Program for radiation and toxic hazards

I worked in my masters thesis with $^{87}Rb$ and $^{40}K$, really small beta emitters. But there are so many other things around in the lab, that I want to keep track on all the things I might get in contact with. Is there any computer program to calculate the dose of the whole decay chain to get a picture of the artificial radiation and supports logging. I don't want to look up all the individual numbers and calculate it manually. Also in my apparatus various clusters (Cr, Ni, Co, Cu, Ag, Pd, Ca....) are produced. E.g. I know chromium(VI) is carcinogen, but the pure metal is not. But in the nanoworld things may change. So is there a database around where I can lookup the toxity of various substances, with an emphasize on nanoparticles? I use nitrile rubber gloves and try to do not inhale something if I clean the apparatus. But this might be not enough precaution. The laser dyes are not healthy too.


Objects made up of electrons?

Say you have a neutral rod, and you bring a positively charged rod beside it (call the side the charged rod is brought near side A and the other side side B). The electrons from the side B will start moving towards side A and the positively charged nuclei in side A will start moving to side B. After a large part of side A consists of only of electrons, the electrons would start repelling each other and the movement of charges will stop. But at this point a very large portion of side A consists of electrons and a large part of side B consists only of positively charged nuclei. The system is at equilibrium. But the rod has now been mostly split into a electron side and a positively charged nuclei side, however it doesn't look different at all? Why? Shouldn't a rod made of atoms look and be completely different than a rod that is mostly made up of electrons on one side and nuclei on the other? I know that electrons are very small, and since side A mostly consists of electrons, shouldn't it be almost invisible?


Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method

I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation between a node $i$ and a node $j$ as a spring system, then $K_{i,j}$ (where $K$ is the stiffness matrix of the system) can be seen as the stiffness constant of the virtual spring between the two nodes. But does there exist a more general interpretation? Perhaps in terms of internal work? Another similar question is: What could be an interpretation of the coefficients $M_{i,j}$ of the mass matrix $M$?


What is the state-of-the-art on spacelike singularities in string theory?

What lessons do we have from string theory regarding the fate of singularities in general relativity? What happens to black hole singularities? What happens to cosmological singularities? Which points of view on string theory yielded results in this respect? String field theory? AdS/CFT? Matrix theory? I suppose perturbative string theory is not applicable in the vicinity of singularities.


Why does nonlinearity in quantum mechanics lead to superluminal signaling?

I recently came across two nice papers on the foundations of quantum mechancis, Aaronson 2004 and Hardy 2001. Aaronson makes the statement, which was new to me, that nonlinearity in QM leads to superluminal signaling (as well as the solvability of hard problems in computer science by a nonlinear quantum computer). Can anyone offer an argument with crayons for why this should be so? It seems strange to me that a principle so fundamental and important can be violated simply by having some nonlinearity. When it comes to mechanical waves, we're used to thinking of a linear wave equation as an approximation that is _always_ violated at some level. Does even the teensiest bit of nonlinearity in QM bring causality to its knees, or can the damage be limited in some sense? Does all of this have any implications for quantum gravity -- e.g., does it help to explain why it's hard to make a theory of quantum gravity, since it's not obvious that quantum gravity can be unitary and linear? > S. Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," 2004, > arXiv:quant-ph/0401062. > > L. Hardy, "Quantum theory from five reasonable axioms," 2001, arXiv:quant- > ph/0101012.


How mirror equation can explain farsightedness correction?

I have a friend who has just show me his medical prescription for hyperopia (farsightedness) correction and he needs glasses with 4,25 diopters for that, which seemed to be weird for me because I had learned, from the mirror equation, that the maximum correction possible for hyperopia is 4 diopters: $$ \frac{1}{f} = \frac{1}{p} + \frac{1}{p'} $$ If we have $0.25m$ for the normal eye distant point and more than $0.25m$ for the farsighted eye distant point (negative sign, because it's a virtual image), then we would have: $$ \frac{1}{f} = \frac{1}{0.25} + \frac{1}{p'} = 4 - \frac{1}{|p'|} \in\quad ]0,4[, \quad\text{since}\quad |p'| \geq 0.25m \quad\text{and}\quad p'<0 $$ I did some google search and find out that, indeed, hyperopia can reach values even greater, such as 20 diopters, but I can't find pages where doctors explain that with equations or physics teachers explain how things really work in ophthalmology. Either I am doing some terrible mistake, or doctors are doing some terrible mistake, or this equation just don't apply to hyperopia at all... Which one is true?


Is there a formal definition of a macroscopic variable in statistical mechanics?

Intuitively it's easy to accept that the usual variables like temperature, internal energy, etc. are 'macroscopic', but does there exist a formal definition of a macroscopic variable? In other words, is there a clear way to separate the set of all observables (and functions of observables) on a system into ones we would describe as 'macroscopic' and ones we would not? EDIT: Since apparently the answer is not completely straightforward, I'm interested in hearing any definitions which have appeared in literature, even if they are only conventions. I'm also interested in any necessary or sufficient conditions.


Entropy of the Sun

* Is it possible to measure or calculate the total entropy of the Sun?   * Assuming it changes over time, what are its current first and second derivatives w.r.t. time?    * What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?


Feynman Diagrams in 2 component notation

When using two component notation people often prefer to refrain from using arrows in Feynman diagrams to denote charge flow as is done in four-component notation. Instead, if understand correctly, they use arrows to denote chirality. I'd like to know what is the prescription to draw out the diagrams. I have read here (pg. 39) that > arrows indicate the spinor index structure, with fields of undotted indices > flowing into any vertex and field of dotted indices flowing out of any > vertices (see the reference above for many examples). However, trying this out on Majorana and Dirac mass terms, this doesn't seem to be correct. A Majorana mass term, $\psi ^\alpha \psi_\alpha +h.c.$, is thus composed only of undotted indices. With the reasoning above, it should have two arrows pointing into the vertex, ![enter image description here](http://i.stack.imgur.com/8B0z7.png) However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass. What am I missing?


How seriously do string theorists take the "landscape"

The string theory landscape seems to this outside observer to be an intermediate step in the intellectual progress toward a more robust theory that explains why our one universe has the particular properties that it has. is this the majority opinion or do most string theorists view the landscape as a plausibly being included in the final form of the theory?


Are all superalgebra's clifford algebra's

I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are clifford algebras, why make a literary distinction?


Does the collapse of the wave function increase entropy of the atomic system itself?

Does wave-function collapse cause the entropy of the atom (ie. the sub-atomic particle system that makes up the atom) to increase?


Hours of light per day based on latitude/longitude formula

I'm looking for a formula that will return the number of hours per day given a specific location. I was thinking that can be calculated as a difference of sunrise and sunset, but I see that there are some other ways, like in this topic. What is the best, fast and correct way to calculate this?


Why are electrons consider waves?

I know the wave nature of electrons was evoked to explain why atoms are stable but I thought waves could be put in the same state like photons yet electrons can not exist in the same state.


Deriving equations of motion of polymer chain with Hamilton's equations

This is related to a question about a simple model of a polymer chain that I have asked yesterday. I have a Hamiltonian that is given as: $H = \sum\limits_{i=1}^N \frac{p_{\alpha_i}^2}{2m} + \frac{1}{2}\sum\limits_{i=1}^{N-1} m \omega^2(\alpha_i - \alpha_{i+1})^2 $ where $\alpha_i$ are generalized coordinates and the $p_{\alpha_i}$ are the corresponding conjugate momenta. I want to find the equations of motion. From Hamilton's equations I get $\frac{\partial H}{\partial p_{\alpha_i}} = \dot{\alpha_i} = \frac{p_{\alpha_i}}{m} \tag{1}$ $- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (\alpha_i - \alpha_{i+1} ) \tag{2}$ , for $i = 2,...,N-1$. Comparing this to my book, (1) is correct, but (2) is wrong. (2) should really be $- \frac{\partial H}{\partial {\alpha_i}} = \dot{p_{\alpha_i}} = -m \omega^2 (2\alpha_i - \alpha_{i+1} - \alpha_{i-1}) \tag{$2_{correct}$}$ Clearly, I am doing something wrong. I suspect that I'm not chain-ruling correctly. But I also don't get, where the $\alpha_{i-1}$ is coming from. Can anybody clarify?


Spin state of electron after measurement

I have a system of two spin 1/2 particles in a superposition of spin states in the z-direction given by: $\psi = \frac{1}{2} |+ +\rangle + \frac{1}{2} |+ -\rangle + \frac{1}{\sqrt{2}} |- -\rangle$ where $+$ designates spin up, $-$ designates spin down and the first particle's state is the first term in each ket and the second particles' state is the second term in each ket. If I measure the spin on the first particle and get a value of $-\hbar / 2$ (corresponding to a spin down state) is the new state of the particles simply $\psi = | - - \rangle$ meaning that the first particle is now "set" to being spin down? And if I determine the spin on the first particle to be spin up, would the subsequent state be $\psi = \frac{1}{\sqrt{2}} |+ +\rangle + \frac{1}{\sqrt{2}} |+ - \rangle$ ? Basically, my question is once I make a measurement of a spin of a particle, does the wavefunction stay collapsed on the spin determined? And does having a second particle affect this in any way?


Does special relativity unify the two phenomena at the base of Faraday's flux law (was Feynman wrong in this case)?

Consider Faraday's flux law for the EMF generated in a conductor loop: $$ \varepsilon = - \frac{d \phi}{dt},$$ where $\varepsilon$ is the EMF, and $\phi$ is the magnetic flux through the loop. There are two possible causes for the flux to variate over time: variations in the magnetic field ("transformer EMF") and variations in the area enclosed by the loop ("motional EMF").   Feynman has noted that this is a unique case where a single rule is explained by two different phenomena: > We know of no other place in physics where such a simple and accurate > general principle requires for its real understanding an analysis in terms > of _two different phenomena_. Usually such a beautiful generalization is > found to stem from a single deep underlying principle. Nevertheless, in this > case there does not appear to be any such profound implication.   >  —Richard P. Feynman, _The Feynman Lectures on Physics_ (Volume II, 17-2). **Is it really true that there is no way to view these two phenomena as one?**   For example, it says in this Wikipedia article that this apparent dichotomy was part of what led Einstein to develop special relativity.   **Does special relativity give us a unifying principle to derive Faraday's law from?**


Pulsar beam radiant intensity distribution

I'm curious about the radiant intensity distribution of pulsars: what's the general dependence of intensity on angle, and what are typical angular beam widths? How much does the beam width vary between pulsars? (Presumably this is tied to magnetic field strength.) Even something as simple as a very sketchy plot of intensity vs. angle would be great. It's easy enough to find plots of observed intensity vs. time for individual pulsars, but it takes a bit to get from those to the distribution at the source.


