model of the electron depicted the electron as a charged conducting sphere, I'd go so far as to say beauty is dangerous.
As regards comment moderation, I notice that I can't comment using my wordpress id. It's The Universe (Google Account) or nothing. You could always try turning comment moderation off.
John Duffield
Sabine Hossenfelder said...
Hi all,
Regarding the comment issue, turns out it's not a problem with my blog, but a blogger-wide issue that will supposedly be fixed next week or such. (See forum thread.) So rather than switching to a different comment widget (which would remove all existing comments), I'll wait this out. Please be warned that this means for the coming week comments will appear even slower than usual.
milkshake said...
I think the elegance aspect has to do with the sparseness of the description and its predictive power - one can get far more phenomena explained and flowing out without fudging than was put in; and preferably it happens in a way that is non-obvious and startling
marten said...
My professor in mathematics used to qualify beautiful equations as horny, because such equations are stimulating the faculty's survival.
Space Time said...
John Duffield,
Dirac seems to be an example of exactly the opposite. Beauty was certainly a very important motivation for his work (one can argue it was the only one). And his contribution to physics is undeniable.
Sabine, if you could have interviewed Dirac, would you have had a different view about beauty?
MartinB said...
I think one additional aspect (or may be you include this with the "Surprise element" under elegance) is that beautiful theories use non-intuitive concepts to explain everyday experience.
Even Newton is rather non-intuitive (compared to Aristoteles).
GR explaining things falling down by time running slower close to a mass gives a totally weird-seeming explanation.
Explanatory power is also important. I remember that I definitely did not find Maxwell's equation beautiful in any way when I first saw them. I appreciated their beauty only after seeing how you can derive things like em-waves from them and how the different terms in the equation conspire to make em-waves possible. So one other aspect of beauty may be only apparent when you find that the equations are easy to operate with and reveal a rich structure of possible things to derive from them. (Complexity from simplicity.)
Sabine Hossenfelder said...
Space Time,
I don't know what you mean. Would I have had a different view about beauty than Dirac? Presumably. Or a different view than presently? Probably not. Or else, I don't know what you mean.
Uncle Al said...
@The Universe Otto Stern’s measured proton magnetic moment showed the Dirac equation is empirically wrong for composite particles. Nobel Prize.
Stern's value was poor but sufficiently far from Dirac's calculated value. Current proton-antiproton values 1.5 ppb diverge re baryogenesis. One hour in a microwave rotational spectrometer measures overall vacuum chiral anisotropy toward hadrons, falsifying simple, natural, elegant.
.... 2.792847350(9)μ_N proton
... -2.7928473441(42)μ_N antiproton
... DOI::10.1038/nature24048
t h ray said...
Well and compactly said.
Space Time,
" ... general relativity is beautiful, modified gravity is not. The orthodox quantum mechanics is beautiful, Bohmian mechanics is not, and so on."
Huh? You're speaking as a mathematician?
Unknown said...
Mathematics is required , but the way it is done to do physics for funds and just survival is wrong. One can go to the screen in " The Man who knew infinity " where Professor Hardy tells S Ramanujan probably in the hospital " I want rigor Ramanujan" when Ramanujan writes the problem and just the right solution without the steps. Ramanujan responds well , he gives rigor ti his solutions with Prof hardy's suggestion. Mathematics can be used purposefully in physics if there is rigor.
Even in engineering and experimental work many do not report error bars, even in high ranking journals which can be done only when they do the experiments atleast thrice. The rush to publish is the primary cause for this.
Patat Je said...
I think you mean that the collapse postulate is removed, and then splitting is added later. Splitting is metaphorical. Splitting never happens. You don't need to know when or where a universe splits. The Schrödinger equation describes everything.
David Bailey said...
Surely the idea of beauty, as applied to a physical theory, only makes sense if it is assumed that you are looking at the fundamental theory. Unless that is the case, ugly equations are the norm.
For example, I was stunned as a teenager by the gas law equation PV=nRT - then I learned it is only an approximation, and more accurate, but vastly uglier equations do better!
Is physics at the depth to find a fundamental theory - who knows, but I'll bet every generation thinks it is!
Space Time said...
"Huh? You're speaking as a mathematician?"
t h ray, are you surprised that the examples I gave were from physics rather than mathematics? Well, it is hard (in my opinion impossible) to find an example of ugly in mathematics.
Rogier Brussee said...
I think an important aspect of beauty that is actually a valid guideline for physics, is that a physical description should be as free as possible from arbitrary choices that are made by us humans to provide for a description. Usually this means being closer to the "Copernican principle" that the world is less centred around you and that if you make a choice at a point in space time (e.g. a frame of reference), there is no trivial way to communicate that choice to the rest of the world. It also usually means symmetric, with the symmetry being the group acting on the possible choices that provide descriptions. This often makes it more technical to write things down (although this is mostly a matter of what you are used to), but it also hides lots of distracting information, that you could use to make write down variants that, however, can't be physically relevant, because they depend on an arbitrary choice you made. It is not unlike abstractions in a programming language.
General relativity is more beautiful than field theory + gravitons, because general relativity does not assume a background metric. Of course in every point of space time, you can _choose_ a frame such that $g_{\mu\nu} = \eta_{\mu\nu}$ but now the description involves a choice. Realising that there is a choice to be made, makes life technically harder but pre relativity you made the tacit assumption that your "inertial frame of reference" was trivially agreed upon in the rest of the universe. Not making that assumption is i.m.o. the heart of general relativity.
Sections of U(1) line bundle are better than wave functions, because only phase differences and absolute values at a point make sense. Once you think about it in this way the operator $\partial_\mu = \frac{\partial}{\partial x^\mu}$ makes no sense anymore: you need a U(1) connection $\nabla$ which in a local trivialisation looks like $\nabla_\mu = \partial_\mu + A_\mu$ with (i times) the vector potential $A$. The curvature is (i times) the Faraday tensor, because EM works in exactly this way on the wave function. Even the Bohm Aharanov effect now falls out naturally as the holonomy of a flat connection. The gauge group $\psi \to e^{i\phi} \psi$ can be seen actively as a symmetry, but more naturally passively as the result of different choices of trivialisation of the line bundle.
The Maxwell equations in Heaviside as written today are more beautiful than the equations written in x,y,z that he wrote down himself because the former are manifestly independent of the choice of a frame of reference. LIkewise, the index notation used by physicists is computationally useful, but it is also horrible in that it makes it impossible to even say what you mean without making a choice of reference frame, not to mention actively encouraging not to say what kind of object you are dealing with "because it is "clear" from the indices, and making people think in terms of components and operations on indices. Weinbergs book on quantum field theory starts by writing down the gamma matrices he uses. It always leaves a lingering feeling of unease about what depends on conventions and what not. Mathematicians (or at least geometers) take pride in writing down coordinate independent intrinsic entities whenever possible, and it greatly helps in only writing down expressions that make sense independent of choices, of which there tend to be very few, so it is a great guiding principle. |
2ee46abfc8334689 | The Holographic Universe
A Vector finds its intuitive counterpart in spectrum with the mechanism of Harmonics.
We are: Vector Convox Spectrum
Convox is a Convex of Harmony. We are the Conductors of the Convention.
‘Con’ is short for Continuum. Because Existence is a Continuum. A vertical Vector in Continuum (spectrum).
From Google.
“In the space-time continuum of General Relativity, events are defined in terms of four dimensions: three of space, and one of time, with one coordinate for each dimension; we continuously “move” along the time dimension.”
So…. Then we’re not actually moving in space at all or anywhere ever. We’re Being Convected while Confecting the Convention.
This is the physics of the Holographic Universe put forth by David Bohm.
Your Face
Your face is a convex of plasma. This plasma has always existed. Meaning, it occupies space with no duration. Your specific convection is the result of duration. We live outside of timelessness phasing into a convex of procedure (space) that’s intuitively fusing Duration into cadence with harmony (time) powered by excitement. How thick is cadence? It now has mass because it occupies space through a harmony convector (you). We are explicating volume in mass and measure in cadence with signature: Time. Happening now through the convex of space-time. Duration is the space between cadent ripples produced by frequency that Vexes (spins) with Procedure (space) to be emphasis; to see itself. Or frequency needs space to repeat and experience the pattern. Frequency repeats and plasma convexs accordingly. Or evolution via expatiation. Enter the need for Empiricism and faces. Those faces need shapes. That’s us.
That’s why we have space time and evolution in progress.
Grist for the Mill
The blue represents Plasma not death. That’s the crucial mistake that’s been made.
Yellow is Nuclear Radioaction or the entire spectrum of excitement. It’s 100% electrons. The highest Octave E that can be fathomed to the infinite power. So, it’s a little hot.
Now for Blue. Plasma is positively charged. It is pure unfiltered Positivity. There is no spectrum. There is just Positivity. Plasma is positivity floating…. Aimlessly…. That’s not excited. That’s weird because we are only used to after it has seen eye-to-eye with its electricity harmonious equal.
This symbol is a good way to explain the convection of the floor of Positivity with its equal in the spectrum of excitement. It’s Grist for the mill Positive Excitement 100% of the time.
Unless the blue represents death or something like that then you’re bound to the low end of the spectrum of excitement, fear and sorrow, with only reflections of this symbol’s by product (nature). Which is just a fleeting harmony of equilibrium. Then it fades as the dark side of this symbol says death to you. Enter: the western disposition.
The dark side is PLASMA.
Cheers to you.
The reason we exist
There are only 2 forms that exist: electricity and plasma. That’s it. Everything is those things. All of it is sentient harmony in all directions.
We are this endless possibilities of Harmony at all times forever. As that, we have already experienced the most beautiful euphoria anyone could imagine ever. We have already experienced every emotion that could be had. Maximum euphoria is known and harmonious in all directions. The same goes as well for the lower spectrum of excitement, fear and sorrow. That’s known, too.
Likewise, we know all the procedures, notes, the steps, the measure that connects it all together immaculately. That’s known!
What we DON’T KNOW is the experience of being BOTH simultaneously. We are expatiating the cadence of Excitement that’s effortlessly ringing out; that’s humming along in the key of E. That Love expatiation, our existence, is a by product of visceral Coherent Resplendent Limitless Joy to the infinite power, that’s Orchestrating coherent excited Pathways that poetically and artistically weave the entire spectrum of electricity (excitement) into a score of notes (emotional tones of excitement) made of Plasma.
Yeah,… 🍻. So… enjoy yourself! You are Love expatiating in cadence with Joy. And everything you do and say is informing the environment (plasma) what note that you want backup instrumentation on. Or, how you want it behave in relation to your emphatic state, e. g. a note in the key of E/Truth.
The Process of Joy
We Think in Musical Color
We have an Electricity guitar in our physical brain. The 5 strings of electricity in our brains are identical to each string on the guitar from (E) Delta (.5 to 4 Hz), (A) Theta (4 to 8 Hz), (D) Alpha (8 to 12 hz), (G) Beta (12 to 40 hz), (B) Gamma (40 to 100 hz).
That little factoid makes the picture above amazing. This is a new way to understand everything we thought we knew. We think in Musical Colors. We literally have musical rainbows coming out of our heads continuously. This is happening right now. Wrap your mind around that!
Conductors of Emphasis in the key of Truth
You are emphasis in the key of Truth.
Parallel is the G chord is emphasis in the Key of E among the polyphony of open chord harmony. Everything is in the key of Einstein’s E, which his E of energy – which is Truth. Just like notes inside E can’t leave the Key of E, we are notes that cannot leave the Truth.
Your body is the screen that’s the extraction method of distinguishing the G chord from the polyphony of flowing sounds; Schrödinger equation is the math of harmonious sounds that are always on and playing. The Heisenberg uncertainty principle, these notes suspended in potential courses of action. The Heisenberg uncertainty principle is Harmony before procedural execution; before you play the song.
In order to emphasize the most exciting chord you can imagine, there needs to be a convection of sound. The G chord needs to be isolated with definition of being. So it can expand in the same space isolated. That’s emphasis.
So we need a cogent mask to filter out all the expatiating harmony of E11#9. We need to single out one of the strains of sound so we can enjoy it by itself. So all harmony is altruistically giving of it’s potential and going with your designs to isolate a specific chord. Existence is changing its course of action because of you.
Why would we want to isolate the G harmony if all harmony is endlessly resplendently expatiating all over the place?
Why do you like some songs and not others? Because we have 5 electric strings in our head and are purely just harmonizing with various songs (people) and reflexively choosing the ones that have the best sound.
We are guided by our Intuition of Truth. We know exactly what key we’re in and we are excited to phase in to people’s songs and jam with them. We do this with everyone through Empathy with Emphasis of likeness. Emphasis being “Emotional Phasing.” more specifically, we create an Emotional Pathway with them and then Emotionally Phase in with them. It’s Fusion. It could be called Emfusis. Or Emphasis. I love language.
We Emotionally Phase in with the emotional stasis of a person whose chord progression that we like. Phasing in, is fusion of music between two or more points. If we don’t like the song being expanded by a person and can’t leave we will refer to the space as hellish. That’s being forced to listen to terrible out of tune music in a closed space for extended duration. Sometimes this is a “job.”
In order to phase in from Schrödinger’s “Phase State” of polyphonic totality, we need a body to convect strains of notes into a song that is exciting to our personal emphasized chord progression.
We need a procedural partition that gives us the ability to access some existing harmony while ordering the totality around it. That’s what your fingers are doing on the fret board. That’s why what your body and mind is doing with the totality of life harmonics.
You are always in the key of Truth. You’ll know truth when you see it because you will immediately harmonize with it. You are an Altruist because you volunteer to harmonize with everything, even if the harmony is weak.
Even if the song is weak you solider on tenaciously in spite of bad music. You are a unconditional love outlet with tacit knowledge of universal harmony.
The suggestion I make is to figure out when the music goes bad or changes key and the having the strength to modulate to your preferred key or just stop playing bad music with them.
The goal is always play music that you like. If you get stuck playing music you don’t like with someone and can’t leave, because of your persistent altruism, then stop playing with them and find a better rock band.
We are the conductor of emphasis in design harmonics.
5 electric strings in your head
This is probably the most beautiful and significant discovery of my life.
The physics of each of the chords on the guitar are
identical to the waves in your brain. Picture above.
E – Delta
A – Theta
D – Alpha
G – Beta
B – Gamma
E – repeats
The ramifications of this is enormous.
Also, the 5 elements that are discharge from nuclear isotopes are equal to the above 5. The 3 sets of 5 are equal.
From the smallest thing to largest thing everything is proportionally connected. The ratio is 1.61803398875, the golden ratio.
More on this to come.
We are made of electricity. You are positively charging the space you occupy.
Cheers to you.
Embrace the Musical
We are Harmony.
We have experienced the harmony and excitement of our unlimited imagination and the structure of how it unfolds as a procedure (space), but we have not experienced both simultaneously – Space with duration (time).
Enter the convex of Space-Time, ladies and gentlemen. This little number is a mere 14.5 billion years old little fusion apparatus. So precious and youthful. Space-time, where you go if you want to simultaneously experience both the procedure and the harmony, a myriad of harmonious excitations, with all your friends. Yay.
Come to space-time and play your life song! Book now and get the gold package of multi harmonious connections and musical fantasia that’s completely full immersion.
That’s right, folks, with the new and improved gold package of full immersion, you will be your own stream of music and you won’t even know it. That’s right, you’ll experience your own amazing tune and remain totally convinced it’s not you. We’re proud of this feature.
Aaahhh… Space-time. Where all the cool kids jam and riff.
This moment, right now, took 14.5 billion years to grow into what it is.
Embrace it.
Your Life is A Song. The physics.
The Heisenberg uncertainty principle is that electrons don’t have a precise size. An electron is considered the most basic building block of physical matter. An Electron is a “thing” that is not a Thing. It’s size undefined. It’s somehwere between real and not real.
Here’s the solution.
The Heisenberg uncertainty principle is equal to looking at your guitar and thinking of playing the Note G. The note your thinking of is the floating electron.
It’s looking at a tuned guitar string set (Schrödinger’s equation) and Imagining the Note (the electron) that you want to play.
Here’s what’s happening. The Note already exists because all Harmony already exists. The emphasis of G is happening all the time, but it doesn’t have a loading dock by which to communicate itself. When you’re in this brainstorming session, what you are doing, is harmonizing with G without a procedure; without the mechanism of 1st, grab guitar, 2nd grab pick, 3rd, place fingers in correct places, 4th strum strings: which then phases in your brain waves/chords, the math of Schrodinger’s equation and the physics of Ohms law. Your mind and body are harmonizing.
A couple of these prodigy musicians, Beethoven, grew def but could still “hear” the music. What they were doing was Heisenberg’s uncertainty principle. They were guided by the suspended action of a Note he knew to belong somewhere exciting.
They decided where the note would go because, “That’s just where it fits the best to get the outcome (sound/ excitement/ excitation) that I want.” This suspended action doesn’t take up space and neither does Heisenberg’s electrons.
The uncertainty principle is the discovery of Harmony always existing without Procedure.
We Are Harmony. – Werner Heisenberg
The double slit experiment.
What’s happening in this experiment is one song is playing then it is interrupted by a sequential harmonious perspective (the electron detector). That new perspective harmonizes with the electron at a different measure and changes the tune of the music (the course of action).
When a song on the radio is interrupted by another song, that’s the intuitive equivalent of the Double Slit experiment.
There’s an experiment called the Quantum Eraser. Briefly, it says that electrons will travel back in time to correct themselves when the path of the electron is not known.
This is equal to when you are playing a chord/note and then…forget what the next note is and then…. stop making note decision and let the already produced sound reverberate in people’s memory.
When no note is being played (emphasized) Existence plays ALL Harmony Simultaneously. We get overwhelmed because we see a lot of stuff but with no obvious structure. A lot of harmony but not a way to connect them.
The Schrodinger equation is the math for ALL Simultaneous Harmony with no emphasis anywhere.
What we do then, as a progressive life choice, is look at all the songs (events) that could be played (experienced/ emphasized) and we choose the song that we want to rock out to, that excites us most, that we harmonize with.
Our job in life is to rock out to our favorite song and explore fully the excitement of this score of notes (insights and experiences) while expatiating (playing the song).
Your life is a song.
Procedural Mask
Your body is a procedural mask(it occupies space). Your mission should you choose to accept it is; harmonize your electric soul with as many contributions from your fellow electrical outlets as you dreamed plausible. Your mask is your sponge. Drink, light sockets, drink!
We are Harmony |
d5a56d5ae81427fb | Atomic structure and spectra
Atomic structure and spectra
The idea that matter is subdivided into discrete building blocks called atoms, which are not divisible any further, dates back to the Greek philosopher Democritus. His teachings of the fifth century b.c. are commonly accepted as the earliest authenticated ones concerning what has come to be called atomism by students of Greek philosophy. The weaving of the philosophical thread of atomism into the analytical fabric of physics began in the late eighteenth and the nineteenth centuries. Robert Boyle is generally credited with introducing the concept of chemical elements, the irreducible units of which are now recognized as individual atoms of a given element. In the early nineteenth century John Dalton developed his atomic theory, which postulated that matter consists of indivisible atoms as the irreducible units of Boyle's elements, that each atom of a given element has identical attributes, that differences among elements are due to fundamental differences among their constituent atoms, that chemical reactions proceed by simple rearrangement of indestructible atoms, and that chemical compounds consist of molecules which are reasonably stable aggregates of such indestructible atoms.
Electromagnetic nature of atoms
The work of J. J. Thomson in 1897 clearly demonstrated that atoms are electromagnetically constituted and that from them can be extracted fundamental material units bearing electric charge that are now called electrons. The electrons of an atom account for a negligible fraction of its mass. By virtue of overall electrical neutrality of every atom, the mass must therefore reside in a compensating, positively charged atomic component of equal charge magnitude but vastly greater mass. See Electron
Thomson's work was followed by the demonstration by Ernest Rutherford in 1911 that nearly all the mass and all of the positive electric charge of an atom are concentrated in a small nuclear core approximately 10,000 times smaller in extent than an atomic diameter. Niels Bohr in 1913 and others carried out some remarkably successful attempts to build solar system models of atoms containing planetary pointlike electrons orbiting around a positive core through mutual electrical attraction (though only certain “quantized” orbits were “permitted"). These models were ultimately superseded by nonparticulate, matter-wave quantum theories of both electrons and atomic nuclei. See Quantum mechanics
The modern picture of condensed matter (such as solid crystals) consists of an aggregate of atoms or molecules which respond to each other's proximity through attractive electrical interactions at separation distances of the order of 1 atomic diameter (approximately 10-10 m) and repulsive electrical interactions at much smaller distances. These interactions are mediated by the electrons, which are in some sense shared and exchanged by all atoms of a particular sample, and serve as an interatomic glue that binds the mutually repulsive, heavy, positively charged atomic cores together. See Solid-state physics
Bohr atom
The hydrogen atom is the simplest atom, and its spectrum (or pattern of light frequencies emitted) is also the simplest. The regularity of its spectrum had defied explanation until Bohr solved it with three postulates, these representing a model which is useful, but quite insufficient, for understanding the atom.
Postulate 1: The force that holds the electron to the nucleus is the Coulomb force between electrically charged bodies.
Postulate 2: Only certain stable, nonradiating orbits for the electron's motion are possible, those for which the angular momentum associated with the motion of an electron in its orbit is an integral multiple of h2&pgr; (Bohr's quantum condition on the orbital angular momentum). Each stable orbit represents a discrete energy state.
Postulate 3: Emission or absorption of light occurs when the electron makes a transition from one stable orbit to another, and the frequency &ngr; of the light is such that the difference in the orbital energies equals h&ngr; (A. Einstein's frequency condition for the photon, the quantum of light).
Here the concept of angular momentum, a continuous measure of rotational motion in classical physics, has been asserted to have a discrete quantum behavior, so that its quantized size is related to Planck's constant h, a universal constant of nature. Velocity v, in rotational motion about a central body, is defined as the product of the component.
Modern quantum mechanics has provided justification of Bohr's quantum condition on the orbital angular momentum. It has also shown that the concept of definite orbits cannot be retained except in the limiting case of very large orbits. In this limit, the frequency, intensity, and polarization can be accurately calculated by applying the classical laws of electrodynamics to the radiation from the orbiting electron. This fact illustrates Bohr's correspondence principle, according to which the quantum results must agree with the classical ones for large dimensions. The deviation from classical theory that occurs when the orbits are smaller than the limiting case is such that one may no longer picture an accurately defined orbit. Bohr's other hypotheses are still valid.
According to Bohr's theory, the energies of the hydrogen atom are quantized (that is, can take on only certain discrete values). These energies can be calculated from the electron orbits permitted by the quantized orbital angular momentum. The orbit may be circular or elliptical, so only the circular orbit is considered here for simplicity. Let the electron, of mass m and electric charge -e, describe a circular orbit of radius r around a nucleus of charge +e and of infinite mass. With the electron velocity v, the angular momentum is mvr, and the second postulate becomes Eq. (1).
The integer n is called the principal quantum number. The possible energies of the nonradiating states of the atom are given by Eq. (2).
Here 0 is the permittivity of free space, a constant included in order to give the correct units to the statement of Coulomb's law in SI units.
The same equation for the hydrogen atom's energy levels, except for some small but significant corrections, is obtained from the solution of the Schrödinger equation, as modified by W. Pauli, for the hydrogen atom. See Quantum numbers
The frequencies of electromagnetic radiation or light emitted or absorbed in transitions are given by Eq. (3),
where E and E are the energies of the initial and final states of the atom. Spectroscopists usually express their measurements in wavelength λ or in wave number &sgr; in order to obtain numbers of a convenient size. The wave number of a transition is shown in Eq. (4). (4) If T = E(hc), then Eq. (5) results. Here T is called the spectral term. (5)
The allowed terms for hydrogen, from Eq. (2), are given by Eq. (6).
The quantity R is the important Rydberg constant. Its value, which has been measured to a remarkable and rapidly improving accuracy, is related to the values of other well-known atomic constants, as in Eq. (6). See Rydberg constant
The effect of finite nuclear mass must be considered, since the nucleus does not actually remain at rest at the center of the atom. Instead, the electron and nucleus revolve about their common center of mass. This effect can be accurately accounted for and requires a small change in the value of the effective mass m in Eq. (6).
In addition to the circular orbits already described, elliptical ones are also consistent with the requirement that the angular momentum be quantized. A. Sommerfeld showed that for each value of n there is a family of n permitted elliptical orbits, all having the same major axis but with different eccentricities. Illustration a shows, for example, the Bohr-Sommerfeld orbits for n = 3. The orbits are labeled s, p, and d, indicating values of the azimuthal quantum number l = 0, 1, and 2. This number determines the shape of the orbit, since the ratio of the major to the minor axis is found to be n(l = 1). To a first approximation, the energies of all orbits of the same n are equal. In the case of the highly eccentric orbits, however, there is a slight lowering of the energy due to precession of the orbit (illus. b). According to Einstein's theory of relativity, the mass increases somewhat in the inner part of the orbit, because of greater velocity. The velocity increase is greater as the eccentricity is greater, so the orbits of higher eccentricity have their energies lowered more. The quantity l is called the orbital angular momentum quantum number or the azimuthal quantum number. See Relativity
Possible elliptical orbits, according to the Bohr- Sommerfeld theoryenlarge picture
Possible elliptical orbits, according to the Bohr- Sommerfeld theory
Multielectron atoms
In attempting to extend Bohr's model to atoms with more than one electron, it is logical to compare the experimentally observed terms of the alkali atoms, which contain only a single electron outside closed shells, with those of hydrogen. A definite similarity is found but with the striking difference that all terms with l > 0 are double. This fact was interpreted by S. A. Goudsmit and G. E. Uhlenbeck as due to the presence of an additional angular momentum of (h2&pgr;) attributed to the electron spinning about its axis. The spin quantum number of the electron is s = .
The relativistic quantum mechanics developed by P. A. M. Dirac provided the theoretical basis for this experimental observation. See Electron spin
Implicit in much of the following discussion is W. Pauli's exclusion principle, first enunciated in 1925, which when applied to atoms may be stated as follows: no more than one electron in a multielectron atom can possess precisely the same quantum numbers. In an independent, hydrogenic electron approximation to multielectron atoms, there are 2n2 possible independent choices of the principal (n), orbital (l), and magnetic (ml, ms) quantum numbers available for electrons belonging to a given n, and no more. Here ml and ms refer to the quantized projections of l and s along some chosen direction. The organization of atomic electrons into shells of increasing radius (the Bohr radius scales as n2) follows from this principle. See Exclusion principle
The energy of interaction of the electron's spin with its orbital angular momentum is known as spin-orbit coupling. A charge in motion through either “pure” electric or “pure” magnetic fields, that is, through fields perceived as “pure” in a static laboratory, actually experiences a combination of electric and magnetic fields, if viewed in the frame of reference of a moving observer with respect to whom the charge is momentarily at rest. For example, moving charges are well known to be deflected by magnetic fields. But in the rest frame of such a charge, there is no motion, and any acceleration of a charge must be due to the presence of a pure electric field from the point of view of an observer analyzing the motion in that reference frame. See Relativistic electrodynamics
A spinning electron can crudely be pictured as a spinning ball of charge, imitating a circulating electric current. This circulating current gives rise to a magnetic field distribution very similar to that of a small bar magnet, with north and south magnetic poles symmetrically distributed along the spin axis above and below the spin equator. This representative bar magnet can interact with external magnetic fields, one source of which is the magnetic field experienced by an electron in its rest frame, owing to its orbital motion through the electric field established by the central nucleus of an atom. In multielectron atoms, there can be additional, though generally weaker, interactions arising from the magnetic interactions of each electron with its neighbors, as all are moving with respect to each other and all have spin. The strength of the bar magnet equivalent to each electron spin, and its direction in space are characterized by a quantity called the magnetic moment, which also is quantized essentially because the spin itself is quantized. Studies of the effect of an external magnetic field on the states of atoms show that the magnetic moment associated with the electron spin is equal in magnitude to a unit called the Bohr magneton.
The energy of the interaction between the electron's magnetic moment and the magnetic field generated by its orbital motion is usually a small correction to the spectral term, and depends on the angle between the magnetic moment and the magnetic field or, equivalently, between the spin angular momentum vector and the orbital angular momentum vector (a vector perpendicular to the orbital plane whose magnitude is the size of the orbital angular momentum). Since quantum theory requires that the quantum number j of the electron's total angular momentum shall take values differing by integers, while l is always an integer, there are only two possible orientations for s relative to l: s must be either parallel or antiparallel to l.
For the case of a single electron outside the nucleus, the Dirac theory gives Eq. (7)
for the spin-orbit correction to the spectral terms. Here α = e2/(2ε0hc) ≅ 1/137 is called the fine structure constant.
In atoms having more than one electron, this fine structure becomes what is called the multiplet structure. The doublets in the alkali spectra, for example, are due to spin-orbit coupling; Eq. (7), with suitable modifications, can still be applied.
When more than one electron is present in the atom, there are various ways in which the spins and orbital angular momenta can interact. Each spin may couple to its own orbit, as in the one-electron case; other possibilities are orbit-other orbit, spin-spin, and so on. The most common interaction in the light atoms, called LS coupling or Russell-Saunders coupling, is described schematically in Eq. (8).
This notation indicates that the l are coupled strongly together to form a resultant L, representing the total orbital angular momentum. The si are coupled strongly together to form a resultant S, the total spin angular momentum. The weakest coupling is that between L and S to form J, the total angular momentum of the electron system of the atom in this state.
Coupling of the LS type is generally applicable to the low-energy states of the lighter atoms. The next commonest type is called jj coupling, represented in Eq. (9).
Each electron has its spin coupled to its own orbital angular momentum to form a ji for that electron. The various ji are then more weakly coupled together to give J. This type of coupling is seldom strictly observed. In the heavier atoms it is common to find a condition intermediate between LS and jj coupling; then either the LS or jj notation may be used to describe the levels, because the number of levels for a given electron configuration is independent of the coupling scheme.
Nuclear magnetism and hyperfine structure
Most atomic nuclei also possess spin, but rotate about 2000 times slower than electrons because their mass is on the order of 2000 or more times greater than that of electrons. Because of this, very weak nuclear magnetic fields, analogous to the electronic ones that produce fine structure in spectral lines, further split atomic energy levels. Consequently, spectral lines arising from them are split according to the relative orientations, and hence energies of interaction, of the nuclear magnetic moments with the electronic ones. The resulting pattern of energy levels and corresponding spectral-line components is referred to as hyperfine structure. See Nuclear moments
Nuclear properties also affect atomic spectra through the isotope shift. This is the result of the difference in nuclear masses of two isotopes, which results in a slight change in the Rydberg constant. There is also sometimes a distortion of the nucleus, which can be detected by ultrahigh precision spectroscopy. See Molecular beams, Particle trap
Doppler spread
In most cases, a common problem called Doppler broadening of the spectral lines arises, which can cause overlapping of spectral lines and make analysis difficult. The broadening arises from motion of the emitted atom with respect to a spectrometer. Several ingenious ways of isolating only those atoms nearly at rest with respect to spectrometric apparatus have been devised. The most powerful employ lasers and either involve saturation spectroscopy, utilizing a saturating beam and probe beam from the same tunable laser, or use two laser photons which jointly drive a single atomic transition and are generated in lasers so arranged that the first-order Doppler shifts of the photons cancel each other. See Doppler effect
Radiationless transitions
It would be misleading to think that the most probable fate of excited atomic electrons consists of transitions to lower orbits, accompanied by photon emission. In fact, for at least the first third of the periodic table, the preferred decay mode of most excited atomic systems in most states of excitation and ionization is the electron emission process first observed by P. Auger in 1925 and named after him. For example, a singly charged neon ion lacking a 1s electron is more than 50 times as likely to decay by electron emission as by photon emission. In the process, an outer atomic electron descends to fill an inner vacancy, while another is ejected from the atom to conserve both total energy and momentum in the atom. The ejection usually arises because of the interelectron Coulomb repulsion. See Auger effect
Cooling and stopping atoms and ions
Despite impressive progress in reducing Doppler shifts and Doppler spreads, these quantities remain factors that limit the highest obtainable spectroscopic resolutions. The 1980s and 1990s saw extremely rapid development of techniques for trapping neutral atoms and singly charged ions in a confined region of space, and then cooling them to much lower temperatures by the application of laser-light cooling techniques. Photons carry not only energy but also momentum; hence they can exert pressure on neutral atoms as well as charged ions. See Laser cooling
Schemes have been developed to exploit these light forces to confine neutral atoms in the absence of material walls, whereas various types of so-called bottle configurations of electromagnetic fields developed earlier remain the technique of choice for similarly confining ions. Various ingenious methods have been invented to slow down and even nearly stop neutral atoms and singly charged ions, whose energy levels (unlike those of most more highly charged ions) are accessible to tunable dye lasers. These methods often utilize the velocity-dependent light pressure from laser photons of nearly the same frequency as, but slightly less energetic than, the energy separation of two atomic energy levels to induce a transition between these levels.
The magnetooptic trap combines optical forces provided by laser light with a weak magnetic field whose size goes through zero at the geometrical center of the trap and increases with distance from this center. The net result is a restoring force which confines sufficiently laser-cooled atoms near the center. Ingenious improvements have allowed cooling of ions to temperatures as low as 180 × 10-9 K.
For more highly ionized ions, annular storage rings are used in which radial confinement of fast ion beams (with speeds of approximately 10% or more of the speed of light) is provided by magnetic focusing. Two cooling schemes are known to work on stored beams of charged particles, the so-called stochastic cooling method and the electron cooling method. In the former, deviations from mean stored particle energies are electronically detected, and electronic “kicks” that have been adjusted in time and direction are delivered to the stored particles to compensate these deviations. In electron cooling, which proves to be more effective for stored heavy ions of high charge, electron beams prepared with a narrow velocity distribution are merged with the stored ion beams. When the average speeds of the electrons and the ions are matched, the Coulomb interaction between the relatively cold (low-velocity-spread) electrons and the highly charged ions efficiently transfers energy from the warmer ions, thereby reducing the temperature of the stored ions.
References in periodicals archive ?
Cowan, The Theory of Atomic Structure and Spectra, Univ. |
ae3a2e183f17e27f | Mathematical Colloquium: Localization of interacting quantum particles with quasi-random disorder
Vieri Mastropietro
Universita’ di Milano
Monday, May 22, 2017 - 16:00 to 17:00
It is well established at a mathematical level that disorder can produce Anderson localization of the eigenvectors of the single particle Schrödinger equation. Does localization survive in presence of many body interaction? A positive answer to such question would have important physical consequences, related to lack of thermalization in closed quantum systems. Mathematical results on such issue are still rare and a full understanding is a challenging problem. We present an example in which localization can be proved for the ground state of an interacting system of fermionic particles with quasi random Aubry-Andre' potential. The Hamiltonian is given by $N$ coupled almost-Mathieu Schrödinger operators. By assuming Diophantine conditions on the frequency and density, we can establish exponential decay of the ground state correlations. The proof combines methods coming from the direct proof of convergence of KAM Lindstedt series with Renormalization Group methods for many body systems. Small divisors appear in the expansions, whose convergence follows exploiting the Diophantine conditions and fermionic cancellations. The main difficulty comes from the presence of loop graphs, which are the signature of many body interaction and are absent in KAM series. V.Mastropietro. Comm Math Phys 342, 217 (2016); Phys Rev Lett 115, 180401 (2015); Comm. Math. Phys. (2017)
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7d4a8e22ca31fb9e | Now we are ready to understand how it is that carbon is such a versatile element. It is at the basis of all organic chemistry and, in particular, biochemistry. The functioning of all living things depends on water and on the versatility of the carbon atom.
We saw that the carbon atom’s electron-shell configuration was
12C: 1s22s22p2
so it has four electrons in its valence shell (n=2). That enables it to share its four electrons with four others from other atoms. The bonds tend to be equally spaced around the carbon atom in the form of a tetrahedron, like those little creamer packets you get in cheap restaurants. For instance, a carbon atom can bond with four hydrogens, sharing each of its four valence electrons with one hydrogen, so each hydrogen has two and the carbon has eight and everybody is happy. This is called methane and looks like this.
"Methane-2D-stereo" by SVG version by Patricia.fidi - Own work. Licensed under Public Domain via Wikimedia Commons.
Methane molecule, CH4 by Patricia.fidi via Wikimedia Commons.
You should see one of the lower-right-hand hydrogens as pointing up out of the page; the other, down into it. The angles between any two adjacent connecting lines (which of course are only imagined by us) are about 109.5°. Carbon’s versatility in binding is illustrated by the examples in this diagram.
Versatiliy of carbon bonding, after Lehninger.
Versatility of carbon bonding, after Lehninger.
The dots represent valence electrons and the right-hand column is another way of looking at the product in terms of bonds rather than electrons. Each line between atoms is a shared pair of electrons. Note the double and triple inter-carbon bonds in the last two examples. This large number of ways of bonding is the key to carbon’s versatility. In fact, compared to the huge number of such molecules possible, only a relatively small number of the same biomolecules occur in living organisms. This is the first example we see of nature using the same set of techniques or tools all over the biosphere.
Single bonds between carbons also exist, of course, and have the particular advantage that the carbons and whatever is bonded to them can rotate around the axis linking the two carbons. This is more important than one might think. It turns out that some proteins function differently in their left-handed and right-handed versions. Since rotation can change the shape of the molecule, this enables biomolecules with hundreds of atoms to take on specific shapes with definite mechanical or fluid properties. (We will see some of this in the biochemistry chapter.)
The importance of water is not just because we drink it. Let’s go look at that.
Atomic energy levels and chemical bonding
Atomic structure is the basis of chemistry. It is explained by Quantum Mechanics, which is part of physics. We will see that physics explains chemistry, which explains physiology, which at least start to explain neurobiology. It’s one thing that leads to another.
In QM, the properties of a system, that is, a given object or set of objects, such as an atom, are given by the solution to the Schrödinger equation for the system. For atoms, there are a set of solutions, corresponding to different energy states of the atoms. What follows may smack of numerology.
Consider the hydrogen atom, composed of one negatively-charged electron in orbit around a nucleus containing one positively-charged proton. (This is an experimental result.) Look out, the orbit is not a well-defined path around the nucleus like those animations you see in TV ads, but rather a cloud of probability which indicates the likelihood that the electron will be found at any given point in the cloud. This is due to the probabilistic character of QM and the Uncertainty Principle. The different solutions to the Schrödinger equation express the possible energy values of the atom. Each one is specified by a set of integer numbers called quantum numbers. In the case of the hydrogen atom, they are the following:
1. The principal quantum number, designated by the symbol n, takes on integer values from 1 on up, but in practice only to 7. It indicates the shell, or level of the cloud, in which the electron is found. The values 1-7 are often indicated by the letters K, L, M…Q.
2. The orbital quantum number, l, indicates a level within the shell which is called the subshell, It can take on values from 0 up to n-1. The values 0-3 are often referred to as s, p, d and f.1The notations s, p, d and f come from spectroscopy and are abbreviated forms of sharp, principal, diffuse and fundamental.
3. The orbital magnetic quantum number, m, refers to the magnetic orientation of the electron. It can range from -l up through +l.
4. The electron spin, ms, can take on only two values, ½ or -½.
So the only allowed values for the quantum numbers are
n = 1, 2, 3, …
l = 0…n-1 (for a given value of n)
m = -l…+l (for a given value of l)
because those are the ones for which the Schrödinger equation has solutions. It is actually quite simple.
The QM exclusion principle forbids two electrons to occupy the same state. So each set of values (n, l, m, ms) can correspond to only one electron. The result is illustrated in the following table.
n (shell)
l (subshell)
m (orbital)
Max no. electrons
1 0 0 2
2 0
3 0
-1, 0, 1
-2. -1, 0, 1, 2
4 0
-1, 0, 1
-2, -1, 0, 1, 2
The fact that the quantum numbers do not vary continuously from, say, 0 to 0.001 and then 0.002 and on, but but jump from one integer value to another means that the energy of the electron in the electric field of the nucleus also takes on non-continuous values. These are called quantum states and are a feature, or if you prefer, a peculiarity, of QM.
The chemical properties of an atom depend only on the number of electrons. This is equal to the number of protons and is called the atomic number. All atoms except hydrogen have nuclei which also contain neutrons. The table summarizes the allowed values of quantum numbers for the first four shells.
In specifying which subshells are occupied by the electrons in an atom, one often uses the format
where l is specified as s, p, d or f and # is the number of electrons in the subshell. In its minimum energy state, called the ground state, the carbon atom (atomic number = 12, nucleus contains 6 protons and 6 neutrons) has the following electron configuration:
12C: 1s22s22p2
which indicates the maximum number of two electrons in shell 1, again in subshell s of shell 2 and the remaining two in subshell p of shell 2. Similarly, oxygen (atomic number = 16, 8 each of protons and neutrons) is
16O: 1s22s22p4
the meaning of which should now be clear.
What is interesting is that, for energetic reasons, each atom would like to have its outside subshell filled. If a few electrons are missing, it wants more; if most are missing, it might be willing to give up the rest in order to have an empty outside shell, referred to as the valence shell. (The number of electrons in this outer shell is called the valence.2Officially, the maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.) For instance, hydrogen
1H: 1s1
wants two electrons or none in its 1s shell, so it could give up its electron or gain one. What happens is, two H atoms share their electrons to make a molecule of H2, so each has two electrons half the time. Better than nothing.3To continue the anthropomorphisms, this is a kind of solidarity in which humans are often lacking.
Since oxygen already has shell 2 half-filled, it would probably prefer to gain electrons to fill it. And carbon… but carbon is special and will be considered in a moment.
Look at sodium (Na, atomic number 11) and chlorine (Cl, atomic number 17):
Na: 1s22s22p63s1
Cl: 1s22s22p63s23p5
Sodium could happily give up that 3s electron and chlorine could use it to fill up its 3p valence shell. And this is what happens in table salt, NaCl. If you put salt in water, it separates (for reasons which will be discussed shortly) into charged ions, Na+ and Cl, because chlorine is greedy and keeps that negative 3s electron it took away from sodium. This attraction for electrons is called electronegativity. This is very important in biochemical reactions in cells, as we shall see.
In brief, it turns out that elements with two, ten or eighteen electrons are particularly stable.
Chemistry is the study of chemical systems (atoms, molecules) and chemical bonding between such objects. In the case of NaCl, the sodium and chlorine have opposite electrical charge and the attractive electric force is what holds the molecule together. This is called ionic bonding. Sometimes, when atoms cannot decide which has more right to an electron, the electron is shared between them, as in H2, making both atoms relatively happy. Bonding based on shared electrons is called covalent bonding; it is a sort of consensus situation, if we may go on with the anthropomorphism.
Elements with the same number of electrons in their outer shells have similar chemical properties. So they are arranged in columns in that wonderful physical/chemical tool, the periodic table of the elements.
Periodic table of the elements
Periodic table from Wikimedia Commons
It is easy to see that each element in the first column is like hydrogen in having one electron in its valence shell.
H: 1s1
Li: 1s22s1
Na: 1s22s22p63s1
K: 1s22s22p63s23p64s1
… and so on.
The extra elements in the middle are rule-breakers. Instead of filling one subshell before moving on to the next, they start one, add a small number (often only one) of electrons to the next, then go back to finish filling the next-to-last.
Columns in the table are called groups; rows, periods.
The subshell configurations we have been giving are for the lowest energy state of the atom, called the ground state, in which subshells are filled from the “bottom” up (with some exceptions, as just mentioned). But if that hydrogen electron is struck by a photon, enough energy may be transferred from the photon to the 1s electron to push it into a higher-energy subshell. The atom is then said to be in an excited state. The electron may then re-descend spontaneously to the lower subshell, emitting a photon of energy equivalent to the difference in energy levels of the subshells. In QM, photons behave like waves whose energy is a function of their frequency, so the frequency – equivalently, the color – of the light emitted is characteristic of the difference in energy of the two subshells. Any atom’s subshells will therefore correspond to a given set of photon frequencies emitted and these are seen as colors, although not all these colors will be visible to a human eye. The set of frequencies constitute the spectrum of the atom and may be used to analyze the identity of a light source. In this way, we can identify the chemical components of light-emitting objects like distant stars.
There are two other types of bonding. We will consider hydrogen bonds very shortly in the discussion of water. The fourth form is due to the shifting electron density distribution around an atom. At times, this may form a temporary dipole even in a neutral atom. This may in turn induce a dipole in nearby atom in such a way that the two dipoles attract each other very weakly. This is London, or van der Waal’s, bonding.
The functioning of all living things depends on water and on the versatility of the carbon atom. So let’s start with carbon.
Notes [ + ]
1. The notations s, p, d and f come from spectroscopy and are abbreviated forms of sharp, principal, diffuse and fundamental.
3. To continue the anthropomorphisms, this is a kind of solidarity in which humans are often lacking.
What atomic physics and chemistry tell us
I am, reluctantly, a self-confessed carbon chauvinist. Carbon is abundant in the Cosmos. It makes marvelously complex molecules, good for life. I am also a water chauvinist. Water makes an ideal solvent system for organic chemistry to work in and stays liquid over a wide range of temperatures. But sometimes I wonder. Could my fondness for materials have something to do with the fact that I am made chiefly of them?
– Carl Sagan, Cosmos
The early stages of the universe and the lives of stars are the matter of physics and astronomy and their offspring, astrophysics and cosmology. By the time the first living things showed up on Earth, processes were occurring which require our knowing about the phenomena described by the science of chemistry. QM is the basis of atomic physics and that is the basis of chemistry, so we are ready for it.
To do even begin a comprehensive survey of chemistry is well beyond the scope of this document. We will illustrate its usefulness and some of its fruits by considering two subjects of great importance not only to Carl Sagan but to all of us – carbon and water.
In order to do that, it is necessary to know about several sujects:
Then we move on to consider, first, the past, starting almost 14 Gya. |
62ee07bbe14fe3b2 | Bohmian Indeterminism
David Bohm
see David Bohm (Photo Credit) Bohmian mechanics is not just an “interpretation” of quantum mechanics. It is a radical revision. In this note, I’d like to point out one reason that it’s an implausible revision: Bohmian mechanics is rampantly indeterministic in a way that quantum mechanics is not.
Setup: Review of Bohmian mechanics
bad cash credit loan quick In Bohmian mechanics, the locations of n particles are described by a point in a real manifold \mathcal{M}, called the configuration space. The trajectory of a system of particles is a curve t\mapsto x(t) through that manifold. The theory also includes a set of square-integrable functions on this space \psi:\mathcal{M}\rightarrow\CC called wavefunctions. A physical system in Bohmian mechanics can be characterized by a configuration space \mathcal{M}, a wavefunction space \mathcal{H}, and also a self-adjoint linear operator H:\mathcal{H}\rightarrow\mathcal{H} called the Hamiltonian. This Hamiltonian generates a one-parameter group of wavefunctions \psi(t) that solves the Schrödinger equation,
1 hour payday loans no credit check direct lender uk \[i\frac{d}{dt}\psi(t) = H\psi(t).\]
payday loans for people who don't work For a given Bohmian system (\mathcal{M},V,H), an initial condition is a pair (x,\psi), with x\in\mathcal{M} and \psi\in \mathcal{H}. The fundamental law of Bohmian mechanics then says that, given an initial (x,\psi), the trajectory of the system x(t) is a solution to the Bohmian Guidance Equation, \[\frac{dx}{dt} = \frac{1}{m}\cdot\mathrm{Im}\;\frac{\nabla\cdot \psi(t)}{\psi(t)},\]
payday loans ace where \psi(t) is the solution to the Schrödinger equation with initial condition \psi=\psi(0).
Rampant Bohmian indeterminism
united cash payday loans Take as simple a Bohmian system (\mathcal{M},V,H) as you can imagine: a free particle confined to a finite space. It turns out that the Bohmian description is rampantly indeterministic.
go site Let the configuration space be \mathcal{M}=(-1,1), which describes the possible locations of a particle on a string of finite length. Let \mathcal{H} be the set of differentiable square integrable functions \psi:(-1,1)\rightarrow\CC. And let H=\frac{m}{2}P^2, the Hamiltonian for a particle free of any forces or interactions, and where P=i\frac{d}{dx}. As a dirt-simple example of indeterminism, choose the initial condition x=0, and \psi(x) given by, \[ \psi(x) = e^{\frac{2m}{3}ix^{3/2}}. \]
This wavefunction is square integrable and differentiable. (Square-integrability follows from the fact that \int_{-1}^{1}\psi^*(x)\psi(x) dx = \int_{-1}^{1}dx = 2 <\infty, and differentiability is obvious.) But let’s calculate what the Guidance Equation looks like for this initial wavefunction. Since \nabla P^2 = \nabla(-\nabla^2) = P^2\nabla, the unitary propagator \UU_t=e^{-itP^2/2m} for our Hamiltonian satisfies \nabla \UU_t = \UU_t \nabla. Therefore,
\[ \frac{dx}{dt} = \frac{1}{m}\mathrm{Im}\; \frac{\nabla\UU_t\psi(x)}{\UU_t\psi(x)} = \frac{1}{m}\mathrm{Im}\;\frac{\UU_t\nabla\psi(x)}{\UU_t \psi(x)} = \frac{1}{m}\mathrm{Im}\;\frac{imx^{1/2}\UU_t\psi(x)}{\UU_t\psi(x)} = x^{1/2}. \]
The differential equation dx/dt = x^{1/2} is well-known to be indeterministic, following a much-discussed example of John Norton (2008 / animated summary). But let me make it explicit: a Bohmian particle with this initial configuration is compatible with a continuum of future trajectories all satisfying the Guidance Equation. Namely,
\[ x(t) = \begin{cases} (1/144)(t - T)^4 & \text{for $t\geq T$}\\ 0 & \text{for $t \leq T$.} \end{cases}\]
for any arbitrary time T. (We restrict our attention to times t during which the particle is in the interval (-1,1), namely (t-T)^4<1.) These solutions correspond to a Bohmian particle that sits at x=0 up until an arbitrary time T, when it randomly begins moving.
Not the dome
As a point of comparison, recall that many have complained about the “unphysical” features of the surface of Norton’s dome, such as the infinite Gaussian curvature at the apex (e.g. here and here). No such complaining need be tolerated in the case of Bohmian mechanics. There is no surface to complain about. There is only the wavefunction \psi(x), which is a perfectly boring, deterministic wavefunction from the perspective of orthodox quantum mechanics. In particular, it is the initial condition for a unique solution \psi(t)=e^{-itH}\psi to the Schrödinger equation, which is defined for all times t. It is only with the addition of the Bohmian Guidance Equation that a pathology occurs.
In order to avoid such pathologies, Bohmian mechanics must somehow excise this class of wavefunctions from the theory. But it’s not clear how to motivate this excision in a non-ad hoc way. And it’s even less clear whether it can be done in a way that avoids doing damage to the ordinary quantum dynamics.
2 thoughts on “Bohmian Indeterminism
1. Jon Clark
This page is full of illogical and erroneous information on Bohm’s incredible interpretation of the Quantum Theory:
1) dx/dt=\sqrt{x} is not “indeterministic” – it’s just a differential equation. Differential equations by themselves say nothing about determinism; they may be chaotic but this one is in fact very simple to solve. You know that x(t) =t^2/4 is a solution which is physically real. The particle starts from rest and then accelerates. That random acceleration solution on the dome is unphysical: if the particle is placed at the very top it will never move. Just because a solution is allowed by the differential equation doesn’t mean it’s physically real.
2) Even if what you said about your wave function ψ is correct, does that wave function correspond to a physically real Hamiltonian? What is the potential energy of this hypothetical system? You’ll find that the wave function you have given is impossible because it is not differentiable twice over [-1,1], what you claimed as the domain of integration. The second derivative of ψ is indeterminate at x=0. Every wave function must be differentiable twice in order for it to solve Schödinger’s equation, and the Guidance equation is a consequence of Bohm’s equations being derived from Schödinger’s equation and reproducing Jacobi’s equation, H(S)+dS/dt=0.
In summary: Bohm’s interpretation is in fact deterministic and experimentally equivalent to standard Quantum theory, albeit while painting a radically different and actually sensible picture of the mysterious atomic world.
1. Bryan Post author
Hi Jon. On 1): “Indeterminism” is just a synonym to say that the DiffEQ has non-unique solutions for some initial data. I do agree that the dome is pathological, although just calling some solutions “not physical” isn’t quite precise enough for me; some of the precise pathologies are spelled out here.
On 2): The QM Hamiltonian for my system here is just the free-particle Hamiltonian; see the second paragraph. There is no potential energy. My initial \psi is square-integrable, so it is a perfectly good quantum state in the Hilbert space. It evolves (in ordinary QM) as a simple free wave according to the equation \psi(t)=e^{-itH}\psi. It is an absolutely trivial and boring quantum system, unless you’re a Bohmian, in which case it is radically pathological.
This is because wavefunctions don’t need to be twice-differential in QM, and in general they cannot all be when we say that the states form a Hilbert space. Schrödinger’s equation does not require any differentiability conditions when it is cast in its integral form \psi(t)=e^{-itH}\psi. The unitary operator U(t)=e^{-itH} is bounded, and therefore defined on the entire Hilbert space of states, even though the Hamiltonian (and thus the differential form of the equation) is not defined for all states.
All of the problems here really do come directly from Bohmian mechanics. In ordinary QM, the system is perfectly non-pathological. I’m afraid I’m not as optimistic as you are about BM!
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8fbb93a30393418d | Pauli exclusion principle
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Wolfgang Pauli
The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously. In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: n, the principal quantum number, , the angular momentum quantum number, m, the magnetic quantum number, and ms, the spin quantum number. For example, if two electrons reside in the same orbital, and if their n, , and m values are the same, then their ms must be different, and thus the electrons must have opposite half-integer spins of 1/2 and −1/2. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin-statistics theorem of 1940.
A more rigorous statement is that the total wave function for two identical fermions is antisymmetric with respect to exchange of the particles. This means that the wave function changes its sign if the space and spin co-ordinates of any two particles are interchanged.
Particles with an integer spin, or bosons, are not subject to the Pauli exclusion principle: any number of identical bosons can occupy the same quantum state, as with, for instance, photons produced by a laser and Bose–Einstein condensate.
The Pauli exclusion principle describes the behavior of all fermions (particles with "half-integer spin"), while bosons (particles with "integer spin") are subject to other principles. Fermions include elementary particles such as quarks, electrons and neutrinos. Additionally, baryons such as protons and neutrons (subatomic particles composed from three quarks) and some atoms (such as helium-3) are fermions, and are therefore described by the Pauli exclusion principle as well. Atoms can have different overall "spin", which determines whether they are fermions or bosons — for example helium-3 has spin 1/2 and is therefore a fermion, in contrast to helium-4 which has spin 0 and is a boson.[1]:123–125 As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms.
"Half-integer spin" means that the intrinsic angular momentum value of fermions is (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states. Bosons include the photon, the Cooper pairs which are responsible for superconductivity, and the W and Z bosons. (Fermions take their name from the Fermi–Dirac statistical distribution that they obey, and bosons from their Bose–Einstein distribution).
In the early 20th century it became evident that atoms and molecules with even numbers of electrons are more chemically stable than those with odd numbers of electrons. In the 1916 article "The Atom and the Molecule" by Gilbert N. Lewis, for example, the third of his six postulates of chemical behavior states that the atom tends to hold an even number of electrons in any given shell, and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube (see: cubical atom).[2] In 1919 chemist Irving Langmuir suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells around the nucleus.[3] 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".[4]:203
Pauli looked for an explanation for these numbers, which were at first only empirical. At the same time he was trying to explain experimental results of the Zeeman effect in atomic spectroscopy and in ferromagnetism. He found an essential clue in a 1924 paper by Edmund C. Stoner, which 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.[5][6]
Connection to quantum state symmetry[edit]
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state and the other in state , and is given by:
and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies A(x,y) = 0 when x = y, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric second-order tensor.
Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component
is necessarily antisymmetric. To prove it, consider the matrix element
This is zero, because the two particles have zero probability to both be in the superposition state . But this is equal to
The first and last terms on the right side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
Pauli principle in advanced quantum theory[edit]
According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin.
In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta-function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space the exclusion principle is valid also for finite repulsion in a Bose gas with delta-function interactions,[7] as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.
Atoms and the Pauli principle[edit]
The Pauli exclusion principle helps explain a wide variety of physical phenomena. One particularly important consequence of the principle is the elaborate electron shell structure of atoms and the way atoms share electrons, explaining the variety of chemical elements and their chemical combinations. An electrically neutral atom contains bound electrons equal in number to the protons in the nucleus. Electrons, being fermions, cannot occupy the same quantum state as other electrons, so electrons have to "stack" within an atom, i.e. have different spins while at the same electron orbital as described below.
An example is the neutral helium atom, which has two bound electrons, both of which can occupy the lowest-energy (1s) states by acquiring opposite spin; as spin is part of the quantum state of the electron, the two electrons are in different quantum states and do not violate the Pauli principle. However, the spin can take only two different values (eigenvalues). In a lithium atom, with three bound electrons, the third electron cannot reside in a 1s state, and must occupy one of the higher-energy 2s states instead. Similarly, successively larger elements must have shells of successively higher energy. The chemical properties of an element largely depend on the number of electrons in the outermost shell; atoms with different numbers of occupied electron shells but the same number of electrons in the outermost shell have similar properties, which gives rise to the periodic table of the elements.[8]:214–218
Solid state properties and the Pauli principle[edit]
In conductors and semiconductors, there are very large numbers of molecular orbitals which effectively form a continuous band structure of energy levels. In strong conductors (metals) electrons are so degenerate that they cannot even contribute much to the thermal capacity of a metal.[9]:133–147 Many mechanical, electrical, magnetic, optical and chemical properties of solids are the direct consequence of Pauli exclusion.
Stability of matter[edit]
The stability of the electrons in an atom itself is unrelated to the exclusion principle, but is described by the quantum theory of the atom. The underlying idea is that close approach of an electron to the nucleus of the atom necessarily increases its kinetic energy, an application of the uncertainty principle of Heisenberg.[10] However, stability of large systems with many electrons and many nucleons is a different matter, and requires the Pauli exclusion principle.[11]
A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered the balance of attractive (electron–nuclear) and repulsive (electron–electron and nuclear–nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.[13][14]
The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or Coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.
Astrophysics and the Pauli principle[edit]
Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in some astronomical objects. In 1995 Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in neutron stars, although at a much higher density than in ordinary matter.[15] It is a consequence of general relativity that, in sufficiently intense gravitational fields, matter collapses to form a black hole.
Astronomy provides a spectacular demonstration of the effect of the Pauli principle, in the form of white dwarf and neutron stars. In both bodies, atomic structure is disrupted by extreme pressure, but the stars are held in hydrostatic equilibrium by degeneracy pressure, also known as Fermi pressure. This exotic form of matter is known as degenerate matter. The immense gravitational force of a star's mass is normally held in equilibrium by thermal pressure caused by heat produced in thermonuclear fusion in the star's core. In white dwarfs, which do not undergo nuclear fusion, an opposing force to gravity is provided by electron degeneracy pressure. In neutron stars, subject to even stronger gravitational forces, electrons have merged with protons to form neutrons. Neutrons are capable of producing an even higher degeneracy pressure, neutron degeneracy pressure, albeit over a shorter range. This can stabilize neutron stars from further collapse, but at a smaller size and higher density than a white dwarf. Neutrons are the most "rigid" objects known; their Young modulus (or more accurately, bulk modulus) is 20 orders of magnitude larger than that of diamond. However, even this enormous rigidity can be overcome by the gravitational field of a massive star or by the pressure of a supernova, leading to the formation of a black hole.[16]:286–287
See also[edit]
1. ^ Kenneth S. Krane (5 November 1987). Introductory Nuclear Physics. Wiley. ISBN 978-0-471-80553-3.
2. ^ [1]
3. ^ Langmuir, Irving (1919). "The Arrangement of Electrons in Atoms and Molecules" (PDF). Journal of the American Chemical Society. 41 (6): 868–934. doi:10.1021/ja02227a002. Retrieved 2008-09-01.
4. ^ Shaviv, Glora. The Life of Stars: The Controversial Inception and Emergence of the Theory of Stellar Structure (2010 ed.). Springer. ISBN 978-3642020872.
5. ^ Straumann, Norbert (2004). "The Role of the Exclusion Principle for Atoms to Stars: A Historical Account". Invited talk at the 12th Workshop on Nuclear Astrophysics. CiteSeerX accessible.
6. ^ Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren W. Pauli, Zeitschrift für Physik, February 1925, Volume 31, Issue 1, pp 765-783
7. ^ A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6, page 283, 1982
9. ^ Kittel, Charles (2005), Introduction to Solid State Physics (8th ed.), USA: John Wiley & Sons, Inc., ISBN 978-0-471-41526-8
10. ^ Elliot J. Lieb The Stability of Matter and Quantum Electrodynamics
11. ^ This realization is attributed by Lieb and by GL Sewell (2002). Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press. ISBN 0-691-05832-6. to FJ Dyson and A Lenard: Stability of Matter, Parts I and II (J. Math. Phys., 8, 423–434 (1967); J. Math. Phys., 9, 698–711 (1968) ).
12. ^ As described by FJ Dyson (J.Math.Phys. 8, 1538–1545 (1967) ), Ehrenfest made this suggestion in his address on the occasion of the award of the Lorentz Medal to Pauli.
14. ^ Dyson, Freeman (1967). "Ground‐State Energy of a Finite System of Charged Particles". J. Math. Phys. 8 (8): 1538–1545. Bibcode:1967JMP.....8.1538D. doi:10.1063/1.1705389.
15. ^ Lieb, E. H.; Loss, M.; Solovej, J. P. (1995). "Stability of Matter in Magnetic Fields". Physical Review Letters. 75 (6): 985–9. Bibcode:1995PhRvL..75..985L. arXiv:cond-mat/9506047Freely accessible. doi:10.1103/PhysRevLett.75.985.
16. ^ Martin Bojowald (5 November 2012). The Universe: A View from Classical and Quantum Gravity. John Wiley & Sons. ISBN 978-3-527-66769-7.
• Dill, Dan (2006). "Chapter 3.5, Many-electron atoms: Fermi holes and Fermi heaps". Notes on General Chemistry (2nd ed.). W. H. Freeman. ISBN 1-4292-0068-5.
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
• Massimi, Michela (2005). Pauli's Exclusion Principle. Cambridge University Press. ISBN 0-521-83911-4.
External links[edit] |
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e66b11912939baef | H for Hydrogen
Hydrogen was first recognized as a distinct element in 1766 by English scientist Henry Cavendish, when he prepared it by reacting hydrochloric acid with zinc.
French scientist Antoine Lavoisier named the element hydrogen (1783). The name comes from the Greek ‘hydro’ meaning water and ‘genes’ meaning forming – hydrogen is one of the two water forming elements.
Here are 12 interesting facts about hydrogen, the simplest and commonest element in the universe.
About 10 percent of the weight of living organisms is hydrogen – mainly in water, proteins and fats.
Liquid hydrogen has the lowest density of any liquid.
Hydrogen is the only element that can exist without neutrons. Hydrogen’s most abundant isotope, Protium, has no neutrons. Some recent theories of particle physics predict that proton decay can occur with a half-life of the order of 1036 years.
Antihydrogen is the only antimatter element made so far, with atoms of antihydrogen synthesized at CERN lasting for as long as 1000 seconds (almost 17 minutes). Each atom of antihydrogen contains a positron (positively charged version of the electron) orbiting an antiproton (negatively charged version of the proton).
Hydrogen is believed to be one of three elements produced in the Big Bang; the others are helium and lithium.
We owe most of the energy on our planet to hydrogen. The Sun’s nuclear fires convert hydrogen to helium releasing a large amount of energy.
Hydrogen is the only atom for which the Schrödinger equation has an exact solution.
The first chain reaction discovered was not a nuclear reaction; it was a chemical chain reaction. It was discovered in 1913 by Max Bodenstein, who saw a mixture of chlorine and hydrogen gases explode when triggered by light. The chain reaction mechanism was fully explained in 1918 by Walther Nernst.
Liquid hydrogen is used as a rocket fuel, for example powering the Space Shuttle’s lift-off and ascent into orbit.
Hydrogen’s two heavier isotopes (deuterium and tritium) are used in nuclear fusion.
Large quantities of hydrogen are used in the production of ammonia, hydrogenation of fats and oils, methanol production, hydrocracking, and hydrodesulfurization. Hydrogen is also used in metal refining.
I am really enjoying these posts. They are learning, relearning and “wow, I did not know that”, kind of experiences for me.
References: wikipedia.com, chemicool.com
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d7c91921ba2dbb4e | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
SIGMA 7 (2011), 113, 11 pages arXiv:1112.2333
Breaking Pseudo-Rotational Symmetry through H+2 Metric Deformation in the Eckart Potential Problem
Nehemias Leija-Martinez a, David Edwin Alvarez-Castillo b and Mariana Kirchbach a
a) Institute of Physics, Autonomous University of San Luis Potosi, Av. Manuel Nava 6, San Luis Potosi, S.L.P. 78290, Mexico
b) H. Niewodniczanski Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Kraków, Poland
Received October 12, 2011, in final form December 08, 2011; Published online December 11, 2011; Misprints are corrected December 24, 2011
The peculiarity of the Eckart potential problem on H+2 (the upper sheet of the two-sheeted two-dimensional hyperboloid), to preserve the (2l+1)-fold degeneracy of the states typical for the geodesic motion there, is usually explained in casting the respective Hamiltonian in terms of the Casimir invariant of an so(2,1) algebra, referred to as potential algebra. In general, there are many possible similarity transformations of the symmetry algebras of the free motions on curved surfaces towards potential algebras, which are not all necessarily unitary. In the literature, a transformation of the symmetry algebra of the geodesic motion on H+2 towards the potential algebra of Eckart's Hamiltonian has been constructed for the prime purpose to prove that the Eckart interaction belongs to the class of Natanzon potentials. We here take a different path and search for a transformation which connects the (2l+1) dimensional representation space of the pseudo-rotational so(2,1) algebra, spanned by the rank-l pseudo-spherical harmonics, to the representation space of equal dimension of the potential algebra and find a transformation of the scaling type. Our case is that in so doing one is producing a deformed isometry copy to H+2 such that the free motion on the copy is equivalent to a motion on H+2, perturbed by a coth interaction. In this way, we link the so(2,1) potential algebra concept of the Eckart Hamiltonian to a subtle type of pseudo-rotational symmetry breaking through H+2 metric deformation. From a technical point of view, the results reported here are obtained by virtue of certain nonlinear finite expansions of Jacobi polynomials into pseudo-spherical harmonics. In due places, the pseudo-rotational case is paralleled by its so(3) compact analogue, the cotangent perturbed motion on S2. We expect awareness of different so(2,1)/so(3) isometry copies to benefit simulation studies on curved manifolds of many-body systems.
Key words: pseudo-rotational symmetry; Eckart potential; symmetry breaking through metric deformation.
pdf (467 Kb) tex (130 Kb) [previous version: pdf (466 kb) tex (130 kb)]
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Previous article Next article Contents of Volume 7 (2011) |
cedcc77a4fbae697 | Off-resonance magnetoresistance spike in irradiated ultraclean 2D electron systems
Nanoscale Research Letters20138:241
DOI: 10.1186/1556-276X-8-241
Received: 11 November 2012
Accepted: 6 April 2013
Published: 16 May 2013
We report on the theoretical studies of a recently discovered strong radiation-induced magnetoresistance spike obtained in ultraclean two-dimensional electron systems at low temperatures. The most striking feature of this spike is that it shows up on the second harmonic of the cyclotron resonance. We apply the radiation-driven electron orbits model in the ultraclean scenario. Accordingly, we calculate the new average advanced distance by the electron in a scattering event which will define the unexpected resonance spike position. Calculated results are in good agreement with experiments.
Off-resonance Microwaves Magnetoresistance
Transport excited by radiation in a two-dimensional electron system (2DES) has been always [13] a central topic in basic and especially in applied research. In the last decade, it was discovered that when a high mobility 2DES in a low and perpendicular magnetic field (B) is irradiated, mainly with microwaves (MW), some striking effects are revealed: radiation-induced magnetoresistance (R x x ) oscillations and zero resistance states (ZRS) [4, 5]. Different theories and experiments have been proposed to explain these effects [618], but the physical origin is still being questioned. An interesting and challenging experimental results, recently obtained [19] and as intriguing as ZRS, consists in a strong resistance spike which shows up far off-resonance. It occurs at twice the cyclotron frequency, w≈2wc[19], where w is the radiation frequency, and wc is the cyclotron frequency.
Remarkably, the only different feature in these experiments [19] is the use of ultraclean samples with mobility μ ∼ 3 × 107 cm2 V s-1 and lower temperatures T∼0.4 K. Yet, for the previous ‘standard’ experiments and samples [4, 5], mobility is lower (μ < 107 cm2 V s-1) and T higher (T ≥ 1.0 K).
In this letter, we theoretically study this radiation-induced R xx spike, applying the theory developed by the authors, the radiation-driven electron orbits model[610, 2025]. According to the theory, when a Hall bar is illuminated, the electron orbit centers perform a classical trajectory consisting in a classical forced harmonic motion along the direction of the current at the radiation frequency, w. This motion is damped by the interaction of electrons with the lattice ions and with the consequent emission of acoustic phonons.
We extend this model to an ultraclean sample, where the Landau levels (LL), which in principle are broadened by scattering, become very narrow. This implies an increasing number of states at the center of the LL sharing a similar energy. In between LL, the opposite happens: the density of states dramatically decreases. This will eventually affect the measured stationary current and R x x .
We obtain that in the ultraclean scenario, the measured current on average is the same as the one obtained in a sample with full contribution to R x x but delayed as if it were irradiated with a half MW frequency (w/2). Accordingly, the cyclotron resonance is apparently shifted to a new B-position around w ≈ 2wc.
The radiation-driven electron orbits model was developed to explain the R x x response of an irradiated 2DEG at low magnetic field [610, 2025]. The corresponding time-dependent Schrödinger equation can be exactly solved. Thus, we first obtain an exact expression of the electronic wave vector for a 2DES in a perpendicular B, a DC electric field, and radiation:
Ψ N ( x , t ) ϕ n ( x - X - x cl ( t ) , t ) ,
where ϕ n is the solution for the Schrödinger equation of the unforced quantum harmonic oscillator. x cl (t) is the classical solution of a forced and damped harmonic oscillator:
x cl = e E o m ( w c 2 - w 2 ) 2 + γ 4 cos wt = A cos wt ,
where E0 is the MW electric field, and γ is a damping factor for the electronic interaction with the lattice ions.Then, the obtained wave function is the same as the standard harmonic oscillator, where the center is displaced by x cl (t). Next, we apply time-dependent first-order perturbation theory to calculate the elastic charged impurity scattering rate between the two oscillating Landau states, the initial Ψ n , and the final state Ψ m [610, 2024]: Wn,m = 1 / τ, with τ being the elastic charged impurity scattering time.
We find that the average effective distance advanced by the electron in every scattering jump [610, 2024],
Δ XMW = Δ X0 + A cosw τ, where Δ X0, is the advanced distance in the dark [26]. Finally, the longitudinal conductivity σ xx is given by,
σ xx dE Δ X MW τ = dE Δ X 0 + A cos w τ τ ,
with E being the energy [26], and Δ X MW τ the average electron drift velocity. To obtain R xx , we use the usual tensor relationships R xx = σ xx σ xx 2 + σ xy 2 σ xx σ xy 2
Importantly, resistance is directly proportional to conductivity: R xx σ xx . Thus, finally, the dependence of the magnetoresistance with radiation is given by:
R xx A cos w τ .
Results and discussion
For ultraclean samples, γ is very small; for experimental magnetic fields [19], Γ < w c This condition will dramatically affect the average advanced distance by electron in every scattering process. In contrast with standard samples where electrons always find available empty states where to be scattered, in ultraclean samples, we can clearly find two different scenarios that are described in Figure 1.
Figure 1
Schematic diagrams of electronic transport for a ultraclean sample (narrow Landau levels and weak overlapping). (a) In the lower part, no MW field is present. (b) The orbits move backwards during the jump, and the scattering ends around the central part of a LL (grey stripes); then, we have full contribution to the current. (c) The scattering jump ends in between LL (white stripes), giving rise to a negligible contribution to the current because the low density of final Landau states. (d) We depict a ZRS situation. Dotted line represents the Fermi level before the scattering jump; white and black circles represent empty and occupied orbits after the jump, respectively.
In the four panels of energy versus distance, the grey stripes are LL tilted by the action of the DC electric field in the x direction. Here, LL are narrow ( Γ < w c and hardly overlap each other, leaving regions with a low density of states in between (white stripes). Therefore, we can observe regularly alternating grey (many states) and white (few states) stripes equally spread out. The first scenario corresponds (see Figure 1b) to an electron being scattered to the central part of a LL. As a result, the scattering can be completed with empty states to be occupied; we obtain full contribution to the conductivity and R x x . In Figure 1c, we describe the second scenario where the electron scatters to a region in between LL with a very low density of states. Obviously, in this case, there is no much contribution to the average or stationary current. In Figure 1d, the scattering is not efficient because the final Landau state is occupied. Both regimes, ‘in-between LL’ and ‘center of LL’, are distributed equally and alternately along one cycle of the MW-driven electron orbit motion; then, only in one-half of the cycle, we would obtain a net contribution to the current or R x x .
This situation is physically equivalent to having a half amplitude harmonic motion of frequency w. On the other hand, it is well known that for a simple harmonic motion, it is fulfilled that averaging in one cycle, A 2 cos wt = A cos w 2 t Adapting this condition to our specific case, our MW-driven (forced) harmonic motion can be perceived on average as a forced harmonic motion of whole amplitude (full scattering contribution during the whole cycle) and half frequency:
A 2 cos w τ A 2 cos w 2 τ ,
being, A 2 = e E o m ( w c 2 - ( w 2 ) 2 ) 2 + γ 4 and A = e E o m ( w c 2 - w 2 ) 2 + γ 4 last equation is only fulfilled when A ≃ A2, which is a good approximation according to the experimental parameters [19], (T = 0.4 K, B ≤ 0.4 T,w=101 GHz and MW power P ∼ 0.4-1 mW). With these parameters, we obtain that the amplitudes A and A2 are similar and of the order of 10-6 to 107 m. The consequence is that the ultraclean harmonic motion (electron orbit center displacement) behaves as if the electrons were driven by the radiation of half frequency. Therefore, applying next the theory [610] for the ultraclean scenario, it is straightforward to reach an expression for magnetoresistance:
R xx e E o m ( w c 2 - ( w 2 ) 2 ) 2 + γ 4 cos w 2 τ.
According to it, now the resonance in R x x will take place at w ≈ 2wc, as experimentally obtained [19]. The intensity of the R xx spike will depend on the relative value of the frequency term, ( w c 2 - ( w 2 ) 2, and the damping parameter γ in the denominator of the latter R xx expression. When γ leads the denominator, the spike is smeared out. Yet, in situations where γ is smaller than the frequency term, the resonance effect will be more visible, and the spike will show up.
The damping parameter γ is given, after some lengthy algebra, by [27]:
γ = 1 τ ac T × 2 eB h m = 0 1 Π Γ ( E n - w ac - E m ) 2 + Γ 2 T × 1 - e - ΠΓ w c 1 + e - ΠΓ w c ,
where wac is the frequency of the acoustic phonons for the experimental parameters [19].For ultraclean samples γ is small [19], and according to the last expression, this makes also the term inside the brackets and γ smaller [2830]. In other words, it makes the damping by acoustic phonon emission and the release of the absorbed energy to the lattice increasingly difficult. Therefore, we have a bottleneck effect for the emission of acoustic phonons. Now, it is possible to reach a situation where ( w c 2 - ( w 2 ) 2 ) 2 γ 4, making a resonance effect visible and, therefore, giving rise to a strong resonance peak at w ≈ 2wc.
In Figure 2, we present a calculated irradiated R xx vs. static magnetic field for a radiation frequency of f = 101 GHz. The curve or a dark situation is also presented. For a temperature T = 0.4 K, we obtain a strong spike at w ≈ 2wc as in the experiments by [19].
Figure 2
Calculated irradiated magnetoresistance versus static magnetic field for a radiation frequency of f = 101 GHz. The dark curve is also presented. For a temperature of 0.4 K, we observe an intense spike at w ≈ 2wc.
Finally, we obtain the usual radiation-induced R x x oscillations and ZRS as in standard samples.
In this letter, we have presented a theoretical approach to the striking result of the magnetoresistance spike in the second harmonic of the cyclotron frequency. According to our model, the strong change in the density of Landau states in ultraclean samples affects dramatically the electron impurity scattering and eventually the conductivity. The final result is that the scattered electrons perceive radiation as of half frequency. The calculated results are in good agreement with experiments.
Authors’ information
JI is an associate professor at the University Carlos III of Madrid. He is currently studying the effect of radiation on two-dimensional electron systems.
This work is supported by the MCYT (Spain) under grant MAT2011-24331 and ITN grant 234970 (EU).
Authors’ Affiliations
Escuela Politécnica Superior, Universidad Carlos III
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© Iñarrea; licensee Springer. 2013
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aeecb0370f6585f3 | Field of Science
Recyle, Reuse
A quick flip through my students' problem sets tells me more about them than just their ability to do quantum mechanics. My greener students make good use of the reverse side of the mountain of pages their colleagues print out every day. The paper drafts and announcements on the back side often catch my eye and offer a window into a student world I don't often get to see. Of course, recycling sheets for writing is hardly a new phenomenon. Palimpsests are parchment or vellum pages that have been erased by various methods and reused. The earlier, generally chemical, methods of erasure left faint traces of the original writing on the sheets. As methods improved, and relied more on mechanical means, such as sanding with pumice, the erasure became more complete.
In my quantum chemistry class we talked about fluorescence today. A common example of fluorescence is the odd luminescence of white t-shirts under a black light. The black light is a source of UV light, which excites some of the molecules in the detergent residue (yep - that bright white shirt is not quite as clean as you think!). The molecules then re-emit light at a slightly lower energy, which happens to be in the visible, and that we perceive as an eerie glow. The glow is present even in daylight, but the amount of visible radiation emitted through fluorscence is so much smaller than what is in incident sunlight that it swamps out the effect. But it does make your whites look subtly brighter, which is why detergent companies include "brighteners" in their formulations.
So what do white shirts have to do with palimpsests? X-rays are just another form of light (albeit very high energy light) and can cause fluorescence, too. Iron in the ink is the source of the fluorescence. Researchers at Stanford have recently uncovered not only an Archimedes manuscript hidden underneath a 13th century Byzantine prayer book, but also a text by Hyperides, a contemporary of Aristotle. The discovery of this text extends the known works of Hyperides by 20%!
A Request to Readers
Half-awake, half-life
I had a moderate allergic reaction to peanuts last night. I took diphenhydramine (Benadryl) and the hives had subsided by this morning. Lecturing on perturbation theory was more challenging. I felt like I was walking in a fog. Which got me to wondering, just what was the half-life of Benadryl? Benadryl has a relatively long half-life, between 8 and 10 hours. A typical 50 mg dose leads to a peak blood level of around 80 nanograms/ml. Most people feel drowsy at blood levels around 30 nanograms/ml. Assuming first order kinetics apply to the breakdown/elimination of Benadryl, a 30 nanogram/ml is not unlikely 10 to 15 hours later. Which would certainly explain my fogged state this morning! But not so foggy as to be unable to work the kinetics....
The Sticking Point - or Weird Words of Science 11: Eutectic
It would be an understatement to say that my youngest son is not looking forward to his annual flu shot. Last week, the Philadelphia Inquirer had a piece on helping kids cope with the pain of immunizations. Son was not particularly impressed with the advice, but he noticed that they refered to a cream which could be applied to diminish the pain of injections. "What is it, Mom?" "EMLA, I think." "EMLA?" "A eutectic mixture of local anesthetics..." Somewhere around mixture, I think I lost him!
What is a eutectic mixture? Eutectic comes from the Greek eutektos for "easily melted"(any resemblence to tectonic is, I believe, purely accidental - tectonic also comes from the Greek, but for building, not melting!). An eutectic mixture is one in which the melting point of the mix is lower than the melting point of either of the components. The binary phase diagram has a "eutectic point". EMLA is a mixture of equal weights of lidocaine and prilocaine, made into an emulsion.
It's apparently quite effective, but requires a lead time of several hours (and the foresight to ask the pediatrician for a prescription!).
"What was to be demonstrated" needed to be demonstrated!
A student in my office hours today asked me what the term QED meant at the bottom of a page, and got a (very short) lesson in Latin. Quod erat demonstrandum, "what was to be demonstrated", is a translation of the Greek hoper edei deixai used by Euclid to close a proof. Modern mathematical publications often substitute other symbols, including a or simply note: proven.
Orion brandy anyone?
I'll admit to being a Trekkie at some time in my life, but Dr. McCoy's stash of Saurian brandy aside, there is alcohol in interstellar space. More than 120 molecules and ions -- including ethanol -- have been identified by radioastronomers in interstellar space. The transitions between different molecular rotational states give rise to very specific lines, which can be used as molecular fingerprints.
The lines which helped identify ethanol are rotations around the carbon-carbon single bonds, rather like little propellers turning. The lines arise from vicinity of the Orion Nebula (a mere 1500 light years away), which can be seen just under Orion's belt.
Isotope Counts
Strands of Life
The Nobel prize in medicine and physiology today went to two American scientists, Andrew Fire and Craig Mello, for their work on gene silencing and double-stranded RNA. When we think double-stranded, we often think of another pair of Nobel laureates (Watston and Crick) and a related molecule, DNA. RNA indeed is typically single-stranded, and uses a modified set of bases relative to DNA, subsitituting uracil for thymine. (Wikipedia has a nice diagram.)
The double-stranded version, dubbed RNAi, interferes with the decoding of genes in cells, hence the "gene-silencing" tag.
A colleagues hazards that Nobel winners are getting younger every year. Is it because the time between discovery and award is shrinking or is it that younger scientists are making more critical discoveries?
Girls Don't Like Hard Science
John Tierney had an op-ed piece in the NY Times on Tuesday about the recent National Academy of Sciences report on bias toward women in science. He dismisses their findings of bias, and pins the reason for the underrepresentation of women in research universities on "they don't want to". Most girls, he opines, like the soft sciences, because they are concrete and people oriented, while boys prefer the abstract and "things". Perhaps, but you can be movivated by the concrete, be people oriented and still do "hard science". Check out the letters in response to his piece (full disclosure, one of the letters is mine). Martha Pollack's response regarding engineering was wonderful - people oriented hard science exists.
Royal Purple Molecules
My quantum mechanics class had a problem last week aimed at figuring out the color of a porphyrin molecules. Porphyrins are nitrogen-containing ring shaped chelating molecules (here is a picture) and are ubiquitous in biological systems. An iron bound to a porphyrin is the heme in hemoglobin, when a magnesium is bound, it is a key piece of chlorophyll.
The color of porphyrin should not be a mystery, as long as you know some Greek. The name comes from the Greek for purple, and indeed these compounds have vivid red-violet hues.
I asked the students to compute the energy needed to excite one of porphyrin's 18 pi electrons from the highest occupied level to the lowest unoccupied level, assuming that they could model the compound as 18 independent electrons trapped in a square 1000 pm on a side. The answer in the back of the book gave the absorbtion wavelength of 588 nm, which is precisely what you would expect for a purple compound (absorbing visible yellow light). It seemed too good to be true, for such a simple model to give such a good anwer and it was! There is an error in the answer, and the actual value is not in the visible at all, suggesting that the porphyrin is colorless!
The problem was an apt one for me to be grading this morning, as I was waiting to donate some of my own hemes in the form of whole blood at the college's blood drive.
Weird Words of Science 10: Eigenvalue
We're looking at the Schrödinger equation to start the term in physical chemistry. It is, of course, an eignevalue equation. The term is really a pastiche of German and English, or perhpas a quasi-translation of the German term, eigenwert. The prefix "eigen" is best translated for quantum mechanics as "characteristic". Chemists often use the eigenvalues to "characterize" or "classify" the wavefunctions or states of a systems. For example, the 1s orbital takes its designation from two eigenvalues of the wavefunction: n=1 and l=0.
"Among those ... trying to acquire a general acquaintance with Schrödinger's wave mechanics there must be many who find their mathematical equipment insufficient to follow his first great problem to determine the eigenvalues and eigenfunctions for the hydrogen atom. " Nature 23 July 192
Culture of Chemistry returns with the new term!
Weird Words of Science 9: Tooth and Claw - Chelation
Orac has been posting about the abuse of chelation therapy for treating autism and other disorders. So what's a chelate and how does it work to remove metal ions from the body? EDTA is shorthand for ethylenediamminetetraacetic acid, which has the structure shown at the left. The disodium calcium salt of EDTA is the usual chemotheraputic form. Lone pairs of electrons on the nitrogens and oxygens of the EDTA (tagged blue and red in the photo) latch onto the metal. This Lewis acid-base reaction results in the metal being sequestered inside the EDTA molecule. Tucked away inside the EDTA, the metal can't accumulate in the body's tissues and is eventually eliminated. EDTA has different affinities for different metal ions, but is a pretty effective scavenger of most metal ions, including iron and calcium. Removal of too much calcium can result in cardiac arrest, so EDTA is not without safety isses, as Orac points out!
The word chelation come from the Greek for claw. Molecules that attach to metals at multiple points, like EDTA, are called multidentate ligands from their capacity to "bite" onto the metal. EDTA makes a hexadentate metal-ligand complex (6 points of attachment) with some ions, a pentadentate complex with others.
Weird Words of Science 8: Ligands, the ties that bind
Many transition metals react with bases (such as ammonia) to produce beautifully colored transition metal-ligand complexes. The word ligand comes from the Latin ligare which means to tie or bind. The same root leads to ligaments, which tie your bones together.
The photo shows green Ni(H2O)62+ and blue green Ni(NH3)62+. The ligands are water and ammonia respectively, "tied" to the Ni(II) center. The ligands form an octahedron around the metal center.
Trojan Horse Molecules: Penicillin
Penicillin was one of the first antibiotics in wide use. It was discovered in the late 19th century by a French medical student (Ernest Duchesne), though his work was never pursued. Fleming independently discovered the antibacterial activity of Penicillium mold derivatives in 1928. The active molecule was difficult to extract. The compound was finally synthesized in 1957 by John Sheehan, a chemist at MIT. This feat was made possible by the determination of penicillin's structure in 1944 by Dorothy Crowfoot Hodgkin, an X-ray crystallographer who won the 1964 Nobel prize in chemistry for that discovery and many others (including B-12 and insulin).
How does penicillin work? It is a Trojan horse molecule. Penicillin disrupts the synthesis of bacterial cell walls, thus inhibiting the bacteria's reproduction. The enzyme responsible for assembling the cell walls picks up penicillin, thinking it can incorporate into the wall. Unfortunately for the bacteria, the penicillin molecule opens up and destroys the enzyme's ability to function.
The key step in this sneak attack is the nucleophilic attack of the enzyme onto an electrophilic site on the four-membered β-lactam ring. We've been discussing these reactions in my general chemistry class this week.
Watch this webcast if you want to see how the reaction works and learn a bit about nucelophilic reactions.
Elemental Tales: Get the lead out!
Workers manufacturing the pigment white lead (Pb(OH)2.2PbCO3 apparently made a habit of adding dilute sulfuric acid to their drinking water to prevent lead poisoning. The reaction of the sulfate ions (SO42-) with the aqueous lead ions (Pb2+) forms an insoluble precipitate of lead sulfate, effectively removing the lead from the water (as long as you let the precipipate settle before drinking!). This risk of lead poisoning in these workers was so high that it was referred to as "painter's colic".
van Gogh's Palette
In an attempt to brighten a dreary Philadelphia day, I pulled out a coffee mug that glows with Vincent van Gogh's sunflowers. Among the most vivid of his favorite pigments is chrome yellow. Chrome yellow was first isolated from a natural source (the mineral crocoite) in the late 18th century by Parisienne chemist Vauquelin. By the late 19th century, when van Gogh's sunflowers took form, the vibrant yellow was one of a series of new and exceptionally vivid colors. Chrome yellow is actually a lead salt, lead chromate (PbCrO4. The pigment isstill used today but it has been replaced in many cases by similarly colored, less toxic organic pigments. Unfortunately chrome yellow degrades over time, so that the once brilliantly glowing sunflowers now appear to be dry, drab ocher shadows of van Gogh's vision.
Perhaps influenced by the mug, this week's webcast general chemistry example problem is based on a simple inorganic synthesis of the chrome yellow pigment. One of my colleague's uses another synthesis. in her course on "The Stuff of Art"
Read more about the history and chemistry of color in Bright Earth: Art and the Invention of Color by Philip Ball.
Making a Mark
My interest in MRI has become less academic. I need an MRI of my hand. The orthopedic surgeon noted in passing that they will mark the spot of interest with a capsule of vitamin E, in the same way that they use lead markers in X-rays. I wondered what was so special about the vitamin E that left a trace in the MRI. Turns out that the spin-lattice relaxation time (T1) of the H's in tocopherol's chain of -CH2s is very short, and provides a high intensity signal which can be used to mark the spot. Mineral oil will work, too, but the vitamin E capsules are convenient.
A Magnetic Moment
The Culture of Chemistry welcomes 2006 - now that the grading is done and vacation has begun for me in earnest.
Graham at "Over My Med Body" notes that the total radiation dose in a year from natural background sources is much larger than the dose from any single test. He notes that ultrasound and MRIs are exceptions: ultrasound uses sound waves, and MRIs use magnets. What exactly do those magnets do?
The nuclei of many atoms have "spin" states. Like quarks which have a property called by physicists "color" but are not actually different colors like socks, spin is an instrinsic property of nuclei but this does not necessarily mean that the atoms are spinning like the earth! Hydrogen atoms, of which there are many in the human body (more than 10 pounds worth) have two spin states. Not every atom has multiple spin states. Carbon-12 (the most common form of carbon) has only one spin state. So what happens in an MRI? Radiation (yes, radiation, just very, very low energy radiation) in the form of radio waves forces the hydrogen nuclei to change state to the higher energy spin state. The time it takes for the hydrogens to relax to their low energy spin state is measured. There are two ways for the hydrogen atom to "lose spin", one is called spin-lattice relaxation (T1), the other is spin-spin relaxation (T2). Hydrogen atoms in different environments relax at different rates. Hydrogens in fatty tissue, for example, have very different relaxation times than watery tissue.
So if the changes happen because of radiation, what are the magnets for? It turns out that the separation between spin states depends on the magnitude of the magnet field, as well as the magnetic moment of the nucleus. In the earth's field, the energy between spin states is too small to do the trick of exciting them up to the higher energy state and watching them fall down. You need a high magnet field to do this. |
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This article is about waves in the scientific sense. For waves on the surface of the ocean or lakes, see Wind wave. For other uses, see Wave (disambiguation).
In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium (space or mass). Frequency refers to the addition of time. Wave motion transfers energy from one point to another, which displace particles of the transmission particles medium -that is with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations (of a physical quantity), around almost fixed locations.
There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.
The second main type of wave, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types of waves vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.
Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields.
A wave can be transverse or longitudinal. Transverse waves occur when a disturbance creates oscillations that are perpendicular to the propagation of energy transfer. Longitudinal waves occur when the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space.
General features[edit]
Surface waves in water showing water ripples
A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line.
The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall 1980, p. 8). However, this motion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic (e.g., light) waves in a vacuum, where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are water waves on the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.
It may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved. For example, in the case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh waves, dispersion; and so on....
Other properties, however, although usually described in terms of origin, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently of their physical origin.[1] For example, based on the mechanical origin of acoustic waves, a moving disturbance in space–time can exist if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion. Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.
Mathematical description of one-dimensional waves[edit]
Wave equation[edit]
Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling
Wavelength λ, can be measured between any two corresponding points on a waveform
animation for 2 wavelength, green wave traverse to the right while blue wave transverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.note that f(x,t) + g(x,t) = u(x,t)
• in the direction in space. E.g., let the positive direction be to the right, and the negative direction be to the left.
• with constant amplitude
• with constant velocity , where is
• with constant waveform, or shape
This wave can then be described by the two-dimensional functions
(waveform traveling to the right)
(waveform traveling to the left)
or, more generally, by d'Alembert's formula:[3]
representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained[4] as the partial differential equation
General solutions are based upon Duhamel's principle.[5]
Wave forms[edit]
Main article: Waveform
Sine, square, triangle and sawtooth waveforms.
The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x-direction at velocity v (and G will propagate at the same speed in the negative x-direction).[6]
In the case of a periodic function F with period λ, that is, F(x + λvt) = F(xvt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(xv(t + T)) = F(xvt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[7]
Amplitude and modulation[edit]
amplitude modulation can be achieved through f(x,t) = 1.00*sin(2*pi/0.10*(x-1.00*t)) and g(x,t) = 1.00*sin(2*pi/0.11*(x-1.00*t))only the resultant is visible to improve clarity of waveform
Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.
Main article: Amplitude modulation
The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form:[8][9][10]
where is the amplitude envelope of the wave, is the wavenumber and is the phase. If the group velocity (see below) is wavelength-independent, this equation can be simplified as:[11]
showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.[11][12]
Phase velocity and group velocity[edit]
Main articles: Phase velocity and Group velocity
There are two velocities that are associated with waves, the phase velocity and the group velocity. To understand them, one must consider several types of waveform. For simplification, examination is restricted to one dimension.
This shows a wave with the Group velocity and Phase velocity going in different directions.
The most basic wave (a form of plane wave) may be expressed in the form:
which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, , makes clear that this expression describes a vibration of wavelength traveling in the x-direction with a constant phase velocity .[13]
The other type of wave to be considered is one with localized structure described by an envelope, which may be expressed mathematically as, for example:
where now A(k1) (the integral is the inverse Fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k1 = k. In exponential form:
with Ao the magnitude of A. For example, a common choice for Ao is a Gaussian wave packet:[14]
where σ determines the spread of k1-values about k, and N is the amplitude of the wave.
The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k1), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.[13] However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of stationary phase for evaluation of such integrals.[15]) The condition for φ to vary slowly is that its rate of change with k1 be small; this rate of variation is:[13]
where the evaluation is made at k1 = k because A(k1) is centered there. This result shows that the position x where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:
The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantum mechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)2/(2m). Consequently, for that wave situation, the group velocity is
showing that the velocity of a localized particle in quantum mechanics is its group velocity.[13] Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.[16] In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wave propagates.
Sinusoidal waves[edit]
Main article: Sinusoidal wave
Sinusoidal waves correspond to simple harmonic motion.
Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude described by the equation:
• is the space coordinate
• is the time coordinate
• is the wavenumber
• is the angular frequency
• is the phase constant.
The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter).
The wavelength is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber , the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation
The period is the time for one complete cycle of an oscillation of a wave. The frequency is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:
In other words, the frequency and period of a wave are reciprocals.
The angular frequency represents the frequency in radians per second. It is related to the frequency or period by
The wavelength of a sinusoidal waveform traveling at constant speed is given by:[17]
where is called the phase speed (magnitude of the phase velocity) of the wave and is the wave's frequency.
Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.[18]
The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.[23][24] In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.[25] When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.
Plane waves[edit]
Main article: Plane wave
Standing waves[edit]
Standing wave in stationary medium. The red dots represent the wave nodes
A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.
The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.
Physical properties[edit]
Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism
Waves exhibit common behaviors under a number of standard situations, e. g.
Transmission and media[edit]
Waves normally move in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:
• A bounded medium if it is finite in extent, otherwise an unbounded medium
• A linear medium if the amplitudes of different waves at any particular point in the medium can be added
• A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space
• An anisotropic medium if one or more of its physical properties differ in one or more directions
• An isotropic medium if its physical properties are the same in all directions
Absorption of waves means, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave with that same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrational motion. If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, then those electrons will absorb the energy of the wave and transform it into vibrational motion.
Main article: Reflection (physics)
When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.
Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.
Main article: Refraction
Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.
Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.
Main article: Diffraction
A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.
Main article: Polarization (waves)
Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg
The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization. it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter.
Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.
Schematic of light being dispersed by a prism. Click to see animation.
A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colours and that these colours cannot be decomposed any further.[26]
Mechanical waves[edit]
Main article: Mechanical wave
Waves on strings[edit]
Main article: Vibrating string
The speed of a transverse wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ):
where the linear density μ is the mass per unit length of the string.
Acoustic waves[edit]
Acoustic or sound waves travel at speed given by
or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).
Water waves[edit]
Shallow water wave.gif
Main article: Water waves
• Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
• Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas;
• Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
• Ocean surface waves, which are perturbations that propagate through water.
Seismic waves[edit]
Main article: Seismic waves
Shock waves[edit]
Formation of a shock wave by a plane.
Main article: Shock wave
• Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves[27]
• Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.
• It is worth noting that the mass-energy equivalence equation can be solved for this form: .
Electromagnetic waves[edit]
Onde electromagnétique.png
An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and Gamma rays.
Quantum mechanical waves[edit]
Main article: Schrödinger equation
See also: Wave function
Schrödinger equation[edit]
The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.
Dirac equation[edit]
The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[28]
de Broglie waves[edit]
Main articles: Wave packet and Matter wave
Louis de Broglie postulated that all particles with momentum have a wavelength
where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10−13 m.
A wave representing such a particle traveling in the k-direction is expressed by the wave function as follows:
where the wavelength is determined by the wave vector k as:
and the momentum by:
However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,[29] a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.
In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.[30] Gaussian wave packets also are used to analyze water waves.[31]
For example, a Gaussian wavefunction ψ might take the form:[32]
at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. It is well known from the theory of Fourier analysis,[33] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.[34] Given the Gaussian:
the Fourier transform is:
The Gaussian in space therefore is made up of waves:
that is, a number of waves of wavelengths λ such that kλ = 2 π.
The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles
Gravity waves[edit]
Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example.
Gravitational waves[edit]
Main article: Gravitational wave
Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016.[35] Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.
WKB method[edit]
Main article: WKB method
In a nonuniform medium, in which the wavenumber k can depend on the location as well as the frequency, the phase term kx is typically replaced by the integral of k(x)dx, according to the WKB method. Such nonuniform traveling waves are common in many physical problems, including the mechanics of the cochlea and waves on hanging ropes.
See also[edit]
Waves in general[edit]
Electromagnetic waves[edit]
In fluids[edit]
In quantum mechanics[edit]
In relativity[edit]
Other specific types of waves[edit]
Related topics[edit]
1. ^ Lev A. Ostrovsky & Alexander I. Potapov (2002). Modulated waves: theory and application. Johns Hopkins University Press. ISBN 0-8018-7325-8.
2. ^ Michael A. Slawinski (2003). "Wave equations". Seismic waves and rays in elastic media. Elsevier. pp. 131 ff. ISBN 0-08-043930-6.
3. ^ Karl F Graaf (1991). Wave motion in elastic solids (Reprint of Oxford 1975 ed.). Dover. pp. 13–14. ISBN 978-0-486-66745-4.
4. ^ For an example derivation, see the steps leading up to eq. (17) in Francis Redfern. "Kinematic Derivation of the Wave Equation". Physics Journal.
5. ^ Jalal M. Ihsan Shatah; Michael Struwe (2000). "The linear wave equation". Geometric wave equations. American Mathematical Society Bookstore. pp. 37 ff. ISBN 0-8218-2749-9.
6. ^ Louis Lyons (1998). All you wanted to know about mathematics but were afraid to ask. Cambridge University Press. pp. 128 ff. ISBN 0-521-43601-X.
7. ^ Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0-470-18590-2.
8. ^ Christian Jirauschek (2005). FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection. Cuvillier Verlag. p. 9. ISBN 3-86537-419-0.
9. ^ Fritz Kurt Kneubühl (1997). Oscillations and waves. Springer. p. 365. ISBN 3-540-62001-X.
10. ^ Mark Lundstrom (2000). Fundamentals of carrier transport. Cambridge University Press. p. 33. ISBN 0-521-63134-3.
11. ^ a b Chin-Lin Chen (2006). "§13.7.3 Pulse envelope in nondispersive media". Foundations for guided-wave optics. Wiley. p. 363. ISBN 0-471-75687-3.
12. ^ Stefano Longhi; Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernández-Figueroa; Michel Zamboni-Rached; Erasmo Recami. Localized Waves. Wiley-Interscience. p. 329. ISBN 0-470-10885-1.
13. ^ a b c d Albert Messiah (1999). Quantum Mechanics (Reprint of two-volume Wiley 1958 ed.). Courier Dover. pp. 50–52. ISBN 978-0-486-40924-5.
14. ^ See, for example, Eq. 2(a) in Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics: An introduction (2nd ed.). Springer. pp. 60–61. ISBN 3-540-67458-6.
15. ^ John W. Negele; Henri Orland (1998). Quantum many-particle systems (Reprint in Advanced Book Classics ed.). Westview Press. p. 121. ISBN 0-7382-0052-2.
16. ^ Donald D. Fitts (1999). Principles of quantum mechanics: as applied to chemistry and chemical physics. Cambridge University Press. pp. 15 ff. ISBN 0-521-65841-1.
17. ^ David C. Cassidy; Gerald James Holton; Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339 ff. ISBN 0-387-98756-8.
18. ^ Paul R Pinet (2009). op. cit. p. 242. ISBN 0-7637-5993-7.
19. ^ Mischa Schwartz; William R. Bennett & Seymour Stein (1995). Communication Systems and Techniques. John Wiley and Sons. p. 208. ISBN 978-0-7803-4715-1.
20. ^ See Eq. 5.10 and discussion in A. G. G. M. Tielens (2005). The physics and chemistry of the interstellar medium. Cambridge University Press. pp. 119 ff. ISBN 0-521-82634-9. ; Eq. 6.36 and associated discussion in Otfried Madelung (1996). Introduction to solid-state theory (3rd ed.). Springer. pp. 261 ff. ISBN 3-540-60443-X. ; and Eq. 3.5 in F Mainardi (1996). "Transient waves in linear viscoelastic media". In Ardéshir Guran; A. Bostrom; Herbert Überall; O. Leroy. Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering. World Scientific. p. 134. ISBN 981-02-4271-9.
21. ^ Aleksandr Tikhonovich Filippov (2000). The versatile soliton. Springer. p. 106. ISBN 0-8176-3635-8.
22. ^ Seth Stein, Michael E. Wysession (2003). An introduction to seismology, earthquakes, and earth structure. Wiley-Blackwell. p. 31. ISBN 0-86542-078-5.
23. ^ Seth Stein, Michael E. Wysession (2003). op. cit.. p. 32. ISBN 0-86542-078-5.
24. ^ Kimball A. Milton; Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Springer. p. 16. ISBN 3-540-29304-3. Thus, an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]...
25. ^ Raymond A. Serway & John W. Jewett (2005). "§14.1 The Principle of Superposition". Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN 0-534-49143-X.
26. ^ Newton, Isaac (1704). "Prop VII Theor V". Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures. 1. London. p. 118. All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these...
27. ^ M. J. Lighthill; G. B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the Royal Society of London. Series A. 229: 281–345. Bibcode:1955RSPSA.229..281L. doi:10.1098/rspa.1955.0088. And: P. I. Richards (1956). "Shockwaves on the highway". Operations Research. 4 (1): 42–51. doi:10.1287/opre.4.1.42.
28. ^ A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0-486-66741-3. (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
29. ^ Ming Chiang Li (1980). "Electron Interference". In L. Marton; Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0-12-014653-3.
30. ^ See for example Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN 3-540-67458-6. and John Joseph Gilman (2003). Electronic basis of the strength of materials. Cambridge University Press. p. 57. ISBN 0-521-62005-8. ,Donald D. Fitts (1999). Principles of quantum mechanics. Cambridge University Press. p. 17. ISBN 0-521-65841-1. .
31. ^ Chiang C. Mei (1989). The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47. ISBN 9971-5-0789-7.
32. ^ Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN 3-540-67458-6.
33. ^ Siegmund Brandt; Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 0-387-95141-5.
34. ^ Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN 0-521-59827-3.
35. ^ "Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". CBC. 11 February 2016.
• Fleisch, D.; Kinnaman, L. (2015). A student's guide to waves. Cambridge, UK: Cambridge University Press. ISBN 978-1107643260.
• Campbell, Murray; Greated, Clive (2001). The musician's guide to acoustics (Repr. ed.). Oxford: Oxford University Press. ISBN 978-0198165057.
• French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 0-393-09936-9. OCLC 163810889.
• Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company. ISBN 0-534-00758-9. .
• Hunt, Frederick Vinton (1978). Origins in acoustics. Woodbury, NY: Published for the Acoustical Society of America through the American Institute of Physics. ISBN 978-0300022209.
• Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 0-8018-5870-4. .
• Griffiths, G.; Schiesser, W. E. (2010). Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with Matlab and Maple. Academic Press. ISBN 9780123846532.
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bdd30e465fb7c313 | picture of mechanics
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Dynamics affects both observables and, dually, states; this is most well known in quantum mechanics but applies equally well to classical mechanics. The different “pictures” of mechanics differ in how the dynamics is explicitly formalized:
The pictures are named after those physicists (Werner Heisenberg, Erwin Schrödinger, and Paul Dirac) who first used or popularised these approaches to quantum physics.
With global time
Let us assume a global notion of time, say a fixed background? spacetime which is globally hyperbolic, so that it admits a foliation into Cauchy surfaces, and choose a time coordinate? for this foliation. The upshot of this is that each event occurs at a time tt, and conversely we can speak of space at any time tt (at least within certain bounds). Thus we may speak sensibly of either the state of the world at time tt or the value of some observable quantity at time tt.
Because this is a picture of dynamics, states or observables (as appropriate to the picture) will vary through time. We therefore have a time evolution? operator U(t,t)U(t,t') between any two times t,tt,t'; actually, we need consider only U(t)U(t,0)U(t) \coloneqq U(t,0), since U(t,t)=U(t)U(t) 1U(t,t') = U(t) \circ U(t')^{-1}.
In the Heisenberg picture, for each observable AA, we speak of AA only at some time tt, so our actual observables are of the form A(t)A(t). We write abstractly
A(t)=A(0)U(t) A(t) = A(0) \cdot U(t)
to show the evolution of the observable through time. However, when it comes to the state of the world, we speak of a single state ψ\psi that describes the world at all times.
In the Schrödinger picture, we instead speak of ψ(t)\psi(t), the state of the world at time tt. We write abstractly
ψ(t)=U(t)*ψ(0) \psi(t) = U(t) \ast \psi(0)
(the Schrödinger equation) to show the evolution of the state through time. However, when it comes to observables, we use only the observable AA across all times.
To see the connection between the two pictures, recall that an observable AA and a state ψ\psi together produce a probability distribution giving the probability that any given value of AA will be observed, given that the world is in state ψ\psi. (This is true throughout mechanics, although it is obscured in non-statistical classical mechanics, since the probability distributions produced by classical pure states are all delta measure?s.) Assuming that AA belongs to an appropriate algebra of observables and the probability measures are sufficiently nice, we may restrict attention to the expectation values A ψ\langle{A}\rangle_\psi of these distributions, since the entire distribution can be recovered from A n ψ\langle{A^n}\rangle_\psi as nn varies over natural numbers.
The connection between the two pictures is then given by
AU(t) ψ=A U(t)*ψ. \langle{A \cdot U(t)}\rangle_\psi = \langle{A}\rangle_{U(t) \ast \psi} .
It remains to say exactly what U(t)U(t) is and what the operations \cdot and *\ast are. Let us use the density matrix formulation of quantum statistical mechanics?, since classical and non-statistical mechanics may be recovered as special cases, by restricting (respectively) the allowed observables or states. In this case, both states and observables are given by linear operators on a Hilbert space HH, and we have
A ψ=tr(Aψ) \langle{A}\rangle_\psi = \tr(A \psi)
(using the trace operation). Each U(t)U(t) is a unitary operator on HH (since time evolution between Cauchy surfaces is a symmetry), and we have
AU(t)=U(t) 1AU(t) A \cdot U(t) = U(t)^{-1} A U(t)
(a right action?) and
U(t)*ψ=U(t)ψU(t) 1 U(t) \ast \psi = U(t) \psi U(t)^{-1}
(a left action?). We then have
AU(t) ψ=tr(U(t) 1AU(t)ψ)=tr(AU(t)ψU(t) 1)=A U(t)*ψ, \langle{A \cdot U(t)}\rangle_\psi = tr(U(t)^{-1} A U(t) \psi) = tr(A U(t) \psi U(t)^{-1}) = \langle{A}\rangle_{U(t) \ast \psi} ,
as desired, using the cyclic property of the trace?.
The time evolution operator U(t)U(t) is often derived from a Hamiltonian and the formula for A(t)A(t) or ψ(t)\psi(t) is further derived from a differential equation involving this Hamiltonian. However, this is unnecessary for the connection between the two pictures.
Without time
If spacetime is not globally hyperbolic, then there is no time coordinate tt, and none of the discussion above makes sense; or if we choose a coordinate tt and call it time regardless, then time evolution is not a symmetry and we do not have the operators U(t)U(t).
In this case, the Heisenberg picture still makes sense, even though we cannot expect to calculate A(t)A(t) from A(0)A(0) (if it even makes sense to discuss such things). This is easily seen in field theory, where the operators called AA above are really of the form A(x,y,z)A(x,y,z). Then the Heisenberg picture's A(t)A(t) is really A(x,y,z,t)A(x,y,z,t), or simply A(p)A(p) where pp indicates an event (a point in spacetime). So even if the coordinates x,y,z,tx,y,z,t do not make sense, still A(p)A(p) does; and even if the equations of physics cannot be thought of as describing evolution through time, still they can be thought of as describing the relationships between observables at different places in spacetime.
In contrast, the Schrödinger picture cannot be so treated. One may be led to the contrary impression by the quantum mechanics of a single particle without any internal structure (not even spin), in which case the Hilbert space of (pure quantum-mechanical) states is naturally identified with L 2( 3)L^2(\mathbb{R}^3) and the state ψ\psi is really ψ(x,y,z)\psi(x,y,z). In this case, the Schrödinger picture's ψ(t)\psi(t) is really ψ(x,y,z,t)\psi(x,y,z,t), that is ψ(p)\psi(p). However, this fails in classical or statistical mechanics; and even in non-statistical quantum mechanics, it breaks down if the particle has internal structure or there is more than one particle in the world. Then we see that the spacial coordinates x,y,zx,y,z generalise to the arbitrary coordinates of configuration space, while tt remains only tt, and there is no way to subsume it into a spacetime coordinate.
Historically, the terms ‘Schrödinger picture’ and ‘Heisenberg picture’ (at least) referred to more than what we discuss above; they referred to the entirety of the differences between Schrödinger's and Heisenberg's approaches to quantum mechanics.
For example, these terms included also Schrödinger's use of typically wave-like functions as pure states (and correspondingly operators in the higher-type-theoretic sense as observables) vs Heisenberg's use of infinite-dimensional matrices as observables (and correspondingly infinite sequences as pure states). This difference was rectified by von Neumann's application of Hilbert space to the problem, showing that (if one suitably restricts the allowed functions and sequences and also identifies equivalent functions a bit) both approaches used Hilbert space (what we would now call the infinite-dimensional separable Hilbert space) as the space of pure states.
This is entirely separate from the question of whether states or observables are taken to evolve with time. Still, there is this connection: Schrödinger evolved states, and his approach was called ‘wave mechanics’ after his representation for states, while Heisenberg evolved observables, and his approach was called ‘matrix mechanics’ after his representation for observables.
duality between algebra and geometry in physics:
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation
See for instance sections 7.19.1–3 in
• Eberhard Zeidler, Quantum field theory. A bridge between mathematicians and physicists – volume I Springer (2009) (web).
To check conventions at least, see Wikipedia:
A note on how the Schrödinger picture in the form of extended FQFT on Lorentzian manifolds is related to the Heisenberg picture in the form of AQFT is in
Revised on September 24, 2014 08:49:46 by Toby Bartels ( |
f005c1372bd4d4c4 | Probability amplitude
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A wave function for a single electron on 5d atomic orbital of a hydrogen atom. The solid body shows the places where the electron's probability density is above a certain value (here 0.02 nm−3): this is calculated from the probability amplitude. The hue on the colored surface shows the complex phase of the wave function.
In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.
Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding (see References), and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger[clarification needed] and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
Neglecting some technical complexities, the problem of quantum measurement is the behaviour of a quantum state, for which the value of the observable Q to be measured is uncertain. Such a state is thought to be a coherent superposition of the observable's eigenstates, states on which the value of the observable is uniquely defined, for different possible values of the observable.
When a measurement of Q is made, the system (under the Copenhagen interpretation) jumps to one of the eigenstates, returning the eigenvalue to which the state belongs. The superposition of states can give them unequal "weights". Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical factor squared. These numerical factors are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the Born rule.
Different observables may define incompatible decompositions of states.[clarification needed] Observables that do not commute define probability amplitudes on different sets.
In a formal setup, any system in quantum mechanics is described by a state, which is a vector |Ψ⟩, residing in an abstract complex vector space, called a Hilbert space. It may be either infinite- or finite-dimensional. A usual presentation of that Hilbert space is a special function space, called L2(X), on certain set X, that is either some configuration space or a discrete set.
For a measurable function , the condition specifies that a finitely bounded integral must apply:
this integral defines the square of the norm of ψ. If that norm is equal to 1, then
It actually means that any element of L2(X) of the norm 1 defines a probability measure on X and a non-negative real expression |ψ(x)|2 defines its Radon–Nikodym derivative with respect to the standard measure μ.
If the standard measure μ on X is non-atomic, such as the Lebesgue measure on the real line, or on three-dimensional space, or similar measures on manifolds, then a real-valued function |ψ(x)|2 is called a probability density; see details below. If the standard measure on X consists of atoms only (we shall call such sets X discrete), and specifies the measure of any xX equal to 1,[1] then an integral over X is simply a sum[2] and |ψ(x)|2 defines the value of the probability measure on the set {x}, in other words, the probability that the quantum system is in the state x. How amplitudes and the vector are related can be understood with the standard basis of L2(X), elements of which will be denoted by |x or x| (see bra–ket notation for the angle bracket notation). In this basis
specifies the coordinate presentation of an abstract vector |Ψ⟩.
Mathematically, many L2 presentations of the system's Hilbert space can exist. We shall consider not an arbitrary one, but a convenient one for the observable Q in question. A convenient configuration space X is such that each point x produces some unique value of Q. For discrete X it means that all elements of the standard basis are eigenvectors of Q. In other words, Q shall be diagonal in that basis. Then is the "probability amplitude" for the eigenstate x|. If it corresponds to a non-degenerate eigenvalue of Q, then gives the probability of the corresponding value of Q for the initial state |Ψ⟩.
For non-discrete X there may not be such states as x| in L2(X), but the decomposition is in some sense possible; see spectral theory and Spectral theorem for accurate explanation.
Wave functions and probabilities[edit]
If the configuration space X is continuous (something like the real line or Euclidean space, see above), then there are no valid quantum states corresponding to particular xX, and the probability that the system is "in the state x" will always be zero. An archetypical example of this is the L2(R) space constructed with 1-dimensional Lebesgue measure; it is used to study a motion in one dimension. This presentation of the infinite-dimensional Hilbert space corresponds to the spectral decomposition of the coordinate operator: x| Q | Ψ⟩ = xx | Ψ⟩, xR in this example. Although there are no such vectors as x |, strictly speaking, the expression x | Ψ⟩ can be made meaningful, for instance, with spectral theory.
Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function ψ is the wave function.
If the function ψL2(X), ‖ψ‖ = 1 represents the quantum state vector |Ψ⟩, then the real expression |ψ(x)|2, that depends on x, forms a probability density function of the given state. The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in X to obtain probability values – as was stated above, the system can't be in some state x with a positive probability. It gives to both amplitude and density function a physical dimension, unlike a dimensionless probability. For example, for a 3-dimensional wave function, the amplitude has the dimension [L−3/2], where L is length.
Note that for both continuous and infinite discrete cases not every measurable, or even smooth function (i.e. a possible wave function) defines an element of L2(X); see #Normalisation below.
Discrete amplitudes[edit]
When the set X is discrete (see above), vectors |Ψ⟩ represented with the Hilbert space L2(X) are just column vectors composed of "amplitudes" and indexed by X. These are sometimes referred to as wave functions of a discrete variable xX. Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator. Components of the vector will be denoted by ψ(x) for uniformity with the previous case; there may be either finite of infinite number of components depending on the Hilbert space. In this case, if the vector |Ψ⟩ has the norm 1, then |ψ(x)|2 is just the probability that the quantum system resides in the state x. It defines a discrete probability distribution on X.
|ψ(x)| = 1 if and only if |x is the same quantum state as |Ψ⟩. ψ(x) = 0 if and only if |x and |Ψ⟩ are orthogonal (see inner product space). Otherwise the modulus of ψ(x) is between 0 and 1.
A discrete probability amplitude may be considered as a fundamental frequency[citation needed] in the Probability Frequency domain (spherical harmonics) for the purposes of simplifying M-theory transformation calculations.
A basic example[edit]
Take the simplest meaningful example of the discrete case: a quantum system that can be in two possible states: for example, the polarization of a photon. When the polarization is measured, it could be the horizontal state | H ⟩, or the vertical state | V ⟩. Until its polarization is measured the photon can be in a superposition of both these states, so its state |ψ could be written as:
The probability amplitudes of |ψ for the states | H ⟩ and | V ⟩ are α and β respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is α2, and the probability of being vertically polarized is β2.
Therefore, a photon in a state would have a probability of 1/3 to come out horizontally polarized, and a probability of 2/3 to come out vertically polarized when an ensemble of measurements are made. The order of such results, is, however, completely random.
In the example above, the measurement must give either | H ⟩ or | V ⟩, so the total probability of measuring | H ⟩ or | V ⟩ must be 1. This leads to a constraint that α2 + β2 = 1; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. If to understand "all the possible states" as an orthonormal basis, that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained above.
One can always divide any non-zero element of a Hilbert space by its norm and obtain a normalized state vector. Not every wave function belongs to the Hilbert space L2(X), though. Wave functions that fulfill this constraint are called normalizable.
The Schrödinger wave equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time. Suppose a wavefunction ψ0(x, t) is a solution of the wave equation, giving a description of the particle (position x, for time t). If the wavefunction is square integrable, i.e.
for some t0, then ψ = ψ0/a is called the normalized wavefunction. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, at a given time t0, ρ(x) = |ψ(x, t0)|2 is the probability density function of the particle's position. Thus the probability that the particle is in the volume V at t0 is
Note that if any solution ψ0 to the wave equation is normalisable at some time t0, then the ψ defined above is always normalised, so that
is always a probability density function for all t. This is key to understanding the importance of this interpretation, because for a given the particle's constant mass, initial ψ(x, 0) and the potential, the Schrödinger equation fully determines subsequent wavefunction, and the above then gives probabilities of locations of the particle at all subsequent times.
The laws of calculating probabilities of events[edit]
A. Provided a system evolves naturally (which under the Copenhagen interpretation means that the system is not subjected to measurement), the following laws apply:
1. The probability (or the density of probability in position/momentum space) of an event to occur is the square of the absolute value of the probability amplitude for the event: .
2. If there are several mutually exclusive, indistinguishable alternatives in which an event might occur (or, in realistic interpretations of wavefunction, several wavefunctions exist for a space-time event), the probability amplitudes of all these possibilities add to give the probability amplitude for that event: .
3. If, for any alternative, there is a succession of sub-events, then the probability amplitude for that alternative is the product of the probability amplitude for each sub-event: .
4. Non-entangled states of a composite quantum system have amplitudes equal to the product of the amplitudes of the states of constituent systems: . See the #Composite systems section for more information.
Law 2 is analogous to the addition law of probability, only the probability being substituted by the probability amplitude. Similarly, Law 4 is analogous to the multiplication law of probability for independent events; note that it fails for entangled states.
B. When an experiment is performed to decide between the several alternatives, the same laws hold true for the corresponding probabilities: .
Provided one knows the probability amplitudes for events associated with an experiment, the above laws provide a complete description of quantum systems in terms of probabilities.
The above laws give way to the path integral formulation of quantum mechanics, in the formalism developed by the celebrated theoretical physicist Richard Feynman. This approach to quantum mechanics forms the stepping-stone to the path integral approach to quantum field theory.
In the context of the double-slit experiment[edit]
Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic double-slit experiment, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that P(through either slit) = P(through first slit) + P(through second slit), where P(event) is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When nature does not have a way to distinguish which slit the electron has gone through (a much more stringent condition than simply "it is not observed"), the observed probability distribution on the screen reflects the interference pattern that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. This is an example of the case A as described in the previous article. The complex amplitudes which represent the electron passing each slit (ψfirst and ψsecond) follow the law of precisely the form expected: ψtotal = ψfirst + ψsecond. This is the principle of quantum superposition. The probability, which is the modulus squared of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
Here, and are the arguments of ψfirst and ψsecond respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term is called the "interference term", and this would be missing if we had added the probabilities.
However, one may choose to devise an experiment in which he observes which slit each electron goes through. Then case B of the above article applies, and the interference pattern is not observed on the screen.
One may go further in devising an experiment in which he gets rid of this "which-path information" by a "quantum eraser". Then, according to the Copenhagen interpretation, the case A applies again and the interference pattern is restored.[3]
Conservation of probabilities and the continuity equation[edit]
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.
Define the probability current (or flux) j as
measured in units of (probability)/(area × time).
Then the current satisfies the equation
The probability density is , this equation is exactly the continuity equation, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where j corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local conservation of charges.[clarification needed]
Composite systems[edit]
For two quantum systems with spaces L2(X1) and L2(X2) and given states 1 and 2 respectively, their combined state 12 can be expressed as ψ1(x1) ψ2(x2) a function on X1×X2, that gives the product of respective probability measures. In other words, amplitudes of a non-entangled composite state are products of original amplitudes, and respective observables on the systems 1 and 2 behave on these states as independent random variables. This strengthens the probabilistic interpretation explicated above.
Amplitudes in operators[edit]
The concept of amplitudes described above is relevant to quantum state vectors. It is also used in the context of unitary operators that are important in the scattering theory, notably in the form of S-matrices. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of matrix elements squared are interpreted as transition probabilities just as in a random process. Like a finite-dimensional unit vector specifies a finite probability distribution, a finite-dimensional unitary matrix specifies transition probabilities between a finite number of states. Note that columns of a unitary matrix, as vectors, have the norm 1.
The "transitional" interpretation may be applied to L2s on non-discrete spaces as well.
See also[edit]
1. ^ The case of an atomic measure on X with μ({x}) ≠ 1 is not interesting, because such x that μ({x}) = 0 are unused by L2(X) and can be dropped, whereas for x of positive measures the value of μ({x}) is virtually the question of rescaling of ψ(x). Due to this trivial fix this case was hardly ever considered by physicists.
2. ^ If X is countable, then an integral is the sum of an infinite series.
3. ^ A recent 2013 experiment seems to give a clue about the correct physical interpretation of such phenomena. The information can actually be obtained, but then it seems like the electron went through all the possible paths simultaneously. (Certain ensemble-alike realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates, L. Ph. H. Schmidt et al., Phys. Rev. Week ending 2013.
1. The Nobel Prize in Physics 1954.
2. The Feynman Lectures on Physics, Volume 3, Feynman, Leighton, Sands. Narosa Publishing House, New Delhi, 2008. |
c40fe3ac70cda59b | Monday, May 25, 2015
Separability and quantum mechanics
Tuesday, Apr 21st
Fernando Barbero, CSIC, Madrid
Title: Separability and quantum mechanics
PDF of the talk (758k)
Audio [.wav 20MB]
by Juan Margalef-Bentabol, UC3M-CSIC, Madrid
Classical vs Quantum: Two views of the world
In classical mechanics it is relatively straightforward to get information from a system. For instance, if we have a bunch of particles moving around, we can ask ourselves: where is its center of mass? What is the average speed of the particles? What is the distance between two of them? In order to ask and answer such questions in a precise mathematical way, we need to know all the positions and velocities of the system at every moment; in the usual jargon, we need to know the dynamics over the state space (also called configuration space for positions and velocities, or phase space when we consider positions and momenta). For example, the appropriate way to ask for the center of mass, is given by the function that for a specific state of the system, gives the weighted mean of the positions of all the particles. Also, the total momentum of the system is given by the function consisting of the sum of the momenta of the individual particles. Such functions are called observables of the theory, therefore an observable is defined as a function that takes all the positions and momenta, and returns a real number. Among all the observables there are some ones that can be considered as fundamental. A familiar example is provided by the generalized position and momenta denoted as and .
In a quantum setting answering, and even asking, such questions is however much trickier. It can be properly justified that the needed classical ingredients have to be significantly changed:
1. The state space is now much more complicated, instead of positions and velocities/momenta we need a (usually infinite dimensional) complex vector space with an inner product that is complete. Such vector space is called a Hilbert space and the vectors of are called states (up to a complex multiplication).
2. The observables are functions from to itself that "behave well" with respect to the inner product (these are called self-adjoint operators). Notice in particular that the outputs of the quantum observables are complex vectors and not numbers anymore!
3. In a physical experiment we do obtain real numbers, so somehow we need to retrieve them from the observable associated with the experiment. The way to do this is by looking at the spectrum of , which consists of a set of real numbers called eigenvalues associated with some vectors called eigenvectors (actually the number that we obtain is a probability amplitude whose absolute value squared is the probability of obtaining as an output a specific eigenvector).
The questions that arise naturally are: how do we choose the Hilbert space? how do we introduce fundamental observables analogous to the ones of classical mechanics? In order to answer these questions we need to take a small detour and talk a little bit about the algebra of observables.
Algebra of Observables
Given two classical observables, we can construct another one by means of different methods. Some important ones are:
• By adding them (they are real functions)
• By multiplying them
• By a more sophisticated procedure called the Poisson bracket
The last one turns out to be fundamental in classical mechanics and plays an important role within the Hamiltonian form of the dynamics of the system. A basic fact is that the set of observables endowed with the Poisson bracket forms a Lie algebra (a vector space with a rule to obtain an element out of two other ones satisfying some natural properties). The fundamental observables behave really well with respect to the Poisson bracket, namely they satisfy simple commutation relations i.e. if we consider the - position observable and "Poisson-multiply" it by the - momentum observable, we obtain the constant function if , or the constant function if .
One of the best approaches to construct a quantum theory associated with a classical one, is to reproduce at the quantum level some features of its classical formulation. One way to do this is to define a Lie algebra for the quantum observables such that some of such observables mimic the behavior of the Poisson bracket of some classical fundamental observables. This procedure (modulo some technicalities) is known as finding a representation of this algebra. In order to do this, one has to choose:
1. A Hilbert space .
2. Some fundamental observables that reproduce the canonical commutation relations when we consider the commutator of operators.
In standard Quantum Mechanics the fundamental observables are positions and momenta. It may seem that there is a great ambiguity in this procedure, however there is a central theorem due to Stone and von Neumann that states that, under some reasonable hypothesis, all the representations are essentially the same.
One of the hypotheses of the Stone-von Neumann theorem is that the Hilbert space must be separable. This means that it is possible to find a countable set of orthonormal vectors in (called Hilbert basis) such that any state -vector- of can be written as an appropriate countable sum of them. A separable Hilbert space, despite being infinite dimensional, is not "too big", in the sense that there are Hilbert spaces with uncountable bases that are genuinely larger. The separability assumption seems natural for standard quantum mechanics, but in the case of quantum field theory -with infinitely many degrees of freedom- one might expect to need much larger Hilbert spaces i.e. non separable ones. Somewhat surprisingly, most of the quantum field theories can be handled with our beloved and "simple" separable Hilbert spaces with the remarkable exception of LQG (and its derivative LQC) where non separability plays a significant role. Henceforth it seems interesting to understand what happens when one considers non separable Hilbert spaces [3] in the realm of the quantum world. A natural and obvious way to acquire the necessary intuition is by first considering quantum mechanics on a non-separable Hilbert space.
The Polymeric Harmonic Oscillator
The authors of [2,3] discuss two inequivalent (among the infinitely many) representations of the algebra of fundamental observables which share a non familiar feature, namely, in one of them (called the position representation) the position observable is well defined but the momentum observable does not even exist; in the momentum representation the roles of positions and momenta are exchanged. Notice that in this setting, some familiar features of quantum mechanics are lost for good. For instance, the position-momentum Heisenberg uncertainty formula makes no sense at all as both position and momentum observables need to be defined.
To improve the understanding of such systems and gain some insight for the application to LQG and LQC, the authors of [1] (re)study the -dimensional Harmonic Oscillator (PHO) in a non separable Hilbert space (known in this context as a polymeric Hilbert space). As the space is non separable, any Hilbert basis should be uncountable. This leads to some unexpected behaviors that can be used to obtain exotic representations of the algebra of fundamental observables.
The motivation to study the PHO is kind of the same as always: the HO, in addition to being an excellent toy model, is a good approximation to any 1-dimensional mechanical system close to its equilibrium points. Furthermore, free quantum field theories can be thought of as ensembles of infinitely many independent HO's. There are however many ways to generalize the HO to a non separable Hilbert space and also many equivalent ways to realize a concrete representation, for instance by using Hilbert spaces based on:
The eigenvalue equations in these different spaces take different forms: in some of them they are difference equations, whereas in others they have the form of the standard Schrödinger equation with a periodic potential. It is important to notice nonetheless that writing Hamiltonian observables in this framework turn out to be really difficult, as only one of the position or momentum observables can be strictly represented. This means that for the other one it is necessary to rely on some kind of approximation (that can be obtained by introducing an arbitrary scale) and choosing a periodic potential with minima corresponding to the one of the quadratic operator. The huge uncertainty in this procedure has been highlighted by Corichi, Zapata, Vukašinac and collaborators. The standard choice leads to an equation known as the Mathieu equation but other simple choices have been explored, as the one shown in the figure.
Energy eigenvalues (bands) of a polymerized harmonic oscillator. The horizontal axis shows the position (or the momentum depending on the chosen representation), the vertical axis is the energy and the red line represents the particular periodic extension of the potential used to approximate the usual quadratic potential of the HO. The other lines plotted in this graph correspond to auxiliary functions that can be used to locate the edges of the bands that define the point spectrum in the present example.
As we have already mentioned, the orthonormal bases in non separable Hilbert spaces are uncountable. A consequence of this is the fact that the orthonormal basis provided by the eigenstates of the Hamiltonian must be uncountable, i.e. the Hamiltonian must have an uncountable infinity worth of eigenvalues (counted with multiplicity). A somewhat unexpected result that can be proved by invoking classical theorems on functional analysis in non-separable Hilbert spaces is the fact that these eigenvalues are gathered in bands. It is important to point out here that only the lowest-lying part of the spectrum is expected to mimic reasonably well the one corresponding to the standard HO, however it is important to keep also in mind the huge difference that persists: even the narrowest bands contain a continuum of eigenvalues.
Some physical consequences
The fact that the spectrum of the polymerized harmonic oscillator displays this band structure is relevant for some applications of polymerized quantum mechanics. Two main issues were mentioned in the talk. On one hand the statistical mechanics of polymerized systems must be handled with due care. Owing to the features of the spectrum, the counting of energy eigenstates necessary to compute the entropy in the microcanonical ensemble is ill defined. A similar problem crops up when computing the partition function of the canonical ensemble. These problems can probably be circumvented by using an appropriate regularization and also by relying on some superselection rules that eliminate all but a countable subset of energy eigenstates of the system.
A setting where something similar can be done is in the polymer quantization of the scalar field (already considered by Husain, Pawłowski and collaborators). As this system can be thought of as an infinite ensemble of harmonic oscillators, the specific features of their (polymer) quantization will play a significant role. A way to avoid some difficulties here also relies on the elimination of unwanted energy eigenvalues by imposing superselection rules as long as they can be physically justified.
[1] J.F. Barbero G., J. Prieto and E.J.S. Villaseñor, Band structure in the polymer quantization of the harmonic oscillator, Class. Quantum Grav. 30 (2013) 165011.
[2] W. Chojnacki, Spectral analysis of Schrodinger operators in non-separable Hilbert spaces, Rend. Circ. Mat. Palermo (2), Suppl. 17 (1987) 135–51.
[3] H. Halvorson, Complementarity of representations in quantum mechanics, Stud. Hist. Phil. Mod. Phys. 35 (2004) 45-56.
Tuesday, May 5, 2015
Cosmology with group field theory condensates
Tuesday, Feb 24th
Steffen Gielen, Imperial College
Title: Cosmology with group field theory condensates
PDF of the talk (136K)
Audio [.wav 39MB]
by Mercedes Martín-Benito, Rabdoud University
One of the most important open questions in physics is how gravity (or in other words, the geometry of spacetime) behaves when the energy densities are huge, of the order of the Planck density. Our most reliable theory of gravity, general relativity, fails to describe the gravitational phenomena in high energy density regimes, as it generically leads to singularities. These regimes are achieved for example at the origin of the universe or in the interior of black holes, and therefore we do not have yet a consistent explanation for these phenomena. We expect quantum gravity effects to be important in such situations, but general relativity, being a theory that treats the geometry of the spacetime as classical, do not take those quantum gravity effects into account. Thus, in order to describe black holes or the very early universe in a physically meaningful way it seems unavoidable to quantize gravity.
The quantization of gravity not only requires attaining a mathematically well-described theory with predictive power, but also the comparison of the predictions with observations to check that they agree. The regimes where quantum gravity plays a fundamental role, such as black holes or the early universe, might seem very far from our observational or experimental reach. Nevertheless, thanks to the big progress that precision cosmology has undergone in the last decades, in the near future we may be able to get observational data about the very initial instants of the universe that could be sensitive to quantum gravity effects. We need to get prepared for that, putting our quantum gravity theories at work in order to extract cosmological predictions from them.
This is the main goal of Steffen's analysis. He bases his research in the approach to quantum gravity known as Group Field Theory (GFT). GFT defines a path integral for gravity, namely, it replaces the classical notion of unique solution for the geometry of the spacetime with a sum over an infinity of possibilities to compute a quantum amplitude. The formalism that it uses is pretty much like the usual quantum field theory formalism employed in particle physics. There, given a process involving particles, the different possible interactions contributing to that process are described by so-called Feynman diagrams, that are later summed up in a consistent way to finally lead to the transition amplitude of the process that we are trying to describe. GFT follows that strategy. The corresponding Feynman diagrams are spinfoams, and represent the different dynamical processes that contribute to a particular spacetime configuration. GFT is thus linked to Loop Quantum Gravity (GFT), since spinfoams are one main proposal for defining the dynamics of LQG. The GFT Feynman expansion extends and completes this definition of the LQG dynamics by trying to determine how these diagrams must be summed up in a controlled way to obtain the corresponding quantum amplitude.
GFT is a fundamentally discrete theory, with a large number of microscopical degrees of freedom. These degrees of freedom might organize themselves, following somehow a collective behavior, to lead to different phases of the theory. The hope is to find a phase that in the continuum limit agrees with having a smooth spacetime as described by the classical theory of general relativity. In this way, we would make the link between the underlying quantum theory and the classical one that explains very well the gravitational phenomena in regimes where quantum gravity effects are negligible. To understand this, let us make the analogy with a more familiar theory: Hydrodynamics.
We know that the fundamental microscopical constituents of a fluid are molecules. The dynamics of this micro-constituents is intrinsically quantum, however these degrees of freedom display a collective behavior that leads to macroscopic properties of the fluid, such as its density, its velocity, etc. In order to study these properties it is enough to apply the classical theory of hydrodynamics. However we know that it is not the fundamental theory describing the fluid, but an effective description coming from an underlying quantum theory (condense matter theory) that explains how the atoms form the molecules, and how these interact among themselves giving rise to the fluid.
The continuum spacetime that we are used to might emerge, in a similar way to the example of the fluid, from the collective behavior of many many quantum building blocks, or atoms of spacetime. This is, in plane words, the point of view employed in the GFT approach to quantum gravity.
While GFT is still under construction, it is mature enough to try to extract physics from it. With this aim, Steffen and his collaborators, are working in obtaining effective dynamics for cosmology starting from the general framework of GFT. The simplest solutions of Einstein equations are those with spatial homogeneity. These turn out to describe cosmological solutions, which approximate rather well at large scales the dynamics of our universe. Then, in order to get effective cosmological equations from their GFT, they postulate very particular quantum states that, involving all the degrees of freedom of the GFT, are states with collective properties that can give rise to a homogeneous and continuum effective description. The similarities between GFT and condense matter physics allows Steffen and collaborators to exploit the techniques developed in condense matter. In particular, based on the experience on Bose-Einstein condensates, the states that they postulate can be seen as condensates.
The collective behavior that the degrees of freedom display leads, in fact, to a homogeneous description in the macroscopic limit. The effective equations that they obtain agree in the classical limit with cosmological equations, but remarkably retaining the main effects coming from the underlying quantum theory. More specifically, these effective equations know about the fundamental discreteness, as they explicitly get corrections (non-present in the standard classical equations) that depend on the number of quanta (spacetime “atoms”) in the condensate. These results form the basis of a general programme for extracting effective cosmological dynamics directly from a microscopic non-perturbative theory of quantum gravity. |
e735eeabc5a31117 | Friday, October 10, 2014 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
The very meaning of "probability" violates the time-reversal symmetry
An exchange with the reader reminded me that I wanted to dedicate a special blog post to one trivial point which is summarized by the title. This trivial issue is apparently completely misunderstood by many laymen as well as some low-quality scientists such as Sean Carroll.
This misunderstanding prevents them from understanding both quantum mechanics and classical statistical physics, especially its explanation for the second law of thermodynamics (or the arrow of time).
Time goes up (up=future, down=past). The right diagram.
What is the issue? For the sake of completeness, let's talk about the spreading of the wave function \(\psi(x,t)\) describing the position of a particle. In the diagram above, time starts at the bottom and it goes up. You see that there are are three stages of "spreading". The wave packet spreads between \(t=0\) and \(t=1\), then it abruptly shrinks because the particle is observed, and then is spreads again from \(t=1\) to \(t=2\), shrinks at \(t=2\), and spreads between \(t=2\) and \(t=3\). The diagram is qualitative and could be applied to the probability distributions for any observable in classical or quantum physics, OK?
You see that the diagram above is self-evidently asymmetric with respect to the upside-down flip. The flipped version looks like a tree
and I will refer to it as "the wrong diagram".
What is going on here? Between \(t=0\) and \(t=1\), and similarly in the other two stages of the correct tree diagram, the probability distribution or the wave function evolve according to some equations that have a mathematical property: they are invariant under the time-reversal symmetry. The wave function \(\psi^*(x,t)\) with the extra complex conjugation evolves in the same way (i.e. obeying the same Schrödinger equation) as \(t\) goes up as \(\psi(x,t)\) evolves if \(t\) goes down.
Similar comments apply to the evolution of the phase space probability distribution in classical statistical physics. The equation is known as the Liouville equation and its fundamental form is symmetric under the time-reversal symmetry, too. These three "continuous segments" of the "green tree" diagram are something that the confused people don't have a problem with.
What they have a problem with are the "discontinuous jumps" at \(t=1\) and \(t=2\) – and lots of their counterparts in the real world. Needless to say, they're the moments of the "measurement". Look at the measurement at \(t=1\), for example: this moment is the horizontal line of the first picture that divides the tree to 1 triangle below the line and 2 triangles above the line. Why did the distribution shrink at that moment? Why didn't it "expand" instead? To answer this simple question, let's first describe the situation near the moment \(t=1\):
At time \(t=1-\epsilon\), i.e. before the measurement, the wave packet was spread. The location of the particle (or any property of a classical or quantum system) was ill-defined or fuzzy or uncertain or partly unknown.
At time \(t=1+\epsilon\), i.e. after the measurement, the wave packet was concentrated. The location of the particle (or any property of a classical or quantum system) became well-defined or sharp or certain or well-known.
I originally wrote the second sentence using the clipboard (via copy-and-paste) but then I had to edit it because the adjectives are different. In fact, they are completely opposite. Note that if you interchange the moments \(1\pm \epsilon\) with one another, you simply obtain propositions that are wrong. One may "suddenly learn" some information but one may never "unlearn it" abruptly after an infinitesimal period of time.
Of course, you may "flip" these definitions – but then you will get an equivalent description of physics in which \(-t\) is used instead of \(+t\), and/or in which the word "past" means the "future" and vice versa. There is no reason to add this extra confusion; you won't gain anything by this proposed chaos in the terminology. The "past" and the "future" are totally and qualitatively different whenever something about learning or observing or probabilities is involved (and it always is).
The probability distribution at the moment \(t=1-\epsilon\) i.e. before the measurement – whether the probability distribution is calculated from a wave function in quantum mechanics, or it is a fundamental object in classical statistical physics (or its informal counterparts: my statements really apply to any form of probability discussed by anyone and anywhere) – determines at what locations \(x\) the wave packet is more likely to be concentrated at \(t=1+\epsilon\), i.e. right after the measurement. Yes, I have used the word "likely" again so the "definition" is circular. It's inevitable because one can't really define the Bayesian probability in terms of anything more fundamental. There is nothing more fundamental than that.
But what's important to notice is that the meaning of the probability always refers to the situation
a property is unknown/blurred at \(t=1-\epsilon\)
it is well-known/sharp at \(t=1+\epsilon\)
The two signs simply cannot be interchanged. The very meaning i.e. the right interpretation of the wave function or the phase space probability distribution is in terms of probabilities so the time-reversal-breaking quote above is inevitably understood in every and any discussion about probability distributions and wave functions.
Their very meaning – their defining property – is to tell us something about the final state of the measurement at \(t=1+\epsilon\), out of some incomplete knowledge at \(t=1-\epsilon\). Again, to stress the point, their very meaning is to tell us something about time-reversal-asymmetric abrupt events. If there were no time-reversal-asymmetric abrupt changes of our knowledge i.e. if there were no learning and no measurements and no observations, there could be no probabilities! In that case, there would be no probability distributions and there would be no wave functions because the very meaning of all these things is to tell us what to expect at \(t=1+\epsilon\).
There is no contradiction connected with the existence of the "event of learning" or "measurement" at \(t=1\). Obviously, we sometimes have to learn some information about something, otherwise we couldn't talk about anything and there could be no science – or ordinary life, for that matter. If the process of learning has some internal structure, if we are measuring something with an apparatus that works for complicated reasons, there is something to discuss.
But if we only want to talk about the general claim that "there are measurements" i.e. events in which we suddenly learn some sharp or sharper information about something, there is really nothing to talk about. It's as elementary and irreducible fact about the human thought as you can get. People learn. Ergo there are these "shrinking discontinuities" in the probability distributions for everything and anything in the world. The "past side" ("before" side) of these measurements always has a more blurry distribution than the "future side" (or "after" side). Whoever writes whole chapters or books or book series about the very existence of "observations" or "learning of the information" is guaranteed to have written meaningless, pompous, vacuous philosophical flapdoodle only.
This behavior of the probabilities around the measurement – where the probability tells us what to expect "after" the measurement – is the source of what I call the logical arrow of time. Stephen Hawking and others use the word "psychological arrow of time" and it's clearly the same thing. Hawking uses the term "psychological" for a good reason – learning about something by seeing it is a "psychological process".
The reason why I prefer to avoid this "psycho-" prefix is that it leads the people to think that an analysis how brains work and whether they have consciousness and what consciousness means is an obligatory ingredient in a complete analysis of the logical arrow of time. It's not and that's why the term "logical arrow of time" is more appropriate. What we really need is just the fact that some information (about an observable, a property of the external world) or the truth value of a proposition is unknown at \(t=1-\epsilon\) but it is known at \(t=1+\epsilon\). I don't need to assume anything whatsoever about "agent" for whom it is known, his or her structure, the mechanisms inside the brain, and so on. I don't need to assume that there is an "agent" that also has some other capabilities aside from knowing or not knowing whether a proposition about Nature is true. The logical arrow of time is about the (logical) truth values of propositions that abruptly change at \(t=1\), and the probabilities tell us what the final product of the change (which "after" state) is reasonable to be expected!
This logical arrow of time is a simple, elementary, and irreducible part of our existence within Nature. But it has consequences. If you think about the comments above and recognize that all these things are as clear as you can get, you should also understand that there is no "measurement problem" in quantum mechanics – the existence of the "a measurement" is tautologically an inseparable part of any statement about the corresponding "probabilities".
And you will understand that there is no problem with the thermodynamic arrow of time, either. The proof of Boltzmann's H-theorem or its variations and generalizations are proofs that the thermodynamic arrow of time (showing the direction of increasing entropy) is inevitably correlated with the logical arrow of time. But the logical arrow of time always exists in any logical framework that talks about probabilities because probabilities are always relevant before a moment when a property is "learned" or "decided" so they are linked to time and the relationship treats the past and the future absolutely asymmetrically!
I don't really believe too much that this clear-as-sky explanation of this issue will make someone new scream "Heureka" because these people love to be confused idiots and they are proud about it. It is probably a part of their self-confidence to think that there is something seriously wrong with statistical physics or quantum mechanics or thermodynamics (and maybe with mathematics, too) and it would hurt their ego if they had to learn that something has been wrong with the (time-reversal-asymmetric) semi-infinite part of their world lines i.e. with their lives up to this moment when they had a nonzero probability to understand what "probability" means and why none of these would-be problems exist. But the probability was too low so it's not surprising that most of them have remained confused morons instead.
And that's the memo.
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reader NikFromNYC said...
Searching for a concrete model of this, I suggest to myself how mere random noise in systems would result in practical information loss that by its very nature is only one way in time for unknown information about earlier conditions has no mechanism to suddenly reappear if I try to turn back time.
reader BobSykes said...
Thank you, yet again, for making sense of physics.
I sometimes wonder (you being all the way out there in the Czech wilderness, on the edge of Conan's steppe) if your position is not somewhat like that of Davy Crockett at the Alamo, surrounded by the likes of Carroll thirsting for the kill.
reader Luboš Motl said...
LOL, thanks, and I am going to relearn what I know about the Texan revolution now. ;-)
reader Luboš Motl said...
I agree, if I understand. ;-) The basic step of the shrinking (throwing dice and learning the result) is irreversible in the normal real world. So despite the T-symmetry of the evolution for the amplitudes, the overall rules still know that the future depends on the past and not vice versa.
reader JollyJoker said...
Isn't this somewhat like checking if there's a deviation from expectations using all the data instead of looking separately at individual channels?
This seems like a pretty smart thing to do. They mention one double check of their method, "It is interesting to note that the distributions observed here are very different from those observed in our analysis of the 7 TeV data" that could indicate they have something real.
reader Shannon said...
And I thought it was going without saying !... Either I was being to simplistic or I am just one clever duda.
reader Luboš Motl said...
Yup, it could be very smart, and far-reaching, unless it is wrong.
When I was now throwing votes in the local and senate elections, I was thinking what could be wrong about it.
If they neglect that the number of events "N" in a bin isn't a continuous real number, but a positive integer, then they may get completely distorted predictions for channels where N is really really low, like 0,1,2,3 etc.
But if their result isn't an artifact of such things, it could be cool and there is some sense in which they accumulate the signal of "new physics Yes/No" from all the places.
reader Simon Phoenix said...
Yes - totally agree with all that - rather trivial as you say.
But isn't it rather missing the point as far as QM is concerned? All of this is fine and dandy **given** a measurement. The problem isn't this interpretation of the change of probability upon measurement - and it is clearly asymmetrical - the issue is the following.
System S, System M - M is our measuring device. They interact. QM applies to everything including the measurement device. S and M interact - QM says this interaction gives a unitary evolution for the state of the S+M system agreed?
The measurement occurs and S is now described by an eigenstate of the observable. This is a **non-unitary** evolution as I'm sure you would agree. How has this occurred if all interactions within QM are described as unitary processes? Is 'measurement' a different kind of interaction obeying different dynamical rules?
One FAPP solution is the decoherence approach in which we derive a master equation for the reduced density operator of S - by a suitable coarse graining procedure. This then gives 'irreversibility' because we've smoothed out over things we don't know (the degrees of freedom of the measuring device + environment).
If we suppose that S starts in a pure state then the entropy of S+M increases after measurement - assuming evolution to a mixture of eigenstates as required - this increase in entropy comes about because we effectively lose information (it's 'stored' in the environment upon which we do some kind of averaging procedure).
Sure, we can just assume a measurement (and yes if we're talking at the level of probabilities only, a measurement is implicit anyway) - and yes we have less uncertainty after the measurement so we get the inverted tree diagram for the probabilities as above - but the question is whether the formalism of QM can describe this process, or whether measurement is an axiom we have to apply (the projection postulate).
reader Luboš Motl said...
You *are* one clever dude! I think that many folks like you have lots of common sense (more than myself) that's been trained in the real world, so if they get the concept of probability or something, it's really immune against some basic confusions.
reader Luboš Motl said...
Everything is fine given a measurement.
If a measurement isn't given, then we can't talk about it, can we? ;-)
You shouldn't be sure about such things. I disagree because the jump from the spread wave function for S+M to a localized one isn't *evolution at all*. It's just an interpretation of the wave function. All systems in Nature *always* evolve unitarily and this "collapse" is just a psychological process. One learns the value of an observable describing S, so one may work with a simplified wave function where all other components of the wave function are set to zero. But one doesn't have to. One may still work with the full wave function for S+M that hasn't collapsed in any way and talk about the conditional and correlated probabilities of all properties of S and M.
reader JollyJoker said...
Presumably they don't have enough information to feel very confident of their results. I hope some people from Atlas/CMS take the time to look at this.
reader Lucas Martins said...
This logical arrow of time could be identified to the non-commutation of the projection operators in the quantum mechanics of closed systems? I mean, quantum mechanics is contemplated with logical arrow of time by this non-commutation?
reader Luboš Motl said...
Dear Lucas, good try but I don't think so. The nonzero commutators don't have any preferences for one direction of time or another. Moreover, these comments about the logical arrow of time apply to probabilities even in classical statistical physics - any contexts where probabilities appear - where the commutators are zero.
reader Luboš Motl said...
It's also plausible that they delay the publication of papers that find too few events... ;-)
reader NikFromNYC said...
Philosophically it goes back to the hidden variable hypothesis, where no information really is lost, except to us macroscopic scientists, and that all this statistics is merely measurement theory imposed upon actual reality. Your promotion of mathematics as the basis for the uncertainty principle helps me take randomness seriously, but I don't understand the basis for that randomness, and if that randomness really is dictated by the uncertainty principle itself. Lacking a math background, I can only ask vaguely.
reader Lucas Martins said...
Dear Lubos, I catched now. When we build a history we put the time evolution operator between this projections. So, the non-commutation of this projections only tell us that some histories couldn't have an usual probabilities.
So, were is the logical arrow of time in quantum mechanics of closed systems? Is the logical arrow of time manifest only in decoherence mechanism? Is the logical arrow of time manifest only in the quasiclassical realm?
Note: With "manifest", I don't mean say an explanation of the logical arrow of time, I mean say only a symptom of this arrow (like the second law).
reader Shannon said...
Thanks to you too Lubos.
reader Luboš Motl said...
Right, Lucas! The nonzero commutators of the projection operators are the reason why it's nontrivial to make the set of histories "consistent".
Decoherence is a complex process - a derivation of the evolution of the density matrix on paper, in certain circumstances. Note that the text about probabilities above didn't mention the word "decoherence" once because the basic arrow of time of the probability doesn't depend on decoherence. It doesn't depend on anything quantum in fact, as I already said. So I don't understand why you keep on promoting the word "decoherence" here that is much less fundamental than the truly primordial source of the arrow of time.
Also, the arrow of time has nothing to do with the question whether a system is open or closed. Both open and closed systems require probabilities to be described and every time a probability is interpreted - when we measure something, new time-reversal asymmetry or irreversibility is introduced.
So I guess that the answer to all your questions - and probably 60 other similar questions you are going to ask - is No, No, No, No. The logical arrow of time appears in absolutely all systems and all theories and all circumstances and every attempt of yours to suggest that it depends on some very special effects or circumstances is just completely wrong, wrong, wrong, wrong.
Isn't it enough to ask this question once instead of 64 times?
reader Simon Phoenix said...
Agreed. But now I would suggest you're skirting perilously close to MWI. In order to be consistent with unitary evolution you'd then have to have a combined entangled wavefunction including the possible outcomes - but somehow only one of those possibilities is actually realized when the measurement is done - according to the probability rule. We only experience one of these outcomes - so a full entangled description including all possible outcomes (unitary evolution) doesn't really accord with this experience (we don't seem to consciously perceive superpositions - not, of course, that this has anything to do with consciousness!)
Keep everything unitary, by all means, but I can't see a way of incorporating new knowledge (the measurement result) in this approach other than non-unitarily 'by hand' - to calculate relevant probabilities for subsequent measurements we'd have to work along those branches of the entangled wavefunction selected by our previous actual measurement results.
We use a new wavefunction to describe the post-measurement state based on our new knowledge (we assign the alternative 'null result' branches a zero probability after measurement)
So when we incorporate new knowledge (an actual result) rather than just calculate a priori probabilities the non-unitarity is implicit. There's no unitary process that allows us to acquire this actual knowledge in the first place.
reader Lucas Martins said...
(You can only change your answering if you want, without submit my question ;))
Forgive me if I'm slow to understand. Are Probabilities in quantum mechanics interpreted in this way due to decoherence mechanism and the construction of the quasiclassical realm?
I saw your twitter about the time asymmetry in the definition of probability. I'm very okay with that. So, this question and the question about symptom of the logical arrow of time in quantum mechanics is strongly correlated.
If the answering is yes, so in an Universe made by few degrees of freedom don't present the logical arrow of time. Of course that in this universe we don't have anything to do or see, because don't exist a emergent reality (quasiclassical realm).
reader Gordon said...
Sean and acolytes think that you misunderstand entropy and the arrow of time and are rather and say so sarcastically and dismissively; and ultimately refer to some famous pedagogue who taught you stat mech at Rutgers who they say would disagree with you. It makes no sense to me--your explanation is clear--their's is muddy hand waving (to mix metaphors). At this point in time, I am not current, but I have taken thermodynamics and statistical mechanics and think, like you, that Boltzmann is basically all that is needed along with some understanding of QM to understand the arrow of time. Sean's book was ok for the first few chapters or so, then launched into lala land.
reader Lucas Martins said...
I need to time to think, this things is very complicated for me. But thanks for your time and patience.
reader Luboš Motl said...
It's complete bullšit. I have always been in complete agreement with my StatMech and Thermodynamic instructors, undergraduate or graduate ones, and I've been arguably the best student in every single class.
Is the lie written somewhere so that I could sue the jerk for libel?
reader Luboš Motl said...
"But now I would suggest you're skirting perilously close to MWI."
I don't know what to do with claims (or accusations?) like that. How do you measure the "distance from MWI"? Not only the distance is ill-defined; MWI itself is ill-defined. So what can any sentence about "distance from MWI" possibly mean? If some sentence sold as "MWI" are closer to correct physics - i.e. to what I say - than other sentences, it's clearly a coincidence.
There is no room for "many worlds" in quantum physics.
We only experience one of these outcomes
We only experience one outcome because that's what physics predicts. When an electron is in the state "0.6 times up plus 0.8 times down", it implies that if we learn the value of the spin (in other words, if we measure it), it will be either up or down. The first option has 36% likelihood, the second one has 64% likelihood. This is the only right interpretation of the implications of the wave function for the question "what spin we will experience and how many options we will experience".
reader Lucas Martins said...
The logical arrow of time is this:
A => B ?
reader Jan Reimers said...
Hi Lubos Thanks again for your efforts on this topic. Sean makes a lot of effort to portray himself as a deep thinker ... but he is NOT.
Why do we have to use the term "measurement problem" instead of just calling it the "interaction problem"? This would subtract humans with laboratories and machines out of the discussion. And then once that step is complete just get rid of the word problem ... interactions in QM are not a "problem".
Also why not put a little bit of slope on the horizontal branch bottoms on your diagram to show schematically the (very tiny) de-coherence time? Then people can stop puzzling about *apparent* non-unitary behaviour.
reader lukelea said...
Dear Lubos, The word knowledge is something that has always confused me about the uncertainty principle. You wrote in one of the comments,
"Nonzero commutators - quantum mechanics' uncertainty principle - means that the maximum knowledge must be smaller than the knowledge of all observables, so something has to be probabilistic. You can't know X and P at the same time.'
What bothers me is that the uncertainty principle is also used to explain why the electron does not spiral into the proton (and why the vacuum state is full of virtual particles too, I think?). Clearly these facts are not about knowledge. They are true irrespective of anything we know. It's more like X and P cannot even simultaneously exist within some limit.
Is this right? thanks,
reader Frank Ch. Eigler said...
That 8:24 snapshot looks superficially similar, but the the rough appearance in the russian reproduction case only appears on the exit side, whereas incoming shrapnel signature is all over the real crash.
reader Jan Reimers said...
Hi Lubos, It looks like you didn't read my 'And then once that step is complete just get rid of the word problem ... interactions in QM are not a "problem"'
My point is that there are processes like your tree taking place far away from earth where particles collide and there is no measuring going on -- just interactions and de-coherence.
"The effect of the measurement on the wave function isn't given by any particular map on the Hilbert space"
That is true for Hilbert space of Psi (the "system" that the tree is describing), but I don't see how this true for the much larger Hilbert space of "system"+environment. That larger Hilbert space is still undergoing unitary evolution which causes the apparent change in probabilities of the Christmas tree "system".
reader john said...
You implicitly think that wave function is something real. This is wrong. Wavefunction is only a tool to calculate probabilities of properties. Also there is no objective description of the system which the reason lies behind your problems with realization of a particular property (during measurement). I suggest you to read the book "Consistent Quantum Histories" Robert Griffths. It is free.
reader Luboš Motl said...
Dear Jan, the laws of quantum physics hold for a "system" just like they hold for "system+environment". There is really no physical difference between the two. The separation of some degrees of freedom to the "system" and "environment" is really just a matter of conventions. So it's clearly wrong to suggest that the general laws of quantum mechanics apply in one case and not another.
reader Jan Reimers said...
Dear Lubos, Yes you are right (what else is new:) and that is were the FAPP (For All Practical Purposes) concept makes its entry. If the "system" is interacting then we cannot apply the laws of QM to just the system and strictly speaking we cannot talk about probabilities of observable s for just the system. But FAPP you can describe the system as isolated with its own unitary evolution between t=1 and t=2, and FAPP you can ask about probabilities of the system. Your errors will exponentially small until you hit t=1 or 2.
reader Luboš Motl said...
Luke, the uncertainty principle has many implications. Everything that was impossible in classical physics but becomes possible in quantum mechanics (or vice versa) is due to the uncertainty principle. You surely don't believe that if a principle has more than 1 implication, it is a contradiction, do you?
Concerning the knowledge, I am just saying that in classical physics, because it is defined as physics where all observables commute with each other, it is possible to "simultaneously diagonalize them" - more ordinarily, to find the values of all of them. The maximum knowledge has certain, error-free values of everything that can be known, and this set of numbers typically evolves according to deterministic laws.
In quantum mechanics, such a complete knowledge of the state of the system is impossible due to the uncertainty principle. If X is known, then P is perfectly unknown, and so on. Or some compromise with uncertainties in both. And similarly for lots of other sets of observables. This is a *consequence* of the nonzero commutators.
You are perfectly right that X and P cannot simultaneously *exist*. It is a stronger description of the uncertainty principle that you said, and it's true, too. However, what "really exists" is a potential minefield and one may still talk about what is "possible to know" too, right?
reader Simon Phoenix said...
Yes John, it is a mathematical object that allows us to calculate probabilities - it is also a mathematical object that evolves unitarily (however we wish to 'interpret' it). When we make a measurement there's a real physical change - a bit is recorded - what's the unitary connection between the states of the world before and after the recording of this bit?
reader Luboš Motl said...
Dear Jan, thanks for your comment which I don't understand. Which exponentially small errors are you talking about? What is the exponent and why the errors aren't zero? Why do you think that there will be errors at t=1 or t=2? We're just not at the same frequency at all.
In physics, we are calculating what we see when we measure something. Science isn't obliged to predict anything that can't be measured, and indeed, quantum mechanics uses this principle maximally because it labels all such questions meaningless.
The picture with the tree was sort of meant to suggest that we only make measurements at t=1 and t=2 etc. so all the state vector at fractional times is just a speculative auxiliary object and it makes no physical sense to talk about "errors" of the wave function at those times because nothing is measured at those times. Physically, errors may only refer to the difference between what we measure when we measure - e.g. at t=1 or t=2 - from the predictions. In a single repetition of the measurement, there will be unavoidable uncertainties from the uncertainty principle etc. If one repeats the same experiment many times to calculate the observed probability distributions, they will agree with the predictions of a correct quantum mechanical theory exactly.
To (almost) exactly measure these distributions etc., one needs a classical device - a device for which the basic laws of classical physics are an excellent approximation. So you may say that it is a gadget that perfectly decoheres, and/or with lots of environment, and so on. The founding fathers of QM chose not to decompose this condition to some smaller criteria. They emphasized what really matters and what really matters is that the object used to measure these distributions in (repeated) experiments has to be a classical object. The more classical it is, the more accurately one may measure the distributions etc.
reader Luboš Motl said...
Simon, it's like talking to a wall. Because the wave function is *not* a real object, as John tried to remind you, the change of the wave function is clearly *not* a real physical process, either.
reader Luboš Motl said...
The logical arrow of time is the principle that all general rules of physics of the form
A(t_1) implies B(t_2)
for some propositions at the given time that is either guaranteed to hold or that has a calculable probability obeys "t_1 is smaller than t_2".
So the future is probabilistically but otherwise "accurately" determined from the past and this relationship cannot be reverted.
reader Luboš Motl said...
The quantum randomness may be shown not to arise from hidden variables.
The nonzero commutators imply that the probabilities of different values can't be just 0% or 100% because if the observables X,P had well-defined c-values x,p at a given moment, they would obey XP-PX=0 because xp-px = 0, like all numbers, instead of the correct XP-PX = i*hbar. So some generalization has to hold instead, and when one looks what the generalization actually says, it says that for a given state, every observable such as X (or P) has some probability for each allowed eigenvalue. The non-existence of the "common eigenstates" of non-commuting observables is enough to allow a nonzero commutator.
reader Simon Phoenix said...
"As I have already explained to you in a way that most dogs must have already understood but you have not, the concept of "unitarity" is only meaningfully defined for actual "maps" from the Hilbert space to the Hilbert space but the measurement isn't defined by any map"
But that's what I've been saying! Measurement is a non-unitary process, i.e. it isn't described by a unitary map!
In a strict function sense a measurement provides a probabilistic 'map' from a set of input states (all possible states of the physical system to be measured) to a set of output states (eigenstates) which is not invertible. The measurement is just a function that takes an input state and gives an output state according to a well-defined probability rule.
This mathematical description is independent of how we choose to interpret what the wavefunction means.
Interactions are described by invertible and deterministic maps. So yes measurements are clearly not described by unitary maps. So measurements in QM are not interactions in the usual sense.
All physical objects are supposed to be described by QM. Measuring devices are physical objects. All physical objects interact with one another according to a unitary map, according to QM. A measurement is an example of an interaction that cannot be described by a unitary map.
Personally I think there's a problem here that's not wholly explained away by decoherence, or by appeal to probability. In physical terms there's a real physical change when a measurement is made - a real bit is recorded. The input and output states are not connected by a unitary map. We must appeal to something different than the usual unitary evolution rule in order to explain this (or introduce some further assumptions if we believe that decoherence or consistent histories provide an adequate explanation).
reader Luboš Motl said...
No, the change of the wave function during measurement is not a map - equivalently, it is not a function - which implies that it makes no sense to ask whether the change is unitary or not, and it implies that everything else you write is gibberish, too.
reader Jan Reimers said...
I was talking about the density matrix elements between the (small) system Hilbert space and (large) environment Hilbert. These decay rapidly towards zero as (exp(-t/decoherence_time) right after the interaction (measurement) takes place. assuming these elements are zero is FAPP a good approximation after t1+eps where eps is a few e-foldings of t/decoherence_time. But you already knew that.
reader Luboš Motl said...
I see, Jan.
You probably mean that the off-diagonal elements of the reduced density matrix for S only quickly decay - they decay much more quickly than exponentially, however (in decoherence). They decay like exp(-D*exp(C*t)). These matrix elements of the reduced density matrix may be computed as a sum of many terms calculated from the whole S+M. Each of these terms is even smaller because it is a product of many tiny numbers.
The usage of the reduced density matrix is absolutely sufficient and *exact* for any predictions as long as we will only measured observables acting on the Hilbert space (tensor factor) of S. If we want (and/or are able) to measure observables of both S and M, there is no point in separating the "environment" M.
reader NikFromNYC said...
Do there must be time within the realm of mathematics itself, then?
reader Simon Phoenix said...
A measurement is a stochastic process that takes an input and produces an output - whether we call it a map or a function is irrelevant. What is relevant, as you keep emphasizing, is that whatever it is it is not describable by any unitary map - therefore it is not an interaction in the usual QM sense - because interactions between physical objects are all describable by unitary maps.
I'm sorry that you think all this is gibberish, that I am more stupid than most dogs, and that my thinking is completely fucked-up, but there must surely be something a bit puzzling with a theory that
(1) stipulates interactions are governed by unitary evolutions
(2) is unable to describe the output of an interaction between a measuring device and the measured system as a unitary evolution of the input state (without recourse to further assumptions such as those required to make decoherence treatments work)
It's one thing being entirely comfortable with the rules of QM (which I am) and quite another trying to fit it all into some kind of intuitive framework. As mentioned earlier - it's not immediately obvious why vectors in a complex Hilbert space should be the mathematical entity we require to describe the 'state' of something - even if we only think of this as an entirely abstract object that represents our state of knowledge (whatever that means precisely). It's even less intuitively obvious why the interaction of 'measurement' is different from other interactions in QM.
reader kashyap vasavada said...
Hi Lubos: Slightly off-topic, but not completely so. There is a report in Physics Christopher Ferrie, Joshua Combes (Physical Review Letters) argue that
such measurements, and their counterparts known as "weak values", might not be inherently quantum mechanical .They say that the results from such measurements can be replicated classically and are therefore not properties of a quantum system. I know, you do not care for the theory of weak measurement. What do you think of this proof?
reader john said...
You can think in the following way. Quantum mechanics tell us what we get when we measure a system using an "infinitely" large apparatus (These are words of a winner of fundamental physics prize, you can trust them). Wave function isn't something corresponding to a point in phase space in classical mechanics. Also note that it is extremely important that measurements disturb the system. If they didn't all of quantum mechanics would collapse. Measurements don't commute <-> corresponding operators don't commute. This approach is totally fine. However I believe there is a better approach at least "aesthetically".
You must have noticed that you talk about closed systems such as universe with this point of view. Consistent Histories or Decoherent Histories approach (created by Omnes, Griffiths, Hartle, Gell-mann) defines what is a physical property and allows you to say something about closed systems. However Copenhagen interpretation can be derived using Consistent Histories, so they are really not different things.
Personally, CH approach made me understand quantum mechanics much better. However as I understood it (at least I believe so :) ), I also understood Copenhagen interpretation better (one can easily derive it in CH) so now I am much more comfortable while using Copenhagen interpretation in daily life. For example you can prove in CH that: "Measurements don't commute <-> corresponding operators don't commute", so you really don't loss anything while you are thinking about what you can measure or not. I did understood what is complementarity about and what really wave function is. CH approach stresses this things very strongly. For example CH approach doesn't call wave function as state of the system, it says pre-probability.
Something I have said above may seem cryptic to you. I can't summarize the CH approach here but I strongly recommend you the read the book by Griffths.
reader de^mol said...
" all over the real crash".
I don't know what you mean. It is first of all only confined to the cockpit area. Also, in the Russian video you see outcoming (7:56 & 8:18) AND incoming holes (8:13 & 8:16 ). You see the same in AND outcoming holes on the MH17:
Here, more outcoming holes:
reader john said...
I don't know what they mean exactly by weak measurement but if they propose that they can simultaneously measure position and momentum, like said here, then they are wrong. As a proof, read
reader Simon Phoenix said...
I get all that, and I'm really not thinking of the wavefunction as some kind of 'real' object - but the fact remains that 'real' (which it clearly isn't I agree) or representative of our probabilities (when mod squared) its evolution is governed by a unitary and deterministic map, unless our object 'represented' by this wavefunction is subjected to a measurement. So we look at 2 quantum systems and we can use our knowledge of the input states (or wavefunctions) and the physical interaction to determine what our probabilities should be for some observable for one of the systems at some subsequent time.
If one of those systems, however, is designated as a 'measuring' device - then we can no longer apply the same mathematical rules to determine the mathematical representation of these changing values at this subsequent time - despite the fact that measurement must be an interaction. The same mathematical rules don't apply when one of the systems performs what we call a measurement - or when the interaction between the two systems is deemed to have constituted a measurement of one of the systems.
If we input a single photon to a beamsplitter then the output state/representation of our probabilities of the 2 arms is described by ~ |10> + |01> which is an entangled state of the 2 spatial modes of the field. Suppose we perform an ideal photon measurement in the output arms and find that we detect the photon in arm 1 (let's not quibble here about the fact that a photon measurement is destructive). It would now be wrong, given this knowledge, to describe our new representation of the probabilities by an entangled wavefunction between the 2 modes. Indeed, no experiment that could be subsequently (individually) be performed on these 2 photons would reveal entanglement (with or without knowledge that a measurement had been performed, or its result).
But all that's happened is that we've interacted things together (our field modes with our detectors) - it's just that we call this special kind of interaction a 'measurement'. If we'd interacted these photons with, say, 2 2-level atoms in high-Q cavities, we'd be using the unitary evolution of the interaction to determine our representation of the probabilities - and we'd also see evidence of entanglement between the photons in subsequent measurements if we tailored things just right.
Our mathematical description, our mathematical representation of things changes according to whether the interaction is deemed to be a measurement or not.
And if a measurement has been performed our mathematical description then changes according to the new knowledge that we get (or is recorded) from this special kind of interaction we call measurement, and yes it can be subjective. So what's different? In one interaction there is a physical record of a bit which requires energy to erase (the change in the state of our knowledge which is a physical change in our brains, or the physical output of the measuring device), in the other there is not.
reader Luboš Motl said...
Hi Simon, I've posted about 10 comments of yours just during the last 10 hours. None of them makes any sense and you show zero potential for learning even if it comes to completely trivial things.
I've wasted at least an hour just with you today and I don't want this to continue so I am placing you on the black list.
reader Luboš Motl said...
Sorry, Jan, "matrix element between a small Hilbert space and large Hilbert space" is just an oxymoron, a nonsense. The density matrix is an operator so its matrix elements are always between two elements of the *same* Hilbert space.
reader Curious George said...
Refreshing to know that it happens in physics. I thought Cook, Lewandowsky, and their Australian teams only handled "climate".
reader Tony said...
Thank you for this explanation Lubos.
As I understand it, before throwing a dice we have a probability distribution that we will get one of the six numbers. That probability is 'sharp' because we always get exactly one of six numbers. That is, excluding unlikely case where a dice lands so that it manages to balance on one of its edges :)
Once a dice is thrown we see the final result. One may say we measured it with our eyes. One may say, if one really likes those words better, that probability distribution collapsed to a single outcome.
reader lcs1956 said...
For the same reason that the precise wavelength of a localized wave packet cannot exist. Just the math of Fourier transforms, nothing more mysterious. The conceptual hurdle is the association of a particle with a wave packet.
reader Luboš Motl said...
Tony, one may talk about the "collapse" of the distribution for the dice if it makes him feel so much better. In some sense, it does collapse.
But the point is that there is no "internal structure" of this collapse to be decomposed or looked for. The collapse is an inevitable part of the term "probability".
reader Luboš Motl said...
Subtleties are only legitimately important if 1) the big picture is at least approximately right, and 2) if these subtleties are not cherry-picked or selectively interpreted and adjusted according to double standards.
Both conditions have to hold. Unfortunately, none of them is being satisfied in the "mainstream" Western reporting about the Ukrainian civil war and many other events.
reader Eclectikus said...
Yep, I agree in that, too.
reader Luboš Motl said...
Great to hear that, Eclectikus!
reader kashyap vasavada said...
Thanks Lubos for a detailed reply. In physics we are learning again and again that the fact that someone does very important work once, does not mean he is right all the time!!
reader anna v said...
Or the excess/deficit may be coming from a part of the variables that is not well explored as far as cuts etc. When one is concentrating on clearing up the Higgs, for example, It could be that the cuts etc that clear up around 125 GeV are not appropriate for other higher intervals. Then, if some excess or deficit appears, they will hold their horses reanalyzing or waiting for more data .
reader Johannes said...
An interesting quote from the preposterous universe:
"As far as I can tell, the revolutionaries make their case by setting up a stripped-down straw-man version of quantum mechanics that nobody really believes (nor ever has, going back to Copenhagen), then proclaiming victory when they show that it’s inadequate, even though nobody disagrees with them."
reader Michael Gersh said...
