title: The Weyl tensor and gravitational waves

text: How exactly is the Weyl tensor is connected with information about gravitational waves? And what are physical reasons for that?


title: Determining wave function for term symbol 1D

text: I am trying to follow a book (Introduction to Ligand Field Theory by Ballhausen in 1962 on pg 15), but it isn't clear how they make a particular leap. **Background** I want to find the wave function for the term $^1 D$. We know that $\psi(L,M_L,S,M_S) = \psi (2,2,0,0)$ is made up of three micro states: $(2^+,0^-)$, $(2^-,0^+)$, $(1^+,1^-)$. Thus, a linear combination must be taken of these micro states. Now in the book they say they must be orthogonal to $\psi(3,2,1,0)$ and $\psi(4,2,0,0)$. Presumably these are picked because we just determined those via lowering operators in the previous section. I don't take it that there is another reason those two wave functions are mentioned. $(2^+,0^−)$ means that electron 1 has $m_s=+1/2$ and $m_l=2$ and that electron 2 has $m_s=−1/2$ and $m_l=0$. We write that $\Psi=|(\psi^+_1)(\psi^−_2)|$ where we have written the short form of the determinantal antisymmetrized normalized wave function, which comes from the diagonal element in the Slater determinant AND separated the orbital- and spin-dependent parts of the wave function (spin is denoted by super plus or minus sign). **Problem** Anyway, I understand that $\psi(2,2,0,0) = a (2^+,0^-) + b (2^-,0^+) + c(1^+,1^-)$. Then they write "and we get $a \sqrt{3} - b \sqrt{3} + c \sqrt{8} = 0$ and $a + b = 0$." I'm totally lost how they made this leap and why these must be equal to zero. Does this have to do with the orthogonal wave functions just mentioned? Where did the numbers in the sqrt come from? And the minus sign! Greatly appreciated.


title: Does someone falling into a black hole see the end of the universe?

text: This question was prompted by Can matter really fall through an event horizon?. Notoriously, if you calculate the Schwarzschild coordinate time for anything, matter or light, to reach the event horizon the result is infinite. This implies that the universe ages by an infinite time before someone falling into the black hole reaches the event horizon, so could that person see the universe age by an infinite time? To be more precise, suppose the observer starts falling from rest at time $t = 0$ and some initial distance $r > r_s$. If we wait for some time $T$ then shine a light ray at the falling observer. Will the light ray always reach the falling observer before they cross the event horizon? If not, what is the formula for the longest time $T$ that we can wait and still be sure the ray will catch the observer? If $T$ is not bounded it implies that observer could indeed see the end of the universe. I can think of a qualitative argument for an upper limit on $T$, but I'm not sure how sound my argument is. The proper time for the observer to fall to the event horizon is finite - call this $\tau$. The proper time for the light ray to release the horizon is zero, therefore the light ray will reach the observer before they cross the event horizon only if $T < \tau$. Hence $T$ is bounded and the observer won't see the end of the universe. I think a more rigorous approach would be to determine the equations of motion (in the Schwarzschild coordinates) for the falling observer and the light ray, and then find the condition for the light to reach the falling observer at some distance $\epsilon$ from the event horizon. Then take the limit as $\epsilon \rightarrow 0$. In principle this seems straightforward, but in practice the algebra rapidly defeated me. Even for a light ray the radial distance:time equation isn't closed form (Wolfram claims it needs the $W$ function) and for the falling observer the calculation is even harder.


title: Modeling Quantum Aspects with Probability

text: This is a question I've had a while about quantum theory. Many times when I look at books and equations about this subject matter I see that the use many concepts in probability. (Correct me if Im wrong) Like how the movement of atom can be modeled using probability distributions. So I was wondering when physics takes this approach to this subject matter is it saying that the movement of an atom is actually random or that the details of explaining may be to cumbersome and be approximated with great accuracy just by using these distributions?


title: Balloon Expansion

text: I would like to do a calculation of an expanding balloon represented by a triangular spring-damper mesh system. Also the behavior of the balloon when it is put in a container, where its shape conform to the shape of the container while expanding. Can anybody point me to any studies being done?


title: Gravity - What happens when two objects of unequal masses fall freely towards the ground? (Revisited)

text: The common perception regarding what happens when two objects of equal sizes but unequal mass are allowed to fall freely towards the ground is that - both the objects make contact with the ground at the same instant. This is attributed to the fact that the acceleration of both the objects towards the earth are the same for a given height. This was the reason for Galileo's expirement. But the above fact ignores the fact that each of the three objects are attracting each other. In particular, the heavier object also attracts the lighter object, thus decreasing the distance between the lighter object and earth. Note : I am ignoring the fact that the lighter object would also attract the heavier object (but to a lesser extent) and since the earth moves approximately in the same direction as a result of the attraction from the other two spheres, that is also ignored. Otherwise the earth can be said to move diagonally towards the heavier object (again to a small extent). * * * I have included an illustration showing the movement of the lighter object towards the heavier one. ![Gravity Free Fall - Scenario 1](http://i.stack.imgur.com/SuIwb.png) * * * **As such the lighter object should fall first is it not?** Note : Generally, the two objects are placed close to each other (the distance-d between the objects is lesser than the height-h from the ground - 'd < h' ). Continuing with the same logic, I have the following understanding : When two spheres of equal size but unequal masses are allowed to fall freely towards the ground - 1.) From a height greater than the distance between the two spheres, the lighter sphere makes contact with the ground first. 2.) From same height as the distance between the two spheres, both the spheres make contact with the ground at the same time. 3.) From a height lesser than the distance between the two spheres, the heavier sphere makes contact with the ground first. Note : a.) The ground is also a sphere. b.) The distance between the spheres are measured from their centers. **Is my understanding correct?** I also found this Phys.SE post. Which speaks of a scenario where in only two objects are involved. My question is similar to it. PS: I have blogged about it before as well, saying it is so and also elaborated on two other cases ($d=h$ and $d>h$).


title: What is the 'quantum-developed' or 'relativistic-developed' equation of the electrostatic force?

text: Quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics that is the first theory where full agreement between quantum mechanics, special relativity and electrodynamics is achieved.   * What is the 'quantum-developed' equation of the classical electrostatic force? $$F=\frac {q^2}{4\pi \epsilon_o r^2}$$   * What is, it's 'relativistic-developed' equation?


title: Resolution of twin paradox using Lorentz velocity addition

text: In the following lecture, starting at minute 29:00 and going further, the professor resolves the Twin Paradox using Lorentz velocity addition. I have a question about this: Isn't the figure given below (taken from a slide in the lecture, at time 37:50) referring to the case where there is an independent object moving relative to the inertial frames S and S'? In the example of the twins that he discusses, there is no object independent of the frames of reference. We simply have the twins, each attached to her respective reference frame. Why, then, does he use the formula for Lorentz velocity addition? What object is he referring to? Link for the lecture: http://www.youtube.com/watch?v=4A5EQaXhCTw ![enter image description here](http://i.stack.imgur.com/qxbMm.jpg)


title: What is lepton number?

text: What exactly is a lepton number of a particle? With the charge (eg proton is just 1, not the exact charge), I can understand because it's a physical property, put a particle with charge + next to another particle with charge + and they will repel. What is the lepton number in similar terms? Or is it just a convention that worked in the observable particle interactions (that it is conserved, like charge)


title: Cause of electromagnetic induction?

text: The rate of change of magnetic flux through a surface (open) is related with the line integral over the closed loop binding the selected surface by one of the Maxwell's equation. But that means even if a changing magnetic flux is present anywhere on the surface such that the electric and magnetic field at the chosen surface's loop is zero. For example, for a perfect toroidal solenoid, all the magnetic field and hence flux is confined within the windings. Therefore, for a wire making a loop surrounding the toroid and passing through the centre, the fields (electric and magnetic) at the wire is zero yet the changing flux produces an emf and a current in the loop-wire! Is this not spookier than the conventional "action at a distance". **So what induces the EMF in this wire?**


title: Renormalization scheme independence of beta function

text: I have some questions about renormalization. To my understanding, in order to deal with infinities that appear in loop integrals, one introduces some kind of regulator (eg, high momentum cutoff, taking $d\to d+\epsilon$, etc.) so that we get a finite answer, which blows up as we remove the regulator. Then we renormalize various coefficients in the Lagrangian in a regulator-dependent way so that, in scattering amplitudes, a finite piece remains as we remove the cutoff. The regulator seems to always require introducing some arbitrary scale (eg, the momentum cutoff or $\mu^\epsilon$ for dim-reg), although I'm not sure why this must be the case. Now, this finite piece is completely arbitrary, and depends on what scheme we want to use (eg, on-shell, minimal subtraction, etc). The beta function is then, roughly speaking, the rate of change of this finite piece under changes in the scale in the regulator. My question is, what exactly is the invariant information in the beta function? Under changing renormalization scheme, it obviously changes, but it seems that this roughly corresponds to a diffeomorphism on the space of couplings (is this always true? for example, in on-shell renormalization, if we set the mass to the physical mass and the coupling to the corresponding exact vertex function at zero momentum, these seem to be independent of scale - do the beta functions vanish here?). Also, it seems to depend on exactly what regulator we are using, and how the dimensionful scale enters into it. There are certain quantities, such as the anomalous dimension of a field at a fixed point, which must be independent of all these choices, but I don't understand why this is the case. Finally, and this is more of a philosophical question, what exactly do the infinities in the loop integrals mean in the first place? Why is it that we can remove them by any of these various regulators and expect to get the same answer? I've read something about Haag's theorem, that interacting theories live in a different Hilbert space than the free-field Fock space, and that this is somehow related, but I'm not sure why. Thank you.


title: Why isn't a mirror more effective in preventing global warming than a solar panel?

text: In many countries, solar panels are heavily subsidized because they replace fossil fuels that apparently cause global warming. A mirror of the same size would - through a completely different effect - also reduce some global warming by reducing the solar energy reaching the earth. Considering that mirrors are much cheaper than solar panels, wouldn't they be more cost- effective in mitigating global warming?


title: Applications of recoil principle in classical physics

text: Are there any interesting, important or (for the non physicist) astonishing examples where the recoil principle (as special case of conservation of linear momentum) is applied beside rockets and guns? I am teaching physics in high school and was just wondering if there are any more interesting examples than the two textbook standard examples above. I am not interested in examples which are not in the domain of macroscopic, classical physics like Mößbauer-effect or something like that.


title: Why does my bicycle brakes squeak in the morning?

