title: Why is th $\hat{r}$ component zero in this integral?

text: I'm trying to evaluate the magnetic field by calculating the Coloumb integral $\overrightarrow{A}$, and later I will take: $$\overrightarrow{B}=\nabla \times \overrightarrow{A}$$ However, in the middle of everything, I get to (cylindrical coordinates): $$\overrightarrow{A}=\frac{\mu_oI}{4\pi} \oint_{C} \frac{a\hat{\phi'}}{[r²+(z-b)^2-2ar\cos(\phi-\phi')]^{1/2}} d\phi'$$ I should show that the only component of this expression is $\hat{\phi}$, where: $$\hat{r'}=\hat{x}\cos{\phi'}+\hat{y}\sin{\phi}$$ $$\hat{\phi'}=-\hat{x}\sin{\phi'}+\hat{y}\cos{\phi'}$$ $$\hat{r}=\hat{x}\cos{\phi}+\hat{y}\sin{\phi}$$ $$\hat{\phi}=-\hat{x}\sin{\phi}+\hat{y}\cos{\phi}$$ and that it's leading to only this integral: $$\overrightarrow{A}=\frac{\mu_oI}{4\pi} \oint_{C} \frac{a\cos{\left(\phi-\phi'\right)}\hat{\phi}}{[r²+(z-b)^2-2ar\cos(\phi-\phi')]^{1/2}} d\phi'$$ But when I rewrite how $\hat{\phi'}$ relates to $\hat{\phi}$ and $\hat{r}$, it's not so obvious that the $\hat{r}$ component disappears. The writer refers to symmetry, but I can't still figure this out. So I basically have this integral instead because I can't show how the $\hat{r}$ component is zero: $$\overrightarrow{A}=\frac{\mu_oI}{4\pi} \oint_{C} \frac{a\sin{\left(\phi-\phi'\right)}\hat{r}+a\cos{\left(\phi-\phi'\right)}\hat{\phi}}{[r²+(z-b)^2-2ar\cos(\phi-\phi')]^{1/2}} d\phi'$$ Any ideas? I've been stuck at this for a while, and I've tried to show it in Cartesian coordinates without no luck.


title: Why is particle number conserved, and what are the bounds on non-conservation?

text: Think of a modified Mott experiment: You place a single particle in the center of an empty perfect detector. The particle is described by a wave function, which will spread outwards and interact at exactly one point P. But if physics is local, how does nature around a different point Q know the particle already interacted at P? **What keeps nature from messing up, and "double-interacting" the particle, or forgetting to let the particle interact?** Historically, people came up with various non-satisfying solutions:   * Hidden variables, the particle takes a classical path after all   * Non-locality, P and Q communicate (in a way that doesn't allow signaling)   * Particle number is just conserved. Nature does magical bookkeeping. ("shut up and calculate") Of course, looking at the math, this is not so mysterious after all! One- particle states belong to a different Hilbert space than multi-particle states (which live in a Fock space). There is no way one state could evolve into the other, so no way for nature to mix up. The way QM attaches a "global variable" named "particle number" to the wavefunction is by putting different $n$-particle wavefunctions in different buckets. But this is still not very satisfying. It is merely a construction that ensures the right outcome. It hard codes particle number conservation, rather than explaining it. This makes me wonder if particle number conservation is really so exact. Small violations might point to an underlying mechanism. Has it ever been tested? **Are there experimental limits on particle number non-conservation?** Like where you put one photon in a box, and in rare cases five, or none, come out. * * * Clarification: I'm of course well aware that particle number is not conserved absolutely. You can have one photon in the initial state, and multiple of lower energy in the final state. You can have particles decay, and you can have them convert. And then there are intermediate, virtual particles. Momentum and energy between initial and final state are always 100% conserved. I'm not talking about any of this (although some of it might be relevant for the answer). What I'm asking about is this: In the Born interpretation of the wave function, the amplitude squared is the probability to detect a given particle somewhere. E.g. it'll be with 40% probability in a certain region, and with 60% outside of the region. _But the probability is zero to detect particles in both disjunct regions._ **Has someone tested if that is really the case?** Prepare a 1 GeV proton, put it somewhere at rest ($x$ and $p$ small, macroscopically). Look a bit later again. On average, there will still be one 1 GeV proton there. But does it happen sometimes that you'll find two, or none? Do we know for sure that the wave function collapse (nature's bookkeeping, particle conservation, or whatever you want to call it) always works as thought? Can one give a bound on how large deviations can maximally be?


title: Diffussivity of gases, molecular weight and Lewis number

text: I am trying to understand the following figure of laminar premixed flames: ![enter image description here](http://i.stack.imgur.com/hCKz4.png) The Lewis number is defined as $Le = \frac{\alpha_{mix}}{D_{fuel}}$ where $\alpha_{mix}$ is the thermal diffusivity of the mixture and $D_{fuel}$ is the mass diffusivity of the fuel. The propane fuel has larger molecular weight than air and methane has lower molecular weight compared to air. It is not clear to me why mass diffusivity increases as the flame gets richer in the case for propane (Lewis number drops) but the converse is true for methane. I know it has to do with the difference in molecular weight, but why does diffusivity increase for propane if you introduce more propane into the system but decrease in the case of methane? I am therefore not sure why Le decreases in case of propane but increases in the case of methane as the mixture gets richer. Maybe the thermal diffusivity is changing?


At CERN - What do you call the moment (event) particles crash together in the particle accelerator? At CERN they crash different particles together and measure what comes out. What is the name of the event (moment) the particles collide in the tunnel? Is there a name for the event itself or the crash itself?


title: Implicit Postulate of Quantum Mechanics

text: Consider the following quantum system: a particle in a one dimensional box (= infinite potential well). The energy eigenstates wave functions all vanish outside the box. But the position eigenstates wave functions don't all vanish outside the box. Each one of them is a delta function at a specific location, and some of these locations are outside the box. So it seems that there is no overlap between certain position eigenstates and all energy eigenstates. So the energy eigenstates don't span the whole Hilbert space! And these position states have zero probability for any energy outcome in a measurement! Now, I know that when speaking about an infinite potential well, it is assumed the particle cannot be outside the well. But I don't see any reason to assume this from the postulates of quantum mechanics. Is there an implicit additional postulates that says: "The Hilbert space of the system is spanned by the Hamiltonian operator's eigenstates (and not other operators, such as position)"? Or is the infinite potential well just an ill defined system because it contains infinities (just like free particle...)?


title: Physical structures that trap information

text: I labeled this question "soft" because it might not make any sense. Anyway, what I'm wondering is if there's a notion in physics that deals with the ability of matter to trap information. For instance, if I get in the way of a red light and see it, then my eye and my brain have taken the information about the wavelength of that light and physically persisted it. This seems like it might be a fundamental characteristic of structures that lead to life as we know it. Is there any current branch of physics that gets into a topic like this or am I getting into philosophy here?


title: Pressure experienced due to magnetic force?

text: > A current $I$ flows in a long thin walled cylinder(parallel to the axis) of > radius $R$. What pressure do the walls experience? This is the 263rd problem in Section III from the book 'Problems In General Physics by IE Irodov'. * * * **My attempt:** Consider a thin segment of the cylinder parallel to the axis of thickness $Rd\theta$(i.e. subtends $d\theta$ at the center). As the current is uniform, the current in this segment will be $$dI = I\frac{d\theta}{2\pi}$$ The magnetic field $B$ near this segment will be $$B=\frac{\mu_0 I}{2\pi R}$$ Force on a length $l$ of this segment will be $$F=BIl=\frac{\mu_0 I}{2\pi R} I\frac{d\theta}{2\pi} l=\frac{\mu_0I^2ld\theta}{4\pi^2R}$$ Pressure will be $$P=\frac{F}{A}=\frac F{lRd\theta}=\frac{\mu_0I^2}{4\pi^2R^2}$$ * * * The answer given in the book is $$\frac{\mu_0I^2}{8\pi^2R^2}$$ There are two problems after this also asking to find the pressure due to magnetic forces, in which my answer differs from the actual answer by a factor of $\frac 12$, just like above. Is there something I am missing in my understanding of pressure?


title: Direct collision on smooth surface

text: Hello I have a problem on my hands an I cannot figure out the answer. Two spheres of equal mass on on a smooth surface. sphere 1 is moving at $2v \ m/s$. Sphere 2 is stationary and 1m from a bollard. In this problem $e=e$ I have found that the speeds after collision 1 are $u(1-e)$ and $u(1+e)$. I need to prove the distance between both spheres is $\frac{2e}{1+e}$ when sphere 2 hits the bollard. What I have done is:$$\frac{1}{u(1+e)} = t$$$$(\frac{1}{u(1+e)})(u(1-e))$$ But that leads to nothing, what is the correct way to solve this problem. Thanks


title: Have general relativistic effects of all of the components of the stress-energy tensor been measured?

text: The stress-energy tensor is: ![Stress energy tensor, image from wikipedia](http://i.stack.imgur.com/qAzuU.png) Have general relativisic effects of all of the components of the stress-energy tensor been measured? I've heard that the accelerating expansion of the universe is due to negative pressure, but I don't know which terms are involved in the precession of mercury's orbit or frame dragging.


title: In QFT, how can it be shown that the field out, ${\phi_{out}}$, is a free field if the field in, ${\phi_{in}}$, is a free field?

text: In the Dirac picture of QFT interacting fields, if the field in, ${\phi_{in}}$, is a free field, then I know that the field out, ${\phi_{out}}$ should also be a free field. How can this be shown? Let's assume that the in and out fields obey their respective Heisenberg equations of motion: ${ i\dot{\phi}_{in}=\left[H_{0}\left(\phi_{in},\pi_{in}\right),\phi_{in}\right]\textrm{;}\\\ i\dot{\phi}_{out}=\left[H_{0}\left(\phi_{out},\pi_{out}\right),\phi_{out}\right]\textrm{.} }$


title: What is the correct Hamiltonian for a system of coupled quantum oscillators?

text: The Hamiltonian (see Eqn. 1 in Appendix 2 of this paper) for a system of coupled quantum oscillators is given as $$H=\frac{1}{2}∑_{i}p^{2}_{i}+\frac{1}{2}∑_{j,k}A_{jk}q_{i}q_{k}$$ Yet, in my QM course, the Hamiltonian for such a system was given as $$H=\frac{1}{2m}∑_{i}p^{2}_{i}+\frac{k}{2}∑_{i}x^{2}_{i}+\frac{K}{2}∑_{i}(x_{i}-x_{i+1})^{2}$$ where the third term represents the coupling between the oscillators. Why aren't these equations equivalent?


title: Intuitive meaning of factor 2 in formula of vertical throw max height $h=v^2/2g$

text: This is a question about a simple thing. The simplified expression for maximum height in vertical throw is $h=\frac{v^2}{2g}$ , could anyone explain **intuitively** (analogies are welcome) why there is a factor 2 on the denominator? And I mean intuitively, so not by transforming formulas etc, but by saying what that constant actually represents in this model.