We can't all be experts in everything, but if you want to call our differences nuance, that's fine with me. But the personal part? We were both personally affected by WWII. My homeland was not only betrayed, it was wiped off the map permanently. That doesn't change a thing. CinC is a term of art with actual meaning, to some. To others, not so much.
reader Luboš Motl said...
Dear Michael, you are forcing me to discuss this personally because your otherwise indefensible claim that Hitler didn't become the commander-in-chief in early 1938 is being justified by mysterious references to your being an expert.
I don't respect that as an argument. I don't respect you as an expert. Show me one paper or something like that which would claim that Hitler was never a commander-in-chief in 1938. You must know that you won't find anything like that because you are making it up and your claims of expertise are just rubbish.
You may have become a great fan of some generals of the German Army in the late 1930s or something like that, and worship them etc. but they were *not* commanders-in-chief e.g. in February 1938.
reader Tony said...
In this simple picture, if one says that the dice may show 3 in this universe, but all the other values may and will occur in other, parallel universes, looks like useless, pot-headed non-sequitur.
I was watching video by David Deutsch and he seems to believe that only many-worlds can account for the additional computational power that quantum computation brings with respect to the classical computation - as if additional computations from parallel universes somehow leak into ours.
reader Luboš Motl said...
Absolutely, Tony! If a many-worlds description is meant to be at least approximately equivalent to quantum mechanics, the different copies of the world must completely decouple - the worlds where Hitler won the Second World War (and the whole "tree" of such worlds) must become forever inaccessible from our world where Hitler lost, otherwise it would be a contradiction.
But if those "other worlds" are completely inaccessible, their large number can't possible influence what is happening in our world!
There is one seemingly different but ultimately equivalent way to say what's wrong. The many-worlds evangelists never say "how many worlds" there are in their list of "many worlds". But the idea is that the worlds get "reproduced" when some measurement is done, i.e. after some amount of decoherence.
But the very point and power of a quantum computation is that there is *no* decoherence during the whole process of computation at all, so the number of worlds doesn't grow at all! So it can't be exponential.
The fairy-tale that the "exponential speedup is possible because the number of many worlds is exponential" may sound OK at the level of this slogan composed of a dozen of words. But it fails every other basic test that it should pass.
Deutsch's claims are actually stronger and therefore even weirder. He doesn't only say that many-worlds give *a* viable explanation why quantum computers may be fast. He says that the quantum computers *prove* that there must be many worlds.
This is not one but at least two major steps away from any defensibility. In order to show that the quantum computer speedup *proves* the many worlds, one would have to say that the many worlds are the *only viable* way to explain such a fast computation. But in reality, not only he can't prove that it is the *only viable* explanation. He doesn't even have *a* viable explanation because of the two contradictions in the first paragraphs.
And this Deutsch guy is often sold as a guru of quantum mechanics. He hasn't penetrated a micron beneath the surface of popular nonsense that the laymen are being served about quantum mechanics. Nothing he says really makes any sense whatever. It's just a sequence of words for people who think that sentences with certain buzzwords and a certain predetermined message are "cool" - even if every piece of logic in those statements is completely defective.
reader etudiant said...
The BUK missile has a continuous rod warhead, basically explosives wrapped in a steel rod that has regular indentations in it, sort of a cocoon of integrally connected steel lumps.
On exploding, this produces a shower of rod fragments of very similar size. It is very difficult to distinguish holes produced in a sheet aluminum skin by this warhead from those produced by a gun. The best clue is the pattern of the holes, but in this case there are few pieces big enough to show patterns.
The cockpit fragments that have been shown in news photos are more suggestive of an explosion imho, simply because the metal is really peppered, which is hard to achieve with a gun.
Given the lack of transparency in the inquiry, I do not see how the eventual report will be credible.
reader Michael Gersh said...
[sigh] I really have no reason to argue with you, but, technically, Hitler was never Commander in Chief of the Wehrmacht. This is a non trivial distinction. If you need a link, try this:
Hitler did, however, usurp the command structure and name himself "Supreme
Commander of the German Armed Forces" and become the de facto CinC at the end of 1941.
I am sure that you could argue convincingly that dark matter is not merely a convenient fudge factor to support a cosmological theory that is no longer supported by observation, and, in that arena, I would never be able to win an argument with you. And I am also sure that my losing argument, like yours here, would be supported by spurious entries in Wikipedia.
reader Luboš Motl said...
The page you linked to doesn't say that Hitler didn't become the commander-in-chief in 1938. You know that very well. No book, paper, or page has ever claimed something of the sort because it's bullshit. Hitler unambiguously became the commander-in-chief in early 1938.
reader Michael Gersh said...
Ban me if you like, but I stand by what I wrote here.
reader Quantum said...
But it's possible to unlearn! It's called "forgetting".
And it's possible for wavefunctions to contract over time. An example would be a Loschmidt reversal. Or let's say we have a simulation of a quantum system on a quantum computer with no interaction between the memory states of the quantum computer and the environment. Then, uncompute. Voila!
What you call a "collapse" due to learning by an agent is actually none other than the creation of an entanglement between the memory states of the agent and the system in question. That analysis was done by von Neumann a long time ago.
You haven't solved the measurement problem in the least bit. Neither have you explained the thermodynamic arrow of time.
reader Luboš Motl said...
No, you're wrong, Quantum. The reversal of the "abrupt shrinking" associated with interpreting a "probability" cannot exist in Nature - or anything that contains the concept of ordered time similar to the physical one.
reader JW Smith said...
I've wondered if collapse is a bit too classical of a notion in that it gives rise to intuitions that there is a measurement problem so I propose you say it narrows to a classical limit or approximation. Nothing changes just the words so its a completely pedagogical value.
reader alejandro rivero said...
It is not only that you may "flip" these definitions; it is that both definitions coexist. There are two functions that collapse: the future probability of measuring, and the past probability of coming from some determinate past state.
That is even classical. If I see a person, whose location I didn't know, crossing a transfer gate in an airport. I have both an estimate of the possible flights he could be going to take and an estimate of the past flights he could be coming from. There is a collapse, but not a violation of time symmetry.
reader john said...
why do you want entropy to be low during big bang ? the only thing you have to assume is big bang, you don't need to assume a special state for big bang for entropy to increase. On the contrary you have to impose a special condition on the big bang for entropy to decrease.
reader TomVonk said...
I think this is indeed a nuance that only matters for those who study military history.
In every country and Germany is not an exception, there is political command and military command.
Also in every country and Germany is not an exception, the latter is subordinated to the former.
So everything is only a question of how is managed the interface and what names are used for the interface managers.
Up to 1938 this was organised Chancellor -> Ministry of Defence (political layer) -> OKH (Oberkommando des Heeres) -> Army groups -> Armies -> Armycorps-> etc (military layer). To that adds separately air and sea.
The top of the military layer (head of OKH) was called Oberbefehlshaber des Heeres (translation in english CiC or supreme commander of the armed forces).
In 1938 Hitler created the OKW (Oberkommando der Wehrmacht) and the chain became :
Kanzler und Führer (political layer) -> OKW -> OKH (Army) + OKL (air) + OKM (sea) (military layer)
The OKW role was coordination and in practice it became the general staff of the Führer.
So one one side you are right - the Kanzler and Führer (political head) has never been head of the OKW so strictly speaking he had never been supreme military CiC even if he had (political) authority on it like in every country even today.
You are also right that end 1941 Hitler took over in own name the head of OKH (not OKW !) so that he became head of the Ground forces directly.
One can note that this created a horrible situation on paper because in the military hierarchy Keitel (chief of OKW) became superior of Hitler (only chief of OKH) :)
On the other side Lubos is not quite wrong either because in practice the OKW was a tool for Hitler taking direct influence on OKH which was up to 1938 rather independent minded. Btw the German generals surnamed Keitel (OKW chief) Lakei what says it all !
So while Hitler was not directly (de iure) supreme army commander (=OKH chief) in name before 1941, he became it in practice (de facto) in 1938 with the creation of the OKW.
While this is important to understand the german army chain of command and strategy for 1938-1945, it is of second order for this thread.
reader Dilaton said...
Concerning people who have a problem with measurements in QM or with the thermodynamic arrow of time, I would say it is their own psychological problem ;-P
And the people who suffer from this deficit and have an ego with a radius bigger than the Hubble radius think everybody else is confused as well, which encourages them to pompously make themself seen andheard in popular media channels, blogs, books, etc ...
reader Tony said...
I think I understand Lubos's response. What about CD ROM degrading after a long time or just getting melted?
Those are two completely different processes that erase information. In principle, I think they are reversible, but it is not time reversal symmetry at work in these instances.
reader Luboš Motl said...
Thanks, Tony! Concerning the differences,
1) obtaining information by observation of previously uncertain information may be said to be instantaneous; forgetting is gradual
2) this is related: the "end side" of the observation really eliminates the probabilities for all other options except for one (collapse); in forgetting, both sides may be generic distributions with no zeroes
3) the likelihood of one answer or another on the "sharp side" of the observation is determined - uniform according to the distribution on the other side; the distribution in the case of forgetting is unconstrained
4) obtaining information by observing (with predictable probabilities) occurs even in a "perfect brain" with a huge excess memory; forgetting etc. only happens when there are imperfections or capacity limitations
5) forgetting is a reducible process that depends on particular smaller processes in the brain etc. and only some of them; obtaining information through observation is irreversible and the information is universal
One may generally feel that something goes up or down, so they're reverse to each other, but all the details are different. The laws governing these two processes are just completely different, T-asymmetric.
The claim that they are T-images of each other is like saying that gas spontaneously spreads from a full kitchen to the other (previously empty) room due to the second law; and the reverse is to close the door and pump all the gas back to the kitchen only. The trend of a particular observable, the ratio of mass of the gas in the two rooms, gets reverted, but all the details about these two processes are completely different from T-images of one another.
reader Maznak said...
It was NOT aircraft gun
reader Luboš Motl said...
The web page you have linked to contains no new information or new images that we haven't discussed yet, and surely no new justification of the proposition in your comment. It is an incoherent summary of a TV program that says exactly the opposite than your comment.
reader Anto said...
Hi Lubos,
So, for me - as a layman - quantum mechanics has always been inherently difficult. Are you saying that the probability distribution (and the expanding range, as time goes on) does not mean that the particle is in all possible places at once (as quantum mechanics is often explained), but that it actually in one place, except that we can't know where it is until we actually measure it?
Second, if that is the case, why doesn't your diagram at t = 1,2,n reduce down to the same certainty as t=0?
reader Luboš Motl said...
Of course I am. The paticle isn't "here AND there". It is "here OR there". The Schrodinger's cat is "dead OR alive". This is true for all probabilistic distributions and those derived from a wave function in quantum mechanics aren't exceptions.
Quantum mechanics only differs by the ability to discuss probabilities for mutually non-commuting observables, and the nonzero commutators (aka the uncertainty principle) are responsible for everything that is new about quantum mechanics.
But once you fix one observable, like the position of a particle, the probability distributions derived for that observable are probability distributions, so the different options are true with the word "OR" in between them, not "AND"!
reader Anto said...
Fantastic! Thank you very much. If every physicist/journalist explained the probability distribution aspect quantum mechanics to the laymen in these terms, it would be far more approachable.
[Mind you, quantum entanglement - spooky action at a distance - is still something to get the mind around.]
reader Anto said...
One more question, if I may. The "Is light a particle, or a wave?" question. Would I be right in assuming that, yes it is a particle, but that it travels along a wave-like path through space/time?
reader Guest said...
surely the actual impact holes with huge variation of size and shape are a lot different from the almost uniform, round aircraft gun impact holes - also, and that is my take, a lot of the impact is in the front of the airplane - cockpit panels etc - which would mean a head-on flying fighter. Not very likely mode of operation when it comes to using aircraft guns. With so many impacts, it would also mean that the gun would have to keep perfect aim for a very long time - not likely when shooting unguided projectiles from a distance from a flying platform that is influenced by turbulence and maneuvering etc.
reader Luboš Motl said...
It is neither a flow of ordinary (classical) particles, nor an ordinary (ciassical) wave.
It is a set of excitations of a quantum field, a thing that may only be properly understood within the formalism of quantum mechanics, and this new entity may either be interpretated as a flow of particles whose probabilities of positions and velocities are only predictable using probability (amplitude) waves; or as a state of wave in the electromagnetic fields whose quantities however don't commute which also means that the energy carried by frequency-f waves is quantized in multiples of E=hf, the energy of a "photon".
reader Anto said...
OK, thank you. I think I understand that. So, why a wave? Why not a straight line? Is the quantum field naturally "wavy"?
reader Luboš Motl said...
Dear Anto, every field (classical or quantum) is naturally wavy. Waves are the only way how a field may differ from the vacuum.
A field is a number at each point, and F=0 means the vacuum state, the configuration with the minimum energy E=0. Everything else with a nonzero F means that the field is excited.
When you excite it to F=A in a small region, the excitation will spread to the rest of space. Far enough, it will always look like waves. More precisely, every solution for the electromagnetic field may be written down as a linear combination (sum) of sine waves with a given direction, frequency, and polarization.
I probably don't understand how you are imagining a "straight line field". It sounds like an oxymoron to me. Field is everywhere, by its definition.
reader Luboš Motl said...
It does *not* imply a head-on flying fighter. SU-34/35/37, for example, are known to have rear-pointing radars and the capability to install rear-pointing guns.
reader Jock M said...
"Franco was a deeply ideological fascist"
Except he wasn't; sure, he was anti-democratic and a traditionalist and nationalist defender of the monarchy, the Church, and the army, but he relegated the real ideological fascists (the Falangists) to a very minor role in his regime. On this see the books of Prof Stanley Payne for a corrective.
As for the Loyalists, they celebrated their commitment to democracy (of Robespierre's totalitarian kind during his carnival of terror and death--see Prof J L Talmon work on this) during the Spanish Civil War by murdering ten thousand unarmed priests, nuns, and monks. And Spanish Stalinists set about methodically murdering those on their own side--namely anarchists and sydicalists, Some "democrats," uh-huh. . . .
reader Luboš Motl said...
Dear Jock, it makes no sense to argue, prolong this war that occurred 80 years ago. I have no doubt that just like democratic Czechoslovakia in the 1930s, I would have sympathized with the republicans.
The point is that it was a civil war where different people stood at each side, that it is primarily a war polarizing the nation (Spain or Ukraine) itself, but that everyone in abroad has some opinion about it, too. But the existence of opinions and interests doesn't change the fact that it is a civil war.
reader Anto said...
Sorry - I was confusing quantum with classical. As laymen do!
However, I think I understand your field explanation. Zero energy/excitation = zero field = flat/straight.
However, energy/mass = excitation = agitation/disruption = wave-like travel?
I guess that, like many other laymen, I'm able to get my mind around the macro-universe, but struggle with the micro-universe.
I think that I've properly understood your explanation of the classical statistical physics, but I'm still struggling with the translation to quantum physics.
I don't want to take up any more of your time. I see your explanation of the quantum field, above.
Something which I find hard to understand is that, with all of the different forces/particles/masses/energies in the universe, that anything is predictable at all. Normally, in say a pond, if you have 5 or 10 people throwing stones into the water, there is chaos.
However, with the universe, despite all of these competing sources of mass/energy, there is still much predictability.
reader Eclectikus said...
Just a side note, what JockM is saying, in the same path of my previous comments, are factual points about the Spanish Civil War. I doubt that you Lubos might defend the republicans, well, at least not as a whole, first because they did represent the worst of the communist point of view, i.e. stalinism (remember for example that Trotsky was murdered by Ramon Mecarder, a Spanish spook), and secondly also because nowadays this point of view is only maintained by the unicornial leftism, under no circumstances I can see you on this front. :)
Also I agree that Stanley G. Payne is a good antidote for the mainstream view about Spanish Civil War that you seem assume, e.g. see this chapter from his book "A History of Spain and Portugal":
reader Luboš Motl said...
Sorry, by the republicans, I mean the whole side of the civil war that was fighting against the nationalists. Be sure that it's not a typo when I write that I would support it. |
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The Information Philosopher
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Information is neither matter nor energy, although it needs matter to be embodied and energy to be communicated. Why should it become the preferred basis for all philosophy?
As most all of us know, matter and energy are conserved. This means that there is just the same total amount of matter and energy today as there was at the universe origin.
But then what accounts for all the change that we see, the new things under the sun? It is information, which is not conserved and has been increasing since the beginning of time, despite the second law of thermodynamics, with its increasing entropy, which destroys order.
What is changing is the arrangement of the matter into what we can call information structures. What is emerging is new information. What idealists and holists see is that emergence of immaterial information.
Living things, you and I, are dynamic growing information structures, forms through which matter and energy continuously flow. And it is information processing that controls those flows!
At the lowest levels, living information structures blindly replicate their information. At higher levels, natural selection adapts them to their environments. At the highest levels, living things develop behaviors, intentions, goals, and agency, introducing purpose into the universe.
Information is the modern spirit, the ghost in the machine, the mind in the body. It is the soul, and when we die, it is our information that perishes, unless the future preserves it. The matter remains.
If we don't remember the past, we don't deserve to be remembered by the future. This is especially true for the custodians of knowledge.
Information can explain the fundamental metaphysical connection between materialism and idealism. Information philosophy replaces the determinism and metaphysical necessity of eliminative materialism and reductionist naturalism with metaphysical possibility.
Information is the form in all concrete objects as well as the content in non-existent, merely possible, thoughts and other
abstract entities. It is the disembodied, de-materialized essence of anything.
Perhaps the most amazing thing about information philosophy is its discovery that abstract and immaterial information can exert an influence over concrete matter, explaining how mind can move body, how our thoughts can control our actions, deeply related to the way the quantum wave function controls the probabilities of locating quantum particles.
Information philosophy goes beyond a priori logic and its puzzles, beyond analytic language and its paradoxes, beyond philosophical claims of necessary truths, to a contingent physical world that is best represented as made of dynamic, interacting information structures.
Knowledge can be defined as information in minds - a partial isomorphism of the information structures in the external world. Information philosophy is a correspondence theory.
Sadly, there is no isomorphism, no information in common, between words and objects. This accounts for much of the failing of analytic language philosophy in the past century.
Although language is an excellent tool for human communication, it is arbitrary, ambiguous, and ill-suited to represent the world directly. Human languages do not picture reality. Information is the lingua franca of the universe.
The extraordinarily sophisticated connection between words and objects is made in human minds, mediated by the brain's experience recorder and reproducer (ERR). Words stimulate neurons to start firing and to play back those experiences that include relevant objects.
Neurons that were wired together in our earliest experiences fire together at later times, contextualizing our new experiences, giving them meaning. And by replaying emotional reactions to similar earlier experiences, it makes then "subjective experiences," giving us the feeling of "what it's like to be me" and solving the "hard problem" of consciousness.
Beyond words, a dynamic information model of an information structure in the world is presented immediately to the mind as a simulation of reality experienced for itself.
Without words and related experiences previously recorded in your mental experience recorder, we could not comprehend words. They would be mere noise, with no meaning.
By comparison, a diagram, a photograph, an animation, or a moving picture can be seen and mostly understood by human beings, independent of their native tongue. The basic elements of information philosophy are dynamical models of information structures. They go far beyond logic and language as a representation of the fundamental, metaphysical, nature of reality.
Visual and interactive models "write" directly into our mental experience recorders.
Computer animated models must incorporate all the laws of nature, from the differential equations of quantum physics to the myriad information processes of biology. Simulations are not only our most accurate knowledge of the physical world, they are among the best teaching tools ever devised. We can transfer knowledge non-verbally to coming generations in most of the world's population via the Internet and nearly ubiquitous smartphones.
Consider the dense information in Drew Berry's real-time animations of molecular biology. These are the kinds of dynamic models of information structures that we believe can best explain the fundamental nature of reality - "beyond logic and language."
If you think about it, everything you know is pure abstract information. Everything you are is an information structure, a combination of matter and energy that embodies, communicates, and most important, processes your information. Everything that you value contains information.
And while the atoms, molecules, and cells of your body are important, many only last a few minutes and most are completely replaced in just a few years. But your immaterial information, from your original DNA to your latest experiences, will be with you for your lifetime.
You are a creator of new information, part of the cosmic creation process. Your free will depends on your unique ability to create freely generated thoughts, multiple ideas in your mind as alternative possibilities for your willed decisions and responsible actions.
Anyone with a serious interest in philosophy should understand how information is created and destroyed, because information is much more fundamental than the logic and language tools philosophers use today. Information philosophy goes "beyond logic and language."
Information is the sine qua non of meaning. This I-Phi website aims to provide a deep understanding of information that should be in every philosopher's toolbox.
We will show why information should actually be the preferred basis for the critical analysis of current problems in a wide range of disciplines - from information creation in cosmology to information in quantum physics, from information in biology (especially evolution) to psychology, where it offers a solution to the classic mind-body problem and the problem of consciousness. And of course in philosophy, where failed language analysis can be replaced or augmented by immaterial information analysis as a basis for justified knowledge, objective values, human free will, and a surprisingly large number of problems in metaphysics.
Above all, information philosophy hopes to replace beliefs with knowledge. Instead of the primitive idea of an other-worldly creator, we propose a comprehensive explanation of the creation of this world that has evolved into the human creativity that invents such ideas.
The "miracle of creation" is happening now, in the universe and in you and by you.
But what is information? How is it created? Why is it a better tool for examining philosophical problems than traditional logic or linguistic analysis? And what are some examples of classic problems in philosophy, in physics, and in metaphysics with information philosophy solutions?
What problems has information philosophy solved?
Why has philosophy made so little progress? Is it because philosophers prefer problems, while scientists seek solutions? Must a philosophical problem once solved become science and leave philosophy? Bertrand Russell thought so. The information philosopher thinks not.
Russell said:
"as soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science...while those only to which, at present, no definite answer can be given, remain to form the residue which is called philosophy."
But in order to remain philosophy, interested philosophers must examine our proposed information-based solutions and evaluate them as part of the philosophical dialogue.
Among the proposed solutions to classic philosophical problems are:
Information analysis also makes significant progress on a number of the classic problems in metaphysics, many of these virtually unchanged since they were identified as puzzles and paradoxes over two millennia ago, such as The Statue and Lump of Clay, The Ship of Theseus, Dion and Theon, or Tibbles, the Cat, The Growing Problem, The Debtor's Paradox, The Problem of the Many, and The Sorites Problem.
Among the metaphysical problems with suggested information philosophy solutions are:
It also turns out that the methodology of information philosophy can be productively applied to some outstanding problems in physics. Philosophers of science might take an interest in the proposed information-based solutions to these problems in the "foundations" of physics.
What is information?
A common definition of information is the act of informing - the communication of knowledge from a sender to a receiver that informs (literally shapes) the receiver. Often used as a synonym for knowledge, information traditionally implies that the sender and receiver are human beings, but many animals clearly communicate. Information theory studies the communication of information.
Information philosophy extends that study to the communication of information content between material objects, including how it is changed by energetic interactions with the rest of the universe.
We call a material object with information content an information structure. While information is communicated between inanimate objects, they do not process information, which we will show is the defining characteristic of living beings and their artifacts.
The sender of information need not be a person, an animal, or even a living thing. It might be a purely material object, a rainbow, for example, sending color information to your eye.
The receiver, too, might be merely physical, a molecule of water in that rainbow that receives too few photons and cools to join the formation of a crystal snowflake, increasing its information content.
Information theory, the mathematical theory of the communication of information, says little about meaning in a message, which is roughly the use to which the information received is put. Information philosophy extends the information flows in human communications systems and digital computers to the natural information carried in the energy and material flows between all the information structures in the observable universe.
A message that is certain to tell you something you already know contains no new information. It does not increase your knowledge, or reduce the uncertainty in what you know, as information theorists put it.
If everything that happens was certain to happen, as determinist philosophers claim, no new information would ever enter the universe. Information would be a universal constant. There would be "nothing new under the sun." Every past and future event could in principle be known by a god-like super-intelligence with to a fixed totality of information (Laplace's Demon).
Physics tells us that the total amount of mass and energy in the universe is a constant. The conservation of mass and energy is a fundamental law of nature. Some mathematical physicists erroneously think that information should also be a conserved quantity, that information is a constant of nature. This includes some leading mathematical physicists.
But information is neither matter nor energy, though it needs matter to be embodied and energy to be communicated. Information can be created and destroyed. The material universe creates it. The biological world creates it and utilizes it. Above all, human minds create, process, and preserve information, the Sum of human knowledge that distinguishes humanity from all other biological species and that provides the extraordinary power humans have over our planet.
We propose information as an objective value, the ultimate sine qua non.
Information philosophy claims that man is not a machine and the brain is not a computer. Living things process information in ways far more complex, if not faster, than the most powerful information processing machines. What biological systems and computing systems have in common is the processing of information, as we must explain.
Whereas machines are assembled, living things assemble themselves. They are both information structures, patterns, through which matter and energy flows, thanks to flows of negative entropy coming from the Sun and the expanding universe. And they both can create new information, build new structures, and maintain their integrity against the destructive influence of the second law of thermodynamics.
Biological evolution began when the first molecule replicated itself, that is, duplicated the information it contained. But duplication is mere copying. Biological reproduction is a much more sophisticated process in which the germ or seed information of a new living thing is encoded in a data or information structure (a genetic code) that can be communicated to processing systems that produce another instance of the given genus and species.
Ontologically random imperfections, along with the deliberate introduction of random noise, for example in sexual recombinations, in the processing systems produce the variations that are selected by evolution based on their reproductive success. Errors are not restricted to the genetic code, occurring throughout the development of each individual up to the present.
Cultural evolution is the creation and communication of new information that adds to the sum of human knowledge. The creation and evolution of information processing systems in the universe has culminated in minds that can understand and reflect on what we call the cosmic creation process.
How is information created?
Ex nihilo, nihil fit, said the ancients, Nothing comes from nothing. But information is no (material) thing. Information is physical, but it is not material. Information is a property of material. It is the form that matter can take. We can thus create something (immaterial) from nothing! But we shall find that it takes a special kind of energy (free or available energy, with negative entropy) to do so, because it involves the rearrangement of matter.
Energy transfer to or from an object increases or decreases the heat in the object. Entropy transfer does not change the heat content, it represents only a different organization or distribution of the matter in the body. Increasing entropy represents a loss of organization or order, or, more precisely, information. Maximum entropy is maximum disorder and minimal information.
As you read this sentence, new information is (we hope) being encoded/embodied in your mind/brain. Permanent changes in the synapses between your neurons store the new information. New synapses are made possible by free energy and material flows in your metabolic system, a tiny part of the negative entropy flows that are coursing throughout the universe. Information philosophy will show you how these tiny mental flows allow you to comprehend and control at least part of the cosmic information flows in the universe.
Cosmologists know that information is being created because the universe began some thirteen billion years ago in a state of minimal information. The "Big Bang" started with the most elementary particles and radiation. How matter formed into information structures, first atoms, then the galaxies, stars, and planets, is the beginning of a story that will end with understanding how human minds emerged to understand our place in the universe.
The relation between matter and information is straightforward. The embodied information is the organization or arrangement of the matter plus the laws of nature that describe the motions of matter in terms of the fundamental forces that act between all material particles.
The relation between information and energy is more complex, and has led to confusion about how to apply mathematical information theory to the physical and biological sciences. Material systems in an equilibrium state are maximally disordered, have maximum entropy, no negative entropy, and no information other than the bulk parameters of the system.
In the case of the universe, the initial parameters were very few, the amount of radiant energy (the temperature) and the number of elementary particles (quarks, gluons, electrons, and photons) per unit volume, and the total volume (infinite?). These parameters, and their changes (as a function of time, as the temperature falls) are all the information needed to describe a statistically uniform, isotropic universe and its evolution.
Information philosophy will explain the process of information creation in three fundamental realms - the purely material, the biological, and the mental.
The first information creation was a kind of "order out of chaos," when matter in the early universe opened up spaces allowing gravitational attraction to condense otherwise randomly distributed matter into highly organized galaxies, stars, and planets. It was the expansion - the increasing space between material objects - that drove the universe away from thermodynamic equilibrium (maximum entropy and disorder) and in some places created negative entropy, a quantitative measure of orderly arrangements that is the basis for all information.
Purely material objects react to one another following laws of nature, but they do not in an important sense create or process the information that they contain. It was the expansion, moving faster than the re-equilibration time, and the gravitational forces, that were responsible for the new structures.
A qualitatively different kind of information creation was when the first molecule on earth to replicate itself went on to duplicate its information exponentially. Here the prototype of life was the cause for the creation of the new information structure. Accidental errors in the duplication provided variations in replicative success. Most important, besides creating their information structures, biological systems are also information processors. Living things use information to guide their actions.
With the appearance of life, agency and purpose appeared in the universe. Although some philosophers hold that life just gives us the "appearance of purpose."
The third process of information creation, and the most important to philosophy, is human creativity. Almost every philosopher since philosophy began has considered the mind as something distinct from the body. Information philosophy can now explain that distinction. The mind can be considered the immaterial information in the brain. The brain, part of the material body, is a biological information processor. The stuff of mind is the information being processed and the new information being created. As some philosophers have speculated,
mind is the software in the brain hardware.
Most material objects are passive information structures.
Living things are information structures that actively process information. They communicate it between their parts to build, maintain, and repair their (material) information structure, through which matter and energy flow under the control of the information structure itself.
Resisting the second law of thermodynamics locally, living things increase entropy globally much faster than non-living things. But most important, living things increase their information content as they develop. Humans learn from their experiences, storing knowledge in an experience recorder and reproducer (ERR).
Mental things (ideas) are pure abstractions from the material world, but they have control (downward causation) over the material and biological worlds. This enables agent causality. Human minds create information structures, but their unique creation is the collection of abstract ideas that are the sum of human knowledge. It is these ideas that give humanity unparalleled extraordinary control over the material and biological worlds.
It may come as a surprise for many thinkers to learn that the physics involved in the creation of all three types of information - material, biological, and mental - include the same two-step sequence of quantum physics and thermodynamics at the core of the cosmic creation process.
The most important information created in a mind is a recording of an individual's experiences (sensations). Recordings are played back (automatically and perhaps mostly unconsciously) as a guide to evaluate future actions (volitions) in similar situations. The particular past experiences reproduced are those stored in the brain located near elements of the current experience (association of ideas).
Just as neurons that fire together wire together, neurons that have been wired together will later fire together.
Sensations are recorded as the mental effects of physical causes.
Sensations are stored as retrievable information in the mind of an individual self. Recordings include not only the five afferent senses but also the internal emotions - feelings of pleasure, pain, hopes, and fears - that accompany an experience. They constitute "what it's like" for a particular being to have an experience.
Volitions are the mental causes of physical effects.
Volitions begin with 1) the reproduction of past experiences that are similar to the current experience. These become thoughts about possible actions and the (partly random) generation of other alternative possibilities for action. They continue with 2) the evaluation of those freely generated thoughts followed by a willful selection (sometimes habitual) of one of those actions.
Volitions are followed by 3) new sensations coming back to the mind indicating that the self has caused the action to happen (or not). This feedback is recorded as further retrievable information, reinforcing the knowledge stored in the mind that the individual self can cause this kind of action (or sometimes not).
Many philosophers and most scientists have held that all knowledge is based on experience. Experience is ultimately the product of human sensations, and sensations are just electrical and chemical interactions with human skin and sense organs. But what of knowledge that is claimed to be mind-independent and independent of experience?
Why is information better than logic and language for solving philosophical problems?
Broadly speaking, modern philosophy has been a search for truth, for a priori, analytic, certain, necessary, and provable truth.
But all these concepts are mere ideas, invented by humans, some aspects of which have been discovered to be independent of the minds that invented them, notably formal logic and mathematics. Logic and mathematics are systems of thought, inside which the concept of demonstrable (apodeictic) truth is useful, but with limits set by Kurt Gödel's incompleteness theorem. The truths of logic and mathematics appear to exist "outside of space and time." Gottfried Leibniz called then "true in all possible worlds," meaning their truth is independent of the physical world. We call them a priori because their proofs are independent of experience, although they were initially abstracted from concrete human experiences.
Analyticity is the idea that some statements, propositions in the form of sentences, can be true by the definitions or meanings of the words in the sentences. This is correct, though limited by verbal difficulties such as Russell's paradox and numerous other puzzles and paradoxes. Analytic language philosophers claim to connect the words with objects, material things, and thereby tell us something about the world. Some modal logicians (cf. Saul Kripke) claim that words that are names of things are necessary a posteriori, "true in all possible worlds." But this is nonsense, because we invented all those words and worlds. They are mere ideas.
Perhaps the deepest of all these philosophical ideas is necessity. Information philosophy can now tell us that there is no such thing as absolute necessity. There is of course an adequate determinism in the macroscopic world that explains the appearance of deterministic laws of nature, of cause and effect, for example. This is because macroscopic objects consist of vast numbers of atoms and their individual random quantum events average out. But there is no metaphysical necessity. At the fundamental microscopic level of material reality, there is an irreducible contingency and indeterminacy. Everything that we know, everything we can say, is fundamentally empirical, based on factual evidence, the analysis of experiences that have been recorded in human minds.
So information philosophy is not what we can logically know about the world, nor what we can analytically say about the world, nor what is necessarily the case in the world. There is nothing that is the case that is necessary and perfectly determined by logic, by language, or by the physical laws of nature. Our world and its future are open and contingent, with possibilities that are the source of new information creation in the universe and source of human freedom.
For the most part, philosophers and scientists do not believe in ontological possibilities, despite their invented "possible worlds," which are on inspection merely multiple "actual worlds." They are "actualists." This is because they cannot accept the idea of ontological chance. They hope to show that the appearance of chance is the result of human ignorance, that chance is merely an epistemic phenomenon.
Now chance, like truth, is just another idea, just some more information. But what an idea! In a self-referential virtuous circle, it turns out that without the real possibilities that result from ontological chance, there can be no new information. Information philosophy offers cosmological and biological evidence for the creation of new information in the universe. So it follows that chance is real, fortunately something that we can keep under control. We are biological beings that have evolved, thanks to chance, from primitive single-cell communicating information structures to multi-cellular organisms whose defining aspect is the creation and communication of information.
The theory of communication of information is the foundation of our "information age." To understand how we know things is to understand how knowledge represents the material world of embodied "information structures" in the mental world of immaterial ideas.
All knowledge starts with the recording of experiences. The experiences of thinking, perceiving, knowing, believing, feeling, desiring, deciding, and acting may be bracketed by philosophers as "mental" phenomena, but they are no less real than other "physical" phenomena. They are themselves physical phenomena.
They are just not material things.
Information philosophy defines human knowledge as immaterial information in a mind, or embodied in an external artifact that is an information structure (e.g., a book), part of the sum of all human knowledge. Information in the mind about something in the external world is a proper subset of the information in the external object. It is isomorphic to a small part of the total information in or about the object. The information in living things, artifacts, and especially machines, consists of much more than the material components and their arrangement (positions over time). It also consists of all the information processing (e.g., messaging) that goes on inside the thing as it realizes its entelechy or telos, its internal or external purpose.
All science begins with information gathered from experimental observations, which are themselves mental phenomena. Observations are experiences recorded in minds. So all knowledge of the physical world rests on the mental. All scientific knowledge is information shared among the minds of a community of inquirers. As such, science is a collection of thoughts by thinkers, immaterial and mental, some might say fundamental. Recall Descartes' argument that the experience of thinking is that which for him is the most certain.
Information philosophy is not the philosophy of information (the intersection of computer science, information science, information technology, and philosophy), just as linguistic philosophy - the idea that linguistic analysis can solve (or dis-solve) philosophical problems - is not the philosophy of language. Compare the philosophy of mathematics, philosophy of biology, etc.
The analysis of language, particularly the analysis of philosophical concepts, which dominated philosophy in the twentieth century, has failed to solve the most ancient philosophical problems. At best, it claims to "dis-solve" some of them as conceptual puzzles. The "problem of knowledge" itself, traditionally framed as "justifying true belief," is recast by information philosophy as the degree of isomorphism between the information in the physical world and the information in our minds. Information psychology can be defined as the study of this isomorphism.
We shall see how information processes in the natural world use arbitrary symbols (e.g., nucleotide sequences) to refer to something, to communicate messages about it, and to give the symbol meaning in the form of instructions for another process to do something (e.g., create a protein). These examples provide support for both theories of meaning as reference and meaning as use.
Note that just as language philosophy is not the philosophy of language, so information philosophy is not the philosophy of information. It is rather the use of information as a tool to study philosophical problems, some of which are today yielding tentative solutions. It is time for philosophy to move beyond logical puzzles and language games.
The Fundamental Question of Information Philosophy
Our fundamental philosophical question is cosmological and ultimately metaphysical.
What are the processes that create emergent information structures in the universe?
Given the second law of thermodynamics, which says that any system will over time approach a thermodynamic equilibrium of maximum disorder or entropy, in which all information is lost, and given the best current model for the origin of the universe, which says everything began in a state of thermodynamic equilibrium some 13.75 billion years ago, how can it be that living beings are creating and communicating vast amounts of new information every day?
Why are we not still in that original state of equilibrium?
Broadly speaking, there are four major phenomena or processes that can reduce the entropy locally, while of course increasing it globally to satisfy the second law of thermodynamics. Three of these do it "blindly," the fourth does it with a built-in "purpose," or telos."
1. Universal Gravitation
2. Quantum Cooperative Phenomena (e.g., crystallization, the formation of atoms and molecules)
3. "Dissipative" Chaos (Non-linear Thermodynamics)
4. Life
None of these processes can work unless they have a way to get rid of the positive entropy (disorder) and leave behind a pocket of negative entropy (order or information). The positive entropy is either conducted, convected, or radiated away as waste matter and energy, as heat, or as pure radiation. At the quantum level, it is always the result of interactions between matter and radiation (photons). Whenever photons interact with material particles, the outcomes are inherently unpredictable. As Albert Einstein discovered ten years before the founding of quantum mechanics, these interactions involve irreducible ontological chance.
Negative entropy is an abstract thermodynamic concept that describes energy with the ability to do work, to make something happen. This kind of energy is often called free energy or available energy.
In a maximally disordered state (called thermodynamic equilibrium) there can be matter in motion, the motion we call heat. But the average properties - density, pressure, temperature - are the same everywhere. Equilibrium is formless. Departures from equilibrium are when the physical situation shows differences from place to place. These differences are information.
The second law of thermodynamics then simply means that isolated systems will eliminate differences from place to place until all properties are uniformly distributed. Natural processes spontaneously destroy information. Consider the classic case of what happens when we open a perfume bottle.
In the late nineteenth century, Ludwig Boltzmann revolutionized thermodynamics with his kinetic theory of gases, based on the ancient assumption that matter is made up of collections of atoms. He derived a mathematical formula for entropy S as a function of the probabilities of finding a system in all the possible microstates of a system. When the actual macrostate is one with the largest number W of microstates, entropy is at a maximum, and no differences (information) are visible.
Boltzmann could not prove his "H-Theorem" about entropy increase. His contemporaries challenged a "statistical" entropy increase on grounds of microscopic reversibility and macroscopic recurrence (both problems solved by information philosophy). He could not prove the existence of atoms.
In the early twentieth century, Just before Boltzmann died, Albert Einstein formulated a statistical mechanics that put Boltzmann's law of increasing entropy on a firmer mathematical basis. Einstein's work predicted the size of miniscule fluctuations around equilibrium, which Boltzmann had expected. Einstein showed that entropy does not, in fact, continually increase. It can decrease randomly in short bursts of local higher densities or organized motions. Though quickly extinguished, Einstein showed that the occasionally correlated motions of invisible atoms explains the visible "Brownian motion" of tiny particles like seed pollen.
Einstein's calculations led to predictions that were confirmed quickly, proving the existence of discrete atoms that had been hypothesized for centuries. Sadly, Boltzmann may not have known of Einstein's proofs for his work. Later Einstein saw the same fluctuation in radiation, proving his revolutionary hypothesis of light quanta, now called photons. Although this is rarely appreciated, it was Einstein who showed that both matter and energy are discrete, discontinuous particles. His most famous equation shows they are convertible into one another, E = mc2. He also showed that the interaction of matter and radiation, of atoms and photons, always involves ontological chance. This bothered Einstein greatly, because he thought his God should not "play dice."
Late in life, Einstein said that if matter and energy cannot be described with the local continuous analytical functions in space and time needed for his field theories, that all his work would be "castles in the air." But the loss of classical deterministic ideas - which have ossified much of philosophy, crippling philosophical progress - is more than offset by the indeterministic of an open future and Einstein's belief in the "free creation of new ideas."
In the middle twentieth century, Claude Shannon derived the mathematical formula for the communication of information. John von Neumann found it to be identical to Boltzmann's formula for entropy, though with a minus sign (negative entropy). Where Boltzmann entropy is the number of possible microstates, Shannon entropy is the number of possible messages that can be communicated.
Shannon found that new information cannot be created unless there are multiple possible messages. This in turn depends on the ontological chance discovered by Einstein. In a deterministic universe, the total information at all times would be a constant. Information would be a conserved quantity, like matter and energy. "Nothing new under the Sun." But it is not constant, though many philosophers, mathematical physicists, and theologians (God's foreknowledge) still think so. Information is being created constantly in our universe. And we are co-creators of the information, including Einstein's "new ideas."
Because "negative" entropy (order or information) is such a positive quantity, we chose in the 1970's to give it a new name - "Ergo," and to call the four phenomena or processes that create negative entropy "ergodic," for reasons that will become clear. But today, the positive name "information" is all that we need to do information philosophy.
Answering the Fundamental Question of Information Philosophy
How exactly has the universe escaped from the total disorder of thermodynamic equilibrium and produced a world full of information?
It begins with the expansion of the universe. If the universe had not expanded, it would have remained in the original state of thermodynamic equilibrium. We would not be here.
To visualize the departure from equilibrium that made us possible, remember that equilibrium is when particles are distributed evenly in all possible locations in space, and with their velocities distributed by a normal law - the Maxwell-Boltzmann velocity distribution. (The combination of position space and velocity or momentum space is called phase space). When we open the perfume bottle, the molecules now have a much larger phase space to distribute into. There are a much larger number of phase space "cells" in which molecules could be located. It of course takes them time to spread out and come to a new equilibrium state (the Boltzmann "relaxation time.")
When the universe expands, say grows to ten times its volume, it is just like the perfume bottle opening. The matter particles must redistribute themselves to get back to equilibrium. But suppose the universe expansion rate is much faster than the equilibration or relaxation time. The universe is out of equilibrium, and in a flat, ever-expanding, universe it will never get back!
In the earliest moments of the universe, material particles were in equilibrium with radiation at extraordinarily high temperatures. When quarks formed neutrons and protons, they were short-lived, blasted back into quarks by photon collisions. As the universe expanded, the temperature cooled, the space per photon increased and the mean free time between photon collisions increased, giving larger particles a better chance to survive. The expansion red-shifted the photons. decreasing the average energy per photon, and eventually reducing the number of high energy photons that disassociate matter. The mean free path of photons was very short. They were being scattered by collisions with electrons.
When temperature declined further, to 5000 degrees, about 400,000 years after the "Big Bang," the electrons and protons combined to make hydrogen and (with neutrons) helium atoms.
At this time, a major event occurred that we can still see today, the farthest and earliest event visible. When the electrons combined into atoms, the electrons could no longer scatter the photons so easily. The universe became transparent for the photons. Some of those photons are still arriving at the earth today. They are now the red-shifted and cooled down cosmic microwave background radiation. While this radiation is almost perfectly uniform, it shows very small fluctuations that may be caused by random difference in the local density of the original radiation or even in random quantum fluctuations.
These fluctuations mean that there were slight differences in density of the newly formed hydrogen gas clouds. The force of universal gravitation then worked to pull relatively formless matter into spherically symmetric stars and planets. Thus is the original order out of chaos, although this phrase is now most associated with the work on deterministic chaos theory and complexity theory, as we shall see.
How information creation and negative entropy flows appear to violate the second law of thermodynamics
In our open and rapidly expanding universe, the maximum possible entropy (if the particles were "relaxed" into a uniform distribution among the new phase-space cells) is increasing faster than the actual entropy. The difference between maximum possible entropy and the current entropy is called negative entropy. There is an intimate connection between the physical quantity negative entropy and abstract immaterial information, first established by Leo Szilard in 1929.
As pointed out by Harvard cosmologist David Layzer, the Arrow of Time points not only to increasing disorder but also to increasing information.
Two of our "ergodic" phenomena - gravity and quantum cooperative phenomena - pull matter together that was previously separated. Galaxies, stars, and planets form out of inchoate clouds of dust and gas. Gravity binds the matter together. Subatomic particles combine to form atoms. Atoms combine to form molecules. They are held together by quantum mechanics. In all these cases, a new visible information structure appears.
In order for these structures to stay together, the motion (kinetic) energy of their parts must be radiated away. This is why the stars shine. When atoms join to become molecules, they give off photons. The new structure is now in a (negative) bound energy state. It is the radiation that carries away the positive entropy (disorder) needed to balance the new order (information) in the visible structure.
In the cases of chaotic dissipative structures and life, the ergodic phenomena are more complex, but the result is similar, the emergence of visible information. (More commonly it is simply the maintenance of high-information, low-entropy structures.) These cases appear in far-from-equilibrium situations where there is a flow of matter and energy with negative entropy through the information structure. The flow comes in with low entropy but leaves with high entropy. Matter and energy are conserved in the flow, but information in the structure can increase (information is not a conserved quantity).
Information is neither matter nor energy, though it uses matter when it is embodied and energy when it is communicated. Information is immaterial.
This vision of life as a visible form through which matter and free energy flow was first seen by Ludwig van Bertlanffy in 1939, though it was made more famous by Erwin Schrödinger's landmark essay What Is Life? in 1945, where he claimed that "life feeds on negative entropy."
Both Bertalanffy and Schrödinger knew that the source of negative entropy was our Sun. Neither knew that the ultimate cosmological source of negative entropy is the expansion of the universe, which allowed ergodic gravitation forces to form the Sun. Note that the positive entropy radiation leaving the Sun becomes diluted as it expands, creating a difference between its energy temperature and energy density. This difference is information (negative entropy) that planet Earth uses to generate and maintain biological life.
Note that the 300K (the average earth temperature) photons are dissipated into the dark night sky, on their way to the cosmic microwave background. The Sun-Earth-night sky is a heat engine, with a hot energy source and cold energy sink, that converts the temperature difference not into mechanical energy (work) but into biological energy (life).
When information is embodied in a physical structure, two physical processes must occur.
Our first process is what John von Neumann described as
irreversible Process 1.
The first process is the collapse of a quantum-mechanical wave function into one of the possible states in a superposition of states, which happens in any measurement process. A measurement produces one or more bits of information. Such quantum events involve irreducible indeterminacy and chance, but less often noted is the fact that quantum physics is directly responsible for the extraordinary temporal stability and adequate determinism of most information structures.
We can call the transfer of positive entropy, which stabilizes the new information from Process 1, Process 1b.
The second process is a local decrease in the entropy (which appears to violate the second law of thermodynamics) corresponding to the increase in information. Entropy greater than the information increase must be transferred away from the new information, ultimately to the night sky and the cosmic background, to satisfy the second law.
Given this new stable information, to the extent that the resulting quantum system can be approximately isolated, the system will deterministically evolve according to von Neumann's Process 2, the unitary time evolution described by the Schrödinger equation.
The first two physical processes (1 and 1b) are parts of the information solution to the "problem of measurement," to which must be added the role of the "observer." We shall see that the observer involves a mental Process 3.
The discovery and elucidation of the first two as steps in the cosmic creation process casts light on some classical problems in philosophy and physics , since it is the same two-step process that creates new biological species and explains the freedom and creativity of the human mind.
The cosmic creation process generates the conditions without which there could be nothing of value in the universe, nothing to be known, and no one to do the knowing. Information itself is the ultimate sine qua non.
The Three Kinds of Information Emergence
Note there are three distinct kinds of emergence:
1. the order out of chaos when the randomly distributed matter in the early universe first gets organized into information structures.
This was not possible before the first atoms formed about 400,000 years after the Big Bang. Information structures like the stars and galaxies did not exist before about 400 million years. As we saw, gravitation was the principal driver creating information structures.
Nobel prize winner Ilya Prigogine discovered another ergodic process that he described as the "self-organization" of "dissipative structures." He popularized the slogan "order out of chaos" in an important book. Unfortunately, the "self" in self-organization led to some unrealizable hopes in cognitive psychology. There is no self, in the sense of a person or agent, in these physical phenomena.
Both gravitation and Prigogine's dissipative systems produce a purely physical/material kind of order. The resulting structures contain information. There is a "steady state" flow of information-rich matter and energy through them. But they do not process information. They have no purpose, no "telos."
Order out of chaos can explain the emergence of downward causation on their atomic and molecular components. But this is a gross kind of downward causal control. Explaining life and mind as "complex adaptive systems" has not been successful. We need to go beyond "chaos and complexity" theories to teleonomic theories.
2. the order out of order when the material information structures form self-replicating biological information structures. Some become information processing systems.
In his famous essay, "What Is Life?," Erwin Schrödinger noted that life "feeds on negative entropy" (or information). He called this "order out of order."
This kind of biological processing of information first emerged about 3.5 billion years ago on the earth. It continues today on multiple emergent biological levels, e.g., single-cells, multi-cellular systems, organs, etc., each level creating new information structures and information processing systems not reducible to (caused by) lower levels and exerting downward causation on the lower levels.
And this downward causal control is extremely fine, managing the motions and arrangements of individual atoms and molecules.
Biological systems are cognitive systems, using internal "subjective" knowledge to recognize and interact with their "objective" external environment, communicating meaningful messages to their internal components and to other individuals of their species with a language of arbitrary symbols, taking actions to maintain themselves and to expand their populations by learning from experience.
With the emergence of life, "purpose" also entered the universe. It is not the pre-existent "teleology" of many idealistic philosophies (the idea of "essence" before "existence"), but it is the "entelechy" of Aristotle, who saw that living things have within them a purpose, an end, a "telos." To distinguish this evolved telos in living systems from teleology, modern biologists use the term "teleonomy."
3. the pure information out of order when organisms with minds generate, store (in the brain), replicate, utilize, and then externalize some non-biological information, communicating it to other minds and storing it in the environment. Communication can be by hereditary genetic transmission or by an advanced organism capable of learning and then teaching its contemporaries directly by signaling, by speaking, or indirectly by writing and publishing the knowledge for future generations.
This kind of information can be highly abstract mind-stuff, pure Platonic ideas, the stock in trade of philosophers. It is neither matter nor energy (though embodied in the material brain), a kind of pure spirit or ghost in the machine. It is a candidate for the immaterial dualist "substance" of René Descartes, though it is probably better thought of as a "property dualism," since information is an immaterial property of all matter.
The information stored in the mind is not only abstract ideas. It contains a recording of the experiences of the individual. In principle every experience may be recorded, though not all may be reproducible/recallable.
The negative entropy (order, or potential information) generated by the universe expansion is a tiny amount compared to the increase in positive entropy (disorder). Sadly, this is always the case when we try to get "order out of order," as can be seen by studying entropy flows at different levels of emergent phenomena.
In any process, the positive entropy increase is always at least equal to, and generally orders of magnitude larger than, the negative entropy in any created information structures, to satisfy the second law of thermodynamics. The positive entropy is named for Boltzmann, since it was his "H-Theorem" that proved entropy can only increase overall - the second law of thermodynamics. And negative entropy is called Shannon, since his theory of information communication has exactly the same mathematical formula as Boltzmann's famous principle;
S = k log W,
where S is the entropy, k is Boltzmann's constant, and W is the probability of the given state of the system.
Material particles are the first information structures to form in the universe.. They are quarks, baryons, and atomic nuclei, which eventually combine with electrons to form atoms and eventually molecules, when the falling temperature becomes low enough. These material particles are attracted by the force of universal gravitation to form the gigantic information structures of the galaxies, stars, and planets.
Microscopic quantum mechanical particles and huge self-gravitating systems are stable and have extremely long lifetimes, thanks in large part to quantum stability. Stars are another source of radiation, after the original Big Bang cosmic source, which has cooled down to 3 degrees Kelvin (3°K) and shines as the cosmic microwave background radiation.
Our solar radiation has a high color temperature (5000K) and a low energy-content temperature (273K). It is out of equilibrium and it is the source of all the information-generating negative entropy that drives biological evolution on the Earth. Note that the fraction of the light falling on Earth is less than a billionth of that which passes by and is lost in space.
A tiny fraction of the solar energy falling on the earth gets converted into the information structures of plants and animals. Most of it gets converted to heat and is radiated away as waste energy to the night sky.
Every biological structure is a quantum mechanical structure. Quantum cooperative phenomena allow DNA to maintain its stable information structure over billions of years in the constant presence of chaos and noise. And biological structures contain astronomical numbers of particles, allowing them to average over the random noise of individual quantum events, becoming "adequately determined."
The stable information content of a human being survives many changes in the material content of the body during a person’s lifetime. Only with death does the mental information (spirit, soul) dissipate - unless it is saved somewhere.
The total mental information in a living human is orders of magnitude less than the information content and information processing rate of the body. But the cultural information structures created by humans outside the body, in the form of external knowledge like this book, and the enormous collection of human artifacts, now rival the total biological information content.
The Shannon Principle - No Information Without Possibilities
In his development of the mathematical theory of the communication of information, Claude Shannon showed that there can be no new information in a message unless there are multiple possible messages. If only one message is possible, there is no information in that message.
We can simplify this to define the Shannon Principle. No new information can be created in the universe unless there are multiple possibilities, only one of which can become actual.
An alternative statement of the Shannon principle is that in a deterministic system, information is conserved, unchanging with time. Classical mechanics is a conservative system that conserves not only energy and momentum but also conserves the total information. Information is a "constant of the motion" in a determinist world.
Quantum mechanics, by contrast, is indeterministic. It involves irreducible ontological chance.
An isolated quantum system is described by a wave function ψ which evolves - deterministically - according to the unitary time evolution of the linear Schrödinger equation.
(ih/2π) ∂ψ/∂t =
The possibilities of many different outcomes evolve deterministically, but the individual actual outcomes are indeterministic.
This sounds a bit contradictory, but it is not. It is the essence of the highly non-intuitive quantum theory, which combines a deterministic "wave" aspect with an indeterministic "particle" aspect.
In his 1932 Mathematical Foundations of Quantum Mechanics, John von Neumann explained that two fundamentally different processes are going on in quantum mechanics (in a temporal sequence for a given particle - not at the same time).
1. Process 1. A non-causal process, in which the measured electron winds up randomly in one of the possible physical states (eigenstates) of the measuring apparatus plus electron.
The probability for each eigenstate is given by the square of the coefficients cn of the expansion of the original system state (wave function ψ) in an infinite set of wave functions φ that represent the eigenfunctions of the measuring apparatus plus electron.
cn = < φn | ψ >
This is as close as we get to a description of the motion of the "particle" aspect of a quantum system. According to von Neumann, the particle simply shows up somewhere as a result of a measurement.
Information physics says that the particle shows up whenever a new stable information structure is created, information that can be observed.
Process 1b. The information created in Von Neumann's Process 1 will only be stable if an amount of positive entropy greater than the negative entropy in the new information structure is transported away, in order to satisfy the second law of thermodynamics.
2. Process 2. A causal process, in which the electron wave function ψ evolves deterministically according to Schrödinger's equation of motion for the "wave"aspect. This evolution describes the motion of the probability amplitude wave ψ between measurements. The wave function exhibits interference effects. But interference is destroyed if the particle has a definite position or momentum. The particle path itself can never be observed.
Von Neumann claimed there is another major difference between these two processes. Process 1 is thermodynamically irreversible. Process 2 is in principle reversible. This confirms the fundamental connection between quantum mechanics and thermodynamics that is explainable by information physics.
Information physics establishes that process 1 may create information. It is always involved when information is created.
Process 2 is deterministic and information preserving.
The first of these processes has come to be called the collapse of the wave function.
It gave rise to the so-called problem of measurement, because its randomness prevents it from being a part of the deterministic mathematics of process 2.
But isolation is an ideal that can only be approximately realized. Because the Schrödinger equation is linear, a wave function | ψ > can be a linear combination (a superposition) of another set of wave functions | φn >,
| ψ > = cn | φn >,
where the cn coefficients squared are the probabilities of finding the system in the possible state | φn > as the result of an interaction with another quantum system.
cn2 = < ψ | φn >2.
Quantum mechanics introduces real possibilities, each with a calculable probability of becoming an actuality, as a consequence of one quantum system interacting (for example colliding) with another quantum system.
It is quantum interactions that lead to new information in the universe - both new information structures and information processing systems. But that new information cannot subsist unless a compensating amount of entropy is transferred away from the new information.
Even more important, it is only in cases where information persists long enough for a human being to observe it that we can properly describe the observation as a "measurement" and the human being as an "observer." So, following von Neumann's "process" terminology, we can complete his admittedly unsuccessful attempt at a theory of the measuring process by adding an anthropomorphic
Process 3 - a conscious observer recording new information in a mind. This is only possible if the local reductions in the entropy (the first in the measurement apparatus, the second in the mind) are both balanced by even greater increases in positive entropy that must be transported away from the apparatus and the mind, so the overall change in entropy can satisfy the second law of thermodynamics.
An Information Interpretation of Quantum Mechanics
Our emphasis on the importance of information suggests an "information interpretation" of quantum mechanics that eliminates the need for a conscious observer as in the "standard orthodox" Copenhagen Interpretation. An information interpretation dispenses also with the need for a separate "classical" measuring apparatus.
There is only one world, the quantum world.
We can say it is ontologically indeterministic, but epistemically deterministic, because of human ignorance
Information physics claims there is only one world, the quantum world, and the "quantum to classical transition" occurs for any large macroscopic object with mass m that contains a large number of atoms. In this case, independent quantum events are "averaged over," the uncertainty in position and momentum of the object becomes less than the observational accuracy as
Δv Δx > h / m and as h / m goes to zero.
The classical laws of motion, with their implicit determinism and strict causality emerge when microscopic events can be ignored.
Information philosophy interprets the wave function ψ as a "possibilities" function. With this simple change in terminology, the mysterious process of a wave function "collapsing" becomes a much more intuitive discussion of possibilities, with mathematically calculable probabilities, turning into a single actuality, faster than the speed of light.
Information physics is standard quantum physics. It accepts the Schrödinger equation of motion, the principle of superposition, the axiom of measurement (now including the actual information "bits" measured), and - most important - the projection postulate of standard quantum mechanics (the "collapse" so many interpretations deny).
But a conscious observer is not required for a projection, for the wave-function "collapse", for one of the possibilities to become an actuality. What it does require is an interaction between (quantum) systems that creates irreversible information.
In less than two decades of the mid-twentieth century, the word information was transformed from a synonym for knowledge into a mathematical, physical, and biological quantity that can be measured and studied scientifically.
In 1929, Leo Szilard connected an increase in thermodynamic (Boltzmann) entropy with any increase in information that results from a measurement, solving the problem of "Maxwell's Demon," a thought experiment suggested by James Clerk Maxwell, in which a local reduction in entropy is possible when an intelligent being interacts with a thermodynamic system.
In the early 1940s, digital computers were invented by von Neumann, Shannon, Alan Turing, and others. Their machines could run a stored program to manipulate stored data, processing information, as biological organisms had been doing for billions of years.
Then in the late 1940s, the problem of communicating digital data signals in the presence of noise was first explored by Shannon, who developed the modern mathematical theory of the communication of information. Norbert Wiener wrote in his 1948 book Cybernetics that "information is the negative of the quantity usually defined as entropy," and in 1949 Leon Brillouin coined the term "negentropy."
Finally, in the early 1950s, inheritable characteristics were shown by Francis Crick, James Watson, and George Gamow to be transmitted from generation to generation in a digital code.
Information is Immaterial
Information is neither matter nor energy, but it needs matter for its embodiment and energy for its communication.
A living being is a form through which passes a flow of matter and energy (with low entropy). Genetic information is used to build the information-rich matter into an information-processing structure that contains a very large number of hierarchically organized information structures.
All biological systems are cognitive, using their internal information structure to guide their actions. Even some of the simplest organisms can learn from experience. The most primitive minds are experience recorders and reproducers.
In humans, the information-processing structures create new actionable information (knowledge) by consciously and unconsciously reworking the experiences stored in the mind.
Emergent higher levels exert downward causation on the contents of the lower levels, ultimately supporting mental causation and free will.
Notice the absurdity of the idea that the random motions of the transfer RNA molecules (green in the video above), each holding a single amino acid (red), are carrying pre-determined information of where they belong in the protein being built.
Determinism is an emergent property and an ideal philosophical concept, unrealizable except approximately in the kind of adequate determinism that we experience in the macroscopic world, where the determining information is part of the higher-level control system.
The total information in multi-cellular living beings can develop to be many orders of magnitude more than the information present in the original cell. The creation of this new information would be impossible for a deterministic universe, in which information is constant.
Immaterial information is perhaps as close as a physical or biological scientist can get to the idea of a soul or spirit that departs the body at death. When a living being dies, it is the maintenance of biological information that ceases. The matter remains.
Biological systems are different from purely physical systems primarily because they create, store, and communicate information. Living things store information in a memory of the past that they use to shape their future. Fundamental physical objects like atoms have no history.
And when human beings export some of their personal information to make it a part of human culture, that information moves closer to becoming immortal.
Human beings differ from other animals in their extraordinary ability to communicate information and store it in external artifacts. In the last decade the amount of external information per person may have grown to exceed an individual's purely biological information.
Since the 1950's, the science of human behavior has changed dramatically from a "black box" model of a mind that started out as a "blank slate" conditioned by environmental stimuli. Today's mind model contains many "functions" implemented with stored programs, all of them information structures in the brain. The new "computational model" of cognitive science likens the brain to a computer, with some programs and data inherited and others developed as appropriate reactions to experience.
The Experience Recorder and Reproducer
The brain should be regarded less as an algorithmic computer, with one or more central processing units addressing multiple data storage systems, than as a multi-channel and multi-track experience recorder and reproducer with an extremely high data rate. Information about an experience - the sights, sounds, smells, touch, and taste - is recorded along with the emotions - feelings of pleasure, pain, hopes, and fears - that accompany the experience. When confronted with similar experiences later, the brain can reproduce information about the original experience (an instant replay) that helps to guide current actions.
The ERR model stands in contrast to the popular cognitive science or “computational” model of a mind as a digital computer. No algorithms, data addressing schemes, or stored programs are needed for the ERR model.
The physical metaphor is a non-linear random-access data recorder, where data is stored using content-addressable memory (the memory address is the data content itself). Simpler than a computer with stored algorithms, a better technological metaphor might be a video and sound recorder, enhanced with the ability to record - and replay - smells, tastes, touches, and critically essential, feelings.
The biological model is neurons that wire together during an organism’s experiences, in multiple sensory and limbic systems, such that later firing of even a part of the wired neurons can stimulate firing of all or part of the original complex.
A conscious being is constantly recording information about its perceptions of the external world, and most importantly for ERR, it is simultaneously recording its feelings. Sensory data such as sights, sounds, smells, tastes, and tactile sensations are recorded in a sequence along with pleasure and pain states, fear and comfort levels, etc.
All these experiential and emotional data are recorded in association with one another. This means that when the experiences are reproduced (played back in a temporal sequence), the accompanying emotions are once again felt, in synchronization.
The ability to reproduce an experience is critical to learning from past experiences, so as to make them guides for action in future experiences. The ERR model is the minimal mind model that provides for such learning by living organisms.
The ERR model does not need computer-like decision algorithms to reproduce past experiences. All that is required is that past experiences “play back” whenever they are stimulated by present experiences that resemble the past experiences in one or more ways.
Where neuroscientists have shown "neurons that fire together wire together," the ERR model of information philosophy simply is "neurons that have been wired together will fire together."
Neuroscientists and philosophers of mind have long asked how diverse signals from multiple locations in the brain over multiple pathways appear so unified in the brain. The ERR model offers a simple solution to this “binding” problem. Experiences are bound at their initial recording. They do not have to be re-associated by some central processing unit looking up where experiences may have been distributed among the various sensory or memory areas.
The ERR model may also throw some light on the problem of "qualia" and of "what it's like to be" a particular organism.
Information Philosophy and Modern Philosophy
Modern philosophy is a story about discovery of timeless truths, laws of nature, a block universe in which the future is a logical extension of the past, a primal moment of creation that starts a causal chain in which everything can be foreknown by an omniscient being. Modern philosophy seeks knowledge in logical reasoning with clear and unchanging concepts.
Its guiding lights are thinkers like Parmenides, Plato, and Kant, who sought unity and identity, being and universals.
Tradition, Modern, and Postmodern In a traditional society, authoritative knowledge is that which has been handed down. Moderns are those who think that all knowledge must be based on reason. Postmoderns recognize that much knowledge has been invented, arbitrarily created
In modern philosophy, the total amount of information in the conceptually closed universe is static, a physical constant of nature. The laws of nature allow no exceptions, they are perfectly causal. Everything that happens is said to have a physical cause. This is called "causal closure". Chance and change - in a deep philosophical sense - are said to be illusions. Every event must have a cause, a reason.
Information philosophy, by contrast, is a story about invention, about novelty, about biological emergence and new beginnings unseen and unseeable beforehand, a past that is fixed but an ambiguous future that can be shaped by teleonomic changes in the present.
Its model thinkers are Heraclitus, Protagoras, Aristotle, and Hegel, for whom time, place, and particular situations mattered.
Information philosophy is built on probabilistic laws of nature. The fundamental challenge for information philosophy is to explain the emergence of stable information structures from primordial and ever-present chaos, to account for the phenomenal success of deterministic laws when the material substrate of the universe is irreducibly chaotic, noisy, and random, and to understand the concepts of truth, necessity, and certainty in a universe of chance, contingency, and indeterminacy.
Determinism and the exceptionless causal and deterministic laws of classical physics are the real illusions. Determinism is information-preserving. In an ideal deterministic Laplacian universe, the present state of the universe is implicitly contained in its earliest moments.
This ideal determinism does not exist. The "adequate determinism" behind the laws of nature emerged from the early years of the universe when there was only indeterministic chaos.
In a random noisy environment, how can anything be regular and appear determined? It is because the macroscopic consequences of the law of large numbers average out microscopic quantum fluctuations to provide us with a very adequate determinism.
Information Philosophy is an account of continuous information creation, a story about the origin and evolution of the universe, of life, and of intelligence from an original quantal chaos that is still present in the microcosmos. More than anything else, it is the creation and maintenance of stable information structures, despite the destructive entropic requirements of the second law of thermodynamics, that distinguishes biology from physics and chemistry.
Living things maintain information in a memory of the past that they can use to shape the future. The "meaning" in the information is their use of it. Some get their information "built-in" via heredity. Some learn it from experience. Others invent it!
Ancient Philosophy, before the advent of Modern Theology with John Duns Scotus and Thomas Aquinas, and Medieval Philosophy, before the beginning of Modern Philosophy with René Descartes, covered the same wide range of questions now addressable by Information Philosophy.
The Development of Information Philosophy
Our earliest work on information philosophy dates from the 1950's, based on suggestions made thirty years earlier by Arthur Stanley Eddington. In his 1928 Nature of the Physical World, Eddington argued that quantum indeterminacy had "opened the door of human freedom," and that the second law of thermodynamics might have some bearing on the question of objective good.
In the 1950's, we studied the then leading philosophies of positivism and existentialism.
Bertrand Russell, with the help of G. E. Moore, Alfred North Whitehead, and Ludwig Wittgenstein, proposed logic and language as the proper foundational basis, not only of philosophy, but also of mathematics and science. Wittgenstein's Tractatus imagined that a set of all true propositions could capture all the knowledge of modern science.
4.11 The totality of true propositions is the whole of natural science
(or the whole corpus of the natural sciences)
Their logical positivism and the variation called logical empiricism developed by Rudolf Carnap and the Vienna Circle proved to be failures in grounding philosophy, mathematics, or science.
On the continent, existentialism was the rage. We read Friedrich Nietzsche, Martin Heidegger, and Jean-Paul Sartre.
The existentialist continentals argued that freedom exists, but there are no objective values. The utilitarian English argued that values exist, but human freedom does not.
We wrote that "Values without freedom are useless. Freedom without values is absurd."
This was a chiasmos like the great figure of Immanuel Kant, rephrased by Charles Sanders Peirce as "Idealism without Materialism is Empty. Materialism without Idealism is Blind."
In the 1960's, we formulated arguments that cited "pockets of low entropy," in apparent violation of the second law, as the possible basis for anything with objective value. We puzzled over the origin of "negative entropy," since the universe was believed to have started in thermodynamic equilibrium and the second law of thermodynamics says that (positive) entropy can only increase.
In the late 1960's, we developed a two-stage model of free will and called it Cogito, a term often associated with the mind and with thought. With deference to Descartes, the first modern philosopher, we called "negative entropy" Ergo. While thermodynamics calls it "negative," information philosophy sees it as the ultimate "positive" and deserving of a better name. We thought that Ergo etymologically suggests a fundamental kind of energy ("erg" zero), e.g., the "Gibbs free energy," G0, that is available to do work because it has low entropy.
In the early 70's, we decided to call the sum of human knowledge the Sum, to complete the triple wordplay on Descartes' proof of his existence.
We saw a great battle going on in the universe - between originary chaos and emergent cosmos. The struggle is between destructive chaotic processes that drive a microscopic underworld of random events versus constructive cosmic processes that create information structures with extraordinary emergent properties that include adequately determined scientific laws - despite, and in many cases making use of, the microscopic chaos.
Since the destructive chaos is entropic, we repurposed a term from statistical mechanics and called the anti-entropic processes creating information structures ergodic. The embedded Ergod resonated.
Created information structures range from galaxies, stars, and planets, to molecules, atoms, and subatomic particles. They are the structures of terrestrial life from viruses and bacteria to sentient and intelligent beings. And they are the constructed ideal world of thought, of intellect, of spirit, including the laws of nature, in which we humans play a role as co-creator.
Information is constant in a deterministic universe. There is "nothing new under the sun." The creation of new information is not possible without the random chance and uncertainty of quantum mechanics, plus the extraordinary temporal stability of quantum mechanical structures.
It is of the deepest philosophical significance that information is based on the mathematics of probability. If all outcomes were certain, there would be no "surprises" in the universe. Information would be conserved and a universal constant, as some mathematicians mistakenly believe. Information philosophy requires the ontological uncertainty and probabilistic outcomes of modern quantum physics to produce new information.
But at the same time, without the extraordinary stability of quantized information structures over cosmological time scales, life and the universe we know would not be possible. That stability is the consequence of an underlying digital nature. Quantum mechanics reveals the architecture of the universe to be discrete rather than continuous, to be digital rather than analog. Digital information transfers are essentially perfect. All analog transfers are "lossy."
Moreover, the "correspondence principle" of quantum mechanics and the "law of large numbers" of statistics ensures that macroscopic objects can normally average out microscopic uncertainties and probabilities to provide the "adequate determinism" that shows up in all our "Laws of Nature."
Information philosophy explores some classical problems in philosophy with deeper and more fundamental insights than is possible with the logic and language approach of modern analytic philosophy.
By exploring the origins and evolution of structure in the universe, information philosophy transcends humanity and even life itself, though it is not a mystical metaphysical transcendence.
Information philosophy uncovers the creative process working in the universe
to which we owe our existence, and therefore perhaps our reverence for its "providence".
Information philosophy locates the fundamental source of all values not in humanity ("man the measure"), not in bioethics ("life the ultimate good"), but in the origin and evolution of information in the cosmos.
Information philosophy is an idealistic philosophy, a process philosophy, and a systematic philosophy, the first in many decades. It provides important new insights into the Kantian transcendental problems of epistemology, ethics, freedom of the will, god, and immortality, as well as the mind-body problem, consciousness, and the problem of evil.
In physics, information philosophy (or information physics) provides new insights into the problem of measurement, the paradox of Schrödinger's Cat, the two paradoxes of microscopic reversibility and macroscopic recurrence that Josef Loschmidt and Ernst Zermelo used to criticize Ludwig Boltzmann's explanation of the entropy increase required by the second law of thermodynamics, and finally information provides a better understanding of the entanglement and nonlocality phenomena that are the basis for modern quantum cryptography and quantum computing.
Finally, a new philosophy of biology should be based on the deep understanding of organisms as information users, information creators, information communicators, and at the higher levels, information processors, including humans who have learned to store information externally and transfer it between the generations culturally. Except for organisms that can extract information by photosynthesis of the negative entropy (free or available energy) streaming from the sun, most living things destroy other cells to extract the information needed to maintain their own low entropy state of organization. Most life feeds on other life.
And most life communicates with other life. Even single cells, before the emergence of multicellular organisms, developed communication systems between the cells that are still visible in slime molds and social amoebae today. In a multicellular organism, every cell has some level of communication with all the others. Most higher level organisms share communal information that makes them stronger as a social group than as independent individuals. The sum of human knowledge has amplified the power of humanity, for better or worse, to a level that can control the environmental conditions on all of planet Earth.
Information biology is the hypothesis that all biological evolution should be viewed primarily as the development of more and more powerful users, creators, and communicators of information. Seen though the lens of information, humans are the current end product of information processing systems. With the emergence of life and mind, purpose (telos) appeared in the universe. The teleonomic goal of each cell is to become two cells, which replicates its information content. The purpose of each species is to improve its reproductive success relative to other populations. The purpose of human populations then is to use, to add to, and to communicate human knowledge in order to maximize the human capital per person.
Like love, the information that is shared by educating others is not used up. Information is not a scarce economic good. The more that information is communicated, the more of it there is, in human minds (not brains), and in the external stores of human knowledge. These include books of course, but in the future they will be the interconnected knowledge bases of the world wide web, including, since books are expensive and inaccessible for many.
The first thing we must do for the young is to teach them how to teach themselves by accessing these knowledge systems with handheld devices that will some day be available for all the world's children, beyond one laptop per child to one smartphone per child.
Based on insights into the discovery of the cosmic creation process, the Information Philosopher proposes three primary ideas that are new approaches to perennial problems in philosophy. They are likely to change some well-established philosophical positions. Even more important, they may reconcile idealism and materialism and provide a new view of how humanity fits into the universe.
The three ideas are
• An explanation or epistemological model of knowledge formation and communication. Knowledge and information are neither matter nor energy, but they require matter for expression and energy for communication. They seem to be metaphysical.
Briefly, we identify knowledge with actionable information in the brain-mind. We justify knowledge by behavioral studies that demonstrate the existence of information structures implementing functions in the brain. And we verify knowledge scientifically.
• A basis for objective value, a metaethics beyond humanism and bioethics, grounded in the fundamental information creation processes behind the structure and evolution of the universe and the emergence of life.
Briefly, we find positive value (or good) in information structures. We see negative value (or evil) in disorder and entropy tearing down such structures. We call energy with low entropy "Ergo" and call anti-entropic processes "ergodic." We recognize that "ergodic" is itself too esoteric and thus not likely to be widely accepted. Perhaps the most positive term for what we value is just "information" itself!
Our first categorical imperative is then "act in such a way as to create, maintain, and preserve information as much as possible against destructive entropic processes."
Our second ethical imperative is "share knowledge/information to the maximum extent." Like love, our own information is not diminished when we share it with others
Our third moral imperative is "educate (share the knowledge of what is right) rather than punish." Knowledge is virtue. Punishment wastes human capital and provokes revenge.
• Watch a 10-minute animated tutorial on the Two-Stage Solution to
the Free Will Problem
A scientific model for free will and creativity informed by the complementary roles of microscopic randomness and adequate macroscopic determinism in a temporal sequence that generates new information.
Briefly, we separate "free" and "will" in a two-stage process - first the free generation of alternative possibilities for action (which creates new information), then an adequately determined decision by the will. We call this two-stage view our Cogito model and trace the idea of a two-stage model in the work of two dozen thinkers back to William James in 1884.
This model is a synthesis of adequate determinism and limited indeterminism, a coherent and complete compatibilism that reconciles
free will with both determinism and indeterminism.
David Hume thought he had reconciled freedom with determinism. We reconcile free will with indeterminism and an "adequate" determinism.
Because it makes free will compatible with both a form of determinism (really determination) and with an indeterminism that is limited and controlled by the mind, the leading libertarian philosopher Bob Kane suggested we call this model "Comprehensive Compatibilism."
The problem of free will cannot be solved by logic, language, or even by physics. Man is not a machine and the mind is not a computer.
Free will is a property of a biophysical information processing system.
All three ideas depend on understanding modern cosmology, physics, biology, and neuroscience, but especially the intimate connection between quantum mechanics and the second law of thermodynamics that allows for the creation of new information structures.
All three are based on the theory of information, which alone can establish the existential status of ideas, not just the ideas of knowledge, value, and freedom, but other-worldly speculations in natural religion like God and immortality.
All three have been anticipated by earlier thinkers, but can now be defended on strong empirical grounds. Our goal is less to innovate than to reach the best possible consensus among philosophers living and dead, an intersubjective agreement between philosophers that is the surest sign of a knowledge advance.
This Information Philosopher website aims to be an open resource for the best thinking of philosophers and scientists on these three key ideas and a number of lesser ideas that remain challenging problems in philosophy - on which information philosophy can shed some light.
Among these are the mind-body problem (the mind can be seen as the realm of information in its free thoughts, the body an adequately determined biological system creating and maintaining information); the common sense intuition of a cosmic creative process often anthropomorphized as a God or divine Providence; the problem of evil (chaotic entropic forces are the devil incarnate); and the "hard problem" of consciousness (agents responding to their environment, and originating new causal chains, based on information processing).
Philosophy is the love of knowledge or wisdom. Information philosophy (I-Phi or ΙΦ) qualifies and quantifies knowledge as meaningful actionable information. Information philosophy reifies information as an immaterial entity that has causal power over the material world!
What is information that merits its use as the foundation of a new method of inquiry?
Abstract information is neither matter nor energy, yet it needs matter for its concrete embodiment and energy for its communication. Information is the modern spirit, the ghost in the machine. It is the stuff of thought, the immaterial substance of philosophy.
Information is a powerful diagnostic tool. It is a better abstract basis for philosophy, and for science as well, especially physics, biology, and neuroscience. It is capable of answering questions about metaphysics (the ontology of things themselves), epistemology (the existential status of ideas and how we know them), and idealism itself.
Information philosophy is now more than the solution to three fundamental problems we identified in the 1960's and '70's. I-Phi is a new philosophical method, capable of solving multiple problems in both philosophy and physics. It needs young practitioners, presently tackling some problem, who might investigate the problem using this new methodology.
Note that, just as the philosophy of language is not linguistic philosophy, Information philosophy is not the philosophy of information, which is mostly about computers and cognitive science, the computational theory of mind.
Philosophers like Ludwig Wittgenstein labeled many of our problems “philosophical puzzles.” Bertrand Russell called them “pseudo-problems.” Analytic language philosophers thought many of these problems could be “dis-solved,” revealing them to be conceptual errors caused by the misuse of language.
Information philosophy takes us past logical puzzles and language games, not by diminishing philosophy and replacing it with science.
Russell insisted that
“questions which are already capable of definite answers are placed in the sciences, while those only to which, at present, no definite answer can be given, remain to form the residue which is called philosophy.”
Information philosophy aims to show that problems in philosophy should not be reduced to “Russell’s Residue.”
The language philosophers of the twentieth century thought that they could solve (or at least dis-solve) the classical problems of philosophy. They did not succeed. Information philosophy, by comparison, now has cast a great deal of light on some of those problems. It needs more information philosophers to join us to make more progress.
To recap, when information is stored in any structure, two fundamental physical processes occur. First is a "collapse" of a quantum mechanical wave function, reducing multiple possibilities to a single actuality. Second is a local decrease in the entropy corresponding to the increase in information. Entropy greater than that must be transferred away from the new information structure to satisfy the second law of thermodynamics.
These quantum level processes are susceptible to noise. Information stored may have errors. When information is retrieved, it is again susceptible to noise. This may garble the information content. In information science, noise is generally the enemy of information. But some noise is the friend of freedom, since it is the source of novelty, of creativity and invention, and of variation in the biological gene pool.
Biological systems have maintained and increased their invariant information content over billions of generations, coming as close to immortality as living things can. Philosophers and scientists have increased our knowledge of the external world, despite logical, mathematical, and physical uncertainty. They have created and externalized information (knowledge) that can in principle become immortal. Both life and mind create information in the face of noise. Both do it with sophisticated error detection and correction schemes. The scheme we use to correct human knowledge is science, a two-stage combination of freely invented theories and adequately determined experiments. Information philosophy follows that example.
If you have read this far, you probably already know that the Information Philosopher website itself is an exercise in information sharing. It has seven parts, each with multiple chapters. Navigation at the bottom of each page will take you to the next or previous part or chapter.
Teacher and Scholar links display additional material on some pages, and reveal hidden footnotes on some pages. The footnotes themselves are in the Scholar section.
Our goal is for the website to contain all the great philosophical discussions of the three original problem areas we identified in the 1970's - COGITO (freedom), ERGO (value), and SUM (knowledge) - plus potential solutions for several classic problems in philosophy and physics, many of which had been designated "pseudo-problems" or relegated to "metaphysics."
We have now shown that information philosophy is a powerful diagnostic tool for addressing metaphysical problems. See The Metaphysicist.
In the left-hand column of all I-Phi pages are links to nearly three hundred philosophers and scientists who have made contributions to these great problems. Their web pages include the original contributions of each thinker, with examples of their thought, usually in their own words, and where possible in their original languages as well.
All original content on Information Philosopher is available for your use, without requesting
permission, under a Creative Commons Attribution License. cc by
Copyrights for all excerpted and quoted works remain with their authors and publishers.
Introduction Knowledge Value Freedom Mind Chance Quantum Afterword
For Teachers
A web page may contain two extra levels of material. The Normal page is material for newcomers and students of the Information Philosophy. Two hidden levels contain material for teachers (e.g., secondary sources) and for scholars (e.g., footnotes, and original language quotations).
Teacher materials on a page will typically include references to secondary sources and more extended explanations of the concepts and arguments. Secondary sources will include books, articles, and online resources. Extended explanations should be more suitable for teaching others about the core philosophical ideas, as seen from an information perspective.
For Scholars
Scholarly materials will generally include more primary sources, more in-depth technical and scientific discussions where appropriate, original language versions of quotations, and references to all sources.
Footnotes for a page appear in the Scholar materials. The footnote indicators themselves are only visible in Scholar mode.
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Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Louise Antony
Thomas Aquinas
David Armstrong
Harald Atmanspacher
Robert Audi
Alexander Bain
Mark Balaguer
Jeffrey Barrett
William Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
Michael Burke
Joseph Keim Campbell
Rudolf Carnap
Ernst Cassirer
David Chalmers
Roderick Chisholm
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Herbert Feigl
John Martin Fischer
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
Sam Harris
William Hasker
Georg W.F. Hegel
Martin Heidegger
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Andrea Lavazza
Keith Lehrer
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James Symposium
Copenhagen Interpretation of Quantum Mechanics
The idea that there was a Copenhagen way of thinking was christened as the "Kopenhagener Geist der Quantentheorie" by Werner Heisenberg in the introduction to his 1930 textbook The Physical Principles of Quantum Theory, based on his 1929 lectures in Chicago (given at the invitation of Arthur Holly Compton).
It is a sad fact that Einstein, who had found more than any other scientist on the quantum interaction of electrons and photons, was largely ignored or misunderstood at this Solvay, when he again clearly described nonlocality
At the 1927 Solvay conference on physics entitled "Electrons and Photons," Niels Bohr and Heisenberg consolidated their Copenhagen view as a "complete" picture of quantum physics, despite the fact that they could not, or would not, visualize or otherwise explain exactly what is going on in the microscopic world of "quantum reality."
From the earliest presentations of the ideas of the supposed "founders" of quantum mechanics, Albert Einstein had deep misgivings of the work going on in Copenhagen, although he never doubted the calculating power of their new mathematical methods. He described their work as incomplete because it is based on the statistical results of many experiments so only makes probabilistic predictions about individual experiments. Einstein hoped to visualize what is going on in an underlying "objective reality."
Bohr seemed to deny the existence of a fundamental "reality," but he clearly knew and said that the physical world is largely independent of human observations. In classical physics, the physical world is assumed to be completely independent of the act of observing the world. In quantum physics, Heisenberg said that the result of an experiment depends on the free choice of the experimenter as to what to measure. The quantum world of photons and electrons might look like waves or look like particles depending on what we look for, rather than what they "are" as "things in themselves."
The information interpretation of quantum mechanics says there is only one world, the quantum world. Averaging over large numbers of quantum events explains why large objects appear to be classical
Bohr thus put severe epistemological limits on knowing the Kantian "things in themselves," just as Immanuel Kant had put limits on reason. The British empiricist philosophers John Locke and David Hume had put the "primary" objects beyond the reach of our "secondary" sensory perceptions. In this respect, Bohr shared the positivist views of many other empirical scientists, Ernst Mach for example. Twentieth-century analytic language philosophers thought that philosophy (and even physics) could not solve some basic problems, but only "dis-solve" them by showing them to be conceptual errors.
Neither Bohr nor Heisenberg thought that macroscopic objects actually are classical. They both saw them as composed of microscopic quantum objects.
The exact location of that transition from the quantum to the classically describable world was arbitrary, said Heisenberg. He called it a "cut" (Schnitt). Heisenberg's and especially John von Neumann's and Eugene Wigner's insistence on a critical role for a "conscious observer" has led to a great deal of nonsense being associated with the Copenhagen Interpretation and in the philosophy of quantum physics. Heisenberg may only have been trying to explain how knowledge reaches the observer's mind. For von Neumann and Wigner, the mind was considered a causal factor in the behavior of the quantum system.
Today, a large number of panpsychists, some philosophers, and a small number of scientists, still believe that the mind of a conscious observer is needed to cause the so-called "collapse" of the wave function. A relatively large number of scientists opposing the Copenhagen Interpretation believe that there are never any "collapses" in a universal wave function.
In the mid 1950's, Heisenberg reacted to David Bohm's 1952 "pilot-wave" interpretation of quantum mechanics by calling his own work the "Copenhagen Interpretation" and the only correct interpretation of quantum mechanics. A significant fraction of working quantum physicists say they agree with Heisenberg, though few have ever looked carefully into the fundamental assumptions of the Copenhagen Interpretation.
This is because they pick out from the Copenhagen Interpretation just the parts they need to make quantum mechanical calculations. Most textbooks start the story of quantum mechanics with the picture provided by the work of Heisenberg, Bohr, Max Born, Pascual Jordan, Paul Dirac, and of course Erwin Schrödinger.
What Exactly Is in the Copenhagen Interpretation?
• The quantum postulates. Bohr postulated that quantum systems (beginning with his "Bohr atom" in 1913) have "stationary states" which make discontinuous "quantum jumps" between the states with the emission or absorption of radiation. Until at least 1925 Bohr insisted the radiation itself is continuous. Einstein said radiation is a discrete "light quantum" (later called a photon) as early as 1905.
Ironically, largely ignorant of the history of quantum mechanics (dominated by Bohr's account), many of today's textbooks teach the "Bohr atom" as emitting or absorbing photons - Einstein light quanta!
Also, although Bohr made a passing reference, virtually no one today knows that discrete energy states or quantized energy levels in matter were first discovered by Einstein in his 1907 work on specific heat.
• Wave-particle duality. The complementarity of waves and particles, including a synthesis of the particle-matrix mechanics theory of Heisenberg, Max Born, and Pascual Jordan, with the wave mechanical theory of Louis deBroglie and Erwin Schrödinger.
Again ironically, wave-particle duality was first described by Einstein in 1909. Heisenberg had to have his arm twisted by Bohr to accept the wave picture.
• Indeterminacy principle. Heisenberg sometimes called it his "uncertainty" principle, which could imply human ignorance, implying an epistemological (knowledge) problem rather than an ontology (reality) problem.
Bohr considered indeterminacy as another example of his complementarity, between the non-commuting conjugate variables momentum and position, for example, Δp Δx ≥ h (also between energy and time and between action and angle variables).
• Correspondence principle. Bohr maintained that in the limit of large quantum numbers, the atomic structure of quantum systems approaches the behavior of classical systems. Bohr and Heisenberg both described this case as when Planck's quantum of action h can be neglected. They mistakenly described this as h -> 0. But h is a fundamental constant.
The quantum-to-classical transition is when the action of a macroscopic object is large compared to h . As the number of quantum particles increases (as mass increases), large macroscopic objects behave like classical objects. Position and velocity become arbitrarily accurate as h / m -> 0.
Δv Δx ≥ h / m.
There is only one world. It is a quantum world. Ontologically it is indeterministic, but epistemically, common sense and everyday experience inclines us to see it as deterministic. Bohr and Heisenberg insisted we must use classical (deterministic?) concepts and language to communicate our knowledge about quantum processes!
• Completeness. Schrödinger's wave function ψ provides a "complete" description of a quantum system, despite the fact that conjugate variables like position and momentum cannot both be known with arbitrary accuracy, as they can in classical systems. There is less information in the world than classical physics implies.
The wave function ψ evolves according to the unitary deterministic Schrödinger equation of motion, conserving that information. When one possibility becomes actual (discontinuously), new information may be irreversibly created and recorded by a measurement apparatus, or simply show up as a new information structure in the world.
By comparison, Einstein maintained that quantum mechanics is incomplete, because it provides only statistical information about ensembles of quantum systems. He also was deeply concerned about nonlocality and nonseparability, things not addressed at all by the Copenhagen interpretation.
• Irreversible recording of information in the measuring apparatus. Without this record (a pointer reading, blackened photographic plate, Geiger counter firing, etc.), there would be nothing for observers to see and to know.
Information must come into the universe long before any scientist can "observe" it. In today's high-energy physics experiments and space research, the data-analysis time between the initial measurements and the scientists seeing the results can be measured in months or years.
All the founders of quantum mechanics mention the need for irreversibility. The need for positive entropy transfer away from the experiment to stabilize new information (negative entropy) so it can be observed was first shown by Leo Szilard in 1929, and later by Leon Brillouin and Rolf Landauer.
• Classical apparatus?. Bohr required that the macroscopic measurement apparatus be described in ordinary "classical" language. This is a third "complementarity," now between the quantum system and the "classical apparatus"
But Born and Heisenberg never said the measuring apparatus is "classical." They knew that everything is fundamentally a quantum system.
Lev Landau and Evgeny Lifshitz saw a circularity in this view, "quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case [correspondence principle], yet at the same time it requires this limiting case for its own formulation.
• Statistical interpretation (acausality). Born interpreted the square modulus of Schrödinger's complex wave function as the probability of finding a particle. Einstein's "ghost field" or "guiding field," deBroglie's pilot or guide wave, and Schrödinger's wave function as the distribution of the electric charge density were similar views in much earlier years. Born sometimes pointed out that his direct inspiration was Einstein.
All the predicted properties of physical systems and the "laws of nature" are only probabilistic (acausal, indeterministic ). All results of physical experiments are statistical.
Briefly, theories give us probabilities, experiments give us statistics.
Large numbers of identical experiments provide the statistical evidence for the theoretical probabilities predicted by quantum mechanics.
Bohr's emphasis on epistemological questions suggests he thought that the statistical uncertainty may only be in our knowledge. They may not describe nature itself. Or at least Bohr thought that we can not describe a "reality" for quantum objects, certainly not with classical concepts and language. However, the new concept of an immaterial possibilities function (pure information) moving through space may make quantum phenomena "visualizable."
Ontological acausality, chance, and a probabilistic or statistical nature were first seen by Einstein in 1916, as Born later acknowledged. But Einstein disliked this chance. He and most scientists appear to have what William James called an "antipathy to chance."
• No Visualizability?. Bohr and Heisenberg both thought we could never produce models of what is going on at the quantum level. Bohr thought that since the wave function cannot be observed we can't say anything about it. Heisenberg said probability is real and the basis for the statistical nature of quantum mechanics.
Whenever we draw a diagram of the waves impinging on the two-slits, we are in fact visualizing the wave function as possible locations for a particle, with calculable probabilities for each possible location.
Today we can visualize with animations many puzzles in physics, including the two-slit experiment, entanglement, and microscopic irreversibility.
• No Path?. Bohr, Heisenberg, Dirac and others said we cannot describe a particle as having a path. The path comes into existence when we observe it, Heisenberg maintained. (Die “Bahn” entsteht erst dadurch, dass wir sie beobachten)
Einstein's "objective reality" hoped for a deeper level of physics in which particles do have paths and, in particular, they obey conservation principles, though intermediate measurements needed to observe this fact would interfere with the experiments.
• Paul Dirac formalized quantum mechanics with these three fundamental concepts, all very familiar and accepted by Bohr, Heisenberg, and the other Copenhageners:
• Axiom of measurement. Bohr's stationary quantum states have eigenvalues with corresponding eigenfunctions (the eigenvalue-eigenstate link).
• Superposition principle. According to Dirac's transformation theory, ψ can be represented as a linear combination of vectors that are a proper basis for the combined target quantum system and the measurement apparatus.
• Projection postulate. The collapse of the wave function ψ, which is irreversible, upon interacting with the measurement apparatus and creating new information.
• Two-slit experiment. A "gedanken" experiment in the 1920's, but a real experiment today, exhibits the combination of wave and particle properties.
Note that what two-slit experiment really shows is
There are many more elements that play lesser roles, some making the Copenhagen Interpretation very unpopular among philosophers of science and spawning new interpretations or even "formulations" of quantum mechanics. Some of these are misreadings or later accretions. They include:
• The "conscious observer." The claim that quantum systems cannot change their states without an observation being made by a conscious observer. Does the collapse only occur when an observer "looks at" the system? How exactly does the mind of the observer have causal power over the physical world? (the mind-body problem).
Einstein objected to the idea that his bed had diffused throughout the room and only gathered itself back together when he opened the bedroom door and looked in.
John von Neumann and Eugene Wigner seemed to believe that the mind of the observer was essential, but it is not found in the original work of Bohr and Heisenberg, so should perhaps not be a part of the Copenhagen Interpretation? It has no place in standard quantum physics today
• The measurement problem, including the insistence that the measuring apparatus must be described classically when it is made of quantum particles. There are actually at least three definitions of the measurement problem.
1. The claim that the two dynamical laws, unitary deterministic time evolution according to the Schrödinger equation and indeterministic collapse according to Dirac's projection postulate are logically inconsistent. They cannot both be true, it's claimed.
The proper interpretation is simply that the two laws laws apply at different times in the evolution of a quantum object, one for possibilities, the other for actuality (as Heisenberg knew):
• first, the unitary deterministic evolution moves through space exploring all the possibilities for interaction,
• second, the indeterministic collapse randomly (acausally) selects one of those possibilities to become actual.
2. The original concern that the "collapse dynamics" (von Neumann Process 1) is not a part of the formalism (von Neumann Process 2) but is an ad hoc element, with no rules for when to apply it.
If there was a deterministic law that predicted a collapse, or the decay of a radioactive nucleus, it would not be quantum mechanics!
3. Decoherence theorists say that the measurement problem is the failure to observe macroscopic superpositions, such as Schrödinger's Cat.
• The many unreasonable philosophical claims for "complementarity:" e.g., that it solves the mind-body problem?,
• The basic "subjectivity" of the Copenhagen interpretation. It deals with epistemological knowledge of things, rather than the "things themselves."
Opposition to the Copenhagen Interpretation
Albert Einstein, Louis deBroglie, and especially Erwin Schrödinger insisted on a more "complete" picture, not merely what can be said, but what we can "see," a visualization (Anschaulichkeit) of the microscopic world. But de Broglie and Schrödinger's emphasis on the wave picture made it difficult to understand material particles and their "quantum jumps." Indeed, Schrödinger and more recent physicists like John Bell and the decoherence theorists H. D. Zeh and Wojciech Zurek deny the existence of particles and the collapse of the wave function, which is central to the Copenhagen Interpretation.
Perhaps the main claim of those today denying the Copenhagen Interpretation (and standard quantum mechanics) began with Schrödinger's (nd later Bell's) claim that "there are no quantum jumps." Decoherence theorists and others favoring Everett's Many-Worlds Interpretation reject Dirac's projection postulate, a cornerstone of quantum theory.
Heisenberg had initially insisted on his own "matrix mechanics" of particles and their discrete, discontinuous, indeterministic behavior, the "quantum postulate" of unpredictable events that undermine the classical physics of causality. But Bohr told Heisenberg that his matrix mechanics was too narrow a view of the problem. This disappointed Heisenberg and almost ruptured their relationship. But Heisenberg came to accept the criticism and he eventually endorsed all of Bohr's deep philosophical view of quantum reality as unvisualizable.
In his September Como Lecture, a month before the 1927 Solvay conference, Bohr introduced his theory of "complementarity" as a "complete" theory. It combines the contradictory notions of wave and particle. Since both are required, they complement (and "complete") one another.
Although Bohr is often credited with integrating the dualism of waves and particles, it was Einstein who predicted this would be necessary as early as 1909. But in doing so, Bohr obfuscated further what was already a mysterious picture. How could something possibly be both a discrete particle and a continuous wave? Did Bohr endorse the continuous deterministic wave-mechanical views of Schrödinger? Not exactly, but Bohr's accepting Schrödinger's wave mechanics as equal to and complementing his matrix mechanics was most upsetting to Heisenberg.
Bohr's Como Lecture astonished Heisenberg by actually deriving (instead of Heisenberg's heuristic microscope argument) the uncertainty principle from the space-time wave picture alone, with no reference to the acausal dynamics of Heisenberg's picture!
After this, Heisenberg did the same derivation in his 1930 text and subsequently completely accepted complementarity. Heisenberg spent the next several years widely promoting Bohr's views to scientists and philosophers around the world, though he frequently lectured on his mistaken, but easily understood, argument that looking at particles disturbs them. His microscope is even today included in many elementary physics textbooks.
Bohr said these contradictory wave and particle pictures are "complementary" and that both are needed for a "complete" picture. He co-opted Einstein's claim to a more "complete" picture of an objective" reality, one that might restore simultaneous knowledge of position and momentum, for example. Classical physics has twice the number of independent variables (and twice the information) as quantum physics. In this sense, it does seem more "complete."
Many critics of Copenhagen thought that Bohr deliberately and provocatively embraced logically contradictory notions - of continuous deterministic waves and discrete indeterministic particles - perhaps as evidence of Kantian limits on reason and human knowledge. Kant called such contradictory truths "antinomies." The contradictions only strengthened Bohr's epistemological resolve and his insistence that physics required a subjective view unable to reach the objective nature of the "things in themselves." As Heisenberg described it in his explanation of the Copenhagen Interpretation,
This again emphasizes a subjective element in the description of atomic events, since the measuring device has been constructed by the observer, and we have to remember that what we observe is not nature in itself but nature exposed to our method of questioning. Our scientific work in physics consists in asking questions about nature in the language that we possess and trying to get an answer from experiment by the means that are at our disposal.
Copenhagen Interpretation on Wikipedia
Copenhagen Interpretation on Stanford Encyclopedia of Philosophy
"Copenhagen Interpretation of Quantum Theory", in Physics and Philosophy, Werner Heisenberg, 1958, pp.44-58
"The Copenhagen Interpretation", American Journal of Physics, 40, p.1098, Henry Stapp, 1972
"The History of Quantum Theory", in Physics and Philosophy, Werner Heisenberg, 1958, pp.30-43
For Teachers
For Scholars
• Born's statistical interpretation - brings in Schrödinger waves, which upset Heisenberg
• uncertainty principle, March 1927
• complementarity - waves and particles, wave mechanics and matrix mechanics, again upsets Heisenberg
• the two-slit experiment
• measurements, observers, "disturb" a quantum system, - Microscope echo
• loss of causality (Einstein knew), unsharp space-time description (wave-packet)
• classical apparatus, quantum system
• our goal not to understand reality, but to acquire knowledge Rosenfeld quote
• Experimenter must choose either particle-like or wave-like experiment - need examples
• Heisenberg uncertainty was discontinuity, intrusion of instruments, for Bohr it was "the general complementary character of description" - wave or particle
• Complementarity a general framework, Heisenberg particle uncertainty a particular example
• Einstein/Schrödinger want a field theory and continuous/waves only? Bohr wants sometimes waves, sometimes particles. Bohr wants always both waves and particles.
• Combines Heisenberg's "free choice" of experimenter as to what to measure, with Dirac's "free choice" of Nature with deterministic evolution of possibilities followed by discontinuous and random appearance of one actual from all the possibles.
Chapter 1.1 - Creation Chapter 1.3 - Information
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8dd675b3db460036 | Waves, Math, & the Creator
By Dr. Adam F. Hannon | Algebra
Jun 27
Part 1 of 3
Note: I’m excited to introduce this series of guest posts from a good friend, physicist, engineer, and data scientist, Dr. Adam F. Hannon. The series explores waves (and I don’t just mean ocean waves!). More technical details are given in the endnotes for those who would like to dig deeper. I hope you’ll be as awed as I’ve been as I’ve edited the posts at God’s handiwork in the waves all around us…which math helps us explore.
– Katherine
No doubt you learned about sines and cosines in school, along with the number pi (π). The amazing thing is these tools help us describe waves, from ocean waves to light waves to particle waves inside atoms.
Here’s the basic mathematical equation that describes almost all waves, known as the Wave Equation:
The Wave Equation
The Wave Equation[i]
Without going into too much detail about the equation (more detail is included in an endnote[ii]), it essentially describes the relationship between how a wave varies over time to how it varies in space. The letters and symbols stand for different quantities. For example, the t stands for time, the Greek letter ψ (psi pronounced “sigh”) is the quantity that is “waving” or oscillating (e.g. for an ocean wave it is the distance from where the water surface is to where it would be without the wave), and the v for velocity (think speed in a specific direction). A schematic of how ψ and v look for a water ocean wave is shown in Figure 1.
Figure 1. A picture of a water wave with the math symbols we use to describe it.
Waves and Matter
Ocean waves are one example of a wave occurring in matter. Matter refers to the materials that make up the world around us. In the case of ocean waves, the matter is the water, and that water is what is making up the wave.
Another example of waves occurring in matter is a vibrating string such as in a piano, guitar, or other musical instrument (here the matter waving is the metallic string). The string goes up and down, much like an ocean wave. These string vibrations can be thought of as waves…and they too can be described mathematically with the same wave equation as the ocean waves.
Now, the vibrating string causes the air molecules near the string to move in a similar way, which creates another wave in the air around us. While we cannot see the matter “waving” here, sound waves are really the same idea as the waves we see in oceans and strings. Air is full of gas particles that are a form of matter and that can move as a pressure wave so we hear sound. The sounds travel as waves through the gas particles in the air until it reaches our ears.
All these waves occurring in matter can be described by the Wave Equation above. What differs between the different waves is the kind of matter that is waving, how the waves are constrained (whether they can move freely as with ocean waves and air sound waves or if they are constrained, as is the case with a vibrating string), and the speed of the waves. If we wanted to explore these different waves mathematically, we would need to use algebra and calculus to solve the Wave Equation for ψ as a function of spatial position x and time t. The final solution we would get would depend on the material variables and boundary constraints of the wave. The solutions are generally made up of combinations of sine and cosine functions. In a future blog, we will look at such solutions for a vibrating string in a bit more detail so we can explore more of the order God placed within sounds.
Waves and Light
A light wave is another kind of wave.
Light waves (a.k.a. electromagnetic waves or photons) are interesting in that they are actually two waves in one. One of the waves is an electric field (the same kind of electric field that gives you electricity in your house to run the computer or other electronic device you’re reading this on) and the other is a magnetic field (the same kind of magnetic field that holds cute baby pictures of your friends and relatives on your refrigerator). The two fields each have their own wave equation and are coupled together such that a light wave has two oscillatory (i.e., wave-like) components traveling at the same time. A schematic of this is shown in Figure 2.
Figure 2: A schematic diagram of a light wave. The electric field is shown with blue arrows and the magnetic field with red arrows. The light wave is traveling in time and space in the direction of the black arrow (which is perpendicular to all the red and blue arrows) at a certain speed, which we’ve represented with a c, known as the speed of light.
Each of these light waves can also be described by the Wave Equation.
Only it’s common to go ahead and rewrite the equation using different letters in order to specify that in the case of light waves, it’s E (the electric field; color coded blue) and B (the magnetic field; color coded red) that are “waving” or oscillating, and that the velocity is the speed of light (represented by a c; color coded purple):
The Wave Equation
Wave Equation[iii]
Wave Equation Rewritten for the Electric Field in Light Waves[iv]
Wave Equation Rewritten for the Magnetic Field in Light Waves[v]
Interesting Note: You’ll notice that throughout physics many equations are basically the same except for the use of different letters. Using an E and a B here make it clearer what exactly is “waving” or oscillating in this case (the electric field and the magnetic field). And we use c instead of a v as that instantly tells us that in this particular case, our velocity (in this case, the speed of light) is constant because the convention is that c represents a constant value. (Because of the consistent way God governs the universe, the speed of light in vacuum is always constant—670,616,629 miles per hour—or 299,792.5 kilometers per hour for our international friends.) Another thing worth noting is the use of B for the magnetic field. You might be wondering why in the world a B is used to describe this? Well, it’s partly because M was used to describe a material property called the magnetization, so scientists had to choose a different letter.
Waves and the Subatomic
Interestingly, wave-like equations can also describe where the very particles that make up the atoms in our bodies (and the rest of the universe) are located!
It turns out that electrons orbit the nucleus of an atom in cloud shaped orbits. Well, the Schrödinger wave equation[vi] (yes, this equation is really describing an electron’s orbit as a wave!) shown below describes the likelihood of finding a given particle (i.e., an electron) at a given location in space and time:
Electrons in an atom occupy spherical orbitals that are known as spherical harmonics. These are in fact electron probability waves that are confined to a spherical region (in this case centered around the nucleus of an atom). The first set of these spherical harmonics can be seen in Figure 3 overlapping for a carbon atom.[vii] The different colors represent the different wave orbitals. Although they may not look like a wave, mathematically they can still be thought of as waves.
Figure 3: Schematic of the first 4 spherical harmonic orbitals of a carbon atom. The inner magenta spherical orbital is hard to see, but the 3 outer dumbbell shaped orbitals (red, green, and blue) can be seen quite easily.
If the electrons in our atoms did not fall into these Wave Equation-based orbitals, everything we know would instantly collapse in on each other. However, God knew what he was doing when he set the math to govern even the smallest of particles so that we could actually sit here and talk about the amazing structure of atoms.
Waves and the Creator
There are many other wave types we could look at (such as the gravitational waves that ripple through the very space-time fabric of our universe), each of which could be described with similar but slightly different equations. (The slight differences are due to the different physics involved.)
The fact that the math of music is so similar to the math of the subatomic is simply stunning. Even more stunning is the incredible order and design we see everywhere—from the oceans’ waves to the way God arranged the electrons in our atoms to make life possible. As we explore God’s creation with math (including algebra and calculus), we see His wisdom, care, and faithfulness.
In Psalm 104, after reflecting on many aspects of God’s creation, the Psalmists cries, “O Lord, how manifold are your works! In wisdom have you made them all; the earth is full of your creatures” (Psalm 104:24 esv). May we join the Psalmist in praising our Creator and resting in His incredible care.
Note: Stay tuned for the next blog on waves, in which we’ll explore light waves in more detail…
[i] Source for wave equation: John R. Taylor, Classical Mechanics (Sausalito, CA: University Science Books, 2005), p. 695, eqn. 16.39.
[ii] In this simplest form, the equation says that for a given physical property that can vary in space and time, the second-order partial derivative/effective acceleration of the property is equal to the velocity of the wave squared times the second-order sum of the spatial derivative/divergence of the spatial gradient of the property.
[iv] Source for equations: David J. Griffiths, Introduction to Electrodynamics, 3rd ed. (USA: Upper Saddle River, NJ: Prentice Hall, 1999), p. 376, eqn. 9.41 and 9.42.
[v] Ibid.
[vi] The keen observer will note this is not exactly the same as the standard wave equation mentioned earlier, but there are many similarities, so let us note the differences. Aside from all the physics constants (h is Planck’s constant, i is the imaginary number, V is an applied potential that can vary in space and time, and μ is the reduced mass of the particle being described), we see the main differences are that the time derivative is first order instead of second and that there is an applied potential term. The main effect of the time part being first order is that that the time part of the solution is a simple exponential, but since there is also an imaginary number, they normally are still oscillatory solutions.
The cool thing with the Schrödinger equation is that depending on what you put in for the V term, you get different kinds of waves. For a simple bound potential (called the particle in a box), you actually get the same free standing waves you would get for a guitar string! You can put a more complicated function in and get something called a harmonic oscillator. If you put in the actual electric potential caused by the protons in the nucleus of an atom, you get the spherical harmonics as discussed.
Source for equation: Stephen Gasiorowicz, Quantum Physics, 3rd ed. (John Wiley & Sons, 2003), p. 31, eqn. 2-23.
[vii] The interesting fact is the spherical harmonic orbitals actually do not overlap in the sense that an electron in one orbital has no likelihood of being in the wave state of another spherical harmonic at a given time (aside from electrons having different spin, but that is just an extra thing to account for that does not affect things too much).
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Dr. Adam F. Hannon is a data scientist for the company FraudScope where he uses advanced algorithms to aid in the detection and prevention of health care fraud. Before working there, he was a postdoctoral research associate at the National Institute of Standards and Technology, where he conducted research in simulating the self-assembly of complex polymer systems (particularly block copolymer blends) and incorporating advanced inverse algorithms and physics based models into X-ray characterization techniques. He obtained his doctor of science (ScD) degree in materials science and engineering from the Massachusetts Institute of Technology and BS degrees in both physics and polymer & fiber engineering from the Georgia Institute of Technology.
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81c745c0c817695e | How to play mathematics
The world is full of mundane, meek, unconscious things embodying fiendishly complex mathematics. What can we learn from them?
Cold and calculating. A Dorid nudibranch (Tritoniella belli) in Antarctica. Photo by Norbert Wu/Minden/National Geographic
Margaret Wertheim writes about the cultural resonances of science and mathematics. Her books include The Pearly Gates of Cyberspace (1999) and Physics on the Fringe (2012). She also creates art and science projects, including Crochet Coral Reef, which has been exhibited at the Hayward Gallery, the Smithsonian, and elsewhere. She lives in Los Angeles.
4,200 words
What does it mean to know mathematics? Since maths is something we teach using textbooks that demand years of training to decipher, you might think the sine qua non is intelligence – usually ‘higher’ levels of whatever we imagine that to be. At the very least, you might assume that knowing mathematics requires an ability to work with symbols and signs. But here’s a conundrum suggesting that this line of reasoning might not be wholly adequate. Living in tropical coral reefs are species of sea slugs known as nudibranchs, adorned with flanges embodying hyperbolic geometry, an alternative to the Euclidean geometry that we learn about in school, and a form that, over hundreds of years, many great mathematical minds tried to prove impossible.
Sea slugs have at least the rudiments of brains; they generally possess a few thousand neurons, whose large size has made these animals a model organism for scientists studying basic neuronal functioning. This tiny number isn’t nearly enough to enable the slug to formulate any representation of abstract signs, let alone an ability to mentally manipulate them, and yet, somehow, a nudibranch materialises in the fibres of its very being a form that genius-level human mathematicians didn’t discover until the 19th century; and when they did, it nearly drove them mad. In this instance, complex brains were an impediment to understanding.
Nature’s love affair with hyperbolic geometry dates to at least the Silurian age, more than 400 million years ago, when sea floors of the early Earth were covered in vast coral reefs. Many species of corals, then and now, also have hyperbolic structures, which we immediately recognise by the frills and crenellations of their forms. Although corals are animals, they have only very simple nervous systems and can’t be said to have a brain. A head of coral is actually a colonial organism made up of thousands of individual polyps growing together; collectively, they grow a vascular system, a respiratory system and a crude gastrointestinal system through which all the individuals of the colony eat and breathe and share nutrients. Nothing like a brain exists, and yet the colony can organise itself into a mathematical surface disallowed by Euclid’s axiom about parallel lines. Strike two against ‘higher intelligence’.
Ask any fifth-grader what the angles of a triangle add up to, and she’ll say: ‘180 degrees’. That isn’t true on a hyperbolic surface. Ask our fifth-grader what’s the circumference of a circle and she’ll say: ‘2π times the radius’. That’s also not true on a hyperbolic surface. Most of the geometric rules we’re taught in school don’t apply to hyperbolic surfaces, which is why mathematicians such as Carl Friedrich Gauss were so disturbed when finally forced to confront the logical validity of these forms, and hence their mathematical existence. So worried was Gauss by what he was discovering about hyperbolic geometry that he didn’t publish his research on the subject: ‘I fear the howl of the Boetians if I make my work known,’ he confided to a friend in 1829. To their universal horror, other mathematicians soon converged on the same conclusion and the genie of non-Euclidean geometry was let loose.
But can we say that sea slugs and corals know hyperbolic geometry? I want to argue here that in some sense they do. Absent the apparatus of rationalisation and without the capacity to form mental representations, I’d like to postulate that these humble organisms are skilled geometers whose example has powerful resonances for what it means for us humans to know maths – and also profound implications for teaching this legendarily abstruse field.
I’m not the first person to have considered the mathematical capacities of non-sentient things. Towards the end of Richard Feynman’s life, the Nobel Prize-winning physicist is said to have become fascinated by the question of whether atoms are ‘thinking’. Feynman was drawn to this deliberation by considering what electrons do as they orbit the nucleus of an atom. In the earliest days of atomic science, atoms were conceived as little solar systems with the electrons orbiting in simple paths around their nuclei much as a planet revolves around its sun. Yet in the 1920s, it became evident that something much more mathematically complex was going on; in fact, as an electron buzzes around its nucleus, the shape it makes is like a diffused cloud. The simplest electron clouds are spherical, others have dumbbell and toroidal shapes. The form of each cloud is described by what’s called a Schrödinger equation, which gives you a map of where it’s possible for the electron to be in space.
Schrödinger equations (after the pioneering quantum theorist Erwin Schrödinger and his hypothetical cat), are so complicated that, when Feynman was alive, the best supercomputers could barely simulate even the simplest orbits. So how could a brainless electron be effortlessly doing what it was doing? Feynman wondered if an electron was calculating its Schrödinger equation. And what might it mean to say that a subatomic particle is calculating?
Electrons don’t follow mathematical instructions any more than Jimi Hendrix followed a musical score
The world is full of mundane, meek, unconscious things materially embodying fiendishly complex pieces of mathematics. How can we make sense of this? I’d like to propose that sea slugs and electrons, and many other modest natural systems, are engaged in what we might call the performance of mathematics. Rather than thinking about maths, they are doing it. In the fibres of their beings and the ongoing continuity of their growth and existence they enact mathematical relationships and become mathematicians-by-practice. By looking at nature this way, we are led into a consideration of mathematics itself not through the lens of its representational power but instead as a kind of transaction. Rather than being a remote abstraction, mathematics can be conceived of as something more like music or dancing; an activity that takes place not so much in the writing down as in the playing out.
Music gives us a rich analogy by which to consider the idea of mathematics as performance, for you don’t need to be able to write down music to be a musician. Maybe if you want to play Mozart, but not in many other cases. Most folk music throughout history has been created by people who are sonically illiterate. Elvis Presley, Michael Jackson, Eric Clapton and Jimi Hendrix all claimed not to read music. In a British TV interview, Paul McCartney said: ‘As long as the two of us know what we’re doing, ie, John and I, we know what chords we’re playing and we remember the melody, we don’t actually ever have the need to write it down or read it.’
Indian classical music, easily as complex as the Western classical cannon, is based on ragas that were generally transmitted aurally from master to student, not traditionally written down. In this millennia-old practice, music is recognised as an innately mathematical form: the Sanskrit word prastara means the ‘study of mathematically arranging’ ragas and rhythms into pleasing compositions. Ragas certainly can be written down (indeed, Indian musical notation dates back more than 2,000 years), and mathematics can be notated, but it doesn’t have to be. There are lots of things doing maths without a formal script, and I’d argue that it makes no sense to say that electrons or sound waves are following mathematical instructions any more than it makes sense to say that Jimi Hendrix was following a musical score. The possibility of writing down music is something apart from its performance, and maths can be considered in a similar way. In short, the notation isn’t the act.
Among my favourite mathematical performers are holograms, which enact a gorgeous operation called the Fourier transform. This extraordinarily complex, elegant equation is named in honour of Joseph Fourier, a mathematician and physicist who advised Napoleon and discovered what we now call the greenhouse effect (he called it the ‘hotbox’ effect). The Fourier transform has been called the most useful piece of mathematics of all time; you rely on its power every time you make a cellphone call or listen to a piece of digitally recorded music. Music synthesis also results from clever applications of Fourier’s equations. We’ll get to the audio part in a moment, but first let’s look at the visual face of this mathematical marvel.
Holograms differ from photographs in a fundamental way: a photo captures a two-dimensional rendering of light and shade and colour, like a very detailed painting; meanwhile, when light shines through a holographic plate, it assembles into a three-dimensional replica of the original object, recreating in light a simulacra of that thing. The image you see with a hologram is sculptural, really occupying 3D space, so you can move around and view it from different angles. Yet when you look at a holographic plate, there’s no image at all, just a blur in which you may be able to discern speckled rings and dots. What’s been captured on the plate is the Fourier transform of the object, which encodes more information and a different kind of information than a photo can.
Every object has a Fourier transform, and in theory we could calculate the transform of any object we desire and make a holographic plate to generate its form even though an actual physical object never existed. The emerging field of computer-generated holography (CGH) is trying to do just this. If it can be made to work, it will revolutionise computer games and animation; we’d be able to watch whole movies akin to the marvellous holographic projection of Princess Leia in the original Star Wars film.
Calculating transforms for complex objects requires vast computational powers and skills as yet unachieved by human CGH practitioners. Nonetheless, simple chemicals interacting with light on a piece of film manage to enact Fourier transforms of complicated scenes. Acting together, wave fronts of light and atoms execute a beautiful piece of mathematical encoding, and when the light plays back through the film they do the de-encoding. As such, where a photograph is a representation, a hologram is a performance.
Fourier came to his equation in the early 1800s, not to describe images (the origin of holograms dates to the 1940s), but to describe heat flow, and it turns out that his mathematics also leads to enormously powerful applications in the audio domain. Why does a piece by Mozart sound so different when played on a flute or a violin? One way of explaining it is that, although both instruments are playing the same sequence of notes, the Fourier transform of the sound produced by each one is different. The transform reveals the sonic DNA of the instrument’s sound, giving us a precise description of its harmonic components (formally, it describes the set of pure sine waves that make up the sound.) With software, audio engineers can analyse the transform of a musical recording and tell you what kind of instrument was playing; moreover, they can tweak the transform to bring out qualities they like and filter out ones they don’t. By fiddling with the maths, one can sculpt the sound to suit particular tastes.
Calculating Fourier transforms of sounds is a lot easier than calculating the transforms of visual scenes, and software engineers have created programs to simulate musical instruments (eg, Apple’s GarageBand), effectively giving users a sim-orchestra on their laptops for the price of an app. Advances in Fourier-based sound simulation have revolutionised the economics of the music business including movie scoring. Now you don’t need an actual orchestra to produce stirring strings to accompany a heroine’s triumph, you can conjure them from the virtual depths, generated through mathematics.
From my perspective, even the chairs can be said to be participating in the mathematical performance enacted in a concert hall
While music synthesis demonstrates how we can employ mathematics to create something powerful out of a vacuum, here I’m more interested in what happens in actual concert halls. Great halls have their own unique ‘sound’, with each room acting as a filter for the music, tweaking and sculpting its Fourier transform. Contemporary acoustic engineers use Fourier techniques when designing new concert halls, manipulating the architecture of the space, for example adding baffles in specific places, all aided by software that simulates how sounds will react within the space. If the engineers do their job well, there will be no ‘dead spots’ and the hall will sing with warmth and resonance. Here we have a mathematical performance between the sound waves, the architecture, and the surfaces of the walls.
Some music schools now have electronic ‘practice rooms’ where, through software, you can dial up a Fourier-based simulation of a cathedral or a tin shed and hear what your playing would sound like in different spaces. However, music connoisseurs will tell you that no sim is a substitute for physical reality, which is why revered concert halls, such as Vienna’s Musikverein, or New York’s Carnegie, won’t be replaced by software any time soon. It’s interesting that most of the best-rated halls were built before 1901, a fact that the acoustic legend Leo Beranek has attributed to their lack of fancy architecture (their resolutely shoe-box shape) and their lightly upholstered seats. From the perspective I’m adopting, even the chairs can be said to be participating in the mathematical performance enacted in a concert hall. Score another home run for non-sentience.
Since at least the time of Pythagoras and Plato, there’s been a great deal of discussion in Western philosophy about how we can understand the fact that many physical systems have mathematical representations: the segmented arrangements in sunflowers, pine cones and pineapples (Fibonacci numbers); the curve of nautilus shells, elephant tusks and rams horns (logarithmic spiral); music (harmonic ratios and Fourier transforms); atoms, stars and galaxies, which all now have powerful mathematical descriptors; even the cosmos as a whole, now represented by the equations of general relativity. The physicist Eugene Wigner has termed this startling fact ‘the unreasonable effectiveness of mathematics’. Why does the real world actualise maths at all? And so much of it? Even arcane parts of mathematics, such as abstract algebras and obscure bits of topology often turn out to be manifest somewhere in nature. Most physicists still explain this by some form of philosophical Platonism, which in its oldest form says that the universe is moulded by mathematical relationships which precede the material world. To Platonists, matter is literally in-formed, and guided by, a pre-existing set of mathematical ideals.
In the Platonic way of seeing, matter (the stuff of everything) is rendered inert, stripped of power and subordinated to ethereal mathematical laws. These laws are given ontological primacy with matter being effectively a sideline to the ‘true reality’ of the equations. Over the past half-century, this vision has been updated somewhat because now matter, or subatomic particles, have themselves been enfolded into the equations. Matter has been replaced by fields – as in electric and magnetic fields – and now it’s the fields that follow the laws. Still, it’s the laws that retain primacy and power; hence the obsession with finding an ultimate law, a so-called ‘theory of everything’.
Platonism has always bothered me as a philosophy in part because it’s a veiled form of theology – mathematics replaces God as the transcendent, a priori power – so if we want to articulate an alternative, we need new ways of interpreting mathematics itself that don’t also slip into deistic modes. Thinking about maths as performative points a way forward, while also offering a powerful pedagogic model.
Corals and sea slugs construct hyperbolic surfaces and it turns out that humans can also make these forms using iterative handicrafts such as knitting and crochet – you can do non-Euclidean geometry with your hands. To crochet a hyperbolic structure, one just increases stitches at a regular rate by following a simple algorithm: ‘Crochet n stitches, increase one, repeat ad infinitum.’ By increasing stitches, you increase the amount of surface area in a regular way, visually moving from a flat or Euclidean plane into a ruffled formation that models the ‘hyperbolic plane’. Mathematically speaking, the hyperbolic plane is the geometric opposite of the sphere: where the surface of a sphere curves towards itself at every point, a hyperbolic surface curves away from itself. We can define these different surfaces in terms of their curvature: a Euclidean plane has zero curvature (it’s flat everywhere), a sphere has positive curvature, and a hyperbolic plane has negative curvature. In this sense, it is a geometric analogue of a negative number.
Knitting, crochet and weaving were the original digital technologies: their algorithmic ‘patterns’ are literally written in code
Just as geometric relationships on a sphere are different to those on a flat plane – think of what you know about the surface of the Earth versus a flat piece of paper – so, they are different again on a hyperbolic surface. Whereas on a flat plane the angles of a triangle add up to 180 degrees, on a sphere they add up to more, and on a hyperbolic surface they add up to less. It’s hard to appreciate this abstractly when you learn it from textbooks, as I did at university, but you can demonstrate it materially on a crocheted hyperbolic plane by stitching triangles onto the surface. You can also demonstrate visually that parallel lines diverge and other apparent absurdities. If Gauss had known how to crochet he mightn’t have been driven so bonkers.
Crochet Coral Reef by Margaret and Christine Wertheim and the Institute For Figuring. Photo © IFF
It took a woman, the mathematician Daina Taimina at Cornell University, to discover hyperbolic crochet and to give mathematicians a tangible model of this form. I have conducted workshops about this with women all over the world delighting in how much geometry can be conveyed through acts of making. There’s also a link here with general relativity, because the discovery of the hyperbolic plane opened up a whole new era in geometric thinking, leading ultimately to generalised Riemannian geometry, which can describe any complexly curved surface, and is the mathematics underlying Albert Einstein’s equations for the cosmos.
Via handicrafts, we can introduce people to concepts about curved spacetime and multidimensional manifolds, leading with our fingers, and out to questions about measuring the structure of the cosmic whole. We can see this as a form of ‘digital intelligence’, and it’s worth noting that iterated handicrafts (knitting, crochet, weaving) were the original digital technologies: their algorithmic ‘patterns’ are literally written in code. It’s no coincidence that computer punch cards were derived from the cards used in automated looms. Here, knowing emerges from hands performing mathematics: it is a kind of embodied figuring.
People talk about playing music but mathematics can also be a form of play. One way of thinking about maths is as a language of pattern and form, so when you play with patterns you are doing maths. A beautiful example of mathematical pattern-play can be seen with the great Islamic mosaicists who decorated mosques and palaces such as the Alhambra Palace in Granada in Spain with intricate tilings whose mathematical complexities are still a source of wonder.
Long before European geometers realised that there are only 17 mathematically distinct tessellations of the plane – different ways of filling an area with a regular tiling pattern – medieval mosaicists working with their hands using the Hasba method knew about them all. Moreover, medieval Islamic tilers had also discovered aperiodic tiling, which is a way of filling a plane where the pattern never repeats. Western mathematicians discovered these tilings only in the 1960s, again after centuries of theorising that such patterns were impossible. One of the magical qualities of aperiodic tilings is that they look simultaneously random and regular; as a geometric form of chaos, they are rule-based yet inherently unpredictable.
Mosaic tiling from the tomb of Hafez in Shiraz. Courtesy Wikipedia
At first, when Western mathematicians (Sir Roger Penrose among them) discovered aperiodic tilings, these formations were thought to be just a mathematical curiosity; like hyperbolic surfaces, they seemed to defy common sense so that no one imagined such things could be present in the physical world. Prejudice was so intense that when the Israeli chemist Dan Shechtman announced in 1982 that he’d created a new type of crystal with an aperiodic structure, many fellow scientists refused to believe him. (Like Gauss, he too delayed publishing because of the supposedly absurd nature of his claims.) Shechtman’s quasicrystals have brought about a paradigm shift in crystallography, in part because now we know that crystals can be chaotic, exhibiting order without repetition.
Aperiodic ‘Penrose’ tiling pattern. Courtesy Wikipedia
Lewis Carroll would have had a field day with this concept, which calls to mind the Red Queen’s exhortation to Alice that, with practice, one can ‘believe six impossible things before breakfast’. In 2009, after an intense search, a naturally occurring example of an aperiodic crystal was also found in the mineral icosahedrite. Strike three against intelligence as a prerequisite for doing mathematics.
Image of an aluminium-palladium-manganese quasicrystal surface. Courtesy Wikipedia
As a nice coda to this story, in 2011 Shechtman was awarded the Nobel Prize in chemistry.
Proof that studying equations isn’t the only path to mathematical insight also comes to us from Africa, where craftsmen discovered fractals centuries ago. A wide variety of fractal patterns are incorporated into African textiles, hairstyling, metalwork, sculpture, painting and architecture. One marvellous Ba-Ila village in southern Zambia is laid out in a fractal design reminiscent of the Mandelbrot set, that swirling icon of 1990s computer-graphic cool. In his book African Fractals: Modern Computing and Indigenous Design (1999), the mathematician Ron Eglash traces the story of the southern continent’s priority in a branch of geometry that came into Western consciousness only around the turn of the 20th century, and didn’t really flourish here until the development of computer graphics chips.
Fractal model for Ba-Ila village. From ‘African Fractals: Modern Computing and Indigenous Design’ by Ron Eglash
First three iterations of fractal model for Ba-Ila village. From ‘African Fractals: Modern Computing and Indigenous Design’ by Ron Eglash
Sea slugs do maths, electrons do maths, minerals do maths. Rainbows do an incredible mathematical performance when you take into account the primary and secondary bows, the dark band between them, and the red and green arcs of light under the primary bow. Next time you see a good rainbow, stop and take a look at the space around it, there’s so much going on; classical geometric optics doesn’t begin to capture its complexity. A stunning piece of mathematical performance is enacted by a peregrine falcon as it hurtles towards its prey; with its head held straight so it can fix one eye steadily on the quarry at a constant angle of 40 degrees, it swoops down at 200 mph in a perfect logarithmic spiral. Leonhard Euler’s 18th-century formula, with its unique mathematical properties, is enacted here by a bird.
All around us, nature is playing mathematical games and we too can join in the fun. Mathematics need not be taught as an abstraction, it can be approached as an embodied practice, like learning a musical instrument. This doesn’t invalidate what goes on in university classrooms or academic textbooks, since society needs professional mathematicians who can work with symbols, people such as Fourier and Bernhard Riemann who developed the maths that assists us to make cellphone calls, or determine the structure of the cosmos, and so much else besides. Because nature does so much mathematics, there will probably never be a time when professional ‘symbolising’ isn’t profoundly useful. In 2016, the Nobel prize in physics was awarded for ‘theoretical discoveries about topological phase transitions in matter’ – astonishing, complex work that emerged out of the discovery of another kind of supposedly impossible object (the quasi-particle) and whose mathematical insights might pave the way for quantum computers.
By thinking about mathematics as performance, we liberate it from the straightjacket of abstraction into which it has been too narrowly confined. If you ask professional mathematicians what they love about their work, a likely answer is its beauty. ‘Euclid alone looked on beauty bare’ wrote the poet Edna St Vincent Millay in 1923, while the mathematician André Weil (brother of Simone) claimed that solving a hard mathematical problem topped sexual pleasure.
The professionals know that mathematics swings; they delight in its playfulness, the plasticity of its forms, and (after some initial shock) the absurdities it throws up. Hyperbolic surfaces, aperiodic tilings, Möbius strips, negative numbers and zero all generated alarm at first, yet were ultimately embraced as gateways to new continents of mathematical wonder.
You don’t have to be a symbol-expert to appreciate this terrain. Just as humans are endowed with an ability to dance and play music (even if education too often crushes this out of us), so we have innate form-making and pattern-playing proclivities. Sea slugs, sound waves and falcons do mathematics; Islamic mosaicists and African architects do it too. So can you.
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9dac2b321295c656 | Theoretical and experimental justification for the Schrödinger equation
From Wikipedia, the free encyclopedia
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The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.
This article is at a postgraduate level. For a more general introduction to the topic see Introduction to quantum mechanics.
Classical electromagnetic waves[edit]
Nature of light[edit]
The quantum particle of light is called a photon. Light has both a wave-like and a particle-like nature. In other words, light can appear to be made of photons (particles) in some experiments and light can act like waves in other experiments. The dynamics of classical electromagnetic waves are completely determined by Maxwell's equations, the classical description of electrodynamics. In the absence of sources, Maxwell's equations can be written as wave equations in the electric and magnetic field vectors. Maxwell's equations thus describe, among other things, the wave-like properties of light. When "classical" (coherent or thermal) light is incident on a photographic plate or CCD, the average number of "hits", "dots", or "clicks" per unit time that result is approximately proportional to the square of the electromagnetic fields of the light. By formal analogy, the wavefunction of a material particle can be used to find the probability density by taking its absolute-value squared. Unlike electromagnetic fields, quantum-mechanical wavefunctions are complex. (Often in the case of EM fields complex notation is used for convenience, but it is understood that in fact the fields are real. However, wavefunctions are genuinely complex.)
Maxwell's equations were completely known by the latter part of the nineteenth century. The dynamical equations for light were, therefore, well-known long before the discovery of the photon. This is not true for other particles such as the electron. It was surmised from the interaction of light with atoms that electrons also had both a particle-like and a wave-like nature. Newtonian mechanics, a description of the particle-like behavior of macroscopic objects, failed to describe very small objects such as electrons. Abductive reasoning was performed to obtain the dynamics of massive objects (particles with mass) such as electrons. The electromagnetic wave equation, the equation that described the dynamics of light, was used as a prototype for discovering the Schrödinger equation, the equation that describes the wave-like and particle-like dynamics of nonrelativistic massive particles.
Plane sinusoidal waves[edit]
Electromagnetic wave equation[edit]
where c is the speed of light in the medium. In a vacuum, c = 2.998 × 108 meters per second, which is the speed of light in free space.
The magnetic field is related to the electric field through Faraday's law (cgs units)
Plane wave solution of the electromagnetic wave equation[edit]
The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units)
Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarised wave propagating from left to right.
for the electric field and
for the magnetic field, where k is the wavenumber,
is the angular frequency of the wave, and is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. In complex notation, the quantity is the amplitude of the wave.
is the Jones vector in the x-y plane. The notation for this vector is the bra–ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector. The angles are the angle the electric field makes with the x axis and the two initial phases of the wave, respectively.
The quantity
is the state vector of the wave. It describes the polarization of the wave and the spatial and temporal functionality of the wave. For a coherent state light beam so dim that its average photon number is much less than 1, this is approximately equivalent to the quantum state of a single photon.
Energy, momentum, and angular momentum of electromagnetic waves[edit]
Energy density of classical electromagnetic waves[edit]
Energy in a plane wave[edit]
For a plane wave, converting to complex notation (and hence dividing by a factor of 2), this becomes
Fraction of energy in each component[edit]
The fraction of energy in the x component of the plane wave (assuming linear polarization) is
with a similar expression for the y component.
The fraction in both components is
Momentum density of classical electromagnetic waves[edit]
The momentum density is given by the Poynting vector
The momentum density has been averaged over a wavelength.
Angular momentum density of classical electromagnetic waves[edit]
The angular momentum density is
For a sinusoidal plane wave the angular momentum is in the z direction and is given by (going over to complex notation)
where again the density is averaged over a wavelength. Here right and left circularly polarized unit vectors are defined as
Unitary operators and energy conservation[edit]
A wave can be transformed by, for example, passing through a birefringent crystal or through slits in a diffraction grating. We can define the transformation of the state from the state at time t to the state at time as
To conserve energy in the wave we require
This implies that a transformation that conserves energy must obey
Hermitian operators and energy conservation[edit]
If is an infinitesimal real quantity , then the unitary transformation is very close to the identity matrix (the final state is very close to the initial state) and can be written
and the adjoint by
The factor of i is introduced for convenience. With this convention, it will be shown that energy conservation requires H to be a Hermitian operator and that H is related to the energy of a particle.
Energy conservation requires
Since is infinitesimal, which means that may be neglected with respect to , the last term can be omitted. Further, if H is equal to its adjoint:
it follows that (for infinitesimal translations in time )
so that, indeed, energy is conserved.
Operators that are equal to their adjoints are called Hermitian or self-adjoint.
The infinitesimal translation of the polarization state is
Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator. While this derivation is classical, the concept of a Hermitian operator generating energy-conserving infinitesimal transformations forms an important basis for quantum mechanics. The derivation of the Schrödinger equation follows directly from this concept.
Quantum analogy of classical electrodynamics[edit]
The treatment to this point has been classical. However, the quantum mechanical treatment of particles follows along lines formally analogous however, to Maxwell's equations for electrodynamics. The analog of the classical "state vectors"
in the classical description is quantum state vectors in the description of photons.
Energy, momentum, and angular momentum of photons[edit]
The early interpretation is based on the experiments of Max Planck and the interpretation of those experiments by Albert Einstein, which was that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation
where is an experimentally determined quantity known as the reduced Planck's constant. If there are photons in a box of volume , the energy (neglecting zero point energy) in the electromagnetic field is
and the energy density is
The average number of photons in the box in a coherent state is then
which implies that the momentum of a photon is
(or equivalently ).
Angular momentum and spin[edit]
Similarly for the angular momentum
which implies that the angular momentum of the photon is
the quantum interpretation of this expression is that the photon has a probability of of having an angular momentum of and a probability of of having an angular momentum of . We can therefore think of the angular momentum of the photon being quantized as well as the energy. This has indeed been experimentally verified. Photons have only been observed to have angular momenta of .
Spin operator[edit]
The spin of the photon is defined as the coefficient of in the angular momentum calculation. A photon has spin 1 if it is in the state and -1 if it is in the state. The spin operator is defined as the outer product
The expected value of a spin measurement on a photon is then
An operator S has been associated with an observable quantity, the angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for angular momentum, but it is in general true for any observable quantity.
Probability for a single photon[edit]
There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation is applicable to thermal or to coherent light (see Quantum optics). The latter interpretation is the option for a single-photon Fock state. Dirac explains this [Note 1] in the context of the double-slit experiment:
Probability amplitudes[edit]
The probability for a photon to be in a particular polarization state depends on the probability distribution over the fields as calculated by the classical Maxwell's equations (in the Glauber-Sudarshan P-representation of a one-photon Fock state.) The expectation value of the photon number in a coherent state in a limited region of space is quadratic in the fields. In quantum mechanics, by analogy, the state or probability amplitude of a single particle contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: (The following quote is from Baym, Chapter 1)
1. The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. ...
de Broglie waves[edit]
Louis de Broglie. De Broglie received the Nobel Prize in Physics in 1929 for his identification of waves with particles.
In 1923 Louis de Broglie addressed the question of whether all particles can have both a wave and a particle nature similar to the photon. Photons differ from many other particles in that they are massless and travel at the speed of light. Specifically de Broglie asked the question of whether a particle that has both a wave and a particle associated with it is consistent with Einstein's two great 1905 contributions, the special theory of relativity and the quantization of energy and momentum. The answer turned out to be positive. The wave and particle nature of electrons was experimentally observed in 1927, two years after the discovery of the Schrödinger equation.
de Broglie hypothesis[edit]
De Broglie supposed that every particle was associated with both a particle and a wave. The angular frequency and wavenumber of the wave was related to the energy E and momentum p of the particle by
The question reduces to whether every observer in every inertial reference frame can agree on the phase of the wave. If so, then a wave-like description of particles may be consistent with special relativity.
Rest frame[edit]
First consider the rest frame of the particle. In that case the frequency and wavenumber of the wave are related to the energy and momentum of the particles properties by
where m is the rest mass of the particle.
This describes a wave of infinite wavelength and infinite phase velocity
The wave may be written as proportional to
This, however, is also the solution for a simple harmonic oscillator, which can be thought of as a clock in the rest frame of the particle. We can imagine a clock ticking at the same frequency as the wave is oscillating. The phases of the wave and the clock can be synchronized.
Frame of the observer[edit]
It is shown that the phase of the wave in an observer frame is the same as the phase of the wave in a particle frame, and also the same as clocks in the two frames. There is, therefore, consistency of both a wave-like and a particle-like picture in special relativity.
Phase of the observer clock[edit]
In the frame of an observer moving at relative speed v with respect to the particle, the particle clock is observed to tick at a frequency
is a Lorentz factor that describes time dilation of the particle clock as observed by the observer.
The phase of the observer clock is
where is time measured in the particle frame. Both the observer clock and the particle clock agree on the phase.
Phase of the observer wave[edit]
The frequency and wavenumber of the wave in the observer frame is given by
with a phase velocity
The phase of the wave in the observer frame is
The phase of the wave in the observer frame is the same as the phase in the particle frame, as the clock in the particle frame, and the clock in the observer frame. A wave-like picture of particles is thus consistent with special relativity.
In fact, we now know that these relations can be succinctly written using special relativistic 4-vector notation:
The relevant four-vectors are:
The relations between the four-vectors are as follows:
The phase of the wave is the relativistic invariant:
Bohr atom[edit]
Niels Bohr. In 1922 the Nobel Prize in Physics was awarded to Niels Bohr for his contributions to the understanding of quantum mechanics.
Inconsistency of observation with classical physics[edit]
The de Broglie hypothesis helped resolve outstanding issues in atomic physics. Classical physics was unable to explain the observed behaviour of electrons in atoms. Specifically, accelerating electrons emit electromagnetic radiation according to the Larmor formula. Electrons orbiting a nucleus should lose energy to radiation and eventually spiral into the nucleus. This is not observed. Atoms are stable on timescales much longer than predicted by the classical Larmor formula.
Also, it was noted that excited atoms emit radiation with discrete frequencies. Einstein used this fact to interpret discrete energy packets of light as, in fact, real particles. If these real particles are emitted from atoms in discrete energy packets, however, must the emitters, the electrons, also change energy in discrete energy packets? There is nothing in Newtonian mechanics that explains this.
The de Broglie hypothesis helped explain these phenomena by noting that the only allowed states for an electron orbiting an atom are those that allow for standing waves associated with each electron.
Balmer series[edit]
The Balmer series identifies those frequencies of light that can be emitted from an excited hydrogen atom:
where R is known as the Rydberg constant and is equal to 13.6 electron volts.
Assumptions of the Bohr model[edit]
The Bohr model, introduced in 1913, was an attempt to provide a theoretical basis for the Balmer series. The assumptions of the model are:
1. The orbiting electrons existed in circular orbits that had discrete quantized energies. That is, not every orbit is possible but only certain specific ones.
2. The laws of classical mechanics do not apply when electrons make the jump from one allowed orbit to another.
3. When an electron makes a jump from one orbit to another the energy difference is carried off (or supplied) by a single quantum of light (called a photon) which has an energy equal to the energy difference between the two orbitals.
4. The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L according to the equation
Where n = 1,2,3,… and is called the principal quantum number.
Implications of the Bohr model[edit]
In a circular orbit the centrifugal force balances the attractive force of the electron
where m is the mass of the electron, v is the speed of the electron, r is the radius of the orbit and
where e is the charge on the electron or proton.
The energy of the orbiting electron is
which follows from the centrifugal force expression.
The angular momentum assumption of the Bohr model implies
which implies that, when combined with the centrifugal force equation, the radius of the orbit is given by
This implies, from the energy equation,
The difference between energy levels recovers the Balmer series.
De Broglie's contribution to the Bohr model[edit]
The Bohr assumptions recover the observed Balmer series. The Bohr assumptions themselves, however, are not based on any more general theory. Why, for instance, should the allowed orbits depend on the angular momentum? The de Broglie hypothesis provides some insight.
If we assume that the electron has a momentum given by
as postulated by the de Broglie hypothesis, then the angular momentum is given by
where is the wavelength of the electron wave.
If only standing electron waves are permitted in the atom then only orbits with perimeters equal to integral numbers of wavelengths are allowed:
This implies that allowed orbits have angular momentum
which is Bohr's fourth assumption.
Assumptions one and two immediately follow. Assumption three follows from energy conservation, which de Broglie showed was consistent with the wave interpretation of particles.
Need for dynamical equations[edit]
The problem with the de Broglie hypothesis as applied to the Bohr atom is that we have forced a plane wave solution valid in empty space to a situation in which there is a strong attractive potential. We have not yet discovered the general dynamic equation for the evolution of electron waves. The Schrödinger equation is the immediate generalization of the de Broglie hypothesis and the dynamics of the photon.
Schrödinger equation[edit]
Analogy with photon dynamics[edit]
The dynamics of a photon are given by
where H is a Hermitian operator determined by Maxwell's equations. The Hermiticity of the operator ensures that energy is conserved.
Erwin Schrödinger assumed that the dynamics for massive particles were of the same form as the energy-conserving photon dynamics.
where is the state vector for the particle and H is now an unknown Hermitian operator to be determined.
Particle state vector[edit]
Rather than polarization states as in the photon case, Schrödinger assumed the state of the vector depended on the position of the particle. If a particle lives in one spatial dimension, then he divided the line up into an infinite number of small bins of length and assigned a component of the state vector to each bin
The subscript j identifies the bin.
Matrix form and transition amplitudes[edit]
The transition equation can be written in matrix form as
The Hermitian condition requires
Schrödinger assumed that probability could only leak into adjacent bins during the small time step dt. In other words, all components of H are zero except for transitions between neighboring bins
Moreover, it is assumed that space is uniform in that all transitions to the right are equal
The same is true for transitions to the left
The transition equation becomes
The first term on the right side represents the movement of probability amplitude into bin j from the right. The second term represents leakage of probability from bin j to the right. The third term represents leakage of probability into bin j from the left. The fourth term represents leakage from bin j to the left. The final term represents any change of phase in the probability amplitude in bin j.
If we expand the probability amplitude to second order in the bin size and assume space is isotropic, the transition equation reduces to
Schrödinger equation in one dimension[edit]
Probability densities for the electron at different quantum numbers in the hydrogen atom.
The transition equation must be consistent with the de Broglie hypothesis. In free space the probability amplitude for the de Broglie wave is proportional to
in the non-relativistic limit.
The de Broglie solution for free space is a solution of the transition equation if we require
The time derivative term in the transition equation can be identified with the energy of the de Broglie wave. The spatial derivative term can be identified with the kinetic energy. This suggests that the term containing is proportional to the potential energy. This yields the Schrödinger equation
where U is the classical potential energy and
Schrödinger equation in three dimensions[edit]
In three dimensions the Schrödinger equation becomes
Hydrogen atom[edit]
The solution for the hydrogen atom describes standing waves of energy exactly given by the Balmer series. This was a spectacular validation of the Schrödinger equation and of the wave-like behaviour of matter.
See also[edit]
1. ^ This explanation is in some sense antiquated or even obsolete, as we now know that the concept of a single-photon wavefunction is disputed [1], that in a coherent state one indeed deals with the probable number of photons, given by coherent-state Poissonian statistics, and that different photons can indeed interfere[2].
• Baym, Gordon (1969). Lectures on Quantum Mechanics. W. A. Benjamin. ISBN 978-0805306675.
• Dirac, P. A. M. (1958). The Principles of Quantum Mechanics (Fourth ed.). Oxford. ISBN 0-19-851208-2. |
e8065e7e9b10ded6 | Schrödinger Equation and Scatter Equation
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The Spatial part of the Time-Independent Schrödinger Equation (TISE) is
\left(- \frac {\hbar^2}{2m} \nabla^2 +V(r) \right) \psi(r) = E \psi(r)
by setting
k^2 = 2m E / \hbar^2 and U(r) = 2 m V(r) /\hbar^2
the equation becomes a wave equation with a source, or scattering equation.
(\nabla^2 +k^2) \psi(r) = U(r) \psi(r)
for solving it, we have to find the Green function such that
(\nabla^2 + k^2 ) G(r,r') = \delta(x-x')
The solution is easy ( i will post it later, you can think the Green function is the inverse of the Operator)
G(r,r') = - \frac{1}{4\pi} \frac {Exp( \pm i k \cdot (r-r') )} { |r - r'| }
the particular solution is
\psi_p(r) = \int {G(r,r') U(r') \psi(r') dr'}
plus the homogeneous solution
but it is odd that the solution contain itself!
Optical Model
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In nuclear physics, the Optical Model means, we are treating the scattering problem is like optical wave problem. due to the incident beam can be treated as a wave-function. and this wave will be scattered by the target.
when the beam is far away from the target, the wave function of the incident beam should satisfy the Schrödinger equation in free space :
\left( \frac {\hbar^2 } {2m} \nabla^2 + V(r) \right) \psi( \vec{r} ) = E \psi ( \vec{r} )
and the plane wave solution is
\psi ( \vec{r} ) \sim Exp ( \pm i \vec{k} \cdot \vec {r} )
after the scattering, there will be some spherical wave come out. the spherical wave should also satisfy the free-space Schrödinger equation.
\psi( \vec{r} ) \sim Y(\theta, \phi) \frac {Exp( \pm i \vec{k} \cdot \vec{r} ) }{r}
Thus, the process of scattering can be think in this way:
where f(θ) is a combination of spherical wave.
one consequence of using Optical Model is, we use complex potential to describe the nuclear potential terms in quantum mechanics.
when using a complex potential, the flux of the incident beam wave function can be non-zero. meanings that the particles in the beam are being absorbed or emitted. This corresponding to the inelastic scattering.
The reason for the “OPTICAL” is come form the permittivity and permeability of the EM field. for metallic matter, their permittivity or permeability may have a imaginary part. and this imaginary part corresponding to the absorption of the light. so, nuclear physics borrow the same idea.
the flux is defined as:
J = \frac { \hbar }{ 2 i m} ( \psi^*(r) \nabla \psi(r) - \psi(r) \nabla \psi^* (r) )
and the gradient of the flux, which is the absorption (sink) or emission ( source ) is:
\nabla J = \frac {\hbar }{ 2 i m }( \psi^* \nabla^2 \psi - \psi \nabla^2 \psi^* )
The Schrödinger equation gives the equation for the wave function:
\nabla^2 \psi(r) = \frac { 2m} {\hbar^2} ( E - V(r)) \psi(r)
when sub the Schrödinger equation in to the gradient of flux, we have:
\nabla J = \frac {1} {i \hbar } ( V(r) - V^*(r) ) | \psi |^2 = \frac { 2} {\hbar } Im ( V) | \psi |^2
we can see, if the source and the sink depend on the complex part of the potential. if the imaginary part is zero, the gradient of the flux is zero, and the wave function of the beam is conserved.
Larmor Precession (quick)
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Magnetic moment (\mu ) :
\mu = \gamma J
\gamma = g \mu_B
Notice that we are using natural unit.
\mu_B is Bohr magneton, which is equal to
Larmor frequency:
the precession can be understood in classical way or QM way.
Classical way:
solving gives the procession frequency is :
\omega = - \gamma B
QM way:
The Tim dependent Schrödinger equation (TDSE) is :
the solution is
However, the rotation operator on z-axis is
Thus, the solution can be rewritten as:
That makes great analogy on rotation on a real vector.
WKB approximation
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I was scared by this term once before. ( the approach an explanation from J.J. Sakurai’s book is not so good) in fact, don’t panic, it is easy. Let me explain.
i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.
The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in Exp( i k x) , wavelength = 2 π / k, when it expressed in Exp( - \kappa x ) , wavelength = 1/κ.
in general, the wavefunction can be expressed as amplitude and phase:
\Psi(x) = A(x)Exp(i \phi(x))
where A(x) and \phi(x) are real function
sub this into the time-independent Schrödinger equation (TISE)
\Psi '' (x) = - \frac {2 m} {\hbar^2 } ( E - V(x) ) \Psi (x)
\Psi ''(x) = ( A''(x)- A(x) \phi'(x)^2 + 2 i A'(x) \phi'(x)+ i A(x)\phi''(x) ) Exp(i \phi (x) )
and separate the imaginary part and real part.
The imaginary part is can be simplified as:
2 A'(x) \phi '(x) + A(x) \phi ''(x) = 0 = \frac {d}{dx} ( A^2(x) \phi '(x)
A(x) = \frac {const.} {\sqrt {\phi '(x)}}
The real part is
A''(x) = \left ( \phi ''(x) - \frac {2m}{\hbar^2 } ( E - V(x) ) \right) A(x)
we use the approx. that A''(x) = 0 , since it varies slowly.
\phi '(x) = \sqrt { \frac {2m}{\hbar^2} (E - V(x) ) }
\Rightarrow \phi(x) = \int \sqrt { \frac {2m}{\hbar ^2} ( E - V(x ) )} dx
if we set,
for clear display and p(x) is the energy different between energy and the potential. the solution is :
\Psi(x) = \frac {const.}{\sqrt {p(x)}} Exp \left( i \int p(x) dx \right)
Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.
This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.
when the energy is smaller then the potential, than, the wavefunction is under decay.
one direct application of WKB approxi is on the Tunneling effect.
if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to
Exp \left( - 2 \sqrt { \frac {2m}{\hbar ^2 } A_{area} ( V(x) - E )} \right)
Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.
Hydrogen Atom (Bohr Model)
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OK, here is a little off track. But that is what i were learning and learned. like to share in here. and understand the concept of hydrogen is very helpful to understand the nuclear, because many ideas in nuclear physics are borrow from it, like “shell”.
The interesting thing is about the energy level of Hydrogen atom. the most simple atomic system. it only contains a proton at the center, um.. almost center, and an electron moving around. well, this is the “picture”. the fact is, there is no “trajectory” or locus for the electron, so technically, it is hard to say it is moving!
why i suddenly do that is because, many text books said it is easy to calculate the energy level and spectrum for it. Moreover, many famous physicists said it is easy. like Feynman, Dirac, Landau, Pauli, etc… OK, lets check how easy it is.
anyway, we follow the usual say in every text book. we put the Coulomb potential in the Schrödinger equation, change the coordinate to spherical. that is better and easy for calculation because the coulomb potential is spherical symmetric. by that mean, the momentum operator (any one don’t know what is OPERATOR, the simplest explanation is : it is a function of function.) automatically separated into 2 parts : radial and angular part. The angular part can be so simple that it is the Spherical harmonic.
Thus the solution of the “wave function” of the electron, which is also the probability distribution of the electron location, contains 2 parts as well. the radial part is not so trivial, but the angular part is so easy. and it is just Y(l,m) .
if we denote the angular momentum as L, and the z component of it is Lz, thus we have,
L^2 Y(l,m) = l(l+1) \hbar^2 Y(l,m)
L_z Y(l,m) = m \hbar Y(l,m)
as every quadratic operator, there are “ladder” operator for “up” and “down”.
L_\pm Y(l,m) =\hbar \sqrt{l(l+1) - m(m\pm 1)} Y(l,m \pm 1)
which means, the UP operator is increase the z-component by 1, the constant there does not brother us.
it is truly easy to find out the exact form of the Y(l,m) by using the ladder operator. as we know, The z component of the a VECTOR must have some maximum. so, there exist an Y(l,m) such that
L_+ Y(l,m) =0
since there is no more higher z-component.
by solve this equation, we can find out the exact form of Y(l,m) and sub this in to L2, we can knowMax(m) = l . and apply the DOWN operator, we can fins out all Y(l,m) , and the normalization constant is easy to find by the normalization condition in spherical coordinate, the normalization factor is sin(\theta) , instead of 1 in rectangular coordinate.
\int_0^\pi \int_0^{2 \pi} Y^*(l',m') Y(l,m) sin(\theta) d\theta d \psi = \delta_{l' l} \delta_{m' m}
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70cad0aa928e14f1 | indico First event Previous event Thesis defense Next event Last event | view: | manage export to personal scheduler |
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PhD Thesis: Theoretical studies of chemical dynamics on excited states, driven by non-adiabatic effects
Thesis defense
Monday 23 May 2016
from 10:00 to 13:00
at FA32
Speaker : Sifiso Musa Nkambule (Stockholm University, Department of Physics)
Abstract : This thesis is based on theoretical studies of molecular collisions occurring at relatively low to intermediate collision energies. The collisions are called dissociative recombination (DR) and mutual neutralization (MN). In a molecular quantum mechanical picture, both reactions involve many highly excited molecular electronic states that are interacting by non-adiabatic couplings with each other. The molecular complexes involved in the collisions are relatively (diatomic or triatomic systems) composed of relative light atoms. This allows for accurate quantum chemistry calculations and a quantum mechanical description of the nuclear motions. The reactions studied here are the MN reaction in collisions of H++ H-, Li++ F-, and He++ H- and the DR reaction of H2O+. Rotational couplings are investigated in the study of MN reaction for He++ H . For some reactions, the electronic resonant states have to be considered. These are not bound states, but are states interacting with the ionization continuum. Electronic structure calculations are combined with electron scattering calculations to accurately compute potential energy curves for the resonant states involved in the DR of H2O+ and the MN of He++ H. From these calculations, the autoionization widths of the resonant states are also obtained. Once the potential energy curves are computed for the systems, the nuclear dynamics are studied either semi-classically, using the Landau-Zener method or quantum mechanically, employing the time-independent and time-dependant Schrödinger equations. Reaction cross section and final states distribution are computed for all the reactions, showing significantly large cross section at low to intermediate collision energies. For the MN processes, studied here, not only total cross sections are calculated but differential cross sections as well. Where possible, comparisons with previous experimental and theoretical results are performed.
AlbaNova | Last modified 02 May 2016 14:46 | HELP |
38ff9dc0533a0b7f | Optics Research in Qatar Gaining Traction and Entering Collaborative Phase
Lasers running through a medium
Lasers running through a medium
While working for Bloomsbury Qatar Foundation Journals’ QScience media organization from 2011 to 2016, we served QNRF as a publisher of their newsletter. Although credits have not been assigned or retained, I researched, interviewed and wrote this article, and it exists in the QNRF newsletter archives. It is linked out to the archives directly before the following text. Researchers and organizations will attest to my work if contacted.
— Emily Alp
ARCHIVE. Compared to studies in the fields of biology and engineering, nonlinear dynamics might not be so obvious in terms of its worth. In reality, it is an area of physics research that permeates the natural world and a field integral to so many others. Dr. Milivoj Belic won the 2012 QNRF Research Team of the Year Award for his prolific contributions in this field, accounting for more than ten percent of Texas A&M at Qatar’s publications. His team’s specific focus is nonlinear optics, wherein they research the behavior of materials and laser light as they interact.
“What we do is manipulate photons, which are particles of light that can also be considered waves,” Dr. Belic said, “and we consider processes that happen in material when you shine laser light on them. So in essence we play with the wave phenomena. This is under the umbrella of quantum mechanics, but we do not do quantum mechanics; we do nonlinear optics.”
In linear optics photons do not “talk to” each other; however, in nonlinear optics they do, through the medium. Understanding the conversations—through the evolving language of nonlinear equations, i.e., nonlinear dynamics—helps researchers understand the material under study.
“In physics, very few things are done and finished once and for all, at least what has been done within the last century,” Dr. Belic said. “Most of those things are a never-ending story. Bit by bit you discover new things. But the problems and topics of research are there … an immense number of unsolved questions and half-baked answers.”
By running lasers through different types of materials such as gases, photo-refractive crystals, and nematic liquid crystals, Dr. Belic and his team observe the entire system as the light propagates, to get an idea of the material's response and the processes at play. The equations describing these processes are linked to waves and light and also with the response of the material—so it is both the response of the material and the behavior of the laser light, together, that are studied.
“We attack nonlinear equations; so it’s a mathematical physics problem. Now with such equations, it’s not like ‘aha, that’s it, we solved it!’—most often, it cannot be solved, at least not analytically. You have to try something different. Still, for many such equations we found ways to treat them analytically and this is something for which my team is becoming known internationally.”
The mathematical language around many physical phenomena is based on differential equations. A classic example would be the Schrödinger equation, which describes how the state of a quantum system changes over time. This is useful in linear systems and quantum mechanics. However, Dr. Belic explained that nonlinear dynamics is even more challenging than quantum mechanics. Specifically, it involves nonlinear Schrödinger equations and relies heavily on computers to crunch numbers because the responses in nonlinear systems are sometimes so erratic, evading analysis through the equations used in more predictable systems. Interestingly, most natural systems and materials require nonlinear thinking.
“Laws of physics are laws of nature,” Belic explained. “You have to master them and you have to know how to apply them. Mathematics is the language of nature. Things in nature are best explained through mathematics. Physics is essentially applied mathematics. In theoretical physics, you have to reason. But then you have experimental physics, so you have to experiment—to make a model, make predictions and test them. This can also turn out the other way around, where somebody finds something experimentally and then explains it.”
Whereas research in many fields is goal or product oriented, Dr. Belic said his team’s research is often curiosity-driven. A co-evolution of experiment and theory, the research requires a constant striving into the unknown.
“We are always trying to understand things, to contribute to a bank of understanding about nature at the basic level,” Dr. Belic explained. “We don’t produce gadgets—we want to know how they work. Here in Qatar, we had to start from scratch, so we started with theory. Some of our experiments are performed in other places such as Australia, the US, Serbia, France and Germany … we have a lot of collaborators.
“This work could contribute to other fields, not tomorrow, not today but in the foreseeable future,” he continued. “Newton formulated his laws in mathematical terms, and at the time people were asking ‘what is this for?’ It was a hundred years before people realized how useful they were.”
What excites Dr. Belic now is the potential to collaborate with researchers in other fields, enriching findings with the basic knowledge of physics and properties of materials.
“Before, physicians were doing their thing, mathematicians were doing their thing, chemists were doing their thing, and that approach was disjointed. But now we realize that if you want to make progress in brain research you cannot do so by the medical profession alone. For example, one of my collaborators is making a mathematical model of a brain cancer tumor. We all have to work together and that is the idea. And that is really the push nowadays with the funding agencies. Our team would like to go and collaborate with the Qatar Foundation institutes and has begun discussions with many of them.”
Establishing homegrown teams that are capable of producing great research requires a long period of cultivation. Qatar Foundation and TAMUQ have chosen this path and have generously supported the creation of high-quality team-oriented research centers. This turns the spotlight toward Qatar Foundation and TAMUQ as well as the whole Middle Eastern region. We greatly appreciate the strong support we have been given by TAMUQ and QNRF, and look forward to a bright future,” Dr. Belic said.
NPRP 25-6-7-2
Nonlinear Photonics for All-optical Telecommunication and Information Technologies. |
ad801546904480c6 | Competencies for an Undergraduate Degree in Computational Physics
This is a draft of a set of competencies for an undergraduate bachelor degree in computational physics. They have been reviewed by a number of physicists interested in this topic. Not everyone who participated in the review fully agreed with all aspects but this draft represents a reasonable consensus on what students need to know.
The competencies and skills outlined are only for the computational part. We assume that the students will also learn the standard canon although presumably in a somewhat different (i.e. computational, context). People will no doubt have suggestions for changes/additions and deletions. Items marked with a * are optional/advanced topics that may not be included in all programs. The Competencies listed below are appropriate for programs that offer a degree in Computational Physics. Most departments will not be able to do this. Programs that offer a minor or certificate in computational physics will of necessity have to make choices about what topics to omit. Departments that offer only a single course in Computational Physics will have to make hard choices about what to include. However it is sometimes possible to weave computation into the standard undergraduate physics curriculum even when you do not have the luxury of teaching dedicated computational courses.
These competencies are meant to be a guide, not a prescriptive canon. They should be used, as a starting point for thinking about what students should learn from a computational physics program.
In order to be successful either in graduate school or industry a student with a degree in computational physics should be able to tackle unfamiliar computational problems with some degree of confidence and independence. To achieve this requires a skill set that goes beyond simply knowing how to integrate numerically a differential equation. Students should have a skill set (call it "meta-skills") that includes:
• Knowing what can be computed and what cannot.
• The ability to extract physics from the results of a computation. After all, the point of computation is scientific insight not computation.
• Validation and Verification: Is the model being solved correctly? If the model is not exact is it good enough for the problem at hand? Are there special cases that can be checked and have they been checked? How accurate are the results? How accurately do we need the results to be for the problem at hand? How else can we check the code?
• How does the computation scale, i.e. how does the CPU time change as the number of cores, amount and types of memory, and number of CPUs are increased?
• Knowing how to communicate the results in a clear and professional manner.
• Knowing how to document code so that others can understand, use and extend it.
• Knowing how to write modular code and understand why this is important.
• Understanding the importance of "time to science". In other words, when should you spend time rewriting your code to run on a GPU or in parallel or with a faster algorithm, and when should you just be willing just to wait longer for a result.
Students also need what we will call high-level skills. These are less overarching than the "meta-skills" but are broader and conceptually more general than any specific topic or algorithm.
"High-level skills":
• Changing from dimensionless "computational units" to physics units and from physical units to dimensionless units.
• Knowing how to estimate the accuracy of solutions for real world problems.
• Awareness/Insight into picking an algorithm suitable for the problem. Understanding the limitations of the algorithm. Understand that all algorithms have limitations.
• Understanding accuracy and precision and the difference between them.
• Knowing how to figure out where your code is "spending its time", and knowing where it is worth spending your time to speed up a code.
In our experience students have a hard time acquiring both the meta-skills and the high-level skills and these have to be emphasized throughout the curriculum.
Topics should when possible be taught in a physics context. The physics topics listed are meant to be suggestive not prescriptive. No one will teach all of the listed topics and everyone will teach some topics that are not in the list. Different institutions will no doubt teach different topics even if the computational topics are the same and a single physics topic may span multiple computational topics. For example the solution of the Laplace equation by finite differences might include a discussion of solutions of systems of linear equations, sparse matrices, and visualization. The list of topics here is minimal. The goal is to develop students that can apply their computational skills to unfamiliar physics problems and learn new numerical methods when needed.
• * Computer arithmetic and errors
• *Physics Topics examples: Error accumulation in upward and downward recursion of spherical Bessel function; accumulation of error in internal reflections within a sphere.
• Linear Algebra of real and complex matrices
1. Solutions of systems of linear equations
2. Eigenvalues and eigenvectors
3. *Sparse vs. normal matrices
• Physics topics examples: Normal modes of coupled oscillators, solution of Laplace's equation, Standing Waves on a drum by finite differences
• Numerical Integration
1. Low dimensional Integrals: Simpson plus at least one other method
2. High Dimensional Integrals: Monte Carlo method
• Physics topics examples: The finite pendulum, diffraction, matrix elements in QM for low dimensional integrals.
• Numerical Differentiation
1. Finite Differences
2. Numerical Differentiation of noisy data
• Physics Topics examples: Laplace's Equation, numerical differentiation of data from a physics lab
• Numerical Solution of Ordinary Differential Equations
1. Methods: Euler, Verlet and Runge-Kutta, Symplectic vs. Non- Symplectic. Note: Symplectic vs. Non-Symplectic is an advanced topic which many programs will omit.
2. Types of ODE's: Initial value problems, boundary value problems, eigenvalue problems
• Physics Topics examples: Celestial Mechanics, Chaotic Systems, The Schrödinger equation, Molecular Dynamics
• Numerical Solution of Partial Differential Equations; types of problems and boundary conditions
1. Finite Difference
• Physics Topics examples: Solution of Laplace, diffusion/heat equation, Wave Packet Dynamics
1. * Finite Elements
• Physics topic example: Poisson Equation
1. * FDTD
• Physics topic example Solution of Maxwell's equations in either one or two spatial dimensions.
• Numerical Root Finding
1. Newton, bisection
2. Secant Method
• Physics Topic examples: Finite square well. Wave on a circular drum, roots of Bessel functions
• Monte-Carlo Methods
• Physics Topics examples: One and Two Dimensional Ising Model. Simulations of fluids such as hard spheres, hard disks, and Lennard-Jones models as well as percolation models. Random walk, spontaneous decay simulation
• Fourier Transforms
• Physics topic examples: Spectrum of data from labs, power spectrum of a chaotic system
• Fourier Series, discrete Fourier transform, fast Fourier transform
• Physics topic example: Waves on a string
• Curve fitting including log-log and log-linear scales
• Physics Topic examples: Fitting real data from labs and/or public available data (NASA/CERN and so on) or data generated from undergraduate research
• Visualization
Emphasis should be on using visualizations to understand the physics/mathematics of the problem
• Physics topics: Almost all
• Symbolic Mathematics
Students should know a symbolic language such as Mathematica, Maple or Sage.
• Physics topics: Throughout the curriculum as appropriate.
Although not listed above, students should also know:
• The basics of a UNIX/LINUX operating system as well as a windows based system (i.e. Windows or OS X or similar)
• Know how to run a batch job
• A compiled language such as C or Java or Python
• Latex
• How to use and create numerical libraries
• * Experience with version control, such as Git
• At some point students should also do a computational project of respectable length and complexity. This could be a summer research project or a senior thesis or capstone project. Such an arrangement is congruent with the overall undergraduate computational science competencies.
• Although the physics topics listed are not meant to be prescriptive it is probably true that there are some physics topics so important that every student should see them. We suggest that these are:
1. Numerical solutions to the Schrödinger equation
2. Numerical solutions to the Ising model
3. Numerical solutions to the Laplace equation
4. Heat Equations
5. Wave equations
6. Numerical solutions to Maxwell's equations, at least in one dimension
7. A chaotic system
8. Fourier Series and the idea of expansion in orthogonal functions and implication of a limited number of data measurements.
9. Analysis of real data
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bf5b2ce9a2dffe1f | Reconciling an Ordinary World
Featured Blogger / by Chad Orzel /
Advances in materials and techniques bring physicists a step closer to observing the oddities of quantum behavior at the real-world scale.
One of the most vexing things about studying quantum mechanics is how maddeningly classical the world is. Quantum physics features all sorts of marvelous things—particles behaving like waves, objects in two places at the same time, cats that are both alive and dead—but we don’t see those things in the world around us. When we look at an everyday object, we see it in a definite classical state and not in any of the strange combinations of states allowed by quantum mechanics. Particles and waves look completely different, dogs can only pass on one side or the other of an obstacle, and cats are stubbornly alive or dead, not both at once.
Over the last 80 or so years, physicists have struggled to discover the origin of this apparent division between the quantum and classical worlds. Niels Bohr treated it as axiomatic in the “Copenhagen interpretation” of quantum theory, but this was an ad hoc addition to the theory. Nothing in the core equations of the theory says that a cat can’t be in two states at once, and physicists have had to work very hard to find possible explanations for why this doesn’t happen, proposing additions to the Schrödinger equation or invoking quantum gravity.
In recent years, new advances in materials and experimental techniques have made it possible to see quantum behavior in larger and larger objects, and a sort of cottage industry has sprung up in looking for quantum behavior of macroscopic objects. The DAMOP meeting in May featured an entire invited session on the subject with speakers from Yale, Vienna, and Munich, and another talk on experiments at LIGO that have pushed gram- and kilogram-scale mirrors toward the quantum limit (article here).
In “Quantum Mechanics in Ordinary Objects,” Veronique Greenwood reports on the latest development in this fast-growing field, a new experiment from the group of Michael Roukes at Caltech. The Roukes group has manufactured a “bridge” two micrometers in length next to an “artificial atom” consisting of a small loop of superconductor. The two are close enough together that their motion is coupled—when the “atom” is in a higher energy state, the “bridge” vibrates at a higher frequency, and vice versa. They have used the vibration of the “bridge” to detect the state of the “atom” and observed the discrete energy steps that give quantum mechanics its name. In the future, they hope to reverse the experiment, and use the state of the “atom” to detect quantized vibrations in the “bridge,” when the whole system is cooled down to low enough energy. Then they can try to prepare the “bridge” in a superposition of two states at once, and see what happens.
The Caltech group is still a long way from observing a cat in two places at once—only a physicist would consider a mass of 40 trillionths of a gram “macroscopic”—but this would be the largest object by far ever to show unambiguous quantum behavior. If they succeed, it could provide new insight into why the world we see is so depressingly ordinary compared to the world of quantum theory.
Discuss this article at ScienceBlogs »
Originally published July 29, 2009
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47b076f5dce05a45 | Take the 2-minute tour ×
We know $\Psi(x,t)$ is complex, but can $\Psi(x)$ be complex? I have seen particle in a box, well and harmonic oscillator. All have real solutions for time-independent Schrödinger equation. Hence, I am curious to know examples where it is complex. This question says that it is possible, hence my request is for examples and references.
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5 Answers
up vote 4 down vote accepted
A charged particle in external magnetic field has the following Hamiltonian:
$$\hat H=\frac1{2m}\left(\hat{\vec p}-q\vec A\right)^2+qV,$$
where $\vec A$ is vector potential, $V$ is scalar potential and $\hat{\vec p}=-i\hbar\nabla$ is momentum operator.
If you set $\vec A\not=0$, you'll get non-trivially complex wavefunction, and there will be no degeneracy due to time inversion symmetry as is the case for usual running wave, because magnetic field breaks time-inversion symmetry.
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Equation 6 in arxiv.org/ftp/arxiv/papers/0712/0712.4201.pdf gives the solution in 2-d polar cordinates. It is complex in the sense there is an $e^{il\theta}$ factor which is a complex sinusoid. – Rajesh D Dec 26 '13 at 10:56
Right, but this time it's not arbitrary. It's a phase which depends on spatial coordinate. You can't just simply multiply the solution by phase coefficient or take linear combination of some eigenstates to form a real wavefunction which would also be eigenfunction. Thus this complexity does have physical significance, and you can't avoid it. – Ruslan Dec 26 '13 at 11:10
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Rajesh: That means time independent Schrodinger equation does not demand complex numbers if one chooses to avoid them. right?
NOTE: This answer is sort of cheating due to the degeneracy, and thus doesn't really answer the OP's question. Hopefully someone with better knowledge can answer Rajesh's answer with real rigor.
Here's an example of a physical system which admits eigenstates which are nontrivially complex-valued. In molecular physics in the limit of the separability of vibrational motion from the other degrees of freedom, polyatomics with doubly-degenerate vibrational eigenbases have stationary states which can be represented via their occupation numbers in $\nu$ and $l$, the total vibrational quantum number and the vibrational angular momentum quantum number, respectively.