text: Every morning, when I ride my bicycle, there is a very loud squeak coming from the brakes. I have an aluminum rim (I think that they are of the sprint type) and rubber pad brakes. A couple of brake events afterward end that and after that the breaks are silent. I suspect it has something to do with the morning plumpness but yet:   1. Why would a damp rim give a noise under friction?   2. Why is it so loud? There is no amplification mechanism I can think of in the braking system. Some photos: A sketch of what I believe is my rim is.   A braking mechanism, Not mine but quite similar. The rim also looks similar.   A closeup on a brake pad _Photos 1 and 2 are taken from wikipedia, photo 3 is taken from a reparto corse_


title: Stimulated emission and No cloning theorem

text: I have a little trouble with the simulated emission. I know of the no-cloning theorem which states that it is not possible to duplicate any state. One the other hand, I know about the stimulated emission which out of a photon produce exactly the same (wavelength, polarisation, etc...). Maybe the fact is that the excited atom do a measure. (Let's say it can stimulate emission with only one direction of polarisation.) But now if I have a statiscal number of atoms with random "polarisation" direction, I should be able to copy any incident photon. I've made a cloning machine. This cannot be true because of the no-cloning theorem. But I can't figure why.


title: Is using a swing an example of normal or of parametric resonance?

text: Parametric resonance is a situation where the driving frequency is a multiple of the eigenfrequency. Various people say that using a swing and propelling it oneself is such a case, with the driving frequency being the double of the eigenfrequency. But when I use a swing, as I did today, my own motion has the same frequency as that of the swing, not twice the frequency. An example description is http://www.hk-phy.org/articles/swing/swing_e.html which claims that self-propelled swinging is normal resonance. The opposite is said here: Wikipedia entry What is the correct view: is self-propelled swinging normal resonance or is it parametric resonance?


title: Simplest derivation of Fourier transform for periodic functions (in crystal lattice)?

text: What is the simplest derivation of the following two well-known formulas that work for crystal lattice [1]: $$ F[f(\mathbf{x})] \equiv \tilde f(\mathbf{G}) = {1\over\Omega_\mathrm{cell}} \int_{\Omega_\mathrm{cell}} f(\mathbf{x}) e^{-i\mathbf{G} \cdot \mathbf{x}}\,d^3 x $$ $$ F^{-1}[\tilde f(\mathbf{G})] = f(\mathbf{x}) = \sum_{\mathbf{G}} \tilde f(\mathbf{G}) e^{+i\mathbf{G} \cdot \mathbf{x}} $$ See the question How to derive inverse Fourier transform for periodic functions (in crystal lattice)? for context. The difference from that question is that here we allow any derivation. [1] Martin, R. M. (2004). Electronic Structure -- Basic Theory and Practical Methods (p. 642). Cambridge University Press.


title: How a plasma reacts with an AC electromagnet

text: Any metal conductor such as aluminium or copper is a conductor because of the free electrons in such metals. Plasma also has free electrons so I would like to know if you can repel plasma the same way as aluminium is repelled when near an AC electro magnet? If so how come when I spin a magnet near my plasma globe there is no noticeable reaction compared to an aluminium can?


title: "Periodic Table" of Particles of the Standard Model?

text: What is a good, single, "periodic table" of all the particles of the Standard Model? I thought Particle Data Group would have a single-page PDF of this, but I couldn't find a single table listing all the particles.


title: $1.7\cdot 10^{-24}$ mole apples a day

text: As the title suggests I was wondering why the International Bureau of Weights and Measures decided a mole to be a standard (SI-)unit. After some research I found I was not alone with this problem. The core of my question is:   1. How is the unit “mole” necessary as a standard unit?   2. If mole is the standard unit, why wouldn't I have a give all numbers in mole? Of course a mole is a convenient unit but I can't see how it is as fundamental as i.e. a meter, as it is clearly based on the concept of counting the atoms. EDIT: Due to the first answer I got I realized my question is probably misleading: by fundamental I do not mean dictated by nature but considered a base unit. The need of units is obvious in the case of meters and seconds: no matter how you want to measure time or length, you will necessarily have to compare it to some standard (in this case meter and seconds, independently of how they are defined). In the case of "number of particles" this is not needed, instead one could say they are compared to "1". This is normally not considered a unit. Is this difference only a personal opinion?


Does anyone take the Wightman axioms seriously? Mainly with respect to quantum gravity or gauge theores, abelian or non-abelian? Anyone doing any research on axiomatization of QFTs in some way?


title: Why does a particle fall in a straight line?

text: In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths connecting the two points and minimise the action. This turns out to be a Brachistone, as first shown by Bernouilli. When we fix the end-point vertically below; the Brachistone is in fact astraight-line. But is there a principle in Lagrangian Mechanics that allows me to choose the point vertically below? Of course, we know from Newtons Mechanics that this must be the case. But how do we determine that end-point entirely within Lagrangian Mechanics?


title: Estimating minimum energy with uncertainty principle

text: I'm currently trying to solve a problem that involves estimating the minimum energy of a particle in the potential: $$ V(x) = \frac{-V_0a}{|{x}|} $$ I'm quite confused about how to handle the absolute value in the potential. So far I know that the total energy of the particle will be $$ E = \frac{p^2}{2m} - \frac{V_0a}{|{x}|} $$ The expectation value of the energy is therefore $$ \langle E\rangle = \frac{\langle p^2\rangle}{2m} - \frac{V_0a}{\langle|{x}|\rangle} $$ Now, can I use the fact that $$\frac{1}{|x|}=\frac{1}{\sqrt{x^2}}$$ and then $\langle|x|\rangle = \sqrt{\langle x^2\rangle} = \Delta x$? Then from the uncertainty principle we know that $\Delta p = \hbar/2\Delta x$ and $\Delta p^2 = \langle p^2\rangle$ ($\langle p\rangle = 0$ and $\langle x\rangle$ = 0 in a minimum energy state) Plugging this back into the energy equation gives $$ \langle E\rangle = \frac{\hbar^2}{8m\Delta x^2} - \frac{V_0a}{\Delta x} $$ When minimising this I get $$ E_{min} = \frac{-2mV_0^2a^2}{\hbar^2} $$ Now this doesn't look wildly wrong but I'm not sure if I've used the right approach with $\langle|x|\rangle = \sqrt{\langle x^2\rangle} = \Delta x$.


title: Formula for critical mass? Critical mass of polonium?

text: I'm looking for the critical mass of Polonium; is there a formula? E.g.:$$\text{Neutrons / Protons}\cdot\text{ Constant = X kg}$$ There is a little table in the Wikipedia. E.g.: $\begin{array}{ccccr} \text{Name}&\text{Symbol}&\text{Neutrons}&\text{Protons}&\text{Crit. mass} \\\\\hline \text{Uranium-233}&{}^{233}\text{U}&141&92&15\:\mathrm{kg} \\\\\text{Plutonium-238}&{}^{238}\text{Pu}&144&94&9\text{–}10\:\mathrm{kg} \\\\\text{Plutonium-240}&{}^{240}\text{Pu}&146&94&40\:\mathrm{kg} \end{array}$ I'm wondering, is there any dependence? What about with a neutron reflector? * * * What is the critical mass of Polonium?


title: Pressure applied to flat plate

text: I have a question about the pressure placed on a plate of material X, how the force is distributed and what would be the material property that would determine its failure. To simplify things "suppose" I wanted to build a vacuum chamber with a viewing window made of material X, that could handle a external pressure of 0.990Bar, and has a dimension of 500mm x 500mm x thickness. How could I calculate a for safety reasons the minimum thickness needed for the window so that it doesn't implode, ignoring safety factors for now. Also could this calculation be then used on a box of some arbitrary dimensions to calculate the minimum thickness of the material? Thanks.


title: Conservation of energy and the 'crazy ball' product

text: Well I'm not sure how many people remember the crazy ball - a small ball made of rubber which bounced like crazy. What I noticed is that the ball seemed to bounce higher than the point from which it was actually dropped. How is this possible? Doesn't this violate the law of energy conservation?


title: Can universe be a closed manifold?

text: I had a question at MSE which gave a rise to another question. Maxwell equations can be written in form $$d\star F = J$$ Then by Stokes theorem we have $$ \int_U J = \int_U d \star F = \int_{\partial U} \star F.$$ If $U$(universe) is closed then $\partial U=\emptyset$ is empty. So $\int_U J=0$. What does says about space time. Is it possible that $\int_U J=0$? Or is it proof that universe cannot be closed?


title: What is the derivation for the exponential energy relation and where does it apply?

text: Very often when people state a relaxation time $\tau_\text{kin-kin}, \tau_\text{rot-kin}$,, etc. they think of a context where the energy relaxation goes as $\propto\text e^{-t/\tau}$. Related is an approach to compute it via $$\tau=E(0)/\left(\tfrac{\text d E}{\text d t}\right)_{E=0}.$$ Both are justified from considering dynamics for which $$\frac{\text d E}{\text d t}=-\frac{1}{\tau}(E-E(0)).$$ My question is: What fundamentally leads to this relation? I conjecture it relates to a Master equation, which mirrors the form "$\dot x=Ax+b$". But I'm not sure how the degrees of freedom in the Master equation translate to the time dependence of the macroscopic energy value. There will also be a derivation from the Boltzmann equation somehow, for some conditions, but what is the general argument and where does it work?


title: Why is water blue (on a quantum level)

text: OK, lets formulate it differently and say water works as a blue passing / red restricting filter. It is actually observable. Just do a dive in a swimming pool with white light (maybe even at night) and a pool with a metal casing or white tiles. The further you look, the more blue it is. So why is this so? Why is water not transparent? And how does it happen on a molecular, subatomar or electron shell level? Do 'red' waves become absorbed and retransmitted as 'blue' light waves or do only the 'red' waves get absorbed and the 'blue' can pass? Can Maxwell help?


title: How does an earthen pot keep water cool?

text: I understand that evaporative cooling takes place thanks to small pores contained in the pot and that allow some water to go through and evaporate. However I couldn't understand clearly whether water inside the pot stays at its original temperature or would it cool further? If it will become cooler then how?


title: Is pressure invariant under Lorentz transformations?

text: In an article from the reference there are following words (page 2): "...pressure P is a Lorentz invariant... the result follows from standart properties of the relativistic stress-energy tensor...". What properties are used by the authors of an article? For example, I used the expression for stress-energy tensor of an isotropic body: $$ T_{\alpha \beta} = (\varepsilon + p)\frac{v_{\alpha }v_{\beta}}{c^{2}} - g_{\alpha \beta }p. $$ If pressure is determined as a 3-trace of this tensor, it's obviously that it isn't Lorentz invariant.


title: Is Newton's universal gravitational constant the inverse of permittivity of mass in vacuum?