General integral to find resistance

My question is: **is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?**. Obviously, if so, what is it, and if not, why? It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive. I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation. To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\rho$ in units of $(\Omega m)$. # Single Volume Framework We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it. This volume may have a constant resistivity $\rho$ while everywhere outside the volume is completely electrically insulating. # Infinite Volume Framework An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\rho(\vec{r})$ and no longer require a boundary condition. In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$. # Known Algebraic Analogs The basic algebraic formulation that I find insufficient is: $$R = \rho \frac{\ell}{A}$$ Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area. Obviously, this is a rather simple equation that won't apply to more complicated geometry. Even more sophisticated academic sources seem to give equations that fall short of what I'm asking. For example: $$R = \rho \int_0^l \frac{1}{A(x)} dx$$ I think it's obvious that an equation such as this is built upon a myriad of assumptions. For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly. Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it. I have some reasons to suspect this might actually be rather difficult. A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking. So maybe this will result in two interconnected calculus equations. # Motivation I had an interest in Squishy Circuits, and it occurred to me that I can't quickly and simply write down the equation for resistance between two points. The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating. However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions. If you make your conductor dough too long and/or too thin, you will encounter dimming of the light you connect with it. Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light.


Moment of inertia of a sector about its center point?

What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane?


Recommended first accessories for starblast 4.5

My son got an orion starblast 4.5 for Christmas. It comes with orion explorer II 17mm and 6mm eyepieces. We are looking at some additional accessories and wondering what you would recommend as "first accessories" to get the most out of the telescope. Our initial inclination is towards:   1. A barlow lens. Probably the orion shorty 2x barlow. Is the shorty-plus twice as good? (It is twice the price.)   2. A 25mm eye piece. From what we've read, a reflector with these specs is best for wide field views.   3. A solar filter. We think it would be cool to look at the sun.


How can I determine whether the mass of an object is evenly distributed?

How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?


In Which of the following situations would force be exerted on on object and no work be done on the object?

Consider the following question: > In which of the following situations would a force be exerted on an object > and no work be done on the object? > > I. a centripetal force is exerted on a moving object > > II. a force in the opposite direction as the object is moving > > III. a force is exerted on an object that remains at rest > > A.) I only > > B.) I and II > > C.) I and III > > D.) II and III > > E.) I, II, and III My response to this question is only II, but this is not a choice. I do not understand how I or III could have work. My logic is that in I, the force is centripetal and therefore is not parallel to the object's path, in any case it would be perpendicular because centripetal is towards the center. Work must be parallel. In III, if an object remains at rest, $d = 0$. And $W = fd$, so $W = 0$. I can't find the flaw in my logic. Any help is appreciated! Thanks!


Eigenvalue problem for differential equations in QM

I have a very simple question with regard to numerical methods in physics. I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I want to find $(E,\psi(x))$ satisfying \begin{align} \left[-\frac{1}{2}\partial_x^2 + V(x) \right]\psi(x)=E \psi(x). \end{align} Now how do I do it exactly? Naively I would implement the following algorithm: 1) Pick some $E$. 2) I want to find $\psi(x)$ which is normalizable. So I could pick a large $L > 0 $, set $\psi(-L) = \epsilon > 0$ and $\psi'(-L) = \epsilon'>0$ and numerically integrate from there using the Schrodinger equation. 3) If I encounter a solution which is exponentially small far to the right of the origin, then I say the solution is normalizable (since it is decaying at $|x|\to\infty$), and I accept the pair $(E,\psi(x))$. 4) I increment $E \to E + dE$ and I repeat the process. In doing so, I should get the spectrum around my starting value of $E$. Does this algorithm actually work?? It also seems to me like a very uncontrolled way of doing it; I have no idea how accurate the spectrum is going to be. For example, would changing $L, \epsilon, \epsilon'$ make a difference? The thing is, I know from Sturm-Liouville theory that the spectrum $E$ is going to be discrete (given $V(x)$ satisfying some nice properties). So the spectrum is going to be a set of measure 0 amongst the entire real line that $E$ lives in. This means that I'm almost surely (i.e. with probability 1) never going to get a solution that is normalizable, and whatever solution I try to numerically integrate from my starting point is always going to blow up having integrated far enough to the right. So, what algorithm do people use to numerically obtain the spectrum and the eigenvalues? How do I also control accuracy of the spectrum generated?


Electric field of a negative charge

How was it discovered that the electric field of a negative charge points towards the charge itself? Is it true? ![field of a negative point charge](http://i.stack.imgur.com/bh9uO.png) (Courtesy of wikipedia)


Frequency Of Light

I am confused on few topics... What is meant by "Frequency of Light"? Does the Photon(s) vibrate, that is known as its frequency? If the Photons vibrate, then they have a specific frequency, then What is meant by "Higher frequency light" as used in Photo- electric Effect? In which directions/axis do they vibrate to have a specific frequency. Why is it that nothing can go faster than the Speed of light. Does these two things have any relation to each other? -Thanks.


Doubt in Kinematics

I know that this isn't the place for such basic questions, but I didn't find the answer to this anywhere else. It's pretty simple: some particle moves in straight line under constant acceleration from one point $x_0$ to another point $x_1$ during the time interval $\Delta t_1$. When the particle reaches the point $x_1$ it reverses it's movement and goes to another point $x_2$ during another time interval $\Delta t_2$. I want to determine $x_2$, however I don't understand how to do it. My try was: Let $\Delta x_1 =x_1 -x_0$ be the first displacement and let $\Delta x_2 = x_2 - x_1$ be the second displacement. Then I can calculate two velocities: $$v_1 = \frac{\Delta x_1}{\Delta t_1}$$ $$v_2 = \frac{\Delta x_2}{\Delta t_2}$$ My thought is then to find the acceleration as: $$a = \frac{v_2-v_1}{\Delta t_2 + \Delta t_1}$$ But I'm not sure it'll work, since the movement reverses at $x_1$ and since I'm assuming the velocities constant on the intervals. Can someone help me how to think with this problem, and how to solve it?


Smaller mass in gravity well?

When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from infinity, radial path) have an effective mass equal to its rest mass, as here potential and kinetic energies are ballanced? The question is based on the concept of equivalence of energy and mass.


Can the field generated by a magnet domain extend to infinity?

As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there be a certain region beyond which the field won't be applicable? The instinctive answer to this question is no, but if you think about it we see the magnet's influence on the space around it as the result of equipotential regions then the contention is that only so many discrete equipotential regions are possible (the fact that something is not countable doesn't automatically mean it's infinite). Hence, that line of thought goes, there should be a limit theoretically and practically until which a field can exist. Can you please clarify this sticking point for me? Am I pushing the analogies we use to understand fields too far? What conceptual mistake am I making over here?


How can we test if something is a wave?

More specifically, I want to understand why a wave is a wave but a wave packet is not considered a wave (as discussed in this question). I would think that if something have these characteristics: 1. Wavelength; 2. Frequency; 3. Period; 4. Amplitude; and 5. Wave velocity; then it must be a wave. Does a wave packet have these characteristics? Do you have a better set of rules to apply to test if something is a wave?


Breakdown voltage of a dielectric

I know that a capacitor with a dielectric can operate normally up till a certain voltage (AFAIK called breakdown voltage) which depends on the strength of the dielectric placed between the plates. After this voltage, the circuit becomes short and current flows between the plates and thus the capacitor breaks down. But i want to know what is exactly happening when we say a dielectric "breaks down" ? What I know about a dielectric is that due to the electric field (because of the plates of the capacitor) the molecules of the dielectric align themselves accordingly and set up an electric field in the opposite direction, thus decreasing the net electric field. So, please can anyone tell me what happens during breakdown?


Is information propagated across a medium in any other way than waves?

Is information propagated in any other way than waves? Please distinguish "propagation across a medium" from information "storage within stable states of matter", which might difuse or interact chemically. Information might be stored in stable configurations of matter, which might diffuse, or interact chemically (odor,DNA), but these might be orders of magnitude weaker, in range and dissipation. Is there a domain in physics comparing wave vs non-wave propagation. The two most known are sensory: sound and electromagnetic propagation. I think gravity probes are still searching for waves in this medium. Why does nature prefer waves for long distance calls? Perhaps because it involves a minimum dissipation of energy?


How to model an apartment's airconditioning?

**Background:**   This reminds situation reminds be of an episode from the _The Big Bang Theory_ , although it is quite different from it. My roommate and I have similar temperature preferences and also share the objective of saving money on electricity. The optimal solution towards this objective is what we have differing opinions about. **The problem:**   In my opinion, during the daytime when no-one's around, the temperature setting should be at least 4 °F more than the normal setting (given an ongoing summer 90+ °F). I would go far as to argue that during the daytime the cooling should be completely switched off. My roommate's counter-argument is that assuming a PID controlled model, the overshoot caused by the cooling in the evening, in an effort to get the temperature down from the day's high, will more than nullify the electricity saved during the day thereby resulting in higher costs. The controller is a programmable digital thermostat with 4 separate settings for `sleep`, `wake`, `leave` and `return` for two regimens namely `weekday` and `weekend`. **Specifically:**   Instead of hand-waving I was looking for a mathematical model to simulate the heating-cooling of a typical apartment. A quick Google search yielded no useful models (most I could get my hands on are macro-models related to multiple commercial units). I am looking for a model — with elements like the `heat exchange` (i.e. the ac), the `heat sink` (i.e. the apartment) and the `environment` — a general model which can be used for a _first-order_ approximation. Assume that the cost of electricity is constant throughout the period under consideration (even if it isn't, a cost vs time graph can be easily incorporated in the model).