In polar coordinates with radius $\rho$ and angle $\theta$, the stationary states of $H$ are given via their expansion in terms of the generating function $$G=\pi^{-1/2}\rho^{1/2}\mbox{exp}\left[-\frac{1}{2}\rho^2+\frac{1}{\sqrt{2}}\rho e^{i\theta }su+\frac{1}{\sqrt{2}}\rho e^{-i\theta}s u^{-1}-\frac{1}{2}s^2\right]$$ $$=\sum_{\nu,l}\frac{\left<\rho,\theta|\nu,l\right>s^\nu u^l}{\left\{2^\nu[\frac{1}{2}(\nu+l)]![\frac{1}{2}(\nu-l)]!\right\}^{1/2}}$$ where $l\in\{-\nu,-\nu+2,...,\nu-2,\nu\}$.
For example, here is a complex plot of the function $|\nu=3,l=1\rangle$ state:
enter image description here
Here is a plot of the $|\nu=6,l=0\rangle$ state:
enter image description here
And here is a plot of the $|\nu=7,l=3\rangle$ state:
enter image description here
The plots are generated in a Hue colorspace, with the brightness proportional to amplitude, and color cyclically coded according to phase angle in the complex plane.
With the exception of zero vibrational angular momentum states, it is visually apparent that none of the wavefunctions can be represented as a real function multiplied by a phase factor $e^{i\alpha}$. Hopefully you find this sufficient as an example where a system has stationary states which are non-trivially complex-valued. I'm sure there are lots of other examples of cases where the time-independent Schrodinger equation has eigenbases which are nontrivially complex-valued, but this is the first example which came to mind.
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Yeah, they are complex-valued... too bad they are also degenerate. You can choose such linear combinations of them (e.g. $\left|+m\right>+\left|-m\right>$ where $m$ is magnetic quantum number) which will become real and still remain eigenstates. Of course, they won't have some observables (like $L_z$) definite, but they will have others. – Ruslan Dec 25 '13 at 20:02
Not that it's worth changing here but I leave this comment any time I see pictures like this -- choose different color maps! If it has red and green in it, particularly close together, those of us who are color-blind have a really tough time with it. Spectrum, rainbow, etc are all poor choices for displaying most data, despite them being the default in most viz. software! Single color changes, saturation changes, and so on make better color maps for everybody, but especially those who are color-blind. Just some friendly advice :) – tpg2114 Dec 25 '13 at 20:04
@Ruslan: Haha, yeah I figured somebody would point that out. Still though, that leaves the OP's question a bit unanswered, ie, whether non-degeneracy implies that the wavefunction can be represented as a real function times a constant factor. I'm curious if there's a theorem which supports or negates this. – DumpsterDoofus Dec 25 '13 at 20:24
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Before starting, let me be pedantic and define the phrase "real up to an overall phase." I will say a function $f(x)$ is real up to an overall phase if I can find a real number $\alpha$ such that $e^{i\alpha}f(x)$ is real everywhere. [Incidentally, note that if a wave function is real up to an overall phase, it is also imaginary up to an overall phase].
Now that we've established that terminology, I claim that the question of whether or not the wave function is real up to a phase must be considered in two cases:
1. If the energy eigenfunction $\Psi(x)$ is associated with a non-degenerate energy eigenvalue will be real up to an overall phase.
2. If the energy eigenfunction $\Psi(x)$ is living in the energy eigenspace associated with a degenerate energy eigenvalue will generically be complex. While there will always be special eigenfunctions living in this space that are real up to an overall phase, the general eigenfunction is complex and you cannot restrict yourself to the "real subspace" without losing the ability to describe some physical systems.
The examples you gave (particle in a box, harmonic oscillator) fall under case 1: there are no degenerate energy levels, so the eigenfunctions are real up to an overall phase. However I can easily give you physical examples that fall under case 2 (and I will later, and DumpsterDoofus and DarenW have already done so).
Before going into some gory detail, I should just say that whether or not the wave function is real up to an overall phase is sort of irrelevant, physically speaking. You should think of the wave function as being a complex valued quantity. For example, if the state is a momentum eigenfunction--which is the case in particle accelerators, which measure the momenta of particles very precisely--then the wave function is complex $e^{ipx}$ and is certainly not real up to an overall phase. In certain cases it happens to turn out that the wave function is real up to an overall phase, but this is frankly something of a mathematical accident and doesn't have any interesting physical consequences. You certainly should not make the leap from "there exist wave functions that are real up to an overall phase" to "all physical situations can be described by a wave function that is real up to an overall phase."
With that said, let's discuss case 1. We are looking for eigenfunctions $\psi(x)$ that obey the energy eigenvalue equation [working in the position representation] \begin{equation} \hat{H}\psi=E\psi \end{equation}
where $\hat{H}=-\frac{\hbar^2}{2m}\nabla^2+V(x)$ is the Hamiltonian operator.
Since the Hamiltonian must be a hermitian operator, we have that $(\hat{H}f)^*=\hat{H}f^*$ for any function $f(x)$. Meanwhile, $E=E^*$ since $\hat{H}$ is hermitian so it has real eigenvalues.
As a result, $\psi^*(x)$ is also an eigenfunction of $\hat{H}$ with eigenvalue $E$.
However, we are assuming that the energy eigenvalue $E$ is non-degenerate. In other words, there is only one eigenfunction with eigenvalue $E$, up to an overall factor. So it must be that there is some $\lambda$ such that
\begin{equation} \psi(x)=\lambda \psi^*(x) \end{equation}
Clearly this equation is only consistent if $|\lambda|=1$, so we may write $\lambda=e^{-2 i\alpha}$ for a real $\alpha$ without any loss of generality. But then $\psi(x)$ is real up to an overall phase, because $\phi(x)=e^{i\alpha}\psi(x)$ obeys $\phi(x)=\phi^*(x)$.
That establishes case 1.
Before going on to case 2 it is interesting to see why the particle in a box falls under this case.
The hamiltonian for a particle in a box is the same as the hamiltonian for a free particle \begin{equation} \hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}, \end{equation} with the boundary conditions \begin{equation} \psi(0)=\psi(L)=0. \end{equation} Since the hamiltonian is the same as for a free particle the eigenfunction must have the form \begin{equation} \psi(x)=Ae^{ikx}+Be^{-ikx} \end{equation} where $A$ and $B$ are complex coefficients. You can see that the general solution is not real up to an overall phase! This is because there are TWO solutions with energy $E$: one is $e^{ikx}$ and the other is $e^{-ikx}$. It is only after we impose the "particle in a box" boundary conditions that we find that the eigenfunction must have the form $\psi(x)=N\sin\left(\frac{\sqrt{2mE}}{\hbar}x\right)$. The boundary conditions essentially "project out" one of the two eigenfunctions with the energy level $E$. We are left with one linearly independent solution that can satisfy the boundary conditions, and by the general argument I gave above this eigenfunction has to be real. (And indeed, explicit calculation shows it is--it is the sine function). If you want me to go into more detail about this part, I would be happy to update the answer to be more precise.
The 1D harmonic oscillator is slightly different. There are two solutions for a given energy eigenvalue $E$. However only one will be normalizable. So for each energy level $E$ there is only 1 energy eigenfunction, and thus this eigenfunction is real up to an overall phase.
OK, case 2. It is easy to see that in general a member of a degenerate eigenspace won't be real. $\psi(x)$ and $\psi^*(x)$ are still both solutions since $\hat{H}$ is hermitian. However there is no reason that $\psi(x)=\lambda\psi^*(x)$.
Even more to the point, since the energy eigenspace is degenerate it means that there are at least two distinct, orthogonal eigenfunctions $\psi(x), \phi(x)$ that have the same energy $E$. Any linear combination of them will also have the energy $E$. So it's very easy to construct a complex energy eigenfunction living in this subspace, even if $\psi$ and $\phi$ happen to be real: I just take $a \psi+b\phi$ with complex $a,b$.
On the other hand it's also easy to see that there are always special eigenfunctions in the eigenspace that are real. For example, $\psi(x)+\psi^*(x)$ is real.
It's useful to give an example. A good example is actually the particle in a periodic box. This is just like the particle in a box in that the Hamiltonian is the same as for a free particle. However, instead of imposing the boundary conditions $\psi(0)=\psi(L)=0$ we will impose the periodic boundary conditions \begin{array} \ \psi\left( 0 \right)&=&\psi \left( L \right) \\ \psi'\left(0\right)&=&\psi'\left(L\right) \end{array} where $\psi'(x)=d\psi/dx$.
These boundary conditions are appropriate in many situations, for example:
1. Electrons in a solid will experience periodic conditions because of the periodic nature of a crystal
2. In extra dimensional models where the extra dimension is compact, the wave function has to be periodic
3. If you work in polar coordinates in 2D or spherical coordinates in 3D, the angle $\theta$ will obey periodic boundary conditions
Starting with the general solution $\psi(x)=A e^{ikx}+Be^{-ikx}$, the boundary conditions amount to the single condition $e^{ikL}=1$, so $k=2\pi n/L$ (which quantizes the energy levels). Crucially, the boundary conditions do not impose any conditions on the coefficients $A$ and $B$. So it for any allowed energy level $E$, there are TWO allowed energy eigenfunctions. Thus in general, a state with energy $E$ is not real up to an overall phase.
There are two special states with energy $E$ that are real. They are \begin{equation} \psi_c(x)=\frac{1}{2}\left(e^{ikx}+e^{-ikx}\right)=\cos kx, \psi_s(x)=\frac{1}{2i}\left(e^{ikx}-e^{-ikx}\right)=\sin kx \end{equation} However there is no reason to think that nature will "prefer" these states over other possible states with the same energy with complex valued wave functions. If we set up a periodic particle in a box in the lab and fixed the energy to be $E$, in general we would get a state that wasn't $\psi_s$ or $\psi_c$.
As I said, there are many interesting examples that fall under case 2. Another one is the 2D harmonic oscillator, with $V(x,y)=m\omega^2(x^2+y^2)$. In general, as you go to dimensions greater than 1, the solvable problems tend to have degenerate eigenvalues. So even though you start off with examples that are of the form of case 1, you should expect to see many more examples in case 2 as you advance in quantum mechanics.
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Thanks for the great answer. But point 2 being complex solution is still not a worry for me. – Rajesh D Dec 26 '13 at 10:59
@Rajesh D Do you mind elaborating on what you mean? I don't mind expanding my answer. I think point 2 is very important. For example, all scattering problems fall under point 2. In scattering it is actually very important that the wave function is complex, and not just real up to a phase. A related statement is that it is very important to use the Feynman propagator, and not the usual retarded propagator, when doing scattering in quantum mechanics. – Andrew Dec 26 '13 at 11:27
i fully agree that it has to be complex. I am working on ways to make $\psi(x)$ a real function all the time by tweaking the theory a bit. Crazy thing to do, but anyway its not relevant to mention here. sorry about that. – Rajesh D Dec 26 '13 at 11:40
I see. It sounds like an interesting thing to try--even if you ultimately fail you will probably learn a lot. One thing you might find interesting if you haven't heard of it is the "wick rotation"--you use analytic continuation to deal with real quantities to do quantum calculations. – Andrew Dec 26 '13 at 23:51
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For example, you can start with any real solution and multiply it by a phase factor $\exp(i\alpha)$, where $\alpha$ is constant and real, but is not a multiple of $\pi$.
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but that has no physical significance! – Rajesh D Dec 25 '13 at 17:50
@RajeshD Yes, and that means that having strictly real solutions also has no physical significance and is preferred only because most people are more comfortable in $\mathbb{R}$ than in $\mathbb{C}$. – dmckee Dec 25 '13 at 17:59
@dmckee : That means time independent Schrodinger equation does not demand complex numbers if one chooses to avoid them. right? – Rajesh D Dec 25 '13 at 18:01
I'm still thinking about that. – dmckee Dec 25 '13 at 18:03
@RajeshD : If you have some degeneracy for a energy $E_n$, you could have different and linearily independent real solutions $\psi_n^i$. Now, because of the linearity of QM, any complex linear combination of the $\psi_n^i$ is also a solution, for instance $\psi_n^1 + i \psi_n^2$ is a complex solution. – Trimok Dec 25 '13 at 18:49
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In one dimension, a running wave, as eq. (2) in that question. There isn't any more one can do in 1D that won't come out seeming contrived.
In two dimensions, a good example is a circular box, or any spherically symmetric potential. The wavefunction can be written as a radial factor times an angular factor. The angular factor may be a linear combination of $\sin(n\theta)$ and $\cos(n\theta)$ or, equivalently, of $\exp(i n \theta)$ and $\exp(-i n \theta)$. The latter are nicer to deal with in terms of eigenvalues and angular momenta. It's the same as in 3D, without 'z', so let's go there.
In three dimension, in atomic physics, the 'm' quantum number for orbitals. You can have an electron in, for example, some mix of $2p_x$, $2p_y$, and $2p_z$ orbitals, or you can use $2p_+$ $2p_-$ and $2p_z$, where $2p_\pm = 2p_x \pm i2p_y$. (likewise for n=3, 4, ... and similar combinations for the d, f ...) The $\pm$ orbitals are better when dealing with magnetic fields, spin-orbit coupling, conservation of angular momentum in scattering, and so on.
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I don't understand the later part. What is your answer? 2-d and 3-d demand complex $\Psi(x,y,z)$? – Rajesh D Dec 25 '13 at 18:07
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a05fa61fa7cf96d1 | About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 982753, 16 pages
Research Article
Estimates for Unimodular Multipliers on Modulation Hardy Spaces
1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
3School of Science, Hangzhou Dianzi University, Hangzhou 310016, China
Received 23 November 2012; Accepted 23 January 2013
Academic Editor: Baoxiang Wang
It is known that the unimodular Fourier multipliers are bounded on all modulation spaces for . We extend such boundedness to the case of all and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study the boundedness for the operator , for the case and Finally, we investigate the boundedness of the operator for and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroup .
1. Introduction
A Fourier multiplier is a linear operator whose action on a test function on is formally defined by The function is called the symbol or multiplier of .
In this paper, we will study the unimodular Fourier multipliers with symbol for . They arise when one solves the Cauchy problem for dispersive equations. For example, for the solution of the Cauchy problem we have the formula . Here is the Laplacian and is the multiplier operator with symbol (see [1] for its definition). The cases are of particular interest because they correspond to the (half-) wave equation, the Schrödinger equation, and (essentially) the Airy equation, respectively.
Unimodular Fourier multipliers generally do not preserve any Lebesgue space , except for . The -spaces are not the appropriate function spaces for the study of these operators and the so-called modulation spaces are good alternative classes for the study of unimodular Fourier multipliers. The modulation spaces were first introduced by Feichtinger [24] to measure smoothness of a function or distribution in a way different from spaces, and they are now recognized as a useful tool for studying pseudodifferential operators [57]. We will recall the precise definition of modulation spaces in Section 2 below.
Recently, the boundedness of unimodular Fourier multipliers on the modulation spaces has been investigated in [1, 815]. Particularly, one has the following results.
Theorem A (see [11]). Let ,, , and . One has, for , where
Here (and throughout this paper), we use the notation to mean that there is a positive constant independent of all essential variables such that .
Theorem B (see [15]). Let , , and . Then is bounded from to if and only if
In this paper, we use a different method from [15] to prove the following theorem, which, in particular, uses the modulation Hardy spaces that will be later defined in Section 2.
Theorem 1. Let , , . For a positive , denote . Let if n is even and if is odd.(i)Assume . If and , one has Particularly, the above inequality holds for all if is a positive even number. (ii) For any , one has for any .
Here (iii) Assume . If , then for all .
We want to make a few remarks on Theorem 1. First, (iii) in Theorem 1 says that when , compared to the case in (i), one obtains a larger range of and a smaller range of . We do not know if there is a unified formula regarding and for all dimension . Second, in the proof we will see that, in the low frequency parts of the definition of , the fractional Schrödinger semigroup has a growth when is growing, but it gains an arbitrary regularity. In the high frequency part, the semigroup can be controlled by at each piece of its decomposition with frequency . This phenomenon was also more precisely observed in [1, 15] (see also [11]). Thirdly, the case was studied in [8, 16].
Since the norm is dominated by the norm and the Riesz transforms are bounded on , by the Riesz transform characterization of the (see Section 2), we easily obtain the following corollary.
Corollary 2. Let , and . One has for where
Our next result shows that the asymptotic factor in Theorem 1 is the best for all , at least for .
Theorem 3. Let . The asymptotic factor in Theorem 1 is the best. Precisely, for , if then
In the next theorem, we state some mixed norm estimates.
Theorem 4. Let and . For , suppose .(i)If , then (ii)If , then
We consider the following linear Cauchy problem with negative power:
We give the grow-up rate of the solution to the above Cauchy problem in the modulation spaces.
Theorem 5. Assume and .(i)Let . One has that for any (ii)For any , one has
Now, we study the following Cauchy problem of the nonlinear dispersive equations (NDE): where for some positive integer . For , the space is defined by
We obtain the quantitative forms about the solution to the above Cauchy problem of the nonlinear dispersive equations.
Theorem 6. Let , , and assume
Assume for any
There exists such that the above Cauchy system (NDE) has a unique solution , where depends on the norm and .
According to the inclusions of modulation space (see Proposition in [13]), we know the space of initial data if .
Theorem 7. Let . Assume and for any
The rest of the paper is organized as follows. In Section 2, we recall or establish some necessary lemmas and known results. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 3, respectively. Finally, in Section 5, we give some applications including the boundedness for the operator in the case and , including negative .
2. Preliminaries
2.1. The Definitions
The modulation space is originally defined by Feichtinper in 1983 on the locally compact Ablian groups . When , the modulation space can be equivalently defined by using the unit-cube decomposition to the frequency space (see Appendix in [13], also [14, 17]). The following definition is based on the unit-cube decomposition introduced in [13].
Let be a fixed nonnegative-valued function in with support in the cube and satisfy for any in the cube . By a standard constructive method, we may assume that for all , where is the -shift of that is defined by
For each , we use as its symbol of a smooth projection on the frequency space. Precisely, for any , we have
Let be a Banach space of measurable functions on with quasi-norm . We define the modulation space where By definition, we have the inclusion It is known that the definition of the modulation space is independent of the choice of functions . In this paper, we are particularly interested in the cases and , where is the Lebesgue space and is the real Hardy space. For all , we call the modulation spaces and the modulation Hardy space. As a usual notation we similarly define By the definition and known properties of , we have that for all , and for all , For simplicity in notation, we denote The following imbedding relation can be found in Proposition of [18]. Let , . If then
2.2. Spaces
It is well known that the Hardy space coincides with the Lebesgue space when . For , the space has many characterizations. We will use its Riesz transform characterization in this paper. For an integer and multi-index , let denote the generalized Riesz transform where each is the Riesz transform of if and . It is known that for and all , where is a sum of finite terms.
The operator is a convolution. We have Also it is well known that is bounded on spaces for any .
2.3. Some Lemmas and Known Results
Lemma 8. Let and . Suppose that there is an integer , such that for all test functions for and for . Here and is a real number. Then for , one has where is an arbitrary positive number.
Proof. The case is proved in [11]. It suffices to show the lemma for . By the Riesz transform characterization of , for , we have By checking the Fourier transform, we have the identity where So for , one has A similar argument shows that for , for any . The rest of the lemma easily follows from the definition of the modulation spaces.
Lemma 9 (see [18, 19]). Let denote an open set and . If and the rank of the matrix is at least for all (), then
Lemma 10. Let and . Suppose that is a function with support in . Then
Proof. The case is known [20]. It then suffices to show that for , for large . Let be a standard bump radial function supported in the set and satisfying, for all , Noting the support condition of , we write where the sets , are defined by For , we use polar coordinates to write where is the induced Lebesgue measure on the unit sphere . When is even, taking integration by parts for times on the inside integral, we obtain When is odd, we use integration by parts for times on the inside integral, Again we obtain that for odd , For , without loss of generality, we assume . Perform integration by parts on the variable for suitable amount of times. We similarly obtain For , invoking Lemma 9, we obtain Noting that contains no more than numbers of , it is easy to check
The lemma is proved.
Lemma 11 (see [21, pages 163–171]). Let and Suppose that is a Fourier multiplier with symbol . If is a bounded function which is of class in and if with , then is a bounded operator on and
Lemma 12. Let and . For all , one has
This lemma can be found in Section 4.2 of [11].
Lemma 13. Let be a compact subset in , and let . There exists a constant depending only on the diameter of and , such that for all satisfying .
This lemma is the Nikol'skij-Triebel inequality, see Proposition in [20] (also Lemma 2.5 in [22]).
Lemma 14. Let and be compact subsets of . Then there exists a constant depending only on the diameters of and , such that for all , satisfying and .
This is Lemma 2.6 in [22] (see also Proposition in [20]).
Lemma 15 (Pitt's theorem). If and , then
Lemma 16. Let and satisfy Then one has
This result is a particular case of Lemma 2.5 in [8].
3. Proof of Theorem 1
The operator is a convolution operator with the symbol . This symbol is a function on with compact support. Clearly for any and , we have that for ,
So Lemma 11 implies the following estimate.
Proposition 17. Let . For any with , one has
By the proof of Lemma 8 and Proposition 17, we have that for all ,
The following proposition extends Lemma 12 to all .
Proposition 18. Let . For any with , for any , one has
Proof. The proof uses the same idea used in proving the case which was represented in [11]. For the convenience of the reader, we present its proof.
Let be the kernel of . Then By Lemma 14 and (46), we have Thus to prove the proposition, it suffices to show
For simplicity, we prove the case . The proof for , is tedious but shares the same idea as that for .
First we study the case . For , and , if we denote If , we denote
Also, for and , we define sets It is easy to check Let We have for , Write where
It is easy to check that if and supp, the phase function satisfies So by Lemma 9, we have
Observe the easy fact that if and supp , for any integer , Perform integration by parts on and variables both for times such that . An easy computation shows that
The estimates for and are exactly the same. We only estimate . Take integration by parts on variable for times with . Again, a simple computation shows that if we chose a suitably large . These estimates on , , indicate provided .
We now turn to show the case . For , and , let be the numbers defined above. For and , we define sets It is easy to check Let Thus, Using the same argument as we used before, we can show We complete the proof of Proposition 18.
We are now in a position to prove Theorem 1.
Proof. By an argument involving interpolation and duality, it suffices to show the case . Using Proposition 18, the inequality in (76) and the definition of the modulation spaces, we easily obtain (ii) in Theorem 1.
To show (i) and (iii) in Theorem 1, by Proposition 18 and the definition of the modulation spaces, it suffices to show Again, by Lemma 14, the proof of the inequality in (101) can be reduced to show that for , We show (iii) first. The proof of may illustrate the method. When By Hölder's inequality and the Plancherel theorem, the first term above For the second term, performing integration by parts, we obtain since
Now we return to show (i) of Theorem 1. We will prove only the case . Write Using Hölder's inequality and the Plancherel theorem, we obtain
For , we denote sets We now write To show (102), it now suffices to show that for each ,
Using the Leibniz rule, for any positive integer , we have Here, an easy induction argument shows that, for , where is a homogeneous function of degree for each . We now write where By the definition, it is easy to see that each is an and function with support in the cube .
Let . Performing integration by parts on variables for times, we have
We first estimate each , . Recall that we assume . Let , so . By the choice of and the assumption it is easy to see . Therefore, by Hölder's inequality, we obtain For each , by the choice of , the assumption on , and an easy computation, it is not difficult to see that we may obtain a number in the interval satisfying By Hölder's inequality and Pitt's theorem, for each , we obtain Combining all the estimates, we have It remains to estimate .
It is easy to see that the choice of and the condition in the theorem imply |
6d029cb57f72e84a | Interpreting Quantum Theory Without Losing Your Marbles: A Look at the Implications of Spontaneous Localization Theories on Physical Reality
As a predictive theory, quantum mechanics has been extraordinarily successful; the successful prediction of the electron's anomalous magnetic moment, for example, is considered one of the greatest triumphs of theoretical physics. However, the nondeterminism at the heart of quantum theory raises some troubling philosophical questions; after all, quantum mechanics does not explicitly prohibit a macroscopic system from being in a superposition of two states, but we never actually see such a macroscopic superposition. Physicists tend to dismiss such concerns; after all, quantum mechanics makes excellent predictions, and physics is, in the end, the business of making predictions about the world. Ghirardi, Rimini, and Weber have proposed a mechanism for avoiding macroscopic superposition states, which they term Spontaneous Localization (SL) theory. It, and its refinements such as Continuous Spontaneous Localization, offer an interesting way of dealing with the problem of macroscopic superpositions, and may lead to other interesting insights into the foundations of quantum reality.
I. The Macro-objectification Problem
The state-vector formalism in quantum mechanics allows for superpositions of states, such that the state vector |Ψ> describing a particle in a box could be, for example, an admixture of two energy eigenstates: |Ψ> = a|ψa> + b|ψb>. While this feature of quantum mechanics undeniably allows accurate predictions on the energy and distance scales where classical mechanics inevitably fails, it also allows, on first glance, a potentially embarassing situation: nothing in quantum theory explicitly prohibits macroscopic objects from being placed in superpositions of two or more perceptually different states. This leads to two separate problems: first, what meaning can we attach to a situation in which an object is in a superposition of perceptually different states; and second, assuming that meaning can be reasonably attached to such a situation, why are these superpositions never observed? After all, it is said that everything not forbidden by the laws of nature is mandatory. This second problem is known as the macro-objectification problem, and it is one that generally does not bother physicists.
Physicists, on the whole, tend to look to the Copenhagen interpretation for a resolution to the macro-objectification problem; this states that, for example, upon measurement of the energy of a particle in the previously-discussed state |Ψ> = a|ψa> + b|ψb>, the particle will be found to have either energy Ea or Eb, with probabilities |a|2 and |b|2, respectively. Further, the act of measurement "collapses" the state vector, so that the particle assumes the pure state associated with the measured energy. But what, exactly, constitutes a "measurement," and what is the mechanism by which the state-vector collapses when one occurs? On this point, standard quantum theory is resolutely silent. Something about this situation seems to have bothered Schrödinger, who put this question in the form of his (in)famous cat paradox:
A cat is placed in a steel chamber, together with the following hellish contraption… in a Geiger counter there is a tiny amount of radioactive substance, so tiny that maybe within an hour one of the atoms decays, but equally probably none of them decays. If one decays then the counter triggers and via a relay activates a little hammer which breaks a container of cyanide. If one has left this entire system for an hour, then one would say the cat is living if no atom has decayed. The first decay would have poisoned it. The wave function of the entire system would express this by containing equal parts of the living and dead cat.1
In the situation described, the cat would be neither alive nor dead until the chamber was opened, and it would be the opening of the chamber which forced the cat to become either alive or dead. Schrödinger felt that this situation was absurd, and not many people would argue with him on that point. The problem lies in the Copenhagen interpretation's use of the word "measurement" – what, exactly, constitutes a measurement? What sort of interactions cause the wavefunction to collapse, and why do no macroscopic superposition states persist? D. J. Griffiths' view on this matter is, I think, representative of the mainstream of the scientific community:
Of course, in some ultimate sense the macroscopic system is itself described by the laws of quantum mechanics. But wave functions, in the first instance, describe individual elementary particles; the wave function of a macroscopic object would be a monstrously complicated composite, built out of all the wave functions of its 1023 constituent particles. Presumably somewhere in the statistics of large numbers macroscopic linear combinations become extremely improbable.2
In other words, the physics community is largely inclined to take quantum mechanics as a ssuccessful predictive theory and leave the macro-objectification problem alone for the time being; the problem, however remains: how to account for the absence of observed macroscopic superposition states, given a quantum theory which does not prohibit their existence?
II. The GRW Spontaneous Localization Theory
Fortunately, several alternatives to the Copenhagen interpretation exist which purport to solve the macro-objectification problem in various ways; perhaps the most interesting of these is the collapse model proposed by Ghirardi, Rimini, and Weber. This model, termed Spontaneous Localization (SL) theory, proposes a modification to the evolution of quantum states described by the Schrödinger equation: every so often, particles experience something called "Gaussian hits," which involve multiplying the wavefunction of the particle by a Gaussian and renormalizing; this has the effect of locating the particle fairly precisely. In this sense, it is analogous to the effect of a position measurement under the Copenhagen interpretation.
Under the GRW spontaneous localization theory, a particle will experience a hit at randomly distributed times according to a Poisson distribution with mean frequency w; upon a hit at location a, the wavefunction is multiplied by the Gaussian G(x) = exp(-(½ d2)(x-a)2) and then normalized. The parameter d describes how closely the particle is localized, and both d and w can be chosen to agree with observations; GRW suggest the values w = 10-15 s-1 and d = 10-5 cm.3 The probability that a hit occurs at location a is proportional to the norm of the wavefunction at a. Since w is miniscule, it is highly improbable that a single particle will experience a hitting within a reasonable time of observation, so the quantum properties of microscopic systems remain intact. In a macroscopic system, however, such as a marble containing on the order of 1023 particles, one of the constituent particles will be hit, on average, 108 times in a second. Since the particles in the marble interact with each other and maintain their average separations, a hit on any particle localizes the entire marble – this leads to the familiar classical behavior for macroscopic systems without recourse to the ill-defined "collapse rule" of standard quantum theory.
SL theory is promising for several reasons. First and foremost, it yields the familiar behavior of objects at the micro- and macroscopic levels; objects containing only a few thousand particles show the behavior standard quantum theory would lead us to expect, and objects containing large numbers of particles behave classically. Second, it is empirically falsifiable; this is of paramount importance for physical theories. SL makes explicit predictions which differ from those of standard quantum mechanics, and tells us where to look for the anomalies. If SL holds, one would expect to see anomalous results in systems comprising on the order of 1015 particles, so that localizations could be expected to happen occasionally on reasonable experimental time scales. Two-slit diffraction experiments have been carried out using objects such as buckyballs, which are still too small to see anomalies related to SL; still on the horizon is a proposed diffraction experiment using virus particles. The virus diffraction experiment has the potential to shed some light on the behavior of systems which lie between the domains in which quantum and classical physics dominate. Finally, although SL appears at first glance to introduce two new physical constants, which is always grounds for skepticism, it has been recently proposed that the gravitational interaction could lead to the localizations, and that the parameters d and w are of the right order of magnitude for a gravitational interaction. If this turns out to be a viable proposition, it would certainly lend weight to the SL hypothesis.
III. The Problem of the "Tails"
SL purports to solve the macro-objectification problem by multiplying the wavefunction of a particle with a Gaussian at randomly determined times. The astute observer will note that, even after localization, the wavefunction of any particle will still have infinite support – that is, there will still be a nonzero probability of finding the particle an arbitrary distance from the position at which it was "localized"! SL, then, avoids the most embarassing sort of superposition, the wavefunctions with two peaks of roughly equal probability at perceptually different locations – after all, the narrow Gaussian profile ensures that there can be only one peak – but it still leaves open the possibility that we could leave a table at location a, look for it, and find it in an arbitrarily distant position. The question is now whether SL has solved the macro-objectification problem at all, or merely swept it under the rug, so to speak.
What meaning can we assign to a wavefunction with tails, in terms of observable reality? After all, we certainly do not see many macroscopic objects spontaneously teleporting themselves around! Part of the reason for this is that the probability for this sort of teleportation is unfathomably small; Pearle gives the following illustrative example: "Let V be a sphere of radius 10-6 cm, and suppose that a hydrogen atom lies with its nucleus at the center of the sphere. Let us ask the question: is the electron in V? We readily calculate that <ψ|P1V|ψ> ? 1- 10-169."4 In other words, you are about 10160 times more likely to win the lottery than to make a position measurement and find your electron anywhere outside that sphere. In this case, it is probably fair to say that the electron is located within that sphere. Similarly, if you place a coin in a jar, the probability of later finding it elsewhere is even smaller than 10-169, and it seems fair to say that the coin is in the jar, the tails of its wavefunction notwithstanding. Unfortunately, "it seems fair" is far from an acceptable argument in this context; electrons, for example, regularly show behavior that would be absurd from a classical viewpoint. We need some justification for our claim that the coin is located in the jar.
The standard solution to the tails problem is unfortunately not much clearer than our claim that "it seems fair" to say that the coin is in the jar, given the miniscule probability of finding it elsewhere. Albert and Loewer propose the following criterion, which they call PosR, for stating that a particle is in a certain location:
"Particle x is in region R" if and only if the proportion of the total squared amplitude of x's wave function which is associated with points in R is greater than or equal to 1-p.5
Any value of p less than 0.5 suffices to prevent us from saying that a particle lies in disjoint regions, but it is clear from the examples above that we can use values of p which are much closer to 0. Essentially, p can be chosen arbitrarily to fit those situations in which we consider the particle localized. Clifton and Monton generalize PosR to multi-particle systems in the following manner, which they term the "fuzzy link":
"Particle x lies in region Rx and y lies in Ry and z lies in Rz and…" if and only if the proportion of the total squared amplitude of y(t, rI, …, rn) that is associated with points in Rx, Ry, Rz … is greater than or equal to 1-p.6
They then show that, given a large enough number of particles, it is possible to create a situation wherein, by PosR, it is possible to say that x lies in region Rx, that y lies in Ry, that z lies in Rz, and so forth, but that according to the fuzzy link, the proposition "Particle x lies in region Rx and y lies in Ry and z lies in Rz and…" is false; in other words, it is possible for counting to fail with macroscopic objects!
Additional arguments related to the failure of the enumeration principle under SL will be discussed below, but one might wonder whether it is possible to avoid the problem entirely by eliminating the tails of the wavefunction. Perhaps the "hits" could multiply the wavefunction by a triangular or boxcar window, rather than by a Gaussian – then the probability of finding the particle outside the region to which it is localized would be zero. Unfortunately, under SL theory, the Schrödinger equation is unmodified, and under the Schrödinger equation, any wavefunction with finite support in position space has an infinite spread in momentum. This means that even if the wavefunction has finite support at the time t of a hit, it will again have infinite support at any time t' > t. Even if a particle is perfectly localized at (x,t), there will be a nonzero probability of finding it arbitrarily far from x at any time afterward. This is no mere quirk of the Schrödinger equation, either – tails form just as quickly in the relativistic Dirac equation. In fact, according to Pearle, a wavefunction without tails is physically meaningless in a relativistic theory:
If you have a tail, no matter how small, and you know the field w(x, t) which the state vector evolved under, you can run the evolution equation backwards and recover the statevector at any earlier time. If, on the other hand, the tail was completely cut off, you get a nonsensical irrelevant earlier statevector, even in standard quantum theory… I cannot see how you could get sensible results in another Lorentz frame without having the tail to tell you how to do it.4
The "tails" of the wavefunction, then, are necessitated by quantum theory – there is no way to avoid having them.
IV. Are Tails Really a Problem?
Since it seems that tails are here to stay, at least until someone comes up with a replacement for the Schrödinger equation, we are forced to consider whether their presence is as problematic as it seems to be. What meaning can we attach, perceptually, to an object whose wavefunction is spread out in space? Fortunately, SL theory eliminates the possibility of macroscopic objects which are spread out by more than ~10-5 cm, so we need not consider situations in which a chair, for example, has equal probabilities of being in two positions 1 meter apart. We need only consider situations wherein a macroscopic object is well-localized, such that its probability of being found very far away from its expected position is miniscule. In this case, the answer is literally right in front of us; after all, Schrödinger tells us that the wavefunctions of everyday objects are spread out in space, but we percieve them as having definite positions. There is no reason to expect our perceptions to be any different now that we have a mechanism by which they could be localized.
Since there is no reason to expect any perceptual problem, we turn to the enumeration problem. As noted earlier, it is possible, given enough marbles, to be able to say with virtual certainty that each marble is in a box, and for the probability that all the marbles are in the box to be close to zero. This would seem to violate the enumeration principle which states that if marble 1 is in the box and marble 2 is in the box and … and marble n is in the box and no marbles are in the box, then there are n marbles in the box. A violation of the enumeration principle would have the extremely unfortunate consequence that arithmetic would not apply to macroscopic objects, and that consequence would be a fatal blow to a theory purporting to describe the macroscopic world. Clearly we should take a closer look at the details of this situation. The argument comes from Peter Lewis and involves a set of n non-interacting marbles in the state
all> = (a|in>1 + b|out>1) r (a|in>2 + b|out>2) r … r (a|in>n + b|out>n)
where |in>i refers to the ith marble being in the box, and |out>i refers to the ith marble being out of the box. Although PosR may require a to be very close to 1 before we can say that the marbles are in the box, a will never be exactly 1 because of the tails. Since the probability of finding any given marble in the box is a2, the probability of finding all n marbles in the box is a2n – and here is where the trouble begins. For sufficiently large n, a2n << 1, and there is a very good chance that if we look, not all of the marbles will be in the box! Clearly something is not right here.
The string of replies and counter-replies on this topic is beyond the scope of this paper, but Clifton and Monton have argued that while it is in principle possible to violate the enumeration principle under SL theory, the interaction of the counter with the marbles guarantees that the enumeration principle can never be experimentally falsified, and thus that arithmetic continues to apply to everyday objects:
To manifest a failure of conjunction introduction, one has to get an … apparatus which measures the system as a whole appropriately correlated with the system, and one has to get … apparatuses which measure the location of each marble appropriately correlated with each marble. Once all this is done, the requisite entanglement between the marbles (or particles) will be established and the dynamics of the GRW theory will guarantee that the system will either be in, or almost instantaneously evolve to, a state where the various apparatuses are in agreement and no failure of arithmetic is ever manifest.6
Thus GRW dodges another bullet; counting can never actually fail when carried out, since the process of counting entangles all of the marbles and ensures that they do not stay in a potentially dangerous state for more than a split second.
V. Conclusions
As we have seen, GRW's spontaneous localization theory offers an attractive solution to the macro-objectification problem. Under SL, objects such as electrons, which familiarly exhibit quantum behavior, continue to do so; likewise macroscopic objects such as tables and chairs continue to behave classically. In short, SL predicts that most things will behave exactly as we see them behave; nevertheless, it represents enough of an alteration to standard quantum mechanics that we might expect to see the difference in objects of sizes intermediate between atoms and marbles. Although SL cannot eliminate wavefunctions' "tails," it is clear under the Schrödinger evolution that even if they were eliminated, they would return after a split second; furthermore, it is not at all clear that the tails involved in SL present any sort of problem, either perceptually or philosophically. In the end, SL and its more complex refinements present an intriguing solution to the macro-objectification problem, and it should be interesting to see whether these collapse theories stand up to the experimental scrutiny which is sure to be forthcoming.
1. E. Schrödinger, quoted in Griffiths, David J. Introduction to Quantum Mechanics, p. 382. (Upper Saddle River: Prentice-Hall, 1995)
2. Griffiths, David J. Introduction to Quantum Mechanics, p. 383. (Upper Saddle River: Prentice-Hall, 1995)
3. Ghirardi, G.C., Rimini, A., and Weber, T. 1986, ‘Unified dynamics for microscopic and macroscopic systems’, Physical Review, D 34, 470
4. Pearle, P. "Tales and Tails and Stuff and Nonsense". Published in R. S. Cohen, M. A. Horne, and J. S. Stachel, ed. Experimental Metaphysics – Quantum Mechanical Studies in Honor of Abner Shimony, volume 1 (Dordrecht: Kluwer, 1997)
5. Albert, D. and Loewer, B. "Tails of Schrödinger’s Cat". Published in R. Clifton, ed. Perspectives on Quantum Reality, (Dordrecht: Kluwer, 1996)
6. Clifton, R. and Monton, B. "Losing Your Marbles in Wavefunction Collapse Theories". The British Journal for Philosophy of Science, December 1999. |
3bca7bbc5072627b |
The concept of self-bending light was inspired by quantum mechanics and the realization in 1979 by Michael Berry and Nandor Balazs that the Schrödinger equation could support "Airy" wavepackets of particles, which accelerate without an external force. Then in 2007, Demetrios Christodoulides and colleagues at the University of Central Florida created the optical equivalent of an Airy wavepacket. This is possible because the equation describing paraxial beams – beams in which the constituent rays all travel almost parallel to the direction of the beam's propagation – is mathematically identical to the Schrödinger equation once several parameters are interchanged, such as mass and refractive index.
The Florida team generated a specially shaped laser beam that could self-accelerate, or bend, sideways. The researchers did not bend the laser beam as a whole but rather the high-intensity regions within it. To do this they passed a centimetre-wide ordinary laser beam through a device known as a spatial light modulator that adjusted the phase of the beam at thousands of points across its width. Rather than acting like a lens and focusing all of the beam's constituent rays to a single point, the modulator instead changed the relative phase of the rays such that their interference produced a region of maximum intensity that curved sideways in the shape of a gentle parabola across the beam as it propagated forward, along with a number of fainter regions on one side.
Intriguing characteristics
In addition to this self-bending, the beam's intensity pattern also has a couple of other intriguing characteristics. One is that it is non-diffracting, which means that the width of each intensity region does not appreciably increase as the beam travels forwards. This is unlike a normal beam – even a tightly collimated laser beam – which spreads as it propagates. The other unusual property is that of self-healing. This means that if part of the beam is blocked by opaque objects, then any disruptions to the beam's intensity pattern could gradually recover as the beam travels forward.
A limitation of the Florida work, however, is that Airy beams can only be bent through relatively small angles up to about 15 degrees. This means that they cannot provide the sharp turns needed for manipulation on the micron or nanometre scale.
But then in April this year, Mordechai Segev and colleagues at the Technion-Israel Institute of Technology derived a set of general solutions to Maxwell's equations showing that a non-diffracting non-paraxial beam should exist and that it should accelerate in a circle. A month later, two teams produced such beams in the lab – each bending light a 60-degree arc. One team was led by Xiang Zhang of the University of California, Berkeley in the US and the other by John Dudley of the University of Franche-Comte in France.
Not just circular motion
Now, two independent teams have shown, both theoretically and experimentally, that non-paraxial acceleration along trajectories other than a circle is possible. One group is led by Berkeley's Zhang and it studied both elliptical and parabolic motions via analytical and numerical 2D scalar analysis. The other team is led by Florida's Christodoulides and it considered elliptical motion using numerical 3D vector analysis. In the experiments, both groups used continuous-wave lasers, with a wavelength of 532 nm for Zhang's group and 633 nm for Christodoulides' group, shining them through spatial light modulators with phase variation calculated using special computer programs. In both cases, the groups were also able to bend the light through about 60 degrees.
According to Berkeley group member Peng Zhang, these latest studies could lead to a number of practical applications. These include particle manipulation and the burning of curved channels through air to guide plasmas for remote sensing. He also says they could be useful in medicine, allowing doctors, for example, to image or destroy a tumour behind an organ without destroying that organ. "The self-healing of the beam would be very useful," he adds, "because it would allow you to send energy deep into tissue even with obstacles in the way."
In addition, Xiang Zhang says that the approach can be generalized to any other kind of wave system, such as matter waves, electron waves or acoustics. In fact, he points out, his group is investigating the bending of sound waves. He believes that it should be possible to transport sound energy around corners by manipulating the phase of acoustic waves with a device equivalent to a spatial light modulator.
Ingenious, but not new?
Jérôme Kasparian of the University of Geneva in Switzerland, who was not involved in the latest work, is enthusiastic, explaining that the two groups have "elaborated a general framework to describe and therefore predict" large-angle bending of light. However, Michael Berry of Bristol University in the UK, is less so. He believes that the authors do not make it clear that in their experiments they are not bending light rays themselves but the rays' envelopes, or "caustics". "The technical details in these papers are ingenious and interesting to specialists, and I hope the renewed emphasis will lead to applications," he says. "But while the papers are technically interesting, they are unsurprising because they contain no fundamental new idea."
The research is described in two papers published in Physical Review Letters. |
f509eacbf994beff | Potential energy
Required math: arithmetic
Required physics: Newton’s law, kinetic energy
As we stated in the page on kinetic energy, energy in physics is the ability to do work, and work, in turn, is defined as the product of a force and the distance over which it acts. If a single force {F} acts on a constant mass {m} over a given distance, it will cause the mass to accelerate with a constant acceleration {a}, where these three quantities are related by Newton’s law: {F=ma}. It is important to realize that this formula is a mathematical expression of Newton’s assumption (based on observations) of how the world works. It is not the end result of some complicated mathematical derivation; it is simply stated as the starting point for Newton’s version of physics.
From the page on kinetic energy, we can see that the result of a force acting on a mass for a certain time is that the mass speeds up (due to its acceleration) and after a time {t}, it will have a velocity {v=at}. The energy transferred to the mass by the force is all kinetic energy (energy of motion), and has the value
\displaystyle E_{K}=\frac{1}{2}mv^{2} \ \ \ \ \ (1)
In order for this to happen, the force has to be completely unopposed, which is virtually impossible to arrange in the real world. A mass falling due to gravity may seem to be unopposed, but the friction with the air works against the gravitational force, so that the velocity after a given time in free fall will be less than {v=at}. In fact, a mass falling through the Earth’s atmosphere has a maximum attainable velocity known as the terminal velocity, whose value depends on the mass and the shape of the object. A sheet of paper weighing a few grams reaches its terminal velocity much faster than a small iron pellet of the same mass.
In the case of objects falling through air, the energy due to the gravitational force that is not converted into kinetic energy of the falling object is transferred to the air molecules through which the mass falls. The energy may show up in the form of heat or turbulence in the air, both of which are forms of kinetic energy since they are due to the motion of the air molecules.
But what happens when we actively oppose a force by moving a mass against the direction in which the force acts? We do this whenever we pick up some object, such as lifting a pencil off a desk. To do this, we are generating a force from the muscles in our arm (which is ultimately electrical force, but never mind that for now). If we raise the pencil at a constant velocity, then the amount of force we are generating is exactly equal and opposite to the gravitational force pulling the pencil down. To see this, remember Newton’s first law: an object with no net force acting on it will either remain at rest or move with a constant velocity. If we are moving the pencil at a constant velocity, it must have no net force acting on it, so the upward force we are generating must exactly balance the downward force due to gravity.
However, the force we are exerting to lift the pencil acts through a certain distance, so by the definition of work, we should be transferring some energy to the pencil. Since there is no acceleration, there is no change in the velocity, so clearly this energy is not showing up as kinetic energy. Where is it going?
This is where the idea of potential energy comes in. Whenever work is done by one force against another force, the mass is ‘storing’ this work as potential energy. If the first force (our arm lifting the pencil) is removed (we let the pencil go), then the second force (gravity) is free to act on the object and convert this stored energy back into kinetic energy (the pencil falls, and accelerates as it does so).
As Galileo famously showed, and as countless high school physics students have verified ever since, the gravitational force on an object near the surface of the Earth is proportional to the object’s mass, and can be written as
\displaystyle F_{g}=mg \ \ \ \ \ (2)
where {g} is the acceleration due to gravity, with a value of approximately 9.8 metres per second per second. What this curious set of units means is that for every second in free fall (ideally in a vacuum), an object’s velocity increases by 9.8 metres per second.
Given the gravitational force, we can find how much work we have to do to lift an object by a height {h}: we need to resist a constant force through a distance {h}, so we are doing an amount of work equal to {mgh} (force times distance, remember). This is the amount of energy that is being ‘stored’ in the object and is therefore the amount that would be released if the object is dropped and allowed to fall through the distance {h}. If the object’s entire store of potential energy is allowed to be converted into kinetic energy (by allowing the object to fall the full distance {h} back to its starting point) then the kinetic energy it will have at that point is
\displaystyle E_{K}=\frac{1}{2}mv^{2}=mgh \ \ \ \ \ (3)
from which we can deduce its velocity as
\displaystyle v=\sqrt{2gh} \ \ \ \ \ (4)
which, by the way, is independent of the mass, so Galileo was right after all: all objects fall at the same rate.
A constant force like gravity near the Earth’s surface is a particularly easy example since we can find the amount of work done against the force by simple multiplication. Most real forces, of course, aren’t as cooperative, and vary with distance. This means that we need to use calculus to find the amount of work done, and thus the potential energy stored, when we move a mass around in such force fields. However, the principle is the same: find the amount of work that is needed to move the mass from point A to point B and the result is the potential energy stored in the mass. When the mass is released, the potential energy will be converted to kinetic energy (or released in some other way if the object is not totally free to move under the influence of the force) if and when it gets back to point A.
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85b57cab09614db2 | Take the 2-minute tour ×
This is a thought experiment, so please don't treat it too harsh :-)
Short: If we could isolate two places A and B in the universe from all and any interaction with the surroundings, is there a physical law which states "if something is dropped in place A, it has to stay there"?
Long version: Let's assume that the energy of the whole universe is fixed. Let's further assume that it is (by some trick) possible to completely isolate a box of 1m^3, say in the center of a planet (all gravitational and centrifugal forces cancel themselves out), the mass of the planet shields against radiation and we use a trick to shield against neutrinos or we ignore them since the rarely interact with matter).
How does an object behave when it has no interaction with the rest of the universe whatsoever? If I put an object in a box described above and I have several such boxes, would it matter in which box the object is?
Is there a law which says "even if no one knows, the object has to stay where it is"? Or is that just our expectation based on everyday experience?
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3 Answers 3
up vote 4 down vote accepted
Classically Sklivvz's answer would be correct. But in quantum world the story is not quite over. In the following I'll talk about particles (because for them quantum effects are more apparent) but it would also be true for bigger objects (although the bigger the object the more improbable the "teleportation" would be).
First, from the point of view of Quantum Mechanics and for a while assuming that you can really cancel all forces on the particle you are essentially investigating a double-well potential. There will always be some tunelling between two boxes. If the boxes are too far away then it will be very improbable (where this improbability increases exponentially with distance) but surely possible that if you look into second box after some time, you'll find your particle there.
Now, in reality the picture is complicated by Quantum Field Theory. First, it stops to make sense to talk about particles because they are indistinguishable and also they can be created out of nothing. Related to this is the fact that you can't ever dispose of all interaction because vacuum itself is very lively place! There are particles created and annihilated all the time and this has observable effects on any object (see Casimir effect).
So you would need you object to be big enough so that it becomes distinguishable (with high probability). But as already stated the bigger the object, the lower the probability that it leaves the box.
The conclusion, as you might have guessed beforehand, is that with extremely high probability nothing interesting will happen at all.
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correct - by "object" I have assumed a classical-sized test object. – Sklivvz Nov 28 '10 at 15:45
I missed vacuum fluctuations; thanks for pointing that out. – Aaron Digulla Nov 28 '10 at 18:02
Interestingly, if the two boxes were close together, a particle-antiparticle pair could be spontaneously created between the two, and the antiparticle annihilate with the original particle. The result would be that the particle has disappeared from one box and appeared in the other - in other words, the teleportation mentioned in the question title. – Phil H Nov 30 '10 at 13:24
Of course: Newton's first law!
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I agree for classical physics. But I don't change it's state of motion, I just change its space-time coordinate in an instant. So the first law doesn't really apply. – Aaron Digulla Nov 28 '10 at 18:04
@Aaron: you can't just change an object's spacetime coordinate by a finite amount instantaneously, unless you have something which can exert an infinite force at your disposal (this doesn't exist in the real world). Doing so would be a violation of Newton's second law in classical physics, or of the Schrödinger equation in quantum mechanics. – David Z Nov 28 '10 at 22:19
@David: So how does an electron tunnel? As I understand it, the same rules could apply to macroscopic objects (only the probability is way too low to experience it). – Aaron Digulla Nov 29 '10 at 14:17
How does an object behave when is has no interaction with the rest of the universe whatsoever? Like every object you ever studied in a physics class. Interaction with the rest of the universe is the hard part that we always abstract away so we can isolate some particular interaction of interest. A perfectly isolated system would always behave in perfect accordance with the rules governing whatever sort of object it was.
In quantum terms (since you've got "quantum" in the question title), what you're describing is an infinite square well. That is, the potential barrier for the system in question is of infinite height, perfectly isolating the system from everything else. The wavefunction at the edges of the box would be exactly zero, and the wavefunction everywhere outside the box would be zero. This is the only system in quantum physics that does not have some probability (however infinitesimal) of turning up at some different location. This is, of course, a textbook idealization and not anything you could actually make in reality.
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c0554184debb9df2 | 2.1.4 Periodic Potentials and Bloch's Theorem
The Bloch Theorem
In the most simplified version of the free electron gas, the true three-dimensional potential was ignored and approximated with a constant potential conveniently put at 0 eV (see quantum mechanics script as well)
The true potential, however, e.g. for a Na crystal including some energy states, is periodic and looks more like this:
Threedimensional potential
Semiconducting properties will not emerge without some consideration of the periodic potential - we therefore have to solve the Schrödinger equation for a suitable periodic potential. There are several (for real potentials always numerical) ways to do this, but as stated before, it can be shown that all solutions must have certain general properties. These properties can be used to make calculations easier and to obtain a general understanding of the the effects of a periodic potential on the behavior of electron waves.
The starting point is a potential V(r) determined by the crystal lattice that has the periodicity of the lattice, i.e.
V(r) = V(r + T)
With T = any translation vector of the lattice under consideration.
We then will obtain some wavefunctions yk(r) which are solutions of the Schrödinger equation for V(r). As before, we use a quantum number "k" (three numbers, actually) as an index to distinguish the various solutions.
The Bloch theorem in essence formulates a condition that all solutions yk(r), for any periodic potential V(r) whatsoever have to meet. In one version it ascertains
yk(r) = uk(r) · exp (i · k · r)
With k = any allowed wave vector for the electron that is obtained for a constant potential, and uk(r) = arbitrary functions (distinguished by the index k that marks the particular solution we are after), but always with the periodicity of the lattice, i.e.
uk(r + T) = uk(r)
Any wavefunction meeting this requirement we will henceforth call a Bloch wave.
The Bloch theorem is quite remarkable, because, as said before, it imposes very special conditions on any solution of the Schrödinger equation, no matter what the form of the periodic potential might be.
We notice that exactly as in the case of the constant potential , the wave vector k has a twofold role: It is still a wave vector in the plane wave part of the solution, but also an index to yk(r) and uk(r) because it contains all the quantum numbers, which ennumerate the individual solutions.
Blochs theorem is a proven theorem with perfectly general validity. We will first give some ideas about the prove of this theorem, and then discuss what it means for real crystals. As always with hindsight, Blochs theorem can be proved in many ways; the links give some examples. Here we only look ageneral outlines of how to prove the theorem:
It follows rather directly from applying group theory to crystals. In this case one looks at symmetry properties that are invariant under translation.
It can easily be proved by working with operator algebra in the context of formal quantum theory mathematics.
It can be directly proved in simple ways - but then only for special cases or with not quite kosher "tricks".
It can be proved (and used for further calculations), by expanding V(r) and y(r) into a Fourier series and then rewriting the Schrödinger equation. This is a particularly useful way because it can also be used for obtaining specific results for the periodic potential. This proof is demonstrated in detail in the link, or in the book of Ibach and Lüth.
Blochs theorem can also be rewritten in a somewhat different form, giving us a second version:
yk(r + T) = yk(r) · exp(ikT)
This means that any function yk(r) that is a solution to the Schrödinger equation of the problem, differs only by a phase factor exp(ikT) between equivalent positions in the lattice.
This implies immediately that the probability of finding an electron is the same at any equivalent position in the lattice since, exactly as we expected, because
[yk(r + T)]2 = [yk(r)]2 · [exp(ikT)]2 = [yk(r)]2
Since [exp(ikT)]2 = 1 for all k and T.
That this second version of Blochs theorem is equivalent to the first one may be seen as follows.
If we write the wave function in the first form yk(r) = uk(r) · exp(ikr) and consider its value at an equivalent lattice position r + T we obtain
yk(r + T) = uk(r + T) · exp [ik · (r + T)] = uk(r) · exp (ikr) · exp (ikT) = yk(r) · exp(ikT)
= uk(r ) = yk(r) q.e.d
Blochs theorem has many more forms and does not only apply for electrons in periodic potentials, but for all kinds of waves, e.g. phonons. However, we will now consider the theorem to be proven and only discuss some of its implications.
Implications of the Bloch Theorem
One way of looking at the Bloch theorem is to interprete the periodic function uk(r) as a kind of correction factor that is used to generate solutions for periodic potentials from the simple solutions for constant potentials.
We then have good reasons to assume that uk(r) for k vectors not close to a Brillouin zone will only be a minor correction, i.e. uk(r) should be close to 1.
But in any case, the quantity k, while still being the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = k, or of its wavelength l = 2p/k, since:
The momentum of the electron moving in a periodic potential is no longer constant (as we will see shortly); for the standing waves resulting from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while k is not.
There is no unique wavelength to a plane wave modulated with some arbitrary (if periodic) function. Its Fourier decomposition can have any spectra of wavelengths, so which one is the one to associate with k?
To make this clear, sometimes the vector k for Bloch waves is called the "quasi wave vector".
Instead of associating k with the momentum of the electron, we may identify the quantity k, which is obviously still a constant, with the so-called crystal momentum P, something like the combined momentum of crystal and electron.
Whatever its name, k is a constant of motion related to the particular wave yk(r) with the index k. Only if V = 0, i.e. there is no periodic potential, is the electron momentum equal to the crystal momentum; i.e. the part of the crystal is zero.
The crystal momentum P, while not a "true" momentum which should be expressible as the product of a distinct mass and a velocity, still has many properties of momentums, in particular it is conserved during all kinds of processes.
This is a major feature for the understanding of semiconductors, as we will see soon enough!
One more difference to the constant potential case is crucial: If we know the wavefunction for one particular k-value, we also know the wavefunctions for infinitely may other k-values, too.
This follows from yet another formulation of Bloch's theorem:
If yk(r) = uk(r) · exp(ikr) is a particular Bloch wave solving the Schrödinger equation of the problem, then the following function is also a solution.
yk + g(r) = uk + g(r) · exp i[k + g]r
With g = arbitrary reciprocal lattice vector as always.
This is rather easy to show and you should attempt it yourself. It has a far reaching consequence:
If yk(r) is a solution of the Schrödinger equation for the system, it will always be associated with a specific energy E(k) which is a constant of the system for the particular sets of quantum numbers embodied by k. Since yk(r) is identical to yk + g(r), its specific energy E(k + g) must be identical to E(k), or
E(k + g) = E(k)
This is a major insight. However, there is also a difficulty:
The equation does not mean that two electrons with wave vectors k and k + g have the same energy (see below), but that any reciprocal lattice point can serve as the origin of the E(k) function.
Lets visualize this for the case of an infinitesimaly small periodic potential - we have the periodicity, but not a real potential. The E(k) function than is practically the same as in the case of free electrons, but starting at every point in reciprocal space:
Energy parabolums
Indeed, we do have E(k + g) = E(k), but for dispersion curves that have a different origin.
We have even more, we now have also many energy values for one given k, and in particular all possible energy values are contained within the first Brilluoin zone (between -1/2g1 and +1/2g1 in the picture).
It thus is sufficient to consider only the first Brillouin zone in graphical representations - it contains all the information available about the system.This is called a reduced representation of the band diagram, which may look like this:
reduced band diagram
The branches outside the 1. BZ have been "folded back" into the 1. BZ, i.e. translated by the approbriate reciprocal lattice vector g. To make band diagrams like this one as comprehensive as possible, the symmetric branch on the left side is omitted; instead the band diagram in a different direction in reciprocal space is shown.
Again, this looks like a specific electron could now have many energies all at once - this is, of course, not the case.
Different energies, formerly distinguished by different k - vectors, are still different energies, but now the branches in the 1st Brillouin zone coming from larger k - vectors belong to different bands. Every energy branch in principle should carry an index denoting the band; this is, however, often omitted.
The identical construction, but now for the energy functions of a periodic potential as given before, now looks like this
Reduced band diagram 2
We now have band gaps - regions with unattainable energies - in all directions of the reciprocal lattice.
A numerical example for the Kroning Penney Model is shown in this link.
What does this mean for a particular electron, say one on the lowest branch of the blue diagram with the wave vector k1? It has a definite energy E associated with it.
But it also could have larger energies: all the values obtained for the same k but in higher branches of the band diagram.
For a transition to the next higher branch the energy DE1 is needed. It has to be supplied from the outside world.
After the transition the electron has now a higher energy, but the wave vector is the same. But wait, in the reduced band diagram, we simply omitted a reciprocal wave vector, so its wave vector is actually k1 + g. If we index the situation after the transition with "2", before with "1", we have the following equations.
E2 = E1 + DE
k2 = k1 + g
|k1| ¹ |k2|
This is simply Braggs law, but now for inelastic scattering, where the magnitude of k may change - but only by a specified amount tied to a reciprocal lattice vector.
Since we interpreted k as crystal momentum, we may consider Braggs law to be the expression for the conservation of momentum in crystals.
The reduced band diagram representations thus allow a very simple graphical representation of allowed transitions of electrons from one state represented by to another state (E2 , k2): the states must be on a vertical line through the diagram, i.e. straight up or down.
An alternative way of describing the states in the spirit of the reduced diagram is to use the same wave vector k1 for all states and a band index for the energy. The transition then goes from (En , k) to (Em, k) with n, m = number of the energy band involved.
The possibility of working in a reduced band diagram, however, does not mean that wave vectors larger than all possible vectors contained in the 1. BZ are not meaningful or do not exist:
Consider an electron "shot" into the crystal with a high energy and thus a large k - e.g. in an electron microscope. If you reduce its wave vector by subtracting a suitable large g vector without regard to its energy and band number, you may also reduce its energy - you move it. e.g., from a band with a high band number to a lower one. While this may happen physically, it will only happen via many transitions from one band to the next lower one - and this takes time!
Most of the time in normal applications the electron will contain its energy and its original wave vector. And it is this wave vector you must take for considering diffraction effects! An Ewald (or Brillouin) construction for diffraction will give totally wrong results for reduced wave vectors - think about it!
If you feel slightly (or muchly) confused at this point, that is as it should be. Blochs theorem, while relatively straightforward mathematically, is not easy to grasp in its implications to real electrons. The representation of the energy - wave vector relationship (the dispersion curves) in extended or reduced schemata, the somewhat unclear role of the wave vector itself, the relation to diffraction via Braggs law, the connection to electrons introduced from the outside, e.g. by an electron microscope (think about it for minute!), and so on, are difficult concepts not easily understood at the "gut level".
While it never hurts to think about these questions, it is sufficient for our purpose to just accept the reduced band structure scheme and its implications as something useful in dealing with semiconductors - never mind the small print associated with it.
However, if you want to dig deeper: These problem are to some extent rooted in the formal quantum mechanics behind Blochs theorem. It has to do with Eigenvectors and Eigenvalues of Operators; a glimpse of these issues can be found in an advanced module.
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© H. Föll (Semiconductor - Script) |
fde3c3afe5607fa8 | onsdag 9 oktober 2013
Nobel Prize in Chemistry Awarded for Not Solving Schrödinger's Equation
Picture from presentation of Nobel Prize in Chemistry 2013: Multiscale Models
• the development of multiscale models for complex chemical systems:
2 kommentarer:
1. I would call this mathematical (or calculational) heuristics, and it is NOT physical science. It is guessing, and guessing by computer at that. They might as well admit they are using fudge factors, and have done with the "progress in science" propaganda, because it is degeneration of science, not progression.
2. Well, a man can only what a man can do, and if solving the Schrödinger equation is beyond human capability, then you have to solve some other equation and that is not necessarily degeneration. It could be just realism, but it could also be fake science. |
a052a4887f130e9b | Monday, January 9, 2017
Are quanta particles or waves?
The title of this post is an age-old question isn't it? Particle or wave? Wave or particle? Many have rightly argued that the so-called "wave-particle duality" is at the very heart of quantum weirdness, and hence, of all of quantum mechanics. Einstein said it. Bohr said it. Feynman said it. Two out of those three are physics heroes of mine, so that's a majority right there.
Feynman, when talking about what we now call the wave-particle duality, was referring to the famous "double-slit experiment". He wrote (in his famous Feynman Lectures, Chapter 37 of Volume 1, to be precise):
Richard Feynman (1918-1988)
Source: Wikimedia
"We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."
So what is Feynman talking about here? Instead of launching on a lengthy exposition of the double-slit experiment, as luck would have it I've already done that, in a blog post about the quantum eraser. That post, incidentally, was No. 6 in the "Quantum measurement" series that starts here. You don't necessarily have to have read all those posts to follow this one, but believe me, it would help a lot. At the minimum, start at No. 6 if you're not already familiar with the double-slit experiment. But you'll get a succinct introduction to the double-slit experiment below anyway.
Alright, back to quantum mechanics. Actually, step back a little bit more, to classical mechanics. In classical physics, there is no duality between waves and particles. Waves are waves, and they would never behave like particles. For example, you can't kick a wave, really, no matter what the surfer types tell you. Particles on the other hand, do not interfere with each other as waves do. You can kick particles (kinda), and you can count them. You can't count waves.
What Bohr, Einstein, and Feynman are trying to tell you is that in quantum mechanics (meaning the real world, because as I have told you before, classical mechanics is an illusion, it does not exist) the same stuff can be either particle OR wave. Not both, mind you. Here's what Einstein said about this, and to tell you the truth, this statement sounds like he's been hanging out with Bohr far too much:
A. Einstein (1879-1955)
Source: Wikimedia
I've used a picture of Einstein in 1904 here, because you've seen far too many pics of him sticking out his tongue and hair disheveled. He wasn't like that most of the time when he made his most important contributions.
Lest you think that the troubles these 20th century physicists had with quantum mechanics is the stuff of history, think again. In 2012, a mere 5 years ago, experimenters from Germany (in the lab of the very eminent Wolfgang Schleich) claimed that they had collected evidence that a quantum system can be both particle and wave at the same time. Such an observation-if true-would run afoul of Bohr's "duality principle", which declared that a quantum system can only be one or the other, depending on the type of experiment used to examine the system. One or the other, but never both
Rest assured though, analyzing results of the Schleich experiment in a different way reveals that all is well with complementarity after all, as was pointed out by a team at the University of Ottawa, led by the equally eminent Robert Boyd. (You can read an excellent summary of that controversy in Tom Siegfried's piece here.) What all this fighting about duality should teach you is that this is not at all a solved problem. As recently as a few days ago, Steven Weinberg (who, full disclosure, has also been in my pantheon of physicists ever after I read his "First Three Minutes" at a very tender age) wrote about the particle-wave duality in the New York Review of Books. I hope that he reads this post, because it may alleviate some of his troubles.
In this piece, entitled "The Trouble with Quantum Mechanics", Weinberg admits to being as puzzled as his predecessors Einstein, Bohr, and Feynman, about the true nature of quantum physics. How can we understand, he muses, that quantum dynamics is governed by a deterministic equation (the Schrödinger equation), yet when we try to measure something, then all we can muster is probabilities? "So we still have to ask", Weinberg writes, "how do probabilities get into quantum mechanics?"
How indeed. You know of course, from reading my diatribes, that this is a question I am interested in myself. I have obliquely hinted that I think I know where the probabilities are coming from (if you can find the relevant post) and that one day I'll write a detailed account of that idea (it's 3/4 written already, actually). But today is not that day. Having convinced you that the particle-wave duality is still a very hot topic in quantum physics, let me take on that particular subject first.
What I want to do in this blog post is to make you think differently about the complementarity principle. What I'm going to tell you is that you should stop thinking in terms of "particle or wave". It is a false dichotomy. It is a false dilemma because quantum systems are neither particle nor wave. Those two are classical concepts, after all. Strictly speaking, quantum systems are quantum fields. But this is not the time to delve into quantum field theory, so instead I will try to marshal the tools of quantum information theory to tell you what is really complementary in quantum measurement, what it is that you can have "only one of", and what it is that is being "traded-off". You don't exchange a bit of particle for a bit of wave, this much I can tell you right here.
To do this, I have to introduce you to some very counter-intuitive quantum stuff. Now, you might argue: "All quantum stuff is counter-intuitive", and I'd have to agree with you if all your intuition is classical. What I am going to tell you is stuff that even baffles seasoned quantum physicists. I'm going to tell you about quantum experiments where the "nature" of the quantum experiment that you perform can be changed after you've already completed the experiment!
Let me remind you right here, that the--also very eminent--Niels Bohr tried to teach us that whether a quantum system appears as a particle or as a wave depends on the type of experiment you subject it to. Here I'm telling you that this is a bunch of hogwash, because I'll show you that when you do an experiment, you can change whether it is a "particle"- or a "wave"-experiment long after the data have been collected!
I know you're not shocked at my dissing Bohr as I have a habit of doing so. But I'm in good company, by the way, if you read what Feynman wrote about Bohr in his "Surely You're Joking" series.
"Alright I bite", one of you readers exclaimed just now, "how do you retroactively change the type of experiment you make?"
Glad you asked. Because now I can talk about John Archibald Wheeler. Wheeler was not a conventional physicist: Even though his early career as a nuclear physicist led to several important contributions to the Manhattan project, he was also interested in many other areas of physics. Indeed, he was a central figure in the "revival" of general relativity theory. (That theory had gone a bit out of fashion when people realized that many predictions of the theory were difficult to measure.) Wheeler co-authored what many (including myself) think is the best book on the topic: "Gravitation" (with Charles Misner and Kip Thorne). That book is often just referred to as "MTW".
John Archibald Wheeler (1911-2008).
Source: University of Texas
I never got to meet Wheeler, perhaps because I entered the field of quantum gravity too late. While Wheeler has been influential in the field of quantum information, it really was his gravity work that had the most lasting impact. He invented the terms "black hole" and "wormhole", after all. His most influential contribution to quantum information science is, undoubtedly, the "delayed choice" gedankenexperiment. Let me explain that to you.
Wheeler's thought experiment examines the question of whether a photon, say, takes on wave or particle nature before it interacts with the experiment, sensing (in a way) what kind of experiment is going to be performed on it. In the simplest version of the delayed choice experiment, the nature of the experiment would be changed "after the photon had made up its mind" whether it was going to play the role of particle, or whether it would make an appearance as a wave. Needless to say, this is of course not how quantum mechanics works, and Wheeler was fully aware of it. His interpretation was that a photon is neither wave nor particle, and that it takes on one of the two "coats" only when it is being observed. I'm going to tell you that I agree with the first part (the photon is neither wave nor particle), but I disagree with the second part: it does not in fact take on either particle or wave nature after it is observed. It never ever takes on such a role.
If you think about it, the idea that a system only "comes into being by being observed" is preposterous (however, such a thought was quite in line with some other of Wheeler's philosophies). Measurements are interactions with other systems just as much as any other interactions are: there is nothing special about measurement. This is, in essence, what I'm going to try to convince you of.
Even though the reasoning behind the delayed-choice experiment is preposterous, it has generated an enormous amount of work. Let's first look at how we may set up such an experiment. Below is an illustration of a double-slit experiment from Feynman's famous lecture, where he replaced photons by electrons shot out of an electron gun (such devices are perfectly reasonable and feasible). Note that Caltech, where Feynman spent the majority of his career, has made these lectures freely available. The particular chapter can be accessed here
Fig. 1: An interference experiment with electrons. (Source: Feynman Lectures on Physics)
Later on, we're going to be using photons instead of electrons for the quantum system, because experiments are much easier with photon beams as opposed to electron beams. In that case, we are going to assume that any light is going to be so faint that it can't be thought of as the classical light waves that give rise to Young's interference fringes. Then, at any point in time, there will be at most one photon between the double-slit and the detector, so you have to think about single photons either taking one or the other, or both paths, through the double-slit experiment.
Quantum mechanics predicts that a single electron takes both paths to create the interference pattern in the figure above at (c). Thus, it must somehow interfere with itself, which is difficult to imagine if you think of the electron as a particle. (Which of course it is not). Can we force it to behave as a particle? Suppose you put a particle detector between the wall and the backstop: one behind slit 1, and one behind slit 2. If you get a "hit" on either detector, then you know which path the electron travelled. (You can do this experiment without actually removing the electron, so that you can still get patterns on the screen.) When you obtain this "which-path" information, the interference pattern disappears: you've forced the electron to behave as a particle.
Wheeler's idea was this: Suppose the distance between the wall and the backstop is very, very large. If you do not put the contraption that will measure which path the electron took (the "which-path detector") into the experiment, the electron would have no choice but to go along both paths, ready to interfere with itself and create the interference pattern on the screen. But suppose you bring in the "which-path" apparatus after the electron has passed the slit, but before it is going to hit the screen. Is the electron wave function that is on the "other path" going to "change its mind", or go backwards? What would happen? The thought experiment very nicely illustrates how preposterous the idea is that the experiment itself determines "what the quantum system is", as changing the experiment mid-flight cannot possibly change the nature of the electron.
The experiment I'm going to describe to you (the delayed-choice quantum eraser experiment) has in fact been carried out several times now, and drives Wheeler's idea to the extreme. The choice of experiment (insert the "which-path" detector or not), can be made after the electron has hit the screen! If you are a reader for whom this is immediately obvious, then congratulations (and consider a career in quantum physics, if this is not already your career). It is indeed completely obvious if you understand quantum mechanics, but let me walk you through it anyway.
First, if it was the experiment that determines the nature of the quantum system (particle or wave), how can you change the experiment after it already has occurred? That this is possible is also due to the peculiarities of quantum physics, and it is also the hardest to explain. I'll do it with photons rather than electrons, as this is the experiment that was carried out, and it is also the description I used in the paper that I'm really writing about. You knew this was coming, didn't you?
We can do double-slit experiments with photons just as with electrons: we just have to turn down the intensity of light such that individual photons can be registered on a phosphorescent screen. When you see the screen light up at a particular spot (or, in more modern times, a pixel on a CCD detector lights up), you interpret it that a photon has hit there. Often, the double-slit is replaced by a Mach-Zehnder interferometer, but you shouldn't worry about such technicalities: you can in fact use either.
To pull off this feat of changing the experiment after the fact, you have to create an entangled pair of photons first. You already know what an entangled pair (a "Bell-state") is, because I wrote about it several times: for example in the context of black holes here, and in the context of quantum teleportation and superdense coding here. This pair of photons is also sometimes called an Einstein-Podolsky-Rosen (EPR) pair, because that trio first described a similar entangled state in a very famous paper in 1935.
Let's create such a pair by entangling the "polarization" degree of freedom of the photon. This is the part that is a bit more complicated: to understand it, you have to understand polarization.
Every photon can come in two different polarization states, but what these states are depends on how you decide to measure them. This will be crucial, because this is in fact how you change the measurement after the fact. The thing to know about an entangled pair is that it is in a superposition of those two states. Suppose we use as basis for the photon polarization the "horizontal/vertical" basis. That means that if a photon is polarized horizontally, and you put a filter in front of it that only allows vertical polarization to go through, then out comes nothing. Polarization is, if you will, a photon's way of wiggling. Below is a picture which shows the photon wiggling in the "vertical" and in the horizontal way. But they can also wiggle in the "circular-left" and "circular-right" way. In fact, it can wiggle in an infinite number of "opposing ways", and these are related to each other by a unitary transformation.
Fig. 2: One way of depicting photon polarization.
The way a photon is polarized can be changed by an optical element (a "wave plate"), and this ability will be key in the experiment. Suppose we begin with a pair of photons A and B in a Bell-state, written in terms of the horizontal $|h\rangle$ and vertical $|v\rangle$ polarization eigenstates:
$|\Psi\rangle_{AB}=\frac1{\sqrt2}(|h\rangle_A|v\rangle_B+|v\rangle_A|h\rangle_B)$ (1)
You notice that neither of the photons has a defined state, but if I measure one of them (say A) and find that my detector says it is in an $|h\rangle$ state, then I can be sure that measuring B will give you "v", no matter whether you do the measurement now, or a year later with a detector placed a light year away. This is precisely what Einstein could not stomach, calling this mysterious bond "spooky action at a distance", but a careful analysis reveals that there is no "action" at all: signals cannot be sent using this bond.
But here's the thing: I can measure photon B either in the $h,v$ coordinate system, or in another one. This will become crucial, so keep this in mind. But for the moment let's forget that a "copy" of photon A (the entangled partner) is flying out there, possibly to a measurement device a light-year away. Actually, there is nothing a light year away from us, so let's say we are far in the future and the detector is on Proxima Centauri, about 4 and a quarter light years away. It'll just be a longer experiment.
Photon A now goes through a double-slit, just as the electrons in Figure 1. Now we'll do the "are you a particle or a wave" measurement. We do this by putting so-called "quarter-wave plates" in the path of the photons. When you do this, you entangle the polarization of the photon with the spatial degree of freedom (namely "left slit" or "right slit"). Once you've done this, you only have to measure the polarization of photon A to know whether it went through the left or right slit. In a way, you've tagged the photon's path by the polarization. After doing this, you will lose the interference pattern. You can either have an interference pattern (and we say that the photon wavefunction is "coherent"), or you can have "which-path" information, which makes the wavefunction incoherent. Or so people thought for a long time. It turns out that you can also have a a little bit of both, but you can't have both full which-path information, and full coherence: there is a tradeoff. And that tradeoff depends on the angle by which you rotate the polarization basis. In the description above, we used "quarter-wave" plates, which give you full information, and zero coherence. Choose something other than 45 degrees (that's the quarter wave), and you can get a little bit of both.
It turns out that there is a simple relationship that quantifies this tradeoff in terms of the angle you choose to do the tagging with. Let's call this angle $\phi$. We can then define the "distinguishability" $D$ and the "visibility" $V$, where $D^2$ measures how well you can distinguish the photon paths (a measure of which-path information), while $V^2$ quantifies the visibility of the interference fringes (a measure of the coherence of the wavefunction). A celebrated inequality (due to Greenberger and Yasin [1]) states that
$D^2+V^2\leq1$ (2)
Now, according to what I just wrote, choosing the angle of the wave plate when performing the which-path entangling operation chooses the experiment for you: Set it at 0 degree and you do not entangle at all, so that no which-path information is obtained (then $D^2=0$ and $V^2=1$). Set it at $\phi=\pi/4$, and you get perfect which-path information, and no visibility. How can you choose the experiment after the fact, when you have to choose the angle when setting up the experiment? How?
So the following is what makes quantum mechanics so beautiful. You can actually do this because when I described the experiment to you, I did not (it turns out) use an entangled EPR pair as the input, I used a photon in a defined polarization state, such as $|h\rangle$. I did not tell you about this because it would have confused you. I needed you to understand how to extract which-path information first, and how doing it gradually will gradually destroy coherence.
Now take a deep breath, and read very slowly.
If the input to the two-slits (and therefore to the "which-path" detector that entangles polarization and path) is the EPR state Eq. (1), you actually do not get any which-path information using the quarter-wave plate. This is because when the photon "comes in", it is not in a defined polarization state. If it was not in a defined state, you extract nothing. So for that setup, $V^2=1$ even though $\phi=\pi/4$.
Now one more deep breath after you digested this bit. Maybe take two, just to be safe.
Whether the state that comes in to the two slits is indeed Eq. (1) is up to the person at Proxima Centauri, a year after that data was recorded on the CCD screen on Earth. This is because of what is $|h\rangle$ and what is $|v\rangle$ is determined by how you measure it. A quantum system does not have a state until you say how you measure it. It will be in the $h,v$ basis if that is the basis of your measurement device. It will be in the $R,L$ (right-circular, left-circular) basis, if that is instead what you will choose to examine it with. Or it could be anything in between.
I wrote about this at length in the blog post about the collapse of the wavefunction, within the "On quantum measurement" series. (Rightfully, the present post really should be "On quantum measurement. Part 8, but I decided to make it stand alone). Please go back to that if the two breaths did not help. There is also an intriguing parallel to how Shannon entropy is not defined until you determine how you will be measuring it, as I wrote about in "What is Information-Part 1". The deeper reason for this is that all of physics is about the relative state of measurement devices. Mark my words.
The reason our person at Proxima Centauri handling photon B actually prepares the state is because photon A is not "projected" at any point of the experiment. This could be done, of course, but that is a different experiment. So now we can see how the delayed-choice experiment works: If Proxima Centauri person (PCP, for short) measures at an angle $\theta=0$ with respect to the preparation Eq. (2), then the photon is in a defined state (no matter whether the outcome is $h$ or $v$) and only then do you actually extract which-path information. In that case, visibility $V^2=0$. If PCP measures at $\theta=\pi/4$ on the other hand, the entanglement operation (the "tagging") does not work: it is as if the measurement by PCP "erased" the tagging, and $V^2=1$ instead. So indeed, a measurement far in the future (well, here more than four years in the future) will determine what kind of an experiment is done on the photon. The event far in the future will determine whether the photon appeared as a particle, or a wave. Weird, right?
What is that you ask? How can an event far in the future affect the data that are stored on a device far in the past?
I didn't say it did, did I? Of course it does not. The truth is much more magical. Without going into all the details here (but which you can read about in any paper about the Bell-state quantum eraser, or indeed my own paper referenced below), the result of the measurement by PCP in the future contains crucial information about how to decode the data in the past, information that is akin to the key in a cryptography procedure.
Yes, cryptographic. That is indeed what I wrote. You will only be able to decipher $D^2$ and $V^2$ when the measurement in the future (which is really a state preparation in the past) is available to you. That is the true magic of quantum mechanics. Without it, you won't be able to see any fringes in the data. But with it, you may be able to reconstruct them to full visibility, if that is how the photon was measured at Proxima Centauri.
How do I know any of this is true? Because we (my student Jennifer Glick and I) analyzed the entire experiment in terms of quantum information theory, and ultimately were able to write down the equations that describe discrimination and visibility (coherence) entirely in terms of entropies and information, in [2] (Jennifer did all the calculations and wrote the first draft of the manuscript). Clearly, "which-path information" should have an obvious information-theoretic rendering, but it turns out that this is actually a little bit tricky because it really is a "conditional information". But it turns out that "coherence" (or "visibility") can also be measured information-theoretically. And lo and behold, the two are related. In our description, they are related by a common information-theoretic identity: the chain rule for entropies. According to that identity, information $I$ and coherence $C$ (as a function of the PCP angle $\theta$) are related so that
$I(\theta)+C(\theta)=1$ (3) .
In a simple qubit model, the information and coherence take on extremely simple forms, namely $I(\theta)=H[\sin^2(\theta+\pi/4)]$ with $C(\theta)=1-H[\sin^2(\theta+\pi/4)]$, where $H[p]$ is the standard Shannon entropy function $H[p]=-p\log(p)-(1-p)\log(1-p)$. And take a look at how our information-theoretic quantities compare to the quantum optical measures of discrimination and visibility in Fig. 3 below. It almost looks like that discrimination and visibility (coherence) should have been defined information-theoretically from the outset, doesn't it?
Fig.3: Top: Which-path information (solid line) and coherence (dashed line) in terms of quantum information theory. Bottom: Discrimination (solid) and visibility (dashed) in quantum optics. $Q$ refers to the quantum state at the beam-splitter, and $D_A$ and $D_B$ refer to polarization detectors. From [2].
So what does all this teach us about quantum mechanics in the end (besides, of course, that quantum mechanics is awesome)? We have learned at least two things. Quantum systems are not either particle or wave. They are in fact neither because both concepts are classical in nature. This, to some extent, I stipulate we knew already. Wheeler knew it. (Bohr, I contend, not so much). But what I've shown you is that quantum systems don't "change their colors" after measurement either, as Wheeler had advocated. They remain "neither", even when we think we pinned them down, because what I've shown you is that you can have them take on this coat or that, or any in between, years after the ink has dried (I mean, after the data were recorded). They (the photons, electrons, etc.) are not one or the other. They appear to you the way you choose you want to see them, when you interrogate a quantum state with classical devices.
Those devices cannot reveal to you the reality of the quantum state, because the devices are classical. Don't hate them because of their limitations. Instead, use them wisely, because what I just showed you is that, if used in a clever manner, they enable you to learn something about the true nature of quantum physics after all. As, for example, the experiment in [3] does.
[1] D.M. Greenberger and A. Yasin, "Simultaneous wave and particle knowledge in a neutron interferometer. Physics Letters A 128 (1988) 391-394.
[2] J.R. Glick and C. Adami, "Quantum information theory of the Bell-state quantum eraser". Phys. Rev. A 95 (2017) 012105. Full text also on arXiv
Note: Jennifer Glick is first author on this paper because she performed all calculations in it and wrote the first draft.
[3] Y.H. Kim, R. Yu, S.P. Kulik, Y.H. Shih, and M.O. Scully, “Delayed “choice” quantum eraser,” Phys Rev Lett 84 (2000) 1-5. |
d79ac78257e18216 | light as both particle and wave Fabrizio Carbone/EPFL
This accomplishment is ushering in a new era of quantum holography, which will give scientists a new way of looking at quantum phenomena.
Quantum holograms
Unlike photography, holography recreates the spatial structure of objects, giving us their 3-D shapes. The technique takes advantage of something called classical interference, which is when two waves meet and form a new wave.
But classical interference is impossible with photons, since their phases (a property of waves) are constantly fluctuating. So the Warsaw physicists tried to give quantum holograms a taste of their own medicine by using quantum interference, in which photons' wave functions (which have to do with the probability of the particle being in a particular state) interact.
"Wave function is a fundamental concept in quantum mechanics and the core of its most important principles, the Schrödinger equation," according to a press release. "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."
So why photons?
While filming the behavior pairs of photons, Radoslaw Chrapkiewicz and Michal Jachura, two of the researchers, noticed something called two-photon interference.
In two-photon interference, pairs of distinguishable photons act randomly when entering a beam splitter (which divides a ray of light). But nondistinguishable photons exhibit quantum interference, which affects their behavior. The pairs are always either transmitted or reflected together.
"Following this experiment, we were inspired to ask whether two-photon quantum interference could be used similarly to classical interference in holography in order to use known-state photons to gain further information about unknown-state photons. Our analysis led us to a surprising conclusion: It turned out that when two photons exhibit quantum interference, the course of this interference depends on the shape of their wavefronts [an imaginary surface joining all adjacent points with the same phase]," Chrapkiewicz said in the press release.
Understanding quantum mechanics
This experiment has huge implications for our understanding of the fundamental laws of quantum mechanics, a field of physics that has been perplexing scientists for more than a century. It allows scientists to gain valuable information about the phase of a photon's wave function.
The researchers hope to apply this method to create holograms of more complex quantum objects, which might have implications that stretch beyond fundamental science into real world applications.
"All of us — I mean physicists — must first get our heads around this new tool," said Konrad Banaszek, a researcher in the experiment. "It's likely that real applications of quantum holography won't appear for a few decades yet, but if there's one thing we can be sure of, it's that they will be surprising." |
f0a7e0edb9cbb551 |
Talk:Complex number/Draft
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Definition Numbers of the form a+bi, where a and b are real numbers and i denotes a number satisfying . [d] [e]
APPROVED Version 1.0
See comments above
I made some comments in the section Talk:Complex number/Draft#What the symbol i means in this article above, marked with bullet points, which have not yet been addressed. (I just wanted to mention them after the page break so they won't be forgotten when editing the next version.) --Catherine Woodgold 20:28, 6 May 2007 (CDT)
I edited in all of the changes I had suggested except for the stuff about quantum physics. I don't have a textbook on the subject handy. --Catherine Woodgold 19:17, 7 May 2007 (CDT)
I'm not entirely happy with my text in the QM section, either. Trying to interpret superposed states in terms of probabilities is dicey at best, anyway. I'll have to think about this and see if I can come up with something better. In any case, I'm intrigued by what Robert Tito had to say about other uses of complex numbers, particularly in Hamiltonian systems (conjugate coordinates with a factor of i?) Anyway, I was just trying to come up with something that would be recognizable to a wide range of readers (albeit not mathematically naïve ones). If nothing else, the Schrödinger Equation has a certain iconic value. I'm certainly open to other suggestions. Greg Woodhouse 19:38, 7 May 2007 (CDT)
Wait! I think the quantum physics stuff is good! It just needs some editing, as I suggested, e.g. defining the symbols used etc. --Catherine Woodgold 19:42, 7 May 2007 (CDT)
It seems to me that saying things like " is Planck's constant divided by " wouldn't really add anything to the article, and I guess that's what bothers me: if you (generic) know what Schrödinger's equation is, this probably doesn't need to be said, and if you don't, the section really doesn't add anything. Greg Woodhouse 22:40, 7 May 2007 (CDT)
Part of chemistry and physics workgroups?
(I hope no one minds if I move this discussion "below the bar". Greg Woodhouse 03:34, 7 May 2007 (CDT))
I am just curious why this article's checklist includes it in the chemistry and physics workgroups. It seems that even though this article has applications in those field, including it in every workgroup it applies could get out of hand. - Jared Grubb 12:32, 6 May 2007 (CDT)
that answer is simple: the need for something like a complex number arose from these sciences not from math. Math formalized it, thats all. Robert Tito | Talk
I disagree. We have an example showing that complex numbers are important in the sciences, too, but complex numbers were introduced in a fundamental way in mathematics (i.e., not just as a notational convenience) long before quantum mechanics had even been thought of. Greg Woodhouse 16:59, 6 May 2007 (CDT)
Then again, since those workgroups are there, maybe you can sign off on it, too. :-) Greg Woodhouse 17:02, 6 May 2007 (CDT)
Physics and chemistry used the notion of complex numbers as from the 18th century - when they needed them to describe things. Euler, Gauss, Fourier were not mathematicians but physicists/chemists that needed a solution for their math problems. the complex number by far didn't start with quantum mechanics. I might mention Hamiltonian mechanics as an example, or canonicals. Robert Tito | Talk 17:35, 6 May 2007 (CDT)
It still seems a little odd to me. But, I suppose Ohm's law would be more at home in the electrical engineering workgroup than the mathematics, even though it is a mathematic equation... - Jared Grubb 23:14, 6 May 2007 (CDT)
The historical development of the concept of complex numbers seems like an interesting topic for an article (albeit a challenging one!), but so far as this article is concerned, I don't think it's really that important. No, that doesn't sound right: I don't mean it's not important, only that I don't think it needs to be addressed in the context of this article. Greg Woodhouse 03:34, 7 May 2007 (CDT)
Does anyone know how to create an archive? Is there an automated, or at least "official" way to do it? I just got the following warning:
WARNING: This page is 87 kilobytes long; some browsers may have problems editing pages approaching or longer than 32kb. Please consider breaking the page into smaller sections.
Greg Woodhouse 03:34, 7 May 2007 (CDT)
Re archiving: you might want to discuss with Chris Day or see Talk:Biology/Draft, but since that template is named "Experimental" I suppose procedures haven't been finalized. (Discussion about it arising from pages like this one may drive the finalization of such procedures.) Maybe it's being discussed on the forum somewhere, or if not someone could start. --Catherine Woodgold 07:45, 7 May 2007 (CDT)
Re being part of chemistry and physics workgroups: Chemistry and physics also need to use 1 + 1 = 2. They also need to use words with syllables to communicate technical concepts; that doesn't mean the linguistics Syllable page has to be in the chemistry and physics workgroups. Those sciences use math -- that doesn't mean math is part of the science. I think perhaps people in the chemistry and physics workgroups should decide whether the article is included or not. It's OK with me either way -- it's not that unreasonable. More justifiable than including a page that presents a proof of 1 + 1 = 2 in those sciences. --Catherine Woodgold 18:23, 7 May 2007 (CDT)
This article was approved by a math editor and currently is listed in the Math Workgroup Approved articles, but not in Chemistry or Physics. Since the article is cross-listed in three workgroups, will there need to be three approval processes? Or will we need editors from all three areas to agree before any one draft gets approved? Or will we declare one "father" workgroup, and the others just raise objections or not... I know this approval process is still in its infancy, but these are questions we really should address at some point. - Jared Grubb 02:41, 8 May 2007 (CDT)
Error in multiplicative property
I've just put the following message on User talk:Nancy Sculerati.
Dear Nancy. Etienne Parizot found and fixed an error in Complex number/Draft which is also present in the approved version, Complex number. The formula
halfway the section "The complex exponential" should read
(with a plus sign added on the left-hand side).
This is a big error so I want it to be fixed as soon as possible. I can't imagine any editor would argue with this change. However, I'm not sure what our options are.
• Some places hint at the possibility to have the constabulary do limited changes to articles without going through the whole approval process (for instance, the section #Copyediting matters above). I couldn't find anything about rules or procedure though. If such a possibility exists, that would be my preference. For the record, the article complex number was nominated by Greg Martin and the nomination was supported by me.
• If this is not possible, I'd like the approval to be revoked. As far as I can see, there is no rule or precedent for this, only an empty section at CZ:Approval Process.
• If neither of the above is possible, or if it would take too long, we can always go for the option of nominating the fixed version for approval. To be honest, it's not that important in the big scheme of things, but it is embarrassing and I feel responsible for it.
Any guidance from you (or anybody who happens to read this) would be much appreciated.
-- Jitse Niesen 08:37, 10 May 2007 (CDT)
Hi All, I have commented out the Approval tag per nominating editor Jitse Niesen who has revoked his approval. During this time, our Approval editor, Nancy Sculerati can make the appropriate changes and she can replace the Approval tag. If more chances are made, then I would suggest giving yourself an additional 24 hours before re-approval to give others a chance to review the changes. --Matt Innis (Talk) 08:46, 10 May 2007 (CDT)
I think this was handled very well. What I saw was: Etienne correctly decided that this problem should be corrected very quickly; Nancy sent me (presumably other editors) an email to alert me to it; by the time I came to CZ, it had already been decided that the change was appropriate and needed, and was made to the approved version as well as to the draft.
Although this might be "outside the rules", I think that here judgment wins the day, concerning what we might call "clear factual mistakes or obvious typos". I certainly take responsibility for nominating the article for approval without seeing this mistake. And sharp eyes Etienne! - Greg Martin 14:29, 10 May 2007 (CDT)
There will always be such mistakes, and a good approval process can take care of them.It is understandable that Jitse, who was shocked by the sudden recognition of such a mistake, wanted it fixed IMMEDIATELY. It was understandable that Matt acted to accomodate him, my only point is -in the future we now know that the approvals editor could have done the copyedit at Jitse's say so. Approval cannot be "revoked" in this manner. Think about it. If it could be, that sets a terrible precedent, you can imagine how in a different circumstance such a precedent could be misused. I cannot add details to the approval process policy without the Editorial Council (of which I am a member) being up and running, with a voting process in place. Right now we are figuring out the process. I said several times that I would take responsibilty for copyedits at this stage with any of the nominating editors.Maybe we should add that, in an emergency the constable can put up a note saying that there is a copyediting problem that is being corrected- in progress. Nancy Sculerati 12:10, 11 May 2007 (CDT)
Style issues
Quoting some examples of style that I consider a bit too informal. In particular, there are many phrases/clauses that make the article verbose. In the first paragraph, there are: Of course, As it happens, At first glance, perhas more importantly.
Is this as per the policy of Citizendium? Should the number of such phrases/clauses be reduced? Vipul Naik 02:19, 8 June 2007 (CDT)
Hi Vipul, and welcome to CZ!, the answer to your first question is "yes, there are style differences here" - see this section of the article mechanics article concerning style. Your input is welcome. Matt Innis (Talk) 07:56, 8 June 2007 (CDT)
Personally, I prefer more casual or informal style. Of course, this doesn't mean the articles need be any less precise or rigorous, only more readable, and maybe a little less intimidating. Greg Woodhouse 11:39, 8 June 2007 (CDT)
Remaining errors in this approved article?
Hi, I came across this Error claims on WP Signpost. I think there are valid points there, especially regarding the interpretation of 1/z and the comment on the potential function (clearly it can't represent some force since it is a scalar). These should be looked at closer. Are there plans to have this article revised in the near future? Thanks. Hendra I. Nurdin 13:55, 20 October 2007 (CDT)
Hi, did you look at the draft? ;-) Except for the potential thing, deleted from the draft some time ago, I don't really think this is as problematic as suggested. But if you feel like, we could find a better wording for some text. Then re-approving looks like a good idea. Aleksander Stos 14:41, 20 October 2007 (CDT)
PS. You may also have a look at my "advanced" draft. At present I gave up the idea behind that work -- but some portions of the article might be useful here. I don't know. Aleksander Stos 16:02, 20 October 2007 (CDT)
I'm merely passing on some criticisms I happened to stumble onto (I don't know how many people have read it before). Of course, if there is anything valid in them then they should be considered. As for "complex division amounting to conjugation with scaling", well it does sound a bit misleading to me (I don't know about other people, which is why I brought it up here :-)). Consider then it does not have anything to do with the conjugate of which is -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of in the article. As for your "advanced" draft (such as the section on roots of complex numbers), perhaps parts of it can go as subpages of the article? Hendra I. Nurdin 19:35, 20 October 2007 (CDT)
P.S. Does the removal of the assertion that the potential function represents some force from the article not warrant a re-approval process? This gives rise again to the issue that some relatively "minor" changes like this to an approved article should be possible to do with ease. Hendra I. Nurdin 19:44, 20 October 2007 (CDT)
Hendra, all it would take is a math editor who has not worked on it to nominate it for re-approval, or three editors who have worked on it to re-approve. If we can get that together, I will be glad to make the draft the approved version. --Matt Innis (Talk) 20:40, 20 October 2007 (CDT)
Well, let's see what others think about this first, as it could be that it's just me being pedantic and perhaps in view of others these changes may not be necessary :-) Anyways, any further changes need to be considered carefully, so that if it does have to go through re-approval, no further minor changes would need to be made afterwards. Hendra I. Nurdin 21:08, 20 October 2007 (CDT)
Okay, sounds like a plan. I was going to leave a message on Jitse's page but you beat me to it! I'll wait and see what develops. If you have any questions, just stop by my talk page. --Matt Innis (Talk) 21:53, 20 October 2007 (CDT)
I hadn't seen the criticisms before. I think they are valid points and that we should revise the article accordingly. Hendra, please change the draft as you see fit (I'm rather busy now so I can't be of much help at the moment, sorry). The approval process is not that much effort, so you shouldn't worry about that. -- Jitse Niesen 22:07, 20 October 2007 (CDT)
Done. I have also added a remark that division by c+di is only defined if c and d are not simultaneously zero in the part of the article that discusses operation on complex numbers. Therefore I invite all authors and editors who had been previously involved in the approved article to check my edits and make any modifications and corrections as deemed necessary. However, I think that the work is not all done yet. There is a bit more to be done on the section about complex numbers in physics. The sentence
"Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ..."
is quite vague and is likely to cause misunderstanding. I guess I know a fair bit about the mathematical formalisms of quantum mechanics, but I'd rather not delete things nor make substantial changes without first soliciting the opinions of those who have worked on this part, and other authors who know the subject quite well, and get their input on what is meant exactly by this sentence and whether it needs to be further elaborated upon for clarity, or changed to avoid misinterpretations. Hendra I. Nurdin 00:25, 21 October 2007 (CDT)
QM again
The article states:
In my view this sentence should be deleted because it has absolutely nothing to do with complex numbers. It gives me the unpleasant WP experience of something that is added by somebody somewhere with some time on his hands, which is why many WP articles are headache-causing kinds of patchwork. At most one could do in this article is a link to quantum mechanics, where the Born postulate for probability of observation can be put in proper setting. --Paul Wormer 07:42, 21 October 2007 (CDT)
PS Most interactions in QM are invariant under time reversal. It can be shown that ψ can be chosen to be real in that case. And indeed, 95% of quantum chemistry deals with real functions. --Paul Wormer 07:45, 21 October 2007 (CDT)
Yes, deletion of the whole section would be one solution. As a replacement application we could instead insert the Laplace and Fourier transforms which use complex numbers in an essential way, or perhaps something on phasors. Let's see what the editors think would be best. Hendra I. Nurdin 08:03, 21 October 2007 (CDT)
There are also certain irreducible representations of some (physically important) groups that inherently are complex (Wigner, Am. J. Math. vol 62, p. 57 1941). These could be mentioned as examples of complex numbers in physics. --Paul Wormer 09:45, 21 October 2007 (CDT)
Paul, would you be interested in putting this in the article to replace what is currently there? Btw, which group does this paper talk about? Perhaps we can work on this section together, I could insert some additional engineering applications. Let me know what you think. Thanks. Hendra I. Nurdin 07:30, 23 October 2007 (CDT)
Dear Hendra, don't you think it would be a good idea to leave it to the approving editors to correct the article? We can signal what we don't like. For instance, the following sentence in the article
is bordering on being wrong; a wave function can be a superposition of eigenstates, see particle in a box for a graphic example. Maybe "see" refers to a collapse of the wave function, but that would be a collapse to an eigenstate of the position operator. Further, the Schrödinger equation mentioned (time-dependent) is not an eigenvalue equation, so the term "operator defined by" is pretty inconclusive.
I am of the opinion that it is better to spend our energy on new articles, given the present vast emptiness of CZ. In Legendre polynomial I linked to orthogonal polynomials. I saw that you wrote Gram-Schmidt, so for you it would be a piece of cake to write a nice article about general orthogonal polynomials, with links to Laguerre, Hermite, Legendre, Jacobi, etc. Best wishes, --Paul Wormer 06:59, 25 October 2007 (CDT)
PS. Upon rereading the Wigner article that I mentioned earlier, I noticed that Wigner does not mention any specific groups, only characteristics of groups. But, complex numbers are essential for irreducible representations of cyclic groups and for the even-dimensional irreps of SU(2). Schur's second lemma requires the solution of a polynomial equation and hence an algebraically closed field. --Paul Wormer 06:59, 25 October 2007 (CDT)
I removed the whole QM section; there are simply too many problems with it. It would be nice if somebody could write a section on applications of complex numbers outside maths. You don't need permission of the approving editors to do so (that's why it's called a draft), but I hereby do give you permission in case you feel happier with it.
I'm not so sure what the best application would be to put in that section. Phasors is relatively easy to explain, but I think it's mainly an organizational tool and it's not essential to use complex numbers - one can just use sine and cosine. However, Laplace transforms may be too difficult, given that we tried hard to make the page understandable with a minimum of prior knowledge. Or perhaps QM is a good example after all when written up properly; we can just show the Schrodinger equation and say that it has an i in there.
By the way, I moved from Australia to England and that's why I haven't been around much lately. Still settling in, and all my books are still en route, but I should be able to spend some more time here soon. -- Jitse Niesen 07:51, 26 October 2007 (CDT)
Representation via matrices?
Just wondering if anyone had already considered an alternative version of the formal definition by defining complex numbers as being a subset of GL_2(R)? Many "university level" people will have seen the basic definition of matrix multiplication, and as such, it may seem less foreign than the definition of multiplication for ordered pairs of real numbers. ...said Barry R. Smith (talk) (Please sign your talk page posts by simply adding four tildes, ~~~~.)
Move some topics to advanced page?
Near the top of the discussion, where a plan for the "complex number" page was sketched, the following comment was made: "I like to introduce complex numbers to my students with the example of the resolution of the cubic equation with the so called Gerolamo Cardano's method (in fact it is due to Scipione del Ferro and Niccolò Tartaglia). Computations are quite easy, and the striking fact is that during them, one has to use some imaginary number which square would be -1, but once the computations are finished, one gets the three real solutions of the equation!" While it is true that this is probably the earliest example where it became clear that complex numbers were necessary even for the study of real quantities, I definitely disagree with the statement "computations are quite easy". I am also surprised that you introduce complex numbers to students with this example. I have given a project to students of working through this example, and I would not say that they found it easy. In another section of this discussion is the comment, "I certainly agree that articles, especially articles about basic topics like complex numbers, shouldn't scare the reader away right off the bat, but perhaps we need to temper our desire to make the article start out slowly and in a non-intimidating fashion with a bit of logical coherence." It seems to me that our first example might scare many readers away right off the bat. What does a mathematician think about this example? Is it a struggle to get through? Does it make you not want to continue?
The initial idea of writing x=u+v where u and v will be specified later is offputting to many people. This is followed by an application of the binomial theorem, and then an unmotivated factoring step. Then, it is stated that "we only required that x = u + v. Hence, we can choose another condition on u and v. We pick this condition to be 3uv − 15 = 0". Again, the average non-specialist, I would imagine, would wonder, why this condition? And why are we allowed to choose another condition? I could go on listing more potential difficulties that I see in this example.
As we now have the advanced subpage option, why not move this example to an advanced subpage, and then refer to it in the actual article. Perhaps say something like, "A common question is why bother with complex numbers when real numbers almost always seem sufficient for applications. Indeed, the ancients would ignore complex solutions to quadratic equations. It wasn't until the 16th century that it began to be clear that sometimes, complex numbers were indispensable even in problems that seemingly only involve real numbers. An example of this can be found on the "advanced subpage".
Even if this approach isn't taken, might I suggest an alternative way to formulate the current example that is probably more palatable to the average reader: do not derive the solution of the "reduced cubic" by introducing u,v, etc. Instead, just give them the formula for the roots of a reduced cubic -- it isn't very complicated, and an analogy can be made with the quadratic formula. Then show that 4 is a solution, but that the formula gives an expression involving a complex number. Finally, show that this complex can be written as 4.
Also, I think that no matter what, some reference should be made to the fact that although complex numbers are typically introduced these days in high school, when the quadratic formula comes up, that the ancients had versions of the quadratic formula but still didn't accept complex numbers. Barry R. Smith 13:34, 30 April 2008 (CDT)
All sounds good. Go for it! J. Noel Chiappa 22:26, 4 May 2008 (CDT)
New philosophical addition
With regards to Christopher Reiss's new philosophical addition, I believe that this material definitely should be moved out of the introductory paragraph -- as far as I know, an introductory paragraph should usually serve as an abstract for the article, giving a concise non-technical summary of what most would consider the most salient features of the topic?
Now the issues raised are partially addressed already in the history section, notably, the fact that ancients did not believe that complex numbers were "real". Furthermore, the long example in the "advanced" subpage describes the first instance in which complex numbers seemed to be necessary for something. If this material could be improved using some of Christopher's ideas, then I propose that the material be integrated into a whole within to history section.
The only content I would take issue with is the statement that "It is now understood that arithmetic is a pure abstraction which we are free to modify", and related declarations. This is probably the most common philosophical view taken by current mathematicians, and probably many people well versed in mathematics. However, this idea is much older -- Platonic realism is an early example, due to Plato millennia ago. So the phrase "it is now" is misleading. On the other hand, the statement "understood that arithmetic is a pure abstraction" is also misleading. Better would be to say that the majority of current mathematicians believe this. But there is still a raging debate about this in the philosophy of mathematics. For alternatives, see "empiricism" and "fictionalism", for example, at the philosophy of math wikipedia site. Even if we add this material to what Christopher wrote, I believe that content would more properly be put in a philosophy of math site, as the result would probably be quite lengthy and too much of a tangent from the topic of "complex numbers".
Any thoughts?Barry R. Smith 12:20, 22 May 2008 (CDT)
I completely agree. I did not see any reply from Christopher, so I simply removed the text. Here it is for future reference:
Complex numbers were once considered 'fictitous' on the grounds "there is no square root of negative one." This misconception is rooted in a philosophical conception of number which is now seen as misguided. The notion behind this 'predjudice' is : arithmetic exists in the physical world, or is an attribute of physical reality. This notion has repeatedly proven a stumbling block in the history of mathematics. It is now understood that arithmetic is a pure abstraction which we are free to modify. It is legitimate to experiment with the abtract system first and then seek real world mechanisms which the abstraction can model. Rather suprisingly, by freeing the abstract system so that it is no longer "real", the abstraction became a much broader and more powerful model of the physical world.
This has happened repeatedly in the development of mathematics. Originally, only the counting numbers, 1,2,3 .. were considered 'real'. This left the result of certain division operations - fractions - 'unreal'. But one can define an arithmetic of fractions which is immensely useful in the physical world, and which also describes the counting numbers as a special case. Similarly, the result of certain subtraction operations yielded 'unreal', negative results. Arithmetic was expanded again to include negative numbers. Yet again, it was found that the square root of two has no solution among the fractions. These 'unreal' entities were eventually admitted into arithmetic as it continued to grow in power.
As Barry says, parts of this may, in less absolute form, be integrated in the History section (or perhaps elsewhere); in fact, I think the question of whether complex numbers exist probably should be treated in the article. I would include a link to an article about philosophical aspects of mathematics where this is discussed in more detail. -- Jitse Niesen 07:19, 2 June 2008 (CDT)
Division and conjugation
Discussion moved from User talk:Jitse Niesen#Complex number page BEGIN Peter Schmitt 23:44, 19 July 2009 (UTC)
The Complex number page still contains what I consider to be a blatant error: "In other words, up to a scaling factor, division by z is just complex conjugation." I think this would be correct if it said, "In other words, up to a scaling factor, division by z is just multiplication by the complex conjugate of z" or if it said "In other words, up to a scaling factor, taking the reciprocal of z is just complex conjugation"; but as it stands (according to the only reasonable interpretation I can see) it's equating two operations which in general involve completely different changes to the angle on the complex plane. As you know, this problem was pointed out on a Wikipedia discussion page in 2007. As a math editor, would you please either ask a constable to correct just this one sentence in the current article, or arrange to have the draft approved? (I haven't looked at the latest draft; I'm just concerned about this particular error.) Thanks. Catherine Woodgold 15:28, 11 July 2009 (UTC)
The formulation may be unfortunate, but it is correct. The "scaling factor" is 1/|z|^2, a real number, and the angles (the argument) of the conjugate and the inverse are the same. Probably it would be better to write "In other words, up to the scaling factor 1/|z|^2, division by z is just complex conjugation." I don't know if in such a case approved version can be corrected. Peter Schmitt 22:07, 11 July 2009 (UTC)
Why don't the two of you, and possibly anyone else you can rope in, work out an *exact* replacement phrase and then put it into this discussion area. If all of you agree that it should replace the Approved version, either I'll change it myself or I'll ask Matt what he things about it. Hayford Peirce 22:30, 11 July 2009 (UTC)
My suggestion:
In other words, up to the scaling factor (a real number), division by z is just complex conjugation.
(Unfortunately, the fraction looks awful in text.) Peter Schmitt 22:57, 11 July 2009 (UTC)
If we use our current approval (and re-approval) rules: since Peter is a mathematics editor, I would suggest that Peter refrain from making any changes to the article and let Jitse or Catherine make the change on the Draft. Then (assuming Peter agrees with the change), he can can re-nominate the draft for approval using the single editor process (since he has not made any content edits to the article)... HELLO CATHERINE! :)
Check with the User:Approvals Manager (Joe) if you want to be sure. D. Matt Innis 23:21, 11 July 2009 (UTC)
Matt's suggestions sound v. feasible to me. Hayford Peirce 23:36, 11 July 2009 (UTC)
As far as I see, the draft differs much from the approved version (a direct comparison seems to be difficult). I thought there is a possibility to edit such things without (re)approval? This is not a correction but only a clarification, and certainly not a change of content. (By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.) Peter Schmitt 23:41, 11 July 2009 (UTC)
I have put a link on the Approval Manager talk page. Peter Schmitt 23:49, 11 July 2009 (UTC)
I looked at the draft and the approved version, they are completely different on this point. Somebody made some drastic changes. Further, I would say division of c by z is multiplication of c by the complex conjugate of z (and division by the modulus square of modulus of z). In the polar representation of complex numbers the issue is completely trivial, as we will all agree. It is , which holds for all real and complex . --Paul Wormer 08:09, 12 July 2009 (UTC)
"and division by the square of the modulus". Yes, this would avoid the displayed fraction. Peter Schmitt 09:53, 12 July 2009 (UTC)
If we want to spell it out completely, then with
--Paul Wormer 11:13, 12 July 2009 (UTC)
It seems that you are saying that replacing the approved version with the draft would add more errors than it would fix. Your choices then would be to 1) fix the errors in the draft and use either the individual editor approval or three editor approval method to change the approved version or 2) revert the draft version to the version you like, then make the change that Catherine and Peter are looking to make and then use the individual or three editor approval methods as above. Does anyone see any other choices.
The idea of the approval rules is to make us work together to come up with the most accurate article possible while, at the same time, allowing the article to remain stable while we do. Hopefully, this reduces the workload on our experts. The errors in the draft are an example of why we want to have an approved version that is difficult to change - without editorial input. Changing our rules for something like this that can be managed within the same rules seems only a means to weaken them. However, it is possible, but would require community input from all the workgroups to consider all the ramifications of such a change.
(By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.) It's not so much about cheating, it shows that more than one editor agrees to the change, thus increasing the likelihood that the change is more accurate - while at the same time allowing only two editors to make a difference (which is easier than finding three - something that you are also asking for). It is a way to both make it easier to make a change and keep one fallible editor from approving his/her own work. I hope that makes sense.
D. Matt Innis 12:33, 12 July 2009 (UTC)
I think I have been misunderstood here, and I'll try to clarify: I noticed the message of Catherine (not addressed at me) and answered it. The challenged sentence - in the approved version - is correct, but might indeed be confusing for some readers. Therefore I suggested a minor edit to the approved version (thinking of CZ:Approval_Process#Overview, last paragraph) because approving the draft version would - in view of the major changes - require more checking and possibly a lot of discussion. If this is not thought as adequate or allowed, then the approved version can stay as it is.
The remark on the approval process was a reaction on the suggestion:
What is the difference between an explicit suggestion by an editor which is dutifully incorporated by some author (possibly a non-editor), and the same change made by the editor himself? The difference is only a formal one -- that was what I meant by cheating. (There need not be another editor involved!) Moreover, I thought that copyedit changes are allowed -- and this I would classify as copyediting.
Peter Schmitt 13:11, 12 July 2009 (UTC)
That does clarify some. I did not understand that the discussion concerned something that was basically correct on the approved page. Unfortunately, I'm not sure that I could have known that if you had not told me :) - which I think makes it different than a copyedit - which anyone could recognize does not change meaning. Because it is not really self-evident that it does not make a content change, I don't think this is something that a constable can or should do without the approvals manager seeing things through (which he very well might do). It's more to protect the editor that has endorsed the article than anything else.
Concerning the addendum, Jitse knows very well where that came from. I actually revoked the approval of one of his articles when he realized it had a math error in it. He'll tell you that I took a pretty good beating over that one! And I think they were right to do that, approved articles need to be hard to change. The addendum makes it clear that the nominating editor can change it with the help of the approvals manager. It is the nominating editor who has his name on the article and therefore has endorsed it. That is why we have given him/her more leeway to make a change. Jitse could still make that change, as you note, I think - with the help of the approvals manager. Of course, the other choices still remain - to re-approve using your credentials if Jitse does not respond.
You make another good point about an author being able to make a change that an editor cannot, but it still requires two heads. Remember that we deal with controversial articles that have competing views even among editors. The concept is to keep one view from eliminating the other view without some oversight. Whether this is successful at keeping that from happening, or if it keeps us from making more important corrections, or if we might be able to come up with a better way, is something that might need discussion elsewhere if it is causing problems. D. Matt Innis 14:33, 12 July 2009 (UTC)
Peter says that the approved version is essentially correct, but I side with Catherine and say that the word multiplication is missing. --Paul Wormer 15:23, 12 July 2009 (UTC)
Because of the formula above the sentence I read this only as "division (of 1) by z" ... if you read it as "division of some u by z, then this would certainly be wrong. Perhaps I did not want to see a serious mistake? Peter Schmitt 15:35, 12 July 2009 (UTC)
Looks like you're getting closer to a more accurate description. Once you've decided what you want to do, let the approvals manager know and we'll go from there. I think our options remain the same. D. Matt Innis 17:07, 12 July 2009 (UTC)
Since no further comments have been added, I suggest to replace
In other words, up to a real scaling factor (the square of the reciprocal of the modulus), division of 1 by z is just complex conjugation.
because this seems to be the least change which repairs the sentence in the approved version (and still fits into the style of the paragraph).
For a reapproval of a new draft certainly more changes (and discussions) will be made.
Peter Schmitt 00:21, 19 July 2009 (UTC)
I suggest that we proceed through a re-approval process anyway. I don't see three editors on that approval tag. D. Matt Innis 18:30, 19 July 2009 (UTC)
Discussion moved from User talk:Jitse Niesen#Complex number page END Peter Schmitt 23:44, 19 July 2009 (UTC)
How to proceed?
Ok, but I do not know what I should do. I do not want to approve the draft as it is now.
But I want to change one sentence of the approved version in order to either
correct an error in a sentence, or
to clarify a sentence that can be misunderstood
(depending how one interprets this sentence in context).
I would approve such a corrected version (and I would make the changes myself, it this is allowed) — but how is this done correctly?
Peter Schmitt 00:04, 20 July 2009 (UTC)
Okay, you can't correct it yourself and approve it yourself. You can still use Jitse, or the other editors as noted above. However, since you are an editor and since there are lots of other changes on the draft, I think it would be possible to revert the entire draft to the approved version then have someone make a change that you can agree to and nominate that version as 2.0. Once the version is approved, you can then revert the draft to the older draft (if you want) or start over again from there - as long as you discuss why on the talk page. If someone disagrees, that gives them plenty of opportunity and options to discuss any issues before it gets re-approved. Again, though, the User:Approvals Manager needs to be kept abreast of this and had veto power if something doesn't look right - because he is my redundant failsafe system! D. Matt Innis 04:16, 20 July 2009 (UTC)
If it's really just fixing an error, then I don't see any reason we can't just insert the correction into the approved version of the article; we've done that before, though mainly just with grammatical or spelling errors. Since this would be a content correction, we would need to make sure that the original approving editors still approve or we'll have to remove their names from the approval. Jitse is definitely still around to agree or disagree with a suggested change but Greg Martin hasn't made a contribution for over 2 years, so he'll be harder to track down.
So these are our options, as I see them: (1) If we have a specific proposal for a change on the table, we can ask Jitse to review it. Then, if he agrees to the change, we'll make the change on the approved version and replace Greg Martin's name with Peter's, assuming that Peter approves of the rest of the article. But then I think we still need one more editor. (2) We can do the same thing and try to track down Greg Martin. (3) We can re-initiate the approval process for some later version of the article. That could be the current version or it could be any other version since the time of approval or it could be a wholly new version. We would need either one un-involved editor or three who have contributed. (4) We can revoke approval based on the error that the current approved version contains. This is the least preferable of our options, and we should avoid it if we can.
By the way, does this topic fall within the field of number theory? If so, Barry Smith might be willing to sign off on approval, either for the old draft with the proposed change or on a new draft.
Apologies for taking so long to jump into the conversation. --Joe (Approvals Manager) 17:48, 22 July 2009 (UTC)
Number theory is notoriously difficult to define. Although "complex number" sounds like it might fall under the field of number theory, I think most mathematicians would agree that the entire scope of the study of complex numbers is not classified as number theory.Barry R. Smith 18:44, 22 July 2009 (UTC)
The elementary theory of complex numbers (which means, not the advanced theory of analytic functions) is within the scope of every mathematician. And moreover, even the advanced (if not too much advanced, maybe) theory of analytic functions is usually within the scope of number theorists. I think so. (Sorry for the late intervention.) Boris Tsirelson 19:02, 23 July 2009 (UTC)
The topic doesn't necessarily have to fall wholly within your field in order for you to approve it. In fact, most of our approved articles span several fields of inquiry and each individual editor who participates in the approval process usually comes from only one or two of the fields that are relevant. Working together, the editors from different fields approve a single topic. If number theory has something important to say about complex numbers, then that is enough for you to be an approving editor. --Joe (Approvals Manager) 16:16, 23 July 2009 (UTC)
I agree with the choices that Joe gives above with the exception of 4, which seems that we should have editors decide to revoke approval, and if we have that, we might as well re-approve it. After looking at the amount of additional edits that have been made to the draft, and considering that most were made by well informed authors and editors, I think it might be a better choice to keep working on this draft until we have something that Peter can agree to endorse. Othewise, it is likely that we will be re-approving again and again until all of the changes have been incorporated anyway. How close are we Peter? D. Matt Innis 23:35, 22 July 2009 (UTC)
(unindent) Allow me to clarify my position in some detail: This discussion was not started by me, but by a concern of Catherine Woodgold placed on Jitse's talk page. I noticed it, and that Jitse did not react. (He is online with some edits on WP, so I am not sure if we may count on him.)
I probably would not have looked at the (approved) Complex number article for a long time, and if so, I would probably not have noticed the sentence under discussion because I would have read it as correct by (unconciously) assuming what the author (probably) had in mind when writing it. (That certainly is also the reason why it was not caught at approval time.) Checking the article, I still was not looking thoroughly enough, and thought that clarifying "scaling factor" would suffice. However, Paul pointed out that I was wrong in reading "division" as "division of 1" -- taken literally, this was not said.
Since this sentence lacks clarity, it should be corrected. I do not see this change as "a content correction", but I won't fight your ruling. And I am with Catherine that it should be corrected as quickly as possible. Since I have not yet thoroughly read the complete article I have no opinion how close to (re)approval the current draft is. Anyway, I think this (and similar cases) should be corrected independently, and be recognized as corrections to the first approved version (I suppose that earlier approved versions will still be accessible as such? They should!) and not be hidden among a lot of changes between two approved versions.
I am not sure if I understand what "(speciality) editor" for number theory means and implies. Basic arithmetic with complex numbers is common knowledge for all mathematicians (and most natural scientists) and should be known by many non-mathematicians, as well. On the other hand, number theory also uses complex numbers, e.g., when discussing algebraic number fields. What I want to say by this: This is not an issue for which any special knowledge is needed.
Peter Schmitt 11:58, 23 July 2009 (UTC)
In my view the needed correction is almost equal to correcting a typographical error, it is really minute. Let's not waste any more time on it, give Peter (or me) the right to fix it in the approved article as follows
Old: "In other words, up to a scaling factor, division by z is just complex conjugation"
New: "In other words, up to a scaling factor, division of one by z is just complex conjugation"
Addition of 5 letters, 2 short words.
--Paul Wormer 12:12, 23 July 2009 (UTC)
Okay, the two of you agree that it is not really a change in content but only a correction/clarification of the presentation. Let's make the change in the approved version and then if the draft develops to the point that we need to reapprove, we will. Matt (or Hayford, if you're watching) could you make the change Paul wrote above? Thanks much.
By the way, there is a precedent for what I'm calling "ex post facto approvals". If there are not already three approving editors listed in the metadata of an approved article and another editor from the relevant workgroup(s) wishes to add his or her name to the approval, he or she may do so. Simply add your name below the other approving editors in the "required for Approved template" section of the metadata and then notify me on the Approval Manager's talk page. So if either Peter or Barry (Barry, see my response above) would like to add his name to the approval of the current version after we make the correction, I would encourage it. --Joe (Approvals Manager) 16:16, 23 July 2009 (UTC)
You are asking me to change an article that was approved by Greg MArtin and Jitse Niesen to something that Peter Schmitt and Paul Wormer think is wrong, but you want me to leave Greg and Jitse's names on it. Whether I agree that the change is a good one is not an argument from an administrative perspective. We have two choices, get Jitse to change it or re-approve a draft version. So, if I can, I will revert all the changes on the draft and then make the change in the new draft (which anyone could have done) and it will be up to Peter to decide if he wants to use the single editor approval for it. That's as far as a constable can go, it is up to you guys to pull the rest together. D. Matt Innis 23:40, 23 July 2009 (UTC)
Okay, this is the new draft version that includes reverting to the original and then changing the sentence in question. While it is possible that other changes may occur, I won't replace the changes that have been made since until after it is approved. Then I will return the changes that were made since the first approval to the new draft version. I do not expect this to be a precedent for how things should be doen in the future without further discussion from the rest of the community and consideration is made for how it would affect other more controversial articles, but at least it doesn't give attribution to Jitse and Greg for something they did not approve, which I think is important from the standpoint of reliability and liability. D. Matt Innis 23:53, 23 July 2009 (UTC)
What you describe is what I was worried about but not what I thought I was asking you to do. The approval process is in place precisely because it stops the content from slipping and sliding away from the approved version. But if this is really just fixing an error and not changing the content, this situation isn't different from other corrections that have been made. I get the impression from the mathematicians here that this is akin to fixing a sentence that mistakenly says "3 divided by 6 is 2" when the truth is that "6 divided by 3 is 2".
I don't want the situation to become acrimonious though. So let's see if we can't simply reapprove. That means we need an editor to start the process. --Joe (Approvals Manager) 00:31, 24 July 2009 (UTC)
Sorry, I shall not be able to join in again before tomorrow (in about 30 hours) -- I say this to avoid the impression that I do no longer follow this. For the moment, just a short remark: The change is similar to changing a "this is" in a "the last mentioned is" to avoid that the "this" is misunderstood. Is this (only) so complicated because it is mathematics? Peter Schmitt 10:08, 24 July 2009 (UTC)
Not exactly because it is mathematics but because it is a topic that neither Matt nor I are expert in. We both want to make sure that we aren't making a change to the approved article that the original approving editors would disapprove of. It seems that that is the case here, but the only way for us to be absolutely sure is to get word from the original approving editors. Spelling or grammar mistakes are easier because anyone can recognize them. I hope we can re-approve the article and improve it even more in other ways while we're at it; that's what re-approval is all about, after all. --Joe (Approvals Manager) 14:19, 24 July 2009 (UTC)
Now you have chosen to let the article persist (for who knows how long) with— what Catherine Woodgold considers to be—a blatant error.--Paul Wormer 15:56, 24 July 2009 (UTC)
Just to be clear, all that needs to be done is for a math editor who has not made an edit to approve this version of the draft. The only thing that is different than the current approved version is the edit that was requested - "In other words, up to a scaling factor, division of one by z is just complex conjugation". D. Matt Innis 01:28, 25 July 2009 (UTC)
Well, I have just filled out the template (hopefully, correct -- is it true that the date should be given in this form???). I have read the article and discovered nothing serious. (Some historical remarks should perhaps be checked.) I also see some things I would change or extend, but this would take time -- and may be a cause for long discussions. Furthermore, if I would start to edit, I would not be able to approve it. So it seems best to leave the article as it is. Peter Schmitt 00:42, 26 July 2009 (UTC)
Peter, thanks for taking the lead. After re-approval, we can keep working on any minor adjustments on the draft version. --Joe (Approvals Manager) 00:50, 26 July 2009 (UTC)
Excellent. I'll wait to get the go-ahead from Joe (Approvals Manager) tomorrow. Sorry for any undue delay, but all in all, this way we have a legitimate single editor approval that has been duly scrutinized and agreed upon. Peter will also then be able to make any 'corrections' to the approved article once his name is on it. Meanwhile, if anyone would like to discuss changing the process to make these things easier, feel free to bring your ideas to the forums or the CZ Talk:Approval Process. D. Matt Innis 02:00, 26 July 2009 (UTC)
Looks good to me. Go ahead with the mechanics!
If other editors still wish to lend their support to the newly approved version, they may do so by following the steps I described above for "ex post facto approvals". --Joe (Approvals Manager) 14:19, 27 July 2009 (UTC)
APPROVED Version 2.0
This version includes one of many changes that were made to the draft since the first approval by other editors. I'll replace the old draft changes now so that they may be considered in any future versions. D. Matt Innis 02:51, 28 July 2009 (UTC)
I've restored the draft. One important thing to remember is that the sentence that was changed in the new version: "In other words, up to a scaling factor, division of one by z is just complex conjugation" does not appear in the draft because the sentence was essentially reworked too much for me to replace. It woul dbe nice for someone to take a look and make sure that what is there is accurate. Thanks all! D. Matt Innis 03:00, 28 July 2009 (UTC)
correcting approved version
Concerning the problems with Complex number#Complex numbers in physics in the approved version of this page (see #QM again Paul Wormer has suggested [1] to delete it completely. After some thinking about the options to resolve the problem, I consider it best to remove this section from the approved version and not to approve the current draft which contains several major revisions including the removal of another section, too. The article is nicely written but lacks some characteristics that an encyclopedia's (main) page should have. Therefore it is best to keep it as it is as an approved page (but correct errors, of course), instead of rewriting it partially.
May I do the necessary steps without being disqualified as approving editor? Peter Schmitt 23:58, 18 December 2009 (UTC)
Peter, BECAUSE you are the editor whose name is on this article, I think it is appropriate that you should be able to change information that you *now* feel is improper. I have reverted the draft to the current approved version. To continue to be within our CZ:Approval Process guidelines on single editor approval, how about asking Paul if he would clean up the physics section in any way that he feels necessary (as we would want a physics special expert to do) and then decide if you want to re-approve that version. D. Matt Innis 01:11, 19 December 2009 (UTC)
I haven't seen any movement here. Is this solution not working for you, or is it just a low priority? D. Matt Innis 16:48, 22 December 2009 (UTC)
Yes, Peter what's happening?--Paul Wormer 17:56, 22 December 2009 (UTC)
Sorry, if I kept you waiting. During the last days I did not do much here. (And today I suddenly lost contact to CZ for several hours.) But the main point is: Since I am expected not to touch the draft I have waited for someone to do the edits (remove the physics section, and - perhaps - make the already corrected "division sentence" still more explicit.
Paul, I have seen your new picture, and I have some comments, but I think that this need not concern us here. (A theoretical question: If I edit the caption, and the picture is used - would this disqualify me as approving single editor?) Since we both think that a new start will be best it is better to have the article (and the draft) as untouched as possible. After correcting its factual errors we can without time pressure think about a good place for it (subpage?) and a replacement. (I shall probably prepare something to start with offline first.)
Off topic: I have corrected the typo in Literature/Draft. There should be now problem to correct it on the Approved page. And the other open cases should also be possible, even if Editors are needed. There are History and Biology Editors active.
Peter Schmitt 00:20, 23 December 2009 (UTC)
I took care of the spelling error on Literature/Draft, thanks! My understanding is that any corrections to content on this page would have to be made by Paul (including image captions) in order for the single editor process to remain a viable option. D. Matt Innis 15:11, 23 December 2009 (UTC)
Nomination for reapproval
Thanks, Paul, for making the corrections! I have now nominated the draft for reapproval. I have set the date to today, so that this step can be performed whenever a Constable feels ready. (For the reasons for this reapproval see above.)
For more information see also my talk page: Complex numbers again. I hope that no further problems will surface in this article. However, as discussed there, Paul and I think that eventually this article should be replaced by a more encyclopedic one, while this colloquial introduction should find another home.
Peter Schmitt 12:46, 25 December 2009 (UTC)
APPROVED Version 2.1
I think there should be a section in this article about complex numbers expressed as phasors, which are commonly used in electrical engineering calculations involving alternating current voltages, currents, and impedance. Phasors are complex numbers expressed in terms of magnitude and phase angle. Henry A. Padleckas 11:29, 28 February 2011 (UTC)
You are talking about the polar form, aren't you? Calling this "phasor" in the context of the complex number would be confusing.
It is rather the other way: Phasors are described with the help of complex numbers (and not: complex numbers are expressed as phasors). --Peter Schmitt 22:24, 28 February 2011 (UTC)
Yes, the polar form, as in . Henry A. Padleckas 22:41, 28 February 2011 (UTC) |
2a12351993448002 | Saturday, July 12, 2008
DNA as topological quantum computer: XIV
I have worked hardly in refining the chapters about quantum biology in TGD Universe. DNA as topological quantum computer was the original idea that has generalized considerably and led to a beautiful unification of various basic ideas. I attach below the abstract of the chapter DNA as topological quantum computer contained in the book "Genes and Memes".
This chapter represents a vision about how DNA might act as a topological quantum computer (tqc). Tqc means that the braidings of braid strands define tqc programs and M-matrix (generalization of S-matrix in zero energy ontology) defining the entanglement between states assignable to the end points of strands define the tqc usually coded as unitary time evolution for Schrödinger equation. One can ends up to the model in the following manner.
There are several problems related to the details of the realization.
1. How nucleotides A,T,C,G are coded to the strand color and what this color corresponds to physically? There are two options which could be characterized as fermionic and bosonic.
i) Magnetic flux tubes having quark and anti-quark at their ends with u,d and uc, dc coding for A,G and T,C. CP conjugation would correspond to conjugation for DNA nucleotides.
ii) Wormhole magnetic flux tubes having wormhole contact and its CP conjugate at its ends with wormhole contact carrying quark and anti-quark at its throats. The latter are predicted to appear in all length scales in TGD Universe.
2. How to split the braid strands in a controlled manner? High Tc super conductivity provides a possible mechanism: braid strand can be split only if the supra current flowing through it vanishes. A suitable voltage pulse induces the supra-current and its negative cancels it. The conformation of the lipid controls whether it it can follow the flow or not.
3. How magnetic flux tubes can be cut without breaking the conservation of the magnetic flux? The notion of wormhole magnetic field could save the situation now: after the splitting the flux returns back along the second space-time sheet of wormhole magnetic field. An alternative solution is based on reconnection of flux tubes. Since only flux tubes of same color can reconnect this process can induce transfer of strand color: "color inheritance": when applied at the level of amino-acids this leads to a successful model of protein folding. Reconnection makes possible breaking of flux tube connection for both the ordinary magnetic flux tubes and wormhole magnetic flux tubes.
4. How magnetic flux tubes are realized? The interpretation of flux tubes as correlates of directed attention at molecular level leads to concrete picture. Hydrogen bonds are by their asymmetry natural correlates for a directed attention at molecular level. Also flux tubes between acceptors of hydrogen bonds must be allowed and acceptors can be seen as the subjects of directed attention and donors as objects. Examples of acceptors are aromatic rings of nucleotides, O= atoms of phosphates, etc.. A connection with metabolism is obtained if it is assumed that various phosphates XMP,XDP,XTP, X=A,T,G,C act as fundamental acceptors and plugs in the connection lines. The basic metabolic process ATP® ADP+Pi allows an interpretation as a reconnection splitting flux tube connection, and the basic function of phosphorylating enzymes would be to build flux tube connections as also of breathing and photosynthesis.
Wednesday, July 09, 2008
A code for protein folding and bio-catalysis
The TGD inspired model for the evolution of genetic code leads to the idea that the folding of proteins obeys a folding code inherited from the genetic code. After some trials one ends up with a general conceptualization of the situation with the identification of wormhole magnetic flux tubes as correlates of attention at molecular level so that a direct connection with TGD inspired theory of consciousness emerges at quantitative level. This allows a far reaching generalization of the DNA as topological quantum computer paradigm and makes it much more detailed. By their asymmetric character hydrogen bonds are excellent candidates for magnetic flux tubes serving as correlates of attention at molecular level.
The constant part of free amino-acid containing O-H, O=, and NH2 would correspond to the codon XYZ in the sense that the flux tubes would carry the "color" representing the four nucleotides in terms of quark pairs. Color inheritance by flux tube reconnection makes this possible. For the amino-adics inside protein O= and N-H would correspond to YZ. Also flux tubes connecting the acceptor atoms of hydrogen bonds are required by the model of DNA as topological quantum computer. The long flux tubes between O= atoms and their length reduction in a phase transition reducing Planck constant could be essential in protein-ligand interaction.
The model predicts a code for protein folding: depending on whether also =O-O= flux tubes are allowed or not, Y=Z or Y=Zc condition is satisfied by the amino-acids having N-H-O= hydrogen bond. For =O-O= bonds Y-Yc pairing holds true. Y=Zc option predicts the average length of alpha bonds correctly. Y=Z rule is favored by the study of alpha helices for four enzymes: the possible average length of alpha helix is considerably longer than the average length of alpha helix if gene is the unique gene allowing to satisfy Y=Z rule. The explicit study of alpha helices for four enzymes demonstrates that the failure to satisfy the condition for the existence of hydrogen bond fails rarely and at most for two amino-acids (for 2 amino-acids in single case only). For beta sheets there ar no failures for Y=Z option.
The information apparently lost in the many-to-one character of the codon-amino-acid correspondence would code for the folding of the protein and similar amino-acid sequences could give rise to different foldings. Also catalyst action would reduce to effective base pairing and one can speak about catalyst code. The DNA sequences associated with alpha helices and beta sheets are completely predictable unless one assumes a quantum counterpart of wobble base pairing meaning that N-H flux tubes are before hydrogen bonding in quantum superpositions of braid colors associated with the third nucleotides Z of codons XYZ coding for amino-acid. Only the latter option works. The outcome is very simple quantitative model for folding and catalyst action based on minimization of energy and predicting as its solutions alpha helices and beta sheets.
I want to express my gratitude for Dale Trenary for interesting discussions, for suggesting proteins which could allow to test the model, as well as providing concrete help in loading data help from protein data bank. Also I want to thank Timo Immonen for loaning the excellent book "Proteins: Structures and Molecular Properties" of Creighton. Also Pekka Rapinoja for writing the program transforming protein data file to a form readable by MATLAB.
For details see the new chapter A Model for Protein Folding and Bio-catalysis of "Genes and Memes". |
a26ef221c5f9f783 | Erwin Schrödinger
From The Infosphere, the Futurama Wiki
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Tertiary character
Erwin Schrödinger
Erwin Schrödinger.png
Facing URL after a car crash (6ACV16).
Date of birth12 August, 1887
Planet of originEarth, Europe, Austria, Vienna
First appearance"Law and Oracle" (6ACV16)
Voiced byMaurice LaMarche
Wikipedia has information unrelated to Futurama
Erwin Schrödinger is a physicist considered one of the fathers of quantum mechanics. While widely believed to have been born in August 1887 and was believed to have died in January 1961, Schrödinger was seen in New New York in July 3011 (6ACV16), so it is possible that he never actually died, but was rather frozen at (for example) Applied Cryogenics.
Schrödinger wears his hair slicked back, a pair of glasses, a black bow tie, a brown jacket, a white shirt, brown pants, and red shoes; speaks with a German accent; and has a malevolent-looking expression. His likeness of today is similar to that of the 1940s. He is described as being "a major violator of the laws of Physics" by Chief O'Mannahan, who claims that "guys like [him] really bust [her] uterus".
19th and 20th centuries[edit]
According to mainstream belief, Erwin Rudolf Josef Alexander Schrödinger was born on 12 August, 1887, in Vienna and died of tuberculosis on 4 January, 1961, also in Vienna. He received the Nobel Prize in Physics in 1933 for the Schrödinger equation and proposed the thought experiment Schrödinger's cat in 1935, among many other conquests.
31st century[edit]
Schrödinger, in July 3011, broke speed limit, possibly in an attempt to secure his box, and it got the attention of two NNYPD officers.
After detecting that Schrödinger's car was going at fifteen miles per hour over the speed of light, policemen Fry and URL engaged in pursuit of the physicist, following him to Circuit City in their motorcycles. Schrödinger did not stop the vehicle and a Tron-like race took place. Although he managed to evade them for a while, Schrödinger was tricked into entering the Fresnel Circle. With the City's light, the Circle created a rainbow of duplicates of Schrödinger's car and of Schrödinger himself, which ultimately caused the car to crash.
Upon crawling out of his car, Schrödinger was approached by the policemen, who learned his identity by examining his DNA and career chip. Schrödinger was then asked what was in the box that he had in the car and responded that it was "a cat, some poison, and a cesium atom", going on to say that the cat was in a superposition of both states. Doubting his words, Fry opened the box and was attacked by the cat, the status of which was therefore confirmed. URL found the poison immediately afterwards.
Fry and URL later took Schrödinger to the NNYPD and Chief O'Mannahan rewarded them with a promotion to the Future Crimes Division.
Image gallery[edit]
Additional Info[edit]
• Prior to appearing in "Law and Oracle", Schrödinger's cat was parodied in "Mars University" as "Witten's Dog" and he was also referenced in "A Clone of My Own", where a club called Schrödinger's Kit-Kat Club is seen. The name of the club also references his thought experiment.
• URL pronounced his first name incorrectly. It is actually pronounced "Er-VIN", and not "Er-WIN".
[Circuit City. Fry and URL are pointing guns at Schrödinger.]
Fry: DNA and career chip, please.
[Schrödinger offers his hand and Fry pierces it with a gun that projects a hologram reading NNY DMV, ERWIN SCHRÖDINGER and showing Schrödinger's profile photograph.]
URL: Erwin Schrödinger, huh? What's in the box, Schrödinger?
Erwin Schrödinger: Um... A cat, some poison, und a cesium atom.
Fry: The cat! Is it alive or dead? [Schrödinger is not given the time to reply.] Alive or dead?!
[URL pushes Schrödinger against his car's door, alarming him.]
URL: Answer him, fool.
Erwin Schrödinger: It's a superposition of both states until you open it and collapse the wave function.
[Fry enters the car.]
Fry: Says you.
[Fry opens the box and a cat jumps out of it, attacking him. Fry screams. URL takes a close look at the box.]
URL: There's also a lotta drugs in there.
[Chief O'Mannahan's office. Chief O'Mannahan is shaving above a drawing of Schrödinger, whom Fry and URL have brought to the NNYPD.]
Chief O'Mannahan: You boys did good. Nailed a major violator of the laws of Physics.
URL: He's goin' down. [URL lifts up Schrödinger's cat.] Cat's gonna testify.
[Chief O'Mannahan lifts up the drawing of Schrödinger, revealing it to read WANTED.]
Chief O'Mannahan: Guys like this really bust my uterus. You're both getting a promotion! Ever heard of the Future Crimes Division? |
db669c3d909b6ab6 | Barely Functional Theories
Musings on science and game design by James Furness.
Vibrational Analysis
A python script to solve the nuclear Schrödinger equation in the Born-Oppenheimer approximation for a diatomic molecule. The script comes in two files, the actual integrator Solver.py, the simple GUI wrapper QuantumWobbler.py and an example script for plugging the output of the solver into matplotlib. All three are provided here along with sample input, packaged as a zip archive shared here under the MIT licence.
Download QuantumWobbler.zip
The first 10 vibrational energy levels and wave functions for molecular hydrogen.
This solver was written as the start of a side project to investigate the vibrational spectra of diatomic molecules in strong magnetic fields bound by the perpendicular paramagnetic bonding mechanism[1,2,3]. This project was sidelined before any magnetic fields could be incorporated, though the field free solver was finished. At its core, the script is simply an implementation of the Cooley-Numerov integrator [4] that solves the nuclear Schrödinger for a given guess energy. The output was polished up for use in a talk and the solver is uploaded here in case anyone would like to use it.
A GUI was added for use in undergraduate labs, but was not necessary in the end. As such it remains a little underdeveloped and buggy in places, though the core solver is solid. Whilst the GUI runs acceptably, it is quite unstable and not recommended for routine use. If anyone would like to use it for teaching please let me know and I will clean it up, otherwise it doesn’t warrant the effort to polish something that will not get used.
A much more refined solver called LEVEL has been developed by R. J. Le Roy, and has a far greater feature set. It is an old program however, and can be a little un-intuitive to use. If your needs are simple energy levels or a plot of the nuclear wave functions however, then my solver might fit your needs with less hassle.
The script is written in Python 2.7 and the core solver requires only the matplotlib and scipy modules to run. The GUI also requires the enthought.traits, chaco and pyface modules to run. The simplest way to acquire and install these modules is to use the package manager in a Python distribution such as Enthought Canopy, setup should be straight forwards.
General Operation
An input file containing information about the system is required for both GUI operation and when the solver is used directly.
It is essential for the solver’s operation that the sampled points on the electronic energy curve have a regular separation, e.g. 0.02,0.04,0.06… The solver is not sophisticated enough to interpolate between the data points provided.
Guess levels are optional and not required. If manual guesses are missing a fitted Morse potential will be used to generate automatic guess levels. Manually setting initial guesses can be useful when the automatic guess converges to the incorrect level.
The input file should have the structure:
M1 = MASS ATOM 1
M2 = MASS ATOM 2
SEPARATION (Bohr radii),TOTAL ENERGY (Hartree)
The GUI should be self explanatory, simply load the potential input file, modify the other data fields as required and hit “Solve the system” to run the solver. Text and graphical output can be saved using the respective buttons.
The GUI runs as expected for the first solution, but can become unstable when trying to load and solve subsequent potentials as the solver is not correctly reset between runs. As such it is recommended to restart the GUI for each system. If anyone finds the GUI routinely useful please let me know and I will fix this issue to allow sequential runs.
Solver as a python module
The Solver.py module can be imported into other scripts as with any other module. A Details object should be created by calling the readInput function providing path to the input file. The Details object returned from this function can be passed as an argument to the driver function which returns the solved system in a new Details object. Full documentation of the Details object and the properties containing the solution eigenvalues and wave functions can be found in the Solver.py file, and the example script.
This code is shared under the MIT liscence Copyright 2016 James Furness.
You are free to use, modify and distribute the code, though recognition of my effort is appreciated! |
b0460e7e0184af0b | Kimball Tutorial 1 Tutorial 2
Content: Using the exact solution of the Schrödinger equation for the ground state of the H atom, derive the Ansatz of G.E.Kimball (für deutsche Leser: die ersten 10 Seiten der Einführung enthalten etwa das gleiche) ES 20 July 2017//2002//1982
Properties of the H1s wavefunction
H atom, 1s ground state (spatial) wavefunction: Proton at origin, r distance of electron from proton
Exact H1s wavefunction in atomic units! e is the base of the natural logarithms.
Graphics:H atom, 1s wavefunction
H atom ground state electron probability density (see end paragraph for a discussion)
Graphics:H atom, 1s 'point' density
H atom ground state probability density summed over all space. Must be 1, meaning: We will find the electron of the atom with certainty somewhere in the universe
The same two functions in 3D:
Graphics:H atom, 1s wavefunction
Graphics:H atom, 1s 'point' density
This shows the electron density cusp at the location of the proton
Plot of the radial density of the H atom ground state: This is the point density at r multiplied with the volume of an infinitesimal spherical shell K_tutorial1a_13.pngdr, plotted as function of r. Dimension wise, it is the charge in that spherical shell at r. Its maximum is at r = 1 a0, the radius of the 1s orbit of the atom model of Niels Bohr.
Graphics:H atom, 1s 'radial' density plotted
The next image is the surface of revolution of the above curve around the r2ψ2-axis
with proton at the origin
Graphics:H atom, 1s 'radial' density
H atom, average distance of electron from proton R, bohr units
Inside this root mean-square distance we find about 58% of the total charge of 1 electron:
and at 5.7 times that distance = 450 pm the atom is nearly complete, hence atoms, while formally infinite, are in practice minute:
Energy components:
H atom, potential energy
H atom, total potential energy
H atom, kinetic energy
H atom, total kinetic energy
H atom, total Energy
H atom, ground state equilibrium energy terms expressed with R, the expectation value of the distance of the electron from the proton
(The next few rows of arithmetic can easily be done by pencil on paper! We let Mathematica do it to make it totally transparent. When you call the notebook with the same name K_tutorial1a2.nb you can check all the calculations online with Mathematica)
Kimball’s “Ansatz”: Total energy Etot = T+V
T has dimension K_tutorial1a_36.png, V K_tutorial1a_37.png, hence:
and should be a minimum! Set its derivative to zero to find the optimal R=x:
Substitute x for R:
We have already determined, that Etot = -1/2 Eh, and R=3/2 a0.
Solve for the constants a,b using these results:
Substitute a,b into the “Ansatz” above and obtain Kimball’s equation for the H atom in its stationary ground state:
This is exact! It is a significant mathematical transformation of Schrödinger’s equation for the H-atom ground state, contains the same information and has been misunderstood by all practitioners of a qualitative Kimball or tangent sphere model:
Kinetic energy Eh: Full dimension K_tutorial1a_47.png
Potential energy Eh:
Virial Theorem
Check the ratio
This should be -2.00... by the Virial Theorem, which is valid in classical and quantum physics. The value -2 of this ratio signifies, that the force maintaining the observed system in a state of equilibrium varies proportional to K_tutorial1a_54.png. It is the electrostatic force between every charged particle pair of the system! If computations of molecules and solid lattices are reasonably complete, they yield Vir of 2.000±0.002. Most texts on quantum chemistry give a proof of this theorem.
Note: In this tutorial I have used the qualifier "exact" for the H1s (spatial) wavefunction and the "Kimball equation". Those two expressions are equivalent. They are "exact" in the usual context of quantum chemical computations, but not "absolutely true". We have neglected the finite kinetic energy of the proton (see Tutorial2 and 6), which is ~1/1836 of the kinetic energy of the electron, and have assumed that the proton is a point charge with a size negligible compared to the "size" of the H-atom. Furthermore, all movements within the atom leave the barycenter fixed. We have neglected relativistic effects, quantum electrodynamic corrections, magnetic interactions between electron and proton spins, and others. They are all of an absolute size 10-5 or less of the values shown.-
Finally, this H1s (spatial) wavefunction is the (real valued) amplitude function which goes through zero once during every period of the "wave", i.e. the particle vanishes and reappears again - addio conservation of mass! This catastrophe is eliminated by the clever mathematics of the complete 1s wavefunction which is complex-valued with a complex timefactor. It is often told, that the "square" of a wavefunction φ at a certain location q (of configuration space) is the probability for finding the particle whose state is φ(q) at q. However, this pertains to the real part of the product φ*(q)φ(q) of two complex conjugate factors. This prevents the particle described by the wavefunction φ(q) to change between death and birth during every period! I do not know of any textbook with "wave mechanics for the high school" which describes this correctly! It would help to tell this and thus avoid the erroneous notion that quantum mechanical "waves" have macroscopic analogues. Hence, "wave mechanics for the highschool" is a fake: The pupils (perhaps with exceptions, like young Wolfgang Pauli) just do not have the mathematical skill to understand Hermitian operators and work with them.- You may find a careful discussion of these topics in the first 37 pages of P.W. Atkins, Molecular Quantum Mechanics, 2nd ed, Oxford, 1983.
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Does the Schrodinger equation describe particles popping in and out of existence?
1. Sep 13, 2012 #1
Hello -
A few questions I have after watching Brian Green’s The Elegant Universe –
Within the video Dr. Green shows a neat way to view the different scales relativity and quantum mechanics are involved with. He takes an elevator to a top floor to show relativity’s applicable scale. He steps out of the elevator to show planets below him. The fabric of space-time is show as a graph paper grid – everything was very calm.
He takes the elevator down (way down) and steps out to show the quantum scale. The environment was very noisy (I compare it to watching an oscilloscope with the voltage scale set way down – lots of jitter and noise.
Within this jitter it was explained that particles and their corresponding anti-particles were briefly popping into and out of existence.
From my very limited experience with the Schrodinger equation – I see that the limits on the integral can be used to set a time range and / or volume range and the solution is a probability of that event happening within that range.
Does Schrodinger equation describe these particles / anti-particles popping in and out of existence?
If one was to solve a problem for a particular particle – an electron popping into existence – is there a parameter within the Schrodinger equation that is particle specific? Meaning how would equation differ by solving the probability of an electron popping into existence versus a different particle popping into existence?
2. jcsd
3. Sep 13, 2012 #2
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2016 Award
Staff: Mentor
In nonrelativistic quantum mechanics, there are no particles appearing/vanishing. You need quantum field theory, there you have operators which can produce and annihilate particles in the hamiltonian. If you use that in the Schrödinger equation, it can handle creation and annihilation of particles.
An electron cannot simply be created, you need an additional positron popping up. And if those do not interact with other particles in the right way, they have to vanish again. However, you can calculate the probability for all allowed processes - at least in theory.
4. Sep 14, 2012 #3
There is a very simple answer: No.
5. Sep 16, 2012 #4
As far as I know all the Schrodinger equation tells you is, if you have a particle or system of particles how the wavefunction of that particle or system behaves spatially and temporally.
Similar Discussions: Does the Schrodinger equation describe particles popping in and out of existence? |
c0d6e1f2d8376ed4 | Complex Exponentials
Complex exponentials are used immensely in math and as a result, in many fields of science. It is also used in abundance throughout this site so it is important to understand what they are for future reference. They show the relationship between exponentials and trigonometry on a fundamental level. The following is the relationship.
This relationship may seem very obscure but it can be shown to be true in many ways. Assume the derivative of both sides are taken twice as shown below.
The second derivatives of both functions are equal to negative of the original function. However, this may seem, in some sense, like a coincidence. There exists an alternative proof of this equivalence. Consider the Taylor series of , , and as shown below.
Assume is substituted with for the definition of . The following is the result.
By regrouping and factoring out, one starts to see the formation of the identity.
The terms grouped in parentheses are simply the Taylor series of sine and cosine as presented above. This was the desired result. The identity can be proven through Taylor series.
One can also redefine sine and cosine in terms of the exponentials as shown below.
This can be tested by substituting in the values for and . Note: because but . is basically the complex conjugate of .
Note their resemblance to the definition of hyperbolic sine and cosine ( and ). In fact, and .
In fact, there exists an interesting connection between this identity and the plotting of complex numbers. Complex numbers are plotted in two dimensional space where their imaginary component is the coordinate and their real component is the coordinate. This means any given coordinate has the value . Now consider a Cartesian space with a unit circle in it. Any point in the unit circle has the coordinates where is the angle the line from the point to the origin makes with the positive x-axis. If one thinks of this Cartesian space as a complex space, the value of any point on the unit circle is . In fact, any point on complex space can be represented using complex exponentials by simply making the coefficient of the exponential the distance from the origin ().
It’s amazing how to see how two totally different concepts are actually intertwined with each other. A complex exponential is just a wave in complex space.
If all this math is still confusing and doesn’t quite capture the relationship between the two concepts for you, watch 3Blue1Brown‘s amazing video on this concept that explains why it works geometrically.
3 thoughts on “Complex Exponentials
1. Even though I understand most of the math, the relation between exponentials and trigonometry still feels mysterious and amazing to me. I think I still didn’t built a full intuitive sense of it.
The fact that it is essential in one of the most fundamental physical laws, the Schrödinger equation, makes it even better. It’s like physics is built on the most beautiful mathematical equation.
Thanks for your great post.
• You should watch the video at the bottom of the post! It explains the connection between exponentiation, complex numbers, and trigonometry very elegantly through visuals without using any of the complex math in this post at all. It might make things a little more clear.
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2. […] path. The phase of any path is (if you don’t know about equations of the form , check out this quick post). The probability of a particle to go from one point to another is determined by adding up all the […]
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NDSolve[eqns, y, x, xmin, xmax ] 用来求解函数y的常微分方程 eqns的数值解，自变量 x 的范围为 xminxmax.
NDSolve[eqns, y, x, xmin, xmax , t, tmin, tmax ] 用来求解偏微分方程eqns的数值解.
NDSolve[eqns, , , ... , x, xmin, xmax ] 用来求解函数 的数值解.
• NDSolve根据InterpolatingFunction对象得到函数的解.
NDSolve[eqns, y[x], x, xmin, xmax ] 给出 y[x] 的解而不是函数 y 自身.
• Differential equations 必须通过使用D的导数比如 y'[x]给出, 而不通过使用 Dt的全导数给出.
• 在常微分方程中函数 必须仅依赖与单变量 x. 在偏微分方程中它们可能依赖于多个变量.
• 微分方程必须包括足够的初始条件或边值条件以完全确定 的解.
• 初始条件和边值条件通常以 y[ ] Equal , y'[ ] Equal ,等形式给出,但也可能由更多复杂的方程构成.
• 出现在初始条件或边值条件的点 不必在解所考察的范围xminxmax内.
• 可以给出下列可选项:
"\!\(\*StyleBox[\"\\\"AccuracyGoal\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Automatic\\\"\", \"MR\"]\) ""解的数字绝对准确度"
"\!\(\*StyleBox[\"\\\"Compiled\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"True\\\"\", \"MR\"]\) "是否编译原来的方程
"\!\(\*StyleBox[\"\\\"InterpolationPrecision\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Automatic\\\"\", \"MR\"]\) "所返回的插值数据的精度
"\!\(\*StyleBox[\"\\\"MaxSteps\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Automatic\\\"\", \"MR\"]\) "要采用的最大步数
"\!\(\*StyleBox[\"\\\"MaxStepSize\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Infinity\\\"\", \"MR\"]\) "每一步的最大步长
"\!\(\*StyleBox[\"\\\"PrecisionGoal\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Automatic\\\"\", \"MR\"]\) "解的数字精确度
"\!\(\*StyleBox[\"\\\"StartingStepSize\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"Automatic\\\"\", \"MR\"]\) "所使用的初始步长
"\!\(\*StyleBox[\"\\\"WorkingPrecision\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"$MachinePrecision\\\"\", \"MR\"]\) "内部计算中所使用的小数数位
• 当达到所指定的AccuracyGoalPrecisionGoal时,NDSolve停止.
• 如果在解的值接近0时要准确地得到该解,AccuracyGoal应被设置得较大，或是Infinity.
• 参见 Mathematica 全书: 3.9.1节 和 3.9.7节.
• 实现注释: 参见 A.9.4节.
• 同时参见: DSolve, NIntegrate.
Further Examples
Ordinary differential equations: Basic usage
This command finds a numerical approximation to a function that is equal to its first derivative at each point x between and , and that has the value when x is . NDSolve returns a rule to replace y by an InterpolatingFunction object.
Mathematica expresses the closed-form solution of this interesting differential equation as a pure function.
We rewrite it as an ordinary algebraic expression. The solution has a singularity at .
Now we solve the same equation approximately, using NDSolve instead of DSolve. As we can see from the closed-form solution, the warning is appropriate.
We can use an InterpolatingFunction object like any other function that evaluates to a number. Here are three ways to check the boundary condition.
The solution was not found all the way to , but the InterpolatingFunction object still allows you to evaluate with arguments outside its range. You can see that the approximate value is quite large, though smaller than the correct value of infinity!
It is also easy to make plots of solutions. Here is how to see a graph of an approximation to the function which is the reciprocal of its derivative. The Evaluate command saves time by substituting the InterpolatingFunction once, instead of for each number used to generate the plot. Not too surprisingly, this plot looks very much like the square root function. What do you think would happen if you tried to solve in the other direction?
Evaluate the cell to see the graphic.
Mathematica can handle higher order equations. Here is a plot of the interesting solution of a third-order equation. Note that you have to specify enough initial conditions to uniquely determine the solution.
Evaluate the cell to see the graphic.
Solving systems of equations works similarly. For systems of two equations a so-called phase plot is often a good way to visualize the solution. Here is a phase plot that describes the motion of a weakly damped pendulum.
Evaluate the cell to see the graphic.
Ordinary differential equations: Options
This way of solving the differential equation for the cosine function doesn't work over such a long interval.
You need to increase the setting of the MaxSteps option. As expected, at the end of the interval the solution is close to .
The Method option allows you to specify which method NDSolve uses to approximate the solution. The default method, Automatic, automatically switches between Gear and Adams, depending on stiffness. Another possibility for equations which are not stiff is RungeKutta. For some problems, the Runge-Kutta method can find the solution using fewer steps.
For the Rössler equations, the Runge-Kutta method needs about half as many steps as the default method.
Evaluate the cell to see the graphic.
One way to get a very precise solution of an ODE is to give a sufficiently high value for the WorkingPrecision option. Note that AccuracyGoal and PrecisionGoal default to 10 less than the value of WorkingPrecision when it is greater than $MachinePrecision.
You need to be careful not to use too high a value for WorkingPrecision, because as working precision increases, not only does the time taken for each arithmetic operation and function evaluation increase, but the number of steps typically increases exponentially also.
These commands compare a known exact solution with solutions computed with different values of WorkingPrecision.
The values of x at which the steps are taken is kept in the third part of the InterpolatingFunction object. This is why the Length command above gives the number of steps.
Ordinary differential equations: Boundary value problems
Simple linear boundary value problems can be solved by constructing the boundary value equations appropriately. If the order of the equation is you need to know the values of some combination of the function or its derivatives at points.
This normalized equation describes the effect of a wave incident on the right edge of an optical medium (where x is ).
Evaluate the cell to see the graphic.
Here is a third-order equation where the function values and combinations of the derivatives are known at three points.
Evaluate the cell to see the graphic.
Not all linear equations with boundary values can be solved by the method that is implemented. Nonlinear equations cannot be solved, either.
Evaluating this generates a lot of messages and no result.
Partial differential equations: Basic usage
For the first three examples, consider the one-dimensional heat equation. With fixed boundary conditions this is a model for diffusion of heat in an insulated rod with the temperatures at the endpoints held fixed.
This command solves the heat equation with the left end ( ) held at fixed temperature , the right end ( ) held at fixed temperature , and an initial heat profile given by the quadratic in x.
Just as for ODE's, the solution is returned as a rule to replace the dependent variable with an InterpolatingFunction object. The only noticeable difference is that the InterpolatingFunction object is two-dimensional. That is to say, it takes two arguments to evaluate, with the arguments in the same order as the variables in the NDSolve command.
The boundary conditions can be a linear combination of Dirichlet and Neumann type conditions. For example, this command solves a model for the diffusion of heat in a one-dimensional insulated rod with the left end held at a constant temperature and the right end radiating into free space. The Neumann boundary condition was entered using the Derivative operator. An equivalent way to give the condition is ((u[x,t] + D[u[x,t],x]) /. x->1) Equal 0.
Evaluate the cell to see the graphic.
An even more interesting case is when the boundary conditions are time-dependent. For example, entering these command produces a plot of a solution with the temperature at the left edge varying sinusoidally.
Evaluate the cell to see the graphic.
It is also possible to compute the solution of some systems of PDE's. These commands solve a system of mixed parabolic-hyperbolic type and produce separate contour plots for each of the dependent variables.
Evaluate the cell to see the graphics.
Periodic boundary conditions are frequently convenient for numerical solutions. You can tell NDSolve to solve with periodic boundary conditions by specifying that the values of the solution are to be equal at the left and right edges of the domain in one independent variable for all values of the other independent variable.
For example, an interesting solution with periodic solutions is the nonlinear Schrödinger equation. These commands set up a periodic initial condition (it happens to be a soliton), and computes the solution, and then produces plots showing the modulus and real and imaginary parts of the solution.
Evaluate the cell to see the graphics.
Partial differential equations: Limitations
NDSolve uses a method for solving PDE's that is called the numerical method of lines. It discretizes in one variable to make a system of ODE's. This system is then solved using the ODE methods built into NDSolve.
For the method to work, an initial function must be specified for one variable and boundary values may be specified for the other variable. The initial function is used to find the initial conditions for the system of the ODE's. Boundary and initial values may be specified on at most three sides of a rectangle.
The method has the advantage that it can solve a reasonably large class of equations. However, there are types of equations which it cannot solve.
Because elliptic problems are ill-posed unless boundary values are specified on all sides of a region, this method cannot find solutions of elliptic problems. A classic example is Laplace's equation. Entering this command indicates what will happen when you try to do something of this sort. Typically ill-posedness will appear as numerical instability, and looking at the scale on the plot produced indicates that you should heed the message seriously!
Evaluate the cell to see the graphic.
Another class of problems that the method cannot currently handle are those that form singularities in the solution. The discretization is done with finite difference methods, so fronts may be incorrectly resolved or completely lost. For example, Burgers' equation is a model for some of the features of gas dynamics, including shock formation. Entering these commands will produce a plot that show the oscillations typical of the interaction between finite difference methods and fronts.
Evaluate the cell to see the graphic.
Partial differential equations: Options
Many of the options that control the ODE solver also apply to the PDE solver, though often in a different way. The solutions for PDE's are computed in two stages. First the equation is discretized, and then the resulting system of ODE's is solved using NDSolve's built-in method. If you want different option values for the two stages, you can specify the option value as a list. (The order of the independent variables in the command determines to what variable the options apply.)
The following options to NDSolve can be used in such a list: AccuracyGoal, PrecisionGoal, DifferenceOrder, MaxSteps, MaxStepSize, StartingStepSize.
In the discretization stage, the default used is fourth-order finite differences. In some cases fourth-order is not optimal; you can control this with the DifferenceOrder option. When the equation has high spatial differential order (eg. Airy's equation), it is better to increase the order.
Entering these commands computes a solution to Airy's equation with periodic boundary conditions and produces plots that you can view as an animation. Since the independent variable is the one with an initial function, it is the one for which we specify the difference order of . (The variable x appears after t in the list of arguments of NDSolve.)
Evaluate the cell to see the graphic.
We clean up by clearing the definitions that were made. |
4da9776d13b926de | Baggott: Quantum Reality
I recently read, and very much enjoyed, Quantum Reality (2020) by Jim Baggot, an author (and speaker) I’ve come to like a lot. I respect his grounded approach to physics, and we share that we’re both committed to metaphysical realism. Almost two years ago, I posted about his 2014 book Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth, which I also very much enjoyed.
This book is one of a whole handful of related books I bought recently now that I’m biting one more bullet and buying Kindle books from Amazon (the price being a huge draw; science books tend to be pricy in physical form).
The thread that runs through them is that each author is committed to realism, and each is disturbed about where modern physics has gone. Me, too!
I have been increasingly disturbed about modern physics ever since I read The Trouble with Physics (2006), by Lee Smolin. Even then, from years of following the well-grounded blogs of Sabine Hossenfelder and Peter Woit, I felt too many physicists had wandered from doing science to doing science fiction. I’m appalled by the notion of “post-empirical science” — a notion that couldn’t be more contrary to the spirit of science.
[See: Fairy Tale Physics (Apr 2020), Our BS Culture (Dec 2020), and Our Fertile Imagination (Jan 2021), for three of my more recent posts discussing the embrace of pure speculation by theorists and culture. It’s a trend that damages and undermines science in a time when science is under fire from politics and society. Proponents of Intelligent Design rightfully ask, “If multiverses that must be taken on faith are ‘science’ then why isn’t ID?” The truth is that both are fantasy bullshit (FBS).]
Speaking of Peter Woit and Lee Smolin, among that aforementioned handful of new books is Woit’s Not Even Wrong (2006), which I’ve been meaning to read for years, and Smolin’s latest book, Einstein’s Unfinished Revolution: The Search for What Lies Beyond the Quantum (2019). I read the former before reading Quantum Reality and am currently enjoying the latter.
Also in the handful is Fashion, Faith, and Fantasy in the New Physics of the Universe (2016), the most recent popular science book from Roger Penrose. It’s a book I’ve been wanting to read since I first heard of it.
Penrose’s books aren’t for the faint of heart. In the 1990s, it took me years and multiple readings to fully absorb his 1989 book, The Emperor’s New Mind: Concerning Computers, Minds and The Laws of Physics. (In large part I owe my skepticism about computationalism to that book. It planted the first seed of discontent about a topic I’d taken for granted until then.) Recently I posted (in fact, twice) about his 2010 book, Cycles of Time: An Extraordinary New View of the Universe, which explores his Conformal Cyclic Cosmology (CCC) hypothesis. (While I enjoyed Cycles of Time, the CCC hypothesis is pure speculation, which Penrose readily admits. See Sabine Hossenfelder’s recent video about it.)
You might be thinking, “Wait, that’s four books. A handful should be five!” I also bought Baggott’s The Quantum Cookbook: Mathematical Recipes for the Foundations of Quantum Mechanics (2020). As the title suggests, it’s for those with mathematical inclinations. It straddles the gap between textbook and popular science book. Each chapter shows how a famous physicist derived a famous physics equation. For examples, the first chapter takes the reader through how Planck derived E=hv, the second how Einstein derived E=mc2 (which is as far as I’ve gotten so far).
In Not Even Wrong, Woit explores why string theory is an interesting idea that’s become fairy tale physics. As he mentions, many physicists see it as pure math because it offers no evidence or way of testing the theory (and has the landscape problem). On the other hand, mathematicians see it as physics because, as math, it lacks the rigor that’s foundational in math. It seems many assume that, if not to them, it somehow makes sense to someone. (Woit, by the way, is primarily a mathematician with a strong interest in physics.)
Two books by Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (1999) and The Fabric of the Cosmos: Space, Time, and the Texture of Reality (2004), got me excited about string theory, but everything I read since then gave me a different outlook (starting with Smolin’s The Trouble with Physics).
In Fashion, Faith, and Fantasy, Penrose echoes that in his first section, Fashion, which is about how string theory became such a strong fashion in physics that many new physicists saw no alternative but to work on it if they wanted a job or any funding. As I already quite agree, I jumped to section two, Faith, which is about the abiding faith many theorists have that quantum mechanics is a complete description of reality.
As I mentioned, Penrose’s books aren’t for the faint of heart. He is far more willing to dive into the technical weeds than most popular science writers. While reading section two, I decided to rest my mind and read Not Even Wrong, Quantum Reality, and currently, Einstein’s Unfinished Revolution.
I’ll get back to Fashion, Faith, and Fantasy in due time. Since he refers to it many times in the book, I also bought his 2004 book, The Road to Reality: A Complete Guide to the Laws of the Universe, so I have plenty of Penrose waiting to challenge my mind. I am looking forward to what his Fantasy section is about!
§ §
Which finally brings me back to Baggott, Quantum Reality, and metaphysical realism.
The heart of the issue is the wavefunction. Is it something real or does it merely encode what we know about a quantum system? And if it is real, exactly what is it?
An issue like this doesn’t exist in classical physics where, at least to a first approximation, the equations “wear” their reality in plain sight and there is no confusion. A good example is Newton’s Second Law, usually expressed as F=ma (force is equal to mass time acceleration).
Above I said, “to a first approximation” because a deeper look turns up a puzzle: what is mass? Newton defined it as volume times density but defines density as mass per volume — which is circular and, thus, not much of a definition. Baggott wrote a whole book about it, Mass: The Quest to Understand Matter from Greek Atoms to Quantum Fields (2017).
Quantum Reality features both Preamble and a Prologue, the latter subtitled Why Didn’t Somebody Tell Me About All This Before? In the Prologue, Baggott touches on his own journey, first learning basic quantum mechanics as part of learning chemistry but then encountering the famous EPR paper from 1935 as well as the experiments by Alain Aspect in 1982. He reports that it sent him into a tailspin and began a 30-year journey to try to understand.
He ends the Prologue by saying: “I can happily attest to the fact that, like charismatic physicist Richard Feynman, I still don’t understand quantum mechanics. But I think I now understand why.”
Part I of Quantum Reality is called The Rules of the Game, and it’s an overview of our understanding of quantum mechanics. Here are the chapter headings:
1. The Complete Guide to Quantum Mechanics (abridged); Everything You’ve Ever Wanted to Know, and a Few Things You Didn’t
2. Just What is This Thing Called ‘Reality’, Anyway; The Philosopher and the Scientist: Metaphysical Preconceptions and Empirical Data
3. Sailing on the Sea of Representation; How Scientific Theories Work (and Sometimes Don’t)
4. When Einstein Came Down to Breakfast; Because You Can’t Write a Book About Quantum Mechanics without a Chapter on the Bohr-Einstein Debate
It’s in this part that he introduces his key metaphor of the Ship of Science sailing the Sea of Representation back and forth between the shores of Empirical Reality and Metaphysical Reality.
Ship of Science sailing the Sea of Representation
From Quantum Reality, drawn by Eugenia Nobati and © Jim Baggot
During its back-and-forth journeys it needs to avoid the rocky shoal of Scylla, which lies close to the shores of Empirical Reality. He defines it as “rather empty instrumentalism,” that, while perfectly valid empirically, is devoid of “any real physical insight and understanding.”
The Ship of Science also needs to avoid the whirlpool of Charybdis, which lies close the beaches of Metaphysical Reality. “It is a whirlpool of wild, unconstrained metaphysical nonsense.”
He has developed this metaphor over time, and I can recall him mentioning it in talks that predate this book. (I do not recall it being mentioned in Farewell to Reality, but my memory can be like Swiss cheese when it comes to some things. Like Sherlock Holmes, I don’t even try to remember things that I don’t deem useful.)
The key point of the metaphor is that science proceeds by moving back and forth between metaphysical speculation and experiment, and that both are crucial to the process. No scientific theory is without some metaphysical assumptions, but those must be grounded in experiment if they are to mean anything.
In Part II, Playing the Game, Baggott covers many of the popular interpretations of quantum mechanics (the only physics in which interpretation is even necessary). Here are the chapter headings:
1. Quantum Mechanics is Complete So Just Shut Up and Calculate; The View from Scylla: The Legacy of Copenhagen, Relational Quantum Mechanics, and the Role of Information.
2. Quantum Mechanics is Complete But We Need to Reinterpret What it Says; Revisiting Quantum Probability: Reasonable Axioms, Consistent Histories, and QBism.
3. Quantum Mechanics is Complete So We Need to Add Some Things; Statistical Interpretations Based on Local and Crypto Non-local Hidden Variables.
4. Quantum Mechanics is Incomplete So We Need to Add Some Other Things; Pilot Waves, Quantum Potentials, and Physical Collapse Mechanisms.
5. Quantum Mechanics is Incomplete Because We Need to Include My Mind (or should that be Your Mind?); Von Neumann’s Ego, Wigner’s Friend, the Participatory Universe, and the Quantum Ghost in the Machine.
6. Quantum Mechanics is Incomplete Because… Okay, I Give Up; The View from Charybdis: Everett, Many Worlds, and the Multiverse
The basic tension here is between the decidedly anti-realist views of Bohr and, hence, the Copenhagen school (which dominated quantum physics) and those, such as Einstein, who preferred a realist approach.
The problem is that, if we take QM as complete, then we’re stuck with either anti-realism or some metaphysical extremes (such as the MWI, which Baggott paints as “magical realism” — a point I’ve made repeatedly).
Baggott discusses how quantum physics seems to have encountered Kant’s noumena, the things-in-themselves that we can only know through their representations in our senses. Anti-realist views accept that we can never know them, only those appearances through their interactions with our experiments. Of course, all our experiments use classical physics. The quantum world, in some sense, is indeed inaccessible to us. We can never actually see a superposition, for instance.
§ §
This has gotten long, so I’ll stop, but I expect I’ll return to this and the other books in future posts. I have strong objections to the fantasy bullshit the science and social world seems to wallow in these days, and while posts may do nothing, expressing my dissatisfaction, I’ve always found, is good for my mental health.