text: Is it possible to consider Newton's universal gravitational constant, $G$, as inverse of vacuum permittivity of mass? $$\epsilon_m=\frac {1}{4\pi G}$$ if so, then vacuum permeability of mass will be: $$\mu_m=\frac {1}{\epsilon_m c^2}$$ then Gravitational force will be: $$F_G=\frac {1}{4\pi \epsilon_m}\frac {m.M}{r^2}$$


title: What are the implications of the Nesvizhevsky experiment and followup experiments with ultracold neutrons?

text: I remember reading about the groundbreaking experiment by Nesvizhevsky (et al. 2001) some 12 years ago using ultra-cold neutrons which showed the first experimental evidence of quantum gravity. It is my understanding that these experiments has been repeated since then in labs around the world and similar results observed(neutrons bouncing off the reflector in a discrete set of quantized energy states rather than varying continuously)and more recently by Kobakhizde(2010), who was testing Erik Verlinde's "entropy theory of gravity". Am I correct about the reproducibility of Nesvizhevsky's results? If so, these experiments seem to suggest that gravity is indeed an actual force after all and not simply an emergent property as posited by GR. But do any of the results of these experiments conflict with the weak/strong equivalence principles? Someone on Yahoo! answers with the handle OzoneGuy claimed that the observations were explainable simply by diffraction. But this could be tested by positioning the diffraction grating to be parallel to the neutron reflector.So it is simply diffraction? Or is there actual evidence of quantum gravity?


title: Topological vs. non-topological noetherian charges

text: What (if any) is the relationship between the conserved (non-topological) noetherian charges and topological charges? Namely, is there any "generalization" of the Noether's first theorem that includes topological charges as subcase and it provides some relationship between these two classes of charge?


title: Gravitational redshift of Hawking radiation

text: How can Hawking radiation with a finite (greather than zero) temperature come from the event horizon of a black hole? A redshifted thermal radiation still has Planck spectrum but with the lower temperature (remember CMB with temperature redshifted by expansion of the universe). Now, redshift at the event horizon is infinite (time is frozen for a distant observer) so the temperature of the radiation would be zero for him/her, that is no radiation is detected


title: Finding Phase angle of Simple Harmonic Motion?

text: A sinusoidal oscillator has : $$x=x_{max} \cos(\omega t - \varphi )$$ Period is 2, initial displacement is 100mm initial velocity is 200mm/s What is the phase angle assuming $-\pi < \varphi < \pi$ How do I go about solving this? Is the phase $(\omega t - \varphi)$? But I do not know what $x_{max}$ is, how am I supposed to solve for the angle?


title: Showing constraint is nonholonomic

text: One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping. These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - a\cos\theta d\phi = 0$ Where $\phi$ is the angle of how far the disk has rotated, $-\theta$ is angle that velocity makes with respect to $x$. We know that these aren't exact differentials because to put it in form $Mdx + Ndy = 0$, we don't have $\partial M/\partial y = \partial N / \partial x$. But that doesn't mean we can't find some integrating factor to multiply it by and make it an exact differential? So I don't see how we know that is it nonintegrable? If I take $f(x,\phi)[ dx - a \sin\theta d\phi] = 0$ and try to manipulate it to get something analogous to $\partial M/\partial y = \partial N / \partial x$, then I get a tricky PDE, which I don't know how to solve.


title: What is the entropy of a system when its volume tends to zero?

text: Say that a closed system has $n$ dimensions and is in the shape of a $n$-ball with a radius of 1, it's volume will be $$\frac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2}+1)}$$ which tends to 0 yet is not empty as n tends to infinity. My question might not make any sense, and my understanding of entropy might make things even worse, but if it's sensible, what will become of the entropy of such a system? If my first exposé really makes no sense, I would reformulate as follows: let's say that an unfathomable force compresses a closed system such that it's volume decreases towards zero, what will become of the entropy of such a system? (I realise it might not be the same problem/question)


title: Why can we allow the speed of light being infinite in case of Surface Plasmons?

text: I have a problem with understanding of these sentences: > We have indicated in the opening paragraph of the Introduction that surface > plasmon polaritons are solutions of Maxwell’s equations in which the effects > of retardation—the finiteness of the speed of light—are taken into account. > An important subclass of surface plasmon polaritons are surface plasmons. > These can be viewed as the limiting case of surface plasmon polaritons when > the speed of light is allowed to become infinitely large. > > Nano-optics of surface plasmon polaritons (1.1.2) - Anatoly Zayats et.al. How can we allow c be infinite? Why is it correct in case of surface plasmons? The article is available online, easy to find with Google Scholar


title: If a photon's emission is detected is it real or virtual?

text: I understand that one can measure a single photon being absorbed using a photomultiplier tube or CCD. Can one measure a single photon being _emitted_ by monitoring the current through an LED or the recoil of an emitting ion? Is it therefore possible to detect the same photon both being emitted and later being absorbed? After the photon's emission has been detected, but before its absorption has been registered, is it real or virtual? I assume a QED calculation would proceed on the assumption that the photon is virtual but if its emission has been measured then surely it should be treated as real? Is there a paradox here?


title: Since when were Loop Quantum Gravity (LQG) and Einstein-Cartan (EC) theories experimentally proven?

text: Can this template at Wikipedia be true? It seems to suggest that Einstein- Cartan theory, Gauge theory gravity, Teleparalleism and Euclidean Quantum Gravity are fully compatible with observation! It also suggests that Loop Quantum Gravity and BEC Vacuum Theory among others, are experimentally constrained whereas string theory/M theory are disputed! What I understand by "Fully compatible with observation" is that all its predictions are confirmed by experiments and it has been found to be more accurate than General Relativity. Has such evidence really been found? Or am I misinterpreting "Fully compatible with observation"? Maybe it means it has been tested only when it reduces to General Relativity? But if that where the case, shouldn't M theory/String theory also be listed under "Fully Compatible" since their predictions also go down to Classical General Relativity at the low-energy, classical limit, if all other forces (other than gravity?) are gotten rid off? What I understand by "Experimentally constrained" is that it is true given certain modifications. However, as far as I know, Loop Quantum Gravity violates Lorentz symmetry and has thus been experimentally "excluded" while BEC Vacuum theory isn't even mainstream? What I understand by "Developmental/Disputed" is that it is still undergoing development OR it has **almost** been experimentally proven wrong but it is still not settled in mainstream physics. If LQG doesn't go to the excluded section, it should at least come here? Since the violation of Lorentz symmetry has been disproven according to this. So my question is "Is this template really reliable?".


title: Measurement and wavefunction collapse. problematic time in quantum mechanics

text: Q: When does the wavefunction collapse? A: When a measurement is made. But when exactly is this? I have a question about the time at which a measurement can be considered to have occurred: what exactly constitutes a measurement? Consider the following experiment: A beam of spin $½$ particles of well defined momentum,p, and pure spin state $\lvert+z\rangle$ enter a Stern gerlach loop like those in Feynman’s lectures lec5 vol3. The loop is oriented to select $\lvert+x\rangle$ and $\lvert-x\rangle$ states depending on which channel of the loop is blocked, if neither is blocked the loop passes the $\lvert+z\rangle$ unchanged. Suppose we block the $\lvert-x\rangle$ channel and measure the spin of the emerging particle; with certainty it is |+x>. We say that the wavefunction $\lvert+z\rangle$= $\lvert+x\rangle$ + $\lvert-x\rangle$ has been collapsed to $\lvert+x\rangle$ by interaction with the filter. Now remove the detector far from the loop, distance D, so that most of the flight time of a particle of mass m, to the detector $(=\frac{Dm}p)$ , occurs after passing the filter. ( we could make D so large that a light signal could not go from the filter to the detector in the time between unblocking a channel and detecting the particle spin). Can we say there is a time after which the particle has passed the filter and is thus definitely in the $\lvert+x\rangle$ state but has yet to be detected? What if at the last moment, just before detection, the blocked channel of the filter was opened so that when the detection takes place both channels were open? Has the wavefunction now been instantly ‘uncollapsed’ ? If so we detect $\lvert+z\rangle$ with certainty rather than $\lvert+x\rangle$. If so we cannot assign a time at which the particle interacted with the original filter apparatus because in a sense the interaction has not finished until the particle is detected and removed from further interaction. Is the state of the whole apparatus which the particle may have passed through only considered at the moment at which the particle is detected and irreversibly removed from play? This seems very odd. In a sense this means the particle has no history. Only the state of the whole apparatus at the time of detection becomes important. Do we in fact detect $\lvert+z\rangle$ with certainty in this experimental situation? Have I missunderstood something here? Is a measurement necessarily irreversible? So if the particle just passes through the filter butis not detected then that is not a measurement?


title: I'm reading this in a senior year physics text book, need to know if this is true, and if so then how? :

text: "Even if a magnet is broken into atoms, each atom shall be a complete magnet. If the atom is further broken into electrons, protons, neutrons, etc. even then each particle shall behave like a complete magnet."


title: How many seconds should you accelerate to complete the run on time?

text: Your goal is to complete your 10,000-m run in less than 30.0 min. Apparently you've started too slowly because at 27.0 min, you still have 1100 m to go! Part A You can accelerate at 0.20 m/s2. For how many seconds should you accelerate (and after this have constant velocity) in order to complete the run on time?


title: How come vibrations?

text: We all know that sound sensation is produced only when sound waves reach upto us. We all know that sound waves are disturbances propagating in air, Vibration is necessary for the generation of sound, but it always forces me to ponder that how was it known or deduced that vibration is necessary for any form of sound wave? Hope someone explains me this.


title: Zero point fluctuation of an harmonic oscillator

text: In a paper, I ran into the following definition of the zero point fluctuation of our favorite toy, the harmonic oscillator: $$x_{ZPF} = \sqrt{\frac{\hbar}{2m\Omega}} $$ where m is its mass and $\Omega$ its natural frequency. However, when I try to derive it with simple arguments, I think of the equality: $$E = \frac12 \hbar\Omega=\frac12 m \Omega^2 x_{ZPF}^²$$ (using the energy eigenvalue of the $n=0$ state) giving me: $$x_{ZPF} = \sqrt{\frac{\hbar}{m\Omega}} $$ differing from the previous one by a factor $\sqrt2$. I am just puzzled, is it a matter of conventions or is there a fundamental misconception in my (too?) naive derivation?


title: Escape velocity from Earth

text: We know the escape velocity from the Earth is 11.2km/s. Isn't it the velocity required to escape from earth if we go normal to the surface of earth? i.e while we derive the formula for the escape velocity from earth we never consider the slanted motion of the object. So when we launch a rocket we need to use very less value of velocity compare to escape velocity to escape from earth because rocket follows a slanted path so curvature of earth has an effect? If what I said above is right, then how we can say escape velocity is 11.2 km/s?


