title: Finding the ground state of the toric code Hamiltonian

text: How do I write by proof, the ground state of the toric code (by Kitaev) Hamiltonian $ H=-\sum_{v}A(v)-\sum_{p}B(p) $ where $A(v)=\sigma_{v,1}^{x}\sigma_{v,2}^{x}\sigma_{v,3}^{x}\sigma_{v,4}^{x}$ and plaquette term $B(p)=\sigma_{p,1}^{z}\sigma_{p,2}^{z}\sigma_{p,3}^{z}\sigma_{p,4}^{z} $ ? Here $v$ are indices of vertices on a lattice with spin-1/2 particles on the edges, $p$ refers to the indices of the plaquettes in the lattice. ![enter image description here](http://i.stack.imgur.com/WUl3V.png)


title: Why Hausdorff and Paracompact manifold in GR?

text: 1. What can we say about the **transition map** if the manifold is a **Hausdorff space**?   2. Why do we need the manifolds to be **Hausdorff** and **paracompact** in General Relativity?


title: Single directional electric field insulator?

text: Is there any material, (kind of like a one way mirror), which allows an Electric Field to pass through from one direction, but not from the other? Thanks. Edit: As Ali has pointed out, one way mirrors do not actually block light from filtering through. Maybe I could reformulate it by asking. Is there anyway to polarize Electric Fields in the way we do for light, ie the Force vector applied to a particle behind said insulator would only act in one direction?


title: Photon Absorption and Emission: Conductors v. Semiconductors

text: I'm having a hard time understanding how photon absorption and emission in metals (conductors) compares to semiconductors. Obviously, in SCs, absorbed photons lead to electron-hole pairs and emitted photons correspond to recombinations. I don't exactly understand how this process works in metals, since we don't even consider holes in the first place (and have essentially no band gap). Do we just consider absorbed photons as promoting electrons to excited states, rather than pair generation? Why exactly do metals emit absorbed light rapidly? I've read that light induces alternating currents on the metal surface and that ACs rapidly emit light, but I have no idea why. I'm not sure what alternating currents have to do with this (I'm sure there is an obvious explanation I should be familiar with). I'm more familiar with band structure in SCs and therefore I'm having a hard time comparing radiative processes in metals to SCs.


title: Which came first, movement or heat?

text: According to my measly understanding of the universe, when particles hit one another, some of their kinetic energy is transformed into heat. But when we heat particles (for instance, putting a bucket of water on a stove) we can see they gain kinetic energy (the water-particles start fidgeting about until they turn to gas, which moves freely). So, heat transforms to movement, movement transforms to heat. Cool. Which came first and started the chain? I tried to solve it by imagining a dude standing inside a bubble, watching the big-bang (he has a very special bubble, called Mr. Bubble). When the universe went all BOOOOM, at that first time-unit, I tried thinking about what would happen to the bubble. Having only a bit of world experience and a bit of logic, my first thought was "it'd heat up". But I hardly think that qualifies as an actual answer. Am I getting it completely wrong? Will Mr. Bubble be blown back but remain at present temperature? Is movement a form of heat? Or are all kinds of energy essentially the same thing, with different manifestations? Or maybe Mr. Bubble was the culprit all along?


title: What is the perturbative vacuum?

text: Will someone please supply me with a good definition of the perturbative vacuum vis a vis quantum cosmology?


title: The amplitude for a particle to be created/annihilated

text: In A. Zee's "QFT in a nutshell", there is such a statement about massive spin-1 particles (Chap I.5). > The three polarization vectors $\xi^{(a)}_\mu(k)$ are simply the three unit > vectors pointing along the $x$, $y$, and $z$ axes, respectively $(a = 1, 2, > 3)$: $\xi^{(1)}_\mu = (0, 1, 0, 0)$, $\xi^{(2)}_\mu = (0, 0, 1, 0)$, > $\xi^{(3)}_\mu = (0, 0, 0, 1)$. In the rest frame $k_μ = (m, 0, 0, 0)$. > > The amplitude for a particle with momentum $k$ and polarization $a$ to be > created at the source is proportional to $\xi^{(a)}_\lambda(k)$, and the > amplitude for it to be absorbed at the sink is proportional to > $\xi^{(a)}_\nu(k)$. Why is the amplitude proportional to polarization vectors? Is there an intuitive explanation? Thanks!