If, like me, you share a sense of disquiet about perceived fantasy bullshit in science, if you are, to your core, a metaphysical realist, then I highly recommend Quantum Reality as a great read. The other books I’ve mentioned here also express the need for grounding in empiricism and realism.
For whatever it’s worth, being a metaphysical realist puts one in the company of Albert Einstein, who spent his life trying (unsuccessfully) to complete quantum mechanics.
And the thing is, it seems almost self-evident that QM is incomplete because it is at odds with our other greatest theory, general relativity (which is an entirely realist theory). It seems to me almost foolish to accept QM as is given that conflict.
As a final note, I’ve never been much taken with Carlo Rovelli, who’ve I’ve seen as not just an anti-realist, but as something of a space cadet (as we used to say). I’ve also never been taken with theories that make relations fundamental (I see them as necessarily secondary). I’ve long thought Leibniz and relationalism is “not even wrong.”
So, I never bothered with Rovelli’s Relational QM, but after reading Baggott’s book I see that Rovelli, at least in RQM, is a realist in seeing quantum objects as real. The theory, however, is still anti-realist in seeing that the only access to those objects is through their relations with our (classical) experiments. In that sense, he is aligned with Bohr.
§ §
Stay realist, my friends! Go forth and spread beauty and light.
About Wyrd Smythe
The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe
24 responses to “Baggott: Quantum Reality
• Wyrd Smythe
Oof, sorry, that went a lot longer than I thought it would (in fact, I thought it would be a short post). There is so much I didn’t cover that I’ll definitely be returning to this book and the others I mentioned.
• Wyrd Smythe
FWIW, I lean towards physical collapse theories, and I’m not entirely convinced the wavefunction is real. (Because, if it is, it lives in very high dimension, in some cases infinite dimension, complex Hilbert space and requires complex numbers. It’s hard to understand how something like that could be real.)
I also lean strongly towards thinking QM is badly in need of a fresh start. I agree with Philip Ball and quantum revisionism. (See: Ball: Beyond Weird)
• Wyrd Smythe
Here, perhaps, is an intuition for what happens when a quantum system interacts with a classical system:
Imagine a system small enough to exhibit quantum behavior as a single song playing. Imagine a system large enough to be “classical” as 10²º songs playing. So, then what happens when that single song encounters 10²º songs? For that matter, what happens to any single song within the 10²º songs?
Individual songs are completely swamped out by the vastly larger collection.
As a further intuition about measurement interactions, think of a mousetrap or a gun with a “hair trigger”. The tiny interaction with the mouse, or the tiny interaction of pressing the gun trigger, results in the release of a much larger amount of the stored energy in the mousetrap’s spring or the gun’s bullet.
All classical devices that can detect a quantum system operate as stored energy systems waiting for a tiny triggering action. For instance, a Geiger counter uses a Geiger-Müller tube in which the stored energy is several hundred volts difference between the tube’s shell and the inner central wire. See the linked article for how a single particle causes a cascading avalanche of events that trigger the release of that stored energy.
• Wyrd Smythe
Here’s a presentation Baggott gave for the Royal Institute about his book:
• Katherine Wikoff
I’m having trouble processing the phrase “post-empirical science.” I know what the words mean, but isn’t the whole point of science to investigate empirical phenomena? Does post-empirical science basically return us to pre-Enlightenment science?
• diotimasladder
Farewell to Reality—catchy title!
I’ve never heard the phrase, “post-empirical science”, but “post-empirical” seems to accurately describe some of the theories you mention, though I’m not sure about scientific. Can there be a post-empirical science? I don’t know. I suppose that all depends on whether science is allowed to change. (Into metaphysics.) 🙂
I have to say, I have little interest in Many Worlds and the like because I see them as entirely mathematical, which means there’s nothing in it I have access to. I’m also realizing that when theories get interpreted for laymen by science writers looking for a clever turn of phrase, something is likely to get screwed up. Sometimes they turn theories into preposterous metaphors that sound like science fiction. Bad science fiction. Upon encountering these stories, for a moment I question whether the problem lies with the interpreter or the theory, then I realize I have more interesting pursuits to attend to.
What is matter?—Now that’s a question! Really, it’s one I’ve often wondered about.
• Wyrd Smythe
The funny thing is that my two-year old post about that book has been getting hits lately. Baggott is a pretty good science writer, and he seems very good at making the material accessible to regular audiences. His avoidance of any math notation in Quantum Reality kinda made me smile. Instead of the very common Dirac notation, e.g. |Ψ⟩, he used little pictures of boxes and nary a Greek letter in sight.
Exactly as you say, post-empirical “science” shouldn’t be seen as science at all, but as metaphysics. One of Baggott’s points is that all science contains metaphysical assumptions, but we need to bracket those as best as we can and allow experimental results to steer us. Yet many well-respected scientists (and some philosophers) have argued in favor of post-empirical science. Given the whole “post-factual” thing going on in society now, and the growing disdain for science, indulging in such fantasies does science harm.
Part of the problem about science writing when it comes to the quantum world is that there aren’t words or intuitions to explain it, and therefore all the preposterous metaphors and analogies that don’t really convey much understanding. Because they can’t — no one understands what the math means. As I’ve gotten into that math, my whole view of the subject has evolved, and I’ve come to realize how empty all those words are. It’s like climbing a mountain and finally seeing above the trees.
And, yeah, while the math isn’t that hard, it absolutely requires a serious interest and a willingness to invest the time. Most people have better ways to spend their time. Only us über-geeks find it worthwhile. 🧐👨🏼🔬
That’s a good point about the MWI being mathematical. It really is in that it’s based on the notion of applying the Schrödinger equation to all of reality. The thing that’s struck me lately is, given that there is no test we can make, nor any evidence in support of the MWI, what is the point, really? We can’t leverage it or prove it, so it really does seem, as Baggott says, like just giving up and accepting a kind of magic. I truly cannot fathom the attraction. It seems almost a cult to me.
What is matter is the key question! The best minds have been chewing at that for hundreds of years and so far, no one knows for sure. The current best answer? Invisibly tiny disturbances (wavelets) in various quantum fields. Or maybe tiny vibrating strings. Or… 🤷🏼♂️
• Lee Roetcisoender
I believe that where we are on the scientific frontier is an analogue of the flat earth syndrome of the middle ages. If one is able to project themselves into Columbus’ situation, one can see that our current moon and space exploration is a tea party compared to what he went through. Space exploration doesn’t require real root expansions of thought. We have no reason to doubt that existing forms of thought are adequate to handle it and therefore, the whole endeavor is just a branch extension of what Columbus did over five hundred years ago.
In contrast, a really new direction in exploration, one that would look to us today the same way the world looked to Columbus would be in a completely different direction. And that direction would be into realms beyond reason. I think present-day reason is an analogue of the flat earth syndrome of the medieval period; if one goes too far beyond the boundaries of acceptable reason it is presumed that one will fall off into insanity; and we are all very much afraid of that.
In summary: the problem is not with the physical sciences, the fundamental problem is rooted in reason itself. It isn’t so much that reason itself is the problem; it is the self-induced fear of moving beyond the boundaries of acceptable norms. It’s a personal choice really, not a choice one can make by reading scientific or religious publications.
Just a thought for the day folks……
• Wyrd Smythe
The day and, as I recall, a number of days in the past. I’ll ask the same question today as I did on those days: Okay, what do you suggest we replace reason with? Reason has a proven track record; how should we approach your “unreasonable” program?
Perhaps this is too much reason, but I’d say the opposite is true of Columbus versus space travel. Perhaps you recall the grade school rhyme, “In fourteen-hundred-and-ninety-two; Columbus sailed the ocean blue.” That was also about the year that the first surviving Earth globe was made (by German mapmaker Martin Behaim). In fact, the ancient Greeks knew the Earth was a sphere as early as the 3rd century B.C. (because they used reason). They even made a pretty good estimate of its size.
So, Columbus was just doing what mariners had been doing for centuries — making an ocean voyage. He had no fear of a flat Earth. He intended goal was to reach India by going around the Earth. (You have perhaps fallen into the popular myth that his voyage was to prove the Earth round, but that was well known at that time. He was only seeking a better route.)
In contrast, if you remember the early days of space exploration, there were serious concerns that humans could not survive in weightless conditions, and some of our first steps into space were tests of whether it was possible. We very successfully used reason to approach a true edge of our intellectual map, and the result is that we’ve sailed past Pluto to beyond the neighborhood of the Solar system.
So, if it’s not too unreasonable, I’ll ask again: How exactly do you see reason failing, and what is the better course we should take?
• Lee Roetcisoender
The fundamental problem with reason is that it’s self-serving; so in that sense it’s a psychical problem not a problem with reasoning itself. Reasoning is a tool; the carpenter builds the house not the hammer.
Leaving the psychology aside; a step in the right direction for metaphysics in general and the physical sciences is to clearly define and establish what constitutes a “proof”. For example: the experiment that Arthur Eddington conducted in 1915 during a solar eclipse “proved” that the sun will bend a beam of light; and that is all this experiment proved.
In contrast, Eddington’s experiment “does not” prove that Einsteins theory of GR is correct; that conclusion is an inference. So, there’s much work that has to be done and the main thrust of that work has to focus on the psychology that utilizes and all too often weaponizes the tool of reasoning. Ukraine is the prevailing example of this type of nonsense…….
• Wyrd Smythe
I would say the fundamental problem, as always, is humans misusing their tools — a purely sociological and psychological issue. What’s going on in the Ukraine right now has to do with a self-isolated narcissistic egomaniac who may well be clinically insane. It has nothing to do with reason; quite the opposite, I’d say. Indeed, it is the carpenter, not the hammer.
No scientist worth their salt would ever say that Eddington’s eclipse observation proved GR. Despite far more accurate tests since then, no scientist worth their salt would ever say GR is proven now. Experiment can only confirm the predictions a theory makes or — better yet — disprove it, because that is the only possible final result.
This is well-known as “the white swans problem.” One can have a theory that “All Swans Are White” but no matter how many white swans we find, we can never be sure about the next one. (And, indeed, the Europeans discovered black swans when they finally got to Australia.) A White Swans theory can only be falsified, never proven (unless one is capable of examining every swan past, present, and future).
There is no need to clearly define what constitutes “proof” — the problem, as I said, is well-known. About the only theories that can be proven are mathematical (which, even then, depend on accepting some basic axioms regarding numbers). In science and physics, theories are only either falsified or contingently accepted over time as lots and lots of tests fail to falsify them. GR is such a theory that has survived all tests that might have falsified it but didn’t. Even so, because it’s not unified with our other well-tested theory, quantum mechanics, we know it can’t possibly be a ground truth.
• Lee Roetcisoender
I think you summed up the limitations of empiricism very well, at least from the perspective of the physical sciences and a posteriori, whereas mathematics is a priori judgements; and for the most part it appears that the ability to make prudent judgements is as good as it gets.
“It has nothing to do with reason; quite the opposite…”
Correct me if I am wrong, but I think what you mean by this assessment is “the ability to reason correctly” is diminished not reason itself. This distinction naturally leads us to the next question: what is reason?
Fundamentally, I see reason as intrinsic power; one can call it capacity or ability but either way it reduces back to power or the ability to act. The ability to think for oneself is power and it is this thing we refer to as power that is abused by the carpenter.
Secondarily, I see rationality which is an expression of reason and/or an expression of power as a discrete binary system; and since rationality is a discrete binary system, as a system it is limited to the patterns of form.
• Wyrd Smythe
I don’t know that I would refer to the a priori nature of mathematics as “judgement” because I associate that term with a choice between competing arguments. The reason we have a judicial system is that laws — which are kind of black-and-white mathematical constructs — often fail to anticipate edge cases, the unexpected, or the need for compassion.
Keep in mind also that mathematics leads to such things as Riemann surfaces and many other constructs with no known physical reality. (One issue with string theory, for instance, is that it seems a mathematical fantasy.) There is also what Cantor showed about uncountable sets and what Gödel showed about math’s incompleteness, which, to some extent, is why laws can never cover all cases.
I think the first paragraph of the Wikipedia article on reason offers a good definition:
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, language, mathematics, and art, and is normally considered to be a distinguishing ability possessed by humans. Reason is sometimes referred to as rationality.
I think equating it with “power” is too reductive (and I don’t see it as the power to act; one can sit on the couch and reason — indeed, that is what philosophy is all about). It definitely is a power, one that seems characteristic of intelligent minds, but so too is our ability with language, music, love, altruism, and others.
To the extent that rational thinking is associated with judgment, through things such as compassion, it evades the binary nature of logic (which is, indeed, strictly about the form of an argument, but reason concerns also with its content).
• Lee Roetcisoender
Like everything else in our culture, the quote from Wiki tell us what reason does but it does not tell us what reason “is”. All too often reason is conflated with rationality but they are two distinctly different things.
Sorry to hear that you are so down on reduction. The power to act comes first in hierarchy, and for conscious beings like ourselves, this power to act is literally the very mentation process or thinking. It is motion in the truest sense resulting in form, form such as language, music, love, altruism, etc. Therefore, the “power to act” lies at the core of motion everywhere within the universe.
So, I really don’t see how one can marginalize this thing called “power”. Kant ferociously defended his metaphysical position that power is a fundamental reality and literally the “thing-in-itself”.
• Wyrd Smythe
I can’t tell if you’re trolling me for fun or just didn’t understand what I said. Hint: I am not down on reduction nor marginalizing power. Until and unless you can show that you understand me, there isn’t much point in this conversation.
• Lee Roetcisoender
Oh, I do understand you Wyrd. To borrow a metaphor; your interests are in the beauty of surface appeal, what goes on in the branches and twigs of the tree whereas my interests are in the beauty of underlying form, what goes on in the root of the tree. The root defines the tree, not its trunk, branches, leaves, flowers or fruit.
Because of our personal preferences and for the sake unity, the two of us really have no business engaging in philosophical discussions…..
• Wyrd Smythe
No, Lee, you clearly don’t. This very post, and so many others of mine, show otherwise. Your very entry into this conversation was about the question of what matter is — not that you’ve provided any kind of answer. Nor have you answered the question I’ve asked you repeatedly now and in the past: If not reason, then what replaces it?
We get to this point every time, and what I find so sad about internet intellectual wannabes is the way they flee when truly challenged. Intellect talks; bullshit walks.
• Lee Roetcisoender
As I’ve stated before and will repeat; there is nothing wrong with reason of and by itself because reason is a tool. Unfortunately, the “power” of this wonderfully magnificent tool is used for self-serving purposes, and that “purpose” is what has to change.
Human beings are the apex predator on this planet, and the power of reason has served the species as predator. But unless or until we are willing individually or collectively to address this self-serving attribute of our intrinsic predator mind-set nothing will change.
We all like to sit in a seat of self-righteous indignation wagging our judgmental finger at the other guy. But make no mistake my friend; no human being on this planet is any better or worse than any other human being; there are no good guys or bad guys, there are no heroes or villains, there are no slime balls or upstanding citizens; there are only individuals of a predatory species doing what a predatory species does. The only thing that separates one individual from another is a matter of scope; and if you find that fact of the matter offensive, then you just don’t get it.
I am no god damned messiah my friend, but I find our intrinsic nature to be very offensive.
• Wyrd Smythe
In my second reply to you I wrote, “I would say the fundamental problem, as always, is humans misusing their tools — a purely sociological and psychological issue.” This has nothing to do with reason (the hammer) but with the carpenter.
Further, the failings of the human species are a keynote of this blog and many of its posts. I absolutely agree that we’re victims of our own success and intrinsic nature, but it seems to me that, if we’re to grow up as a species, it will be reason that helps us accomplish that.
I believe that morality and high intelligence are correlated, and frankly, I see the intrinsic problem as our own stupidity. Our powers (by which I mean abilities) of reason are too frequently overwhelmed by our greed and tribal natures. As I also suggested, the problem isn’t reason so much as its lack and failure.
(FWIW, having met both true slime balls as well as true upstanding citizens, I’d have to disagree that they don’t exist. There are, indeed, those who rise above and those who sink below. We’re a richly varied species.)
• Lee Roetcisoender
This is what I consider to be the quintessential challenge for reason:
On an individual level, do you think that reason would be willing to forsake everything that reason believes to be true for that which is “unknown”? In other words; would reason be willing to place a life changing or perceived life threatening bet not knowing in advance or even having a clue what it would get in return?
Note: I’m not referencing religious experience because all religions “supposedly” know in advance what the return on investment will actually be…
• Wyrd Smythe
Well, I think we make those (non-religious) “leaps of faith” all the time. Some are small, some are huge. My own marriage is a good example. Had I approached it with more logic and less heart, or had I been able to know the future, I wouldn’t have done it, and my life would have been much better for it. As a friend of mine once said, life is a crapshoot. You roll the dice and hope for the best. To one extent or another, we’re all Bayesians, and explorers, scientists, artists, and storytellers, crave the unknown.
I would add that, while reason uses logic, it also uses judgement, compassion, hope, and faith.
(One reason I don’t consider myself a musician is that I know too many real musicians, and I’ve seen them produce beautiful music from crude or even broken instruments. The true artist never blames the tool because art comes from the heart. Hammers can destroy or build depending on who is wielding them.)
• Wyrd Smythe
Finished Lee Smolin’s Einstein’s Unfinished Revolution (post coming Sunday Monday). Now I need to dive back into Penrose’s Fashion, Faith, and Fantasy. And after that, his The Road to Reality. I’m getting a strong dose of quantum realism here (and loving it).
• Smolin: Einstein’s Unfinished Revolution | Logos con carne
[…] Earlier this month I posted about Quantum Reality (2020), Jim Baggott’s recent book about quantum realism. Now I’ve finished another book with a very similar focus, Einstein’s Unfinished Revolution: The Search for What Lies Beyond the Quantum (2019), by Lee Smolin. […]
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8839bea5d1ec5c72 | We gratefully acknowledge support from
the Simons Foundation and member institutions.
Authors and titles for math.PR in May 2020, skipping first 175
[ total of 315 entries: 1-50 | 26-75 | 76-125 | 126-175 | 176-225 | 226-275 | 276-315 ]
[176] arXiv:2005.12689 [pdf, ps, other]
Title: Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\geq 3$
Comments: 47 pages
[177] arXiv:2005.12706 [pdf, ps, other]
Title: Edwards-Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d \geq 3$
Comments: 46 pages, 2 figures, final version to appear at Ann. Inst. Henri Poincare
[178] arXiv:2005.12733 [pdf, ps, other]
Title: Stein's method of exchangeable pairs in multivariate functional approximations
Journal-ref: Electron. J. Probab. 26, 1 - 50, 2021
Subjects: Probability (math.PR)
[179] arXiv:2005.12738 [pdf, ps, other]
Title: Quasi-ergodic limits for finite absorbing Markov chains
Comments: 30 pages
Journal-ref: Linear Algebra and its Applications 609 (2021), pp. 253-288
Subjects: Probability (math.PR)
[180] arXiv:2005.12824 [pdf, ps, other]
Title: The CS decomposition and conditional negative correlation inequalities for determinantal processes
Authors: André Goldman
Subjects: Probability (math.PR)
[181] arXiv:2005.12845 [pdf, ps, other]
Title: Higher order terms of the spectral heat content for killed subordinate and subordinate killed Brownian motions related to symmetric α-stable processes in R
Authors: Hyunchul Park
Subjects: Probability (math.PR)
[182] arXiv:2005.12970 [pdf, ps, other]
Title: The continuous-time frog model can spread arbitrarily fast
Comments: 8 pages
Journal-ref: Statistics & Probability Letters, Volume 172, May 2021
Subjects: Probability (math.PR)
[183] arXiv:2005.13036 [src]
Title: Coupled McKean-Vlasov stochastic differential equations with jumps
Authors: Huijie Qiao
Comments: Some results are similar to that in [21]
Subjects: Probability (math.PR)
[184] arXiv:2005.13051 [pdf, ps, other]
Title: Ergodicity and steady state analysis for Interference Queueing Networks
Comments: 12 pages
Subjects: Probability (math.PR)
[185] arXiv:2005.13141 [pdf, ps, other]
Title: Probability of consensus in the multivariate Deffuant model on finite connected graphs
Comments: 12 pages, 1 figure
Subjects: Probability (math.PR)
[186] arXiv:2005.13327 [pdf, ps, other]
Title: A note on the Fredrickson-Andersen one spin facilitated model in stationarity
Authors: Assaf Shapira
Subjects: Probability (math.PR)
[187] arXiv:2005.13376 [pdf, other]
Title: Random discrete concave functions on an equilateral lattice with periodic Hessians
Comments: 56 pages. arXiv admin note: substantial text overlap with arXiv:1909.08586
Subjects: Probability (math.PR)
[188] arXiv:2005.13383 [pdf, other]
Title: Choquet random sup-measures with aggregations
Authors: Yizao Wang
Comments: major revision with a new title; 27 pages
Subjects: Probability (math.PR)
[189] arXiv:2005.13437 [pdf, ps, other]
Title: Limit Profiles for Reversible Markov Chains
Comments: v2. Minor typos corrected. Title updated. Second author's name updated from "Thomas" to "Olesker-Taylor". To appear in PTRF
Journal-ref: Probab. Theory Relat. Fields 182, 157-188 (2022)
Subjects: Probability (math.PR); Combinatorics (math.CO); Group Theory (math.GR); Representation Theory (math.RT)
[190] arXiv:2005.13491 [pdf, other]
Title: Success probability for selectively neutral invading species in the line model with a random fitness landscape
Comments: 25 pages, 4 figures
Journal-ref: Stud. Appl. Math. 2021; 1- 27
[191] arXiv:2005.13505 [pdf, ps, other]
Title: On the Gromov-Prohorov distance
Authors: Svante Janson
Comments: 10 pages. References added
Subjects: Probability (math.PR)
[192] arXiv:2005.13533 [pdf, ps, other]
Title: Inhomogeneous Circular Law for Correlated Matrices
Comments: 51 pages. In the this version, we relaxed the regularity assumption on the test function f in the local law, Theorem 2.7. Moreover, we expanded the union bound argument in the proof of Corollary 2.8 and corrected some typos and formulations throughout the paper
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Functional Analysis (math.FA); Operator Algebras (math.OA)
[193] arXiv:2005.13570 [pdf, other]
Title: The distance exponent for Liouville first passage percolation is positive
Comments: 15 pages, 1 figure; to appear in PTRF
[194] arXiv:2005.13576 [pdf, other]
Title: Tightness of supercritical Liouville first passage percolation
Comments: 72 pages, 9 figures; to appear in JEMS
[195] arXiv:2005.13587 [pdf, ps, other]
Title: Central limit theorems for stochastic wave equations in dimensions one and two
Comments: Ver1: 19 pages
Journal-ref: Stochastics and Partial Differential Equations: Analysis and Computations 2021
Subjects: Probability (math.PR)
[196] arXiv:2005.13676 [pdf, ps, other]
Title: Feynman-Kac formula for iterated derivatives of the parabolic Anderson model
Subjects: Probability (math.PR)
[197] arXiv:2005.13758 [pdf, ps, other]
Title: $L^p$-Kato class measures and their relations with Sobolev embedding theorems for Dirichlet spaces
Authors: Takahiro Mori
Comments: 22 pages; title of paper changed, to appear in Journal of Functional Analysis
Subjects: Probability (math.PR)
[198] arXiv:2005.13824 [pdf, ps, other]
Title: Poisson limit theorems for the Robinson-Schensted correspondence and for the multi-line Hammersley process
Comments: version 2: 41 pages, the proofs are now more detailed
Subjects: Probability (math.PR); Combinatorics (math.CO)
[199] arXiv:2005.13832 [pdf, ps, other]
Title: Tree limits and limits of random trees
Authors: Svante Janson
Comments: 50 pages
Subjects: Probability (math.PR)
[200] arXiv:2005.13834 [pdf, ps, other]
Title: On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices
Authors: Félix Parraud
Comments: arXiv admin note: text overlap with arXiv:1912.04588
Subjects: Probability (math.PR); Operator Algebras (math.OA)
[201] arXiv:2005.13875 [pdf, other]
Title: The $β$-Delaunay tessellation I: Description of the model and geometry of typical cells
Subjects: Probability (math.PR)
[202] arXiv:2005.13878 [pdf, ps, other]
Title: On hyperbolic characteristic functions from an analytic and a free-probability point of view
Subjects: Probability (math.PR)
[203] arXiv:2005.13942 [pdf, ps, other]
Title: Analysis of an infinite-buffer batch-size-dependent service queue with discrete-time Markovian arrival process: D-$MAP/G_n^{(a,b)}/1$
Comments: 26 pages, 1 figure
Subjects: Probability (math.PR)
[204] arXiv:2005.13961 [pdf, ps, other]
Title: On the joint moments of the characteristic polynomials of random unitary matrices
Comments: Minor revision according to referee reports. To appear IMRN
[205] arXiv:2005.14011 [pdf, other]
Title: Homological Percolation: The Formation of Giant k-Cycles
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Algebraic Topology (math.AT)
[206] arXiv:2005.14033 [pdf, other]
Title: Backwards semi-martingales into Burgers' turtulence
Comments: 21 pages, 5 figures
Subjects: Probability (math.PR)
[207] arXiv:2005.14056 [pdf, ps, other]
Title: On $r$-to-$p$ norms of random matrices with nonnegative entries: Asymptotic normality and $\ell_\infty$-bounds for the maximizer
Comments: 51 pages
Subjects: Probability (math.PR); Combinatorics (math.CO)
[208] arXiv:2005.14102 [pdf, ps, other]
[209] arXiv:2005.14177 [pdf, ps, other]
Title: Trajectorial dissipation and gradient flow for the relative entropy in Markov chains
Comments: 34 pages. References added
Subjects: Probability (math.PR)
[210] arXiv:2005.14179 [pdf, ps, other]
Title: Variance Reduction in Simulation of Multiclass Processing Networks
Comments: 25 pages
Subjects: Probability (math.PR)
[211] arXiv:2005.14180 [pdf, other]
Title: Delocalization transition for critical Erdős-Rényi graphs
[212] arXiv:2005.14393 [pdf, ps, other]
Title: Moderate Deviations for the SSEP with a Slow Bond
Comments: 24 pages
Subjects: Probability (math.PR)
[213] arXiv:2005.14460 [pdf, ps, other]
Title: Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
Comments: 38 pages
Subjects: Probability (math.PR)
[214] arXiv:2005.14526 [pdf, ps, other]
Title: Large deviation principle for the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity
Subjects: Probability (math.PR)
[215] arXiv:2005.14566 [pdf, other]
Title: Heavy-Traffic Universality of Redundancy Systems with Assignment Constraints
Comments: 53 pages, 4 figures
Subjects: Probability (math.PR)
[216] arXiv:2005.14591 [pdf, other]
Title: Gaussian fluctuations from random Schrödinger equation
Comments: v2, 26 pages, minor revision
[217] arXiv:2005.14610 [pdf, ps, other]
Title: Critical Brownian multiplicative chaos
Authors: Antoine Jego
Comments: Published version. 49 pages
Journal-ref: Probability Theory and Related Fields (2021)
Subjects: Probability (math.PR)
[218] arXiv:2005.14701 [pdf, ps, other]
Title: Pinning for the critical and supercritical membrane model
Comments: Corrected some typos
Journal-ref: Prob. Math. Phys. 2 (2021) 745-820
Subjects: Probability (math.PR)
[219] arXiv:2005.00291 (cross-list from math.AP) [pdf, ps, other]
Title: Dissipative martingale solutions of the stochastically forced Navier--Stokes--Poisson system on domains without boundary
Comments: 66 pages
[220] arXiv:2005.00511 (cross-list from math.ST) [pdf, other]
Title: How to reduce dimension with PCA and random projections?
Comments: 56 pages, 12 figures
Subjects: Statistics Theory (math.ST); Probability (math.PR)
[221] arXiv:2005.00530 (cross-list from math.CO) [pdf, ps, other]
Title: A random analogue of Gilbreath's conjecture
Authors: Zachary Chase
Comments: 14 pages, 1 figure
[222] arXiv:2005.00755 (cross-list from math-ph) [pdf, ps, other]
Title: Energy Growth of Infinite Harmonic Chain under Microscopic Random Influence
Authors: A. Lykov
Journal-ref: Markov Processes And Related Fields, v. 26, 2, 287-304, 2020
[223] arXiv:2005.01032 (cross-list from math-ph) [pdf, ps, other]
Title: Uniformly Bounded Initial Chaos in Large System Often Intensifies Infinitely
Authors: A. Lykov, V. Malyshev
Journal-ref: Markov Processes And Related Fields, v. 26, 2, 213-231, 2020
[224] arXiv:2005.01114 (cross-list from math.DS) [pdf, ps, other]
Title: Almost Sure Invariance Principle for Random Distance Expanding Maps with a Nonuniform Decay of Correlations
Subjects: Dynamical Systems (math.DS); Probability (math.PR)
[225] arXiv:2005.01165 (cross-list from math.NA) [pdf, ps, other]
Title: Milstein schemes and antithetic multi-level Monte-Carlo sampling for delay McKean-Vlasov equations and interacting particle systems
Comments: 35 pages, 4 figures
Subjects: Numerical Analysis (math.NA); Probability (math.PR)
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e726382a00ab214e | Determinism vs Intrinsic randomness which one is preferred by Buddhist philosophy?
Ok good point. I mean falsification of the claim that rebirth exist. If we flip it to falsification of the notion of there’s nothing after death, just one rebirth evidence case does that already.
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Go to the blog for earlier parts.
Now is the time to recap on what concepts are at stake in various quantum interpretations, and the comments involving Buddhism, whether Buddhism prefers an interpretation to have this or that quality. You’ll have familiarity with most of them by now after reviewing so many experiments.
I will mainly discuss the list on the table of comparisons taken from wikipedia. Table at the interlude: A quantum game.
Meaning: results are not probabilistic in principle. In practice, quantum does look probabilistic (refer to Stern-Gerlach experiment), but with a certain interpretation, it can be transformed back into deterministic nature of things. This determinism is a bit softer than super-determinism, it just means we can in principle rule out intrinsic randomness. The choice is between determinism and intrinsic randomness.
Classical preference: deterministic. Many of the difficulties some classical thinking people have with quantum is the probabilistic results that we get from quantum. In classical theories, probability means we do not know the full picture, if we know everything that there is to know to determine the results of a roll of a dice, including wind speed, minor variation in gravity, the exact position and velocity of the dice, the exact rotational motion of the dice, the friction, heat loss etc, we can in principle calculate the result of a dice roll before it stops. The fault of probability in classical world is ignorance. In quantum, if we believe that the wavefunction is complete (Copenhagen like interpretations), then randomness is intrinsic, there’s no underlying mechanism which will guarantee this or that result, it’s not ignorance that we do not know, it’s nature that doesn’t have such values in it.
A Buddhist’s comment (basically me lah): On the one hand, we do not admit the existence of fatalism or fate, on the other hand, we don’t believe things happen for no reason. There was a heretical teacher back in Buddha’s time called Makkhali Gosala. Makkhali teaches the doctrine of fatalism. Everything is fixed, predetermined, there’s no role for effort in morality.
From sutta DN2, we get a glimpse of his teachings, which seems to include both fatalism and no causes.
“Great king, there is no cause or condition for the corruption of sentient beings. Sentient beings are corrupted without cause or condition. There’s no cause or condition for the purification of sentient beings. Sentient beings are purified without cause or condition. One does not act of one’s own volition, one does not act of another’s volition, one does not act from a person’s volition. There is no power, no energy, no manly strength or vigor. All sentient beings, all living creatures, all beings, all souls lack control, power, and energy. Molded by destiny, circumstance, and nature, they experience pleasure and pain in the six classes of rebirth. There are 1.4 million main wombs, and 6,000, and 600. There are 500 deeds, and five, and three. There are deeds and half-deeds. There are 62 paths, 62 sub-eons, six classes of rebirth, and eight stages in a person’s life. There are 4,900 Ājīvaka ascetics, 4,900 wanderers, and 4,900 naked ascetics. There are 2,000 faculties, 3,000 hells, and 36 realms of dust. There are seven percipient embryos, seven non-percipient embryos, and seven embryos without attachments. There are seven gods, seven humans, and seven goblins. There are seven lakes, seven winds, 700 winds, seven cliffs, and 700 cliffs. There are seven dreams and 700 dreams. There are 8.4 million great eons through which the foolish and the astute transmigrate before making an end of suffering. And here there is no such thing as this: “By this precept or observance or mortification or spiritual life I shall force unripened deeds to bear their fruit, or eliminate old deeds by experiencing their results little by little,” for that cannot be. Pleasure and pain are allotted. Transmigration lasts only for a limited period, so there’s no increase or decrease, no getting better or worse. It’s like how, when you toss a ball of string, it rolls away unraveling. In the same way, after transmigrating the foolish and the astute will make an end of suffering.”
Here’s the Buddha’s critique on him, from the sutta AN1:319
“Mendicants, I do not see a single other person who acts for the hurt and unhappiness of the people, for the harm, hurt, and suffering of many people, of gods and humans like that silly man, Makkhali. Just as a trap set at the mouth of a river would bring harm, suffering, calamity, and disaster for many fish, so too that silly man, Makkhali, is a trap for humans, it seems to me. He has arisen in the world for the harm, suffering, calamity, and disaster of many beings.”
In practise terms, we should acknowledge that there are causes which we can built to attain to liberation from suffering, effort is important. Causes are important. So morality observance is not wasted, it is encouraged. The law of kamma does argue against suffering happening for no cause and against suffering is fated to happen. It gives hope in that in the present moment, we can plant good seeds to ripen to good results. The patterns from old kamma by itself doesn’t predetermine all future, the input from present moment is important too.
So in choosing between determinism and intrinsic randomness, it is a toss up. If we can be assured that this kind of determinism does not lead to superdeterminism (which is basically fatalism), it’s a better choice. If not, intrinsic randomness of quantum can be made to not contradict Buddhism. The results of individual experiments cannot be pointed to have one cause or another. To see this, refer to Stern-Gerlach experiment, same set up, that is same cause and conditions, different results of up and down for each identically prepared particles. Remember the exercise in ruling out hidden variables, there’s no underlying difference between one particle and the next already, if we believe that wavefunction is complete. So this seems to violate cause plus conditions equals results in kammic teaching. Yet, it allows for the future to have different paths even if the past is exactly the same.
Richard A. Muller in his book the Physics of Now, argues that physics cannot rule out free will based on quantum phenomenon. His work as an experimental physicists allowed him to analyse pions (one of the many subatomic particles in particle physics) in particle accelerators. Two pions he had observed interfere with each other, that shows that their wavefunctions are exactly the same. So same cause and condition. However, the pions disintegrated at different times, so different results. Thus it would seem that quantum rules out fatalism if we interpret wavefunction as the complete description of the quantum system. The price we pay is, we cannot point to a reason why this pion decay faster than that one. If we light up two dynamites, they explode at the same time, not so with two identically created pions which are born in the same instance.
Also, when quantum is decohered up to the Newtonian physics, this quantum randomness hardly shows up in the macroscopic realm, well except for the radioactive decay which is used in the popular example of Schrödinger’s cat. So we cannot claim that there’s no cause for things based on mere acceptance of intrinsic quantum randomness. The results of quantum experiments are also pretty well defined to be in a range, eg. The spin result in Stern Gerlach is only up or down in the measurement axis. There’s no unpredictable thing like the electrons suddenly group together to become a dragon for no reason. So the cause-condition-result relationship can be restored, if we expand the definition of result to be quantum probabilistic range of result, and the probability distribution function is well defined and deterministic based on the experimental set up. For example, pion is created as cause and condition, result is, pion will decay. The time for the decay of pion matters not much.
Thus, there might be a stronger push to reject determinism.
Ontic wavefunction
Meaning: taking the wavefunction as a real physical, existing thing as opposed to just representing our knowledge. This is how Jim Baggott split up the various interpretations in his book Quantum reality.
Realist Proposition #3: The base concepts appearing in scientific theories represent the real properties and behaviours of real physical things. In quantum mechanics, the ‘base concept’ is the wavefunction.
Classical preference: classically, if the theory works and it has the base concepts in it, we take the base concept of the theory seriously as real. For example, General relativity. Spacetime is taken as dynamic and real entities due to our confidence in seeing the various predictions of general relativity being realized. We even built very expensive gravitational wave detectors to detect ripples in spacetime (that’s what gravitational waves are), and observed many events of gravitational waves via LIGO (Laser Interferometer Gravitational-Wave Observatory) from 2016 onwards. We know that spacetime is still a concept as loop quantum gravity denies that spacetime is fundamental, but build up from loops of quantum excitations of the Faraday lines of force of the gravitational field. Given that quantum uses wavefunction so extensively, some people think it’s really real out there.
A Buddhist’s comment: Well, the Buddha never mentioned wavefunctions as far as I know, so doesn’t really matters either way. Repeating the response in the motivation, the base concepts in classical theories just live in the heads of the physicists. Nature works as it is, the understanding of nature is also dependently arising, empty of inherent nature. This is how we can let go of even physics theories. Those who are more keen on practise may lean more towards seeing wavefunction as mere reflection of our knowledge rather than a real thing. As anything we deem real, we tend to attach to as we have the mistaken notion that real means permanent, reliable. As attachment causes suffering, we can save ourselves the trouble of suffering by not taking the reality of wavefunctions too seriously.
Unique History
Meaning: The world has a definite history, not split into many worlds, for the future or past. I suspect this category is created just for those few interpretations which goes wild into splitting worlds.
Classical preference: Yes, classically, we prefer to refer to history as unique.
A Buddhist’s comment: The past and future strictly speaking exist only in our minds. We can only have access to the present moment, the here and now. We remember the past (and due to light speed delay, we essentially see the past light cones reaching our eyes, but in practise we call it present), we can project the future. So having split past or split futures doesn’t really matter. However, the Buddha when he relates to his past lives didn’t change his story everytime, and he didn’t acknowledge a fixed future as the discussion on fatalism above shows. Thus the Buddhist philosophy of time also fits in growing box theory of time with the past is fixed, present exist, but future is free.
So we are more comfortable with splitting the future rather than the past.
Hidden Variables
Meaning: The wavefunction is not a complete description of the quantum system, there are some other things (variables) which are hidden from us and experiments and might be still underlying the mechanism of quantum, but we do not know. Historically, the main motivation to posit hidden variables is to oppose intrinsic randomness and recover determinism. However, Stochastic interpretation is not deterministic yet have hidden variables, and many worlds and many mind interpretations are deterministic yet do not have hidden variables.
Classical preference: Yes for hidden variables, if only to avoid intrinsic randomness, and to be able to tell what happens under the hood, behind the quantum stage show.
A Buddhist’s comment: this seems like a good opportunity to insert the influence of mind on matter. We can even put kamma as a nice touch on how kammic actions, generated by the mind (intentions) can have physical effects in the world, not just mental results. However, there’s no reason to insist on it. The variables are hidden anyway, thus no way to test for it.
Collapsing wavefunction
Meaning: That the interpretation admits the process of measurement collapses the wavefunction. This collapse is frown upon by many because it seems to imply two separate processes for quantum evolution
1.The deterministic, unitary, continuous time evolution of an isolated system (wavefunction) that obeys the Schrödinger equation (or a relativistic equivalent, i.e. the Dirac equation).
2.The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement, the collapse of wavefunction, which is only there to link the quantum formalism to observation.
Further problem includes that there’s nothing in the maths to tell us when and where does the collapse happens, usually called the measurement problem. A further problem is the irreversibility of the collapse.
Classical preference: Well, classically, we don’t have two separate process of evolution in the maths, so there’s profound discomfort if we don’t address what exactly is the collapse or get rid of it altogether. No clear choice. Most classical equations, however, are in principle reversible, so collapse of wavefunction is one of the weird non classical parts of quantum.
A Buddhist’s comment: This doesn’t really concern Buddhism. Irreversible, reversible, all part of impermanence.
Observer’s role
Meaning: do observers like humans play a fundamental role in the quantum interpretation? If not, physicists can be comfortable with a notion of reality which is independent of humans. If yes, then might the moon not be there when we are not looking? What role do we play if any in quantum interpretations?
Classical preference: Observer has no role. Reality shouldn’t be influenced just by observation.
A Buddhist’s comment: Just by studying quantum physics, we are participating in it. We cannot verify if reality is independent of observer as an observer. As highlighted in the motivation, an universe without sentient beings in there is metaphysics to us, as we are limited to observation from the vantage point of a sentient being. However, this is more of a tautology we would say the same to classical physics. Does observer play a fundamental role in quantum? Maybe, maybe not. Good if there is, then there can be more serious consideration to observe what does the observer do. In meditation, we call this mindfulness of the mind. Too often we don’t observe the observer. That’s where a lot of trouble starts.
Local dynamics
Meaning: is quantum local or nonlocal? Local here means only depends on surrounding phenomenon, limited by speed of light influences. Nonlocal here implies faster than light effect, in essence, more towards the spooky action at a distance. This is more towards the internal story of the interpretations. In practice, instrumentally, we use the term quantum non-locality to refer to quantum entanglement and it’s a real effect, but it is not signalling. Any interpretations which are non-local may utilise that wavefunction can literally transmit influences faster than light, but overall still have to somehow hide it from the experimenter to make sure that it cannot be used to send signals faster than light.
Classical preference: Local. This is not so much motivated by history, as Newtonian gravity is non-local, it acts instantaneously, only when gravity is explained by general relativity does it becomes local, so only from 1915 onward did classical physics fully embrace locality. Gravitational effects and gravitational waves travel at the speed of light, the maximum speed limit for information, mass, and matter. Quantum field theories, produced by combining quantum physics with special relativity is strictly local and highly successful, thus it also provides a strong incentive to prefer local interpretations by classically thinking physicists.
A Buddhist’s comment: Local or non local doesn’t really matter to Buddhists. There are many instances in the suttas where the devas and brahmas disappear from their realm and appear on earth to meet the Buddha. Depending on the nature of these beings, we might have claims that Buddhism allows for faster than light or not. However, the strongest motivation to disallow faster than light is the time travel conundrum. Since speed of light limits are not important or even hinted at in the suttas, there seems to be no reason for Buddhists to insist on locality, other than to adopt the concern of physicists.
Counterfactually definite
Meaning: Reality is there. There are definite properties of things we did not measure. Example, the Heisenberg uncertainty principle says that nature does not have 100% exact values for both position and momentum of a particle at the same time. Measuring one very accurately would make the other have much larger uncertainty. The same is true of Stern Gerlach experiments on spin. An electron does not have simultaneously a definite value for spin for both x-axis and z-axis. These are the experimental results which seem to show that unmeasured properties do not exist, rejecting counterfactual definiteness. We had also seen how Leggett’s inequality and Bell’s inequality together hit a strong nail on reality existing. Yet, some quantum interpretations still managed to recover this reality as part of the story of how quantum really works.
Classical preference: of course we prefer reality is there. The moon is still there even if no one is looking at it.
A Buddhist’s comment: It’s not hard for Buddhists to reject counterfactual definiteness. After all, the measurement is not done, why should we expect the properties to be underlying there in waiting? This is one of the strongest thing people identify intuitively when they read about quantum physics and then Buddhism or the other way around. Similar comments from the observer role can apply here too. We cannot verify if reality is independent of observer as an observer. We cannot say reality is there without measuring it. This is also a strong push for investigation by the Buddha. He asked us to come and see, investigate his words. He even showed a method to attain to the 4th Jhana, then develop the powers to recollect past lives to verify rebirth, the powers of divine eye to see the life, action, death, results of various beings to verify kamma. Thus Buddhism is not interested in metaphysics. And insisting on properties of things to be there without measuring it seems to be metaphysics. Of course, still, we believe that kamma and rebirth still works as usual even if we don’t develop those powers to directly verify it. Thus Buddhists do believe in counterfactual definiteness for these properties as well. Perhaps we are being too hasty in abandoning this concept?
Extant universal wavefunction
Meaning: If we believe that quantum is complete, it is fundamental, it in principle describes the whole universe, then might not we combine quantum systems descriptions say one atom plus one atom becomes wavefunction describing two atoms, and combine all the way to compass the whole universe? Then we would have a wavefunction describing the whole universe, called universal wavefunction. If we believe in the axioms of quantum, then this wavefunction is complete, it contains all possible description of the universe. It follows the time-dependent Schrödinger equation, thus it is deterministic unless you’re into consciousness causes collapse or consistent histories. No collapse of wavefunction is possible because there’s nothing outside the universe to observe/ measure this wavefunction and collapse it, unless you’re into the consciousness causes collapse interpretation or Bohm’s pilot wave mechanics. It feels like every time I try to formulate a general statement some interpretations keeps getting in the way by being the exceptions.
Classical preference: Well, hard to say, there’s no wavefunction classically, but I am leaning more towards yes, if quantum is in principle fundamental and describing the small, then it should still be valid when combined to compass the whole universe.
A Buddhist’s comment: There are things outside of the universe. In sutta DN27:
There comes a time when, Vāseṭṭha, after a very long period has passed, this cosmos contracts. As the cosmos contracts, sentient beings are mostly headed for the realm of streaming radiance. There they are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.
There comes a time when, after a very long period has passed, this cosmos expands. As the cosmos expands, sentient beings mostly pass away from that host of radiant deities and come back to this realm. Here they are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.
In sutta DN 1, there’s more details on the first being to be reborn back into the universe.
There comes a time when, after a very long period has passed, this cosmos expands. As it expands an empty mansion of Brahmā appears. Then a certain sentient being—due to the running out of their life-span or merit—passes away from that host of radiant deities and is reborn in that empty mansion of Brahmā. There they are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.
But after staying there all alone for a long time, they become dissatisfied and anxious: ‘Oh, if only another being would come to this state of existence.’ Then other sentient beings—due to the running out of their life-span or merit—pass away from that host of radiant deities and are reborn in that empty mansion of Brahmā in company with that being. There they too are mind-made, feeding on rapture, self-luminous, moving through the sky, steadily glorious, and they remain like that for a very long time.
Now, the being who was reborn there first thinks: ‘I am Brahmā, the Great Brahmā, the Undefeated, the Champion, the Universal Seer, the Wielder of Power, the Lord God, the Maker, the Author, the Best, the Begetter, the Controller, the Father of those who have been born and those yet to be born. These beings were created by me! Why is that? Because first I thought:
“Oh, if only another being would come to this state of existence.” Such was my heart’s wish, and then these creatures came to this state of existence.’
And the beings who were reborn there later also think: ‘This must be Brahmā, the Great Brahmā, the Undefeated, the Champion, the Universal Seer, the Wielder of Power, the Lord God, the Maker, the Author, the Best, the Begetter, the Controller, the Father of those who have been born and those yet to be born. And we have been created by him. Why is that? Because we see that he was reborn here first, and we arrived later.’
And the being who was reborn first is more long-lived, beautiful, and illustrious than those who arrived later.
It’s possible that one of those beings passes away from that host and is reborn in this state of existence. Having done so, they go forth from the lay life to homelessness. By dint of keen, resolute, committed, and diligent effort, and right focus, they experience an immersion of the heart of such a kind that they recollect that past life, but no further.
They say: ‘He who is Brahmā—the Great Brahmā, the Undefeated, the Champion, the Universal Seer, the Wielder of Power, the Lord God, the Maker, the Author, the Best, the Begetter, the Controller, the Father of those who have been born and those yet to be born—is permanent, everlasting, eternal, imperishable, remaining the same for all eternity. We who were created by that Brahmā are impermanent, not lasting, short-lived, perishable, and have come to this state of existence. This is the first ground on which some ascetics and brahmins rely to assert that the self and the cosmos are partially eternal.
Thus, there’s no issue with universal wavefunction, the Brahms from the realm of streaming radiance (2nd Jhana Brahma realm) might act as the observer to collapse the wavefunction of the universe if need be. By the way, the above quote shows the Buddhist conception of how the ideal of a creator God comes to be.
Anyway this universal wavefunction along with the unique history are usually not a thorny issue that people argue about when they discuss preferences for interpretations, unless they have nothing much else to talk about.
Now that we have covered the relevant concepts, the classical preferences for them and a Buddhist’s comment about them, here’s some reflection. Buddhism is generally more open compared to classical thinking in accepting many strange features of various quantum interpretations. Buddhism is also less decisive in placing bets on what the “real” interpretation should look like or have certain properties, except for the clear rejection of superdeterminism.
Thus, from here we can dash out any hope of trying to use Buddhism as a guide to select interpretations. Still, certain interpretations will resonate with Buddhist concepts more strongly compared to others, but the preliminary analysis here seems to suggest that we do not place hope on advancing the physics interpretation cases via philosophical inputs from Buddhism. What about the payoff for Buddhists? We can still go through the interpretations and then Buddhists can realise that we cannot make simple statements like quantum supports Buddhist philosophical concepts. Many of the interpretations might not be relevant or resonate with Buddhist concepts and some might resonate strongly. It’s important to keep in mind that as interpretations, experiments had not yet been able to rule one or another out yet, and it’s a religion (personal preferences) for physicists to choose one over another based on which classical concepts they are more attached to.
I would agree that there is no such thing as causality in the EBTs. Dependent origination, for instance, is about dependency and conditionality, not causality. At the same time, I am not sure that there are only “patterns”. The EBTs do present certain kinds of conditional relationships as if they were invariable laws, not merely descriptions of nature.
Perhaps the most important example of this is the dependent nature of the factors in dependent origination. “Willed actions” (saṅkhārā) depend on ignorance (avijjā), and so it goes all the way to the end of DO. This is specifically stated to be an invariable law. There is no possibility of “willed actions” if ignorance comes to an end. This is not causality, yet it more than a “pattern”.
I am wondering whether there is a distinction to be made between the physical and the mental. The physical world as conceived in physics, although ultimately based on experience, is really just an abstraction of the experiential world we live in. It is hard to see how we could ever be directly aware of causal relationships in such a world. But in the mental world of direct experience, where the clarity is proportional to the stillness of the mind, it is not unreasonable to think that we should be able, based on deep meditation, to infer laws that are more than just correlation. It seems to me that this is why the Buddha can state that the dependency in DO is a fixed law of nature.
Physicist views it the other way around. Because of lack of training of stillness, their mind causal relationships is not clear to them.
But having models of physical world and experiments verify them to a very high degree of significant figures and able to predict new phenomena and discover it renders those models as super real to them and the story the model tells is to be taken seriously.
Hence another reason for the interest in the interpretations, the story behind the successful quantum theory. We know it works very well, we cannot agree upon what story it tells us about nature. That to me is the real quantum weirdness. I plan to end this part of quantum interpretation and buddhism comparison Physics and Buddhism
By noticing that there’s no quantumness of quantum, except maybe contextuality. It is empty of inherent nature. |
cf8393c7d06bfc99 | How to project qubits faster using quantum feedback
Kurt Jacobs 65 Abraham heights, Nelson 7001, Auckland, NZ
When one performs a continuous measurement, whether on a classical or quantum system, the measurement provides a certain average rate at which one becomes certain about the state of the system. For a quantum system this is an average rate at which the system is projected onto a pure state. We show that for a standard kind of continuous measurement, for a qubit this rate may be increased by applying unitary operations during the measurement (that is, by using Hamiltonian feedback), in contrast to the equivalent measurement on a classical bit, where reversible operations cannot be used to enhance the rate of entropy reduction. We determine the optimal feedback algorithm and discuss the Hamiltonian resources required.
03.67.-a, 03.65.Ta, 02.50.Tt, 02.50.Ey
It was discovered recently (DJJ ; FJ ) that the average amount by which the quantum state of a system is purified during a measurement depends, in general, on the basis in which the measurement is made. Since changing the basis of a measurement is equivalent to performing a unitary transformation on the system, we can re-state this property by saying that a unitary transformation may be applied to a system, so as to increase the average amount of information that the measurement provides about the final (post-measurement) state. This restatement makes it particularly clear what is being changed about the measurement in order to enhance the information: merely the addition of a separate Hamiltonian evolution. Taking this point of view, the ability to perform unitary transformations, or equivalently, Hamiltonian evolution, can be considered as a resource which can be used to enhance the properties of a fixed measurement process.
Consider now a continuous measurement of, for example, a qubit. This may be represented by a sequence of identical ‘finite strength’ measurements, each of which partially projects the qubit onto the basis DJJ (We will refer to this as the computational basis, by which is simply meant the basis in which information is encoded). As more measurements are made, the purity of the state of the qubit increases, until eventually the qubit ends up in one of the basis states. Examining this process, we find that the state that results from a measurement in the sequence is not ideal for the purposes of purification for the next measurement in the sequence. It is therefore possible to perform a unitary transformation at the end of each measurement (and depending on the measurement result) to increase the average rate at which the state is purified. In the continuum limit this becomes a continuous feedback process.
This fact, while interesting as it provides a technique for enhancing a continuous measurement, is, if anything, more interesting from a fundamental point of view, due to the fact that the same measurement process performed on a classical bit cannot be enhanced by a reversible transformation, since the necessary sequence of transformations requires that superposition states of the computational basis must be available. Thus, this constitutes an example of something which the quantum nature of an object makes possible.
Since this feedback algorithm is designed to enhance the properties of the measurement itself, the resulting measurement process is an example of an adaptive measurement, a term first introduced, we believe, by Wiseman Wadapt . Wiseman’s adaptive measurement scheme, realized recently by Armen et. al. Armen , was designed to make a canonical phase measurement optimally well, given that, in practice, one only has arbitrary quadrature measurements at ones disposal. The topic discussed here is therefore a different kind of application for adaptive measurements.
Before we begin the development of the feedback algorithm, let us say a few words about quantum and classical measurements, and the relationship between them. Classical measurements on classical systems are described by Bayesian inference, and may be written in the same form as quantum measurements. If one writes the classical probability distribution of the quantity one is measuring as a diagonal density matrix, then classical measurements are in fact a subset of quantum measurements: While quantum measurements are described by a set of operators , where the only restriction is that , classical measurements have the further restriction that all the commute with the density matrix describing the classical system (for a fuller discussion see KJ ).
The behavior of a quantum measurement with commuting operators therefore reduces to that of a classical measurement when either unitary transformations are not available to rotate the qubit out of the computational basis, or there is a rapid decoherence process which immediately decoheres the qubit in the computational basis. In that case the unitary transformation becomes merely a classical diffusion process acting on the bit.
Here we will be interested in a widely applicable model of continuous measurement, where the measurement record is a Wiener process. We will measure in the z-basis of a spin-half system (that is, measure the observable ), denoting the basis states as (thus the final result of the measurement will either be or ). (Any further reference to ‘the computational basis’ will refer to these states.) To describe the continuous measurement one applies a POVM in each small time interval , where the POVM is chosen to scale with time in such a way that a sensible continuum limit exists CM . The resulting continuous measurement is not merely a mathematical curiosity, as it corresponds to real measurements on physical systems PhysMeasGen ; PMQbit1 ; PMQbit2 ; PMQbit3 ; PMQbit4 (in particular, Korotkov (PMQbit1 ; PMQbit2 ; PMQbit3 ; PMQbit4 ) gives explicit examples for solid-state qubits). While the approach in CM uses a POVM in each time step which has an infinite number of outcomes, for measurements on a two-state system one can alternatively employ a POVM with two-outcomes. A two-outcome measurement which provides information about the computational basis is
We will refer to this measurement in what follows as . When provides no information about the quantum system, leaving the state of knowledge unchanged, and when or the operators are rank one projectors, so that provides the maximal amount of information about the final state; this case is an ‘infinite strength’ measurement by the terminology of DJJ ; FJ .
Setting , and taking the limit of repeated measurements as , the evolution of the density matrix describing the observers state of knowledge, , is given by the stochastic Schrödinger equation
where is the Gaussian stochastic Wiener increment satisfying .
We are interested here in how the observer’s uncertainty of the quantum state reduces over time. In order to keep the calculations tractable, we will use as our measure of uncertainty the so called “linear entropy”, . (That this is a useful measure of uncertainty is due to its concavity FJ .) For a completely mixed state, , and for a pure state, .
In general, the reduction in the linear Entropy will depend upon the outcome of the measurement. As such, a sensible measure of the purifying power of the measurement is therefore the average reduction in the linear entropy, over the two outcomes. From FJ we know that the amount by which, on average, the two-outcome measurement purifies the state (that is, reduces ) is
where we have written the initial density matrix as , with being the angle between and the -axis, and . The reduction in uncertainty is maximized when . That is, when the basis in which the density matrix is diagonal is maximally different from the -basis, being the basis in which the measurement is made, the measurement is most effective in purifying the state. This somewhat curious result means that, if the observer’s state-of knowledge is not maximally complementary to the measurement basis, and the observer’s objective is purification, then a unitary transformation should be applied to the state prior to the measurement. It is also worth noting that, when the average reduction in the entropy is greatest, this reduction is the same for both measurement outcomes. Thus, in this case, the entropy reduction is deterministic, not random.
Now let us consider the consequences of this for a continuous measurement, performed on an initially completely mixed state. Approximating the continuous measurement by a sequence of measurements , we see that when we make the first measurement, is maximal. However, the result of the first measurement does not produce a state diagonal in a basis complementary to the basis, and this is generally true of the state resulting from a measurement in the -basis. As a result, the average purification will not be optimal for at least the majority of the measurements , and thus for essentially all of the duration of the continuous measurement process.
This suggests, therefore, the following procedure: After each measurement we apply a unitary transformation to rotate the state appropriately, so as to achieve maximal for each measurement in the sequence (each measurement step). Taking the continuum limit, this results in a continuous feedback algorithm which increases the rate of projection during the continuous measurement. Calculating the behavior of the linear entropy, , as a function of time is straightforward, because, as mentioned above, when is maximal, the change in is the same for both outcomes of the measurement. Hence, for a finite sequence of steps we have, for the step,
where . In the continuum limit this becomes
Without the feedback process, the evolution of depends on the measurement outcomes, and as a result is stochastic. Thus, not only does the Hamiltonian feedback algorithm increase the rate at which the state is purified, but also changes the entropy from being a random function of time to a deterministic one. (In fact, it is clear that this is true for any entropy, linear, von Neumann, or otherwise.)
In the absence of feedback, due to its stochastic nature, the behavior of is much harder to obtain, even for the simple canonical continuous (classical!) measurement we have here. Using the technique in JK , one obtains, for an initially completely mixed state,
where we use the subscript to denote the fact that this is also the result for a classical continuous measurement on a classical bit. An analytic solution for the integral in Eq.(6) does not appear to exist, and we therefore evaluate it numerically. In figure 1 we plot the speed-up factor provided by the feedback algorithm in obtaining a given final level of purity, when the initial state is completely mixed. This factor is independent of , and increases with the final purity, tending to 2 as the final entropy tends to zero.
The speed-up factor provided by the optimal Hamiltonian feedback algorithm, as a function of final purity, for a continuous measurement in a given basis of a qubit. The case displayed is for an initially completely mixed state.
The Hamiltonian feedback algorithm above has been obtained simply by optimizing the increase in purity of the final state at each measurement step. We now show that this is, in fact, the optimal feedback algorithm for obtaining the maximum purity of the state at any particular future time. First, let us denote the map which takes us from an initial linear entropy , to the final linear entropy , for a single time step, when we use the optimal unitary transformation, as . Thus we have , so is linear in .
Now, let us consider a single measurement step , where the initial entropy is . If we use a sub-optimal procedure, then one of two states results, and we can label the linear entropy of these as and , respectively. On the other hand, the entropy that results from the optimal procedure is , and the three entropies satisfy
where and are the respective probabilities that the two sub-optimal entropies were obtained. If we apply the optimal procedure to both results (both sides) then since is linear, we have
Alternatively, if we were to apply a non-optimal procedure to both cases (both sides of Eq.(7)), then we would have, for each of the outcomes ,
Thus, by the end of the second step, two sequential non-optimal procedures would give
where the first line uses Eq.(9), and the second uses the linearity of . Thus, after two steps the result of non-optimal measurements in both steps gives an average entropy which is higher than the result of two optimal steps. Clearly this procedure can be repeated times to obtain the result for measurement steps. Thus, we can write the final result of non-optimal steps as
for some and . In order for the procedure which uses all optimal steps to render a greater entropy than a procedure that uses some non-optimal steps, then it would have to be possible to apply an optimal step to the left hand side of Eq.(11), such that . However, since is linear, this is impossible. We can therefore conclude that the Hamiltonian feedback algorithm presented above gives the maximum possible entropy reduction for any number of steps, or, in the continuum limit, the maximum possible entropy reduction for a measurement of any given duration.
So far we have discussed the effects of the feedback algorithm, but not given explicitly the algorithm itself. The unitary transformation required after each measurement step must be such so as to rotate the qubit so that the Bloch vector lies in the -plane. The first thing to note is that the minimum angle of rotation required to do this is achieved by rotating such that the and elements of the Bloch vector remain in the same proportions (i.e. by keeping the angle that the Bloch vector makes with the and axes the same). The second is that an application of the measurement also keeps these angles the same. Assuming that the initial state is either completely mixed, or has been rotated prior to the measurement so that the Bloch vector lies in the -plane (which is the optimal thing to do), then the initial , and the state after a measurement is
The unitary transformation required to rotate such a state so that once again , using the minimum angle of rotation, is , with
and where we have used the fact that , and defined by . Since remains the same throughout the sequence of measurements, remains unchanged throughout the feedback process, and it is merely that changes at each feedback step. After the measurement is given by Eq.(4), and hence
Note that, when the initial state is completely mixed, the rotation angle , after the first measurement is always , regardless of the strength of (i.e. regardless of the value of ).
In the continuum limit, the feedback angle becomes
Since the Hamiltonian required to rotate through is proportional to , the Hamiltonian required for the feedback is proportional to the measurement noise . Thus the feedback required to obtain an optimal projection rate is Wiseman-Milburn type Markovian feedback, with the addition of a time dependent factor. However, this kind of feedback is strictly speaking an idealization valid in the limit of a large Hamiltonian, since is infinite. In addition, the feedback Hamiltonian diverges for an initially completely mixed state, since . Thus, a real continuous feedback procedure will provide a lower rate of projection, depending on the available Hamiltonian resources. We note that the divergence of the feedback Hamiltonian for has an analogue in Wiseman’s adaptive phase measurement; in that scheme, for this state, it is the rate of change of the phase estimate which diverges. It is interesting that for both schemes this divergence is associated with the fact that the measurement must break the symmetry of the state.
While the feedback algorithm projects a qubit onto a final pure state with maximal speed, it is fairly clear from the construction of the process that the operations corresponding to the many possible final outcomes will be mutually non-orthogonal. As a result, this adaptive measurement will almost certainly not provide full information regarding the initial preparation of the qubit. This is why we refer to the algorithm as projecting, rather than measuring the qubit, for the sense of the term “measurement” contains a certain ambiguity: it could mean either obtaining information about the initial preparation, useful in classical and quantum communication, or the final state, useful in quantum feedback control and quantum state preparation. While in this case the quantum state is projected maximally fast, this is at the expense of losing information about the initial preparation. This raises the question of whether there is a trade-off between speed of projection, and loss of initial information, in the kind of measurements considered here. In addition, optimal algorithms for projecting higher dimensional systems, and optimal rates obtainable with fixed Hamiltonian resources, are also open questions for further work.
The author would like to thank Howard Wiseman for helpful comments on the manuscript.
For everything else, email us at [email protected]. |
0702d2dbf9bf181a | Recent zbMATH articles in MSC 76 2022-09-13T20:28:31.338867Z Werkzeug Approximation methods in science and engineering 2022-09-13T20:28:31.338867Z "Jazar, Reza N." Publisher's description: \textit{Approximation Methods in Engineering and Science} covers fundamental and advanced topics in three areas: Dimensional Analysis, Continued Fractions, and Stability Analysis of the Mathieu Differential Equation. Throughout the book, a strong emphasis is given to concepts and methods used in everyday calculations. Dimensional analysis is a crucial need for every engineer and scientist to be able to do experiments on scaled models and use the results in real world applications. Knowing that most nonlinear equations have no analytic solution, the power series solution is assumed to be the first approach to derive an approximate solution. However, this book will show the advantages of continued fractions and provides a systematic method to develop better approximate solutions in continued fractions. It also shows the importance of determining stability chart of the Mathieu equation and reviews and compares several approximate methods for that. The book provides the energy-rate method to study the stability of parametric differential equations that generates much better approximate solutions. \begin{itemize} \item Covers practical model-prototype analysis and nondimensionalization of differential equations; \item Coverage includes approximate methods of responses of nonlinear differential equations; \item Discusses how to apply approximation methods to analysis, design, optimization, and control problems; \item Discusses how to implement approximation methods to new aspects of engineering and physics including nonlinear vibration and vehicle dynamics. \end{itemize} Book review of: R. J. Hosking and R. L. Dewar, Fundamental fluid mechanics and magnetohydrodynamics 2022-09-13T20:28:31.338867Z "Roberts, A. J."|roberts.anthony-john Review of [Zbl 1332.76001]. Book review of: M. Asadzadeh, An introduction to the finite element method for differential equations 2022-09-13T20:28:31.338867Z "Sachs, Ekkehard" Review of [Zbl 1446.65001]. Partial differentials with applications to thermodynamics and compressible flow 2022-09-13T20:28:31.338867Z "Braga da Costa Campos, Luis Manuel" "Vilela, Luís António Raio" Publisher's description: This book is part of the series ``Mathematics and Physics Applied to Science and Technology.'' It combines rigorous mathematics with general physical principles to model practical engineering systems with a detailed derivation and interpretation of results. The book presents the mathematical theory of partial differential equations and methods of solution satisfying initial and boundary conditions. It includes applications to acoustic, elastic, water, electromagnetic and other waves, to the diffusion of heat, mass and electricity, and to their interactions. The author covers simultaneously rigorous mathematics, general physical principles and engineering applications with practical interest. The book provides interpretation of results with the help of illustrations throughout and discusses similar phenomena, such as the diffusion of heat, electricity and mass. The book is intended for graduate students and engineers working with mathematical models and can be applied to problems in mechanical, aerospace, electrical and other branches of engineering. Homogenization of a coupled incompressible Stokes-Cahn-Hilliard system modeling binary fluid mixture in a porous medium 2022-09-13T20:28:31.338867Z "Lakhmara, Nitu" "Mahato, Hari Shankar" Summary: A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes-Cahn-Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter \(\varepsilon\) in the considered model. At first, the existence of solution of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach. Bifurcation structure and stability of steady gravity water waves with constant vorticity 2022-09-13T20:28:31.338867Z "Dai, Guowei" "Li, Fengquan" "Zhang, Yong"|zhang.yong.12|zhang.yong.13|zhang.yong.9|zhang.yong.15|zhang.yong.2|zhang.yong.1|zhang.yong.11|zhang.yong.8|zhang.yong|zhang.yong.14|zhang.yong.4|zhang.yong.5|zhang.yong.7 Summary: This paper studies the local bifurcation direction, stability properties and global structure for a nonlinear pseudodifferential equation, which describes the periodic travelling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation points. In particular, their signs can be strictly judged when constant vorticity vanishes. Furthermore, we present the stability analysis for the travelling water waves that have small vorticity and amplitude. We also show that the global bifurcation curves can't form a loop. Moreover, if the total head is bounded, the existence of waves of all amplitudes from zero up to that of Stokes' highest wave has been established. Asymptotic behavior of solution of Whitham-Broer-Kaup type equations with negative dispersion 2022-09-13T20:28:31.338867Z "Bedjaoui, Nabil" "Kumar, Rajesh" "Mammeri, Youcef" Summary: In this work, we discuss the long time behavior of solutions of the Whitham-Broer-Kaup system with Lipschitz nonlinearity and negative dispersion term. We prove the global well-posedness when \(\alpha+\beta^2<0\) as well as the convergence to 0 of small solutions at rate \(\mathcal{O}(t^{-1/2})\). Improved regularity criteria for the MHD equations in terms of pressure using an Orlicz norm 2022-09-13T20:28:31.338867Z "Choe, Hi Jun" "Neustupa, Jiří" "Yang, Minsuk" Summary: We present new regularity criteria in terms of the negative part of the pressure \(p\) or the positive part of the extended Bernoulli pressure \(\mathcal{B} := p + \frac{ 1}{ 2} | \mathbf{u} |^2 + \frac{ 1}{ 2} | \mathbf{b} |^2\), where \(\mathbf{u}\) is the velocity, and \(\mathbf{b}\) is the magnetic field. The criteria extend the previously known results, and the extension is enabled by the use of an appropriate Orlicz norm. On the wave interactions for the drift-flux equations with the Chaplygin gas 2022-09-13T20:28:31.338867Z "Li, Shuangrong" "Shen, Chun" The authors study solutions to the system (modeling two-phase flows) \[ \left\{ \begin{array}{l} \partial_t\rho_1 +\partial_x(\rho_1 u)=0,\\ \partial_t\rho_2 +\partial_x(\rho_2 u)=0,\\ \partial_t\big((\rho_1+\rho_2) u\big) +\partial_x\Big((\rho_1+\rho_2)u^2- (\frac{1}{\rho_1}+\frac{1}{\rho_2})\Big)=0, \end{array}\right. \] emerging from piecewise-constant inital data. The system is strictly hyperbolic with all three characteristic fields being linearly degenerate. The first part of the work deals with the Riemann problem. The authors exhibit two distinct situations: in one case the solution is described through three contact discontinuities, while the other case produces a less standard ``delta shock wave'' solution. In the second part, the authors investigate the interactions between contact discontinuities and delta shock waves, contact discontinuities and contact discontinuities, delta shock waves and delta shock waves, emerging from piecewise constant initial data with three states. Reviewer: Vincent Duchêne (Rennes) The Riemann problem for the nonisentropic Baer-Nunziato model of two-phase flows 2022-09-13T20:28:31.338867Z "Thanh, Mai Duc" "Vinh, Duong Xuan" Summary: The Riemann problem for the well-known Baer-Nunziato model of two-phase flows is solved. The system consists of seven partial differential equations with nonconservative terms. The most challenging problem is that this model possesses a double eigenvalue. Although characteristic speeds coincide, the curves of composite waves associated with different characteristic fields can be still constructed. They will also be incorporated into composite wave curves to form solutions of the Riemann problem. Solutions of the Riemann problem will be constructed when initial data are in supersonic regions, subsonic regions, or in both kinds of regions. A unique solution and solutions with resonance are also obtained. Regularization estimates and hydrodynamical limit for the Landau equation 2022-09-13T20:28:31.338867Z "Carrapatoso, Kleber" "Rachid, Mohamad" "Tristani, Isabelle" Summary: In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and we obtain the instantaneous expected anisotropic gain of regularity (see [\textit{M. Rachid}, ``Hypoelliptic and spectral estimates for the linearized Landau operator'', Preprint, \url{arXiv:2004.09300}] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system. A stability result for the identification of a permeability parameter on Navier-Stokes equations 2022-09-13T20:28:31.338867Z "Aguayo, Jorge" "Osses, Axel" Secondary flows from a linear array of vortices perturbed principally by a Fourier mode 2022-09-13T20:28:31.338867Z "Chen, Zhi-Min" Summary: In the understanding of primary bifurcating flows of a linear array of electromagnetically forced vortices in an experimental fluid motion, a theoretical study on the nonlinear instability is presented. The existence of the bifurcating flows is obtained from a Fourier mode perturbation. This large-scale perturbation, leading to the primary bifurcation observed in a laboratory experiment, was found to be generated principally from a single vortex mode. An exact solution for the semi-stationary compressible Stokes problem 2022-09-13T20:28:31.338867Z "Dong, Jianwei" Summary: In this note, we present an exact solution for the semi-stationary compressible Stokes problem in \(\mathbb{R}^N\). In the case of radial symmetry, an exact solution with velocity of the form \(c(t)r^s\) is obtained for \(s=\frac{1-N\gamma +\gamma}{\gamma +1}\), where \(\gamma >1\) is the adiabatic index and \(r=|x|\). Some interesting properties of the exact solution are analyzed. Low Mach number limit for the full compressible magnetohydrodynamic equations without thermal conductivity 2022-09-13T20:28:31.338867Z "Guo, Liang" "Li, Fucai" Summary: In this paper we consider the low Mach number limit of the full compressible magnetohydrodynamic equations for the polytropic ideal gas with zero thermal conductivity coefficient in the whole space \(\mathbb{R}^n\) (\(n=2, 3\)). We focus on the case that the pressure varies near its equilibrium state. It means that the density and the temperature may change around their limit functions, and hence generalize the case on the perturbation of the constant states for the density and the temperature. We establish this limit process rigorously when the initial data is well-prepared. Moreover, we also obtain the convergence rates. Stability and periodicity of solutions to Navier-Stokes equations on non-compact Riemannian manifolds with negative curvature 2022-09-13T20:28:31.338867Z "Nguyen, Thieu Huy" "Vu, Thi Ngoc Ha" "Nguyen, Thi Van" Summary: Let \((M, g)\) be a non-compact Riemannian manifold having negative Ricci curvature tensor. Then, we consider the Navier-Stokes Equations (NSE) for vector fields on \((M, g)\) and prove the existence of a bounded solution to NSE on \((M, g)\). Moreover we show the stability on a small neighborhood for such a solution. Then, using such a local stability we show the existence of a time-periodic solution to NSE under the action of a time-periodic external force. Our result can be considered as a Serrin-type theorem for the case of non-compact Riemannian manifolds with negative curvature tensors. Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data 2022-09-13T20:28:31.338867Z "Paicu, Marius" "Zhang, Ping" Summary: We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using the Cattaneo type law instead of the Fourier law, evolving in a thin strip \(\mathbb{R} \times (0, \epsilon)\). The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data. Compared with [\textit{M. Paicu} et al., Adv. Math. 372, Article ID 107293, 41 p. (2020; Zbl 1446.35105)] for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data, here the initial data belongs to the Gevrey class 2, which is very sophisticated even for the well-posedness of the classical Prandtl system (see [\textit{H. Dietert} and \textit{D. Gérard-Varet}, Ann. PDE 5, No. 1, Paper No. 8, 51 p. (2019; Zbl 1428.35355)] and [\textit{C. Wang}, \textit{Y. Wang,} and \textit{P. Zhang}, ``On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class'', Preprint, \url{arXiv:2103.00681}]); furthermore, the estimate of the pressure term in the hyperbolic Prandtl system give rise to additional difficulties. Existence and uniqueness result for a fluid-structure-interaction evolution problem in an unbounded 2D channel 2022-09-13T20:28:31.338867Z "Patriarca, Clara" Summary: In an unbounded 2D channel, we consider the vertical displacement of a rectangular obstacle in a regime of small flux for the incoming flow field, modelling the interaction between the cross-section of the deck of a suspension bridge and the wind. We prove an existence and uniqueness result for a fluid-structure-interaction evolution problem set in this channel, where at infinity the velocity field of the fluid has a \textit{Poiseuille flow} profile. We introduce a suitable definition of weak solutions and we make use of a penalty method. In order to prevent the obstacle from going excessively far from the equilibrium position and colliding with the boundary of the channel, we introduce a \textit{strong force} in the differential equation governing the motion of the rigid body and we find a unique global-in-time solution. On numerical approximations to fluid-structure interactions involving compressible fluids 2022-09-13T20:28:31.338867Z "Schwarzacher, Sebastian" "She, Bangwei" Summary: In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions. New thought on Matsumura-Nishida theory in the \(L_p\)-\(L_q\) Maximal regularity framework 2022-09-13T20:28:31.338867Z "Shibata, Yoshihiro" Summary: This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier-Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to \textit{A. Matsumura} and \textit{T. Nishida} [Commun. Math. Phys. 89, 445--464 (1983; Zbl 0543.76099)]. In [loc. cit.], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to \(L_2\) in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the \(L_\infty\) norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [loc. cit.] in \(L_p\) (\(1 <p \le 2\)) in time and \(L_2\cap L_6\) in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [loc. cit.] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal \(L_p\) (\(1 <p \le 2\)) in time and \(L_2\cap L_6\) in space regularity and \(L_p\)-\(L_q\) decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach's fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the \(L_p\)-\(L_q\) maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal \(L_p\)-\(L_q\) regularity and \(L_p\)-\(L_q\) decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics. Inviscid limit of the inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\) 2022-09-13T20:28:31.338867Z "Wang, Dixi" "Yu, Cheng" "Zhao, Xinhua" Summary: In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in \(\mathbb{R}^3\). In particular, this limit is a weak solution of the corresponding Euler equations. We first deduce the Kolmogorov-type hypothesis in \(\mathbb{R}^3\), which yields the uniform bounds of \(\alpha^{th}\)-order fractional derivatives of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2_x\) for some \(\alpha > 0\), independent of the viscosity. The uniform bounds can provide strong convergence of \(\sqrt{\rho^\mu} \mathbf{u}^\mu\) in \(L^2\) space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations. Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion 2022-09-13T20:28:31.338867Z "Wang, Wenjun" "Wen, Huanyao" Summary: We consider the full compressible Navier-Stokes equations with reaction diffusion. A global existence and uniqueness result of the strong solution is established for the Cauchy problem when the initial data is in a neighborhood of a trivially stationary solution. The appearance of the difference between energy gained and energy lost due to the reaction is a new feature for the flow and brings new difficulties. To handle these, we construct a new linearized system in terms of a combination of the solutions. Moreover, some optimal time-decay estimates of the solutions are derived when the initial perturbation is additionally bounded in \(L^1\). It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solution so that the long time behavior for the hyperbolic-parabolic system is exactly the same as that for the heat equation. As a byproduct, the above time-decay estimate at the highest order is also valid for compressible Navier-Stokes equations. The proof is accomplished by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity. Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6 2022-09-13T20:28:31.338867Z "Wu, Bian" Summary: By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force \(f \in L_{\mathrm{loc}}^q \cap L^{3/2}\), \(q>3\) in general open subdomains of \(\mathbb{R}^6\). These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in \(\mathbb{R}^4 \times [0, \infty[\). Global solutions to 3D incompressible Navier-Stokes equations with some large initial data 2022-09-13T20:28:31.338867Z "Yu, Yanghai" "Li, Jinlu" "Yin, Zhaoyang" Summary: In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. More precisely, we prove that if \[ \begin{aligned} \Bigg(&\| u_0^1 + u_0^2 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}} + \| u_0^3 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}}\Bigg) \Bigg(\| u_0^1 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}} + \| u_0^2 \|_{\dot{B}_{p, 1}^{\frac{ 3}{ p} - 1}}\Bigg)\\ &\times \exp \Bigg(C \Big(\| u_0 \|_{\dot{B}_{\infty, 2}^{- 1}}^2 + \| u_0 \|_{\dot{B}_{\infty, 1}^{- 1}} \Big)\Bigg) \end{aligned} \] is small enough, the Navier-Stokes equations have a unique global solution. As an application, we construct two examples of initial data satisfying the smallness condition, but whose \(\dot{B}_{\infty, \infty}^{- 1} (\mathbb{R}^3)\) norm can be arbitrarily large. Rayleigh-Taylor instability for viscous incompressible capillary fluids 2022-09-13T20:28:31.338867Z "Zhang, Zhipeng" Summary: We investigate the linear and nonlinear instability of a smooth Rayleigh-Taylor steady state solution to the three-dimensional incompressible Navier-Stokes-Korteweg equations in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution and find that for any capillary coefficient \(\kappa >0\), we can construct the solutions of the linearized problem that grow in time in Sobolev space \(H^m\), thus leading to the linear instability. However, with the help of the constructed unstable solutions of the linearized problem, we just establish the nonlinear instability for small enough capillary coefficient \(\kappa >0\). Conjugate points in \(\mathcal{D}_\mu^s(S^2)\) 2022-09-13T20:28:31.338867Z "Benn, J." Summary: Rossby-Haurwitz waves on the sphere \(S^2\) form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the \(L^2\) metric in the volume preserving diffeomorphism group of \(S^2\). Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves. Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems 2022-09-13T20:28:31.338867Z "Wang, Guodong" Summary: This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in \(L^p\) norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem. On a higher integral invariant for closed magnetic lines, revisited 2022-09-13T20:28:31.338867Z "Akhmet'ev, Peter M." Summary: We recall a definition of an asymptotic invariant of classical link, which is called \(M\)-invariant. \(M\)-invariant is a special Massey integral, this integral has an ergodic form and is generalized for magnetic fields with open magnetic lines in a bounded \(3D\)-domain. We present a proof that this integral is well defined. A combinatorial formula for \(M\)-invariant using the Conway polynomial is presented. The \(M\)-invariant is a higher invariant, it is not a function of pairwise linking numbers of closed magnetic lines. We discuss applications of \(M\)-invariant for MHD. Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces 2022-09-13T20:28:31.338867Z "Azanzal, Achraf" "Allalou, Chakir" "Melliani, Said" Summary: This paper establishes the existence and uniqueness, and also presents a blow-up criterion, for solutions of the quasi-geostrophic (QG) equation in a framework of Fourier type, specifically Fourier-Besov-Morey spaces. If it is assumed that the initial data \(\theta_0\) is small and belonging to the critical Fourier-Besov-Morrey spaces \(\mathscr{F} {\mathscr{N}}_{p, \lambda, q}^{3-2 \alpha +\frac{\lambda -2}{p}} \), we get the global well-posedness results of the QG equation (1). Moreover, we prove that there exists a time \(T > 0\) such that the QG equation (1) admits a unique local solution for large initial data. Mixing solutions for the Muskat problem 2022-09-13T20:28:31.338867Z "Castro, A." "Córdoba, D." "Faraco, D." Summary: We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type \(H^5\) initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed. Hamiltonian description of internal ocean waves with Coriolis force 2022-09-13T20:28:31.338867Z "Cullen, Joseph D." "Ivanov, Rossen I." Summary: The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a nearly Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface. On uniqueness and helicity conservation of weak solutions to the electron-MHD system 2022-09-13T20:28:31.338867Z "Dai, Mimi" "Krol, Jacob" "Liu, Han" Summary: We study weak solutions to the electron-MHD system and obtain a conditional uniqueness result. In addition, we prove conservation of helicity for weak solutions to the electron-MHD system under a geometric condition. Travelling waves in the Boussinesq type systems 2022-09-13T20:28:31.338867Z "Dinvay, Evgueni" Summary: Considered herein are a number of variants of the Boussinesq type systems modelling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified. Uniform regularity for a density-dependent incompressible Hall-MHD system 2022-09-13T20:28:31.338867Z "Fan, Jishan" "Zhou, Yong" Summary: This paper proves uniform regularity for a density-dependent incompressible Hall-MHD system with positive density. Energy considerations for nonlinear equatorial water waves 2022-09-13T20:28:31.338867Z "Henry, David"|henry.david.1 Summary: In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived. Continued gravitational collapse for gaseous star and pressureless Euler-Poisson system 2022-09-13T20:28:31.338867Z "Huang, Feimin" "Yao, Yue" Global well-posedness of classical solutions to the Cauchy problem of two-dimensional barotropic compressible Navier-Stokes system with vacuum and large initial data 2022-09-13T20:28:31.338867Z "Huang, Xiangdi" "Li, Jing" Optimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balance 2022-09-13T20:28:31.338867Z "Ji, Ruihong" "Yan, Li" "Wu, Jiahong" Summary: This paper focuses on the three-dimensional (3D) incompressible anisotropic Boussinesq system with horizontal dissipation. The goal here is to assess the stability property and pinpoint the precise large-time behavior of perturbations near the hydrostatic balance. Important tools such as Schonbek's Fourier splitting method have been developed to understand the large-time behavior of PDE systems with full dissipation, but these tools may not apply directly when the systems are only partially dissipated. This paper solves the stability problem and designs an effective approach to obtain the optimal decay rates for the anisotropic Boussinesq system concerned here. The tool developed in this paper may be useful for many other partially dissipated systems. Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem 2022-09-13T20:28:31.338867Z "Lepe, Felipe" "Rivera, Gonzalo" "Vellojin, Jesus" Orbital stability of the sum of smooth solitons in the Degasperis-Procesi equation 2022-09-13T20:28:31.338867Z "Li, Ji" "Liu, Yue" "Wu, Qiliang" Summary: The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the \(L^2 \cap L^\infty\) orbital stability of a wave train containing \(N\) smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the \(L^2\)-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce \textit{a priori} estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation. On the effect of fast rotation and vertical viscosity on the lifespan of the \(3D\) Primitive equations 2022-09-13T20:28:31.338867Z "Lin, Quyuan" "Liu, Xin"||||| "Titi, Edriss S." Summary: We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation \(|\Omega|\), we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only \(L^2\) in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large \(|\Omega|\), we show that the existence time of the solution can be prolonged, with ``well-prepared'' initial data. Finally, in the case of two spatial dimensions with \(\Omega =0\), we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables. Global well-posedness of 3d axisymmetric MHD-Boussinesq system with nonzero swirl 2022-09-13T20:28:31.338867Z "Liu, Qiao" "Yang, Yixin" Summary: In this paper, we consider the 3d axisymmetric MHD-Boussinesq system with nonzero swirl, and prove that the system, with initial data \((u_0, h_0, \rho_0) = (u^r_0 e_r + u^\theta_0 e_\theta + u^z_0 e_z, h^\theta_0 e_\theta, \rho_0)\) which satisfies some small nonlinear condition, admits a global unique solution \((u, h, \rho)\). Furthermore, some continuation criteria that imply regularity of axisymmetric solutions are also obtained. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation 2022-09-13T20:28:31.338867Z "Nazarov, Fedor" "Zumbrun, Kevin" Summary: We establish an instantaneous smoothing property for decaying solutions on the half-line \((0, +\infty)\) of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \(H^1\) stable manifolds of such equations, showing that \(L^2_{loc}\) solutions that remain sufficiently small in \(L^\infty\) (i) decay exponentially, and (ii) are \(C^\infty\) for \(t>0 \), hence lie eventually in the \(H^1\) stable manifold constructed by Pogan and Zumbrun. Global existence in critical spaces for non Newtonian compressible viscoelastic flows 2022-09-13T20:28:31.338867Z "Pan, Xinghong" "Xu, Jiang" "Zhu, Yi"|zhu.yi.1|zhu.yi.3|zhu.yi.2 Summary: We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the ``div-curl'' structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from \textit{combination} of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations. A ternary Cahn-Hilliard-Navier-Stokes model for two-phase flow with precipitation and dissolution 2022-09-13T20:28:31.338867Z "Rohde, Christian" "von Wolff, Lars" Sharp convergence rates for Darcy's law 2022-09-13T20:28:31.338867Z "Shen, Zhongwei"|shen.zhongwei Summary: This article is concerned with Darcy's law for an incompressible viscous fluid flowing in a porous medium. We establish the sharp \(O(\sqrt{\varepsilon})\) convergence rate in a periodically perforated and bounded domain in \(\mathbb{R}^d\) for \(d\geq 2\), where \(\varepsilon\) represents the size of solid obstacles. This is achieved by constructing two boundary layer correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values between the solution and the leading term in its asymptotic expansion. One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data. Compactness and large-scale regularity for Darcy's law 2022-09-13T20:28:31.338867Z "Shen, Zhongwei"|shen.zhongwei.1 Summary: This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior \(C^{1,\alpha}\) and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform \(W^{k,p}\) estimates for \(1<p<\infty\). The MHD equations in the Lorentz space with time dependent external forces 2022-09-13T20:28:31.338867Z "Tan, Zhong" "Zhou, Jianfeng" Summary: We are concerned with the well-posedness of the incompressible Magneto-hydrodynamical (MHD) equations in \(\mathbb{R}^n\) (\(n\ge 3\)). First, by assuming the smallness of the external force in Lorentz spaces, we prove the existence, uniqueness and the time regularity of periodic mild solution of an integral form of MHD Eqs. (1.1). Next, we prove the local existence and uniqueness of mild solution of the Cauchy problem of MHD Eqs. (1.2). Finally, appealing to the existence and uniqueness of the mild solution of (1.2), we show that the obtained solution \((u, b)\) of (1.1) becomes the time periodic strong solution, derived from the strong solvability of the inhomogeneous Stokes equation and heat equation by an additional assumption of the external force. Inexact GMRES iterations and relaxation strategies with fast-multipole boundary element method 2022-09-13T20:28:31.338867Z "Wang, Tingyu" "Layton, Simon K." "Barba, Lorena A." Summary: Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, \(p\). We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing \(p\). In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between \(1.5 \times\) and \(2.3 \times\) for Laplace problems and between \(2.7 \times\) and \(3.3 \times\) for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, \url{}, and we share reproducibility packages for all results in \url{}. A stochastic approach to enhanced diffusion 2022-09-13T20:28:31.338867Z "Zelati, Michele Coti" "Drivas, Theodore D." Summary: We provide examples of initial data which saturate the enhanced diffusion rates proved for general shear flows which are Hölder regular or Lipschitz continuous with critical points, and for regular circular flows, establishing the sharpness of those results. Our proof makes use of a probabilistic interpretation of the dissipation of solutions to the advection diffusion equation. Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model 2022-09-13T20:28:31.338867Z "Zhang, Qinglong" Summary: The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of \(\delta\)-waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a \(\delta\)-shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research. Stabilization and exponential decay for 2D Boussinesq equations with partial dissipation 2022-09-13T20:28:31.338867Z "Zhong, Yueyuan" Summary: This paper focuses on a special 2D Boussinesq equation with partial dissipation, for which the velocity equation involves no dissipation and there is only damping in the horizontal component equation. Without buoyancy force, the corresponding vorticity equation is a 2D Euler-like equation with an extra Calderon-Zygmund-type term. Its stability is an open problem. Our results reveal that the buoyancy force exactly stabilizes the fluids by the coupling and interaction between the velocity and temperature. In addition, we prove the solution decays exponentially to zero in Sobolev norm. Convergence toward the steady state of a collisionless gas with Cercignani-Lampis boundary condition 2022-09-13T20:28:31.338867Z "Bernou, Armand" Summary: We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence toward it, in the \(L^1\) norm. The strategy is an application of a deterministic version of Harris' subgeometric theorem, in the spirit of the recent results of Cañizo-Mischler and of the previous study of Bernou. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem. Transport equations with inflow boundary conditions 2022-09-13T20:28:31.338867Z "Scott, L. Ridgway" "Pollock, Sara" Summary: We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late \textit{C. J. Amick} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 473--513 (1977; Zbl 0367.76027)], we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions. Time-fractional Moore-Gibson-Thompson equations 2022-09-13T20:28:31.338867Z "Kaltenbacher, Barbara" "Nikolić, Vanja" The non-Lipschitz stochastic Cahn-Hilliard-Navier-Stokes equations in two space dimensions 2022-09-13T20:28:31.338867Z "Sun, Chengfeng" "Huang, Qianqian" "Liu, Hui"|liu.hui.2|liu.hui.1|liu.hui.4 Large deviation principles for a 2D stochastic Allen-Cahn-Navier-Stokes driven by jump noise 2022-09-13T20:28:31.338867Z "Tachim Medjo, Theodore" Ergodic theory for energetically open compressible fluid flows 2022-09-13T20:28:31.338867Z "Fanelli, Francesco" "Feireisl, Eduard" "Hofmanová, Martina" Summary: The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier-Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff-Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any \(t \in R\)). In particular, the ergodic averages converge for \textit{any} trajectory provided its \(\omega\)-limit set in the trajectory space supports a unique (in law) stationary solution. Emergent behaviors of relativistic flocks on Riemannian manifolds 2022-09-13T20:28:31.338867Z "Ahn, Hyunjin" "Ha, Seung-Yeal" "Kang, Myeongju" "Shim, Woojoo" Summary: We present a relativistic counterpart of the Cucker-Smale (CS) model on Riemannian manifolds (manifold RCS model in short) and study its collective behavior. For Euclidean space, the \textit{relativistic Cucker-Smale} (RCS) model was introduced in [\textit{S.-Y. Ha} et al., Arch. Ration. Mech. Anal. 235, No. 3, 1661--1706 (2020; Zbl 1439.35397)] via the method of a rational reduction from the relativistic gas mixture equations by assuming space-homogeneity, suitable ansatz for entropy and principle of subsystem. In this work, we extend the RCS model on Euclidean space to connected, complete and smooth Riemannian manifolds by replacing usual time derivative of velocity and relative velocity by suitable geometric quantities such as covariant derivative and parallel transport along length-minimizing geodesics. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using LaSalle's invariance principle. As concrete examples, we further analyze the RCS models on the unit sphere \(\mathbb{S}^d\) and the hyperbolic space \(\mathbb{H}^d\). More precisely, we show that the RCS model on \(\mathbb{S}^d\) exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in \(\mathbb{H}^d\). For the hyperbolic space \(\mathbb{H}^d\), we also rigorously justify smooth transition from the RCS model to the CS model in any finite time interval, as speed of light tends to infinity. Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness 2022-09-13T20:28:31.338867Z "Novack, Michael" "Yan, Xiaodong" Summary: We consider the 3D smectic energy \[ \mathcal{E}_\varepsilon(u) = \frac{1}{2}\int_\Omega \frac{1}{\varepsilon} \left( \partial_z u-\frac{(\partial_x u)^2+(\partial_y u)^2}{2}\right)^2 +\varepsilon \left(\partial_x^2u + \partial_y^2u\right)^2dx\,dy\,dz. \] The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on \(\mathcal{E}_\varepsilon\) as \(\varepsilon \rightarrow 0\) by introducing 3D analogues of the Jin-Kohn entropies [\textit{W. Jin} and \textit{R. V. Kohn}, J. Nonlinear Sci. 10, No. 3, 355--390 (2000; Zbl 0973.49009)]. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for \(\varepsilon_n\rightarrow 0\) and an energy-bounded sequence \(\{u_n\}\) with \(\Vert\nabla u_n\Vert_{L^p(\Omega)}\), \(\Vert \nabla u_n\Vert_{L^2(\partial\Omega)}\le C\) for some \(p>6\), we obtain compactness of \(\nabla u_n\) in \(L^2\) assuming that \(\Delta_{xy}u_n\) has constant sign for each \(n\). The measurement and analysis of shapes. An application of hydrodynamics and probability theory 2022-09-13T20:28:31.338867Z "Benn, James" "Marsland, Stephen" Summary: A de Rham \(p\)-current can be viewed as a map (the current map) between the set of embeddings of a closed \(p\)-dimensional manifold into an ambient \(n\)-manifold and the set of linear functionals on differential \(p\)-forms. We demonstrate that, for suitably chosen Sobolev topologies on both the space of embeddings and the space of \(p\)-forms, the current map is continuously differentiable, with an image that consists of bounded linear functionals on \(p\)-forms. Using the Riesz representation theorem, we prove that each \(p\)-current can be represented by a unique co-exact differential form that has a particular interpretation depending on \(p\). Embeddings of a manifold can be thought of as shapes with a prescribed topology. Our analysis of the current map provides us with representations of shapes that can be used for the measurement and statistical analysis of collections of shapes. We consider two special cases of our general analysis and prove that: (1) if \(p=n-1\) then closed, embedded, co-dimension one surfaces are naturally represented by probability distributions on the ambient manifold and (2) if \(p=1\) then closed, embedded, one-dimensional curves are naturally represented by fluid flows on the ambient manifold. In each case, we outline some statistical applications using an \({\dot{H}}^1\) and \(L^2\) metric, respectively. Kolmogorov's theory of turbulence and its rigorous 1d model 2022-09-13T20:28:31.338867Z "Kuksin, Sergei" The author summarizes the main results (with some sketched proofs) of the book [One-dimensional turbulence and the stochastic Burgers equation. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1486.60002)] coauthored with \textit{A. Boritchev} and \textit{S. Kuksin}. The author considers the 1D viscous Burgers equation with periodic boundary condition and additive noise which is spatially smooth. When the viscosity is small enough (equivalently, the Reynolds number is sufficiently big), he is able to rigorously estimate some quantities like dissipation scale, structure function and energy spectrum; the purpose is to compare these quantities with the predictions of Kolmogorov's turbulence theory, abbreviated as the K41 theory. The author concludes that the statistical properties of stochastic 1D Burgers equation with small viscosity are close analogues of the main laws of the K41 theory, which supports the belief that K41 theory is ``close to the truth''. Reviewer: Dejun Luo (Beijing) Comparison of gradient approximation methods in schemes designed for scale-resolving simulations 2022-09-13T20:28:31.338867Z "Bakhné, S." "Bosnyakov, S. M." "Mikhaĭlov, S. V." "Troshin, A. I." Summary: Various methods for improved accuracy approximation of the gradients entering the diffusion fluxes are considered. Linear combinations of 2nd order difference schemes for a non-uniform grid that transform into 4th order schemes in the uniform case were investigated. We also considered 3rd and 4th order schemes for approximating gradients on a non-uniform grid in the normal and tangent directions to the cell face, respectively, based on Lagrange polynomials. The initial testing was carried out on one-dimensional functions: a smooth Gauss function and a piecewise linear function. Next, the schemes were applied in direct numerical simulation of the Taylor-Green vortex. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems 2022-09-13T20:28:31.338867Z "Allwright, Amy" "Atangana, Abdon" Summary: The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established. Non-overlapping Schwarz algorithms for the incompressible Navier-Stokes equations with DDFV discretizations 2022-09-13T20:28:31.338867Z "Goudon, Thierry" "Krell, Stella" "Lissoni, Giulia" The authors consider the numerical resolution of the unsteady incompressible Navier-Stokes problem. They first establish the well-posedness of DDFV (Discrete Duality Finite Volume) schemes on the whole spatial domain with general convection fluxes defined by \(B\)-schemes. Subsequently, they propose two non-overlapping DDFV Schwarz algorithms. DDFV discretizations are constructed with suitable transmission conditions. When using standard convection fluxes in the domain decomposition method, the iterative process converges to a system with modified fluxes at the interface. However, it is possible to modify the fluxes of the domain decomposition algorithm so that it converges to the reference scheme on the entire domain. Some numerical tests are presented to illustrate the behavior and the performances of the algorithms Reviewer: Abdallah Bradji (Annaba) Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model 2022-09-13T20:28:31.338867Z "Remesan, Gopikrishnan Chirappurathu" The author considers the finite volume approximation, on nonuniform Cartesian grids, of the nonlinear scalar conservation law \(\partial_t \alpha +\operatorname{div}(u f(\alpha )) = 0\) in two and three spatial dimensions with an initial data of bounded variation. A uniform estimate on total variation of discrete solutions is proved. The standard assumption which states that the advecting velocity vector is divergence free is relaxed. Since the underlying meshes are nonuniform Cartesian, it is possible to adaptively refine the mesh on regions where the solution is expected to have sharp fronts. A uniform BV estimate is also obtained for finite volume approximations of conservation laws that has a fully nonlinear flux on nonuniform Cartesian grids. Some numerical tests are presented to support the theoretical results. Reviewer: Abdallah Bradji (Annaba) Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn-Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity 2022-09-13T20:28:31.338867Z "Chen, Chuanjun" "Yang, Xiaofeng" Summary: We construct a fully-discrete finite element numerical scheme for the Cahn-Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier-Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn-Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh-Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically. Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise 2022-09-13T20:28:31.338867Z "Feng, Xiaobing" "Qiu, Hailong" Summary: This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite element methods. Local transparent boundary conditions for wave propagation in fractal trees. I: Method and numerical implementation 2022-09-13T20:28:31.338867Z "Joly, Patrick" "Kachanovska, Maryna" A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot's model 2022-09-13T20:28:31.338867Z "Khan, Arbaz" "Zanotti, Pietro" The paper analyzes the numerical method for Biot's model describing the elastic wave propagation inside a porous medium saturated with a fluid. Variables in this model represent the displacement of the medium and the fluid pressure. In addition, there are several material parameters. However, spurious oscillations or volumetric locking may occur for specific values of these parameters. The authors focus on overcoming this problem and propose a method that is robust in the sense that it is uniformly stable with respect to all parameters. First, the authors establish a novel nonsymmetric variational setting, where the norm measuring the data is not dual to the norm for measuring the solution. Then, they show the well-posedness of the setting and derive stability estimates. Furthermore, the authors propose a method that uses the backward Euler scheme for temporal discretization combined with the finite element method using first-order nonconforming Crouzeix-Raviart elements for the displacement and first-order discontinuous piecewise affine functions for the fluid pressure. The presented analysis of stability and error estimates leads to the conclusion that the method is robust and quasi-optimal. Finally, possible generalizations of the results are discussed. Reviewer: Dana Černá (Liberec) Second-order convergence of the linearly extrapolated Crank-Nicolson method for the Navier-Stokes equations with \(H^1\) initial data 2022-09-13T20:28:31.338867Z "Li, Buyang" "Ma, Shu" "Wang, Na" Summary: This article concerns the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with \(H^1\) initial data. By utilizing special locally refined temporal stepsizes, we prove that the linearly extrapolated Crank-Nicolson scheme, with the usual stabilized Taylor-Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to support the theoretical analysis. Local and parallel efficient BDF2 and BDF3 rotational pressure-correction schemes for a coupled Stokes/Darcy system 2022-09-13T20:28:31.338867Z "Li, Jian" "Wang, Xue" "Al Mahbub, Md. Abdullah" "Zheng, Haibiao" "Chen, Zhangxin" This paper extends authors earlier work [\textit{J. Li} et al., Comput. Math. with Appl. 79, 337--353 (2020; Zbl 1443.65187); Numer. Methods Partial Differential Equations 35, 1873--1889 (2019; Zbl 1423.76253)] where first- and second-order (in time) BE (backward Euler) and BDF2 schemes with the rotational pressure-correction methods introduced in [\textit{J. Guermond} et al., SIAM J. Numer. Anal. 43, 239--258 (2005; Zbl 1083.76044)] are studied for a coupled Stokes/Darcy system. These temporal schemes are developed, and the BDF2/BDF3 rotational pressure-correction methods are studied for the Stokes/Darcy system. It was proven that the BDF2/BDF3 rotational pressure-correction methods are unconditionally stable, long-time accurate with a uniform-in-time error bound, and efficient in that only two decoupled equations are required to solve at each time step. At each time step, only one linear system of equations has to be solved, which thus significantly reduces the computational time and memory costs in practice. The presented projection methods are combined with the local and parallel methods based on full overlapping decoupled techniques for the coupled Stokes/Darcy system which increases the computational efficiency further. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed methods. Reviewer: Bülent Karasözen (Ankara) A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergence 2022-09-13T20:28:31.338867Z "Masri, Rami" "Liu, Chen" "Riviere, Beatrice" Summary: A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the proposed scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates. New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous media 2022-09-13T20:28:31.338867Z "Sun, Weiwei" Summary: Numerical methods and analysis for compressible miscible flow in porous media have been investigated extensively in the last several decades. Amongst those methods, the lowest-order mixed method is the most popular one in practical applications. The method is based on the linear Lagrange approximation for the concentration and the lowest order (zero-order) Raviart-Thomas mixed approximation for the Darcy velocity/pressure. However, the existing error analysis only provides the first-order accuracy in \(L^2\)-norm for all three physical components in spatial direction, which was proved under certain extra restrictions on both time step and spatial meshes. The analysis is not optimal for the concentration mainly due to the strong coupling of the system and the drawback of the traditional approach which leads to serious pollution to the numerical concentration in analysis. The main task of this paper is to present a new analysis and establish the optimal error estimate of the commonly-used linearized lowest-order mixed FEM. In particular, the second-order accuracy for the concentration in spatial direction is proved unconditionally. Moreover, we propose a simple recovery technique to obtain a new numerical Darcy velocity/pressure of second-order accuracy by re-solving an elliptic pressure equation. Also we extend our analysis to a second-order time discrete scheme to obtain optimal error estimates in both spatial and temporal directions. Numerical results are provided to confirm our theoretical analysis and show the efficiency of the method. A two-grid combined mixed finite element and discontinuous Galerkin method for an incompressible miscible displacement problem in porous media 2022-09-13T20:28:31.338867Z "Yang, Jiming" "Su, Yifan" Summary: An incompressible miscible displacement problem is investigated. A two-grid algorithm of a full-discretized combined mixed finite element and discontinuous Galerkin approximation to the miscible displacement in porous media is proposed. The error estimate for the concentration in \(H^1\)-norm and the error estimates for the pressure and the velocity in \(L^2\)-norm are obtained. The analysis shows that the asymptotically optimal approximation can be achieved as long as the mesh size satisfies \(h = O(H^2)\), where \(H\) and \(h\) are the sizes of the coarse mesh and the fine mesh, respectively. Meanwhile, the effectiveness of the presented algorithm is verified by numerical experiments, from which it can be seen that the algorithm is spent much less time. Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell 2022-09-13T20:28:31.338867Z "Yang, Xiaofeng" Summary: We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the ``zero-energy-contribution'' feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension. The convergence analysis of semi- and fully-discrete projection-decoupling schemes for the generalized Newtonian models 2022-09-13T20:28:31.338867Z "Zhou, Guanyu" Summary: We propose two linear schemes (1st- and 2nd-order) for the generalized Newtonian flow with the shear-dependent viscosity, which combine the decoupling techniques with the projection methods. The linear stabilization terms mimic \(-k\partial_t \Delta{\boldsymbol{u}}\) and \(-k\partial_{tt} \Delta{\boldsymbol{u}}\) from the PDE point of view. By our schemes, each velocity component can be computed in parallel efficiently using the same solver \((I-\alpha^{-1}k\Delta)^{-1}\) at every time level. We analyze the convergence rates of the (temporally) semi- and the fully-discrete schemes. The theoretical results are testified by the numerical experiments. An efficient DWR-type a posteriori error bound of SDFEM for singularly perturbed convection-diffusion PDEs 2022-09-13T20:28:31.338867Z "Avijit, D." "Natesan, S." Summary: This article deals with the residual-based a posteriori error estimation in the standard energy norm for the streamline-diffusion finite element method (SDFEM) for singularly perturbed convection-diffusion equations. The well-known dual-weighted residual (DWR) technique has been adopted to elevate the accuracy of the error estimator. Our main contribution is finding an efficient computable DWR-type robust residual-based a posteriori error bound for the SDFEM. The local lower error bound has also been provided. An adaptive mesh refinement algorithm has been addressed and lastly, some numerical experiments are carried out to justify the theoretical proofs. Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow 2022-09-13T20:28:31.338867Z "Burman, Erik" "Puppi, Riccardo" Summary: We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as \(\mathcal{O} (h^{-1})\), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as \(\mathcal{O} (h^{- k -1})\), \(k\) being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual \(L^2\)-norm. However, we are still able to recover the optimal a priori \(L^2\)-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited. A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equations 2022-09-13T20:28:31.338867Z "Caucao, Sergio" "Gatica, Gabriel N." "Ortega, Juan P." Summary: We propose and analyze a new mixed finite element method for the nonlinear problem given by the coupling of the steady Brinkman-Forchheimer and double-diffusion equations. Besides the velocity, temperature, and concentration, our approach introduces the velocity gradient, the pseudostress tensor, and a pair of vectors involving the temperature/concentration, its gradient and the velocity, as further unknowns. As a consequence, we obtain a fully mixed variational formulation presenting a Banach spaces framework in each set of equations. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and Babuška-Brezzi's theory in Banach spaces, are applied to prove the unique solvability of the continuous and discrete systems. Appropriate finite element subspaces satisfying the required discrete inf-sup conditions are specified, and optimal \textit{a priori} error estimates are derived. Several numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. Analysis of a stabilized finite element approximation for a linearized logarithmic reformulation of the viscoelastic flow problem 2022-09-13T20:28:31.338867Z "Codina, Ramon" "Moreno, Laura" Summary: In this paper we present the numerical analysis of a finite element method for a linearized viscoelastic flow problem. In particular, we analyze a linearization of the logarithmic reformulation of the problem, which in particular should be able to produce results for Weissenberg numbers higher than the standard one. In order to be able to use the same interpolation for all the unknowns (velocity, pressure and logarithm of the conformation tensor), we employ a stabilized finite element formulation based on the Variational Multi-Scale concept. The study of the linearized problem already serves to show why the logarithmic reformulation performs better than the standard one for high Weissenberg numbers; this is reflected in the stability and error estimates that we provide in this paper. An embedded discontinuous Galerkin method for the Oseen equations 2022-09-13T20:28:31.338867Z "Han, Yongbin" "Hou, Yanren" Summary: In this paper, the \textit{a priori} error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in the \(L^2 (\Omega)\) norm, has an optimal error bound with convergence order \(k+1\), where the constants are dependent on the Reynolds number (or \(\nu^{-1})\), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence order \(k+1/2\). Here, \(k\) is the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results. Numerical upscaling for heterogeneous materials in fractured domains 2022-09-13T20:28:31.338867Z "Hellman, Fredrik" "Målqvist, Axel" "Wang, Siyang" Summary: We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise \textit{e.g.} in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An \textit{a priori} error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems. A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems 2022-09-13T20:28:31.338867Z "He, Xiaoxiao" "Song, Fei" "Deng, Weibing" The paper deals with the numerical solution of the Stokes interface problem by a stabilized extended finite element method on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming \(P_1\) velocity and elementwise \(P_0\) pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size \(h\), the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in \(L_2\) norm for pressure are uniform to the viscosity and the location of the interface. Two numerical examples are presented to support the theoretical analysis. Reviewer: Vit Dolejsi (Praha) Error analysis of higher order trace finite element methods for the surface Stokes equation 2022-09-13T20:28:31.338867Z "Jankuhn, Thomas" "Olshanskii, Maxim A." "Reusken, Arnold" "Zhiliakov, Alexander" Summary: The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in \(\mathbb{R}^3\). The method employs parametric \(\mathbf{P}_k\)-\(P_{k-1}\) finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin-Helmholtz instability problem on the unit sphere. Coupled iterative analysis for stationary inductionless magnetohydrodynamic system based on charge-conservative finite element method 2022-09-13T20:28:31.338867Z "Zhang, Xiaodi" "Ding, Qianqian" Summary: This paper considers charge-conservative finite element approximation and three coupled iterations of stationary inductionless magnetohydrodynamics equations in Lipschitz domain. Using a mixed finite element method, we discretize the hydrodynamic unknowns by stable velocity-pressure finite element pairs, discretize the current density and electric potential by \(\boldsymbol{H}(\operatorname{div},\varOmega)\times L^2(\varOmega)\)-comforming finite element pairs. The well-posedness of this formula and the optimal error estimate are provided. In particular, we show that the error estimates for the velocity, the current density and the pressure are independent of the electric potential. With this, we propose three coupled iterative methods: Stokes, Newton and Oseen iterations. Rigorous analysis of convergence and stability for different iterative schemes are provided, in which we improve the stability conditions for both Stokes and Newton iterative method. Numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed methods. Port-Hamiltonian formulations of poroelastic network models 2022-09-13T20:28:31.338867Z "Altmann, R." "Mehrmann, V." "Unger, B."|unger.benjamin Summary: We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by (formally) setting the second-order term zero. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations. Fluid-structure coupling simulation of explosive flow field in heterogeneous porous media based on fractal theory 2022-09-13T20:28:31.338867Z "Zhu, Qingyong" "Lu, Shun" "Lai, Dongheng" "Sun, Junjun" Dispersion of waves in two and three-dimensional periodic media 2022-09-13T20:28:31.338867Z "Godin, Yuri A." "Vainberg, Boris" Summary: We consider the propagation of acoustic time-harmonic waves in homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius \(a\) of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two- and three-dimensional cases as \(a\to0\) and first several terms of the expansions are evaluated explicitly. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Dirichlet, Neumann, and transmission boundary conditions are considered. In the former case, we estimate the cutoff wavelength \(\lambda_{\mathrm{max}}\) supported by the periodic medium in two and three dimensions. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of the asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs. Open-channel flow 2022-09-13T20:28:31.338867Z "Chaudhry, M. Hanif" Publisher's description: Now in the third edition, this text on open-channel flow presents introductory material on the topic as well as up-to-date information for the planning, design, and operation of water-resource projects. It has a very strong emphasis on the application of efficient solution techniques, computational procedures, and modern methods of analysis and provides detailed coverage of steady and unsteady flows. The new edition includes a new chapter on the modeling of levee breach, new sections on hydraulic models and velocity measurements, new and updated references and problem sets and exercises, and a solutions manual. The material has been class tested over many years and the book is an ideal textbook for students in senior-level undergraduate and graduate courses on open-channel flow and hydraulic engineering, and as a reference for civil engineers needing up-to-date information on the latest developments in open-channel flow. \begin{itemize} \item Detailed coverage of steady and unsteady flows; \item Includes practical examples and sample computer programs in Python; \item New problem sets, some designed especially for take-home tests, and a solution manual for instructors. \end{itemize} See the review of the 2nd edition in [Zbl 1149.76001]. For the 1st edition see [Zbl 1138.76002]. Internal waves in the ocean. Theory and practice 2022-09-13T20:28:31.338867Z "Stastna, Marek" Publisher's description: This monograph provides a concise overview of nonlinear internal wave theory. It serves as a self-contained reference for both students of mathematics as well as scientific professionals by presenting the material in two parts, isolating the narrative analysis from the mathematical detail. This unique format allows the text to remain accessible to oceanographers and researchers outside of mathematics by presenting a range of relevant theories on their own terms. Conversely, it enables applied mathematicians to understand how the conversation between mathematics and sciences proceeds in a field that has developed through a combination of the two. In addition, the text is supplemented by hands-on Matlab software, as the book incorporates a collection of working codes that enable readers to reproduce all theoretical figures in the text, with modification potential to fit a range of applications including a number of mini-projects outlined throughout the text. Designing complex fluids 2022-09-13T20:28:31.338867Z "Ewoldt, Randy H." "Saengow, Chaimongkol" Summary: Taking a small step away from Newtonian fluid behavior creates an explosion in the range of possibilities. Non-Newtonian fluid properties can achieve diverse flow objectives, but the complexity introduces challenges. We survey useful rheological complexity along with organizing principles and design methods as we consider the following questions: How can non-Newtonian properties be useful? What properties are needed? How can we get those properties? For the entire collection see [Zbl 1489.76002]. Entropy generation in Casson nanofluid flow past an electromagnetic stretching Riga plate 2022-09-13T20:28:31.338867Z "Oyelakin, I. S." "Ghosh, R." "Mondal, S." "Sibanda, P." Summary: This paper investigates entropy generation in a Casson nanofluid flow past an electromagnetic stretching Riga plate. Entropy generation is a measure of irreversibility factors in thermodynamic processes. It is a common feature in heat transfer studies, and as such, the study includes the effect of viscous dissipation. We solve the model equations using the spectral local linearization method. The study considers the impact of some other physical parameters like the Casson, velocity ratio, and electromagnetic parameters. A good correlation is achieved when the present results are compared with published literature. The results indicate that the velocity ratio parameter significantly influences the fluid flow, temperature, and concentration profiles. The entropy generation increases with an increase in concentration and Brinkmann number, whereas an opposite behavior is observed for increasing the value of the modified Hartmann number. Again, increasing the Casson parameter increases the temperature and concentration profiles, whereas the velocity profile reduces. Computational investigation of Stefan blowing effect on flow of second-grade fluid over a curved stretching sheet 2022-09-13T20:28:31.338867Z "Punith Gowda, R. J." "Baskonus, Haci Mehmet" "Naveen Kumar, R." "Prasannakumara, B. C." "Prakasha, D. G." Summary: Non-Newtonian fluids have extensive range of applications in the field of industries like plastics processing, manufacturing of electronic devices, lubrication flows, medicine and medical equipment. Stimulated from these applications, a theoretical analysis is carried out to scrutinize the flow of a second-grade liquid over a curved stretching sheet with the impact of Stefan blowing condition, thermophoresis and Brownian motion. The modelled governing equations for momentum, thermal and concentration are deduced to a system of ordinary differential equations by introducing suitable similarity transformations. These reduced equations are solved using Runge-Kutta-Fehlberg fourth fifth order method (RKF-45) by adopting shooting technique. The solutions for the flow, heat and mass transference features are found numerically and presented with the help of graphical illustrations. Results reveal that, curvature and Stefan blowing parameters have propensity to rise the heat transfer. Further, second grade fluid shows high rate of mass and heat transfer features when related to Newtonian fluid for upsurge in values of Brownian motion parameter. Global well posedness for a Q-tensor model of nematic liquid crystals 2022-09-13T20:28:31.338867Z "Murata, Miho" "Shibata, Yoshihiro" Summary: In this paper, we prove the global well posedness and the decay estimates for a \(\mathbb{Q}\)-tensor model of nematic liquid crystals in \(\mathbb{R}^N\), \(N \ge 3\). This system is a coupled system by the Navier-Stokes equations with a parabolic-type equation describing the evolution of the director fields \(\mathbb{Q}\). The proof is based on the maximal \(L_p\)-\(L_q\) regularity and the \(L_p\)-\(L_q\) decay estimates to the linearized problem. Retraction of thin films coated by insoluble surfactants 2022-09-13T20:28:31.338867Z "De Corato, Marco" "Tammaro, Daniele" "Maffettone, Pier Luca" "Fueyo, Norberto" Summary: We investigate the retraction of a circular thin film coated with insoluble surfactants, which is punctured at its centre. We assume that the surface pressure of the liquid-gas interface is related to the number density of surfactants through a linear equation of state, which is characterized by a single parameter: the Gibbs dilation modulus. To solve the governing equations and track the deformation of the domain, we use the finite element method with an arbitrary Lagrangian-Eulerian approach where the film surface is sharp. Our simulations show that the surface elasticity introduced by the surfactants slows down the retraction and introduces oscillations at early times. In agreement with previous experiments and theoretical analysis, we find that the presence of surfactants introduces perturbations of the film thickness over progressively larger distances as the surface elasticity increases. The surface perturbations travel faster than the retracting edge of the film at a speed proportional to the Gibbs modulus. For large values of the Gibbs modulus, the interface behaviour approaches that of an incompressible two-dimensional solid. Our analysis sheds light on the effect of insoluble surfactants on the opening of a circular hole in a thin film and can be extended to investigate the onset of surface cracks and fractures. Multiscale modelling and splitting approaches for fluids composed of Coulomb-interacting particles 2022-09-13T20:28:31.338867Z "Geiser, Jürgen" Summary: We consider fluids composed of Coulomb-interacting particles, which are modelled by the Fokker-Planck equation with a collision operator. Based on modelling the transport and collision of the particles, we propose new, computationally efficient, algorithms based on splitting the equations of motion into a global Newtonian transport equation, where the effects of an external electric field are considered, and a local Coulomb interaction stochastic differential equation, which determines the new velocities of the particle. Two different numerical schemes, one deterministic and the other stochastic, as well as an Hamiltonian splitting approach, are proposed for coupling the interactionand transport equations. Results are presented for two- and multi-particle systems with different approximations for the Coulomb interaction. Methodologically, the transport part is modelled by the kinetic equations and the collision part is modelled by the Langevin equations with Coulomb collisions. Such splitting approaches allow concentrating on different solver methods for each different part. Further, we solve multiscale problems involving an external electrostatic field. We apply a multiscale approach so that we can decompose the different time-scales of the transport and the collision parts. We discuss the benefits of the different splitting approaches and their numerical analysis. Comparison of two higher accuracy unstructured scale-resolving approaches applied to dual-stream nozzle jet simulation 2022-09-13T20:28:31.338867Z "Bosnyakov, S. M." "Volkov, A. V."|volkov.aleksei-vasilevich "Duben', A. P." "Zapryagaev, V. I." "Kozubskaya, T. K." "Mikhaĭlov, S. V." "Troshin, A. I." "Tsvetkova, V. O." Summary: Dual-stream nozzle jet computations conducted using different numerical algorithms developed in TsAGI and KIAM RAS are presented. Scale-resolving approaches of DES family based on higher accuracy numerical methods are applied. The flow considered was studied experimentally at ITAM SB RAS. The jet was axisymmetric up to the influence of the supporting pylons, cold, subsonic at the inner nozzle exit and supersonic at the outer nozzle exit. The computational data is compared with the experiment and with each other. One-dimensional unsteady flow from a cylindrical draining tank 2022-09-13T20:28:31.338867Z "Marotta, Sebastian M." "Geeter, Chris" "Huynh, Richard" Summary: We study the differential equation that corresponds to the one-dimensional frictionless unsteady flow model of a cylindrical draining tank. We survey previous results, solve the equation applying new changes of variables and procedures, and present new exact elementary solutions. The problem provides an excellent example of application that is accessible to undergraduate students after a first course on differential equations. Flood inundation prediction 2022-09-13T20:28:31.338867Z "Bates, Paul D." Summary: Every year flood events lead to thousands of casualties and significant economic damage. Mapping the areas at risk of flooding is critical to reducing these losses, yet until the last few years such information was available for only a handful of well-studied locations. This review surveys recent progress to address this fundamental issue through a novel combination of appropriate physics, efficient numerical algorithms, high-performance computing, new sources of big data, and model automation frameworks. The review describes the fluid mechanics of inundation and the models used to predict it, before going on to consider the developments that have led in the last five years to the creation of the first true fluid mechanics models of flooding over the entire terrestrial land surface. For the entire collection see [Zbl 1489.76002]. Wave breaking in undular bores with shear flows 2022-09-13T20:28:31.338867Z "Bjørnestad, Maria" "Kalisch, Henrik" "Abid, Malek" "Kharif, Christian" "Brun, Mats" Summary: It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of \textit{H. Favre} [Ondes de translation. Paris: Dunod (1935)] that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg-de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent. Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system 2022-09-13T20:28:31.338867Z "Gao, Xiao-Tian" "Tian, Bo" Summary: Studies on the water waves are undertaken in hydrodynamics. In this Letter, a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system describing the water waves in an infinitely narrow channel of constant depth is taken into consideration. Through symbolic computation, concerning the horizontal velocity and elevation of the water wave, this Letter presents two branches of the similarity reductions. Symbolic computation on a \((2+1)\)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system for the water waves 2022-09-13T20:28:31.338867Z "Gao, Xin-Yi" "Guo, Yong-Jiang" "Shan, Wen-Rui" Summary: Water waves attract people's attention. For the water waves, a \((2+1)\)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system is hereby studied. As for the horizontal velocity and elevation of the water wave, on the one hand, with the scaling transformations and symbolic computation, a set of the hetero-Bäcklund transformations is constructed, linking the original system with a known generalized variable-coefficient Burgers equation. As for the horizontal velocity and elevation of the water wave, on the other hand, with symbolic computation, a set of the similarity reductions is constructed, from the original system to a known ordinary differential equation. All our results depend on the variable coefficients in the original system. Water wave scattering by a thin vertical submerged permeable plate 2022-09-13T20:28:31.338867Z "Gayen, Rupanwita" "Gupta, Sourav" "Chakrabarti, Aloknath" Summary: An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock's expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov's book [\textit{F. D. Gakhov}, Boundary value problems. Oxford-London-Edinburgh-New York-Paris-Frankfurt: Pergamon Press (1966; Zbl 0141.08001)]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock's theorems. On the structure of steady parasitic gravity-capillary waves in the small surface tension limit 2022-09-13T20:28:31.338867Z "Shelton, Josh" "Milewski, Paul" "Trinh, Philippe H." Summary: When surface tension is included in the classical formulation of a steadily travelling gravity wave (a Stokes wave), it is possible to obtain solutions that exhibit parasitic ripples: small capillary waves riding on the surface of steep gravity waves. However, it is not clear whether the singular small surface tension limit is well posed. That is, is it possible for an appropriate travelling gravity-capillary wave to be continuously deformed to the classic Stokes wave in the limit of vanishing surface tension? The work of \textit{B. Chen} and \textit{P. G. Saffman} [Stud. Appl. Math. 62, 1--21 (1980; Zbl 0446.76023)] had suggested smooth continuation was not possible, while the numerical study of \textit{L. W. Schwartz} and \textit{J.-M. Vanden-Broeck} [J. Fluid Mech. 95, 119--139 (1979; Zbl 0419.76014)] used an amplitude parameter that made it difficult to understand the structure of solutions for small values of the surface tension. In this paper we numerically explore the low surface tension limit of the steep gravity-capillary travelling-wave problem. Our results allow for a classification of the bifurcation structure that arises, and serve to unify a number of previous numerical studies. Crucially, we demonstrate that different choices of solution amplitude can lead to subtle restrictions on the continuation procedure. When wave energy is used as a continuation parameter, solution branches can be continuously deformed to the zero surface tension limit of a travelling Stokes wave. Dynamics of nonlinear waves in a Burridge and Knopoff model for earthquake with long-range interactions, velocity-dependent and hydrodynamics friction forces 2022-09-13T20:28:31.338867Z "Nkomom, Théodule Nkoa" "Ndzana, Fabien II" "Okaly, Joseph Brizar" "Mvogo, Alain" Summary: We investigate the dynamics of nonlinear waves in a long-range extension of the Burridge and Knopoff model for earthquake. We consider the dissipative hydrodynamics forces. The spatio-temporal dynamics of the system is found by introducing in the coupling spring and hydrodynamics forces a linear term that decays as a power-law, with an exponent \(s\) such that \(1<s\leq 3\). The theoretical framework for the analysis is presented in the rotative wave approximation. Due to the non analytic properties of the dispersion relation, we use the discrete derivative operator technique. The dynamics of the system is governed by the complex Ginzburg-Landau equation, allowing breather-like soliton solutions. We use the relevant case \(s=2\), and results show that the magnitude, the velocity, and area of propagation of nonlinear waves strongly depend on the frictions forces. Our analytical results are in good agreement with numerical experiments and confirm the correctness of the method. Vortex collapses for the Euler and quasi-geostrophic models 2022-09-13T20:28:31.338867Z "Godard-Cadillac, Ludovic" Summary: This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by \textit{C. Marchioro} and \textit{M. Pulvirenti} [Mathematical theory of incompressible nonviscous fluids. New York, NY: Springer-Verlag (1994; Zbl 0789.76002)] concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case. Laminar flow of a viscous liquid in the entrance region of a circular pipe 2022-09-13T20:28:31.338867Z "Kazakov, L. I." Summary: An approximate theory of stationary axisymmetric laminar flow of a viscous incompressible fluid in the entrance region of a circular pipe is presented. It gives correct (within \(\pm 2\)\%) calculated values of different physical characteristics of the established flow, which coincide with the known calculated and experimental data. Instead of the traditional Bernoulli equation for the entire length of the entrance region, the work at hand uses the equation of the axial pressure gradient averaged over the pipe section to determine the pressure value. Navier-Stokes equations, the algebraic aspect 2022-09-13T20:28:31.338867Z "Zharinov, V. V." Summary: We present an analysis of the Navier-Stokes equations in the framework of an algebraic approach to systems of partial differential equations (the formal theory of differential equations). \(L^p\)-strong solution for the stationary exterior Stokes equations with Navier boundary condition 2022-09-13T20:28:31.338867Z "Dhifaoui, Anis" Let \(\Omega \subset \mathbb{R}^3\) be an unbounded domain with compact boundary of class \(C^{2,1}\) such that \(\mathbb{R}^3\setminus \overline \Omega \) is connected. The paper studies the Stokes system with Navier boundary condition \( -\Delta u+\nabla p=f\), \( \nabla \cdot u=0 \) in \( \Omega \), \( u_n=g\), \( [T(u,p)n^\Omega +\alpha u]_\tau = h \) on \( \partial \Omega \). A solution \( (u,p)\) is from the weighted Sobolev spaces \( W^{2,q}_{k+1}(\Omega )\times W^{1,q}_{k+1}(\Omega )\). Reviewer: Dagmar Medková (Praha) An alternative proof of \(L^q-L^r\) estimates of the Oseen semigroup in higher dimensional exterior domains 2022-09-13T20:28:31.338867Z "Hishida, Toshiaki" Summary: \(L^q-L^r\) decay estimates of the Oseen semigroup in \(n\)-dimensional exterior domains were well established by \textit{T. Kobayashi} and \textit{Y. Shibata} [Math. Ann. 310, No. 1, 1--45 (1998; Zbl 0891.35114)] \((n=3)\), \textit{Y. Enomoto} and \textit{Y. Shibata} [J. Math. Fluid Mech. 7, No. 3, 339--367 (2005; Zbl 1094.35097)] \((n\ge 3)\) and \textit{Y. Maekawa} [J. Inst. Math. Jussieu 20, No. 3, 859--891 (2021; Zbl 1465.76032)] \((n=2)\). The same result has been recently proved by the present author [Math. Ann. 372, No. 3--4, 915--949 (2018; Zbl 1405.35139); Arch. Ration. Mech. Anal. 238, No. 1, 215--254 (2020; Zbl 1446.35100)] for a generalized Oseen evolution operator in 3-dimensional exterior domains, where rotation as well as translation of a rigid body is taken into account and, moreover, both translational and angular velocities can be time-dependent. The approach developed there can be considerably simplified if both the non-autonomous character and rotation are absent. As a consequence, an alternative short proof of decay estimates of the Oseen semigroup can be available without relying on analysis of the resolvent and the argument works for \(n\ge 3\) as well. I thus believe that the presentation of the proof would be worth publishing here. Variable viscosity effect on boundary layer flow along continuously moving plate with the thermal boundary condition of the third kind 2022-09-13T20:28:31.338867Z "Jha, Basant K." "Samaila, Gabriel" Summary: An incompressible and viscous fluid flow past a constant moving plate with convective boundary condition considering the variable viscosity effect is fully presented. The solution to the governing equation is obtained by Runge Kutta Ferberg four-fifth order (RKF45) method in Maple software. Four fluids namely; mercury, air, sulphur oxide and water whose respective Prandtl numbers are 0.044, 0.72, 2 and 7 are considered during the computation. The effect of the controlling parameters such Biot number \((Bi)\), Prandtl number (Pr), reference temperature \(( \theta_r)\) and exponential constant (N) on the temperature distribution, velocity profile, Nusselt number and the Skin friction is presented using tables and line graphs. It is found that the temperature distribution is inversely proportional to Biot number \((Bi)\) augment whereas the velocity profile decreases as the reference temperature \(( \theta_r)\) propagates. The results also revealed that the thickness of the thermal boundary layer decrease as Prandtl number (Pr) increases. For liquids fluid, the skin friction increases with the exponential constant \((N)\) propagation whereas decreases for gases fluid. The effect of the Biot number on the skin friction exhibits opposite behaviour with that of the exponential constant. Regarding the Nusselt number, the exponential constant augment increases the Nusselt number for both gases and liquids fluid. Vortex reconnection and turbulence cascade 2022-09-13T20:28:31.338867Z "Yao, Jie" "Hussain, Fazle" Summary: As a fundamental topology-transforming event, reconnection plays a significant role in the dynamics of plasmas, polymers, DNA, and fluids -- both (classical) viscous and quantum. Since the 1994 review by \textit{S. Kida} and \textit{M. Takaoka} [Annu. Rev. Fluid Mech. 26, 169--189 (1994; Zbl 0802.76016)], substantial advances have been made on this topic. We review recent studies of vortex reconnection in (classical) viscous flows, including the physical mechanism, its relationship to turbulence cascade, the formation of a finite-time singularity, helicity dynamics, and aeroacoustic noise generation. For the entire collection see [Zbl 1489.76002]. Flow control for unmanned air vehicles 2022-09-13T20:28:31.338867Z "Greenblatt, David" "Williams, David R." Summary: The pervasiveness of unmanned air vehicles (UAVs), from insect to airplane scales, combined with active flow control maturity, has set the scene for vehicles that differ markedly from present-day configurations. Nano and micro air vehicles, with characteristic Reynolds numbers typically less than \(10^5\), rely on periodically generated leading-edge vortices for lift generation, propulsion, and maneuvering. This is most commonly achieved by mechanical flapping or pulsed plasma actuation. On larger UAVs, with Reynolds numbers greater than \(10^5\), externally driven and autonomous fluidic systems continue to dominate. These include traditional circulation control techniques, autonomous synthetic jets, and discrete sweeping jets. Plasma actuators have also shown increased technological maturity. Energy efficiency is a major challenge, whether it be batteries and power electronics on nano and micro air vehicles or acceptably low compressor bleed on larger UAVs. Further challenges involve the development of aerodynamic models based on experiments or numerical simulations, as well as flight dynamics models. For the entire collection see [Zbl 1489.76002]. Cattaneo-LTNE effects on the stability of Brinkman convection in an anisotropic porous layer 2022-09-13T20:28:31.338867Z "Hema, M." "Shivakumara, I. S." "Ravisha, M." Summary: The stability of Brinkman local thermal nonequilibrium anisotropic porous convection under the impact of Cattaneo law of heat conduction in solid is investigated. In the analysis, anisotropies in permeability and thermal (solid and fluid phases) conductivities are highlighted. Condition for stationary onset and oscillatory onset is obtained by carrying out linear instability analysis. A novel result is that the instability occurs through oscillatory mode against the stationary convection perceived in the absence of Cattaneo effect. The relative magnitudes of governing parameters on the initiation of oscillatory instability are delineated in detail. The thermal and mechanical anisotropies inflict stabilizing and destabilizing effects on the onset, respectively. The influence of mechanical anisotropy, thermal anisotropy of fluid, thermal relaxation time parameters and the Darcy number is to broaden the size of convection cells whereas thermal anisotropy of the solid and the Darcy-Prandtl number demonstrate a mixed behaviour. A first order amplitude equation is derived separately for steady and overstable modes by performing a weak nonlinear stability analysis using a modified multiscale method. Depending on the values of governing parameters, it is seen that the stationary mode bifurcates subcritically and supercritically, while the oscillatory mode always bifurcates supercritically. Finite rotating and translating vortex sheets 2022-09-13T20:28:31.338867Z "Protas, Bartosz" "Llewellyn Smith, Stefan G." "Sakajo, Takashi" Summary: We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse [\textit{A. E. H. Love}, Proc. Lond. Math. Soc. 25, 18--42 (1894; JFM 25.1467.02); \textit{T. B. Mitchell} and \textit{L. F. Rossi}, Phys. Fluids 20, No. 5, Paper No. 054103, 12 p. (2008; Zbl 1182.76523)] and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches. Advection versus diffusion in Richtmyer-Meshkov mixing 2022-09-13T20:28:31.338867Z "Doss, Forrest W." Summary: The Richtmyer-Meshkov (RM) instability is one of the most severe degradation mechanisms for inertial confinement fusion (ICF), and mitigating it has been a priority for the global ICF effort. In this Letter, the instability's ability to atomically mix is linked to its background decay of residual turbulent energy. We show how recently derived inequalities from the mathematical theory of PDEs constrain the evolution. A model RM process at leading order may diffusively mix or retain imprints of its initial structures indefinitely, depending on initial conditions, and there exists a theoretical range of zero-mixing for certain values of parameters. The results may apply to other systems resembling scalar transport in decaying turbulence. Effects of head loss, surface tension, viscosity and density ratio on the Kelvin-Helmholtz instability in different types of pipelines 2022-09-13T20:28:31.338867Z "Yang, X. C."|yang.xu-chen|yang.xia-chun|yang.xuecheng|yang.xuchi|yang.xiaochun|yang.xuechao|yang.xiaochen|yang.xiaochuan|yang.xiaochao|yang.xuechang|yang.xinchao|yang.xiuchun|yang.xiaocheng "Cao, Y. G."|cao.yonggang|cao.yigang Summary: We report the effects of head loss, surface tension, viscosity and density ratio on the Kelvin-Helmholtz instability (KHI) in two typical pipelines, i.e., straight pipeline with different cross-sections and bend pipeline. The dynamic governing equations for upper and lower fluids in the two pipes are solved analytically. We find in the straight pipeline with different cross-sections that the relative tangential velocity of fluid decreases with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor decreases with the increase of the head loss and surface tension but increases with the density ratio of upper and lower fluids; the higher the height of fluid interface is, the more both the relative tangential velocity of fluid and the amplification factor are depressed. In the bend pipeline, the critical tangential velocity of fluid is found to decrease with the increase of the head loss, viscosity and density ratio of upper and lower fluids, but it increases with the surface tension; the amplification factor increases with the head loss and density ratio of upper and lower fluids, but it decreases with the increase of the surface tension; when the elbow angle is close to \(80^\circ\), the head loss reaches its maximum. The results provide guidance for pipeline design and theoretical prediction for flooding velocity in different types of tubes. Rayleigh-Taylor and Richtmyer-Meshkov instabilities: a journey through scales 2022-09-13T20:28:31.338867Z "Zhou, Ye" "Williams, Robin J. R." "Ramaprabhu, Praveen" "Groom, Michael" "Thornber, Ben" "Hillier, Andrew" "Mostert, Wouter" "Rollin, Bertrand" "Balachandar, S." "Powell, Phillip D." "Mahalov, Alex" "Attal, N." Summary: Hydrodynamic instabilities such as Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities usually appear in conjunction with the Kelvin-Helmholtz (KH) instability and are found in many natural phenomena and engineering applications. They frequently result in turbulent mixing, which has a major impact on the overall flow development and other effective material properties. This can either be a desired outcome, an unwelcome side effect, or just an unavoidable consequence, but must in all cases be characterized in any model. The RT instability occurs at an interface between different fluids, when the light fluid is accelerated into the heavy. The RM instability may be considered a special case of the RT instability, when the acceleration provided is impulsive in nature such as that resulting from a shock wave. In this pedagogical review, we provide an extensive survey of the applications and examples where such instabilities play a central role. First, fundamental aspects of the instabilities are reviewed including the underlying flow physics at different stages of development, followed by an overview of analytical models describing the linear, nonlinear and fully turbulent stages. RT and RM instabilities pose special challenges to numerical modeling, due to the requirement that the sharp interface separating the fluids be captured with fidelity. These challenges are discussed at length here, followed by a summary of the significant progress in recent years in addressing them. Examples of the pivotal roles played by the instabilities in applications are given in the context of solar prominences, ionospheric flows in space, supernovae, inertial fusion and pulsed-power experiments, pulsed detonation engines and Scramjets. Progress in our understanding of special cases of RT/RM instabilities is reviewed, including the effects of material strength, chemical reactions, magnetic fields, as well as the roles the instabilities play in ejecta formation and transport, and explosively expanding flows. The article is addressed to a broad audience, but with particular attention to graduate students and researchers who are interested in the state-of-the-art in our understanding of the instabilities and the unique issues they present in the applications in which they are prominent. Effect of water vorticity on wind-generated gravity waves in finite depth 2022-09-13T20:28:31.338867Z "Abid, Malek" "Kharif, Christian" Summary: The generation of wind waves at the surface of an established underlying vertically sheared water flow, of constant vorticity, is considered. A particular attention is paid to the role of the vorticity in water on wind-wave generation in finite depth. The present theoretical results are compared with experimental data obtained by \textit{I. R. Young} and \textit{L. A. Verhagen} [``The growth of fetch limited waves in water of finite depth. I: Total energy and peak frequency'', Coastal Eng. 29, No. 1--2, 47--78 (1996; \url{doi:10.1016/S0378-3839(96)00006-3})], in the shallow Lake George (Australia), and the least squares fit of these data by \textit{I. R. Young} [``The growth rate of finite depth wind-generated waves'', ibid. 32, No. 2--3, 181--195 (1997; \url{doi:10.1016/S0378-3839(97)81749-8})]. It is shown that without vorticity in water, there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth. Transverse bifurcation of viscous slow MHD shocks 2022-09-13T20:28:31.338867Z "Barker, Blake" "Monteiro, Rafael" "Zumbrun, Kevin" Summary: We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model and setting) an open question of \textit{K. Zumbrun} and \textit{D. Serre} [Indiana Univ. Math. J. 48, No. 3, 937--992 (1999; Zbl 0944.76027)]. Convective, absolute and global azimuthal magnetorotational instabilities 2022-09-13T20:28:31.338867Z "Mishra, A."|mishra.abinash|mishra.arijit|mishra.ankit|mishra.akshat|mishra.arvind-kumar|mishra.anwesha|mishra.awadhesh-kumar|mishra.amardeep|mishra.amrita|mishra.amarendra|mishra.aditya-mani|mishra.amit-kumar|mishra.abhishek-c|mishra.amitabh|mishra.apurva|mishra.aseem-k|mishra.ashok-kumar|mishra.asim-kumar|mishra.arti|mishra.ajit|mishra.akansha|mishra.alok-kumar|mishra.amiya|mishra.aditi|mishra.aurosish|mishra.anju|mishra.apoorva|mishra.anshuman|mishra.arunima|mishra.anurag|mishra.anil-kumar|mishra.asitav|mishra.arabinda|mishra.amruta|mishra.anand-k|mishra.ambuj-kumar|mishra.akash-k|mishra.avdesh|mishra.akshaya-kumar|mishra.ashish|mishra.asmita|mishra.asha-s|mishra.amar-p|mishra.alpna|mishra.arindam|mishra.aashwin-ananda|mishra.ajay-k|mishra.akhilesh-k|mishra.arunodaya-raj "Mamatsashvili, G." "Galindo, V." "Stefani, F." Summary: We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a cylindrical Taylor-Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using direct numerical simulation and find that its emerging butterfly-type structure -- a spatio-temporal variation in the form of axially upward and downward travelling waves -- is in a very good agreement with the linear analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern. Breathers, cascading instabilities and Fermi-Pasta-Ulam-Tsingou recurrence of the derivative nonlinear Schrödinger equation: effects of `self-steepening' nonlinearity 2022-09-13T20:28:31.338867Z "Yin, H. M."|yin.huiming|yin.hui-min "Chow, K. W."|chow.ka-wing|chow.kong-wing Summary: Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrödinger equation, which incorporates second order dispersion, cubic nonlinearity and self-steepening effect. By insisting on periodic boundary conditions, a cascading process will occur where initially small higher order Fourier modes can grow alongside with lower order modes. Typically a breather is first observed when all modes attain roughly the same order of magnitude. Beyond the formation of the first breather, analytical formula of spatially periodic but temporally localized breather ceases to be a valid indicator. However, numerical simulations display Fermi-Pasta-Ulam-Tsingou type recurrence. Self-steepening effect plays a crucial role in the dynamics, as it induces motion of the breather and generates chaotic behavior of the Fourier coefficients. Theoretically, correlation between breather motion and the Lax pair formulation is made. Physically, quantitative assessments of wave profile evolution are made for different initial conditions, e.g. random noise versus modulation instability mode of maximum growth rate. Potential application to fluid mechanics is discussed. Nozzle dynamics and wavepackets in turbulent jets 2022-09-13T20:28:31.338867Z "Kaplan, Oğuzhan" "Jordan, Peter" "Cavalieri, André V. G." "Brès, Guillaume A." Summary: We study a turbulent jet issuing from a cylindrical nozzle to characterise coherent structures evolving in the turbulent boundary layer. The analysis is performed using data from a large-eddy simulation of a Mach 0.4 jet. Azimuthal decomposition of the velocity field in the nozzle shows that turbulent kinetic energy predominantly resides in high azimuthal wavenumbers; the first three azimuthal wavenumbers, that are important for sound generation, contain much lower, but non-zero amplitudes. Using two-point statistics, low azimuthal modes in the nozzle boundary layer are shown to exhibit significant correlations with modes of the same order in the free-jet region. Spectral proper orthogonal decomposition is used to distill a low-rank approximation of the flow dynamics. This reveals the existence of tilted coherent structures within the nozzle boundary layer and shows that these are coupled with wavepackets in the jet. The educed nozzle boundary-layer structures are modelled using a global resolvent analysis of the mean flow inside the nozzle to determine the most amplified flow responses using the linearised Navier-Stokes system. It is shown that the most-energetic nozzle structures can be successfully described with optimal resolvent response modes, whose associated forcing modes are observed to tilt against the nozzle boundary layer, suggesting that the Orr mechanism underpins these organised, turbulent, boundary-layer structures. On a mechanism of near-wall reverse flow formation in a turbulent duct flow 2022-09-13T20:28:31.338867Z "Zaripov, Dinar" "Ivashchenko, Vladislav" "Mullyadzhanov, Rustam" "Li, Renfu" "Mikheev, Nikolay" "Kähler, Christian J." Summary: We address the issue of the generation mechanism of near-wall reverse flow (NWRF) events in a fully developed turbulent duct flow using direct numerical simulations and particle image velocimetry at a relatively low Reynolds number \(Re_\tau \simeq 200\). The analysis demonstrates the existence of a large-scale high-momentum flow structure originating upstream of a NWRF region. We propose a conceptual model of the NWRF formation and suggest that they are caused by intensive hairpin vortices incipient at the interface between large-scale high- and low-momentum flow regions identified using a conditional averaging procedure. The similarity of a flow topology associated with the NWRF region for \(Re_\tau \simeq 200\) with those for \(Re_\tau \simeq 1000\) [\textit{R. C. Chin} et al., ``Conditionally averaged flow topology about a critical point pair in the skin friction field of pipe flows using direct numerical simulations'', Phys. Rev. Fluids 3, No. 11, Article ID 114607, 13 p. (2018; \url{doi:10.1103/PhysRevFluids.3.114607})] and \(550 \leqslant Re_\tau \leqslant 2000\) [\textit{J. I. Cardesa} et al., J. Fluid Mech. 880, Paper No. R3, 11 p. (2019; Zbl 1430.76291)] indicates the generality of the proposed mechanism. Theoretical and numerical analysis of a simple model derived from compressible turbulence 2022-09-13T20:28:31.338867Z "Gavrilyuk, Sergey" "Hérard, Jean-Marc" "Hurisse, Olivier" "Toufaili, Ali" Summary: Turbulent compressible flows are encountered in many industrial applications, for instance when dealing with combustion or aerodynamics. This paper is dedicated to the study of a simple turbulent model for compressible flows. It is based on the Euler system with an energy equation and turbulence is accounted for with the help of an algebraic closure that impacts the thermodynamical behavior. Thereby, no additional PDE is introduced in the Euler system. First, a detailed study of the model is proposed: hyperbolicity, structure of the waves, nature of the fields, existence and uniqueness of the Riemann problem. Then, numerical simulations are proposed on the basis of existing finite-volume schemes. These simulations allow to perform verification test cases and more realistic explosion-like test cases with regards to the turbulence level. Mixing and combustion in a laminar shear layer with imposed counterflow 2022-09-13T20:28:31.338867Z "Sirignano, William A." Summary: Three-dimensional, steady laminar flow structures with mixing, chemical reaction, normal strain and shear strain representative of turbulent combustion are analysed. A mixing layer is subjected to counterflow in the transverse \(y\)- and \(z\)-directions providing the important practical interaction of shear-strain rate with normal-strain rate. Larger consequences for mixing rates and burning rates occur than would appear with shear strain or normal strain alone. The three characteristic times for chemical reaction, normal strain and shear strain are cast through two ratios: a Damköhler number based on rate of shear strain and a ratio of rate of normal strain to rate of shear strain. Reduction to a one-dimensional similar form is obtained with density and property variations. A generalization is found extending the Crocco integral for non-unitary Prandtl number and for imposed normal strain. A diffusion flamelet model with combined shear and normal strains is developed. Another similar solution is obtained for a configuration with a dominant diffusion flame and a weaker fuel-rich premixed flame. A conserved scalar is cast as the independent variable giving an alternative description. The imposed normal strain decreases mixing-layer thickness and increases scalar gradients and transport rates. Diffusion control is possible for partially premixed flames in the multi-branched flame situation. The imposition of shear strain and thereby vorticity on the counterflow can have a substantial consequence, indicating the need for flamelet models with both shear strain and normal strain. Evolution equations for the decomposed components of displacement speed in a reactive scalar field 2022-09-13T20:28:31.338867Z "Yu, R." "Nilsson, T." "Fureby, C." "Lipatnikov, A. N." Summary: The study of a turbulent premixed flame often involves analysing quantities conditioned to different iso-surfaces of a reactive scalar field. Under the influence of turbulence, such a surface is deformed and translated. To track the surface motion, the displacement speed (\(S_d\)) of the scalar field respective to the local flow velocity is widely used and this quantity is currently receiving growing attention. Inspired by the apparent benefits from a simple decomposition of \(S_d\) into contributions due to (i) curvature, (ii) normal diffusion and (iii) chemical reaction, this work aims at deriving and exploring new evolution equations for these three contributions averaged over the reaction surface. Together with a previously obtained \(S_d\)-evolution equation, the three new equations are presented in a form that emphasizes the decomposition of \(S_d\) into three terms. This set of equations is also supplemented with a curvature-evolution equation, hence providing a new perspective to link the flame topology and its propagation characteristics. Using two direct numerical simulation databases obtained from constant-density and variable-density reaction waves, all the derived equations and the term-wise decomposition relations are demonstrated to hold numerically. Comparison of the simulated results indicates that the thermal expansion weakly affects the key terms in the considered evolution equations. Thermal expansion can cause variations in the averaged \(S_d\) and its decomposed parts through multiple routes more than introducing a dilatation term. The flow plays a major role to influence the key terms in all equations except the curvature one, due to a cancellation between negatively and positively curved surface elements. Ensemble gradient for learning turbulence models from indirect observations 2022-09-13T20:28:31.338867Z "Ströfer, Carlos A. Michelén" "Zhang, Xin-Lei" "Xiao, Heng" Summary: Training data-driven turbulence models with high fidelity Reynolds stress can be impractical and recently such models have been trained with velocity and pressure measurements. For gradient-based optimization, such as training deep learning models, this requires evaluating the sensitivities of the RANS equations. This paper explores the use of an ensemble approximation of the sensitivities of the RANS equations in training data-driven turbulence models with indirect observations. A deep neural network representing the turbulence model is trained using the network's gradients obtained by backpropagation and the ensemble approximation of the RANS sensitivities. Different ensemble approximations are explored and a method based on explicit projection onto the sample space is presented. As validation, the gradient approximations from the different methods are compared to that from the continuous adjoint equations. The ensemble approximation is then used to learn different turbulence models from velocity observations. In all cases, the learned model predicts improved velocities. However, it was observed that once the sensitivity of the velocity to the underlying model becomes small, the approximate nature of the ensemble gradient hinders further optimization of the underlying model. The benefits and limitations of the ensemble gradient approximation are discussed, in particular as compared to the adjoint equations. The attachment angle of a sonic line to the streamlined surface 2022-09-13T20:28:31.338867Z "Sizykh, G. B." Summary: This work considers the angle of the sonic-line attachment to the surface in flows with uniform fields of entropy and total enthalpy. A rigorous study of the Euler equations (without the use of asymptotic, numerical, and other approximate methods) is carried out. Plane-parallel and nonswirling axisymmetric flows are considered. It is shown that the attachment angle of the sonic line depends on the curvature of the surface. If the surface is convex towards the flow, then the attachment angle on the subsonic side is strictly greater than \(90 \degree \). If the surface is concave towards the flow, then the attachment angle on the subsonic side is strictly less than \(90 \degree \). The attachment to a straight section of the surface in a plane-parallel flow always occurs along the normal. Similarly, only the attachment along the normal is possible if the sonic line attaches to a straight generatrix parallel to the axis of symmetry in nonswirling axisymmetric flows. For the case when the straight generatrix is not parallel to the axis of symmetry, it is shown that the attachment angle from the sonic side can only be either \(90 \degree \) (attachment along the normal) or \(0 \degree \) (attachment along the tangent). A kinetic shock layer in the spreading plane of a lifting-body apparatus 2022-09-13T20:28:31.338867Z "Ankudinov, A. L." Summary: In this paper, we propose an effective computational mathematical interpretation of the problem of the nonequilibrium flow of a polyatomic gas in a kinetic thin viscous shock layer near a blunt body in the plane of its symmetry. The correlation of flows in the kinetic and Navier-Stokes thin viscous shock layer on the frontal spreading line, which allows constructing the solution of the kinetic problem based entirely on the Navier-Stokes equations, is indicated. Using the proposed approach, heat transfer on the wall along the entire spreading line of a model of an aerospace aircraft of lifting-body type was numerically studied. The calculation results are compared with the data of the tunnel experiment. Transition study for asymmetric reflection between moving incident shock waves 2022-09-13T20:28:31.338867Z "Wang, Miao-Miao" "Wu, Zi-Niu" Summary: The transition criteria seen from the ground frame are studied in this paper for asymmetrical reflection between shock waves moving at constant linear speed. To limit the size of the parameter space, these criteria are considered in detail for the reduced problem where the upper incident shock wave is moving and the lower one is steady, and a method is provided for extension to the general problem where both the upper and lower ones are unsteady. For the reduced problem, we observe that, in the shock angle plane, shock motion lowers or elevates the von Neumann condition in a global way depending on the direction of shock motion, and this change becomes less important for large shock angle. The effect of shock motion on the detachment condition, though small, displays non-monotonicity. The shock motion changes the transition criteria through altering the effective Mach number and shock angle, and these effects add for small shock angle and mutually cancel for large shock angle, so that shock motion has a less important effect for large shock angle. The role of the effective shock angle is not monotonic on the detachment condition, explaining the observed non-monotonicity for the role of shock motion on the detachment condition. Furthermore, it is found that the detachment condition has a wavefunction form that can be approximated as a hybrid of a sinusoidal function and a linear function of the shock angle. Structure-preserving discretization of a coupled heat-wave system, as interconnected port-Hamiltonian systems 2022-09-13T20:28:31.338867Z "Haine, Ghislain" "Matignon, Denis" Summary: The heat-wave system is recast as the coupling of port-Hamiltonian subsystems (pHs), and discretized in a structure-preserving way by the partitioned finite element method (PFEM) [\textit{F. L. Cardoso-Ribeiro} et al., IMA J. Math. Control Inf. 38, No. 2, 493--533 (2021; Zbl 1475.93051); ``A structure-preserving Partitioned Finite Element Method for the 2D wave equation'', IFAC-PapersOnLine 51, No. 3, 119--124 (2018; \url{doi:10.1016/j.ifacol.2018.06.033})]. Then, depending on the geometric configuration of the two domains, different asymptotic behaviours of the energy of the coupled system can be recovered at the numerical level, assessing the validity of the theoretical results of \textit{X. Zhang} and \textit{E. Zuazua} [Arch. Ration. Mech. Anal. 184, No. 1, 49--120 (2007; Zbl 1178.74075)]. For the entire collection see [Zbl 1482.94007]. The macroelement analysis for axisymmetric Stokes equations 2022-09-13T20:28:31.338867Z "Lee, Young-Ju" "Li, Hengguang" Summary: We consider the mixed finite element approximation of the axisymmetric Stokes problem (ASP) on a bounded polygonal domain in the \(rz\)-plane. Standard stability results on mixed methods do not apply due to the singular coefficients in the differential operator and due to the singular or vanishing weights in the associated function spaces. We develop new finite element analysis in these weighted spaces, and propose macroelement conditions that are sufficient to ensure the well-posedness of the mixed methods for the ASP. These conditions are local, relatively easy to verify, and therefore will be useful for validating the stability of a variety of mixed finite element methods. These new conditions can not only re-verify existing stable mixed methods for the ASP, but also lead to the discovery of new stable conservative mixed methods. We report numerical test results that confirm the theory. A study of several artificial viscosity models within the discontinuous Galerkin framework 2022-09-13T20:28:31.338867Z "Yu, Jian" "Hesthaven, Jan S." Summary: Dealing with strong shocks while retaining low numerical dissipation traditionally has been one of the major challenges for high order methods like discontinuous Galerkin (DG). In the literature, shock capturing models have been designed for DG based on various approaches, such as slope limiting, (H)WENO reconstruction, a posteriori sub-cell limiting, and artificial viscosity, among which a subclass of artificial viscosity methods are compared in the present work. Four models are evaluated, including a dilation-based model, a highest modal decay model, an averaged modal decay model, and an entropy viscosity model. Performance for smooth, non-smooth and broadband problems are examined with typical one- and two-dimensional cases. Evaluation of selected finite-difference and finite-volume approaches to rotational shallow-water flow 2022-09-13T20:28:31.338867Z "Holm, Håvard H." "Brodtkorb, André R." "Broström, Göran" "Christensen, Kai H." "Sætra, Martin L." Summary: The shallow-water equations in a rotating frame of reference are important for capturing geophysical flows in the ocean. In this paper, we examine and compare two traditional finite-difference schemes and two modern finite-volume schemes for simulating these equations. We evaluate how well they capture the relevant physics for problems such as storm surge and drift trajectory modelling, and the schemes are put through a set of six test cases. The results are presented in a systematic manner through several tables, and we compare the qualitative and quantitative performance from a cost-benefit perspective. Of the four schemes, one of the traditional finite-difference schemes performs best in cases dominated by geostrophic balance, and one of the modern finite-volume schemes is superior for capturing gravity-driven motion. The traditional finite-difference schemes are significantly faster computationally than the modern finite-volume schemes. A consistent and conservative phase-field method for multiphase incompressible flows 2022-09-13T20:28:31.338867Z "Huang, Ziyang" "Lin, Guang" "Ardekani, Arezoo M." Summary: In the present study, a consistent and conservative Phase-Field method, including both the model and scheme, is developed for multiphase flows with an arbitrary number of immiscible and incompressible fluid phases. The \textit{consistency of mass conservation} and the \textit{consistency of mass and momentum transport} are implemented to address the issue of physically coupling the Phase-Field equation, which locates different phases, to the hydrodynamics. These two consistency conditions, as illustrated, provide the ``optimal'' coupling because (i) the new momentum equation resulting from them is Galilean invariant and implies the kinetic energy conservation, regardless of the details of the Phase-Field equation, and (ii) failures of satisfying the second law of thermodynamics or the \textit{consistency of reduction} of the multiphase flow model only result from the same failures of the Phase-Field equation but are not due to the new momentum equation. Physical interpretation of the consistency conditions and their formulations are first provided, and general formulations that are obtained from the consistency conditions and independent of the interpretation of the velocity are summarized. Then, the present consistent and conservative multiphase flow model is completed by selecting a reduction consistent Phase-Field equation. Several novel techniques are developed to inherit the physical properties of the multiphase flows after discretization, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, and the general theorems to preserve the consistency conditions on the discrete level. Equipped with those novel techniques, a consistent and conservative scheme for the present multiphase flow model is developed and analyzed. The scheme satisfies the consistency conditions, conserves the mass and momentum, and assures the summation of the volume fractions to be unity, on the fully discrete level and for an arbitrary number of phases. All those properties are numerically validated. Numerical applications demonstrate that the present model and scheme are robust and effective in studying complicated multiphase dynamics, especially for those with large-density ratios. Numerical solver for the Boltzmann equation with self-adaptive collision operators 2022-09-13T20:28:31.338867Z "Cai, Zhenning" "Wang, Yanli" Efficient spectral method for stable stratified power-law fluid flows with dispersion over convectively heated truncated cone in a non-Darcy porous medium 2022-09-13T20:28:31.338867Z "RamReddy, Ch."| "Srivastav, Abhinava" Summary: This problem deals with the power-law fluid flow with thermally stable stratification in a non-Darcy porous medium over a convectively heated truncated cone and this work is very useful in actual and applied circumstances due to presence of non-linear Boussinesq approximation. The combined thermal diffusivity is taken as the addition of molecular diffusivity and diffusivity related to mechanical dispersion. Local non-similarity technique and spectral local linearization method are applied to solve the governing equations. A convergence test for this scheme is performed and validation of methodology is given by comparing the results in special cases with already established results. It is noted that the proposed combined scheme is an efficient algorithm with faster convergence and it acts as an alternative tool for regular numerical techniques to solve non-linear boundary value problems that occur frequently in industrial and engineering applications. The major conclusion of this study is that the magnitude of skin friction coefficient and Nusselt number are very much influenced with the presence and absence of Biot number, thermal stratification and thermal dispersion for the power-law fluids and strongly depend on the non-linear density temperature parameter. A hybrid immersed boundary-lattice Boltzmann method for simulation of viscoelastic fluid flows interaction with complex boundaries 2022-09-13T20:28:31.338867Z "Sedaghat, M. H."|sedaghat.maral-khadem "Bagheri, A. A. H." "Shahmardan, M. M." "Norouzi, M." "Khoo, B. C." "Jayathilake, P. G." Summary: In this study, a numerical technique based on the Lattice Boltzmann method is presented to model viscoelastic fluid interaction with complex boundaries which are commonly seen in biological systems and industrial practices. In order to accomplish numerical simulation of viscoelastic fluid flows, the Newtonian part of the momentum equations is solved by the Lattice Boltzmann Method (LBM) and the divergence of the elastic tensor, which is solved by the finite difference method, is added as a force term to the governing equations. The fluid-structure interaction forces are implemented through the Immersed Boundary Method (IBM). The numerical approach is validated for Newtonian and viscoelastic fluid flows in a straight channel, a four-roll mill geometry as well as flow over a stationary and rotating circular cylinder. Then, a numerical simulation of Oldroyd-B fluid flow around a confined elliptical cylinder with different aspect ratios is carried out for the first time. Finally, the present numerical approach is used to simulate a biological problem which is the mucociliary transport process of human respiratory system. The present numerical results are compared with appropriate analytical, numerical and experimental results obtained from the literature. Bayesian learning of stochastic dynamical models 2022-09-13T20:28:31.338867Z "Lu, Peter" "Lermusiaux, Pierre F. J." Summary: A new methodology for rigorous Bayesian learning of high-dimensional stochastic dynamical models is developed. The methodology performs parallelized computation of marginal likelihoods for multiple candidate models, integrating over all state variable and parameter values, and enabling a principled Bayesian update of model distributions. This is accomplished by leveraging the dynamically orthogonal (DO) evolution equations for uncertainty prediction in a dynamic stochastic subspace and the Gaussian Mixture Model-DO filter for inference of nonlinear state variables and parameters, using reduced-dimension state augmentation to accommodate models featuring uncertain parameters. Overall, the joint Bayesian inference of the state, model equations, geometry, boundary conditions, and initial conditions is performed. Results are exemplified using two high-dimensional, nonlinear simulated fluid and ocean systems. For the first, limited measurements of fluid flow downstream of an obstacle are used to perform joint inference of the obstacle's shape, the Reynolds number, and the \(\mathcal{O}(10^5)\) fluid velocity state variables. For the second, limited measurements of the concentration of a microorganism advected by an uncertain flow are used to perform joint inference of the microorganism's reaction equation and the \(\mathcal{O}(10^5)\) microorganism concentration and ocean velocity state variables. When the observations are sufficiently informative about the learning objectives, we find that our posterior model probabilities correctly identify either the true model or the most plausible models, even in cases where a human would be challenged to do the same. A fast convergent semi-analytic method for an electrohydrodynamic flow in a circular cylindrical conduit 2022-09-13T20:28:31.338867Z "Abukhaled, Marwan" "Khuri, S. A." Summary: A semi-analytical solution of the nonlinear boundary value problem that models the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit is presented. An integral operator expressed in terms of Green's function is constructed then followed by an application of fixed point theory to generate a highly accurate semi-analytical expression of the fluid velocity for all possible values of relevant parameters. A proof of convergence for the proposed method, based on the contraction mapping principle, is presented. Numerical simulations and comparison with other analytical methods confirm that the proposed approach is convergent, stable, and highly accurate. Multisymplectic variational integrators for fluid models with constraints 2022-09-13T20:28:31.338867Z "Demoures, François" "Gay-Balmaz, François" Summary: We present a structure preserving discretization of the fundamental spacetime geometric structures of fluid mechanics in the Lagrangian description in 2D and 3D. Based on this, multisymplectic variational integrators are developed for barotropic and incompressible fluid models, which satisfy a discrete version of Noether theorem. We show how the geometric integrator can handle regular fluid motion in vacuum with free boundaries and constraints such as the impact against an obstacle of a fluid flowing on a surface. Our approach is applicable to a wide range of models including the Boussinesq and shallow water models, by appropriate choice of the Lagrangian. For the entire collection see [Zbl 1482.94007]. Metriplectic integrators for dissipative fluids 2022-09-13T20:28:31.338867Z "Kraus, Michael" Summary: Many systems from fluid dynamics and plasma physics possess a so-called metriplectic structure, that is the equations are comprised of a conservative, Hamiltonian part, and a dissipative, metric part. Consequences of this structure are conservation of important quantities, such as mass, momentum and energy, and compatibility with the laws of thermodynamics, e.g., monotonic dissipation of entropy and existence of a unique equilibrium state. For simulations of such systems to deliver accurate and physically correct results, it is important to preserve these relations and conservation laws in the course of discretisation. This can be achieved most easily not by enforcing these properties directly, but by preserving the underlying abstract mathematical structure of the equations, namely their metriplectic structure. From that, the conservation of the aforementioned desirable properties follows automatically. This paper describes a general and flexible framework for the construction of such metriplectic structure-preserving integrators, that facilitates the design of novel numerical methods for systems from fluid dynamics and plasma physics. For the entire collection see [Zbl 1482.94007]. Off-grid DOA estimation based on alternating iterative weighted least squares for acoustic vector hydrophone array 2022-09-13T20:28:31.338867Z "Wang, Weidong" "Zhang, Qunfei" "Shi, Wentao" "Tan, Weijie" "Mao, Linlin" Summary: In this paper, an alternating iterative weighted least squares method is proposed to handle the off-grid issue in sparsity-based direction of arrival (DOA) estimation for acoustic vector hydrophone (AVH) array. Firstly, the off-grid model via AVH array is formulated by introducing a bias parameter into the signal model. Secondly, the reconstructed interference plus noise covariance matrix is calculated as the weighting term. Then, a novel objective function with respect to the sparse signal and the unknown bias parameter is developed based on weighted least squares. Finally, the closed-form solutions of the sparse signal and the unknown bias parameter are deduced. Simulation results reveal that compared with the state-of-the-art algorithms, the proposed method improves the DOA estimation accuracy in the presence of a coarse sample grid and has a faster convergence speed. Furthermore, the effectiveness and robustness of the proposed method are verified by the underwater experimental results. An adaptive virtual element method for incompressible flow 2022-09-13T20:28:31.338867Z "Wang, Ying"|wang.ying.2|wang.ying.8|wang.ying.4|wang.ying.5|wang.ying.6|wang.ying.3|wang.ying.1 "Wang, Gang"|wang.gang.4|wang.gang.1|wang.gang.3|wang.gang|wang.gang.2 "Wang, Feng"|wang.feng.4|wang.feng.1|wang.feng.2 Summary: In this paper, we firstly present and analyze a residual-type a posteriori error estimator for a low-order virtual element discretization for the Stokes problem on general polygonal meshes. We prove that this estimator yields globally upper and locally lower bounds for the discretization error. Then, we extend the estimator to the Navier-Stokes problem. In order to deal with the case of small viscosity, we modify the discrete bilinear form following the idea of variational multiscale method. Since the virtual element method naturally handles hanging nodes, the mesh refinement can exploit them without any local refinement to recover mesh conformity. A series of benchmark tests are reported to verify the effectiveness and flexibility of the designed error estimator when it is combined with adaptive mesh refinement. A Cartesian-to-curvilinear coordinate transformation in modified ghost fluid method for compressible multi-material flows 2022-09-13T20:28:31.338867Z "Xu, Liang" "Lou, Hao" "Yang, Wubing" "Liu, Tiegang" Summary: Modified ghost fluid method (MGFM) provides us an effective manner to simulate compressible multi-material flows. In most cases, the applications are limited in relatively simple geometries described by Cartesian grids. In this paper, the MGFM treatment with the level set (LS) technique is extended to curvilinear coordinate systems. The chain rule of differentiation (applicable to general curvilinear coordinates) and the orthogonal transformation (applicable to orthogonal curvilinear coordinates) are utilized to deduce the Cartesian-to-curvilinear coordinate transformation, respectively. The relationship between these two transformations for the extension of the LS/MGFM algorithm is analyzed in theory. It is shown that these two transformations are equivalent for orthogonal curvilinear grids. The extension of the LS/MGFM algorithm using the chain rule has a wider range of applications, as there is essentially no requirement for the orthogonality of the grids. Several challenging problems in two- or three-dimensions are utilized to validate the developed algorithm in curvilinear coordinates. The results indicate that this algorithm enables a simple and effective implementation for simulating interface evolutions, as in Cartesian coordinate systems. It has the potential to be applied in more complex computational domains. Maximal regularity for compressible two-fluid system 2022-09-13T20:28:31.338867Z "Piasecki, Tomasz" "Zatorska, Ewelina" The authors studied a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure. They showed that regular solutions in a \(L^p-L^q\) maximal regularity setting exist both locally and globally in time, under additional smallness assumptions on the initial data. The interesting proof relies on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated linear problem. Reviewer: Teng Wang (Beijing) Global strong solution to the Cauchy problem of 1D viscous two-fluid model without any domination condition 2022-09-13T20:28:31.338867Z "Gao, Xiaona" "Guo, Zhenhua" "Li, Zilai" Summary: In this paper, we consider the Cauchy problem to the compressible two-fluid Navier-Stokes equations in one-dimensional space allowing vacuum. It is shown that the compressible two-fluid Navier-Stokes equations admit global strong solution with the large initial value and no the domination condition1 which was posed in [\textit{A. Vasseur} et al., J. Math. Pures Appl. (9) 125, 247--282 (2019; Zbl 1450.76033)], when the initial vacuum can be permitted inside the region. Note that this result is proved without any smallness conditions on the initial value. Homogenization of the evolutionary compressible Navier-Stokes-Fourier system in domains with tiny holes 2022-09-13T20:28:31.338867Z "Pokorný, Milan" "Skříšovský, Emil" Summary: We study the homogenization of the evolutionary compressible Navier-Stokes-Fourier system in a bounded three-dimensional domain perforated with a large number of very tiny holes. We show that under suitable assumptions on the smallness and distribution of the holes the limit system remains the same in the unperforated domain. One of the main novelty in the paper consists in the treatment of the entropy inequality and thus the paper also improves the related result in the steady case from \textit{Y. Lu} and \textit{M. Pokorný} [J. Differ. Equations 278, 463--492 (2021; Zbl 1458.35043)]. A simplified lumped parameter model for pneumatic tubes 2022-09-13T20:28:31.338867Z "Kamiński, Zbigniew" Summary: Tubes are commonly used in pneumatic systems for transferring energy and control signals. Using the control volume method, a mathematical tube model has been developed, which takes into account the effect of resistance, capacitance and inertance on the dynamic properties of control and supply circuits of pneumatic systems. The adequacy of the computer model developed in Matlab/Simulink was verified by comparing the results of simulation studies with the results of experimental tests of airflow through tubes of varying diameter and length. The advantage of the computer model is the capability to model pneumatic systems under varying conditions of heat exchange with the environment by changing the coefficient of the polytropic process coefficient. Physical modelling of a long pneumatic transmission line: models of successively decreasing complexity and their experimental validation 2022-09-13T20:28:31.338867Z "Kern, Richard" Summary: There exist a significant number of models, which describe the dynamics of pneumatic transmission lines. The models are based on different assumptions and, thereby, vary in the physical phenomena they incorporate. These assumptions made are not always stated clearly and the models are rarely validated with measurement data. The aim of this article is to present multiple distributed parameter models that, starting from a physical system description, successively decrease in complexity and finally result in a rather simple system representation. Data, both from simulation studies as well as from a pneumatic test bench, serve as a quantitative validation of these assumptions. Based on a detailed discussion of the different models, this article aims at facilitating the choice of an appropriate model for a given task where the effect of long pneumatic transmission lines cannot be neglected and a trade-off between accuracy and complexity is required. Accuracy of a low Mach number model for time-harmonic acoustics 2022-09-13T20:28:31.338867Z "Mercier, J.-F." A method to enhance the noise robustness of correlation velocity measurement using discrete wavelet transform 2022-09-13T20:28:31.338867Z "Son, Pong-Chol" "Kim, Kyong-Il" "Choe, Kyong-Chol" "Kye, Hyok-Il" Correlation velocity measurement techniques are efficiently used to measure the velocity of underwater vehicles, and the maximum correlation coefficient is an important parameter of correlation velocity measurement. In this paper, the relationship between the signal-to-noise ratio (SNR) of the received signal and the maximum value of the correlation matrix is considered, and the maximum correlation coefficient equation according to SNR is presented. Especially, a wavelet thresholding denoising method is successfully used to improve SNR so that the noise robustness of correlation velocity measurement is proposed, and the modified maximum correlation coefficient equation according to SNR is provided accordingly. Simulation results show that the new method of correlation velocity measurement using wavelet thresholding could get the largest maximum correlation coefficients according to SNR steadily compared with the classical method. In particular, the performance improvements in correlation velocity log (CVL) operating under low SNR below 6 dB are more significant. Reviewer: Yankui Sun (Beijing) Mixed convection fluid flow over a vertical cone saturated porous media with double dispersion and injection/suction effects 2022-09-13T20:28:31.338867Z "Meena, Om Prakash" "Janapatla, Pranitha" "Srinivasacharya, D."|srinivasacharya.darbhasayanam Summary: This study reflects the combined impact of double dispersion and injection/suction on mixed convection flow over a vertical cone in an incompressible viscous fluid-saturated porous medium. The governing equations of the model are non-dimensionalized throughout the appropriate transformations and received non-similarity equations are solved numerically via bivariate Chebyshev spectral collocation quasi-linearization method. Computations are reported here graphically to analyze the impact of governing parameters at the different stream-wise locations on the velocity, temperatures, and concentration profiles, like Prandtl number, Schmidt number, buoyancy parameter, injection and suction parameter, thermal dispersion, and Solutal dispersion parameters. Skin friction, heat, and mass transfer rates are also reported in graphical and tabular form. To establish the efficiency of the adopted numerical technique, we have made a comparison with the earlier published results and found them to be of great consent. The residual analysis study also illustrated, which proves the convergence of the present results. A route to chaos in Rayleigh-Bénard heat convection 2022-09-13T20:28:31.338867Z "Hsia, Chun-Hsiung" "Nishida, Takaaki" Summary: We use numerical methods to study the global bifurcation diagrams of the Bénard convection problem. In our computations, we include a huge number of Fourier modes of stream function and temperature function so that our results reflect more reality of the dynamics of the Rayleigh-Bénard heat convection. Our results confirm that the period doubling scenario is a route to chaos. Thermophoresis on free convective unsteady/steady Couette fluid flow with mass transfer 2022-09-13T20:28:31.338867Z "Jha, Basant K." "Sani, Hadiza N." Summary: This article reports analytical as well as numerical solutions for fully developed unsteady natural convection in a Couette flow with transfer of mass due to thermophoresis. The time dependent model describes a physical situation which is solved using a finite difference Scheme that is implicit, backward in time and centered in space. The steady state version of the present physical situation has been solved exactly. The influence of controlling parameters on dimensionless velocity, concentration, skin-frictions and Sherwood number are demonstrated through graphs and tables. Graphical results show that Schmidt number should be greater than 0.6 for thermophoresis to be effective in air. The numerical results reveal that the ratio of convective mass transfer to diffusive mass transfer increases with increases in thermophoresis coefficient and time until it finally reaches its steady-state value. Effect of non-inertial acceleration on Brinkman-Bénard convection in water-copper nanoliquid-saturated porous enclosures 2022-09-13T20:28:31.338867Z "Siddheshwar, P. G." "Veena, B. N." Summary: In the present paper we have considered rotating porous tall, square and shallow enclosures heated from below. Linear and non-linear analyses are made using a minimal representation by Fourier trigonometric series. The study is done for realistic boundary condition. Thermophysical properties of water-copper nanoliquid as a function of properties of water as base liquid, copper as nanoparticle and 30\% glass fiber reinforced polycarbonate as porous medium are obtained from either phenomenological laws or mixture theory. Non-existence of oscillatory convection is discussed. The range for the existence of unicellular convection is mentioned. The effects of Brinkman number (\( \varLambda \)), porous parameter (\( \sigma^2\)), aspect ratio (\(A\)) and volume fraction (\( \chi \)) in the presence of rotation on the onset of convection and heat transfer are studied and illustrated graphically. The analytically intractable Lorenz model is derived and transformed into the tractable Ginzburg-Landau equation using the multiscales method. The definition of Ozoe heat transfer parameter is introduced to discuss the rate of heat transfer enhancement or reduction. It is observed that \(Ta\), \( \varLambda\) and \(\sigma^2\) have stabilizing effect on the system and thereby leading to diminished heat transfer whereas \(A\) and \(\chi\) have destabilizing effect on the system and thereby leading to increased heat transfer. Among the three enclosures considered in the study enhanced heat transfer takes place in tall enclosure followed by square and shallow enclosures respectively. It is further observed that presence of nanoparticles advances the onset of convection and enhances the heat transfer. The results of the paper are compared with previous existing results in the absence of rotation and the good agreement is found between them. Influence of electroosmosis mechanism and chemical reaction on convective flow over an exponentially accelerated plate 2022-09-13T20:28:31.338867Z "Vijayaragavan, R." "Bharathi, V."|bharathi.v-vijaya "Prakash, J."|prakash.j-ravi|prakash.jitendra|prakash.j-s|prakash.jyoti|prakash.jai Summary: This article is primarily attained to study the electric double layer (EDL) phenomena and chemical reaction effects on unsteady natural convection flow through exponentially accelerated plate. The Poisson Boltzmann equation is used to derive the electroosmosis mechanism. The special effect of Lorentz and Darcy forces are considered in the proposed mathematical model. Governing equations of proposed model is linearized through Debye-Hückel linearization and dimensionless analysis. The system of nonlinear partial differential equations are solved with the help of Laplace transform technique. Further, the Nusselt number and Sherwood number are also derived. The graphical outcomes for velocity, temperature, concentration, Nusselt number and Sherwood number are illustrated with the help of Matlab software. Validation of the present solution is obtained by Laplace transform method which is compared with numerical solution obtained by finite difference with the help of MATLAB code. It is seen that the velocity profile is unequivocally reliance with attractive field and EDL thickness. It is also found that chemical reaction parameter significantly pretends on temperature distribution. This idea can be equipped for being applied in different complex frameworks where the electroosmosis stream can be moved by CPUs gadget. Persisting asymmetry in the probability distribution function for a random advection-diffusion equation in impermeable channels 2022-09-13T20:28:31.338867Z "Camassa, Roberto" "Ding, Lingyun" "Kilic, Zeliha" "McLaughlin, Richard M." Summary: In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein-Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies [\textit{R. Camassa} et al., Physica D 400, Article ID 132124, 32 p. (2019; Zbl 1453.60116)] numerically demonstrated that the decaying passive scalar's long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed [\textit{R. M. McLaughlin} and \textit{A. J. Majda}, Phys. Fluids 8, No. 2, 536--547 (1996; Zbl 1023.76560)]. Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the \(N\)-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in [\textit{J. C. Bronski} and \textit{R. M. McLaughlin}, Phys. Fluids 9, No. 1, 181--190 (1997; Zbl 1185.76678)]. Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in [Camassa et al., loc. cit.]. Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein-Uhlenbeck case to the white noise case in the limit \(\gamma \to \infty\) of the OU damping parameter, which generalizes the results for free space in [\textit{S. G. Resnick}, Dynamical problems in non-linear advective partial differential equations, The University of Chicago (PhD Thesis) (1996)] to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of \(\gamma\), which is in contrast to the conclusion in [\textit{E. Vanden Eijnden}, Commun. Pure Appl. Math. 54, No. 9, 1146--1167 (2001; Zbl 1036.76025)] for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the \(N\)-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of \textit{J. C. Bronski} et al. [J. Stat. Phys. 128, No. 4, 927--968 (2007; Zbl 1185.76693)]. The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations. Various formulations and approximations of incompressible fluid motions in porous media 2022-09-13T20:28:31.338867Z "Brenier, Yann" Summary: We first recall various formulations and approximations for the motion of an incompressible fluid, in the well-known setting of the Euler equations. Then, we address incompressible motions in porous media, through the Muskat system, which is a friction dominated first order analog of the Euler equations for inhomogeneous incompressible fluids subject to an external potential. The combined effects of wall properties and space porosity on MHD two-phase peristaltic slip transport through planar channels 2022-09-13T20:28:31.338867Z "Eldesoky, I. M." "Abumandour, R. M." "Kamel, M. H." "Abdelwahab, E. T." Summary: In this article, a theoretical investigation is analyzing the effects of the complaint wall properties, the slip conditions, the space porosity, and the transverse magnetic field on the magnetohydrodynamic peristaltic transport of viscous compressible flow carrying out some rigid spherical suspension particles flowing through space porous medium in a horizontal elastic rectangular channel. The flexible channel walls are taken as a sinusoidal wave. The expressions describing the peristaltic transport are mathematically analyzed using the perturbation technique with a small amplitude wave ratio. The analytical study describes the influence of various wall parameters such as damping force, wall tension, and wall elasticity and flow parameters as compressibility parameter, slip parameter, suspension parameter, Reynolds number, space porosity, and magnetic field parameter on the net axial velocity. The reversal flow occurs at the channel core and boundaries due to the slip and the magnetic field effects. Biological, geophysical, and industrial fluid dynamics applications are important models for the peristaltic transport described in this work. Coupled hydro-mechanical modeling of gas flow in shale matrix considering the fractal characteristics of nanopores 2022-09-13T20:28:31.338867Z "Gao, Qi" "Cheng, Yuanfang" "Han, Songcai" "Li, Yang"|li.yang.8|li.yang.6|li.yang.7 "Yan, Chuanliang" "Han, Zhongying" On the influence of state selection on mass conservation in dynamic vapour compression cycle models 2022-09-13T20:28:31.338867Z "Laughman, Christopher R." "Qiao, Hongtao" Summary: Many dynamic models of vapour compression systems experience nonphysical variations in the total refrigerant mass contained in the system when common modelling approaches are used. Rather than using the traditional state variables of pressure and specific enthalpy, the use of density as a state variable can eliminate these variations. The reasons for these variations are explained, and a set of test models is developed to study the effect of the state variable selection on the overall system charge. Results from both a simplified cycle model and a realistic air-source heat pump model indicate that this alternative approach has significant benefits for maintaining a fixed mass of refrigerant in the cycle. Fundamental fluid dynamics challenges in inkjet printing 2022-09-13T20:28:31.338867Z "Lohse, Detlef" Summary: Inkjet printing is the most widespread technological application of microfluidics. It is characterized by its high drop productivity, small volumes, and extreme reproducibility. This review gives a synopsis of the fluid dynamics of inkjet printing and discusses the main challenges for present and future research. These lie both on the printhead side -- namely, the detailed flow inside the printhead, entrained bubbles, the meniscus dynamics, wetting phenomena at the nozzle plate, and jet formation -- and on the receiving substrate side -- namely, droplet impact, merging, wetting of the substrate, droplet evaporation, and drying. In most cases the droplets are multicomponent, displaying rich physicochemical hydrodynamic phenomena. The challenges on the printhead side and on the receiving substrate side are interwoven, as optimizing the process and the materials with respect to either side alone is not enough: As the same ink (or other jetted liquid) is used and as droplet frequency and size matter on both sides, the process must be optimized as a whole. For the entire collection see [Zbl 1489.76002]. On the shape of air-liquid interfaces with surface tension that bound rigidly rotating liquids in partially filled containers 2022-09-13T20:28:31.338867Z "Ramé, Enrique" "Weinstein, Steven J." "Barlow, Nathaniel S." Summary: The interface shape of a fluid in rigid body rotation about its axis and partially filling the container is often the subject of a homework problem in the first graduate fluids class. In that problem, surface tension is neglected, the interface shape is parabolic and the contact angle boundary condition is not satisfied in general. When surface tension is accounted for, the shapes exhibit much richer dependencies as a function of rotation velocity. We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. We pay special attention to shapes near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation at certain specific rotational speeds. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross-section and a meniscus forming a bubble. In each case, we develop exact solutions for the respective axial lengths as infinite series in powers of appropriate rotation parameters, and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally, we apply the method of asymptotic approximants to yield analytical expressions for the axial lengths of the menisci over the whole range of rotation speeds. In this application, the analytical solution is employed to examine errors introduced by the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension. The retraction of jetted slender viscoelastic liquid filaments 2022-09-13T20:28:31.338867Z "Sen, Uddalok" "Datt, Charu" "Segers, Tim" "Wijshoff, Herman" "Snoeijer, Jacco H." "Versluis, Michel" "Lohse, Detlef" Summary: Long and slender liquid filaments are produced during inkjet printing, which can subsequently either retract to form a single droplet, or break up to form a primary droplet and one or more satellite droplets. These satellite droplets are undesirable since they degrade the quality and reproducibility of the print, and lead to contamination within the enclosure of the print device. Existing strategies for the suppression of satellite droplet formation include, among others, adding viscoelasticity to the ink. In the present work, we aim to improve the understanding of the role of viscoelasticity in suppressing satellite droplets in inkjet printing. We demonstrate that very dilute viscoelastic aqueous solutions (concentrations \(\sim\) 0.003 \% wt. polyethylene oxide, corresponding to nozzle Deborah number \(De_n\sim 3\)) can suppress satellite droplet formation. Furthermore, we show that, for a given driving condition, upper and lower bounds of polymer concentration exist, within which satellite droplets are suppressed. Satellite droplets are formed at concentrations below the lower bound, while jetting ceases for concentrations above the upper bound (for fixed driving conditions). Moreover, we observe that, with concentrations in between the two bounds, the filaments retract at velocities larger than the corresponding Taylor-Culick velocity for the Newtonian case. We show that this enhanced retraction velocity can be attributed to the elastic tension due to polymer stretching, which builds up during the initial jetting phase. These results shed some light on the complex interplay between inertia, capillarity and viscoelasticity for retracting liquid filaments, which is important for the stability and quality of inkjet printing of polymer solutions. Continuum and molecular dynamics studies of the hydrodynamics of colloids straddling a fluid interface 2022-09-13T20:28:31.338867Z "Maldarelli, Charles" "Donovan, Nicole T." "Ganesh, Subramaniam Chembai" "Das, Subhabrata" "Koplik, Joel" Summary: Colloid-sized particles (10 nm-\(10 \mu\) m in characteristic size) adsorb onto fluid interfaces, where they minimize their interfacial energy by straddling the surface, immersing themselves partly in each phase bounding the interface. The energy minimum achieved by relocation to the surface can be orders of magnitude greater than the thermal energy, effectively trapping the particles into monolayers, allowing them freedom only to translate and rotate along the surface. Particles adsorbed at interfaces are models for the understanding of the dynamics and assembly of particles in two dimensions and have broad technological applications, importantly in foam and emulsion science and in the bottom-up fabrication of new materials based on their monolayer assemblies. In this review, the hydrodynamics of the colloid motion along the surface is examined from both continuum and molecular dynamics frameworks. The interfacial energies of adsorbed particles is discussed first, followed by the hydrodynamics, starting with isolated particles followed by pairwise and multiple particle interactions. The effect of particle shape is emphasized, and the role played by the immersion depth and the surface rheology is discussed; experiments illustrating the applicability of the hydrodynamic studies are also examined. For the entire collection see [Zbl 1489.76002]. Development of mathematical modeling of multi-phase flow of Casson rheological fluid: theoretical approach 2022-09-13T20:28:31.338867Z "Nazeer, Mubbashar" "Hussain, Farooq" "Hameed, M. K." "Ijaz Khan, M." "Ahmad, Fayyaz" "Malik, M. Y." "Shi, Qiu-Hong" Summary: Theoretical stud of a rheological fluid suspended with two types of nanoparticles through a steep channel is presented in this article. Each suspension is formed by using the non-Newtonian Casson fluid model as the base liquid. Particulate flows are generated mainly due to the effects of gravitational force. In addition to this, the contribution of transversely applied magnetic fields is also considered. Further, the flow dynamics of Casson multiphase flows are compared with the ones suspended with the Newtonian fluid model. A closed-form solution is obtained for the mathematical modeled nonlinear partial differential equations which are transformed into a set of the ordinary differential equation. Separate expressions for volumetric flow rate and pressure gradient have been formulated, as well. Numerical results computed in the different tables show that Hafnium particles gain more momentum than crystal particles. Owing to, many engineering applications of highly thick multiphase flows, such as in chemical and textile industries, it is evident that Casson multiphase suspensions are quite suitable for coating purposes. Moreover, magnetized multiphase flows are compared with the previous investigation as the limiting case for the validation. Irreversibility analysis for axisymmetric nanomaterial flow towards a stretched surface 2022-09-13T20:28:31.338867Z "Song, Ying-Qing" "Shah, Faqir" "Khan, Sohail A." "Khan, M. Ijaz" "Malik, M. Y." "Sun, Tian-Chuan" Summary: Magnetohydrodynamic axisymmetric flow of viscous nanoliquid towards a variable stretching sheet is scrutinized. Flow is generated due to nonlinear stretching. Joule heating, heat flux and dissipation are analyzed in heat expression. Random and thermophoresis diffusions are considered. Physical description of entropy generation is also accounted. Entropy generation and heat transfer analysis are scrutinized through thermodynamics laws. Furthermore chemical reaction along with Arrhenius activation energy is addressed. Ordinary differential system is obtained through suitable variables. Homotopic convergent solutions for nonlinear system is developed. Influence of flow variables on entropy rate, velocity, Bejan number, concentration and temperature are analyzed. Further velocity gradient and heat and mass transfer rates are discussed. Reduction in velocity is noticed for magnetic variable. Thermal field has an enhancing trend for magnetic and Eckert number. Larger thermophoresis variable rises the temperature. An increment in entropy rate is observed for magnetic parameter. An increment in drag force is seen for magnetic variable. Theoretical analysis of linearized non-isothermal two-dimensional model of liquid chromatography columns packed with core-shell particles 2022-09-13T20:28:31.338867Z "Uche, Ugochukwu David" "Uche, Mercy" Summary: A linearized single-solute two-dimensional general rate model of non-isothermal liquid chromatography for columns of cylindrical geometry packed with core-shell particles is formulated and solved analytically to investigate the effects of temperature changes. A linear system of convection-diffusion partial differential equations is developed by the model equations. The solutions of the equations are obtained by applying Hankel transformation, Laplace transformation, Eigen-decomposition method and a general method for solving ordinary differential equations. The coupling between the concentration fronts and thermal waves is illustrated and key parameters that influence the chromatography column's performance are identified. For the same system of equations for both linear and nonlinear isotherms, a finite volume scheme is applied. Moreover, the ranges of validity of the analytical results are found using the same finite volume scheme. An analysis of the unified formulation for the equilibrium problem of compositional multiphase mixtures 2022-09-13T20:28:31.338867Z "Ben Gharbia, Ibtihel" "Haddou, Mounir" "Tran, Quang Huy" "Vu, Duc Thach Son" Summary: In this paper, we conduct a thorough mathematical analysis of the unified formulation advocated by \textit{A. Lauser} et al. [``A new approach for phase transitions in miscible multi-phase flow in porous media'', Adv. Water Res. 34, No. 8, 957--966 (2011; \url{doi:10.1016/j.advwatres.2011.04.021})] for compositional multiphase flows in porous media. The interest of this formulation lies in its potential to automatically handle the appearance and disappearance of phases. However, its practical implementation turned out to be not always robust for realistic fugacity laws associated with cubic equations of state, as shown by the first author and \textit{E. Flauraud} [``Study of compositional multiphase flow formulation using complementarity conditions'', Oil Gas Sci. Technol. 74, Article No. 43, 15 p. (2019; \url{doi:10.2516/ogst/2019012 })]. By focusing on the subproblem of phase equilibrium, we derive sufficient conditions for the existence of the corresponding system of equations. We trace back the difficulty of cubic laws to a deficiency of the Gibbs functions that comes into play due to the ``unifying'' feature of the new formulation. We propose a partial remedy for this problem by extending the domain of definition of these functions in a natural way. Besides, we highlight the crucial but seemingly unknown fact that the unified formulation encapsulates all the properties known to physicists on phase equilibrium, such as the tangent plane criterion and the minimization of the Gibbs energy of the mixture. Moisture in textiles 2022-09-13T20:28:31.338867Z "Duprat, C."|duprat.camille Summary: The interactions of textiles with moisture have been thoroughly studied in textile research, while fluid mechanists and soft matter physicists have partially investigated the underlying physics phenomena. A description of liquid morphologies in fibrous assemblies allows one to characterize the associated capillary forces and their impact on textiles, and to organize their complex moisture transport dynamics. This review gathers some of the common features and fundamental mechanisms at play in textile-liquid interactions, with selected examples ranging from knitted fabrics to nonwoven paper sheets, associated with experiments on model systems. For the entire collection see [Zbl 1489.76002]. Design and simulation of mechanical ventilators 2022-09-13T20:28:31.338867Z "El-Hadj, Abdellah" "Kezrane, Mohamed" "Ahmad, Hijaz" "Ameur, Houari" "Bin Abd Rahim, S. Zamree" "Younsi, Abdelhakime" "Abu-Zinadah, Hanaa" Summary: During this period of COVID-19 pandemic, the lack of medical equipment (like ventilators) leads to complications arising in the medical field. A low-cost ventilator seems to be an alternative substitute to fill the lacking. This paper presents a numerical analysis for predicting the delivered parameters of a low-cost mechanical ventilator. Based on several manufactured mechanical ventilators, two proposed designs are investigated in this study. Fluid-structure interaction (FSI) analysis is used for solving any problems with the first design, and computational fluid dynamic (CFD) analysis with moving boundary is used for solving any issues with the second design. For this purpose, ANSYS Workbench platform is used to solve the set of equations. The results showed that the Ambu-bag-based mechanical ventilator exhibited difficulties in controlling ventilation variables, which certainly will cause serious health problems such as barotrauma. The mechanical ventilator based on piston-cylinder is more satisfactory with regards to delivered parameters to the patient. The ways to obtain pressure control mode (PCM) and volume control mode (VCM) are identified. Finally, the ventilator output is highly affected by inlet flow, length of the cylinder, and piston diameter. Theoretical analysis of rolling fluid turbines 2022-09-13T20:28:31.338867Z "Kincl, Ondřej" "Pavelka, Michal" "Maršík, František" "Sedláček, Miroslav" Double Magnus type wind turbine 2022-09-13T20:28:31.338867Z "Klimina, L. A." "Shalimova, E. S." "Dosaev, M. Z." "Selyutskiy, Yu. D."|selyutskii.yu-d Summary: A closed mathematical model of a double Magnus type wind turbine with a horizontal axis is constructed. The propellers of the turbine are supposed to rotate in opposite directions. For such a system, equations of motion are derived. In the case of dimensions of the front propeller being two times smaller than dimensions of the rear propeller, operating modes and a trapped power coefficient are found numerically. Fluid dynamics of axial turbomachinery: blade- and stage-level simulations and models 2022-09-13T20:28:31.338867Z "Sandberg, Richard D." "Michelassi, Vittorio" Summary: The current generation of axial turbomachines is the culmination of decades of experience, and detailed understanding of the underlying flow physics has been a key factor for achieving high efficiency and reliability. Driven by advances in numerical methods and relentless growth in computing power, computational fluid dynamics has increasingly provided insights into the rich fluid dynamics involved and how it relates to loss generation. This article presents some of the complex flow phenomena occurring in bladed components of gas turbines and illustrates how simulations have contributed to their understanding and the challenges they pose for modeling. The interaction of key aerodynamic features with deterministic unsteadiness, caused by multiple blade rows, and stochastic unsteadiness, i.e., turbulence, is discussed. High-fidelity simulations of increasingly realistic configurations and models improved with help of machine learning promise to further grow turbomachinery performance and reliability and, thus, help fluid mechanics research have a greater industrial impact. For the entire collection see [Zbl 1489.76002]. Exact viscous compressible flow describing the dynamics of the atmosphere 2022-09-13T20:28:31.338867Z "Ionescu-Kruse, Delia" Summary: We focus on the Navier-Stokes equations for a compressible viscous fluid -- allowing variations of the dynamic eddy viscosity only in the vertical direction -- and the continuity equation. Our problem is written in spherical coordinates, in a non-inertial rotating frame. For zonal flows, with no variations in the longitudinal direction, and in a neighbourhood of the Equator, we get a linear parabolic evolution equation that we solve by the method of separation of variables. For a dynamic eddy viscosity which decreases with height above the ground level, the velocity field obtained has an azimuthal component which depends on time and has a nonlinear dependence on the radial coordinate, and a vertical component which depends linearly on the radial coordinate. Wave propagation in rotating shallow water in the presence of small-scale topography 2022-09-13T20:28:31.338867Z "Goldsmith, E. J." "Esler, J. G." Summary: The question of how finite-amplitude, small-scale topography affects small-amplitude motions in the ocean is addressed in the framework of the rotating shallow water equations. The extent to which the dispersion relations of Poincaré, Kelvin and Rossby waves are modified in the presence of topography is illuminated, using a range of numerical and analytical techniques based on the method of homogenisation. Both random and regular periodic arrays of topography are considered, with the special case of regular cylinders studied in detail, because this case allows for highly accurate analytical results. The results show that, for waves in a \(\beta\)-channel bounded by sidewalls, and for steep topographies outside of the quasi-geostrophic regime, topography acts to slow Poincaré waves slightly, Rossby waves are slowed significantly and Kelvin waves are accelerated for long waves and slowed for short waves, with the two regimes being separated by a narrow band of resonant wavelengths. The resonant band, which is due to the excitation of trapped topographic Rossby waves on each seamount, may affect any of the three wave types under the right conditions, and for physically reasonable results requires regularisation by Ekman friction. At larger topographic amplitudes, for cylindrical topography, a simple and accurate formula is given for the correction to the Rossby wave dispersion relation, which extends previous results for the quasi-geostrophic regime. Reacting multi-component fluids: regular solutions in Lorentz spaces 2022-09-13T20:28:31.338867Z "Mucha, Piotr Bogusław" "Piasecki, Tomasz" Summary: The paper deals with the analysis of a model of a multi-component fluid admitting chemical reactions. The flow is considered in the incompressible regime. The main result shows the global existence of regular solutions under the assumption of suitable smallness conditions. In order to control the solutions a special structure condition on the derivatives of chemical production functions determining the reactions is required. The existence is shown in a new critical functional framework of Lorentz spaces of type \(L_{p, r}(0, T; L_q)\), which allows to control the integral \(\int_0^\infty \|\nabla u(t)\|_\infty dt\). Asymptotic shallow models arising in magnetohydrodynamics 2022-09-13T20:28:31.338867Z "Alonso-Orán, Diego" Summary: In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in [\textit{A. Castro} and \textit{D. Lannes}, J. Fluid Mech. 759, 642--675 (2014; Zbl 1446.76077)] to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model. Dynamo action between two rotating discs 2022-09-13T20:28:31.338867Z "Arslan, A."|arslan.a-muzaffer|arslan.ayse-n|arslan.ahmet-faruk|arslan.a-v|arslan.abdullah-n|arslan.ali|arslan.aykut "Mestel, A. J." Summary: Dynamo action is considered in the region between two differentially rotating infinite discs. The boundaries may be insulating, perfectly conducting or ferromagnetic. In the absence of a magnetic field, various well-known self-similar flows arise, generalising that of von Kármán. Magnetic field instabilities with the same similarity structure are sought. The kinematic eigenvalue problem is found to have growing modes for \(Re_m > R_c\simeq 100\). The growth rate is real for the perfectly conducting and ferromagnetic cases, but may be complex for insulating boundaries. As \(Re_m\to\infty\) it is shown that the dynamo can be fast or slow, depending on the flow structure. In the slow case, the growth rate is governed by a magnetic boundary layer on one of the discs. The growing field saturates in a solution to the nonlinear dynamo problem. The bifurcation is found to be subcritical and nonlinear dynamos are found for \(Re_m\gtrsim0.7R_c\). Finally, the flux of magnetic energy to large \(r\) is examined, to determine which solutions might generalise to dynamos between finite discs. It is found that the fast dynamos tend to have inward energy flux, and so are unlikely to be realised in practice. Slow dynamos with outward flux are found. It is suggested that the average rotation rate should be non-zero in practice. Inclined MHD and radiative Maxwell slip fluid flow and heat transfer due to permeable melting surface with a non-linear heat source 2022-09-13T20:28:31.338867Z "Dadheech, Amit" "Parmar, Amit" "Olkha, Amala" Summary: The study analyzed a non-Newtonian Maxwell fluid flow past a permeable and melting surface with non-linear thermal radiation, inclined magnetic field chemical reaction with higher-order and non-uniform heat sources effects numerically. The governing PDEs are transformed into non-linear ODEs and solved by the shooting technique based on Runge Kutta with MATLAB toolbox. The results are shown graphically and in tabular form. The apprehensions of pictorial and tabular notations are used to analyze the effect of physical parameters governing velocity, energy, and mass. The obtained result thus confirms that an excellent agreement is achieved with those available in the open literature. The outcomes are represented as a magnetic parameter, porosity parameter and Maxwell fluid parameter have reduced the momentum boundary layer thickness. Nonstationary flow of a viscous incompressible electrically conductive fluid on a rotating plate 2022-09-13T20:28:31.338867Z "Gurchenkov, A. A." Summary: In this work, the evolution of a flow of a viscous electrically conductive fluid on a rotating plate in the presence of a magnetic field is studied. The analytical solution of three-dimensional unsteady equations of magnetohydrodynamics is presented. The velocity field and the induced magnetic field in the flow of a viscous electrically conductive fluid filling a half-space bounded by a flat wall are determined. The fluid, together with the bounding plane, rotates as a whole with a constant angular velocity around a direction not perpendicular to the plane. An unsteady flux is induced by suddenly beginning vibrations of the wall and an applied magnetic field directed perpendicular to the plane. A number of special cases of the wall motion are considered. Based on the results obtained, the individual structures of the boundary layers near the wall are investigated. Melting heat transfer of MHD micropolar fluid flow past an exponentially stretching sheet with slip and thermal radiation 2022-09-13T20:28:31.338867Z "Mandal, Iswar Chandra" "Mukhopadhyay, Swati" "Vajravelu, Kuppalapalle" Summary: The effects of velocity slip and radiation on MHD flow and melting heat transfer of a micropolar fluid due to an exponentially stretched sheet are presented. By means of similarity transformations the leading partial differential equations are changed to a set of ordinary differential equations which are nonlinear. Numerical solutions of the nonlinear system of equations are then obtained by changing the boundary value problem first to an initial value problem. It is observed that the pertaining parameters have significant effects on the flow and heat transfer characteristics, which are presented and talked about in detail through their illustrations. Due to boost in the melting parameter, the fluid velocity, angular velocity and temperature are found to decrease. Fluid velocity and angular velocity both decrease with a rise in slip at the boundary but quite opposite is the effect on the temperature. Numerical simulation of radiative MHD Sutterby nanofluid flow through porous medium in the presence of Hall currents and electroosmosis 2022-09-13T20:28:31.338867Z "Ramesh, K."|ramesh.k-v|ramesh.k-t|ramesh.kasilingam|ramesh.kiran "Rawal, Madhav" "Patel, Aryaman" Summary: Analysis of thermal and fluid phenomena based on the fluid dynamics theory leads to understanding of fundamental mechanisms in modern technologies. Thermal/fluid transport is critical to many applications, such as photothermal cancer therapy, solar thermal evaporation and polymer composites. The current study focusses to investigate the effect of magnetohydrodynamics, Hall currents and electroosmosis on the propulsion of Sutterby nanofluids in a porous microchannel. The Brownian motion and thermophoresis effects have also been considered. The governing equations for the momentum, temperature and nanoparticle volume fraction have been modified under the suitable non-dimensional quantities. The resulting dimensionless system of equations have been solved using bvp4c package in computational software MATLAB. The pictorial representations have been presented for various flow quantities with respect to sundry fluid parameters. It is noted from the investigation that, there is a decrease in fluid velocity with an increase in Hartmann number, temperature decreases with the increment in radiation parameter and nanoparticle volume fraction reduces with the increment of Prandtl number and thermophoresis parameter. The results obtained for the Sutterby nanofluid propulsion model reveal many engrossing behaviors and has many applications such as disease diagnostics and cancerous tissues destruction, and that provide a further dimension to investigate the nanofluid flow problems with thermophysical properties in two/three dimensions. On the inverse problem for Channell collisionless plasma equilibria 2022-09-13T20:28:31.338867Z "Allanson, Oliver" "Troscheit, Sascha" "Neukirch, Thomas" Summary: Vlasov-Maxwell equilibria are described by the self-consistent solutions of the time-independent Maxwell equations for the real-space dynamics of electromagnetic fields and the Vlasov equation for the phase-space dynamics of particle distribution functions (DFs) in a collisionless plasma. These two systems (macroscopic and microscopic) are coupled via the source terms in Maxwell's equations, which are sums of velocity-space `moment' integrals of the particle DF. This paper considers a particular subset of solutions of the broad plasma physics problem: `the inverse problem for collisionless equilibria' (IPCE), viz. \textit{`given information regarding the macroscopic configuration of a collisionless plasma equilibrium, what self-consistent equilibrium DFs exist?'} We introduce the constants of motion approach to IPCE using the assumptions of a `modified Maxwellian' DF, and a strictly neutral and spatially one-dimensional plasma, and this is consistent with `\textit{P. J. Channell}'s method' [``Exact Vlasov-Maxwell equilibria with sheared magnetic fields'', Phys. Fluids 19, No. 10, 1541--1545 (1976; \url{doi:10.1063/1.861357})]. In such circumstances, IPCE formally reduces to the inversion of Weierstrass transformations [\textit{G. G. Bilodeau}, Duke Math. J. 29, 293--308 (1962; Zbl 0154.38003)]. These are the same transformations that feature in the initial value problem for the heat/diffusion equation. We discuss the various mathematical conditions that a candidate solution of IPCE must satisfy. One method that can be used to invert the Weierstrass transform is expansions in Hermite polynomials. Building on the results of \textit{O. Allanson} et al. [``From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials'', J. Plasma Phys. 82, No. 3, Article ID 905820306, 28 p. (2016; \url{doi:10.1017/S0022377816000519})], we establish under what circumstances a solution obtained by these means converges and allows velocity moments of all orders. Ever since the seminal work by \textit{I. B. Bernstein} et al. [Phys. Rev., II. Ser. 108, 546--550 (1957; Zbl 0081.44904)] on `stationary' electrostatic plasma waves, the necessary quality of non-negativity has been noted as a feature that any candidate solution of IPCE will not \textit{a priori} satisfy. We discuss this problem in the context of Channell equilibria, for magnetized plasmas. Bulk viscosity in relativistic fluids: from thermodynamics to hydrodynamics 2022-09-13T20:28:31.338867Z "Gavassino, L." "Antonelli, M."|antonelli.michele|antonelli.marco|antonelli.miranda-j|antonelli.michela|antonelli.massimo "Haskell, B." Taylor dispersion in non-Darcy porous media with bulk chemical reaction: a model for drug transport in impeded blood vessels 2022-09-13T20:28:31.338867Z "Roy, Ashis Kumar" "Bég, O. Anwar"|beg.osman-anwar "Saha, Apu Kumar" "Murthy, J. V. Ramana" Summary: The present article discusses the solute transport process in steady laminar blood flow through a non-Darcy porous medium, as a model for drug movement in blood vessels containing deposits. The Darcy-Brinkman-Forchheimer drag force formulation is adopted to mimic a sparsely packed porous domain, and the vessel is approximated as an impermeable cylindrical conduit. The conservation equations are implemented in an axisymmetric system \((R, Z)\) with suitable boundary conditions, assuming constant tortuosity and porosity of the medium. Newtonian flow is assumed, which is physically realistic for large vessels at high shear rates. The velocity field is expanded asymptotically, and the concentration field decomposed. Advection and dispersion coefficient expressions are rigorously derived. Extensive visualization of the influence of effective Péclet number, Forchheimer number, reaction parameter on velocity, asymptotic dispersion coefficient, mean concentration, and transverse concentration at different axial locations and times is provided. Increasing reaction parameter and Forchheimer number both decrease the dispersion coefficient, although the latter exhibits a linear decay. The maximum mean concentration is enhanced with greater Forchheimer numbers, although the centre of the solute cloud is displaced in the backward direction. Peak mean concentration is suppressed with the reaction parameter, although the centroid of the solute cloud remains unchanged. Peak mean concentration deteriorates over time since the dispersion process is largely controlled by diffusion at the large time, and therefore the breakthrough curve is more dispersed. A similar trend is computed with increasing Péclet number (large Péclet numbers imply diffusion-controlled transport). The computations provide some insight into a drug (pharmacological agents) reacting linearly with blood. Optimal swimmers can be pullers, pushers or neutral depending on the shape 2022-09-13T20:28:31.338867Z "Daddi-Moussa-Ider, Abdallah" "Nasouri, Babak" "Vilfan, Andrej" "Golestanian, Ramin" Summary: The ability of microswimmers to deploy optimal propulsion strategies is of paramount importance for their locomotory performance and survival at low Reynolds numbers. Although for perfectly spherical swimmers minimum dissipation requires a neutral-type swimming, any departure from the spherical shape may lead the swimmer to adopt a new propulsion strategy, namely those of puller- or pusher-type swimming. In this study, by using the minimum dissipation theorem for microswimmers, we determine the flow field of an optimal nearly spherical swimmer, and show that indeed depending on the shape profile, the optimal swimmer can be a puller, pusher or neutral. Using an asymptotic approach, we find that amongst all the modes of the shape function, only the third mode determines, to leading order, the swimming type of the optimal swimmer. Propagation of a terahertz Bessel vortex beam through a homogeneous magnetized plasma slab 2022-09-13T20:28:31.338867Z "Li, Haiying" "Ding, Wei" "Liu, Jiawei" "Ying, Ci" "Bai, Lu" "Wu, Zhensen" Summary: This paper provides an analytic method to study propagation characteristics of a linearly polarized Bessel vortex beam through a homogeneous magnetized plasma slab. The incident Bessel vortex beam, as well as the reflected, transmitted and internal fields are expanded in terms of cylindrical vector wave functions (CVWFs). The effects of plasma thickness, electron density and magnetic induction strength on the contour profiles of the reflected and transmitted beams and orbital angular momentum (OAM) spectra are analyzed and discussed in detail. In particular, the magnetic induction strength has a significant impact on the polarization of the transmitted beam, but not on OAM state distribution. The channel capacity of THz OAM multiplexing decreases with an increase of plasma thickness and electron density. On the temperature equation in classical irreversible thermodynamics 2022-09-13T20:28:31.338867Z "Ciancio, Vincenzo" Summary: Abstract.In this paper, by using a procedure of classical irreversible thermodynamics with internal variables, some possible interactions among heat conduction and viscous-elastic flows for rheological media are studied. By introducing as an vectorial internal variable \(\boldsymbol \xi\), which influences thermal and diffusion phenomena, phenomenological equation for these variables are derived. A general vector, \(\boldsymbol J\), is introduced which assumes the role of heat flux and it is shown that, in isotropic media, \(\boldsymbol J\) can be composed of two parts and this allows to obtain a heat equation that generalizes both the Fourier equation and the Maxwell-Cattaneo-Vernotte (M-C-V) equation. A general temperature equation and the energy balance equation for viscoelastic media are obtained. Basic space plasma physics 2022-09-13T20:28:31.338867Z "Baumjohann, Wolfgang" "Treumann, Rudolf A." Publisher's description: This textbook describes Earth's plasma environment from single particle motion in electromagnetic fields, with applications to Earth's magnetosphere, up to plasma wave generation and wave-particle interaction. The origin and effects of collisions and conductivities are discussed in detail, as is the formation of the ionosphere, the origin of magnetospheric convection and magnetospheric dynamics in solar wind-magnetosphere coupling, the evolution of magnetospheric storms, auroral substorms, and auroral phenomena of various kinds. The second half of the book presents the theoretical foundation of space plasma physics, from kinetic theory of plasma through the formation of moment equations and derivation of magnetohydrodynamic theory of plasmas. The validity of this theory is elucidated, and two-fluid theory is presented in more detail. This is followed by a brief analysis of fluid boundaries, with Earth's magnetopause and bow shock as examples. The main emphasis is on the presentation of fluid and kinetic wave theory, deriving the relevant wave modes in a high temperature space plasma. Plasma instability is the most important topic in all applications and is discussed separately, including a section on thermal fluctuations. These theories are applied to the most interesting problems in space plasma physics, collisionless reconnection and collisionless shock waves with references provided. The Appendix includes the most recent developments in the theory of statistical particle distributions in space plasma, the Kappa distribution, etc, also including a section on space plasma turbulence and emphasizing on new observational developments with a dimensional derivation of the Kolmogorov spectrum, which might be instructive for the student who may worry about its origin. The book ends with a section on space climatology, space meteorology and space weather, a new application field in space plasma physics that is of vital interest when considering the possible hazards to civilization from space. See the reviews of the first and second editions in [Zbl 0971.82040; Zbl 1252.82001]. Parametrising non-linear dark energy perturbations 2022-09-13T20:28:31.338867Z "Hassani, Farbod" "L'Huillier, Benjamin" "Shafieloo, Arman" "Kunz, Martin" "Adamek, Julian" (no abstract) Consistent Blandford-Znajek expansion 2022-09-13T20:28:31.338867Z "Armas, Jay" "Cai, Yangyang" "Compère, Geoffrey" "Garfinkle, David" "Gralla, Samuel E." (no abstract) Cosmic microwave background anisotropy numerical solution (CMBAns). I: An introduction to \(C_l\) calculation 2022-09-13T20:28:31.338867Z "Das, Santanu" "Phan, Anh"|phan.anh-vu|phan.anh-huy (no abstract) Boundary stabilization of an elastic body surrounding a viscous incompressible fluid 2022-09-13T20:28:31.338867Z "Do, K. D." Summary: This paper considers the problem of boundary feedback stabilization of an elastic body surrounding a viscous incompressible fluid described by Navier-Stokes equations in three dimensional space. The paper gives a proof of global existence of a weak solution of the closed-loop system via the Galerkin method. Due to consideration of less regular initial values of the fluid velocity, the forces induced by the fluid on the elastic body are not able to bound. Therefore, the paper handles ``fluid work and fluid power'' on the elastic body in stability and convergence analysis of the closed-loop system. |
b944c6b3dfb97990 | More on Quantum things
English: Schrödinger equation of quantum mecha...