title: Heat transfer between two surfaces

text: Suppose I have surface A in contact with surface B, if I apply Fourier's law of heat transfer, which $K$ should I use, $K_a$ or $K_b$? Essentially asking whether the same block of material heats faster in 300 degree water or 300 degree air or the same.


title: How to derive the Time Dilation formula given:

text: Since I do not have 10 reputation I will fit this all in one image: ![enter image description here](http://i.stack.imgur.com/XYbVO.png) If you have any questions please dont hesitate to ask. **EDIT:** http://img16.imageshack.us/img16/8604/myquestion2.png


title: why does a mirror show what is in front of it?

text: the only answer I can think of is that light is reflected from the objects in front of the mirror (visible color) and then reflects again off of the mirror to our eye, but im not quite satified with this answer


title: Vertical velocity to reach a certain point given a flat horizontal velocity?

text: Suppose I have two points such as $a(4,6,9)$ and $b(32,5,12)$. If I have a flat velocity pointing $a$ to $b (b - a)$ which has an arbitrarily defined magnitude. Given gravity $g$,how can I calculate the horizontal upwards velocity required to ensure that an object can be propelled from point $a$ and land on point $b$?


title: Entangled particles

text: So we have two particles (A and B) that are entangled. From what I understand, entanglement isn't destroyed, it is only obscured by subsequent interactions with the environment. Particle A goes zooming off into outer space. 10 years later, Particle B becomes incorporated into my brain. The day after that, an alien scientist measures the entangled property on Particle A. This will have some immediate non-local effect on Particle B won't it? And since B's state has been altered (in some sense), and it is part of my brain, then my brain state has been altered as well, hasn't it? Maybe only a tiny amount, obscured by the many environmental interactions that the two particles have been subjected to since the initial entanglement, but in a way that is real and at least conceivably significant. And if that is true, then to the extent that mental states supervene on brain states, my mental state would also have been altered by non-local effects. Or is that wrong? Thanks!


title: How can I calculate Vapor Pressure Deficit from Temperature and Relative Humidity?

text: I have a series of measurements of temperature and relative humidity (RH), together with mosquito capture data. Because mosquitoes are sensitive to desiccation, it's reasonable that RH may be useful in predicting their activity. But I recently heard of a value called saturation deficit or vapor pressure deficit which has been well correlated with evapotranspiration in plant leaves. Is it accurate to say saturation deficit is the "perceived dryness" or "drying power" of the air? Can saturation deficit be calculated directly from temperature and RH? Is it dependent on elevation, air pressure or anything else (all measurements are <600m above sea level). How do we calculate it? I have found two methods online which give wildly different results.


title: Other possible theories (other than string theory) which are generalizations of the standard model with incorporation of gravity

text: The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena).


title: How does throttling effect the upstream flow?

text: Let's assume we have a simple duct with a fan (compressor) situated somewhere along it. The fan runs at the same speed always. At the end of the duct, downstream of the fan, there is a throttle valve. The throttle valve is simply a plate facing normal to the flow in the duct, and can move horizontally (towards and away from the duct). Effectively the plate can completely block the duct outlet, giving the ducted flow no where to exit, or it can be placed sufficiently far away such that the flow "exhausts" to the atmosphere freely. My question is now: **Why does throttling (moving the plate closer to the duct exit) reduce the mass flow rate of the flow in the duct?** Will the same result be observed in a potential (irrotational) flow? I understand that the throttling process is isenthalpic, which results in a loss of stagnation pressure across it, if that helps from the start. Edit: I would also like to understand the causality of the process: does the throttling cause a reduced mass flow which means the fan will now achieve a higher head (pressure increase) at the same speed or does the throttling cause an increased pressure which reduces the mass flow?


title: Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

text: Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum mechanics. $$ \langle x|p|y \rangle = -i \delta'(x-y). \tag{1} $$ The first one is from the canonical commutator $$[x,p]=i \tag{2} $$ As shown in Dirac's principles of quantum mechanics 4th edition section 22, actually there is an ambigulity in this procedure. $$ \langle x|p|y \rangle = -i \delta'(x-y) + F' \delta (x-y) \tag{3} $$ where $F':=dF/dx$ is a derivative of a general function $F(x)$ (I slightly modify the equation in Dirac's book). Eq. (3) satisfies the commutator (1), but implies arbitary value of expectation of momentum. As noticed by Dirac, this ambigulity can be removed from a local phase factor $\psi \rightarrow e^{-iF(x)} \psi$. The second one is regarding momentum operator is the generator of translation, as given in Sakurai's modern quantum mechanics 1st edition p54, > $$ ( 1- \frac{ i p \Delta x'}{\hbar} ) | \alpha \rangle = \int dx' > \mathcal{F} (\Delta x') | x ' \rangle \langle x | \alpha \rangle $$ $$ = > \int dx' | x' + \Delta x' \rangle \langle x' | \alpha \rangle $$ $$ = \int > dx' | x' \rangle \langle x' - \Delta x' | \alpha \rangle $$ $$ = \int dx' | > x' \rangle \left( \langle x'| \alpha \rangle - \Delta x' \frac{ > \partial}{\partial x'} \langle x' | \alpha \rangle \right). \tag{1.7.15} $$ Comparision both sides yields > $$ p | \alpha \rangle = \int dx' | x' \rangle \left( - i \frac{ > \partial}{\partial x'} \langle x'| \alpha \rangle \right) \tag{1.7.16} $$ $$ > \langle x' | p | \alpha \rangle = - i \frac{ \partial }{ \partial x'} > \langle x'| \alpha \rangle \tag{1.7.17} $$ It seems that the arbitrariness as Eq. (3) is hidden in the second approach. My questions are: (i) Did Dirac already discover the origin of gauge invariance? If we do not consider from the commutator, I may say, there is no gauge invariance. A local phase factor would modify the expectation value of momentum, therefore we can only have global phase factor corresponds to the same state. However, since local phase factor comes from commutator, it has to be a redundency. (ii) Is the first method, from commutator, more fundamental than translation generator? Since we can find gauge invariance from the first method.


title: $\mathrm{d} \Omega_{CM}$ for a $1\rightarrow 2$ particle decay?

text: The differential solid angle is described in e.g. Srednicki's QFT text but only for the case of scattering. Because in the case of scattering it's defined with respect to the incoming three-momentum ${\bf k_1}$ and it's scattered three-momentum ${\bf k_1'}$. This definition would not make much sense for a particle decay (I can't at least see how it would easily carry over). What about the case of **particle decay**? What is $$\mathrm{d} \Omega_{CM}$$ in the case of particle decay? Any good references where it is properly defined?


title: Would the temperature of a gas change when accelerated in a train?

text: I was thinking about a situation where some gas is enclosed inside a container and kept in a train at rest. The train accelerates, gains a maximum speed and then suddenly stops. Would the temperature of the gas change during this process? I know that temperature is directly proportional to average kinetic energy of gas molecules but would the train during this process change the kinetic energy of the gas molecules inside the closed container?


title: Electric field inside a metal nanoparticle?

text: In localized surface plasmon resonance, the conduction electrons of a metal nanoparticle oscillate in tune with the frequency of the incident electromagnetic wave . The effect is dominated by electrons present on the surface of the particle as it is a nanostructure and surface atoms constitute a major portion of total number of atoms. As a dipole mode is created , electric field strength nearby the particle on outside is increased but what happens inside the volume of the nanoparticle ? For a small metallic nanoparticle , let us say 20 nm , the field (considereing a 500nmEM wave)totally penetrates the particle and the electrons of the whole particle (volume + surface ) will resonate at plasma frequency, then there will be a field gradient generated inside the particle too but inside metals , a field cannot exist. I am a little confused in this aspect.


title: Questions on electric field formed in conductor and the magnetice field it moves through

text: When a conductor (e.g. a wire) is moving through a uniform magnetic field, it will create a electric field in the wire as the magnetic field exerts a force on the moving conduction electrons in the wire and create a potential difference. ![the wire is so thick!](http://i.stack.imgur.com/2kBWr.gif) And here is my questions:   1. (If we can get the magnetic field to exert a super strong force on the electrons) Is it ever possible for the electrons in the wire to get separated from the wire like what happens when high energy photon hits an atom and knock off its electrons?   2. Since $F_e$ and $F_b$ always act in the opposite direction, why don't they cancel off one another (so the net force is 0) when the charges are in the electromagnetic equilibrium ($qE = qvB$) ?   3. What's so significant about the this equilibrium ($qE = qvB$)? The formula for motional emf $emf = vBl$ is derived from the fact that $E = vb$ and that was because the charges are in the electromagnetic equilibrium. But wouldn't this formula $emf = vBl$ be inaccurate if we have a motional emf that are caused by charges not in the electromagnetic equilibrium ($qE = qvB$)? Why when there is a motional emf we assume that the chargers are in this equilibrium? I'm still learning secondary school physics and I'm very new to this topic and so please explain it in laymen terms if possible. Thank you so much.


title: Determine whether the light will reach certain points or not

text: I've got an interesting question which goes like that: suppose we have a long cylindrical rod made of a transparent material (see illustration below). A point source $O$ which sits on rod's axis emits light in all directions. The light is partially reflected and partially refracted (i.e. goes to the air). The question is - which points will the light rays reach? (1) A only, (2) A and B only (3) All the four points (4) Not enough information ![enter image description here](http://i.stack.imgur.com/ARRM0.jpg) According to the book the right answer is (3), but I'm not so sure that this is correct - one the one hand, the light is emitted in all possible directions but on the other hand, my friend thinks there could be a total internal reflection at some point. In both cases there's no strict proof for either answer. By the way, there's no information about positions, indices of refractions and so on.


title: Radiative cooling time for a black body

text: Okay, so this is confusing me a bit. How can I calculate the time it takes to cool a perfect black body from an initial temperature $T$ down to equilibrium temperature (say, 3 K for space)? I know that $$\frac{P}{A}=\sigma T^4,$$ but I'm stuck at getting $t$ out of there.


title: Any closed system?

text: Is there any physical example of a real closed system? I am aware that the whole universe can be considered as a closed system, but I am looking for a smaller example.


title: Light in fibre optic cables

text: Imagine a fibre optic cable stretching from London to New York (5571 km) carrying a data signal This data signal is split into two parts, red and blue light Both signals start travelling down the wire at the same time. Would they reach the other end at the same time? If not, what would be the distance and time delay between the two signals at the other end? (what is the equation to work this out?)


title: speed of light observation

text: I was on a ship at night travelling parallel to the shore watching a lighthouse beam hitting the concrete sea wall. As the beam hit the wall at ninety degrees and continued to rotate I saw what appeared to be the light splitting into its spectrum and streaking away at incredible speed. Was I seeing the actual speed of light as the beam's direction changed?


title: Derivation of the Thermal Noise Spectrum

text: The thermal noise spectrum is given by: $$\mathcal{S}(f) = \frac{\hbar f}{2(e^{\frac{\hbar f}{kT}} - 1)}$$ This equation seems really similar to the Dirac-Fermi distribution but where does it come from?