title: Constructing the space of quantum states

text: I want to learn how to construct spaces of quantum states of systems. As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is the Hamiltonian of the harmonic oscillator with the quadratic term: $\hat{H}=\hat{H}_{0}+\hat{H}_{1}$, where $\hat{H}_{0}=\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+1/2\right)$, $\hat{H}_{1}=\dot{\imath}\gamma\left(\hat{a}^{\dagger}\right)^{2}-\dot{\imath}\gamma\left(\hat{a}\right)^{2}$; $\hat{a}$, $\hat{a}^{\dagger}$-ladder operators, $\gamma$-real parameter _For this purpose, we should define a complete set of commuting observables (CSCO)_. As for the harmonic oscillator, we can define a "number" operator $N=\hat{a}^{\dagger}\hat{a}$. _We can prove the following statement:_ Let be $a$ and $a^{\dagger}$ Hermitian conjugated operators and $\left[a,a^{\dagger}\right]=1$. Define operator $N=aa^{\dagger}$. Then we can prove that $\left[N,a^{p}\right]=-pa^{p}, \left[N,a^{\dagger p}\right]=pa^{\dagger p}$ and that **the only algebraic functions of $a$ and $a^{\dagger}$, which commute with $N$, are the functions of $N$**. (For example, see Messiah, Quantum Mechanics, exercises after chapter $12$) Using this statement, we conclude(am i right?) that **operator $N$ forms a CSCO**. So, sequence of eigenvectors of operator $N$ forms the basis of the space of states. So, I've come to the conclusion that _the space of states of the described system is the same as the space of states of harmonic oscillator_. **But** operators $a$, $a^{\dagger}$ can always be determined (as i think), so this arguments will be valid, so I've come to the conclusion, that **spaces of states of all systems will be the same**. **After that I realized that I am mistaken**. Would you be so kind to explain where is a mistake in the arguments above? And can you give some references/articles/books where i can read some additional information about constructing spaces of states for different systems?


title: Schrodinger's equation (explanation to non physicist)

text: For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the explanation. However I'm not so stupid to think I can understand this landmark without some years of physics. I'll just be happy to be able to _read_ the equation and recognize it in its various forms. To be more precise, here are my questions. Hyperphysics tell me that Shrodinger's equation _"is a wave equation in terms of the wavefunction"_.   1. Where is the wave equation in the most general form of the equation? $$\mathrm{i}\hbar\frac{\partial}{\partial t}\Psi=H\Psi$$ I thought wave equation should be of the type $$\frac{\partial^2}{\partial^2t}u=c^2\nabla^2u$$ It's the difference in order of of derivation that is bugging me. From Wikipedia > _"The equation is derived by partially differentiating the standard wave > equation and substituting the relation between the momentum of the particle > and the wavelength of the wave associated with the particle in De Broglie's > hypothesis."_   2. Can somebody show me the passages in a simple (or better general) case?   3. I think this questions is the most difficult to answer to a newbie. What is the Hamiltonian of a state? How much, generally speaking, does the Hamiltonian have to do do with the energy of a state?   4. What assumptions did Schrödinger make about the wave function of a state, to be able to write the equation? Or what are the important things I should note in a wave function that are basilar to proof the equation? With both questions I mean, what are the passages between de Broglie (yes there are these waves) and Schrödinger (the wave function is characterized by)?   5. It's often said _"The equation helps finds the form of the wave function"_ as often as _"The equation helps us predict the evolution of a wave function"_ Which of the two? When one, when the other?


title: Net Electrostatic Field inside a Conductor is $\vec{0}$

text: I've heard an explanation for this, and my professor wasn't really able to clarify my questions, so I was hoping someone could help: Suppose there is an electrostatic field $\vec{E_{net}}\ne\vec{0}$, then the electrons on/in the conductor would be free to move, and then $\vec{E}_{net}$ wouldn't be an electrostatic field. Contradiction. **Question** : It seems like we are considering the electrons as both the source and test charges, because electrostatic field means that the source charges are stationary (and the contradiction is that the electrons are moving), but this supposed $\vec{E_{net}}$ is also moving the electrons. How is this different from just having a collection of source charges? If you have a collection of stationary charges, and you're trying to find the electrostatic field, then by the argument above, any supposed electrostatic field would cause these charges to move, and so this would contradict the field being electrostatic.


title: What is the difference of work $W$ and thermal energy $Q$ in thermodynamic Stirling-process for ideal gas?

text: What is the difference of work $W$ and thermal energy $Q$ in thermodynamic Stirling-process (in simple form) for ideal gas? I think that you need work to preserve this process and you bring thermal energy so that temperature $T$ rises in the process.


title: Why laundry dry up also in cold/frost?

text: Why laundry dry up also in cold/frost? When you have frost, water in the clothes should freeze, but if clothes are dry, then it should be possible that steam in the clothes does not have time to freeze.


> **Possible Duplicate:**   >  Non-linear dynamics of classical hydrogen atom I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom. Taking into account Newton equations for the electron and the proton and Maxwell equations for the electromagnetic field produced by these charges one obtains a higly non-linear set of coupled equations. In such a nonlinear system could some feedback effects between proton and electron take place so to make possible a stable dynamics (or at least a dynamics unstable on such long time scales longer that we can consider hydrogen to be stable)? In this way the system's stability already obtained through quantum mechanics could be reproduced by a full classical approach! P.s.: Please, as I know of the great successes that quantum theory has had since its birth, try not to answer the question telling how quantum mechanics wonderfully solves the problem. P.p.s.: I'm also aware of the fact that electron should lose energy and that this should cause it to fall on the proton in a very short time, so please try to avoid also this argument. I asked this question to understand if the oversymplifing hypothesis', which are fundamental in solving this problem (neglect proton's motion and, as a consequence, magnetic effects) and are quiet ubiquitous in physics, wouldn't mask the potential richness that could arise from mathematical complexity.


title: What does the wind speed have to be to blow away a person?

text: With the approaching hurricane, I am curious about what would happen if I go outside, in particular whether the wind gusts might be fast enough to blow me away. How fast would the wind have to be to blow away a person?


title: How quarks converted into leptons

text: ![Pion Plus Decay](http://hyperphysics.phy- astr.gsu.edu/hbase/particles/imgpar/pidec1.gif) Since the charged pions decay into two particles, a muon and a muon neutrino, seems quarks disappeared!, The decay proceeds by the weak interaction $W^{+}$ and can be visualized in terms of Feynman diagrams. Isn't it why Quarks are not directly observed!? I read somewhere: > If you are consistent thinker you can go even further and question existence > of quarks themselfs then you will not have a problem with fractional charge


title: Derivation of the "Bethe sum rule"

text: I am trying to work out the steps of the proof of the expression: $$\sum_n (\mathcal{E_n}-\mathcal{E_s})|\langle n|e^{i\mathbf{q}\cdot\mathbf{r}}|s \rangle|^2 = \frac{\hbar^2q^2}{2m}$$ from Eq. (5.48) in the book _Principles of the Theory of Solids_ by Ziman. In the book it is mentioned that this can be shown by expanding out: $$[[\mathcal{H},e^{i\mathbf{q}\cdot\mathbf{r}}],e^{-i\mathbf{q}\cdot\mathbf{r}}]$$ where $$\mathcal{H} = -\frac{\hbar^2}{2m}\nabla^2+\mathcal{V}(\mathbf{r})$$ with $\mathcal{V}(\mathbf{r})$ being periodic. What I did is a simple expansion: $$[[\mathcal{H},e^{i\mathbf{q}\cdot\mathbf{r}}],e^{-i\mathbf{q}\cdot\mathbf{r}}] = 2\mathcal{H}-e^{i\mathbf{q}\cdot\mathbf{r}}\mathcal{H}e^{-i\mathbf{q}\cdot\mathbf{r}}-e^{-i\mathbf{q}\cdot\mathbf{r}}\mathcal{H}e^{i\mathbf{q}\cdot\mathbf{r}}$$ Then I took the inner product with the eigenstates of the Hamiltonian $\mathcal{H}|s\rangle = E_s |s\rangle$ to get $$\langle s|[[\mathcal{H},e^{i\mathbf{q}\cdot\mathbf{r}}],e^{-i\mathbf{q}\cdot\mathbf{r}}]|s\rangle = 2E_s - \langle s|e^{i\mathbf{q}\cdot\mathbf{r}}\mathcal{H}e^{-i\mathbf{q}\cdot\mathbf{r}}|s\rangle - \langle s|e^{-i\mathbf{q}\cdot\mathbf{r}}\mathcal{H}e^{i\mathbf{q}\cdot\mathbf{r}}|s\rangle$$ Now, what's obstructing my calculation is the fact that I cannot justify the last two terms in the above expression being equal. I really need them to be equal to show the top identity (also known as the Bethe sum rule). The main obstacle is the fact that $e^{i\mathbf{q}\cdot\mathbf{r}}$ is non-Hermitian. I have found this identity in many books and journal articles. But I cannot find a satisfactory proof anywhere. One such example is this article: > Sanwu Wang. Generalization of the Thomas-Reiche-Kuhn and the Bethe sum > rules. _Phys. Rev. A_ **60** no. 1, pp. 262–266 (1999). Th result is stated in Eq. (3) and proof is given in section III. A. At the end of the proof they set $F = e^{i.\mathbf{q}.\mathbf{r}}$ to recover Eq. (3). The fact that they did not assume any form for $F$ means it must hold for _any_ function. There is, however, one step that I cannot justify. In Eq. (9) how can they write: $$\langle 0|F(x)|l \rangle = \langle l|F(x)|0 \rangle$$ If I can even justify the above equality (for my case) then I'm set.


title: Kinetic Energy of Stone

text: If a stone is launched upward, of which is subject to gravity and air resistance, which of the following will have a greater kinetic energy? The stone at a point on its way up, or the stone at the same point on its way down? I know $KE=1/2mv^2$, and $PE=mgy$. If I had to guess I would say the kinetic energy is equal up and down, but I'm not sure. Could someone explain this concept to me?


title: Calculating threshold energy of particle reactions

text: {..everything that follows is in the domain of relativistic kinematics..} Say a particle A collides with a particle B at rest and produces particles C and D. What exactly is the definition of "threshold energy" for a reaction ? Is it the energy that A should have so that so that the heavier of C and D is produced at rest? Given the masses of all the particles, how does one calculate the threshold energy that A must have to cause this reaction ? I intuitively feel that a threshold energy scenario would mean that C and D are moving along the same direction in which A was coming in. I think it is an unnecessary "waste" of energy for C and D to develop momentum in the two transverse directions of A's motion. But I can't prove this in general. I would like to know how this situation is analyzed.