title: QFT: differential cross section from center of mass to lab frame

text: I have the following process: two ingoing particles, a photon hitting a nucleus, and two outgoing particles, the nucleus and a pion. I have computed $|M|^2$ and the differential cross section in the center of mass frame $\frac{d \sigma}{d \Omega_{CM}}$; I now have to go into the lab frame, where the nucleus is initially at rest, and consider the limit of a infinite massive nucleus $M_N \to \infty$, and compute $\frac{d \sigma}{d \Omega_{lab}}$. Is there a general procedure to go from the first to the second? I first wrote $\frac{d \sigma}{dt}$ and then multiplied it for a rather complicated expression that I found on a book to obtain $\frac{d \sigma}{d \Omega_{lab}}$. However, taking the infinite massive nucleus limit, the result I get is not what I'm supposed to.


title: Why is the world sheet of an open string a cylinder?

text: I went to a lecture a few weeks ago and was told the following: The world sheet of a closed string is a normal, standing cylinder. The world sheet of an open string is a cylinder on its side. This shows clear duality, therefore unifying gravity with gauge theories, as closed strings are there for gravity and open strings for gauge theories. (Bear in mind this is incredibly simplified as it would be unnecessary for me to go into great detail) What puzzled me is why the open string produces a world sheet that is a cylinder. Surely it would produce a 2-dimensional square/rectangle, as this is the shape that's produced when drawing out a world sheet of any horizontally straight line. The world sheet being a cylinder would mean (according to my intuition) that the string would have to sort of split in 2 (the two strings produced being the same length as the previous lone string) and then rejoin again, producing a cylindrical shape. This explanation seems extremely non-intuitive, which forces me to question it. So why is the world sheet of an open string a cylinder as opposed to a 2-dimensional square/rectangle?


title: Boussinesq approximation for the Navier Stokes' equation - discrepancy

text: In the Navier Stokes' equation: > $\rho_0 \left( \frac{\partial v}{\partial t} + v \cdot \nabla v\right) = > -\nabla p + \mu \nabla^2 v + \hat{f}$ I included the temperature variation of density as per the Boussinesq approximation as > $\hat{f} = \rho_0(1-\alpha\Delta T)g\hat{k}$. Choosing a viscous time scale $t^* = \frac{t}{d^2/\nu}$ where $d$ is a measure of length (in this case, film thickness) and non-dimensionalizing length and velocity with: $x^*=x/d$ and $v^*=\frac{v}{\nu/d}$, the Navier Stokes' equation is modified to: > $\rho_0\left(\frac{\nu^2}{d^3} \frac{\partial v}{\partial t} + > \frac{\nu^2}{d^3} v\cdot\nabla v \right) = -\frac{1}{d}\nabla p + > \frac{\mu\nu}{d^3} \nabla^2 v + \rho_0(1 - \alpha \Delta T) g\hat{k}$. Dividing throughout by $\frac{\rho_0 \nu^2}{d^3}$, while realizing that the non-dimensional pressure falls out as $\frac{\rho_0 \nu^2}{d^2}$, I get: > $\frac{\partial v}{\partial t} + v\cdot\nabla v = -\nabla p + \nabla^2 v + > \frac{gd^3}{\nu}(1 - \alpha \Delta T)g\hat{k}$ Further splitting these terms for simplification: > $\frac{\partial v}{\partial t} + v\cdot\nabla v = -\nabla p + \nabla^2 v + > \frac{g d^3}{\nu^2}\hat{k} - \frac{gd^3}{\nu^2} \alpha \Delta T \hat{k}$ Recognizing the following non dimensional numbers:   1. Galileo number, $Ga = \frac{g d^3}{\nu^2}$   2. Grashoff number, $Gr = \frac{gd^3}{\nu^2} \alpha \Delta T = \frac{Ra}{Pr}$ Where $Ra$ is the Rayleigh number and $Pr$ is the Prandtl number. I rewrite my now non dimensional Navier Stokes' equation as: > $\frac{\partial v}{\partial t} + v \cdot \nabla v = -\nabla (p - Ga z > \hat{k} + Gr z \hat{k}) + \nabla^2 v$ Where $z$ is the z coordinate and $\hat{k}$ is the unit normal in the $z$ direction. Is this approach flawed since when I compare this with page 2 of this, equation 4 in Bestehorn et al. and equations 5 thru 7 in Ybarra et al., they all have a temperature difference multiplied with the Rayleight number which doesn't make sense. Did I do something wrong?


title: Why do electromagnetic waves become weaker with distance?