title: Recent missed opportunities à la Freeman Dyson

text: There is an excellent paper by Freeman Dyson from 1972 (here) and therein the author cites old talks by Hilbert (here) and Minkowski (chapter 2 here) speaking about similar topics, namely how opportunities for discoveries could be better if mathematicians and physicist worked more closely. I wanted to ask if there are recent publications (roughly > 2000) of that type?


title: How to calculate pressure loss due to water leakage from a hole in a pressurized unit

text: I am trying to develop a method for calculting the rate of pressure loss from a small hole in a pressurized vessel full of water (small air pocket likely at top of vessel). I've found a formula for calculting the flow rate of a liquid through a small hole (link), but I don't know how to relate this to a rate of pressure decay as it is kind of a cyclical process. As water leaves the vessel, the water pressure would decrease, thus also decreasing the flow rate. This would probably happen until surface tension stopped the leak. My thought is that, assuming a rigid vessel, pressure would decrease as a function of the volume of water in the vessel, and the pressure and the flow rate as some sort of differential equation perhaps, but I'm not sure how to go about deriving that. The application in question is the hydrostatic testing of plate heat exchangers. When completing unbalanced tests, water is pumped into one side of the heat exchanger and held at this pressure for a period of time. My thinking is that given a leak, the pressure should drop quite quickly thus indicating a failure. Alternatively, the rate at which the pressure drops should allow us to back calculate the rate of water leakage and maybe even hole size. Any help or direction would be super appreciated. Thanks!


title: Is the Standard Model consistent (UV complete)?

text: This is a question about the self-consistency of the Standard Model - which I believe is the same as asking whether it is UV complete - in other words, can it be used to predict experimental results at arbitrarily high (and low) energy scales? Note I am asking for a rigorously defensible statement _about_ the Standard Model, not a technical explanation of why said statement is true. (An aside - I understand that the term "Standard Model" can include the original version (massless neutrinos), or various extensions that allow massive neutrinos.) I understand that the Standard Model is certainly not correct at the Planck mass, and is not able to explain cosmological observations, so it is observationally falsified, but my question relates to internal self- consistency. This note by Rubakov gives an attempted answer, > Standard Model is a well-defined theory, in the sense that everything is > calculable, at least in principle, within this theory in terms of finite > number of parameters (some quantities are hard and even impossible to > calculate in practice because of strong coupling in the low-energy QCD). > With $m_H \lesssim200 GeV$ this theory can be extended up to Planck > energies. In fact the Higgs mass $m_H$ is indeed less than 200 GeV. The Wikipedia article equivocates: > The Standard Model is renormalizable and mathematically self-consistent(1) > > (1) In fact, there are mathematical issues regarding quantum field theories > still under debate (see e.g. Landau pole), but the predictions extracted > from the Standard Model by current methods are all self-consistent. For a > further discussion see e.g. R. Mann, chapter 25. What is the best current answer?


title: What would happen when a material with Inhomogeneously broadened profile interacts with a light beam?

text: I have something which I am sure that I understand it correctly. Would you please told me if I explain this correctly or not. Sorry in advance for my English. When a light beam interacts with an atoms of a material with an inhomogeneously broadened profile, an electron got excited and emits a photon. Because of the inhomogeneouly broadened profile, for every different frequency of the light beam, the material emits a photon with different linewidth. Do I say it correctly?


title: Electromagnetic black hole?

text: So I was thinking about something for the past while Consider a large spherical foam-ball with homogeneous density. Where a foam ball is defined as an object that can absorb matter with 0 friction (example: a gravitational well without an object inside). This is a purely theoretical construct. If the foam-ball has a radius R and a charge Q. What charge must the foam-ball have, such that there is well defined sphere or horizon such that any object with a negative charge (even if it is the slightest bit) must travel faster than C in order to escape the field of the ball. Aka what charge would turn this into a black-hole for all objects with opposite sign on their charge? Once again I assume the foam-ball is a single particle, that doesn't repel itself... its just a very large homogenous 'thing' with charge.


title: Transport theorem derivation question

text: I am trying to rigorously go through some fluid mechanics proofs and theorems. I am currently going through a proof related to the transport theorem and I am having trouble with a step. The steps in question are the following, first the variables are transformed according to: $$ \frac{d}{dt}\int_{W_t} \rho \mathbf{u} dV = \frac{d}{dt}\int_W (\rho \mathbf{u})(\phi (\mathbf{x},t),t)J(\mathbf{x},t)dV $$ where $\phi(\mathbf{x},t)$ is the trajectory function of the particle found at $\mathbf{x}$ at time $t=0$. $W$ is the volume of the fluid under consideration at $t=0$ and $W_t$ is that volume at time $t$. $J$ is the Jacobian determinant of the transformation, followed by differenciating under the integral sign (RHS) $$ \frac{\partial}{\partial t}(\rho \mathbf{u}) (\phi(\mathbf{x},t),t) = \left( \frac{D}{Dt} \rho \mathbf{u} \right) (\phi (\mathbf{x},t),t) $$ In the context of the first equation, why is the second equation true? When I try to prove it to myself I get: $$ \frac{D}{Dt}(\rho \mathbf{u}) = \frac{\partial}{\partial t}(\rho \mathbf{u}) + \mathbf{u}\cdot \nabla (\rho \mathbf{u}) $$ and I don't know how to eliminate the second term. I think it has something to do with the trajectory function $\phi$ but I am not familiar enough with this idea to know how to proceed.


title: Quantum Excitations

text: In the context of quantum field theory, is the schrodinger or dirac equation actually describing some sort of an actual wave in some field like light in EM field ? So all particles are actually waves in their respective fields ?


title: Do electromagnetic waves have endpoints?

text: When learning about electromagnetic waves at school we never talked about any endpoints as we did with standing waves, so I assumed that light has an endless length, but that doesn't make sense. So my question is, does it make sense for em waves to have specific length? If yes, how does it differ from one to another?


title: Magnetic field inside and outside cylinder with varying current density

text: I am reading through _Introduction to Electrodynamics_ by David J. Griffiths and came across the following problem: > A steady current $I$ flows down a long cylindrical wire of radius $a$. Find > the magnetic field, both inside and outside the wire if the current is > distributed in such a way that $J$ is proportional to $s$, the distance from > the axis. If $J$ is proportional to the distance from the axis $r$, then we have: $$\vec{J}(\vec{r})=kr\,\boldsymbol{\hat{z}}$$ We also have: $$\iint_{\Sigma} \vec{J}(\vec{r})\cdot\:\mathrm{d}\vec{A}=I$$ We therefore have: $$\int_{0}^{2\pi}\int_{0}^{a}kr^{2}\:\mathrm{d}r\:\mathrm{d}\theta=\frac{2\pi k a^{3}}{3}=I $$ And therefore we have: $$\vec{J}(\vec{r})=\frac{3Ir}{2\pi a^{3}}\,\boldsymbol{\hat{z}}$$ Using Ampére's law we have: $$2\pi r B = \mu_{0}I \implies \vec{B}(\vec{r})=\frac{\mu_{0} I}{2\pi r}\,\boldsymbol{\hat{\theta}}$$ If we take a Ampérian loop inside the cylinder, we have: \begin{align} 2\pi r B &= \mu_{0} \iint_{\Sigma}\vec{J}(\vec{r})\cdot\:\mathrm{d}\vec{A} \\\ &= \mu_{0}\int_{0}^{2\pi}\int_{0}^{r}\frac{3 I r'^{2}}{2\pi a^{3}}\:\mathrm{d}r'\:\mathrm{d}\theta \\\ &= \frac{\mu_{0} I r^{3}}{a^{3}} \end{align} And therefore we have: $$\vec{B}(\vec{r})=\begin{cases}\frac{\mu_{0} I r^{2}}{2\pi a^{3}} \,\boldsymbol{\hat{\theta}} & r < a \\\ \frac{\mu_{0} I}{2\pi r}\,\boldsymbol{\hat{\theta}} & r \geq a \end{cases}$$ Is this the correct magnitude and direction of the magnetic field?


title: How would you explain the cause of earths magnetism to an 8th grade student

text: What would happen if I freely suspend a bar magnet at north pole, will it become vertical? I am the 8th grade kid.What would happen if earth brakes into two pieces,will there be new north and south poles developed?


title: Is the Earth negatively or positively charged?

text: The Earth carries a negative electric charge of roughly 500 thousand Coulombs. Does that mean the Earth is negatively charged?


title: Where do these formulas for an object being pushed up an incline come from?

text: An object on an incline with an angle $\theta$ is being pushed at constant speed. Constant speed implies $a = 0$. Because $F=ma$ and the object must have mass greater than 0.   1. If I want to find the force of the push, and it is being pushed parallel to the incline then the formula to use is: $F=w\sin\theta$, where $w$ is the object's weight. **Why?**   2. If it is being pushed parallel to the floor the formula is $F\cos\theta=w\sin\theta$. **Why?** * * * I tried to work them out to find the logic behind the formulas, but maybe I am making too many assumptions. So can someone explain how to arrive at these formulas?   1. $$\sum F = w\sin\theta-F = ma = 0$$ This is because of $x^2+y^2=r^2$, so I used that to find F.   2. $$\begin{align} \sum F &= w\sin\theta + w\cos\theta+ F\sin\theta + F\cos\theta=0\\\ \sum F &= w\sin\theta + 0+ 0 + F\cos\theta=0 \end{align}$$ So because $a = 0$ I make my equations equal 0. Then I say that $w\cos\theta = 0$ because gravity is on the y axis, not on the x axis. Then I say $F\sin\theta = 0$ because the push is parallel to the ground.


title: Twin paradox - observers counter orbiting Earth

text: Imagine three observers - one (A) stationary on the surface of Earth (latitude 0 deg) and two others orbiting the planet in the same circular equatorial orbit just in the opposite direction. When the orbiting observers B and C meet each other just above the observer A all theisr clocks are synchronized. When their meet again what will their clock show? They move relative to each other, but experience equal acceleration and gravitational potential. Will the orbiting observers have their clock synchronized all the time? If the time intervals differ who measures greater time and why? Suppose Earth in non- rotating itself and with perfect spherical symmetry.


title: Why doesn't anomalous dispersion allow faster-than-light propagation?

text: It seems that the phase velocity of light could be greater than $c$, if $\sqrt{\epsilon \mu} < 1/c$, i.e. for anomalous dispersion.   1. Are there examples of such media? For diamagnetics it seems possible since $\mu < \mu_0$.   2. And what is the physical meaning of phase velocity in this case, besides being speed of wave surface? Why it cannot be seen as a physical entity that is traveling faster than light?


title: Why is radiation at the relativistic limit characterized by 1/gamma angle?

text: Trying to think of a reason as to why? Also, the factor that radiation is zero on axis? I haven't been able to resolve these two fundamental principles in my head :(


title: How do I know an initial speed of a thrown object using the max height

text: > The simulation being referred to is in box2d An object is thrown to the max height of $h$ with gravity of $g$, what is it initial speed?   I tried the following:   $v = v_0 - g t$   $0 = v_0 - g t$   $t = \frac{v_0}{g}$ $h = v_0 * t - \frac{1}{2}g * t^2$   $v_0 = \sqrt{2 * g * h}$ But putting it into physical simulation gives different max height, is the equation wrong or its a simulation artifact?