Question about moment of inertia and velocity

First off, I swear this is not homework. I'm doing some practice problems because I got an exam coming up. I'm stuck on this one: ![alt text](http://i.stack.imgur.com/w7KMz.png) I figured I would use energy conservation for this problem. So since the thing is not moving initially, I tried doing $mgh=1/2 I\omega^2+1/2 mv^2$, but that doesnt give me the right answer. Any ideas?


Will the Hubble constant reach zero asymptotically in the far future?

In the current accelerated expansion universe model will the Hubble constant reach zero asymptotically in the far future?


Is mass of a particle changed when it is charged?

If a particle of mass $M$ is given an electric charge $Q$, will its mass change?


Surface normal on the earth to the sun at a given point in time

How complicated is it to calculate a surface normal on the spherical approximation of the earths surface pointing towards the sun at a given point in time? What I try do is to highlight a small area on a world map where the radiation from the sun tangential hits a imaginary sphere around the earth. This is just to get an idea where this area was at the given time, so this information has not to be very accurate. As a mater of fact I will just incorporated this information in the map if it is easy enough to calculate.


Scale invariance plus unitarity implies conformal invariance?

What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?


Gauss' Law for Magnetism Derivative Form: With or without volume integral?

I've been reading through FLP Vol. II, and he has proven that as the flux through a closed surface is: $\ \int_{surface} \mathbf{F} \space \mathrm{d}\mathbf{a} $, according to the divergence theorem, the flux through a surface can be defined as: $\ \int_{volume} \nabla \cdot \mathbf{F} \space \mathrm{d}V $, where $\ \mathbf{F} $ is any vector field, and the volume is that which is enclosed by the surface. Previously he had stated as a word equation that: $\ \text{Flux of } \mathbf{B} \text{ through any closed surface}=0. $ I would therefore assume that $\ \int_{volume} \nabla \cdot \mathbf{B} \space \mathrm{d}V = 0$, however Gauss' law for magnetism states that: $\ \nabla \cdot \mathbf{B} = 0$. Does that mean that $\ \nabla \cdot \mathbf{B} = 0$ and $\ \int_{volume} \nabla \cdot \mathbf{B} \space \mathrm{d}V = 0$ are equivalent statements, or am I making a fundamental error somewhere?


When did we learn that stars die?

As we all know, the stars we see in the night sky might already be dead. I was wondering though, when was this fact or conclusion commonly established? Today, most people (let's assume with an above average education) would probably be aware of this fact. When is the earliest time when the same could be said? I am particularly interested if the same could be said for the time period revolving around the period 1850 - 1900. I know that the speed of light was approximated fairly accurately in the 17th century. Knowing this (finite) speed, it's not hard for me to draw the conclusion that the source of the light I see may not be there anymore. Would this be an easy conclusion to draw a hundred years ago however? Maybe they thought stars don't die?


Ambiguity in Beta Functions (2-loop)

Beyond one-loop, the beta function of a QFT is scheme dependent. I would like to understand better this ambiguity. The easiest thing to say is that you haven't calculated something physical, so of course it does not need to be scheme independent. However, the anomalous dimension of operators is I think an observable quantity since we may measure critical exponents in the lab, and the anomalous dimension results from the same sort of calculation. Moreover, I can relate the beta functions to the trace anomaly. Schematically, $\langle \partial_\mu j_{dilation}^\mu \rangle=\langle T^\mu_\mu \rangle \approx \beta$ (See Peskin 19.5 for the case of QED). If i couple some field to the trace of $T$ I think I should be able to turn this anomaly into a cross-section for some process which would be measurable (think of ABJ anomaly and $\pi^0 \to \gamma \gamma$ for instance). So the questions is: 1) Is it known how the terms in the beta function may differ between regularization schemes? If I try to calculate the couplings at the fixed point using different schemes, will I get the same answer (I am aware the location of the fixed point is not physical, but if I use the same field variables I could imagine this being scheme independent)? How may I see that although the beta function and location of the fixed point are ambiguous, the anomalous dimensions are not? 2) How would this ambiguity cancel out if I have a theory where I can turn the trace anomaly into a prediction of a scattering amplitude? Or can this simply not be done? Any clarification or suggestion for references is appreciated.


Using Lorentz Invariance of Charge To Calculate Current Density

I'm attempting a problem from Zwiebach: A First Course in String Theory and am completely stuck. Could anyone give me a hint? The problem is as follows. Consider $S$, $S'$ two Lorentz frames with $S'$ boosted along the $+x$ axis. In frame $S$ we have a cubic box with sides of length $L$ at rest. The box is filled by a material, also at rest, of uniform charge density $\rho$. In $S$ we assume that the charge density $\underline{j}=0$. Use the Lorentz invariance of charge to calculate the charge density $\rho'$ and current density $\underline{j}'$ in $S'$. Verify that $(c\rho,\underline{j})$ a 4-vector. The charge density is easy. Indeed $L^3\rho = Q = Q' = L'^3\rho'=\frac{L^3}{\gamma}\rho'$ so $\rho' = \gamma \rho$. I know I'm right here because this agrees with what we'd expect from a 4-vector under Lorentz boost. To do the current density I tried to use $0=\frac{\textrm{d}Q}{\textrm{d}t}=\frac{\textrm{d}Q'}{\textrm{d}t'}=\int_S\underline{j}'.\textrm{d}\underline{a}=j'_xL^2$ so $\underline{j}'=0$ since $j'_y=j'_z=0$ clearly must be zero. I know this is wrong though, because it doesn't agree with what I'd expect from a 4-vector! What am I doing wrong? And is this the right way to go about this question? Many thanks in advance!


Is dimensional analysis used outside fluid mechanics and transport phenomena?

Most dimensionless numbers (at least the ones easily found) used for dimensional analysis are about fluid dynamics, or transport phenomena, convection and heat transfer - arguably also sort of fluid mechanics. My understanding of dimensional analysis is the following: Derive dimensionless numbers from the description of a system, find the ones physically meaningful, and use them to compare different situations or to scale experiments. Is this possible in other fields, like classical mechanics, and their engineering applications? Example: describe a horizontal beam by: $$ X=\frac {\text{forces acting on the beam}} {\text{forces beam can withstand without plastic deformation}} $$ Both parts of the ratio being functions of shape, density, gravity, material constants etc. My assumption is yes, it's possible, but most fields outside the sort-of fluid mechanics described above are easy enough to calculate without dimensional analysis.


Measurements for thermal diffusivity of graphene?

We have known for a long time that graphene has in-plane thermal conductivity ranging between 2000 and 4000 $W m^{-1} K^{-1}$. But in order to model heat transport on a sheet of graphene, we need more than the conductivity: we also need specific heat in order to obtain the thermal diffusivity that is used in the equation. I couldn't find any measurement results online for this quantity. I've only seen some crude estimates based on phonon transport, but even so, no specific figures.


design of the heat exchanger...in chimney

![heat exchanger diagram](http://i.stack.imgur.com/w04j8.jpg) I want to design a heat exchanger in a chimney in order to utilize heat from chimney. I have done several experiments, but I could not determine the exact length of tube (carrying water), such that its inlet temp is ambient temperature and outlet temperature is expected to be 100 degree Celsius. I would be very thankful if some one would help me determine the length of the tube. tube diameter is 10mm (almunium), chimney diameter is(120 mm). Can anyone help me with formulas involved in calculating the heat transfer and length of the tube... I already calculated the lenght experimentally but I could not do it mathematically.


Tighten rope around cylindricaly shaped space

Imagine we live on cylinder(we are 2d creature), put a rope around that cylinder and start pulling both ends of the rope against each another. Will the space get deformed? I guess it will, I have to put some energy to the rope and energy deforms the space. Is it possible to calculate shape of the space based on the force I put in the rope? In order to calculate this I guess you have to make 3d analogy and use laws of general relativity.


What really goes on in a vacuum?

I've been told that a vacuum isn't actually empty space, rather that it consists of antiparticle pairs spontaneously materialising then quickly annihilating, which leads me to a few questions. Firstly, is this true? And secondly, if so, where do these particles come from?... (do the particles even have to come from anywhere?)


Neutron decay and electron anti neutrino $n\to p + e + \bar{\nu}_e$

Why do we need neutrino to explain neutron decay? Is there any evidence regarding existence neutrinos in the context of $n\to p + e + \bar{\nu}_e$?


Why does the nature always prefer low energy and maximum entropy? I've just learned electrostatics and I still have no idea why like charges repel each other. http://in.answers.yahoo.com/question/index?qid=20061106060503AAkbIfa I don't quite understand why U has to be minimum. Can someone explain?


Simple explanation of Quantum Zeno Effect

I'm a student and I had to give a talk on seminar about Quantum Zeno effect and Anti-Zeno effect to my colleagues (all listeners have had a course in quantum physics, but not a heavy one with all the bra and ket stuff). My first idea to give a simple explanation of Zeno's effect was this: Let's take a look at exponential decay where the chance for particle or state to survive some time $t$ is $P_S=e^{-t/\tau}$. If I measure it after time $\tau$ I have chance $P_S=1/e$ that it will still be intact. If I instead allow it to do it's things only for time $\tau/N$ and then measure it, the survival chance will be $P_S=e^{-1/N}$ which approaches $1$ as $N$ increases. To achieve the same total time, I have to repeat this procedure $N$ times and the total survival probability is... $P_S=(e^{-1/N})^N=1/e$. So it obviously doesn't work, I get no Zeno's effect in this way. It's interesting that after I gave the talk professor rose and said "Well, this can be easily understood if we look at the exponential decay". Then he started drawing exponent and another exponent that's repeatedly interrupted and reset to initial state after small intervals. Later we agreed that this won't actually work, but the question is - why? Why doesn't this intuitively obvious way doesn't work and what would be the correct law out of which one could see the Zeno's effect? Is there any elegant way to explain this effect without heavy math and angles of state vectors? ADDITIONAL QUESTION (related): Is it correct to use name "Quantum Zeno Effect" for turning of polarization by series of inclined polarizers or the thing that is done in this article?