text: From what I know, In electromagnetic waves, the components electric field and magnetic field generate each other i.e. magnetic field generated from moving electric charge generates another electric field and which also generates another magnetic field and so on thus, creating an electromagnetic field. So each of them gets weaker with distance which results in weaker EMF. Am I correct here?


title: Do commutation relations exist between superfields?

text: To quantize a theory, Klein gordon field for example, commutation relations are stablished. Or anticommuting ones in the fermionic case. If I have the Wess.Zumino model or the free model: $$S~=~\int\Phi\Phi^\dagger d^8z.$$ Can I obtain in some way relations (commutation or anticommutation, I don't know) like this: $$[\Phi(z),\Phi^\dagger(z^\prime) ]~=~?$$


title: Can cellular automata be reconcilied with quantum mechanics?

text: CAs are deterministic representations of the universe, which, according to the Bell's inequality are not entirely accurate. Cells interact "locally" (only with the closest neighbours), while quantum entanglement proposes the opposite. So, in layman terms, what changes should be made in cellular automata in order to -if this is even possible- make them represent the universe realistically?


title: computing wavefront deviations for off-axis mirrors

text: Suppose you have two off-axis aligned mirrors which are nearly planar, with small deformations such that a wavefront at plane $P_I$ at distance $-D$ from the mirrors, is refocused at distance $D$ at plane $P_O$. I only care about monochromatic waves of wavelength $\lambda$ I want to compute how much orders of Zernike mode expansion are required to account in the deformations of the mirrors, such that an input gaussian beam with intensity $I_I$ is projected to a gaussian beam with intensity $I_O$, such that $\frac{I_O}{I_I} > 1 - f$. That is, only a fraction $f$ of the beam is lost at most from the $T_{00}$ mode I've been looking into the literature but most of it does the wavefront calculations for on-axis elements. I wanted to find some proper literature for the off-axis case that would suit the particulars of my problem


title: experimental bounds on microcausality violation

text: In "The Great Soviet Encyclopedia", 3rd Edition from 1970-1979, (evidently an old book), some V. I. Grigor’ev has a well-written little note on microcausality. Towards the end he states an experimental bound on the vialotation of microcausality. According to that ancient encyclopedia article, if such violation occurs at all then it must be on scales smaller than $10^{-15} cm$. I don't see a citation for that number in his text (I don't have the original book in my hand though, to be frank) and I am wondering which experiment Grigor’ev might have been thinking of. And: if there are decent such experiments indeed, then probably there are more recent numbers on whatever that experiment actually measured. My question is if anyone might be able to help me with finding whatever useful there might be to say about experimental bounds of the kind Grigor’ev seems to be referring to, or anything similar. Maybe I should say that I am not trying to find such violations. Rather, I am in some discussion with some mathematicians interested in axiomatic QFT, and I'd find it entertaining to be able to accompany the statement of the microcausality axiom (or a similar axiom) with something substantial that has the word "experiment" in it.


title: Supernovae and black holes?

text: I think i am correct in saying that a supernova ($Type$ $II$) is caused by the collapse of the core of a giant star. This contraction of the core is stopped by the Pauli exclusion principle and the core becomes rigid. The outer layers now rebound of the rigid core and are thrown into space as a supernovae. But presumably this leaves behind a neutron star and not a black hole. For a black hole to form the Pauli exclusion principle would not be strong enough to stop the collapse? If so what causes the outer layers to rebound, as there is no ridged object for them to rebound off? What I am basically asking is when a black hole forms what mechanism causes the supernovae?


title: Dark matter and mass-energy conservation

text: I have been thinking recently about dark matter, and it lead me to the following question. Consider a hypothetical particle which, like the photon, is chargless and massless. However, unlike the photon, it is absorbed by nothing and reflected by nothing. Its refractive index through any material is always 1. It does not interact with anything by any means other than the gravitational effect caused by its energy. Let us call this a 'dark photon'. I am sure it does not exactly fit in current definitions of dark matter or dark energy, it is just something inspired by them. Let us now assume that sometimes, for some particles, a particle/antiparticle annihilation can yield these dark photons rather than normal photons. Is there any way to distinguish this situation from one in which the particle and antiparticle just disappear into nothingness, with their mass and energy removed from the universe in violation of mass energy conservation? I, with my admittedly quite Newtonian view of gravity, cannot see one. For example, an observer 'looking' at the particle and antiparticle gravitationally would 'see' a point particle with their total mass-energy at their centre of mass-energy. This would remain true after they have been converted to dark photons, except that the gravitational force would disappear once the dark photons have moved past the observer. The time taken for this would be r/c, if r is the distance from the observer to the centre of mass- energy. The same is true after they disappear into nothingness, the observer would still feel the force for a time r/c, since the effects of gravity are not instantaneous. The same is of course also true for normal photons, except that this case can be distingushed from the previous two by detecting the photons, or using some material to slow them down. But how could dark photons be distinguished from a violation of mass-energy conservation? Is it possible? If not, does that mean mass-energy conservation prohibits the existence of such particles, even though they don't actually violate mass-energy conservation?