title: Cylindrical charge distribution

text: I'll just list the problem first and then explain what I need help with. > Consider a cylindrical charge distribution extending from $r = 0$ to $r = > 8.2 $cm of charge density $\rho = r/a_0$, where $a_0 = 21 cm · m^3/µC$. This > cylindrical charge distribution is surrounded by a dielectric shell of > dielectric constant $6.72$ whose inner radius is $17.7 cm$ and outer radius > is $25.9 cm$. I cannot seem to find a formula for finding the electric field at a given distance. Any suggestions are where to even start?


title: Proper Time with a non-zero Initial Velocity

text: I have a question regarding how to find the proper time for a body with an initial velocity to slow down to 0. For example, the equation I have been working with looks like: $$\int^\tau_0 d\tau=\int^t_0\frac{1}{\sqrt{1+\frac{(\alpha t+v)^2}{c^2}}}dt$$ Where $t$ is equal to the coordinate time for the rocket to slow down, $\alpha$ is the proper acceleration and $v$ is the intial velocity. Integrating gives: $$\tau=\frac{c}{a}arcsinh(\frac{\alpha t+v}{c})$$ I believe, however, that this is incorrect as the values it returns are far too small to be a legitimate value for proper time. I do believe that this is on the right track to deriving the correct value.


title: Understanding Well Defined States

text: I am self-studying from a text in QM. Well defined states are mentioned several times. By and large these are consistent and seem to be readily apparent:   * states of well defined energy are basis kets.   * if $Q$ is an observable, any arbitrary ket can be represented as a linear combination of states {$q_i$} in which values of $Q$ are well defined.   * whenever $[Q,H] = 0$, a state of well defined $Q$ evolves into another such state. So far so good. Then comes the "bombshell." The text later on says: > States of w-d energy are **unphysical** and **never occur in Nature**. They > are incapable of changing in any way, and hence it is impossible to get a > system into such a state. I would appreciate any explanations reconciling these two apparently contradictory characterizations of well defined states. And, in particular, how then to understand, e.g., a creation operator in the case of a harmonic oscillator in a stationary state. **EDIT** Hopefully there might be any answer regarding the last part of the question regarding harmonic oscillators. From the perspective of a naive beginner, it looks like the analysis is a representation of reality. As I think about this, one thing that comes to mind is that the creation and annihilation operators are not Hermetian, with my presumed inference that this undermines the "reality" of the model. But in view of the highlighted quote, it seems that it is but an exquisite mathematical exercise. To the extent this is accurate, then what is the relevance of studying it. I would hazard a guess that it might be a framework for studying the anharmonic oscillator.


title: Investigation of a pendulum's period, problem creating equation to sum the dynamic velocity

text: I am investigating the period of a pendulum swing. This is a simple harmonic pendulum and I am already aware of the common, but slightly inaccurate, $2\pi \sqrt{\frac{L}{G}}$ formula. My problem is that I cannot figure out how to find the total distance traveled by the bob since the angle between $F_{g}$ and the string is constantly changing as the bob falls and therefore the acceleration is not constant. **Question:** So, my question is, how do I calculate the distance when the acceleration is not constant? **Notes:**   * $F_{g}$ is the force of gravity pulling the bob downward.   * $\sin\theta$ is the tangential force of the pendulum and it represents the force applied to the bob causing a the swing. It is the projection of $F_{g}$ onto the string.   * $\theta$ is the angle between $F_{g}$ and the string   * The movement of this bob is being considered as falling being dropped from 0 degrees (with respect to the unit circle) and continuing until it reaches a distance of $\frac{r*\pi}{2}$ where $r$ is the length of the string and the radius of the circular path.   * The distance formula used is $d=v_{1}*t + \frac{1}{2}*a*t^{2}$ but since the initial velocity is 0 it is not being considered. I also believe this formula to be inaccurate since the acceleration is not constant. **Diagram:** ![My Considerations](http://i.stack.imgur.com/baO45.jpg)


title: Lorentz transformations and a time dilation don't return the same result

text: Lets deal with a problem like this: > Lets say we have a twin A in coordinate system $x'y'$ who does a **round** > trip to a star which is $12$ light years away and he is travelling with a > speed $u=0.6c$. Twin B stays on Earth with coordinate system $xy$. How many > years will pass for the twin B for a twin A to return. How many years will > it pass for the Twin B? * * * **In this case i ll set myself in a coordinate system $xy$** I calculate the time $\Delta t$ (this is easy because $u$ is a constant): $$u = \frac{\Delta x}{\Delta t} \longrightarrow \Delta t = \frac{\Delta x}{u} = \frac{2 \cdot 12~l.y.}{0.6c}=\frac{2\cdot 12\cdot c \cdot 1 y}{0.6c} = 40y$$ So for twin B it will pass 40 years. **According to time dilation every clock that is not at rest in the chosen system doesn't measure proper time**. In the curent chosen coordinate system $xy$ only clock of the observer B is at rest so we choose a proper time like this $\boxed{\Delta t \equiv \tau}$. From this it follows that: \begin{align} \Delta t' = \gamma \Delta t = 1.25 \cdot 40y = 50y \end{align} So the observer in coordinate system $xy$ would say that for his brother it has passed 50 years. Now i will try to calculate the same result using the Lorentz transformations for coordinate system $xy$: \begin{align} \Delta t' &= \gamma \left(\Delta t - \Delta x \frac{u}{c^2}\right)\\\ \Delta t' &= 1.25\left(40y - 2\cdot 12 \cdot c \cdot 1y \frac{0.6c}{c^2}\right)\\\ \Delta t' &= 1.25\left(40y - 2\cdot 12 \cdot 1y\cdot 0.6\right)\\\ \Delta t' &= 32y \end{align} **The result is not the same...** * * * The results would be the same if i chose the proper time differently $\boxed{\Delta t' \equiv \tau}$ from this it would then folow: $$\Delta t = \gamma \Delta t \longrightarrow \Delta t' = \frac{1}{\gamma} \Delta t = \frac{1}{1.25}40y = 32y$$ But i can't do this because the clock which measures time $\Delta t'$ is not at rest in the coordinate system $xy$. * * * **So we have a problem.** I realy don't know what i did wrong. Is my perception of a proper time wrong or is the Lorentz transformation wrong. What am i missing to fully understand this?


title: Polarization selection rules

text: So, I am just reviewing some of my physics stuff, and I have read something about polarization selection rules in inter-band-transitions in quantum wells. If I calculate them, I get a strength of 1/2 for heavy holes-transitions (HH) in x- and y-directions, and a strength of 1/6 for light holes-transitions (LH). My problem is now: What does this mean? Do the HH-transitions happen three times more than the LH-transitions, or are they the probability of each transition? But if they are the probability, they have to add up to 1, but they don't. So, where is my mistake?


title: Why does leaning back make a kicked ball go higher?

text: It's relatively common knowledge that when one kicks a football, if they lean back the ball usually goes higher, and if they lean forward slightly it goes at a more desirable height. Why does leaning back make the ball go higher? I'm guessing that it could be something to do with the centre of mass or angular momentum.


title: Light Rays that are Perfectly Parallel

text: I just heard this simple reasoning in a documentary film: > Light rays from distant stars are perfectly parallel. This is pretty interesting thought. In nature, it is hard to find something really perfect. For example, there are no isolated systems -- only quasi- isolated. But here it looks like indeed the light rays are perfectly parallel. **Maybe this can be confirmed by calculations?** \-- e.g. that angle between two light rays cannot be infinitely small (Planck constant will be probably involved?).


title: What's the lower limit for energy usage to desalinate water?

text: Consider a desalination process where you enter sea water and receive fresh water and brine (or maybe pure salt). How do I compute the least amount of energy per mass? I think this has something to do with the entropy of the different mixtures, mostly dependent on the salt concentration in the brine. Is this correct? How to calculate?


title: Lorentz transformations in Dirac equation

text: Let's denote a spinor $\xi$. If $(\theta ,\phi)$ are the parameters of a rotation and pure Lorentz transformation, then how $\xi$ could be written as $$\xi ~\rightarrow~ \exp\left(\ i \frac{\bf{\sigma}}{2}\cdot \theta +\frac{\bf{\sigma}}{2} \cdot \phi\right) \xi~?$$


title: Geometric interpretation of perturbation theory in quantum field theory

text: I am studying GR right now, and one interesting thing I learned about vectors is that they are defined to have the same properties as derivatives. With this in mind, can I make a differential geometric interpretation of ordinary perturbation theory used in quantum field theory in the following way:?   1. The set of all field configurations makes a (very complicated) manifold.   2. We do perturbation theory about a background field configuration corresponding to a point on this manifold.   3. The perturbations are like expanding functions out to first order -- can be viewed like tangent vectors Am I on some track to enlightenment, or is this a dead-end train of thought?


title: Would magnetic flux be necessary for analogous systems?

text: I learned about the analogousness between mechanical and electrical systems a few months back (with the help of Feynman). Yesterday, my professor was lecturing about this topic, when she told that electrical systems can be written in two ways - either by using the voltage source, or by using the current source (drawing the equivalent analogs, and writing differential equations). I've always thought current sources to be abstract constructs for the electrical engineers to solve problems in their boring way, node analysis and stuff, since it's voltage that drives the free electrons (hence, current). Anyways, my confusion was about something called "Force-Current" analogous system, where the situation has gone _crazy_. Here's my (rather ugly) drawing of her example. We were asked to draw the equivalent electrical analogs. (It's a convention that whenever we make use of current source, we analyze them using nodes) ![](http://i.stack.imgur.com/Y4DUu.jpg) I can understand the $F\to V$ system alright. Because, it makes use of the analog:   * $x\to q$   * $m\to L$   * $c\to R$   * $k\to 1/C$ But, it becomes hard to perceive, when current source $F\to I$ comes into play. Now, the analogs are _crazy_... ( **Note:** This analog wasn't discussed by Feynman)   * $x\to \phi$   * $m\to C$   * $c\to 1/R$   * $k\to 1/L$ Is it right, actually? I'm skeptical about this kind of mapping. In this case, $G=1/R$ is conductance (which is the thing analogous to the damping factor $c$). Would this mean that conductance is the actual _resistance_ here? And, more weirdo - capacitance is analogous to mass. * * * The DEs for this system are somewhat long. I'll choose an RLC circuit to express my point. In terms of $F\to V$, (using $x\to q$) $$V_R=R \frac{dq}{dt},\ \ V_L=L\frac{d^2 q}{dt^2},\ \ V_C=\frac{q}C$$ Now, using $F\to I$, (using $x\to \phi$) $$I_R=\frac{1}R\frac{d\phi}{dt},\ \ I_L=\frac{\phi}{L},\ \ I_C=C\frac{d^2 \phi}{dt^2}$$ The $x\to q$ was fine, because motion of charge is current, and that's the basis for all electrical systems. But, the $x\to \phi$ is inconceivable. The magnetic flux, how $\phi$ changes, induced emf, etc. are needed for inductors. Yeah, but how can they be applied to resistors or capacitors? With this framework for writing the differential equations, it seems as if magnetic flux is the clockwork behind the working of such a system. This horror made me think that the $F\to I$ analog (using $\phi$) is just another abstract mathematical construct for writing differential equations. Am I right? Is it reasonable at all?