title: Explicit Variation of Gibbons-Hawking-York Boundary Term

text: Are there any references that present the explicit variation of the Hilbert- Einstein action plus the Hawking-Gibbons-York boundary term, and demonstrate the cancellation of the normal derivatives of metric variations? I have tried to read the original papers by York and Gibbons&Hawking, but they are not that pedagogical to me... Thanks to anyone who can help me!


title: Least-action classical electrodynamics without potentials

text: Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which depends only on the electric and magnetic fields and which will have Maxwell's equations as its Euler-Lagrange equations?


title: How thermal imaging cameras work

text: As far as I see from wiki, 'consumer'-grade(non-cryogenic) thermal imaging cameras use microbolometer sensors to get integrated IR intensity over some 5-12um range. But I had an impression, that having just integrated IR intensity you can't get surface temperature precisely, especially with 0.1C precision we see on consumer cameras. IR thermometers (pyrometers) for example use 2-band IR measurement, but I don't see this principle being used in thermal imaging. Or there is some sort of 'know-how' matrix of IR band filters in front of the sensor?


title: Electric field caused by magnetic field

text: Does electric field caused by time varying magnetic field form closed loops(electric field starts from a positive charge and ends at a negative charge)? and are they conservative or non-conservative fields?


title: Why do prescription glasses that darken in sunlight fail to darken when the light travels through a car windshield?

text: I was going to buy this add on, as I almost exclusively use my sunglasses for driving. I'm just stacking them over my normal ones now... but it's quite annoying that I almost paid the extra $$$ for this feature. Found out at the last second, luckily.


title: continuity equation for a fluid with variable density

text: I am trying to derive the equation for buoyancy frequency in a stratified fluid and am struggling with some of the equations. I have a limited background in fluid dynamics so I basically just need someone to break down the continuity equation for me, in terms that are easily understood. From the set of notes I am looking at: ![enter image description here](http://i.stack.imgur.com/x3SJV.png) This is the first step of the method for deriving the equation, can anyone explain this in simplistic terms?


title: Induced voltage by a rotating ring

text: Two concentric rings dielectrics and uniformly charged are suspended on the same floor. The outer ring has a radius R and mass M, while the other has radius r << R and mass M. The outer ring that has overall charge Q, is made to rotate around the axis passing through the center and perpendicular to the plane of the ring itself, with angular acceleration α. Calculate the angular acceleration α' of thr inner ring. * * * My solution is the following: The outer ring produce a variation of magnetic flux on the inner ring equal to               (μ * Q * π^2 * r^2 ) / ( α * R )  (*)      The inner ring produce a self-induced variation of magnetic flux for the law of Faraday-Lenz equal to               (μ * Q * π^2 * r' ) / α'   (**)      Comparing the ( * ) and ( ** ), I obtain               α' = α * R / r      Is it my solution right?


title: Water evaporation

text: It is known that molecules at the surface are strongly attached to each other (more attraction less repulsion) than those within the bulk attraction and repulsion are balanced). This is the molecular description of the surface tension and its net direction i.e., parallel to the water surface. Briefly, considering energy (total energy cost) - the cost of excluding a surface molecule is less, than to break a molecule from bulk and evaporate it from the surface! Now, considering evaporation - the question is - why evaporation occurs towards molecules of the surface (the stronger attachment than those of the bulk) and not through molecules from the bulk (like assumed in boiling)?


title: Questions about angular momentum and 3-dimensional(3D) space?

text: **Q1:** As we know, in classical mechanics(CM), according to Noether's theorem, there is always one conserved quantity corresponding to one particular symmetry. Now consider a classical system in a $n$ dimensional general coordinates space described by the Lagrangian $L(q,\dot{q})$, where $q=(q_1,...,q_n)$ and $\dot{q}=(\dot{q_1},...,\dot{q_n})$ are general coordinates and general velocities, respectively. If the system has $SO(n)$ spatial rotation-symmetry, e.g.$\forall A\in SO(n),L(Aq^T,A\dot{q}^T)=L(q,\dot{q})$, then we can get $\frac{n(n-1)}{2}$(number of the generators of group $SO(n)$) conserved angular momentum. **My question is as follows** , now note that our spatial dimension is $n$, when $n=3\rightarrow $ number of angular momentum$(\frac{n(n-1)}{2})$$=$spatial dimension$(n)$$=$3, otherwise, number of angular momentum$\neq $spatial dimension, so why spatial dimension 3 is so special? Is there any deep reason for number 3 or it's just an accidental event? Or even does this phenomena have something to do with the fact that we "live" in a 3D world? **Q2:** In quantum mechanics(QM), a Hermitian operator $J=(J_x,J_y,J_z)$ is called an angular momentum iff $[J_x,J_y]=iJ_z,[J_y,J_z]=iJ_x,[J_z,J_x]=iJ_y$. **And my question is as follows** , in QM, we can have 1-component momentum operator $\hat{p}$ , or 2-component momentum operator$(\hat{p_x},\hat{p_y})$, and so on. But Why we have never encountered a angular momentum with only two components $J=(J_x,J_y)$? Can we define a 2-component angular momentum? Like in the **CM** case, again the **number 3** is special in **QM** case, why? Thanks in advance. **By the way:** More questions concerning the **definition of rotation groups** for angular momentum can be found here , who are interested in may have a look, thanks.


title: Transmission/Equilibrium Question

text: Suppose there are two wheels linked by some kind of apparatus with a transmission, in a vacuum. Wheel A is spinning, wheel B is not. Is it empirically possible for the transmission to transfer the energy from wheel A to wheel B such that wheel A stops spinning and wheel B spins at a speed very close to that which wheel A was originally spinning? If so, how would such a transmission look? If I understand correctly, this should be a question of gear ratios and how close the gear ratio can come to infinity. Please help me understand this better. Thanks!


title: The effects of heat on gravitational fields

text: In boiling soapy water, globs of soap coalesce as the temperature increases to boiling. Does this mean that temperature increases the gravitational pull of bodies?


title: What reflects/scatters sunlight most?

text: The difference between a white object and a mirror is mainly scattering versus reflection. If we neglect this difference and define the reflectivity by input energy minus transmitted energy minus absorbed energy, what is the material with the highest reflectivity? With material, I mean something like a single element mirror, any multi-layer mix in a dielectric mirror or any sort of white-colored surface. As the reflectivity depends on the wavelength, let's assume, our input energy is distributed according to the sun's spectrum (on the earth's ground). Let's further assume, there is no polarization. With these definitions, it should be possible to define a reflection coefficient as a scalar. I am interested in theoretical aspects of the problem, laboratory results/examples and home-made solutions (alumium foil, wall color, etc)


title: Difference in chemical potential in supersaturated solutions

text: I have been more or less struggeling to understand an equation that is apparently used in almost all books covering crystals in any way. Basically every book that I have found explains the following: In a system with a one-component vapour at the phase boundary vapour-solid, the difference in chemical potential between the two phases is given by: $\Delta\mu = \Delta\mu_v - \Delta\mu_c,$ where $\Delta\mu_v$ and $\Delta\mu_s$ are the deviaitions of the chemical potentials of the vapor and solid states from their equilibrium positions, both being functions of pressure only (T is being kept constant). If those deviations are small then we can write the $\Delta$s as partial derivatives which are integrated from $P_0$ (equilibrium pressure) to $P$ (actual pressure). By treating the vapor as an ideal gas, they finally arrive at: $\Delta\mu = kT\ln(\frac{P}{P_0}),$ which explains how the change in chemical potential leading to nucleation is affected by the deviation of the pressure from the equilibrium point. This all is quite easy to understand and fine with me, but now every book says the following: In case of solution crystallization the equation may be rewritten to give: $\Delta\mu = kT\ln(\frac{C}{C_0}),$ where $C$ and $C_0$ are the actual and equilibrium (saturated) concentrations of the solute, however, no book that I have found (and I searched through a lot) actually derives or even explains why the pressures can be swapped by concentrations in case of solutions. My explanation would be that thinking of pressure being linearly dependent on the amount of particles crossing unit area in unit time in the vapor and comparing that to the concentration which gives a linear dependence on the amount of particles crossing unit area in unit time in the solution, the two equations could be the same. However, I don't know if that is correct, nor have I read this explanation anywhere but I need some proper source that can be quoted for the last equation or a derivation for the latter from first principals. Could somebody please help me with that?


title: Tension in an Atwoods machine conceptual?

text: ![enter image description here](http://i.stack.imgur.com/jLunv.gif) Assuming $T_1$ is the force that acts on box $1$ and $T_2$ is the force that acts on box $2$. Exactly what causes the Tension? Why does $T_1 = T_2$? The problem is we are told to memorize that $T_1 = T_2$ for mass less ropes, but I do not understand why, especially in an Atwoods machine. When I think about it, I know it has something to do with the force $m_2g$ which causes the box with mass $m_1$ to rise, and vice versa. But I cannot apply physics terminology to this, or really understand whats going on.


title: Geometric Interpretation of these equations of motion?

text: I was reading my Engineering Mechanics book, and it derived some strange looking integrals I'll have to apply. I could memorize them, but I'd rather understand them - then I won't have to memorize. A few words mentioning Leibniz notion, and viola - we have these relationships: $$\int_{v_0}^vdv=\int_0^ta_cdt$$ $$\int_{s_0}^sds=\int_0^t(v_0+a_ct)dt$$ $$\int_{v_0}^vvdv=\int_{s_0}^sa_cds$$ I can manipulate them given appropriate data, but they don't really _mean_ much to me. (I _do_ easily understand things like $a=\frac{dv}{dt}$,however). I anyone could point me to website that discuss or even just show me how to derive these, I would really appreciate it - I find it easier to understand geometric interpretations, but I would appreciate any knowledge you would be willing to share.


title: Expanding universe and the peculiar velocity

text: Hubble's law states that the universe is expanding with a velocity equals Hubble's constant*distance from earth. But, recent findings show that the Andromeda galaxy is actually blueshifting towards us and nearby stars and galaxies do show motion with respect to the Earth with so called peculiar velocities. What's the catch here? I am a beginner in this subject matter. Any help would be duly appreciated.


title: Aside from Noether's theorem, what other concepts would explain energy conservation?