English: Schrödinger equation of quantum mechanics (1927). (Photo credit: Wikipedia)
Schrodinger’s wave equation describes how the quantum state of a quantum system changes with time. Everett’s insight was that the observer of a quantum state was as much part of the system as the observed part of the system. Therefore they were “entangled” in the quantum sense and would be covered by a single quantum state equation.
If the observer and the observed are thus entangled, then so must be an observer who observes the quantum state of the observer and the observed. One can then extend this to the whole universe, which leads to the concept of a wave equation or function which describes the whole Universe.
English: Quantum mechanics travelling wavefunc...
English: Quantum mechanics travelling wavefunctions (Photo credit: Wikipedia)
That there is an equation for the universe is not really surprising and indeed, it is not surprising that it could be a quantum wave equation as the quantum world seems to form the basis of the physical, apparently classically described, world that we see.
I base this idea on the fact that everything that we sees appears to be describable in terms of a deterministic equation. It has been argued that such things as “psi phenomena“, but such claims are yet to be conclusively verified, with many putative examples having been discredited.
Embed from Getty Images
Some people argue for a soul or mind as an example of a non-physical entity, but any such concept leaves a lot of questions to be asked. A non-physical entity cannot, by definition almost, be measured in any way, and there is difficulty in showing how such a non-physical entity can interact with physical ones, and therefore be noticed or detected.
By definition almost, a physical entity, such a body, is only influenced by physical things. If this were not the case we would see physical entities not following the laws of physics. For example, if it is possible to move an object by mind power or telekinesis, one would see the object disobeying fundamental scientific laws, like Newton’s First Law of Motion.
English: Isaac Newton Dansk: Sir Isaac Newton ...
The mind is a curious example of a physical entity which is often thought of as being non-physical. After all, a mind does not have a physical location, apart from the skull of the person whose mind it is, and it can’t be weighed as such.
The mind however is a pattern, on the brain, made up of the state of trillions of neurones. It is made up of information, and is much like a computer program which is made up of the state of a few billion physical logic circuits in the guts of the computer.
Vista de la Motherboard
Vista de la Motherboard (Photo credit: Wikipedia)
Open a computer and you won’t see “an image” anywhere. You will see patterns of bits of data in the memory, or on the hard disk, or maybe in transit, being sent to a computer screen. Similarly if you open someone’s skull you will not see an image there either. Just a bunch of neurones in particular states.
The one glaring exception to all the above, is, perhaps, consciousness. It’s hard to describe consciousness in terms of a pattern or patterns of the states of our neurones, but I believe that that is fundamentally what it is.
Schéma d'un neurone , commenté en francais
Schéma d’un neurone , commenté en francais (Photo credit: Wikipedia)
Some people argue that we are conscious beings, (true), and that we consciously make choices (false, in my opinion). When we look closely at any choice that we make, it appears to be that choice is in fact illusory, and that our actions are determined by prior factors.
People seem to realise this, although they don’t acknowledge it. When questioned, there is always some reason that they “choose” in a particular way. Perhaps they don’t have enough cash to choose the luxury option when out shopping, or their desire outweighs their financial state. When pushed people can always think of a reason.
English: A choice of which way to go The choices are a path to Greengore or Intack or the Old Clitheroe Road (Photo credit: Wikipedia)
To be sure, many “reasons” are actually post choice rationalisations, and choices may be based more on emotions than valid rational reasons, but whatever the emotions (such as the desire for an object), the emotions precede decision.
If, as sometimes happens, a person has to make a choice between two alternatives, that person can be almost paralysed with indecision. Even then, when a decision is finally made, it can be either a random choice, or maybe the person may say that they made a particular choice because they had decided a different way in another situation, or similar (e.g. they like the colour blue!).
English: Choose your leaders and place your trust
English: Choose your leaders and place your trust (Photo credit: Wikipedia)
If there is no non-physical component to the Universe, as appears very likely, and psi phenomenon do not exist, then everything has a cause. I don’t mean this in the sense that event A causes event B which causes C, but more in the sense that the slope that a marble is on causes it to move in a particular direction.
Causality seems to be a continuum thing, rather than the discrete A causes B case. We can only get an approximation of the discrete case if we exclude all other options. There is a latin term for this : ceteris paribus – all other things being kept the same. “Ceteris paribus” would exclude the case where a wind blowing up or across the slope changes the path of the marble.
English: Picture of marbles from my collection
English: Picture of marbles from my collection (Photo credit: Wikipedia)
For this reason I dislike the Many Worlds Interpretation of Quantum Physics, as it is usually stated. The usual metaphor is a splitting movie film, which results in two distinct tracks in the future. I feel that a better picture would be a marble on a slope with a saddle.
The marble may go left, or it may go right, or it may even follow the line of the saddle. We still require “ceteris paribus” to exclude crosswinds, but there is no split as such. In a quantum model, the marble goes both left and right (and traverses the peak of the saddle with vanishing probability).
Monkey saddle
Monkey saddle (Photo credit: Wikipedia)
The probability that it goes left or right is determined by the wave equation for the system, and has a real physical meaning, which it doesn’t (so far as my knowledge goes) in the splitting metaphor.
I don’t know how my speculations stack up against the realities of quantum mechanics, but I like my interpretation, purely on aesthetic grounds, even if it is far from the mark!
Embed from Getty Images
Predicting the future
Future car!
Future car! (Photo credit: Little Black Cherry)
Zabriskie Point at sunrise in Death Valley
New Scientist
New Scientist (Photo credit: Wikipedia)
Mathematical induction can be informally illus...
cubed earth theory
cubed earth theory (Photo credit: Joelstuff V4)
English: Mathematical induction as domino effe...
The End Of Certainty?
The End Of Certainty? (Photo credit: minifig) |
f65f094a09964d10 | 2022-11-30T03:46:56Z https://oai.zbmath.org/v1/
oai:zbmath.org:7489559 2022-03-15T14:10:40Z 26 35 65 81
Erfanian, M.; Zeidabadi, H.; Rashki, M.; Borzouei, H. 2020 7489559 English Springer International Publishing (SpringerOpen), Cham https://zbmath.org/07489559 Content generated by zbMATH Open, such as reviews, classifications, software, or author disambiguation data, are distributed under CC-BY-SA 4.0. This defines the license for the whole dataset, which also contains non-copyrighted bibliographic metadata and reference data derived from I4OC (CC0). Note that the API only provides a subset of the data in the zbMATH Open Web interface. In several cases, third-party information, such as abstracts, cannot be made available under a suitable license through the API. In those cases, we replaced the data with the string 'zbMATH Open Web Interface contents unavailable due to conflicting licenses.' Adv. Difference Equ. 2020, Paper No. 344, 20 p. (2020). 65D07; 35R11; 81Q05; 35Q55; 26A33 Solving a nonlinear fractional Schrödinger equation using cubic B-splines j |
260db5201f0c5c38 | When the cosmological “constant” is derived from modern five-dimensional relativity, exact solutions imply that for small systems it scales in proportion to the square of the mass. However, a duality transformation implies that for large systems it scales as the inverse square of the mass.
1. Introduction
The cosmological “constant” as it appears in Einstein’s general relativity has several puzzling aspects, and it is a serious problem to understand why its value as inferred from cosmology is much smaller than its magnitude as implied by particle physics. However, it has been known for a long time that the cosmological “constant” appears more naturally when the world is taken to be five-dimensional [1], and recently there has been intense work on the modern versions of 5D relativity where the extra dimension is not compactified [24]. The purpose of the present paper is to draw together various results in the literature which indicate that there may be simple scaling relations between the values of the cosmological “constant” and the mass of the system concerned. Tentatively, we identify for small systems and for large, gravitationally-dominated systems. While these relations cannot be rigorously established with our present level of understanding, we believe that it is useful to point them out as guides for future research.
The subjects which indicate possible relations are diverse and include the embedding of -dominated solutions of 4D general relativity in the so-called 5D canonical metric [58]; the embeddings which lead to variable values of [913]; the equations of motion for canonical and related metrics [1420]; conformal transformations which affect and possibly [21, 22]; the vacuum and gauge fields associated with elementary particles [23, 24]; and the wave-particle duality connected with certain -dominated 5D metrics [2527]. Most of our results are in Section 2. There we will reexamine the meaning of , reinterpret two classes of known solutions, and present a new class with interesting properties. Section 3 is a conclusion.
To streamline the work, we will often absorb the speed of light , the gravitational constant , and the quantum of action , except in places where they are made explicit to aid in understanding. As usual, uppercase Latin letters run for time, space and the extra dimension. We label the last to avoid confusion. Lowercase Greek letters run . Other notation is standard.
2. The Cosmological “Constant” and Possible Scaling Relations
In this section, we will examine certain subjects which involve the cosmological “constant” of a spacetime and the mass of a test particle moving in it. That these parameters may be linked can be appreciated by noting that 5D relativity is broader than Einstein’s 4D theory, being in general an account of gravity, electromagnetism, and a scalar field, where the last is widely believed to be concerned with how particles acquire mass [24]. However, in 5D neither nor are in general constants. Rather, they depend on the field equations and solutions of them. It is common to take the field equations to be given in terms of the Ricci tensor by These apparently empty 5D equations actually contain Einstein’s 4D equations with a finite energy-momentum tensor, a result guaranteed by Campbell’s embedding theorem [57]. This means that the 4D theory is smoothly contained in the 5D one and that the latter can be brought into agreement with observations at some level.
In Einstein’s theory, the cosmological “constant” is usually introduced by adding a term to the field equations: Here, is the metric tensor, whose covariant derivative is zero, hence the acceptability of the noted term. We recognize that the term is a kind of a gauge term. It is sometimes moved to the right-hand side of Einstein’s equations, where it can be viewed as a vacuum fluid with density and equation of state . However, it should be recalled that the coupling constant between the left-hand (or geometrical) side of the Einstein equations and the right-hand (or matter) side is . This, therefore, cancels the similar coefficient of the vacuum density, leading us back to the realization that is really a stand-alone parameter insofar as general relativity is concerned (this is in line with the fact that its physical dimensions or units are , matching those of the rest of the field equations, which involve the second derivatives of the dimensionless metric coefficients with respect to the coordinates.) An implication of this is that when is derived from a 5D as opposed to a 4D theory, it may be connected not with gravity but with the scalar field, a possibility we will return to later.
The quantum vacuum, as opposed to the classical one, is frequently attributed an energy density which is calculated in terms of many simple harmonic oscillators and expressed in terms of an effective value of [23]. This energy density is formally divergent, unless it is cut off by introducing a minimum wavelength or equivalently a maximum wave number . With this being understood, there results . If the cutoff in is chosen to be the inverse of the Planck length, this has the size of erg cm−3. For comparison, the cosmologically determined value of (~10−56 cm−2) corresponds to an energy density of order 10−8 erg cm−3. The discrepancy, of order 10120, is the crux of the cosmological-constant problem.
An alternative interpretation of the result in the preceding paragraph is to imagine that the quantum vacuum does not spread through ordinary 3D space but is concentrated in particles of mass . It is reasonable to suppose that the stuff of each particle occupies a volume whose size is given by the Compton wavelength, . Then, the average density is approximately This expression is formally identical to the one above. But the high-density vacuum is now confined to the particle, as expected if it is the product of a scalar field which couples to matter (see below). There is no conflict between (3) and the all-pervasive cosmological vacuum discussed above, so the cosmological-constant problem is avoided.
The best way to incorporate a scalar field into physics is to take its potential to be the extra, diagonal element of an extended 5D metric tensor. Then, following Kaluza the extra, nondiagonal elements can be identified with the potentials of electromagnetism, while the 4D block remains as a description of the 4D Einsteinian gravity. Since we are here mainly interested in the scalar field, we can eliminate the electromagnetic potentials by a suitable use of the coordinate degrees of freedom of the metric, so the interval for the gravitational and scalar fields is Here and depend in general on both the coordinates of spacetime () and the extra dimension (). The symbol indicates whether the extra dimension is spacelike or timelike, both being allowed in modern 5D theory (the extra dimension does not have the physical nature of an extra time, so for there is no problem with closed timelike paths). Many solutions are known of the field equations (1) for the metric (4) [24]. It transpires that the easiest way to approach the field equations is by splitting the 4D part of the metric into two functions; thus, Here, is a gauge function which determines the behavior in , while depends only on the spacetime coordinates . While the form (5) provides a mathematical advantage, it involves a physical quandary: does an observer experience the whole 4D space or only the spacetime-dependent subspace ? This question is akin to the argument for the so-called Jordan frame versus the Einstein frame in old 4D scalar-tensor theory, where a scalar function was applied to the 4D metric with no fifth dimension. It did not find a definitive answer then and has not done so today. There is a difference in the physics between the two frames, but so long as the function is slowly varying, this will be minor. Cosmological observations may one day reveal the difference between the two frames, but for now we proceed with the view that they yield complementary physics.
An instructive case of the metric (5) has and , where is any solution of the Einstein equations without ordinary matter but with a vacuum fluid whose density is measured by . This is known as the (pure) canonical metric. There is a large literature on this case (see [8] for a review). It includes the Schwarzschild-de Sitter metric for the sun and the solar system and the de Sitter metric for the universe in its inflationary stage. It turns out that the equations of motion for a test particle in the 5D metric (5) are the same as those in the 4D theory, a result which enforces agreement with the classical tests of relativity [28, 29]. The dynamics may be obtained either by using the 5D geodesic equation or by putting in (5). The latter is based on the fact that null paths in 5D with reproduce the timelike paths of massive particles in 4D with , as well as the paths of photons with . The definition of dynamics and causality by matches the null nature of the field equations (1). It turns out that the nature of the motion in the extra dimension depends on the choice of in the metric (5), as does the sign of . Thus introducing a constant , we findThe second of these equations is of particular interest, because it is the same as the expression for the wave function in old wave mechanics. In fact, it may be shown that the 5D geodesic equation for the (pure) canonical metric reproduces the Klein-Gordon equation with in place of and in place of [2527]. We will meet the Klein-Gordon equation again below. Here, we note that the (pure) canonical metric suggests the possibility that Here, has been written in terms of the Compton wavelength. This identification presupposes that the observer experiences the 4D spacetime in (5) rather than the composite spacetime defined by . This is a subtle issue, as noted above, and we will return to it below.
The next most simple case of (5) is when a shift is applied to the extra coordinate in the canonical metric. This may appear to be close to trivial, but it is not because of the way in which the 4D Ricci scalar transforms and with it [9, 10, 21, 22]. The equations of motion and the mass of a test particle for the shifted canonical metric were worked out by Ponce de Leon [1620]. He used the principle of the least action and the eikonal equation for massive and massless particles, as opposed to the geodesic equation used by Mashhoon et al. [14, 15]. As before, it turns out that for a spacelike extra dimension () and for a timelike one (). The metric and the expressions for and areThe second line here requires lengthy calculations for and [9, 10, 1620], so the fact that we again find is significant.
The third case we present is more complicated than the canonical metrics studied in the two preceding paragraphs. In (5), we put ), where . This may be shown to satisfy the field equations (1), which break down into sets: ten relations which determine the energy-momentum tensor necessary to balance Einstein’s equations; four conservation-type relations which fix a 4-tensor that has an associated scalar ; and one wave equation for the scalar field . The work is tedious (see [24]; indices are raised and lowered using of (5)). The metric and final results of the field equations read as follows: Here, a comma denotes the partial derivative, a semicolon denotes the (4D) covariant derivative, and where .
There are scalar quantities associated with the above which are of physical interest. For example, can be obtained by contracting (9b) and using (9d) to simplify it; as given by the contraction of (9c) is a conserved quantity; and the (4D) Ricci or curvature scalar can be expressed in its general form and in the special form it takes for the metric (9a). Thus,These relations and (9a), (9b), (9c), and (9d) can be given physical interpretations along the lines of what has been done for other solutions in the literature [24]. The energy-momentum tensor (9b) shows that the source consists of the scalar field plus a term which, because of its proportionality to , would usually be attributed to a vacuum fluid with cosmological constant . The conserved tensor of (9c) obeys by the field equations, and its scalar has in other works been linked to the rest mass of a test particle, which here is [2527]. This is confirmed by the wave equation (9d), which deserves some discussion.
Relation (9d), depending on the choice for , is known either as the Helmholtz equation or as the Klein-Gordon equation. Many solutions to it are known with applications to problems in atomic physics (like diffusion) and elementary particle physics (like wave mechanics). There are different modes of behavior, depending on whether or , which correspond to the monotonic and oscillatory modes (6a) and (6b) of the canonical metric discussed before. For the present metric (9a), the scalar field may be real or complex, and in the latter case for the wave equation (9d) is identical to the Klein-Gordon equation, with being the Compton wavelength of the test particle. This is similar to a previous interpretation based on the shifted-canonical metric [2527]. (In (9d), the oscillation is in , whereas in the corresponding equation of [2527] it is in , because in the canonical metric it is presumed that , so the physical behavior is moved from one parameter to the other. In (9a), the problem can be made explicitly complex by writing , if so desired.)
It may seem strange that a classical field theory yields an equation typical of (old) quantum theory, but it should be recalled that the wave equation (9d) comes from the field equation , which does not exist in standard general relativity. In fact, the present interpretation of the metric (9a) is fully consistent with the approach to noncompactified 5D relativity known as Space-Time-Matter theory, where matter on the macroscopic and microscopic scales is taken to be the result of higher-dimensional geometry [24]. By contrast, while the metric (9a) may resemble the warp metric of the alternative approach to 5D relativity known as the Membrane theory, in that approach, the “” in the exponent of the 4D part of the metric is absent, which means that the metric does not satisfy the field equations in the simple form (1). Our view is that (9a), (9b), (9c), and (9d) show the wave-mechanical properties of matter. The scalars (10a), (10b), and (10c) associated with the solution bear this out. With conventional units restored, the conserved quantity is inversely proportional to the Compton wavelength of a test particle moving in the spacetime. Viewed as a wave which couples to matter, we expect that the Compton wavelength should be consistent with the radius of curvature of the spacetime, and this is confirmed by the relation for . Lastly, we note that the aforementioned relation shows once again that .
This relation is common to the three classes of solutions examined above, which come from the different choices of the gauge function in (5). They involve which gives (6a), (6b), which gives (8a), (8b), and which gives (9a), (9b), (9c), and (9d). By comparison with known physics, we infer that the constant length is inversely proportional to the particle mass , which we can write in terms of the Compton wavelength as . The exponential gauge, in particular, leads from the field equation to the Klein-Gordon equation, which is the basic relation in wave mechanics (its low-energy limit is the Schrödinger equation which underlies the physics of the hydrogen atom). The implication is that the scalar field of 5D relativity is connected to the mass of a particle, and with the phenomenon of wave-particle duality ([2527]; the Klein-Gordon equation can have real or complex forms). These comments are in accordance with the longstanding view that theories of Kaluza-Klein type provide a way of unifying the interactions of particles with gravity. What is, however, of the latter interaction? It is natural to wonder if there is not a complementary relation to what we have found above, but for macroscopic gravity-dominated systems.
This subject will require detailed analysis, but some comments of a preliminary type may be made. It is useful, in this context, to reconsider the traditional distinction between inertial mass () and gravitational mass (). The Kaluza-Klein equation involves the former, so our previous considerations have concerned and as the scaling relation for the cosmological “constant”. It is clear that this scaling rule cannot persist to arbitrarily large masses without leading to excessive curvature of empty spacetime (). We expect, therefore, that it might pass over to some other scaling relation for large gravitational masses.
Such a relation is actually implicit in certain works on the canonical metric [24, 822]. We recall that the 4D part of the 5D canonical metric involves the combination . This can be compared to the element of action for classical mechanics, . Two obvious identifications are possible: and . We have already explored the former, so attention is focused on the latter. In fact the possibility has been considered, mainly in relation to cosmology, and cannot be ruled out [24, 1113]. As regards , we note that its behavior depends on the coordinate frame experienced by an observer (see above). To illustrate this, consider a vacuum spacetime with the (pure) canonical metric, where the 4D part of the interval is . The effective value of can be obtained from either the Ricci scalar or the Einstein tensor and depends on whether the observer experiences only or the full . The results are, respectively, and , and both appear in the literature. Let us take the second alternative and combine it with the physical identification noted above. The obvious parameter with which to geometrize the gravitational mass is , the Schwarzschild radius. Then we find that in total, . That is, for large gravitationally dominated systems we expect to scale as the inverse square of the mass.
The argument of the preceding paragraph is tentative, but can be checked by combining it with the more detailed work concerning the inertial mass which went before. For simplicity, we take the numerical factors to be those of the canonical case and consider a proton (inertial mass ) and the observable part of the universe (gravitational mass ). Then, the scaling relations for the cosmological “constant” read and . These can be combined to give the number of baryons in the observable universe as In this, we substitute the quantum field theoretical value of cm−2 and the cosmological value of cm−2 (obtained from , where and =, together with current observational data giving km s−1 Mpc−1 and ). The result is , which is in agreement with conventional estimates.
The two scaling relations considered in this section should be regarded as complementary. The first is better based on theory than the second, since it can be examined in three gauges rather than one. However, there is in principle no conflict between them, and in practice we expect the first to grade into the second. The rule should be dominant on the particle scale (~10−13 cm), and the rule should be dominant on the cosmological scale (~1028 cm). Theoretically, they should be comparable on scales of order 100 km, which in practice is rough where quantum interactions and solid-state forces are superseded by the effects of gravity.
3. Conclusion
We have seen in the preceding section that the cosmological constant is open to reinterpretation, particularly as a measure of the energy density of the vacuum fields of particles. It is somewhat better understood in cosmology, where its theoretical status is relatively clear in Einstein’s equations, and where observations establish its approximate value. Unfortunately, there is a very large mismatch between the microscopic and the macroscopic domains. This can in principle be alleviated by using a five-dimensional theory, of the kind indicated by unification, where in general is not a universal constant but a variable. This is shown most clearly by the 5D canonical gauge, where scales according to the size of the potential well () or the value of the extra coordinate (). Since the mass () of a test particle also depends on these parameters, we are tentatively led to suggest scaling relations of the form . For the canonical gauge in its pure and shifted forms, the scaling relation is for small and has the form . This is also the form derived from the exponential gauge, which has the advantage of showing that the extra field equation resembles the Klein-Gordon equation of wave mechanics, implying that the scalar field is connected with particle mass.
There is, however, an alternative interpretation of the canonical gauge and others like it. The 4D part of this involves a term , and to match the classical action , it is possible to use the gravitational mass with rather than the inertial mass with . The implication is that when gravity is dominant, for large , there is a scaling relation of the form . This macroscopic relation should be viewed as complementary to the microscopic one, the changeover occurring at a length scale of order 100 km. When the two relations are combined, it is possible to obtain an expression for the number of baryons in the observable universe. This result (11) agrees with conventional estimates, which may be seen as provisional support for the idea that the cosmological “constant” varies with scale.
Thanks for comments are due to members of the Space-Time-Matter group (5Dstm.org). |
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7. Fujitsu Supercomputer Achieves World Record in Computational Quantum Chemistry
Fujitsu Supercomputer Achieves World Record in Computational Quantum Chemistry
Solves optimization problem to reveal the behavior of 3 key molecules, contributes to research in science and technology
Fujitsu Limited,Chuo University
Tokyo, May 28, 2010
Fujitsu Limited and Chuo University of Japan today announced that a team of researchers(1) from Chuo University, Kyoto University, Tokyo Institute of Technology and Japan's Institute of Physical and Chemical Research (known as Riken) employed the T2K Open Supercomputer - which was delivered by Fujitsu to Kyoto University's Academic Center for Computing and Media Studies - to successfully compute with high precision, as a world first, an optimization problem to reveal the molecular behavior of methyl radical (CH3), ammonia (NH3) and oxygen (O2).
This accomplishment paves the way for computing the behavior of complicated molecules that cannot be seen by the human eye, by enabling researchers to gain a greater understanding of the behavior of water molecules, the properties of proteins, photosynthesis, and the mechanisms of superconductivity would also contribute to the development of new medicines and new materials. Furthermore, a wide range of potential applications is expected to emerge from this research, not only in the fields of physics and chemistry, but also in engineering and social sciences areas such as natural sciences, control design and signal/image processing.
Potential of Supercomputers
Supercomputers are computers capable of quickly performing large-scale and advanced computations that are difficult to solve using average computers. Supercomputers have received a great deal of attention as a tool for solving important issues facing human society, such as environmental problems and challenges in the medical and manufacturing fields.
One reason why supercomputers have become so important is attributable to their role in computer simulations. Computer simulations, which use computers to compute and reproduce various phenomena, have been called the "third pillar of science" alongside theory and experimentation. Computer simulation is becoming an indispensable tool in all fields of research and development, from basic research to manufacturing.
The T2K Open Supercomputer (Figure 1), which was delivered by Fujitsu to Kyoto University's Academic Center for Computing and Media Studies, is a computer equipped for handling large-scale advanced scientific computation.
Figure 1: T2K Open Supercomputer and specifications
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Many of the physical and chemical phenomena surrounding us today are governed by an equation called the Schrödinger equation (Figure 2). By being able to solve the Schrödinger equation, one is able to determine the state and energy of atoms and molecules, thereby allowing for an understanding of various phenomena.
For example, the Schrödinger equation enables scientists to determine how carbon dioxide (CO2) is transformed into oxygen (O2), what happens when two forms of matter are mixed, and how to formulate effective medicines. Through the computation of the Schrödinger equation, it is possible to explain the mechanisms of such chemical phenomena without the need for experimentation.
In reality, however, if the Schrödinger equation is precisely applied, it can become extremely complex and can turn into an enormous equation that holds little hope of being computable. Thus far, the equation has only been employed in cases where it can be relatively easily computed.
Figure 2: Schrödinger equation
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Previous Challenges
In 2001, Maho Nakata of Kyoto University (presently a researcher at Riken) and Professor Hiroshi Nakatsuji (presently of the Quantum Chemistry Research Institute) proposed a computational method for solving the optimization problem of the direct variational calculation of reduced density matrices, instead of solving the massive Schrödinger equation.
This computational method involved the use of an optimization problem computational technique called Semidefinite Programming (SDP)(2). However, the results were limited to small atoms and molecules, and faster computation of SDP became the key to performing computations for larger molecules with complicated behavior in a short amount of time.
The research team from Chuo University, led by Professor Katsuki Fujisawa, developed the SDPARA software package, based on an advanced optimization algorithm, as a high-speed SDP computational method. By running large-scale tests of SDPARA on the T2K Open Supercomputer, the team was successfully able for the first time ever to precisely compute the behavior of methyl radical (CH3), ammonia (NH3) and oxygen (O2).
During the actual computation, the matrix for the largest molecule employed in this study - ammonia (NH3) - reached a size of 19,640 × 19,640, and therefore had too many elements to be processed in a practical amount of time using average computer systems (Figure 3). By employing a supercomputer, the team was successfully able to solve the matrix in the computing time shown in Figure 4. For this computation, the T2K Open Supercomputer employed 128 nodes for its computations, utilizing a total memory volume of 4 terabytes and 2048 cores.
Figure 3: Successfully calculated massive semidefinite programming (SDP)
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Figure 4: Computation time for massive-scale semidefinite programming (SDP) in the field of quantum chemistry
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Potential Applications
As a world first, the research team succeeded for the first time ever in precisely computing the optimization problem (using SDP) to reveal the behavior of the molecules methyl radical (CH3), ammonia (NH3) and oxygen (O2). Because the methodology can compute the behavior of complicated molecules without the need for experimentation, it has the potential to be applied in a variety of fields, such as the development of new drugs and new materials, as well as applications in physics, chemistry and engineering.
In addition, this research has opened up the possibility of using supercomputers for computations in the field of superconductivity - a feat which thus far no computer has been able to accomplish yet. Furthermore, the research is expected to contribute to the development of innovations that are presently impossible in the area of energy storage (power storage), and in the medical and electronics fields.
Future Initiatives
The research team will strive to contribute to the advancement of science and technology through research that leverages high-speed supercomputers.
The large-scale supercomputer computations for this research have been supported by the Collaborative Research Program for Large-Scale Computation of ACCMS and IIMC, Kyoto University. In addition, partial software development has been made possible through the Chuo University Grant for Special Research.
• [1] Team of researchers:
Research team members: Katsuki Fujisawa, Associate Professor, Department of Industrial and Systems Engineering, Chuo University; Makoto Yamashita, Assistant Professor, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology Graduate School of Information Science and Engineering; Maho Nakata, Advanced Center for Computing and Communication, RIKEN; Kinji Kimura, Associate Professor, Graduate School of Informatics, Kyoto University.
• [2] Semidefinite Programming (SDP):
A currently evolving field that originated from linear programming, a mathematical technique. Active research on SDP is underway globally.
About Fujitsu
About Chuo University
Chuo University was founded as Igirisu Horitsu Gakko (the English Law School) in 1885 and have six faculties and their graduate schools, as well as three professional graduate schools (Chuo Graduate School of International Accounting, Chuo Law School, Strategic Management Course). The Faculty of Science and Engineering was founded in 1949 and have nine departments (Mathematics, Physics, Civil Engineering, Precision Mechanics, Electrical Electronic and Communication Engineering, Applied Chemistry, Industrial and Systems Engineering, Information and Systems Engineering, Biological Science). Number of Students of the Faculty of Science and Engineering: 4,154 (as of May 1, 2010)
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Date: 28 May, 2010
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eedeb2d86f17eaa4 | Vibrational Lab of Hcl
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Physical Chemistry Laboratory II, CHEM 3155.001
April 20, 2012
Introduction and Objective
The experimental objective of this lab was to collect an IR spectrum of gaseous HCl and from it the experimental rotational constant, B, and fundamental vibration frequency, v0, can be calculated(1). The concept of infrared spectroscopy deals with the infrared region of the electromagnetic spectrum. Molecules absorb at specific resonant frequencies that are characteristic of their structure. It will match the frequency of the bond or group that vibrates because molecules are constantly in motion, both intermolecular vibrational and molecular rotational motion. The different modes are only IR active if it is associated with a dipole. Also, infrared absorption or emission can only occur at allowed transition levels. The frequencies of electromagnetic radiation absorbed or emitted by the transition between two of these levels for a diatomic molecule fall within the range of the infrared wavelengths. This allows the transitions to be measured using the method of IR for diatomic molecules, such as HCl. Because only specific transition levels are allowed, it is concluded that the values are quantized and quantum mechanical results are related to molecular motion. To understand the information contained in the HCl spectrum, we must take into account the vibrational and rotational energy levels of the molecule. One way the energy contributions from the various sources within the molecule can be separated and treated as independent contributions, equation 1, is through the Born-Oppenheimer approximation. E=EVIB+EROT= Ev+EJ (1)
Now each of the separate energies can be modeled independently. The rotational energy of the molecule can be modeled as a rigid rotor, which treats molecules as fixed masses on a spinning bar. The exact expression of rotational energy levels for rigid rotors can be obtained by solving the Schrödinger equation, equation 2, where J is the rotational quantum number, I is the moment of inertia, and B is the rotational constant, equation 3. EJ= h28π2IJ+1=ħ22IJJ+1=BhcJJ+1 (2)
B=h8π2cI 3
The other form of the energy is vibrational energy, and it can be modeled by the harmonic oscillator, which treats molecules as balls on a spring. For a classical system of a classical harmonic oscillator, the potential function for the vibration of diatomic molecules is equation 4. V= kx22=kR-Re22 (4)
Using equation 4 in the one-dimensional Schrödinger equation gives the energy of a quantum-mechanical harmonic oscillator, equation 5, where v is the vibrational quantum number, k is the spring constant, μ is the reduced mass, and υ0 is the wavenumber, equation 6. Ev=v+12ħk/μ=v+12hυ0 (5)
υ0= 12πkμ (6)
Using equation 1, equation 2 and equation 5 can be combined into equation 7. Ev,J=v+12hυ0+BJJ+1hc (7)
The change in energy due to the change in transition levels can be measured using equation 8, where in the excited state v=1 and J=J+1 and in the ground state v=0 and J=J. ∆E=Eexcited-Eground (8)
Plugging equation 7 into equation 8 with the appropriate variables gives equation 9, which is the equation for the R branch, or left side, of the experimental spectrum, which reduces to equation 10. ∆E=1+12hv0+BhJ+1J+1+1-0+12hv0-BhJJ+1 (9)
∆E=hv0+Bh2J+2 (10)
Equation 10 can then be rearranged into equation 11, where v = E/hc and v0= v0/h. v=v0+2BJ+1 (11)
Equation 11 is a linear plot of v, the wavenumber versus J+1, where J is the rotational quantum number. The slope of the graph is 2B and the y-intercept is v0. Equation 8 can also be used where in the excited state v=1 and J=J-1 and in the ground state v=0 and J=J. Plugging these variables into equation 7 and equation 8 gives equation 12, which is the equation for the P branch, or right side, of the experimental...
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7186481d9a61e81d | This Quantum World/Implications and applications/Why energy is quantized
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Why energy is quantized[edit]
Limiting ourselves again to one spatial dimension, we write the time independent Schrödinger equation in this form:
Since this equation contains no complex numbers except possibly itself, it has real solutions, and these are the ones in which we are interested. You will notice that if then is positive and has the same sign as its second derivative. This means that the graph of curves upward above the axis and downward below it. Thus it cannot cross the axis. On the other hand, if then is negative and and its second derivative have opposite signs. In this case the graph of curves downward above the axis and upward below it. As a result, the graph of keeps crossing the axis — it is a wave. Moreover, the larger the difference the larger the curvature of the graph; and the larger the curvature, the smaller the wavelength. In particle terms, the higher the kinetic energy, the higher the momentum.
Let us now find the solutions that describe a particle "trapped" in a potential well — a bound state. Consider this potential:
Potential energy well.svg
Observe, to begin with, that at and where the slope of does not change since at these points. This tells us that the probability of finding the particle cannot suddenly drop to zero at these points. It will therefore be possible to find the particle to the left of or to the right of where classically it could not be. (A classical particle would oscillates back and forth between these points.)
Next, take into account that the probability distributions defined by must be normalizable. For the graph of this means that it must approach the axis asymptotically as
Suppose that we have a normalized solution for a particular value If we increase or decrease the value of the curvature of the graph of between and increases or decreases. A small increase or decrease won't give us another solution: won't vanish asymptotically for both positive and negative To obtain another solution, we must increase by just the right amount to increase or decrease by one the number of wave nodes between the "classical" turning points and and to make again vanish asymptotically in both directions.
The bottom line is that the energy of a bound particle — a particle "trapped" in a potential well — is quantized: only certain values yield solutions of the time-independent Schrödinger equation: |
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Letter Box
Fallacy of the Assumption of Statistical Independence Between Successive Indeterminant Events
by Thomas V. Gillman ’49
As a preface I would like to mention a concept that bears on the relation between physics as encompassed by science and metaphysics the branch of philosophy that treats of the ultimate nature of existence, reality, and experience.
Recent research in cosmology indicates that there exists a universal wave function that determines everything in the universe. This unique field gives being to the recognized physical fields-gravitational and electromagnetic forces, the strong force that among other things is responsible for the sun shining, and the weak force instrumental in radioactive decay. The further possibility exists that life is a direct expression of the effects of such a "force" field and that the evolutionary properties of living matter provide evidence of the indeterminate nature of the universal wave function.
There is a simple experiment that I believe demonstrates the existence and the workings of such a universal wave function. The results, at the very least, represent an instance of the indeterminate nature of the probability aspect of wave mechanics at the macro level.
A Heuristic Experiment
The procedure is simply a matter of flipping a coin and recording the result of each toss, head or tail, as well as the sequence of the results for an arbitrarily large number of tosses- something in the order of 100 tosses. No attempt is made to maintain a uniform time interval between tosses, since time apparently does not enter in.
The expectation is that in the long term the number of resulting heads and tails will be approximately equal, since the likelihood of a head or tail is 1/2. According to probability theory each toss of the coin is an independent event, therefore, there is not supposed to be any relation between successive tosses of the coin. The probability of any particular sequence of heads or tails is therefore the product of their "independent" probabilities. For example, the probability of tossing seven heads in succession would be (112)7 or 1/128-a fairly unlikely sequence of events but well within the range of expectation. On the other hand, it is common knowledge that in many games of chance players often experience "runs of luck" in which the outcome temporarily favors them. How are such unlikely courses of events to be explained?
In conducting this experiment it is common that over the course of 100 or so tosses a sequence of at least six or more heads or tails will occur. Even longer sequences are interrupted by only one or two inverse events, thus establishing a trend or a "run" as it is often described.
Graphic Results
If one plots the sequence of tosses on the horizontal axis and the algebraic results of the coin tossing on the vertical axis, with the simple assumption that each toss represents a unit gain or loss of some sort of "potential" from one toss to the next, some fascinating patterns emerge. These correspond to the so-called "runs" of good or bad luck that gamblers experience. A more interesting finding is that these deviations tend to propagate or persist. That is, the number of heads or tails sometimes does not even out for long sequences. [Note that these sequences are time independent and therefore do not represent periods, but they do seem to indicate a "progression".]
A question that comes to mind is whether or not the cumulative "potential" indicated on the graphs provides evidence for the existence of a deterministic element that enters into these results. The gambler will tell you that when he is on a "roll" he is able to "influence" the course of events. Who is to say? What is obvious is that if one bets in concert with one of these "potential" swings-these apparent "drifts" of the probability function-one is going to be ahead of the odds for an indeterminate but substantial number of events!
Whereas in the previous plot, the number of opposing events (heads and tails) is not far from the expected proportion, viz., 50:50, this is not always the case, as shown in the following graph.
While there is little basis for conclusion at this stage, the results do lead to speculation. The partially determinate nature of the outcomes of events that have traditionally been treated as indeterminate, random, and independent may point to the possible fluctuations of something of the nature of a general field which "determines" the course of events. Events at the macro level are usually analyzed in terms of cause and effect, but there are many events where the outcomes cannot be predicted but can only be described statistically.
Another instance is radioactive decay at the subatomic level. Anyone who has listened to a Geiger counter knows that these events occur randomly but in time-wise bursts that exhibit no regularity. And there is no way of predicting the decay of a particular radioactive atom in terms of time or location. The best we can do is to establish a so-called half-life — a period during which half of the atoms originally present will have decayed. We have no way of knowing what conditions, if any, "cause" the occurrence of the decay event.
The most that we know, at present, is something about the elementary particles of matter that are involved in the process. It has been discovered that, in the vernacular of elementary particle physics, the '"weak" interaction causes radioactive decay of nuclear constituents and unstable leptons, and is mediated by the massive W and Z bosons. Whatever precipitates (causes) these weak interactions within a space-time frame is unknown, at least as far as I am aware.
Further speculation
Now wouldn't it be interesting if it were found that what we call probability is nothing more than the way in which the occurrence of events is modulated by something in the nature of a general wave? Further, wouldn't it be a kick if the effects of a general field are reflected in the activities of living matter, the primary characteristic of which is purposiveness or goal-oriented behavior?
Suppose that living matter has the power to causally influence the outcome of events. This would help to explain the apparent evolutionary discontinuities that are reflected in the geologic record. This leads to the further possibility that evolution occurs, not as a result of environmental change, but reflects the implicit capability of living matter to affect change as a way of adapting to changing environmental requirements or opportunities? This is in direct contrast to Darwin's theory of the survival of the fittest, or the occurrence of natural selection among the chance variants or "sports" that are speculated to arise spontaneously.
Running parallel to such speculation is the heuristic work of the engineer and behaviorist William Powers. (William T. Powers Living Control Systems: Selected Papers (Gravel Switch, KY: The Control Systems Group, Inc., 1989).) He shows that control in living systems is neither subject to chance nor to the causal control of outside agencies. Behavior is not a direct response to external stimuli but is under the direct and nonprobabilistic control of feedback mechanisms built into the organism. We find that this cybernetic mechanism is typical of living organisms and, therefore, is a major design aspect implicit in the life functions.
Returning to a consideration of the results of the coin-toss experiment, the necessary next experiment would be to look to the identification of something in the nature of a bifurcation that will predict the onset of another probability swing or trend in the course of action. That appears to be the nature of evolutionary change, indeed of all change. If such change can be controlled, as we attempt to do through planning, then we have evidence for the intervention of a life force (willpower?) in the determination of the outcome of events.
The Statistical Postulate of Quantum Mechanics
In a discussion of quantum mechanics, physicist Victor J. Stenger (ViCtor J. Stenger, The Unconscious Quantum (Amherst, NY: Prometheus Books, 1995), pp 56-60) indicates:
"In 1926 Max Born proposed what was to become a primary postulate of quantum mechanics in the von Neumann scheme. According to this postulate, the wave function is used to compute the probability P for a particle to be found in a particular state. This probability was to be proportional to II2, the square of the magnitude of the wave function .... This postulate was extended by Wolfgang Pauli to include the probability for finding a particle at a particular position.
"Pauli proposed that the probability P for finding a particle in an infinitesimal volume element AV located in a specific region of space is equal to the square of the magnitude of the wave function computed at that point multiplied by V: P = II2 AV. Since we can measure volume in any units we wish, no loss of generality occurs if we assume a unit volume, V = 1, and simply write P = II2 and understand it to mean probability per unit volume, that is, probability density.
Paraphrasing Pauli's postulate in terms of my conjecture about a universal wave: The probability of an event, that is, conversion of the energy of the universal wave into a material state at a particular place (the result of the toss of a coin is thought of as equivalent to the conversion of energy into a particle) is equal to the square of the wave function,. Mathematically, squaring of the quantum "state" converts it from imaginary and complex to real and rational, and this would be analogous to the occurrence of a real event. A positive value may correspond to a constructive (energy-binding) event typical of the action of living systems, while a negative value would represent a destructive (entropic) event.
Stenger mentions that the role of statistics in quantum mechanics was supported by Einstein's calculation of the probabilities for atomic transitions; however, the uncertain nature of its predictions was one of the aspects that Einstein found unsatisfying about quantum mechanics. Einstein is well known for having said, 'God does not play dice,' What he was really objecting to was the notion that statistics was the final word. He found it hard to accept that no underlying causal laws determined the behavior of individual quantum particles at the most fundamental level. As Stenger explains,
"Actually, statistics enters quantum mechanics only in an indirect way. The time-dependent Schrödinger equation predicts the exact value of the wave function at future times given its value at some initial time. Probability enters with the Bom postulate when the time comes to make a prediction on the expected value of some measurement."
When such a measurement is attempted, however, we are attempting to calculate at some instant in time when in fact the application of the Schr6dinger equation varies constantly over time. So we are left with a probability distribution rather than a specific value at tx.
Thereby, quantum mechanics is often said to be "deterministic" in that its basic equation, the Schrödinger equation, precisely determines the time evolution of the wave function. However, it is indeterministic in the sense that knowledge of the wave function is not always sufficient to predict the outcome of a measurement — or of an event.
By the probability postulate, the wave function allows for the prediction of the average motion of a system [the probable outcome of a coin toss] but not the outcome in any particular instance, which is what the above experiment demonstrates. I believe that we are approaching the time when some deterministic theory will evolve that goes beyond quantum mechanics and which applies to individual quantum systems as well as to causal events at the macro level.
According to the recent theoretical development of Dr. Frank Tipler, we have an all-pervasive physical field which gives being to all being — which gives life to all living things-and which itself is generated by the ultimate life which it defines. Through this "physical" field we humans are apparently capable of superimposing our own wills on the ordinarily indeterminate laws of probability and the chaotic physical laws that prevail throughout the universe. This we do when we exercise our intellect and creativity.
This is some of the speculative thinking that can serve as a precursor to further theoretical development — and it comes from reaction to a portion of an article by Billy Goodman '80 in the January 29, 2003, issue of PAW entitled "Thinking about Thinking." Now, what are the odds against that development?
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3e9b91bde4f0a32f | Copyright © 2004 jsd
How to Draw Molecules ...
Just Like Lewis Dot Diagrams, Only Easier & Better
1 Introduction
Our primary goal for today is to be able to draw diagrams of molecules, such as we see in figure 1.
Figure 1: A Few Small Molecules
These diagrams tell us that the F2 molecule has a single bond, the CO2 molecule has two double bonds, and the HCN molecule has one single bond plus one triple bond.
There are two main methods for constructing such diagrams.
* Contents
1 Introduction
2 How to Draw Molecules
2.1 Electron-Rich Molecules
2.1.1 Fluorine
2.1.2 Nitrogen
2.1.3 Oxygen
2.1.4 Cyanic Acid and Isocyanic Acid; Formal Charge
2.1.5 Acetic Acid
2.1.6 Sulfur Dioxide
2.1.7 Sulfate Ion and Sulfuric Acid
2.1.8 Methanesulfonic Acid
2.1.9 Biomolecules
2.1.10 Summary
2.2 How to Explain Hole-Bonding
2.3 Spectroscopy and Energy Levels
2.4 Limitations of the Naive Approach, and How to Overcome Them
3 Some Supporting Ideas
3.1 The Aufbau Principle
4 Approximate Bond Angles
4.1 Basic Ideas
4.2 Some Examples
4.2.1 Methane
4.2.2 Ammonia
4.2.3 Water
4.2.4 CO2 and the Energy of Double Bonds
4.2.5 HCN, C2H2, and the Energy of Triple Bonds
4.2.6 CO3=
4.2.7 Nitrogen Dioxide Radical
4.2.8 Other Cases
5 Some Peculiar Examples
5.1 The Nitro Group
5.2 Sulfur Hexafluoride
5.3 Sulfur Tetrafluoride
6 Overview of the Contrast
7 Fun With Lewis Dot Diagrams ... Or Not
7.1 Hydrides and Other Successes
7.2 Oxygen
7.3 Sulfate Ion; Expanded Octets
7.4 Methanesulfonic Acid
7.5 Divergent Predictions
7.6 Additional Remarks
8 Conceptual Foundations (or lack thereof)
8.1 Contrast: Molecular Orbitals versus Lewis
8.2 Contrast : Hole Bonding versus Lewis
8.3 What Implies What
8.4 The Aufbau Principle (or not)
9 Discussion
10 Additional Background and Context
10.1 The Hole Concept
10.2 Coulombic Models
11 Summary
12 References
* PART I
2 How to Draw Molecules
In this section, we discuss a way of diagramming molecules that has many advantages. It is in all ways preferable to Lewis dot diagrams.
The diagrams provide predictions about the number of bonds. This is important, because it implies, among other things, predictions about the existence, non-existence, and reactivity of molecules. That is, when we learn there are four single bonds in CH4 and two double bonds in CO2, we also learn that reactions that produce molecules of CH4 are vastly more plausible than reactions that produce molecules of CH2, and similarly molecules of CO2 are more plausible than molecules of CO4.
2.1 Electron-Rich Molecules
Our first goal is to draw (and understand!) some simple molecules. Before we start talking about rules, let alone theories, let’s look at some data.
N≡N triple bond 5 electrons in the N=2 shell
O÷O two units of bond strength 6 electrons in the N=2 shell
F−F single bond 7 electrons in the N=2 shell
Ne Ne no bond 8 electrons in the N=2 shell
The trend is clear: In this part of the periodic table, more electrons means less bonding. We shall see that this trend holds true for a broad class of interesting and important molecules (but not for all molecules).
We can construct a simple rule for drawing such molecules and predicting the number of bonds. We will begin by stating the rule and applying it to some examples, and then in section 2.2 we will explain the rationale behind the rule.
The procedure begins with figure 2, which can be considered a fragment of a simplified periodic table. Hydrogen is shown with one dot, representing one hole, because it is one electron shy of the helium closed-shell configuration. Similarly fluorine has one hole, because it is one electron shy of neon.
Figure 2: Hole-Count of Some Common Atoms
This is just the venerable idea of valence, with the twist that the dots represent hole-valence, not electron-valence. The elements in figure 2 all have the property of being “electron rich” which means they have relatively few holes.
This meaning of the word “hole” has been in use for many decades. You may have encountered it in connection with P-type semiconductors. It is pretty much synonymous with “vacancy”. It means nothing more or less than a place where an electron could have been, but is not. In particular, holes could perfectly well be high-lying vacancies, not just low-lying vacancies. (This stands in contrast to ordinary back-yard holes, which always extend downward, below the general terrain.)
2.1.1 Fluorine
We can use the fluorine molecule as our first example. To diagram the F2 molecule, we start by drawing the atoms. Figure 3 shows two independent fluorine atoms. Each carries one hole, as we know from figure 2.
Figure 3: Two Fluorine Atoms
We are free to slide the holes around. Figure 4 shows the same thing, with the holes rearranged to face each other.
Figure 4: Two Fluorine Atoms, Rearranged
Finally, we connect the dots. Two holes make one chemical bond, as shown in figure 5.
Figure 5: Diagram of the Fluorine Molecule
Unlike a Lewis dot diagram, this figure does not portray electrons, but rather holes. (The notion of holes has been part of physics since 1930, as discussed in section 10.1.) That is, each little dot in the diagram represents the absence of an electron from an antibonding orbital.
Notice the logic here: In electron-rich molecules, what we casually call “bonding” might more logically be called absence of antibonding. It’s a double negative.
2.1.2 Nitrogen
As our next example, we turn to nitrogen. The procedure is the same. To diagram the N2 molecule, we start by drawing the atoms. Figure 6 shows two independent nitrogen atoms. Each carries three holes, as we know from figure 2.
Figure 6: Two Nitrogen Atoms
Next, we slide the holes around so that they face each other. Finally, we connect the dots. Two holes make one chemical bond. The nitrogen molecule contains a triple bond, as shown in figure 7.
Figure 7: Diagram of the Nitrogen Molecule
Figure 7 is the normal, conventional way of diagramming the N2 molecule. Meanwhile, figure 8 shows the same thing with a little extra detail. It shows three antibonding orbitals (in color) each of which bears two holes.
Figure 8: Extra-Detailed Diagram of the Nitrogen Molecule
2.1.3 Oxygen
Let’s consider oxygen, which is a slightly tricky case. If you blindly follow the simplified procedure outlined above, you will wind up with a double bond. That is, the molecule will be diagrammed as O=O.
However, we are allowed to use additional information.
For any (or all!) of those reasons, we can conclude that oxygen contains unpaired electrons. Therefore diagramming the molecule as O÷O is vastly preferable to O=O. The preferred diagram is also shown in figure 9.
Figure 9: Diagram of Oxygen Molecule
Figure 9 shows the O2 molecule with a moderate level of detail. Meanwhile, figure 10 shows the same thing with a little extra detail. It is useful to compare this against figure 8. In both cases we have three antibonding orbitals. The difference is that in nitrogen, the orbitals are fully populated, for a total of three units of bond strength, while in oxygen, there is only fully-populated orbital and two half-populated orbitals, for a total of two units of bond strength.
Figure 10: Extra-Detailed Diagram of Oxygen Molecule
To repeat: hole-counting alone does not suffice to tell us whether O÷O or O=O is the preferred representation of the oxygen molecule. We need additional evidence from observation and/or theory to tell us that O÷O is the right answer. This is discussed more fully in section 2.4.
2.1.4 Cyanic Acid and Isocyanic Acid; Formal Charge
To arrive at the most appropriate structure, a good rule of thumb is to distribute the bonds so as to minimize formal charge. This tends to minimize the electrostatic potential energy, which is an energetically favorable thing to do, other things being equal.1
The procedure is to break each bond apart into two holes. (That is, break each dash into two dots.) For the bond between atom A and atom B, assign one of the dots to each atom. Then compare the number of dots assigned this way to the number of dots the atom “normally” has, where normal is defined by the table in figure 2. If the assignments are the same, the atom has zero formal charge. If the atom has more holes than normal, it bears a positive formal charge, and if it has fewer holes than normal, it bears a negative formal charge. Remember, these dots represent holes, not electrons, and each hole makes a positive contribution to the formal charge.
The idea of assigning half of each bond to each of the atoms makes perfect sense for symmetrical molecules, but is only a first approximation in non-symmetrical molecules. It is certainly not a law of nature.
Applying this rule gives us:
H−O−C≡N cyanic acid
H−N=C=O isocyanic acid
It is certainly possible to imagine something like H−O=C=N, but it would be disfavored due to needlessly high electrostatic potential energy. It would have positive formal charge on the oxygen, and negative formal charge on the nitrogen ... unlike the formulas in equation 1, which have zero formal charge on each of the atoms.
It is amusing to note that both of these acids have the same conjugate base, namely the cyanate ion:
N≡C−O((−)) cyanate ion
The symbol to the right of the O in equation 2 is not a dash; it is a superscript minus sign. I know a minus sign looks a lot like a dash, but in this case it is a minus sign. You can recognize it as such partly because it is up in the exponent, and partly because it does not join two atoms.
You could imagine something like O=C=N((−)) (which could be called the isocyanate ion), but since oxygen is more electronegative than nitrogen, it makes sense to put the negative charge on the oxygen.
Note: Electrons are very much more mobile than protons. The isocyanate ion would convert immediately to the cyanate ion. In contrast, unionized isocyanic acid converts slowly, if at all, to cyanic acid.
2.1.5 Acetic Acid
Let’s move on to a larger and more complicated molecule, namely acetic acid. The first step, as always, is to figure out what molecule we’re talking about. In this case its systematic name is ethanoic acid, and its formula may be written CH3COOH.
The easiest way to proceed is to draw the molecule, placing next to each atom one dot for each hole that it contributes – counting down from the appropriate atomic closed-shell configuration – as indicated in figure 2. Therefore the acetic acid molecule looks like
H O
. ..
. ..
H. .C. .C
. .
. .
H O. .H
Then we just connect the dots. Observation tells us this molecule is non-paramagnetic, so the simple rule “two dots make a dash” is all we need. The result is:
H O
| //
H - C - C
| \
H O - H
Each dash represents one bond. For electron-rich molecules such as this, each bond is due to holes (i.e. the absence of electrons) in a high-lying antibonding orbital.
With a little experience, you discover that it is possible to save some work by treating the methyl functional group as a black box, like this:
(CH ) - C
3 \
O - H
or equivalently
Me - C
O - H
It is also interesting to see what happens when the acetic acid molecule reacts with water. In the classical approximation, the result can be drawn as:
// (+)
Me - C H - O - H
\ ((-)) |
O H
Acetate Hydronium
Ion Ion
We see that on the acetate ion, there is an oxygen with a formal negative charge, and only one hole associated with it, while on the hydronium ion, there is an oxygen with a formal positive charge, and three holes associated with it.
In this diagram, the formal charges are indicated in double parentheses, because they don’t tell you anything you don’t already know. If the indications were missing, you could easily reconstruct them based on the valence of each atom and the number of bonds it has.
The rest of this subsection is tangential to our main discussion, but it has tremendous practical ramifications, so we’d better mention it: In the classical approximation there are two different ways of drawing the acetate ion, namely
O O
/ //
Me - C <--> Me - C
\\ \ (-)
O O
Either one of the oxygens could get the double bond, while the other one gets the formal negative charge. Actually neither of these alternatives is really correct, for the same reason that neither of the Kekulé structures correctly describes benzene. In fact you get a quantum-mechanical resonance. That is, the superposition of the two alternatives has a lower energy than either of them separately. The process of resonance involves moving an electron from one oxygen to the other. This only works in the ion. In contrast, in the neutral acid molecule, there are two alternatives, but the rate of quantum-mechanical tunneling from one to the other is negligibly small, since it would require moving a proton, not just an electron. This is important, because it means the ion’s energy is resonantly lowered while the neutral molecule’s energy is not lowered. This makes the acid much stronger than it would otherwise be, by making it energetically favorable for the acid to get rid of its proton.
2.1.6 Sulfur Dioxide
Now let’s examine sulfur dioxide. As usual, the starting point is to draw the atoms with the right number of dots, i.e. the right number of holes. This gives us a rough idea of what’s going on, as shown in figure 11.
Figure 11: Sulfur Dioxide : Starting Point
Next, as usual, we see if we can combine the dots into bonds, to create a ball-and-stick model of the molecule. The two possibilities are shown in figure 12.
Figure 12: Sulfur Dioxide : Two Ball-and-Stick Models
The final answer will be some sort of resonance i.e. a quantum mechanical superposition of the two ball-and-stick models. This is shown in figure 13.
Figure 13: Sulfur Dioxide : Symmetric
If you want to attribute half a unit of negative formal charge to each oxygen atom, and a full unit of positive charge to the sulfur atom, that is not necessary but is harmless provided you don’t take it too literally. There is no well-defined dividing line that determines which part of the electron cloud “belongs” to one atom or another.
In any case we know from experiment that the SO2 molecule is bent and has a nonzero electric dipole moment. (Contrast this with CO2 which is linear and has no dipole moment.)
Also, spectroscopy and molecular orbital calculations tell us that the 3d orbitals make no observable contribution to the bonding in SO2.
As always, diagrams of the sort we are constructing here describe covalent bonding. A bond between dissimilar atoms will exhibit some percentage of ionic character. A single bond between sulfur and oxygen should have roughly 22% ionic character.
2.1.7 Sulfate Ion and Sulfuric Acid
Another example that illustrates the simplicity and power of the hole-bonding approach is the sulfate ion, SO4=. As usual, we start by drawing the atoms, with the correct number of dots, where each dot represents a hole in an antibonding orbital. We started with ten dots, because each of the five atoms is divalent according to figure 2, but we erased two of the dots because of the double negative charge carried by the ion. Remember the dots represent holes, and the two negative charges fill those holes.
Figure 14: Sulfate Ion
We then replace pairs of dots with dashes representing a two-hole bond. The diagram practically draws itself. The molecule is tetrahedral and completely symmetric:
Figure 15: Sulfate Ion : Hole-Pair Bonds
If you want to think of this as having a +2 charge on the central sulfur atom, and a −1 charge on each of the four oxygens, that’s optional but harmless. That is consistent with basic electronegativity ideas, and semi-quantitatively consistent with the electron-density data from detailed quantum chemistry calculations.
There is no trouble drawing a sensible diagram for the sulfuric acid molecule, as shown in figure 16.
Figure 16: Sulfuric Acid
This stands in contrast to the usual Lewis dot approach, which has some problems, as discussed in section 7.3.
2.1.8 Methanesulfonic Acid
A diagram of the methanesulfonate ion is shown in figure 17. In the ion, all three oxygens are equivalent. In contrast, in the methanesulfonic acid molecule, as shown in figure 18, the acid hydrogen breaks this symmetry.
Figure 17: Methanesulfonate Ion
Figure 18: Methanesulfonic Acid
When we check the formal charge, we find a formal charge of +2 on the sulfur atom, and a formal charge of −1 on each of the three oxygen atoms in the methanesulfonate ion. This is consistent with the overall charge (−1) on the ion. This is also consistent with what we know about electronegativity; oxygen is considerably more electronegative than sulfur.
Similarly in the acid molecule, we find a formal charge of +2 on the sulfur atom, zero formal charge on the acid oxygen, and a formal charge of −1 on the other two oxygens. Again this is consistent with the overall charge balance, and consistent with what we know about electronegativity.
For a discussion of how our diagrams differ from the prior art, see section 7.4.
It must be emphasized that the minus sign in double parentheses in figure 17 is strictly parenthetical; you could erase it and the diagram would have the same meaning. You can verify and/or reconstruct the parenthetical minus sign by adding up the formal charge on each atom in the drawing. It would be quite wrong to calculate the formal charge according to the drawing and then add the parenthetical charge; that would be counting the same charge twice.
Such a parenthetical charge indication stands in contrast to a formula such as H, which is an ion, unlike H which is a neutral atom. The minus sign in the H formula is not parenthetical. Similarly OCN is an empirical formula, and the minus sign is necessary to indicate its charge, in contrast to N≡C−O which is a structural diagram, and the charge can be inferred from the structure, namely N≡C−O((−)).
In mathematics, parentheses are used for grouping, with no implication that the contents are merely parenthetical. In written English, parentheses are used for parenthetical remarks.
I use double parentheses when I want to make it clear that something is parenthetical, not a grouping.
2.1.9 Biomolecules
Figure 19 shows a sketch of a molecule with a fair bit of biological significance, namely glucose.
Figure 19: Glucose Molecule
To sketch this, you need not only the empirical formula, but also some clue about which isomer is desired. The isomer shown here is quite common in nature. This figure is meant to show the number of bonds and the topology (i.e. what is connected to what); it is not meant to show every detail of the three-dimensional shape.
Most biomolecules are sufficiently electron-rich that the simplest hole-counting notions suffice.
Figure 19 is what you get by using the hole-counting method. It is consistent with the way professional biologists and biochemists sketch this molecule. If you doubt this, go to http://www.google.com/search?q=glucose and look at the results.
We are quite aware that hole-counting is not the only way of predicting the topology and the number of bonds. There is another method that is almost universally taught in high-school chemistry textbooks ... but it is not consistent with the known facts about molecules, and is not consistent with the way professionals draw things. This is discussed in Part II.
Hole-counting agrees with real-world practice.
2.1.10 Summary
To summarize this section:
Hole-bonding diagrams correctly predict the number of bonds in every case where Lewis dot diagrams had any hope of making correct predictions ... and they have the advantage that they can be extended to additional cases, plus the advantage of being consistent with the paramagnetism data, the spectroscopic data, et cetera.
2.2 How to Explain Hole-Bonding
At the simplest level, we can explain the hole-bonding method by saying there is abundant observational and theoretical evidence that antibonding orbitals exist.
If somebody asks what, exactly, is the evidence, we can either tell them, or (if necessary) say that the full details are beyond the scope of the course.
If you want a mental picture that folks can use to make the antibonding idea more concrete, you can model each antibonding orbital as being a spring in compression, trying to destroy the molecule by pushing the atoms apart. In particular, it behaves like an air-spring, i.e. a parcel of gas under pressure, pushing the atoms apart. Each bond that we draw represents a net attraction, because it portrays the absence of a repulsive force.
For a truly introductory treatment, stop here!
To a more-sophisticated audience, I would point out that the air-spring analogy is really rather good. The pressure of the gas comes from the kinetic energy of the particles ... as does, ultimately, the repulsive force of the antibonding orbitals.
If people can accept that the sun’s size is determined by a tradeoff between an attractive interaction (gravitational PE) and an outward pressure (due to the KE of the particles), they ought to be able to tolerate the idea that atoms also make a tradeoff between an attractive interaction (electrostatic PE) and an outward pressure (KE again).
At the next level of detail, I remark that because of degeneracy, electrons produce a greater amount of pressure than you would have expected based on a naïve classical model. That is, an atom is more like a neutron star than like the sun, as discussed in reference 1.
Also at the not-quite-introductory level, I would point out that compared to F2, O2 has fewer electrons but more bonding. The same remark applies at every step in the sequence Ne2 → F2 → O2 → N2, in the sense that progressively fewer electrons means progressively more bonding. The number of bonds is observable in terms of the nonexistence of Ne2 and the progressively increasing binding energy in the rest of the sequence. This is direct evidence in support of the antibonding idea: we see progressively fewer electrons and progressively more bond-strength. So ... this basic reactivity data provides evidence in support of the hole-bonding idea. What could be more appropriate than that? It’s not necessary to make it more complicated than that.
2.3 Spectroscopy and Energy Levels
At the next level of detail, I would point to the spectroscopic data, which has been around for over 100 years, as strikingly direct evidence for molecular orbitals.
A good way to visualize the orbitals, and the filling of orbitals, is to draw energy level diagrams such as figure 20 through figure 23. If you want more information about what these diagrams mean, see e.g. reference 2.
In all such diagrams, energy increases vertically. The energy is not shown to scale, but the ordering of the levels is shown correctly. The electrons in the core levels are shown in gray, because they almost never participate in chemical reactions.
The bonding versus antibonding character of each level is encode three ways in the diagram: The tips of the level turn up for bonding levels, and turn down for antibonding orbitals. The name of the level has a “*” in it for antibonding orbitals, and not for bonding orbitals. Finally, off to the right, the levels are explicitly labeled “antibonding” or “bonding”.
The name of each level tells us something about the symmetry of the molecular orbital wavefunction, and about how it can be derived from the atomic orbitals of the atoms that make up the molecule.
In all of these four cases, there are three high-lying antibonding orbitals.
Figure 20: F2 Energy Levels
Figure 21: O2 Energy Levels
Figure 22: N2 Energy Levels
Figure 23: C2 Energy Levels
Figure 24: B2 Energy Levels
2.4 Limitations of the Naive Approach, and How to Overcome Them
The naïve hole-counting procedure as described above is more-or-less mechanical and mindless. It correctly predicts the number of bonds in all situations where the Lewis fairy-tale had any hope of giving the right answer, namely in all situations where the molecule is sufficiently electron-rich that all its bonding orbitals are filled. (That means, among other things, that adding an electron to the molecule – for instance by replacing C with N, or replacing N with O – will reduce the net number of bonds by half a unit, by occupying an antibonding orbital.)
For molecules that are not so electron-rich, naïve counting doesn’t suffice. The Lewis approach has no hope of getting the right answer for such molecules, and cannot be repaired or extended. In pleasant contrast, the molecular-orbital description does of course get the right answer. The canonical illustration of non-naïve counting is dicarbon. The octet fairy-tale predicts that it should have a quadruple bond, but the real molecule has only a double bond. As we move along the series F−F, O÷O, N≡N, every time we remove two electrons we increase the number of bonds, by depopulating antibonding orbitals, as shown in figure 20 through figure 22. But when we extend the series, stepping from dinitrogen to dicarbon, we remove two electrons from a bonding orbital, as shown in figure 23. Therefore the number of bonds goes down to 2, not up to 4. If all we cared about was bond-strength, we would blissfully write C=C ... but that would give a misleading picture of the hole-count. In some cases, we need a picture that assigns the correct formal charge in dicarbon. In such cases, the preferred picture is C/=\C, where
The symbol C/=\C tells us we have a total of eight holes, i.e. four holes per carbon, i.e. a net neutral charge ... and a net bond strength of two.
To repeat: you can’t get the right answer for dicarbon just by naïve counting; you have to actually know something about the molecule. That is, you have to look at the energy levels of the molecular orbitals, as shown in figure 23. For more information, see e.g. reference 2.
Similarly, for very electron-poor molecules such as Li2, you are better off counting electrons than counting holes. We depict this differently, namely Li∼Li in which the curly ∼ symbol represents the presence of bonding electrons (counting up from helium) not the absence of antibonding electrons (counting down from neon).
Amusingly, dicarbon can be represented in two ways, either as C/≈\C (counting up from helium) or as C/=\C (counting down from neon):
As a separate issue, naïve counting does not suffice to predict that O÷O is a better description of the oxygen molecule than O=O. We start by considering both possibilities, and then use the available experimental and/or theoretical information to decide which version is better. One option is to observe that liquid oxygen is attracted by a magnet. (In contrast, liquid nitrogen shows no comparable attraction.) Alternatively, you don’t even need a magnet, and you don’t need a fancy spectrometer: If you put liquid oxygen and liquid nitrogen in transparent flasks, you can see from across the room that they are different. The oxygen has a pale blue color, due to the unpaired electrons. A third option is to look at the molecular orbital energy-level diagram, where you discover that there are two degenerate orbitals into which the highest-lying electrons can be placed. In accordance with Hund’s rule, the electrons will unpaired whenever there are enough energetically-accessible orbitals to make unpairing possible. All in all, we have multiple lines of evidence telling us that O÷O is vastly preferable to O=O.
Similar remarks apply to B2. A glance at figure 24 allows us to predict that it has one unit of bond strength, that it is paramagnetic (which it is), and that its spectrum shows a triplet Zeeman splitting of the ground state (which it does). To figure out what is going on in this molecule, it is easier to count electrons rather than holes, counting up from helium rather than counting down from neon (although counting down would give the right answer also). To depict this molecule in a way that shows the bonding as well as the antibonding, you can write B\;/B. Here we use a semicolon instead of a colon to indicate that we are counting electrons, not holes. Similarly we write the slanted lines the other way (i.e. \ / instead of / \) to indicate electrons not holes.
In general, if you observe that a molecule is paramagnetic, it tells something about the degeneracy of the highest molecular energy level, and conversely if you know the energy levels you can predict the paramagnetism. This is in stark contrast to the Lewis approach, which makes irreparably wrong predictions about O2, and cannot describe B2 at all.
Consider the contrast:
Here’s the central unifying idea: In general, you need to think about the molecular orbitals. You need to figure out which bonding orbitals are occupied and which antibonding orbitals are occupied. For electron-rich molecules, all the bonding orbitals are filled, so a simple mechanical hole-counting procedure suffices. Hole-counting is not fully general, but its domain of applicability covers a remarkably large number of molecules, including most of the molecules commonly encountered in organic chemistry.
Some peculiar additional examples are discussed in section 5.
3 Some Supporting Ideas
3.1 The Aufbau Principle
Aubau is the German word for “building up”.
The Aufbau principle has been known since the earliest days of quantum mechanics. It is a rule of thumb, not a rigorous theorem. Among other things, it tells us that when we go from O2 to F2, we expect the two molecules to have qualitatively similar energy-level diagrams, which is just what we see in figure 21 and figure 20. The main difference is that in F2, the levels are more fully occupied. Same levels, more occupation.
There are exceptions to the Aufbau principle. You can see that in going from N2 to O2, there is one pair of levels that re-order themselves. But the rest of the energy-level diagram is pretty much undisturbed.
It is worth noting that as we go from O÷O to F-F, the number of bonds decreases. F-F has more electrons but less bonding. That is consistent with the Aufbau principle, because we are not depopulating pre-existing bonding orbitals; we are populating a new antibonding orbital.
For more on this, see section 8.4.
4 Approximate Bond Angles
4.1 Basic Ideas
It is important to know the shape of molecules. For instance, the well-known properties of water depend crucially on the fact that the molecule is bent, not linear in shape.
Lewis dot structures are often praised as being a stepping-stone toward VSEPR (Valence Shell Electron Pair Repulsion) which is a convenient method for predicting bond angles. Like Lewis dot diagrams, VSEPR is based on the unjustifiable idea of molecular octets, and an unjustifiable focus on bonding orbitals (to the neglect of antibonding orbitals).
The purpose of this section is to point out that approximate bond angles can be predicted without basing the prediction on “molecular octets”. This is the final nail in the coffin of “molecular octet” notions.
This will be presented as an empirical rule of thumb, without any deep derivations.
The ideas here in section 4 are much less well-founded than in other sections. When we say to use a wooden molecule of a carbon atom, pre-drilled with four holes in a tetrahedral pattern, you may reasonably ask “why” we should start there. The answer is that empirically and retrospectively, starting there leads to qualitatively-correct predictions in a goodly number of simple cases.
Disclaimer: Let’s be clear: The models presented here in section 4 are partly bogus and partly fortuitous.
You shouldn’t stake your life on any predictions made by such models … but let’s keep things in perspective: These models are in no ways worse, and are in some ways better, than models based on Lewis “octets” in molecules. The a posteriori justification is about the same, and the a priori wrong assumptions are fewer.
So here it is: The rule-of-thumb is that for elements in the N=2 and N=3 rows of the periodic table, the bonding pattern will be tetrahedral, or a subset of tetrahedral, unless there is compelling reason for it to be otherwise.
4.2 Some Examples
4.2.1 Methane
As a first example, methane is tetrahedral. That’s not very tricky. Why shouldn’t it be tetrahedral? It is easy to build a hands-on model of this, using wooden balls for atoms and tiny coil springs for the bonds. The carbon atom is pre-drilled with four holes in a tetrahedrally-symmetric pattern. Pictures can be found in reference 3.
4.2.2 Ammonia
An important, interesting example is ammonia. Let the vertices of the basic tetrahedron be called A, B, C, and D. In the classical approximation,2 sites B, C, and D are occupied by ligands, while the A site is unoccupied. This is an example of what I call a subset of the tetrahedral configuration. This particular subset is called trigonal pyramidal ... but that’s just another way of saying tetrahedral with one vacancy. Again it is easy to build a hands-on model of this. Again the carbon atom is pre-drilled with four holes in a tetrahedral pattern. Three of the holes receive bonds, while the fourth one goes vacant. Pictures can be found in reference 3.
4.2.3 Water
The next example is water. The hydrogen ligands occupy sites C and D, while sites A and B are unoccupied. I am not saying that A and B are occupied by “lone pairs”(which is what VSEPR would say); I’m just saying they are unoccupied. This configuration is properly called bent ... but that’s just another way of saying tetrahedral with two vacancies.
I am explicitly ducking the question of detailed bond angles. The theory, at its present state of development, predicts symmetry, not details. In the case of water, it says that the molecule is bent, not straight. It says that the bond angle is approximately the tetrahedral angle, but does not predict whether the angle is slightly greater or slightly less than the mathematically-perfect tetrahedral angle.
4.2.4 CO2 and the Energy of Double Bonds
Similarly carbon dioxide is linear. It is conventional to diagram the molecule as O=C=O ... but that is somewhat misleading, because it suggests that all four bonds are straight and parallel to each other. A somewhat more informative diagram is shown in figure 25.
Figure 25: Carbon Dioxide : Linear Molecule, Bent Bonds
At this level of approximation, we can say that the four bonds depart the central carbon atom in the four tetrahedral directions, and then bend as necessary to terminate on the oxygen atoms. Two of the bonds lie are in the xy plane, while the other two lie in the xz plane.
Again it is easy to build a ball-and-spring model of this. The model depends on the fact that the bonds are bendable and springy. Pictures of such models can be found in reference 3.
Here’s why this is important: We expect a double bond to be stronger than a single bond, but nowhere near twice as strong (other things being equal). We can explain this by saying in a double bond, the bonds are bent and under stress, which costs energy. We can understand this energy as follows: The holes bear a positive charge, so there is some electrostatic energy involved in bending them closer together. There is also an increase in kinetic energy, since momentum depends directly on the curvature of the wavefunction, and kinetic energy depends on momentum squared. A tetrahedral arrangement of straight bonds has minimal potential energy (since the bonds are as far apart as possible) and minimal kinetic energy (since the bonds are straight).
4.2.5 HCN, C2H2, and the Energy of Triple Bonds
The HCN molecule is linear, not bent, and it’s easy to see why. There is a triple bond to the nitrogen, specifically H−C≡N. Here (as always) we use a dash to represent two holes, i.e. the absence of antibonding electrons, not the presence of bonding electrons. We think about the shape as follows: start with the default tetrahedral configuration. The hydrogen is bonded to the A vertex. The nitrogen is bonded to the B, C, and D vertices. The nitrogen ion core sits at a location that is the average or the sum of the B, C, and D vectors. If you look at the geometry of the tetrahedron, you see that this puts the nitrogen directly opposite the hydrogen ... so we have an easily-visualized model of why HCN is linear.
Similarly, acetylene (H−C≡C−H) is linear.
A triple bond is even more highly stressed, more energetic, and more reactive than a double bond. Among other things, this explains why acetylene has remarkably high fuel value.
4.2.6 CO3=
Consider the carbonate ion, CO3=. In the classical approximation, you can draw it with one double-bonded oxygen and two single-bonded oxygens. From this you can easily (and correctly) predict that it will be planar. At the next level of sophistication, moving beyond the classical approximation, there will be resonance, i.e. the three oxygens will “take turns” having the double bond, so the molecule is actually threefold symmetric. This configuration is called trigonal planar. In this case you can build three different not-quite-correct classical models; the correct symmetry is obtained by averaging these three.
4.2.7 Nitrogen Dioxide Radical
Let’s look at NO2. The constituent atoms are shown in figure 26.
Figure 26: Nitrogen Dioxide – Constituent Atoms
There are seven holes, which is enough to make three and a half bonds. Note that any molecule where not all the electrons are paired is called a radical. There are many ways of drawing the bonds in such a molecule. At this level of detail, it is not at all obvious whether we should draw the molecule as linear or bent. Here are some facts that we might use as the basis for an analogy:
Hole-counting theory does not provide enough information to determine whether NO2 is linear or bent. The situation is somewhere between NOCl and CO2, so it could go either way. However, experiment tells us that it has a nonzero permanent dipole moment, so it must be bent. Therefore we choose to draw basis states as shown in figure 27.
Figure 27: Nitrogen Dioxide – Basis States
The actual molecule is shown in figure 28. It is a resonant superposition of the two basis states shown in figure 27. Each oxygen is bonded to the nitrogen by one ordinary bond plus a partial bond with 3/4ths of a unit of bond strength.
Figure 28: Nitrogen Dioxide Molecule
We can contrast this with figure 29, which shows a hypothetical linear molecule. It has four “partial bonds”, each of which has 7/8ths of a unit of bond strength (for a total of 3.5 units). Hole-counting does not rule this out, but experiment tells us that this shape is disfavored relative to the bent shape.
Figure 29: Nitrogen Dioxide – Wrong
The physics here can be partially rationalized as follows: There are many contributions to the overall energy. Resonance makes a contribution that tends to favor the linear structure. Meanwhile, bond-bending (as discussed in section 4.2.4) makes a contribution that tends to favor the bent structure. Hole-counting cannot tell you which of these contributions will win out.
4.2.8 Other Cases
Things get trickier if we consider bonding that involves elements in later rows of the periodic table, beyond N=3 ... but let’s not worry about that right now.
Some harder-to-explain examples (including SF6 and SF4) are discussed in section 5.
5 Some Peculiar Examples
Here are some examples that don’t fit the simple pattern. It should be emphasized that there are millions of molecules for which you can draw a satisfactory bonding diagram without thinking very hard, but there are a few where you can’t.
5.1 The Nitro Group
One example that requires special handling is the nitro group, as found for instance in nitromethane (CH3)−NO2. Mindlessly playing the connect-the-dots game might produce the following diagram:
O - O
\ /
which is straightforward, elegant, symmetric, and wrong. The main problem is that it’s inconsistent with the IR spectroscopy data, which shows floppy, low-frequency bending of the O−N−O angle, inconsistent with closure of the three-member ring via an O−O bond. Instead it supports the following ringless structure:
(-) (-)
O O O O
\\ / \ //
N(+) <--> N(+)
| |
HCH HCH
H H
I imagine there’s also NMR, NQR, and reactivity data confirming there’s a positive charge on the nitrogen.
To understand this result, you need to consider bond angles, as discussed in section 4. The natural angle for the O−N−O and O−O−N bonds should be close to the tetrahedral angle (109 degrees). Distorting these angles to form a 60−60−60 degree triangle is not energetically feasible.
To say the same thing more simply: In the real molecule, the oxygen atoms are significantly farther apart than the drawings above might suggest.
5.2 Sulfur Hexafluoride
As a second peculiar example, consider SF6. Naive hole-counting tells us this molecule has 8 holes, which is enough for four ordinary bonds ... but there’s a problem, because there are six ligands. It is totally unacceptable to have 6 bonds with 2/3rds of a unit of bond-strength apiece, because QM tells us that using s and p orbitals gives us only four bonds, and only four places to put bonds. The only possible explanations for SF6 must involve contributions from some “supplementary” orbitals in addition to the conventional valence orbitals 3s and 3p. Hypotheses to be considered include 2p, 3d, 4p, and perhaps others. The energy levels of these supplementary orbitals is high, but not so wildly high as to make them completely inaccessible. This completely changes the hole-counting process, because there is a whole new set of bonding and antibonding molecular orbitals to be considered.
5.3 Sulfur Tetrafluoride
As a third peculiar example, consider SF4. Naive hole-counting tells us this molecule has six holes. That means it cannot have four ordinary chemical bonds; it can have at most three ... even though there are four ligands. That’s actually the right answer. Apparently the four fluorines take turns being bonded, with each one being bonded three quarters of the time. This is not different in principle from the nitro group discussed in section 2.4, or the carbonate ion discussed in section 4, except that in those cases the ligands take turns having a second bond, while in SF4 the ligands take turns having their first bond. You might imagine this makes SF4 somewhat unstable against disintegration, and you’d be right.
I don’t know of any useful convention for drawing a 3/4-strength bond.
We should also consider the hypothesis that SF4 has two ordinary bonds (each using a pair of holes) and two half-bonds (each using one unpaired hole). However, the lack of paramagnetism indicates this is not what happens. I don’t have any deep theoretical basis for predicting this, so for now let’s just consider it an observed fact.
Old-fashioned VSEPR predicts a remarkable “see-saw” shape for SF4. This would be considered a triumph for VSEPR, except that molecular dynamics studies indicate that that it’s not really a good description of what’s going on. I don’t know how to describe the shape of this molecule, and I certainly don’t have a simple way of predicting the shape.
We must also consider the hypothesis that d-orbitals are involved, which would throw off the hole count.
6 Overview of the Contrast
The relationship between the various available methods is shown in figure 30, which is a plot of “what you get out” (results) as a function of “what you put in” (effort).
Figure 30: Results versus Work for Various Methods
The figure makes several important points, including:
Compared to Lewis dot structures, it appears that BABE and hole counting are better in every way, for the following reasons:
1. In every case where the Lewis dot method had any hope of getting the right answer, hole counting gives the right answer.
2. In every case where the Lewis dot method had any hope of getting the right answer, hole counting is no harder. It is usually easier.
3. The whole idea of filled Lewis octets in molecules is an impediment to deeper understanding. It must be completely unlearned before you do any sort of spectroscopy or molecular physics. In contrast, hole counting is a stepping stone on the way towards deeper understanding.
4. There are simple cases (including the familiar O2 molecule) where the Lewis dot approach stubbornly makes wrong predictions. In contrast, hole counting makes it possible to draw diagrams that correspond to reality.
We shall see that the Lewis dot method is limited both as to domain and to range. By domain, we mean the class of molecules that it applies to. By range, we mean what it says about a given molecule. The tradeoff between domain and range is shown in table 1. In the table, domain runs horizontally, while range runs vertically.
Molecule CH4 CO O2 C2 B2
Number of Bonds Yes Yes Yes no no
All Electrons Paired Yes Yes no Yes no
Filled “Octet” Yes no no no no
Table 1: Lewis Scorecard
Remember that this section is just an overview; supporting evidence will be presented in section 8 and later sections.
In table 1, you can that if we severely restrict the domain, keeping only the first column (namely simple hydrides such as CH4, NH3 etc.), then over this domain the Lewis method makes quite a range of successful predictions. If you want to sell the Lewis method as a theory of simple hydrides, that would be OK. Alas, people often try to sell it as something much more than that.
The second row of the table is important, because it represents not just the CO molecule but a wide class of electron-rich molecules, including most biomolecules. The Lewis method can predict the number of bonds for such molecules. However, the idea that atoms in such molecules have filled octets “just like neon” is hogwash, as we know from the spectroscopy data, from modern theory, and from many other lines of evidence.
Note that C2 is just barely outside the domain of electron-rich molecules. CO is sufficiently electron-rich, while C2 is not.
We now switch switch from a column-by-column discussion to a row-by-row discussion of the table.
As you can see from the top row of table 1, if we restrict the range to predictions as to number of bonds and nothing else, the domain gets bigger. If you want to sell the Lewis method as an unexplained, unprincipled mnemonic for predicting only bond order in electron-rich molecules made from certain elements in the upper-right corner of the periodic table, I might even buy it.
As a minor point of terminology, the term bond order is sometimes used to denote the distinction between single, double, and triple bonds. In this context the order is just a number, as in order-of-magnitude, or the order of a polynomial; it does not refer to the sequencing (or “ordering”) of the bonds left-to-right or anything like that.
The situation soon becomes ugly because most textbook authors cannot resist the temptation to “explain” the Lewis method in terms of atomic physics “principles”. They throw around physics ideas such as the energy of filled shells in neon, and then claim that “sharing” allows each atom in (say) the O2 molecule or the F2 molecule to have a filled shell “just like neon”. Scare quotes are necessary because usually nearly every word of such an “explanation” is false.3
There are of course other limitations. The basic Lewis method obviously has no hope of explaining a molecule such as SF6. Some authors try to patch this up in terms of an “expanded octet”.4 However, given that there aren’t really any octets in typical molecules, it seems pointless to talk about expanding something that doesn’t exist.
7 Fun With Lewis Dot Diagrams ... Or Not
This section serves to put hole-bonding in context, relative to the widely-taught Lewis dot method.
Lewis dot diagrams have lots of problems, and it is possible to do much, much better, with zero additional work, using hole-counting and related methods as discussed in section 2.
7.1 Hydrides and Other Successes
Of course the Lewis dot method is not entirely without merit. Sometimes it makes correct predictions. For starters, as far as I can tell, it gives a satisfactory description of molecules in the sequence CH4, NH3, H2O, HF, Ne, which are all well-behaved molecules ... and conversely it correctly predicts that related entities such as CH3 are not well-behaved molecules under ordinary conditions. Another reasonably satisfactory example is NaCl, as discussed in section 10.2.
7.2 Oxygen
There are, alas, other cases where Lewis dot diagrams stubbornly and irreparably make wrong predictions.
Reference 4, from UC Berkeley, shows a conventional Lewis dot structure for the oxygen molecule, O2, which can also be called dioxygen. This is the way chemistry is taught in a lot of places, not just Berkeley. Figure 31 portrays the same idea, using a style I prefer because it is slightly more explicit. The black dots represent electrons. The reddish shaded areas represent orbitals. The proponents of such diagrams like to call attention to the following good points:
Figure 31: Conventional Unsatisfactory Lewis Dot Diagram for O2
However, this diagram is nonsense. It cannot possibly be correct. The most obvious problem is that O2 is well known to be paramagnetic. If you pour liquid oxygen into the gap of a magnet, it will stick to the magnet. There is an animated GIF image of this in reference 6 and a nice still image in reference 7. Paramagnetism arises from unpaired electrons. Figure 31 categorically predicts that all electrons are paired. Therefore:
There have been various attempts to solve this problem. One attempt, from the University of Illinois, is shown in reference 8. Figure 32 portrays the same idea, using my preferred style.
Figure 32: Single-Bond : Unsatisfactory Lewis Dot Diagram for O2
This solves the paramagnetism problem, but creates several others.
So let’s try again. Hope springs eternal.
Reference 9 shows an oxygen molecule with one regular bond and two half-bonds. Figure 33 portrays the same idea, using my preferred style.
Figure 33: Two Half-Bonds : Unsatisfactory Lewis Dot Diagram for O2
Alas this, too, has its problems.
We would prefer not to have a theory of chemistry based on rote dogma ("four legs good, two legs bad, eight dots good, ..."). Instead we would like to have a rule based on physics, mathematics, and logic.
For isolated atoms in the N=2 and N=3 rows, there is a valid octet rule, to wit: there are four orbitals near each atom, and each orbital can be occupied by at most two electrons (because of the exclusion principle). So far so good. The problem arises only when we try to extend this rule to the atoms in molecules. There arises a question of how to interpret the word “near” in this rule. The Lewis method chooses an interpretation that leads to DCBO (double counting of bonding orbitals) and this is irreparably wrong.
The five orbitals in figure 33 means either that the figure is wrong, or we need to entirely abandon DCBO in general and Lewis dot diagrams in particular.
7.3 Sulfate Ion; Expanded Octets
Figure 34 shows the Lewis dot diagram for the sulfate ion. Remember, in this section, the dots in the diagrams represent electrons, not holes.
Figure 34: Sulfate Ion – Lewis Dot Diagram
This isn’t particularly terrible. However, it is unnecessarily uglier and more complicated than the conventional diagram we saw in figure 15.
Within the Lewis formalism, just as in other formalisms, it is conventional to represent a bond by a dash, so when a pair of dots represents a bond we can replace it with a dash. In the case of the sulfate ion, the result is shown in figure 35.
Figure 35: Sulfate Ion – Lewis Structure
This can be compared and contrasted with figure 15.
If you are interested in formal charge, figure 35 can be interpreted as assigning a +2 formal charge to the central sulfur atom, and −1 formal charge to each of the oxygen atoms. This is the same interpretation as we saw in section 2.1.7. It is consistent with the known electronegativity, and consistent with the known symmetry.
For some reason, figure 36 is commonly (albeit not universally) presented as “the” Lewis structure for the sulfate ion. It can be found in the Encyclopedia Britannica, on the wikipedia site, and in many textbooks.
Figure 36: Sulfate Ion – Another Lewis Structure?
This has the small advantage of having zero formal charge on the sulfur atom. In return it incurs the large disadvantage of violating the “Lewis octet” rule. There are six bonds to the central sulfur atom. Another disadvantage is that it violates the known symmetry; the molecule is observed to be tetrahedral, all four oxygen atoms are observed to be equivalent. It seems odd that even articles that say in the text that the oxygens are equivalent and tetrahedral persist in using a non-symmetric drawing.
You could try to salvage the symmetry by saying the final structure is a quantum mechanical superposition of all the various ways of arranging the single and double bonds, leading to an average bond strength of 1.5 everywhere.
No matter what you do with the symmetry, we still have a violation of the “Lewis octet rule” since there are still six bonds to the central sulfur atom. Sometimes people try to wave away this violation by talking about an “expanded octet”, presumably involving the sulfur atom’s d orbitals. This is troublesome because it doesn’t explain why an octet might be “expanded” in some cases but not others.
In particular, the symmetry tells us there can’t be any d orbitals involved. The tetrahedral shape has the symmetry of an sp3 hybrid; any nontrivial admixture of d orbitals would change the symmetry.
Since there was never a rational physical basis for Lewis octets to begin with, don’t hold your breath waiting for a physical basis for the expanded octets.
One possible Lewis structure for sulfuric acid follows the same pattern, as shown in figure 37.
Figure 37: Sulfuric Acid – Lewis Structure
Just as figure 36 is a modification of figure 35, there is a widely-seen alternative to figure 37, namely figure 38.
Figure 38: Sulfuric Acid – Another Lewis Structure?
Again this violates the “Lewis octet” rule.
The “expanded octet” problem is not confined to sulfates; the various sulfonic acids are almost universally drawn with six bonds to the central sulfur atom. See section 7.4.
7.4 Methanesulfonic Acid
It is worth noting that the drawing of methanesulfonic acid in figure 18 (in section 2.1.8) shows single bonds everywhere, and only four bonds to the central sulfur atom.
That’s remarkable, because almost all references show this molecule with double bonds to two of the oxygen atoms, for a total of six bonds to the sulfur atom, as shown in figure 39.
Figure 39: Methanesulfonic Acid Lewis Structure?
Let’s consider the pros and cons of figure 39, according to the usual Lewis notions. We find (a) there is formal charge neutrality everywhere, which is good; (b) there are filled “Lewis octets” on the oxygen atoms on the outside of the molecule; although (c) the so-called octet rule has been seriously violated on the central sulfur atom.
Again as mentioned in section 7.3, sometimes people try to wave away the octet violation by talking about an “expanded octet”, presumably involving d orbitals.
In any case, in the BABE scheme, there is no problem, as discussed in section 2.1.8. Since we know that “filled Lewis octets” never existed to begin with, since we are not double-counting bonding orbitals, we have no fear of unfilled octets (on the oxygens or anywhere else).
7.5 Divergent Predictions
Recall that one place where the BABE method diverges from Lewis octets is that the Lewis method stubbornly predicts that the O2 molecule has the O=O structure. It cannot accommodate the O÷O structure.
Here we have a second place where the methods diverge. The BABE method stubbornly predicts single bonds (and nothing but single bonds) in the sulfate ion, sulfuric acid, and methanesulfonic acid. In contrast, the Lewis method (as practiced by experienced professionals) usually makes a different prediction than the BABE method does concerning sulfate, and always (as far as I can tell) makes a different prediction regarding methanesulfonic acid.
This should have observable consequences in the electron density. The double bond has zero electron density along its midline. Is this observed in real life, or not? I don’t know for sure, but I predict not.
Here’s one reason why the BABE method cannot predict a double bond in these molecules. In going from figure 35 figure 36, we re-arranged some electrons. We took some electrons from a “lone pair” and used them to make another bond. This cannot happen in the BABE approach, because there are no lone pairs. The concept of lone pairs does not exist, so there’s nothing to re-arrange. You can’t just add another bond by pulling it out of the air, since that would change the overall electrical charge of the molecule.
In case you’re wondering whether there might be holes in a d orbital that could be used to form the additional bond, that’s a clever idea, but it won’t work. In these electron-rich molecules, bonds are holes in antibonding orbitals. All the low-lying d orbitals are bonding, and you can’t make bonds by putting holes in such orbitals. There are lots of games you can play with d orbitals, but none of them (so far as I can tell) increase the bond order in these molecules.
7.6 Additional Remarks
Note that the easy method of counting formal charge by counting bonds, as suggested in section 2.1.4, only works within the BABE formalism, i.e. when the bonds represent pairs of holes in antibonding orbitals in electron-rich molecules. In contrast, in the Lewis scheme, counting formal charge would be trickier, because we would need to account for lone pairs as well as bonds. This is another of the reasons why the BABE method is just plain easier.
If you are only interested in bond order, and you are sure you will never do any spectroscopy or any molecular structure calculations, you might conclude that Lewis dot diagrams are OK, for electron-rich compounds (with a few exceptions such as O2).
Please keep in mind that except for the hydrides mentioned in section 7.1, Lewis dot diagrams make false predictions about the spectroscopy and molecular orbitals of almost all molecules, not just oxygen. O2 is just the canary in the coal mine, in the sense that it is where we first noticed the problem. But it did not cause the problem, and ignoring the dead canary will not solve the problem.
As Kim Philby said: “To betray, you must first belong.” Applying this to the notion of filled Lewis octets, the fact that this fundamentally wrong notion makes some correct predictions about bond-order is a big part of what makes it so pernicious.
8 Conceptual Foundations (or lack thereof)
8.1 Contrast: Molecular Orbitals versus Lewis
The bond-strength diagrams discussed in section 2 are firmly based on sound theory, as we now discuss.
This is called Molecular Orbital theory, or MO theory for short. Like most theories, it seems simple if you understand it, and formidably complicated if you don’t understand it. Like any good theory, watered-down, approximate, qualitative versions are available ... and that’s where we should start. Here then are some of the qualitative things we know about MO theory:
The MO explanation is consistent with the paramagnetism data. The Lewis dot diagram stubbornly predicts no paramagnetism, contrary to observations.
The MO explanation is consistent with the optical spectroscopy data. The Lewis dot diagram has little to say about spectroscopy, and what it does say is categorically wrong.
The MO explanation is consistent with reactivity data, including things like B2 and Be2. There are many cases – including B2 and Be2 – where Lewis dot diagrams make predictions that cannot be reconciled with observations.
The MO explanation is consistent with theory: atomic physics, quantum mechanics, and all that. The Lewis dot approach was published about ninety years ago (reference 10), considerably before there was a comprehensive quantum-mechanical understanding of molecular bonding (reference 11). The Lewis approach cannot be reconciled with present-day theoretical understanding.
A reasonably-accessible discussion of molecular orbitals, including a sketch of the energy-level diagram for O2, can be found in reference 2. Another useful source is reference 12, which contains energy-level diagrams for some interesting molecules (water, hydrogen fluoride, ammonia, methane, ethane, ethene, ethyne).
Among the things we learn is that the Lewis approach is a dead duck. We shouldn’t be surprised by this; it made the wrong predictions in figure 31, it was violated in figure 32, and it was toppled off its foundations in figure 9. The MO explanation tells us that these are not isolated or superficial mistakes, but instead result from a systematic and fundamental misconception. DCBO is just wrong.
The MO explanation tells us that there should be eight molecular orbitals (not all of which will be filled). We know from classical physics and mathematics (Liouville’s theorem) that eight is the right answer. In the Lewis dot approach, e.g. figure 31, we see six molecular orbitals, 2 unshared on the left, 2 shared in the middle, and 2 unshared on the right. We know (experimentally and theoretically) that this is the wrong number.
Ironically enough, this means that while figure 9 has too many orbitals to be a valid Lewis structure (7 instead of 2+2+2), according to the molecular orbital explanation it doesn’t have too many – just the wrong kind.
This leads us to a key point: The data and the theory tell us that antibonding orbitals exist (as well as bonding orbitals of course). In particular, the two electrons that are responsible for the paramagnetism of O2 sit in antibonding orbitals, as shown in figure 21.
This in turn means that there is no hope of fixing the Lewis dot formalism, except by throwing it out and starting over, as was done in section 2. A proper theory cannot be based on DCBO. A proper theory must represent antibonding electrons (not just bonding and non-bonding electrons). A proper theory must have a way to take into account the energy levels of the molecular orbitals, to provide a foundation for applying Hund’s rules.
8.2 Contrast : Hole Bonding versus Lewis
Consider the contrast:
The old Lewis approach requires folks to believe in the unjustified – and unjustifiable – DCBO rule. It requires them to believe that O2 has six molecular orbitals (two on the left, two shared, and two on the right). The new hole-bonding approach requires folks to believe in the well-founded notion of antibonding orbitals (as well as bonding orbitals) and degrees of occupation thereof. In fact O2 has eight molecular orbitals (four bonding and four antibonding).
At this level of detail, the hole-bonding story is more truthful and less complicated than the Lewis story.
At this point,
8.3 What Implies What
Let us examine what does – and what doesn’t – constitute evidence for the Lewis octet rule. This will help us understand its domain of validity. A high-level summary of the situation is shown in figure 40. The rest of this section is devoted to explaining the details.
Figure 40: Inference Diagram
When applied to a molecule like O2, the Lewis octet rule says that the number of actual valence electrons, plus the number of electrons involved in bonds, must add up to 16, i.e. one octet per atom. The point is that the electrons involved in bonds are intentionally double-counted. Since there are two electrons per bond, this leads to the bond-order formula:
bond order =
16 − # of valence electrons
Now let us contrast this with a proper molecular-orbital explanation.
So let’s see how the contestants have scored:
1. When applied to pairs of atoms from the N=2 row, equation 3 is right about half the time. This comes from comparing the formula to observed results, and is independent of whatever derivation or rationale (if any) you think lies behind the formula.
2. The Lewis octet rule has less validity than the bond-order formula. That is, if we try to interpret equation 3 in terms of octets, we are right less than half of the time, because the octet picture predicts not just bond-order, but also falsely predicts that all the electrons in O2 are paired. And the Lewis octet picture is grossly incompatible with the spectroscopic data.
3. If we interpret equation 3 terms of molecular orbitals, we realize it applies if-and-only-if all four of the molecule’s bonding orbitals are full, i.e. when we are considering putting electrons into zero or more of the three antibonding orbitals that lie at the top of the energy ladder. So in some sense it’s an antibonding formula, not a bonding formula.
4. More importantly, the MO explanation tells us when we should and when we shouldn’t pay attention to equation 3. The MO explanation gives us a way of understanding the whole sequence He2, Li2, Be2, B2, C2, N2, O2, F2, Ne2 (including the spectroscopic data and the paramagnetism data).
To repeat: equation 3 summarizes the data over a narrow range, but is not (by itself) an explanation. Explaining the formula in terms of octets only makes things worse. MO theory explains equation 3, explains its limitations, and explains lots more besides.
8.4 The Aufbau Principle (or not)
This continues the discussion that began in section 3.1.
It is worth noting that the Aufbau principle is utterly incompatible with the Lewis octet idea. Let’s do an example, starting with O2. The Lewis “theory” tells us that each atom in the molecule will have an octet of electrons “around” it. If we now construct F2 in accordance with the Aufbau principle, we leave the oxygenic electrons alone, and add two more electrons. There are now more than eight electrons “around” each atom.
For most molecules, this is an inescapable conflict. Either the Aufbau principle is wrong, or the Lewis octet idea is wrong. (The wrongness of the Lewis octet idea is discussed in section 7.)
The Lewis model explains bond order in terms of bonding pairs and non-bonding pairs. MO theory explains bond order in terms of bonding and antibonding.
If you think in terms of bonding and non-bonding, there is no Aufbau-compatible way to explain how F2 can have more electrons but less bond strength. It is perfectly compatible with the Aufbau principle to say that when we go from O÷O to F-F, we leave the bonding levels alone but add electrons to an antibonding level.
There is a big difference between antibonding and non-bonding. There is a big difference between BABE and BNBE (i.e. Lewis dot structures).
9 Discussion
Almost all scientific results, experimental and theoretical, are approximate and imperfect.
One criterion for judging scientific approximations is that they ought to be controlled approximations. That is, the error ought to be small, we ought to have an upper bound on how big the error is, and we would like to have a way of making the error smaller if and when we need to.
Lewis dot diagrams do not do well under this criterion. The Lewis octet rule seems to be an all-or-nothing proposition. Either the O2 molecule has just six molecular orbitals, or it does not. In fact it does not, as we know from the paramagnetism data, the spectroscopy data, et cetera. So Lewis dot diagrams must be put into the same category as the oyster/R mnemonic.
Another rule of good scientific practice is to be up-front about whatever approximations are being made. This is where the most immediate improvements can be made. If you feel you must draw Lewis dot diagrams, go ahead, but make sure the audience understands that:
Another rule of good scientific practice asserts that it is bad manners to criticize a result for being imperfect unless you’ve got something better to suggest. In this case the suggestion is to use the bond-strength diagrams described in section 2, which are preferable to Lewis dot diagrams in every way.
Similarly, the bond-order formula equation 3 is valid over a limited range, but still has more validity than the Lewis octet picture. It can be seen either as a narrow corollary of the molecular orbital explanation, or it can be used on its own merits as an empirical / phenomenological / mnemonic / numerological rule.
It is perfectly possible to visualize the results of the bond-order formula. You can draw pictures of molecules including F−F, O÷O, N≡N, C/=\C, B\;/B, and Li∼Li ... without connecting any of this to “molecular octets”. We connect this to correct theory by saying that each − or ∼ represents one unit of bond strength. A ∼ represents a filled bonding orbital, while a − represents an unfilled antibonding orbital.
Things go awry when people try to use the successes of the bond-order formula as evidence for the Lewis octet rule, by back-tracking along arrow (2) in figure 40. It is not logical to follow arrow (2) while ignoring arrows (1), (3), and (4). It is not fair to pay attention to the occasional fortuitous successes of the DCBO rule, while ignoring its failures and ignoring the many successes of competing hypotheses.
To say the same thing in other words: What many people claim as successes for the molecular octet rule are nothing of the sort; they are really just successes of the bond-order formula for electron-rich molecules – equation 3 – with no necessary connection to the molecular octet rule.
10 Additional Background and Context
10.1 The Hole Concept
The Hall effect (reference 13) was discovered in 1879. It was found that some metals (e.g. aluminum and magnesium) have positive Hall coefficients, indicating positively-charged majority carriers – what we would nowadays call holes. Note that this was many years before the date conventionally assigned to the “discovery” of the electron (reference 14).
The notion of holes is central to any discussion of solid-state electronics. This goes back to Peierls’s 1929 explanation of the Hall effect (reference 15).
Holes have been part of quantum field theory since Dirac’s epochal work in 1930 (reference 16, page 293):
The annihilation of a negative-energy electron is to be understood as the creation of a hole in the sea of negative-energy electrons, or the creation of a positron.
The idea of holes in the context of bonds between atoms is not new, either. In 1969, Baym (reference 17, page 454) remarked that a halogen could be regarded
... as having a weakly bound “hole” which also easily forms chemical bonds.
I’m not quite sure why he put the word hole in scare quotes.
10.2 Coulombic Models
Let’s consider a crystal of NaCl. It exhibits ionic bonding, in contrast to the covalent bonding in (say) a silicon crystal. It can be described reasonably well in classical terms: the Na+ ion and the Cl ion are modeled as classical hard spheres (of appropriate sizes), and they are held together by Coulomb interactions.
This is consistent with the idea of Lewis octets, in the sense that the Na+ ion and the Cl ion can be explained in terms of filled atomic octets. (Remember, atomic octets actually have some basis in fact, whereas molecular octets almost never do.)
It must be emphasized that the Coulomb interaction between ions is quite different from the covalent bond. In the real world, some bonds exhibit a mixture of ionic and covalent character. NaCl crystals are near the ionic extreme, while silicon crystals are at the covalent extreme.
Coulomb’s law can be expressed in a variety of equivalent forms. These include:
V(r) = [
4 π є0
where V(r) is the voltage (i.e. electrical potential) at location r due to a charge Q at the origin. The quantity in square brackets is a universal constant, called Coulomb’s constant. It is occasionally written kC but more commonly as 1/(4 π є0).
An equivalent expression is:
F(r) = [
4 π є0
q Q
where F(r) is the force on a charge q at location r due to a charge Q at the origin. The r/|r| factor is a unit vector in the r direction. The force is directed radially outward if Q and q have the same sign.
The Coulomb interaction depends on the charge (but not the mass) of the particles. It is a long-range force, in the sense that the interaction energy falls off like 1/r. In contrast, the size and strength of the covalent bond depend on the mass (as well as charge) of the electron. The covalent bond is necessarily a short-range interaction: at large distances the wavefunction falls off exponentially, at a rate determined by the mass of the electron.
Coulomb’s equation tells us the electrostatic potential energy, but we must also include the kinetic energy. We must include the kinetic energy if we are to have any hope of explaining many of the most well-known and important chemical facts, including:
Charles-Augustin de Coulomb died in 1806, more than 100 years before the advent of quantum mechanics. Therefore speaking of a “Coulombic” model rather strongly implies a classical model. We know that any correct description of covalent bonding requires non-classical ideas – far, far beyond what you can get from equation 4.
Some people who are fed up with Lewis dot diagrams try to solve their problems by suggesting we can describe everything (or almost everything) in terms of “Coulombic” models. This is not a good idea. Let’s be clear: Talking about a “Coulombic” model of the water molecule or the ethylene molecule is at best misleading. It is at best an abuse of the terminology.
Coulomb’s law is part of the description of any chemical bond (including covalent as well as ionic), but is equally true that bolts are part of a car. We don’t talk about the “bolt” model of how cars work, and we shouldn’t talk about the “Coulombic” model of bonding, except perhaps for ionic compounds such as NaCl. Coulomb’s law is a law of physics, but it is only one law among many; it is not a complete description.
(In contrast, it is OK to talk about a “quantum mechanical” model. That’s because QM explicitly incorporates all the other laws of physics. On the RHS of the Schrödinger equation we find the Hamiltonian, which includes all the contributions to the energy. In contrast, the Coulomb energy calculated using equation 4 is just one contribution to the energy, one contribution among many.)
11 Summary
There is nothing you can do via Lewis dot structures that you can’t do at least as easily and at least as accurately via hole-counting. Hole counting, within its domain of applicability, is based on sound theory, whereas Lewis dot diagrams are based on a fairy tale.
There are many cases where Lewis dot diagrams predict bond-order in agreement with the facts. However, this agreement must be considered fortuitous, because there are also many disagreements that ought not be overlooked. The method recommended here – accounting for antibonding orbitals as well as bonding orbitals – retains all the good features of Lewis dot diagrams while getting rid of the disagreements.
There is an octet rule for atoms that is reasonably well founded in atomic physics for a single row-2 atom. Truly there is something special about the complete octet (i.e. closed shell) in neon. Chemistry is primarily about molecules, not isolated atoms, especially not noble-gas atoms. For molecules, the Lewis approach couples the octet idea to DCBO (double counting of bonding orbitals), which is a disaster.
In reality, there are eight valence-shell molecular orbitals in molecules such as O2. This is the starting point for the theory of chemistry described here. This is the right answer. The Lewis approach says O2 has only six molecular orbitals: two on the left, two shared, and two on the right. This is the wrong answer.
Naive hole-counting won’t tell you whether O÷O or O=O is preferable. Making no prediction about this question is better than making a wrong prediction. We leave the door open for other theoretical and/or experimental information to decide the question. Lewis octets stubbornly predict O=O, in defiance of the well-known facts. There is no way to draw a Lewis octet that fits the facts.
Hole-counting gives a good accounting of its own limitations. It is easy to understand that naïve hole-counting works in electron-rich molecules, including a very wide range of important biomolecules. Meanwhile, it is easy to see that slightly less-naïve accounting is needed to explain C=C and other systems that are not electron-rich ... without in any way contradicting what was said about electron-rich systems. Lewis “theory” does not explain its own limitations. There is no way to explain C=C in terms of Lewis octets, nor any good way to explain why not. Octets are either a fundamental principle, or they’re not. If the octet “principle” explains N2, why doesn’t it explain O2 or C2?
Anybody who says my approach is identical to the Lewis approach isn’t paying attention.
In every case where Lewis dot diagrams make correct predictions, hole-counting makes the same predictions (i.e. the correct predictions) with the same amount of effort, or less. Lewis dot diagrams are, of course, better than nothing. The successes of Lewis dot diagrams area a subset of the successes of hole-counting.
Hole-counting is based on molecular orbital theory. It is a shorthand, summarizing (not supplanting) some well-established facts. The Lewis dot diagram scheme is based on DCBO, and on the neglect of antibonding, and on other assumptions that are inconsistent with what’s really going on in the molecule. It is a miracle that Lewis dot diagrams make as few wrong predictions as they do.
Within MO theory we can visualize bond order just as easily as we could using Lewis dot diagrams. For instance we can draw N≡N, where each dash represents an unfilled antibonding orbital. In any case where you could have drawn a Lewis dot diagram, you can draw a bonding diagram consistent with MO theory – and consistent with the facts – with the same amount of effort. In other cases, the correct diagram is slightly more work, but it’s a small price to pay for correctness. Lewis dot diagrams are easy to draw, easy to visualize, compact, and elegant ... but rife with error.
12 References
John Denker, “Pressure, Degeneracy, Exchange Interaction, Neutron Stars, Atoms, Etc.” ./degeneracy.htm
Mark R. Leach “Diatomics : Molecular Orbital Theory” http://www.meta-synthesis.com/webbook/39_diatomics/diatomics.html
John L. Park, “Model Building” http://dbhs.wvusd.k12.ca.us/webdocs/Bonding/Lab-ModelBuilding/
especially http://dbhs.wvusd.k12.ca.us/webdocs/Bonding/Lab-ModelBuilding/H2O-NH3.jpg
and http://dbhs.wvusd.k12.ca.us/webdocs/Bonding/Lab-ModelBuilding/CO2-HCN.jpg
The lower-left corner of the image shows the conventional Lewis dot diagram for O2;
used in a UC Berkeley chemistry course circa 1995:
John Denker, “Models and Pictures of Atomic Wavefunctions” http://www.av8n.com/physics/wavefunctions.htm
Liquid O2 pouring into the gap of a magnet. http://www.chem.wisc.edu/deptfiles/genchem/demonstrations/Movies/o2parasm.gif
Liquid O2 poured into the gap of a magnet. http://demoroom.physics.ncsu.edu/multimedia/images/demos/5G3020.jpg
Single bond in O2 molecule. http://www.chem.uiuc.edu/clcwebsite/jpeg/O2.jpg
Part B of the figure shows one bond plus two half-bonds in O2 molecule;
used in a chemistry course at Penn State:
Gilbert N. Lewis "The Atom and the Molecule" JACS 38 762–786 (1916) http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Lewis-1916/Lewis-1916.html
Linus Pauling, The Nature of the Chemical Bond (1939).
W. Locke, “Introduction to Molecular Orbital Theory” http://www.ch.ic.ac.uk/vchemlib/course/mo_theory/main.html
Edwin Hall, “On a New Action of the Magnet on Electric Currents” American Journal of Mathematics 2 (1879).
J.J. Thompson, Philosophical Magazine 44 293 (1897).
Rudolf Peierls, “On the theory of the Hall effect”, Physikalisches Zeitschrift 30 273–274 (1929).
Paul A. M. Dirac, Principles of Quantum Mechanics (1930).
Gordon Baym, Lectures on Quantum Mechanics (1969).
Of course other things are not equal, and the kinetic energy of the wavefunction is at least as important as the potential energy, so the charge-distribution rule will remain a rule of thumb, not a law of nature.
The classical approximation works OK for liquid ammonia, and for ammonia in aqueous solution. It fails spectacularly for isolated ammonia molecules, because the protons tunnel from one side to the other. For almost all molecules heavier than ammonia, the classical approximation works OK for describing the position of the nuclei, and quantum mechanics is only needed for describing the electrons. But for the isolated ammonia molecule, even the nuclei must be described quantum mechanically. There is a wonderful discussion of this in Feynman volume III chapter 9.
It could be argued that the Lewis “theory” works for simple hydrides such as CH4, NH3 etc., but a theory that only works for a small number of obviously-exceptional cases can’t be considered a respectable theory, especially when it claims to work for a wide class of cases where it does not.
As a minor point, we note in passing that “expanded octet” is a contradiction in terms. Octet means 8, and 8 is not “expandable”. I assume they meant to say “expanded shell” or something like that.
Copyright © 2004 jsd |
9a8476a06fbc592a | Quantum field theory
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"Relativistic quantum field theory" redirects here. For other uses, see Relativity.
In theoretical physics, quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. A QFT treats particles as excited states of an underlying physical field, so these are called field quanta.
In quantum field theory, quantum mechanical interactions between particles are described by interaction terms between the corresponding underlying fields.
Quantum electrodynamics (QED) has one electron field and one photon field; quantum chromodynamics (QCD) has one field for each type of quark; and, in condensed matter, there is an atomic displacement field that gives rise to phonon particles. Edward Witten describes QFT as "by far" the most difficult theory in modern physics.[1]
QFT interaction terms are similar in spirit to those between charges with electric and magnetic fields in Maxwell's equations. However, unlike the classical fields of Maxwell's theory, fields in QFT generally exist in quantum superpositions of states and are subject to the laws of quantum mechanics.
Because the fields are continuous quantities over space, there exist excited states with arbitrarily large numbers of particles in them, providing QFT systems with an effectively infinite number of degrees of freedom. Infinite degrees of freedom can easily lead to divergences of calculated quantities (i.e., the quantities become infinite). Techniques such as renormalization of QFT parameters or discretization of spacetime, as in lattice QCD, are often used to avoid such infinities so as to yield physically meaningful results.
Fields and radiation[edit]
There is currently no complete quantum theory of the remaining fundamental force, gravity. Many of the proposed theories to describe gravity as a QFT postulate the existence of a graviton particle that mediates the gravitational force. Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will behave like Einstein's general theory of relativity in the low-energy limit. Quantum field theory of the fundamental forces itself has been postulated to be the low-energy effective field theory limit of a more fundamental theory such as superstring theory.
Most theories in standard particle physics are formulated as relativistic quantum field theories, such as QED, QCD, and the Standard Model. QED, the quantum field-theoretic description of the electromagnetic field, approximately reproduces Maxwell's theory of electrodynamics in the low-energy limit, with small non-linear corrections to the Maxwell equations required due to virtual electron–positron pairs.
In the perturbative approach to quantum field theory, the full field interaction terms are approximated as a perturbative expansion in the number of particles involved. Each term in the expansion can be thought of as forces between particles being mediated by other particles. In QED, the electromagnetic force between two electrons is caused by an exchange of photons. Similarly, intermediate vector bosons mediate the weak force and gluons mediate the strong force in QCD. The notion of a force-mediating particle comes from perturbation theory, and does not make sense in the context of non-perturbative approaches to QFT, such as with bound states.
The early development of the field involved Dirac, Fock, Pauli, Heisenberg and Bogolyubov. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.
Gauge theory[edit]
Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed by the 1970s through the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.
Grand synthesis[edit]
Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics, which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s, which led to the seminal reformulation of quantum field theory by Kenneth G. Wilson in 1975.
Classical and quantum fields[edit]
A classical field is a function defined over some region of space and time.[3] Two physical phenomena which are described by classical fields are Newtonian gravitation, described by Newtonian gravitational field g(x, t), and classical electromagnetism, described by the electric and magnetic fields E(x, t) and B(x, t). Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom.[3]
Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—photons—rather than continuous fields. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a quantum field.
It is not immediately clear how to write down such a quantum field, since quantum mechanics has a structure very unlike a field theory. In its most general formulation, quantum mechanics is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically observable quantities and the state space represents the possible states of the system under study.[4] For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators \hat{x} and \hat{p}. Field theory, in contrast, treats x as a way to index the field rather than as an operator.[5]
There are two common ways of developing a quantum field: the path integral formalism and canonical quantization.[6] The latter of these is pursued in this article.
Lagrangian formalism[edit]
Quantum field theory frequently makes use of the Lagrangian formalism from classical field theory. This formalism is analogous to the Lagrangian formalism used in classical mechanics to solve for the motion of a particle under the influence of a field. In classical field theory, one writes down a Lagrangian density, \mathcal{L}, involving a field, φ(x,t), and possibly its first derivatives (∂φ/∂t and ∇φ), and then applies a field-theoretic form of the Euler–Lagrange equation. Writing coordinates (t, x) = (x0, x1, x2, x3) = xμ, this form of the Euler–Lagrange equation is[3]
\frac{\partial}{\partial x^\mu} \left[\frac{\partial\mathcal{L}}{\partial(\partial\phi/\partial x^\mu)}\right] - \frac{\partial\mathcal{L}}{\partial\phi} = 0,
where a sum over μ is performed according to the rules of Einstein notation.
By solving this equation, one arrives at the "equations of motion" of the field.[3] For example, if one begins with the Lagrangian density
\mathcal{L}(\phi,\nabla\phi) = -\rho(t,\mathbf{x})\,\phi(t,\mathbf{x}) - \frac{1}{8\pi G}|\nabla\phi|^2,
and then applies the Euler–Lagrange equation, one obtains the equation of motion
4\pi G \rho(t,\mathbf{x}) = \nabla^2 \phi.
This equation is Newton's law of universal gravitation, expressed in differential form in terms of the gravitational potential φ(t, x) and the mass density ρ(t, x). Despite the nomenclature, the "field" under study is the gravitational potential, φ, rather than the gravitational field, g. Similarly, when classical field theory is used to study electromagnetism, the "field" of interest is the electromagnetic four-potential (V/c, A), rather than the electric and magnetic fields E and B.
Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields. These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.
Single- and many-particle quantum mechanics[edit]
In quantum mechanics, a particle (such as an electron or proton) is described by a complex wavefunction, ψ(x, t), whose time-evolution is governed by the Schrödinger equation:
-\frac{{\hbar}^2}{2m}\frac{{\partial}^2}{\partial x^2}\psi(x,t) + V(x)\psi(x,t) = i \hbar \frac{\partial}{\partial t} \psi(x,t).
Here m is the particle's mass and V(x) is the applied potential. Physical information about the behavior of the particle is extracted from the wavefunction by constructing expected values for various quantities; for example, the expected value of the particle's position is given by integrating ψ*(x) x ψ(x) over all space, and the expected value of the particle's momentum is found by integrating ψ*(x)dψ/dx. The quantity ψ*(x)ψ(x) is itself in the Copenhagen interpretation of quantum mechanics interpreted as a probability density function. This treatment of quantum mechanics, where a particle's wavefunction evolves against a classical background potential V(x), is sometimes called first quantization.
This description of quantum mechanics can be extended to describe the behavior of multiple particles, so long as the number and the type of particles remain fixed. The particles are described by a wavefunction ψ(x1, x2, …, xN, t), which is governed by an extended version of the Schrödinger equation.
Often one is interested in the case where N particles are all of the same type (for example, the 18 electrons orbiting a neutral argon nucleus). As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. This is achieved by using a Slater determinant as the wavefunction of a fermionic system (and a Slater permanent for a bosonic system), which is equivalent to an element of the symmetric or antisymmetric subspace of a tensor product.
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_{p\in S_N} |\phi_{p(1)}\rang \otimes \cdots \otimes |\phi_{p(N)} \rang,
where |\phi_i\rang 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. \sqrt{\frac{\prod_j N_j!}{N!}} is a normalizing factor.
There are several shortcomings to the above description of quantum mechanics, which are addressed by quantum field theory. First, it is unclear how to extend quantum mechanics to include the effects of special relativity.[7] Attempted replacements for the Schrödinger equation, such as the Klein–Gordon equation or the Dirac equation, have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions not having a well-defined probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept. The second shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation;[8] this is crucial for describing phenomena such as pair production, which result from the conversion between mass and energy according to the relativistic relation E = mc2.
Second quantization[edit]
Main article: Second quantization
In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics.
Several other approaches exist, such as the Feynman path integral,[9] which uses a Lagrangian formulation. For an overview of some of these approaches, see the article on quantization.
For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states which are possible in the system by |\phi_1\rang, |\phi_2\rang, |\phi_3\rang, and so on. For example, the 3-particle state with one particle in state |\phi_1\rang and two in state |\phi_2\rang is
\frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang
|\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang
|\phi_2\rang |\phi_1\rang \right].
The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states |\phi_1\rang, |\phi_2\rang, etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as
|1, 2, 0, 0, 0, \dots \rangle.
An N-particle state belongs to a space of states describing systems of N particles. The next step is to combine the individual N-particle state spaces into an extended state space, known as Fock space, which can describe systems of any number of particles. This is composed of the state space of a system with no particles (the so-called vacuum state, written as |0\rang), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. States describing a definite number of particles are known as Fock states: a general element of Fock space will be a linear combination of Fock states. There is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.
At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number j indicating which of the single-particle states |\phi_1\rang, |\phi_2\rang,\dots,|\phi_j\rang,\dots it refers to:
| N_1, N_2, N_3, \dots, N_j, \dots \rang .
The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to ladder operators in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator a_2 and creation operator a_2^\dagger are easily defined in the occupation number representation as having the following effects:
a_2 | N_1, N_2, N_3, \dots \rang = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \dots \rang,
a_2^\dagger | N_1, N_2, N_3, \dots \rang = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \dots \rang.
It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation
\left[a_i , a_j \right] = 0 \quad,\quad
\left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad
\left[a_i , a_j^\dagger \right] = \delta_{ij},
where \delta 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.
Applying an annihilation operator a_k followed by its corresponding creation operator a_k^\dagger returns the number N_k of particles in the kth single-particle eigenstate:
a_k^\dagger\,a_k|\dots, N_k, \dots \rangle=N_k| \dots, N_k, \dots \rangle.
The combination of operators a_k^\dagger a_k is known as the number operator for the kth eigenstate.
The Hamiltonian operator of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, for a field of free (non-interacting) bosons, the total energy of the field is found by summing the energies of the bosons in each energy eigenstate. If the kth single-particle energy eigenstate has energy E_k and there are N_k bosons in this state, then the total energy of these bosons is E_k N_k. The energy in the entire field is then a sum over k:
E_\mathrm{tot} = \sum_k E_k N_k
This can be turned into the Hamiltonian operator of the field by replacing N_k with the corresponding number operator, a_k^\dagger a_k. This yields
H = \sum_k E_k \, a^\dagger_k \,a_k.
It turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers Ni can only take on the value 0 or 1. The fermionic annihilation operators c and creation operators c^\dagger are defined by their actions on a Fock state thus
c_j | N_1, N_2, \dots, N_j = 0, \dots \rangle = 0
c_j | N_1, N_2, \dots, N_j = 1, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 0, \dots \rangle
c_j^\dagger | N_1, N_2, \dots, N_j = 0, \dots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \dots, N_j = 1, \dots \rangle
c_j^\dagger | N_1, N_2, \dots, N_j = 1, \dots \rangle = 0.
These obey an anticommutation relation:
\left\{c_i , c_j \right\} = 0 \quad,\quad
\left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad
\left\{c_i , c_j^\dagger \right\} = \delta_{ij}.
One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.
Field operators[edit]
We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states. However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position.
To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator \phi(\mathbf{r}) is
\phi(\mathbf{r}) \ \stackrel{\mathrm{def}}{=}\ \sum_{j} e^{i\mathbf{k}_j\cdot \mathbf{r}} a_{j}.
The bosonic field operators obey the commutation relation
\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad
\left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad
\left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})
where \delta(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.
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 a quantum-mechanical amplitude for finding a particle in some position. 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 (in the non-relativistic limit and for negligible self-interactions) is
H = - \frac{\hbar^2}{2m} \int d^3\!r \ \phi^\dagger(\mathbf{r}) \nabla^2 \phi(\mathbf{r}) + \frac{1}{2}\int\!d^3\!r \int\!d^3\!r' \; \phi^\dagger(\mathbf{r}) \phi^\dagger(\mathbf{r}') 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 \phi 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.
Once the Hamiltonian operator is obtained as part of the canonical quantization process, the time dependence of the state is described with the Schrödinger equation, just as with other quantum theories. Alternatively, the Heisenberg picture can be used where the time dependence is in the operators rather than in the states.
Unification of fields and particles[edit]
The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon.[10] In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. For free (non-interacting) quantum fields, the quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations.
Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated as excitations of an underlying electron field), so long as one can treat interactions as "perturbations" of free fields. There are still unsolved problems relating to the more general case of interacting fields that may or may not be adequately described by perturbation theory. For more on this topic, see Haag's theorem.
Physical meaning of particle indistinguishability[edit]
The second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom.
Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental.
Particle conservation and non-conservation[edit]
During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles (N), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations N is an important and perfectly well-defined quantity, e.g. if we are describing a gas of atoms sealed in a box. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator \hat{N}, which measures the total number of particles present. As with any quantum mechanical observable, \hat{N} is conserved if it commutes with the Hamiltonian. In that case, the quantum state is trapped in the N-particle subspace of the total Fock space, and the situation could equally well be described by ordinary N-particle quantum mechanics. (Strictly speaking, this is only true in the noninteracting case or in the low energy density limit of renormalized quantum field theories)
For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator a_k is immediately put back by the creation operator a_k^\dagger.
On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of \hat{N}, which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of N. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as
H_I = \sum_{k,q} V_q (a_q + a_{-q}^\dagger) c_{k+q}^\dagger c_k,
where a_k^\dagger and a_k denotes the bosonic creation and annihilation operators, c_k^\dagger and c_k denotes the fermionic creation and annihilation operators, and V_q is a parameter that describes the strength of the interaction. This "interaction term" 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 type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. The interaction between electrons and photons is treated in a similar way, but 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 typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.
In condensed matter physics, states with ill-defined particle numbers are particularly 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. In addition, the concept of a coherent state (used to model the laser and the BCS ground state) refers to a state with an ill-defined particle number but a well-defined phase.
Axiomatic approaches[edit]
The preceding description of quantum field theory follows the spirit in which most physicists approach the subject. However, it is not mathematically rigorous. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.
The first class of axioms, first proposed during the 1950s, include the Wightman, Osterwalder–Schrader, and Haag–Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others.
During the 1980s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. However, most of the physically relevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang–Mills theory—is linked to this issue.
Associated phenomena[edit]
In the previous part of the article, we described the most general features of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics involve additional special ideas, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.
Main article: Renormalization
Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances that each give a finite contribution which results in a divergent series.
Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field that it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf with help from Furry, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.
The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, comes from recognizing that all the infinities in the interactions of photons and electrons can be isolated into redefining a finite number of quantities in the equations by replacing them with the observed values: specifically the electron's mass and charge: this is called renormalization. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by 't Hooft and Veltman, is a symmetry-preserving cutoff for perturbation theory (this process is called regularization). There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice.
On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit.
The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories. A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The Standard Model of particle physics is perturbatively renormalizable, and so are its component theories (quantum electrodynamics/electroweak theory and quantum chromodynamics). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined.
The renormalization group describes how renormalizable theories emerge as the long distance low-energy effective field theory for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.
Haag's theorem[edit]
See also: Haag's theorem
From a mathematically rigorous perspective, there exists no interaction picture in a Lorentz-covariant quantum field theory. This implies that the perturbative approach of Feynman diagrams in QFT is not strictly justified, despite producing vastly precise predictions validated by experiment. This is called Haag's theorem, but most particle physicists relying on QFT largely shrug it off.
Gauge freedom[edit]
A gauge theory is a theory that 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. Consequently, 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 the shift may be different at every point in space-time. 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 gauge of variables is termed gauge transformation. It is worth noting that by Noether's theorem, for every such symmetry there exists an associated conserved current. The aforementioned symmetry of the wavefunction under global phase changes implies the conservation of electric charge.
In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics.
The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non-unitary and again inconsistent (see optical theorem).
In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis).
All the fundamental interactions in nature are described by gauge theories. These are:
Multivalued gauge transformations[edit]
The gauge transformations which leave the theory invariant involve, by definition, only single-valued gauge functions \Lambda(x_i) which satisfy the Schwarz integrability criterion
\partial_{x_i x_j} \Lambda = \partial_{x_jx_i} \Lambda.
An interesting extension of gauge transformations arises if the gauge functions \Lambda(x_i) are allowed to be multivalued functions which violate the integrability criterion. These are capable of changing the physical field strengths and are therefore not proper symmetry transformations. Nevertheless, the transformed field equations describe correctly the physical laws in the presence of the newly generated field strengths. See the textbook by H. Kleinert cited below for the applications to phenomena in physics.
Main article: Supersymmetry
Supersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa. It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory.
The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider. The Higgs particle has been detected at the LHC, and no such superparticles have been discovered.
See also[edit]
1. ^ "Beautiful Minds, Vol. 20: Ed Witten". la Repubblica. 2010. Retrieved 22 June 2012. See here.
2. ^ J. J. Thorn et al. (2004). Observing the quantum behavior of light in an undergraduate laboratory. . J. J. Thorn, M. S. Neel, V. W. Donato, G. S. Bergreen, R. E. Davies, and M. Beck. American Association of Physics Teachers, 2004.DOI: 10.1119/1.1737397.
3. ^ a b c d David Tong, Lectures on Quantum Field Theory, chapter 1.
4. ^ Srednicki, Mark. Quantum Field Theory (1st ed.). p. 19.
5. ^ Srednicki, Mark. Quantum Field Theory (1st ed.). pp. 25–6.
6. ^ Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 61.
7. ^ David Tong, Lectures on Quantum Field Theory, Introduction.
8. ^ Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 3.
9. ^ Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0-19-851997-4. Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.
10. ^ Newton, T.D.; Wigner, E.P. (1949). "Localized states for elementary systems". Reviews of Modern Physics 21 (3): 400–406. Bibcode:1949RvMP...21..400N. doi:10.1103/RevModPhys.21.400.
Further reading[edit]
General readers
Introductory texts
Advanced texts
External links[edit] |
448594a76a397f09 | Is quantum mechanics deterministic or random?
Quantum mechanics is intrinsically random. If we look at a radioactive element, it will decay with a specific half-life time, but there is no way to predict exactly when it is going to decay. It is a random process.
On the other hand, the Schrödinger equation that describes the time evolution of a quantum system is fully deterministic. Given well defined initial conditions for the wave-function at some point in time, we can determine the wave-function in any other time, in the future or in the past
\Psi(t) = e^{-\frac{i}{\hbar} \int_0^t H(t) dt} \Psi(0) .
How can these two statements coexist in a single consistent theory?
The answer is that it is just a question of perspective. If you look at the entire wave-function, you would realize that the theory is deterministic. But if you write down the wave-function as a combination of eigen-states, then each state has a probability associated with it depending on its amplitude, according to the Born rule
P(x) = \left|\Psi(x)\right| ,
and therefore from the perspective of the eigen-state the system is random.
For example, take a spin which is in a superposition of up state and down state
\frac{1}{\sqrt{2}} \left ( \left|\uparrow\right> + \left|\downarrow\right> \right) .
There is a 50% chance that it is an up spin and 50% chance that it is a down spin. This randomness/probability is intrinsic in quantum mechanic. The randomness is there even without measurement, but it is easier to describe when we talk about a detector that made a measurement. Before the measurement the spin and the detector are uncorrelated.
\Psi_{\text{initial}} = \frac{1}{2} \left ( \left|\uparrow\right> + \left|\downarrow\right> \right) \left ( \left|\text{detect}\uparrow\right> + \left|\text{detect}\downarrow\right> \right) .
When our detector measures the spin, it also gets into a superposition of states. There is one state in which the spin is up and the detector measured spin up. There is a second state in which the spin is down and the detector measured spin down
\Psi_{\text{final}} = \frac{1}{\sqrt{2}} \left ( \left|\uparrow\right> \left|\text{detect}\uparrow\right> + \left|\downarrow\right> \left|\text{detect}\downarrow\right> \right) .
This measurement process is unitary. The wave-function of the entire system after the measurement can be calculated deterministically. Yet from the point of view of the detector, a purely random event occurred. It is impossible to predict what spin will be detected, because in the final state both spins are detected. But from the point of view of the detector, only one spin is detected, and therefore from its perspective it is a purely random event.
Moreover, random processes are irreversible. If a ball bounces of a wall at a random angle, there is no way to trace back in time the motion of the ball before it hit the wall. Indeed, from the point of view of the detector, the measurement is irreversible. If you start from a state of a detector that measured spin up, there is no way to trace back time to the original state of the spin before measurement. On the other hand, if you start with the full final state that includes the superposition of the two states, you can reverse time and recover the original spin state.