title: How much energy (eV) is in a mosquito's wing beat?

text: We're all familiar with the annoying sound of a mosquito near our ear. I estimate the wing beat frequency to be in the 400 to 800Hz range (to accomodate the Bruce Willis and Danny de Vito of mosquitos :-) I was wondering if we can guesstimate the energy in a single wing beat (say, to within a factor of 10). I have no intuitive understanding about picojoules, which is why I think a value in eV might be more instructive. From this value, can we derive how many molecules of oxygen the mosquito's metabolism must use per second? If you make estimates for various parameters (like mosquito mass, ...), please also estimate an error.


title: Why is renormalization necessary in finite theories?

text: Michael Brown made the following comment here: > The modern understanding of renormalization (due to Kadanoff, Wilson and > others) is hardly controversial and has nothing really to do with > infinities. It is necessary even in completely finite theories, but the fact > that it fixes infinities in QFT is a bonus. Can anyone explain what this means?


title: Circular motion

text: > _A small block slides down from the top of a hemisphere of radius $r$. It is > assumed that there is no friction between the block and the hemisphere. At > what height $h$ will be the block lose contact with the surface of sphere._


title: Optimal slit width in Young's double slit experiment

text: I'm trying to do Young's double slit experiment at home. Note that I don't have a laser, only a torch. I could get a bulb or use a candle though, if it helps I built the slits by cutting into a black chart paper with a knife. I tried to build a setup by placing a single slit, double slits and a screen one after the other, and shine a torch through the single slit. I tried varying distance between single-slit/double-slit and double-slit/screen which did not help. **_I did not observe any interference pattern._** All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits. > I think that is precisely the problem, that the slits are pretty thick. > **What is the optimal slit width I should have to observe an interference > pattern? How do I build that with a chart paper?** I have seen videos of this experiment online, where people use pencil leads and _hair_ (!!), but they both use a laser. _I don't want to use a laser_ (just because I don't have one, and I'm probably too lazy to go and get one!). Note that you may suggest using any other materials that might be easily available at home.


When it comes to maths collaboration online, the mathematicians seem to be very well organised. A recent example is Polymath8 from Terence Tao's blog: > The main objectives of the polymath8 project, initiated back in June, were > to understand the recent breakthrough paper of Zhang establishing an > infinite number of prime gaps bounded by a fixed constant {H}, and then to > lower that value of {H} as much as possible. After a large number of > refinements, optimisations, and other modifications to Zhang’s method, we > have now lowered the value of {H} from the initial value of {70,000,000} > down to (provisionally) {4,680}, as well as to the slightly worse value of > {14,994} if one wishes to avoid any reliance on the deep theorems of Deligne > on the Weil conjectures. Googling "polymath project" returns results dedicated to math collaboration, whereas "polyphysics project" returns nothing of interest. Are there examples of online physics collaboration similar to polymath for mathematicians?


title: Why does music have a different tonality when heard from earphones a distance away?

text: Why when I have music on in earphones and I have them resting on a desk, the music sounds different, like in a different tonality/pitch to when I am actually using them?


title: Rigorous distinction between quasiparticles and collective excitations

text: I would like to hear your opinion on the question whether there is an accepted distinction between both concepts in condensed matter physics. I would tend to use the word quasiparticle for dressed fermionic particles and the term collective excitation in the case of bosonic excitation. But considering for example magnons in ferromagnets makes it very arbitrary kind of definition since magnons can be viewed (if I am not wrong) both as quasiparticles and also as collective excitations.


title: Can Pauli exclusion be described locally?

text: Is it possible, in principle, to define the exclusion principle in a "local" sense, as a property of the tangent space at a point, or a single fiber of a spin bundle? Or does it necessitate a global description, e.g. coordinate-free field operators? As a hypothetical example to capture the spirit of the question, could infinitesimal probability and spin currents interact in a manifestly exclusive way? I'm not holding my breath, because for all I know, the "locality" I allude to is ill-defined, up to interpretation and precluded by ten different no-go theorems. :)


title: Will negatively charged strangelets be produced by the LHC?

text: Witten (and earlier, Bodner) hypothesized that strange matter (up, down, strange quarks) should be more stable than "regular" nuclear matter(The strange matter hypothesis). That is that the typical hadrons in matter are metastable, and will eventually decay into strange matter because it can exist at lower energy configuations. There was some publicity about strangelets as a disaster scenario for the RHIC and the LHC, and a very interesting paper was commissioned by the Brookhaven National Laboratory to evaluate risks of various disasters possible in the RHIC: The authors are Busza, Jaffe, Sandweiss, and Wilczek. A conclusion is that positively charged and neutral strangelets would be no problem, and that negatively charged ones, which would be a huge problem because they would convert all the nuclear matter to strange matter, would be very unlikely based on cosmic ray evidence (ie lack of them) and other considerations. There are theories that strange matter occurs in the cores of neutron stars and may convert them completely resulting in denser, smaller stars than predicted for nuclear matter, but I am not sure. Do we expect the LHC to produce negatively charged strangelets? I think the answer to the question is NO, none will be found or produced, but the reasoning from cosmic rays is not definitive imo( I guess I should have said, "but asymptotically close to it"--G). The Jaffe paper above has merit I believe not just as a physics document, but also as public relations, and has historical value, so I wanted to draw attention to it.


title: Wicks Theorem and Gaussian Integrals

text: I am trying to complete A. Zee's book _QFT in a Nutshell_ and at page 15 he mentions Wick's theorem and Wick contractions. (apologies for the huge page- snip). Why does he mean by connecting the '$2n$ points' and how does the diagram of the $6x$'s prove wick's theorem? Also why is the final result for just $a^{-3}$ times the number of distinct Wick contractions? How does one derive this? ![enter image description here](http://i.stack.imgur.com/QgjUQ.png)


title: Is there a reaction force on the ball in a vertical circular motion?

text: You have a light string. At one end of the string there is a ball modelled as a particle. The string has negligible mass. The other end of the string is fixed at a point and the ball is undergoing circular motion. At the bottom, the centripetal force = mg - T. At the top, the centripetal force = T + mg. m = mass of ball g = gravitational field strength T = tension If an exam asks you: What is the reaction force at the bottom? What is the reaction force at the top? Ball goes in a vertical circular path. Resistive forces = 0. Is the reaction force always 0? The ball isn't in contact with anything but the string. Isn't the centrifugal force fictitious? Thanks. Edit: Velocity is not constant due to conservation of energy, so the centripetal force is not constant. I was wondering if there is a reaction force acting on the ball, and if so, what is the value of that reaction force.


title: Does a man on the moon experience day?

text: Does a man on the moon (or any other similar body) experience daytime like we do? By daytime I mean looking upward and seeing a bright "sky," not dark space. If not, then what is the reason behind it?


title: Why is the intensity of Hawking radiation dependent on the size of the black hole it comes from?

text: I am asking for a non mathematical answer. I think I have a pretty good understanding of physics without the maths, which unfortunately eludes my understanding. It may be the answer would be obvious if I understood the maths, but I don't. As I see it, the event horizon has the same property for any size of black hole; it is the point from which light cannot escape. Given that this property is the same regardless of the size of the hole, it seems reasonable to suppose that the intensity of the Hawking radiation would be constant per unit area of the horizon. I know I am wrong, but I don't know why. A non mathematical explanation would be appreciated.


title: Evidence of why the Standard Model is a successful theory of particle physics

text: When discussing physics with laypersons, I'm often in the situation where I have to explain what the Standard Model is, and why it's a successful theory of particle physics. To help in such situations, I'm looking for explicit examples of where the Standard Model has been successful - i.e. the prediction of certain particles, incredible agreement between experimental/theoretical results and so on. Conversely, what, if any, are the failures of the Standard Model (apart from the the obvious not-including-gravity one). Are there theoretical predictions which differ significantly from experimental results?


title: Entangled Electrons

text: What attribute would help determine in a spaghetti of tangled-entangled electrons, which two are the respective entangled electron pairs, since obviously there is an attribute other than charge or spin which entangles them.. As the charges are separated, a field force is presumably created between them. There's also the communication of 'spin attributes' between distanced entangled electrons, might this be related to a fluctuation in the lines of force (whether magnetic or dielectric)? This may be unanswerable, since manipulating instantaneous transmission signals as vibrations in lines of force between entangled electrons would be worthy of a Nobel, and I expect no Nobelists here, just thinkers.. Plus, we may not have measurements of I'm asking about.


title: What indicates if object will be reflected - certain example

text: If I throw a small rock(1kg) at a big rock(100kg) the small rock is reflected; Let's say my weight is 80kg - if I would jump into a big rock instead of being reflected I would move in the same direction as a big rock. The big rock is heavier but it is not reflecting me - Why is that?


title: Calculating equipotential lines and current density in a rectangular conductor

text: (This isn't homework, I'm trying to make an illustration for an article I'm writing.) Let's say that I have a thin rectangular bar of uniform conductivity, and I have point probes at various places: ![](https://farm4.staticflickr.com/3860/14574870780_288fc527bc_o.png) The bar has width $w$ and thickness $t$ where $t \ll w $. I am going to inject current into the bar between points $\boldsymbol A$ and $\boldsymbol D$. (let's just say 1 ampere enters at point $\boldsymbol A$ and leaves at point $\boldsymbol D$) These are centered along the bar's width and are a distance $d$ apart. How would I figure out the equipotential lines and current density? * * * edit: Vague memories of college electrostatics are coming back - it's Laplace's equation that is relevant here, I need to find a solution to $\nabla^2 V = 0$, then the electric fields are just the gradient of $V$ so $J = \sigma E = \sigma \nabla V$, and I know the boundary conditions at the outside of the rectangle are that the perpendicular component of $E$ is $0$, but I'm not sure what to do next.


title: Proof of Loss of Lorentz Invariance in Finite Temperature Quantum Field Theory

text: In the standard quantum field theory we always take the vacuum to be a invariant under Lorentz transformation. For simple cases, at least for free fields, is very simple to actually prove this. Now consider the thermal state at a given inverse temperature $\beta$ in a QFT, namely the one given by the density operator $\rho = \frac{e^{-\beta H}}{Z(\beta)}$. There is an old heuristic argument by which we loose Lorentz covariance at finite temperature: because our system is coupled to a heat bath we do have a preferred frame of reference, viz. the one in which the heat bath is static, so to ensure thermodynamical equilibrium. Although I find the argument very reasonable I have yet to see a detailed proof of this fact. None of the usual textbooks (Kapusta, Le Bellac, etc...) furnish even a hint, nor did a keyword search for papers. Does anyone know a reference for this, or the proof itself? To be very clear, the proof should be able to show this: given a quantum field $\phi(t)$ (I'm suppressing space coordinates for simplicity), one can define the thermal state as the one that satisfies the KMS condition $\langle \phi(t)\phi(t')\rangle_\beta= G(t-t')=G(t'-t-i\beta)$ or in words it is the state such that the Greens function is periodic (or anti-periodic for fermionic fields) in imaginary time with period $\beta$. Now perform a Lorentz transformation to go to new coordinates. Then the Green function in the new frame is **not** periodical in imaginary time. Therefore the state given by the density operator above only is a thermal state with inverse temperature $\beta$ in one frame. Now I would ideally be interested in an "elementary proof", that is one using the usual tools of QFT. If you happen to know a proof in a more sophisticated framework, like Algebraic QFT, I would appreciate if along with the reference you could give a brief idea behind the proof.