An alternative definition of the creation and annihilation operators?

Suppose we have a system of bosons represented by their occupation numbers $$\tag{1} | n_1, n_2, ..., n_\alpha, ... \rangle$$ Then we can define creation and annihilation operators $$\tag{2} a_\alpha^\dagger| n_1, n_2, ..., n_\alpha, ... \rangle = \sqrt{n_\alpha+1} | n_1, n_2, ..., n_\alpha+1, ... \rangle$$ $$\tag{3} a_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = \sqrt{n_\alpha} | n_1, n_2, ..., n_\alpha-1, ... \rangle$$ This is nice because the number operator is just $a_\alpha^\dagger a_\alpha$. However, would it be sensible to define an alternate set of operators to work with? $$\tag{4} b_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = | n_1, n_2, ..., n_\alpha+1, ... \rangle$$ $$\tag{5} c_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = \begin{cases} | n_1, n_2, ..., n_\alpha-1, ... \rangle & n_\alpha>0 \\\ 0 & n_\alpha=0 \end{cases}$$ $$\tag{6} N_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle = n_\alpha| n_1, n_2, ..., n_\alpha, ... \rangle $$ Why don't we work with these operators? The bosonic creation and annihilation operators $a_\alpha^\dagger$ and $a_\alpha$ were defined to mimic the harmonic oscillator's raising and lowering operators ($x \pm i p$), but is there any compelling reason to keep the $\sqrt{n_\alpha+1}$ and $\sqrt{n_\alpha}$ factors? I suppose $a_\alpha^\dagger$ and $a_\alpha$ obey nice properties such as $[a_\alpha,a_\alpha^\dagger]=1$ and the fact that they are Hermitian adjoints of each other. What are the analogous relationships that $b_\alpha$ and $c_\alpha$ would obey?


Is there an efficiency-optimizing layout of items in a refrigerator?

Is there some optimal layout of items, or types of items (characterized by size, shape, density, solid/liquid, etc) that will allow a common household refrigerator to function with optimal energy efficiency? I know this is a very broad and any answer will have to make several assumptions, not the least of which include how often the refrigerator is opened and for how long. I'd be perfectly happy with a "steady-state" answer that assumes, e.g. the refrigerator is never opened. But more is always better. If nothing else, I'm curious about what considerations might go into conducting a more thorough analysis.


Why do white dwarfs shine white?

The hotter something is glowing the more white/blue it appears. A dying medium sized star expands, cools and becomes a red giant for a while, but eventually it is going to gravitationally collapse (once enough Iron (Fe) is accumulated in the core). Then it blows the outer layers away and what is left collapses into a white dwarf. What makes the dwarf shine? and why is it white? Does the luminosity decreases as the object cools down, or is there some other reaction that keeps it glowing for a long time? Can a white dwarf turn brown or black never to be seen again? Do all white dwarfs turn into Neutron stars eventually?


Matter in superconductive state

We distinguish between the states of matter: gas, liquid and solid. Possibly we could add the plasma state and/or the superconductive state as new states of matter. Phase transistions at certain temperature perhaps with some other conditions should have to exist. What do you think, does it make sense?


What happens to entangled particles when momentum is measured?

In Wikipedia it is mentioned that position and momentum can be entangled as well as spin and polarization etc. I assume etc. is charge etc. I understand how if you measure spin up on one of a pair you get spin down on the second of the pair. What happens to the other particle in an entangled pair if I measure the momentum, position or charge of one of the particles? Is there a momentum up and down or charge up and down analog? http://en.wikipedia.org/wiki/Quantum_entanglement


Can physics (ever) explain intrinsic properties of nature?

I may be totally off with this quite abstract (?) question(s). But still, here are some closely related sub-questions:   * Is there a list of currently "known" intrinsic properties of nature?   * How exactly is an intrinsic property of nature defined? Is it defined as " _it is so, because it is so_ " or " _it is so, because it cannot be otherwise (in our Universe)_ "? For example, Wikipedia about mass as intrinsic property: > _An intrinsic property is a property that an object or a thing has of > itself, independently of other things, including its context. An extrinsic > (or relational) property is a property that depends on a thing's > relationship with other things. For example, **mass is an intrinsic property > of any physical object**._ Is this a correct explanation of an intrinsic property?


Does light change color on its way through a window?

Looking at the refractive index of glass, it's around $1.6$. Then the speed of light $x$ through light should be given by $$ 1.6 = \frac{3.0\times10^8}{x}, $$ so $x$ is about $2\times10^8~\mathrm{m}~\mathrm{s}^{-1}$ The frequency is kept constant, so the wavelength must adapt to suit the slower speed, giving a wavelength of $2/3$ the original. Does this mean that when passing through glass, say red light (wavelength $650~\mathrm{nm}$) changes to indigo ($445~\mathrm{nm}$), as $650 \times 2/3 = 433~\mathrm{nm}$, or is my logic flawed somewhere?


What determines color -- wavelength or frequency?

What determines the color of light -- is it the wavelength of the light or the frequency? (i.e. If you put light through a medium other than air, in order to keep its color the same, which one would you need to keep constant: the wavelength or the frequency?)


Pn junction voltage drop?

This image from wikipedia, explains that there occurs a potential drop across a pn semiconductor junction, and an electric field confined to the depletion region.![enter image description here](http://i.stack.imgur.com/7VaN3.png) I already know the reason for the existence of this drop and the calculation of the difference but I have two questions regarding this drop. **1) If then and p doped regions are externally connected using a perfectly conducting wire, why will not any current flow?**   Connecting the two regions should equalize their potentials (since wire is a resistance less) conductor, and therefore the gradient at the junction is destroyed resulting in a diffusion current which is obviously against conservation of energy as the semiconductor has non-zero resistivity. Where will extra potential drops will be created so that Kirchoff's voltage rule holds without any current and the built-in potential difference $V_o$ of the juction persists? **2)** (probably naive) If an external Bias is applied, of say $|V|<|V_o|$, then the pn potential difference across the junction will just reduce by that amount ($|V|$). Assuming that the external voltage source is ideal without any resistance, **what will be the potential drops which would sum to zero in this case? Will drop due to the resistance of semiconductor play a par in it? If yes, then there cannot be any ideal resistance-less semiconductor junction, can there?**


Some questions on the Wilson loop in the projective construction?

Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we say that _$H$ and $H'$ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces_. And the Wilson loop $W(C)$ can be defined as the trace of matrix-product $P(C)$(see the notations here ). Now my questions are: (1)"$H$ and $H'$ are gauge equivalent" if and only if "$W(C)=W'(C)$ for all loops $C$ on the 2D lattice ". Is this true? How to prove or disprove it? (2)If the system is on a 2D torus, is $W(L)$ always a _positive real number_ ? Which means that the 'total flux'(the phase of $W(L)$) through the torus is quantized as $2\pi\times integer$, where $L$ is the boundary of the 2D lattice. (3)If the Hamiltonian contains extra terms, say $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+\psi_i^T\eta_{ij}\psi_j+H.c.+\psi_i^\dagger h_i\psi_i)$, is the Wilson loop still defined as $W(C)=tr(P(C))$? Thanks a lot.


Problem with Velocity of efflux

I am stuck in this problem- ![The Problem](http://i.stack.imgur.com/CBC4d.png) I need to find the velocity of efflux at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the container]. Here's my approach Velocity of liquid at the upper-surface = $v_2$ Velocity of efflux (velocity of water at the hole, right?) = $v_1$ Using Bernoulli's equation for the surface and the hole - $$ P_{atm} + \rho_2 g (h_1 + h _2) + \frac{1}{2}\rho_2 v_2^2 = P_{atm} + \rho_1 g h_1 + \rho_2 g h_2 + \frac{1}{2}\rho_1 v_1^2 \\\ \implies \rho_2 g h_1 + \rho_2 g h_2 - \rho_1 g h_1 - \rho_2 g h_2 = \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) \\\ \implies \frac{1}{2}(\rho_1 v_1^2 - \rho_2 v_2^2) = g h_1 (\rho_2 - \rho_1) $$ Now, let area of the base be $A_2$ and that of the hole be $A_1$ then, using equation of continuity, $$ A_1 v_1 = A_2 v_2 \\\ \implies v_2 = \frac{A_1}{A_2} v_1 \\\ \implies v_2 \approx 0 (\because {A_1 << A_2}) $$ Using this value in the previous Bernoulli's relation $$ \frac{1}{2}(\rho_1 v_1^2) = g h_1 (\rho_2 - \rho_1) \\\ \implies \frac{1}{2} \rho_1 v_1^2 = g h_1 (\rho_2 - \rho_1) \\\ \implies v_1 = \sqrt {\frac {2gh_1(\rho_2 - \rho_1)}{\rho_1}} $$ Which is not the correct answer. I did get a correct answer in the chat room, but it was using a different method. What's wrong with my method?


Ramsey Interactions

What are Ramsey interactions? I am researching atomic clocks and am not sure why the atoms need to be exposed twice to an electromagnetic field in order to cause excitation.


What is "Symmetric Fission"?

Dose anyone has a clue what Symmetric Fission is? I couldn't find any explanation on what is it on internet.


Bose-Einstein condensate and nonlinear waves

Can Bose-Einstein condensate be written as non-linear wave equation (in terms of mean field approximation theory)? the equation is: ![enter image description here](http://i.stack.imgur.com/X8kEs.png) source: http://xxx.tau.ac.il/abs/1308.2288 > What I do understand by the Bose-Einstein condensate is, it is the state of > atoms at very low temperatues and at the moment the atoms forget their > previous identity. Therefore all the atoms stay in the same quantum state. Am I right?


future light cones and light paths

I understand that an event, in a four dimensional space-time, produces a light cone. As time increases the cones gets larger on either side of the event (past and future). For example the if the sun where to "go out" it would take 8 minuets for the earth to be affected by it simply because it takes approximately 8 minuets for light from the sun to reach the earth due to its location in the future light cone of the event (the sun being the event). Einstein made a suggestion that space-time is warped (vs flat) by the distribution of mass and energy and that bodies (like earth, jupiter, etc.) are meant to follow straight paths but cannot in a warped space (or appear not to because space is warped). However light supposedly follows these warped paths (called geodesics) as well even tho light is energy. How can space bend light? and why does light have to follow a specific path which is warped by space? Things with mass (like water) have to go around things (like rocks), but light can go through certain things or expand, but it doesn't move to the side like water does to a rock. Does it?