This picture is also consistent in the context of thermodynamics. In thermodynamics, entropy is not an intrinsic property of the physical system. Entropy is a measure of how much we know about the system, it depends on our perspective.
For example, take a classical ideal gas in a box. What happens if you double the volume of the box? If you know the exact position and momentum of every particle in the gas at a certain time, you can predict their future position. Therefore the entropy of the system does not change when the volume increases. But, if you are not aware of the microscopical state, your conclusion would be that the entropy increased with the change in volume.
Ideal gas, about to double in volume.
Back to our quantum system, from the perspective of the entire wave-function, the entropy is constant. As long as you look at the microscopic details of a closed system, the entropy does not increase over time. This is consistent with the fact that the process is reversible.
From the perspective of the detector, there was one bit of random information in the measurement, and therefore the entropy increased exactly by log(2). This is again consistent with the fact that from the point of view of the detector the process is irreversible.
We described the exact same physical system from the point of view of two different actors with different knowledge about the system. One description is deterministic and one is random. Both are correct at the same time, it is just a question of what is your perspective.
Understanding quantum mechanics
For almost a century now, there is an on going debate on the interpretations of quantum mechanics. Two such interpretations are the Copenhagen Interpretation (CI) and the Many Worlds Interpretation (MWI).
Beyond that, there is a meta-debate going on for almost as long about whether there is any value in debating these interpretations. Physicists that think this is a pointless endeavor consist of the Shut Up And Calculate (SUAC) camp.
In a recent article in the New York Times, Sean Carroll writes that Even Physicists Don’t Understand Quantum Mechanics. The New York Times is not a scientific journal, so normally I would not cite it as an authoritative scientific source. But it is a good source to demonstrate sentiments. This is an example of a physicist studying the foundation of quantum mechanics who feels that too many physicist do not care that they do not understand quantum mechanics, as long as they know how to make calculations.
Much of what is written here was inspired by Sabine Hossenfelder’s blog. Specifically, by her post on quantum measurement and the long comment thread that followed. This is also where I was referred to a paper by Lev Vaidman on the Ontology of the wave function and the many-worlds interpretation. My views are closely aligned with the views he presents in this paper. Vaidman’s entire career was researching the foundations of quantum mechanics. It would be hard to blame him that he does not care about the subject.
Let me try to convince you that at least some of us in the SUAC camp do care about understanding quantum mechanics. The point is that we actually understand quantum mechanics fairly well. While we might not understand everything, our understanding is deep enough that we are convinced that we do not need to worry about quantum interpretations. We also understand the measurement process. The “measurement problem” is a solved problem.
As a baseline, all parties in this debate actually agree about almost everything. We all agree that the theoretical formulation of quantum mechanics is based on Schrodinger’s equation (and all its extensions) together with the Born rule that interprets the wave-function as a probability amplitude. We also agree about all the experimental results that confirm the validity of quantum mechanics.
The only disagreement is about the process of measurement, which you could say, is somewhat crucial for connecting theory with experiment. So lets delve deeper into the Measurement Postulate.
A measurement involves an observable operator which is an Hermitian operator. Examples of Hermitian operators are position, momentum, energy, angular momentum and spin. An Hermitian operator has real eigen-values that correspond to the real values that we measure. Each eigen-value has a corresponding eigen-state.
This is where the problem begins. The Measurement Postulate states that a measurement would result in the system being in the eigen-state corresponding to the eigen-value that was measured. This is a non-unitary, non-local, non-deterministic, irreversible process. Therefore, it can not be described using the standard Hamiltonian time evolution of quantum mechanics.
The Copenhagen Interpretation (CI) claims that the wave-function collapses during the measurement. Therefore, the measurement process cannot be described using the standard Hamiltonian time evolution of quantum mechanics. The problem with this interpretation is that no one ever came up with a mechanism for this collapse that is consistent with existing experiments and that makes any new unique predictions.
Many Worlds Interpretation (MWI) claims that all branches of the measurement keep on existing, there is no collapse. One could say that the world “splits” during measurement to several worlds, but what MWI really says is that everything is encoded in the wave-function, so there is no special event during measurement. This solves the unitarity problem. We now have a reversible, deterministic, local model.
One criticism of MWI is that without collapse the measurement has no effect on our system. Choosing an observable Hermitian operator is equivalent to choosing a basis, it does not change the physics. This is a legitimate criticism, but the problem is not in quantum mechanics. The problem is that in the formulation of the Measurement Postulate, people forget to mention that a measurement requires an interaction between the observable operator and our system.
For example, if we want to measure the spin of a particle, we use the Stern–Gerlach experiment as a detector. Such a detector is essentially equivalent to adding an interaction term in the Hamiltonian that describes the coupling between the particle’s spin and a non-uniform magnetic field. A measurement is not just a change of basis. We can model how our detector deflects different spins in different directions.
The Stern-Gerlach experiment
The Stern-Gerlach experiment
To illustrate the effect of the measurement, think about an experiment that starts with a spin pointing up (\left|\uparrow\right>). The spin is first measured in the X axis and then it is measured in the Y axis.
A Stern–Gerlach experiment where the initial state is spin up. First the spin is measured in the X direction and then in the Y direction.
If, like the critics claim, the measurement in the X axis had no effect on the spin, it would stay in the up state. Then the measurement in the Y direction would just be an up state (\left|\uparrow\right>). If the measurement “collapses” the wave-function, there would be a mix of two pure spin states, one pointing left (\left|\leftarrow\right>) and one pointing right (\left|\rightarrow\right>). Then, the measurement in the Y axis would measure each such pure state as a combination of up (\left|\uparrow\right>) and down (\left|\downarrow\right>) spins, resulting in a mixed state of 4 pure spin states.
Four different outcome of our experiment where we vary the strength of the measurement in the X direction. (a) there is no X measurement, we only measure spin up. (b) weak X measurement, there is small separation in X and therefore a small chance of measuring spin down. (c) stronger X measurement, better separation in X and therefore a higher change of measuring spin down. (d) Full separation in X and therefore all four possible outcomes have equal probability.
Conceptually, one could calculate the results of this experiment using the Schrödinger equation, modelling explicitly the elusive wave-function collapse. There is no need for any extra measurement postulate. I must admit that I did not perform these calculations. Solving the Schrödinger equation for this case can be done analytically. But matching the boundary conditions and setting initial condition for the wave-packet is a bit more complicated. If I’ll get around to it, I will post a detailed explanation of the calculation. Meanwhile, the plot above illustrates the expected results.
Another criticism of MWI is that we cannot have any evidence about other worlds because these worlds have no effect on our world. But we do have evidence. In the double slit experiment the particle goes through one slit in one world and through the other slit in the other world. The interference pattern in the double slit experiment is the manifestation of the effect that different worlds have on each other.
Some worry that scaling up the measurement to macroscopic scales would require giving up reductionism. Yet the Stern-Gerlach experiment is an explicit example of how a microscopic spin quantity transforms into a macroscopic spatial separation.
Others worry that there is something mysterious going on during decoherence. Again, the Stern-Gerlach experiment shows us exactly how it works. The calculation is time-reversible. Still, the incoming wave-packet is a combination of energy states that come out of the detector with very specific phases for each state. Reversing the experiment by sending in two spins and trying to get out a pure spin state is clearly not practical.
The last criticism I can think of is that in MWI probabilities are meaningless if we claim that all outcomes exist. I do not see how this different from CI where all outcomes are possible. Moreover, it is not different from classical probabilities.
To summarize, since the introduction of the Schrödinger equation in 1925, and the Born rule in 1926, we have made some progress in our understanding of quantum mechanics. The EPR experiment, Bell’s inequalities and Everett’s many worlds interpretation contributed towards this understanding. This progress convinces us that nature behaves exactly according to the rules quantum mechanics. There is just no shred of evidence for a missing ingredient in our understanding.
As physicists we always hope that nature would challenge us with fascinating riddles. Unfortunately, I am fairly convinced that interpretations of quantum mechanics and the Measurement Postulate are not such riddles. |
5182c7fd1368b095 | Friday, April 29, 2011
Octonions and quantum physics
Peter Woit reports in "This Week's Hype" string model hype about classical number fields, in particular the possible role of octonions. It would be nice to write a comment and tell how elegantly classical number fields appear in TGD framework and make dual descriptions in terms of 8-D Minkowski space (sub-space of complexified octonions) and M4× CP2 unique. Unfortunately, Peter Woit wathces over his territory so jealously against invasion of anything which stinks like a good idea that it does not bother to take the risk of getting biten. Therefore John Baez - who as a Name is allowed to make intelligent comments- must continue to live in the illusion that no-one does anything to understand the role of octonions in physics.
The non-associativity of octonions is the basic problem if one attempts to build octonionic quantum mechanics. Nothing like this is tried in TGD. Instead, classical number fields appear at the level of classical physics (see this). Space-time surfaces as classical correlates of quantum physics are conjectured to decompose to associative (/quaternionic/Minkowskia)n and co-associative (/co-quaternionic/Euclidian) regions so that the weakness of octonionic quantum mechanics would turn into a strength making classical physics completely unique purely number theoretical. More precisely, the induced spinor structure for 8-D imbedding space has a special representation in terms of octonionic gamma matrices and the induced gamma matrices (not strictly speaking matrices anymore) are conjectured to span a quaternionic or co-quaternionic subspace of octonions over complex numbers at each point of the preferred extremal of Kähler action.
Addition: Also Motl has comments about octonions. The usual flood of insults and extremely arrogant super-stringy attitude towards anyone who does not regard superstrings as the laws of Moses for physics and dares to ask whether some aspects of super-strings might be part of a more successful physical theory. John Baez was the target of the aggression at this time. Maybe it is high time to Lubos to realize that the glamour of Harward does not last forever: we also remember that the exit of Lubos from Harward was not graceful. Some real output would be desperately needed if Lubos wants to keep his position as a blog authority and we have been waiting for years. My comment about the role of classical number fields in physics of course goes un-noticed: Lubos reads nothing which he has decided to represent crackpot theory. In any case, Lubos does a valuable work: he teaches us to tolerate people behaving like complete idiots. Learning this is after all the only manner to build a better world;-).
Thursday, April 28, 2011
About GRT limit of TGD
TGD should have General Relativity type theory as an appropriate limit. Therefore it is interesting to see what one obtains when one applies TGD picture by replacing space-times as 4-surfaces with abstract geometries as in Einstein's theory and assumes holography in the sense that space-times satisfy besides Einstein-Maxwell equations also conditions guaranteeing Bohr orbit like property. The resulting picture could be also regarded as quantized GRT type limit of quantum TGD obtained by dropping the condition that space-times are surfaces. This limit could also provide totally new insights to the quantization of GRT.
Several pleasant surprises were in store.
1. Essentially the same formalism could apply to GRT limit of TGD as TGD itself meaning that Einstein-Maxwell system can be described as almost topological QFT with holography implying that action reduces to 3-D Chern-Simons action with a metric dependent constraint term expressing the weak form of electric-magnetic duality and quantizing electric charge.
2. The existence of this limit gives valuable information also about TGD itself. In particular, the interpretation of the weak form of electric-magnetic duality is sharpened. The space-time regions with Minkowskian signature would be those in which only electromagnetic and gravitational interactions make themselves visible and regions with Euclidian signature would be the interiors of generalized Feynman graphs in which electroweak and color interactions become manifest. In particular, Weinberg angle should vanish in the Minkowskian phase so that electromagnetic field reduces to induced Kähler field identifiable as Maxwell field of Einstein-Maxwell system. This conforms with the finding that Kähler coupling strength equals to fine structure constant within the very tight constraints available.
3. The limit also suggests how one could understand the extremely small value of cosmological constant characterizing the cosmology according to GRT in terms of CP2 geometry providing idealization for the space-time region with Euclidian signature of metric representing generalized Feynman graphs also in GRT framework.
4. Non-Euclidian regions could correspond also to blackhole like regions in TGD framework, where only part of the interior of black hole is imbeddable. Black holes would naturally correspond to gigantic values of gravitational Planck constant implying that the Compton length of black-hole is of order of Schwartschild radius. Black hole would be elementary parton with very large fermion and antifermion numbers and large Planck constant and consist of dark matter in TGD sense. This picture is mathematically consistent since at event horizon the determinant of four-metric vanishes so that it is light-like just as it is at wormhole throats. Consistency with experimental factors is also achieved: about the interiors of blackholes we know nothing so that nothing prevents from assuming that it has Euclidian signature of metric: especially so if this explains the mysterious cosmological constant and standard model quantum numbers.
GRT is a more general theory than TGD in the sense that much more general space-times are allowed than in TGD - this leads also to difficulties - and one could also argue that the mathematical existence of WCW Kähler geometry actually forces the restriction of these geometries to those imbeddable in M4× CP2 so that the quantization of GRT type theory would lead to TGD.
1. The conceptual framework of TGD
There are several reasons to expect that something analogous to thermodynamics results from quantum TGD. The following summarizes the basic picture, which will be applied to a proposal about how to quantize (or rather de-quantize!) Einstein-Maxwell system with quantum states identified as the modes of classical WCW spinor field with spinors identifiable in terms of Clifford algebra of WCW generated by second quantized induced spinor fields of H.
1. In TGD framework quantum theory can be regarded as a "complex square root" of thermodynamics in the sense that zero energy states can be described in terms of what I call M-matrices which are products of hermitian square roots of density matrices and unitary S-matrix so that the moduli squared gives rise to a density matrix. The mutually orthogonal Hermitian square roots of density matrices span a Lie algebra of a subgroup of the unitary group and the M-matrices define a Kac-Moody type algebra with generators proportional to powers of S assuming that they commute with S. Therefore this algebra acts as symmetries of the theory.
What is nice that this algebra consists of generators multi-local with respect to partonic 2-surfaces and represents therefore a generalization of Yangian algebra. The algebra of M-matrices makes sense if causal diamonds (double light-cones) have sizes coming as integer multiples of CP2 size. U-matrix has as its rows the M-matrices. One can look how much of this structure could make sense in GRT framework.
2. In TGD framework one is forced to geometrize WCW consisting of 3-surfaces to which one can assign a unique space-time surfaces as analogs of Bohr orbits and identified as preferred extremals of Kähler action (Maxwell action essentially). The 3-surfaces could be identified as the intersections space-time surface with the future and past light-like boundaries causal diamond (CDs analogous to Penrose diagrams). The preferred extremals associated with the preferred 3-surfaces allow to realize General Coordinate Invariance (GCI) and its natural to assign quantum states with these.
GCI in strong sense implies even stronger form of holography. Space-time regions with Euclidian signature of metric are unavoidable in TGD framework and have interpretation as particle like structure and are identified as lines of generalized Feynman diagrams. The light-like 3-surfaces at which the signature of the induced metric changes define equally good candidates for 3-surfaces with which to assign quantum numbers. If one accepts both identifications then the intersections of the ends of space-time surfaces with these light-like surfaces should code for physics. In other words, partonic 2-surfaces plus their 4-D tangent space-data would be enough and holography would be more or less what the holography of ordinary visual perception is!
In the sequel the 3-surfaces at the ends of space-time and and light-like 3-surfaces with degenerate 4-metric will be referred to as preferred 3-surfaces.
3. WCW spinor fields are proportional to a real exponent of Kähler function of WCW defined as Kähler action for a preferred extremal so that one has indeed square root of thermodynamics also in this sense with Kähler essential one half of Hamiltonian and Kähler coupling strength playing the role of dimensionless temperature in "vibrational" degrees of freedom. One should be able to identify the counterpart of Kähler function also in General Relativity and if one has Einstein-Maxwell system one could hope that the Kähler function is just the Maxwell action for a preferred extremal and therefore formally identical with the Kähler function in TGD framework.
Fermionic degrees of freedom correspond to spinor degrees of freedom and are representable in terms of oscillator operators for second quantized induced spinor fields. This means geometrization of fermionic statistics. There is no quantization at WCW level and everything is classical so that one has "quantum without quantum" as far as quantum states are considered.
4. The dynamics of the theory must be consistent with holography. This means that the Kähler action for preferred extremal must reduce to an integral over 3-surface. Kähler action density decomposes to a sum of two terms. The first term is jαAα and second term a boundary term reducing to integral over light-like 3-surfaces and ends of the space-time surface. The first term must vanish and this is achieved if the Kähler current jα is proportional to Abelian instanton current
jα ∝ *jαα β γ δAβJγ δ
since the contraction involves Aα twice. This is at least part of the definition of preferred extremal property but not quite enough. Note that in Einstein-Maxwell system without matter jα vanishes identically so that the action reduces automatically to a surface term.
5. The action would reduce reduce to terms which should make sense at light-like 3-surfaces. This means that only Abelian Chern-Simons term is allowed. This is guaranteed if the weak form of electric-magnetic duality stating
at preferred at light-like throats with degenerate four-metric and at the ends of space-time surface. These conditions reduce the action to Chern-Simons action with a constraint term realizing what I call weak form of electric-magnetic duality. One obtains almost topological QFT since the constraint term depends on metric. This is of course what one wants.
Here the constant k is integer multiple of basic value which is proportional to gK2 from the quantization of Kähler electric charge which corresponds to U(1) part of electromagnetic charge. Fractional charges for quarks require k=ngK2/3. Physical particles correspond to several Kähler magnetically charged wormhole throats with vanishing net magnetic charge but with non-vanishing Kähler electric proportional to the sum ∑i εi kiQm,i, with εi=+/- 1 determined by the direction of the normal component of the magnetic flux for i:th throat.
The first guess is that the length of magnetic flux tube associated with the particle is of order Compton length or perhaps corresponds to weak length scale as was the original proposal. The screening of weak isospin can be understood as magnetic confinement such that neutrino pair at the second end of magnetic flux tube screens the weak charged leaving only electromagnetic charge. Also color confinement could be understood in terms of flux tubes of length of order hadronic size scales. Compton length hypothesis is enough to understand color confinement and weak screening.
Note that 1/gK2 factor in Kähler action is compensated by the proportionality of Chern-Simons action to gK2. This need not mean the absence of non-perturbative effects coming as powers of 1/gK2 since the constraint expressing electric magnetic duality depends on gK2 and might introduce non-analytic dependence on gK2.
6. In TGD the space-like regions replace black holes and a concrete model for them is as deformations of CP2 type vacuum extremals which are just warped imbeddings of CP2 to M4× CP2 with random light-like random curve as M4 projection: the light-like randomness gives Virasoro conditions. This reflects as a special case the conformal symmetries of light-like 3-surfaces and those assignable to the light-like ends of the CDs.
One could hope that this picture more or less applies for the GRT limit of quantum TGD.
2. What one wants?
What one wants is at least following.
1. Euclidian regions of the space-time should reduce to metrically deformed pieces of CP2. Since CP2 spinor structure does not exist without the coupling of the spinors to Kähler gauge potential of CP2 one must have Maxwell field. CP2 is gravitational instanton and constant curvature space so that cosmological constant is non-vanishing unless one adds a constant term to the Maxwell action, which is non-vanishing only in Euclidian regions. It is matter of taste, whether one regards V0 as term in Maxwell action or as cosmological constant term in gravitational part of the action. CP2 radius is determined by the value of this term so that it would define a fundamental constant.
This raises an interesting question. Could one say that one has a small value of cosmological constant defined as the average value of cosmological constant assignable to the Euclidian regions of space-time? The average value would be proportional to the fraction of 3-space populated by Euclidian regions (particles and possibly also macroscopic Euclidian regions). The value of cosmological constant would be positive as is the observed value. In TGD framework the proposed explanation for the apparent cosmological constant is different but one must remain open minded. In fact, I have proposed the description in terms of cosmological constant also as a proper description in the approximation to TGD provided by GRT like theory. The answer to the question is far from obvious since the cosmological constant is associated with Euclidian rather than Minkowskian regions: all depends on the boundary conditions at the wormhole throats where the signature of the metric changes.
2. One can also consider the addition of Higgs term to the action in the hope that this could allow to get rid of constant term which is non-vanishing only in Euclidian regions. It turns turns out that only free action for Higgs field is possible from the condition that the sum of Higgs action and curvature scalar reduces to a surface term and that one must also now add to the action the constant term in Euclidian regions. Conformal invariance requires that Higgs is massless.
The conceptual problem is that the surface term from Higgs does not correspond to topological action since it is expressible as as flux of Φ∇ Φ. Hence the simplest possibility is that Kähler action contains a constant term in Euclidian regions just as in TGD, where curvature scalar is however absent. Einstein-Maxwell field equations however apply that it vanishes and is effectively absent also in GRT quantized like TGD.
3. Reissner-Nordström solutions are obtained as regions exterior to CP2 type regions. In black hole horizon the metric becomes light-like and the solution can be glued to a deformed CP2 type region with metric becoming degenerate at the 3-surface involved. This surface corresponds to wormhole throat in TGD framework. Blackhole is replaced with CP2 type region. In TGD black hole solutions indeed fail to be imbeddable at certain radius so that deformed CP2 type vacuum extremal is much more natural object than black hole. In the recent framework the finite size of CP2 means that macroscopic size for the Euclidian regions requires large deformation of CP2 type solution.
Remark: In TGD framework large value of hbar and space-time as 4-surface property changes the situation. The generalization of Nottale's formula for gravitational Planck constant in the case of self gravitating system gives hbargr= GM2/v0, where v0/c<1 has interpretation as velocity type parameter perhaps identifiable as a rotation velocity of matter in black hole horizon. This gives for the Compton length associated with mass M the value LC= hbargr/M= GM/v0. For v0=c/2 one obtains Scwartschild radius as Compton length. The interpretation would be that one has CP2 type vacuum extremal in the interior up to some macroscopic value of Minkowski distance. One can whether even the large voids containing galaxies at their boundaries could correspond to Euclidian blackhole like regions of space-time surface at the level of dark matter.
4. The geometry of CP2 allows to understand standard model symmetries when one considers space-times as surfaces. This is not necessarily the case for GRT limit.
1. In the recent case one has different situation color quantum numbers make sense only inside the Euclidian regions and momentum quantum numbers in Minkowskian regions. This is in conflict with the assumption that quarks can carry both momentum and color. On the other, color confinement could be used to argue that this is not a problem.
2. One could assume that spinors are actually 8-component M4× CP2 spinors but this would be somewhat ad hoc assumption in general relativistic context. Also the existence of this kind of spinor structure is not obvious for general solutions of Einstein-Maxwell equations unless one just assumes it.
3. It is far from clear whether the symplectic transformations of CP2 could be interpreted as isometries of WCW in general relativity like theory. These symmetries certainly act in non-trivial manner on Euclidian regions but it is highly questionable whether this could give rise to a genuine symmetry. Same applies to Kac-Moody symmetries assigned to isometries of M4× CP2 in TGD framework. These symmetries are absolutely essential for the existence of WCW Kähler geometry in infinite-D context as already the uniqueness of the loop space Kähler geometries demonstrates (maximal group of isometries is required by the existence of Riemann connection).
Note that a generalization of Equivalence Principle follows in TGD framework from the assumption that coset representations of super-conformal symplectic algebra and super Kac-Moody algebra define conformally invariant physical states. The equality of gravitational and inertial masses follows from the condition that the actions of the super-generators of two algebras are identical. This also justifies the use p-adic thermodynamics for the scaling generator of either super-conformal algebra without a loss of conformal invariance.
5. One could argue that GRT limit does not make sense since in Minkowskian regions the theory knows nothing about the color and electroweak quantum numbers: there is only metric and Maxwell field. On the other hand, in TGD one has color confinement and weak screening by magnetic confinement. If the functional integral over Euclidian regions representing generalized Feynman diagrams is enough to construct scattering amplitudes, pure Einstein-Maxwell system in Minkowskian regions might be enough. All experimental data is expressible in terms of classical em and gravitational fields. If Weinberg angle vanishes in Minkowskian regions, electromagnetic field reduces to Kähler form and the interpretation of the Maxwell field as em field should make sense. The very tight empirical constraints on the value of Kähler coupling strength αK indeed allow its identification as fine structure constant at electron length scale.
6. One can worry about the almost total disappearance of the metric from the theory. This is not a problem in TGD framework since all elementary particles correspond to many-fermion states. For instance, gauge bosons are identified as pairs of fermion and antifermion associated with opposite throats of a wormhole connecting two space-time sheets with Minkowskian signature of the induced metric. Similar picture should make sense also now.
7. TGD possesses also approximate super-symmetries and one can argue that also these symmetries should be possessed by the GRT limit. All modes of induced spinor field generate a badly broken SUSY with rather large value of N (number of spinor modes) and right-handed neutrino and its antiparticle give rise to N=2 SUSY with R-parity breaking induced by the mixing of left- and right handed neutrinos induced by the modified Dirac equation. This picture is consistent with the existing data from LHC and there are characteristic signatures -such as the decay of super partner to partner and neutrino- allowing to test it. These super-symmetries might make sense if one replaces ordinary space-time spinors with 8-D spinors.
Note that the possible inconsistency of Minkowskian and Euclidian 4-D spinor structures might force the use of 8-D Minkowskian spinor structure.
3. Preferred extremal property for Einstein-Maxwell system
Consider now the preferred extremal property defined to be such that the action reduces to Chern-Simons action at space-like 3-surfaces at the ends of space-time surface and at light-like wormhole throats.
1. In Maxwell-Einstein system the field equations imply
jα=0 .
so that the Maxwell action for extremals reduces automatically to a surface term assignable to the preferred 3-surfaces. Note that Higgs field could in principle serve as a source of Kähler field but its presence does not look like a good idea since it is not present in the field equations of TGD and because the resulting boundary term is not topological.
2. The condition
J=k× *J
at preferred 3-surfaces guarantees that the surface term from Kähler action reduces to Abelian Chern-Simons term and one has hopes about almost topological QFT.
Since CP2 type regions carry magnetic monopole charge and since the weak form of electric-magnetic duality implies that electric charge is proportional to the magnetic charge, one has electric charge without electric charge as Wheeler would express it. The identification of elementary building blocks as magnetic monopoles leads in TGD context to the picture about particle as Kähler magnetic flux tubes having opposite magnetic charges at their ends. It is not quite clear what the length of the tubes is. One possibility is Compton length and second possibility is weak length scale and the color confinement length scale. Note that in TGD the physical charges reside at the wormhole throats and correspond to massless fermions.
3. CP2 is constant curvature space and satisfies Einstein equations with cosmological constant. The simplest manner to realize this is to add to the action constant volume term which is non-vanishing only in Euclidian regions. This term could be also interpreted as part of Maxwell action so that it is somewhat a matter of taste whether one speaks about cosmological constant or not. In any case, this would mean that the action contains a constant potential term
V= V0× (1+sign(g))/2 ,
where sign(g)=-1 holds true in Minkowskian regions and sign(g)=1 holds true in Euclidian regions.
Note that for a piece of CP2 V0 term can be expressed is proportional to Maxwell action and by self-duality this is proportional to instanton action reducible to a Chern-Simons term so that V0 is indeed harmless from the point of view of holography.
4. For Einstein-Maxwell system with similar constant potential in Euclidian regions curvature scalar vanishes automatically as a trace of energy momentum tensor so that no interior or surface term results and the only surface term corresponds to a pure Chern-Simons term for Maxwell field. This is exactly the situation also in quantum TGD. The constraint term guaranteeing the weak form of electric-magnetic duality implies that the metric couples to the dynamics and the theory does not reduce to a purely topological QFT.
5. In TGD framework a non-trivial theory is obtained only if one assumes that Kähler function corresponds apart from sign to either the Kähler action in the Euclidian regions or its negative in Minkowskian regions. This is required also by number theoretic vision. This implies a beautiful duality between field descriptions and particle descriptions.
This also guarantees that the Kähler function reducing to Chern-Simons term is negative definite: this is essential for the existence of the functional integral and unitarity of the theory. This is due to the fact that Kähler action density as a sum of magnetic and electric energy densities is positive definite in Euclidian regions. This duality would be very much analogous to that implied by the possibility to perform Wick rotation in QFTs. Therefore it seems natural to postulate similar duality also in the proposed variant of quantized General Relativity.
6. The Kähler function of the WCW would be given by Chern-Simons term with a constraint expressing the weak form of electric-magnetic duality both in TGD and General Relativity. One should be able regard also in GRT framework WCW as a union of symmetric spaces with Kähler structure possessing therefore a maximal group of isometries. This is an absolutely essential prerequisite for the existence of WCW Kähler geometry. The symmetric spaces in the union are labelled by zero modes which do not contribute to the line element and would represent classical degrees of freedom essential for quantum measurement theory. In TGD the induced CP2 Kähler form would represents such degrees of freedom and the quantum fluctuating degrees of freedom would correspond to symplectic group of δ M4+/-× CP2.
The difference between TGD and GRT would be that light-like 3-surfaces for all possible space-times containing Euclidian and Minkowskian regions would be considered for GRT type theory. In TGD these space-times are representable as surfaces of M4× CP2. In TGD framework the imbeddability assumption is crucial for the mathematical existence of the theory since it eliminates space-times with non-physical characteristics. The problem posed by arbitrarily large values of cosmological constants is one of the basic problems solved by this assumption. Also mass density is sub-critical for cosmologies with infinite duration and critical cosmologies are unique apart from their duration and quantum critical cosmologies replace inflationary cosmologies.
7. Note that one could consider assigning the gravitational analog of Chern-Simons term with the preferred 3-surfaces: this kind of term is discussed by Witten in this classic work about Jones polynomial. This term is a non-abelian version of Chern-Simons term and one must replace curvature tensor with its contraction with sigma matrices so that 4-D spinor structure is necessarily involved. The objection is that this term contains second derivatives. In TGD spinor structure is induced from that of M4× CP2 and this kind of term need not make sense as such since gamma matrices are expressed in terms of imbedding space gamma matrices: among other things this resolves the problems caused by the non-existence of spinor structure for generic 4-geometries. The coupling to the metric however results from the constraint term expressing weak form of electric-magnetic duality.
The difference between TGD and GRT would be basically due to the factor of scattering amplitudes coming from the duality expressing electric-magnetic duality and due to the fact that induced metric in terms of H-coordinates and Maxwell potential is expressible in terms of CP2 coordinates. The latter implies topological field quantization and many-sheeted space-time crucial for the interpretation of quantum TGD.
4. Could ZEO and the notion of CD make sense in GRT framework?
The notion of CD is crucial in ZEO and one can ask whether the notion generalizes to GRT context. In the previous arguments related to EG the notion of ZEO plays a fundamental role since it allows to replace S-matrix with M-matrix defining "complex square root" of density matrix.
1. In TGD framework CDs are Cartesian products of Minkowskian causal diamonds of M4 with CP2. The existence of double light-cones in curved space-time would be required and its is not clear whether this makes sense generally. TGD suggest that the scales of these diamonds defined in terms of the proper time distance between the tips are integer multiples of CP2 scale defined in terms of the fundamental constant V0 (the more restrictive assumption allowing only 2n multiples would explain p-adic length scale hypothesis but would not allow the generalization of Kac-Moody algebra spanned by M-matrices). The difference between boundaries of GRT CDs and wormhole throats would be that four-metric would not be degenerate at CDs.
2. The conformal symmetries of light-cone boundary and light-like wormhole throats generalize also now since they are due to the metric 2-dimensionality of light-like 3-surfaces. It is however far from clear whether one can have anything something analogous to conformal variants of symplectic algebra of δ M4+/-× CP2 and isometry algebra of M4× CP2.
Could one perhaps identify four-momenta as parameters associated with the representations of the conformal algebras involved? This hope might be unrealistic in TGD framework: the basic idea behind TGD indeed is that Poincare invariance lost in GRT is retained if space-times are surfaces in H=M4× CP2. The reason is that that super-Kac-Moody symmetries correspond to localized isometries of H whereas the super-conformal algebra associated with the symplectic group is assignable to the light-like boundaries δ M4+/-× CP2 of CD of H rather than space-time surface.
3. One could of course argue that some physical conditions on GRT -most naturally just the highly non-trivial mathematical existence of WCW Kähler geometry and spinor structure- could force the representability of physically acceptable 4-geometries as surfaces M4× CP2. If so, then also CDs would the same CDs as in TGD and quantization of GRT would lead to TGD and all the huge symmetries would emerge from quantum GRT alone.
The first objection is that the induced spinor structure in TGD is not consistent with that natural in GRT. Second objection is that in TGD framework Einstein-Maxwell equations are not true in general and Einstein's equations can be assumed only in long length scales for the vacuum extremals of Kähler action. The Einstein tensor would characterize the energy momentum tensor assignable to the topologically condensed matter around these vacuum extremals and neither geometrically nor topologically visible in the resolution defined by very long length scale. If Maxwell field corresponds to em field in Minkowskian regions, the vacuum extremal property would make sense in scales where matter is electromagnetic neutral and em radiation is absent.
5. What can one conclude?
The previous considerations suggest that a surprisingly large piece of TGD can be applied also in GRT framework and raise the possibility about quantization of Einstein-Maxwell system in terms of Kähler geometry of WCW consisting of 3-geometries instead of 3-surfaces. One can even consider a new manner to understand TGD as resulting from the quantization of GRT in terms of WCW Kähler geometry in the space of 3-metrics realizing holography and making classical theory an exact part of quantum theory. Since the space-times allowed by TGD define a subset of those allowed by GRT one can ask whether the quantization of GRT leads to TGD or at least sub-theory of TGD. The arguments represented above however suggest that this is not the case.
The generalization of S-matrix to a complex of U-matrix, S-matrix and algebra of M-matrices forced by ZEO gives a natural justification for the modification of EG allowing gravitons and giving up the rather nebulous idea about emergent space-time. Whether ZEO crucial for EG makes sense in GRT picture is not clear. A promising signal is that the generalization of EG to all interactions in TGD framework leads to a concrete interpretation of gravitational entropy and temperature, to a more precise view about how the arrow of geometric time emerges, to a more concrete realization of the old idea that matter antimatter asymmetry could be due to different arrows of geometric time for matter and antimatter, and to the idea that the small value of cosmological constant could correspond to the small fraction of non-Euclidian regions of space-time with cosmological constant characterized by CP2 size scale.
The above considerations were inspired by the attempt to understand what is good and what is bad in the entropic gravity scenario of Verlinde in TGD framework with the basic idea being that quantum TGD as a square root of thermodynamics must predict something analogous to thermalization of the lines of generalize Feynman graphs. The above interpretation for the lines of Feynman graphs as analogs of blackholes indeed allows to understand blackhole temperature and entropy as a manifestation of this underlying thermodynamics. The generalization of blackhole thermodynamics implies that both virtual gravitons and gauge bosons are thermalized. For details see the article TGD inspired vision about entropic gravity.
Tuesday, April 26, 2011
D0 reports a new 3 sigma bump with mass around 325 GeV
It seems that experimentalists have gone totally crazy. Maybe new physics is indeed emerging from LHC and they want to publish every data bit in the hope of getting paid visit to Stockholm. CDF and ATLAS have told about bumps and now Lubos tells about a new 3 sigma bump reported by D0 collaboration at mass 325 GeV producing muon in its decay producing W boson plus jets. The proposed identification of bump is in terms of decay of t' quark producing W boson.
Lubos mentions also second mysterious bump at 324.8 GeV or 325.0 GeV reported by CDF collaboration and discussed by Tommaso Dorigo towards the end of the last year. The decays of these particles produce 4 muons through the decays of two Z bosons to two muons. What is peculiar is that two mass values differing by .2 GeV are reported. The proposed explanation is in terms of Higgs decaying to two Z bosons. TGD based view about new physics suggests strongly that the three of four particles forming a multiplet is in question.
One can consider several explanations in TGD framework without forgetting that these bumps very probably disappear. Consider first the D0 anomaly alone.
1. TGD predicts also higher generations but there is a nice argument based on conformal invariance and saying that higher particle families are heavy. What "heavy" means is not clear. It could of mean heavier that intermediate gauge boson mass scale. This explanation does not look convincing to me.
2. Another interpretation would be in terms of scaled up variant of top quark. The mass of top is around 170 GeV and p-adic length scale hypothesis would predict that the mass should equal to a multiple of half octave of top quark mass. Single octave would give mass of 340 GeV. The deviation from predicted mass would be 5 per cent. This quark could correspond to t quark of scaled up hadron physics predicted by TGD and discussed in previous postings (see this, this, abd this).
The prediction of the scaled up hadron physics allows to ask whether a common explanation for all these particles as decay products of kaons of M89 hadron physics could exist. Could charged kaon produceneutral pion and single W boson and therefore muon just as the 300 GeV charged pion would produce W boson plus neutral pion decaying to two jets. This explanation excludes the the interpretation of ATLAS bump as neutral pion and CDF bump as charged kaon but CDF and D0 bumps could live peacefully together.
If there indeed are two slightly different masses one can can ask whether the two different masses could be due to CP breaking. The mass difference between short-lived and long-lived ordinary kaon is however extremely small- 3.5×10-12 MeV- and scaling by a factor 512 would give quite too small mass difference. That CP (or even CPT) breaking should be so large for the scaled up version of hadron physics looks odd. As a matter fact, the splitting is of the same order as electromagnetic splitting between mesons with different charges obtained by scaling with factor 512 from the mass splitting of order 1 MeV for ordinary mesons.
Addition: The newest rumor is that ATLAS rumor about too photo-philic Higgs with exactly the same mass of 115 GeV as the hegemony wanted it to have was not more than a rumor. Sorry Lubos;-).
For me the newest rumor is a relief since it makes it more easier to find room for the remaining rumors in zoomed up hadron physics. The CDF rumor about 145 GeV bump would be interpreted in terms of charged pion. The latest D0 rumor weighting 325 GeV and producing W bosons, and the earlier CDF rumor having two slightly different masses around 325 GeV and producing two Z bosons would in turn be interpreted in terms of scaled up charged and neutral kaons.
However, if a strict scaling would hold true for the meson masses, one could conclude that either 145 GeV or 325 rumour is only a humor since the mass scale ratio 325/145 ≈ 2.24 is smaller than the mass scale ratio for ordinary kaon and pion about 490/140≈ 3.5. This would leave only one or two rumors to be killed. Probably they suffer a natural death within week or two in any case. I have not taken main stream theorists seriously for decades but believed that experimentalists are somehow more rooted in reality. Has the hype disease infected also experimentalists? This would be sad. Addition: The latest rumor about Atlas by Peter Woit tells that New Science has received inner information that ATLAS bump has not been found in other experiments. Tommaso in turn claims that this cannot be true! From which some reader concludes between lines that ATLAS has observed photo-philic Higgs after all!! When physics blogs came, I thought that they would provide forums for a genuine discussion about new ideas and could also serve some kind of educational function: for instance, about statistical methods of particle physics. I was wrong: they are forums for a chat about what names have said, for boosting the ego of the blogger, for the endlessly boring n sigma talk, and speculations around rumors and counter rumors. Does the situation in the web of so called respected blogs reflect the situation also in experimental particle physics? I sincerely hope that this is not the case.
Objection against zero energy ontology and quantum classical correspondence
How arrow of geometric time is selected at quantum level?
I have discussed in the chapter About the Nature of Time of "Matter, Mind, Quantum" how the arrow of geometric time as a correlate for the experienced arrow of geometric time might be selected in TGD Universe. The discussion does not touch the question what arrow of time means at the level of quantum states. Therefore the notion of negative energy signal propagating backwards in geometric time crucial for TGD inspired quantum biology remains somewhat fuzzy.
The recent progress in the understanding of the basic properties of zero energy states makes it possible to understand what arrow of geometric time and the notion of negative energy state and signals propagating to the direction of geomeric past mean at the level of zero energy states. This understanding has surprisingly non-trivial philosophical implications. In the following I shall briefly the quantum view about arrow of time.
Arrow of time as an inherent property of zero energy states?
The basic idea can be expressed in very conscise form. In positive energy ontology arrow of time characterizes dynamics. In zero energy ontology arrow of time characterizes quantum states.
1. The breaking of time reversal invariance (see this) means that zero energy states can be localized with respect to particle number and other quantum numbers only for future or past light-like boundary of CD but not both. M-matrix generalizing S-matrix provides the time-like entanglement coefficients expressing the state at the second boundary as quantum superposition of states with well-defined particle numbers and other quantum numbers. But only at the second end of CD since one cannot choose freely the states at both boundaries: if this were the case the counterpart of Schrödinger equation would be completely non-deterministic. This is what the breaking of time reversal symmetry means. It occurs spontaneously and assigns to the arrow of subjective time geometric arrow of time.
This picture gives a precise meaning to the arrow of geometric time and therefore also for the otherwise fuzzy notion of negative energy signals propagating backwards in space-time playing key role in TGD based models of memory, metabolism, and intentional action (see this).
2. Quantum jump begins with the unitary U-process between zero energy states generating a superposition of zero energy states. After that follows state function reduction cascade proceeding from the level of CD to the level of sub-CDs forming a fractal hierarchy. The reductions cannot take independently at both light-like boundaries of CD as is also clear from the fact that scattering state leads from a prepared state to a quantum superposition of prepared states.
The first guess is that the cascade takes place for the second boundary of CD only so that the arrow of geometric time would be same in all scales. This need not be the case always: the geometric arrow of time seems to change in some situations: phase conjugate laser light and spontaneous self-assembly of bio-molecules are good examples about this (see this and this). In fact, one of the defining properties of living matter could be just the possibility that the arrow of geometric time is not same in all scales (size scales of CDs) so that memory, metabolism, and intentional action become possible. In any case, the second end remains a superposition of quantum states.
The lack of quantum measurements at the second end of space-times could explain why the conscious percepts are sharply localized in time at the second end of CD. This could also allow to understand memories as reductions occurring at the second, non-standard, end of sub-CDs in the geometric past.
3. The correspondence between the reduced state and the quantum superposition of states at the opposite boundary of CD allows an interpretation in terms of logical implication arrow with all statements present in the superposition implying the statement represented by the reduced state. Only implication arrow rather than equivalence is possible unless the M-matrix is diagonal meaning that there are no interactions. If it is possible to diagonalize M-matrix then in diagonal basis one has equivalences. It must be however emphasized that the physically preferred state basis fixed as in terms of eigenstates of density matrix does not allow diagonal M-matrix. Number theoretic conditions required that the density matrix corresponds to fixed algebraic extension of rationals can also make possible the diagonalization without leaving the extension and this condition might be highly relevant in the TGD inspired view about cognition relying on p-adic number fields and their algebraic extensions (see this).
4. In classical logic implication corresponds to the inclusion of subset by subset. In quantum case it corresponds to the inclusion for sub-space of state space. The inclusions of hyper-finite factors (WCW spinors define HFF of type II1) realize the notion of finite measurement resolution, which would suggest that inclusion arrow has also interpretation in terms of finite measurement resolution.
All quantum states equivalent with a given state in the resolution used imply it. Finite measurement resolution would mean that there would infinite number of instances always in the quantum superposition representing the rule A → B. Ironically, both finite measurement resolution and dissipation implying the arrow of geometric time and usually regarded as something negative from the point of view of information processing would be absolutely essential element of logical thinking in this framework.
5. Conscious theorem proving would has as correlate to building of sequences zero energy states representing A → B, B→ C, C → D with basic building bricks representing simple basic rules. These sequences would represent more complex truths.
Does state function-state preparation sequence correspond to alternating arrow of geometric time?
The state function reduction at light-like boundary of CD implies delocalization at the opposite boundary. This inspires so fascinating questions.
1. Could the state function reduction process take place alternately at the two boundaries of CD so that a kind of flip-flop in which the arrow of geometric time changes back and forth would result, and have interpretation as an alternating sequence of state function reductions and state preparations in the framework of positive energy ontology?
2. State function reductions are needed for sensory percepts. Could the sleep-wake-up period correspond to this kind of process so that during what we call sleep the past boundary of our personal CD would be in wake-up state? Could dreams and memories represent sharing of mental images of this kind of consciousness? Could it be that in the time scale of entire life cycle death is accompanied by birth at the second boundary of personal CD? Could this quantum physics representation for endless sequence of deaths and rebirths? Could the fact that old people often spend they last years in childhood have interpretation in this framework?
3. State preparation-reduction cycle might characterize only living matter whereas for inanimate matter second choice for the arrow of time would be dominant between two U-processes. TGD based reformulation of entropic gravity idea of Verlinde in terms of ZEO does not assume the absence of gravitons and the emergence of space-time (see this). The formulation leads to the proposal that thermodynamical stability selects the arrow of the geometric time and that it could be different for matter and antimatter implying that matter and antimatter reside at different space-time sheets. This would explain the apparent absence of antimatter and also support the view that the arrow alternates only in living matter.
The arrow of geometric time and the arrow of logical implication
If physics is mathematics in the sense that there is nothing behind quantum states regarded as purely mathematical objects, Boolean logic must have a direct manifestation in the structure of physical states. Physical states should represent quantal Boolean statements which get their meaning via quantum jumps. In TGD framework WCW ("world of classical worlds") spinor fields represent quantum states of the Universe and WCW spinors correspond to fermionic Fock states for second quantized induced spinor fields at space-time surface. Fock state basis has interpretation in terms of Boolean algebra. In positive energy ontology the problem is that fermion number as a super-selection rule would allow very limited number of Boolean statements to be represented. In ZEO the situation changes.
The fermionic parts of positive and negative energy parts can be seen as quantum superpositions of Boolean statements with fermion number in given mode (equal to 0 or 1) representing yes/no or true/false. Also various spin like quantum numbers associated with oscillator operators have same interpretation. Zero energy state could be seen as quantum superposition of pairs of elements of Boolean algebras associated with positive and negative energy parts of the zero energy state.
The first - and incorrect - interpretation is that zero energy state represents a quantum superposition of equivalent statements a↔ b and thus abstraction A<---> B involving several instances of A and B. The breaking of time reversal invariance allowing localization to definite fermionic quantum numbers at single end of CD only however implies that quantum states can only represent abstraction of logical implication to A→ B rather than equivalence. p-Adic physics for various primes p (see this) would represent correlates for cognition and intentionality.
For background see the chapter About the Nature of Time of "Matter, Mind, Quantum".
Monday, April 25, 2011
Water memory made visible
Water memory is one of those phenomena crucially important for understanding living matter whose existence is stubbornly forbidden by skeptics who say that water is just H2O and nothing else since this is what they learned in the elementary school.
The latest demonstrations of water memory is by the research group of HIV Nobelist Montagnier giving also strong support for a completely new realization of genetic realized somehow by water (see this) but finnish skeptics concluded that Montagnier and his group are either swindlers or that the group knows nothing about basics of experimental biology. Only a complete idiot can have the self-confidence and ignorance possessed by the most pathological finnish skeptics.
In our neighboring country Sweden skeptics have totally different attitude towards truth. For instance, two swedish physics professors admitted that the recent demonstrations of cold fusion by Italian researchers (see this) suggest strongly that new kind of nuclear reactions taking place at low temperatures are involved. The other professor leads swedish skeptics society (see this).
TGD based view about dark matter leads to a model of dark nucleon with size of order DNA triplet in which nucleon states consisting of three quarks are in one-one correspondence with DNA,RNA,tRNA, and aminoacids and vertebrate genetic code has a simple and beatiful realization (see this). This supports the view that genetic code is realized at the nuclear physics level for dark matter in water, and that the chemical realization emerged much later. This would have profound implications for the understanding of evolution and also for what happens in the cellular water of living organisms also now.
Fischer Gabor sent me Youtube video making water memory directly visible. For instance, water droplets remember the person who prepared them or the flower dropped to the water by the structure of the droplets made visible by the method used by the researchers. Essentially holographic memory is in question. Maybe this video might open some eyes to see the fascinating reality in all its beauty. Enjoy!
Thursday, April 21, 2011
TGD based view about entropic gravity
I discussed entropic gravity of Verlinde for some time ago in rather critical spirit but made also clear that quantum TGD in the framework of zero energy ontology could be called square root of thermodynamics so that thermodynamics- or its square root- should emerge at the level of the lines of generalized Feynman diagrams. The intolerable-to-me features of entropic gravity idea are the claimed absence of gravitons and the nonsense talk about the emergence of dimensions assuming at the same time basic formulas of general relativity.
I returned to the topic later again with a boost given by one of the few people in the finnish academic establishment who have regarded me as a life form with some indications about genuine intelligence. What demonstrates the power of a good idea is that just posing some naturally occurring questions led rapidly to a TGD inspired phenomenology of EG allowing to see what is good and what is bad in EG hypothesis and also to see possible far reaching connections with apparently completely unrelated basic problems of recent day physics.
Consider first the phenomenology of EG in TGD framework.
1. Gravitating bodies can be seen as sources of virtual and real gravitons propagating along flux tubes. The gravitons at flux tubes are thermalized and thus characterized by temperature and entorpy when the wavelength is much shorter than the distance between the source and receiver. One can say that massive object serves as a heat source. One could also say that the pair of bodies connected by flux tubes serves as a heat source for the flux tubes with temperature determined by reduced mass so that their is a complete symmetry between the two bodies.
2. The expression for the gravitonic entropy of the flux tube is naturally proportional to the length of flux tube at a given "holographic screen" - and for the gravitonic temperature-naturally proportional to the inverse of distance squared in absence of other heat sources from standard Laplace equation- are consistent with their forms at the non-relativistic limit discussed by Sabine Hossenfelder in very transparent manner. In general case, the stringy slicing for the preferred extremals of Kähler action provide the preferred coordinates in which gravitational potential and the counterpart of the radial coordinate can be identified.
3. EG generalizes to all interactions but negative temperatures mean a severe problem. This in turn suggests a direct connection with matter-antimatter asymmetry. Could thermally stable matter and antimatter correspond in zero energy ontology to different arrows of geometric time and appear therefore in different space-time regions? I have made this question also earlier but with a motivation coming directly from the formalism of quantum TGD.
This approach leads to the question whether the mathematical formalism of quantum TGD could make sense also in General Relativity when appropriately modified. In particular, do the notions of zero energy ontology and causal diamond and the identification of generalized Feynman diagrams as space-time regions of Euclidian signature of the metric make sense? Does the Kähler geometry for world of classical worlds realizing holography in strong sense lead to a formulation of GRT as almost topological QFT characterized by Chern-Simons action with a constraint depending on metric?
1. Einstein-Maxwell theory generalizes Kähler action and the conditions guaranteing reduction of action to 3-D "boundary term" are realized automatically by Einstein-Maxwell equations and the weak form of electric-magnetic duality leads to Chern-Simons action.
2. One distinction beween GRT and TGD is the possibility of space-time regions of Euclidian signature of the induced metric in TGD representing the lines of generalized Feynman diagrams. The deformations of CP2 type vacuum extremals with Euclidian signature of the induced metric represent these lines replace black holes in TGD Universe. Black hole horizons are big particles and are suggested to possess gigantic effective value of Planck constant for which Schwartshild radius is essentially the Compton length for gravitational Planck constant so that black hole becomes indeed a particle in quantum sense. Blackholes represent dark matter in TGD sense.
3. CP2 type vacuum extremals are solutions of Einstein's equations with a unique value of cosmological constant fixing CP2 radius and this constant can be non-vanishing only in regions of Euclidian signature. The average value of the cosmological constant would be proportional to the ratio of the three-volume of Euclidian regions to the whole volue of 3-space and therefore very small. Could this be equivalent with the smallness of the actual cosmological constant? To answer the question one should understand the interaction between Euclidian and Minkowskian regions. I have proposed alternative manners to understand apparent cosmological constant in TGD Universe. Negative pressure could be understood in terms of the magnetic energy of magnetic flux tubes. On the other hand, quantum critical cosmology replacing inflation in TGD framework characterized by single parameter - its duration- corresponds to "negative pressure". These explanations need not be mutually exclusive.
At the formal level the formalism for WCW Kähler geometry generalizes as such to almost topological quantum field theory but the conditions of mathematical existence are extremely powerful and the conjecture is that this requires sub-manifold property.
1. The number of physically allowed space-times is much larger in GRT than in TGD framework and this leads to space-time with over-critical and arbitrarily large mass density and other problems plaguing GRT. M-theory exponentiates the problem and leads to landscape misery. The natural conjecture is that one cannot do without assuming that physically acceptable metrics are representable as surfaces in M4× CP2.
2. CP2 type regions give rise to electroweak quantum numbers and Minkowskian regions to four-momentum and spin. This almost gives standard model quantum numbers just from Einstei-Maxwell system! It is however far from clear whether one obtains both of them at the wormhole throats between the Minkowskian and Euclidian regions (perhaps from the representations of super-conformal algebras associated with light-like 3-surfaces by their geometric 2-dimensionality). Since both are needed it seems that one must replace geometry with sub-manifold geometry. Also electroweak spin is obtained naturally only if spinors are induced spinors of the 8-D imbedding space rather than 4-D spinors for which also the existence of spinor structure poses problems in the general case.
For more details see the article TGD inspired vision about entropic gravitation.
Friday, April 08, 2011
New 150 GeV boson stimulates emotions and bad rhetorics
This bump at 150 GeV manages to generate strong emotional responses. Very understandable. All predictions of the hegemony which has dominated particle physics the last thirty years seem to fail and anomalies suggesting unexpected new physics are emerging. It is intolerable that the theory of this TGD guy who has tried to talk sense for thirty years and been ruthlessly silenced and ridiculed can explain the 150 GeV anomaly elegantly using 15 years old predictions of his theory.
The latest example about highly emotional response is from Lubos. I glue below my two comments to the posting of Lubos, the response of Lubos and my response to it: I do not know whether Lubos allows it to appear in the blog. Note that Lubos carefully avoids of saying anything about the contents of my comments since he simply cannot make any reasonable counter argument. Lubos also argues against completely nonsensical statements that he has put to my mouth: another telltale signature of the rhetoric of a poor loser. Draw your own conclusions.
My first comment
TGD suggests two explanations for the possible new particle. Exotic octet of weak bosons is the first guess: it fails because there is no preference to decays to quarks.
Second explanation is in terms of a decay of charged pion of scaled up variant of ordinary hadron physics: scaling factor is 512 for the mass scale from the ratio for the Mersenne primes M_107 and M_89 labeling corresponding p-adic mass scales. According to the recent view not identical with the original one, pions would be produced abundantly and ρ meson or the first p-adic octave of charged pion would decay to W and neutral pion in turn producing quark jets. One signature of the new hadron physics would be monochromatic photon pairs with photon energy in the range 60-80 GeV. The naive scaling argument from ordinary pion mass would give mass of 71.4 GeV. p-Adic scaling with 2 is possible and produces mass 143.4 GeV to be compared with 145 GeV mentioned by Lubos.
Maybe the most dramatic prediction of TGD will be verified within next years! For details see my blog posting .
My second comment
Internal consistency arguments force to conclude that new physics is in TeV scale and the people in CDF are high rank professionals. Therefore I would be cautious in making skeptic or even cynical comments about their skills and even motivations unless I were a similar top professional myself.
Those who predict take this kind of potential discoveries quite seriously for understandable reasons. Both the forward-backward asymmetry in ttbar production and the new particle candidate can be understood in terms of scaled up variant of hadron physics predicted by TGD as I explain in detail at my blog.
Personally I cannot take seriously any model postulating ad hoc particle with hoc couplings to explain single anomaly. Principle is needed. I though for decades ago that after the advent of superstrings theorists would start to predict entire new branches of physics instead of single particle with couplings tinkered to explain single experimental anomaly.
In any case, LHC will certainly tell within few years what is the truth. We can only wait.
Don't be silly, Matti. Most of similar 3-sigma bumps supporting "previously unexpected physics" that have ever been promoted by similar teams turned have been showed to be flukes or mistakes. I have surely done similar things at the top global level so if you ask me whether I consider myself competent to judge the likelihood that this is just hogwash resulting from some rather silly errors, the answer is a resounding Yes. Your encouragement to irrationally worship people who are at least as fallible as I am and who have done lots of very problematic and complex manipulations doesn't belong to science. Science just doesn't operate and cannot operate in this way, by intimidating researchers by the "expertise" of other experts. Science can only get settled if the arguments are being verified and multiplied, not by mindless agreement with some people who are promoted to infallible holy fathers (and, in this case, also mothers).
Your TGD crackpot junk will be left without comments.
Also, it's nonsense that the LHC will need "years" to decide about similar effects. First of all, the D0 Collaboration - the second team at the Fermilab - will publicize its own verdict within weeks. And the LHC could already have the answer in their collected data, too. If it doesn't, it will have the answer this year. The more likely answer is that the effect is bunk. But if it is not bunk, it is not because of infallibility of the CDF folks who have contributed to this paper.
My response
Dear Lubos,
I am just saying that those people who have theories able to predict something (not very many of them!) are quite interested in these bumps, at I have a high respect to the work of the people doing the hard work with experiments and analyzing their results, and that I do not see why this respect could be somehow crackpottish. Certainly this respect does not mean a blind belief to the correctness of their analysis. We are all human beings and most of us are doing their best.
The person who takes the scientific discussion as a battle rather than exchange and comparison of ideas must fight against the temptation to use as the last weapon the crackpot claim. I can understand that for a fanatic string model aficionado the failure of the cherished theory is extremely traumatic experience. But still: I am disappointed that you could not resist this temptation. You are one of the *very* few blog physicists whom I can take seriously and I would respect you much more if you would make at least a single argument about the explanation provided by TGD. Why it is wrong? Why it is nonsensical? No emotional bursts: just answers to these questions in the spirit of normal scientific argumentation. Just arguments about content instead of crackpot rhetorics.
Dear Matti, apologies but your comment that followed my comment above was so atrocious that I had to use it to ban you.
My comment
Dear Lubos,
it is amusing that you are telling someone that his posting is too atrocius;-)! It was not. I just told that your put my mouth something I never said as anyone can directly check. I also asked you tell tell why my proposal is wrong instead of labeling me as a crackpot. This is just ordinary scientific discussion.
I added to my blog the comments including the comment that you deleted so that anyone can see what is involved: see .
I added also a simple estimate of the decay width of the pion of M89 hadron physics (dominating contribution comes from the box diagram with 3 gluons and one quark decaying to W at edges). The order of magnitude for the decay rate is around 20 GeV as required if one assumes flavor octet explaining also top quark asymmetry.
If you respect the rules of normal scientific debate you should tell what is wrong with the proposed mechanism for associated production of W boson and quark pair from pions produced abundantly. You could also tell what is wrong with the estimate for the decay width: what makes standard calculation crackpottish? The estimate can be found at .
You can of course delete also this posting but I will add it to my blog so that everyone can see what is involved.
With Best Regards.
Matti Pitkanen
I could not get this comment through. The blog program told that it has more than 3000 characters. It had about 1000. Perhaps this is the manner to realize the ban. I am really surprised. The briliant Lubos Motl who has been talking about intellectual honesty is afraid of a real scientific debate and uses this kind of tricks to avoid it?! Why so? If the opponent is just a miserable crackpot it should be extremely easy to demonstrate that his arguments are wrong!
Wednesday, April 06, 2011
New particle at mass about 150 GeV?
Tommaso tells about the newest result from CDF. The eprint of CDF collaboration (the first name in the long list of names is T. Aaltonen who comes from Finland) reports evidence for a new resonance like state, presumably a boson with mass around 150 GeV. The interpretation as Higgs is definitely excluded. Nature seems to be mercilessly humiliating the arrogant theoreticians;-). Tommaso promised to represent further comments already today and we are eagerly waiting! Also this posting is expected to develop in steps.
This posting has been updated a couple of times and reflects the evolution of my confused picture about what is involved. As I said: Nature seems to mercilessly humiliate arrogant theorists, me included. I shall confess below all my silly mistakes: enjoy!
First impressions
For the inhabitant of the TGD Universe the most obvious identification of the new particle would be as an exotic weak boson. The TGD based explanation of family replication phenomenon predicts that gauge bosons come in singlets and octets of a dynamical SU(3) symmetry associated with three fermion generations (fermion families correspond to topologies of partonic wormhole throats characterized by the number of handles attached to sphere). Exotic Z or W boson could be in question.
If the symmetry breaking between octet and singlet is due to different value of p-adic prime alone then the mass would come as an multiple of half-octave of the mass of Z or W. For W boson one would obtain 160 GeV consistent with 150 GeV. Z would give 180 GeV mass which is perhaps too high. The Weinberg angle could be however different for the singlet and octet so that the naive p-adic scaling need not hold true exactly.
Note that the strange forward backward asymmetry in the production of top quark pairs might be understood in terms of exotic gluon octet whose existence means neutral flavor changing currents.
One day later
Bloggers have reacted intensively to the possibility of a new particle. Tommaso has now a nice detailed analysis about the intricacies of the analysis of the data leading to the identification of the bump. Also Lubos and Resonaances have commented the new particle. Its existence have been actually known for months in physics circles. The flow of eprints to arXiv explaining the new particle has begun.
People are already now talking about an entirely new interaction. I have done this for more than decade! Actually I have talked about entire hierarchy of scaled up variants of hadron physics (Aaaarrrrgggghhh!; do not get scared: it was an expression of extreme irritation by some colleague who believes that physics proceeds by infinitesimal steps) associated with Mersenne primes and strongly suggested by p-adic length scale hypothesis!
Why an exotic weak boson a la TGD cannot be in question
From the additional data bits leaking via the blogs I can conclude that the new particle cannot be exotic weak boson but more plausibly the basic signature for what I call M89 hadron physics and for which the proton mass is by a factor 512 higher than for the ordinary hadron physics. Pions are abundantly produced in any hadron physics and the signature of any hadron physics are the weak and electromagnetic decays of pions.
The extremely important data bit that I did not have yesterday is that the decays to two jets favor quark pairs over lepton pairs. A model assuming exotic Z -called Z'- produced together with W and decaying preferentially to quark pairs has been proposed as an explanation. Neither ordinary nor the exotic weak gauge bosons of TGD Universe have this kind of preference to decay to quark pairs so that my first guess was wrong.
The resonance appears to be produced in association with W boson. Now comes the confession! This led on my side to an extremely stupid misunderstanding lasting for weeks. I thought that it is the 150 GeV bump which decays to W boson and dijet and forgot to check this when more data came. Stupid me! Ironically, it turned out that later evidence for the production of Wjj state in a decay of resonance with mass slightly below 150 GeV emerged so that the stupid error might have contained a seed of truth.
Remark: It has turned out that bump does not disappear and the most recent analysis assigns 4.1 sigma signicance to it. The mass of the bump would be at 147+/- 5 GeV. Also some evidence that the entire Wjj system results in the decay of a resonance with mass slightly below 300 GeV has emerged.
Is a scaled up copy of hadron physics in question?
The natural explanation for preference of quark pairs would be that strong interactions are somehow involved. This suggests a state analogous to a charged pion decaying to W boson and two gluons annihilating to the quark pair (box diagram). This kind of proposal is indeed made in Technicolor at the Tevatron and has as its analog second fundamental prediction of TGD that p-adically scaled up variants of hadron physics should exist and one of them is waiting to be discovered in TeV region. This prediction emerged already for about 15 years ago as I carried out p-adic mass calculations and discovered that Mersenne primes define fundamental mass scales (see this).
Sidestep: Also colored excitations of leptons and therefore leptohadron physics are predicted (see this). What is amusing that CDF discovered towards the end of 2008 what became known as CDF anomaly giving support for tau-pions. The evidence for electro-pions and mu-pions had emerged already earlier (for details see the link above). All these facts have been buried underground because they simply do not fit to the standard model wisdom. TGD based view about dark matter is indeed needed to circumvent the fact that the lifetimes of weak bosons do not allow new light particles. There is a long series of postings in my blog about CDF anomaly: see for instance this. At that time I did of course my best to inform colleagues about the predicted scaled up version of hadron physics. The only visible outcome of my efforts was that I lost my right to use the computer of Helsinki University since finnish colleagues got really angry! In any case, it would be nice if CDF would have discovered two new hadron physics without even knowing it!
Back to the topic: TGD indeed predicts p-adically scaled up copy of hadron physics in TeV region and the lightest hadron of this physics is a pion like state produced abundantly in the hadronic reactions. Ordinary hadron physics corresponds to Mersenne prime M107=2107-1 whereas the scaled up copy would correspond to M89. The mass scale would be 512 times the mass scale 1 GeV of ordinary hadron physics so that the mass of M89 proton should be about 512 GeV. The mass of the M89 pion would be by a naive scaling 71.7 GeV and about two times smaller than the observed mass in the range 120-160 GeV and with the most probable value around 145 GeV as Lubos reports. 2*71.7 GeV = 143.4 GeV would be the guess of the believer in the p-adic scaling hypothesis and the assumption that pion mass is solely due to quarks. It is important to notice that this scaling works precisely only if CKM mixing matrix is same for the scaled up quarks and if charged pion consisting of u-d quark pair is in question. The well-known current algebra hypothesis that pion is massless in the first approximation would mean that pion mass is solely due to the quark masses whereas proton mass is dominated by other contributions if one assumes that also valence quarks are current quarks with rather small masses. The alternative which also works is that valence quarks are constituent quarks with much higher mass scale.
The killer prediction for the scaled up hadron physics hypothesis are gamma pairs with gamma energy in the range 60-80 GeV. The naivest assumption would give gamma energy of 71.7 GeV. My guess based on deep ignorance about the experimental side is that this signature should be easily testable: one should scan the energy range 60-80 GeV for mono-chromatic gamma pairs.
The simplest identification of the 150 GeV resonance
The picture about CDF resonance has become clearer during the last weeks (see the postings Theorists vs. the CDF bump and More details about the CDF bump. One of the results is that leptophobic Z' can explain only 60 per cent of the production rate.
Situation is coming also clearer for me. A really cold shower came as I found an incredibly silly misunderstanding in my earlier model which assumed that Wjj results from 150 GeV resonance that I identified as charged pion of M89 hadron physics. It is of course jj which results from 150 GeV bump. This is unforgivable sloppiness. Ironically, there is now however evidence that my erratic assumption was correct in the sense that the entire Wjj might results from a resonance with mass slightly below 300 GeV. This suggests that its mass is in good accuracy two times the mass of 150 GeV bump for which best estimate is 147+/-5 GeV.
This brings in mind the explanation for the two and half year old CDF anomaly in which tau-pions with masses coming as octaves of basic tau-pion played a key role (masses were in good approximation 2k× m(&piτ), m(&piτ)≈ 2m&tau:, k=1,2. The same mechanism would explain the discrepancy between the DAMA and Xenon100 experiments. Could this mechanism be at work also now so that 300 GeV bump would correspond to the first octave M89 pion which would have mass 150 GeV. This would mean that the one first octave of charged M89 pion decaying to W and neutral M89 pion with mass slightly below 150 GeV in turn decaying to two jets. Parity conservation would force the decay via emission of W boson. Parity conservation would prevent the decays to two pions. The nasty question is why the octaves of pion are realized as resonances in ordinary hadron physics. One could indeed imagine the mother particle to be ρ meson of M89 hadron physics: in this case derivative coupling would make the decay rate small near the threshold. One can also ask whether the lightest state of M89 pion could be actually around 73 GeV as the naivest possible scaling of pion mass predicts. If so then the situation would be very similar to that in the case of tau-pion.
Connection with the top pair backward-forward asymmetry?
The predicted exotic octet of gluons proposed as an explanation of the anomalous backward-forward asymmetry in top pair production could actually correspond to the gluons of the scaled up variant of hadron physics. M107 hadron physics would correspond to ordinary gluons only and M89 only to the exotic octet of gluons only so that a strict scaled up copy would not be in question. Could it be that given Mersenne prime tolerates only single hadron physics or leptohadron physics?
In any case, this would give a connection with the TGD based explanation of the backward-forward asymmetry in the production of top pairs. In the collision incoming quark of proton and antiquark of antiproton would topologically condense at M89 hadronic space-time sheet and scatter by the exchange of exotic octet of gluons: the exchange between quark and antiquark would not destroy the information about directions of incoming and outgoing beams as s-channgel annihilation would do and one would obtain the large asymmetry.
Yesterday I generated irritation in learned colleagues by writing: "It would be nice if LHC would add to the Particle Data Tables both gluonic and electroweak octets and TGD to the text books;-)". Remaining in super-optimistic mood I would like to induce even more irritation by writing: "It would be nice if LHC would add to the Particle Data Tables not only exotic gluonic and electroweak octets but entire new hadron physics - and as a side product TGD to the text books;-;.). Good physics is fun! Enjoy!
For more about new physics predicted by TGD see the chapter p-Adic mass calculations: New Physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy". For reader's convenience I have added a short pdf article Is the new boson reported by CDF pion of M89 hadron physics? at my homepage. |
d16ece9ef61f1e27 | Complex Spherical Harmonics
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Spherical harmonic functions arise for central force problems in quantum mechanics as the angular part of the Schrödinger equation in spherical polar coordinates. They are given by , where are associated Legendre polynomials and and are the orbital and magnetic quantum numbers, respectively. The allowed values of the quantum numbers, which follow from the boundary conditions of the problem, are . The complex function is shown on the left, where the shape is its modulus and the coloring corresponds to its argument, the range 0 to corresponding to colors from red to magenta. The center and right graphics show the corresponding real and imaginary parts.
Contributed by: Porscha McRobbie and Eitan Geva (March 2011)
Open content licensed under CC BY-NC-SA
S. M. Blinder, Introduction to Quantum Mechanics: In Chemistry, Materials Science and Biology, Burlington, MA: Elsevier Academic Press, 2004.
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7d27c791553d83a5 | reviews of neither left nor right
-former U.S. Attorney General John Ashcroft
-Dennis Miller
"A must read for all who value freedom."
-Penny Nance, Concerned Women for America
Charles Konia, M.D.
Reprinted from the Journal of Orgonomy
The American College of Orgonomy
The purpose of this article is to show the sharp distinction between the science of orgonomy and the mechanistic approach to the study of nature. Materialistic science postulates that all natural phenomena are based on material process. The question becomes: what is the nature of material processes? Functional thinking asserts that they are based on mass-free energy functions whereas mechanistic materialism states that they are based on mechanical principles. In this article, both of these views are compared with each other to show which one better corresponds to observations of the real world.
Despite the many positive contribution that modern science has made, considering the many destructive consequences that it has wrought on social life, is modern science as it is currently conducted somehow an expression of the emotional If so, how are we to make sense of the fact that today’s science has both constructive and destructive consequences on human life?
With the development of bi-pedal locomotion, the upper limbs of the ancestors of the human race became freed to fashion an endless variety of tools. Other organisms, for example, birds and insects, can use their body as tools. Some non-human primates birds can even fashioned simple tools, but only human beings are able to construct and to fashion tools from a wide range of materials other than their own body. The ability to make tools has extended human functioning in his environment almost without limit. Humans can even build tools in the form of machines and computers to make other tools in endless fashion to adapt to almost any kind of environment. It is this tool making capacity that must have impressed on our early ancestors the materialistic view of the world, a view that emphasized the importance of matter in the day-to-day survival of human beings.
The materialistic view of nature was correct insofar as it did not contain any distortion. What happened around the end of the 18th century, however, was the introduction of a tragic error. The constituents of matter were no longer only material particles, but now, all matter, including living matter, was believed to behave in a mechanical, machine-like fashion. The materialistic view of nature was supplanted by mechanistic materialism. In reciprocal fashion, this view had a destructive mechanizing effect on human beings. Reich states: “All the concepts of himself which man has developed our borrowing from the machines which he has created. Building and handling machines has given man the belief that, through the machines and beyond them, he is developing to a ‘higher’ plane. On the other hand, his machines show the appearance and the mechanics of man. … The product of his mechanistic technique thus is an expression of man himself. The machines, are, in fact, an enormous expression of his biological organization. They enable him to master nature to a far higher degree than he could with his hands alone. They get him mastery of time and space. Thus, the machine has become a part of man himself, a beloved and highly esteemed part. He has the perennial dream that the machines will make life easier for him and will give him an increase enjoyment of life. And the reality? In reality, the machine has become man’s worst enemy. It will remain his worst enemy unless he differentiates himself from the machine (1). Today, the victory of the mechanistic – materialistic view of nature in every one of the natural sciences is complete. It has attained the status of complete legitimacy and reigns supreme. To question it is tantamount to scientific heresy.
How did this distortion come about and what are its practical consequences? These questions cannot be answered without knowledge of orgonomic bio-psychiatry and an understanding that ocular armor results in perceptual and cognitive rigidity and distortions. For example, the rigid application of the second law of thermodynamics, which states that in nature, energy always flows down-hill from higher to lower potentials, is an example distorted thinking, a product of the armored functioning of the mechanistic scientist That energy does flow down-hill in certain cases is correct. But the statement that all energy functions in nature flow in this manner is not only untrue, but it leads to support the application to mechanistic principles in all of nature since, in the area of mechanics, all energy functions actually do flow from higher to lower potentials. Furthermore, the application of mechanistic principles in natural science and the view that nature is no different than a machine is highly destructive to human life since it is an obstacle to recognizing the essential functions of life, to seeing the uniqueness of each individual and to understanding that functions such as self- perception and spontaneous movement have nothing whatsoever to do with mechanics.
The idea that a vital force exists in nature (“vitalism”) was introduced in the 17th century in an attempt to explain these unknown natural functions but this concept was repeatedly rejected because the energy functions responsible for this “life force” were not understood at the time. Today, the idea of a life force is in disfavor in scientific circles but nevertheless mysticism continues to creep unawares into scientific thinking.
With the introduction to mechanistic thinking into the life sciences, all observations and theories about nature were made to fit into this preconceived view. Since all machines and machine made products are identical down to the minutest detail, and since human beings are distortedly viewed as no different than machines, it logically follows in medical scientific circles, despite all observational evidence the contrary, that the body is composed of the number of replaceable parts identical in every respect. Thus, the idea that the triumphs of modern surgery are true advances in medical science is correct from a mechanistic, but not from a biological point of view, since the causes of disease that are treated with these impressive surgical interventions continue to be eluded and will always remain a closed book to the mechanistic physician (see later).
As recently as 50 years ago, biology textbooks defined the essential properties of the living in functional terms such as its reactivity to stimuli, excitability and so on. Today, the criteria defining life have been replaced by structural concepts such as the ability of an organism to form a membrane, growth, reproductive capacity and so on. These structural concepts go hand-in-hand with the structural tenants of molecular biology that the DNA molecule is identical to life. If observations of the living cannot be made to fit into the mechanical paradigm they are either ignored or discarded. Some of these properties that cannot be understood are bio-lumination functions such as mitogenic radiation, pulsatory functions, and spontaneous motion. Mystically distorted ideas of purposefulness are then introduced into mechanistic thinking to fill in the gap. The idea of purposefulness or teleological thinking also originates from the tool making ability of humans. Since the purpose of tools is to extend human functioning in their environment, it is believed by some that nature, in turn, must also have a purpose in relation to humans. This anthropocentric view of nature is a projection, and is responsible for many erroneous notions such as, for example, that human functions must be purposeful. “What is the purpose of sex?” is an oft -quoted question raised in the mechanistic science literature. A teleological (mystical) explanation such as “sex makes natural selection more effective by increasing genetic variation” (2) is then supplied which seems to answer the question but this explains nothing. The correct answer is different and quite simple: there is no purpose to sex. Since life simply functions, sex is one of the many functions that constitute life. In the sexual orgasm, the organism functions to regulate its own energy metabolism.
Mechanistic and mystical thinking are expressions of the ocular armor in the natural scientist. Both operate together to maintain his distorted worldview. This view is passed on as gospel to the credulous public via the media. The question logically arises: what is the function of the mechano-mystical thinking of the armored natural scientist? Before we can answer this question, it is necessary to discuss the characteristics of this form of thought.
The Characteristics of Mechanistic Thinking
Mechanistic thinking is perfectly valid in the fields of machines and machine technology. Mechanistic thinking in natural science is thinking about nature as if it were no different than a machine. The following are some of the axioms of mechanistic thinking in the natural sciences.
1. All realms of nature are predictable at least in principle.
Since the mechanistic scientist views nature as if it were a machine, and since in order for a machine to operate it must be flawless, it follows that if nature is machine-like, it must be predictable and perfect in all respects. According to classical physics, if an event in nature is not mechanically repeatable and predictable, it is not capable of being understood. If the phenomenon nevertheless must be investigated for practical reasons, such as the important fields of quantum physics and meteorology, then statistical techniques are used in order to achieve some degree of predictability. Using statistical methods such as the probability -based Schrödinger equation and weather models to solve problems in quantum physics or meteorology gives the mechanistic scientist partial control over his object of study, and, in some cases, even provides the illusion that the phenomenon under investigation is, in fact, understood. Nevertheless, the natural functions underlying quantum phenomena and meteorology are in the realm of orgonomic science. They are outside the realm of mechanistic science.
The science of mechanics belongs in a more superficial realm and is therefore less inclusive than the broader realm of the natural, including the biological sciences. The flaw in mechanistic thinking is that it views mechanistic materialism as the basis for all natural events instead of recognizing that mechanical science covers only a small part of the world. There are areas in nature where energy actually does flow downhill and therefore physical processes can be mechanically understood according to the principles of electromagnetic or thermodynamic theory, but they are only a part of nature. There is another sphere where energy flows uphill from lower to higher potentials involving the creation of non- living and living systems that is off-limits to the mechanistic view of nature. In these areas, the Creationists and others with their mystical framework come in and try to fill the gap.
2. All forms of energy result from matter.
The mechanistic scientist views matter as being primary in the universe. Therefore, all forms of energy are secondary to material processes. With this axiom, the mechanistic scientist denies the existence of spontaneous movement. However, spontaneous movement is the essential characteristic of living systems. Denial of this movement is dramatic and conclusive evidence for the presence of ocular armor in the mechanistic scientist. Armor gives rise to his distortions in perception and thinking. The mechanistic biologist is literally unable to observe without immobilizing the life that he sees in front of him and feels within his organism. Unable to perceive the spontaneous quality of this movement, he cannot recognize the difference between living and non- living matter. Furthermore, perception of spontaneous movement would force upon him the conclusion that the second law of thermodynamics cannot be universally valid since the existence of spontaneous movement is, itself, a violation of the universal application of this law. He would then have to conclude further that there must be two directions of energy movement in nature, one that moves from lower to higher energy levels — which Reich designated as the orgonomic potential -and one that flows from higher to lower levels — the mechanical potential of classical science.
Finally, from this simple observation, the mechanistic scientist would have to be forced to acknowledge that there must be another kind of energy in the universe to account for this direction of movement from lower to higher-levels, an energy that is mass -free. Wilhelm Reich first identified the spontaneous movement of the energy of living systems as one of its essential characteristics. Reich named his discovery orgone energy, and went on to identify many of the qualitative and quantitative functions which distinguish it from secondary, mechanical forms of energy. As an example, the following diagram depicts the relationship between the orgonomic potential of orgone energy and the mechanical potential of secondary energy:
Orgonotic capacity level
Orgonomic potential Mechanical potential
Since the movement of orgone energy from lower to higher levels is spontaneous, it is unpredictable. An example is the growth of an orgonotic system such as a cloud or living organism. In contrast, the movement of energy from higher to lower potentials is, by its very nature mechanical and, since it follows the second law of thermodynamics, it is predictable.
When the mechanistic psychiatrist misapplies the second law to biological systems, he arrives at many highly erroneous and destructive conclusions. Ideas such as that the brain controls and directs all biological functions, that thinking itself originates from the brain rather than from the body or that disturbances in the body have nothing, whatsoever, to do with psychiatric disorders are some examples. This view of the brain provides the mechanistic psychiatrist with a rationale for treating psychiatric disorders exclusively with medication targeting the brain and it serves to deny the crucial importance of emotional factors in psychiatric disorders. When the mechanistic sociologist misapplies the second law to social systems (nations) he is, in effect, in favor of the idea that all nations in the world should be governed ( controlled) “from the top” by a centralized entity, the United Nations. He is opposed to any attempt by individual nations (particularly theUnited States) to determine their own destiny. Again, this view of the world is an accurate reflection of the armored biophysical state of the mechanistic sociologist, in whom the brain controls and directs the body.
3. The direction of natural research leads to ever increasing complexity.
There are only two directions of investigation in natural science: toward the common functioning principal (CFP) or toward the variations of the common functioning principal as shown in the following diagram:
Toward the Common Functioning Principal.
Toward the Variations.
Research in the direction of common functions leads to greater comprehensiveness of understanding. Research in the direction of the variations of the leads to greater complexity. This is the direction taken by mechanistic science. Since the mechanistic believes matter is primary in nature, his thinking and research is always in the direction of identifying the endless differences (variations) of the material systems under investigation. Mechanistic natural scientists are compulsive gatherers of facts. Since nature is functionally ordered, the uncovered facts have little or no functional relationship to each other. The direction of research is toward ever increasing degrees of complexity and greater detail and there is no possibility of providing a unified functional understanding of nature. The mechanistic psychiatrist focuses on the differences between a hodge-podge of symptom -based diagnostic entities. The mechanistic sex -researcher investigates the limitless variety of disturbed sexual behaviors of armored humans. The mechanistic cancer researcher looks at the differences between one type of cancer cell and another and overlooks what all types of cancers have in common The molecular biologist examines the differences between one DNA molecule and another or one type of virus and another while overlooking their common function. Since he is confined to studying the interactions of material particles and not energy functions, he is limited to the investigation to the structural elements of life, hence his focus on molecular biology. The mechanistic researcher cannot delve into or integrate the common energy functions that unite a particular area of nature. Nor can he bring antithetical functions, such as individual and society, religion and sexuality or psyche and soma into harmony by investigating what these functions have in common. This characteristic of the mechanistic researcher is the reason that rigid boundaries are set up between the various branches of science, and why it is impossible to arrive at unifying principles that integrate them into a single comprehensive body of knowledge.
In contrast, the functional approach to natural science deals primarily with energy functions. Material structure is seen to arise secondarily from energy functions through the processes of freezing or superposition. As a result, functional thinking can proceed in both directions in natural research, either toward the direction of the more complex variations in nature or toward the underlying CFP that integrates these variations depending on the research objective.
4. Space Is Empty.
According to mechanistic- materialistic thinking the ultimate constituents of the universe are material particles. In the absence of matter, space is empty. The discovery of electromagnetism in the 19th century questioned this conclusion since the propagation of electromagnetic waves requires the presence of a medium through which the radiation travels. This was a perfectly rational assumption. For example, sound waves require a material medium through which they travel. Sound waves do not travel in a vacuum. Therefore, some form of medium was postulated (the ether) through which electromagnetic radiation traveled.
The predictive successes of Einstein’s theories made it possible to ignore the existence of a medium through which electromagnetic radiation was propagated. This rejection of the ether theory, which went against the time honored principles of materialistic thinking, amounted to a mystical leap of faith, to a belief in something unknowable. It meant that a physical medium was not required to account for the physical process of electromagnetic propagation and that these waves traveled “in empty space.” For the first time in the history of modern science, mystical thinking became fully incorporated into mechanistic science.
The ether problem was solved in 1939 by Wilhelm Reich when he demonstrated the existence of an all pervasive physical energy in space. The discovery of this new kind of energy made it possible to understand many physical phenomena that were unexplainable to mechanistic science. By demonstrating that space was not empty, Reich opened the way to an understanding of the energetic properties of space. Space is neither empty nor is it an abstraction, as in “space-time”. It has physical properties that can be experimentally investigated.
5. Quantitative properties are more important than qualitative ones.
Quality and quantity are two properties of all natural functions. The mechanistic scientist consistently ignores or overlooks qualitative properties in favor of quantifying nature. If as much effort were spent teaching the qualitative properties of physical laws, as is regularly done focusing on their quantitative properties, the student in physics would have a more pleasurable time learning and have a greater understanding of the material taught in physics classes. In the medical sciences, qualitative properties related to health and disease such as functions associated with vitality, putrefaction and stagnation are not commonly recognized. Mechanistic sociologists do not permit using qualitative terms such as good and evil. Their focus is exclusively on quantitative statistical factors. To them, vital statistics are more important than the energetic vitality of a social system. Qualitative properties of nature are relegated to mystically oriented psychologists and other mystics, people who deal with the subjective states of the mind. Not being fully in contact with energy movement within his armored organism, the mechanistic psychologist cannot accurately observe or objectively measure the subjective movement of emotions and sensations which flow through his body. Since he cannot trust his observations, (he uses double-blind studies), his measurements are confined to the most superficial level of human functioning, such as behavior, intelligence and aptitude testing all the while ignoring the underlying biological basis for these functions. This limitation in his ability to rely on his senses is also the reason that an understanding of the emotional problems of humanity lie outside of his framework of thinking.
In contrast to the mechanistic natural scientist, and the unarmored functional scientists has the ability with the tool of orgonometry to study both quantitative and qualitative properties of nature with equal facility. With sufficient knowledge, he can formulate complete orgonometric equations which include both properties.
The function of mechanistic thinking in the natural sciences
The perceptual functioning of the researcher in natural science is of crucial importance in determining the result of scientific investigation. According to Reich:
“The inclusion of the structure of the observer in the judgment of natural phenomena is a very important, if not decisive, step forward toward the integration of the subjective and the objective, the psychic and the physical. It is chiefly ignorance on the part of mechanistically oriented scientists of the biophysical and depth — psychological functioning of the observer, which has led them into the dead end street, where theoretical physics finds itself today. These scientists, who otherwise have demonstrated such an excellent critical sense of inquiry,… are unaware of the great progress which has been made during the first part of this century, in connecting the functions of perception with the functions of the emotions, and in connecting the emotions with bioenergetic, i.e., truly physical processes in the observing and reasoning organism. Natural-scientific research is an activity which rests on the interaction between observer and nature, or, expressed differently, between orgonomic functions within and the same functions without the observer. Thus, the character structure and the senses of perception in the observer are major, if not decisive, tools of natural research” (3).
We are forced to conclude that the application of mechanistic thinking in the natural sciences functions as an ideology, a belief system about nature arising from the distortions in the perceptual apparatus of the mechanistic scientist. The articles of faith in mechanistic-materialistic ideology are defended with the ferociousness of religious doctrine. Anyone seriously challenging these principles is either made marginal, ridiculed or turned into an object of an emotional plague attack.
As with all ideological systems, mechanistic materialism has three components. 1) a rational desire for knowledge about nature, which is a core function; 2) this original impulse for knowledge splits up into two components, one retains the original impulse, the desire for knowledge, and the other functions as a defense against knowledge. Arising from ocular armor, the defensive impulse originates from the terror of observing nature without distortion. It takes the form of not accurately seeing (psychologically, denying) the object of observation. 3) Finally, there are the predetermined mechanistic ideas about nature, the axioms of mechanistic ideology that are applied to science. Words serve as rationalizations justifying the distorted mechanistic perceptions of nature. This is shown as follows:
Mechanistic ideology (3)
Ocular armor (2)
Desire for knowledge (1)
The ideology of the mechanistic scientist, functions as a defense against experiencing anxiety and hatred that inevitably results when the natural phenomenon under observation (such as the spontaneous movement of the living) is actually observed. For him or her, organ sensations, which are the tools of natural research for the functional scientists have been deadened by armor. The protective function of mechanistic thinking is recognized by its defensive avoidance of observations that do not conform to the mechanistic view of nature. Terror of spontaneous movement is the reason that he must first immobilize the living before he can study or deal with it. It happens, when he observes living specimens under the microscope by staining the preparation, when he routinely uses medication to suppress symptoms as the treatment of choice of psychiatric disorders, or when he breaks the contact between the newborns mother in the delivery room by separation. The reasons given for these practices are merely rationalizations justifying his actions, never the real reason, which is to avoid coming into contact with his terror of the spontaneous movement of the living. The mechanistic scientist must exclude sensation, emotion and perception from his object of study because he cannot fully grasp or tolerate them without introducing distortions in thinking. The spontaneous, pulsatory movement of living systems goes completely unnoticed and this is why life must remain forever, a close book.
Mechanistic Thinking in Natural Science and the Emotional Plague.
The emotional plague is defined as human destructiveness on the social scene. Since the mechanistic scientist’s reasons given for the interpretation of any natural function are defensive originating from his or her character structure, not the true reason, and since this defensive practice of supplying different reasons than the actual ones are an indication that the emotional plague is operative, the question arises is mechanistic ideology in natural science itself a manifestation of the emotional plague? Evidence of the destructive consequences of mechanistic thinking in the life sciences provides the conclusive answer. The following interaction between the pharmaceutical industry, the medical profession and the public is given as an example to illustrate that mechanistic thinking in nature is, indeed, a manifestation of the emotional plague. This relationship functions to instill and actively perpetuate the mechanized view of nature in the public’s mind.
The public at large is no and indoctrinated by the pharmaceutical industry and the medical profession to believe in the principles of allopathic (mechanistic) medicine. Mechanistic medicine asserts that the signs and symptoms of disease are synonymous with medical illness and that prescribing the appropriate medication is the treatment of choice. This often gives rise to the mistaken notion that these medical illnesses are, in fact, understood and that they result exclusively from biochemical abnormalities in the patient. The possibility of recognizing the underlying bioenergetic source of the illnesses is evaded. Therefore, the administration of chemical substances such as pain killers, tranquilizers, and so on, either naturally or artificially produced by the pharmaceutical industry becomes the treatment of choice.
This relation between the public and the pharmaceutical industry and that of the public and the medical profession is one of attractive opposition based on their mechanistic -mystical orientation to medicine:
Public Pharmaceutical Industry.
Public Medical Profession
The same relationship exists between the medical profession and the pharmaceutical industry. Freelance medical writers are hired by pharmaceutical companies to write articles promoting their product. This practice is an open secret in medicine. Many of the articles written in scientific journals are actually written by ghost writers in the pay of drug companies. The seemingly objective articles, which doctors around the world use to guide the care of their patients, are often part of a marketing campaign by drug companies to promote a product or play up the condition it treats (4).
The pharmaceutical industry is concerned primarily with its own economic survival. It manufactures biochemical substances that are generally targeted to suppress the distressing symptom and this practice is commonly believed by all to be curative. Since it focuses on eliminating symptom, it is not concerned with the emotional health of the patient or the underlying disease process. As a result, the disease can reappear in another form once the symptom has been suppressed by the medication. As an example, consider the effects of anxiolytic medication, which eliminates anxiety, but often produces a loss of sexual drive and emotional liveliness in the patient. Many of the undesirable side effects of allopathic medications are manifestations of the underlying disease process taking other forms.
The pharmaceutical industry uses highly aggressive advertisements to promote its products on prime time television and in print to create. Sanctioned by the Food and Drug Administration, it creates the false and highly destructive impression that pharmacologically produced medication is the treatment of choice for all medical and emotional illnesses. To make matters worse, with the support of the government, the medical profession sets strict guidelines regarding what medications are considered acceptable medical practice. In effect, the government controls how medicine should be practiced.
Thus, mechanistic thinking in the natural sciences that is perpetrated on the public by the pharmaceutical industry in collusion with the government and the medical establishment serves as an expression of the organized emotional plague. It operates by misleading and confusing the public and it has brought progress toward understanding the origin of illnesses such as cancer and heart disease to a complete standstill. A report by the International Consortium on the Human Genome states:
“Despite the ever — accelerating pace of biomedical research, the root causes of common human diseases remain largely unknown, preventative measures are generally inadequate, and available treatments are seldom curative”(5).
Mechanistic thinking has all but destroyed the spirit of the natural sciences. This aspect of the emotional plague is responsible for discouraging generations of young talented people from entering into a career in science because they are turned off by the deadening effect of mechanistic science. According to mechanistic materialism, everything in the natural sciences is already known at least in principle. There are no exciting new frontiers in science left to be discovered. All that still needs to be uncovered are the details. What gives the appearance that today’s science is robust and prospering are the numerous technological advances that continue to be made not from new discoveries, but from pre- existing knowledge. However, since these advances do not deal with the core functions of nature and they have no bearing on improving the quality of life or on genuine scientific progress.
Mechanistic thinking in the field of mechanics and machines is perfectly rational because machines function mechanically, but it cannot have any place in the natural sciences. Educating the public regarding the true nature of science is of only partial help because the underlying problem confronting humanity has to do with the fact that armored people think in an armored fashion. They have lost the ability to know, deep down, that mechanistic- materialistic thinking in the natural sciences is inimical to life. Until this pathological condition is recognized and addressed, there can be no hope of developing a genuine natural science, one that is in the service of protecting and furthering unarmored life. Nor can there be any real hope of addressing the emotional and medical illnesses of humanity from a preventative standpoint.
For almost one hundred years the development of natural science has been progressively coming to a halt because mechanistic thinking has crept into every one of its branches. Today, various mystical alternatives are appearing as a reaction to mechanistic science to no avail. Mysticism is as one-sided in its rigid view of the world as is mechanism. The only possible way that the scientific spirit can return to natural science is by replacing mechanistic thinking with a functional approach to the understanding nature. Before this can happen, however, a major restructuring of the individuals in their early developmental years is required. Children must be allowed the freedom to observe and to sense their outer and inner worlds without distortion. Only by applying our knowledge of the emotional plague into this field of human endeavor will this restructuring come about.
1. Reich, W. The Mass Psychology of Fascism
2. Hoekstra, R. “Why Sex Is Good.” Nature, 434, 31 March 2004. pp. 571-573.
3. Reich, W.; Ether, God and Devil. p. 124.
4. Mathews, A, W.: At Medical Journals, Writers Paid by Industry Play Big Role.
The Wall Street Journal, 13 December, 2005.
5. A haploid map of the human Genome, The International HapMap Consortium Nature, and VOL. 437, 27 Oct. 2005, pp 1299-1305.
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Physics with Astrophysics (BSc) (Full-Time, 2020 Entry)
Physics with Astrophysics (BSc)
Physics with Astrophysics (BSc)
• UCAS Code
• F3F5
• Qualification
• BSc
• Duration
• 3 years full-time
• Entry Requirements
• (See full entry
• requirements below)
On our Physics with Astrophysics (BSc) degree, you will join one of our two astrophysics groups. You will be mentored by, and work on projects with, astrophysicists.
Astrophysics has a special flavour. With the arrival of space-based instrumentation and gravitational wave detection, some of the most exciting discoveries in your lifetime are likely to come in astrophysics. However, we can’t conduct experiments on stars or galaxies as they’re too far away and too big. Instead we need to piece together explanations of what we see. This involves understanding the fundamental physics – mechanics, quantum theory, relativity, thermodynamics – and trying to work out what they imply for exoplanets, galaxies, stars and the universe as a whole.
The course covers the principles of physics and their application to explain astrophysical phenomena. In your first year, you will study the classification of astrophysical objects and how we observe them. During the second year, you will study the solar system and stars in some detail. In the third and fourth years, you can study a range of topics including cosmology, exoplanets, the physics of compact objects (black holes, neutron stars and white dwarfs), general relativity and our Sun.
In the first two years you cover the fundamentals that apply throughout physics, such as mechanics and quantum theory, and meet the major phenomena observed in stars and space. There are also practical classes to develop laboratory and observational skills.
In later years look more closely at the phenomena that we can observe as well as those we would like to observe. Examples include star and galaxy formation, cosmology (how the universe was formed and where it may be going), the structure our sun and the formation of planets and other solar systems.
In the final year, you complete a year-long research project, which can be observational, theoretical or some combination of these.
In lectures, laboratory classes, time on telescopes, skills classes.
Class size
Lecture size will naturally vary from module to module. The first year core modules may have up to 350 students in a session, whilst the more specialist modules in the later years will have fewer than 100. The core modules in the first year are supported by weekly classes, at which you and your fellow students meet in small groups with a member of the research staff or a postgraduate student. Tutorials with your personal tutor is normally with a group of 5 students.
Contact hours
You should expect to attend around 12 lectures a week and spend 7 hours on supervised practical (mainly laboratory and computing) work. For each 1 hour lecture, you should expect to put in a further 1-2 hours of private study.
In any year, about 30% of the overall mark is assigned to coursework.
The weighting for each year's contribution to your final mark is 10:30:60 for the BSc course and 10:20:30:40 for the MPhys course.
A level: A*AA to include A in Mathematics (or Further Mathematics) and Physics
IB: 38 to include 6 in Higher Level Mathematics and Physics
BTEC: We welcome applications from students taking a BTEC qualification alongside A level Mathematics and A level Physics. A BTEC qualification in a relevant Science/Engineering subject may be considered alongside A level Mathematics only on an individual basis.
Our standard GCSE requirements
All applicants must possess a minimum level of competence in the English Language and in Mathematics/Science. A pass at Grade C or above, or Grade 4 or above in GCSE English Language and in Mathematics or a Science, or an equivalent qualification, satisfies this University requirement.
Contextual data and differential offers
• Warwick International Foundation Programme (IFP)
Taking a gap year
Applications for deferred entry welcomed.
Open Days
Year One
Quantum Phenomena
This module begins by showing you how classical physics is unable to explain some of the properties of light, electrons and atoms. (Theories in physics, which make no reference to quantum theory, are usually called classical theories.) You will then deal with some of the key contributions to the development of quantum physics including those of: Planck, who first suggested that the energy in a light wave comes in discrete units or 'quanta'; Einstein, whose theory of the photoelectric effect implied a 'duality' between particles and waves; Bohr, who suggested a theory of the atom that assumed that not only energy, but also angular momentum, was quantised; and Schrödinger who wrote down the first wave-equations to describe matter.
Electricity and Magnetism
You will largely be concerned with the great developments in electricity and magnetism, which took place during the nineteenth century. The origins and properties of electric and magnetic fields in free space, and in materials, are tested in some detail and all the basic levels up to, but not including, Maxwell's equations are considered. In addition the module deals with both dc and ac circuit theory including the use of complex impedance. You will be introduced to the properties of electrostatic and magnetic fields, and their interaction with dielectrics, conductors and magnetic materials.
Electronics Workshop
Electronic instrumentation is widely used in virtually all areas of experimental physics. Whilst it is not essential for all experimental physicists to know, for example, how to make a low noise amplifier, it is extremely useful for them to have some knowledge of electronics. This workshop introduce some of the basic electronics which is used regularly by physicists.
Physics Foundations
You will look at dimensional analysis, matter and waves. Often the qualitative features of systems can be understood (at least partially) by thinking about which quantities in a problem are allowed to depend on each other on dimensional grounds. Thermodynamics is the study of heat transfers and how they can lead to useful work. Even though the results are universal, the simplest way to introduce this topic to you is via the ideal gas, whose properties are discussed and derived in some detail. You will also cover waves. Waves are time-dependent variations about some time-independent (often equilibrium) state. You will revise the relation between the wavelength, frequency and velocity and the definition of the amplitude and phase of a wave.
Introduction to Astronomy
The Universe contains a bewildering variety of objects - black holes, red giants, white dwarfs, brown dwarfs, gamma-ray bursts and globular clusters. You will study how, with the application of physics, we have come to know their distances, sizes, masses and natures. The module starts with the Sun and planets and moves on to the Universe as a whole.
Classical Mechanics and Relativity
You will study Newtonian mechanics emphasizing the conservation laws inherent in the theory. These have a wider domain of applicability than classical mechanics (for example they also apply in quantum mechanics). You will also look at the classical mechanics of oscillations and of rotating bodies. It then explains why the failure to find the ether was such an important experimental result and how Einstein constructed his theory of special relativity. You will cover some of the consequences of the theory for classical mechanics and some of the predictions it makes, including: the relation between mass and energy, length-contraction, time-dilation and the twin paradox.
Mathematics for Physicists
All scientists use mathematics to state the basic laws and to analyse quantitatively and rigorously their consequences. The module introduces you to the concepts and techniques, which will be assumed by future modules. These include: complex numbers, functions of a continuous real variable, integration, functions of more than one variable and multiple integration. You will revise relevant parts of the A-level syllabus, to cover the mathematical knowledge to undertake first year physics modules, and to prepare you for mathematics and physics modules in subsequent years.
Physics Programming Workshop
You will be introduced to the Python programming language in this module. It is quick to learn and encourages good programming style. Python is an interpreted language, which makes it flexible and easy to share. It allows easy interfacing with modules, which have been compiled from C or Fortran sources. It is widely used throughout physics and there are many downloadable free-to-user codes available. You will also look at the visualisation of data. You will be introduced to scientific programming with the help of the Python programming language, a language widely used by physicists.
Astrophysics Laboratory I
The Laboratory introduces experimental science. There are experiments in physics and astronomy. The experiments can help give a different and more 'tangible' perspective on material treated theoretically in lectures. They illustrate the importance of correct handling of data and the estimation of error. They provide experience in using a range of equipment.
Year Two
Quantum Mechanics and its Applications
In the first part of this module you will use ideas, introduced in the first year module, to explore atomic structure. You will discuss the time-independent and the time-dependent Schrödinger equations for spherically symmetric and harmonic potentials, angular momentum and hydrogenic atoms. The second half of the module looks at many-particle systems and aspects of the Standard Model of particle physics. It introduces the quantum mechanics of free fermions and discussing how it accounts for the conductivity and heat capacity of metals and the state of electrons in white dwarf stars.
Electromagnetic Theory and Optics
You will develop the ideas of first year electricity and magnetism into Maxwell's theory of electromagnetism. Maxwell's equations pulled the various laws of electricity and magnetism (Faraday's law, Ampere's law, Lenz's law, Gauss's law) into one unified and elegant theory. The module shows you that Maxwell's equations in free space have time-dependent solutions, which turn out to be the familiar electromagnetic waves (light, radio waves, X-rays, etc.), and studies their behaviour at material boundaries (Fresnel Equations). You will also cover the basics of optical instruments and light sources.
Thermal Physics II
Any macroscopic object we meet contains a large number of particles, each of which moves according to the laws of mechanics (which can be classical or quantum). Yet, we can often ignore the details of this microscopic motion and use a few average quantities such as temperature and pressure to describe and predict the behaviour of the object. Why we can do this, when we can do this and how to do it are the subject of this module. The most important idea in the field is due to Boltzmann, who identified the connection between entropy and disorder. The module shows you how the structure of equilibrium thermodynamics follows from Boltzmann's definition of the entropy and shows you how, in principle, any observable equilibrium quantity can be computed.
People have been studying stars for as long as anything else in science. Yet, the subject is advancing faster now than almost every other branch of physics. With the arrival of space-based instruments, the prospects are that the field will continue to advance and that some of the most exciting discoveries reported in physics during our lifetimes will be in astrophysics. In this module, you will study the physics of stars and learn how we explain their behaviour. The module covers the main classifications of stars by size, age and distance from the earth and the relationships between them.
Mathematical Methods for Physicists
You will review the techniques of ordinary and partial differentiation and ordinary and multiple integration. You will develop you understanding of vector calculus and discuss the partial differential equations of physics. (Term 1) The theory of Fourier transforms and the Dirac delta function are also covered. Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. Fourier transforms are a powerful tool in physics and applied mathematics. The examples used to illustrate the module are drawn mainly from interference and diffraction phenomena in optics. (Term 2)
The Solar System
The study of the Solar System has been one of the most important in the history of physics with ramifications beyond science - Galileo was convicted of heresy for arguing that the earth moved round the Sun. Newton developed his theory of gravitation to explain Kepler's observations of the Solar System planets and effectively established what we now call the scientific method. In this module, we will introduce some of the intriguing phenomena observed in our Solar System. Questions we will touch on include: How does the Sun work? How do planets move and form? Do they have atmospheres? While the answers to some of these questions are complicated and still not completely known, we will construct convincing, qualitatively correct and appealing explanations of many of these phenomena using physics studied in the first year.
Astrophysics Laboratory II and Skills
This module develops experimental skills in a range of areas of physics and astrophysics. The module introduces the concepts involved in controlling remote instuments using computers and the collection and analysis of astrophysical data. The module explores information retrieval and evaluation, and the oral and written presentation of scientific material.
Year Three
Quantum Physics of Atoms
The basic principles of quantum mechanics are applied to a range of problems in atomic physics. The intrinsic property of spin is introduced and its relation to the indistinguishability of identical particles in quantum mechanics discussed. Perturbation theory and variational methods are described and applied to several problems. The hydrogen and helium atoms are analysed and the ideas that come out from this work are used to obtain a good qualitative understanding of the periodic table. In this module, you will develop the ideas of quantum theory and apply these to atomic physics.
You will revise the magnetic vector potential, A, which is defined so that the magnetic field B=curl A. We will see that this is the natural quantity to consider when exploring how electric and magnetic fields transform under Lorentz transformations (special relativity). The radiation (EM-waves) emitted by accelerating charges will be described using retarded potentials and have the wave-like nature of light built in. The scattering of light by free electrons (Thompson scattering) and by bound electrons (Rayleigh scattering) will also be described. Understanding the bound electron problem led Rayleigh to his celebrated explanation of why the sky is blue and why sunlight appears redder at sunrise and sunset.
Black Holes, White Dwarfs and Neutron Stars
In this module, you study the compact objects - white dwarfs, neutron stars and black holes (BH) - that can form when burnt out stars collapse under their own gravity. The extreme conditions in their neighbourhood mean that they affect strongly other objects and even the structure of the space-time around them. Compact objects can accrete material from surrounding gases and nearby stars. In the case of BHs this can lead to the supermassive BHs thought to be at the centre of most galaxies. In the most extreme events (mergers of these objects), the gravitational waves (GW) that are emitted are now beginning to be detected on earth (the first GW detection was reported in 2015 almost exactly 100 years after their prediction by Einstein).
Questions about the origin of the Universe, where it is going and how it may get there are the domain of cosmology. In this module, we will ask whether the Universe will continue to expand or ultimately contract. Relevant experimental data include those on the Cosmic Microwave Background radiation, the distribution of galaxies and the distribution of mass in the Universe. Starting from fundamental observations, such as that the night sky is dark and, by appealing to principles from Einstein's General Theory of Relativity, you will develop a description of the Universe and the Big Bang Model.
Plasma Electrodynamics
Plasmas are 'fluids' of charged particles. The motion of these charged particles (usually electrons) is controlled by the electromagnetic fields which are imposed from outside and by the fields which the moving charged particles themselves set up. This module will cover the key equations which describe such plasmas. It will examine some predictions derived on the basis of these equations and compare these with results from laboratory experiments and with observations from in situ measurements of solar system plasmas and remote observations of astrophysical systems. It will also be important to look at instabilities in plasmas and how electromagnetic waves interact with the plasmas.
Communicating Science
Employers look for many things in would-be employees. Sometimes they will be looking for specific knowledge, but often they will be more interested in general skills, frequently referred to as transferable skills. One such transferable skill is the ability to communicate effectively, both orally and in writing. Over the past two years you may have had experience in writing for an academic audience in the form of your laboratory reports. The aim of this module is to introduce you to the different approaches required to write for other audiences. This module will provide you with experience in presenting technical material in different formats to a variety of audiences.
Examples of optional modules/options for current students
Computational Physics, The Distant Universe, Geophysics, Hamiltonian Mechanics, Nuclear Physics, Physics of Electrical Power Generation, Physics of Fluids, Planets, Exoplanets and Life, Solar Magnetohydrodynamics.
Graduates from these courses have gone on to work for employers including: Deloitte Digital, Brunei Shell Petroleum, British Red Cross, EDF Energy, Civil Service, and Deutsche Bank.
They have pursued careers within areas such as physical scientists, finance and investment analysts, programmers and software development professionals, graphic designers, and researchers.
Helping you find the right career
• Career options with a Physics Degree
• Careers in Science
• Warwick careers fairs throughout the year
• Physics Alumni Evening
• Careers and Employer networking event for Physics students
Find out more about our Careers & Skills Services here.
UCAS code
Bachelor of Physics (BSc)
3 years full-time
Start date
28 September 2020
Location of study
University of Warwick, Coventry
Tuition fees
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Additional course costs
This information is applicable for 2020 entry.
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Make it make sense (Photo credit: edmittance)
Things pretty much make sense. If they don’t we feel that there is a reason that they don’t. We laughingly make up goblins and poltergeist to explain how the keys came to be in the location in which they are finally found, but we, mostly, have an underlying belief that there are good, physical reasons why they ended up there.
Things appear to get a little murkier at the level of the quantum, the incredibly small, but even there, I believe that scientists are looking for an explanation of the behaviour of things, no matter how bizarre. One of the concepts that appears to have to be abandoned is that of every day causality, although scientists appear to be replacing that concept with a more probabilistic version of the concept of causality. But I’m not going to go there, as quantum physics has to be spelled out in mathematics or explained inaccurately using analogies. I note that there is still discussion about what quantum physics means.
English: Schrödinger equation of quantum mecha...
English: Schrödinger equation of quantum mechanics (1927). (Photo credit: Wikipedia)
We strive for meaning when we consider why things happen. When a stone is dropped it accelerates towards the earth. This is observation. We also observe the way in which it accelerates and Sir Isaac Newton, who would have known from his mathematics the equation which governed this acceleration, had the genius to realise that the mutual attraction of the earth and the stone followed an inverse square law and, even more importantly, that this applied to any two objects which have mass in the entire universe.
English: Mural, Balfour Avenue, Belfast Mural ...
English: Mural, Balfour Avenue, Belfast Mural on a gable wall on Balfour Avenue in Belfast (see also 978903). The mural “How can quantum gravity help explain the origin of the universe?” was created by artist Liam Gillick and is part of a series of contemporary art projects designed to alert people to the 10 remaining unanswered questions in science at public sites across Belfast. (Photo credit: Wikipedia)
So, that’s done. We know why stones fall and why the earth unmeasurably and unnoticeably jumps to meet it. It is all explained, or is it? Why should any two massy objects experience this attraction? Let’s call it ‘gravity’, shall we? How can we explain gravity?
Well, we could say that it is a consequence of the object having mass, or in other words, it is an intrinsic property of massy objects, which if you think about it, explains nothing, or we can talk about curvature of space, which is interesting, but again explains nothing.
Curved Spaces
Curved Spaces (Photo credit: Digitalnative)
Can you see where I am going with this? Every concept that we consider is either ‘just the way things are’ or requires explanation. Every explanation that we can think up either has to be taken as axiomatic or has to be explained further. Nevertheless most people act as if they believe that there is a logical explanation for things and that things ultimately make sense.
It is possible that there is no logical explanation of things, and that the apparent relationships between things is an illusion. I once read a science fiction story where someone invented a time machine. Everywhere the machine stopped there was chaos, because there were no laws of nature and our little sliver of time was a mere statistical fluke. When they tried to return to the present they could not find it. This little story demonstrates that although we appear to live in a universe that is logical and there appears to be a structure to it, this may just be an illusion.
English: Illustration of the difference betwee...
English: Illustration of the difference between high statistical significance and statistical meaningfulness of time trends. See Wikipedia article “Statistical meaningfulness test” for more info (Photo credit: Wikipedia)
If we do live in a logical universe we not be able to access and understand the basis and structure of it. We may see things “through a glass darkly”. We may be like the inhabitants of Plato’s Cave. Everything we experience we experience through our senses, so our experience of the world is already second-hand and for many purposes we use tools and instruments to view the world around us. Also, our sense impressions are filtered, modified and processed by our brains in the process of experiencing something. We can take prescribed or non-prescribed drugs which alter our view of the world. So how can we know anything about the universe.
Alternatively there may be order to the universe. There may be ‘laws of nature’ and we may be slowly discovering them. I like the analogy of the blanket – a blanket is held between us and the universe but we are able to poke holes in it. Each hole reveals a metaphoric pixel of information about what lies behind the blanket. Over the years, decades, centuries and millennia we have poked an astronomical number of holes in the blanket, so we have a good idea of the shape of what lies behind it.
Cámara estenopéica / Pinhole camera
Cámara estenopéica / Pinhole camera (Photo credit: RubioBuitrago)
So why do things make sense? Is it because there is a structure to the universe that we are either discovering or fooling ourselves into believing that we are discovering, or is there no structure whatsoever and any beliefs that there are illusions. Maybe there’s another possibility. Maybe the universe does have the structure but it is an ‘ad hoc’ structure with no inherent logic to it all!
Highly Illogical
Highly Illogical (Photo credit: Wikipedia)
Banding together
Flag ~ Romania, Roumanie
Flag ~ Romania, also by chance the Tawa colours.
Our local rugby team has made it to the final of a competition (they won!) and naturally supporters are getting ready for the final match. They are organising coaches to take people to the match and no doubt there will be a good turn out. This got me thinking about how humans like to form bands and groups and supporter groups.
I think that banding together is at heart a self-protection thing. A human who belongs to a group gets supported by the group and reciprocally supports the group himself. In many cases the group is in competition against other groups of humans for a scarce resource such as food or territory, or in the case of sport points on the board of the elusive trophy. There is a synergy when people work together.
A rugby union scrum between the British and Ir...
A rugby union scrum between the British and Irish Lions and the All Blacks. (Photo credit: Wikipedia)
It’s not always humans versus humans though. A group may be formed to overcome some physical difficulty or to provide something that an individual can’t provide or achieve by themselves. That’s why travellers form caravans to cross deserts and a group of individuals might be able to buy a bigger boat together than they could have bought alone and take turns using it. Musicians of all genres usually form groups, at least to get started.
Les Rolling Stones à l'Olympia Stadion de Müni...
Les Rolling Stones à l’Olympia Stadion de Münich, alors qu’une partie de la scène avançait dans la foule (Photo credit: Wikipedia)
Forming a group allows individuals to specialise – in a hamlet or village one person becomes the smith, another the baker, another the mayor and another the constable, each person his or her particular skills in the role.
The role of supporters is to encourage and assist but not to actually take part in the contest or enterprise, but sometimes the line is blurred. For example the coach and trainer might not take part in a game, but in some ways they are part of the team. The supporters on the sidelines, yelling encouragement and advice, are even less part of the team, but they can certainly help out, and they form a larger group surrounding the team.
English: Greece - Russia Euro 2008
English: Greece – Russia Euro 2008 (Photo credit: Wikipedia)
Sometimes, of course, two groups of supporters clash. This is generally agreed to be a bad thing, but if you take a step back and think about it, it is to be expected, but not encouraged. It is an unwritten but basic rule of sport that the conflict, physically at least, stay on the field of play. Non-physical conflict, such as chants, banners and team regalia, is permitted between opposing spectators and even encouraged. “Get behind the team” is a rousing call for supporters. No wonder the non-physical conflict fairly often becomes physical.
The biggest ‘teams’ are countries, which strike me as being somewhat artificial in this day and age. Can one supergroup really speak for people who might be thousands of miles away? There may be an aboriginal population in a country that has far more inhabitants of immigrant origins, and these people may not consider themselves to be truly part of the nation in which they reside. Some nomadic people may travel through several countries, and may not consider themselves to be a part of any of them. The sheer size of modern countries almost invites the formation of ethnically or geographically ‘seperatists’ groups.
Matthes -- Separatists at Coblenz (LOC)
Matthes — Separatists at Coblenz (LOC) (Photo credit: The Library of Congress)
Mankind probably started out as family groups, and were probably nomadic. When they settled down (perhaps as a result of developing agriculture) it would seem natural to settle down in larger groups, maybe two or three families to provide defence against those still travelling around. As mankind spread and became more numerous these little settlements would grow into towns, with inhabitants specialising into roles like the smith or baker mentioned above.
At some stage strong leaders became feudal lords. This appears to have been common, but was possibly not universal. Eventually the lords and barons gave their allegiance to a king or overlord and a number of small (by current standards) states were formed, sometimes based around a city as in Sparta in Greece or sometimes based in a geographical area. The debatably mythical Arthur around the 5th or 6th centuries in Britain was supposedly king of Britain, although at that time there were probably several kingdoms in what is now Britain, and Athelstan is usually considered the first true English king.
English: King
English: King (Photo credit: Wikipedia)
The nations of the world are these days largely static in shape and size, but they do still change now and then. Czechoslovakia split apart in 1993, and the Soviet Union (USSR) formed in 1922 and split up in 1991.
The next logical step in this process, one would have expected, would be the formation of a global entity, grouping the whole of mankind into one huge group, but this has not happened. There are a number of global entities, notably the United Nations, but they tend to concentrate on specific areas of endeavour rather than being the World Government that would have been expected. There are ‘blocs’ of similarly inclined countries but these also don’t have the spread of activities that would make them a ‘super-government’.
English: Global map of noted supranational uni...
English: Global map of noted supranational unions. Based roughly upon http://www.towardsunity.org/. (Photo credit: Wikipedia)
It may be that the only thing that would cause the formation of a super-group encompassing all of humanity would be an encounter with hostile and destructive aliens, but the chances of that would be very small.
Take me to your leader. iPhone 3GS
Take me to your leader. iPhone 3GS (Photo credit: Kimb0lene) |
d361a777991818ea | The "curse of dimensionality" is an ubiquitous issue arising in both electronic structure and quantum molecular dynamics, which refers to the exponential scaling of computational cost with the number of degrees of freedom of the systems of interest.
This problem is manifested in many applications of computational studies (e.g. determining transition states of chemical reactions, geometry optimization of large molecular systems, simulating spectra of molecular systems, etc). The "curse of dimensionality" dramatically restricts the practical size of systems that can be studied computationally.
Assuming that only classical computers are available, what are the "smart algorithms" that have been devised to partially solve this "curse of dimensionality" problem and make the subset of the full problem be close to linear scaling?
This is a very broad question, so I am going to give a very brief overview of typical exponentially-scaling problems. I am not an expert in most of these areas, so any suggestions or improvements will be welcome.
Solving the Schrödinger equation
In order to solve the Schrödinger equation numerically, you need to diagonalise a rank $3N$ tensor -- as you can see, a pretty impossible operation not only in terms of CPU power, but also in terms of memory. The main problem in fact is that the wavefunction has to be antisymmetric with respect to all electrons, which is the main reason for combinatorial explosion. An alternative way is to expand the wavefunction as a multivariable Taylor series of antisymmetric functions (determinants), and if you were to do it exactly (full configuration interaction), it also scales exponentially. So at this point you can either solve the equation by ignoring most of the correlation between different degrees of freedom (Hartree-Fock, Moller-Plesset perturbation theory, truncated configuration interaction), project the $3N$-dimensional problem onto a 3-dimensional one, where the exact solution is unknown, but able to be approximated (density functional theory), or solve the correlated problem exactly for an idealised approximate infinite sum (coupled cluster theory). Another way to solve the equation is to convert it to a sampling problem (diffusion quantum Monte Carlo), which is exact for bosons, but needs an approximation for fermions (fixed node approximation), so that it doesn't scale exponentially. There is a lot of literature on making a lot of the above methods linear-scaling using clever approximations or making the formally exact full configuration interaction method more efficient (full configuration interaction quantum Monte Carlo), but in general, the more computational time you throw in, the larger the class of problems your method can tackle and some of the above approximations are better (and slower) than others.
Exploring potential energy surfaces
This is related to the sampling problem which I will address later. Here you convert a $3N$-dimensional sampling problem into a 1, 2 or 3-dimensional one, where you only care about particular nonlinear degrees of freedom (reaction coordinates, collective variables). This gets rid of the exponential scaling, but also needs a certain knowledge of the best/relevant collective variables, which are typically unknown. So this approach is similar in spirit to density functional theory - you convert your problem into a simple one, for which you don't know the exact method and you have to make an educated guess. In terms of sampling nuclear quantum effects, the problem is particularly badly scaling and common methods to estimate typical correlation functions/constants of interest is to either approximate them as simpler classical problems (semi-classical transition state theory), or to convert them into a sampling problem (ring polymer molecular dynamics). The latter is very similar in spirit to diffusion Monte Carlo for electronic structure.
Geometry optimisation
As with all optimisation algorithms, finding a global minimum is an exponentially scaling problem, so to my knowledge, most minimisation algorithms in computational chemistry provide local minima, which scale much better but are also more approximate. In classical computational chemistry you could afford to go one step further and explore much wider conformational space by heating your system up and slowly cooling it down to find some other better minima (simulated annealing). However, as you can see, the result you obtain from this will be highly dependent on chance and convergence will still be exponentially scaling -- there is no way around this.
This is one of the biggest unsolved problems in classical computational chemistry. As usual, local sampling is straightforward and typically scales as $3N\log 3N$ (Markov chain Monte Carlo, leapfrog/any other integrator), whereas enhanced sampling either resorts to using collective variables (metadynamics, umbrella sampling) or providing "locally global" sampling, by smoothening kinetic barriers (replica exchange, sequential Monte Carlo). Now, kinetic barriers slow down local sampling exponentially, but the above methods smoothen these linearly, resulting in cheaper locally enhanced sampling. However, there is no free lunch and global convergence will still be exponential, no matter what you do (e.g. protein folding problem).
Partition function calculation
The partition function is a $3N$-dimensional integral (I am going to focus on the classical case, as the quantum one is even more difficult). One way is to try to estimate the partition function (nested sampling, sequential monte carlo), where your convergence will typically scale exponentially but still much, much more efficient than regular quadrature (see exact diagonalisation of the Schrödinger equation, similar problem). This is very difficult, so we typically only try to calculate ratios of partition functions, which are much more nicely behaved. In these cases you can convert the integration problem into a sampling problem (free energy perturbation, thermodynamic integration, nonequilibrium free energy perturbation) and all above sampling issues still apply, so you never really escape the curse of dimensionality, but you get some sort of local convergence, which is still better than nothing :)
So in conclusion, there is no free lunch in computational chemistry and there are various classes of approximations suitable for different problems and in general, the better scaling your problem is, the more approximate and less applicable in general it is. In terms of "best value" nearly exact methods, my vote is on path integral methods (diffusion Monte Carlo, ring polymer molecular dynamics, sequential Monte Carlo), which convert the exponentially scaling problems into polynomially scaling ones (but still with convergence problems) -- although not perfect, at least you won't need all the atoms of the universe to run these and you won't need to know the answer to get the answer, which is sadly an overwhelming problem in many subfields of computational chemistry.
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$\begingroup$ +10 for sure! Another good one from Godzilla! $\endgroup$ – Nike Dattani May 25 at 7:51
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$\begingroup$ Thank you, Godzilla. This is exact kind of answer that I expect. I want to retrospect the whole subject in a certain small point view. As for me, the general idea to avoid exponential scaling in quantum chemistry is to narrow the operations from full Hilbert space to its small subspace, where localization as well as empirical knowledge plays a important role. $\endgroup$ – Paulie Bao May 25 at 8:57
• $\begingroup$ @PaulieBao I agree, this is the most applicable approach with current computers. An interesting avenue which is becoming more widely explored is formalising our prior knowledge / intuition / data about our system and linking it up into a neural network, instead of purely relying on gut feelings / hearsay / our supervisors' favourite method :) It will be exciting to see whether data science will be able to transform computational chemistry in this regard in the near future. $\endgroup$ – Godzilla May 25 at 9:05
• $\begingroup$ Your Schrödinger estimate is way too optimistic, see my answer below. $\endgroup$ – Susi Lehtola May 25 at 9:50
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$\begingroup$ I partially agree that data science could contribute the subject but I do not like the superficial level applications. As been state by a famous physicist “ if one have four parameters , he could fit any functions to an elephant”. I think need some insight is required. While on the other hand it is far from perfect for the “classical” quantum chemistry algorithms and could be potentially optimized. Another aspect that might be promising is quantum computing which is radically different from classical algorithms. $\endgroup$ – Paulie Bao May 25 at 10:43
The curse of dimensionality is indeed a huge problem in quantum chemistry, since the possible ways N electrons can occupy K orbitals is a binning problem whose computational cost grows factorially (almost as fast as x^x!) with the size of the system. Moreover, for accurate results you need K>>N in order to account for the so-called dynamical correlation, highlighting the computational challenge of the problem.
A huge breakthrough to the curse of dimensionality was proposed by Walter Kohn: instead of the exponentially difficult problem of describing the anti-symmetric wave function, density functional theory (DFT) shows that it is enough to describe just the electron density n(r), which is just a scalar function. The only problem is that we don't know the exact exchange-correlation functional, which describes how the movement of the electrons is correlated. Still, DFT has been hugely successful in both chemistry and materials science, since in many cases it yields sufficiently accurate results. You can also make DFT linear scaling, if you are smart about the algorithm; however, as far as I am aware, many people are still using the polynomially scaling O(N^3) algorithms since for many systems the lower-order terms are still dominating the cost...
The main problem with DFT is that you don't know the accuracy a priori, and DFT doesn't allow a systematic approach to the exact solution. Wave function based methods to the rescue! It turns out that by being smart, in many cases you can avoid the exponential scaling of exact wave function theory. The exact solution is given by diagonalizing the Hamiltonian in the basis of the possible electronic configurations (given by distributing the N electrons into K orbitals, or the K choose N problem); the size of this Hamiltonian is then (K choose N) x (K choose N) although it is extremely sparse. This is known in chemistry as the configuration interaction problem, and in physics as exact diagonalization.
The problem is extremely hard even for K=N. For example, the 16 electrons in 16 orbitals problem, or (16e,16o), if you are looking at the singlet state you have 8 spin-up and 8 spin-down electrons, yielding (16 choose 8)^2 = 165 million possible configurations. If you go to (18e,18o), you get 2.4 billion configurations. (20e,20o) has 34 billion configurations. (22e,22o) has 500 billion configurations. (24e,24o) has 73 trillion configurations. The (18e,18o) is still practical on a desktop computer, but the (24e,24o) is extremely hard even with a huge supercomputer.
The coupled-cluster method re-expresses the problem with an exponential ansatz, which yields a much more rapidly converging expansion for the wave function; you go down from exponential scaling to polynomial cost - assuming that you don't need to include all possible "excitations". The "gold standard" of quantum chemistry, the CCSD(T) method, scales as O(N^7). It's not cheap, but it yields amazingly accurate results for well-behaved molecules. The density matrix renormalization group could also be mentioned here; it is polynomially scaling for "easy" systems, but reduces to exponential scaling for hard ones....
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• $\begingroup$ Good answer which expands on quantum calculations, but $O(N!)$ is cheaper than $O(N^N)$, if that's what you meant in your first paragraph. $\endgroup$ – Godzilla May 25 at 10:26
• $\begingroup$ Thanks for you answer. I think you are emphasis on the problem of pure electronic structure in your answer. However, I think single point electronic structure calculation is not satisfactory in many application level computational studies. If one consider the dimension of the full PES or even beyond Born-Oppenheimer, more complexity is added to the problem. $\endgroup$ – Paulie Bao May 25 at 10:30
• $\begingroup$ @Godzilla123 is it? Stirling's approximation n! ~ sqrt(n) n^n... $\endgroup$ – Susi Lehtola May 25 at 10:58
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$\begingroup$ @SusiLehtola Yes but you are missing a division by $e^n$. Also, it is straightforward to see that $4*4*4*4$ is larger than $4*3*2*1$ and this difference only becomes worse for larger $n$. $\endgroup$ – Godzilla May 25 at 11:01
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$\begingroup$ @PaulieBao K choose N already includes the fermionicity: you can't fit two electrons onto the same orbital, and the configuration is invariant to permutations of the electrons. $\endgroup$ – Susi Lehtola May 26 at 9:39
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Chemistry LibreTexts
13.4: Ions: Electron Configurations and Sizes
• Page ID
• The electron configuration of an atomic species (neutral or ionic) allows us to understand the shape and energy of its electrons. Many general rules are taken into consideration when assigning the "location" of the electron to its prospective energy state, however these assignments are arbitrary and it is always uncertain as to which electron is being described. Knowing the electron configuration of a species gives us a better understanding of its bonding ability, magnetism and other chemical properties.
The electron configuration is the standard notation used to describe the electronic structure of an atom. Under the orbital approximation, we let each electron occupy an orbital, which can be solved by a single wavefunction. In doing so, we obtain three quantum numbers (n,l,ml), which are the same as the ones obtained from solving the Schrodinger's equation for Bohr's hydrogen atom. Hence, many of the rules that we use to describe the electron's address in the hydrogen atom can also be used in systems involving multiple electrons. When assigning electrons to orbitals, we must follow a set of three rules: the Aufbau Principle, the Pauli-Exclusion Principle, and Hund's Rule.
The wavefunction is the solution to the Schrödinger equation. By solving the Schrödinger equation for the hydrogen atom, we obtain three quantum numbers, namely the principal quantum number (n), the orbital angular momentum quantum number (l), and the magnetic quantum number (ml). There is a fourth quantum number, called the spin magnetic quantum number (ms), which is not obtained from solving the Schrödinger equation. Together, these four quantum numbers can be used to describe the location of an electron in Bohr's hydrogen atom. These numbers can be thought of as an electron's "address" in the atom.
To help describe the appropriate notation for electron configuration, it is best to do so through example. For this example, we will use the iodine atom. There are two ways in which electron configuration can be written:
I: 1s22s22p63s23p64s23d104p65s24d105p5
I: [Kr]5s24d105p5
In both of these types of notations, the order of the energy levels must be written by increased energy, showing the number of electrons in each subshell as an exponent. In the short notation, you place brackets around the preceding noble gas element followed by the valence shell electron configuration. The periodic table shows that kyrpton (Kr) is the previous noble gas listed before iodine. The noble gas configuration encompases the energy states lower than the valence shell electrons. Therefore, in this case [Kr]=1s22s22p63s23p64s23d104p6.
Quantum Numbers
Principal Quantum Number (n)
The principal quantum number n indicates the shell or energy level in which the electron is found. The value of n can be set between 1 to n, where n is the value of the outermost shell containing an electron. This quantum number can only be positive, non-zero, and integer values. That is, n=1,2,3,4,..
For example, an Iodine atom has its outmost electrons in the 5p orbital. Therefore, the principle quantum number for Iodine is 5.
Orbital Angular Momentum Quantum Number (l)
The orbital angular momentum quantum number, l, indicates the subshell of the electron. You can also tell the shape of the atomic orbital with this quantum number. An s subshell corresponds to l=0, a p subshell = 1, a d subshell = 2, a f subshell = 3, and so forth. This quantum number can only be positive and integer values, although it can take on a zero value. In general, for every value of n, there are n values of l. Furthermore, the value of l ranges from 0 to n-1. For example, if n=3, l=0,1,2.
So in regards to the example used above, the l values of Iodine for n = 5 are l = 0, 1, 2, 3, 4.
Magnetic Quantum Number (ml)
The magnetic quantum number, ml, represents the orbitals of a given subshell. For a given l, ml can range from -l to +l. A p subshell (l=1), for instance, can have three orbitals corresponding to ml = -1, 0, +1. In other words, it defines the px, py and pzorbitals of the p subshell. (However, the ml numbers don't necessarily correspond to a given orbital. The fact that there are three orbitals simply is indicative of the three orbitals of a p subshell.) In general, for a given l, there are 2l+1 possible values for ml; and in a n principal shell, there are n2 orbitals found in that energy level.
Continuing on from out example from above, the ml values of Iodine are ml = -4, -3, -2, -1, 0 1, 2, 3, 4. These arbitrarily correspond to the 5s, 5px, 5py, 5pz, 4dx2-y2, 4dz2, 4dxy, 4dxz, and 4dyz orbitals.
Spin Magnetic Quantum Number (ms)
The spin magnetic quantum number can only have a value of either +1/2 or -1/2. The value of 1/2 is the spin quantum number, s, which describes the electron's spin. Due to the spinning of the electron, it generates a magnetic field. In general, an electron with a ms=+1/2 is called an alpha electron, and one with a ms=-1/2 is called a beta electron. No two paired electrons can have the same spin value.
Out of these four quantum numbers, however, Bohr postulated that only the principal quantum number, n, determines the energy of the electron. Therefore, the 3s orbital (l=0) has the same energy as the 3p (l=1) and 3d (l=2) orbitals, regardless of a difference in l values. This postulate, however, holds true only for Bohr's hydrogen atom or other hydrogen-like atoms.
When dealing with multi-electron systems, we must consider the electron-electron interactions. Hence, the previously described postulate breaks down in that the energy of the electron is now determined by both the principal quantum number, n, and the orbital angular momentum quantum number, l. Although the Schrodinger equation for many-electron atoms is extremely difficult to solve mathematically, we can still describe their electronic structures via electron configurations.
General Rules of Electron Configuration
There are a set of general rules that are used to figure out the electron configuration of an atomic species: Aufbau's Principle, Hund's Rule and the Pauli-Exclusion Principle. Before continuing, it's important to understand that each orbital can be occupied by two electrons of opposite spin (which will be further discussed later). The following table shows the possible number of electrons that can occupy each orbital in a given subshell.
subshell number of orbitals total number of possible electrons in each orbital
s 1 2
p 3 (px, py, pz) 6
d 5 (dx2-y2, dz2, dxy, dxz, dyz) 10
f 7 (fz3, fxz2, fxyz, fx(x2-3y2), fyz2, fz(x2-y2), fy(3x2-y2)
Using our example, iodine, again, we see on the periodic table that its atomic number is 53 (meaning it contains 53 electrons in its neutral state). Its complete electron configuration is 1s22s22p63s23p64s23d104p65s24d105p5. If you count up all of these electrons, you will see that it adds up to 53 electrons. Notice that each subshell can only contain the max amount of electrons as indicated in the table above.
Aufbau Principle
The word 'Aufbau' is German for 'building up'. The Aufbau principle, also called the building-up principle, states that electron's occupy orbitals in order of increasing energy. The order of occupation is as follows:
Another way to view this order of increasing energy is by using Madelung's Rule:
Madelungs Rule.jpg
Figure 1. Madelung's Rule is a simple generalization which
dictates in what order electrons should be filled in the
however there are exceptions such as
copper and chromium.
This order of occupation roughly represents the increasing energy level of the orbitals. Hence, electrons occupy the orbitals in such a way that the energy is kept at a minimum. That is, the 7s, 5f, 6d, 7p subshells will not be filled with electrons unless the lower energy orbitals, 1s to 6p, are already fully occupied. Also, it is important to note that although the energy of the 3d orbital has been mathematically shown to be lower than that of the 4s orbital, electrons occupy the 4s orbital first before the 3d orbital. This observation can be ascribed to the fact that 3d electrons are more likely to be found closer to the nucleus; hence, they repel each other more strongly. Nonetheless, remembering the order of orbital energies, and hence assigning electrons to orbitals, can become rather easy when related to the periodic table.
To understand this principle, let's consider the bromine atom. Bromine (Z=35), which has 35 electrons, can be found in Period 4, Group VII of the periodic table. Since bromine has 7 valence electrons, the 4s orbital will be completely filled with 2 electrons, and the remaining five electrons will occupy the 4p orbital. Hence the full or expanded electronic configuration for bromine in accord with the Aufbau principle is 1s22s22p63s23p64s23d104p5. If we add the exponents, we get a total of 35 electrons, confirming that our notation is correct.
Hund's Rule
Hund's Rule states that when electrons occupy degenerate orbitals (i.e. same n and l quantum numbers), they must first occupy the empty orbitals before double occupying them. Furthermore, the most stable configuration results when the spins are parallel (i.e. all alpha electrons or all beta electrons). Nitrogen, for example, has 3 electrons occupying the 2p orbital. According to Hund's Rule, they must first occupy each of the three degenerate p orbitals, namely the 2px orbital, 2py orbital, and the 2pz orbital, and with parallel spins (Figure 2). The configuration below is incorrect because the third electron occupies does not occupy the empty 2pz orbital. Instead, it occupies the half-filled 2px orbital. This, therefore, is a violation of Hund's Rule (Figure 2).
nitrogen energy diagram.png
Figure 2. A visual representation of the Aufbau Principle and Hund's Rule. Note that the filling of electrons in each orbital
(px, py and pz) is arbitrary as long as the electrons are singly filled before having two electrons occupy the same orbital.
(a)This diagram represents the correct filling of electrons for the nitrogen atom. (b) This diagramrepresents the incorrect
filling of the electrons for the nitrogen atom.
Pauli-Exclusion Principle
Wolfgang Pauli postulated that each electron can be described with a unique set of four quantum numbers. Therefore, if two electrons occupy the same orbital, such as the 3s orbital, their spins must be paired. Although they have the same principal quantum number (n=3), the same orbital angular momentum quantum number (l=0), and the same magnetic quantum number (ml=0), they have different spin magnetic quantum numbers (ms=+1/2 and ms=-1/2).
Electronic Configurations of Cations and Anions
The way we designate electronic configurations for cations and anions is essentially similar to that for neutral atoms in their ground state. That is, we follow the three important rules: Aufbau's Principle, Pauli-exclusion principle, and Hund's Rule. The electronic configuration of cations is assigned by removing electrons first in the outermost p orbital, followed by the s orbital and finally the d orbitals (if any more electrons need to be removed). For instance, the ground state electronic configuration of calcium (Z=20) is 1s22s22p63s23p64s2. The calcium ion (Ca2+), however, has two electrons less. Hence, the electron configuration for Ca2+ is 1s22s22p63s23p6. Since we need to take away two electrons, we first remove electrons from the outermost shell (n=4). In this case, all the 4p subshells are empty; hence, we start by removing from the s orbital, which is the 4s orbital. The electron configuration for Ca2+ is the same as that for Argon, which has 18 electrons. Hence, we can say that both are isoelectronic.
The electronic configuration of anions is assigned by adding electrons according to Aufbau's building up principle. We add electrons to fill the outermost orbital that is occupied, and then add more electrons to the next higher orbital. The neutral atom chlorine (Z=17), for instance has 17 electrons. Therefore, its ground state electronic configuration can be written as 1s22s22p63s23p5. The chloride ion (Cl-), on the other hand, has an additional electron for a total of 18 electrons. Following Aufbau's principle, the electron occupies the partially filled 3p subshell first, making the 3p orbital completely filled. The electronic configuration for Cl- can, therefore, be designated as 1s22s22p63s23p6. Again, the electron configuration for the chloride ion is the same as that for Ca2+ and Argon. Hence, they are all isoelectronic to each other.
1. Which of the princples explained above tells us that electrons that are paired cannot have the same spin value?
2. Find the values of n, l, ml, and ms for the following:
a. Mg
b. Ga
c. Co
3. What is a possible combination for the quantum numbers of the 5d orbital? Give an example of an element which has the 5d orbital as it's most outer orbital.
4. Which of the following cannot exist (there may be more than one answer):
a. n = 4; l = 4; ml = -2; ms = +1/2
b. n = 3; l = 2; ml = 1; ms = 1
c. n = 4; l = 3; ml = 0; ms = +1/2
d. n = 1; l = 0; ml = 0; ms = +1/2
e. n = 0; l = 0; ml = 0; ms = +1/2
5. Write electron configurations for the following:
a. P
b. S2-
c. Zn3+
1. Pauli-exclusion Principle
2. a. n = 3; l = 0, 1, 2; ml = -2, -1, 0, 1, 2; ms can be either +1/2 or -1/2
b. n = 4; l = 0, 1, 2, 3; ml = -3, -2, -1, 0, 1, 2, 3; ms can be either +1/2 or -1/2
3. n = 5; l = 3; ml = 0; ms = +1/2. Osmium (Os) is an example.
4. a. The value of l cannot be 4, because l ranges from (0 - n-1)
b. ms can only be +1/2 or -1/2
c. Okay
d. Okay
e. The value of n cannot be zero.
5. a. 1s22s22p63s23p3
b. 1s22s22p63s23p6
c. 1s22s22p63s23p64s23d7
1. Atkins, P. W., & De Paula, J. (2006). Physical Chemistry for the Life Sciences. New York, NY: W. H. Freeman and Company.
3. Shagoury, Richard. Chemistry 1A Lecture Book. 4th Ed. Custom Publishing. 2006. Print
• Lannah Lua, Andrew Iskandar (University of California Davis, Undergraduate) Mary Magsombol (University of California Davis) |
ea5958846356c80e | Nonlinear Schroedinger Solitons in Massive Yang-Mills Theory and Partial Localization of Dirac Matter
Journal of Modern Physics
Vol.3 No.8(2012), Article ID:21675,8 pages DOI:10.4236/jmp.2012.38087
Xanthos N. Maintas, Charilaos E. Tsagkarakis, Fotios K. Diakonos, Dimitrios J. Frantzeskakis
Department of Physics, University of Athens, Athens, Greece
Email: fdiakono@phys.uoa.gr
Received April 20, 2012; revised May 25, 2012; accepted June 18, 2012
Keywords: Yang-Mills Solitons; Non-Linear Schroedinger Equation; Dirac Fermions; Localization
We investigate the classical dynamics of the massive SU(2) Yang-Mills field in the framework of multiple scale perturbation theory. We show analytically that there exists a subset of solutions having the form of a kink soliton, modulated by a plane wave, in a linear subspace transverse to the direction of free propagation. Subsequently, we explore how these solutions affect the dynamics of a Dirac field possessing an SU(2) charge. We find that this class of YangMills configurations, when regarded as an external field, leads to the localization of the fermion along a line in the transverse space. Our analysis reveals a mechanism for trapping SU(2) charged fermions in the presence of an external Yang-Mills field indicating the non-abelian analogue of Landau localization in electrodynamics.
1. Introduction
Over the last decades, the classical dynamics of YangMills (YM) field theory has been thoroughly investigated in the literature, both in Minkowski and in Euclidean space (see, e.g., [1] and references therein). The motivation for this study has been mainly the effort to understand the vacuum structure of non-abelian gauge theories like Quantum Chromodynamics (QCD). In a spatially homogeneous description, one can show that the YM classical dynamics possesses a chaotic component attributed to the nonlinear form of the YM self-interaction [1-6]. Generalizing to the case of inhomogeneous solutions, the conformal structure of the YM Lagrangian and the associated absence of a characteristic scale does not permit the presence of localized solutions [7], and complicated patterns with fractal characteristics may appear [8,9]. Recently, it has been argued that classical YangMills solutions may have impact on the properties of the quantum gauge fields. In particular, in [10-12], it was shown that periodic solutions of a special choice for the YM field configuration (Smilga’s choice [1]) after quantization lead to a description of the gauge field propagator compatible with the calculations performed in lattice gauge theories.
On the other hand, localized inhomogeneous solutions could permit a particle interpretation of the YM-field, which may be relevant for several applications where quasi-particles are involved. Such a scenario appears, for example, when the YM-field is coupled to a condensate, breaking spontaneously the underlying gauge symmetry, or when the YM-field itself condensates at particular thermodynamic conditions. In these cases the gauge field can acquire a mass introducing a scale in the YM-theory and bypassing the restrictions of the Coleman theorem [7]. This allows for spatially inhomogeneous localized classical solutions—at least at the level of an effective theory.
In the present work, we follow this line of thoughts trying to explore the space of classical solutions in massive SU(2) Yang-Mills theory. Our primary interest is to display the capacity of the theory in terms of possible classical dynamical behavior, as well as the influence of the choice for the YM-field initial configuration on this dynamics. In particular we will show that at a given combination of scales the classical Yang-Mills theory contains the non-linear Schrödinger equation regime. We start our considerations with a Langrangian describing the interaction of the Yang-Mills field with a scalar field. Then we assume, at the level of the Langrangian, that the scalar field is constant and we remain with a massive Yang-Mills theory. The effect of the spatio-temporal fluctuations of the scalar field is considered in [13]. As a next step, making a choice similar to Smilga’s [1] , we are able to construct within the framework of a multiscale perturbation theory a class of solutions which are localized along a line in the plane transverse to the momentum of the gauge field.
Furthermore, we study the dynamics of Dirac fields in the presence of such a gauge field configuration, considering the latter as an external classical field. We show that the Dirac field becomes bound in the subspace where the external gauge field is localized.
The paper is organized as follows: in section 2 we present the Lagrangian of the considered SU(2) YM field theory, we discuss the multiple scale approach used to solve the corresponding equations of motion and we obtain the associated solutions for the gauge field. We also give an interpretation of the involved parameters. In Section 3 we use the solution found in Section 2 as an external field for the Dirac dynamics of an SU(2)-charged matter field. Finally we end up, in section 4, with a summary and perspectives of our work.
2. Soliton-Like Solutions in the Massive Yang-Mills Dynamics
We start our analysis by considering the Lagrangian describing the interaction of the SU(2) Yang-Mills fieldwith a charged scalar field:
where g is a dimensionless coupling and is the self-interaction potential of the scalar field which we need not to specify more. We only assume that the potential possesses at least one stable equilibrium point. As usual, we use greek letters to denote the space-time components and latin letters to denote the Lie group components of the YM fields. For the SU(2) case take the values 1, 2, 3. Let us now further assume that the scalar field is constant (independent of space-time) and equal to a value corresponding to a stable equilibrium point of V. Then the Lagrangian in Equation (1), up to the constant term which can be neglected, becomes:
In Equation (2), is the mass matrix of the YM field components which is diagonal in the group indicesThe corresponding evolution equations are given by:
where and are the Kronecker delta and the full antisymmetric tensor in SU(2) space, respectively. We use the multiple-scale perturbation theory [14] to solve the nonlinear Equation (3): first, we introduce the new space-time independent variables, , as well as the partial derivatives thereof:
and we assume that the corresponding field variables are expanded into an asymptotic series of the form:
where is a formal small parameter (connected to the kink soliton amplitude and inverse width—see below). Substituting the above expressions into the equations of motion, and equating coefficients of the same powers of, we obtain a set of equations from which can be successively determined. Notice that each field is to be determined so as to be bounded (nonsecular) at each stage of the perturbation.
In order to solve the evolution equations arising at various orders in one can make an appropriate choice for the gauge field components, allowing for their decoupling—at least in the lowest orders in the perturbation expansion. Here, we will use the following configuration for the gauge fields:
which allows us to decouple the corresponding equations of motion up to the order. This configuration is in fact a generalization of the Smilga’s choice [1] for spatial non-homogeneous fields (see Appendix A).
The resulting simplified equations for the component are given as follows:
where we have used the notation:
Here we should note that there is no summation over repeated latin indices in Equations (7)-(9). The equations of the remaining components are obtained in a similar way. Equation (9) still contains a coupling between and, due to the nonlinear term, which can be resolved using the further assumption: [1].
Equations (7)-(9) can be solved self-consistently, leading to the following equations satisfied by the unknown component:
where. After some simple algebraic manipulations, the nonlinear evolution equation (12) takes the usual form of a nonlinear Schroedinger (NLS) equation with a repulsive (self-defocusing) nonlinearity (due to in the nonlinear term):
which has been studied extensively in various branches of physics and, especially, in nonlinear optics [15] and atomic Bose-Einstein condensates [16]. The above NLS equation possesses a stationary kink-type (alias “dark”) soliton solution [17], given by:
where and. Details on the derivation of Equation (13) are provided in Appendix A.
In Figure 1 we show a plot of the solution (14) using the parameter values: a = 53 MeV, ε = 0.1, k0 = 550 MeV and F0 = 3.2 MeV1/2. It can be seen that the obtained form is characterized by a free propagation in z-direction and a kink-soliton profile in the ξ-direction, with.
It is obvious that Equation (13), due to the presence of a first derivative in time, breaks the Lorentz invariance of the initial Lagrangian density; this is in accordance to the assumptions made to obtain the consistent solution (14) decomposing space-time in two inequivalent subspaces (and). This property is inevitably expected to hold for gauge field solutions varying over a finite space interval. Additionally, gauge invariance is violated from the very beginning due to the presence of the gauge field mass term. However, the validity of the solution (14) is restricted to specific space-time scales and, therefore, there is no apparent contradiction with first principles.
After suitable rescaling in order to introduce dimensionless quantities, we have checked the validity of the solution (14) through numerical integration of Equation (3). Adapting the choice (6) for the configuration of the gauge fields we concentrate on the equations of motion for the diagonal components (). The results of our numerical treatment in 1 + 1 dimensions for is shown in the contour plot of Figure 2. Notice that holds for all considered times in accordance with our choice [1]. The solution (14) holds for more than 100 field oscillations indicating its remarkable stability and supporting the validity of our perturbative scheme.
Figure 1. The kink-type solution of Equation (14) for the components of the SU(2) gauge field, using the parameter values: a = 53 MeV, ε = 0.1, k0 = 550 MeV and F0 = 3.2 MeV1/2.
Figure 2. Contour plot of the numerical solution for using as initial condition the analytically obtained form given by Equation (14). The length scale is. We have also used ε = 0.1.
3. Partial Localization of Dirac Matter
In this section we will investigate the dynamics of an SU(2) charged Dirac field in the presence of an external gauge field which has the form found in Equation (14). The corresponding Dirac equation is written as follows:
where are the Pauli spin matrices, are the Dirac matrices, and
is the SU(2) doublet for the fermionic field. For the fermionic mass matrix we assume a diagonal form with. Due to the non-abelian character of the gauge group, the equations describing the dynamics of the two charged fields and, after expanding (15) and substituting Equation (14) for the non-abelian gauge field, take the following coupled form:
Taking into account that the expression (14) for the gauge field is non-covariant, it is consistent to consider the dynamics implied by Equations (16) and (17) in the nonrelativistic limit. For that purpose, it is necessary to write the bispinors and in terms of their components. In that regard, we introduce the following notation:
Applying the standard procedure [18] for obtaining the non-relativistic limit of Equations (16) and (17) (details of the calculations are given in Appendix B), we find the following set of coupled Schroedinger-type equations for the fermionic components
while are determined through as follows:
Equations (19)-(22) can be consistently reduced, using and to the following two equations:
where and.
Without loss of generality we can choose (using the rest frame of the massive gauge field as reference frame) to further simplify the above expressions. Furthermore, in order to allow for non-trivial dynamics in the fermionic field, the corresponding mass has to be small (of order) as compared to the gauge field mass. In this case, writing, we obtain the following system of two equations
where is a mass scale of the order of.
Let us now introduce the length scale and the time scale to express Equations (29) and (30) in a dimensionless form. In these units, the dimensionless frequency of the oscillating YM-field becomes:
It also straightforward to define dimensionless variables and. In these variables, we seek for solutions of the system (29) and (30) having the form:
where F and G are slowly-varying functions of, while is the energy eigenvalue. In this limit, Equations (29) and (30) become:
For, Equations (32) and (33) can be integrated with respect to over a period
since in this time interval F and G are practically constant. Following this procedure, Equations (32) and (33) decouple and obtain the following form:
allowing as a solution a fermionic state which is bound in the direction and has the form [19]:
where N is a normalization constant. The state (36) resembles the Landau levels of a particle in an external magnetic field in quantum electrodynamics. In the YM case under consideration, the magnetic field is generated by the term proportional to in Equation (30). The difference here is that we have a single level independently of the strength of the external Yang-Mills field. In addition, the Dirac particle is trapped only in the - direction, where the external field is also localized. It should be noticed that the condition, necessary for the existence of the solution (36), can be justified by either using a large value or a large value (or both).
It is illuminating to give an example of the energy and length scales involved in this solution. Assuming a gauge field mass of 500 MeV and a much smaller fermionic mass i.e., of order of, we find that the SU(2) charged fermions are trapped in a region of radius of in the (x, y)-plane with energy eigenvalue for an external field of amplitude. It must be noted that for this choice of parameter values the non-relativistic approximation is valid within an error of 15% estimated by the relative magnitude of the first relativistic correction term. In Figure 3 we show the effective potential responsible for the trapping of the Dirac particle using the above mentioned parameter values. The dashed line indicates the energy of the associated bound state in the ξ-space. The fact that this state is very close to the continuum threshold explains the absence of a second bound state. In Figure 4 we show the ξ-dependent wave function corresponding to the bound state displayed in Figure 3. The broad spatial extension of this state is attributed to the small exponent in Equation (36).
Figure 3. The effective potential responsible for the trapping of a Dirac particle with SU(2) charge emerging from the a time-dependent external Yang-Mills field of the type shown in Figure 1. The parameters values used are:,. The dashed line indicates the energy of the bound state.
Figure 4. The ξ-dependent normalized wavefunction of the bound state shown in Figure 3 calculated using the same parameter values.
4. Conclusions and Discussion
We have investigated classical solutions of the SU(2) massive Yang-Mills equations in the framework of multiple scale perturbation theory. Due to the presence of the mass term, conformal symmetry is explicitly broken and the Coleman theorem does not apply [7]. Therefore, the YM dynamics in this case admit soliton-like solutions localized in a subspace of the transverse space.
Such solutions of the Yang-Mills field break both Lorentz and gauge invariance in higher orders of the perturbation expansion, in consistency with the presence of a mass term as well as the appearance of partial localization. Dirac fermions with non-vanishing SU(2) charge, when exposed in an external YM field having the form of these soliton-like solutions, become trapped in a similar way as electrons in a transverse magnetic field (Landau levels). However, the trapping of the SU(2) colored fermions is a pure dynamical effect occurring in the nonadiabatic limit of very fast oscillations of the external YM field, and occurs only along the (x + y)-direction.
Our analysis reveals a mechanism for the occurrence of localized fermionic states with SU(2) charge based on the interaction with a massive Yang-Mills field. The simplifying assumptions made in our approach (two non-vanishing equal components of the gauge field at the leading order) may restrict the profile of the found solutions allowing, on the other hand, for an analytical treatment. Despite this restriction, the main ingredients of the present study could be used as a guide to obtain more general inhomogeneous classical solutions of the massive SU(2) field. However, such a task is a subject for future investigations.
5. Acknowledgements
We thank N. G. Antoniou, E. G. Floratos and A. Tsapalis for helpful discussions. This work was partially supported by the Special Account for Research Grants of the University of Athens.
1. A. Smilga, “Lectures on Quantum Chromodynamics,” World Scientific, Singapore City, 2001. doi:10.1142/9789812810595
2. S. G. Matinyan, G. K. Savvidy and N. G. Ter-ArutyunyanSavvidy, “Classical Yang—Mills Mechanics. Nonlinear Color Oscillations (in Russian),” Journal of Experimental and Theoretical Physics, Vol. 80, 1981, pp. 830-838.
3. B. V. Chirikov and D. L. Shepelyanskii, “Stochastic Oscillations of Classical Yang-Mills Fields (in Russian),” Journal of Experimental and Theoretical Physics Letters, Vol. 34, No. 4, 1981, pp. 171-175.
4. S. G. Matinyan, G. K. Savvidy and N. G. Ter-ArutyunyanSavvidy, “Stochasticity of Classical Yang-Mills Mechanics and Its Elimination by Using the Higgs Mechanism (in Russian),” Journal of Experimental and Theoretical Physics Letters, Vol. 34, No. 11, 1981, pp. 613- 616.
5. S. G. Matinyan, “Dynamical Chaos of Nonabelian Gauge Fields (in Russian),” Fizika Elementarnykh Chastits I Atomnoya Yadra, Vol. 16, 1985, pp. 522-570.
6. S. G. Matinyan, E. P. Prokhorenko and G. K. Savvidy, “Non-Integrability of Time Dependent Spherically Symmetric Yang-Mills Equations,” Nuclear Physics B, Vol. 258, No. 2, 1988, pp. 414-428. doi:10.1016/0550-3213(88)90273-8
7. S. Coleman, “There Are No Classical Glueballs,” Communications in Mathematical Physics, Vol. 55, No. 2, 1977, pp. 113-116. doi:10.1007/BF01626513
8. M. Wellner, “Evidence for a Yang-Mills Fractal,” Physical Review Letters, Vol. 68, No. 12, 1992, pp. 1811-1813. doi:10.1103/PhysRevLett.68.1811
9. M. Wellner, “The Road to Fractals in a Yang-Mills System,” Physical Review E, Vol. 50, No. 2, 1994, pp. 780- 789. doi:10.1103/PhysRevE.50.780
10. M. Frasca, “Strongly Coupled Quantum Field Theory,” Physical Review D, Vol. 73, No. 4, 2006, Article ID: 027701. doi:10.1103/PhysRevD.73.049902
11. M. Frasca, “Infrared Gluon and Ghost Propagators,” Physics Letters B, Vol. 670, No. 1, 2008, pp. 73-77. doi:10.1016/j.physletb.2008.10.022
12. M. Frasca, “Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case,” Modern Physics Letters A, Vol. 24, No. 30, 2009, pp. 2425-2432. doi:10.1142/S021773230903165X
13. V. Achilleos, F. K. Diakonos, D. J. Frantzeskakis, G. C. Katsimiga, X. N. Maintas, C. E. Tsagkarakis and A. Tsapalis, “A Multi-Scale Perturbative Approach to SU(2)- Higgs Classical Dynamics: Stability of Nonlinear Plane Waves And Bounds of the Higgs Field Mass,” Physical Review D, Vol. 85, No. 2, Article ID: 027702. doi:10.1103/PhysRevD.85.027702
14. A. Jeffrey and T. Kawahara, “Asymptotic Methods in Nonlinear Wave Theory,” Pitman, London, 1982.
15. Yu. S. Kivshar and B. Luther-Davies, “Dark Optical Solitons: Physics and Applications,” Physics Reports, Vol. 298, No. 2-3, 1998, pp. 81-197. doi:10.1016/S0370-1573(97)00073-2
16. D. J. Frantzeskakis, “Dark Solitons in Atomic Bose-Einstein Condensates: From Theory to Experiments,” Journal of Physics A-Mathematical and Theoretical, vol. 43, no. 21, 2010.
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19. L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Pergamon Press, Oxford, 1991.
Appendix A
Using the classification of the gauge fields in orders of as stated in Equations (6), we can write the equations of motion for the components, as follows:
where and.
The non-diagonal equations, as well as the equations for the case, are obtained in a similar way and their consistency with the choice in Equation (6) implies the following condition:
for every. Thus, Equation (A3) becomes:
In Equation (A5) the fields and are still coupled due to the presence of the nonlinear term; nevertheless, we can readily resolve this problem by assuming that. Equation (A1) reveals the dependence on the normal scales (in the first order of the perturbation expansion) of the gauge field, as it admits a harmonic solution for of the form:
The function, which is for the moment an arbitrary complex function will be consistently determined by solving the equations arising at higher orders of.
Next, considering Equation (A2), it is clear that the homogenous part of the solution is similar to the one in Equation (A6), due to the fact that the linear operators in Equation (A2) and in Equation (A3) are identical. As a result, the term is secular, as will contain terms of the form.
The condition for nonsecularity, namely, leads to the following two equations [valid at order]:
Since does not depend on x and y [cf. Equation (A6)], one has for furthermore, the condition, introduces an important restriction for the function in Equation (A6): it is necessary to assume that
i.e., is independent of and, a fact which sustains the decomposition of space-time in two inequivalent subspaces, as mentioned in Section 2.
Finally, Equation (A5) decomposes in three independent equations. The first of them reads:
where “nsp” stands for the nonsecular part. The remaining two equations are found by eliminating all secular terms producing divergence of in Equation (A5). This way, we have:
which is treated in the same way as Equation (A8) for the field, and
where “sp” stands for the secular part.
Our assumption that implies that
and, as a result, Equation (A11) should be of the same form for. This requirement is satisfied if
Consequently, Equation (A11) is reduced to the form:
As far as Equation (A11) is concerned, it is important to note that the second term is the contribution of the non-diagonal terms [cf. Equations (A3) and (A4)]. Note that Equation (A12) is actually the NLS equation presented in Section 2 (see Equation (13)).
Appendix B
We start by rewriting Equations (16) and (17) in the following form:
where and are the two components of the bispinor defined in Equation (18). In the following, we will apply the standard procedure [11] in order to obtain the non-relativistic limit of Equations (B1) and (B2). The necessity of this emerges by the violation of the covariance of the gauge field which we have imposed. Eventually, it is consistent to study the non-relativistic case.
Taking into account that, for, Equation (B1) transforms into
From Equation (B3) we obtain the following equations for the doublets and of the field:
and similarly for the field:
where and are slowly varying functions of timewhile or, with F, G being slowly varying functions of time as well, and. Using the relations
Εquation (B4) becomes
and since we have
while for the component, similarly we have
where ,
Finally, we expand Equations (B10) and (B11) in their components resulting in Equations (19)-(22) for the fields. |
0684ed257c0dde00 | JMP Vol.7 No.11 , July 2016
Ion Acoustic Soliton and the Lambert Function
The Sagdeev potential method is employed to compute the width of (Ion-acoustic) soliton propagated in a cold plasma. The computation indicates that the soliton width is a continuous function (of the Mach number M), which is expressed in terms of the Lambert Function. Despite the (fairly) complex form of the function, the numerical plotting makes sense about its changes.
Received 13 May 2016; accepted 24 July 2016; published 27 July 2016
1. Introduction
Over the last decades, there has been a great deal of interest and significant progress in the study of nonlinear plasma theories and many of works and researches in plasma physics devote much attention to these theories. The nonlinear theories include a large number of effects and phenomena such as the nonlinear coherent structures as shock waves, solitary waves (solitons), vortices, etc.
The collective electrical and magnetic properties of plasmas could produce interactions that take the place of collisions and permit shocks and solitons to form. A shock wave is a sudden transition (a type of propagating disturbance) in the properties of a fluid medium (liquid or gas), involving a difference in flow velocity across a narrow (ideally, abrupt) transition. In high-energy density physics, nearly any experiment involves at least one shock wave. Such shock waves may be also produced by applying pressure to a surface or by creating a collision between two materials.
Nonlinear effects in plasmas occur when a large amplitude wave is excited by an external means. Soliton (shock) waves are formed as a result of a balance between the nonlinearity and dispersion (dissipation) of the medium. For example, the dispersion in the ion acoustic wave can be counter-balanced by nonlinearity and an ion acoustic soliton can propagate. From the mathematical point of view, solitons are the stationary solutions to the Korteweg-de Vries equation:
where is a function of position x and time t. Some historical events and discoveries led to the soliton theory and Equation (1) is not the only equation that has solitonic solution. The sine-Gordon and the nonlinear Schrödinger equations are among them. The solutions to the Equation (1) are traveling wave with constant profile in time and they describe the different types of models in various branches of physics and natural science (for instance, Equation (1) is modeled to describe the waves on shallow water surfaces).
As mentioned, Equation (1) may arise in the study of such diverse physical systems as fluids and plasmas. In plasma ambient, Equation (1) may be due to the balancing between nonlinearity and dispersion and for this reason its study is of special interest.
One of the possible approaches to study of the nonlinear effects is the so-called reductive perturbation technique [1] - [5] . This technique is widely employed to investigate the asymptotic behavior of nonlinear excitations and is more convenient to study of small-amplitude nonlinear perturbations, or to treat of plasma waves in a state very close to thermodynamic equilibrium.
Another successful approach to study of the electrostatic solitons and shock waves has been the Sagdeev Potential (SP) or Pseudo Potential (PP) method. The SP is one particular notion that has become immensely important in soliton and shock research [6] . The main advantage of this method over reductive perturbation technique is that, it is appropriate for arbitrary amplitude waves and one can derive all the soliton results of perturbation methods and compare it with the exact results obtained by the SP method [7] .
In the present work, our aim is to compute the soliton width (SW) for an ion acoustic solitary wave propagated in a cold plasma. The computation is based on analogy between pseudo-particle in PP well and real particle in conservative potential well. Knowing the angular frequency of (small) oscillation of real particle about its equilibrium position and comparing this with corresponding quantity in the plasma system, one attains a formula to compute the SW. To understand briefly, the SW is defined as spatial length corresponding to a (complete) spatial oscillation of psuedo-particle in the PP well; comparing this with the definition of period of temporal oscillation for real particle in the well, the computation is straightforward. Also, for better understanding of the changes, the graph of the width function is visualized, with the help of numerical plotting (computer algebra).
The work is organized as follows:
The model and the main calculation are presented in the next section, and conclusions are given in the last section.
2. The Soliton Width as a Function of the Mach Number
We will consider a one dimensional model for propagation of ion acoustic wave in a cold plasma. The solitary wave is generated by spatial oscillation of electrostatic potential which is the so-called psuedo-particle. The wave is travelling to the left in the (to say) x direction with a constant speed. Therefore, in the wave- frame position, the SP is given by [8] :
This PP is subjected to the boundary condition, and the following dimensionless parameters are used
where to be electron temperature, is the ion Debye length and M is the Mach number.
The form of the PP would determine whether soliton like solutions may exist or not. The conditions for the existence of solitary waves are:
where is the maximum value of beyond which the PP becomes imaginary1 and it is often called amplitude of the soliton (or shock). By virtue of above conditions, it is easily to show that the values of the Mach number are confined to the interval
It may be useful to plot the graph of the PP (2) versus its argument. The graph is illustrated in Figure 1 for three values of M. As the figure shows, the depth of the potential well and amplitude of the soliton increase with increasing Mach number.
The PP (2) satisfies the energy condition:
which is analogous to the principle of energy conservation
for a real particle of mass m moving in a (conservative) potential. In view of this analogy, Equation (4) can be regarded as an energy integral of moving pseudo-particle of unite mass, pseudo-position, pseudo- velocity and pseudo-potential. Hence, by the following replacements:
one can obtain similar formulas and results in the plasma system, as discussed below.
We know that for a real particle moving in the potential, the angular frequency of small oscillations, about the equilibrium position (a minimum at) is given by
Figure 1. Psuedo-potential curves V(χ) corresponding to three values of M2 = 1.2, 1.3, 1.4.
and as a result, the period of oscillation motion T becomes
The natural question, then, is what conclusions can be drawn from this for our plasma system, more clearly, what quantities are given by similar formulas which are obtained by the above replacements in (5) and (6), that is
where we also use and the two replacement in (5), in (6)2.
It is easily to check that has the dimensions of the Length, and in below, we see that is SW.
We first remind the following definitions:
T: a time interval corresponding to one complete temporal oscillation of real particle (in the well).
On the other hand the width of the solitary wave is equal to the length that disturbance of the solitary wave takes place, or in SP terminology,
SW: a distance interval corresponding to one complete spatial oscillation of pseudo-particle (in the PP well), thus, it is natural to drawing the result
To use the above formula, we need to know the point, that is minimum of PP. This is given by vanishing of the first derivative of PP (2), namely
solving this equation for, we get
where W is the Lambert function. Substituting into the second derivative
and using Equation (9), we obtain the final expression for the SW as function of the Mach Number, that is
Due to the presence of Lambert function, we have no clear idea of the behavior of the width function (13), then it is instructive to graph the function. Because of the transcendental form of the function, we have to use the numerical plotting and the corresponding graph is illustrated in Figure 2. It represents a monotonically decreasing function of M. The function takes on arbitrarily large values when, and decreases rapidly over the (allowable) range.
As mentioned, the soliton width is the length of a complete sweep of the pseudo-particle in the potential well and as is clear from the Figure 1, the potential well corresponding to the larger values of M is more narrow than
Figure 2. The graph of the soliton width λs versus M: rapidly monotonic decreasing function.
the smaller one, then, the solitons corresponding to larger M have shorter width.
In order to have a quantitative measure of the changes of the width function, let us compute the ratio
where and are near the upper and lower bounds of M respectively, that is,. From Figure 2, one finds
inserting these values in to the ratio, we obtain
This number is the average decreasing rate of the width per mach number, that is the change in the width divided by the change in Mach number. The negative sign indicates the decreasing nature of the function. As it is evident the amount of decreasing is relatively large.
In the above discussion, the width changes were described in terms of the Mach number itself, but, due to definition of the Mach number, the following easy corollary can be deduced:
1) A positive (negative) change in the soliton wave velocity leads to a negative (positive) change in the SW, and
2) A positive (negative) change in the ratio () leads to a positive (negative) change in the SW.
In the end of this section, it is necessary to say that our ability to plot the width function (as continues function) is because of the analytic form of the SP (2). Indeed, the method presented here may be employed for any SP of analytic form [9] . When the SP has not analytic form, there is no such ability, but we may still calculate numerically at each allowable equilibrium (a minimum) position. This is based on estimation of equilibrium position and substituting into the second derivative of SP. In this case, one has a (discrete) point diagram, instead of a continues curve. One such computation has been represented in [10] for an electron-positron plasma including stationary ions in the background.
3. Conclusions and Results
The shock and soliton waves occur most likely because of the nonlinear disturbances, namely discontinuities in various variables as energy density, pressure, temperature, etc, in plasmas and other mediums. In addition to amplitude and velocity of disturbance, the determination of spatial scales within the collision less shock or soliton may be of particular interest. For example, for a shock disturbance which obeys the Burgers equation
with the initial condition
it can be shown [11] that the width and speed of the shock are respectively
In the case of ion acoustic wave which is mediated by electric potential in the plasma, the width of the shock or soliton may be particularly important in its relation to the width of the electrostatic potential drop across the shock.
In this work, the PP approach is employed to calculate the width of (one dimensional) ion acoustic solitary wave propagating in a cold plasma. The calculation shows that the soliton width is a continuous function of the Mach number and the function is expressed in terms of the Lambert function. Because of transcendental form of the width function, we have to use the numerical method (computer algebra) to graph of the function over the (short) allowable range of the Mach number (). The graphical representations help us to understand the behavior of the width function. The main feature of the graph is monotonic rapidly decreasing one, such that the average rate of change is about 82 unit of the length per Mach number. Contrary to the width, the amplitude of solitary wave is an increasing function of the Mach number.
Cite this paper
Maleko lkalami, B. , Mohammadi, T. and Ghamari, K. (2016) Ion Acoustic Soliton and the Lambert Function. Journal of Modern Physics, 7, 1345-1350. doi: 10.4236/jmp.2016.711120.
[1] Salahuddin, M., Saleem, H. and Saddiq, M. (2005) Physical Review E, 66, Article ID: 036407.
[2] Esfandyari-Kalejahi, A., Meh-dipoor, M. and Akbari-Moghanjoughi, M. (2009) Physics of Plasmas, 16, Article ID: 052309.
[3] Tiwari, R.S. (2008) Physics Letters A, 372, 3461-3466.
[4] Tiwari, R.S., Kaushik, A. and Mishra, M.K. (2007) Physics Letters A, 365, 335-340.
[5] Mushtaq, A. and Shah, H.A. (2005) Physics of Plasmas, 12, Article ID: 072306.
[6] Sagdeev, R.Z. (1966) Review of Plasma Physics, 4, 23.
[7] Roychoudhury, R. (2000) Proceedings of Institute of Mathematics of NAS of Ukraine, 30, 510-515.
[8] Chen, F.F. (1984) Introduction to Plasma Physics and Controlled Fusion. 2nd Edition, Plenum, New York.
[9] Malakolkalami, B. and Mohammadi, T. (2014) The Open Plasma Physics Journal, 7, 199.
[10] Malakolkalami, B. and Mohammadi, T. (2012) Journal of Plasma Physics, 78, 05.
[11] Ablowitz, M.J. (2011) Nonlinear Dispersive Waves. Cambridge University Press, New York, 27. |
17d8e32767e44861 | tisdag 19 december 2017
US National Security Strategy: Energy to the Poor
The US National Security Strategy released yesterday, states:
• Climate policies will continue to shape the global energy system.
We read that the US will counter the anti-growth energy agenda of the rich world embraced by Obama-Merkel and now also Macron along with rich Sweden (see previous post), which is detrimental to US interests as well as to those of the developing world in need of fossil fuels to lift their people out of poverty.
This is Trumps Christmas present to the world and represents a truly remarkable case of independent thinking, probably based on a gut feeling that the scientific support of CO2 alarmism is very weak and thus it is pointless and immoral to require the poor people of the world to stay poor.
torsdag 14 december 2017
Sweden as the First Fossil Free Society
The Independent reported in May 2016 that
More Voodoo Physics
In the preceding post I gave an example of inventing fictional physics by misinterpreting the Fundamental Theorem of Calculus to have a physical meaning as a Stefan-Boltzmann radiation law in the form
• $\int_{T_1}^{T_2}f(\nu )\, d\nu =F(T_2)-F(T_1)$
where $F(T)=\sigma T^4$ is a primitive function of a Planck spectrum $f(\nu )>0$ depending on frequency $\nu$ satisfying $\frac{dF}{d\nu}=f$. Here
• $Q=\int_{T_1}^{T_2}f(\nu )\, d\nu$
is the one-way physical radiative heat energy flux from a warm body at temperature $T_2$ to a colder body at temperature $T_2$ expressed as an integral of a spectral radiance, with integration limits scaling with temperature reflecting Wien's displacement law. The misinterpretation is to say that
thus expressing the one-way flux $Q$ from warm to cold, as the difference between two-way heat fluxes $\sigma T_2^4$ from warm to cold and $\sigma T_1^4$ from cold to warm, thus freely inventing two-way heat fluxes from one-way physical heat flux.
The arbitrariness of this invention is expressed by the fact the primitive function $F$ is only determined up to a constant (which cancels in the subtraction). Two-way back-and-forth heat fluxes of any size can thus be freely invented, which by itself is too good to be true physics.
Another example is Einsteins misinterpretation of the Lorentz transformation of mathematical space-time coordinates to have direct physical meaning as distortions of physical space and time, in direct violation of the dictum by Lorentz when introducing his transformation as a mathematical formality without physical meaning. The consequences of this misinterpretation are far-reaching as the revolution (distortion) of our concepts of space and time being forced upon us by the modernity of Einstein's physics. The Nobel Prize in Physics this year to the LIGO recording over a fraction of second of "ripples in the fabric of space-time" of size a fraction of an atomic nucleus over
a distance of the diameter of the Earth from the supposed merger of two black holes 1.3 billion years ago, is a recent example of the grip of a model over reality overwhelming a whole physics community.
The idea of the quantum computer is similarly based on giving the multi-dimensional statistical non-physical Schrödinger equation a direct physical meaning, again in direct violation to the dictum of Schrödinger when introducing his equation. And the quantum computer is still fictional despite major efforts to make it into any from of reality...