text: Energy is defined more in the mathematical sense, and tends to show true with observations in the physical world. But why is energy conserved aside from "Noether's theorem"? In a closed system that has an energy $E$, we know that system will always have this much energy, but why? In the cosmological scale I heard that energy is not conserved, why? Is the universe creating more energy? OR Is the expansion of the universe due to the energy at the beginning that caused it's creation? Could anyone explain their point in layman terms? Without using difficult Mathematics, I'm not as "qualified" to understand such rigorous equations yet.


title: Fundamentals of electrostatics

text: Suppose I have a Gold Leaf Electroscope and the leaves are observed to diverge by a certain amount. Now if I send a beam of X-rays and allow it to fall upon the electroscope for a very short period of time, what exactly will happen?


title: Why do aqueous solutions always "have to be" electrically neutral?

text: I was reviewing some analytical chemistry and stumbled upon a section that explained the imperfection of using a salt bridge. It said that the using dissimilar ions is a problem because in, for example, the case of KCl the $K^+$ and $Cl^-$ have different mobility and so you get regions that are rich in $K^+$ and others rich in $Cl^-$. This means that some specific parts of solution (even though it's small) to be not electrically neutral i.e. electrically charged. So I imagine in theory at least it must be possible to somehow extract these charge-rich areas and put them into a beaker. So what is all this I've been learning about solutions having to be electrically neutral? Physics people?


title: Joule heating due to the (slow) electron drift velocity?

text: I understand the concept of why the signal speed is higher than the electron drift velocity, but I can't understand the concept of joule heating. If electrons move slow then how do they produce a lot of heat when they hit the nucleus. Besides my friend once told me that the drift velocity is the net movement and electrons move fast in all directions, if that is the case why do they move like that?


title: Contact angle of liquid drop on surface

text: How do you explain point 44 of the attached pdf document on surface tension? Here's the link. How is the direction of surface tension found out? (I know it tangential but in which direction along the tangent?) Also why is surface tension for solid-air surface considered when balancing the forces on liquid? Any help will be appreciated.


title: Would HgTe be a topological insulator?

text: In "Quantum Spin Hall Insulator State in HgTe Quantum Wells", researchers observed a 2D topological insulator by sandwiching HgTe between CdTe. Is the CdTe really necessary? Would Vacuum/HgTe/Vacuum itself be a topological insulator?


title: How to design an experiment that shows that a rectangular pulse can be expressed as a series of infinite sinusoids?

text: Is it possible to design a physical experiment that shows that a time limited signal, such as a rectangular pulse is composed of infinite continuous sine/cosine waves?


title: MotoGP motorcycles lift front wheel while exiting curve

text: I've seen multiple times now in MotoGP that motorcycles were leaving a curve, and because of the acceleration the front wheel is slightly leaving the ground. But at the same time they were still leaning to the inside of the curve, just following the curve to the end and then going straight again. Now I understand why the can lift the front wheel. But what I don't get is why they can still finish the curve on one wheel, instead of just going straight and leaving the track. I thought the front wheel is supposed to use its traction on the surface of the ground to lead the motorcycle around the curve, but they keep going around the curve while the front wheel is in the air. Which force makes the motorcycle do a curve while the front wheel is not touching the ground? I'm not sure if my explanation was understandable, here are some pictures of what I mean. They still lean into the curve and follow the curve while already lifting the front wheel: http://50to70.com/wp-content/uploads/2012/07/jorge-lorenzo- motogp-2012-mugello-test.jpg http://farm3.static.flickr.com/2224/2331275819_ccbe52129f.jpg


title: Intuition for Path Integrals and How to Evaluate Them

text: I'm just starting to come across path integrals in quantum field theory, and want to get the right intuition for the them from the start. The amplitude for propagation from $x_a$ to $x_b$ is typically written $$U(x_a,x_b)=\int\mathcal{D}x(t) e^{\frac{i}{\hbar}S[x(t)]}$$ where $S$ is the classical action functional. Some books I read seem to treat $\int \mathcal{D}x(t)$ as a formal sum over all paths connecting $x_a$ and $x_b$. Is this the right way to think about it? I come from quite a pure maths background so I've been trying to imagine this as some kind of measure on the space of smooth curves $x(t)$. I'm having trouble visualising this though. Does anyone have an intuitive argument as to how to picture this perspective? Finally, how does one go about evaluating a general path integral in practice? I know that one can get an approximation by breaking the path up into piecewise linear segments. Is this a method that's used? Presumably also one can do the formal sum, and I assume this is the origin of many infinities. I'd be happy to be told that there is no 'general' method for solution, or 'right' intuition. I'd just be interested to hear a range of ideas on the subject! Many thanks in advance!


title: Which is more fundamental, Fields or Particles?

text: I hope that I am using appropriate terminology. My confusion about quantum theory (beyond my obvious unfamiliarity with its terminology) is basically twofold:   1. I lack an adequate understanding of how the _mathematics_ of quantum theory is supposed to correspond to phenomena in the physical world   2. I still have an incomplete picture in my mind of how cause and effect relationships occur at the quantum level of reality.  This is why phenomena such as "entanglement" make absolutely no sense to me. So, in an attempt to come to some understanding of all of this, I would like to know that if what we conceptualize as a "field" is merely an interaction among particles (bosons and fermions in the case of quantum fields), and particles (themselves) are actually fluctuations in "fields", then which comes first in the hierarchy of cause and effect relationships, particles or "fields"?


title: EM field of free moving electrons?

text: if I stretch two wires parallel to each over and in a little distance and I let through them slow moving electrons, do I measure a current in the wires? Of course the electrons don't hit the wire. Some sources where such experiments described?


title: SIngle slit diffraction and location of dark spots

text: In single slit diffraction, why (according to the equation for $\theta$) do successive dark spots exist at whole number intervals ($n=1,2,3, \ldots$) and not half? The correct path difference for destructive interference is $1/2$ a wavelength, so why in $\theta = n(\lambda)/(\mbox{slit width})$ do we use $n = 1,2,3,\ldots$?


title: Is there a substance that doesn't reflect OR absorb light from the visible light spectrum?

text: Is there a substance that doesn't reflect or absorb visible light but may reflect light from another spectrum? Is there a theoretical substance that would have these properties? **EDIT** :   Sorry I wasn't quite clear with my original question. I've updated it to express more what I was thinking. Would a substance be "invisible" if it didn't reflect or absorb light? Does a real or theoretical substance like that exist? I assume we can see glass and so on because it refracts light and to a certain extent reflects it.


title: Convert an angle

text: Can someone show me a formula in order to convert an angle from [V] to degree or rad? (i use aPotentiometer measuring steering angle) Edit: Well let me exlpain it. I photo of my experiment is shown below. I apply a force to my material and measure the torsion angle with a potentiometer. For example for P=1,18 bar , θ=1,67 V. I want to convert V into degrees or rad. My only clue is that at the full extent of the piston the angle indication in V is 5,33 V – ![enter image description here](http://i.stack.imgur.com/tAaxq.png)


title: The charm of the gyroscope

text: I am fascinated by the gyroscope, like everyboy who was so lucky as to get such a toy as a kid. But probably also grown-ups are not immune to its charm. I am not asking for a theoretical explanation. My question is more practical: **I am looking for a clear and simple way to determine quantitatively the reaction** to a tilt to the spinning axle of a gyro. Here at 0:50 we can see a concrete example. We can guess the girl weighs G= 44/45 Kg, the wheel W = 1 Kg and the platform P_0 = 0.5 kg, her arms are (presumably) R= 40 cm long and the radius of the wheel r = 25 cm. and it is spinning roughly at 4/5 rps. If you have better extimates, please plug in your figures. Can you you explain how you derive/ predict the angular velocity of the system G+P+W = P = 46 Kg when she at 0:56 tilts the wheel by 90° and then at 180° $ L_w = 1*.25^2* 4 = 0.25 $, right?


title: Significance of magnetic translation operator defined in fractional QHE's description

text: What is the significance of the magnetic translation operator used in describing the Fractional Quantum hall effect? I was following Anthony Leggett's lecture video in which he defines these operators and describes their commutation relation in order to find the degeneracy of the ground state. Does it have any physical importance as such or is it just a mathematical construct in order to prove a result about the degeneracy of the ground state ?


title: Mathematically-oriented Treatment of General Relativity

text: Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would   1. Prove all theorems used   2. Use modern "mathematical notation" as opposed to "physics notation", especially with respect to linear algebra and differential geometry    3. Have examples that illustrate both computational and theoretical aspects   4. Have a range of exercises with varying degrees of difficulty, with answers An ideal text would read a lot more like a math book than a physics book and would demand few prerequisites in physics. Bottom line is that I would like a book that provides an axiomatic development of general relativity clearly and with mathematical precision works out the details of the theory. Thank you. Addendum (1): I did not intend to start a war over notation. As I said in one of the comments below, I think indicial notation together with the summation convention is very useful. The coordinate-free approach has its uses as well and I see no reason why the two can't peacefully coexist. What I meant by "mathematics notation" vs. "physics notation" is the following: Consider, as an example, one of the leading texts on smooth manifolds, John Lee's Introduction to Smooth Manifolds. I am very accustomed to this notation and it very similar to the notation used by Tu's Introduction to Manifolds, for instance, and other popular texts on differential geometry. On the other hand, take Frankel's Geometry of Physics. Now, this is a nice book but it is very difficult for me to follow it because 1) Lack of proofs and 2)the notation does not agree with other math texts that I'm accustomed to. Of course, there are commonalities but enough is different that I find it really annoying to try to translate between the two... Addendum (2): For the benefit of future readers, In addition to suggestions below, I have found another text that also closely-aligns with the criteria I stated above. It is, _Spacetime: Foundations of General Relativity and Differential Geometry_ by Marcus Kriele. The author begins by discussing affine geometry, analysis on manifolds, multilinear algebra and other underpinnings and leads into general relativity at roughly the midpoint of the text. The notation is also fairly consistent with the books on differential geometry I mentioned above.


title: Reference request: Introductions to current mathematics derived from / related to gauge theories (in physics)

text: I was searching for introductions to current mathematics derived from / related to gauge theories in physics. Can someone suggest some good references? E.g. _Topics in Physical Mathematics by K. Marathe_ _Audio, video, and other non-text resources are also welcome._


title: Is there a book that discusses General Relativity in terms of Modern Differential Geometry?

text: All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general coordinate change. Modern Differential geometry dispenses with coordinates. Is there a book out there that discusses GR wholly in terms of differential geometry?


title: Literature request for books / review papers on gravitation, gauge theories and related mathematics

text: Similar to this reference, are there more such references / works [including textbooks] available in the literature? (A list would be greatly welcomed and appreciated.) With great appreciation.