Electromagnetic radiation and black body radiation

I was taught today that the Electromagnetic wave Theory is unable to explain black body radiation. The example that was given to me: When a metal is heated, it emits different frequencies of light as it gets hotter. If electromagnetic wave theory was correct, it would not be so, the frequency (color) of light would remain the same, but only the intensity will change. I don't understand why this is so. My logic: Electromagnetic waves occur when a charged body oscillates in a electric and magnetic field. If the metal is provided with more energy (in the form of heat) won't the charged body vibrate faster, thus changing the frequency of the light emitted?


What is the direction of the friction force on a rolling ball?

Suppose you have a solid ball on a horizontal table.   1. What is the direction of friction force when the ball I pushed horizontally and **starts rolling**?   2. Why is the direction of friction as it is?    3. Which forces acts at the contact point between delta time t0 to t1? (If we divide friction force in sub forces) V=1Vx m/s Fx=?


Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not conformal? General Relativity is invariant under diffeomorphisms, but it certainly is not invariant under conformal transformations, if conformal transformations where a subgroup of diff, you would have a contradiction. Or I am overlooking something important?


Induced EMF in axle of car

A car moves on a plane road in east-west direction.At what latitude of earth should it move so that induced EMF in the axle connecting it's wheels is maximum? A. At the poles B. At the equator C. At a latitude inclined at 45 degree to the equator


How does a knife cut things at the atomic level?

As the title says. It is common sense that sharp things cut, but how do they work at the atomical level?


Why is velocity of outermost point on a rotating wheel double the velocity of centre of mass?

'In the answers to one of the questions based on rotation of a disc in my physics book the answer includes the statement 'As we know that the velocity of outermost point on a rotating disc is double the velocity of center of mass'. But, I didn't know that and why is it like that? My thinking: I know that as we go away from the center of mass the tangential velocity of particles of the disc increases according to $v=wr$. But, how does that result in the above result?


Does the 4/3 problem of classical electromagnetism remain in quantum mechanics?

In Volume II Chapter 28 of the Feymann Lectures on Physics, Feynman discusses the infamous 4/3 problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ (uniformly distributed on the surface). If you integrate the energy density of the electromagnetic field over all space outside the particle, you'll get the total electromagnetic energy, which is an expression proportional to $c^2$. The energy divided by $c^2$ is what we usually call the mass, so if we calculate the "electromagnetic mass" in this manner we'll get $m = \frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$. If, on the other hand, you took the momentum density of the electromagnetic field and integrated it over all space outside the particle, you'd get the total electromagnetic momentum, which turns out (for $v<<c$) to be proportional to the velocity of the particle. The constant of proportionality of momentum and velocity is what we call mass, so if we calculated the electromagnetic mass in this way we would get $m = \frac{2}{3}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$, which is $\frac{4}{3}$ times the value we got before! That is the 4/3 problem. Feynman claims that this fundamental issue remains when we move to quantum electrodynamics. Was he right, and if so has the situation changed since the 1960's when he was writing? I've seen claims on the Internet (I don't have the links) that the 4/3 problem is still there in QED, but instead of $\frac{4}{3}$ the coefficient is something closer to 1. Is that true, and if so what's the coefficient? All of this is of course related to issues of self- energy and renormalization. Any help would be greatly appreciated.


Calculating Length Scales from Passive Scalar Field

I have a set of PLIF images of a passive scalar advected in a turbulent flow. I'm wondering if it's possible to estimate the integral length-scale based on the images of the passive scalar, and if so, how would I go about it. Any references to journal articles about this are welcome. It seems that a 2D cross-correlation of the images would help, with defined peaks at various length-scales, but then you'd have to filter out a lot of noise.


Universality of Quantum Operations

Does an analog of the Solovay-Kitaev theorem exist for quantum operations, a generalization of quantum gates that also includes all completely positive maps?


Degeneracies of the first excited state

I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are $\hbar^2\pi^2\over2ml^2$$(n_1^2+n_2^2+n_3^2)$. What is the degeneracy of the 1st excited energy level? By "1st excited energy level" does that mean 1 of the $n_k$'s, say $n_1$, =2 while $n_2=n_3=1$? Or does it mean all 3 dimensions are in their 1st excited state -- $n_k=2$ $ \forall k\in${$1,2,3$}? Also, how does one find the degeneracy? I am guessing that it is 3? Thanks for any help.


What is use of inertial frames

In class of Newton's laws of motion, it was explained that Newton's second law is valid only in inertial frames. Teacher give us a example by considering a lift which is going downwards with acceleration$=a$ then a man watching the lift from outside will write $mg-t=ma$ (where $m$ is mass of the lift and $t$ is upward force on the lift) while a man inside the lift will write $mg-t=0$ (for him there is no acceleration of lift) both equations will give us different value of $t$ so we have to have a frame specified for work ie inertial frame but nearly after 5 months I have a problem. In the observation of the man inside lift acceleration due to gravity is stated as $g$ but I think that it should have value $g-a$ (if a stone fall on Earth with acceleration $g$ then inside the lift it should fall with acceleration $g-a$) and then by substituting this value for acceleration due to gravity in equation of man inside lift we will get the same value of $t$ as we got in equations of man outside the lift. So if both frames give same value of $t$ then why we prefer one frame after other.


Is terahertz radiation new?

Regarding the future applications of terahertz technology, I thought we already knew the entire electromagnetic spectrum. In fact we do not, terahertz lies between microwaves and infrared radiation. The future applications of this is astonishing. For example, faster communications and next generation scanning and imaging technologies. The TSA will pretty much be out of a job, because T-Ray(terahertz) scanners will automatically detect chemical weapons and bombs. I know this because they leave a trace on the terahertz spectrum. The downside with communications is terahertz is absorbed easily by the Earth's atmosphere, so the communications cannot be too high in the sky. What I am asking is, are there any other frequencies on the electromagnetic spectrum we might discover? I was taught in school, that there are radio waves, microwaves, infrared, visible, ultra violent, x-rays, and gamma rays. Now the books have to be rewritten because of the existence of terahertz or t-rays. Remember it lies right between microwaves and infrared radiation. Another question is what about sound? So far there is infra sound, sound, and ultrasound. Do you think in the future we will discover other types of sound? Sorry for the length of my question, as you can see I know a lot about science and technology.


What are bootstraps?

I've heard occasional mentions of the term "bootstraps" in connection with the S Matrix. I believe it applies to an old approach that was tried in the 1960s, whereby - well I'm not sure - but it sounds like they tried to compute the S Matrix without the interaction picture/perturbation theory approach that we currently use. I'm aware that the approach was abandoned, but my question is: how was it envisaged to work ? What was the input to the calculation supposed to be and how did the calculation proceed ? I know it's got something to do with analyticity properties in terms of the momenta, but that's all I know.... For example, from the wikipedia article: > Chew and followers believed that it would be possible to use crossing > symmetry and Regge behavior to formulate a consistent S-matrix for > infinitely many particle types. The Regge hypothesis would determine the > spectrum, crossing and analyticity would determine the scattering > amplitude--- the forces, while unitarity would determine the self-consistent > quantum corrections in a way analogous to including loops. For example - I can't really understand how you would hope to compute the scattering amplitude just given crossing symmetry and assuming analyticity


Capacitance of a free conductor

Consider a ideal conductor in free space.For all purposes here,the zero of the potential is taken at infinity. Suppose I give a charge $Q$ to the conductor.As a result,the conductor will have a potential $V$. The question is can we say that $$Q=CV$$ where $C$ is a constant which depends only on **the shape and size on the conductor and not on the charge Q**?   1. If yes, how shall we prove it mathematically.   2. Also then how can we find the proportionality constant explicitly given the shape and size of the conductor?


what about doing the laser beam in a moving reference frame but with a ball

I realize the situation where a laser beam moves vertically in a moving vehicle![time dilation proof](http://i.stack.imgur.com/qoDgm.gif) but what if the laser beam was a normal ball If we do the same steps of the proof considering that the velocity of the ball is not absolute and will have different velocities in different reference frames there will be no time dilation what is wrong in my understanding because according to special relativity there should be time dilation whether the event is a laser beam bouncing or a ball.


Equilibrium Condition

In classical thermodynamics, equilibrium conditions means maximum entropy for a closed state. However, people always talk about equilibrium for open systems as well. How can one say that an open system has reached equilibrium with out negating the definition.


How does light bend around my finger tip?

When I close one eye and put the tip of my finger near my open eye, it seems as if the light from the background image bends around my finger slightly, warping the image near the edges of my blurry finger tip. What causes this? Is it the heat from my finger that bends the light? Or the minuscule gravity that the mass in my finger exerts? (I don't think so.) Is this some kind of diffraction? ![Light bending around my finger](http://i.stack.imgur.com/e32lY.jpg) To reproduce: put your finger about 5 cm from your open eye, look through the fuzzy edge of your finger and focus at something farther away. Move your finger gradually through your view and you'll see the background image shift as your finger moves. * * * For all the people asking, I made another photo. This time the backdrop is a grid I have on my screen (due to lack of grid paper). You see the grid deform ever so slightly near the top of my finger. Here's the setup: ![Setup](http://i.stack.imgur.com/t2ioO.png)![Finger on a grid](http://i.stack.imgur.com/biR3w.jpg) Note that these distances are arbitrary. It worked just as well with my finger closer to the camera, but this happens to be the situation that I measured. * * * Here are some photos of the side of a 2 mm thick flat opaque plastic object, at different aperture sizes. Especially notice hoe the grid fails to line up in the bottom two photos. ![Object photographed from the side](http://i.stack.imgur.com/33jVJ.jpg)


Energy-momentum conservation without translation symmetry?