All these examples can be viewed to represent Voodoo Physics in the sense of being based on misinterpreting operations on a a doll model to have real physical results.
Here are two quotes by Max Born (Nobel Prize in Physics 1954) expressing the non-physical aspects of Einstein's special theory of relativity expressed in the Lorentz transformation:
• It is hardly possible to illustrate Einstein’s kinematics by means of models.
The counter argument is that Maxwell predicted the existence of electro-magnetic waves from the presence of waves in his model equations, and so Voodoo physics can be real physics. That is right, but it does not say that all Voodoo physics is real physics, as for example:
Update of Talk at Climate Sense 2018: Voodoo Physics
I have put up an new version of my upcoming talk at Climate Sense 2018 with the title:
Of particular focus is the Fundamental Theorem of Calculus stating that if $F(x)$ is a primitive function of $f(x)$, that is $\frac{dF}{dx}=f$, then
• $\int_a^bf(x)\, dx = F(b) - F(a)$.
For example, if $f(x)=1$, then $F(x)=x$, and so we can formally write
• $1 = \int_{100}^{101} f(x)\, dx = F(101)-F(100)=101 - 100$
expressing the positive quantity $\int_{100}^{101} f(x) dx = 1$ as the difference between the two (large) numbers 101 and 100. This is mathematics and not yet physics.
The radiative flux of heat energy $Q$ from a warm body of temperature $T_2$ to a colder body of temperature $T_1 < T_2$ is (essentially) given by the integral
where $f(\nu )$ is the Planck spectrum with $\nu$ frequency. Here the limits of the integral scale with temperature reflecting the cut-off in frequency expressed by Wien's displacement law giving the
warm body the "overkill" spectrum $f(\nu )$ above the cut-off $T_1$ for the cold body, with the overkill effectively causing the heating while the shared spectrum below cut-off has no heating effect, as explained in more detail here.
With $F(T)=\sigma T^4$ and $F$ acting as a primitive function of the Planck function $f$, the radiative flux $Q$ can now according to the Fundamental Theorem formally be expressed in the form of the Stefan-Boltzmann Law
thus formally expressing the positive quantity $Q$ as the difference between two (large) numbers $\sigma T_2^4$ and $\sigma T_1^4$.
The Voodoo Physics of the Greenhouse Effect is the result of giving the mathematical identity
• $\int_{T_1}^{T_2}f(\nu )\,d\nu = \sigma T_2^4 - \sigma T_1^4$
a physical meaning with the physical one-way flux $Q=\int_{T_1}^{T_2}f(\nu )\,d\nu$ expressed as the difference between the entities $F(T_2)=\sigma T_2^4$ and $F(T_1)=\sigma T_1^4$, now freely invented to be forms of two-way "radiative fluxes" back-and-forth between the bodies. From one-way physical flux are thus created two-way fluxes from a mathematical identity without any proper physical correspondence. Note in particular that $F$ as primitive function of $f$ is undetermined up to a constant, thus allowing fictitious back-and-forth fluxes of any magnitude.
Note that $F(T)=\sigma T^4$ can be interpreted physically as the one-way radiative flux from a body of temperature $T$ into a background at 0 K. But the physics is missing of viewing the one-way radiative flux between two bodies as a difference between two separate fluxes into a background at 0 K.
A property (identity) of a mathematical model is thus freely interpreted to be real physics, in the same way as an operation of a voodoo doll is believed to be able to have a real effect on a real person.
This is nothing but Voodoo Physics, and this is the nature of the Greenhouse Effect based on Back Radiation from cold to warm underlying the CO2-alarmism so forcefully preached by IPCC with now Macron as ardent follower.
Macron, despite (or maybe thanks to) his education in French elite schools with all its mathematics, thus appears to be overwhelmed by the Fundamental Theorem of Calculus and cannot separate model from reality.
Voodoo = operation on model believed to have real effect on real person.
PS The Pyrgeometer is a perfect example of a voodoo doll, reporting (non-physical) back radiation $\sigma T_1^4$ from a cold atmosphere at $T_1$ to warmer Earth surface at $T_2$, by measuring $Q$ and (erroneously) viewing $\sigma T_2^4$ to be radiation from the Earth surface, as
if the Earth surface was radiating directly to surrounding space at 0 K and not to the atmosphere at $T_1$ K.
You can buy a pyrgeometer from Kipp and Zonen and play with it as a voodoo doll believing it reports real physics if you want to sell CO2-alarmism. From the above analysis you may understand that in fact it represents a symbiosis of science and commercial industry serving CO2-alarmism by supplying fictional physics. |
8a98d53271ff0501 | Fundamental systems
Fig.: Water molecule with relevant fundamental systems.
The Schrödinger equation has been set up and solved for a number of fundamental systems. The purpose of the fundamental systems is to simplify solving the Schrödinger equation by focussing on only one particular property of a real system at a time. In many cases, good approximate solutions for real systems can be obtained by linear combination of a small number of relevant fundamental systems. For example, in a water molecule, we have two hydrogen atoms and another atom, which we can try to treat as hydrogen-like to start with. In addition, there are two bonds which can vibrate and the molecule can rotate around its two-fold symmetry axis.
Fig.: The relationships between the fundamental systems.
The fundamental systems:
The harmonic oscillator can be developed further to explain the vibrational states of solids (phonons), while the combination of many atoms with their Coulomb potentials leads to the electronic band structure observed in solids. Molecules, on the other hand, are best described by a combination of harmonic oscillators (one for each bond) and rigid rotors (one for each symmetry axis).
Solving the Schrödinger equation
To solve the Schrödinger equation for a particular system, we need:
The solutions are pairs of wave functions and their energy eigenvalues.
Properties of a wave function
The wave function of a particle is a complex function (in the sense of having real and imaginary parts). Its complex square (i.e. the product of the function and its complex conjugate) is a real and positive function which represents the probability density of finding the particle as a function of the spatial coordinates.
Therefore, only those solutions of the Schrödinger equation which satisfy the following conditions are physically sensible:
Next we want to find the wave functions of the hydrogen atom. To do this, we'll need the expression for the del operator in spherical coordinates. |
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Quantum Mechanics
There are different ways of describing quantum mechanics. Each has its individual strength and weaknesses, but in terms of observable predictions they are all equivalent. This situation is analogous to how we can describe classical mechanics in terms of Newtonian mechanics, Lagrangian mechanics, Hamiltonian mechanics or Koopmann-von-Neumann mechanics.
The transition from classical mechanics to quantum mechanics is known as quantization.
The most famous descriptions of quantum mechanics are
Both Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulations both belong to the description known as canonical quantum mechanics. The relevant mathematical stage for both formulations is Hilbert space. The connection between them lies in the identification of Heisenberg's infinite matrices $p_j$ and $q^i$ ($i,j=1,2,3$), representing the momentum and position of a particle moving in $\mathbb{R}^3$, with Schrödinger's operators $-i\hbar\partial/\partial x^j$ and $x^i$ (seen as a multiplication operator) on the Hilbert space $\mathcal H=L^2(\mathbb{R}^3)$, respectively. The key to this identification lies in the canonical commutation relations $$ [p_i,q^j]=-i\hbar \delta^j_i. $$ We usually call these two formulations the "Heisenberg picture" and the "Schrödinger picture", since, both descriptions are actually equivalent. In some sense, the transformation between them is "just a basis change in Hilbert space"1).
Instead of using a Hilbert space, we can use the corresponding configuration space and its associated tangent bundle. In this formulation, our main object is the Lagrangian. This Lagrangian formulation of quantum mechanics is known as path integral quantum mechanics.
Another possibility is to use the corresponding phase space. Our main interest in phase space is the Hamiltonian and the algebraic structure of quantum mechanics is given by the Moyal bracket. The Moyal bracket is the appropriate deformation of the Poisson bracket, which governs the phase space formulation of classical mechanics, i.e. Hamiltonian mechanics. This way of describing quantum mechanics is simply known as phase space formulation of quantum mechanics.
Finally, we can also focus on the individual trajectories in real space ($\mathbb{R}^3$) of the objects in our system. This formulation of quantum mechanics is known as Bohmian mechanics and is analogous to the Newtonian formulation of classical mechanics.
There is no general consensus as to what the fundamental principles of quantum mechanics are and what it really "means". While almost any physicist can do calculations2) in quantum mechanics, the stories that are told about what we really do when we perform these calculations vary wildly. For example, a common question is whether a particle in quantum mechanics already has well-defined properties before we measure it or if they only take on definite values as soon as we measure them.
The thing is that experimentally outcomes stay the same no matter which interpretation we believe in3). In this sense, discussions about the interpretation of quantum mechanics are mostly a matter of taste.
Important notions regarding the interpretation of quantum mechanics are
• the EPR paradox,
• the no-clone theorem,
• Schrödinger's cat,
• the quantum Zeno paradox.
The standard (orthodox) interpretation of quantum mechanics is presented in almost every textbook and known as the Copenhagen interpretation.
According to this interpretation, particles do not possess specific dynamical properties (momentum, position, angular momentum, energy, etc.) until we perform a measurement.
The wave function is interpreted statistically and it collapses once we measure it. Therefore, if we immediately repeat a measurement, we will get the same result again.
Regarding the question, whether a particle already has a definite momentum etc. before we measure it, the Copenhagen interpretation states that
"observations not only disturb what has to be measured, they produce it!" - Pascual Jordan.
In contrast, hidden variable interpretations which are also called realist interpretations, state that
“the position of the particle was never indeterminate, but was merely unknown to the experimenter.” - Bernard d'Espagnat.
A third popular interpretation is called the agnostic interpretation states that it makes no sense to ask such a question since how can we discuss anything that we can never measure. By definition, a property like momentum is undetermined until we measure it and a discussion about its value before measurement makes no sense:
An amazing discussion of the Copenhagen interpreation and how it came about can be found in Quantum Dialogue by Mara Beller.
There are dozens of other interpretations of what quantum mechanics really means:
Recommended Resources:
“If you are not confused by quantum mechanics, then you haven’t really understood it.” Niels Bohr
“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman
Recommended Background
A solid understanding of classical mechanics is certainly helpful to understand quantum mechanics. The standard formulation of quantum mechanics makes use of the Hamiltonian, and hence an understanding of Hamiltonian mechanics is necessary.
A solid understanding of calculus and a rudimentary understanding of linear algebra is essential. You need to know what derivatives, integrals, and Taylor expansions are and how to multiply matrices + what eigenvalues/eigenvectors are. Moreover you should know how to solve ordinary differential equations. Since quantum mechanics is all about probabilities a basic understanding of probability theory is a must-have.
The Traditional Roadmap
The state of a system is described by an object called wave function. The time evolution of states is determined by the Schrödinger equation. Observables are described by operators. Eigenvalues of these operators are possible measurement outcomes. By acting with an operator on the wave function we can calculate the probability for different measurement outcomes. Some observables cannot be determined at the same time with arbitrary precision. For example, we can't determine the position of a particle and its momentum at the same time with arbitrary precision. This is called an uncertainty relation. Because we describe systems in probabilistic terms, the wave function must be normalized. This means that free particles must be described in terms of wave packets, because plane waves cannot be normalized.
The most important experiment that encodes most mysteries of quantum mechanics is the double slit experiment. To quote Feynman: "We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."
Essential Experiments:
To really understand how quantum mechanics works in practice it is crucial to understand a few canonical examples. The particle in a box examples demonstrates nicely the quantization of energy levels and the particle in a potential well how classically impossible things become possible in quantum mechanics (tunneling).
To grasp the deeper structure of quantum mechanics a solid understanding of group theory is crucial. The most important aspect of group theory for quantum mechanics is representation theory. Moreover, to understand what is really going on in many calculations some knowledge of functional analysis and complex analysis are essential.
Essential Math:
The machinery of quantum mechanics is nicely exposed by using the Dirac notation. The wave mechanical description can then be understood as just one special case. The state of a system is then no longer described by a wave function but by an abstract vector in Hilbert space. One of the most important observables is angular momentum and the closely related "internal angular momentum", called spin. For many real-world problems perturbation theory is crucial, because almost no problem in quantum mechanics can be solved exactly. To prepare for quantum field theory, which is mostly about scattering theory, learning the basics in the quantum mechanical context makes sense. Moreover, to understand some of the subtler aspects of quantum mechanics and to see that there is a different but equally powerful formulation, getting some understanding of the path integral formulation is a smart idea.
One of the most important subtle aspects of quantum mechanics, spin, is best understood by having a look at the famous Stern-Gerlach experiment. The role of another crucial notion, called gauge potentials, is exposed by the Ahoronov-Bohm experiment.
To understand the many advanced concepts of quantum mechanics getting a solid understanding of the harmonic oscillator is absolutely crucial. One of the triumphs of quantum mechanics is the correct description of the energy levels of the hydrogen atom. Thus, calculating them, including spin-orbit corrections etc., is something every serious student of quantum mechanics should be able to do.
Essential Problems:
Quantum mechanics is technically difficult. Only a few extremely artificial textbook examples can be solved exactly. For everything else, we need to use approximation techniques to tackle realistic systems.
The most important approximation schemes in quantum mechanics are
• time-independent perturbation theory,
• WKB approximation
• time-dependent perturbation theory,
• adiabatic approximation,
• semi-classical approximation.
To describe scattering processes in quantum mechanics additional tools are needed, especially
• partial wave analysis and
• the Born approximation,
• Fermi's golden rule.
In the beginning quantum mechanics was only a set of heuristic rules derived from experimental observations like the infrared catastrophe in the black-body radation. This set of heuristic rules is known as the "Old Quantum Theory"6).
Historically, Heisenberg's matrix mechanics7) was the first complete formulation of quantum mechanics. Heisenberg's main focus was the noncommutative structure of the quantum mechanical algebra of observables.
Soon after, Schrödinger developed his wave mechanics8) whose main focus is the classical geometric structure of configuration space and all about wave functions.
Only one year afterward, Neumann9) unified the two approaches by introducing the abstract concept of a Hilbert space. Schrödinger's wave functions are vectors living in Hilbert space and Heisenberg’s observables are linear operators acting on these vectors.
Heisenberg's matrix mechanics was developed into its current complete full set of equations by Born, Jordan, and Heisenberg 10)
Around the same time, Dirac discovered independently the same structure11)).
The spectrum of the Hydrogen atom was first calculated by Pauli12).
Only some time afterward - in 1926 - Erwin Schrödinger introduced the concept of the wave function and also calculated the hydrogen spectrum. Since in principle, quantum mechanics was already complete before Schrödinger's famous paper it is reasonable to ask: "So, what did Schrödinger do, in his 1926 paper?"
With hindsight, he took a technical and a conceptual step. The technical step was to change the algebraic language of the theory, unfamiliar at the time, into a familiar one: differential equations. This brought ethereal quantum theory down to the level of the average theoretical physicist. The conceptual step was to introduce the notion of “wave function” ψ, soon to be evolved into the notion of “quantum state” ψ, endowing it with heavy ontological weight."Space is blue and birds fly through it" by Carlo Rovelli
Since Heisenberg and Co. were much earlier than Schrödinger it also makes sense to ask why ultimately Schrödinger "won". Conventionally Students are introduced to quantum mechanics by starting with Schrödinger's "wave mechanics".
Heisenberg lost the political battle against Schrödinger, for a number of reasons. First, all this was about “interpretation” and for many physicists this wasn’t so interesting after all, once the equations of quantum mechanics begun producing wonders. Differential equations are easier to work with and sort of visualise, than non-commutative algebras. Third, Dirac himself, who did a lot directly with non-commutative algebras, found it easier to make the calculus concrete by giving it a linear representation on Hilbert spaces, and von Neumann followed: on the one hand, his robust mathematical formulation of the theory brilliantly focused on the proper relevant notion: the non-commutative observable algebra, on the other, the weight given to the Hilbert space could be taken by some as an indirect confirmation of the ontological weight of the quantum states. Fourth, and most importantly, Bohr —the recognised fatherly figure of the community— tried to mediate between his two brilliant bickering children, by obscurely agitating hands about a shamanic “wave/particle duality”. "Space is blue and birds fly through it" by Carlo Rovelli
Recommended Resources
• Jim Baggott; The Quantum Story
• Abraham Pais, Inward Bound: of Matter and Forces in the Physical World
• Uncertainty by David Lindley
• See also Ron Maimon's answer, Good book on the history of Quantum Mechanics?, URL (version: 2011-12-23):
• W. a. Fedak and J. J. Prentis, “The 1925 Born and Jordan paper ’On quantum mechanics’,” American Journal of Physics 77 (2009) 128.
• B. van der Waerden, Sources of quantum mechanics. North Holland, 1967.
For many more questions and answers see:
What makes a theory Quantum?
What is coherence in quantum mechanics?
What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?
At least in the standard, Hilbert space formulation
This is similar to the statement that it doesn't matter which formulation we use. But here it makes at least some difference since some scenarios can be calculated more easily in a specific formulation.
Heisenberg, W.: Uber die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys. 33 (1925) 879-893. English translation in Sources of quantum mechanics, ed. B.L. van der Waerden
Schrödinger, E.: Quantisierung als Eiegnwertproblem. Ann. d. Physik 79 (1926), 361–376, 489–527
Neumann, J. von: Mathematische Grundlagen der Quantenmechanik. Springer, Heidelberg, 1932
M. Born, P. Jordan, and W. Heisenberg, “Zur Quantenmechanik II,” Zeitschrift f¨ur Physik 35 (1926) 557–615.
P. A. M. Dirac, “The fundamental equations of quantum mechanics,” c. London, Ser. A 645-653 (1925
W. Pauli, “Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik [On the hydrogen spectrum from the standpoint of the new quantum mechanics],” Zeitschrift f¨ur Physik 36 (1926) 336–363
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In physics, quasiparticles and collective excitations (which are closely related) are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with all of the other electrons and nuclei; however it approximately behaves like an electron with a different mass traveling unperturbed through free space. This "electron" with a different mass is called an "electron quasiparticle".[1] In another example, the aggregate motion of electrons in the valence band of a semiconductor is the same as if the semiconductor contained instead positively charged quasiparticles called holes. Other quasiparticles or collective excitations include phonons (particles derived from the vibrations of atoms in a solid), plasmons (particles derived from plasma oscillations), and many others.
These particles are typically called "quasiparticles" if they are related to fermions (like electrons and holes), and called "collective excitations" if they are related to bosons (like phonons and plasmons),[1] although the precise distinction is not universally agreed upon.[2]
The quasiparticle concept is most important in condensed matter physics, since it is one of the few known ways of simplifying the quantum mechanical many-body problem.
General introduction
Solids are made of only three kinds of particles: Electrons, protons, and neutrons. Quasiparticles are none of these; instead they are an emergent phenomenon that occurs inside the solid. Therefore, while it is quite possible to have a single particle (electron or proton or neutron) floating in space, a quasiparticle can instead only exist inside the solid.
Motion in a solid is extremely complicated: Each electron and proton gets pushed and pulled (by Coulomb's law) by all the other electrons and protons in the solid (which may themselves be in motion). It is these strong interactions that make it very difficult to predict and understand the behavior of solids (see many-body problem). On the other hand, the motion of a non-interacting particle is quite simple: In classical mechanics, it would move in a straight line, and in quantum mechanics, it would move in a superposition of plane waves. This is the motivation for the concept of quasiparticles: The complicated motion of the actual particles in a solid can be mathematically transformed into the much simpler motion of imagined quasiparticles, which behave more like non-interacting particles.
In summary, quasiparticles are a mathematical tool for simplifying the description of solids. They are not "real" particles inside the solid. Instead, saying "A quasiparticle is present" or "A quasiparticle is moving" is shorthand for saying "A large number of electrons and nuclei are moving in a specific coordinated way."
Relation to many-body quantum mechanics
The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains around 1017 atoms and 1018 electrons. Each of these attracts or repels every other by Coulomb's law. In quantum mechanics, a system is described by a wavefunction, which, if the particles are interacting (as they are in our case), depends on the position of every particle in the system. So, each particle adds three independent variables to the wavefunction, one for each coordinate needed to describe the position of that particle. Because of this, directly approaching the many-body problem of 1018 interacting electrons by straightforwardly trying to solve the appropriate Schrödinger equation is impossible in practice, since it amounts to solving a partial differential equation not just in three dimensions, but in 3x1018 dimensions – one for each component of the position of each particle.
Therefore, using quasiparticles / collective excitations, instead of analyzing 1018 particles, one needs only to deal with only a handful of somewhat-independent elementary excitations. It is therefore a very effective approach to simplify the many-body problem in quantum mechanics. This approach is not useful for all systems however: In strongly correlated materials, the elementary excitations are so far from being independent that it is not even useful as a starting point to treat them as independent.
Distinction between quasiparticles and collective excitations
There is a difference in the way that quasiparticles and collective excitations are intuitively envisioned.[2] A quasiparticle is usually thought of as being like a dressed particle: It is built around a real particle at its "core", but the behavior of the particle is affected by the environment. A standard example is the "electron quasiparticle": A real electron particle, in a crystal, behaves as if it had a different mass. On the other hand, a collective excitation is usually imagined to be a reflection of the aggregate behavior of the system, with no single real particle at its "core". A standard example is the phonon, which characterizes the vibrational motion of every atom in the crystal.
Effect on bulk properties
Examples of quasiparticles and collective excitations
This section contains examples of quasiparticles and collective excitations. The first subsection below contains common ones that occur in a wide variety of materials under ordinary conditions; the second subsection contains examples that arise in particular, special contexts.
More common examples
• An exciton is an electron and hole bound together.
More specialized examples
• Composite fermions arise in a two-dimensional system subject to a large magnetic field, most famously those systems that exhibit the fractional quantum Hall effect.[6] These quasiparticles are quite unlike normal particles in two ways. First, their charge can be less than the electron charge e. In fact, they have been observed with charges of e/3, e/4, e/5, and e/7.[7] Second, they can be anyons, an exotic type of particle that is neither a fermion nor boson.[8]
• Stoner excitations in ferromagnetic metals
• Skyrmions
See also
2. ^ a b c , by Richard D. Mattuck, p10A guide to Feynman diagrams in the many-body problem. "As we have seen, the quasi particle consists of the original real, individual particle, plus a cloud of disturbed neighbors. It behaves very much like an individual particle, except that it has an effective mass and a lifetime. But there also exist other kinds of fictitious particles in many-body systems, i.e. 'collective excitations'. These do not center around individual particles, but instead involve collective, wavelike motion of all the particles in the system simultaneously."
3. ^ Principles of Nanophotonics by Motoichi Ohtsu, p205 google books link
4. ^ A. Gelfert, 'Manipulative Success and the Unreal', International Studies in the Philosophy of Science Vol. 17, 2003, 245–263
6. ^ Physics Today Article
7. ^ Cosmos magazine June 2008
8. ^ Nature article
10. ^ J. E. Hoffman; McElroy, K; Lee, DH; Lang, KM; Eisaki, H; Uchida, S; Davis, JC; et al. (2002). "Imaging Quasiparticle Interference in Bi2Sr2CaCu2O8+". Science 297 (5584): 1148–51.
Further reading
• Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). Theory of Heavy-Fermion Compounds - Theory of Strongly Correlated Fermi-Systems. Springer.
External links
• – Scientists find new 'quasiparticles'
• Curious 'quasiparticles' baffle physicists by Jacqui Hayes, Cosmos 6 June 2008. Accessed June 2008
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4e43983983199e6f | Schrödinger’s equation as an energy conservation law
Original post:
Schrodinger's equation
heat diffusion 2
heat diffusion 3
Poynting vector
Quantum-mechanical operators
We climbed a mountain—step by step, post by post. 🙂 We have reached the top now, and the view is gorgeous. We understand Schrödinger’s equation, which describes how amplitudes propagate through space-time. It’s the quintessential quantum-mechanical expression. Let’s enjoy now, and deepen our understanding by introducing the concept of (quantum-mechanical) operators.
The operator concept
We’ll introduce the operator concept using Schrödinger’s equation itself and, in the process, deepen our understanding of Schrödinger’s equation a bit. You’ll remember we wrote it as:
schrodinger 5
However, you’ve probably seen it like it’s written on his bust, or on his grave, or wherever, which is as follows:
It’s the same thing, of course. The ‘over-dot’ is Newton’s notation for the time derivative. In fact, if you click on the picture above (and zoom in a bit), then you’ll see that the craftsman who made the stone grave marker, mistakenly, also carved a dot above the psi (ψ) on the right-hand side of the equation—but then someone pointed out his mistake and so the dot on the right-hand side isn’t painted. 🙂 The thing I want to talk about here, however, is the H in that expression above, which is, obviously, the following operator:
That’s a pretty monstrous operator, isn’t it? It is what it is, however: an algebraic operator (it operates on a number—albeit a complex number—unlike a matrix operator, which operates on a vector or another matrix). As you can see, it actually consists of two other (algebraic) operators:
1. The ∇operator, which you know: it’s a differential operator. To be specific, it’s the Laplace operator, which is the divergence (·) of the gradient () of a function: ∇= · = (∂/∂x, ∂/∂y , ∂/∂z)·(∂/∂x, ∂/∂y , ∂/∂z) = ∂2/∂x2 + ∂2/∂y+ ∂2/∂z2. This too operates on our complex-valued function wavefunction ψ, and yields some other complex-valued function, which we then multiply by −ħ2/2m to get the first term.
2. The V(x, y, z) ‘operator’, which—in this particular context—just means: “multiply with V”. Needless to say, V is the potential here, and so it captures the presence of external force fields. Also note that V is a real number, just like −ħ2/2m.
Let me say something about the dimensions here. On the left-hand side of Schrödinger’s equation, we have the product of ħ and a time derivative (is just the imaginary unit, so that’s just a (complex) number). Hence, the dimension there is [J·s]/[s] (the dimension of a time derivative is something expressed per second). So the dimension of the left-hand side is joule. On the right-hand side, we’ve got two terms. The dimension of that second-order derivative (∇2ψ) is something expressed per square meter, but then we multiply it with −ħ2/2m, whose dimension is [J2·s2]/[J/(m2/s2)]. [Remember: m = E/c2.] So that reduces to [J·m2]. Hence, the dimension of (−ħ2/2m)∇2ψ is joule. And the dimension of V is joule too, of course. So it all works out. In fact, now that we’re here, it may or may not be useful to remind you of that heat diffusion equation we discussed when introducing the basic concepts involved in vector analysis:
diffusion equation
That equation illustrated the physical significance of the Laplacian. We were talking about the flow of heat in, say, a block of metal, as illustrated below. The in the equation above is the heat per unit volume, and the h in the illustration below was the heat flow vector (so it’s got nothing to do with Planck’s constant), which depended on the material, and which we wrote as = –κT, with T the temperature, and κ (kappa) the thermal conductivity. In any case, the point is the following: the equation below illustrates the physical significance of the Laplacian. We let it operate on the temperature (i.e. a scalar function) and its product with some constant (just think of replacing κ by −ħ2/2m gives us the time derivative of q, i.e. the heat per unit volume.
heat flow
In fact, we know that is proportional to T, so if we’d choose an appropriate temperature scale – i.e. choose the zero point such that T (your physics teacher in high school would refer to as the (volume) specific heat capacity) – then we could simple write:
∂T/∂t = (κ/k)∇2T
From a mathematical point of view, that equation is just the same as ∂ψ/∂t = –(i·ħ/2m)·∇2ψ, which is Schrödinger’s equation for V = 0. In other words, you can – and actually should – also think of Schrödinger’s equation as describing the flow of… Well… What?
Well… Not sure. I am tempted to think of something like a probability density in space, but ψ represents a (complex-valued) amplitude. Having said that, you get the idea—I hope! 🙂 If not, let me paraphrase Feynman on this:
“We can think of Schrödinger’s equation as describing the diffusion of a probability amplitude from one point to another. In fact, the equation looks something like the diffusion equation we introduced when discussing heat flow, or the spreading of a gas. But there is one main difference: the imaginary coefficient in front of the time derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Schrödinger’s equation are complex waves.”
That says it all, right? 🙂 In fact, Schrödinger’s equation – as discussed here – was actually being derived when describing the motion of an electron along a line of atoms, i.e. for motion in one direction only, but you can visualize what it represents in three-dimensional space. The real exponential functions Feynman refer to exponential decay function: as the energy is spread over an ever-increasing volume, the amplitude of the wave becomes smaller and smaller. That may be the case for complex-valued exponentials as well. The key difference between a real- and complex-valued exponential decay function is that a complex exponential is a cyclical function. Now, I quickly googled to see how we could visualize that, and I like the following illustration:
The dimensional analysis of Schrödinger’s equation is also quite interesting because… Well… Think of it: that heat diffusion equation incorporates the same dimensions: temperature is a measure of the average energy of the molecules. That’s really something to think about. These differential equations are not only structurally similar but, in addition, they all seem to describe some flow of energy. That’s pretty deep stuff: it relates amplitudes to energies, so we should think in terms of Poynting vectors and all that. But… Well… I need to move on, and so I will move on—so you can re-visit this later. 🙂
Now that we’ve introduced the concept of an operator, let me say something about notations, because that’s quite confusing.
Some remarks on notation
Because it’s an operator, we should actually use the hat symbol—in line with what we did when we were discussing matrix operators: we’d distinguish the matrix (e.g. A) from its use as an operator (Â). You may or may not remember we do the same in statistics: the hat symbol is supposed to distinguish the estimator (â) – i.e. some function we use to estimate a parameter (which we usually denoted by some Greek symbol, like α) – from a specific estimate of the parameter, i.e. the value (a) we get when applying â to a specific sample or observation. However, if you remember the difference, you’ll also remember that hat symbol was quickly forgotten, because the context made it clear what was what, and so we’d just write a(x) instead of â(x). So… Well… I’ll be sloppy as well here, if only because the WordPress editor only offers very few symbols with a hat! 🙂
In any case, this discussion on the use (or not) of that hat is irrelevant. In contrast, what is relevant is to realize this algebraic operator H here is very different from that other quantum-mechanical Hamiltonian operator we discussed when dealing with a finite set of base states: that H was the Hamiltonian matrix, but used in an ‘operation’ on some state. So we have the matrix operator H, and the algebraic operator H.
Yes and no. First, we’ve got the context again, and so you always know whether you’re looking at continuous or discrete stuff:
1. If your ‘space’ is continuous (i.e. if states are to defined with reference to an infinite set of base states), then it’s the algebraic operator.
2. If, on the other hand, your states are defined by some finite set of discrete base states, then it’s the Hamiltonian matrix.
There’s another, more fundamental, reason why there should be no confusion. In fact, it’s the reason why physicists use the same symbol H in the first place: despite the fact that they look so different, these two operators (i.e. H the algebraic operator and H the matrix operator) are actually equivalent. Their interpretation is similar, as evidenced from the fact that both are being referred to as the energy operator in quantum physics. The only difference is that one operates on a (state) vector, while the other operates on a continuous function. It’s just the difference between matrix mechanics as opposed to wave mechanics really.
But… Well… I am sure I’ve confused you by now—and probably very much so—and so let’s start from the start. 🙂
Matrix mechanics
Let’s start with the easy thing indeed: matrix mechanics. The matrix-mechanical approach is summarized in that set of Hamiltonian equations which, by now, you know so well:
If we have base states, then we have equations like this: one for each = 1, 2,… n. As for the introduction of the Hamiltonian, and the other subscript (j), just think of the description of a state:
So… Well… Because we had used already, we had to introduce j. 🙂
Let’s think about |ψ〉. It is the state of a system, like the ground state of a hydrogen atom, or one of its many excited states. But… Well… It’s a bit of a weird term, really. It all depends on what you want to measure: when we’re thinking of the ground state, or an excited state, we’re thinking energy. That’s something else than thinking its position in space, for example. Always remember: a state is defined by a set of base states, and so those base states come with a certain perspective: when talking states, we’re only looking at some aspect of reality, really. Let’s continue with our example of energy states, however.
You know that the lifetime of a system in an excited state is usually short: some spontaneous or induced emission of a quantum of energy (i.e. a photon) will ensure that the system quickly returns to a less excited state, or to the ground state itself. However, you shouldn’t think of that here: we’re looking at stable systems here. To be clear: we’re looking at systems that have some definite energy—or so we think: it’s just because of the quantum-mechanical uncertainty that we’ll always measure some other different value. Does that make sense?
If it doesn’t… Well… Stop reading, because it’s only going to get even more confusing. Not my fault, however!
The ubiquity of that ψ symbol (i.e. the Greek letter psi) is really something psi-chological 🙂 and, hence, very confusing, really. In matrix mechanics, our ψ would just denote a state of a system, like the energy of an electron (or, when there’s only one electron, our hydrogen atom). If it’s an electron, then we’d describe it by its orbital. In this regard, I found the following illustration from Wikipedia particularly helpful: the green orbitals show excitations of copper (Cu) orbitals on a CuOplane. [The two big arrows just illustrate the principle of X-ray spectroscopy, so it’s an X-ray probing the structure of the material.]
So… Well… We’d write ψ as |ψ〉 just to remind ourselves we’re talking of some state of the system indeed. However, quantum physicists always want to confuse you, and so they will also use the psi symbol to denote something else: they’ll use it to denote a very particular Ci amplitude (or coefficient) in that |ψ〉 = ∑|iCi formula above. To be specific, they’d replace the base states |i〉 by the continuous position variable x, and they would write the following:
Ci = ψ(i = x) = ψ(x) = Cψ(x) = C(x) = 〈x|ψ〉
In fact, that’s just like writing:
φ(p) = 〈 mom p | ψ 〉 = 〈p|ψ〉 = Cφ(p) = C(p)
What they’re doing here, is (1) reduce the ‘system‘ to a ‘particle‘ once more (which is OK, as long as you know what you’re doing) and (2) they basically state the following:
If a particle is in some state |ψ〉, then we can associate some wavefunction ψ(x) or φ(p)—with it, and that wavefunction will represent the amplitude for the system (i.e. our particle) to be at x, or to have a momentum that’s equal to p.
So what’s wrong with that? Well… Nothing. It’s just that… Well… Why don’t they use χ(x) instead of ψ(x)? That would avoid a lot of confusion, I feel: one should not use the same symbol (psi) for the |ψ〉 state and the ψ(x) wavefunction.
Huh? Yes. Think about it. The point is: the position or the momentum, or even the energy, are properties of the system, so to speak and, therefore, it’s really confusing to use the same symbol psi (ψ) to describe (1) the state of the system, in general, versus (2) the position wavefunction, which describes… Well… Some very particular aspect (or ‘state’, if you want) of the same system (in this case: its position). There’s no such problem with φ(p), so… Well… Why don’t they use χ(x) instead of ψ(x) indeed? I have only one answer: psi-chology. 🙂
In any case, there’s nothing we can do about it and… Well… In fact, that’s what this post is about: it’s about how to describe certain properties of the system. Of course, we’re talking quantum mechanics here and, hence, uncertainty, and, therefore, we’re going to talk about the average position, energy, momentum, etcetera that’s associated with a particular state of a system, or—as we’ll keep things very simple—the properties of a ‘particle’, really. Think of an electron in some orbital, indeed! 🙂
So let’s now look at that set of Hamiltonian equations once again:
Looking at it carefully – so just look at it once again! 🙂 – and thinking about what we did when going from the discrete to the continuous setting, we can now understand we should write the following for the continuous case:
Of course, combining Schrödinger’s equation with the expression above implies the following:
Now how can we relate that integral to the expression on the right-hand side? I’ll have to disappoint you here, as it requires a lot of math to transform that integral. It requires writing H(x, x’) in terms of rather complicated functions, including – you guessed it, didn’t you? – Dirac’s delta function. Hence, I assume you’ll believe me if I say that the matrix- and wave-mechanical approaches are actually equivalent. In any case, if you’d want to check it, you can always read Feynman yourself. 🙂
Now, I wrote this post to talk about quantum-mechanical operators, so let me do that now.
Quantum-mechanical operators
You know the concept of an operator. As mentioned above, we should put a little hat (^) on top of our Hamiltonian operator, so as to distinguish it from the matrix itself. However, as mentioned above, the difference is usually quite clear from the context. Our operators were all matrices so far, and we’d write the matrix elements of, say, some operator A, as:
Aij ≡ 〈 i | A | j 〉
The whole matrix itself, however, would usually not act on a base state but… Well… Just on some more general state ψ, to produce some new state φ, and so we’d write:
| φ 〉 = A | ψ 〉
Of course, we’d have to describe | φ 〉 in terms of the (same) set of base states and, therefore, we’d expand this expression into something like this:
operator 2
You get the idea. I should just add one more thing. You know this important property of amplitudes: the 〈 ψ | φ 〉 amplitude is the complex conjugate of the 〈 φ | ψ 〉 amplitude. It’s got to do with time reversibility, because the complex conjugate of eiθ = ei(ω·t−k·x) is equal to eiθ = ei(ω·t−k·x), so we’re just reversing the x- and tdirection. We write:
〈 ψ | φ 〉 = 〈 φ | ψ 〉*
Now what happens if we want to take the complex conjugate when we insert a matrix, so when writing 〈 φ | A | ψ 〉 instead of 〈 φ | ψ 〉, this rules becomes:
〈 φ | A | ψ 〉* = 〈 ψ | A† | φ 〉
The dagger symbol denotes the conjugate transpose, so A† is an operator whose matrix elements are equal to Aij† = Aji*. Now, it may or may not happen that the A† matrix is actually equal to the original A matrix. In that case – and only in that case – we can write:
〈 ψ | A | φ 〉 = 〈 φ | A | ψ 〉*
We then say that A is a ‘self-adjoint’ or ‘Hermitian’ operator. That’s just a definition of a property, which the operator may or may not have—but many quantum-mechanical operators are actually Hermitian. In any case, we’re well armed now to discuss some actual operators, and we’ll start with that energy operator.
The energy operator (H)
We know the state of a system is described in terms of a set of base states. Now, our analysis of N-state systems showed we can always describe it in terms of a special set of base states, which are referred to as the states of definite energy because… Well… Because they’re associated with some definite energy. In that post, we referred to these energy levels as En (n = I, II,… N). We used boldface for the subscript n (so we wrote n instead of n) because of these Roman numerals. With each energy level, we could associate a base state, of definite energy indeed, that we wrote as |n〉. To make a long story short, we summarized our results as follows:
1. The energies EI, EII,…, En,…, EN are the eigenvalues of the Hamiltonian matrix H.
2. The state vectors |n〉 that are associated with each energy En, i.e. the set of vectors |n〉, are the corresponding eigenstates.
We’ll be working with some more subscripts in what follows, and these Roman numerals and the boldface notation are somewhat confusing (if only because I don’t want you to think of these subscripts as vectors), we’ll just denote EI, EII,…, En,…, EN as E1, E2,…, Ei,…, EN, and we’ll number the states of definite energy accordingly, also using some Greek letter so as to clearly distinguish them from all our Latin letter symbols: we’ll write these states as: |η1〉, |η1〉,… |ηN〉. [If I say, ‘we’, I mean Feynman of course. You may wonder why he doesn’t write |Ei〉, or |εi〉. The answer is: writing |En〉 would cause confusion, because this state will appear in expressions like: |Ei〉Ei, so that’s the ‘product’ of a state (|Ei〉) and the associated scalar (Ei). Too confusing. As for using η (eta) instead of ε (epsilon) to denote something that’s got to do with energy… Well… I guess he wanted to keep the resemblance with the n, and then the Ancient Greek apparently did use this η letter for a sound like ‘e‘ so… Well… Why not? Let’s get back to the lesson.]
Using these base states of definite energy, we can write the state of the system as:
|ψ〉 = ∑ |ηi〉 C = ∑ |ηi〉〈ηi|ψ〉 over all (i = 1, 2,… , N)
Now, we didn’t talk all that much about what these base states actually mean in terms of measuring something but you’ll believe if I say that, when measuring the energy of the system, we’ll always measure one or the other E1, E2,…, Ei,…, EN value. We’ll never measure something in-between: it’s eitheror. Now, as you know, measuring something in quantum physics is supposed to be destructive but… Well… Let us imagine we could make a thousand measurements to try to determine the average energy of the system. We’d do so by counting the number of times we measure E1 (and of course we’d denote that number as N1), E2E3, etcetera. You’ll agree that we’d measure the average energy as:
E average
However, measurement is destructive, and we actually know what the expected value of this ‘average’ energy will be, because we know the probabilities of finding the system in a particular base state. That probability is equal to the absolute square of that Ccoefficient above, so we can use the P= |Ci|2 formula to write:
Eav〉 = ∑ Pi Ei over all (i = 1, 2,… , N)
Note that this is a rather general formula. It’s got nothing to do with quantum mechanics: if Ai represents the possible values of some quantity A, and Pi is the probability of getting that value, then (the expected value of) the average A will also be equal to 〈Aav〉 = ∑ Pi Ai. No rocket science here! 🙂 But let’s now apply our quantum-mechanical formulas to that 〈Eav〉 = ∑ Pi Ei formula. [Oh—and I apologize for using the same angle brackets 〈 and 〉 to denote an expected value here—sorry for that! But it’s what Feynman does—and other physicists! You see: they don’t really want you to understand stuff, and so they often use very confusing symbols.] Remembering that the absolute square of a complex number equals the product of that number and its complex conjugate, we can re-write the 〈Eav〉 = ∑ Pi Ei formula as:
Eav〉 = ∑ Pi Ei = ∑ |Ci|Ei = ∑ Ci*CEi = ∑ C*CEi = ∑ 〈ψ|ηi〉〈ηi|ψ〉E= ∑ 〈ψ|ηiEi〈ηi|ψ〉 over all i
Now, you know that Dirac’s bra-ket notation allows numerous manipulations. For example, what we could do is take out that ‘common factor’ 〈ψ|, and so we may re-write that monster above as:
Eav〉 = 〈ψ| ∑ ηiEi〈ηi|ψ〉 = 〈ψ|φ〉, with |φ〉 = ∑ |ηiEi〈ηi|ψ〉 over all i
Huh? Yes. Note the difference between |ψ〉 = ∑ |ηi〉 C = ∑ |ηi〉〈ηi|ψ〉 and |φ〉 = ∑ |ηiEi〈ηi|ψ〉. As Feynman puts it: φ is just some ‘cooked-up‘ state which you get by taking each of the base states |ηi〉 in the amount Ei〈ηi|ψ〉 (as opposed to the 〈ηi|ψ〉 amounts we took for ψ).
I know: you’re getting tired and you wonder why we need all this stuff. Just hang in there. We’re almost done. I just need to do a few more unpleasant things, one of which is to remind you that this business of the energy states being eigenstates (and the energy levels being eigenvalues) of our Hamiltonian matrix (see my post on N-state systems) comes with a number of interesting properties, including this one:
H |ηi〉 = Eii〉 = |ηiEi
Just think about what’s written here: on the left-hand side, we’re multiplying a matrix with a (base) state vector, and on the left-hand side we’re multiplying it with a scalar. So our |φ〉 = ∑ |ηiEi〈ηi|ψ〉 sum now becomes:
|φ〉 = ∑ H |ηi〉〈ηi|ψ〉 over all (i = 1, 2,… , N)
Now we can manipulate that expression some more so as to get the following:
|φ〉 = H ∑|ηi〉〈ηi|ψ〉 = H|ψ〉
Finally, we can re-combine this now with the 〈Eav〉 = 〈ψ|φ〉 equation above, and so we get the fantastic result we wanted:
Eav〉 = 〈 ψ | φ 〉 = 〈 ψ | H ψ 〉
Huh? Yes! To get the average energy, you operate on |ψ with H, and then you multiply the result with ψ|. It’s a beautiful formula. On top of that, the new formula for the average energy is not only pretty but also useful, because now we don’t need to say anything about any particular set of base states. We don’t even have to know all of the possible energy levels. When we have to calculate the average energy of some system, we only need to be able to describe the state of that system in terms of some set of base states, and we also need to know the Hamiltonian matrix for that set, of course. But if we know that, we can calculate its average energy.
You’ll say that’s not a big deal because… Well… If you know the Hamiltonian, you know everything, so… Well… Yes. You’re right: it’s less of a big deal than it seems. Having said that, the whole development above is very interesting because of something else: we can easily generalize it for other physical measurements. I call it the ‘average value’ operator idea, but you won’t find that term in any textbook. 🙂 Let me explain the idea.
The average value operator (A)
The development above illustrates how we can relate a physical observable, like the (average) energy (E), to a quantum-mechanical operator (H). Now, the development above can easily be generalized to any observable that would be proportional to the energy. It’s perfectly reasonable, for example, to assume the angular momentum – as measured in some direction, of course, which we usually refer to as the z-direction – would be proportional to the energy, and so then it would be easy to define a new operator Lz, which we’d define as the operator of the z-component of the angular momentum L. [I know… That’s a bit of a long name but… Well… You get the idea.] So we can write:
Lzav = 〈 ψ | Lψ 〉
In fact, further generalization yields the following grand result:
If a physical observable A is related to a suitable quantum-mechanical operator Â, then the average value of A for the state | ψ 〉 is given by:
Aav = 〈 ψ | Â ψ 〉 = 〈 ψ | φ 〉 with | φ 〉 = Â ψ 〉
At this point, you may have second thoughts, and wonder: what state | ψ 〉? The answer is: it doesn’t matter. It can be any state, as long as we’re able to describe in terms of a chosen set of base states. 🙂
OK. So far, so good. The next step is to look at how this works for the continuity case.
The energy operator for wavefunctions (H)
We can start thinking about the continuous equivalent of the 〈Eav〉 = 〈ψ|H|ψ〉 expression by first expanding it. We write:
e average continuous function
You know the continuous equivalent of a sum like this is an integral, i.e. an infinite sum. Now, because we’ve got two subscripts here (i and j), we get the following double integral:
double integral
Now, I did take my time to walk you through Feynman’s derivation of the energy operator for the discrete case, i.e. the operator when we’re dealing with matrix mechanics, but I think I can simplify my life here by just copying Feynman’s succinct development:
Done! Given a wavefunction ψ(x), we get the average energy by doing that integral above. Now, the quantity in the braces of that integral can be written as that operator we introduced when we started this post:
So now we can write that integral much more elegantly. It becomes:
Eav = ∫ ψ*(xH ψ(x) dx
You’ll say that doesn’t look like 〈Eav〉 = 〈 ψ | H ψ 〉! It does. Remember that 〈 ψ | = ψ 〉*. 🙂 Done!
I should add one qualifier though: the formula above assumes our wavefunction has been normalized, so all probabilities add up to one. But that’s a minor thing. The only thing left to do now is to generalize to three dimensions. That’s easy enough. Our expression becomes a volume integral:
Eav = ∫ ψ*(rH ψ(r) dV
Of course, dV stands for dVolume here, not for any potential energy, and, of course, once again we assume all probabilities over the volume add up to 1, so all is normalized. Done! 🙂
We’re almost done with this post. What’s left is the position and momentum operator. You may think this is going to another lengthy development but… Well… It turns out the analysis is remarkably simple. Just stay with me a few more minutes and you’ll have earned your degree. 🙂
The position operator (x)
The thing we need to solve here is really easy. Look at the illustration below as representing the probability density of some particle being at x. Think about it: what’s the average position?
average position
Well? What? The (expected value of the) average position is just this simple integral: 〈xav = ∫ P(x) dx, over all the whole range of possible values for x. 🙂 That’s all. Of course, because P(x) = |ψ(x)|2 =ψ*(x)·ψ(x), this integral now becomes:
xav = ∫ ψ*(x) x ψ(x) dx
That looks exactly the same as 〈Eav = ∫ ψ*(xH ψ(x) dx, and so we can look at as an operator too!
Huh? Yes. It’s an extremely simple operator: it just means “multiply by x“. 🙂
I know you’re shaking your head now: is it that easy? It is. Moreover, the ‘matrix-mechanical equivalent’ is equally simple but, as it’s getting late here, I’ll refer you to Feynman for that. 🙂
The momentum operator (px)
Now we want to calculate the average momentum of, say, some electron. What integral would you use for that? […] Well… What? […] It’s easy: it’s the same thing as for x. We can just substitute replace for in that 〈xav = ∫ P(x) dformula, so we get:
pav = ∫ P(p) dp, over all the whole range of possible values for p
Now, you might think the rest is equally simple, and… Well… It actually is simple but there’s one additional thing in regard to the need to normalize stuff here. You’ll remember we defined a momentum wavefunction (see my post on the Uncertainty Principle), which we wrote as:
φ(p) = 〈 mom p | ψ 〉
Now, in the mentioned post, we related this momentum wavefunction to the particle’s ψ(x) = 〈x|ψ〉 wavefunction—which we should actually refer to as the position wavefunction, but everyone just calls it the particle’s wavefunction, which is a bit of a misnomer, as you can see now: a wavefunction describes some property of the system, and so we can associate several wavefunctions with the same system, really! In any case, we noted the following there:
• The two probability density functions, φ(p) and ψ(x), look pretty much the same, but the half-width (or standard deviation) of one was inversely proportional to the half-width of the other. To be precise, we found that the constant of proportionality was equal to ħ/2, and wrote that relation as follows: σp = (ħ/2)/σx.
• We also found that, when using a regular normal distribution function for ψ(x), we’d have to normalize the probability density function by inserting a (2πσx2)−1/2 in front of the exponential.
Now, it’s a bit of a complicated argument, but the upshot is that we cannot just write what we usually write, i.e. Pi = |Ci|2 or P(x) = |ψ(x)|2. No. We need to put a normalization factor in front, which combines the two factors I mentioned above. To be precise, we have to write:
P(p) = |〈p|ψ〉|2/(2πħ)
So… Well… Our 〈pav = ∫ P(p) dp integral can now be written as:
pav = ∫ 〈ψ|ppp|ψ〉 dp/(2πħ)
So that integral is totally like what we found for 〈xav and so… We could just leave it at that, and say we’ve solved the problem. In that sense, it is easy. However, having said that, it’s obvious we’d want some solution that’s written in terms of ψ(x), rather than in terms of φ(p), and that requires some more manipulation. I’ll refer you, once more, to Feynman for that, and I’ll just give you the result:
momentum operator
So… Well… I turns out that the momentum operator – which I tentatively denoted as px above – is not so simple as our position operator (x). Still… It’s not hugely complicated either, as we can write it as:
px ≡ (ħ/i)·(∂/∂x)
Of course, the purists amongst you will, once again, say that I should be more careful and put a hat wherever I’d need to put one so… Well… You’re right. I’ll wrap this all up by copying Feynman’s overview of the operators we just explained, and so he does use the fancy symbols. 🙂
Well, folks—that’s it! Off we go! You know all about quantum physics now! We just need to work ourselves through the exercises that come with Feynman’s Lectures, and then you’re ready to go and bag a degree in physics somewhere. So… Yes… That’s what I want to do now, so I’ll be silent for quite a while now. Have fun! 🙂
Dirac’s delta function and Schrödinger’s equation in three dimensions
Feynman’s rather informal derivation of Schrödinger’s equation – following Schrödinger’s own logic when he published his famous paper on it back in 1926 – is wonderfully simple but, as I mentioned in my post on it, does lack some mathematical rigor here and there. Hence, Feynman hastens to dot all of the i‘s and cross all of the t‘s in the subsequent Lectures. We’ll look at two things here:
1. Dirac’s delta function, which ensures proper ‘normalization’. In fact, as you’ll see in a moment, it’s more about ‘orthogonalization’ than normalization. 🙂
2. The generalization of Schrödinger’s equation to three dimensions (in space) and also including the presence of external force fields (as opposed to the usual ‘free space’ assumption).
The second topic is the most interesting, of course, and also the easiest, really. However, let’s first use our energy to grind through the first topic. 🙂
Dirac’s delta function
When working with a finite set of discrete states, a fundamental condition is that the base states be ‘orthogonal’, i.e. they must satisfy the following equation:
ij 〉 = δij, with δij = 1 if i = j and δij = 0 if ij
Needless to say, the base states and j are rather special vectors in a rather special mathematical space (a so-called Hilbert space) and so it’s rather tricky to interpret their ‘orthogonality’ in any geometric way, although such geometric interpretation is often actually possible in simple quantum-mechanical systems: you’ll just notice a ‘right’ angle may actually be 45°, or 180° angles, or whatever. 🙂 In any case, that’s not the point here. The question is: if we move an infinite number of base states – like we did when we introduced the ψ(x) and φ(p) wavefunctions – what happens to that condition?
Your first reaction is going to be: nothing. Because… Well… Remember that, for a two-state system, in which we have two base states only, we’d fully describe some state | φ 〉 as a linear combination of the base states, so we’d write:
| φ 〉 =| I 〉 CI + | II 〉 CII
Now, while saying we were talking a Hilbert space here, I did add we could use the same expression to define the base states themselves, so I wrote the following triviality:
M1Trivial but sensible. So we’d associate the base state | I 〉 with the base vector (1, 0) and, likewise, base state | II 〉 with the base vector (0, 1). When explaining this, I added that we could easily extend to an N-state system and so there’s a perfect analogy between the 〈 i | j 〉 bra-ket expression in quantum math and the ei·ej product in the run-of-the-mill coordinate spaces that you’re used to. So why can’t we just extend the concept to an infinite-state system and move to base vectors with an infinite number of elements, which we could write as ei =(…, 0, ei = 1, 0, 0,,…) and ej =(…, 0, 0, ej = 1, 0,…), thereby ensuring 〈 i | j 〉 = ei·ej = δijalways! The ‘orthogonality’ condition looks simple enough indeed, and so we could re-write it as:
xx’ 〉 = δxx’, with δxx’ = 1 if x = x’ and δxx’ = 0 if if x ≠ x’
However, when moving from a space with a finite number of dimensions to a space with an infinite number of dimensions, there are some issues. They pop up, for example, when we insert that 〈 xx’ 〉 = δxx’ function (note that we’re talking some function here of x and x’, indeed, so we’ll write it as f(x, x’) in the next step) in that 〈φ|ψ〉 = ∫〈φ|x〉〈x|ψ〉dx integral.
Huh? What integral? Relax: that 〈φ|ψ〉 = ∫〈φ|x〉〈x|ψ〉dx integral just generalizes our 〈φ|ψ〉 = ∑〈φ|x〉〈x|ψ〉 expression for discrete settings for the continuous case. Just look at it. When substituting φ for x’, we get:
x’|ψ〉 = ψ(x’) = ∫ 〈x’|x〉 〈x|ψ〉 dx ⇔ ψ(x’) = ∫ 〈x’|x〉 ψ(x) dx
You’ll say: what’s the problem? Well… From a mathematical point of view, it’s a bit difficult to find a function 〈x’|x〉 = f(x, x’) which, when multiplied with a wavefunction ψ(x), and integrated over all x, will just give us ψ(x’). A bit difficult? Well… It’s worse than that: it’s actually impossible!
Huh? Yes. Feynman illustrates the difficulty for x’ = 0, but he could have picked whatever value, really. In any case, if x’ = 0, we can write f(x, 0) = f(x), and our integral now reduces to:
ψ(0) = ∫ f(x) ψ(0) dx
This is a weird expression: the value of the integral (i.e. the right-hand side of the expression) does not depend on x: it is just some non-zero value ψ(0). However, we know that the f(x) in the integrand is zero for all x ≠ 0. Hence, this integral will be zero. So we have an impossible situation: we wish a function to be zero everywhere but for one point, and, at the same time, we also want it to give us a finite integral when using it in that integral above.
You’re likely to shake your head now and say: what the hell? Does it matter? It does: it is an actual problem in quantum math. Well… I should say: it was an actual problem in quantum math. Dirac solved it. He invented a new function which looks a bit less simple than our suggested generalization of Kronecker’s delta for the continuous case (i.e. that 〈 xx’ 〉 = δxx’ conjecture above). Dirac’s function is – quite logically – referred to as the Dirac delta function, and it’s actually defined by that integral above, in the sense that we impose the following two conditions on it:
• δ(x‘) = 0 if x ≠ x’ (so that’s just like the first of our two conditions for that 〈 xx’ 〉 = δxx’ function)
• δ(x)ψ(x) dx = ψ(x’) (so that’s not like the second of our two condition for that 〈 xx’ 〉 = δxx’ function)
Indeed, that second condition is much more sophisticated than our 〈 xx’ 〉 = 1 if x = x’ condition. In fact, one can show that the second condition amounts to finding some function satisfying this condition:
δ(x)dx = 1
We get this by equating x’ to zero once more and, additionally, by equating ψ(x) to 1. [Please do double-check yourself.] Of course, this ‘normalization’ (or ‘orthogonalization’) problem all sounds like a lot of hocus-pocus and, in many ways, it is. In fact, we’re actually talking a mathematical problem here which had been lying around for centuries (for a brief overview, see the Wikipedia article on it). So… Well… Without further ado, I’ll just give you the mathematical expression now—and please don’t stop reading now, as I’ll explain it in a moment:
I will also credit Wikipedia with the following animation, which shows that the expression above is just the normal distribution function, and which shows what happens when that a, i.e. its standard deviation, goes to zero: Dirac’s delta function is just the limit of a sequence of (zero-centered) normal distributions. That’s all. Nothing more, nothing less.
But how do we interpret it? Well… I can’t do better than Feynman as he describes what’s going on really:
“Dirac’s δ(xfunction has the property that it is zero everywhere except at x = 0 but, at the same time, it has a finite integral equal to unity. [See the δ(x)dx = 1 equation.] One should imagine that the δ(x) function has such fantastic infinity at one point that the total area comes out equal to one.”
Well… That says it all, I guess. 🙂 Don’t you love the way he puts it? It’s not an ‘ordinary’ infinity. No. It’s fantastic. Frankly, I think these guys were all fantastic. 🙂 The point is: that special function, Dirac’s delta function, solves our problem. The equivalent expression for the 〈 ij 〉 = δij condition for a finite and discrete set of base states is the following one for the continuous case:
xx’ 〉 = δ(x − x’)
The only thing left now is to generalize this result to three dimensions. Now that’s fairly straightforward. The ‘normalization’ condition above is all that’s needed in terms of modifying the equations for dealing with the continuum of base states corresponding to the points along a line. Extending the analysis to three dimensions goes as follows:
• First, we replace the x coordinate by the vector r = (x, y, z)
• As a result, integrals over x, become integrals over x, y and z. In other words, they become volume integrals.
• Finally, the one-dimensional δ-function must be replaced by the product of three δ-functions: one in x, one in y and one in z. We write:
r | r 〉 = δ(x − x’) δ(y − y’)δ(z − z’)
Feynman summarizes it all together as follows:
What if we have two particles, or more? Well… Once again, I won’t bother to try to re-phrase the Grand Master as he explains it. I’ll just italicize or boldface the key points:
Suppose there are two particles, which we can call particle 1 and particle 2. What shall we use for the base states? One perfectly good set can be described by saying that particle 1 is at xand particle 2 is at x2, which we can write as | xx〉. Notice that describing the position of only one particle does not define a base state. Each base state must define the condition of the entire system, so you must not think that each particle moves independently as a wave in three dimensions. Any physical state | ψ 〉 can be defined by giving all of the amplitudes 〈 xx| ψ 〉 to find the two particles at x1 and x2. This generalized amplitude is therefore a function of the two sets of coordinates x1 and x1. You see that such a function is not a wave in the sense of an oscillation that moves along in three dimensions. Neither is it generally simply a product of two individual waves, one for each particle. It is, in general, some kind of a wave in the six dimensions defined by x1 and x1Hence, if there are two particles in Nature which are interacting, there is no way of describing what happens to one of the particles by trying to write down a wave function for it alone. The famous paradoxes that we considered in earlier chapters—where the measurements made on one particle were claimed to be able to tell what was going to happen to another particle, or were able to destroy an interference—have caused people all sorts of trouble because they have tried to think of the wave function of one particle alone, rather than the correct wave function in the coordinates of both particles. The complete description can be given correctly only in terms of functions of the coordinates of both particles.
Now we really know it all, don’t we? 🙂
Well… Almost. I promised to tackle another topic as well. So here it is:
Schrödinger’s equation in three dimensions
Let me start by jotting down what we had found already, i.e. Schrödinger’s equation when only one coordinate in space is involved. It’s written as:
schrodinger 3
Now, the extension to three dimensions is remarkably simple: we just substitute the ∂/∂xoperator by the ∇operator, i.e. ∇= ∂/∂x2 + ∂/∂y+ ∂/∂z2. We get:
schrodinger 4
Finally, we can also put forces on the particle, so now we are not looking at a particle moving in free space: we’ve got some force field working on it. It turns out the required modification is equally simple. The grand result is Schrödinger’s original equation in three dimensions:
schrodinger 5
V = V(x, y, z) is, of course, just the potential here. Remarkably simple equations but… How do we get these? Well… Sorry. The math is not too difficult, but you’re well equipped now to look at Feynman’s Lecture on it yourself now. You really are. Trust me. I really dealt with all of the ‘serious’ stuff you need to understand how he’s going about it in my previous posts so, yes, now I’ll just sit back and relax. Or go biking. Or whatever. 🙂
The Uncertainty Principle
New notations
amplitude continuous
The momentum wavefunction
φ(p) = 〈 mom p | ψ 〉
integral 2
ψ(x) = K·ex2/4σ2
η = ħ/2σ
ΔpΔx = ħ/2
ΔpΔx ≥ ħ/2
Schrödinger’s equation: the original approach
Of course, your first question when seeing the title of this post is: what’s original, really? Well… The answer is simple: it’s the historical approach, and it’s original because it’s actually quite intuitive. Indeed, Lecture no. 16 in Feynman’s third Volume of Lectures on Physics is like a trip down memory lane as Feynman himself acknowledges, after presenting Schrödinger’s equation using that very rudimentary model we developed in our previous post:
“We do not intend to have you think we have derived the Schrödinger equation but only wish to show you one way of thinking about it. When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature.”
So… Well… Let’s have a look at it. 🙂 We were looking at some electron we described in terms of its location at one or the other atom in a linear array (think of it as a line). We did so by defining base states |n〉 = |xn〉, noting that the state of the electron at any point in time could then be written as:
|φ〉 = ∑ |xnCn(t) = ∑ |xn〉〈xn|φ〉 over all n
The Cn(t) = 〈xn|φ〉 coefficient is the amplitude for the electron to be at xat t. Hence, the Cn(t) amplitudes vary with t as well as with x. We’ll re-write them as Cn(t) = C(xn, t) = C(xn). Note that the latter notation does not explicitly show the time dependence. The Hamiltonian equation we derived in our previous post is now written as:
iħ·(∂C(xn)/∂t) = E0C(xn) − AC(xn+b) − AC(xn−b)
Note that, as part of our move from the Cn(t) to the C(xn) notation, we write the time derivative dCn(t)/dt now as ∂C(xn)/∂t, so we use the partial derivative symbol now (∂). Of course, the other partial derivative will be ∂C(x)/∂x) as we move from the count variable xto the continuous variable x, but let’s not get ahead of ourselves here. The solution we found for our C(xn) functions was the following wavefunction:
C(xn) = a·ei(k∙xn−ω·t) ei∙ω·t·ei∙k∙xn ei·(E/ħ)·t·ei·k∙xn
We also found the following relationship between E and k:
E = E0 − 2A·cos(kb)
Now, even Feynman struggles a bit with the definition of E0 and k here, and their relationship with E, which is graphed below.
Indeed, he first writes, as he starts developing the model, that E0 is, physically, the energy the electron would have if it couldn’t leak away from one of the atoms, but then he also adds: “It represents really nothing but our choice of the zero of energy.”
This is all quite enigmatic because we cannot just do whatever we want when discussing the energy of a particle. As I pointed out in one of my previous posts, when discussing the energy of a particle in the context of the wavefunction, we generally consider it to be the sum of three different energy concepts:
1. The particle’s rest energy m0c2, which de Broglie referred to as internal energy (Eint), and which includes the rest mass of the ‘internal pieces’, as Feynman puts it (now we call those ‘internal pieces’ quarks), as well as their binding energy (i.e. the quarks’ interaction energy).
2. Any potential energy it may have because of some field (i.e. if it is not traveling in free space), which we usually denote by U. This field can be anything—gravitational, electromagnetic: it’s whatever changes the energy of the particle because of its position in space.
It’s obvious that we cannot just “choose” the zero point here: the particle’s rest energy is its rest energy, and its velocity is its velocity. So it’s not quite clear what the E0 in our model really is. As far as I am concerned, it represents the average energy of the system really, so it’s just like the E0 for our ammonia molecule, or the E0 for whatever two-state system we’ve seen so far. In fact, when Feynman writes that we can “choose our zero of energy so that E0 − 2A = 0″ (so the minimum of that curve above is at the zero of energy), he actually makes some assumption in regard to the relative magnitude of the various amplitudes involved.
We should probably think about it in this way: −(i/ħ)·E0 is the amplitude for the electron to just stay where it is, while i·A/ħ is the amplitude to go somewhere else—and note we’ve got two possibilities here: the electron can go to |xn+1〉, or, alternatively, it can go to |xn−1〉. Now, amplitudes can be associated with probabilities by taking the absolute square, so I’d re-write the E0 − 2A = 0 assumption as:
E0 = 2A ⇔ |−(i/ħ)·E0|= |(i/ħ)·2A|2
Hence, in my humble opinion, Feynman’s assumption that E0 − 2A = 0 has nothing to do with ‘choosing the zero of energy’. It’s more like a symmetry assumption: we’re basically saying it’s as likely for the electron to stay where it is as it is to move to the next position. It’s an idea I need to develop somewhat further, as Feynman seems to just gloss over these little things. For example, I am sure it is not a coincidence that the EI, EIIEIII and EIV energy levels we found when discussing the hyperfine splitting of the hydrogen ground state also add up to 0. In fact, you’ll remember we could actually measure those energy levels (E= EII = EIII = A ≈ 9.23×10−6 eV, and EIV = −3A ≈ −27.7×10−6 eV), so saying that we can “choose” some zero energy point is plain nonsense. The question just doesn’t arise. In any case, as I have to continue the development here, I’ll leave this point for further analysis in the future. So… Well… Just note this E0 − 2A = 0 assumption, as we’ll need it in a moment.
The second assumption we’ll need concerns the variation in k. As you know, we can only get a wave packet if we allow for uncertainty in k which, in turn, translates into uncertainty for E. We write:
ΔE = Δ[E0 − 2A·cos(kb)]
Of course, we’d need to interpret the Δ as a variance (σ2) or a standard deviation (σ) so we can apply the usual rules – i.e. var(a) = 0, var(aX) = a2·var(X), and var(aX ± bY) = a2·var(X) + b2·var(Y) ± 2ab·cov(X, Y) – to be a bit more precise about what we’re writing here, but you get the idea. In fact, let me quickly write it out:
var[E0 − 2A·cos(kb)] = var(E0) + 4A2·var[cos(kb)] ⇔ var(E) = 4A2·var[cos(kb)]
Now, you should check my post scriptum to my page on the Essentials, to see how the probability density function of the cosine of a randomly distributed variable looks like, and then you should go online to find a formula for its variance, and then you can work it all out yourself, because… Well… I am not going to do it for you. What I want to do here is just show how Feynman gets Schrödinger’s equation out of all of these simplifications.
So what’s the second assumption? Well… As the graph shows, our k can take any value between −π/b and +π/b, and therefore, the kb argument in our cosine function can take on any value between −π and +π. In other words, kb could be any angle. However, as Feynman puts it—we’ll be assuming that kb is ‘small enough’, so we can use the small-angle approximations whenever we see the cos(kb) and/or sin(kb) functions. So we write: sin(kb) ≈ kb and cos(kb) ≈ 1 − (kb)2/2 = 1 − k2b2/2. Now, that assumption led to another grand result, which we also derived in our previous post. It had to do with the group velocity of our wave packet, which we calculated as:
= dω/dk = (2Ab2/ħ)·k
Of course, we should interpret our k here as “the typical k“. Huh? Yes… That’s how Feynman refers to it, and I have no better term for it. It’s some kind of ‘average’ of the Δk interval, obviously, but… Well… Feynman does not give us any exact definition here. Of course, if you look at the graph once more, you’ll say that, if the typical kb has to be “small enough”, then its expected value should be zero. Well… Yes and no. If the typical kb is zero, or if is zero, then is zero, and then we’ve got a stationary electron, i.e. an electron with zero momentum. However, because we’re doing what we’re doing (that is, we’re studying “stuff that moves”—as I put it unrespectfully in a few of my posts, so as to distinguish from our analyses of “stuff that doesn’t move”, like our two-state systems, for example), our “typical k” should not be zero here. OK… We can now calculate what’s referred to as the effective mass of the electron, i.e. the mass that appears in the classical kinetic energy formula: K.E. = m·v2/2. Now, there are two ways to do that, and both are somewhat tricky in their interpretation:
1. Using both the E0 − 2A = 0 as well as the “small kb” assumption, we find that E = E0 − 2A·(1 − k2b2/2) = A·k2b2. Using that for the K.E. in our formula yields:
meff = 2A·k2b2/v= 2A·k2b2/[(2Ab2/ħ)·k]= ħ2/(2Ab2)
2. We can use the classical momentum formula (p = m·v), and then the 2nd de Broglie equation, which tells us that each wavenumber (k) is to be associated with a value for the momentum (p) using the p = ħk (so p is proportional to k, with ħ as the factor of proportionality). So we can now calculate meff as meff = ħk/v. Substituting again for what we’ve found above, gives us the same:
meff = 2A·k2b2/v = ħ·k/[(2Ab2/ħ)·k] = ħ2/(2Ab2)
Of course, we’re not supposed to know the de Broglie relations at this point in time. 🙂 But, now that you’ve seen them anyway, note how we have two formulas for the momentum:
• The classical formula (p = m·v) tells us that the momentum is proportional to the classical velocity of our particle, and m is then the factor of proportionality.
• The quantum-mechanical formula (p = ħk) tells us that the (typical) momentum is proportional to the (typical) wavenumber, with Planck’s constant (ħ) as the factor of proportionality. Combining both combines the classical and quantum-mechanical perspective of a moving particle:
v = ħk
I know… It’s an obvious equation but… Well… Think of it. It’s time to get back to the main story now. Remember we were trying to find Schrödinger’s equation? So let’s get on with it. 🙂
To do so, we need one more assumption. It’s the third major simplification and, just like the others, the assumption is obvious on first, but not on second thought. 😦 So… What is it? Well… It’s easy to see that, in our meff = ħ2/(2Ab2) formula, all depends on the value of 2Ab2. So, just like we should wonder what happens with that kb factor in the argument of our sine or cosine function if b goes to zero—i.e. if we’re letting the lattice spacing go to zero, so we’re moving from a discrete to a continuous analysis now—we should also wonder what happens with that 2Ab2 factor! Well… Think about it. Wouldn’t it be reasonable to assume that the effective mass of our electron is determined by some property of the material, or the medium (so that’s the silicon in our previous post) and, hence, that it’s constant really. Think of it: we’re not changing the fundamentals really—we just have some electron roaming around in some medium and all that we’re doing now is bringing those xcloser together. Much closer. It’s only logical, then, that our amplitude to jump from xn±1 to xwould also increase, no? So what we’re saying is that 2Ab2 is some constant which we write as ħ2/meff or, what amounts to the same, that Ab= ħ2/2·meff.
Of course, you may raise two objections here:
1. The Ab= ħ2/2·meff assumption establishes a very particular relation between A and b, as we can write A as A = [ħ2/(2meff)]·b−2 now. So we’ve got like an y = 1/x2 relation here. Where the hell does that come from?
2. We were talking some real stuff here: a crystal lattice with atoms that, in reality, do have some spacing, so that corresponds to some real value for b. So that spacing gives some actual physical significance to those xvalues.
Well… What can I say? I think you should re-read that quote of Feynman when I started this post. We’re going to get Schrödinger’s equation – i.e. the ultimate prize for all of the hard work that we’ve been doing so far – but… Yes. It’s really very heuristic, indeed! 🙂 But let’s get on with it now! We can re-write our Hamiltonian equation as:
= (E0−2A)C(xn) + A[2C(xn) − C(xn+b) − C(xn−b) = A[2C(xn) − C(xn+b) − C(xn−b)]
Now, I know your brain is about to melt down but, fiddling with this equation as we’re doing right now, Schrödinger recognized a formula for the second-order derivative of a function. I’ll just jot it down, and you can google it so as to double-check where it comes from:
second derivative
Just substitute f(x) for C(xn) in the second part of our equation above, and you’ll see we can effectively write that 2C(xn) − C(xn+b) − C(xn−b) factor as:
formula 1
We’re done. We just iħ·(∂C(xn)/∂t) on the left-hand side now and multiply the expression above with A, to get what we wanted to get, and that’s – YES! – Schrödinger’s equation:
Schrodinger 2
Whatever your objections to this ‘derivation’, it is the correct equation. For a particle in free space, we just write m instead of meff, but it’s exactly the same. I’ll now give you Feynman’s full quote, which is quite enlightening:
“We do not intend to have you think we have derived the Schrödinger equation but only wish to show you one way of thinking about it. When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature. The purpose of our discussion is then simply to show you that the correct fundamental quantum mechanical equation [i.e. Schrödinger’s equation] has the same form you get for the limiting case of an electron moving along a line of atoms. We can think of it as describing the diffusion of a probability amplitude from one point to the next along the line. That is, if an electron has a certain amplitude to be at one point, it will, a little time later, have some amplitude to be at neighboring points. In fact, the equation looks something like the diffusion equations which we have used in Volume I. But there is one main difference: the imaginary coefficient in front of the time derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Schrödinger’s equation are complex waves.”
So… That says it all, I guess. Isn’t it great to be where we are? We’ve really climbed a mountain here. And I think the view is gorgeous. 🙂
Oh—just in case you’d think I did not give you Schrödinger’s equation, let me write it in the form you’ll usually see it:
schrodinger 3
Done! 🙂
Quantum math in solid-state physics
electron moving
hamiltonian equations
hamiltonian equations - 2
Ea= E0an − Aan+1 − Aan−1
a(xn) = eikxn
The effective mass of an electron
Fourier_series_and_transform beats
group velocity
solution velocity
formula for m eff
Well… There you go. 🙂
Systems with 2 spin-1/2 particles (II)
In our previous post, we noted the Hamiltonian for a simple system of two spin-1/2 particles—a proton and an electron (i.e. a hydrogen atom, in other words):
After noting that this Hamiltonian is “the only thing that it can be, by the symmetry of space, i.e. so long as there is no external field,” Feynman also notes the constant term (A) depends on the level we choose to measure energies from, so one might just as well take E= 0, in which case the formula reduces to H = Aσe·σp. Feynman analyzes this term as follows:
If there are two magnets near each other with magnetic moments μe and μp, the mutual energy will depend on μe·μp = |μe||μp|cosα = μeμpcosα — among other things. Now, the classical thing that we call μe or μp appears in quantum mechanics as μeσand μpσrespectively (where μis the magnetic moment of the proton, which is about 1000 times smaller than μe, and has the opposite sign). So the H = Aσe·σp equation says that the interaction energy is like the interaction between two magnets—only not quite, because the interaction of the two magnets depends on the radial distance between them. But the equation could be—and, in fact, is—some kind of an average interaction. The electron is moving all around inside the atom, and our Hamiltonian gives only the average interaction energy. All it says is that for a prescribed arrangement in space for the electron and proton there is an energy proportional to the cosine of the angle between the two magnetic moments, speaking classically. Such a classical qualitative picture may help you to understand where the H = Aσe·σequation comes from.
That’s loud and clear, I guess. The next step is to introduce an external field. The formula for the Hamiltonian (we don’t distinguish between the matrix and the operator here) then becomes:
H = Aσe·σp − μeσe·B − μpσp·B
The first term is the term we already had. The second term is the energy the electron would have in the magnetic field if it were there alone. Likewise, the third term is the energy the proton would have in the magnetic field if it were there alone. When reading this, you should remember the following convention: classically, we write the energy U as U = −μ·B, because the energy is lowest when the moment is along the field. Hence, for positive particles, the magnetic moment is parallel to the spin, while for negative particles it’s opposite. In other words, μp is a positive number, while μe is negative. Feynman sums it all up as follows:
Classically, the energy of the electron and the proton together, would be the sum of the two, and that works also quantum mechanically. In a magnetic field, the energy of interaction due to the magnetic field is just the sum of the energy of interaction of the electron with the external field, and of the proton with the field—both expressed in terms of the sigma operators. In quantum mechanics these terms are not really the energies, but thinking of the classical formulas for the energy is a way of remembering the rules for writing down the Hamiltonian.
That’s also loud and clear. So now we need to solve those Hamiltonian equations once again. Feynman does so first assuming B is constant and in the z-direction. I’ll refer you to him for the nitty-gritty. The important thing is the results here:
He visualizes these – as a function of μB/A – as follows:
The illustration shows how the four energy levels have a different B-dependence:
• EI, EII, EIII start at (0, 1) but EI increases linearly with B—with slope μ, to be precise (cf. the EI = A + μB expression);
• In contrast, EII decreases linearly with B—again, with slope μ (cf. the EII = A − μB expression);
• We then have the EIII and EIV curves, which start out horizontally, to then curve and approach straight lines for large B, with slopes equal to μ’.
Oh—I realize I forget to define μ and μ’. Let me do that now: μ = −(μep) and μ’ = −(μe−μp). And remember what we said above: μis about 1000 times smaller than μe, and has opposite sign. OK. The point is: the magnetic field shifts the energy levels of our hydrogen atom. This is referred to as the Zeeman effect. Feynman describes it as follows:
The curves show the Zeeman splitting of the ground state of hydrogen. When there is no magnetic field, we get just one spectral line from the hyperfine structure of hydrogen. The transitions between state IV and any one of the others occurs with the absorption or emission of a photon whose (angular) frequency is 1/ħ times the energy difference 4A. [See my previous post for the calculation.] However, when the atom is in a magnetic field B, there are many more lines, and there can be transitions between any two of the four states. So if we have atoms in all four states, energy can be absorbed—or emitted—in any one of the six transitions shown by the vertical arrows in the illustration above.
The last question is: what makes the transitions go? Let me also quote Feynman’s answer to that:
The transitions will occur if you apply a small disturbing magnetic field that varies with time (in addition to the steady strong field B). It’s just as we saw for a varying electric field on the ammonia molecule. Only here, it is the magnetic field which couples with the magnetic moments and does the trick. But the theory follows through in the same way that we worked it out for the ammonia. The theory is the simplest if you take a perturbing magnetic field that rotates in the xy-plane—although any horizontal oscillating field will do. When you put in this perturbing field as an additional term in the Hamiltonian, you get solutions in which the amplitudes vary with time—as we found for the ammonia molecule. So you can calculate easily and accurately the probability of a transition from one state to another. And you find that it all agrees with experiment.
Alright! All loud and clear. 🙂
The magnetic quantum number
At very low magnetic fields, we still have the Zeeman splitting, but we can now approximate it as follows:
magnetic quantum number
This simplified representation of things explains an older concept you may still see mentioned: the magnetic quantum number, which is usually denoted by m. Feynman’s explanation of it is quite straightforward, and so I’ll just copy it as is:
As he notes: the concept of the magnetic quantum number has nothing to do with new physics. It’s all just a matter of notation. 🙂
Well… This concludes our short study of four-state systems. On to the next! 🙂
Systems with 2 spin-1/2 particles (I)
I agree: this is probably the most boring title of a post ever. However, it should be interesting, as we’re going to apply what we’ve learned so far – i.e. the quantum-mechanical model of two-state systems – to a much more complicated problem—the solution of which can then be generalized to describe even more complicated situations.
Two spin-1/2 particles? Let’s recall the most obvious example. In the ground state of a hydrogen atom (H), we have one electron that’s bound to one proton. The electron occupies the lowest energy state in its ground state, which – as Feynman shows in one of his first quantum-mechanical calculations – is equal to −13.6 eV. More or less, that is. 🙂 You’ll remember the reason for the minus sign: the electron has more energy when it’s unbound, which it releases as radiation when it joins an ionized hydrogen atom or, to put it simply, when a proton and an electron come together. In-between being bound and unbound, there are other discrete energy states – illustrated below – and we’ll learn how to describe the patterns of motion of the electron in each of those states soon enough.
Not in this post, however. 😦 In this post, we want to focus on the ground state only. Why? Just because. That’s today’s topic. 🙂 The proton and the electron can be in either of two spin states. As a result, the so-called ground state is not really a single definite-energy state. The spin states cause the so-called hyperfine structure in the energy levels: it splits them into several nearly equal energy levels, so that’s what referred to as hyperfine splitting.
[…] OK. Let’s go for it. As Feynman points out, the whole model is reduced to a set of four base states:
1. State 1: |++〉 = |1〉 (the electron and proton are both ‘up’)
2. State 2: |+−〉 = |2〉 (the electron is ‘up’ and the proton is ‘down’)
3. State 3: |−+〉 = |3〉 (the electron is ‘down’ and the proton is ‘up’)
4. State 4: |−−〉 = |4〉 (the electron and proton are both ‘down’)
The simplification is huge. As you know, the spin of electrically charged elementary particles is related to their motion in space, but so we don’t care about exact spatial relationships here: the direction of spin can be in any direction, but all that matters here is the relative orientation, and so all is simplified to some direction as defined by the proton and the electron itself. Full stop.
You know that the whole problem is to find the Hamiltonian coefficients, i.e. the energy matrix. Let me give them to you straight away. The energy levels involved are the following:
• E= EII = EIII = A ≈ 9.23×10−6 eV
• EIV = −3A ≈ 27.7×10−6 eV
So the difference in energy levels is measured in ten-millionths of an electron-volt and, hence, the hyperfine splitting is really hyper-fine. The question is: how do we get these values? So that is what this post is about. Let’s start by reminding ourselves of what we learned so far.
The Hamiltonian operator
We know that, in quantum mechanics, we describe any state in terms of the base states. In this particular case, we’d do so as follows:
|ψ〉 = |1〉C1 + |2〉C2 + |3〉C3 +|4〉C4 with Ci = 〈i|ψ〉
We refer to |ψ〉 as the spin state of the system, and so it’s determined by those four Ci amplitudes. Now, we know that those Ci amplitudes are functions of time, and they are, in turn, determined by the Hamiltonian matrix. To be precise, we find them by solving a set of linear differential equations that we referred to as Hamiltonian equations. To be precise, we’d describe the behavior of |ψ〉 in time by the following equation:
hamiltonian operator
In case you forgot, the expression above is a short-hand for the following expression:
hamiltonian operator 2The index would range over all base states and, therefore, this expression gives us everything we want: it really does describe the behavior, in time, of an N-state system. You’ll also remember that, when we’d use the Hamiltonian matrix in the way it’s used above (i.e. as an operator on a state), we’d put a little hat over it, so we defined the Hamiltonian operator as:
So far, so good—but this does not solve our problem: how do we find the Hamiltonian for this four-state system? What is it?
Well… There’s no one-size-fits-all answer to that: the analysis of two different two-state systems, like an ammonia molecule, or one spin-1/2 particle in a magnetic field, was different. Having said that, we did find we could generalize some of the solutions we’d find. For example, we’d write the Hamiltonian for a spin-1/2 particle, with a magnetic moment that’s assumed to be equal to μ, in a magnetic field B = (Bx, By, Bz) as:
sigma matrices
In this equation, we’ve got a set of 4 two-by-two matrices (three so-called sigma matrices (σx, σy, σz), and then the unit matrix δij = 1) which we referred to as the Pauli spin matrices, and which we wrote as:
You’ll remember that expression – which we further abbreviated, even more elegantly, to H = −μσ·B – covered all two-state systems involving a magnetic moment in a magnetic field. In fact, you’ll remember we could actually easily adapt the model to cover two-state systems in electric fields as well.
In short, these sigma matrices made our life very easy—as they covered a whole range of two-state models. So… Well… To make a long story short, what we want to do here is find some similar sigma matrices for four-state problems. So… Well… Let’s do that.
First, you should remind yourself of the fact that we could also use these sigma matrices as little operators themselves. To be specific, we’d let them ‘operate’ on the base states, and we’d find they’d do the following:
You need to read this carefully. What it says that the σz matrix, as an operator, acting on the ‘up’ base state, yields the same base state (i.e. ‘up’), and that the same operator, acting on the ‘down’ state, gives us the same but with a minus sign in front. Likewise, the σy matrix operating on the ‘up’ and ‘down’ states respectively, will give us i·|down〉 and −i·|up〉 respectively.
The trick to solve our problem here (i.e. our four-state system) is to apply those sigma matrices to the electron and the proton separately. Feynman introduces a new notation here by distinguishing the electron and proton sigma operators: the electron sigma operators (σxe, σye, and σze) operate on the electron spin only, while – you guessed it – the proton sigma operator ((σxp, σyp, and σzp) acts on the proton spin only. Applying it to the four states we’re looking at (i.e. |++〉, |+−〉, |−+〉 and |−−〉), we get the following bifurcation for our σx operator:
1. σxe|++〉 = |−+〉
2. σxe|+−〉 = |−−〉
3. σxe|−+〉 = |++〉
4. σxe|−−〉 = |+−〉
5. σxp|++〉 = |+−〉
6. σxp|+−〉 = |++〉
7. σxp|−+〉 = |−−〉
8. σxp|−−〉 = |−+〉
You get the idea. We had three operators acting on two states, i.e. 6 possibilities. Now we combine these three operators with two different particles, so we have six operators now, and we let them act on four possible system states, so we have 24 possibilities now. Now, we can, of course, let these operators act one after another. Check the following for example:
σxeσzp|+−〉 = σxezp|+−〉] = –σxe|+−〉 = –|–−〉
[I now realize that I should have used the ↑ and ↓ symbols for the ‘up’ and ‘down’ states, as the minus sign is used to denote two very different things here, but… Well… So be it.]
Note that we only have nine possible σxeσzp-like combinations, because σxeσz= σzpσxe, and then we have the 2×3 = six σe and σp operators themselves, so that makes for 15 new operators. [Note that the commutativity of these operators (σxeσz= σzpσxe) is not some general property of quantum-mechanical operators.] If we include the unit operator (δij = 1) – i.e. an operator that leaves all unchanged – we’ve got 16 in total. Now, we mentioned that we could write the Hamiltonian for a two-state system – i.e. a two-by-two matrix – as a linear combination of the four Pauli spin matrices. Likewise, one can demonstrate that the Hamiltonian for a four-state system can always be written as some linear combination of those sixteen ‘double-spin’ matrices. To be specific, we can write it as:
We should note a few things here. First, the E0 constant is, of course, to be multiplied by the unit matrix, so we should actually write E0δij instead of E0, but… Well… Quantum physicists always want to confuse you. 🙂 Second, the σeσis like the σ·notation: we can look at the σxe, σye, σze and σxp, σyp, σzp matrices as being the three components of two new (matrix) vectors, which we write as σand σrespectively. Thirdly, and most importantly, you’ll want proof of that equation above. Well… I am sorry but I am going to refer you to Feynman here: he shows that the expression above “is the only thing that the Hamiltonian can be.” The proof is based on the fundamental symmetry of space. He also adds that space is symmetrical only so long as there is no external field. 🙂
Final question: what’s A? Well… Feynman is quite honest here as he says the following: “A can be calculated accurately once you understand the complete quantum theory of the hydrogen atom—which we so far do not. It has, in fact, been calculated to an accuracy of about 30 parts in one million. So, unlike the flip-flop constant A of the ammonia molecule, which couldn’t be calculated at all well by a theory, our constant A for the hydrogen can be calculated from a more detailed theory. But never mind, we will for our present purposes think of the A as a number which could be determined by experiment, and analyze the physics of the situation.”
So… Well… So far so good. We’ve got the Hamiltonian. That’s all we wanted, actually. But, now that we have come so far, let’s write it all out now.
Solving the equations
If that expression above is the Hamiltonian – and we assume it is, of course! – then our system of Hamiltonian equations can be written as:
[Note that we’ve switched to Newton’s ‘over-dot’ notation to denote time derivatives here.] Now, I could walk you through Feynman’s exposé but I guess you’ll trust the result. The equation above is equivalent to the following set of four equations:
We know that, because the Hamiltonian looks like this:
How do we know that? Well… Sorry: just check Feynman. 🙂 He just writes it all out. Now, we want to find those Ci functions. [When studying physics, the most important thing is to remember what it is that you’re trying to do. 🙂 ] Now, from my previous post (i.e. my post on the general solution for N-state systems), you’ll remember that those Ci functions should have the following functional form:
Ci(t) = ai·ei·(E/ħ)·t
If we substituting Ci(t) for that functional form in our set of Hamiltonian equations, we can cancel the exponentials so we get the following delightfully simple set of new equations:
The trivial solution, of course, is that all of the ai coefficients are zero, but – as mentioned in my previous post – we’re looking for non-trivial solutions here. Well… From what you see above, it’s easy to appreciate that one non-trivial but simple solution is:
a1 = 1 and a2 = a3 = a4 = 0
So we’ve got one set of ai coefficients here, and we’ll associate it with the first eigenvalue, or energy level, really—which we’ll denote as EI. [I am just being consistent here with what I wrote in my previous post, which explained how general solutions to N-state systems look like.] So we find the following:
E= A
[Another thing you learn when studying physics is that the most amazing things are often summarized in super-terse equations, like this one here. 🙂 ]
But – Hey! Look at the symmetry between the first and last equation!
We immediately get another simple – but non-trivial! – solution:
a4 = 1 and a1 = a2 = a3 = 0
We’ll associate the second energy level with that, so we write:
We’ve got two left. I’ll leave that to Feynman to solve:
feDone! Four energy levels En (n = I, II, III, IV), and four associated energy state vectors – |n〉 – that describe their configuration (and which, as Feynman puts it, have the time dependence “factored out”). Perfect!
Now, we mentioned the experimental values:
• EIV = −3A ≈ 27.7×10−6 eV
How can scientists measure these values? The theoretical analysis gives us the A and −3A values, but what about the empirical measurements? Well… We should find those values as the hydrogen atoms in state I, II or III should get rid of the energy by emitting some radiation. Now, the frequency of that radiation will give us the information we need, as illustrated below. The difference between E= EII = EIII = A and EIV = −3A (i.e. 4A) should correspond to the (angular) frequency of the radiation that’s being emitted or absorbed as atoms go from one energy state to the other. Now, hydrogen atoms do absorb and emit microwave radiation with a frequency that’s equal to 1,420,405,751.8 Hz. More or less, that is. 🙂 The standard error in the measurement is about two parts in 100 billion—and I am quoting some measurement done in the early 1960s here!]
Bingo! If = ω/2π = (4A/ħ)/2π = 1,420,405,751.8 Hz, then A = f·2π·ħ/4 ≈ 9.23×10−6 eV.
So… Well… We’re done! I’ll see you tomorrow. 🙂 Tomorrow, we’re going to look at what happens when space is not symmetric, i.e. when we would have some external field! C u ! Cheers !
N-state systems
On the 10th of December, last year, I wrote that my next post would generalize the results we got for two-state systems. That didn’t happen: I didn’t write the ‘next post’—not till now, that is. No. Instead, I started digging—as you can see from all the posts in-between this one and the 10 December piece. And you may also want to take a look at my new Essentials page. 🙂 In any case, it is now time to get back to Feynman’s Lectures on quantum mechanics. Remember where we are: halfway, really. The first half was all about stuff that doesn’t move in space. The second half, i.e. all that we’re going to study now, is about… Well… You guessed it. 🙂 That’s going to be about stuff that does move in space. To see how that works, we first need to generalize the two-state model to an N-state model. Let’s do it.
You’ll remember that, in quantum mechanics, we describe stuff by saying it’s in some state which, as long as we don’t measure in what state exactly, is written as some linear combination of a set of base states. [And please do think about what I highlight here: some state, measureexactly. It all matters. Think about it!] The coefficients in that linear combination are complex-valued functions, which we referred to as wavefunctions, or (probability) amplitudes. To make a long story short, we wrote:
These Ci coefficients are a shorthand for 〈 i | ψ(t) 〉 amplitudes. As such, they give us the amplitude of the system to be in state i as a function of time. Their dynamics (i.e. the way they evolve in time) are governed by the Hamiltonian equations, i.e.:
The Hij coefficients in this set of equations are organized in the Hamiltonian matrix, which Feynman refers to as the energy matrix, because these coefficients do represent energies indeed. So we applied all of this to two-state systems and, hence, things should not be too hard now, because it’s all the same, except that we have N base states now, instead of just two.
So we have a N×N matrix whose diagonal elements Hij are real numbers. The non-diagonal elements may be complex numbers but, if they are, the following rule applies: Hij* = Hji. [In case you wonder: that’s got to do with the fact that we can write any final 〈χ| or 〈φ| state as the conjugate transpose of the initial |χ〉 or |φ〉 state, so we can write: 〈χ| = |χ〉*, or 〈φ| = |φ〉*.]
As usual, the trick is to find those N Ci(t) functions: we do so by solving that set of N equations, assuming we know those Hamiltonian coefficients. [As you may suspect, the real challenge is to determine the Hamiltonian, which we assume to be given here. But… Well… You first need to learn how to model stuff. Once you get your degree, you’ll be paid to actually solve problems using those models. 🙂 ] We know the complex exponential is a functional form that usually does that trick. Hence, generalizing the results from our analysis of two-state systems once more, the following general solution is suggested:
Note that we introduce only one E variable here, but N ai coefficients, which may be real- or complex-valued. Indeed, my examples – see my previous posts – often involved real coefficients, but that’s not necessarily the case. Think of the C2(t) = i·e(i/ħ)·E0·t·sin[(A/ħ)·t] function describing one of the two base state amplitudes for the ammonia molecule—for example. 🙂
Now, that proposed general solution allows us to calculate the derivatives in our Hamiltonian equations (i.e. the d[Ci(t)]/dt functions) as follows:
d[Ci(t)]/dt = −i·(E/ħ)·ai·ei·(E/ħ)·t
You can now double-check that the set of equations reduces to the following:
Please do write it out: because we have one E only, the ei·(E/ħ)·t factor is common to all terms, and so we can cancel it. The other stuff is plain arithmetic: i·i = i2 = 1, and the ħ constants cancel out too. So there we are: we’ve got a very simple set of N equations here, with N unknowns (i.e. these a1, a2,…, aN coefficients, to be specific). We can re-write this system as:
The δij here is the Kronecker delta, of course (it’s one for i = j and zero for j), and we are now looking at a homogeneous system of equations here, i.e. a set of linear equations in which all the constant terms are zero. You should remember it from your high school math course. To be specific, you’d write it as Ax = 0, with A the coefficient matrix. The trivial solution is the zero solution, of course: all a1, a2,…, aN coefficients are zero. But we don’t want the trivial solution. Now, as Feynman points out – tongue-in-cheek, really – we actually have to be lucky to have a non-trivial solution. Indeed, you may or may not remember that the zero solution was actually the only solution if the determinant of the coefficient matrix was not equal to zero. So we only had a non-trivial solution if the determinant of A was equal to zero, i.e. if Det[A] = 0. So A has to be some so-called singular matrix. You’ll also remember that, in that case, we got an infinite number of solutions, to which we could apply the so-called superposition principle: if x and y are two solutions to the homogeneous set of equations Ax = 0, then any linear combination of x and y is also a solution. I wrote an addendum to this post (just scroll down and you’ll find it), which explains what systems of linear equations are all about, so I’ll refer you to that in case you’d need more detail here. I need to continue our story here. The bottom line is: the [Hij–δijE] matrix needs to be singular for the system to have meaningful solutions, so we will only have a non-trivial solution for those values of E for which
Det[Hij–δijE] = 0
Let’s spell it out. The condition above is the same as writing:
So far, so good. What’s next? Well… The formula for the determinant is the following:
det physicists
That looks like a monster, and it is, but, in essence, what we’ve got here is an expression for the determinant in terms of the permutations of the matrix elements. This is not a math course so I’ll just refer you Wikipedia for a detailed explanation of this formula for the determinant. The bottom line is: if we write it all out, then Det[Hij–δijE] is just an Nth order polynomial in E. In other words: it’s just a sum of products with powers of E up to EN, and so our Det[Hij–δijE] = 0 condition amounts to equating it with zero.
In general, we’ll have N roots, but – sorry you need to remember so much from your high school math classes here – some of them may be multiple roots (i.e. two or more roots may be equal). We’ll call those roots—you guessed it:
EI, EII,…, En,…, EN
Note I am following Feynman’s exposé, and so he uses n, rather than k, to denote the nth Roman numeral (as opposed to Latin numerals). Now, I know your brain is near the melting point… But… Well… We’re not done yet. Just hang on. For each of these values E = EI, EII,…, En,…, EN, we have an associated set of solutions ai. As Feynman puts it: you get a set which belongs to En. In order to not forget that, for each En, we’re talking a set of N coefficients ai (= 1, 2,…, N), we denote that set not by ai(n) but by ai(n). So that’s why we use boldface for our index n: it’s special—and not only because it denotes a Roman numeral! It’s just one of Feynman’s many meaningful conventions.
Now remember that Ci(t) = ai·ei·(E/ħ)·t formula. For each set of ai(n) coefficients, we’ll have a set of Ci(n) functions which, naturally, we can write as:
Ci(n) = ai(nei·(En/ħ)·t
So far, so good. We have N ai(n) coefficients and N Ci(n) functions. That’s easy enough to understand. Now we’ll define also define a set of N new vectors, which we’ll write as |n〉, and which we’ll refer to as the state vectors that describe the configuration of the definite energy states En (n = I, II,… N). [Just breathe right now: I’ll (try to) explain this in a moment.] Moreover, we’ll write our set of coefficients ai(n) as 〈i|n〉. Again, the boldface n reminds us we’re talking a set of N complex numbers here. So we re-write that set of N Ci(n) functions as follows:
Ci(n) = 〈i|n〉·ei·(En/ħ)·t
We can expand this as follows:
Ci(n) = 〈 i | ψn(t) 〉 = 〈 i | 〉·ei·(En/ħ)·t
which, of course, implies that:
| ψn(t) 〉 = |n〉·ei·(En/ħ)·t
So now you may understand Feynman’s description of those |n〉 vectors somewhat better. As he puts it:
“The |n〉 vectors – of which there are N – are the state vectors that describe the configuration of the definite energy states En (n = I, II,… N), but have the time dependence factored out.”
Hmm… I know. This stuff is hard to swallow, but we’re not done yet: if your brain hasn’t melted yet, it may do so now. You’ll remember we talked about eigenvalues and eigenvectors in our post on the math behind the quantum-mechanical model of our ammonia molecule. Well… We can generalize the results we got there:
So… Well… That’s it! We’re done! This is all there is to it. I know it’s a lot but… Well… We’ve got a general description of N-state systems here, and so that’s great!
Let me make some concluding remarks though.
First, note the following property: if we let the Hamiltonian matrix act on one of those state vectors |n〉, the result is just En times the same state. We write:
We’re writing nothing new here really: it’s just a consequence of the definition of eigenstates and eigenvalues. The more interesting thing is the following. When describing our two-state systems, we saw we could use the states that we associated with the Eand EII as a new base set. The same is true for N-state systems: the state vectors |n〉 can also be used as a base set. Of course, for that to be the case, all of the states must be orthogonal, meaning that for any two of them, say |n〉 and |m〉, the following equation must hold:
n|m〉 = 0
Feynman shows this will be true automatically if all the energies are different. If they’re not – i.e. if our polynomial in E would accidentally have two (or more) roots with the same energy – then things are more complicated. However, as Feynman points out, this problem can be solved by ‘cooking up’ two new states that do have the same energy but are also orthogonal. I’ll refer you to him for the detail, as well as for the proof of that 〈n|m〉 = 0 equation.
Finally, you should also note that – because of the homogeneity principle – it’s possible to multiply the N ai(n) coefficients by a suitable factor so that all the states are normalized, by which we mean:
n|n〉 = 1
Well… We’re done! For today, at least! 🙂
Addendum on Systems of Linear Equations
It’s probably good to briefly remind you of your high school math class on systems of linear equations. First note the difference between homogeneous and non-homogeneous equations. Non-homogeneous equations have a non-zero constant term. The following three equations are an example of a non-homogeneous set of equations:
• 3x + 2y − z = 1
• 2x − 2y + 4z = −2
• −x + y/2 − z = 0
We have a point solution here: (x, y, z) = (1, −2, −2). The geometry of the situation is something like this:
One of the equations may be a linear combination of the two others. In that case, that equation can be removed without affecting the solution set. For the three-dimensional case, we get a line solution, as illustrated below. Intersecting_Planes_2
Homogeneous and non-homogeneous sets of linear equations are closely related. If we write a homogeneous set as Ax = 0, then a non-homogeneous set of equations can be written as Ax = b. They are related. More in particular, the solution set for Ax = b is going to be a translation of the solution set for Ax = 0. We can write that more formally as follows:
If p is any specific solution to the linear system Ax = b, then the entire solution set can be described as {p + v|v is any solution to Ax = 0}
The solution set for a homogeneous system is a linear subspace. In the example above, which had three variables and, hence, for which the vector space was three-dimensional, there were three possibilities: a point, line or plane solution. All are (linear) subspaces—although you’d want to drop the term ‘linear’ for the point solution, of course. 🙂 Formally, a subspace is defined as follows: if V is a vector space, then W is a subspace if and only if:
1. The zero vector (i.e. 0) is in W.
2. If x is an element of W, then any scalar multiple ax will be an element of W too (this is often referred to as the property of homogeneity).
3. If x and y are elements of W, then the sum of x and y (i.e. x + y) will be an element of W too (this is referred to as the property of additivity).
As you can see, the superposition principle actually combines the properties of homogeneity and additivity: if x and y are solutions, then any linear combination of them will be a solution too.
The solution set for a non-homogeneous system of equations is referred to as a flat. It’s a subset too, so it’s like a subspace, except that it need not pass through the origin. Again, the flats in two-dimensional space are points and lines, while in three-dimensional space we have points, lines and planes. In general, we’ll have flats, and subspaces, of every dimension from 0 to n−1 in n-dimensional space.
OK. That’s clear enough, but what is all that talk about eigenstates and eigenvalues about? Mathematically, we define eigenvectors, aka as characteristic vectors, as follows:
• The non-zero vector v is an eigenvector of a square matrix A if Av is a scalar multiple of v, i.e. Av = λv.
• The associated scalar λ is known as the eigenvalue (or characteristic value) associated with the eigenvector v.
Now, in physics, we talk states, rather than vectors—although our states are vectors, of course. So we’ll call them eigenstates, rather than eigenvectors. But the principle is the same, really. Now, I won’t copy what you can find elsewhere—especially not in an addendum to a post, like this one. So let me just refer you elswhere. Paul’s Online Math Notes, for example, are quite good on this—especially in the context of solving a set of differential equations, which is what we are doing here. And you can also find a more general treatment in the Wikipedia article on eigenvalues and eigenstates which, while being general, highlights their particular use in quantum math. |
c0d52771fc68bc4b | Physics » Condensed Matter Physics » Free Electron Model of Metals
Free Electron Model of Metals
Free Electron Model of Metals
Metals, such as copper and aluminum, are held together by bonds that are very different from those of molecules. Rather than sharing and exchanging electrons, a metal is essentially held together by a system of free electrons that wander throughout the solid. The simplest model of a metal is the free electron model. This model views electrons as a gas. We first consider the simple one-dimensional case in which electrons move freely along a line, such as through a very thin metal rod. The potential function U(x) for this case is a one-dimensional infinite square well where the walls of the well correspond to the edges of the rod. This model ignores the interactions between the electrons but respects the exclusion principle. For the special case of \(T=0\;\text{K},\)N electrons fill up the energy levels, from lowest to highest, two at a time (spin up and spin down), until the highest energy level is filled. The highest energy filled is called the Fermi energy.
The one-dimensional free electron model can be improved by considering the three-dimensional case: electrons moving freely in a three-dimensional metal block. This system is modeled by a three-dimensional infinite square well. Determining the allowed energy states requires us to solve the time-independent Schrödinger equation
\(-\cfrac{{h}^{2}}{2{m}_{\text{e}}}\left(\cfrac{{\partial }^{2}}{\partial {x}^{2}}+\cfrac{{\partial }^{2}}{\partial {y}^{2}}+\cfrac{{\partial }^{2}}{\partial {z}^{2}}\right)\psi (x,y,z)=E\;\psi (x,y,z),\)
where we assume that the potential energy inside the box is zero and infinity otherwise. The allowed wave functions describing the electron’s quantum states can be written as
\(\psi (x,y,z)=(\sqrt{\cfrac{2}{{L}_{x}}}\;\text{sin}\;\cfrac{{n}_{x}\pi x}{{L}_{x}})(\sqrt{\cfrac{2}{{L}_{y}}}\;\text{sin}\;\cfrac{{n}_{y}\pi y}{{L}_{y}})(\sqrt{\cfrac{2}{{L}_{z}}}\;\text{sin}\;\cfrac{{n}_{z}\pi z}{{L}_{z}}),\)
where \({n}_{x},{n}_{y},\) and \({n}_{z}\) are positive integers representing quantum numbers corresponding to the motion in the x-, y-, and z-directions, respectively, and \({L}_{x},{L}_{y},\;\text{and}\;{L}_{z}\) are the dimensions of the box in those directions. This equation is simply the product of three one-dimensional wave functions. The allowed energies of an electron in a cube \((L={L}_{x}={L}_{y}={L}_{z})\) are
Associated with each set of quantum numbers \(({n}_{x},{n}_{y},{n}_{z})\) are two quantum states, spin up and spin down. In a real material, the number of filled states is enormous. For example, in a cubic centimeter of metal, this number is on the order of \({10}^{22}.\) Counting how many particles are in which state is difficult work, which often requires the help of a powerful computer. The effort is worthwhile, however, because this information is often an effective way to check the model.
Example: Energy of a Metal Cube
Consider a solid metal cube of edge length 2.0 cm. (a) What is the lowest energy level for an electron within the metal? (b) What is the spacing between this level and the next energy level?
An electron in a metal can be modeled as a wave. The lowest energy corresponds to the largest wavelength and smallest quantum number: \({n}_{x},{n}_{y},{n}_{z}=(1,1,1).\) This equation supplies this “ground state” energy value. Since the energy of the electron increases with the quantum number, the next highest level involves the smallest increase in the quantum numbers, or \(({n}_{x},{n}_{y},{n}_{z})=(2,1,1),(1,2,1),\) or (1, 1, 2).
The lowest energy level corresponds to the quantum numbers \({n}_{x}={n}_{y}={n}_{z}=1.\) From this equation, the energy of this level is
\(\begin{array}{cc}E(1,1,1)\hfill & =\cfrac{{\pi }^{2}{h}^{2}}{2{m}_{e}{L}^{2}}\;({1}^{2}+{1}^{2}+{1}^{2})\hfill \\ & =\cfrac{3{\pi }^{2}\;{(1.05\;×\;10-34\;\text{J}·\text{s})}^{2}}{2\;(9.11\;×\;{10}^{-31}\;\text{kg})\;{(2.00\;×\;{10}^{-2}\;\text{m})}^{2}}\hfill \\ & =4.48\;×\;{10}^{-34}\;\text{J}=2.80\;×\;{10}^{-15}\;\text{eV}\text{.}\hfill \end{array}\)
The next-higher energy level is reached by increasing any one of the three quantum numbers by 1. Hence, there are actually three quantum states with the same energy. Suppose we increase \({n}_{x}\) by 1. Then the energy becomes
\(\begin{array}{cc}E(2,1,1)\hfill & =\cfrac{{\pi }^{2}{h}^{2}}{2{m}_{\text{e}}{L}^{2}}({2}^{2}+{1}^{2}+{1}^{2})\hfill \\ & =\cfrac{6{\pi }^{2}{(1.05\;×\;10-34\;\text{J}·\text{s})}^{2}}{2(9.11\;×\;{10}^{-31}\;\text{kg}){(2.00\;×\;{10}^{-2}\;\text{m})}^{2}}\hfill \\ & =8.96\;×\;{10}^{-34}\;\text{J}=5.60\;×\;{10}^{-15}\;\text{eV}\text{.}\hfill \end{array}\)
The energy spacing between the lowest energy state and the next-highest energy state is therefore
This is a very small energy difference. Compare this value to the average kinetic energy of a particle, \({k}_{\text{B}}T\), where \({k}_{\text{B}}\) is Boltzmann’s constant and T is the temperature. The product \({k}_{\text{B}}T\) is about 1000 times greater than the energy spacing.
Often, we are not interested in the total number of particles in all states, but rather the number of particles dN with energies in a narrow energy interval. This value can be expressed by
where n(E) is the electron number density, or the number of electrons per unit volume; g(E) is the density of states, or the number of allowed quantum states per unit energy; dE is the size of the energy interval; and F is the Fermi factor. The Fermi factor is the probability that the state will be filled. For example, if g(E)dE is 100 available states, but F is only \(5\%\), then the number of particles in this narrow energy interval is only five. Finding g(E) requires solving Schrödinger’s equation (in three dimensions) for the allowed energy levels. The calculation is involved even for a crude model, but the result is simple:
where V is the volume of the solid, \({m}_{e}\) is the mass of the electron, and E is the energy of the state. Notice that the density of states increases with the square root of the energy. More states are available at high energy than at low energy. This expression does not provide information of the density of the electrons in physical space, but rather the density of energy levels in “energy space.” For example, in our study of the atomic structure, we learned that the energy levels of a hydrogen atom are much more widely spaced for small energy values (near than ground state) than for larger values.
This equation tells us how many electron states are available in a three-dimensional metallic solid. However, it does not tell us how likely these states will be filled. Thus, we need to determine the Fermi factor, F. Consider the simple case of \(T=0\;\text{K}\). From classical physics, we expect that all the electrons \((\sim {10}^{22}\;\text{/}\;{\text{cm}}^{3})\) would simply go into the ground state to achieve the lowest possible energy. However, this violates Pauli’s exclusion principle, which states that no two electrons can be in the same quantum state. Hence, when we begin filling the states with electrons, the states with lowest energy become occupied first, then states with progressively higher energies. The last electron we put in has the highest energy. This energy is the Fermi energy \({E}_{\text{F}}\) of the free electron gas. A state with energy \(E<{E}_{\text{F}}\) is occupied by a single electron, and a state with energy \(E>{E}_{\text{F}}\) is unoccupied. To describe this in terms of a probability F(E) that a state of energy E is occupied, we write for \(T=0\;\text{K}\):
\(\begin{array}{ccc}F(E)=1\hfill & & (E<{E}_{\text{F}})\hfill \\ F(E)=0\hfill & & (E>{E}_{\text{F}}).\hfill \end{array}\)
The density of states, Fermi factor, and electron number density are plotted against energy in this figure.
Figure a is a graph of g in parentheses E versus E. The curve starts at zero and goes up and right. It is labeled g in parentheses E is proportional to E raised to half. Figure b is a graph of f in parentheses E versus E. There is a horizontal line at y value I and a vertical line at x value E subscript F. These, along with the axes form a rectangle in the first quadrant. Figure c is a graph of g in parentheses E, f in parentheses E versus E. The curves from figure a and b are superimposed here. The point on the curve with an x value of E subscript F has a y value of g in parentheses E subscript F.
(a) Density of states for a free electron gas; (b) probability that a state is occupied at \(T=0\;\text{K}\); (c) density of occupied states at \(T=0\;\text{K}\).
A few notes are in order. First, the electron number density (last row) distribution drops off sharply at the Fermi energy. According to the theory, this energy is given by
Fermi energies for selected materials are listed in the following table.
Conduction Electron Densities and Fermi Energies for Some Metals
Note also that only the graph in part (c) of the figure, which answers the question, “How many particles are found in the energy range?” is checked by experiment. The Fermi temperature or effective “temperature” of an electron at the Fermi energy is
Example: Fermi Energy of Silver
Metallic silver is an excellent conductor. It has \(5.86\;×\;{10}^{28}\) conduction electrons per cubic meter. (a) Calculate its Fermi energy. (b) Compare this energy to the thermal energy \({k}_{\text{B}}T\) of the electrons at a room temperature of 300 K.
1. From this equation, the Fermi energy is
\(\begin{array}{cc}{E}_{\text{F}}\hfill & =\cfrac{{h}^{2}}{2{m}_{e}}{(3{\pi }^{2}{n}_{e})}^{2\text{/}3}\hfill \\ & =\cfrac{{(1.05\;×\;{10}^{-34}\;\text{J}·\text{s})}^{2}}{2(9.11\;×\;{10}^{-31}\;\text{kg})}\;×\;{[(3{\pi }^{2}(5.86\;×\;{10}^{28}\;{\text{m}}^{-3})]}^{2\text{/}3}\hfill \\ & =8.79\;×\;{10}^{-19}\;\text{J}=5.49\;\text{eV}.\hfill \end{array}\)
This is a typical value of the Fermi energy for metals, as can be seen from this table.
2. We can associate a Fermi temperature \({T}_{\text{F}}\) with the Fermi energy by writing \({k}_{\text{B}}{T}_{\text{F}}={E}_{\text{F}}.\) We then find for the Fermi temperature
which is much higher than room temperature and also the typical melting point \((\sim {10}^{3}\;\text{K})\) of a metal. The ratio of the Fermi energy of silver to the room-temperature thermal energy is
\(\cfrac{{E}_{\text{F}}}{{k}_{\text{B}}T}=\cfrac{{T}_{\text{F}}}{T}\approx 210.\)
To visualize how the quantum states are filled, we might imagine pouring water slowly into a glass, such as that of this figure. The first drops of water (the electrons) occupy the bottom of the glass (the states with lowest energy). As the level rises, states of higher and higher energy are occupied. Furthermore, since the glass has a wide opening and a narrow stem, more water occupies the top of the glass than the bottom. This reflects the fact that the density of states g(E) is proportional to \({E}^{1\text{/}2}\), so there is a relatively large number of higher energy electrons in a free electron gas. Finally, the level to which the glass is filled corresponds to the Fermi energy.
Photograph of a martini glass half filled with water. The water is labeled electron gas and the water line is labeled Fermi energy E subscript F.
An analogy of how electrons fill energy states in a metal. As electrons fill energy states, lowest to highest, the number of available states increases. The highest energy state (corresponding to the water line) is the Fermi energy. (credit: modification of work by “Didriks”/Flickr)
Suppose that at \(T=0\;\text{K}\), the number of conduction electrons per unit volume in our sample is \({n}_{e}\). Since each field state has one electron, the number of filled states per unit volume is the same as the number of electrons per unit volume.
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92e02475a1c1a12a | NSU Scientists Publish Article on Nonlinear Fourier Transform
NSU graduates and researchers at the NSU Nonlinear Photonics Laboratory, Igor Chekhovskoy, Olga Shtyrina, Mikhail Fedoruk, Sergey Medvedev, and Sergey Turitsyn, published their article, “Nonlinear Fourier Transform for Analysis of Coherent Structures in Dissipative Systems”, in one of the most prestigious physics journals, the “Physical Review Letters”. The Journal’s impact factor is 8.839.
The work is devoted to a new application of the inverse scattering method (IST), also known as nonlinear Fourier transform (NFT). In the 1970’s NSU scientists Vladimir Zakharov and Alexei Shabat demonstrated that you can integrate one of the main models of nonlinear physics, the nonlinear Schrödinger equation (NSE), with the help of IST but you must solve the Zakharov – Shabat spectral problem (ZSh). The new method, analogous to the usual Fourier transform, allows us to simplify the analysis and reduce the complex nonlinear dynamics to a simple evolution in a specific base, the nonlinear signal spectrum. The use of NFT, in contrast to the usual Fourier transform, implies finding the continuous and discrete spectrum of the ZSh operator. Returning to the example of the NSE, the discrete spectrum here will describe the soliton part of the signal, and the continuous represents the dispersive waves signal.
The difference between the conventional Fourier transform and NFT can be understood using the example of waves at sea. The Fourier transform gives the dependence of the waves amplitude on their length, which is called the spectrum. The NFT makes it possible to determine not only the wave spectrum, but also the presence of “ships at sea”, i.e. solitons. Continuing the analogy, we can say that NFT allows us to determine the relative magnitude of waves. If the wave is small, then the “ships” (solitons) are clearly visible, and if the wave is strong, then the NFT can detect and determine their movement.
The application of NFT to integrable Hamiltonian equations, such as the NSE, is well researched and represents the classical field of mathematical physics. In this paper, the authors investigated the potential of its application to dissipative nonintegrable systems using the Ginzburg-Landau equation (UGL) as an example. Although NFT cannot be used to solve these systems, the authors showed that the evolution of an optical signal obeying an UGL can be described with good accuracy using a finite number of variables using NFT in cases where the discrete component of the spectrum of the ZSh operator for the corresponding NSE is dominant. This corresponds to cases where the ratio of the signal energy associated with the discrete spectrum to the total energy is close to unity.
Thus, it has been shown that NFT can reduce the number of effective degrees of freedom when coherent structures, such as solitons, dominate the dynamics of an optical signal, even when their evolution is unstable. The stationary solutions of the UGL, which are dissipative solitons, are analyzed and the parameters of these solutions are found when the approach to describing the dynamics based on NFT is applicable.
The figure on the left shows the evolution of the pulse, and on the right the evolution of the continuous and discrete spectrum. It is clear that during propagation, the initial optical pulse passes into asymptotic stability, which can be described with great accuracy by only three points in the discrete spectrum. The energy of the dispersion waves corresponding to the continuous spectrum is small compared to the discrete spectrum.
According to the scientists, this approach can provide new opportunities for the study of complex laser radiation, that consists of a mixture of coherent pulses and dispersive waves. |
eabf67f50ecb2f87 | Let us consider the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the vortex filament equation related to the Cauchy problem for the linear transport equation $$ \partial_t u + \operatorname{div}(\boldsymbol{b} \,u) =0, $$ where $\boldsymbol b$ is a suitably chosen vector field?
In other words, the vortex filament equation comes from the Euler equation under suitable assumptions and can be transformed in the Schrödinger equation with a certain transformation, but does it also have any transport-like structure of the kind displayed above?
Note 1. I've asked a more general question at Surveys/monographs on the vortex filament equation.
• 2
$\begingroup$ apologies for interrupting, but wouldn't it make more sense to read some of the pointers to the literature that have been given in the previous questions you asked on this very same equation, in particular in the answer to your "survey" question, and then come back later when you have a more specific question on something that remains unclear after studying the literature? Firing off a series of closely related questions is not likely to be a productive way to engage with Mathoverflow. $\endgroup$ – Carlo Beenakker May 15 at 20:19
• 1
$\begingroup$ @CarloBeenakker Thank you for the suggestion, I'll be more careful. I chose so single out three questions from the most general one on surveys/monographs just because they are three specific issues that have come up (as opposed to a more general query) that are both related and different enough to be of interest for different communities. The present question particularly is related to the other two (on Schroedinger and Euler equations) because I wonder: other than coming from Euler and being transformed into Schroedinger, does the vortex filament equation have any transport-like structure? $\endgroup$ – Kei May 15 at 21:07
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Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Claude Shannon
David Shiang
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
William Thomson (Kelvin)
Giulio Tononi
Peter Tse
Vlatko Vedral
Heinz von Foerster
John von Neumann
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Stephen Wolfram
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky
Free Will
Mental Causation
James Symposium
Quantum Physics, Thermodynamics, and Information
The core creative process in the universe involves quantum mechanics and thermodynamics.
To understand information creation, information physics provides new insights into the puzzling "problem of measurement" and the mysterious "collapse of the wave function" in quantum mechanics.
Information physics also probes deeply into the second law of thermodynamics to establish the irreversible increase of entropy on a quantum mechanical basis.
"Information physics" is not a new "interpretation" of quantum mechanics. It is not an attempt to alter the standard quantum mechanics, extending it to theories such as "hidden variables," for example. Information physics simply follows the quantum mechanical and thermodynamic implications of cosmic information structures, especially those that were created before the existence of human observers.
Information physics explains the origin of information structures in the universe.
Quantum Mechanics
In classical mechanics, the material universe is thought to be made up of tiny particles whose motions are completely determined by forces that act between the particles, forces such as gravitation, electrical attractions and repulsions, etc.
The equations that describe those motions, Newton's laws of motion, were for many centuries thought to be perfect and sufficient to predict the future of any mechanical system. They provided support for many philosophical ideas about determinism.
In classical electrodynamics, electromagnetic radiation (light, radio) was known to have wave properties such as interference. When the crest of one wave meets the trough of another, the two waves cancel one another.
In quantum mechanics, radiation is found to have some particle-like behavior. Energy comes in discrete physically localized packages. Max Planck in 1900 made the famous assumption that the energy was proportional to the frequency of radiation ν.
E = hν
For Planck, this assumption was just a heuristic mathematical device that allowed him to apply Ludwig Boltzmann's work on the statistical mechanics and kinetic theory of gases. Boltzmann had shown in the 1870's that the increase in entropy (the second law) could be explained if gases were made up of enormous numbers of particles. Where Boltzmann's statistics assumed that the gas particles are distinguishable from one another, Planck's counting did not.
Planck applied his modified form of Boltzmann's statistics of many particles to radiation and derived the distribution of radiation at different frequencies (or wavelengths) just as James Clerk Maxwell and Boltzmann had derived the distribution of velocities (or energies) of the gas particles. But Planck did not think he was describing light particles. It was Einstein who first realized this is what his mathematics was doing.
Note the mathematical similarity of Planck's radiation distribution law (light particles) and the Maxwell-Boltzmann velocity distribution (material particles). Both curves have a power law increase on one side to a maximum and an exponential decrease on the other side of the maximum. When energy is added to matter, it speeds up all the gas particles, but preserves their number. The molecular velocity curves for different temperatures cross one another because the total number of molecules is the same. With increasing temperature T, however, the number of photons increases at all wavelengths.
Planck did not actually believe that radiation came in discrete particles, at least until a dozen years later, and even then he had his doubts. In the meantime, Albert Einstein's 1905 paper on the photoelectric effect showed that light came in discrete particles, subsequently called "photons," analogous to electrons.
Planck was not happy about the idea of light particles, because his use of Boltmann's statistics implied that chance was real. Boltzmann himself had qualms about the reality of chance. Although Einstein also did not like the idea of chancy statistics, he did believe that energy came in packages of discrete "quanta." It was Einstein, not Planck, who quantized mechanics and electrodynamics. Nevertheless, it was for the introduction of the quantum of action h that Planck was awarded the Nobel prize in 1918.
Louis de Broglie argued that if photons, with their known wavelike properties, could be described as particles, electrons as particles might show wavelike properties with a wavelength λ inversely proportional to their momentum p = mev.
p = h/2πλ
Experiments confirmed de Broglie's assumption and led Erwin Schrödinger to derive a "wave equation" to describe the motion of de Broglie's waves. Schrödinger's equation replaces the classical Newton equations of motion.
Note that Schrödinger's equation describes the motion of only the wave aspect, not the particle aspect, and as such it implies interference. Note also that it is as fully deterministic an equation of motion as Newton's equations.
Schrödinger attempted to interpret his "wave function" for the electron as a probability density for electrical charge, but charge density would be positive everywhere and unable to interfere with itself.
Max Born shocked the world of physics by suggesting that the absolute values of the wave function ψ squared (|ψ|2) could be interpreted as the probability of finding the electron in various position and momentum states - if a measurement is made. This allows the probability amplitude ψ to interfere with itself, producing highly non-intuitive phenomena such as the two-slit experiment.
Despite the probability amplitude going through two slits and interfering with itself, experimenters never find parts of electrons. They always are found whole.
In 1932 John von Neumann explained that two fundamentally different processes are going on in quantum mechanics.
cn = < φn | ψ >
This is as close as we get to a description of the motion of the particle aspect of a quantum system. According to von Neumann, the particle simply shows up somewhere as a result of a measurement. Information physics says it shows up whenever a new stable information structure is created.
2. A causal process, in which the electron wave function ψ evolves deterministically according to Schrödinger's equation of motion for the wavelike aspect. This evolution describes the motion of the probability amplitude wave ψ between measurements.
(ih/2π) ∂ψ/∂t =
Information physics establishes that process 1 may create information. Process 2 is information preserving.
Collapse of the Wave Function
Physicists calculate the deterministic evolution of the Schrödinger wave function in time as systems interact or collide. At some point, they make the ad hoc assumption that the wave function "collapses." This produces a set of probabilities of finding the resulting combined system in its various eigenstates.
Although the collapse appears to be a random and ad hoc addition to the deterministic formalism of the Schrödinger equation, It is very important to note that the experimental accuracy of quantum mechanical predictions is unparalleled in physics, providing the ultimate justification for this theoretical kluge.
Moreover, without wave functions collapsing, no new information can come into the universe. Nothing unpredicatable would ever emerge. Determinism is "information-preserving." All the information we have today would have to have already existed in the original fireball.
The "Problem" of Measurement
Quantum measurement (the irreducibly random process of wave function collapse) is not a part of the mathematical formalism (a perfectly deterministic process) of wave function time evolution. It is an ad hoc heuristic description and method of calculation that predicts the probabilities of what will happen when an observer makes a measurement.
In many standard discussions of quantum mechanics, and most every popular treatment, it is said that we need the consciousness of a physicist to collapse the wave function. Eugene Wigner and John Wheeler sometimes describe the observer as making up the "mind of the universe."
Von Neumann contributed a lot to this confusion by his location of the cut (Schnitt) between the microscopic system and macroscopic measurement being anywhere including inside an observer's brain.
Measurement requires the interaction of something macroscopic, assumed to be large and adequately determined. In physics experiments, this is the observing apparatus. But in general, measurement does not require a conscious observer. It does require information creation or there will be nothing to observe.
In our discussion of Schrödinger's Cat, the cat can be its own observer.
The second law of thermodynamics says that the entropy (or disorder) of a closed physical system increases until it reaches a maximum, the state of thermodynamic equilibrium. It requires that the entropy of the universe is now and has always been increasing. (The first law is that energy is conserved.)
This established fact of increasing entropy has led many scientists and philosophers to assume that the universe we have is running down. They think that means the universe began in a very high state of information, since the second law requires that any organization or order is susceptible to decay. The information that remains today, in their view, has always been here. This fits nicely with the idea of a deterministic universe. There is nothing new under the sun. Physical determinism is "information-preserving."
But the universe is not a closed system. It is in a dynamic state of expansion that is moving away from thermodynamic equilibrium faster than entropic processes can keep up. The maximum possible entropy is increasing much faster than the actual increase in entropy. The difference between the maximum possible entropy and the actual entropy is potential information.
Creation of information structures means that in parts of the universe the local entropy is actually going down. Reduction of entropy locally is always accompanied by radiation of entropy away from the local structures to distant parts of the universe, into the night sky for example. Since the total entropy in the universe always increases, the amount of entropy radiated away always exceeds (often by many times) the local reduction in entropy, which mathematically equals the increase in information.
"Ergodic" Processes
We will describe processes that create information structures, reducing the entropy locally, as "ergodic."
This is a new use for a term from statistical mechanics that describes a hypothetical property of classical mechanical gases. See the Ergodic Hypothesis.
Ergodic processes (in our new sense of the word) are those that appear to resist the second law of thermodynamics because of a local increase in information or "negative entropy" (Erwin Schrödinger's term). But any local decrease in entropy is more than compensated for by increases elsewhere, satisfying the second law. Normal entropy-increasing processes we will call "entropic".
Encoding new information requires the equivalent of a quantum measurement - each new bit of information produces a local decrease in entropy but requires that at least one bit (generally much much more) of entropy be radiated or conducted away.
Without violating the inviolable second law of thermodynamics overall, ergodic processes reduce the entropy locally, producing those pockets of cosmos and negative entropy (order and information-rich structures) that are the principal objects in the universe and in life on earth.
Entropy and Classical Mechanics
Ludwig Boltzmann attempted in the 1870's to prove Rudolf Clausius' second law of thermodynamics, namely that the entropy of a closed system always increases to a maximum and then remains in thermal equilibrium. Clausius predicted that the universe would end with a "heat death" because of the second law.
Boltzmann formulated a mathematical quantity H for a system of n ideal gas particles, showing that it had the property δΗ/δτ ≤ 0, that H always decreased with time. He identified his H as the opposite of Rudolf Clausius' entropy S.
In 1850 Clausius had formulated the second law of thermodynamics. In 1857 he showed that for a typical gas like air at standard temperatures and pressures, the gas particles spend most of their time traveling in straight lines between collisions with the wall of a containing vessel or with other gas particles. He defined the "mean free path" of a particle between collisions. Clausius and essentially all physicists since have assumed that gas particles can be treated as structureless "billiard balls" undergoing "elastic" collisions. Elastic means no motion energy is lost to internal friction.
Shortly after Clausius first defined the entropy mathematically and named it in 1865, James Clerk Maxwell determined the distribution of velocities of gas particles (Clausius for simplicity had assumed that all particles moved at the average speed 1/2mv2 = 3/2kT).
Maxwell's derivation was very simple. He assumed the velocities in the x, y, and z directions were independent. [more...]
Boltzmann improved on Maxwell's statistical derivation by equating the number of particles entering a given range of velocities and positions to the number leaving the same volume in 6n-dimensional phase space. This is a necessary state for the gas to be in equilibrium. Boltzmann then used Newtonian physics to get the same result as Maxwell, which is thus called the Maxwell-Boltzmann distribution.
Boltzmann's first derivation of his H-theorem (1872) was based on the same classical mechanical analysis he had used to derive Maxwell's distribution function. It was an analytical mathematical consequence of Newton's laws of motion applied to the particles of a gas. But it ran into immediate objections. The objection is the hypothetical and counterfactual idea of time reversibility. If time were reversed, the entropy would simply decrease. Since the fundamental Newtonian equations of motion are time reversible, this appears to be a paradox. How could the irreversibile increase of the macroscopic entropy result from microscopic physical laws that are time reversible?
Lord Kelvin (William Thomson) was the first to point out the time asymmetry in macroscopic processes, but the criticism of Boltzmann's H-theorem is associated with his lifelong friend Joseph Loschmidt. Boltzmann immediately agreed with Loschmidt that the possibility of decreasing entropy could not be ruled out if the classical motion paths were reversed.
Boltzmann then reformulated his H-theorem (1877). He analyzed a gas into "microstates" of the individual gas particle positions and velocities. For any "macrostate" consistent with certain macroscopic variables like volume, pressure, and temperature, there could be many microstates corresponding to different locations and speeds for the individual particles.
Any individual microstate of the system was intrinsically as probable as any other specific microstate, he said. But the number of microstates consistent with the disorderly or uniform distribution in the equilibrium case of maximum entropy simply overwhelms the number of microstates consistent with an orderly initial distribution.
About twenty years later, Boltzmann's revised argument that entropy statistically increased ran into another criticism, this time not so counterfactual. This is the recurrence objection. Given enough time, any system could return to its starting state, which implies that the entropy must at some point decrease. These reversibility and recurrence objections are still prominent in the physics literature.
The recurrence idea has a long intellectual history. Ancient Babylonian astronomers thought the known planets would, given enough time, return to any given position and thus begin again what they called a "great cycle," estimated by some at 36,000 years. Their belief in an astrological determinism suggested that all events in the world would also recur. Friedrich Nietzsche made this idea famous in the nineteenth century, at the same time as Boltzmann's hypothesis was being debated, as the "eternal return" in his Also Sprach Zarathustra.
The recurrence objection was first noted in the early 1890's by French mathematician and physicist Henri Poincaré. He had found an analytic solution to the three-body problem and noted that the configuration of three bodies returns arbitrarily close to the initial conditions after calculable times. Even for a handful of planets, the recurrence time is longer than the age of the universe, if the positions are specified precisely enough. Poincaré then proposed that the presumed "heat death" of the universe predicted by the second law of thermodynamics could be avoided by "a little patience." Another mathematician, Ernst Zermelo, a young colleague of Max Planck in Berlin, is more famous for this recurrence paradox.
Boltzmann accepted the recurrence criticism. He calculated the extremely small probability that entropy would decrease noticeably, even for gas with a very small number of particles (1000). He showed the time associated with such an event was 101010 years. But the objections in principle to his work continued, especially from those who thought the atomic hypothesis was wrong.
It is very important to understand that both Maxwell's original derivation of the velocities distribution and Boltzmann's H-theorem showing an entropy increase are only statistical or probabilistic arguments. Boltzmann's work was done twenty years before atoms were established as real and fifty years before the theory of quantum mechanics established that at the microscopic level all interactions of matter and energy are fundamentally and irreducibly statistical and probabilistic.
Entropy and Quantum Mechanics
A quantum mechanical analysis of the microscopic collisions of gas particles (these are usually molecules - or atoms in a noble gas) can provide revised analyses for the two problems of reversibility and recurrence. Note this requires more than quantum statistical mechanics. It needs the quantum kinetic theory of collisions in gases.
There are great differences between Ideal, Classical, and Quantum Gases.
Boltzmann assumed that collisions would result in random distributions of velocities and positions so that all the possible configurations would be realized in proportion to their number. He called this "molecular chaos." But if the path of a system of n particles in 6n-dimensional phase space should be closed and repeat itself after a short and finite time during which the system occupies only a small fraction of the possible states, Boltzmann's assumptions would be wrong.
What is needed is for collisions to completely randomize the directions of particles after collisions, and this is just what the quantum theory of collisions can provide. Randomization of directions is the norm in some quantum phenomena, for example the absorption and re-emission of photons by atoms as well as Raman scattering of photons.
In the deterministic evolution of the Schrödinger equation, just as in the classical path evolution of the Hamiltonian equations of motion, the time can be reversed and all the coherent information in the wave function will describe a particle that goes back exactly the way it came before the collision.
But if when two particles collide the internal structure of one or both of the particles is changed, and particularly if the two particles form a temporary larger molecule (even a quasi-molecule in an unbound state), then the separating atoms or molecules lose the coherent wave functions that would be needed to allow time reversal back along the original path.
During the collision, one particle can transfer energy from one of its internal quantum states to the other particle. At room temperature, this will typically be a transition between rotational states that are populated. Another possibility is an exchange of energy with the background thermal radiation, which at room temperatures peaks at the frequencies of molecular rotational energy level differences.
Such a quantum event can be analyzed by assuming a short-lived quasi-molecule is formed (the energy levels for such an unbound system are a continuum of, so that almost any photon can cause a change of rotational state of the quasi-molecule.
A short time later, the quasi-molecule dissociates into the two original particles but in different energy states. We can describe the overall process as a quasi-measurement, because there is temporary information present about the new structure. This information is lost as the particles separate in random directions (consistent with conservation of energy, momentum, and angular momentum).
The decoherence associated with this quasi-measurement means that if the post-collision wave functions were to be time reversed, the reverse collision would be very unlikely to send the particles back along their incoming trajectories.
Boltzmann's assumption of random occupancy of possible configurations is no longer necessary. Randomness in the form of "molecular chaos" is assured by quantum mechanics.
The result is a statistical picture that shows that entropy would normally increase even if time could be reversed.
This does not rule out the kind of departures from equilibrium that occur in small groups of particles as in Brownian motion, which Boltzmann anticipated long before Brown's experiments and Einstein's explanation. These fluctuations can be described as forming short-lived information structures, brief and localized regions of negative entropy, that get destroyed in subsequent interactions.
Nor does it change the remote possibility of a recurrence of any particular initial microstate of the system. But it does prove that Poincaré was wrong about such a recurrence being periodic. Periodicity depends on the dynamical paths of particles being classical, deterministic, and thus time reversible. Since quantum mechanical paths are fundamentally indeterministic, recurrences are simply statistically improbable departures from equilibrium, like the fluctuations that cause Brownian motion.
Entropy is Lost Information
Entropy increase can be easily understood as the loss of information as a system moves from an initially ordered state to a final disordered state. Although the physical dimensions of thermodynamic entropy (joules/ºK) are not the same as (dimensionless) mathematical information, apart from units they share the same famous formula.
S = ∑ pi ln pi
To see this very simply, let's consider the well-known example of a bottle of perfume in the corner of a room. We can represent the room as a grid of 64 squares. Suppose the air is filled with molecules moving randomly at room temperature (blue circles). In the lower left corner the perfume molecules will be released when we open the bottle (when we start the demonstration).
What is the quantity of information we have about the perfume molecules? We know their location in the lower left square, a bit less than 1/64th of the container. The quantity of information is determined by the minimum number of yes/no questions it takes to locate them. The best questions are those that split the locations evenly (a binary tree).
For example:
• Are they in the upper half of the container? No.
• Are they in the left half of the container? Yes.
• Are they in the upper half of the lower left quadrant? No.
• Are they in the left half of the lower left quadrant? Yes.
• Are they in the upper half of the lower left octant? No.
• Are they in the left half of the lower left octant? Yes.
Answers to these six optimized questions give us six bits of information for each molecule, locating it to 1/64th of the container. This is the amount of information that will be lost for each molecule if it is allowed to escape and diffuse fully into the room. The thermodynamic entropy increase is Boltzmann's constant k multiplied by the number of bits.
If the room had no air, the perfume would rapidly reach an equilibrium state, since the molecular velocity at room temperature is about 400 meters/second. Collisions with air molecules prevent the perfume from dissipating quickly. This lets us see the approach to equilibrium. When the perfume has diffused to one-sixteenth of the room, the entropy will have risen 2 bits for each molecule, to one-quarter of the room, four bits, etc.
Let's look at a computer visualization of the equilibration process in a new window.
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Part Six - Chance Part Eight - Afterword
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-30Section 9:
Partial derivatives: Differentiate with respect to one variable, treating the others as constants:
example: if z = x + x2y3
z = 1 + 2x y3
z = 0 + x2 (3y2)
3-Dimensional Square Well: I will solve Schrödinger's
equation for
0 if 0<x<L, and 0<y<L, and 0<z<L
U(x,y,z) =
elsewhere
Main points to notice in the solution:
- The one equation containing x, y and z can be separated into three equations with one
variable each.
- ψ is the product of the solutions of the three equations.
- Separation introduces an arbitrary constant into each equation, the sum of whose squares is
related to energy.
- Imposing the boundary conditions (ψ = 0 at each wall of the box) restricts these constants,
and therefore the energy, to certain values.
- The separation constants are proportional to positive integers called the system's quantum
numbers. These appear in the system's wave function, so different numbers give you
different ψs, which define different states.
Ex. 9-1: An electron is confined to a cube 2.00 Å on a side. Find
a. the energy of the two lowest energy levels, and
b. how many states have each of those energies.
The solution of Schrödinger's equation for a particle in a 3-d box, and the kind of behavior found in
the answer, includes all the main features of real hydrogen. I will not show you the details, because
they are more complicated than for a particle in a box. To summarize:
Put U =
into Schrödinger's equation.
Separate and solve, something like 3-d square well. Apply boundary conditions (different
conditions this time).
-31ψ = nasty looking function which includes E (energy) and n, l and ml , constants analogous
to nx, ny and nz of square well.
Boundary conditions restrict these numbers to values I will put on board.
"Subshells" are sometimes lettered: s state means l = 0, p state means l = 1, d state means l = 2, etc.
Ex. 9-2: Can a 2d state exist?
Ex. 9-3: Neglecting spin, how many states are there with n = 3? What is their energy?
Meaning of the quantum numbers:
n describes the quantization of energy.
Angular momentum, a vector, is also quantized:
- l describes the quantization of ⃑ ’s magnitude.
- ml describes the quantization of ⃑ ’s direction.
Allowed values for L, Lz and θ.
Ex. 9-4: Find L and the allowed values of Lz and θ for a p state.
Hydrogen’s wave functions: The general expression giving for any n, l and ml is too big and
ugly. Just look at a 1s state:
a0 = Bohr radius
Probability distribution: = probability per unit volume.
Probability per unit radius, P(r), is often more useful:
Probability it’s in volume dV = dV
= (4r 2 dr )
1 2a r
P(r ) 3 e 0
a 0
P(r)dr = Probability it’s in interval dr wide.
4r 2
Example: 1s state:
More accurate than the Bohr model is to think of the electron as a charged cloud whose density is
given by . You could think of a pointlike electron darting around so fast that it looks like a
cloud, similar to a fan’s blades looking like a disk when going fast. But unlike a fan, an electron's
position at some moment is unpredictable. The reality of something unknowable is highly
questionable, so you might as well just think of the electron as spreading out into a cloud when
confined to an atom. (And temporarily collapsing to a point if you measure its position.)
Ex. 9-5: Find the most probable distance of the electron from the nucleus in a 1s state.
- An electron has angular momentum, as if it was a
spinning sphere.
Magnitude of ⃑ : Same formula as for L, but quantum number corresponding to l can only be ½:
√( ) (
Direction of ⃑ : ms, the
quantum number
corresponding to ms, can only
be +1/2 or -1/2.
Magnetic moment, ⃑ : (⃑ = I⃑ from PHY 132. Spinning charge amounts to current
loops.) It can be shown that
The z component:
= +9.27 x 10-24 J/T (lab 9)
- Explains why some spectral lines are actually two close lines, and experiments such as Stern &
Gerlach’s (See text.)
Ex 9-6: Not neglecting spin, how many states are there with n = 3?
-34Section 10: Electromagnetic Waves
EM waves, such as light, work by induction: A changing electric field induces a changing magnetic
field which induces a changing electric field which ...
The process is set off by accelerating charges.
Example: Oscillating dipole. (The left side of the diagrams has been omitted.) Turn on AC source
at t = 0, then:
-35Maxwell's Equations: The basic equations of electromagnetism, from which everything else can be
By analogy to y = A cos(kx - ωt) from last fall, the fields making up this wave would be given by:
(1) E = Emax cos (kx - ωt)
k = 2π/λ
(2) B = Bmax cos (kx - ωt)
ω = 2πf
(⃑ and ⃑ are both to the direction of propagation.)
Is this model consistent with Maxwell's equations?
The two Gauss's laws describe static fields, and so don't apply here. Convert the other two to a
differential form:
(Er - El)h = - (dB/dt)(h dx)
dE = – dB
Notation correction: E and B are functions of more than one variable, so these are partial
-36derivatives. (Differentiate with respect to one variable, treating others as constants. Notes p.31)
E = – B
B = –μ0ε0 E
Similarly, the Ampere-Maxwell equation
(The point of that was to convert the relevant Maxwell equations from the form using intergals into
this differential form.)
I will show in class how these follow from equations 1 – 4:
Ex. 10-1: The E vector in an electromagnetic wave varies according to
E = (600 V/m)cos[(1.2 x 107 m-1)x – (3.6 x 1015 s-1)t]. Find:
a. the frequency
b. the wavelength
c. the expression for B.
Electromagnetic Spectrum:
low f (or long λ): Radio waves
X - rays
high f (short λ): gamma rays
Roy G. Biv
Energy in EM waves:
Wave's energy = energy of the E and B fields it's made of.
Notation: U = energy, u = energy/volume
Adding the expressions from Phy 132 sec 11, u = ½ε0E2 + B2/(2μ0)
Substitute in B = E/c and c2 = 1/(μ0ε0), do some algebra:
u = ½ε0E2 + ½ε0E2
-37u = ε0E2
(instantaneous value)
As with RMS current & voltage (PHY 132), (E2)av = ½Emax2, so
uav = ½ε0Emax2
definition: Intensity: I =
EM wave energy density (J/m3)
U = energy, A = area, t = time
This suggested (to John Poynting in 1884) a vector whose magnitude is the wave's intensity, and
whose direction is the wave's direction of propagation:
The Poynting vector
Sav = intensity
Ex. 10-2: 1.0 watt of power is radiated
by the bulb as light. At point P, find
the average values of I, S, E and B.
-38Momentum and Radiation Pressure:
When the wave hits this stationary
charge, the electric force, FE, pushes it
up and down. Once in motion, it also
feels a magnetic force, FM. Over a
cycle, cancels , and there is no net
electric force. But, with the magnetic force,
Thus, the charge gains momentum, which must have come from the wave.
If you put ⃑
⃑ into
(from Phy 131) and make some other substitutions:
Momentum of an EM wave: p = U/c
A perfect absorber
A perfect reflector
gains momentum = U/c
feels pressure = S/c
U = wave's energy
S = Poynting vector
gains momentum = 2U/c
feels pressure = 2S/c
Ex. 10-3: If the acceleration due to the Sun's gravity is 5.93 x 10-3 m/s2, and Sav = 1340 W/m2, what
is the maximum radius of a dust particle which is repelled by radiation pressure at least as strongly
as it is attracted by the Sun's gravity? (Assume the particle's density is 1.0 g/cm2, and that it is very
-39Section 11:
Multi-electron atoms:
There are similar states to hydrogen’s, described by the same four quantum numbers. But now,
more than one of those states has an electron in it at a given time.
Pauli Exclusion Principle: No two electrons in an atom can be in the same state. (Each electron
must have a different set of quantum numbers.)
So, electrons settle, one per state, into the lowest energies available.
A set of values for n, l and ml is called an orbital.
Hund's rule: Given a choice of orbitals with equal energy, electrons usually arrange themselves for a
maximum number of unpaired spins.
For example, nitrogen is
These rules explain the periodic table: similar chemical properties recur as similar outer electron
configurations repeat.
Ex. 11-1: a. What is the electronic configuration for Carbon (Z = 6)? b. What are n, l, ml and ms for
each electron?
A gas of individual atoms has just translational and electronic energy. Molecules can also rotate and
vibrate. Like the electronic energy, rotational and vibrational energies are quantized, which caused
classical physics to fail in dealing with them (Equipartition Theorem).
Rotational Energy (½Iω2):
Diatomic molecule:
(simplest example)
Reduced mass
Moment of inertia
Energy levels and the ΔE between them
-40Ex. 11-2: The J = 4 to J = 3 rotational transition in HCl produces a line at 120.3 µm. Find the
distance between the H and Cl nuclei. (mCl = 5.81 x 10-26kg, mH = 1.67 x 10-27kg)
(Atomic mass unit: 1 u = 1/12 mass of a 12C atom = 1.66 x 10-27 kg)
Vibrational Energy:
ΔE between energy levels
Ex. 11-3: An HCl molecule gives off an 8.66 x 1013 Hz photon when it goes from v = 1 to v = 0.
Find the maximum speed of the H atom relative to the Cl in the ground state.
Solids. (Many atoms in a regular array.)
Energy bands: When close, atoms disturb each other's energy levels. The atoms’ slightly different
levels, taken together, form bands. (Each band is actually many very close levels.)
example, sodium at 0 K:
N = number of atoms
This partly filled band is what
makes metals good electrical
conductors. If electrons are going
to flow when you turn on an
electric field, they have to be able
to gain kinetic energy:
In a conductor, there are higher energy states they can move into.
In an insulator, no states of the right energy are
(Conductor: Highest band partly full. Insulator: one full, next empty.)
Above 0 K, some electrons gain thermal energy and jump above EF:
Fermi - Dirac distribution function (Probability that a state of energy E has an electron in it.):
f(E) =
≈ 1 if E significantly < EF,
≈ 0 if E significantly > EF.
EF = Fermi Energy, k = Boltzmann Constant, T = absolute temperature
Density of States (No. of States Per Unit Volume with Energy Between E and E + dE):
g(E)dE = CE½dE
8 2m 2
= 1.062 x 1056 J-3/2·m-3 = 6.812 x 1027 ev-3/2·m-3 for electrons
N(E)dE = number of electrons per unit volume, between E and E+dE
= (no. of states per V between E and E+dE)(fraction of states containing e-s)
N(E)dE = f(E)g(E)dE
n = density of charge carriers. (Number of free electrons per unit volume, in a metal.)
n= ∫
I will use this to show that EF =
( )
Ex. 11-4: Aluminum’s density is 2.70 g/cm3, and its atomic weight is 26.97. Each atom contributes
three free electrons. Find (a) the charge carrier density, (b) the Fermi energy, (c) the Fermi velocity.
Section 12:
Ex. 12-1: In .05 m3 of aluminum at 300 K, approximately how many free electrons have energies
between 11.900 and 11.901 eV?
-42Ex. 12-2: In the same piece of aluminum, calculate the number of conduction electrons with
energies below 11.0 eV at 300 K.
Semiconductors (ex: silicon)
Doping (adding impurity atoms):
Donor: An atom with too many electrons to fit into lattice gives one off.
n-type semiconductor: Most charge carriers are electrons.
Acceptor: an atom which takes an electron on.
p-type semiconductor: Most charge carriers are holes.
p-n junction:
Free charges leave junction. Very small I because few free
charges are left to flow.
Lots of charges flow in, so I is big.
The junction is a diode, conducting in only one direction. (It rectifies: converts AC into pulses
of DC.)
Some holes get stuck on donor atoms in the thin center layer. They repel additional current. The
more Ib drains off, the larger Ie and Ic. So small base current controls large e to c current.
-43Used as amplifiers, and as "switches" for digital circuitry.
Normal resistivity is due to scattering of individual electrons (by lattice defects and phonons).
At low temperatures, electrons can bind into Cooper pairs:
The pair's total spin is zero ( + ), so Pauli principle doesn't apply. They all go into the same state,
and act collectively. Defects, etc, unable to scatter them all at once.
There is also a critical magnetic field, above which resistance
conducting returns.
Magnetic effects, such as Bc, also limit the current density to a critical value, Jc. (J = current/area)
Meissner effect:
A superconductor expels magnetic fields from its interior.
This is done by developing currents on its surface, which
cancel the external field.
ex: Magnetic levitation:
-44The Nucleus:
Discovery - Rutherford scattering (1911): Number of alpha particles scattered through large angles
by atoms in a gold foil indicated most of the atom's mass is concentrated in a small positive nucleus.
r = r0A1/3
Nuclear radius:
where r0 = 1.2 fm
A = mass number = Z + N
Z = atomic number = number of protons
N = number of neutrons
Isotopes of an element have same Z but different A.
(ordinary H)
Ex 12-3: Find the radius and density of an 56Fe nucleus.
Binding energy = (c2)(difference between mass of nucleus and particles making it up)
Ex: Binding energy of a deuteron - see section 5.
Eb (in MeV) = (ZmH + Nmn - matom)(931.5)
(masses in atomic mass units: 1 u = 1/12 mass of a 12C atom)
mH = mass of 11H = 1.007825 u
mn = mass of neutron = 1.008665 u
Ex 12-4: Find the average binding energy per nucleon of 168O if the mass of one atom is 15.994915
-45Review for Exam 3:
1. What would be the average intensity of an electromagnetic wave in which each cubic meter
contained 5.00 J of energy?
2. When a CO molecule changes from the J = 4 to J = 3 rotational state, it emits a 1.91 x 10-3eV
photon. What is this molecule's rotational kinetic energy in the J = 3 state?
3. In a metal where the Fermi energy is something greater than 2 eV, how many electrons per unit
volume have energies between 0 eV and 2 eV at T = 0 K?
4. The ground state wave function of hydrogen is
ψ(r) = (π a03)-1/2e-r/a, where a0 is the Bohr radius. From this wave function, show that the most
probable distance of the electron from the nucleus is equal to the Bohr radius, a0.
5. Short answer, 5 points each:
a. The energy bands of a certain solid, near absolute
zero, are as shown. Is this material an insulator,
semi-conductor, or a conductor?
b. Consider a free electron in empty space with light falling on it.
i. At an instant when the electron is at rest, is the force on it in the direction of ⃑ , ⃑ , or
ii. Averaged over many cycles of the wave, is the force on the electron in the direction of
⃑ , ⃑ , or ⃑ ?
c. Which quantum number (n, l, ml or ms) describes the quantization of the direction of an
electron's orbital angular momentum?
d. When the electrons in a substance bind into Cooper pairs, what does the substance become?
e. Classical physics and the equipartition theorem predict the specific heat of H2 gas (at
constant volume) to be larger than what is actually measured at low temperatures. Why is it
less than predicted?
-46Section 13:
Nucleons (protons & neutrons) attract each other by the strong force.
Alpha decay: Strong force's range is very short; can't hold back Coulomb repulsion if nucleus is too
alpha particle
Mass numbers add up to same thing on both sides. (conservation of mass)
Atomic numbers add up to same thing on both sides. (conservation of charge)
Ex 13-1: Write the equation for the alpha decay of Radium 226.
Neutrons and protons can decay into each other by the weak force. (Therefore, N Z)
Beta decay:
( : Greek "nu")
Beta particle is an electron or positron (anti-electron).
Neutrino: Does not feel strong force, electromagnetism (no charge), or gravity (little or no
rest mass). Therefore, it interacts very weakly with matter.
Gamma decay: α or β process, or a collision, excites a nucleus.
level by giving off a photon:
( * for excited state.)
γ rays are the most penetrating, α's the least.
Decay rate, R.
Relationships between N (number of nuclei), R, and time.
Half-life (T1/2) = time for half
of original nuclei to decay.
In class, I will show that
T1/2 = (ln 2)/λ
Then, it drops to a lower energy
-47Ex 13-2: Radioactive dating: The ratio of 14C to 12C in all living things is 1.3 X 10-12. The half life of
C is 5730 years. If a 100 gram piece of charcoal has an activity of 17 Bq, how old is it?
Disintegration energy/ reaction energy:
Disintegration: X Y + α, X Y + β + ν, etc.
Reaction: Shooting some particle, a, at a nucleus can transmute it into another element:
a + X Y + b + ... example: Bombarding uranium, the heaviest natural element, with
neutrons can build it up into heavier elements which no one had seen before the 1930's.
Q = (total m before - total m after)c2
(c2 = 931.5 MeV/u)
(Q = KE of Y & other particles, energy of γ rays, etc.)
Ex 13-3: Find the Q value for the α decay of
222.017574 u, 4He: 4.002603 u.
Ra. Mass of
Ra: 226.025406 u,
If Q is positive, the process can happen spontaneously.
If Q is negative, the process can’t happen spontaneously.
Nuclear reactions:
Fission: Split a nucleus into smaller ones.
Fusion: Combine nuclei.
example: In 1939, it was discovered that a bombarding neutron sometimes splits a uranium nucleus,
releasing energy:
+ 23592U
lasts about 10-12s
X + Y + neutrons
fission fragments
(Mass no. & atomic no. must add up to same thing on both sides.)
Chain reaction: The neutrons released go on to split other 235Us, releasing even more neutrons, etc.
examples: original "A" bomb, nuclear reactors.
Critical mass: The minimum mass needed to sustain a chain reaction. (If too few 235Us are around
to be split, neutrons escape from sample faster than new ones are released.)
example: Proton - proton chain (The Sun's main process. Requires great temperature and/or
H + 1H 2H + e+ +
( = neutrino)
H + 2H 3He + γ
(γ = photon)
H + 3He 4He + e+ +
He + 3He 4He + 1H + 1H
Other examples: H - bomb. Maybe someday, fusion reactors.
Effects of radiation:
Ionizing radiation damages molecules in cells. If the cell can't repair the damage:
- Radiation sickness: Many dead/damaged cells, causing blood abnormalities at a certain dosage;
nausea, hair loss, etc at a higher dosage; death at a still higher dose. (It’s kind of like a sunburn that
goes more than skin deep.)
- Cancer: Damaging a cell's genetic material makes it divide out of control.
-Birth defects: Damage to genes in reproductive cells can cause a mutation which is passed down to
all future generations.
There may be no "safe" level of radiation. Some people claim there is even a small chance of the
above from natural background radiation. (Others disagree.)
unit: 1 rad = .01 J
amount of energy absorbed
amount of material absorbing it
The same number of rads of different kinds of radiation causes different amounts of damage:
RBE = Relative Biological Effectiveness = How many rads of x-rays would produce the same
damage as 1 rad of the radiation being used.
Effective dose in REM = (dose in rad)(RBE)
Ex 13-4: A tumor which is ordinarily given a dose of 1000 rad from a Co-60 source (γ rays with
RBE = .7) is to be treated with neutrons having an RBE of 3.0. How many rads are needed?
-49Section 14:
Elementary particles:
Starting in the 1930's, hundreds of subatomic particles were discovered, using cosmic rays and
particle accelerators.
Leptons (ex: e-): Do not feel the strong force. (low mass)
Hadrons: Do feel the strong force.
Mesons: spin = 0 or 1 (medium mass)
Baryons (ex: p+ & n): spin = 1/2 or 3/2 (heaviest)
Conservation Laws: Charge, momentum, etc, and:
Baryon number:
B=1 for baryons, B= –1 for antibaryons, B =0 for other particles
Lepton numbers:
Le = 1 for e- and
, –1 for e+ and
, 0 for others.
L = 1 for e- and
, –1 for + and
, 0 for others.
L = 1 for e- and
, –1 for + and
, 0 for others.
(Bar over the top means antiparticle.)
Ex 14-1: Which reactions can occur, and why?
a. p + n p + p + e +
b. p + n p + p + ̅
c. p + n p + p + e
d. p + n p + p + e +
Since hadrons
- are very numerous,
- are heavy, with measurable diameters,
- have patterns in their properties (see text),
this suggests they are made of still smaller quarks.
So, the most basic building blocks of matter would then be:
(And each has an antiparticle: Same mass, but charge and some other properties opposite.)
Forces between particles: Due to exchange of field particles.
(Analogy: Two kids on skateboards, with boomerangs. To repel, they throw the
boomerangs at each other and catch them. To attract, throw them away from each other,
they boomerang around, come up behind each kid, and they catch them.)
“feeling” interaction
Gauge Boson
(field particle)
Graviton (undetected)
Electric charge
Weak charge
W+, W- & Z
(At close range, gravitation is weakest, then Weak, EM, and Strong is strongest.)
Example: Electromagnetic repulsion between
(Exists only as much time as uncertainty
principle allows "violation" of conservation of energy.)
Weak interaction: Affects hadrons and leptons. (Ex: Causes β decay)
Strong (or "color") force: Binds quarks into hadrons, and hadrons into nuclei.
Color (charge-like property): Red, green & blue.
Analogous to negative charge: antired, antigreen & antiblue.
-51Whole hadrons are colorless:
Meson = a quark & antiquark. (Colors cancel)
Baryon = 3 quarks, or 3 antiquarks. (R + B + G = white)
Ex 14-2: Give the color and flavor of the other quark in a
a. π+ containing a red u quark
b. antiproton containing an antiblue u & antired d.
Unification of forces: Considerable work is currently being done in trying to develop a single
"Theory of Everything." (Something like the unification of electricity with magnetism in the
1800's.) A successful electroweak theory was published in the 1970's. Grand unification theories
which include the strong force exist, but await experimental evidence. Including gravity is hardest.
General Relativity:
Principle of equivalence: Being in a gravitational field is equivalent to being in an accelerated
frame of reference:
No force acts on the objects on the left. No experiment can distinguish that situation from the one in
the center. So, gravity is a pseudo force, similar to a centrifugal “force”.
The “force” of gravity is due to the way a mass distorts spacetime around it:
Consequences of G. R.:
- Bending of light in strong gravitational fields.
- Slowing of time in strong gravitational fields.
- Gravitational redshift.
- Other.
-52Example: Clocks on GPS satellites:
, an object raised 20 200 km above Earth gains an energy per unit mass of
ΔU/m = 4.77 x 10
Fraction its energy increases:
It’s the same for photons:
By T = 1/f, period changes by same factor as f. Any other kind of vibration speeds up
similarly, so a clock gains 5.3 x 10-10 day each day due to G. R. (46 μs) Nanosecond accuracy
is needed, so this must be taken into account.
Universe began with a big bang 13.8 billion years ago (+ ½%), and has been expanding since.
Hubble's law:
v = HR
v = speed object moves away from us
R = distance
H = Hubble constant .022 m/s per light-year
Ex 14-3: The Doppler shift of a certain galaxy's spectrum indicates a speed of 3.0 x 106 m/s. How
far is it from Earth?
The Universe cools as it expands.
t = 10-40s: Universe was probably an ultrahot "quark soup," with the four forces indistinguishable.
The forces became distinct, one by one, as it cooled. By about 10-12s, temperature & density were
down to where the laws of physics are well understood.
t = a few minutes: Temperature low enough for hadrons to bind into nuclei. Elements predicted by
model match observation.
t = a few hundred thousand years: Electrons and nuclei bind into neutral atoms, making universe
transparent. Radiation moving freely since then has cooled into the cosmic microwave background.
(Observation of this was confirmation of big bang model.)
-53Review for Final Exam:
1. Consider the nuclear reaction 2H + 6Li 24He. The masses of these nuclei are: 2H, 3.344019 x
10-27 kg; 6Li, 9.98664 x 10-27 kg; and 4He, 6.64558 x 10-27 kg. Assume the hydrogen and lithium
nuclei collide at very low speeds, meaning that their initial energies are entirely in the form of mass.
a. What is the total kinetic energy, in joules, of the helium nuclei after the reaction?
b. If each 4He gets half of this energy, what is their speed?
2. Assume that a clarinet is a cylindrical tube full of air at room temperature (speed of sound = 343
m/s), 60 cm long, open at one end and closed at the other.
a. Find the frequency and wavelength of the lowest note which can be played on it with all
the side holes shut.
b. Find the frequency and wavelength of the next lowest note. (It’s not just twice the
frequency from a.)
3. None of the following reactions and decays can actually occur. In each case, state a physical
principle which is being violated.
a. p n p p p
b. μ- e- + e
c. p p e
d. π- + p p + π+
4. In a metal where the Fermi energy is 8 eV, about how many free electrons per unit volume have
energies above 9 eV at 300K? (The integral is too ugly to do by hand. Use a calculator or a
website. You still need this simplification: Because the exponential is much larger than 1
throughout the range of integration, e(E-Ef)/kT + 1 ≈ e(E-Ef)/kT. Search for “definite integral calculator.”
At the time I wrote this, I got good results from, and; gave an error message. You may need to use sqrt( ) for √
and exp( ) for an exponential. If it won’t take infinity as a limit of integration, just use 1000.)
5. A light bulb gives off electromagnetic radiation at a rate of 4.50 joules per second, uniformly in
all directions. Find the radiation's average intensity 5.00 m away, and the average electric field
6. A beam of electrons is incident on a crystal,
parallel to the horizontal rows of atoms shown. If first
order diffraction from the planes indicated by dashed
lines is observed when the electron's speed is 2.65 x
107 m/s, what is d?
7. A ray of sunlight is refracted and reflected by a spherical raindrop as shown. (Some geometric
facts are built into the diagram. For example, the two angles labeled "a" are equal because they are
-54base angles of an isosceles triangle.)
a. What is angle a?
b. What is angle b?
c. What is angle c?
d. What is θ, the angle
between the ray's original and
final directions?
(Although a complete treatment
requires considering incident angles
other than 60, you have just
calculated the angular radius of a
8. Short answer, 5 points each:
a. Give an example of a situation in which quantum mechanical tunneling takes place.
b. A π+ meson contains an antired
quark. Give the color and flavor of the other quark(s) it
c. The wave function for a 2p state in hydrogen is (using spherical coordinates)
e r / 2 a0 cos . For this state, what is ψ2 dV, integrated over all places the
5/ 2
4 2 a0
electron can reach?
d. The rod connecting the two
objects has neglegible mass.
For which system, A or B, is the
reduced mass smaller, or are
they equal?
e. The bubble is 1/4 of a wavelength thick (using the
wavelength in the liquid). Will the reflected rays interfere
constructively, destructively, or do something in between? |
280231194532e960 | Skip to main content
Chemistry LibreTexts
13: Harmonic Oscillators and Rotation of Diatomic Molecules
• Page ID
• Recap of Lecture 12
Last lecture addressed three aspects. The first is the introduction of the commutator which is meant to evaluate is two operators commute (a property used extensively in basic algebra courses). Not every pair of operators will commute meaning the order of operations matter. The second aspect is to redefine the Heisenberg Uncertainty Principle now within the context of commutators. Now, we can identify if any two quantum measurements (i.e., eigenvalues of specific operators) will require the Heisenberg Uncertainty Principle to be addressed when simultaneously evaluating them. The third aspect of the lecture was the introduction of vibrations, including how many vibrations a molecule can he (depending on linearity) and the origin of this. The solutions to the harmonic oscillator potential were qualitatively shown (via Java application) with an emphasis of the differences between this model system and the particle in the box (important).
Atoms (very symmetric) Linear molecules (less symmetric) Non-linear molecules (most unsymmetric)
Translation (x, y, and z) 3 3 3
Rotation (x, y, and z) 0 2 3
Vibrations 0 3N − 5 3N − 6
Total (including Vibration) 3 3N 3N
Last lecture laid the groundwork for understanding spectroscopy. We first introduce bra-key notation as a means to simplify the manipulation of integrals. We introduced a qualitative discussion of IR spectroscopy and then focused on "selection rules" for what vibrations are "IR-active" and can be seen in IR spectra. The two criteria we got were that the vibration requires a changing dipole moment (a static dipole is not needed as discussed for \(CO_2\)) and that \(\Delta v = \pm 1\) for the transition. We were setting the ground work for explaining how to derive the second selection rule using the concept of a transition moment and symmetry. We will pick up that discussion there.
Quantum Mechanical Vibrations
The simplified potential discussed in general chemistry is the potential energy function used in constructing the Hamiltonian. From solving the Schrodinger equations, we get eigenfunctions, eigenvalues (energies) and quantum numbers. Combining these on the potential like we did for the particle in a box give a more detailed (quantum) picture
General Potential (not an approximation) of a vibration with associated eigenenergies.
Zero-point energy
Zero-point energy is the lowest possible energy that a quantum mechanical system may have,i.e. it is the energy of the system's ground state. The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously.
For the particle in the 1D box of length \(L\), with energies
\[E_n = \dfrac{h^2 n^2 }{8m L^2}\]
the zero-point energy (\(n=1\)) is
\[E_{ZPE} = \dfrac{h^2}{8m L^2}\]
For the Harmonic Oscillator with energies
the zero-point energy (\(v=0\)) is
\[E_v = \dfrac{h\nu}{2} = \dfrac{\hbar\omega }{2} \]
Many people have make cookie ideas of tapping into the zero-point energy to drive our economy. This is a silly idea and impossible, since this energy cannot never be tapped (see for more discussion
Infrared Spectroscopy
Infrared (IR) spectroscopy is one of the most common and widely used spectroscopic techniques employed mainly by inorganic and organic chemists due to its usefulness in determining structures of compounds and identifying them. Chemical compounds have different chemical properties due to the presence of different functional groups. A molecule composed of n-atoms has 3n degrees of freedom, six of which are translations and rotations of the molecule itself. This leaves 3N-6 degrees of vibrational freedom (3N-5 if the molecule is linear). Vibrational modes are often given descriptive names, such as stretching, bending, scissoring, rocking and twisting. The four-atom molecule of formaldehyde, the gas phase spectrum of which is shown below, provides an example of these terms.
The spectrum of gas phase formaldehyde, is shown below.
Gas Phase Infrared Spectrum of Formaldehyde, \(H_2C=O\)
Characteristic normal modes (vibrations) in formaldehyde
• \(CH_2\) Asymmetric Stretch
• \(CH_2\) Symmetric Stretch
• \(C=O\) Stretch
• \(CH_2\) Scissoring
• \(CH_2\) Rocking
• \(CH_2\) Wagging
More on Dirac's Bra-Ket notation
In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket \( \langle \, | \, \rangle\)) notation. Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics.
• kets: \(| \, \rangle\)
• bras: \(\langle \, | \)
• Bra-Ket Pairs (dot products): \(\langle Φ|Ψ \rangle\), consisting of a left part, \(\langle Φ|\), (the bra), and a right part, \(|Ψ\rangle\), (the ket).
For the ground state of the well-known particle-in-a-box of with length \(L=1\).
\[\langle x | \psi \rangle = \psi(x) = 2^{1/2} \sin (\pi x) \]
However, if we wish to express \( \psi \) in momentum space we would write
\[ \langle p | \psi \rangle = \psi(p) = 2^{1/2} \dfrac{e^{-ip} +1}{\pi^2 - p^2} \]
How one finds this latter expression will be discussed later.
Transition Moment Integrals gives Selection rules
Spectroscopy is a matter-light interaction. You first need to know the results of the Schrödinger equation of a specific system. This includes both eigenstates (wavefuctions), eigenvalues (energies), and quantum numbers and You need to understand how to couple the eigenstates with electromagnetic radiation. This is done via the transition moment integral
\[\langle \psi_i | \hat{M}| \psi_f \rangle \label{26}\]
The transition moment integral gives information about the probability of a transition occurring. For IR of a single harmonic oscillator, \(\hat{M}\) can be set to \(x\). A more detailed discussion will be presented later. So the probability for a transition in HO is \[P_{i \rightarrow f} = \langle \psi_i | x | \psi_f \rangle\]
• From Equation \(\ref{26}\) comes general rules for absorption. For IR, the transition is allowed only if the molecule has a changing dipole moment.
• From Equation \(\ref{26}\) comes selection rules (what possible transitions are allowed). For IR this results in \(\Delta v = \pm 1\).
The vibration must change the molecular dipole moment to have a non-zero (electric) transition dipole moment. Molecules CAN have a zero net dipole moment, yet STILL UNDERGO transitions when stimulated by infrared light.
Dipole Moments (rehash from gen chem)
\[ \vec{\mu} = \sum_i q_i \, \vec{r}_i \label{d1}\]
The dipole moment acts in the direction of the vector quantity. An example of a polar molecule is \(H_2O\). Because of the lone pair on oxygen, the structure of H2O is bent (via VEPSR theory) that the vectors representing the dipole moment of each bond do not cancel each other out. Hence, water is polar.
Example \(\PageIndex{1}\):
Which molecules absorb IR radiation?
No: Vibration does not change the dipole moment of the molecule due to symmetry.
Yes: Vibration does change the dipole moment of the molecule since there is a difference in electronegativity so the distance between the two atoms affects the dipole moment (Equation \ref{d1}).
Yes.: A vibration does change the dipole moment of the molecule since there is a difference in electronegativity so the distance between the two atoms affects the dipole moment. This is not the symmetric stretch, but the other modes.
Energies of Harmonic Oscillators and IR Transitions
Using the harmonic oscillator and wave equations of quantum mechanics, the energy can be written in terms of the spring constant and reduced mass as
\[E = \left(v+\dfrac{1}{2}\right) \dfrac{h}{2\pi} \sqrt{\dfrac{k}{\mu}} \label{16}\]
where \(h\) is Planck's constant and \(v\) is the vibrational quantum number and ranges from 0,1,2,3.... infinity.
\[E = \left(v+\dfrac{1}{2}\right)h \nu_m \label{17}\]
where \(\nu\) is the vibrational frequency vibration. Transitions in vibrational energy levels can be brought about by absorption of radiation, provided the energy of the radiation exactly matches the difference in energy levels between the vibrational quantum states and provided the vibration causes a change in dipole moment. This can be expressed as
\[\Delta E = h\nu_m = \dfrac{h}{2\pi} \sqrt{\dfrac{k}{\mu}} \label{18}\]
At room temperature, the majority of molecules are in the ground state \(v = 0\), so the energies of these states are (via Equation \ref{18} with \(v=0\)):
\[E_o = \dfrac{1}{2}h \nu_m \label{19}\]
when a molecule absorbs energy, there is a promotion to a higher lying energy state. Let consider the first excited state with energy (via Equation \ref{18} with \(v=1\))::
\[ E_1 = \dfrac{3}{2} h\nu_m \label{20}\]
The energy associated with this transition is
\[\begin{align} \Delta E &= E_1 - E_0 \\[4pt] &= \dfrac{3}{2} h\nu_m - \dfrac{1}{2} h\nu_m \\[4pt] &= h\nu_m \label{21} \end{align}\]
The frequency of radiation \(\nu\) that will bring about this change is identical to the classical vibrational frequency (\(\nu_m\) of the bond and it can be expressed as
\[ \begin{align} E_{radiation} = h\nu &= \Delta E \\[4pt] &= h\nu_m \\[4pt] &= \dfrac{h}{2\pi} \sqrt{\dfrac{k}{\mu}} \label{22} \end{align}\]
Equation \ref{22} can be expressed in wavenumbers (by dividing by \(c\)).
\[ \widetilde{\nu} = \dfrac{1}{2\pi c} \sqrt{\dfrac{k}{\mu}} \label{23}\]
• \(c\) is the velocity of light (cm s-1) and
• \(\widetilde{\nu}\) is the wave number of an absorption maximum (cm-1)
Corresponding probability densities. Image used with permission (Public domain; Allen McC.)
Java simulation of particles in boxes :
Selection Rules
Photons can be absorbed or emitted, and the harmonic oscillator can go from one vibrational energy state to another. Which transitions between vibrational states are allowed? If we take an infrared spectrum of a molecule, we see numerous absorption bands, and we need to relate these bands to the allowed transitions involving different normal modes of vibration.
The selection rules are determined by the transition moment integral.
\[ \begin{align} \mu_T &= \int \limits _{-\infty}^{\infty} \psi_{v'}^* (Q) \hat {\mu} (Q) \psi _v (Q) dQ \\[4pt] &= \langle \psi_{v'} |\hat {\mu} |\psi _v \rangle \label {6.6.1} \end{align}\]
To evaluate this integral we need to express the dipole moment operator, \(\hat {\mu}\), in terms of the magnitude of the normal coordinate \(Q\). Evaluating the integral in Equation \(\ref{6.6.1}\) can be difficult depending on the complexity of the wavefunctions used. We can often (although not always) take advantage of the symmetries of the wavefunction (and \(\hat {\mu}\) too) to make things easier.