title: Solving an indeterminate triangle truss structure (statics)

text: From the setup of two trusses as shown in this illustration, how can I solve for the axial reaction forces in the trusses, and in the points A and B (Ax, Ay, Bx, By)? It is assumed that the joints exert no torque/bending moment, and the lengths don't really matter, only the angles alpha and beta. I've drawn two slightly different setups, while the top one reflects the true setup, and the lower one might be easier to solve: ![http://imageshack.us/photo/my- images/16/dsc00374h.jpg/#](http://img16.imageshack.us/img16/4303/dsc00374h.jpg) I'm primarily interested in solving this by hand and calculator, and I'm a little rusty on constraint equations and Lagrange multipliers (though that might be overkill).


title: The shape of speaker cones

text: This is related to another question I just asked, but they are different enough I thought it deserved its own spot. Speaker elements seem to always be shaped like a cone with a portion of a sphere at the center. What is the physics reasoning behind this shape being so standard? Is it a precise shape that all speaker elements share (e.g. parabaloid + sphere), or just kind of a general shape that "works well"? Does it work just as well for any light solid, or is it actually tuned to the speed at which sound propogates through the speaker element?


title: Uncoupling a coupled oscillator Hamiltonian by change of variables

text: I'm working on the problem of two entangled harmonic oscillators with Hamiltonian: $$H = \frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2].$$ Introducing the variables $x_± = \frac{1}{\sqrt{2}} (x_1 ± x_2)$, and frequencies $\omega_+ = k_0^{1/2}$, $\omega_- = (k_0 + 2k_1)^{1/2}$ I've tried to write this Hamiltonian in an uncoupled way. I have found that $(x_+^2 + x_-^2) = (x_1^2 + x_2^2)$ and $2x_-^2 = (x_1 - x_2)^2$. Furthermore, since $ p_{1,2}^2 = \frac{\partial^2}{\partial x_{1,2}^2}$ when ignoring constants, I find $$p_±^2 = \frac{\partial^2}{\partial x_{1}^2} + \frac{\partial^2}{\partial x_{2}^2} ± 2 \frac{\partial}{\partial x_{1}} \frac{\partial}{\partial x_{2}}.$$ Then $\frac{1}{2} (p_+^2 + p_-^2) = p_1^2 + p_2^2$. And: $$H = \frac{1}{2} [ \frac{1}{2} (p_+^2 + p_-^2) + k_0 x_+^2 + (k_0 + 2k_1)x_-^2] = \frac{1}{2} [ \frac{1}{2} (p_+^2 + p_-^2) + \omega_+^2 x_+^2 + \omega_-^2 x_-^2]. $$ The paper I'm working from gives the normalised ground state as $$\psi_0(x_1, x_2) = \pi^{-1/2} (\omega_+ \omega_-)^{1/4} e^{-(\omega_+ x_+^2 + \omega_- x_-^2) / 2}.$$ However, when I try to check that $H \psi_0(x_1, x_2) = E \psi_0(x_1, x_2)$ I don't find anything sensible ($E$ is negative in the constant terms, and still dependent on $x_±$). I suppose then that my Hamiltonian is wrong, but I can't find my error.


title: Where are the centers of mass for a baseball bat vs. a cricket bat?

text: My friend and I were comparing the rightness of a cricket bat versus a baseball bat for general violence purposes. (Eg, rioting.) Where are the centers of mass for these two instruments? Which would be better for an average adult human male to, say, smash in a window or other such similar action? Would the answer be different for an average adult human female?


title: Ampere's Law and net current?

text: ![enter image description here](http://i.stack.imgur.com/3dNQs.png) If the current through a closed Amperian curve is zero, the net circulation around the closed curve should be zero. In the image, above the entire field is perpendicular to the Amperian curve when it enters the curve, but is not entirely perpendicular to the closed Amperian curve when it leaves. Therefore the dot product will be greater when the lines enter than when they leave, so the net circulation wouldn't be zero would it? However, the net current is zero, so the net circulation should be zero right? What am I misunderstanding?