title: Number of Stars vs Value of Omega (Crtitical Density of the Universe)

text: I may be badly mixing things up here. If I am, please kindly correct me. As I understand it, if the universe was too dense at the start of the big bang, it would have collapsed back in on itself. Too sparse, and it would have expanded violently leaving no galaxies. I also read that the ratio of the actual density to the critical density (Omega) leaves little room for error, if you want to have a universe that lasts. The value seems to not be able to vary more than one part in 10^62. Now I've also read that a recent estimate says there are about 10^23 stars (300 sextillion). I noticed that the number of stars is far fewer than the amount of variance allowed for in Omega. So my question is: If you took away one solar mass worth of matter from the early universe, would that have thrown off the balance of the expansion of the universe? Would the universe have not been here if one star was missing? * * * **The TL;DR:** The Acccepted Answer is "Yes, one star would make a difference. It would take even less to throw off Omega." (For the benefit of others coming across this question)


title: Is passing on knowledge/information from one body to another and the resultant state is ever studied as any branch of physics?

text: Does any branch of physics, study the passing off of information/knowledge from one body to another ( body would obviously be living beings here, as they only can do so ). And resultant changes in the physical state of the universe because of that information. For example, whatever physical state, a living being or nature exists, largely depends upon, how information has flowed to that particular point of time. Based on that information only, humans and other living beings act and behave. They react accordingly, resulting into chain of other events that ultimately decide state of non-living things in the nature. A small example can be like two states of universe possible, when : 1) A person A tells ( informs ) B, there is a path-hole ahead. ( Thus B is careful, and does not fall) 2) A person A does not tell B , about the path hole. ( B falls into the path- hole).


title: How are the CPU power and temperature caculated/estimated?

text: 1. From Wikipedia > The power consumed by a CPU, is approximately proportional to CPU frequency, > and to the square of the CPU voltage: $$ P = C V^2 f $$ (where C is > capacitance, f is frequency and V is voltage). I wonder how that is derived from basic circuit theory? How is a CPU modeled as a circuit? Why is it modeled as a capacitance, how about a mixture of resistance, capacitance, and inductance? Is the above formula for $P$ related to that the energy/work of a capacitance is $$ W = \frac{C V^2}{2}? $$ Do we have to distinguish between AC and DC circuits here?   2. From another source, the temperature of a CPU is estimated as a constant factor $$ \text{Processor Temperature} = ( \text{C/W Value} \times \text{Overclocked Wattage}) + \text{Case Temperature} $$ where, if I understand correctly, $\text{Overclocked Wattage}$ is the $P$ in my first formula, and $\text{C/W Value}$ is the constant factor multiplied to $P$. I wonder why we can model the temperature as a linear function of $P$? Specifically, why is there a _constant_ factor $\text{C/W Value}$?   3. In practice, I have encountered two cases. When I scale down the CPU frequency, the CPU temperature decreases. If the CPU frequency is $f$ in my first part (is it?), then the first formula explains this case well. But there is another case that I cannot find explanation from the above parts. When I am running a heavy program, if I use another program called `cpulimit` in Linux to limit the percentage of CPU usage to for example $50\%$ for the program's process (originally there is no limitation, i.e. CPU usage percentage can be 100% for the program), the CPU temperature can also go down. How will you explain this? I posted my questions on http://superuser.com/questions/432377/whats-more- harmful-to-a-cpu-high-load-or-high-temperature, but replies (especially the one by Dennis) there don't seem convincing. Thanks and regards! ?


title: quantum explanation of doppler effect

text: How would quantum mechanics explain doppler effect? And just for curiosity, is there any effect similar to doppler effect occuring at quantum level?


title: Naturalness and experiments

text: Is there an example where model building that is motivated only by Naturalness, has led to experimentally verified observations? If the question is unclear, or if the reader wants more elaboration, then continue reading. Naturalness is defined by 't Hooft as: "A theory with a parameter $m$ is natural if the limit where $m \rightarrow 0$ results in an enhancement of the symmetry of the theory." For example, take fermions in the Standard Model, if you set their masses to zero then you restore chiral symmetry. And the log-divergent radiative corrections will vanish. Now, take the Higgs boson (a scalar), if you calculate the loop correction the Higgs squared mass parameter, you'll get quadratic divergence. If you set the mass to zero, this won't help as you still have quadratic divergence. And no symmetry is restored. If the scale of new physics is at the Planck scale or GUT scale, then you'll have to tune the mass of the Higgs by something like $10^{30}$. If Supersymmetry exists, then the quadratic divergence cancels out. Also, the limit where the Higgs mass goes to zero will result in restoring SUSY. So, SUSY will give you a "natural" model. However, I came across this article: "Is Naturalness Unnatural?" by Nobel laureate Prof. Burton Richter. It addresses three things: Naturalness (which what I'm concerned about), the cosmological anthropic principle, and the landscape in string theory. Concerning Naturalness, in his view, in balancing between having a natural theory with extra parameters, and having a theory with huge fine-tuning (or as he says: somethings are simply initial conditions) but fewer parameter, the latter is more important, while the former can be misleading. Hence, I'm tempted to ask, are there examples in the past where naturalness considerations led to experimentally verified results?


i heard that fission activation energy of (235)U is less than of neutron separation energy of (236)U **so this must the reason that (235)U is fission able** $$E_s+(236)U\to (235)U+n$$ in this interaction $E_s$ is neutron separation energy ( energy required to separate neutron ) ![activation-energy](http://www.files.chem.vt.edu/RVGS/ACT/notes/activation- energy.gif)   1. how to measure activation energy of fission?   2. is there any relation or equation?


title: Finding the initial vertical and horizontal velocity

text: Basically I need to get an initial vertical velocity to make the ball reach a specific height, but also I need to get a horizontal verlocity so that the ball will travel the distance. For an example lets say the distance is 100m and the max height is 50m starting at 0m. Any help is welcome appreciated. Also correct me if I have the wrong tag or am in the wrong part of stack overflow...


title: The multiverse of eternal inflation

text: I don't have a clear picture of what is "by definition" the multiverse appearing in the models of eternal inflation. A long time ago I heard that it is a quantum state, but in the cosmology books I have looked into (all treating the subject superficially, may I suspect it's because the object is still not clearly defined?) they define a non spatially homogeneous scalar field on it, entering in the slow-roll inflation regime in some areas, and creating a nucleation this way... So it must also be a manifold, so we can define a coordinate chart on it. Then how many dimensions is it? Because they is also this picture where it solves the landscape problem of string theory and each bubbles becomes a particular vacua of ST (but I am not sure about that neither as I have not read enough of string theory so far). Well to summarize I have a really blurry picture of this mysterious multiverse, so if someone could share his clear definition of it (assuming it exists) I would be really thankful. ps: is it even a physical object? in the sense obeying itself to some physical laws (which ones?) not like Newton's universe for example? is it dynamical?


title: How to "read" the temperature of an abstract system?

text: How can I interpret the parameter _temperature_ $T$, if I'm not given the description of the system in terms of the equation of state, $E(S,V\ )$ or $S(E,V\ )$ and so on. In many systems it makes sense to think of it as an "energy" itself, e.g. when the entropy is such, that $T$ basically represents the mean kinetic energy of particles. The general definition goes like $$\frac 1 T=\small{\left(\frac{\partial S(E,V\ )}{\partial E}\right)_V}.$$ However, very often, especially when I'm reading about phase transitions, examples for systems with critical exponents and so on, they usually talk about the parameter $T$ and an associated critical $T_c$ without taking any specific system into accound. There is usually some abstract free energy $F$, which pops out abstract quantities. Or in the Boltzman distribution and derived quantities, often something gets activated when the value of $kT$ catches up with some system specific energy value $E_0$. And it gets more complicated when it takes QFT like form. What _is_ the temperature in a general setting. How do I read this kind of things and what should I have in mind when reading these kinds of texts?


title: Trapping an electron

text: Imagine that one could theoretically trap a single electron in a small box, with walls that somehow prevent the electron from passing through and out of the box. Now, the box begins to move in on itself, gradually cornering and trapping the electron. The box continues trapping the electron until it perfectly encases it, disallowing the electron to move anywhere. I know that this scenario is seemingly impossible, but intriguing to think about. It has raised two questions in my mind:   1. Would we then be able to know with exact certainty the electrons velocity and location, thereby negating the Uncertainty Principle?   2. What would happen if the box kept on caving in even after perfectly encasing the electron? Could you actually crush an electron?


title: Role of the canonical ensemble and electric charge in AdS/CFT

text: If we consider a charged black hole in AdS spacetime, we can either do thermodynamics in the grand canonical or the canonical ensemble. In the former, we fix the electrostatic potential $\Phi=A_t(r=\infty)$ at the boundary of the bulk such that $\left<Q\right>=-\frac{1}{\beta}\left(\frac{\partial S}{\partial\beta}\right)_{\beta}$, where $S$ is the Euclidean action. In the latter, we fix the charge $Q$ of the black hole and we do not consider $\Phi$ at all. The phase diagram of the black hole is highly dependent on the choice of ensemble, see for example this paper by Chamblin, Emparan, Johnson and Myers. One could therefore expect that this choice also has an influence on the CFT side. In the AdS/CFT dictionary, the charged black hole gives a global $U(1)$ symmetry on the CFT side. Here $\Phi$ in the bulk corresponds to a chemical potential $\mu$ on the CFT, so that we usually consider the grand canonical ensemble when using the correspondence. My questions are as follows:   * Do we ever consider the _canonical_ ensemble in AdS/CFT?    * If so, what would $Q$ determine on the CFT side (in the same way $\Phi$ determines $\mu$)?    * If we work in the grand canonical ensemble, does $\left<Q\right>$ play any role on the CFT side, or do we only need $\Phi$? EDIT: For my last question: I just looked into holographic superconductors and it seems that from $\Phi$ one can derive both the chemical potential and the charge density of the CFT. The charge density seems to coincide with $\left<Q\right>$ in this case (up to some constant factors), but we don't need $Q$ to calculate it as we can derive it from an asymptotic expansion of $\Phi$. Specifically see page 8 of this paper by Horowitz. I don't think this answers my first 2 questions though.