title: force of hail falling straight

text: There is a hail storm and it's falling straight down (0 wind factor) will the hail meet a moving car with greater force than a car standing still?


title: Dirac's remark that inspired Feynman when formulating path integral

text: When Feynman was trying to formulate path integral of quantum mechanics, he was inspired by Dirac's remark which roughly states that $e^{i\frac{S}{\hbar}}$corresponds to the transition amplitude, where $S$ is the action. I'm wondering what was Dirac's original argument behind this statement.


title: Is the coldest place in the universe likely to be artificial or natural?

text: The Boomerang Nebula is often cited as one of the coldest _natural_ places in the universe at 1 K, but that's about 10 orders of magnitude off from what's been created in a lab. Are there theoretical mechanisms that can inherently trump our (apparent) mastery of cold or is the coldest place in the universe likely to be some scientist's lab (here or elsewhere)?


title: Gauss' law and ions?

text: My text book says that with we have a singly ionized sodium atom net charge +e and if we choose a spherical surface centered on the ion and large enough to contain it all we do not need to know the distribution of charge inside the ion. And using Gauss' law can find the the electric field outside the sphere radius R as $E(R)=-\frac{e}{4\pi \varepsilon_0R^2}$. Here is my problem surly we need to know that the charge is distributed evenly around the center?? If we this was not the case then all the charge could, say, be all at one side of the sphere and thus the electric field at that side will be greater then the electric field at the other side of the sphere will it not??


title: Work and transfer of energy

text: A businesswoman is applying force of 12 N [or more] to carry her briefcase for a horizontal distance of 200 m. How much work is she doing on the briefcase?. I tried multiplying force by distance but the answer I got is different from the answer in the answers sheet. The correct answer is 0. I don't know why We have to use this formula: W = F * D


title: Power due to dipole radiation and time reversal symmetry in classical E&M

text: The dipole formula for the power loss emitted by a time varying electric dipole is (in natural units) $P = \frac{\dot d_i^2}{6 \pi}$. This is clearly even under time reversal symmetry $T$, but a power should be odd under $T$. The same puzzle appears in the power loss in gravitational radiation as described in the quadrupole formula $P = \frac{G}{5} {\dddot I}_{ij}^2$, which also appears even under $T$. What am I missing?


title: Correlation function in relaxation in NMR

text: I am new in this community, I am from a chemistry background. I want to know a detailed solution of a density matrix for a singlet state using the concept of spin lattice relaxation in NMR. I will be happy if anyone can explain in detail or give some references.


title: Does leaning (banking) help cause turning on a bicycle?

text: I think it's clear enough that if you turn your bicycle's steering wheel left, while moving, and you don't lean left, the bike will fall over (to the right) as you turn. I figure this is because the bike's momentum keeps it moving in the direction you were going, and since your wheels have friction against the ground, the top of the bike moves forward relative to the bottom of the wheels. The top of the bike going north while the bottom of the wheels go northwest will understandable cause you to topple. So to counteract this and keep you from falling over, leaning into the turn is necessary. But is there also a causal relationship -- that leaning will **cause** the bike to start to turn? If I start leaning left, I will turn left... but maybe that's because I know that if I don't turn the steering wheel left, the bike will fall over (to the left). I experimented with unruly turns of the steering wheel when I was a kid, and got my scrapes and bruises. Now that I'm a cautious and sedate adult I'm not anxious to experiment that way. :-) (I also want to ask why airplanes bank into a turn... they don't have the same issues as a bike, i.e. the bottom part has no special friction against the ground. But that would probably make the question too broad.)


title: What is the mechanism for fast scrambling of information by black holes?

text: Sekino and Susskind have argued that black holes scramble information faster than any quantum field theory in this paper. What is the mechanism for such scrambling?


title: Formula for getting energy required to accelerate to a certain speed?

text: I am curious how many newtons it takes to accelerate an object with a given coefficient of drag, reference area, weight, and air pressure? I know this may sound like a simple equation but I have no idea where to find this formula. Thanks.


title: Is mass of the parent star is equal to the mass of the born blackhole

text: I want to know whether mass of the black hole which has born from a star is similar to its parent star or not. whether the born black hole will have high or low mass comparing to its parent star. Whether in its life time black hole will increase mass by eating neighboring stars? Advance thanks for your help


title: Eigenfunctions v.s. eigenstates

text: Is there a difference between "eigenfunction" and "eigenstate"? They seem to be used interchangeably in texts, which is confusing. My guess is that an "eigenfunction" has an explicit representation/argument, e.g. in terms of position, $x$, whereas an "eigenstate" is abstract. Thanks in advance for any clarification! :) But then again, some texts violate this rule, so I'm prolly wrong...


title: Medical/Physics Question - Cut Off Finger Preservation

text: This question might require some knowledge of medical/tissue, but physicists seem to know everything, so figured I've give it a shot here ;) When people put cut off fingers they are told to put the fingers on ice. I assume the cold temperature does something to help preserve the finger. If so, my question would be, wouldn't it make more sense to put the cut off finger in a bucket of ice then put water in it, so that the temperature transfer happens faster? If the water runs a risk diluting the blood or anything, then perhaps putting the finger in a zip lock bag then doing the same process with the zip lock bag. So to recap:   1. Does my solution transfer the heat faster.   2. Is there any benefit to either approach mentioned?


Was Planck's constant $h$ the same when the Big Bang happened as it is today? Planck's constant : $$h= 6.626068 × 10^{-34}\, m^2 kg / s,$$ $$E=n.h.\nu,$$ $$\epsilon=h.\nu$$


title: What inspired Schrödinger to derive his equation?

text: I have almost no background in physics and I had a question related to Schrodinger's Equation. I think, it is not really research level so feel free to close it, but I would request you to kindly suggest some existing literature which can help me develop a better understanding for the same. While reading up about it from an introductory text on Quantum Physics, I wondered for a little while how does one derive this equation. Very soon, the author (HC Verma) added the detail that he just took it on faith when he was himself a student. He goes on to say Schrodinger equation rightly predicts the behavior of atomic transitions etc and people believe that it is a fundamental law of nature for quantum systems. Then he raises the question himself which I wanted to know the answer to. Namely > What made Schrodinger write such an equation which became a fundamental > equation, not to be derived from more fundamental equations? He then adds that it will be an interesting topic for students of history of science which does not answer my question. Could you please try answering this question (or is this question really useless) ? Thanks for your time


title: Totally antisymetric wavefunction: clarification about terminology

text: Pauli's Principle says: "The wavefunction of two identical fermions must be totally antisymmetric". I know that, for a antisymmetric wavefunction, $(-1)^L*(-1)^{S+1}*(-1)^{I+1}=-1$ "totally antisymmetric" means this relation or it means that these 3 relations: $(-1)^L=-1$ and $(-1)^{S+1}=-1$ and $(-1)^{I+1}=-1$ must be verified simultaneusly?


With these two elements how can the isotope half-life be determined using first order kinetics? nobelium-259 half-life recorded @ 58 minutes   fermium-253 half-life recorded @ 3 days Although I do not know, there is probably an average starting concentration that can probably be determined by the name of the element. Maybe in the comments someone can hook me up with some estimate averages so the expansion of the math could be given in the answer.


title: A question on spin algebra

text: In scattering theory, one can form a lorentz invariant quantity by $\epsilon_{\mu 1 2\nu}P^{\mu}_{1}P^{\nu}_{2}$ which is really $1\otimes 1$ 's spin 0 state. Is there such a kind of argument to show $\epsilon_{\alpha\beta\mu\nu}$ is also invariant under lorentz transformation without showing the determinant of lorentz transformation being 1. I just want a fancier proof of this.


title: Supersymmetry breaking and Goldstino

text: I would be glad if someone can explain to me the argument as to why supersymmetry breaking is necessarily accompanied by appearance of a massless fermion, namely the goldstino. (and also why this is a non-perturbative effect) Let me quote here the two lines from the third volume of the QFT books by Weinberg where he tries to explain this phenomenon,   * " _(when supersymmetry is broken)..any n-particle state is accompanied with 2 states of the same energy and momentum and opposite statistics, containing an additional 0-momentum goldstino of spin up or down and with another state of the same energy and momentum and the same statistics, containing two additional 0-momentum goldstinos of opposite spin._ "   * " _In particular when supersymmetry is spontaneously broken the vacuum state has non-zero energy, so it must be paired with a fermionic state of the same energy and zero-momentum; more precisely, the vacuum and the state containing two zero-momentum goldstinos are paired with the two states of a single zero-momentum goldstino._ "  I am unable to understand the above two arguments and will be glad if someone can help.


title: Question about Weinberg's derivation of a one-particle states under the Poincare group

text: I'm reading QFT: Vol 1 by Weinberg and I have a (perhaps trivial) question about a statement he makes on page 63. I can follow him to his derivation of equation (2.5.2): \begin{equation} P^\mu U(\Lambda) |p,\sigma \rangle = \Lambda^\mu{}_\rho p^\rho U(\Lambda) |p,\sigma \rangle \end{equation} where the label $\sigma$ denotes all other degrees of freedom in addition to the four-momentum. Now, I can see that, in light of equation (2.5.1), the above equation implies: \begin{equation} U(\Lambda) |p,\sigma \rangle \propto |\Lambda p,\sigma \rangle \end{equation} and so I would write: \begin{equation} U(\Lambda) |p,\sigma \rangle = C |\Lambda p,\sigma \rangle \end{equation} where $C$ is normalization constant to be determined. However, according to Weinberg, equation (2.5.2) implies: \begin{equation} U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,p)|\Lambda p, \sigma'\rangle \end{equation} Now, I do not understand what the above equation exactly means. What does $\sigma'$ represent and why are we summing over it? Thanks in advance.


title: Is it possible for a human being to travel at the speed of light?

text: Can our human body travel at the speed of light in a space shuttle? Indeed, can any material object with nonzero mass travel at the speed of light with respect to a valid inertial frame? If not, why not?


title: How to quote answer when error is smaller than significant figures of data?

text: We have these rules:   1. Can't quote answer to more significant figures than original data.   2. Error quoted to one significant figure unless first significant digit is a one. In which case quote error to 2 significant figure. But with original data to 3 significant figure, I get the answer **13.15(6)**. i.e. **13.15 +/- 0.06** How do I quote this to 3 significant figure without breaking the rules? If I round the error to 1, I still have to have a trailing 0: **13.20(10)**


title: The role of representation theory in QM/QFT?