As I checked, the energy-momentum tensor defined as ${T^\mu}_\nu=\frac{\partial {\cal L}}{\partial(\partial_\mu \phi)}\partial_\nu \phi-{\cal L}{\delta^\mu}_\nu$ at the solution $\phi$ of equation of motion(Euler-Lagrange equation) satisfies automatically the conservation law: $\partial_\mu{T^\mu}_\nu=0$, without any reference to the translation symmetry under $x^\mu\rightarrow x^\mu-a^\mu$.   So, what is the need of this symmetry?   Or, could there be something wrong with my calculation or conceptual issues?


Speed of light affecting appearance of galaxies viewed not face on?

So this has been really bugging me over the past few days (and forgive me if the answer is so simple). Let's say we're observing the Sombrero galaxy. It is about 29 million light years away and 50 thousand light yyears in diameter. So we should be observing the "front" of it at what it looked like 29 million years ago, and the "back" of it 29.05 million years ago. Why doesn't this extra distance change the galaxy's shape? If, for example, the galaxy was moving directly away from us in a straight line (not that it is), wouldn't the galaxy be compressed? Hope this makes sense.


How radio waves penetrate through buildings?

For example how radio signals of a base transceiver station (BTS) penetrate through buildings?


Why don't charges move transverse to an EM wave?

Image we have an ultra-high intensity, ultra low frequency laser, with wattage on the order of terawatts and a wavelength on the order of a lightsecond. We rotate it that the electric field component is oriented on the $\hat z$ axis, then fire it at a macroscopic block with a positive electric charge. Because of the low frequency the block will experience an electric that doesn't immediately change direction, and because of the high intensity the field will be very strong. So from this naive understanding of classical physics, the block will briefly levitate. Except this blatantly contradicts both QM (Compton scattering) and multiple macroscopic experiments (like solar sails), which both say that the block will be pushed in the direction of the laser. What assumptions in the original problem are missing/wrong?


when we rub objects together, what determines which material will pick up electrons?

For example We know glass when rubbed by silk will become positively charged while the silk will be charged negative. What exactly makes glass appropriate for losing electrons in that experiment? (


Unitary transformation behind gauge transformation

It is very well-known that for bosonic operators a Gauge transformation can always be associated with it $$a\rightarrow e^{i\phi}a.$$ Obviously this is a Unitary transformation. Something like $$a^{\prime}=\mathcal{U}^{\dagger}a\mathcal{U}$$ I want to know what is $\mathcal{U}$?


Weightlessness for astronauts

Well, this question has been puzzling me for kinda long time, many people believe that orbiting astronauts feel weightless because they are "beyond the pull of Earth's gravity"...How far from the Earth would a spacecraft have to travel to be truly beyond the Earth's gravitational influence? If a spacecraft were really unaffected by Earth's gravity would it remain n orbit? If so, what is the real reason for weightlessness in orbit?


argument about fallacy of diff(M) being a gauge group for general relativity

I want to outline a solid argument (or bulletpoints) to show how weak is the idea of diff(M) being the gauge group of general relativity. basically i have these points that in my view are very solid but i want to understand if there are misconceptions on my part that i'm simply not getting and if its so, i ask for help to make the case more solid, or understanding why it doesn't apply (to gravity):   * gauge groups are not the same as a symmetry group (thanks to Raymond Streater for making that point completely clear)   * gauge invariance in electrodynamics is an observation that physical observables are unchanged after a gauge transformation **without changing coordinate frame** ( we are ask to believe that in gravity someone did the same? that is, someone made the observation that physical observables are unchanged after a diffeomorphism-gauge-transformation, only to later argue that because of this, that there are no physical observables to begin with, that doesn't make a lot of sense, to not say that its just plain stupid circular argument )   * classic electrodynamics is also invariant (as in symmetry invariant, not as gauge-invariant) under Diff(M). The invariance is of course broken when the theory is quantized and $\hbar$ makes an appearance, because it assumes a preferred scale for certain energies. The key point here is: **classical gravity is not special regarding having diff(M) as a symmetry group**   * from bulletpoints 2 and 3, if i cannot infer that Diff(M) is a gauge-invariance of electrodynamics, the same should apply to gravity For this question, i would say that a valid answer would either disprove any of the arguments as fallacies themselves (hence showing a solid argument why gravity is special and diff(M) is without a doubt its gauge group), or improve the argument for making it bullet-proof (sorry for the pun)


Einstein's postulates $\leftrightarrow$ Minkowski space for a Layman

What's the cleanest/quickest way to go between Einstein's postulates [1] of   1. Relativity: Physical laws are the same in all inertial reference frames.   2. Constant speed of light: "... light is always propagated in empty space with a definite speed $c$ which is independent of the state of motion of the emitting body." to Minkowski's idea [2] that space and time are united into a 4D spacetime with the indefinite metric $ds^2 = \vec{dx}^2 - c^2 dt^2$. Related to the question of what is the best derivation of the correspondence are:   Is the correspondence 1:1? (Does the correspondence go both ways?)   and are there any hidden/extra assumptions? * * * ## Edit Marek's answer is really good (I suggest you read it and its references now!), but not quite what I was thinking of. I'm looking for an answer (or a reference) that shows the correspondence using only/mainly simple algebra and geometry. An argument that a smart high school graduate would be able to understand.


Is it possible to derive the invariant spacetime interval from Einstein's two postulates for SR?

In many textbooks, the interval $$ I = -(c\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$ is taken for granted as the same for two events in any reference frame. Is it possible to derive this just from the two postulates,   1. That the laws of physics are the same in all reference frames   2. That the speed of light _c_ is constant for all reference frames?


2D. Force applied at angle to body, where translational vector will be directed?

I'm not a physicist and just making some research by the way of creating simple physics simulator, because of that, sorry if this is very dumb question, but I really need help with it. Let's assume that some body (rectangle, square, N-polygon, etc.), exists in 2D world in rest (no friction, gravity, etc.) and can be freely moved in any direction. If some pushing / pulling force will be applied to center of mass, then only translational force will exist, this case is very clear to me. But what if force will be applied on the edge of body? What will be if force will be applied at some angle to edge? I understand that this will involve a rotational forces. But how can I calculate the resulting translational vector in this case? Here is image, demonstrating the problem. Force F2 will not involve any rotational forces, I can calculate net force (= F2) and get acceleration vector. All this question is about F and finding resulting translational vector after apply of F. ![example of forces-related question](http://i.stack.imgur.com/v8UwH.png)


Expression for kinetic energy of gas per molecule

The average kinetic energy (KE) per molecule of a gas is $\frac{3}{2}kT$. While finding this we do $$ \text{ Average KE} =\frac{1}{2} M \frac{1}{N}\sum v^2=\frac{3}{2}kT$$ But why do we not add rotational kinetic energy here?


Will the "Vacuum Wine Saver" suck the bubbles out of Champagne?

The "Vacuum Wine Saver" comes with the following "warning": > Not for sparkling wines Intuitively and naively, I imagine that the bubbles (or the "bubble- potential"—my made-up terminology) will be sucked out of the wine by the pump and that this is also the reason for the "warning". What is the better-formulated physical/chemical description and explanation? In other words: What happens and why? * * * Extra: Is this (your answer) also a reason for not semi-compressing half-full flexible plastic cola bottles before closing them?


What happens to water level when ice XII melts?

There exists this famous idea that _if all floating icebergs melt, water level will stay the same_ (because the water replaced by ice is the volume of the melted ice). Now,   1. Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ?   2. Is this always so **on/in earth** ? **EDIT:** I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. "Ice" is always floating or sunken(if possible). To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)? **EDIT 2:** Wikipedia: "Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)". And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the "lake" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen?


Should I learn Classical Physics if I want to learn Quantum Physics?

I don't remember anything from school, so please what do you say about the title?


What is the mathematical background needed for quantum physics?

I'm a computer scientist with a huge interest in mathematics. I have also recently started to develop some interest about quantum mechanics and quantum field theory. Assuming some knowledge in the areas of topology, abstract algebra, linear algebra, real/complex analysis, and probability/statistics, what should I start to read to understand the math behind quantum mechanics and quantum field theory. If you could offer me books, and list them according to the order that I should read them, I would be glad.


A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool).


Basic maths theories for good understanding of the standard model

I want to know what mathematical theories I should be aware of for a deep understanding of the standard particles model.


Mathematical background for Quantum Mechanics

What are some good sources to learn the mathematical background of Quantum Mechanics? I am talking functional analysis, operator theory etc etc...


What is the math knowledge necessary for starting Quantum Mechanics?

Could someone experienced in the field tell me what the minimal **math knowledge** one must obtain in order to grasp the introductory Quantum Mechanics book/course? I do have math knowledge but I must say, currently, kind of a poor one. I did a basic introductory course in Calculus, Linear algebra and Probability Theory. Perhaps you could suggest some books I have to go through before I can start with QM? Thx.


Study Basic Quantum Mechanics

What is the appropriate mathematical background someone must attain in order to enroll in a quantum physics course for beginners?


Mathematics for Quantum Mechanics

What math should I study if I want to get a basic understanding of quantum mechanics and especially to be able to use the Schrodinger's equation.


Will a boiled egg or a raw egg stop rolling first?

If we roll a normal egg and a boiled egg at the same time on a floor 1) with friction 2) without friction which one will come to stop first (if they will stop at all) and why? Can anyone tell me reason for this?


Integrating for velocity

Trying to determine velocity of a falling body with respect to traveled distance and initial speed. I've been provided with the following equation for acceleration as a function of distance and the grav. parameter(constant) of the attracting body : $a= GM/r^2$ Where: a - acceleration.   GM - gravitational parameter(constant).   r - distance to the attracting body. I have entered inputs for GM and r and integrated this equation with respect to $r$. This obviously yielded total acceleration per traveled distance, in other words $m^2/s^2$ at the given altitude. How do I proceed to determine velocity at this altitude?


Total number of electric field lines coming out of a proton?