title: Is it true that any system of accelerating charges will radiate?

text: I was recently told by a physics teacher that "any system of charges in which at least some of the charges are executing some sort of accelerated motion, will radiate and lose energy". This refers to classical electrodynamics (I'm not entering into quantum mechanics), that is, this statement should either be provable or disprovable from Maxwell's equations. I tried to think of a simple system of charges that would disproof this statement, since I intuitively think that it isn't true, but I haven't come up with any. The one example I could think of was a positive charge sitting on top of a negative charge, so that the net charge is zero. No matter how they move it is obvious that there will be no radiation. But my teachers did not accept this trivial example. So my question is whether the statement is true or not, plus a proof or a counterexample of some charges executing accelerated motion without radiating. Thanks.


title: What happens to gas molecules after ionization?

text: I know that gas molecules conduct electricity after they get ionized but what will happen if we keep increasing the voltage even after ionization? Will it explode? If it will then how much energy will be released? Assume I have neon gas in Crookes tube.


On a calm smooth lake, or even a large rain puddle, I've seen transient rough patches on the surface suddenly appear and disappear, and sometimes move across the water some distance before disappearing. The amplitude and wavelengths of the wavelets are centimeter scale, and the motion of the water at the surface appears jumbled and random, almost like boiling, but only at the surface. The disturbance does not seem to propagate far outside the rough patch. I am pretty certain these are being driven by small turbulent gusts of wind (which I can sometimes feel pass around me) and that there are no moving submerged objects. Can wind blowing on smooth water create speckle interference patterns? Is it necessary that the wavelets all be of the same wavelength for a speckle pattern to develop? ANSWER SUMMARY My conclusion is that the phenomenon I observed is not caused by 'speckle interference' although some interference of the ripples is clearly taking place as well. Chris White rightly pointed out that what I described is commonly referred to as capillary waves, so I marked his answer as the correct one. My search for 'capillary waves' on the web turned up this video which is a vigorous example of what I was seeing on my lake. There is a minimum in the phase-velocity of capillary-gravity waves on a air- water interface at a wavelength of 1.7 cm which probably provides a better physical explanation for the unique qualitative appearance than 'speckle interference.' Capillary waves are also thought to have a critical role in the initiation of ocean waves. The complete mathematical theory of capillary-gravity waves does not appear simple. However, I also found this paper describing Capillary Waves Understood by an Elementary Method for anyone interested in a physical explanation.


title: Polaris distance correction and implications for cosmological measurements

text: Polaris has been a guiding light of navigation for centuries. But Polaris also happens to be the closest cepheid variable to our sun. These, together with the type II-A supernovae constitute standard units of luminosity, used to estimate distances for far-away galaxies Recently, it was discovered that our distance measurement of Polaris was off by 30%. I'm not familiar though which measurements rely more on cepheids over IIa supernovae, so i'm curious what cosmological measurements will be affected by this adjustment It is undoubtedly romantic that a star that has been a lighthouse for some many generations of sea explorers, it also represents a beacon for interstellar distances, and might still tell us things that we didn't know


title: What are anomalous threshold singularities

text: In the papers of the 1950s and 1960s, I see reference to anomalous threshold singularities. What are these? Is there a good reference that discusses this subject?


title: Calculate Rotational Intertia

text: If a can of soup, and a can of beans (tightly packed), are set in a race down a rough hill (has friction), the soup wins, because the inside of the can (soup) is not drawing energy from the system. If the can of beans is of mass 16oz (0.454kg) evenly spread, and the radius is say .05m, how can I calculate the difference in acceleration between the two. I assume I would have to calculate normal force on the inline, and then factor in the force used to accelerate the beans. How can I achieve this?


title: What is the 'bump' near $M_{\mu\mu}\approx 30\text{ GeV}$

text: In this (attached) Summer 2011 plot from CMS (twiki page), they have a plot of the dimuon invariant mass spectrum across 3 orders of magnitude in energy. There seems to be a 'bump' near $M_{\mu\mu}\approx 30\text{ GeV}$ which I have indicated in the plot. Does anyone know where the gentle but distinct rise is coming from? ![enter image description here](http://i.stack.imgur.com/PJ9Zu.png)


title: How does the introduction of living things into a closed system affect the rate of change of entropy?