title: Bullet hits block. Show that linear momentum is not conserved

text: > "..." First, we will familiarize us with a collision between a homogeneous > beam of length, $L$, mass $M$ attached with a frictionless hinge in point > $O$ at one of the endpoints of the beam. The moment of inertia of the beam > for rotations around the point $O$ in the direction given by the hinge is > $I$. The beam is macroscopic and aﬀected by gravity. The acceleration of > gravity is $g$ in the negative $y$-direction. Initially, the beam hangs > straight down in the negative $y$-direction, as illustrated in ﬁgure 0.12. A > small bullet of mass $m$ is ﬁred along the $x$-axis, hitting the beam at the > bottom. The bullet has an initial velocity $v_0$ and remains lodged in the > beam after the collision. ![enter image description here](http://i.stack.imgur.com/o9P7V.png) **One of the exercises is:** Show that the linear momentum of the system is generally not conserved during the collision. In what special case would the linear momentum of the system be conserved? As $\vec{G}$ is the only external force acting on the system, and it is perpendicular to both the linear velocity $v$ and the angular velocity $\omega$ during the collision, $\vec{G}$ surely does no work. I'm able to show that $p_o \neq p_1$: $$ m v_0 = (M + m) \omega r$$ $r = \frac{(1/2M+m)L}{M + m}$ , $\omega = \frac{mv_0 - constant*(1-\cos{\theta_2})}{(1/3M+m)L}$ found in earlier exercises $$ \Rightarrow m v_0 \neq \frac{2}{3}m v_0$$


title: What is the precise definition of state of a quantum system?

text: On studying quantum states of the hydrogen atom, I came with a question: what is the precise definition of a quantum state? Particularly, when solving the Schroedinger eq for the H using separation of variables, the quantum numbers and wavefunctions for the states are clear. However, mathematically speaking, any combination (linear) of wavefuntions is still a solution for the H atom; is this solution a state? Moreover, what are the quantum numbeers of this solution? It blows my mind, since this solution has 6 quantum numbers, instead of 3. More generally, my question is: is there a precise definition of state of system?


title: Derivation of matrix element

text: I have tried to understand paragraph 10.7 (Kallen-Lehmann Representation) in Weinberg's Quantum theory of fields (vol.1). He calculated matrix element $$\langle0|\Phi(0)|p\rangle =(2\pi)^{-3/2}\left(2\sqrt{p^{2}+m^{2}}\right)^{-1/2}N.\tag{formula 10.7.19}$$ $N$ is constant. I can't understand how it was obtained. I don't even undestand why it depends on $p$. If we consider the Lorenz invariance of vacuum and operator $\Phi(0)$: $$U_{\Lambda}|0\rangle=|0\rangle\ \ \text{ and } \ \ U_{\Lambda}\Phi(0)U_{\Lambda}^{-1}=\Phi(0)$$ then we have: $$\langle0|\Phi(0)|p\rangle=\langle0|U_{\Lambda}^{\dagger}\Phi(0)|p\rangle=\langle0|U_{\Lambda}^{-1}\Phi(0)|p\rangle=\langle0|\Phi(0)U_{\Lambda}^{-1}|p\rangle=\langle0|\Phi(0)|U_{\Lambda}^{-1}p\rangle,$$ so $$\langle0|\Phi(0)(p)=\langle0|\Phi(0)(U_{\Lambda}^{-1}p).$$ So my first guess was that this matrix element does not depend on $p$ and equals to constant. Can you help me?


title: Is there a soft Goldstino theorem?

text: For ordinary spontaneously broken symmetries, you can demonstrate relations between S-matrix elements with a soft goldstone emission and another S-matrix element without the emission. If I break SUSY I get Nambu-Goldstone fermions. Does there exist a relation between matrix elements with a soft goldstino emission and a matrix element without the emission? Basically, what is the complication because the soft particle is a fermion? References are also useful.


title: Difference between the two equations for acceleration

text: I came upon this while studying S.H.M. Well,is there a difference between writing $$a=\frac{dv}{dt}\;$$ and $$a=v\frac{dv}{dx}\;$$ do they differ on the basis of one being a vector and the other being a scalar equation?Please explain.


are sub-atomic particles really particles or mere concepts in our minds? do they exist independently of human thought? > In the tenth century, Ibn al-Haytham initiated the view that light proceeds > from a source, enters the eye, and is perceived. This picture is incorrect > but is still what most people think occurs, including, unless pressed, most > physicists. To come to terms with the Universe, we must abandon such views. > The world is quantum mechanical: we must learn to perceive it as such. One > benefit of switching humanity to a correct perception of the world is the > resulting joy of discovering the mental nature of the Universe. We have no > idea what this mental nature implies, but — the great thing is — it is true. > Beyond the acquisition of this perception, physics can no longer help. You > may descend into solipsism, expand to deism, or something else if you can > justify it — just don’t ask physics for help. There is another benefit of > seeing the world as quantum mechanical: someone who has learned to accept > that nothing exists but observations is far ahead of peers who stumble > through physics hoping to find out ‘what things are’. If we can ‘pull a > Galileo,’ and get people believing the truth, they will find physics a > breeze. The Universe is immaterial — mental and spiritual. Live, and enjoy. ■ Richard Conn Henry is a Professor in the Henry A. Rowland Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA


title: Corrections and Normalization for Power Spectrum Calculation

text: So I'm hoping I can get some help. I have a 2d image and need to get the 1d power spectrum. I know the basic steps: take fft, take fft^2 to get power, then take average power in radial bins to get 1d power- but it's in the details that I'm unsure. The image has been smoothed/convolved with a Gaussian beam. I have also applied a taper in the image plane that serves the purpose of keeping the noise constant across the image and makes it so that the edges fall to zero. So my issues/concerns are: 1) correcting for the beam smoothing. I know I need to correct for this and that can be done by dividing P_image by P_beam, but do I do the division with the 2d power spectras or the 1d? And should there be any kind of normalization?? Like sum(P_beam)=1 or something?? Just doing the division does not conserve the amplitude of P_image and in not sure if that's ok or if I should do something like P_corrected/sum(P_corrected)*sum(P_image) ?? 2) correcting for the taper. I'm not sure how to do this as the taper was applied as multiplication in the image plane which is convolution in Fourier, so how would I deconvolve from the power? Or do I need to? And once all corrections are done (or even if none were needed) do people apply any normalizations to the power spectrum, as I am interested not just in the shape of the curve but the amplitude as well?


title: Hawking radiation and entropy

text: What is your opinion of the hawking radiation mechanism, does that actually lower the entropy of the black hole?


title: Deriving the relativistic Larmor equation

text: I have derived the Larmor equation as $$P = \frac{q^2}{6\pi \epsilon_0 c^3} |\ddot{r}|^2.$$ How do I make this relativistic? Apparently, I have to consider the acceleration parallel and perpendicular to the velocity of the accelerating charge like: $$P = \frac{q^2}{6\pi \epsilon_0 c^3} (|\ddot{r}'_{\perp}|^2 + |\ddot{r}'_{\parallel}|^2).$$ But how does this lead to the answer: $$P = \frac{q^2}{6\pi \epsilon_0 c^3} \gamma^4(\gamma^2|\ddot{r}_{\perp}|^2 + |\ddot{r}_{\parallel}|^2).$$ I understand that parallel to the velocity, the will be relativistic effects, so where does this extra $\gamma^4$ come from?


title: A variable mass system (cart of sand with a hole)

text: Imagine we have a frictionless semicircular skateboard ramp and we release a cart filled with sand from one edge of the ramp. The cart has a hole and it loses sand at a constant rate. My question is will the cart reach the other extreme edge of the ramp or will it stop midway and slide back down? I am confused because I am not too sure whether the potential energy will constantly decrease because of decrease of mass and so it should not reach the other end


title: Angles in cosmology

text: We have several definitions of distances in cosmology. Take angular diameter distance. $$d_A = a(t)\int \frac{dt}{a(t)}\;.$$ If an object has proper length $L$ then the angle $\theta$ it is seen at is $$\theta = \frac{L}{d_A}$$ This is similar to the way an angle $\theta$ and arc length $s$ are related in a circle with radius $r$ $$ \theta = \frac{s}{r}$$ On cosmological scales distances are so vast that the arc length $s$ is indistinguishable from the proper length $L$ of the star or galaxy observed. All this is fine. But when it comes to the cosmic microwave background, we look at temperature correlations between two directions $\vec{n_1}$ and $\vec{n_2}$ with an angle $\theta$ between them on the sky today. $$ C(\theta) = C(\vec{n_1}\cdot\vec{n_2})$$ These angles are not small necessarily. For the quadrupole there is $90$ degrees between the two directions. Subsequent Higher multipoles have smaller angles. With such wide angles we should no longer use $$\theta = \frac{L}{d_A}$$ but instead use $$tan(\theta) = \frac{L}{d_A}$$ Where am I going wrong with this line of thought?


title: Is a lens with a focal point that doesn't shift as sun shines from different angles possible?

text: My question: Is it possible to manufacture a lens whose focal point doesn't shift in any direction during the day when the sun shines from different angles? It is understood that the focused energy will reach a maximum only if the lens is in the ideal angle to the sun, but I need to build a system that always collects the rays in one point, even if the angle is not perfect. I can cope with a lower energy as long as at least some light is focused constantly at the same spot. I do NOT want to move the lens or the setup in any way. If it is possible: Can you please let me know what kind of lens would be needed for this? ![enter image description here](http://i.stack.imgur.com/xDdH4.jpg)


title: What is the gamma five matrix $\gamma_5$?

text: This Wikipedia page explains that for each of the four main gamma matrices $\gamma^{\mu}$, you can find the covariant matrices $\gamma_{\mu}$ with the equation $\gamma_{\mu} = \eta_{\mu\nu}\gamma^{\mu}$. But that formula doesn't make any sense for $\gamma^5$ because $\eta_{\mu\nu}$ does not have that many indices. So what is $\gamma_5$?


title: Consistent application of Microscopic evolution with the definition of Macrostates?

text: How does one use the concept of microstate evolution and definition of macrostate in a consistent manner. As my understanding goes the thermodynamic concepts are defined in equilibrium. How does one apply these concepts in situations out of equilibrium. I am looking for a simple example involving thermodynamics out of equilibrium. Such as 2 system previously in equilibrium with Temperatures T1 and T2, are coupled. Given the details microscopic coupling, initial statistical distribution of micro-states, what can one say about the evolution and definition Temperature in such a system. I know this is a rather broad subject, and any pointers to the existing body of knowledge will be much appreciated. It would be lovely if an example is discussed for illustration purposes.


title: Has relativity of simultaneity been directly observed?

text: I know that thought experiment about trains when a flash of light in the middle reaches the both end simultaneously for a passenger but different times for the bystander. So were there (non-thought) experiments that directly checked this?


title: Lattice Gauge and Spin Network

text: I see the similarity between the Lattice Gauge and Spin Network . (For example , the both theories depict the node part as quantum (the latter is explained as spin)) Are there any other mathematical,Physical similarity between the two theories ?