Figure: (left) \(f(x) = x^2\) is an example of an even function. (right) \(f(x) = x^3\) is an example of an odd function. Images used with permission from Wikipedia.
While functions exhibit this symmetry, the product of functions inherent the symmetries of the constituent components via a "product table." The one below is in terms of odd/even symmetry, but as you will learn in other classes (especially group theories), there are several symmetries that 3D objects have. The product tables constructed that take into account all such symmetries are more complicated.
Product table Odd Function (anti-symmetric) Even Function (symmetric) No symmetry (neither odd nor even)
Odd Function (anti-symmetric) Even Function (symmetric) Odd Function (anti-symmetric) who knows
Even Function (symmetric) Odd Function (anti-symmetric) Even Function (symmetric) who knows
No symmetry (neither odd nor even) who knows who knows who knows
These symmetries are important since the integrand of an integral (over all space) of an odd function is ALWAYS zero. So you do not need to solve it. |
133468840a250353 | Should creationists accept quantum mechanics?
Published: 25 November 2011 (GMT+10)
Spectrum rainbow
The spectrum in a rainbow
Credit: Wikipedia
Quantum mechanics is one of the brand new ideas to emerge in physics in the 20th century. But is it something creationists should believe? I argue “yes” for two reasons:
1. The evidence supports it: QM solved problems that baffled classical physics, and has passed numerous scientific tests.
2. Fighting against an operational science idea would mean fighting a battle on two fronts. So there is nothing to be gained by diverting our energies, in an area that does nothing to further the creation cause.
Although quantum mechanics is rather outside the scope of our ministry, since it concerns operational science rather than origins, we do receive questions about QM quite often. And we also sometimes receive requests to sponsor various critics of this field. This paper tries to summarize, with as little technical detail as possible, why QM was developed, the overwhelming evidence for it, as well as the lack of any viable alternative. Finally, the pragmatic issue: jumping on an anti-QM bandwagon would just make our job harder and provide not the least benefit to the creation cause.
Backdrop: Classical (Newtonian) physics
Sir Isaac Newton (1642/3–1727) was probably the greatest scientist of all time, discovering the spectrum of light as well as the laws of motion, gravity, and cooling; and also inventing the reflecting telescope and jointly inventing calculus. Yet he wrote more about the Bible than science, and was a creationist1 (and nothing discovered after Darwin would change that).2
Newton’s prowess in science was such that English poet Alexander Pope (1688–1744) wrote the famous epitaph:
The Creation/Fall/Flood is a historical framework taught by the Bible; classical physics is at best just a model to explain how God upholds His creation, not a direct teaching of Scripture. So disagreements with classical physics are in no way like the contradictions of biblical history by uniformitarian geologists and evolutionary biologists.
Nature and nature’s laws lay hid in night;
God said “Let Newton be” and all was light.
Such was his influence that Albert Michelson (1852–1931), the first American to win the Nobel Prize in physics, asserted:
Rather, all that remained, he thought, was more and more precise measurements. He quoted the creationist physicist William Thomson, 1st Baron Kelvin (1824–1907): “the future truths of physical science are to be looked for in the sixth place of decimals.”
Now such statements mainly produce mirth. Even Kelvin himself recognized two “dark clouds” hanging over classical physics, which known theories could not explain:
1. The experiment of Michelson and Morley (1838–1923) that showed effectively no difference in the measured speed of light regardless of direction—to be solved by Einstein’s theory of special relativity, which is outside the scope of this article. Suffice it to say, Einstein made it clear that he deduced many of his ideas from the electromagnetism equations of the great James Clerk Maxwell, a great creationist classical physicist.4 Furthermore, Relativity hasn’t the slightest thing to do with moral relativism: Relativity replaces absolute time and space with another absolute: the speed of light in a vacuum. To underscore this point, Einstein himself preferred the term ‘Invariance Theory’. Finally, creationist physicist Dr Russell Humphreys showed that relativity was an ally of creation, not a foe, and most creationist physicists since then have agreed.
2. Black body radiation, which as will be shown, was one of the mysteries to be solved by quantum mechanics.
Three clouds
Actually, there were three main problems that stumped Newtonian ‘classical’ physics, and quantum mechanics solved them. Despite what some claim, QM is totally unlike Darwinian evolution: QM was driven by unsolved problems and supported by the evidence, and not with any hidden agenda against a Creator. Furthermore, most of the pioneers were reluctant to abandon classical physics.
Another point which seems to be forgotten by some QM critics: the Creation/Fall/Flood is a historical framework taught by the Bible; classical physics is at best just a model to explain how God upholds His creation, not a direct teaching of Scripture. So disagreements with classical physics are in no way like the contradictions of biblical history by uniformitarian geologists and evolutionary biologists.
We also should notice how many of the discoveries that led to QM were rewarded with a Nobel Prize for Physics. By contrast, one gripe of evolutionists is the lack of an award for evolutionary biology;5 Nobel Prizes are awarded only for practical, testable science.6
1. Blackbody radiation
A blackbody is an idealized perfect absorber of all radiation, and as a consequence, is also a perfect emitter. The best approximation to this is a material called super-black, with tiny cavities, actually modeled on the wing rims of certain butterflies.7
Max Planck
Max Planck (1858–1947)
Classical physics predicted that the black body would be a ‘vibrator’ with certain modes, which had different frequencies. And it also predicted that every mode would have the same energy, proportional to temperature (called the Equipartition Theorem). The problem is that there would be more modes at short wavelengths, thus high frequencies, so these modes would have most of the energy. Classical physics led to the Rayleigh–Jeans Law,8 which stated that the energy emitted at a given frequency was proportional to the fourth power of that frequency.
This worked well for low frequencies, but predicted that the radiation would be more and more intense at higher frequencies, i.e. the ultraviolet region of the spectrum and beyond. In fact, it would tend towards infinite energy—clearly this is impossible, hence the term ‘ultraviolet catastrophe’.
Max Planck (1858–1947) solved this problem. Instead of the classical idea, that any mode of oscillation could have any energy, he proposed that they could have only discrete amounts—packets of energy proportional to the frequency. That is, E = hν, where E is energy, ν (Greek letter nu) is frequency, and h is now called Planck’s constant.9 This meant that a mode could not be activated unless it had this minimum amount of energy. The new Planck’s Law matched the observations extremely well at both high and low frequencies. He won the 1919 physics Nobel “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta.”10
Actually, Planck himself was not thinking that he had solved a catastrophe, just that his idea fitted the data well. Rather, he rightly realized that the equipartition theorem was not applicable.11 Interestingly enough, he was sympathetic to Christianity and critical of atheism.12
2. Photo-electric effect
We all know about solar cells now, but over a century ago, the photo-electric effect behind them was a mystery. It was discovered that light could knock electrons out of a material, but the electron energy had nothing to do with intensity of the light, but rather with the frequency. Furthermore, light below a certain threshold frequency had no effect. Very curiously: bright red light (low-frequency) would not work, while faint ultraviolet light (high-frequency) would, even though the energy of the red light was far greater in such cases.
Einstein solved this by proposing that light itself was quantized: came in packets of energy:
Only if the energy packet were greater than the binding energy of the electron would it be emitted. The resulting electron energy would be the difference of the light packet energy and binding energy. So while Planck proposed quantized oscillators, Einstein proposed that electromagnetic radiation was quantized.
It was explicitly for this discovery, not relativity, that Einstein was awarded the 1921 Nobel Prize for Physics:
… for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.14
Einstein called this Lichtquant or light quantum, but the American physical chemist Gilbert Newton Lewis (1875–1946) coined the term photon15 which stuck.
Ironically, like Planck, Einstein didn’t conceive himself as anything more than a classicist. He later vocally opposed the prevailing quantum mechanical interpretations by the Dane Niels Bohr (1885–1962), now called the Copenhagen Interpretation.
3. Atoms
Newton’s discoveries in the spectrum of light presumed that colour was continuous. But when the spectra of individual atoms were measured, they emitted light at discrete frequencies (or absorbed it—dark lines in a “white light” spectrum).
Furthermore, the New Zealander physicist Ernst Rutherford (1871–1937) showed that most of the mass of the atoms was concentrated in a tiny positively charged nucleus, and proposed that electrons orbited like the planets around the sun. The Rutherford model is iconic—it’s what most people imagine when they think of atoms, and is even used in the logo of the United States Atomic Energy Commission and the flag of the International Atomic Energy Agency. Rutherford inexplicably missed out on the Nobel Prize for Physics—instead, the Nobel Prize committee magically transformed him into a chemist, awarding him the Chemistry Prize instead, “for his investigations into the disintegration of the elements, and the chemistry of radioactive substances.”16
However, classical physics predicted that orbiting charged particles like electrons would lose energy to electromagnetic radiation. So their orbits would decay. This, of course, is not what is observed.
To solve this problem, Bohr proposed in 1913 that electrons could only move in discrete orbits, and that these orbits were stable indefinitely. Energy was gained or lost only when the electrons changed orbits, absorbing or emitting electromagnetic radiation—photons of frequency ν = E/h, where E is the energy difference between the states. For electrons in higher energy or ‘excited’ states, this transition would mostly be spontaneous.
Stimulated emission and lasers
In 1917, Einstein realized that a photon with the same energy as the energy difference could increase the probability of this transition.17 Such stimulated emission would produce another photon with the same energy, phase, polarization and direction of travel as the incident photon. This was the first paper to show that atomic transitions would obey simple statistical laws, so was very important for the development of QM. On the practical side, it is immensely valuable, because it is also the basis for masers and lasers. These words were acronyms for Microwave/Light Amplification by Stimulated Emission of Radiation. As a result:
The Nobel Prize in Physics 1964 was divided, one half awarded to Charles Hard Townes, the other half jointly to Nicolay Gennadiyevich Basov and Aleksandr Mikhailovich Prokhorov “for fundamental work in the field of quantum electronics, which has led to the construction of oscillators and amplifiers based on the maser-laser principle.”18
My own green laser pointer relies on an additional QM effect called “second harmonic generation” or “frequency doubling”. Here, two photons are absorbed in certain materials with non-linear optics, and a photon with the combined energy is emitted. In this case, an infrared source with a wavelength of 808 nm pumps an infrared laser with a lower energy of 1064 nm, and this frequency is doubled to produce a green laser beam of 532 nm.
Bohr mocel
Rutherford–Bohr model of the hydrogen atom. Credit: Wikipedia
Bohr’s model strictly applied only to one-electron atoms such as H, He+, Li2+ etc., but he extended it to multi-electron atoms. He proposed that these discrete energy levels could hold only a certain number of electrons—electron shells. This explains the relative inertness of the ‘noble gases’: they already have full shells, so no need to chemically react with another atom to achieve them. It also explains the highly reactive alkali metals: they have one electron over, so can lose it relatively easily to achieve the all-full shell configuration; and the halogens are one electron short, so vigorously try to acquire that one remaining electron from another atom. An illustration of both is the alkali metal halide sodium chloride.
High-school chemistry typically doesn’t go past the Bohr model approach. University chemistry tends to go deeper into more modern quantum mechanics (atomic and molecular orbital theory), of which the Bohr model was a pioneering attempt. Bohr won the physics Nobel in 1922 “for his services in the investigation of the structure of atoms and of the radiation emanating from them.”19
Like Heisenberg and Einstein, Bohr was not happy with aspects of quantum mechanics. In Bohr’s case, for a long time, he was a determined opponent of the existence of photons, trying to preserve continuity in electromagnetic radiation. Bohr also introduced the ‘correspondence principle’: that the new quantum theory must approach classical physics in its predictions when the quantum numbers are large (similarly, relativity theory collapses to ordinary Newtonian physics with velocities that are much smaller than that of light).
Wave-particle duality
The French historian-turned-physicist Louis-Victor-Pierre-Raymond, 7th duc de Broglie (1892–1987) provided another essential concept of quantum mechanics. Just as energy of vibrators and electromagnetic radiation was quantized into discrete packets with particle-like properties, de Broglie proposed that all moving particles had an associated wave-like nature. The wavelength was inversely proportional to momentum, again using Planck’s Constant: λ = h/p, where λ (Greek letter lambda) is wavelength, and p = momentum. This was the subject of his Ph.D. thesis in 1924.20 His own examiners didn’t know what to think, so they asked Einstein. Einstein was most impressed, so de Broglie was awarded his doctorate. Only five years later, he was awarded the Physics Nobel “for his discovery of the wave nature of electrons.”21
It is notable that this prize was awarded before the wave nature of electrons was proven. This happened beyond reasonable doubt when Clinton Joseph Davisson (1881–1958) and George Paget Thomson (1892 –1975) were awarded the 1937 Physics Nobel “for their experimental discovery [made independently of each other] of the diffraction of electrons by crystals.”22 Thomson was the son of J.J. Thomson (1856–1940), who discovered the electron itself. For example, electrons can produce the classic ‘double slit’ interference pattern of alternating ‘light’ and ‘dark’ bands. This pattern is produced even when only one electron goes through a slit at a time.
The discovery of matter waves was instrumental for electron microscopes. These allow smaller objects to be seen than optical microscopes, because the electrons have a smaller wavelength than visible light. The same principle is used for probing atomic arrangements with neutron diffraction—neutrons are almost 2,000 times more massive than electrons, so normally have much more momentum, thus an even smaller wavelength.
Thus de Broglie showed that at a foundational level, both radiation and matter behave as both waves and particles. Writing almost half a century later, he recalled:
Mathematical formulations
In 1925, Werner Heisenberg (1901–1976) formulated a mathematical model to explain the intensity of hydrogen spectral lines. He was then the assistant of Max Born (1882–1970), who recognized that matrix algebra would best explain Heisenberg’s work. Heisenberg was recognized with the 1932 physics Nobel for “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen.”24
The following year, Erwin Schrödinger (1887–1961) developed de Broglie’s ideas of matter waves into the eponymous Schrödinger equation. This describes a physical system in terms of the wavefunction (symbol ψ or Ψ—lower case and capital psi), and how it changes over time. For a system not changing over time, ‘standing wave’ solutions allow the calculation of the possible allowable stationary states and their energies. This brilliantly predicted the energy levels of the hydrogen atom. Later these stationary states were called atomic orbitals. Applied to molecules, they are molecular orbitals, without which much of modern chemistry would be impossible. Other applications of this equation included the calculation of molecular vibrational and rotational states.
Schrödinger’s treatment, as he showed, was equivalent to Heisenberg’s: the stationary states correspond to eigenstates, and the energies to eigenvalues (eigen is the German word for ‘own’ in the sense of ‘peculiar’ or ‘characteristic’). The overall wavefunction could be considered as a superposition of the eigenstates. As Einstein warmly embraced de Broglie’s idea, he did the same to Schrödinger’s, as a more ‘physical’ theory than Heisenberg’s matrices. In 1930, Paul Dirac (1902–1984) combined the two into a single mathematical treatment. Schrödinger and Dirac shared the 1933 Nobel Prize for Physics “for the discovery of new productive forms of atomic theory.”25
Schrödinger was another reluctant convert to QM—he hoped that his wave equation would avoid discontinuous quantum jumps. But he was due to be disappointed: in 1926, Max Born showed that Ψ didn’t have a physical nature; rather, the square of its magnitude |Ψ|2 (or Ψ*Ψ) is proportional to the probability of finding the particle localised in that place. For political reasons, with the developing turmoil of the rise of National Socialism in his country, Germany, Born wasn’t awarded the Nobel Prize for physics until 1954, a half share “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.”26
Weird things
Here is where we find the root of much opposition: the apparently strange things that quantum mechanics predicts.
Uncertainty principle
Heisenberg recognized a fundamental limit to what could be measured. E.g. try to measure the position and momentum of an electron as finely as possible by shining a light photon on it. To finetune the position better, we need a small wavelength. But as de Broglie showed, the shorter the wavelength, the larger the momentum, thus the more that can be transferred to the electron. Thus the electron’s momentum cannot be known precisely. And if we reduce the momentum of the photon to avoid disturbing the electron too much, the wavelength increases, so its position becomes less certain—it is smeared out in space. Thus as Heisenberg said: “It is impossible to determine accurately both the position and the direction and speed of a particle at the same instant.”27 To be precise, the uncertainty in position and momentum is related to Planck’s Constant ΔxΔp ≥ h/4π. The same applies to energy and time: ΔEΔt ≥ h/4π.
Actually, there was a precedent for this in the remarkably productive mind of Einstein: he had recognized that there would be a residual energy even at absolute zero, which he called Nullpunktsenergie,28 or in English zero-point energy. It is easily explained in terms of the uncertainty principle: if there were a zero-energy state in some crystal lattice with fixed atomic positions, it would entail that the atoms’ positions and momenta could be known with total precision. To avoid this, there must be some residual energy.
This is actually proved by the inability to solidify helium no matter how cold, except under very high pressures (25 atmospheres): the zero-point energy would shake any solid lattice apart.
But despite Einstein’s contribution, he detested the uncertainty principle. In the years around 1930, he debated Bohr on various ways around it. These two admired each other greatly, but most physicists thought that Bohr had the better of the arguments—in one famous riposte, he used Einstein’s own theory of general relativity to defeat an ingenious thought experiment.
Interpretations of QM
Thus some creationist (and non-creationist) physicists accept QM but propose a more realist interpretation just as Einstein and Schrödinger advocated. E.g. physicist Dr Russell Humphreys explains (personal communication):
But many of the creationist critics of QM confuse QM with interpretations of QM.
Another strange effect is “entanglement”: two particles interact and thus share the same quantum state until a measurement is made. But we do know something about them, say that their ‘spins’ must be opposite, just that we don’t know which one has which spin. Then the particles go their separate ways. Then we measure one of them, and find that it has, say, anticlockwise spin. This means that the other one must instantly adopt clockwise spin—and so it will prove when it’s measured at any later time, as long as the entanglement is not otherwise disrupted. Both Einstein and Schrödinger disliked the apparent implication that this correlation would travel much faster than light. But many experiments are consistent with this implication, for example one with entangled photons:
The results also set a lower bound on the ‘speed of quantum information’ to 2/3 ×107 and 3/2 ×104 times the speed of light in the Geneva and the background radiation reference frames, respectively.31
To put this into perspective, Newton’s conception of gravitation was criticized at the time for postulating an ‘occult’ action-at-a-distance force which he thought acted instantly (under General Relativity, the force of gravity moves at the speed of light). There is no reason why God’s upholding of His creation (cf. Colossians 1:15) should be limited by the speed of light, especially as God is the creator of time itself.
More evidence
I could not have worked in my own specialist area of spectroscopy unless molecules had quantized energy states, especially in vibrational energy in my case, but electronic states and rotational states as well.
Superconductors and superfluids
Other interesting evidences include superconductors, which I have also researched,32 and superfluids. These are substances with exactly zero resistivity and zero viscosity, respectively.
These are rare examples of quantum behaviour on the macro level. They are related to yet another prediction by Einstein, this time with Satyendra Nath Bose (1894–1974): they realized that at very low temperatures, the wavefunctions of identical particles could overlap to form a single quantum state, now called a Bose–Einstein Condensate.
This easily explains why it’s possible to have zero resistance and viscosity. A current of electrons or fluid usually loses energy to the surrounding materials, but if they are in one quantum state, any possible energy loss would be quantized, thus could not occur below this threshold. Superfluids also exhibit quantized vortices.
Woodward–Hoffmann rules for electrocyclic reactions
One class of organic reactions is electrocyclic, where a conjugated unsaturated “straight” chain hydrocarbon closes into a ring, or the reverse. To do this, there must be some rotation—either the two ends must rotate both clockwise/both anticlockwise, or conrotatory; or one clockwise and the other anticlockwise, or disrotatory. Whether it’s conrotatory or disrotatory turns out to be completely determined. Robert Burns Woodward (1917–1979) and Roald Hoffmann (1937– ) worked out the eponymous rules, based on the conservation of symmetry of the molecular orbitals, which no known classical model could predict.
In particular, the lobes of the molecular orbital can form a bond only if the wavefunction has the same sign (positive or negative), and this can be achieved only by rotation in one of the two possible types (conrotatory or disrotatory). Furthermore, a photochemical reaction turns out to have the opposite symmetry, also explained because the photon excites an electron into another orbital with a different symmetry.
Hoffmann shared the 1981 Nobel with Kenichi Fukui (1918–1998) “for their theories, developed independently, concerning the course of chemical reactions.” Woodward had died before he could be awarded his second Nobel Chemistry Prize.
Designs in nature using QM
Another good reason to support QM is that it is proving to be an ally of the creation model. Some time ago I wrote on how our sense of smell works in accordance with vibrational spectroscopy and quantum mechanical tunneling:
Luca Turin, a biophysicist at University College, London, proposed a mechanism [33,34] where an electron tunnels from a donor site to an acceptor site on the receptor molecule, causing it to release the g-protein. Tunnelling requires both the starting and finishing points to have the same energy, but Turin believes that the donor site has a higher energy than the receptor. The energy difference is precisely that needed to excite the odour molecule into a higher vibrational quantum state. Therefore when the odour molecule lands, it can absorb the right amount of the electron’s energy, enabling tunnelling through its orbitals. This means the smell receptors actually detect the energy of vibrational quantum transitions in the odour molecules, as first proposed by G.M. Dyson in 1937.35
More recent support comes from studies in bird navigation. For some time now, it has been known that birds and many other creatures use the earth’s magnetic field.36 But in European robins, red and yellow light somehow disorients their magnetic sense. So some researchers proposed that light causes one of the eye proteins to emit a pair of ‘entangled’ electrons with opposite spins. Again, we don’t know which is which until a measurement occurs, and here this ‘measurement’ is caused by some difference in the earth’s magnetic field. Thus the other electron must instantly adopt the opposite spin, which the bird detects and somehow computes the information about the magnetism. The birds are disoriented by weak oscillating magnetic field, which could not affect a macro-magnet like a magnetite crystal, but would disrupt an entangled pair.37
The history and practice of QM shows no hidden motivation to attack a biblical world view, in contrast to uniformitarian geology and evolutionary biology. Any proposed replacement theory needs to explain at least all the observations that QM does. This is not a specifically creationist project.
A recent paper paid its usual vacuous homage to evolution:
In artificial systems, quantum superposition and entanglement typically decay rapidly unless cryogenic temperatures are used. Could life have evolved to exploit such delicate phenomena? Certain migratory birds have the ability to sense very subtle variations in Earth’s magnetic field. Here we apply quantum information theory and the widely accepted “radical pair” model to analyze recent experimental observations of the avian compass. We find that superposition and entanglement are sustained in this living system for at least tens of microseconds, exceeding the durations achieved in the best comparable man-made molecular systems. This conclusion is starkly at variance with the view that life is too “warm and wet” for such quantum phenomena to endure.38
Of course, this is more evidence of a Designer whose techniques far exceed the best that man can do—in this case, maintaining quantum entanglement far longer than we can!39
Also, supposedly primitive purple bacteria exploit quantum mechanics to make their photosynthesis 95% efficient. They use a complex of tiny antennae to harvest light, but this complex can be distorted which could harm efficiency. However, because of the wave and particle nature of light and matter, although it absorbs a single photon at a time, the wave nature means that the photon is briefly everywhere in the antenna complex at once. Then of all possible pathways, it is absorbed in the most efficient manner, regardless of any shape changes in the complex. As with the previous example, quantum coherence is normally observable at extremely low temperatures, but these bacteria manage at ordinary temperatures.40
It seems wise for creationists to adopt the prevailing theories of operational science unless there are good observational reasons not to. Otherwise it could give the impression that we are anti-establishment for its own sake, rather than pro-Bible and opposing the establishment only when it contradicts biblical history. Fighting on two fronts has usually been a losing battle strategy. Rather, as previously with relativity, it makes more sense to co-opt it as an ally of creation, as with some of the design features in nature.
1. LaMont, A., Sir Isaac Newton (1642/3 1727): A Scientific Genius, Creation 12(3):48–51, 1990; Return to text.
2. See Sarfati, J., Newton was a creationist only because there was no alternative? (response to critic), 29 July 2002. The critic I was replying to later wrote thanking CMI for the response, and to say that he no longer agreed with the sentiments of his original letter. He was happy for his original letter and response to remain as a teaching point for others who might need correcting. Return to text.
3. Michelson, A.A., Light Waves And Their Uses, pp. 23–25, University of Chicago Press, 1903. Return to text.
4. Lamont, A., James Clerk Maxwell, Creation 15(3):45–47, 1993; Maxwell argued that an oscillating electrical field would generate an oscillating magnetic field, which in turn would generate an oscillating electrical field, and so on. Thus it would be related to the core electromagnetic constants: the permittivity (ε0) and permeability (µ0) of free space, which relate the strengths of electric and magnetic attractions. E.g. Coulomb’s Law is F =-1/(4πε0) q1q2 ⁄ r². Maxwell showed that this radiation would propagate at a speed c² = 1 ⁄ ε0µ0. When the speed of light was found to match this, Maxwell deduced that light must be an electromagnetic wave. Einstein reasoned that since permittivity and permeability are constant for every observer, the speed of light must also be invariant, and instead time and length vary. Return to text.
5. Call for new Nobel prizes to honour ‘forgotten’ scientists, 30 September 2009, archived at Return to text.
6. Except for the 2006 Nobel Prize for physics, which involved proof of the unobserved big bang involving unobserved dark matter. See Sarfati, J., Nobel Prize for alleged big bang proof,, 7–8 October 2006. Return to text.
7. Sarfati, J., Beautiful black and blue butterflies, J. Creation 19(1):9–10, 2005; to text.
8. After John William Strutt, 3rd Baron Rayleigh, OM (1842–1919) and James Hopwood Jeans (1877–1946). Return to text.
9. h = 6.62606957(29)×10−34 J.s. Return to text.
10. Return to text.
11. Galison, P., “Kuhn and the Quantum Controversy”, British J. Philosophy of Science 32(1): 71–85, 1981 P-I-P-E doi:10.1093/bjps/32.1.71 Return to text.
12. Seeger, R., Planck: Physicist, J. American Scientific Affiliation 37:232–233, 1985. Return to text.
13. Einstein, A. Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (On a Heuristic Viewpoint Concerning the Production and Transformation of Light), Annalen der Physik 17(6):132–148, 1905 P-I-P-E doi:10.1002/andp.19053220607. Return to text.
14. Return to text.
15. From phōs (φῶς) light and ōn (ὢν) = being/one. Return to text.
16. Return to text.
17. Einstein, A., Zur Quantentheorie der Strahlung (On the Quantum Mechanics of Radiation), Physikalische Zeitschrift 18:121–128, 1917. Return to text.
18. Return to text.
19. Return to text.
20. Recherches sur la théorie des quanta (Research on the Theory of the Quanta). Return to text.
21. Return to text.
22. Return to text.
23. de Broglie, L., The reinterpretation of wave mechanics, Foundations of Physics 1(1), 1970. Return to text.
24. Return to text.
25. Return to text.
26. Return to text.
27. Heisenberg, W., Die Physik der Atomkerne, Taylor & Francis, 1952, p. 30. Return to text.
28. Einstein, A.; Stern, O., Einige Argumente für die Annahme eine molekular Agitation bein absoluten Nullpunkt (Some arguments in support of the assumption of molecular vibration at the absolute zero), Ann. del Physick 4:551–560, 1913. Return to text.
29. Compare Sarfati, J. Loving God with all your mind: logic and creation, J. Creation 12(2):142–151, 1998; Return to text.
30. Holland, P.R., The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation, Cambridge University Press, 1993. Figure 5.7 on page 184, for example, shows the possible paths of a particle going through the two-slit experiment. Return to text.
31. Zbinden, H. et al., Experimental test of relativistic quantum state collapse with moving reference frames, J. Phys. A: Math. Gen. 34:7103, 2001 P-I-P-E doi:10.1088/0305-4470/34/35/334 Return to text.
32. Mawdsley, A., Trodahl, H.J., Tallon, J., Sarfati, J.D., and Kaiser, A.B., Thermoelectric power and electron-phonon enhancement in YBa2Cu3O7-δ, Nature 328(6127):233–234, 16 July 1987. Return to text.
33. Turin, L., A spectroscopic mechanism for primary olfactory reception, Chemical Senses 21:773, 1996. Return to text.
34. See also Turin, L., The Secret of Scent: Adventures in Perfume and the Science of Smell, 2006. Return to text.
35. Sarfati, J., Olfactory design: smell and spectroscopy, J. Creation 12(2):137–138, 1998; Return to text.
36. See for example Sarfati, J., By Design, ch. 5: Orientation and navigation, CBP, 2008. Return to text.
37. Ritz, T., et al., Resonance effects indicate a radical-pair mechanism for avian magnetic compass, Nature 429:177–180, 13 May 2004 P-I-P-E doi:10.1038/nature02534 Return to text.
38. Gauger, E.M. et al., Sustained Quantum Coherence and Entanglement in the Avian Compass, Physical Rev. Lett. 106: 040503, 2011 P-I-P-E doi: 10.1103/PhysRevLett.106.040503. Return to text.
39. See also Wile, J., Birds Use Quantum Mechanics to Navigate?, 26 March 20l1. Return to text.
40. Hildner, R. et al., Quantum coherent energy transfer over varying pathways in single light-harvesting complexes, Science 340:1448–1451, 2013 | doi:10.1126/science.1235820. See also Wile, J., “Ancient” Bacteria Use Quantum Mechanics!, 11 July 2013. Return to text.
Readers’ comments
Vernon K.
While Quantum Mechanics and Quantum Theory woudl APPEAR to have taken us forward, it would also seem to me that it has takein us into a cul-de-sac from which we cannot extricate ourselves without reworking or throwing it out.
Have you read any of Mile Mathis' reworking of our physics and maths?? I am slogging through his work and find him very persuasive. I would also mention I see nothing in his work which conflicts with literal readings of the Bible or with Electric Universe Theory which I also find compatible with the Bible.
[Links deleted as per feedback rules]
I am certainly not an educated expert so could be missing many things and being misled.
Jonathan Sarfati
From to time, we have been asked about the electric universe theory. As I say to all the enquirers, CMI can't adapt maverick theories in operational science otherwise we would be fighting on two fronts, as explained in my paper above.
Mathis, a leading proponent of the electric universe theory, is certainly not sympathetic to creation, and has had interaction with creationists showing that he is not very well informed. As I explained to another inquirer (and you can make a fair guess what Mathis was claiming from the answer):
Mathis makes a couple of errors. One involves his lack of understanding of facts v interpretations (see for example Evolution & creation, science & religion, facts & bias), and he holds to the myth of neutrality (Myth of neutrality).
Also, it is rather presumptuous to say, “he doesn't know that we have a constant recycling of the field.” Dr Humphreys is well aware of the evolutionary dynamo idea. But he has pointed out that they haven't a viable working model for the Earth's field, and that it doesn't explain fields in bodies that couldn't have a liquid core if they were as old as evolutionists claim.
Note that exponential winding down is basic physics of an circuit with resistance and inductance. My article The earth’s magnetic field: evidence that the earth is young is basically a summary of Dr Humphreys’ research, and includes objections to skeptical arguments. …
The cited article addresses the magnetic field reversals. The overall decay is inescapable even if there is a sinusoidal component, which likely occurred during the Flood year (see the diagram at the top). As for being hot enough,you may well be right. Dr John Baumgardner thinks that the earth was in even more upheaval than previously thought.
Nick W.
Very helpful, very informative article. I especially appreciated the explanation of Schrödinger’s Cat as a reductio ad absurdum, pointing out that the Copenhagen interpretation necessarily implies violations of the law of non-contradiction.
This was helpful because it allows us to then move on to the causal interpretation. Up until this point I had only heard QM spoken of in terms of the copenhagen interpretation and it seemed like madness. I therefore sympathise with those Christians who instinctively oppose QM in an effort to maintain the law of non-contradiction.
Thank you so much for clarifying the difference between the observations and the interpretations.
Graham P.
Magnificent: A very useful précis of quantum physics history; extremely well written.
David H.
This is an excellent brief survey of QM and its history, with some sensible lessons for creationists, and includes some useful examples of recent discoveries. Fascinating, even for someone like me with a background in physical sciences and electronics.
Andrei T.
Thank you so much. My exams are starting this December and I have to know QM for chemistry! My textbook is pretty ‘thick’ on this subject, so this is a great opportunity to study it from different angles!
John T.
A terrific article; Dr. Sarfati is very good on scientific issues.
Philip C.
Very nice article! It does a great job of discussing the history and the main issues. The acceptance of QM is the acceptance of a theory that has very good explanatory power. I especially like the statement “But many of the creationist critics of QM confuse QM with interpretations of QM.” I have indeed found this to be true on many occasions. My area of research is fluorescence spectroscopy and computational chemistry. Electronic structure theory and QM play a big role in everyday life for me. I whole heartedly support your article and think it does a great job at explaining the issues. It should be kept in mind that no scientist accepts QM blindly. We are always told and always work within the framework of—this is a useful idea that has great explanatory power; the implications can seem strange at times but this is just a theory that allows us to make a lot of sense of what we observe. We are not obligated to swallow down every interpretation and every oddity to use and support the theory. QM is quite elegant and supremely useful, it makes sense of the data and observations and has led to many real advances in science! This should not be swept under the rug because we “don’t like” a particular interpretation of it. Thanks again Dr. Sarfati and keep up the good work. I support your work even if it is a little more vibrational and not as optical as I prefer!! God Bless!
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01519a56330381d8 | Hi there! I am Amarashiki (a new “doctor” project). And I have a new upgraded The Spectrum Of Riemannium website!
I will keep this free site though, as backup material. But everything has been moved to the new URL: http://thespectrumofriemannium.com
ALERT: If you was reading my stuff via e-mail, RSS or any other magic device/too, upgrade my URL, please. I will NOT post here anymore. After all, I am paying for the new domain! And it brings new abilities or superpowers to my web! 🙂
I wish you will enjoy my new (improved) site, since even when this beginning trip has ended, a new one is coming! And I hope It will be satisfactory for all of us!
I will be happy to hear from any other suggestion or idea related to my site! Any desiderata? XD
TSOR is just beginning!!!!!!!!
LOG#105. Einstein’s equations.
In 1905, one of Einstein’s achievements was to establish the theory of Special Relativity from 2 single postulates and correctly deduce their physical consequences (some of them time later). The essence of Special Relativity, as we have seen, is that all the inertial observers must agree on the speed of light “in vacuum”, and that the physical laws (those from Mechanics and Electromagnetism) are the same for all of them. Different observers will measure (and then they see) different wavelengths and frequencies, but the product of wavelength with the frequency is the same. The wavelength and frequency are thus Lorentz covariant, meaning that they change for different observers according some fixed mathematical prescription depending on its tensorial character (scalar, vector, tensor,…) respect to Lorentz transformations. The speed of light is Lorentz invariant.
By the other hand, Newton’s law of gravity describes the motion of planets and terrrestrial bodies. It is all that we need in contemporary rocket ships unless those devices also carry atomic clocks or other tools of exceptional accuracy. Here is Newton’s law in potential form:
4\pi G\rho = \nabla ^2 \phi
In the special relativity framework, this equation has a terrible problem: if there is a change in the mass density \rho, then it must propagate everywhere instantaneously. If you believe in the Special Relativity rules and in the speed of light invariance, it is impossible. Therefore, “Houston, we have a problem”.
Einstein was aware of it and he tried to solve this inconsistency. The final solution took him ten years .
The apparent silly and easy problem is to develop and describe all physics in the the same way irrespectively one is accelerating or not. However, it is not easy or silly at all. It requires deep physical insight and a high-end mathematical language. Indeed, what is the most difficult part are the details of Riemann geometry and tensor calculus on manifolds. Einstein got private aid from a friend called Marcel Grossmann. In fact, Einstein knew that SR was not compatible with Newton’s law of gravity. He (re)discovered the equivalence principle, stated by Galileo himself much before than him, but he interpreted deeper and seeked the proper language to incorporante that principle in such a way it were compatible (at least locally) with special relativity! His “journey” from 1907 to 1915 was a hard job and a continuous struggle with tensorial methods…
Today, we are going to derive the Einstein field equations for gravity, a set of equations for the “metric field” g_{\mu \nu}(x). Hilbert in fact arrived at Einstein’s field equations with the use of the variational method we are going to use here, but Einstein’s methods were more physical and based on physical intuitions. They are in fact “complementary” approaches. I urge you to read “The meaning of Relativity” by A.Einstein in order to read a summary of his discoveries.
We now proceed to derive Einstein’s Field Equations (EFE) for General Relativity (more properly, a relativistic theory of gravity):
Step 1. Let us begin with the so-called Einstein-Hilbert action (an ansatz).
S = \int d^4x \sqrt{-g} \left( \dfrac{c^4}{16 \pi G} R + \mathcal{L}_{\mathcal{M}} \right)
Be aware of the square root of the determinant of the metric as part of the volume element. It is important since the volume element has to be invariant in curved spacetime (i.e.,in the presence of a metric). It also plays a critical role in the derivation.
Step 2. We perform the variational variation with respect to the metric field g^{\mu \nu}:
\delta S = \int d^4 x \left( \dfrac{c^4}{16 \pi G} \dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} + \dfrac{\delta (\sqrt{-g}\mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}} \right) \delta g^{\mu \nu}
Step 3. Extract out the square root of the metric as a common factor and use the product rule on the term with the Ricci scalar R:
\delta S = \int d^4 x \sqrt{-g} \left( \dfrac{c^4}{16 \pi G} \left ( \dfrac{\delta R}{\delta g^{\mu \nu}} +\dfrac{R}{\sqrt{-g}}\dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} \right) +\dfrac{1}{\sqrt{-g}}\dfrac{\delta ( \sqrt{-g}\mathcal{L}_{\mathcal{M}})}{\delta g^{\mu\nu}}\right) \delta g^{\mu \nu}
Step 4. Use the definition of a Ricci scalar as a contraction of the Ricci tensor to calculate the first term:
\dfrac{\delta R}{\delta g^{\mu \nu}} = \dfrac{\delta (g^{\mu \nu}R_{\mu \nu})}{\delta g^{\mu \nu} }= R_{\mu\nu} + g^{\mu \nu}\dfrac{\delta R_{\mu \nu}}{\delta g^{\mu \nu}} = R_{\mu \nu} + \mbox{total derivative}
A total derivative does not make a contribution to the variation of the action principle, so can be neglected to find the extremal point. Indeed, this is the Stokes theorem in action. To show that the variation in the Ricci tensor is a total derivative, in case you don’t believe this fact, we can proceed as follows:
Check 1. Write the Riemann curvature tensor:
R^{\rho}_{\, \sigma \mu \nu} = \partial _{\mu} \Gamma ^{\rho}_{\, \sigma \nu} - \partial_{\nu} \Gamma^{\rho}_{\, \sigma \mu}+ \Gamma^{\rho}_{\, \lambda \mu} \Gamma^{\lambda}_{\, \sigma \nu} - \Gamma^{\rho}_{\, \lambda \nu} \Gamma^{\lambda}_{\, \sigma \mu}
Note the striking resemblance with the non-abelian YM field strength curvature two-form
F=dA+A \wedge A = \partial _{\mu} A_{\nu} - \partial _{\nu} A_{\mu} + k \left[ A_\mu , A_{\nu} \right].
There are many terms with indices in the Riemann tensor calculation, but we can simplify stuff.
Check 2. We have to calculate the variation of the Riemann curvature tensor with respect to the metric tensor:
\delta R^{\rho}_{\, \sigma \mu \nu} = \partial _{\mu} \delta \Gamma^{\rho}_{\, \sigma \nu} - \partial_\nu \delta \Gamma^{\rho}_{\, \sigma \mu} + \delta \Gamma ^{\rho}_{\, \lambda \mu} \Gamma^{\lambda}_{\, \sigma \nu} - \delta \Gamma^{\rho}_{\lambda \nu}\Gamma^{\lambda}_{\, \sigma \mu} + \Gamma^{\rho}_{\, \lambda \mu}\delta \Gamma^{\lambda}_{\sigma \nu} - \Gamma^{\rho}_{\lambda \nu} \delta \Gamma^{\lambda}_{\, \sigma \mu}
One cannot calculate the covariant derivative of a connection since it does not transform like a tensor. However, the difference of two connections does transform like a tensor.
Check 3. Calculate the covariant derivative of the variation of the connection:
\nabla_{\mu} ( \delta \Gamma^{\rho}_{\sigma \nu}) = \partial _{\mu} (\delta \Gamma^{\rho}_{\, \sigma \nu}) + \Gamma^{\rho}_{\, \lambda \mu} \delta \Gamma^{\lambda}_{\, \sigma \nu} - \delta \Gamma^{\rho}_{\, \lambda \sigma}\Gamma^{\lambda}_{\mu \nu} - \delta \Gamma^{\rho}_{\, \lambda \nu}\Gamma^{\lambda}_{\, \sigma \mu}
\nabla_{\nu} ( \delta \Gamma^{\rho}_{\sigma \mu}) = \partial _\nu (\delta \Gamma^{\rho}_{\, \sigma \mu}) + \Gamma^{\rho}_{\, \lambda \nu} \delta \Gamma^{\lambda}_{\, \sigma \mu} - \delta \Gamma^{\rho}_{\, \lambda \sigma}\Gamma^{\lambda}_{\mu \nu} - \delta \Gamma^{\rho}_{\, \lambda \mu}\Gamma^{\lambda}_{\, \sigma \nu}
Check 4. Rewrite the variation of the Riemann curvature tensor as the difference of two covariant derivatives of the variation of the connection written in Check 3, that is, substract the previous two terms in check 3.
\delta R^{\rho}_{\, \sigma \mu \nu} = \nabla_{\mu} \left( \delta \Gamma^{\rho}_{\, \sigma \nu}\right) - \nabla _{\nu} \left(\delta \Gamma^{\rho}_{\, \sigma \mu}\right)
Check 5. Contract the result of Check 4.
\delta R^{\rho}_{\, \mu \rho \nu} = \delta R_{\mu \nu} = \nabla_{\rho} \left( \delta \Gamma^{\rho}_{\, \mu \nu}\right) - \nabla _{\nu} \left(\delta \Gamma^{\rho}_{\, \rho \mu}\right)
Check 6. Contract the result of Check 5:
g^{\mu \nu}\delta R_{\mu \nu} = \nabla_\rho (g^{\mu \nu} \delta \Gamma^{\rho}_{\mu\nu})-\nabla_\nu (g^{\mu \nu}\delta \Gamma^{\rho}_{\rho \mu}) = \nabla _\sigma (g^{\mu \nu}\delta \Gamma^{\sigma}_{\mu \nu}) - \nabla_\sigma (g^{\mu \sigma}\delta \Gamma ^{\rho}_{\rho \mu})
Therefore, we have
g^{\mu \nu}\delta R_{\mu \nu} = \nabla_\sigma (g^{\mu \nu}\delta \Gamma^{\sigma}_{\mu\nu}- g^{\mu \sigma}\delta \Gamma^{\rho}_{\rho\mu})=\nabla_\sigma K^\sigma
Step 5. The variation of the second term in the action is the next step. Transform the coordinate system to one where the metric is diagonal and use the product rule:
\dfrac{R}{\sqrt{-g}} \dfrac{\delta \sqrt{-g}}{\delta g^{\mu \nu}}=\dfrac{R}{\sqrt{-g}} \dfrac{-1}{2 \sqrt{-g}}(-1) g g_{\mu \nu}\dfrac{\delta g^{\mu \nu}}{\delta g^{\mu \nu}} =- \dfrac{1}{2}g_{\mu \nu} R
The reason of the last equalities is that g^{\alpha\mu}g_{\mu \beta}=\delta^{\alpha}_{\; \beta}, and then its variation is
\delta (g^{\alpha\mu}g_{\mu \nu}) = (\delta g^{\alpha\mu}) g_{\mu \nu} + g^{\alpha\mu}(\delta g_{\mu \nu}) = 0
Thus, multiplication by the inverse metric g^{\beta \nu} produces
\delta g^{\alpha \beta} = - g^{\alpha \mu}g^{\beta \nu}\delta g_{\mu \nu}
that is,
\dfrac{\delta g^{\alpha \beta}}{\delta g_{\mu \nu}}= -g^{\alpha \mu} g^{\beta \nu}
By the other hand, using the theorem for the derivation of a determinant we get that:
\delta g = \delta g_{\mu \nu} g g^{\mu \nu}
\dfrac{\delta g}{\delta g^{\alpha \beta}}= g g^{\alpha \beta}
because of the classical identity
g^{\alpha \beta}=(g_{\alpha \beta})^{-1}=\left( \det g \right)^{-1} Cof (g)
Cof (g) = \dfrac{\delta g}{\delta g^{\alpha \beta}}
and moreover
\delta \sqrt{-g}=-\dfrac{\delta g}{2 \sqrt{-g}}= -g\dfrac{ \delta g_{\mu \nu} g^{\mu \nu}}{2 \sqrt{-g}}
\delta \sqrt{-g}=\dfrac{1}{2}\sqrt{-g}g^{\mu \nu}\delta g_{\mu \nu}=\dfrac{1}{2}\sqrt{-g}g_{\mu \nu}\delta g^{\mu \nu}
Step 6. Define the stress energy-momentum tensor as the third term in the action (that coming from the matter lagrangian):
T_{\mu \nu} = - \dfrac{2}{\sqrt{-g}}\dfrac{(\sqrt{-g} \mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}}
or equivalently
-\dfrac{1}{2}T_{\mu \nu} = \dfrac{1}{\sqrt{-g}}\dfrac{(\sqrt{-g} \mathcal{L}_{\mathcal{M}})}{\delta g^{\mu \nu}}
Step 7. The extremal principle. The variation of the Hilbert action will be an extremum when the integrand is equal to zero:
\dfrac{c^4}{16\pi G}\left( R_{\mu \nu} - \dfrac{1}{2} g_{\mu \nu}R\right) - \dfrac{1}{2} T_{\mu \nu} = 0
\boxed{R_{\mu \nu} - \dfrac{1}{2}g_{\mu \nu} R = \dfrac{8\pi G}{c^4}T_{\mu\nu}}
Usually this is recasted and simplified using the Einstein’s tensor
G_{\mu \nu}= R_{\mu \nu} - \dfrac{1}{2}g_{\mu \nu} R
\boxed{G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}}
This deduction has been mathematical. But there is a deep physical picture behind it. Moreover, there are a huge number of physics issues one could go into. For instance, these equations bind to particles with integral spin which is good for bosons, but there are matter fermions that also participate in gravity coupling to it. Gravity is universal. To include those fermion fields, one can consider the metric and the connection to be independent of each other. That is the so-called Palatini approach.
Final remark: you can add to the EFE above a “constant” times the metric tensor, since its “covariant derivative” vanishes. This constant is the cosmological constant (a.k.a. dark energy in conteporary physics). The, the most general form of EFE is:
\boxed{G_{\mu\nu}+\Lambda g_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}}
Einstein’s additional term was added in order to make the Universe “static”. After Hubble’s discovery of the expansion of the Universe, Einstein blamed himself about the introduction of such a term, since it avoided to predict the expanding Universe. However, perhaps irocanilly, in 1998 we discovered that the Universe was accelerating instead of being decelerating due to gravity, and the most simple way to understand that phenomenon is with a positive cosmological constant domining the current era in the Universe. Fascinating, and more and more due to the WMAP/Planck data. The cosmological constant/dark energy and the dark matter we seem to “observe” can not be explained with the fields of the Standard Model, and therefore…They hint to new physics. The character of this new physics is challenging, and much work is being done in order to find some particle of model in which dark matter and dark energy fit. However, it is not easy at all!
May the Einstein’s Field Equations be with you!
LOG#103. Numbers: the list.
numbersHello, eager earthlings! Today, my list is about fascinating number types. I love them all so much…
1) Natural numbers.
2) Integer numbers.
3) Fractional numbers.
4) Irrational numbers.
5) Real numbers.
6) Complex numbers.
7) Quaternions.
8) Octonions/Octaves/Cayley numbers. (John C. Baez’s favourite numbers!)
9) Sedenions.
10) Hypernumbers (Cayley-Dickson algebras).
11) Grassmann numbers/Grassmannian numbers/anticommuting c-numbers/supernumbers.
12) Clifford numbers.
13) p-adic numbers.
14) Adelic/idelic numbers (The adelic ring).
15) Ternary numbers and n-ary numbers.
16) q-numbers, or xy-qxy, and their q-deformed generalizations.
17) Tropical numbers.
18) Polygonal numbers.
19) Modular numbers.
20) Surreal numbers.
21) Transfinite numbers.
Do you know a cool type of number I should add to “my list”? Let me know…
May the numbers be with you!!!!!
LOG#102. Superstuff: the list.
Hello, Earth planet! Hello, earthlings!
I want to share with you my list of favourite “superstuff”…Enjoy it:
1) Supersymmetry.
2) Superspace.
3) Supermatrix.
4) Superdeterminant.
5) Supergravity.
6) Superstrings.
7) Super p-branes. p=-1,0,1,2,\ldots. Question: What about p=-1,-2,\ldots or “fractional branes”?
8) Supertwistors.
9) Supermatrices.
10) Superstatistics.
11) Supertime.
12) Superheterodyne. (lol, I know, I know)
13) Superextendon.
14) Superconnection.
15) Superconductivity and superconductors.
16) Superfluids and superfluidity.
17) Superinsulators.
18) Superalloy.
19) Supermassive black hole.
20) Supersymmetric object (=algebra, particle, gauge theory, quantum mechanics, field theory, …)
21) Superconformal symmetry/group/algebra.
22) Supergroups.
23) Supermanifolds.
24) Super linear algebra.
25) Superluminal.
26) Superbradyons/elvisebrions.
27) Supercomputer.
28) Supercapacitor.
29) Supernovae.
30) Superoxide.
31) Superorganism.
32) Superposition principle.
33) Superpower.
34) Superpotential.
35) Superpolynomial.
36) Supergraph.
37) Superreal number.
38) Superresolution.
39) Superring.
40) Superset.
41) SuperEarth.
42) Superstrong force.
43) Superuser.
44) Superunification.
45) Supervisor.
46) Supervectors, supervector spaces.
47) Supervoid.
48) Superworld, superwarp, superforce, supermathematics.
49) Superzoom.
50) (Just for fun) Superman, supergeek, supergeeknerd, superfriends, superstars, superhero, superstudent (übermensch, übergeek, übergeeknerd, überfriends, überstars, überhero, überstudent).
Would you add something else to this list? Let me know…
May the Superforce be with you!
LOG#051. Zeta Zoology.
This log-entry is an exploration journey… To boldly go, where no zeta function has gone before…
Riemann zeta function
The Riemann zeta function is an object related to prime numbers. In general, it is a function of complex variable defined by the next equation:
\boxed{\displaystyle{\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}\dfrac{1}{n^s}=\prod_{p=2}^{\infty}\dfrac{1}{1-p^{-s}}=\prod_{p}\dfrac{1}{1-p^{-s}}}}
\boxed{\displaystyle{\zeta (s)=\dfrac{1}{1-2^{-s}}\dfrac{1}{1-3^{-s}}\ldots\dfrac{1}{1-137^{-s}}\ldots}}
The Jacobi’s theta function is the Mellin transform of Riemann zeta function Jacobi theta function is
\boxed{\displaystyle{\theta (\tau)=\sum_{n=-\infty}^{\infty}e^{\pi i n^2\tau}}}
and then
\boxed{\displaystyle{\zeta (s)=\dfrac{\pi^{s/2}}{2\Gamma (\frac{s}{2})}\int_0^\infty \theta (it)t^{s/2-1}dt}}
Applications: number theory, mathematics, physics, physmatics.
Related ideas: Hilbert-Polya approach, Riemann hypothesis, riemannium, primon gas/free Riemann gas, functional determinant, prime number distribution, Jacobi’s theta function.
Dirichlet eta function
This function is indeed the Riemann zeta function with alternating plus/minus signs. In other words:
\boxed{\displaystyle{\eta (s)=\sum_{n=1}^{\infty}(-1)^{n+1}n^{-s}=\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^s}=\left(1-2^{1-s}\right)\zeta (s)}}
Applications: physmatics.
Related ideas: Riemann zeta function.
Reciprocal Riemann zeta function
Reciprocal zeta function is the following modification of the Riemann zeta function:
\boxed{\displaystyle{\dfrac{1}{\zeta (s)}=\sum_{n=1}^{\infty}\mu (n)n^{-s}=\sum_{n=1}^{\infty}\dfrac{\mu (n)}{n^s}}}
where the Möbius function \mu (n) is defined as follows
\mu (n)=\begin{cases}1\;\; \mbox{if n is a square-free positive integer with even number of prime factors}\\ -1\;\; \mbox{if n is a square-free positive integer with odd number of prime factors}\\ 0\;\; \mbox{if n is not square-free }\end{cases}
A number is said to be square-free if it is not divisible by a number which is a perfect square (excepting the number one). An alternative definition of the Möbius function is given by:
\mu (n)=\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}\;\; \mbox{if}\;\;\omega (n)=\Omega (n)\\ 0\;\;\mbox{if}\;\;\omega (n)<\Omega (n)\end{cases}
and where \omega (n) is the number of different primes dividing the number n and \Omega (n) is the number of prime factors of n, counted with multiplicities. Clearly, the inequality \omega (n)\leq \Omega (n) is satisfied. Moreover, note that \mu (1)=1 and \mu (0) is undefined.
Indeed, we also have:
\boxed{\displaystyle{\dfrac{1}{\zeta (s)}=\left( \prod_p^\infty \dfrac{1}{1-p^{-s}} \right)^{-1}=\prod_p^\infty \left( 1-\dfrac{1}{p^s}\right)}}
This result is important for the so-called Dirichlet generating series:
\boxed{\displaystyle{\dfrac{\zeta (s)}{\zeta (2s)}=\sum_{n=1}^{\infty} \dfrac{\vert\mu (n)\vert }{n^{s}}=\prod_p^\infty \left(1+p^{-s}\right)}}
By the other hand, since
\boxed{\displaystyle{\dfrac{1}{\zeta(s)}=\prod_{p}^\infty (1-p^{-s}) = \sum_{n=1}^{\infty} \dfrac{\mu (n)}{n^{s}}}}
taking the ratio between these last two results, we obtain the beautiful equation
\boxed{\displaystyle{\dfrac{\zeta(s)^2}{\zeta(2s)}=\prod_{p} \left(\dfrac{1+p^{-s}}{1-p^{-s}}\right) = \prod_{p} \left(\dfrac{p^{s}+1}{p^{s}-1}\right)}}
The Liouville function \lambda (n) is defined similarly to the Möbius function. If n is a positive integer, it is:
\lambda (n)=(-1)^{\Omega (n)}
Using the sum of the geometric series, we get:
\boxed{\displaystyle{\zeta(s)=\prod_{p} (1-p^{-s})^{-1}=\prod_{p} \left(\sum_{n=0}^{\infty}p^{-ns}\right) =\sum_{n=1}^{\infty} \dfrac{1}{n^{s}}}}
while if we use the Liouville function, we could write
\boxed{\displaystyle{\dfrac{\zeta(2s)}{\zeta(s)}=\prod_{p} (1+p^{-s})^{-1} = \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^{s}}}}
There is other remarkable family of infinite products
\boxed{\displaystyle{\prod_{p} (1+2p^{-s}+2p^{-2s}+\cdots) = \sum_{n=1}^{\infty}2^{\omega(n)} n^{-s} = \dfrac{\zeta(s)^2}{\zeta(2s)}}}
where again \omega(n) counts the number of distinct prime factors of n and 2^{\omega(n)} is the number of square-free divisors. Furthermore, if \chi (n) is a Dirichlet character of conductor N, so that \chi is totally multiplicative and \chi (n) only depends on n \;(mod N), and \chi (n)=0 if n is not coprime to N, then the following identity holds
\boxed{\displaystyle{\prod_{p} (1- \chi(p) p^{-s})^{-1} = \sum_{n=1}^{\infty}\chi(n)n^{-s}}}
Here it is convenient and common to omit the primes p dividing the conductor N from the product.
Hurwitz zeta function
It is the the generalization of Riemann zeta function given by the next sum:
\boxed{\displaystyle{\zeta (s,Q)=\sum_{n=0}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}=\dfrac{1}{Q^s}+\sum_{n=1}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}}}
Remark: the mathematica code for this function is Zeta[s,Q].
Multiple zeta value/Euler sum/Polyzeta
Multiple zeta values, also called polyzeta function or Euler sums are certain “coloured” generalizations (in several variables) of the Riemann zeta function:
\boxed{\displaystyle{\zeta (s_1,s_2,\ldots,s_m)=\sum_{n_1>n_2>\ldots>n_m>0}^\infty\dfrac{1}{n_1^{s_1}n_2^{s_2}\cdots n_m^{s_m}}=\sum_{n_1>n_2>\ldots>n_m>0}^\infty \prod_{j=1}^m \dfrac{1}{n_j^{s_j}}}}
Polylogarithm/Coloured polylogarithm
The polygogarithm is the following generalization of Riemann zeta function:
\boxed{\displaystyle{\mbox{Li}_s (z)=\sum_{n=1}^{\infty}\dfrac{z^n}{n^s}=\sum_{n=1}^{\infty}z^n n^{-s}}}
There are coloured versions of the polylogarithm:
\boxed{\displaystyle{\mbox{Li}_{ (s_1,s_2,\ldots,s_m) }(z_1,z_2,\ldots,z_m)=\sum_{n_1>n_2>\ldots>n_m>0}^\infty\dfrac{z_1^{s_1}z_2^{s_2}\cdots z_m^{s_m}}{n_1^{s_1}n_2^{s_2}\cdots n_m^{s_m}}=\sum_{n_1>n_2>\ldots>n_m>0}^\infty \prod_{j=1}^m \dfrac{z_j^{s_j}}{n_j^{s_j}}}}
Lerch-zeta function/Lerch-trascendent
The Lerch-zeta function is defined with the sum:
\boxed{\displaystyle{L(\lambda, Q,s)=\sum_{n=0}^{\infty}\dfrac{e^{2\pi i \lambda n}}{(n+Q)^s}}}
The Lerch trascendent is the function
\boxed{\displaystyle{\Phi (z,s,Q)=\sum_{n=0}^{\infty}\dfrac{z^n}{(n+Q)^s}}}
Lerch-zeta function and Lerch trascendent are related through the functional equation
\Phi ( e^{2\pi i\lambda},s,Q)=L( \lambda ,Q,s)
Mordell-Tornheim zeta values
Defined by Matsumoto in 2003, these zeta functions are:
\boxed{\displaystyle{\zeta_{MT,r} (s_1,s_2,\ldots,s_r; s_{r+1})=\sum_{m_1>\cdots>m_r>0}\dfrac{1}{m_1^{s_1}\cdots m_r^{s_r} (m_1+\ldots+m_r )^{s_{r+1}}}}}
Barnes zeta function
This function is the sum
\boxed{\displaystyle{\zeta_N ( s,\omega\vert a_1,\ldots,a_N)=\sum_{n_1\ldots n_N\geq 0}\dfrac{1}{\left(\omega+n_1a_1+\cdots+n_N a_N\right)^s}}}
where \omega, a_j are numbers such that Re(\omega)>0, Re(a_j)>0 and the sum is defined for all complex number s whenever Re(s)>N.
Airy zeta function
Let a_i \forall i=1,2,\ldots,\infty be the zeros of the Airy function \mbox{Ai} (x). Then, the Airy zeta function is the sum:
\boxed{\displaystyle{\zeta_{Ai} (s)=\sum_{i=1}^{\infty}\dfrac{1}{\vert a_i\vert ^s}}}
Arithmetic zeta function
The arithmetic zeta function over some scheme X is defined to be the sum:
\displaystyle{\zeta _X (s)=\prod_x \left(1-N(x)^{-s}\right)^{-1}}
where the product is taken on every closed point of the scheme X.
The generalized Riemann hypothesis over the scheme X is the hypothesis that the zeros of such arithmetic function, i.e, the feynmanity \zeta_X (s)=0, and its poles are found in the next way:
\boxed{\zeta_X (s)=0\leftrightarrow \begin{cases}\mbox{Zeroes at}\;\;\mbox{Re}(s)=\dfrac{1}{2},\dfrac{3}{2},\ldots,\infty\\ \mbox{Poles at}\;\;\mbox{Re}(s)=0,1,2,\ldots,\infty\end{cases}}
inside the critical strip.
Artin-Mazur zeta function
Let us define:
1st. \mbox{Fix}(f^n) is the the set of fixed points of the nth iterated function f^n of f.
2nd. \mbox{Card(Fix)}(f^n) is the cardinality of the set \mbox{Fix}(f^n), i.e., the number of elements of such a set.
Then, the Artin-Mazur zeta function is the zeta function given by the next formula:
\boxed{\displaystyle{\zeta_f (s)=\exp \left(\sum_{n=1}^{\infty} \mbox{Card(Fix)}\left[ f^n\right]\dfrac{z^n}{n}\right)}}
Dedekind zeta function
Let us define:
1st. K is an algebraic number field.
2nd. I is the range of non zero ideals of the ring of integers \mathcal{O}_K of K.
3rd. N_{K/Q} (I) is the aboslute norm of I. When K=\mathbb{Q} we get the usual Riemann zeta function.
Then, the Dedekind zeta function is the sum
\boxed{\displaystyle{\zeta_K (s)=\sum_{I\subseteq \mathcal{O}_K}\dfrac{1}{\left(N_{K/\mathbb{Q}}(I)\right)^s}}}
where \mbox{Re}(s)>1.
Epstein zeta function/Eisenstein series
\boxed{\displaystyle{\zeta_Q (s)=\sum_{(m,n)\neq (0,0)}\dfrac{1}{Q(m,n)^s}}}
where we have defined Q(m,n) as the quadratic form Q(m,n)=cm^2+bmn+an^2. A related concept is the Eisenstein (not confuse with Einstein, please)
\boxed{\displaystyle{E(z,s)=\dfrac{1}{2}\sum_{(m,n)=1}\dfrac{y^s}{\vert mz+n\vert^{2s}}}}
where \mbox{Re}(s)>1 and the sum is taken on every pari of coprime integers. Two integers A and B are said to be coprime (also spelled co-prime) or relatively prime if the only positive integer that evenly divides both of them is 1.
There is a relation with modular forms/automorphic forms as well. Let \tau be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G_{2k}(\tau) of weight 2k, where k\geq 2 is an integer, by the series:
\boxed{\displaystyle{G_{2k}(\tau) = \sum_{ (m,n)\in\mathbb{Z}^2\backslash(0,0)} \dfrac{1}{(m+n\tau )^{2k}}}}
It is absolutely convergent to a holomorphic function of \tau in the upper half-plane and its Fourier expansion given below shows that it can be extended to a holomorphic function at \tau=i\infty. It is a remarkable and surprising fact that the Eisenstein series is a modular form. Indeed, the key property is its SL_2(\mathbb{Z})-invariance. Explicitly if a,b,c,d \in \mathbb{Z} and ad-bc=1 then the next group property is satisfied
\displaystyle{G_{2k} \left( \dfrac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)}
and G_{2k} is therefore a modular form of weight 2k.
Remark: it is important to assume that k\geq 2, otherwise it would be illegitimate to change the order of summation, and the SL_2(\mathbb{Z})-invariance would not remain. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for k=1, although it would only be what mathematicians call a quasimodular form.
Ihara zeta function
This zeta function appears in graph theory and it has an amazing set of useful identities. The Ihara zeta function is the sum:
\boxed{\displaystyle{\zeta_G (u)=\prod_{p}\left( 1-u^{L(p)}\right)^{-1}}}
where the product runs over every prime walk p of the graph G(E,V), i.e., it is taken over closed cycles p=(u_0,u_1,\ldots,u_{L(p)-1};u_0) such as (u_i,u_{(i+2)\mbox{mod}}\; L(p))\in E with u_i\neq u_{(i+2)\mbox{mod}\;L(p)} and L(p) is equal to the length of the cycle p.
The Ihara formula is a key result in graph theory
\boxed{\zeta_G (u)=\dfrac{\left(1-u^2\right)^{\chi (G)}}{\det \left(I-Au-(k-1)u^2(I)\right)}}
and there \chi (G) is the circuit rank, i.e., it is the cyclomatic number of an undirected graph G or the minimum number r of edges necessary to remove from G all its cycles, making it into a forest (graph without cycles, a fores is only a disjoint union of “trees”). Finally, if T is the Hashimoto’s edge adjacency operator, then
\boxed{\displaystyle{\zeta_G (u)=\dfrac{1}{\det (1-Tu)}}}
Lefschetz zeta function
Given a map f, the Lefschetz zeta function is defined as the series
\boxed{\displaystyle{\zeta_f (s)=\exp \left[\sum_{n=1}^\infty L(f^n)\dfrac{z^n}{n}\right]}}
Here, L(f^n) is the Lefschetz number of the n-th iterated f^n of the function f. To see what the Lefschetz number is, click here http://en.wikipedia.org/wiki/Lefschetz_number
Matsumoto zeta function
A class of zeta functions defined by Matsumoto around 1990. They are functions
\boxed{\displaystyle{\phi (s)=\prod_p\dfrac{1}{A_p (p^{-s})}}}
where p is a prime number and A_p is certain polynomial.
Minakshisundaram-Pleijel zeta function
A type of zeta function encoding the eigenvalues of a Lapalacian of a compact riemannian manifold \mathcal{M}. If \mbox{dim}\mathcal{M}=N and the eigenvalues of the Laplace-Beltrami operator are the set \left(\lambda_1,\lambda_2,\ldots\right), then the Minakshisundaram-Pleijel zeta function is defined as the following series (where we removed the zero eigenvalues from the sum and \mbox{Re}(s)>>1, i.e., the real part of s is large enough):
\boxed{\displaystyle{\mathcal{Z} (s)=\mbox{Tr}(A^{-s})=\sum_{n=1, \lambda_n\neq 0}^\infty \vert \lambda_n\vert^{-s}}}
Prime Zeta function
The next function was defined by Fröberg, Cohen and Glaisher, with the only subtle point of being careful to consider 1 as a prime in the sum or not and the notation they used:
\boxed{\displaystyle{P(s)=\sum_p\dfrac{1}{p^s}=\sum_p p^{-s}}}
Note that such a function is a “prime” version of the Riemann zeta function:
\displaystyle{\zeta (s)=\sum_{k=1}^\infty k^{-s}}
Remark: Cohen used a different notation for P(s). He used P(s)=S_s instead of the Fröberg’s and Glaisher notation.
Remark (II): Interestingly, the prime zeta function has the following behaviour close to the axis s=1
P(1+\varepsilon)=-\ln \varepsilon+C+\mathcal{O}(\varepsilon)
\displaystyle{C=\sum_{n=2}^\infty \dfrac{\mu (n)}{n}\ln \zeta (n)\approx -0.315718452\ldots}
This prime zeta function is related to the Riemann zeta function:
\displaystyle{\ln \zeta (s)=-\sum_{p\geq 2}\ln \left(1-p^{-s}\right)=\sum_{p\geq 2}\sum_{k=1}^\infty\dfrac{p^{-ks}}{k}}
\boxed{\displaystyle{\ln \zeta (s)=\sum_{k=1}^\infty\dfrac{1}{k}\sum_{p\geq 2}p^{-ks}=\sum_{k=1}^\infty\dfrac{P(ks)}{k}=\sum_{n>0}\dfrac{P(ns)}{n}=\sum_{n=1}^\infty\dfrac{P(ns)}{n}}}
This equation and definition can be inverted (the original inversion procedure was carried by Glaisher around 1891, it is recalled by Fröberg about 1968, and it was studied later by Cohen, circa 2000):
\boxed{\displaystyle{P(s)=\sum_{k=1}^\infty \dfrac{\mu (k)}{k}\ln \left( \zeta (ks)\right)}}
Remark: the mathematica code for the prime zeta function is PrimeZetaP[s] and Zeta[s] for the Riemann zeta function.
Remark (II): \displaystyle{P(1)=\sum \dfrac{1}{p}=\infty}
Remark (III): Fröberg (1968) stated that very little is known about the prime zeta function zeroes in the complex plane, i.e., the solutions to P(s)=0. Unlike the Riemann zeroes, it seems that prime zeta function zeroes are not on a straight line, but there is no known pattern, if any.
Remark (IV): Despite the divergence of P(1), dropping the initial term and adding the Euler-Mascheroni constant \gamma_E\approx 0.577\cdots provides a new constant! It is called Mertens constant. That is,
\displaystyle{\mbox{MERTENS CONSTANT}=B_1=\gamma_E+\sum_{m=2}^\infty\dfrac{\mu (m)}{m}\ln \left(\zeta (m)\right)\approx 0.2614972128\ldots}
Remark (V): The Artins constant C_{A} is related to P(n) as well
\displaystyle{\ln C_A=-\sum_{n=2}^\infty \dfrac{(L_n-1)P(n)}{n}}
and where L_n is the n-th Lucas number.
Remark (VI): The prime zeta function has the next asymptotical behaviour close to s=1
P(s)\rightarrow P(s)\approx \ln \zeta (s)\sim \ln \left(\dfrac{1}{s-1}\right)
Ruelle zeta function
Let’s define the following concepts:
1st. f is certain function or map on a manifold M.
2nd. \mbox{Fix}(f^n) is the set of fixed points of the nth iterated function f^n of f, being such an iterated function a finite value.
3rd. \phi is certain function on M with values or entries in d\times d complex matrices. The case d=1, \phi=1 corresponds to the Artin-Mazur zeta function.
The Ruelle zeta function is the object defined with the series
\boxed{\displaystyle{\zeta (z)=\exp \left(\sum_{m>\geq 1}\dfrac{z^m}{m}\sum_{x\in \mbox{Fix}(f^m)}\mbox{Tr}\left(\prod_{k=0}^{m-1}\phi \left[ f^k(x)\right]\right)\right)}}
Selberg zeta function
This zeta function is related to a compact ( of finite volume) Riemannian manifold. Assuming that certain manifolf M has constant curvature -1, it can be realized as a quotient of the Poincaré upper half plane
H=\{x+iy\vert x, y\in \mathbb{R},y>0\}
The Poincaré arc length is defined in this space as
and it can be shown to be invariant under fractional linear transformations
z\rightarrow z'=\dfrac{az+b}{cz+d}
with a,b,c,d\in \mathbb{R} and ad-bc>0. Indeed, it is not hard to prove that the geodesics (curves minimizing the Poincaré arc length) are half lines and semicircles in H orthogonal to the real axis. Calling these lines as geodesics creates a model of hyperbolic geometry, i.e., a non-euclidean model for geometry where the 5th Euclid postulate is not longer valid. In fact, there are infinitely many geodesics through a fixed point not meeting a given geodesic. The fundamental group \Gamma of M acts as a discrete group of transformations preserving distances between points. The favourite group between number theorists is called the modular group \Gamma =SL(2,\mathbb{Z}) of 2\times 2 matrices of determinant one and integer entries in the quotien space \overline{\Gamma}=\Gamma/\{\pm I\}. However, the Riemann surface M=SL(2,\mathbb{Z})/H is noncompact, although it does have finite volume. Selberg introduced “prime numbers” in the compact surface M=\Gamma/H to be “primitive cycles” or more precisely “primitive closed geodesics” C in M. There, the word “primitive” means that you can only go around the curve once. Furthermore, the Selberg zeta function, for \text{Re} (s) large enough, is defined to be the sum
\boxed{\displaystyle{Z(s)=\prod_{\left[C\right]}\prod_{j\geq 1}\left(1-e^{(s+j)\nu (C)}\right)}}
and where the product is extended over every primitive closed geodesics C in M=\Gamma/H of Poincaré length \nu (C). By the Selberg trace formula (which we are not goint to discuss here today), there is a duality between the lengths of the primes and the spectrum of the Laplace operator on M. Here, the Laplacian on M is
\Delta =y^2\left(\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2}\right)
Indeed, it shows that one can show that the Riemann hypothesis (suitably modified to fit the situation) can be proved for Selberg zeta functions of compact Riemann surfaces! The closed geodesics in M=\Gamma/H correspond to geodesics in H itself. One can show that the endpoints of such geodesics in the real line \mathbb{R} (note that the real line is the boundary of the set H) are fixed by hyperbolic elements of \Gamma. That is, they are matrices
with trace a+d>2. Primitive closed geodesics correspond to hyperbolic elements that generate their own centralizer in \Gamma.
Shimizu zeta function
We define:
1st. K, a totally algebraid number field.
2nd. M, certain lattice in the field K.
3rd. V, the subgroup of maximal rank of the group of the totally positive units preserving the lattice structure.
Then, the Shimizu zeta function arises in the form
\boxed{\displaystyle{L(M,V,s)=\sum_{p\in \left[M-0\right]}\dfrac{\mbox{sign}N(\mu)}{\vert N(\mu)\vert^s}}}
Shintani zeta function
It is a generalized zeta series with the following formal definition
\boxed{\displaystyle{\zeta (s_1,s_2,\ldots,s_m)=\sum_{n_1,n_2,\ldots,n_m\geq 0}\dfrac{1}{L_1^{s_1}L_2^{s_2}\cdots L_m^{s_m}}}}
where L_j^{s_j} are inhomogeneous functions of (n_1,n_2,\ldots,n_m). Special cases of Shintani zeta function (or Shintani L-series, as they are also called by the mathematicians) are the Barnes zeta function or the Riemann zeta function.
Witten zeta function
Let G be a semisimple Lie group. The Witten L-series or Witten zeta function is defined by
\boxed{\displaystyle{\zeta_W (s)=\sum_{R}\dfrac{1}{\mbox{dim}(R)^s}}}
This sum is taken over the equivalence classes of irreducible representations R of G. Considering a root system \Delta of rank equal to r and with n positive roots in \Delta^+, being all simple without loss of generality, the simple roots \lambda_i allow us to define the Witten zeta function as a function of several variables:
\boxed{\displaystyle{\zeta_W (s_1,s_2,\ldots,s_n)=\sum_{m_1,m_2,\ldots,m_r> 0}\prod_{\alpha \in \Delta^+}\left[\dfrac{1}{\left(\alpha^V,m_1\lambda_1+m_2\lambda_2+\ldots+m_r\lambda_r\right)}\right]}}
Zeta function of an operator
The zeta function of any (pseudo)-differential operator \mathcal{P}, or more generally any operator, can be defined as the following functional series:
\boxed{\displaystyle{\zeta_{\mathcal{P}} (s)=\mbox{Tr}_\zeta (\mathcal{P}^{-s})}}
and where the trace \mbox{Tr}_\zeta is taken over the values s where such number exists (i.e., the zero modes are removed). In fact, the zeta function of an arbitrary operator, that we can call the zetor, is the formal series:
\boxed{\displaystyle{\zeta_{\mathcal{P}} (s)=\sum_{\lambda_i}\lambda_i^{-s}}}
It allow us to define the generalization of the determinant to \infty-dimensional operators in the following non-trivial way:
\boxed{\displaystyle{\det_{\zeta} \mathcal{P}=e^{-\zeta_{\mathcal{P}}^{'}(0)}}}
Dirichlet L-function/L-series
They are the formal series
\boxed{\displaystyle{L(\chi,s)=\sum_n\dfrac{\chi (n)}{n^s}=\prod_{p\;\; prime}\dfrac{1}{1-\chi (p)p^{-s}}}}
where \chi is a Dirichlet character with conductor f, i.e.,
\displaystyle{\sum_ {n=0}^\infty B_{n,\chi}\dfrac{t^n}{n!}=\sum_{n=1}^f\dfrac{\chi (n)te^{nt}}{e^{ft}-1}}
There, the generalized Bernoulli numbers are related to the L-series through the generating function above, and they satisfy the identity
p-adic zeta function
The p-adic analogue of the zeta function is defined with the following equation:
\zeta_p (s)\equiv \dfrac{1}{1-p^{-s}}
Moreover, we also define the zeta function at the infinite real prime:
\zeta_\infty(s) \equiv \pi^{-s/2}\Gamma \left(\dfrac{s}{2}\right)
The p-adic zeta function and the “real” prime zeta function (zeta function in the so-called “infinite prime”) satisfy the important adelic identity:
\displaystyle{\zeta_\infty (s)\prod_{p=2}^\infty \zeta_p (s)=\zeta_{\mathbb{A}}(s)}
where \zeta_{\mathbb{A}} (s)=\zeta_\infty (s)\zeta (s), and \zeta (s) is the classical Riemann zeta function. This adelic identity is just a special case of the adelic-type identity:
\displaystyle{\vert x\vert_\infty \prod_p\vert x\vert_p=1}
Stay tuned…The great adventure of Physmatics is just beginning!
LOG#050. Why riemannium?
This special 50th log-entry is dedicated to 2 special people and scientists who inspired (and guided) me in the hard task of starting and writing this blog.
These two people are
1st. John C. Baez, a mathematical physicist. Author of the old but always fresh/brand new This Week Finds in Mathematical Physics, and now involved in the Azimuth blog. You can visit him here
and here
I was a mere undergraduate in the early years of the internet in my country when I began to read his TWF. If you have never done it, I urge to do it. Read him. He is a wonderful teacher and an excellent lecturer. John is now worried about global warming and related stuff, but he keeps his mathematical interests and pedagogical gifts untouched. I miss some topics about he used to discuss often before in his hew blog, but his insights about virtually everything he is involved into are really impressive. He also manages to share his entusiastic vision of Mathematics and Science. From pure mathematics to physics. He is a great blogger and scientist!
2nd. The professor Francis Villatoro. I am really grateful to him. He tries to divulge Science in Spain with his excellent blog ( written in Spanish language)
He is a very active person in the world of Spanish Science (and its divulgation). In his blog, he also tries to explain to the general public the latest news on HEP and other topics related with other branches of Physics, Mathematics or general Science. It is not an easy task! Some months ago, after some time reading and following his blog (as I do now yet, like with Baez’s stuff), I realized that I could not remain as a passive and simple reader or spectator in the web, so I wrote him and I asked him some questions about his experience with blogging and for advice. His comments and remarks were incredibly useful for me, specially during my first logs. I have followed several blogs the last years (like those by Baez or Villatoro), and I had no idea about what kind of style/scheme I should addopt here. I had only some fuzzy ideas about what to do, what to write and, of course, I had no idea if I could explain stuff in a simple way while keeping the physical intuition and the mathematical background I wanted to include. His early criticism was very helpful, so this post is a tribute for him as well. After all, he suggested me the topic of this post! I encourage you to read him and his blog (as long as you know Spanish or you can use a good translator).
Finally, let me express and show my deepest gratitude to John and Francis. Two great and extraordinary people and professionals in their respective fields who inspired (and yet they do) me in spirit and insight in my early and difficult steps of writing this blog. I am just convinced that Science is made of little, ordinary and small contributions like mine, and not only the greatest contributions like those making John and Francis to the whole world. I wish they continue making their contributions in the future for many, many years yet to come.
Now, let me answer the question Francis asked me to explain here with further details. My special post/log-entry number 50…It will be devoted to tell you why this blog is called The Spectrum of Riemannium, and what is behind the greatest unsolved problem in Number Theory, Mathematics and likely Physics/Physmatics as well…Enjoy it!
The Riemann zeta function is a device/object/function related to prime numbers.
In general, it is a function of complex variable s=\sigma+i\tau defined by the next equation:
Generally speaking, the Riemann zeta function extended by analytical continuation to the whole complex plane is “more” than the classical Riemann zeta function that Euler found much before the work of Riemann in the XIX century. The Riemann zeta function for real and entire positive values is a very well known (and admired) series by the mathematicians. \zeta (1)=\infty due to the divergence of the harmonic series. Zeta values at even positive numbers are related to the Bernoulli numbers, and it is still lacking an analytic expression for the zeta values at odd positive numbers.
The Riemann zeta function over the whole complex plane satisfy the following functional equation:
\boxed{\pi^{-\frac{s}{2}}\Gamma \left(\dfrac{s}{2}\right)\zeta (s)=\pi^{-\frac{(1-s)}{2}}\Gamma \left(\dfrac{1-s}{2}\right)\zeta (1-s)}
Equivalently, it can be also written in a very simple way:
\boxed{\xi (s)=\xi (1-s)}
where we have defined
\xi (s)=\pi^{-\frac{s}{2}}\Gamma \left(\dfrac{s}{2}\right)\zeta (s)
Riemann zeta values are an example of beautiful Mathematics. From \displaystyle{\zeta (s)=\sum_{n=1}^{\infty}n^{-s}}, then we have:
1) \zeta (0)=1+1+\ldots=-\dfrac{1}{2}.
2) \zeta (1)=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots =\infty. The harmonic series is divergent.
3) \zeta (2)=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\ldots =\dfrac{\pi^2}{6}\approx 1.645. The famous Euler result.
4) \zeta (3)=1+\dfrac{1}{2^3}+\dfrac{1}{3^3}+\ldots \approx 1.202. And odd zeta value called Apery’s constant that we do not know yet how to express in terms of irrational numbers.
5) \zeta (4)=\dfrac{\pi^4}{90}\approx 1.0823.
6) \zeta (-2n)=-\dfrac{\pi^{-n}}{2\Gamma (-n+1)}=0,\;\;\forall n=1,2,\ldots ,\infty. Trivial zeroes of zeta.
7) \zeta (2n)=\dfrac{(-1)^{n+1}(2\pi)^{2n}B_{2n}}{2(2n)!}\;\;\forall n=1,2,\ldots ,\infty, where B_{2n} are the Bernoulli numbers. The first 13 Bernoulli numbers are:
B_0=1, B_1=-\dfrac{1}{2}, B_2=\dfrac{1}{6}, B_3=0, B_4=-\dfrac{1}{30}, B_5=0, B_6=\dfrac{1}{42}
B_7=0, B_8=-\dfrac{1}{30}, B_9=0, B_{10}=\dfrac{5}{66}, B_{11}=0, B_{12}=-\dfrac{691}{2730}, B_{13}=0
8) We note that B_{2n+1}=0,\;\; \forall n\geq 1.
9) \zeta (-2n+1)=-\dfrac{B_{2n}}{2n}, \;\; \forall n=1,2,\ldots ,\infty.
For instance, \zeta (-1)=-\dfrac{1}{12}=1+2+3+\ldots, \zeta (-3)=\dfrac{1}{120}, and \zeta (-5)=-\dfrac{1}{252}. Indeed, \zeta (-1) arises in string theory trying to renormalize the vacuum energy of an infinite number of harmonic oscillators. The result in the bosonic string is \dfrac{2}{2-D}. In order to match with Riemann zeta function regularization of the above series, the bosonic string is asked to live in an ambient spacetime of D=26 dimensions. We also have that
\sum \vert n\vert^3=-\dfrac{1}{60}
10) \zeta (\infty)=1. The Riemann zeta value at the infinity is equal to the unit.
11) The derivative of the zeta function is \displaystyle{\zeta '(s)=-\sum_{n=1}^{\infty}\dfrac{\log n}{n^s}}. Particularly important of this derivative are:
\displaystyle{\zeta '(0)=-\sum_{n=1}^\infty \log n=-\log \prod_{n=1}^\infty n=\zeta (0)\log (2\pi)=-\dfrac{1}{2}\log (2\pi)=-\log \sqrt{2\pi}=\log \dfrac{1}{\sqrt{2\pi}}}
or \zeta '(0)=\log \sqrt{\dfrac{1}{2\pi}}
This allow us to define the factorial of the infinity as
\displaystyle{\infty !=\prod_{n=1}^{\infty}n=1\cdot 2\cdots \infty=e^{-\zeta '(0)}=\sqrt{2\pi}}
and the renormalized infinite dimensional determinant of certain operator A as:
\det _\zeta (A)=a_1\cdot a_2\cdots=\exp \left(-\zeta_A '(0)\right), with \displaystyle{\zeta _A (s)=\sum_{n=1}^\infty \dfrac{1}{a_n^s}}
12) \zeta (1+\varepsilon )=\dfrac{1}{\varepsilon}+\gamma_E +\mathcal{O} (\varepsilon ). This is a result used by theoretical physicists in dimensional renormalization/regularization. \gamma_E\approx 0.577 is the so-called Euler-Mascheroni constant.
The alternating zeta function, called Dirichlet eta function, provides interesting values as well. Dirichlet eta function is defined and related to the Riemann zeta fucntion as follows:
This can be thought as “bosons made of fermions” or “fermions made of bosons” somehow. Special values of Dirichlet eta function are given by:
\eta (0)=-\zeta (0)=\dfrac{1}{2} \eta (1)=\log 2 \eta (2)=\dfrac{1}{2}\zeta (2)=\dfrac{\pi^2}{12}
\eta (3)=\dfrac{3}{4}\zeta (3)\approx \dfrac{3}{4}(1.202) \eta (4)=\dfrac{7}{8}\zeta (4)=\dfrac{7}{8}\left(\dfrac{\pi^4}{90}\right)
Remark(I): \zeta(2) is important in the physics realm, since the spectrum of the hydrogen atom has the following aspect
and the Balmer formula is, as every physicist knows
\Delta E(n,m)=K\left(\dfrac{1}{n^2}-\dfrac{1}{m^2}\right)
Remark (II): The fact that \zeta (2) is finite implies that the energy level separation of the hydrogen atom in the Böhr level tends to zero AND that the sum of ALL the possible energy levels in the hydrogen atom is finite since \zeta (2) is finite.
Remark(III): What about an “atom”/system with spectrum E(n)=\kappa n^{-s}? If s=2, we do know that is the case of the Kepler problem. Moreover, it is easy to observe that s=-1 corresponds to tha harmonic oscillator, i.e., E(n)=\hbar \omega n. We also know that s=-2 is the infinite potential well. So the question is, what about a n^{-3} spectrum and so on?
In summary, does the following spectrum
with energy separation/splitting
\boxed{\Delta E(n,m;s)=\mathbb{K}\left(\dfrac{1}{n^{s}}-\dfrac{1}{m^{s}}\right)}
exist in Nature for some physical system beyond the infinite potential well, the harmonic oscillator or the hydrogen atom, where s=-2, s=-1 and s=2 respectively?
It is amazing how Riemann zeta function gets involved with a common origin of such a different systems and spectra like the Kepler problem, the harmonic oscillator and the infinite potential well!
The Riemann Hypothesis (RH) is the greatest unsolved problem in pure Mathematics, and likely, in Physics too. It is the statement that the only non-trivial zeroes of the Riemann zeta function, beyond the trivial zeroes at s=-2n,\;\forall n=1,2,\ldots,\infty have real part equal to 1/2. In other words, the equation or feynmanity has only the next solutions:
\boxed{\mbox{RH:}\;\;\zeta (s)=0\leftrightarrow \begin{cases} s_n=-2n,\;\forall n=1,\ldots,\infty\;\;\mbox{Trivial zeroes}\\ s_n=\dfrac{1}{2}\pm i\lambda_n, \;\;\forall n=1,\ldots,\infty \;\;\mbox{Non-trivial zeroes}\end{cases}}
I generally prefer the following projective-like version of the RH (PRH):
\boxed{\mbox{PRH:}\;\;\zeta (s)=0\leftrightarrow \begin{cases} s_n=-2n,\;\forall n=1,\ldots,\infty\;\;\mbox{Trivial zeroes}\\ s_n=\dfrac{1\pm i\overline{\lambda}_n}{2}, \;\;\forall n=1,\ldots,\infty \;\;\mbox{Non-trivial zeroes}\end{cases}}
The Riemann zeta function can be sketched on the whole complex plane, in order to obtain a radiography about the RH and what it means. The mathematicians have studied the critical strip with ingenious tools an frameworks. The now terminated ZetaGrid project proved that there are billions of zeroes IN the critical line. No counterexample has been found of a non-trivial zeta zero outside the critical line (and there are some arguments that make it very unlikely). The RH says that primes “have music/order/pattern” in their interior, but nobody has managed to prove the RH. The next picture shows you what the RH “say” graphically:
If you want to know how the Riemann zeroes sound, M. Watkins has done a nice audio file to see their music.
You can learn how to make “music” from Riemann zeroes here http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/munafo-zetasound.htm
And you can listen their sound here
Riemann zeroes are connected with prime numbers through a complicated formula called “the explicit formula”. The next equation holds \forall x\geq 2 integer numbers, and non-trivial Riemann zeroes in the complex (upper) half-plane with \tau>0:
\boxed{\displaystyle{\pi (x)+\sum_{n=2}^\infty \dfrac{\pi \left( x^{1/n}\right)}{n}=\text{Li} (x)-\sum_{\lambda =\sigma+i\tau }\left(\text{Li}(x^\lambda)+\text{Li}\left( x^{1-\lambda}\right)\right)+\int_x^\infty\dfrac{du}{u(u^2-1)\ln u}-\ln 2}}
and where \pi (x) is the celebrated Gauss prime number counting function, i.e., \pi (x) represents the prime numbers that are equal than x or below. This explicit formula was proved by Hadamard. The explicit formula follows from both product representations of \zeta (s), the Euler product on one side and the Hadamard product on the other side.
The function \text{Li} (x), sometimes written as \text{li} (x), is the logarithmic integral
\displaystyle{\text{Li} (x) =\text{li} (x)= \int_2^x\dfrac{du}{\ln x}}
The explicit formula comes in some cool variants too. For instance, we can write
\pi (x)=\pi_0 (x)+\pi_1 (x)=\pi_{\mbox{smooth}}+\pi_{\mbox{osc-chaotic}}
\displaystyle{\pi_0 (x)=\sum_{n=1}^\infty\dfrac{\mu (n)}{n}\left[\mbox{Li}(x^{1/n})-\sum_{k=1}^\infty\mbox{Li}(x^{-2k/n})\right]}
\displaystyle{\pi_1 (x)=-2\mbox{Re}\sum_{n=1}^\infty\dfrac{\mu (n)}{n}\sum_{\alpha=1}^\infty\mbox{Li}(x^{(\sigma_\alpha+i\tau_\alpha)/n})}
For large values of x, we have the asymptotics
\pi_0 (x)\approx \mbox{Li} (x)
\displaystyle{\pi_1 (x)\approx -\dfrac{2}{\ln x}\sum_{\alpha=1}^\infty\dfrac{x^{\sigma_\alpha}}{\sigma_\alpha^2+\tau_\alpha^2}\left(\sigma_\alpha\cos (\tau_\alpha \ln x)+\tau_\alpha \sin (\tau_\alpha \ln x)\right)}
Remark: Please, don’t confuse the logarithmic integral with the polylogarithm function \text{Li}_x (s).
Gauss also conjectured that
\pi (x)\sim \text{Li} (x)
Date: January 3, 1982. Andrew Odlyzko wrote a letter to George Pólya about the physical ground/basis of the Riemann Hypothesis and the conjecture associated to Polya himself and David Hilbert. Polya answered and told Odlyzko that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann Hypothesis should be true, and suggested that this would be the case if the imaginary parts, say T of the non-trivial zeros
of the Riemann zeta function corresponded to eigenvalues of an unbounded and unknown self adjoint operator \hat{T}. That statement was never published formally, but it was remembered after all, and it was transmitted from one generation to another. At the time of Pólya’s conversation with Landau, there was little basis for such speculation. However, Selberg, in the early 1950s, proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula shared a striking resemblance to the explicit formula of certain L-function, which gave credibility to the speculation of Hilbert and Pólya.
Dialogue(circa 1970). “(…)Dyson: So tell me, Montgomery, what have you been up to? Montgomery: Well, lately I’ve been looking into the distribution of the zeros of the Riemann zeta function. Dyson: Yes? And? Montgomery: It seems the two-point correlations go as….(…) Dyson: Extraordinary! Do you realize that’s the pair-correlation function for the eigenvalues of a random Hermitian matrix? It’s also a model of the energy levels in a heavy nucleus, say U-238.(…)”
A step further was given in the 1970s, by the mathematician Hugh Montgomery. He investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called Montgomery’s pair correlation conjecture. The Riemann zeros tend not to cluster too closely together, but to repel. During a visit to the Institute for Advanced Study (IAS) in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices. Dyson realized that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random and “very big/large” Hermitian matrix with size NxN. These distributions are of importance in physics and mathematics. Why? It is simple. The eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the theoyr of the so-calle Gaussian unitary ensemble, and both are now believed to obey the same statistics. Thus the conjecture of Pólya and Hilbert now has a more solid fundamental link to QM, though it has not yet led to a proof of the Riemann hypothesis. The pair-correlation function of the zeros is given by the function:
R_2(x)=1-\left(\dfrac{\sin \pi x}{\pi x}\right)^2
In a posterior development that has given substantive force to this approach to the Riemann hypothesis through functional analysis and operator theory, the mathematician Alain Connes has formulated a “trace formula” using his non-commutative geometry framework that is actually equivalent to certain generalized Riemann hypothesis. This fact has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements. However, the mysterious operator believed to provide the Riemann zeta zeroes remain hidden yet. Even worst, we don’t even know on which space the Riemann operator is acting on.
However, some trials to guess the Riemann operator has been given from a semiclassical physical environtment as well. Michael Berry and Jon Keating have speculated that the Hamiltonian/Riemann operator H is actually some kind of quantization of the classical Hamiltonian XP where P is the canonical momentum associated with the position operator X. If that Berry-Keating conjecture is true. The simplest Hermitian operator corresponding to XP is
H = \dfrac1{2} (xp+px) = - i \left( x \dfrac{\mathrm{d}}{\mathrm{d} x} + \dfrac{1}{2} \right)
At current time, it is still quite inconcrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Germán Sierra, the latter in collaboration with P.K.Townsed, have conjectured that since this operator is invariant under dilatations perhaps the boundary condition f(nx)=f(x) for integer n may help to get the correct asymptotic results valid for big n. That it, in the large n we should obtain
s_n=\dfrac{1}{2} + i \dfrac{ 2\pi n}{\log n}
Indeed, the Berry-Keating conjecture opened another striking attack to prove the RH. A topic that was popular in the 80’s and 90’s in the 20th century. The weird subject of “quantum chaos”. Quantum chaos is the subject devoted to the study of quantum systems corresponding to classically chaotic systems. The Berry-Keating conjecture shed light further into the Riemann dynamics, sketching some of the properties of the dynamical system behind the Riemann Hypothesis.
In summary, the dynamics of the Riemann operator should provide:
1st. The quantum hamiltonian operator behind the Riemann zeroes, in addition to the classical counterpart, the classical hamiltonian H, has a dynamics containing the scaling symmetry. As a consequence, the trajectories are the same at all energy scale.
2nd. The classical system corresponding to the Riemann dynamics is chaotic and unstable.
3rd. The dynamics lacks time-reversal symmetry.
4th. The dynamics is quasi one-dimensional.
A full dictionary translating the whole correspondence between the chaotic system corresponding to the Riemann zeta function and its main features is presented in the next table:
In 2001, the following paper emerged, http://arxiv.org/abs/nlin/0101014. The Riemannium arxiv paper was published later (here: Reg. Chaot. Dyn. 6 (2001) 205-210). After that, Brian Hayes wrote a really beautiful, wonderful and short paper titled The Spectrum of Riemannium in 2003 (American Scientist, Volume 91, Number 4 July–August, 2003,pages 296–300). I remember myself reading the manuscript and being totally surprised. I was shocked during several weeks. I decided that I would try to understand that stuff better and better, and, maybe, make some contribution to it. The Spectrum of Riemannium was an amazing name, an incredible concept. So, I have been studying related stuff during all these years. And I have my own suspitions about what the riemannium and the zeta function are, but this is not a good place to explain all of them!
The riemannium is the mysterious physical system behind the RH. Its spectrum, the spectrum of riemannium, are given by the RH and its generalizations.
Moreover, the following sketch from Hayes’ paper is also very illustrative:
What do you think? Isn’t it suggestive? Is it amazing?
Riemann zeta function also arises in the renormalization of the Standard Model and the regularization of determinants with “infinite size” (i.e., determinants of differential operators and/or pseudodifferential operators). For instance, the \infty-dimensional regularized determinant is defined through the Riemann zeta function as follows:
\displaystyle{\det _\zeta \mathcal{P}=e^{-\zeta_{\mathcal{P}}^{'}(0)}}
The dimensional renormalization/regularization of the SM makes use of the Riemann zeta function as well. It is ubiquitous in that approach, but, as far as I know, nobody has asked why is that issue important, as I have suspected from long time ago.
Riemann zeta function is also used in the theory of Quantum Statistics. Quantum Statistics are important in Cosmology and Condensed Matter, so it is really striking that Riemann zeta values are related to phenomena like Bose-Einstein condensation or the Cosmic Microwave Background and also the yet to be found Cosmic Neutrino Background!
Let me begin with the easiest quantum (indeed classical) statistics, the Maxwell-Boltzmann (MB) statistics. In 3 spatial dimensions (3d) the MB distribution arises ( we will work with units in which \hbar =1):
f(p)_{MB}=\dfrac{1}{(2\pi)^3}e^{\frac{\mu -E}{k_BT}}
Usually, there are 3 thermodynamical quantities that physicists wish to compute with statistical distributions: 1) the number density of particles n=N/V, 2) the energy density \varepsilon=U/V and 3) the pressure P. In the case of a MB distribution, we have the following definitions:
\displaystyle{n=\dfrac{1}{(2\pi)^3}\int d^3p e^{\frac{\mu -E}{k_BT}}}
\displaystyle{\varepsilon =\dfrac{1}{(2\pi)^3}\int d^3p Ee^{\frac{\mu -E}{k_BT}}}
\displaystyle{\varepsilon =\dfrac{1}{(2\pi)^3}\int d^3p \dfrac{1}{3}\dfrac{\vert\mathbf{p}\vert^2}{E}e^{\frac{\mu -E}{k_BT}}}
We can introduce the dimensionless variables $late z=\dfrac{mc^2}{k_BT}$, \tau =\dfrac{E}{k_BT}=\dfrac{\sqrt{p^2+m^2c^4}}{k_BT}. In this way,
\vert p\vert=\dfrac{k_BT}{c}\sqrt{\tau^2-z^2}
c^2\vert\mathbf{p}\vert d\vert \mathbf{p}\vert=k_B^2T^2\tau d\tau
With these definitions, the particle density becomes
\displaystyle{n=\dfrac{4\pi k_B^3T^3}{(2\pi)^3}e^{\frac{\mu}{k_BT}}\int_z^\infty d\tau (\tau^2-z^2)^{1/2}\tau e^{-\tau}}
This integral can be calculated in closed form with the aid of modified Bessel functions of the 2th kind:
K_n (z)=\dfrac{2^nn!}{(2n)!z^n}\int_z^\infty d\tau (\tau^2-z^2)^{n-1/2}e^{-\tau} or equivalently
K_n (z)=\dfrac{2^{n-1}(n-1)!}{(2n-2)!z^n}\int_z^\infty d\tau (\tau^2-z^2)^{n-3/2}\tau e^{-\tau}
K_{n+1} (z)=\dfrac{2nK_n (z)}{z}+K_{n-1} (z)
\displaystyle{K_2 (x)=\dfrac{1}{z^2}\int_z^\infty (\tau^2-z^2)^{1/2}\tau e^{-\tau}d\tau}
And thus, we have the next results (setting c=1 for simplicity):
\mbox{Particle number density}\equiv n=\dfrac{N}{V}=\dfrac{k_B^3T^3}{2\pi^2}z^2K_2 (z)=\dfrac{k_B^3T^3}{2\pi^2}\left(\dfrac{m}{k_BT}\right)^2K_2\left(\dfrac{m}{k_BT}\right)e^{\frac{\mu}{k_BT}}
\mbox{Energy density}\equiv\varepsilon=\dfrac{k_B^4T^4}{2\pi^2}\left[ 3\left(\dfrac{m}{k_BT}\right)^2K_2\left(\dfrac{m}{k_BT}\right)+\left(\dfrac{m}{k_BT}\right)^3K_1\left(\dfrac{m}{k_BT}\right)\right]e^{\frac{\mu}{k_BT}}
Even entropy density is easiy to compute:
\mbox{Entropy density}\equiv s=\dfrac{m^3}{2\pi^2}e^{\frac{\mu}{k_BT}}\left[ K_1\left(\dfrac{m}{k_BT}\right)+\dfrac{4k_BT-\mu}{m}K_2\left(\dfrac{m}{k_BT}\right)\right]
These results can be simplified in some limit cases. For instance, in the massless limit z=m/k_BT\rightarrow 0. Moreover, we also know that \displaystyle{\lim_{z\rightarrow 0}z^nK_n (z)=2^{n-1}(n-1)!}. In such a case, we obtain:
n\approx \dfrac{k_B^3T^3}{\pi^2}e^{\frac{\mu}{k_BT}}
\varepsilon \approx \dfrac{3k_B^4T^4}{\pi^2}e^{\frac{\mu}{k_BT}}
P\approx \dfrac{k_B^4T^4}{\pi^2}e^{\frac{\mu}{k_BT}}
We note that \varepsilon=3P in this massless limit.
Remark (I): In the massless limit, and whenever there is no degeneracy, \varepsilon =3P holds.
Remark (II): If there is a quantum degeneracy in the energy levels, i.e., if g\neq 1, we must include an extra factor of g_j=2j+1 for massive particles of spin j. For massless photons with helicity, there is a g=2 degeneracy.
Remark (III): In the D-dimensional (D=d+1) Bose gas with dispersion relationship \varepsilon_p=cp^{s}, it can be shown that the pressure is related with the energy density in the following way
\mbox{Pressure}\equiv P=\dfrac{s}{d}\dfrac{U}{V}=\dfrac{s}{d}\varepsilon
Remark (IV): Let us define p^s (n) as the number of ways an integer number can be expressed as a sum of the sth powers of integers. For instance,
p^1 (5)=7 because 5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1
p^2 (5)=2 because 5=2^2+1^2=1^2+1^2+1^2+1^2+1^2
If E_n=n^s with n\geq 1 and s>0, then x=e^{-\beta} and the partition function is
\displaystyle{Z=\prod_{k}\left( 1+e^{\frac{\mu-E}{k_BT}}\right)}
We will see later that \displaystyle{\sum_{N=0}^\infty x^N=\begin{cases}1+x, FD \\ \dfrac{1}{1-x}, BE\end{cases}}
with \mu =0 is nothing but the generatin function of the partitions p^s (n)
\displaystyle{Z(x=e^{-\beta})=\prod_{n=1}^\infty \dfrac{1}{1-x^{n^s}}=\sum_{n=1}^\infty p^s (n) x^n\approx \int_1^\infty dn p^s (n) e^{-\beta n}}
The Hardy-Ramanujan inversion formula reads (for the case s=1 only):
p(n) \approx \dfrac{1}{4\sqrt{3}N}e^{\pi\sqrt{2N/3}}
Remark (V): There are some useful integrals in quantum statistics. They are the so-called Bose-Einstein/Fermi-Dirac integrals
\displaystyle{\int_0^\infty dx \dfrac{x^{n-1}}{e^x\mp 1}=\begin{cases}\Gamma (n) \zeta (n), \;\; BE\\ \Gamma (n)\eta (n)=\Gamma (n) (1-2^{1-n})\zeta (n),\;\; FD\end{cases}}
The BE-FD quantum distributions in 3d are defined as follows:
where the minus sign corresponds to FD and the plus sign to BE.
We will firstly study the BE distribution in 3d. We have:
\displaystyle{n=\dfrac{1}{(2\pi)^3}\int d^3p \left(e^{\frac{\mu-E}{k_BT}}-1\right)^{-1}=\dfrac{1}{(2\pi)^3}\int d^3p \sum_{n=1}^{\infty}(+1)^{n+1}e^{\frac{n\mu-nE}{k_BT}}}
Introducing a scaled temperature T'=T/n, we get
\displaystyle{n=\sum_{n=1}^{\infty}\left[\dfrac{1}{(2\pi)^3}\int d^3p e^{\frac{n\mu-nE}{k_BT'}}\right]=\sum_{n=1}^{\infty}\dfrac{k_B^3T^3}{2\pi^2}\dfrac{1}{n^3}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{n\mu}{k_BT}}}
Again, we can study a particularly simple case: the massless limit m\rightarrow 0 with \mu\rightarrow 0. In this case, we get:
\displaystyle{n=\dfrac{k_B^3T^3}{\pi^2}\sum_{n=1}^\infty \dfrac{1}{n^3}=\dfrac{k_B^3T^3}{\pi^2}\zeta (3)\approx 1.202\dfrac{k_B^3T^3}{\pi^2}}
\displaystyle{\varepsilon=\sum_{n=1}^\infty\dfrac{3(k_BT)^4}{\pi^2}\dfrac{1}{n^4}=\dfrac{3(k_BT)^4\zeta (4)}{\pi^2}=\dfrac{\pi^2}{30}(k_BT)^4}
\displaystyle{P=\sum_{n=1}^\infty\dfrac{(k_BT)^4}{\pi^2}\dfrac{1}{n^4}=\dfrac{(k_BT)^4\zeta (4)}{\pi^2}=\dfrac{\pi^2(k_BT)^4}{90}}
The FD distribution in 3d can be studied in a similar way. Following the same approach as the BE distribution, we deduce that:
\displaystyle{n=\sum_{n=1}^\infty (-1)^{n+1}\dfrac{(k_BT)^3}{2\pi^2n^3}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{\mu n}{k_BT}}}
\displaystyle{\varepsilon= \sum_{n=1}^\infty (-1)^{n+1}\dfrac{(k_BT)^4}{2\pi^2}\left[3\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)+\left(\dfrac{nm}{k_BT}\right)^3K_1\left(\dfrac{nm}{k_BT}\right)\right]e^{\frac{\mu n}{k_BT}}}
and again the massless limit m=0 and \mu\rightarrow 0 provide
\displaystyle{n\approx \dfrac{(k_BT)^3}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^3}=\dfrac{(k_BT)^3}{\pi^2}\eta (3)=\dfrac{(k_BT)^3}{\pi^2}\left(\dfrac{3}{4}\right)\zeta (3)}
\displaystyle{\varepsilon\approx \dfrac{3(k_BT)^4}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^4}=3(k_BT)^4\eta (4)=3(k_BT)^4\dfrac{7}{8}\zeta (4)=\dfrac{\pi^2(k_BT)^4}{30}\left(\dfrac{7}{8}\right)}
\displaystyle{P\approx \dfrac{(k_BT)^4}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^4}=\left(\dfrac{7}{8}\right)\dfrac{\pi^2(k_BT)^4}{90}}
Remark (I): For photons \gamma with degeneracy g=2 we obtain
n_\gamma =\dfrac{2\zeta (3) (k_BT)^3}{\pi^2}
\varepsilon_\gamma= 3P_\gamma =\dfrac{\pi^2 (k_BT)^4}{15}
s_\gamma =P'(T)=\dfrac{4}{3}\left(\dfrac{\pi^2}{15}\right)(k_BT)^3=\dfrac{2\pi^4}{45\zeta (3)}n
Remark (II): In Cosmology, Astrophysics and also in High Energy Physics, the following units are used
1eV=1.602\cdot 10^{-19}J
\hbar=1=6.58\cdot 10^{-22}MeVs=7.64\cdot 10^{-12}Ks
\hbar c=1=0.19733GeV\cdot fm=0.2290 K\cdot cm
1 K=0.1532\cdot 10^{-36}g\cdot c^2
The Cosmic Microwave Background is the relic photon radiation of the Big Bang, and thus it has a temperature due to photons in the microwave band of the electromagnetic spectrum. Its value is:
T_\gamma \approx 2.725K
Indeed, it also implies that the relic photon density is about n_\gamma =410\dfrac{1}{cm^3}
It is also speculated that there has to be a Cosmic Neutrino Background relic from the Big Bang. From theoretical Cosmology, it is related to the photon CMB temperature in the following way:
T_\nu =\left(\dfrac{4}{11}\right)^{1/3}2.7K or equivalently
T_\nu\approx 1.9K
This temperature implies a relic neutrino density (per species, i.e., with g_\nu=1) about
The cosmological density entropy due to these particles is
s_0=\dfrac{S_0}{V}=\dfrac{4\pi^2}{45}\left[1+\dfrac{2\cdot 3}{2}\left(\dfrac{7}{8}\right)\left(\dfrac{4}{11}\right)\right]T_{0\gamma}^3=2810\dfrac{1}{cm^3}\left( \dfrac{T_{0\gamma}}{2.7K}\right)^3
and then we get
s_0\approx 7.03n_{0\gamma}
Remark (III): In Cosmology, for fermions in 3d ( note that BE implies \varepsilon=3P, and that we must drop the factors \left( 7/8\right), \left( 3/4\right), \left( 7/6\right) in the next numerical values) we can compute
n=\begin{cases}\left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)\dfrac{2\zeta (3)}{\pi^2}(k_BT)^3\\ \left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)31.700\left(\dfrac{k_BT}{GeV}\right)^3\dfrac{1}{fm^3}\\ \left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)20.288\left(\dfrac{T}{K}\right)^3\dfrac{1}{cm^3}\end{cases}
\varepsilon=3P=\begin{cases}\left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(\dfrac{\pi^2}{15}\right)(k_BT)^4\\ \left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)(85.633)\left(\dfrac{k_BT}{GeV}\right)\dfrac{GeV}{fm^3}\\ \left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(0.841\cdot 10^{-36}\right)\left(\dfrac{T}{K}\right)^4\dfrac{g}{cm^3}\end{cases}
s=\dfrac{S}{V}=\left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(\dfrac{4\pi^2}{45}\right)(k_BT)^3=\dfrac{7}{6}\left[\dfrac{2\pi^4}{45\zeta (3)}\right] n
Remark (IV): An example of the computation of degeneracy factor is the quark-gluon plasma degeneracy g_{QGP}. Firstly we compute the gluon and quark degeneracies
g_g=(\mbox{color})(\mbox{spin})=2^3\cdot 2=8\cdot 2=16
g_q=(p\overline{p})(\mbox{spin})(\mbox{color})(\mbox{flavor})=2\cdot 2\cdot 3\cdot N_f=12N_f
Then, the QG plasma degeneracy factor is
g_{QGP}=g_g+\dfrac{7}{8}g_q=16+\dfrac{7}{8}12N_f=16+\dfrac{21}{2}N_f \leftrightarrow \boxed{g_{QGP}=16+\dfrac{21}{2}N_f}
In general, for charged leptons and nucleons g=2, g=1 for neutrinos (per species, of course), and g=2 for gluons and photons. Remember that massive particles with spin j will have g_j=2j+1.
Remark (V): For the Planck distribution, we also get the known result for the thermal distribution of the blackbody radiation
\displaystyle{I(T)=\int_0^\infty f(\nu ,T)d\nu=\dfrac{8\pi h}{c^3}\int_0^\infty \dfrac{\nu^3d\nu}{e^{\frac{h\nu}{k_BT}}-1}=\dfrac{8\pi^5k_B^4T^4}{15c^3h^3}}
Remark (VI): Sometimes the following nomenclature is used
i) Extremely degenerated gas if \mu>>k_BT
ii) Non-degenerated gas if \mu <<-k_BT
iii) Extremely relativistic gas ( or ultra-relativistic gas) if p>> mc
iv) Non-relativistic gas if p<<mc
Let us define the following shift operator \hat{T}:
where \sigma\in \mathbb{R}. Moreover, there is certain isomorphism between the shift operator space and the space of functions through the map \hat{T}\leftrightarrow x^\sigma.
We define the generalized logarithm as the image under the previous map of \hat{T}. That is:
\displaystyle{\mbox{Log}_G(x)\equiv \dfrac{1}{\sigma}\sum_{n=l}^{m}k_n x^{\sigma n}}
where l,m\in \mathbb{Z}, with l<m, m-l=r and x>0. Furthermore, the next contraints are also given for every generalized logarithm:
1st. \displaystyle{\sum_{n=1}^m k_n=0}.
2nd. \displaystyle{\sum_{n=l}^m nk_n=c}, k_m\neq 0, and k_l\neq 0.
3rd. \displaystyle{\sum_{n=l}^m\vert n\vert^l k_n=K_l}, \forall l=2,3,\ldots ,m-l and where K_l \in \mathbb{R}.
With these definitions we also have that
A) \mbox{Log}_G(x)=\ln (x)
B) \mbox{Log}_G(1)=0
Examples of generalized logarithms are:
1) The Tsallis logarithm.
2) The Kaniadakis logarithm.
3) The Abe logarithm.
\mbox{Log}_A(x)=\dfrac{x^{\sigma -1}-x^{\sigma^{-1}-1}}{\sigma-\sigma^{-1}}
4) The biparametric logarithm.
with a=\sigma-1 and b=\sigma^{-1}-1 in the case of the Abe logarithm.
Group entropies are defined through the use of generalized logarithms. Define some discrete probability distribution \left[ p_i\right]_{i=1,\ldots,W} with normalization \displaystyle{\sum_{i=1}^Wp_i=1}. Therefore, the group entropy is the following functional sum:
\boxed{\displaystyle{S_G=-k_B\sum_{i=1}^{W}p_i \mbox{Log}_G \left(\dfrac{1}{p_i}\right)}}
where we have used the previous definition of generalized logarithm and the Boltzmann’s constant k_B is a real number. It is called group entropy due to the fact that S_G is connected to some universal formal group. This formal group will determine some correlations for the class of physical systems under study and its invariant properties. In fact, the Tsallis logarithm itself is related to the Riemann zeta function through a beautiful equation! Under the Tsallis group exponential, the isomorphism x\leftrightarrow e^t is defined to be e_G^t=\dfrac{e^{(1-q)t}-1}{1-q}, and thus we easily get:
\displaystyle{\dfrac{1}{\Gamma (s)}=\int_0^\infty\dfrac{1}{\dfrac{e^{(1-q)t}-1}{1-q}}t^{s-1}dt=\dfrac{\zeta (s)}{(1-q)^{s-1}}}
\forall s such as Re (s)>1 and q<1.
The primon gas/free Riemann gas is a statistical mechanics toy model illustrating in a simple way some correspondences between number theory and concepts in statistical physics, quantum mechanics, quantum field theory and dynamical systems.
The primon gas IS a quantum field theory (QFT) of a set of non-interacting particles, called the “primons”. It is also named a gas or a free model because the particles are non-interacting. There is no potential. The idea of the primon gas was independently discovered by Donald Spector (D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239-252) and Bernard Julia (Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293). There have been later works by Bakas and Bowick (I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881) and Spector (D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp.1919-1927) in which it was explored the connection of such systems to string theory.
This model is based on some simple hypothesis:
1st. Consider a simple quantum Hamiltonian, H, having eigenstates \vert p\rangle labelled by the prime numbers “p”.
2nd. The eigenenergies or spectrum are given by E_p and they have energies proportional to \log p. Mathematically speaking,
H\vert p\rangle = E_p \vert p\rangle with E_p=E_0 \log p
Please, note the natural emergence of a “free” scale of energy E_0. What is this scale of energy? We do not know!
3rd. The second quantization/second-quantized version of this Hamiltonian converts states into particles, the “primons”. Multi-particle states are defined in terms of the numbers k_p of primons in the single-particle states p:
|N\rangle = |k_2, k_3, k_5, k_7, k_{11}, \ldots, k_{137},\ldots, k_p \ldots\rangle
This corresponds to the factorization of N into primes:
N = 2^{k_2} \cdot 3^{k_3} \cdot 5^{k_5} \cdot 7^{k_7} \cdot 11^{k_{11}} \cdots 137^{k_{137}}\cdots p^{k_p} \cdots
The labelling by the integer “N” is unique, since every number has a unique factorization into primes.
The energy of such a multi-particle state is clearly
\displaystyle{E(N) = \sum_p k_p E_p = E_0 \cdot \sum_p k_p \log p = E_0 \log N}
4th. The statistical mechanics partition function Z IS, for the (bosonic) primon gas, the Riemann zeta function!
\displaystyle{Z_B(T) \equiv\sum_{N=1}^\infty \exp \left(-\dfrac{E(N)}{k_B T}\right) = \sum_{N=1}^\infty \exp \left(-\dfrac{E_0 \log N}{k_B T}\right) = \sum_{N=1}^\infty \dfrac{1}{N^s} = \zeta (s)}
with s=E_0/k_BT=\beta E_0, and where k_B is the Boltzmann’s constant and T is the absolute temperature. The divergence of the zeta function at the value s=1 (corresponding to the harmonic sum) is due to the divergence of the partition function at certain temperature, usually called Hagedorn temperature. The Hagedorn temperature is defined by:
This temperature represents a limit beyond the system of (bosonic) primons can not be heated up. To understand why, we can calculate the energy
E=-\dfrac{\partial}{\partial \beta}\ln Z_B=-\dfrac{E_0}{\zeta (\beta E_0)}\dfrac{\partial \zeta (\beta E_0)}{\partial \beta}\approx \dfrac{E_0}{s-1}
A similar treatment can be built up for fermions rather than bosons, but here the Pauli exclusion principle has to be taken into account, i.e. two primons cannot occupy the same single particle state. Therefore m_i can be 0 or 1 for all single particle state. As a consequence, the many-body states are labeled not by the natural numbers, but by the square-free numbers. These numbers are sieved from the natural numbers by the Möbius function. The calculation is a bit more complex, but the partition function for a non-interacting fermion primon gas reduces to the relatively simple form
Z_F(T)=\dfrac{\zeta (s)}{\zeta (2s)}
The canonical ensemble is of course not the only ensemble used in statistical physics. Julia extended the Riemann gas approach to the grand canonical ensemble by introducing a chemical potential \mu (Julia, B. L., 1994, Physica A 203(3-4), 425), and thus, he replaced the primes p with new primes pe^{-\mu}. This generalisation of the Riemann gas is called the Beurling gas, after the Swedish mathematician Beurling who had generalised the notion of prime numbers. Examining a boson primon gas with fugacity -1, it shows that its partition function becomes
\overline{Z}_B=\dfrac{\zeta (2s)}{\zeta (s)}
Remarkable interpretation: pick a system, formed by two sub-systems not interacting with each other, the overall partition function is simply the product of the individual partition functions of the subsystems. From the previous equation of the free fermionic riemann gas we get exactly this structure, and so there are two decoupled systems. Firstly, a fermionic “ghost” Riemann gas at zero chemical potential and, secondly, a boson Riemann gas with energy-levels given by E(N)=2E_0\ln p_N. Julia also calculated the appropriate Hagedorn temperatures and analysed how the partition functions of two different number theoretical gases, the Riemann gas and the “log-gas” behave around the Hagedorn temperature. Although the divergence of the partition function hints the breakdown of the canonical ensemble, Julia also claims that the continuation across or around this critical temperature can help understand certain phase transitions in string theory or in the study of quark confinement. The Riemann gas, as a mathematically tractable model, has been followed with much attention because the asymptotic density of states grows exponentially, \rho (E)\sim e^E, just as in string theory. Moreover, using arithmetic functions it is not extremely hard to define a transition between bosons and fermions by introducing an extra parameter, called kappa \kappa, which defines an imaginary particle, the non-interacting parafermions of order \kappa. This order parameter counts how many parafermions can occupy the same state, i.e. the occupation number of any state falls into the interval \left[0,\kappa-1\right], and therefore \kappa=2 belongs to normal fermions, while \kappa\rightarrow\infty are the usual bosons. Furthermore, the partition function of a free, non-interacting κ-parafermion gas can be defined to be (Bakas and Bowick,1991, in the paper Bakas, I., and M. J. Bowick, 1991, J. Math. Phys. 32(7), 1881):
Z_\kappa=\dfrac{\zeta (s)}{\zeta (\kappa s)}
Indeed, Bakas et al. proved, using the Dirichlet convolution \star, how one can introduce free mixing of parafermions with different orders which do not interact with each other
\displaystyle{f\star g=\sum_{d\vert n}f(d)g\left(\dfrac{n}{d}\right)}
where the symbol d\vert n means d is a divisor of n. This operation preserves the multiplicative property of the classically defined partition functions, i.e., Z_{\kappa_1\star \kappa_2}=Z_{\kappa_1}\star Z_{\kappa_2}. It is even more intriguing how interaction can be incorporated into the mixing by modifying the Dirichlet convolution with a kernel function or twisting factor
\displaystyle{f\odot g=\sum_{d\vert n}f(d)g\left( \dfrac{n}{d}\right) K(n,d)}
Using the unitary convolution Bakas establishes a pedagogically illuminating case, the mixing of two identical boson Riemann gases. He shows that
Z_\infty\star Z_\infty=\zeta ^2(s)\zeta(2s)=\dfrac{\zeta (s)}{\zeta(2s)}\zeta (s)=Z_2Z_\infty=Z_FZ_B
This result has an amazing meaning. Two identical boson Riemann gases interacting with each other through the unitary twisting, are equivalent to mixing a fermion Riemann gas with a boson Riemann gas which do not interact with each other. Therefore, one of the original boson components suffers a transmutation/mutation into a fermion gas!
Remark (I): the Möbius function, which is the identity function with respect to the \star operation (i.e. free mixing), reappears in supersymmetric quantum field theories as a possible representation of the (-1)^F operator, where F is the fermion number operator! In this context, the fact that \mu (n)=0 for square-free numbers is the manifestation of the Pauli exclusion principle itself! In any QFT with fermions, (-1)^F is a unitary, hermitian, involutive operator where F is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, i.e., F=B+L, for all particles in the Standard Model and some (most of) SUSY QFT. The action of this operator is to multiply bosonic states by 1 and fermionic states by -1. This is always a global internal symmetry of any QFT with fermions and corresponds to a rotation by an angle 2\pi. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (-1)^F whereas fermionic operators anticommute with it. This operator really is, therefore, more useful in supersymmetric field theories.
Remark (II): potential attacks on the Riemann Hypothesis may lead to advances in physics and/or mathematics, i.e., progress in Physmatics!
Remark (III): the energy of the ground state is taken to be zero and the energy spectrum of the excited state is E(n)=E_0\ln (p_n), where p_n, n=2,3,5,\ldots, runs over the prime numbers. Let N and E denote now the number of particles in the ground state and the total energy of the system, respectively. The fundamental theorem of arithmetic allows only one excited state configuration for a given energy
E=\ln (n) \;\; mod E_0
where n is an integer. It immediately means that this gas preserves its quantum nature at any temperature, since only one quantum state is permitted to be occupied. The number fluctuation of any state (even the ground state) is therefore zero. In contrast, the changes in the number of particles in the ground state \delta n_0 predicted by the canonical ensemble is a smooth non-vanishing function of the temperature, while the grand-canonical ensemble still exhibits a divergence. This discrepancy between the microcanonical (combinatorial) and the other two ensembles remains even in the thermodynamic limit.
One could argue that the Riemann gas is fictitious/unreal and its spectrum is unrealisable/unphysical. However, we, physicists, think otherwise, since the spectrum E_N=\ln (N) does not increase with N more rapidly than n^2, therefore the existence of a quantum mechanical potential supporting this spectrum is possible (e.g., via inverse scattering transform or supplementary tools). And of course the question is: what kind of system has such an spectrum?
Some temptative ideas for the potential based on elementary Quantum Mechanics will be given in the next section.
Instead of considering the free Riemann gas, we could ask to Quantum Mechanics if there is some potential providing the logarithmic spectrum of the previous section. Indeed, there exists such a potential. Let us factorize any natural number in terms of its prime “atoms”:
N=p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}
Take the logarithm
\log N=\log \left(p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}\right)=n_1\log p_1+n_2\log p_2+\ldots+n_m\log p_m
\displaystyle{\log N=\sum_{i=1}^{m}n_i\log p_i}
where p_i are prime numbers (note that if we include “1” as a prime number it gives a zero contribution to the sum).
Now, suppose a logarithmic oscillator spectrum, i.e.,
\varepsilon_i=\log p_i with p_i=(1),2,3,5,7,11,13,\ldots,137,\ldots,\infty
with i=0,1,2,3,4,\ldots,\infty. In order to have a “riemann gas”/riemannium, we impose an spectrum labelled in the following fashion
\varepsilon_s =\log (2s+1) \forall s=0,1,2,3,\ldots,\infty
Equivalently, we could also define the spectrum of interacting riemannium gas as
\varepsilon_s=\log (s) \forall s=1,2,3,\ldots,\infty
In addition to this, suppose the next quantum postulates:
1st. Logarithmic potential:
V(x)=V_0\ln\dfrac{\vert x\vert}{L} with positive constants V_0, L>0
From the physical viewpoint, the positive constant V_0 means repulsive interaction (force).
2nd. Bohr-Sommerfeld quantization rule:
a) \displaystyle{I=\dfrac{1}{2\pi}\oint pdx=\hbar \left(s+\dfrac{1}{2}\right)}\; \forall s=0,1,\ldots,\infty
or equivalently we could also get
b) \displaystyle{I=\dfrac{1}{2\pi}\oint pdx=\hbar s}\; \forall s=1,2,\ldots,\infty
3rd. Turning point condition:
x_s=L\exp \left(\dfrac{\varepsilon_s}{V_0}\right)
In the case of 2a) we would deduce that
\displaystyle{\dfrac{\hbar \pi}{2}\left(s+\dfrac{1}{2}\right)=\int_0^{x_s}dx\sqrt{2m\left(\varepsilon_s-V_0\ln \dfrac{x}{L}\right)}}
\displaystyle{\dfrac{\hbar \pi}{2}\left(s+\dfrac{1}{2}\right)=\int_0^{x_x}dx\sqrt{-\ln \left(\dfrac{x}{x_s}\right)}=\sqrt{2mV_0}x_s\Gamma \left(\dfrac{3}{2}\right)}
and then
x_s=\sqrt{\dfrac{\pi}{2mV_0}}\hbar \left( s+\dfrac{1}{2}\right)
Then, using the turning point condition in this equation, we finally obtain
\boxed{\dfrac{\varepsilon_s}{V_0}=\ln (2s+1)+\ln \left(\dfrac{\hbar}{2L}\sqrt{\dfrac{\pi}{2mV_0}}\right)} \forall s=0,1,\ldots,\infty
In the case of 2b) we would obtain
\boxed{\dfrac{\varepsilon_s}{V_0}=\ln (s)+\ln \left(\dfrac{\hbar}{L}\sqrt{\dfrac{\pi}{2mV_0}}\right)} \forall s=1,2,\ldots,\infty
In summary, the logarithmic potential provides a model for the interacting Riemann gas!
Massive elementary particles (with mass m) can be understood as composite particles made of confined particles moving with some energy pc inside a sphere of radius R. We note that we do not define futher properties of the constituent particles (e.g., if they are rotating strings, particles, extended objects like branes, or some other exotic structure moving in circular orbits or any other pattern as trajectory inside the composite particle).
Let us make the hypothesis that there is some force F needed to counteract the centrifugal force F_c=\dfrac{\kappa c^2}{R}. The centrifugal force is equal to pc/R, i.e., the balancing force F is F=pc/R. Then, assuming the two forces are equal in magnitude, we get
where A_1 is some constant, and that equation holds regardless the origin of the interaction. The potentail energy U necessary to confine a constituent particle will be, in that case,
\displaystyle{U=\int \dfrac{A_1}{R}dR=A_1\int \dfrac{1}{R}dR=A_1\ln \dfrac{R}{R_\star}}
with R_\star some integration constant to be determined later. The center of mass of the “elementary particle”, truly a composite particle, from the external observer and the mass assinged to the composited system is:
The logarithmic potential energy is postulated to be proportional to m/R, and it provides
U=\dfrac{A_2 m}{R}
with A_2 is another constant. In fact, A_1, A_2 are parameters that don’t depend, a priori, on the radius R but on the constitutent particle properties and coupling constants, respectively. Indeed, for instance, we could set and fix the ratio A_2/A_1 to the constant c^2/G_N, where G_N is the gravitational constant. However, such a constraint is not required from first principles or from a clear physical reason. From the following equations:
m=\dfrac{\hbar}{cR} and U=\dfrac{A_2 m}{R}
we get \boxed{U=\dfrac{A_2 \hbar}{cR^2}}
Quantum Mechanics implies that the angular momentum should be quantized, so we can make the following generalization
U=\dfrac{A_2 m}{cR^2}\rightarrow U_n=\dfrac{A_2 \hbar}{cR_n^2}=\dfrac{A_2 (n+1)\hbar}{cR_0^2}
\forall n=0,1,2,\ldots,\infty
so R_n^2=\dfrac{R_0^2}{n+1}\leftrightarrow R_n=\dfrac{R_0}{\sqrt{n+1}}
Using the previous integral and this last result, we obtain
\ln \left(\dfrac{R_\star}{R_0}\right)=-(n+1)\dfrac{R_\star^2}{R_0^2}
This is due to the fact that U_n=A_2\dfrac{\hbar}{cR_n^2}=\dfrac{A_2\hbar (n+1)}{cR_0^2} and U=A_2\ln \dfrac{R}{R_\star}
Combining these equations, we deduce the value of R_\star as a function of the parameters A_1,A_2
\boxed{R_\star=\sqrt{\dfrac{A_2\hbar}{A_1 c}}}
The ratio R_\star/R_0 can be calculated from the above equations as well, since
\ln \left(\dfrac{R_\star}{R_0}\right)=-(n+1)\dfrac{R_\star^2}{R_0^2} for the case n=0 implies that
\ln \left(\dfrac{R_\star}{R_0}\right)=-\dfrac{R_\star^2}{R_0^2}, and after exponentiation, it yields
Introducing the variable x=\dfrac{R_\star}{R_0} we have to solve the equation x=e^{-x^2}
The solution is \phi=\dfrac{1}{x}=1.53158 from which the relationship between R_\star and R_0 can be easily obtained. Indeed, we can make more deductions from this result. From \ln \phi=1/\phi^2, then
R_n=R_\star e^{(n+1)\ln\phi}
If we take R_\star=\alpha R_0, with R_0=\hbar/mc, then
\alpha=m_0\sqrt{\dfrac{A_2 c}{A_1\hbar}} so
R_n=R_0e^{K\varphi_n} with K=\dfrac{1}{2\pi}\ln \phi and \varphi_n=2\pi (n+1)+\varphi_s \varphi_s=2\pi \left(\dfrac{\ln \alpha}{\ln \phi}\right)
Equivalently, the masses would be dynamically generated from the above equations, since
m_n=\dfrac{\hbar}{R_nc} and m_0=\dfrac{\hbar}{R_0c}
so we would deduce a particle spectrum given by a logarithmic spiral, through the equation
Remark: The shift K\rightarrow -K implies that the spiral would begin with m_0 as the lowest mass and not the biggest mass, turning the spiral from inside to the outside region and vice versa.
In summary, the logarithmic oscillator is also related to some kind of confined particles and it provides a toy model of confinement!
Is the link between classical statistical mechanics and Riemann zeta function unique or is it something more general? C. Tsallis explained long ago the connection of non-extensive Tsallis entropies an the Riemann zeta function, given supplementary arguments to support the idea of a physical link between Physics, Statistical Mechanics and the Riemann hypothesis. His idea is the following.
A) Consider the harmonic oscillator with spectrum
E_n=\hbar\omega n
E(n),\;\forall n=0,1,2,\ldots,\infty, are the H.O. eigenenergies.
B) Consider the Tsallis partition function
\displaystyle{Z_q (\beta )=\sum_{n=0}^{\infty}e_q^{-\beta E_n}=\sum_{n=0}^{\infty}e_q^{-\beta\hbar\omega n}}
where q>1 and the deformed q-exponential is defined as
e_q^z\equiv \left[1+(q-1)z\right]_+^{\frac{1}{1-q}}
and \left[\alpha\right]=\begin{cases}\alpha, \alpha>0\\ 0,otherwise\end{cases}
and the inverse of the deformed exponential is the q-logarithm
\ln_q z=\dfrac{z^{1-q}-1}{1-q}
It implies that
\boxed{\displaystyle{Z_q=\sum_{n=0}^{\infty}\dfrac{1}{\left[1+(q-1)\beta\hbar\omega n\right]^{\frac{1}{q-1}}}=\dfrac{1}{\left[(q-1)\beta\hbar \omega\right]^{\frac{1}{q-1}}}\sum_{n=0}^{\infty}\dfrac{1}{\left[\left(\dfrac{1}{(q-1)\beta\hbar\omega}\right)+n\right]^{\frac{1}{q-1}}}}}
Now, defining the Hurwitz zeta function as:
the last equation can be rewritten in a simple and elegant way:
\boxed{\displaystyle{Z_q=\dfrac{1}{\left[(q-1)\beta\hbar\omega\right]^{\frac{1}{q-1}}}\zeta \left(\dfrac{1}{q-1},\dfrac{1}{(q-1)\beta\hbar\omega}\right)}}
This system can be called the Tsallis gas or the Tsallisium. It is a q-deformed version (non-extensive) of the free Riemann gas. And it is related to the harmonic oscillator! The issue, of course, is the problematic limit q\rightarrow 1.
In the limit Q\rightarrow 1 we get the Riemann zeta function from the Hurwitz zeta function:
The above equation, the partition function of the Tsallis gas/Tsallisium, connects directly the Riemann zeta function with Physics and non-extensive Statistical Mechanics. Indeed, C.Tsallis himself dedicated a nice slide with this theme to M.Berry:
Remark (I): The link between Riemann zeta function and the free Riemann gas/the interacting Riemann gas goes beyond classical statistical mechanics and it also appears in non-extensive statistical mechanics!
Remark (II): In general, the Riemann hypothesis is entangled to the theory of harmonic oscillators with non-extensive statistical mechanics!
For readers not familiarized with Tsallis generalized entropies, I would like to expose you the main definitions of such a generalization of classical statistical entropy (Boltzmann-Gibbs-Shannon), in a nutshell! I have to discuss more about this kind of statistical mechanics in the future, but today, I will only anticipate some bits of it.
Tsallis entropy (and its Statistical Mechanics/Thermodynamics) is based on the following entropy functionals:
1st. Discrete case.
\boxed{\displaystyle{S_q=k_B\dfrac{1-\displaystyle{\sum_{i=1}^W p_i^q}}{q-1}=-k_B\sum_{i=1}^Wp_i^q\ln_q p_i=k_B\sum_{i=1}^Wp_i\ln_q \left(\dfrac{1}{p_i}\right)}}
plus the normalization condition \boxed{\displaystyle{\sum_{i=1}^Wp_i=1}}
2nd. Continuous case.
\boxed{\displaystyle{S_q=-k_B\int dX\left[p(X)\right]^q\ln_q p(X)=k_B\int dX p(X)\ln_q\dfrac{1}{p(X)}}}
plus the normalization condition \boxed{\displaystyle{\int dX p(X)=1}}
3rd. Quantum case. Tsallis matrix density.
\boxed{\displaystyle{S_q=-k_BTr\rho^q\ln _q\rho\equiv k_BTr\rho \ln_q\dfrac{1}{\rho}}}
plus the normatlization condition \boxed{Tr\rho=1}
In all the three cases above, we have defined the q-logarithm as \ln_q z\equiv\dfrac{z^{1-q}-1}{q-1}, \ln_1 z\equiv \ln z, and the 3 Tsallis entropies satisfy the non-additive property:
\boxed{\dfrac{S_q(A+B)}{k_B}=\dfrac{S_q (A)}{k_B}+\dfrac{S_q (B)}{k_B}+(1-q)\dfrac{S_q (A)}{k_B}\dfrac{S_q (B)}{k_B}}
Theoretical physicsts suspect that Physics of the spacetime at the Planck scale or beyond will change or will be meaningless. There, the spacetime notion we are familiarized to loose its meaning. Even more, we could find those changes in the fundamental structure of the Polyverse to occur a higher scales of length. Really, we don’t know yet where the spacetime “emerges” as an effective theory of something deeper, but it is a natural consequence from our current limited knowledge of fundamental physics. Indeed, it is thought that the experimental device making measurements and the experimenter can not be distinguished at Planck scale. At Planck scale, we can not know at this moment how the framework of cosmology and the Hilbert space tool of Quantum Mechanics could be obtained with some unified formalism. It is one of the challenges of Quantum Gravity.
Many people and scientists think that geometry and topology of sub-Planckian lengths should not have any relation with our current geometry or topology. We say and believe that geometry, topology, fields and the main features of macroscopic bodies “emerge” from the ultra-Planckian and “subquantum” realm. It is an analogue to the colours of the rainbow emerging from the atoms or how Thermodynamics emerge from Statistical Mechanics.
There are many proposed frameworks to go beyond the usual notions of space and time, but the p-adic analysis approach is a quite remarkable candidate, having several achievements in its favor.
Motivations for a p-adic and adelic approaches as the ultimate substructure of the microscopic world arise from:
1) Divergences of QFT are believed to be absent with such number structures. Renormalization can be found to be unnecessary.
2) In an adelic approach, where there is no prime with special status in p-adic analysis, it might be more natural and instructive to work with adeles instead a pure p-adic approach.
3) There are two paths for a p-adic/adelic QM/QFT theory. The first path considers particles in a p-adic potential well, and the goal is to find solutions with smoothly varying complex-valued wavefunctions. There, the solutions share certain kind of familiarity from ordinary life and ordinary QM. The second path allows particles in p-adic potential wells, and the goal is to find p-adic valued wavefunctions. In this case, the physical interpretation is harder. Yet the math often exhibits surprising features and properties, and some people are trying to explores those novel and striking aspects.
Ordinary real (or even complex as well) numbers are familiar to everyone. Ostroswski’s theorem states that there are essentially only two possible completions of the rational numbers ( “fractions” you do know very well). The two options depend on the metric we consider:
1) The real numbers. One completes the rationals by adding the limit of all Cauchy sequences to the set. Cauchy sequences are series of numbers whose elements can be arbitrarily close to each other as the sequence of numbers progresses. Mathematically speaking, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. Real numbers satisfy the triangle inequality \vert x+y\vert \leq \vert x\vert +\vert y\vert.
2) The p-adic numbers. The completions are different because of the two different ways of measuring distance. P-adic numbers satisfy an stronger version of the triangle inequality, called ultrametricity. For any p-adic number is shows
\vert x+y\vert _p\leq \mbox{max}\{\vert x\vert_p ,\vert y \vert_p\}
Spaces where the above enhanced triangle inequality/ultrametricity arises are called ultrametric spaces.
In summary, there exist two different types of algebraic number systems. There is no other posible norm beyond the real (absolute) norm or the p-adic norm. It is the power of Mathematics in action.
Then, a question follows inmediately. How can we unify such two different notions of norm, distance and type of numbers. After all, they behave in a very different way. Tryingo to answer this questions is how the concept adele emerges. The ring of adeles is a framework where we consider all those different patterns to happen at equal footing, in a same mathematical language. In fact, it is analogue to the way in which we unify space and time in relativistic theories!
Adele numbers are an array consisting of both real (complex) and p-adic numbers! That is,
\mathbb{A}=\left( x_\infty, x_2,x_3,x_5,\ldots,x_p,\ldots\right)
where x_\infty is a real number and the x_p are p-adic numbers living in the p-adic field \mathbb{Q}_p. Indeed, the infinity symbol is just a consequence of the fact that real numbers can be thought as “the prime at infinity”. Moreover, it is required that all but finitely many of the p-adic numbers x_p lie in the entire p-adic set \mathbb{Z}_p. The adele ring is therefore a restricted direct (cartesian) product. The idele group is defined as the essentially invertible elements of the adelic ring:
\mathbb{I}=\mathbb{A}^\star =\{ x\in \mathbb{A}, \mbox{where}\;\; x_\infty \in \mathbb{R}^{\star} \;\; \mbox{and} \;\; \vert x_p\vert _p=1,\; \mbox{for all but finitely many primes p.}\}
We can define the calculus over the adelic ring in a very similar way to the real or complex case. For instance, we define trigonometric functions, e^X, logarithms \log (x) and special functions like the Riemann zeta function. We can also perform integral transforms like the Mellin of the Fourier transformation over this ring. However, this ring has many interesting properties. For example, quadratic polynomials obey the Hasse local-global principle: a rational number is the solution of a quadratic polynomial equation if and only if it has a solution in \mathbb{R} and \mathbb{Q}_p for all primes p. Furthermore, the real and p-adic norms are related to each other by the remarkable adelic product formula/identity:
and where x is a nonzero rational number.
Beyond complex QM, where we can study the particle in a box or in a ring array of atoms, p-adic QM can be used to handle fractal potential wells as well. Indeed, the analogue Schrödinger equation can be solved and it has been useful, for instance, in the design of microchips and self-similar structures. It has been conjectured by Wu and Sprung, Hutchinson and van Zyl,here http://arXiv.org/abs/nlin/0304038v1 , that the potential constructed from the non-trivial Riemann zeroes and prime number sequences has fractal properties. They have suggested that D=1.5 for the Riemann zeroes and D=1.8 for the prime numbers. Therefore, p-adic numbers are an excellent method for constructing fractal potential wells.
By the other hand, following Feynman, we do know that path integrals for quantum particles/entities manifest fractal properties. Indeed we can use path integrals in the absence of a p-adic Schrödinger equation. Thus, defining the adelic version of Feynman’s path integral is a necessary a fundamental object for a general quantum theory beyond the common textbook version. However, we need to be very precise with certain details. In particular, we have to be careful with the definition of derivatives and differentials in order to do proper calculations. Indeed we can do it since both, the adelic and idelic rings have a well defined translation-invariant Haar measure
Dx=dx_\infty dx_2dx_3\cdots dx_p\cdots and Dx^\star=dx_\infty^\star dx_2^\star dx_3^\star\cdots dx_p^\star\cdots
These measures provide a way to compute Feynman path integrals over adelic/idelic spaces. It turns out that Gaussian integrals satisfy a generalization of the adelic product formula introduced before, namely:
\displaystyle{\int_{\mathbb{Q}_p}\chi_\infty (ax_\infty^2+bx_\infty)dx_\infty \prod_p \int_{\mathbb{Q}_p}\chi_p (ax_p^2+bx_p)dx_p=1}
where \chi is an additive character from the adeles to complex numbers \mathbb{C} given by the map:
\displaystyle{\chi (x)=\chi_\infty (x_\infty)\prod_p \chi_p (x_p)\rightarrow e^{-2\pi ix_\infty}\prod_p e^{2\pi i\{p\}_p}}
and \{x_p\}_p is the fractional part of x_p in the ordinary p-adic expression for x. This can be thought of as a strong generalization of the homomorphism \mathbb{Z}/\mathbb{Z}_n\rightarrow e^{2\pi i/n}.Then, the adelic path integral, with input parameters in the adelic ring \mathbb{A} and generating complex-valued wavefunctions follows up:
\displaystyle{K_{\mathbb{A}} (x'',t'';x',t') =\prod_\alpha \int_{(x' _\alpha ,t' _\alpha)}^{(x'' _\alpha ,t'' _\alpha)}\chi_\alpha \left(-\dfrac{1}{h}\int_{t' _\alpha}^{t''_\alpha}L(\dot{q} _\alpha ,q_\alpha ,t_\alpha )dt_\alpha \right) Dq_\alpha}
The eigenvalue problem over the adelic ring is given by:
U(t) \psi_\alpha (x)=\chi (E_\alpha (t))\psi_\alpha (x)
where U is the time-development operator, \psi_\alpha are adelic eigenfunctions, and E_\alpha is the adelic energy. Here the notation has been simplified by using the subscript \alpha, which stands for all primes including the prime at infinity. One notices the additive character \chi which allows these to be complex-valued integrals. The path integral can be generalized to p-adic time as well, i.e., to paths with fractal behaviour!
How is this p-adic/adelic stuff connected to the Riemannium an the Riemann zeta function? It can be shown that ground state of adelic quantum harmonic oscillator is
\displaystyle{\vert 0\rangle =\Psi_0 (x)=2^{1/4}e^{-\pi x_\infty^2}\prod_p \Omega (\vert x_p\vert_p)}
where \Omega \left(\vert x_p \vert _p\right) is 1 if \vert x_p\vert_p is a p-adic integer and 0 otherwise. This result is strikingly similar to the ordinary complex-valued ground state. Applying the adelic Mellin transform, we can deduce that
\Phi (\alpha)=\sqrt{2}\Gamma \left(\dfrac{\alpha}{2}\right)\pi^{-\alpha/2}\zeta (\alpha)
where \Gamma, \zeta are, respectively, the gamma function and the Riemann zeta function. Due to the Tate formula, we get that
\Phi (\alpha)=\Phi (1-\alpha).
and from this the functional equation for the Riemann zeta function naturally emerges.
In conclusion: it is fascinating that such simple physical system as the (adelic) harmonic oscillator is related to so significant mathematical object as the Riemann zeta function.
The Veneziano amplitude is also related to the Riemann zeta function and string theory. A nice application of the previous adelic formalism involves the adelic product formula in a different way. In string theory, one computes crossing symmetric Veneziano amplitudesA(a,b) describing the scattering of four tachyons in the 26d open bosonic string. Indeed, the Veneziano amplitude can be written in terms of Riemann zeta function in this way:
A_\infty (a,b)=g_\infty^2 \dfrac{\zeta (1-a)}{\zeta (a)}\dfrac{\zeta (1-b)}{\zeta (b)}\dfrac{\zeta (1-c)}{\zeta (c)}
These amplitudes are not easy to calculate. However, in 1987, an amazingly simple adelic product formula for this tachyonic scattering was found to be:
\displaystyle{A_\infty (a,b)\prod_p A_p (a,b)=1}
Using this formula, we can compute and calculate the four-point amplitudes/interacting vertices at the tree level exactly, as the inverse of the much simpler p-adic amplitudes. This discovery has generated a quite a bit of activity in string theory, somewhat unknown, although it is not very popular as far as I know. Moreover, the whole landscape of the p-adic/adelic framework is not as easy for the closed bosonic string as the open bosonic strings (note that in a p-adic world, there is no “closure” but “clopen” segments instead of naive closed intervals). It has also been a source of controversy what is the role of the p-adic/adelic stuff at the level of the string worldsheet. However, there is some reasearch along these lines at current time.
Another nice topic is the vacuum energy and its physical manifestations. There are some very interesting physical effects involving the vacuum energy in both classical and quantum physics. The most important effects are the Casimir effect (vacuum repulsion between “plates”) , the Schwinger effect ( particle creation in strong fields) , the Unruh effect ( thermal effects seen by an uniformly accelerated observer/frame) , the Hawking effect (particle creation by Black Holes, due to Black Hole Thermodynamcis in the corresponding gravitational/accelerated environtment) , and the cosmological constant effect (or vacuum energy expanding the Universe at increasing rate on large scales. Itself, does it gravitate?). Riemann zeta function and its generalizations do appear in these 4 effects. It is not a mere coincidence. It is telling us something deeper we can not understand yet. As an example of why zeta function matters in, e.g., the Casimir effect, let me say that zeta function regularizes the following general sum:
\boxed{\displaystyle{\sum_{n\in \mathbb{Z}}\vert n\vert^d =2\zeta (-d)}}
Remark: I do know that I should have likely said “the cosmological constant problem”. But as it should be solved in the future, we can see the cosmological constant we observe ( very, very smaller than our current QFT calculations say) as “an effect” or “anomaly” to be explained. We know that the cosmological constant drives the current positive acceleration of the Universe, but it is really tiny. What makes it so small? We don’ t know for sure.
Remark(II): What are the p-adic strings/branes? I. Arefeva, I. Volovich and B. Dravogich, between other physicists from Russia and Eastern Europe, have worked about non-local field theories and cosmologies using the Riemann zeta function as a model. It is a relatively unknown approach but it is remarkable, very interesting and uncommon. I have to tell you about these works but not here, not today. I went too far, far away in this log. I apologize…
I have explained why I chose The Spectrum of Riemannium as my blog name here and I used the (partial) answer to explain you some of the multiple connections and links of the Riemann zeta function (and its generalizations) with Mathematics and Physics. I am sure that solving the Riemann Hypothesis will require to answer the question of what is the vibrating system behind the spectral properties of Riemann zeroes. It is important for Physmatics! I would say more, it is capital to theoretical physics as well.
Let me review what and where are the main links of the Riemann zeta function and zeroes to Physmatics:
1) Riemann zeta values appear in atomic Physics and Statistical Physics.
2) The Riemannium has spectral properties similar to those of Random Matrix Theory.
3) The Hilbert-Polya conjecture states that there is some mysterious hamiltonian providing the zeroes. The Berry-Keating conjecture states that the “quantum” hamiltonian corresponding to the Riemann hypothesis is the corresponding or dual hamiltonian to a (semi)classical hamiltonian providing a classically chaotic dynamics.
4) The logarithmic potential provides a realization of certain kind of spectrum asymptotically similar to that of the free Riemann gas. It is also related to the issue of confinement of “fundamental” constituents inside “elementary” particles.
5) The primon gas is the Riemann gas associated to the prime numbers in a (Quantum) Statistical Mechanics approach. There are bosonic, fermionic and parafermionic/parabosonic versions of the free Riemann gas and some other generalizations using the Beurling gas and other tools from number theory.
6) The non-extensive Statistical Mechanics studied by C. Tsallis (and other people) provides a link between the harmonic oscillator and the Riemann hypothesis as well. The Tsallisium is the physical system obtained when we study the harmonic oscillator with a non-extensive Tsallis approach.
7) An adelic approach to QM and the harmonic oscillator produces the Riemann’s zeta function functional equation via the Tate formula. The link with p-adic numbers and p-adic zeta functions reveals certain fractal patterns in the Riemann zeroes, the prime numbers and the theory behind it. The periodicity or quasiperiodicity also relates it with some kind of (quasi)crystal and maybe it could be used to explain some behaviour or the prime numbers, such as the one behind the Goldbach’s conjecture.
8) A link between entropy, information theory and Riemann zeta function is done through the use of the notion of group entropy. Connections between the Veneziano amplitudes, tachyons, p-adic numbers and string theory arise after the Veneziano amplitude in a natural way.
9) Riemann zeta function also is used in the regularization/definition of infinite determinants arising in the theory of differential operators and similar maps. Even the generalization of this framework is important in number theory through the uses of generalizations of the Riemann zeta function and other arithmetical functions similar to it. Riemann zeta function is, thus, one of the simplest examples of arithmetical functions.
10) There are further links of the Riemann zeta function and “vacuum effects” like the Schwinger effect ( pair creating in strong fields) or the Casimir effect ( repulsive/atractive forces between close objects with “nothing” between them). Riemann zeta function is also related to SUSY somehow, either by the striking similarity between the Dirichlet eta function used in Fermi-Dirac statistics or directly with the explicit relationship between the Möbius function and the (-1)^F operator appearing in supersymmetric field theories.
In summary, Riemann zeta function is ubiquitious and it appears alone or with its generalizations in very different fields: number theory, quantum physics, (semi)classical physics/dynamics, (quantum) chaos theory, information theory, QFT, string theory, statistical physics, fractals, quasicrystals, operator theory, renormalization and many other places. Is it an accident or is it telling us something more important? I think so. Zeta functions are fundamental objects for the future of Physmatics and the solution of Riemann Hypothesis, perhaps, would provide such a guide into the ultimate quest of both Physics and Mathematics (Physmatics) likely providing a complete and consistent description of the whole Polyverse.
Then, the main unanswered questions to be answered are yet:
A) What is the Riemann zeta function? What is the riemannium/tsallisium and what kind of physical system do they represent really? What is the physical system behind the Riemann non-trivial zeroes? What does it mean for the Riemann zeroes arising from the Riemann zeta function generalizations in form of L-functions?
B) What is the Riemann-Hilbert-Polya operator? What is the space over the Riemann operator is acting?
C) Are Riemann zeta function and its generalization everywhere as they seem to be inside the deepest structures of the microscopic/macroscopic entities of the Polyverse?
I suppose you will now understand better why I decided to name my blog as The Spectrum of Riemannium…And there are many other reasons I will not write you here since I could reveal my current research.
However, stay tuned!
Physmatics is out there and everywhere, like fractals, zeta functions and it is full of lots of wonderful mathematical structures and simple principles!
LOG#049. Ludicrous speed.
We are going to learn about the different notions of velocity that the special theory of relativity provides.
The special theory of relativity is a simple wonderful theory, but it comes with many misconceptions due to bad teaching/science divulgation. It is not easy to master the full theory of relativity without the proper mathematical background and physical insight. In the internet era where knowledge is shared, a fundamental issue is to understand things properly. There are many people who thinks they understand the theory of relativity when they don’t. Even at the academia.
Moreover, you can find many people in the blogsphere/websphere trying to sell false theories and wrong theories. It is the same like the so-called alternative medicine: they are not medicine at all. Bad science is not science, it is simply a lie and not science at all. It is religion. Science can be critized, but nobody can critize that Earth revolves around the Sun, it is common knowledge and truth. So, we can make critics to scientist, but not the scientific method and well established theories. We can try to understand better or in a novel way, but we can not deny facts and experiments. Gerard ‘t Hooft, Nobe Prize, explain it in his web page www.phys.uu.nl/~thooft/.
It is important to remark that Science revolutions come when we extend the theories we know they are correct, like special relativity and not with a full destruction of the current and well-tested theories. Newtonian relativity is a limit of General Relativity. Galilean relativity is a limit of Special Relativity. Quantum Mechanics is a limit of QFT and so on. The issue is not that. Said these words, I am quite sure that scientists and particularly physicists wish to overcome current theories with new ones. However, the process to create a new theory is not easy. Specially, if you don’t understand the traps and theories that have passed every known test till now.
What is velocity? Classically, the answer is short and very clear/neat: velocity is the rate of change of position with respect to time. It is a vector magnitude. Mathematically speaking is the quotient between the displacement vector and the time interval, or in the infinitesimal limit, the derivative of the position vector with respect to time.
\boxed{\mathbf{v_m}=\dfrac{\Delta \mathbf{r}(t)}{\Delta t}\leftrightarrow \mbox{Average velocity}}
\boxed{\mathbf{v}=\dfrac{d\mathbf{r}(t)}{dt}\leftrightarrow \mbox{Instantaneous velocity}}
In the special theory of relativity, due to the fact that time is not universal but relative we can build different notions of velocity. And it matters. There are some clear concepts from relativity you should master till now:
a) You can attach a clock to any yardstick you could physically use for measurements of space and time.
b) You must distinguish the notions of coordinate velocity (map coordinate is another commonly used notion/concept) and proper velocity. The latter is sometimes called hyperbolic (or imaginary) velocity. These two notions are caused by the presence of two “natural” elections of time: the proper time and the coordinate time.
c) Due to the previous two facts, you must also distinguish between proper acceleration and geometric acceleration. Proper-accelerations caused by the tug of external forces and geometric accelerations caused by choice of a reference frame that’s not geodesic i.e. a local reference coordinate-system that is not ”in free-fall”. Proper-accelerations are felt through their points of action e.g. through forces on the bottom of your feet. On the other hand geometric accelerations give rise to inertial forces that act on every ounce of an object’s being. They either vanish when seen from the vantage point of a local free-float frame, or give rise to non-local force effects on your mass distribution that cannot be made to disappear. Coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by that connection term, and thus when physical and inertial forces add to zero.
People who are not aware of the previous comments, don’t understand relativity and the physics behind it. They even don’t undertand what experiments and their data say.
Let me review the main magnitudes, 3-vectors and 4-vectors which the special theory of relativity studies in the next tables:
The two notions of 3-velocity we do have from the special theory of relativity, i.e., from the 4-velocity \mathbb{U}=\dfrac{d\mathbb{X}}{d\tau}, are:
1) Coordinate velocity, \mathbf{v}:
It is the common notion of 3-velocity, measured from an inertial observer with respect to the coordinate time t. Note that the coordinate time is not a true invariant in SR!
2) Proper velocity (or the hyperbolic velocity/imaginary angle velocity related to it):
\mathbf{w}\equiv \dfrac{d\mathbf{r}}{d\tau}=\gamma \mathbf{v}
where \tau is the proper time. This velocity can intuitively defined as the distance per unit traveler-time, retains many of the properties that ordinary velocity loses at high speed. In addition to these two definitions, we also have:
1)Proper-acceleration \alpha, is the acceleration experienced relative to a locally co-moving free-float-frame, and it helps when we are accelerating, speeding, and in curvy space-time.
2) How some of the space-like effect of sideways ”felt” forces moves into the reference-frame’s time-domain at high speed, making the relatively unknown bound (from special relativity!)
\dfrac{dp}{dt}\leq m\alpha
With the above definitions, the relativistic momentum can be expressed in termns of coordinate velocity or proper velocity as follows:
\mathbf{P}=m\mathbf{w}=M\mathbf{v}=m\gamma \mathbf{v}
is the Lorentz factor. The last equal sign in the previous equation can be easily derived from the relativistic relationship:
and the definition of \gamma above.
Thanks to the metric-equation’s assignment of a frame-invariant traveler or proper-time \tau to the displacement between events in context of a single map-frame of comoving yardsticks and synchronized clocks, proper velocity becomes one of three related derivatives in special relativity (coordinate velocity \mathbf{v}, proper-velocity \mathbf{w}, and Lorentz factor \gamma) that describe an object’s rate of travel. For unidirectional motion, in units of lightspeed c (i.e. c=1 if we want to) each of these is also simply related to a traveling object’s hyperbolic velocity angle or rapidity \eta by the next set of equations:
\eta=\sinh^{-1}\left( \dfrac{w}{c}\right)=\tanh^{-1}\left(\dfrac{v}{c}\right)=\pm \cosh^{-1}\left(\gamma\right)
The next table illustrates how the proper-velocity of w_0 \equiv c or “one map-lightyear per traveler-year” is a natural benchmark for the transition from a sub-relativistic coordinate frame to a (fake) auxiliary super-relativistic motion (in imaginary units of i=\sqrt{-1}). Note that the velocity angle or pseudorapidity \eta and the proper-velocity w run from 0 to infinity and track the physical coordinate-velocity when w<<c. On the other hand when w>>c, the (hyperbolic or imaginary) proper-velocity tracks Lorentz factor \gamma while velocity angle \eta is logarithmic and hence increases much more slowly:
Hyperbolic velocities CAN exceed c! They can reach even the ludicrous speed of \infty when the coordinate velocity approaches c! However, you must never forget the fact that the velocity-angle/hyperbolic velocity IS imaginary in value. It is quite clear from the above table. Indeed, being somehow “trekkie” or a Sci-Fi “romantic” person, you could “define” warp-speeds as “imaginary/hyperbolic” velocities, i.e., in terms of proper velocity. In that case, you could get the correspondence
\mbox{WARP}0.25=\mbox{WARP}1/4=\dfrac{\sqrt{17}}{17}c\approx 0.24c
\mbox{WARP}0.5=\mbox{WARP}1/2=\dfrac{\sqrt{5}}{5}c\approx 0.45c
\mbox{WARP}1=\dfrac{\sqrt{2}}{2}c\approx 0.71c
\mbox{WARP}2=\dfrac{2\sqrt{5}}{5}c\approx 0.89c
\mbox{WARP}3=\dfrac{3\sqrt{10}}{10}c\approx 0.95c
\mbox{WARP}7=\dfrac{7\sqrt{2}}{10}c\approx 0.99c
\mbox{WARP}9=\dfrac{9\sqrt{82}}{82}c\approx 0.994c
\mbox{WARP}10=\dfrac{10\sqrt{101}}{101}c\approx 0.995c
\mbox{WARP}\infty\equiv c
In general, we can define the WARP speed as W=w/c and so, the proper velocity can be expressed in terms of the warp speed W in a very simple way w=Wc. Thus, the real or coordinate velocity would be connected with warp-speed through the relativistic equation:
Of course, the point is that, unlike the Sci-Fi franchise, the real velocity has never exceeded c, only the hyperbolic velocity and the proper velocity (note that in terms of SR, velocities approaching c imply very boosted frames, so despite we could travel to any point of the Universe in SR only approaching c very closely with respect to the traveler proper time-one human life-, but in terms of the “Earth” (or rest) reference frame millions of years would have passed away!).
When the coordinate-speeds approach c, the respective coordinate velocities deviate from this simple addition rule in that rapidities (hyperbolic velocity angle boosts) add instead of velocities, i.e. \eta_{12}=\eta_1+\eta_2. Coordinate velocities add non-linearly. And it is a well-tested consequence of the Special Theory of relativity. For highly relativistic objects (i.e. those with momentum per unit mass much larger than lightspeed) the result of the coordinate-velocity expression familiar from most textbooks is rather uninteresting since the coordinate-velocities all peak out at c, i.e., as everybody knows, in special relativity 1c\boxplus 1c=1c, because applying the relativistic addition of velocities rule, we get
c\boxplus c=\dfrac{ (c + c)}{(1 + 1)}=c
And it is a fact from both theory and experiment! It will remain as long as SR remains a valid theory. SR holds yet with an astonishing degree of precision and accuracy. So, you can not deny every data and experiment that confirms SR. That is completely nonsense but there are some people and pseudo-scientists out there building their own theories AGAINST the achievements and explanations that SR provides to every experiment we have done until the current time. I am sorry for all of them. They are totally wrong. Science is not what they say it is. Any theory going beyond SR HAS to explain every experiment and data that SR does explain, and it is not easy to build such a theory or to say, e.g., why we have not observed (apparently) superluminal objects. I will discuss more superluminal in a forthcoming post/log entry, some posts after the special 50th post/log that is coming after this one! Stay tuned!
Coming back to our discussion…Why is all this stuff important? High Energy Physics is the natural domain of SR! And there, SR has not provided ANY wrong result till, in spite that some researches going beyond the Standard Model include modified dispersion relationships that reduce to SR in the low energy regime, we have not seen yet ANY deviation from SR until now.
For unidirectional motion, at low speeds the coordinate velocity v_{13} of object 1 from the point of view of oncoming object 3 might be described as the sum of the velocity v_{12} of object 1 with respect to lab frame 2 plus the velocity v_{23} of the lab frame 2 with respect to object 3, that is:
Compare this expression to the previously obtained expression for rapidities! Rapidities always add, coordinate velocities add (linearly) only at low velocities. In conclusion, you must be careful by what you mean by velocity is a boosted system!
By the other hand, for relative proper-velocity, the result is:
This expression shows how the momentum per unit mass as well as the map-distance traveled per unit traveler time of object 1, as seen in the frame of oncoming particle 3, goes as the sum of the coordinate-velocities times the product of the gamma (energy) factors. The proper velocity equation is especially important in high energy physics, because colliders enable one to explore proper-speed and energy ranges much higher than accessible with fixed-target collisions. For instance each of two electrons (traveling with frames 1 and 3) in a head-on collision traveling in the lab frame (2) at
or equivalenty w_{12}=w_{23}=\gamma v\approx 88000 lightseconds per traveler second would see the other coming toward them at coordinate velocity v_{13}\approx c and w_{13}=88000^2(1+1) \approx 1.55\cdot 10^{10} lightseconds per traveler second or \gamma_{13}mc^2\approx 7.9 \mbox{PeV}. From the target’s view, that is an incredible increase in both energy and momentum per unit of mass.
Other magnitudes and their frame dependence in SR can be read from the following table:
CAUTION: These results don’t mean that the “real” energy is that. Energy is relative and it depends on the frame! The fact that in colliders, seen from the target reference frame, the energy can be greater than the center of mass energy is not an accident. It is a consequence of the formalism of special relativity. A similar observation can be done for velocities. Coordinate velocities, IN THE FRAMEWORK OF SPECIAL RELATIVITY, can never exceed the speed of light. As long as SR holds, there is no particle whose COORDINATE velocity can overcome the speed of light. However, we have seen that PROPER velocities are other monsters. They serve as a tool to handle rotations along the temporal axis, i.e., to handle boosts mixing space and time coordinates. Proper (or hyperbolic) velocities CAN be greater than speed of light. But, it does not contradict the special theory of relativity at all since hyperbolic velocities ARE NOT REAL since they are imaginary quantities and they are not physical. We can only measure momentum and real quantities! Moreover, remember that, in fact, group or phase velocities we have found before can ALSO be greater than c. So, you must be careful by what do you mean by velocity in SR or in any theory. Furthermore, you must distinguish the notion of particle velocity with those of the relative velocity between two inertial frames, since the particle velocities ( coordinate or proper) always refer to some concrete frame! In summary, be aware of people saying that there are superluminal particles in our colliders or astrophysical processes. It is simply not true. Superluminal objects have observable consequences, and they have failed to be observed ( the last example was the superluminal neutrino affair by the OPERA collaboration, now in agreement with SR).
Remark (I): From the last table we observe that in SR, the rotation angle is imaginary. Therefore, we are forced to use this gadget of hyperbolic velocity in order to avoid “imaginary velocities”.
Remark (II): Hyperbolic velocities would become imaginary velocities if we used the imaginary formalism of SR, the infamous ict=x_4.
Remark (III): Hyperbolic velocities are not coordinate velocities, so they are not physical at all. They are just a tool to provide the right answers in terms of rapidities, or the hyperbolic angle, whose units are imaginary radians! Hyperbolic velocities are measured in imaginary units of velocity!
Remark (IV): About the imaginary issues you can have now. The spacetime separation formula s^2=-c^2t^2+x^2+y^2+z^2 means that the time t can often be treated mathematically as if it were an imaginary spatial dimension. That is, you can define ct=iw so -c^2t^2=w^2, where i is the square root of -1, and w is a “fourth spatial coordinate”. Of course it is not at all. It is only a trick to treat the problem in a clever way. By the other hand, a Lorentz boost by a velocity v can likewise be treated as a rotation by an imaginary angle. Consider a normal spatial rotation in which a primed frame is rotated in the wx-plane clockwise by an angle \varphi about the origin, relative to the unprimed frame. The relation between the coordinates (w',x') and (w,x) of a point in the two frames is:
\begin{pmatrix}w'\\ x'\end{pmatrix}=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}w\\ x\end{pmatrix}
Now set ct=iw and \theta=i\varphi, with t,\theta both real. In other words, take the spatial coordinate w to be imaginary, and the rotation angle \varphi likewise to be imaginary. Then the rotation formula above becomes
\begin{pmatrix}ct'\\ x'\end{pmatrix}=\begin{pmatrix}\cosh\theta & -\sinh\theta\\ -\sinh\theta & \cosh\theta\end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}
This agrees with the usual Lorentz transformation formulat if the boost velocity v and boost angle \theta are related by the known formula \tanh\theta=v/c=\beta. We realize that if we identify the imaginary angle with the rapidity, we are back to Special Relativity. Indeed, it is only the rotations involving the time axis which can cause confusion because they are so different from our everyday experience. That is, we experience rotations along some direction in our daily experience, so we are familiarized with rotations and their (real) rotation angles. However, rotations along a time axis mixing space and time is a weird creature. It uses imaginary numbers or, if we avoid them, we have to use hyperbolic (pseudo)-rotations.
A) Lorentz factor \gamma=\dfrac{E}{mc^2}
\boxed{\gamma \equiv \frac{dt}{d\tau}= \sqrt{1+\left(\frac{w}{c}\right)^2} = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} = \cosh[\eta] \equiv \frac{e^{\eta} + e^{-\eta}}{2}}
B) Proper-velocity or momentum per unit mass.
\boxed{\frac{w}{c}\equiv \frac{1}{c} \frac{dx}{d\tau}=\frac{v}{c} \frac{1}{\sqrt{1-(\frac{v}{c})^2}}=\sinh[\eta]\equiv \frac{e^{\eta} - e^{-\eta}}{2} =\pm\sqrt{\gamma^2 - 1}}
C) Coordinate velocity v\leq c.
\boxed{\frac{v}{c} \equiv \frac{1}{c}\frac{dx}{dt}=\frac{w}{c}\frac{1}{\sqrt{1 + (\frac{w}{c})^2}} = \tanh[\eta] \equiv \frac{e^{2\eta} - 1} {e^{2\eta} + 1}= \pm \sqrt{1 - \left(\frac{1}{\gamma}\right)^2}}
D) Hyperbolic velocity angle or rapidity.
\boxed{\eta =\sinh^{-1}[\frac{w}{c}] = \tanh^{-1}[\frac{v}{c}] = \pm \cosh^{-1}[\gamma]}
or in terms of logarithms:
\boxed{\eta = \ln\left[\frac{w}{c} + \sqrt{\left(\frac{w}{c}\right)^2 + 1}\right] = \frac{1}{2} \ln\left[\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\right] = \pm \ln\left[\gamma + \sqrt{\gamma^2 - 1}\right]}
E) Warp speed (just for fun): |
49e8cd0d7a806d18 | I am slightly confused about hybridisation and how it relates to molecular and atomic orbitals, despite having pored through many sources online. I was hoping someone could verify whether my current understanding is correct, in particular regarding what hybridisation actually is/does because I have not read this explicitly, but am assuming it is the case from what I have read so far:
• The Schrödinger equation can be solved to give the atomic orbitals (at least, some simplification involving the effective nuclear charge can be used to find the outer (and inner?) atomic orbitals.
• In hybridisation, we first consider the shape of a molecule and then we consider each atom seperately and how we could combine the atomic orbitals so that the geometry about each atom is correct. Now what I am particularly unsure about is: Are the hybridised orbitals entirely a conceptual construct, or are they mathematical solutions to the Schrödinger equation? I was thinking that maybe the hybrid atomic orbitals that are created are a linear combination of the atomic orbitals, and thus this linear combination also solved the Schrödinger equation and can exist but would have a different energy to the individual atomic orbitals? I am finding it difficult to think how a linear combination of the atomic orbitals could produce something with a completely different shape though (although I suppose the orbitals are orientated and essentially vectorial, so I think it could work). Perhaps the hybrid atomic orbital form could also be, say, the product of the atomic orbital forms? Although then I do not think this would solve the Schrödinger equation anymore. But I am not sure. And anyway, all of these ideas assume that hybridisation has a mathematical basis in the Schrödinger equation, which I am not sure about at all!
• Then molecular orbital theory takes the orbitals of each atom (I think theoretically it takes every orbital in every atom?) in the molecule and combines them to form a molecular orbital for the whole molecule - no longer simply an overlap between two atoms. This certainly also solves the Schrödinger equation. From what I understand, theoretically the calculation to form MOs should involve only the AOs of each atom, but to simplify the situation sometimes hybrid atomic orbitals are used in MO construction? If so, surely the hybrid atomic orbitals must be a linear combination of the atomic orbitals that solve the Schrödinger equation, so that this can be used in molecular obrital theory which is based in the solution to the Schrödinger equation? Also, where in the calulcation are the coefficients for each atomic orbital arising? Is it from the boundary conditions including things like molecule geometry, or perhaps that we know the energy of a molecule and when we put this particular energy in to the Schrödinger equation it gives us the coefficients?
I apologise for the long post and I realise there seem to be many questions within here, however they are all linked and about the mathematical underpinnings of hybridisation/valence bond theory and molecular orbitals, so I thought they belonged in one post.
• $\begingroup$ When you say hybridization, are you referring to LCAO (linear combination of atomic orbitals)? Or are you referring to the hybridization taught in organic chemistry classes (where they only deal with the valence orbitals)? $\endgroup$ – CoffeeIsLife May 19 '17 at 19:42
• $\begingroup$ @QuantumAMERICCINO I think the latter (sp/sp2/sp3 mixing etc)- i.e. the hybridisation where you want it to match up with the shape of the molecule $\endgroup$ – 21joanna12 May 19 '17 at 20:31
• $\begingroup$ An important point is that the total electron density described by the hybrid orbitals is the same as the density of the unhybridized orbitals. We're just carving that space up differently in terms of names like drawing different boundaries on a map. The land isn't changed by the change in names. $\endgroup$ – Andrew Feb 12 at 2:15
Mathematically, atomic/molecular orbitals are 1-electron wavefunctions (hydrogen-like wavefunctions) that are used as a basis with-which the total N-election wavefunction is expanded. The N-election wavefunction is a determinant (or a linear combination of determinants) built from these 1-electron wavefunctions: an anti-symmetric (Fermi statistics) linear combination of Hartree products (usually a product of MOs expanded in a basis of AOs).
For simplicity's sake, lets assume we are interested only in 1-determinant wavefunctions. The Hartree-Fock method will (when used where HF is appropriate) result in a variationally-optimal 1-determinant wavefunction that is an energy eigenfunction of the applicable Hamiltonian operator. Not only is the final, self-consistently optimized HF total N-electron wavefunction an energy eigenfunction, but the individual HF canoncial MOs are also eigenfunctions of the Hamiltonian/Fock operator, with eigenvalues that are the so-called 'orbital energies' (canonical orbitals are defined as the optimzed orbitals which diagonalized the Fock matrix).
However, there is nothing special about these canonical orbitals in the context of the total system energy. Any linear, norm-preserving transformation of the occupied orbital space will result in an N-electron wavefunction that is still an eigenfunction of the Fock operator, while the individual MOs are no longer eigenfunctions of the Fock operator (for example, localized MOs, LMOs).
So, once we have a HF wavefunction and optimized MOs, we can linearly transform the MOs however we want. Such a linear transformation might have the effect of producing orbitals that resemble hybridized orbitals. Or maybe we are interested in generating localized MOs. The point is, a linear transformation within the occupied orbital space results in a wavefunction that has the exact same electon density and energy eigenvalue as the wavefunction HF originally provides us in terms of Canonical orbitals.
Orbitals transformed to look like hybrid-orbitals are on the exact same footing as any other choice of orbital.
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