I have to calculate the total number of electric field lines through a proton. I tried using Gauss' Law, i.e, $$\phi = \oint\boldsymbol E.d\boldsymbol s = {\frac{q}{\epsilon_0}} $$ $$So, \phi = {\frac{q}{\epsilon_0}} $$ $$ \textrm{Therefore}, \phi = \frac{1.6\times10^{-19}}{8.854\times10^{-12}}$$, as $q=1.6\times10^{-19}C$ and $\epsilon_0 = 8.854\times10^{-12}C^2N^{-1}m^{-2}$. But the answer is $0.18\times10^{-7}$ or something like that. How can that be the number of field lines? How can that be the number of anything? I mean it's a fractional number. Any help is appreciated. Thanks.


How does flux tubes between quarks bind them together?

If you have, say, a proton it has gluon field fluctuations around it. Those flux tubes between the quarks suppresses the gluon field fluctuations and create a true vacuum between them(correct me if I'm wrong), but how does that bind the quarks together? I've read that it costs energy to clear the vacuum out, but I still don't quite get it. Thank you!


Special Relativity and current in wire

If I am a stationary observer and the electrons are moving relative to me,then shouldn't its density increase according to special relativity and thereby create an altogether negative net charge.


Explain what happens to object in lift

In the following image,three cases have been mentioned. $N$ is the normal force acting on the object inside the lift and $mg$ is the force of attraction due to gravity.   In case 1, $N = mg$.   In case 2, $N = m(g+a)$ and   in case 3, $N = m(g-a)$. Why is it so in 2nd and 3rd case? ![enter image description here](http://i.stack.imgur.com/xdA80.png)


How to calculate cord tension in a vertical circle?

Mass m is connected to the end of a cord at length R above its rotational axis (the axis is parallel to the horizon, the position of the mass is perpendicular to the horizon). It is given an initial velocity, V0, at a direction parallel to the horizon. The initial state is depicted at position A in the image. The forces working on the mass are MG from the earth and T the tension of the cord. How can I calculate the tension of the cord when the mass is at some angle $\theta$ from its initial position (position B in the image)? ![image](http://img535.imageshack.us/img535/8167/76300324.png). Here's what I thought: Since the mass is moving in a circle then the total force in the radial direction is T - MG*$\cos\theta$ = M*(V^2)/R and so T = MG*$\cos\theta$+M*(V^2)/R but since MG applies acceleration in the tangential direction then V should also be a function of $\theta$ and that is where I kind of got lost. I tried to express V as the integration of MG*$\sin\theta$, but I wasn't sure if that's the right approach.


The final death of a black hole

What are the different death scenarios for a black hole? I know they can evaporate through Hawking radiation - but is there any other way? What if you just kept shoveling more and more mass and energy into the black hole?


Potential energy curve for intermolecular distance

![potential energy vs intermolecular distance r](http://www.a-levelphysicstutor.com/images/matter/E-r-graph.jpg) I'm trying to understand this curve better, but I can't quite figure out what "negative potential energy" means. The graph should describe a molecule oscillating between $A$ and $B$, however where I'm stuck in reasoning this is that the PE is equal in $A$ and $B$, but then why does this mean $r$ will increase in $A$ (repel) and decrease in $B$?


Doubt concerning centripetal acceleration

What is the centripetal acceleration and angular velocity of a child located 8.2 m the center of a carousel? The speed (size of the tangential velocity) of the child is 2.1 m / s A train moves in a straight path north until it turns to west. If the road segment used to change direction is shaped like a quarter circle of radius 30 m and the train takes 30 seconds to traverse that part of the road, What is it the speed (size of the velocity vector) and the centripetal acceleration acts on the train as it traverses the curve. I am reviewing some concepts like centripet force, `ar = ( v^2 ) / r` also this: > The direction of the centripital acceleration is always inwards along the > radius vector of the circular motion. The magnitude of the centripetal > acceleration is related to the tangential speed and angular velocity as > follows: ![enter image description here](http://i.stack.imgur.com/XGojw.png) Can you please guide me to solve the 2 problems above? for the fisrt one is it only : `(2.1 m / s)^2 / 8.2 m` ?


Working of a p-n junction diode when forward biased

If p-type semiconductor and n-type semiconductor of a diode are equally doped, and if the diode is forward biased, then holes will move toward the n-type semiconductor and electrons will move toward the p-type semiconductor and they will diffuse with each other. Then will there be any electron that will go to the positive terminal of the battery if all of them have diffused with each other? I can't understand, please help me!


Where did $\mathcal{M}^\mu(k) = \int \mathrm{d^4}x \; \exp(\mathrm{i}k \cdot x)\langle f | j^\mu(x) | i\rangle$, in Peskin and Schroeder, come from?

On page 160 Peskin & Schroeder, they say: > Therefore we expect $\mathcal{M}^\mu(k)$ to be given by a matrix element of > the Heisenberg field $j^\mu$: $$\mathcal{M}^\mu(k) = \int \mathrm{d^4}x \; > \exp(\mathrm{i}k \cdot x)\langle f | j^\mu(x) | i\rangle.$$ Why do they expect that? Where does this come from? I've seen similar expressions with e.g. two Lorentz indices and so on. I've been taught (obviously not enough) qft through path integrals and have a hard time to make connections between the different formalisms.


Is there an analogue of a geodesic for the evolution of the electromagnetic field?

For a charged particle moving in free space, we can say from the homogeneity of space-time, that it moves along a geodesic. Is there an analogous principle for the evolution of the electromagnetic field in space-time?


Minimum seperation between two Airy disks as a function of the distance between two point sources of coherent light passing through the same objective

I have two coherent point sources of light, $A$ and $B$, separated by a distance $L$, which I focus down to the diffraction limit using a high-powered objective (e.g. a $\approx 100x$ objective). If I turn on $A$ and turn off $B$, I have an Airy disk at position $c_1$, and I turn off $A$ and I turn on $B$, I have an Airy disk at position $c_2$. Given that both light sources are sent through the same objective, what is the minimum distance between $c_1$ and $c_2$? Is it simply $L$ scaled down by the objective (i.e. $\frac{L}{100}$)? Or does something odd happen because of e.g. curvature of the lens in the objective? EDIT: A restatement of this question would be the following - Assuming all of the optics are perfect, if I shine a laser at a point (x,y) on an objective, and then shine the laser at a point (x2,y2), will the peak of the Airy disk move the same distance?


Law of reversibility of light and total internal reflection

When a light passes from a denser to a rarer medium at critical angle of incidence the light rays graces through the surface of the denser medium.According to the law of reversibility of light same thing should happen when we reverse the direction of light.But, how can that be? How does the light know when to go into the denser medium?


Does a nonzero Poynting vector mean that there is propagation of energy?

I don't know how this "paradox" can be solved. I'm given the following system: A permanent magnet with a magnetic field given by ($\hat{a}$ are unit vectors in the x and y directions) $$\vec{H}=H_0\hat{a}_y$$ and a parallel plane capacitor with an electric field $$\vec{E}=E_0\hat{a}_x$$ Poynting's vector is given by: $$\vec{S}=\vec{E}\times\vec{H}=H_0E_0\hat{a}_z \neq 0$$ The funny part comes when the professor told that in a system like that "clearly" there is not propagation (wich I know will imply some short of energy flux) in the $z$ direction, hence the "paradox". Is there or is there not propagation of energy? Any hint will be appreciated, thank you for your time.


Using the boundary states, is there a precise way to write down a planar open string multi-loop amplitude as a closed string tree amplitude?

The only explicit computation I have seen is the planar 1-loop one, but there should be a way to write the multi-loop case in terms of boundary states as well.


What is the "discrete" analogue to "continuum" mechanics?

If I wanted to explore a discrete mathematics approach to continuum mechanics, what textbooks should I look into? I suppose a ready answer to the question might be: "computational continuum mechanics", but usually textbooks that discuss such a subject are usually focused upon applying numerical analysis to continuous theories (i.e. the base is continuous), whereas I would like to know if there is a treatment of the subject that builds up from a base that is discrete.


Electric field generated by a point charge moving at the speed of light

![enter image description here](http://i.stack.imgur.com/sUvHk.png) As you see, this is the electric field generated by a point charge moving at constant speed v. I know that when $v$ -> 0, $E$ is just the Coloumb Law. But how do you interpret $E$ when $v$ -> $c$ ? Can I just interpret it as the field of electromagnetic wave, because it moves at the speed of light? ![enter image description here](http://i.stack.imgur.com/LvWkF.png)


Imagine a long bar floating in space. What force does it exert on itself in the middle due to gravity?

# Problem If you had a long bar floating in space, what would be the compressive force at the centre of the bar, due to the self-weight of both ends? Diagram - what is the force at point X in the middle of the bar?:               <----------------------L--------------------->, total mass M     =======================X======================   <- the bar                      F---> X <---F      # Summary You should be able to simplify by cutting the bar into pieces, but that gives a different answer depending on how many pieces you use (see below). So the simplification must be wrong - but why? # My approach ### Split bar in two So, one approximation would be to cut the bar in half - two pieces of length L/2, mass M/2:                        (M/2)<-------L/2------->(M/2)                #1          X           #2   <- bar approximated as blobs #1 and #2      Force at X is G(M1.M2)/(R^2) = G (M/2)^2 / (L/2)^2 = G M^2 / L^2 Or Fx / (G. M^2 / L^2) = **1** But is that really valid? If so, shouldn't you get the same answer if you split the bar into four pieces? ### Split bar into four                  (M/4)<-L/4->(M/4)<-L/4->(M/4)<-L/4->(M/4)          #1          #2    X     #3          #4      My assumption is that the force at X is the sum of the attractions of each blob on the left to every blob on the right. Force at X = #1<>#3 + #1<>#4 + #2<>#3 + #2<>#4 ('<>' being force between blobs #x and #y). Fx / (G.M^2 / L^2) = (2/4)^-2 + (3/4)^-2 + (1/4)^-2 + (2/4)^-2 = **1.61** This is bigger than the previous result (1.61 vs 1). ### Split bar into six Similarly, if you split into 6 blobs, the total force comes out as: Fx / (G.M^2 / L^2) = (3/6)^-2+(4/6)^-2+(5/6)^-2 + (2/6)^-2+(3/6)^-2+(4/6)^-2 + (1/6)^-2+(2/6)^-2+(3/6)^-2 Fx / (G.M^2 / L^2) = **2.00** ### So what's wrong with my approach? And what is the real answer? So it seems the more pieces we split the bar into, the larger the result gets. There's clearly something wrong with my assumptions! - but what? I'd be very glad if someone here could explain this. Thanks! **EDIT** As Peter Shor pointed out, my calculations had some dodgy algebra and I'd calculated $$L^2/M^2$$ values rather than $$M^2/L^2$$. I've now corrected that - the value still increases as you divide into more masses. I'll do a bit more work with more divisions and see if this leads to convergence or not.