text: I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are studying. This collection of of symmetries forms a group $G$. However, there are multiple Hilbert spaces that we can use to represent said system; so we would like to somehow represent $G$ as a collection of isomorphisms on whatever Hilbert space we choose to use. And to do so, we use a representation. Firstly, is this a proper interpretation? Is there more to it? Secondly, representations are typically group homomorphisms, not necessarily isomorphisms. Can we represent a group (in physics) with a representation that is not an isomorphism? And what does it mean physically if we can do so?


title: Microwaves and state of matter

text: Does the heating efficiency of a microwave oven depend on the state of matter that it heats? If yes, how?


title: Takhatajan's mathematical formulation of quantum mechanics

text: So I began skimming L. Takhatajan's _Quantum Mechanics For Mathematicians,_ and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.) I've only taken a basic physics-style quantum mechanics class and was wondering if someone could explain how the operator definition of states (Axiom 3) corresponds to what is typically taught in a physics class (e.g. the $L^{2}$ functions). I recently saw the post about the rigged Hilbert space formulation of QM and am also interested in how these axioms compare to Takhatajan's. Does Takhatajan's definition of states somehow include (or can be identified with) the eigenfunctions of the position and momentum operators?


title: Force felt by capacitor mounted on printed circuit board

text: I have a printed circuit board with a surface mount electrolytic capacitor soldered to it. It has two spindly legs that hold up this can, so it can move back and forth on the legs' axis just slightly. This is enough for it to break off of the board during vibration testing. ![enter image description here](http://i.stack.imgur.com/99FCO.jpg) The test is done by bolt mounting the PCB to a metal sled that is then vibrated back and forth randomly with a controlled acceleration (and I'm assuming deceleration) that does not exceed 3g. My capacitor broke off its leads at a much lower acceleration than the manufacturers' test data shows. This brings me to my actual physics question. Because the acceleration of the sled is held constant, I'm curious if the mass of the sled used makes a difference in the force felt by the capacitor. If test house 1 used a 1 lb sled but test house 2 used a 40 lb sled, would the force felt by the capacitor be greater at test house 2 because the mass of the sled would require more force to reach that acceleration?


title: What is the X(t) dependence of the ends of stretched Nambu-Goto string?

text: I'd like to have an expression for the velocity of an end of an open string being contracted due to it's tension. Not that of transverse oscillations, which seems to be the velocity of light, but rather that of movement of an end in spacetime. **Task** : Initially I have an open Nambu-Goto string of length $\sigma_0$. I've stretched it from $\sigma_0$ to $\sigma_l$ and let to relax. Because it is strained with tension $T_0$ it wants to decrease ist length. **Question** : What is the $X(t)$ dependence of string's endpoints (t-time, X-coordinates)? Could I regard the tension $T_0$ as a constant force acting on charges at the Ends of the string (Chan-Patton charges to my knowledge)? If it is so, I could describe the Motion of the strings Ends as a Motion of a relativistic particle: $X(t)=\frac{c^2}{F/m}(\sqrt{1+\frac{(t-t_0)F/m}{c}})^2-1)$? Or it it is not allowed?


title: Why is ground state $| 0 \rangle$ of harmonic oscillator a coherent state?

text: Is ground state $| 0 \rangle$ of harmonic oscillator a coherent state just because it minimize the uncertainty product? What is the intuition of this. I don't quite understand the significance of the ground state being a coherent state. Any explanation would be appreciated


title: Verlet timestep

text: I am using velocity Verlet in molecular dynamics. Is just a simple question, if a simulation using time-step femtosecond, in velocity Verlet is just necessary $dt = 10^{-15}$ to use femtosecond?


title: What do physicists mean by ${g^{i}}_j$?

text: Maybe this is an idiot question, but in relativity I see a lot of ${g^{i}}_j$ for a metric tensor $g$. Is this just $$\delta^i_j ~=~ g(dx^i \sharp, \partial_{ x^j})~?$$


title: How can the Higgs have so many different Yukawa couplings?

text: How can the Higgs have so many Yukawa couplings? Isn't this about the same as saying the Higgs has a force or charge for each different coupling? Is this some indication of substructure for the Higgs or some kind of mixing with other scalar neutral particles that have not been found yet?


title: Is Uranium renewable, or will this science fiction scenario become reality?

text: In my imagination, nuclear power could disapear when we use up all the uranium generating electricity, or exploding nuclear bombs. Is uranium a renewable resourse? if not, can we prove that this element could not be created by throwing neutrons and protons together?


title: Increase in Wavelength of an LED immersed in liquid nitrogen

text: "Hi, I've been working on an experiment where I dip LEDs into liquid nitrogen and note the color change. All online sources I've seen predict that the wavelength of the light emitted by the LED will decrease upon immersion. What I've been observing, however, is a sudden decrease in wavelength followed by a definite increase in wavelength. So, for example, a yellow LED turned green- ish at first, and then rapidly changed to yellow-orange. I've been looking around for an explanation of why I am getting this (opposite) effect, but haven't had any luck. Does anyone have any ideas?" Quoted from Re: Increase in Wavelength of an LED Immersed in Liquid Nitrogen


title: Why do green lasers appear brighter and stronger than red and blue lasers?

text: This is mostly for my own personal illumination, and isn't directly related to any school or work projects. I just picked up a trio of laser pointers (red, green, and blue), and I notice that when I project them, the red and the blue appear to be dimmer to my eye than the green one. I had a fleeting suspicion that, perhaps this is an effect of blue and red being at the periphery of the visual light scale, but I honestly have no idea if this is the case or if it's just my eyes playing tricks on me. All three lasers have the same nominal strength, in this case.


title: Is it fabric of space or fabric of space-time?

text: Very quick question, in the general relativity is it fabric of space, fabric of space-time or both mentioned?


title: Linearized Einstein Equations

text: I've been trying to prove this equation: $$ \delta G_{\alpha\beta}=-\frac{1}{2}\left(\square\bar{h}_{\alpha\beta}+2R{}_{\gamma\alpha\delta\beta}\bar{h}^{\gamma\delta}\right)+\frac{1}{2}\left(\bar{h}_{\alpha;\gamma\beta}^{\gamma}+\bar{h}_{\beta;\gamma\alpha}^{\gamma}-g_{\alpha\beta}\bar{h}^{\rho\delta}{}_{;\rho\delta}\right), $$ for the first order Einstein tensor, where $\bar{h}_{\alpha\beta}:=h_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\left[g_{\gamma\delta}h^{\gamma\delta}\right]$. I've tried it a dozen times and always get this expression: $$ \delta G_{\alpha\beta}=-\frac{1}{2}\left(\square\bar{h}_{\alpha\beta}-2R{}_{\gamma\alpha\delta\beta}\bar{h}^{\gamma\delta}\right)+\frac{1}{2}\left(\bar{h}_{\alpha;\gamma\beta}^{\gamma}+\bar{h}_{\beta;\gamma\alpha}^{\gamma}-g_{\alpha\beta}\bar{h}^{\rho\delta}{}_{;\rho\delta}\right). $$. Can anyone prove the right expression from: $$ \delta G{}_{\alpha\beta}=\frac{1}{2}\left(-h_{\gamma;\beta\alpha}^{\gamma}+h_{\beta;\alpha\gamma}^{\gamma}+h_{\alpha;\beta\gamma}^{\gamma}-h_{\alpha\beta}{}^{;\gamma}{}_{;\gamma}\right)-\frac{1}{4}g{}_{\alpha\beta}\left[g{}^{\gamma\delta}\left(-h_{\rho;\delta\gamma}^{\rho}+h_{\delta;\gamma\rho}^{\rho}+h_{\gamma;\delta\rho}^{\rho}-h_{\delta\gamma}{}^{;\rho}{}_{;\rho}\right)\right]. $$ Thank you


title: Is any apparent horizon a minimal surface?

text: I faced "any apparent horizon is a minimal surface", but I don't know how I can relate a physical concept (apparent horizon) to pure mathematical concept (minimal surface). How can I prove it?


title: Potential Energy Concept

text: Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out and the book moves with a constant velocity. Consider the situation after the book has been lifted, and it has come to rest once again. According to the work and kinetic energy laws $\Delta W = \Delta K$ This seems to hold here, since both are zero. That's oka,y but where did the increase in the potential energy come from ? Is energy not conserved?


title: Work done by a conservative force

text: Wikipedia: > _A conservative force is a force with the property that the work done in > moving a particle between two points is independent of the taken path. > Equivalently, if a particle travels in a closed loop, the net work done (the > sum of the force acting along the path multiplied by the distance travelled) > by a conservative force is zero._ Does the work done by the force remain 0 even if it varies at all points on the loop? From the definition given in wikipedia it seems as if it is defined for work done by a constant conservative force. Is the work done over a loop 0 for variable forces as well? Can it be somehow proven for a variable force?


title: Tension on either side of a massive pulley?

text: When a pulley has mass, why is tension on both the sides different? Why don't we consider rotation of pulley when it is massless?


title: Tension in vertical circular motion

text: In vertical circular motion we conserve energy for calculating velocities at a point (if initial velocity given). But, energy can only be conserved when forces are conservative. Tension is not a conservative force. Does it not affect the particles velocity? Is the tension's role only to provide centripetal acceleration?


title: What distance does one travels in his lifetime?

text: I think this isn't answerable without having point of reference from which speed at which earth travels in space, however if I am wrong the please give absolute distance.   Considering that earth spins around it's axis and around the sun and solar system rotates around centre of the galaxy. The circumference of the Earth at the equator is 25,000 miles. The Earth rotates in about 24 hours. Therefore, if you were to hang above the surface of the Earth at the equator without moving, you would see 25,000 miles pass by in 24 hours, at a speed of 25000/24 or just over 1000 miles per hour. Earth is also moving around the Sun at about 67,000 miles per hour. Using speed measurements of the gas at different distances from the Galactic centre, the Sun appears to be cruising along at 200 kilometres per second and it takes 240 million years to complete the grand circuit around the Galaxy. I am not sure if it's ok to just add numbers or some formula should be used since all the movements are rotational. **If average person lives 80 years, what distance does one travels in his lifetime?** EDIT: Point of reference should be centre of our galaxy, unless you believe there is better point of reference possible.   EDIT2: Is this collect calculation or am I missing something. > 1000 mph = 447.04 m/s, 67000 mph = 29951.68 m/s   >  447.04+29951.68 + 200000 = 230398.72 m/s   >  80 years = 2.52455e9 seconds   >  230398.72 m/s * 2.52455e9 seconds = 5.8165309e+14 meters   >  5.8165309e+14 metres = 3888.11076 Astronomical Units OR 0.0614820933 Light > Years. Edit3: Following answer from @Johannes > 371000 m/s * 2.52455e9 seconds = 9.3660805e+14 metres   >  9.3660805e+14 metres = 6260.83811 Astronomical Units OR 0.0990016635 Light > Years. Edit4: I just realised that I need to add all 4 (earth rotation + rotation around sun + rotation around galaxy + and galactic cluster movement speed relative to CMB). > 5.8165309e+14 meters + 9.3660805e+14 metres = 1.5182611e+15 metres   >  1.5182611e+15 metres = 10148.9489 Astronomical Units = 0.160483757 Light > Years One travels ~16% of light year in a lifetime.