text: Does the introduction of living things into a closed system increase or decrease the overall **rate of change of entropy** of a system?


title: About voltage and charge of van de graff generator

text: I have read that in case of Van de graff generator $V=kQ/r$ where $r$ is radius of the sphere. If that's the case, does the same voltage results in bigger charges in bigger radii?


title: Measurement of speed of static electric field propagation?

text: Could one measure the time delay between charging up an electrode and measuring the static electric field at some distance? For example I've looked up the specifications of an Electric Field Meter (JCI140 Field Mill Static Monitor from Chilworth Technology) and the manufacturers say that it can measure a surface voltage of 1 volt at a distance of 1/10 meter. Therefore if one charged up an electrode to 1 MV then according to the inverse square law it should be just detectable at a distance of 100 meters. Is this right? The time delay for light to travel 100 meters is about $3\times 10^{-7}$ secs which would be easily measurable with a GHz electronic oscillator. Would this experimental setup be possible?


title: Probability of measuring momentum

text: Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. **Attempt** For a momentum operator, $\hat p |\psi\rangle = p |\psi \rangle $ so $-i\hbar \frac{\partial \psi}{\partial x} = p \psi $. This implies that in momentum space, $\psi_{(p)} \propto e^{i\frac{p}{\hbar}} $ and $p = \hbar k$ The wavefunction given is: $$ \psi = \frac{A}{2} \left( e^{-i\frac{p}{\hbar}x} + e^{i\frac{p}{\hbar}x} \right) + \frac{A}{2} \left( e^{-i\frac{2p}{\hbar}x} + e^{i\frac{2p}{\hbar}x} \right) $$ Thus possible momentum measurements are $\pm \hbar k$ and $\pm 2\hbar k$. Probabilities are $\frac{A^2}{4}$ each?


title: Choice of sign of exponential argument affecting tractability of wave reflection at a boundary

text: It is possible to show that functions of the form $f_{1}(kx-\omega t)$, $g_{1}(kx+\omega t)$, $f_{2}(\omega t-kx)$ and $g_{2}(\omega t+kx)$ are all solutions of the wave equation $\dfrac{\partial^{2}y}{\partial x^{2}}=\dfrac{1}{v^{2}}\dfrac{\partial^{2}y}{\partial t^{2}}$ for $v=\frac{\omega}{k}$, although we may write a general solution to the wave equation as a superposition of either the first two functions, or the latter two functions. There is, as far as I can tell, no real difference between the two choices. Suppose we have a semi-infinite string extending from $x=-\infty$ to $x=0$, with a mass $m$ terminating the string at $x=0$. If the oscillations in the string are small, then we may take the tension $T$ in the string to be constant, in which case we have the boundary condition: $m\dfrac{\partial^{2}y}{\partial t^{2}}\vert_{x=0}=T\dfrac{\partial y}{\partial x}\rvert_{x=0}$. If we want to find the relative amplitudes of the incident and the transmitted waves, then it is typical to substitute a function of the form $y(x,t)=Ae^{i(\omega t -kx)}+A^{\prime}e^{i(\omega t+kx)}$ into the boundary condition and then solve for $\frac{A^{\prime}}{A}$. Although this is perfectly doable, what I don't understand is why instead substituting $y(x,t)=Ae^{i(kx-\omega t)}+A^{\prime}e^{i(kx+\omega t)}$ into the boundary condition doesn't produce an equation involving $A$ and $A^{\prime}$ that doesn't involve $t$, as the other choice of function did. If there isn't a difference in how they describe a wave propagating along a string, why should it make a difference to analysing wave reflection at a boundary?


title: Question Concerning Position Of A Particle At Any Given Time

text: After years of procrastinating i've decided not to "move ahead" with physics without getting this ridiculously trivial question clear!*I know i had asked a similar question as silly and stupid as this one, however for some reason this equation keeps haunting me cause i see it almost everywhere in kinematics! * We know that the position of a particle at any time is given by $x = x_{0} + v_{0}t + \frac{1}{2}at^2$.I'am aware that $x_{0}$ and $v_{0}$ are obtained by solving for the constants of integration at $t = 0$. But why should i even care what the velocity or the acceleration is?What purpose do they serve?What has these two quantities got to do with anything? Ok.to be more "technical" Why should one "involve" velocity and acceleration when the equation is for determining postion at any given point in time? To be honest i've tried to digest what this equation is really telling me by watching and reading numerous materials both online and offline regarding fundamentals of physics.Nothing seems to be working.I've decided to leave it to you as i seriously lack intuition.Let me reiterate by saying i've no problem with deriving this equation.