If gravity bends space time, could gravity be manipulated to freeze time?

You age at a different rate depending on the force of gravity. Astronauts age fractions of fractions of fractions of a second less than earthlings. If you took a sphere of equal masses, separated by space, then found the exact center of the gravitational pulls of all masses. How would time react?


Mario Livio's book on symmetry and the relationship between gravity and acceleration

In his book: _The Equation That Couldn't be Solved_ Mario Livio explains the equivalence principle in laymen's terms. I took the statement on page 209: _The force of gravity and the force resulting from acceleration are in fact the same._ to mean that since our universe is accelerating, gravity is the result. In other words, we have gravity because we have acceleration. This made a lot of sense to me and seemed to explain things (like warps in space- time due to gravity of very large objects. They 'have' gravity because they are accelerating masses. In my beginner's head I extended that conclusion to mean if the acceleration of the universe was zero we would have mass but no gravity. Based on what I've read since, this does not seem to be true...is it?


How to show the invariant nature of some value by the group theory representations?

Let's have Dirac spinor $\Psi (x)$. It transforms as $\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)$ representation of the Lorentz group: $$ \Psi = \begin{pmatrix} \psi_{a} \\\ \kappa^{\dot {a}}\end{pmatrix}, \quad \Psi {'} = \hat {S}\Psi . $$ Let's have spinor $\bar {\Psi} (x)$, which transforms also as $\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)$, but as cospinor: $$ \bar {\Psi} = \begin{pmatrix} \kappa^{a} & \psi_{\dot {a}}\end{pmatrix}, \quad \bar {\Psi}{'} = \bar {\Psi} \hat {S}^{-1}. $$ How to show formally that $$ \bar {\Psi}\Psi = inv? $$ I mean that if $\Psi \bar {\Psi}$ refers to the direct product (correct it please, if I have done the mistake) $$ \left[\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right) \right]\otimes \left[\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right) \right], $$ what group operation corresponds to $\bar {\Psi} \Psi$? This question is strongly connected with this one.


Potentiometer at null pointer!

Consider the situation when a cell of an unknown emf is being measured using a potentiometer. We slide the jockey so as to obtain the null point. Now, is there any current in the potentiometer wire at the null point? Since we know that there is no current in the arm containing the unknown cell, its terminals have acquired equal potentials,how is it possible that there is any current in the potentiometer wire when that is in parallel to that cell. Potential difference across AB= Potential difference across CD? ![enter image description here](http://i.stack.imgur.com/YcUOl.gif)


Donors/Acceptors in Metal Oxides

Can anyone explain to me why most articles describe chromium as an acceptor in titanium dioxide? In TiO2, titanium has the charge state Ti$^{4+}$ and oxygen has the charge state O$^{2-}$. When Cr substitutes for Ti, it does so as Cr$^{3+}$. Now, at first glance, Cr has atomic number 24 and Ti 22. Cr therefore has two more valence electrons and is a donor. In TiO2, Cr$^{3+}$ actually has _three more_ valence electrons than the Ti$^{4+}$ ([Ar]$3d^34s^0$ vs [Ar]$3d^04s^0$). It should therefore be a donor, right? The thing is, it forms a deep impurity level near the valence band. TiO2 has an energy gap of around 3.2 eV, and the impurity state is about 1.0 eV from the valence band maximum. To me, that makes it a **deep donor**. For some reason, journals almost always describe it as an acceptor. Can someone help me make sense of this? My understanding has always been simply this: more electrons than host $\Rightarrow$ donor, fewer electrons than host (more holes) $\Rightarrow$ acceptor. The position of the impurity level, to my (perhaps incorrect) knowledge, does not determine whether or not the impurity is actually a donor or acceptor, but rather whether it is a recombination center or trap. We can have localized states near the middle of the bandgap that are technically donors/acceptors but function as recombination centers, so I'm not sure what I'm missing here.


Special relativity and electromagnets

This Veritasium video explains how electromagnets can be explained by special relativity, and how the magnetic field surrounding a current-carrying wire can also be viewed as an electric field, if your frame of reference is moving with respect to the wire. The example they use is a positively-charged cat, moving along a current- carrying wire in the same direction as the electron drift: ![positively charged cat next to a current-carrying wire](http://i.stack.imgur.com/HHUiP.png) If you view this from the rest frame of the cat, then the electron's drift velocity is zero, while the protons are moving to the left. Because the protons are moving, length contraction makes it look like (to the cat) there are more of them, giving the wire a net _positive_ charge, _repelling_ the cat. This makes sense and is all kinds of elegant and intuitive. It explains electromagnets in a way that depends on only three simple concepts:   1. motion is relative   2. things contract in their direction of apparent motion   3. opposite charges attract, like charges repel Groovy. Now back up in the video. Derek says: > Now the number of protons is equal to the number of negative electrons, so > overall the wire is neutral. So if there were a positively-charged cat > nearby, it would experience no force from the wire at all. And even if there > were a current in the wire, the electrons would just be drifting in one > direction, but the density of positive and negative charges would still be > the same, and so the wire would be neutral, so no force on the kitty. ![Derek standing next to a current-carrying wire](http://i.stack.imgur.com/Gb796.png) Wait... _what_? Why is it that in the cat's frame, the protons are moving, are contracted, and the wire is charged, but in Derek's frame, the electrons are moving, _but are not contracted_ , and the wire is still neutral? How can you say "well, length contraction creates charge imbalances, allowing magnetic forces to be explained as electrical ones if you choose the right reference frame", but simultaneously say "but length contraction doesn't happen sometimes"? That's not elegant at all. Is there an elegant, intuitive1 explanation? 1: meaning, I've seen the math on Wikipedia and it's over my head. There is also current in wire + special relativity = magnetism where the answer to my question seems to be "the Lorentz force". OK, but that negates the elegance of the explanation above, with only three simple axioms. Are they not sufficient? If so, why?


Intuitively, why is a reversible process one in which the system is always at equilibrium?

A process is reversible if and only if it's always at equilibrium during the process. Why? I have heard several specific example of this, such as adding weight gradually to a piston to compress the air inside reversibly, by why should it be true in general? EDIT: Here is something that would firmly convince me of this: Suppose I have a reversible process that is not always in equilibrium. Describe a mechanism for exploiting this process to create a perpetual motion machine.


An entropy of the Wigner function

Is there an entropy that one can use for the Wigner quasi-probability distribution? (In the sense of a phase-space probability distribution, not - just von Neumann entropy.) One cannot simply use $\int - W(q,p) \ln\left[ W(q,p) \right] dq dp$, as the Wigner function is not positively defined. The motivation behind the question is the following: A paper I. Białynicki-Birula, J. Mycielski, Uncertainty relations for information entropy in wave mechanics (Comm. Math. Phys. 1975) (or here) contains a derivation of an uncertainty principle based on an information entropy: $$-\int |\psi(q)|^2 \ln\left[|\psi(q)|^2\right]dq-\int |\tilde{\psi}(p)|^2 \ln\left[|\tilde{\psi}(p)|^2\right]dp\geq1+\ln\pi.$$ One of the consequences of the above relation is the Heisenberg's uncertainty principle. However, the entropic version works also in more general settings (e.g. a ring and the relation of position - angular momentum uncertainty). As $|\psi(q)|^2=\int W(q,p)dp$ and $|\tilde{\psi}(p)|^2=\int W(q,p)dq$ and in the separable case (i.e. a gaussian wave function) the Winger function is just a product of the probabilities in position an in momentum, it is tempting to search for an entropy-like functional fulfilling the following relation: $$1+\ln\pi\leq\text{some_entropy}\left[ W\right]\leq -\int |\psi(q)|^2 \ln\left[|\psi(q)|^2\right]dq-\int |\tilde{\psi}(p)|^2 \ln\left[|\tilde{\psi}(p)|^2\right]dp.$$


Massive excitations in Conformal Quantum Field Theory

Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of quantum states into irreducible representations of the Poincare group). Classification of irreducible unitary representations of the Poincare group leads to the notions of mass and spin. Now, suppose we have a conformal QFT and are doing the same trick with the conformal group. Which irreducible representations do we have? We still have the massless particles (at least I'm pretty sure we do although I don't immediately see the action of special conformal transformations). However, all representations for a given spin s and any mass m > 0 combine into a single irreducible representation. > What sort of physical object corresponds to this representation? Is it > possible to construct a scattering theory for such objects? Is it possible > to define unstable objects of this sort?


range of the difference of two-qubit density matrix determinants

The determinant of a two-qubit (4 x 4) density matrix lies between 0 and (1/2)^8. (A pure state has determinant zero, and the fully mixed [classical] state, determinant (1/2)^8.) The determinant of the partial transpose (transpose in place the four 2 x 2 blocks) of such a matrix (nonnegative values indicating separability) lies between -(1/2)^4 and (1/2)^8. (The minimum is achieved by a Bell state and the maximum, again by the fully mixed state.) What is the range (upper and lower limits) of the difference of these two determinants?


Is the opening of the NOVA program a Calabi-Yau space?

Is the opening of the NOVA program on PBS a Calabi-Yau space?


Phase shift in electromagnetic potential

In Aharonov-Bohm effect, how to derive that the wave function of a electric charge $q$ acquires a phase shift $\phi=\frac{q}{\hbar}\int \mathbf{A} \cdot d\mathbf{x}$ after travelling in the non-zero magnetic vector potential $\mathbf{A}$?


How non-abelian gauge coupling runs below confinement or QCD scale?

I know the famous beta function of asymptotic free, but that seems describe the running coupling beyond confinement/QCD scale so that a perturbative analysis can apply. But how coupling runs below that scale? Any comment or references are greatly appreciated.