title: Proving one field equation leads to the other

text: Assume that the universe is homogenous and isotropic, and the following equation holds: \begin{equation}R_{00}-\frac{1}{2}g_{00}R=8\pi GT_{00}; \space \space \nabla_{\mu}T^{\mu 0}=0.\end{equation} How do I prove that the following equations are identically satisfied provided that the above two are satisfied? \begin{equation}R_{0i}-\frac{1}{2}g_{0i}R=8\pi GT_{0i}; \space \space R_{ij}-\frac{1}{2}g_{ij}R=8\pi GT_{ij}; \space \space \nabla_{\mu}T^{\mu i}=0.\end{equation} My approach was to write $g_{00}=1$ and $g_{ij}=-a^2\gamma_{ij}$ and evaluate the Ricci tensors and so on, but I know this is not the way to do it. Can anyone suggest me the way?


title: Why are waves, the means, of information transfer over long distances, excluding difusion or contact of info stored in matter

text: **Practical** long distance communication, which does not rely on the movement of encoded configurations of matter, from source to destination(odor,books,DNA,floppy disk), always involves waves (EM, Sound). Whether acoustic or electromagnetic, we need to wiggle some matter, which radiates some of the energy to the field, to wiggle something else, inverse squared, at the sensor receiver! Maybe this should be obvious, but it was a nice surprise for me. And maybe a motivator for explorers of waves.


title: How to test cutoff frequency of IR filter on camera?

text: Modern cell phones seems to come with IR filters on their cameras. I want to do an experiment to figure out what wavelengths these filters allow to pass and which they block. How would I go about doing this? Also, what sort of equipment would I need? I am a university student so I have access to a nicely supplied physics lab. Thanks.


title: How many stars within 5 parsecs?

text: I've ran some scripts on some star catalogs, which counted the number of stars with a parallax greater than 200 mas, which should mean everything closer than 5 pc. The results are as follows: original Hipparcos catalog 40 original Tycho catalog **10776** revised Hipparcos catalog 43 HYG database 64 How the heck are there that many stars according to the Tycho catalog? (I've checked the code many times, checked the values it was accepting and I used it as the base for the others.) Also, how can there be so much difference between the other three? **EDIT:** If I discard negative parallaxes, the Tycho catalog has 5273 stars with parallaxes greater than 200 mas. **EDIT EDIT** : Discarding all stars with the "dubious parallax" flag, 4213 remain. **EDIT EDIT EDIT** : Discarding all stars with a quality of medium or worse, 161 stars remain.


title: Extremal black hole with no angular momentum and no electric charge

text: A black hole will have a temperature that is a function of the mass, the angular momentum and the electric charge. For a fixed mass, Angular momentum and electric charge are bounded by the extremality condition $$M^2 - a^2 - Q^2 \gt 0$$ Exactly at the extremality boundary, both entropy and temperature are zero. Suppose i create a black hole with a spherically symmetric, incoming wavefront of electromagnetic radiation **in a pure quantum state** (that is, the density matrix satisfies the property $\rho^2 = \rho$). The wavefront is shaped in a way such that the whole energy of the packet will be inside the Schwarzschild radius, which will form an event horizon. Since the wavepacket is as nearly as pure as it is physically possible to create, the quantum (Von Neumann) entropy is zero or nearly zero. But the formation of the black hole does not create nor destroy entropy, so the black hole must contain zero or nearly zero entropy as well. So the black hole seems to be extremal (it has zero temperature) but it nonetheless does not have any angular momentum (it is formed from a wavefront with zero net polarization over the whole sphere) and it does not have any charge (electromagnetic radiation is neutral). > **Question:** what "hair" does have a black hole formed from such a pure > state, so that it can be extremal and still do not have angular momentum or > electric charge (which are the _classical_ hair that we come to expect from > classical general relativity) _This question is a mutation ofthis question, but while that specific question tries to look what input black hole states create specific output (Hawking) radiation states that are far from thermal from a statistical point of view, this question is specific about extremality that is unrelated to angular momentum and charge_


title: Calculating the expectation value for kinetic energy $\langle E_k \rangle$ for a known wave function

text: I have a wavefunction ($a=1nm$): $$\psi=Ax\exp\left[\tfrac{-x^2}{2a}\right]$$ for which I already calculated the normalisation factor (in my other topic): $$A = \sqrt{\frac{2}{a\sqrt{\pi a}}} = 1.06\frac{1}{nm\sqrt{nm}}$$ What I want to know is how to calculate the expectation value for a kinetic energy. I have tried to calculate it analyticaly but i get lost in the integration: \begin{align} \langle E_k \rangle &= \int\limits_{-\infty}^{\infty} \overline\psi\hat{T}\psi \,dx = \int\limits_{-\infty}^{\infty} Ax \exp \left[{-\tfrac{x^2}{2a}}\right]\left(-\tfrac{\hbar^2}{2m}\tfrac{d^2}{dx^2}Ax \exp \left[{-\tfrac{x^2}{2a}}\right]\right)\,dx =\dots \end{align} At this point I go and solve the _second derivative_ and will continue after this: \begin{align} &\phantom{=}\tfrac{d^2}{dx^2}Ax \exp \left[{-\tfrac{x^2}{2a}}\right] = A\tfrac{d^2}{dx^2}x \exp \left[{-\tfrac{x^2}{2a}}\right]= A\tfrac{d}{dx}\left(\exp \left[{-\tfrac{x^2}{2a}}\right]-\tfrac{2x^2}{2a}\exp \left[{-\tfrac{x^2}{2a}}\right]\right)= \\\ &=A \left(-\tfrac{2x}{2a}\exp \left[{-\tfrac{x^2}{2a}}\right] - \tfrac{1}{a}\tfrac{d}{dx}x^2\exp \left[{-\tfrac{x^2}{2a}}\right]\right) = \\\ &=A \left(-\tfrac{x}{a}\exp \left[{-\tfrac{x^2}{2a}}\right] - \tfrac{2x}{a}\exp \left[{-\tfrac{x^2}{2a}}\right] + \tfrac{x^3}{a^2}\exp \left[{-\tfrac{x^2}{2a}}\right]\right) = \\\ &= A \left(-\tfrac{3x}{a}\exp \left[{-\tfrac{x^2}{2a}}\right] + \tfrac{x^3}{a^2}\exp \left[{-\tfrac{x^2}{2a}}\right]\right) \end{align} Ok so now I can continue the integration: \begin{align} \dots &= \int\limits_{-\infty}^{\infty} Ax \exp \left[{-\tfrac{x^2}{2a}}\right]\left(-\tfrac{\hbar^2}{2m} A \left(-\tfrac{3x}{a}\exp \left[{-\tfrac{x^2}{2a}}\right] + \tfrac{x^3}{a^2}\exp \left[{-\tfrac{x^2}{2a}}\right]\right)\right)\,dx = \\\ &= \int\limits_{-\infty}^{\infty} -\frac{A^2\hbar^2}{2m}x\exp\left[-\tfrac{x^2}{2a}\right] \left(-\tfrac{3x}{a}\exp \left[{-\tfrac{x^2}{2a}}\right] + \tfrac{x^3}{a^2}\exp \left[{-\tfrac{x^2}{2a}}\right]\right) \,dx\\\ &= \int\limits_{-\infty}^{\infty} \frac{A^2\hbar^2}{2m}\left(\tfrac{3x^2}{a}\exp \left[{-\tfrac{x^2}{a}}\right] - \tfrac{x^4}{a^2}\exp \left[{-\tfrac{x^2}{a}}\right]\right) \,dx\\\ &= \frac{A^2\hbar^2}{2m} \underbrace{\int\limits_{-\infty}^{\infty}\left(\tfrac{3x^2}{a}\exp \left[{-\tfrac{x^2}{a}}\right] - \tfrac{x^4}{a^2}\exp \left[{-\tfrac{x^2}{a}}\right]\right) \,dx}_{\text{How do i solve this?}}=\dots\\\ \end{align} This is the point where I admited to myself that I was lost in an integral and used the WolframAlpha to help myself. Well I got a weird result. My professor somehow got this ($m$ is a mass of an electron) but I don't know how: \begin{align} \dots = \frac{\hbar^2}{2m}\cdot\frac{3}{2a} = \frac{3\hbar^2}{4ma} = 0.058eV \end{align} Can anyone help me to understand the last integral? How can I solve it? Is it possible analyticaly (it looks like professor did it, but i am not sure about it)?


title: Sound - for purposes of vibration

text: What is the best way to distribute noise from more than one source (I'm envisioning a system with many), within a dome, with the ground as its primary target, at optimal frequencies and volumes to create maximum vibrations on the ground? Think, this picture - http://hyperphysics.phy- astr.gsu.edu/hbase/acoustic/imgaco/foc3.gif \- but with more than one source. ![enter image description here](http://i.stack.imgur.com/Zt4xK.gif) I'm assuming you would want the dome to be perfectly circular, and have the point at which the dome wall meets the ground at a slight incline (as opposed to vertical) so all the sound stays directed at the ground. Imagine a tennis ball which is cut in half, but was split just before the equator. You take the side that is smaller, and flip it over. Thats what I am imagining. I'm just judging from the above drawing that you would want the sound sources crammed as close as possible in the center of the dome, in as perfect as a circle as you could design it, and maybe directed so that they are all pointed to the center of their own, independent, equally sized areas (the dome being the total area of the surface that the sound is being pointed at). It seems like if you could cram all the sound into a point of singularity and put it at the center of the dome, and then release it somehow - that would be ideal (although that sounds pretty complicated). Not sure about frequencies or volumes.


title: Reasons for violation of universality in statistical mechanics

text: The Universality in statistical mechanics is nicely explained by the renormalization group theory. However, there are fair amount of numerical and theoretical studies show that it can be violated in models such as Ising model, spin glass, polymer chain and percolation (Example references: 1, 2, 3, 4, 5, 6, 7) and some violations are pretty strong. So, my questions are:   * What are the general reasons of the violation of the universality?   * Also, why can't they be explained by the renormalization group theory and scaling hypothesis?   * Is it always true that there are some "core elements" for each universality class? Whenever they are missing, we can definite say they are not in the same universality class.   * Is there an example of commonly recognized "universality class" (such as 2D Ising model) that show different critical exponents by changing the microscopic details? Note, it should not happen by the definition of universality class, but there is no proof for all universality class. I would accept answers discussing solid examples for some universality class about these questions. * * * Note: Universality refers to a large class of different systems exhibiting the same properties regardless of its microscopic details such as the lattice type (square, hexagonal, triangular, and kagome lattices). In statistical mechanics, the universality class are usually determined by the symmetry of the order parameter (such as up-down symmetry in Ising model and spherical symmetry in Heisenberg model) and the topological structure (such as dimensionality).