title: M-Theory: Are the branes still colliding and matter is being produced into our universe?

text: In M-Theory membranes are rippling in the 11th dimension and when they collided the bing bang happened and the ripples produced matter, is it still happening and matter is still being produced? Pardon me if I miss something, also would like to understand 11 dimensions and M-theory, any good resource for a computer engineer who has a hobby of studying physics(can't really solve and understand those complex mathematical equations, but will try every bit to learn and understand them).


title: What phenomenon is used is Optical fibre?

text: I want to know which phenomenon is used in fibre optics to transmit light rays. I visited http://en.wikipedia.org but did not find it. I also tried it on google but all in vain. Can anyone please tell me?      **ADDED:**   How is it possible to maintain a degree greater than critical angle?


title: Consistently oscillating micron sized particles in a liquid

text: I have an idea that would involve the measurement of an oscillating substance whilst the substance is immersed in a liquid. However, the method that I would use to measure the frequency of these oscillations is a novel one. Therefore, I need a benchmark. Thus, can anyone direct me to a paper that gives the details - frequency, amplitude, characteristic size of the particles, etc. - of some kind of micron sized particle that will undergo oscillations when being excited by some kind of light source? I've spoken to others in my department, and they have mentioned gold particles as such a benchmark. However, I have been unable to find a good paper detailing the types of vibrations I would need (i.e. the creation of longitudinal waves) under laser irradiation. Details on gold would, therefore, do the job nicely, but any such sized micron sized particle details would be great. Spherical particles would be preferable given the geometry of my setup.


title: Thermodynamic relations from Gibbs-Duhem

text: Given the Gibbs-Duhem relation $V dp = S dT - N d \mu$, I am having trouble deriving the following identity: $\ (\frac{\partial N}{\partial \mu})_{V,T} = N (\frac{\partial \rho}{\partial p})_T$ The problem is that the variables $N$ and $\rho$ don't appear as infinitisemals in the equation. How can I proceed?


title: Why is introductory physics not taught in a more "axiomatic" way?

text: I am engineer who was taught the standard four semester, two-year physics courses from Halliday and Resnick's book. However, after reading the insightful answers here (e.g. Ron Maimon's awesome explanation of $v^2$ term in kinetic energy) and some good books (e.g. _QED_ and Weyl's fantastic _Space-Time-Matter_ ), my eyes have opened to the possibilities of explaining deep physical knowledge from simple first principles. So, my question is: Why is undergraduate Physics not taught more in an axiomatic fashion, e.g. rigorously introduce the concepts of time and space (like Weyl does), then introduce the Galilean transformation and the concept of fields. Then proceed to show some powerful consequences of these. One can argue the classic approach is more suited for engineers and related disciplines, which probably is true. But _future physicists learn from the same books, too_ , which I find to be amazing.


title: How can we describe the trajectory of an object projected into a viscous fluid?

text: Can we quantitatively describe the motion of a particle inside a fluid? Let us consider the case when an object (say a sphere, so that stoke's law holds true) is projected into a fluid of viscosity $\mu$, with a velocity $v$, at an angle $\theta$ with the horizontal. I myself approached it through Newtonian mechanics. First off the problem is when the object enters the fluid. The buoyant force as well as the viscous force are varying. There will be some threshold velocity for which it enters the fluid. Let's assume that $v$ given to the ball is enough to take it into the fluid. But how do we describe the motion of the ball while it is midway into the fluid? I tried writing the forces (Buoyant, Viscous and Gravitational) with respect to displacement and form some sort of a differential equation but it din't work for me. So i skipped that part and considered that the ball has entered the fluid and has a velocity $v'$ and makes an angle $\phi$ with the horizontal. This part i think is simpler. At an arbitrary time, i assumed the $x$ component of velocity to be $v_x$ and the $y$ component to be $v_y$. $$\vec{F_x}=-6 \pi \eta r \vec{v_x}$$ &$$\vec{F_y}=-6 \pi \eta r \vec{v_y}+m\vec{g}+\vec{F_{B}}$$ I do not know what to do next. Help will be appreciated, in both the cases- first, when the ball is entering the fluid, and the second after entering the fluid.


title: solution of pendulum equation

text: I have the pendulum expression $$\ddot{\theta}+\omega_{o}^{2}\sin(\theta)=0,$$ where I used a Taylor expansion for the sine term: $$\ddot{\theta}+\omega_{o}^{2}\left(\theta-\frac{1}{6}\theta^3\right)=0$$ How do I solve this differential equation for a simple pendulum?


title: Dimensionless numbers or parameters that are $\ll 1$ or $\gg 1$

text: Many times, in various fields, authors use the notation: $$C\ll 1$$ or $$C\gg 1,$$ where $C$ is some parameter related to the system being studied. I know this is highly dependent on application, but is there a general range of $C$ that satisfies the relations above? I'm asking because I have a parameter that characterizes two types of behaviors of my system. For $C = 0.074, 0.005, 0.086$ type A behavior is observed and for $C = 0.78, 1.24$ type B behavior is observed. If I say type A behavior is observed for $C \ll 1$, is that consistent with the data presented above?


title: calculating the solar declination

text: I was looking at this page http://www.illustratingshadows.com/www-formulae-collection.pdf and found the equation used for estimating the solar declination (sun declination). One problem, though, I wish to cite this equation in my work and was hoping there would be a textbook that shows this same equation? Can anyone suggest a reference for this other than a website?


title: How fast does light travel through a fibre optic cable?

text: The principle behind a fibre optic cable is that light is reflected along the cable until it reaches the other side, like in this diagram: ![Fibre optic cable diagram](http://i.stack.imgur.com/JT79Y.gif) Although I know that the light is slowed down somewhat because it's not going through air, I've always wondered about another factor: what about the fact that the light path is zig-zagged rather than straight? Doesn't that significantly increase the distance that the light has to travel? If so, by how much does it slow down the time the light takes to travel through the cable?


title: What are circles on broth (eating soup) surface?

text: Think about broth in the soup, usually it has circles on its surface. What are their properties? Why there are many of them (not a few big blobs)? Are they depended on liquid's temperature? What needs to be added to water so these kind of circles appear? I can only guess that it is because of oil in the broth, and shape is due to tensions on the surface; is that correct? ![enter image description here](http://i.stack.imgur.com/fuoXV.jpg)


title: Is it correct to assume that a stretched rubber-band has negative entropy change?

text: If so, how could we express it in equations connecting S,T,Q? I was wondering if the net change is heat transfer was positive; Since we could feel the heat when it is stretched.


title: How are matrices used to represent quantities, and what is the meaning of a matrix?

text: So I'm reading this text on Quantum Mechanics, and it goes through a few chapters that I understand fairly well including probability. But then it says that all quantities, like position and energy of an object, are represented in a matrix, and that quantities have associated probability distributions. I kind of get this, although I'm a little unclear about whether we're talking about full m-by-n matrices or just vectors. If it's vectors, yeah, I'm sort of familiar with that. But if not, how do you use a full m-by-n matrix to represent a quantity? And then further along, it says that is the mean of a matrix, but doesn't say what that is. Is it the average of all the coordinates in the matrix, so it's $\displaystyle \frac{1}{mn}\sum_{(i,j)\in \ulcorner m \urcorner \times \ulcorner n \urcorner}a_{ij}$? Or are each of the columns supposed to represent separate quantities and then I guess the mean is a vector of the means of the columns? The only guidance the text gives in this regard is "Some of the basic rules of quantum mechanics involve simple relations between quantities, expressed in terms of matrices, and corresponding relations between mean values. Consider a quantity represented by a matrix $M$. Let $<M>$ denote its mean value. For any number $z$, the matrix $zM$ represents the original quantity multiplied by $z$. Its mean value is $<zM>=z<M>$." And so on. But nowhere does it define the mean of a matrix, it just jumps into this notation. Some quick websearching showed that there doesn't seem to be any consensus on what is meant by the mean of a matrix representing a quantity.


title: In a radioactive Bose-Einstein condensate, would all the atoms disintegrate simultaneously?

text: They're all supposed to do the same thing - so I suppose they would? Also, would the resulting half-life be the same as that of the individual atoms it is made up of?


title: How is the direction of a compass affected by a current in a wire?

text: If I place a compass over a wire(such that the wire is positioned north-south) with charge flowing through it, and it points northeast, how can I determine the direction of the electron current flowing through the wire?


title: Derivation of rest energy in Landau & Lifshitz

text: In Landau & Lifshitz _The Classical Theory Of Fields_ there's a statement: > $$\mathscr E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}\tag{9.4}$$ This very > important formula shows, in particular, that in relativistic mechanics the > energy of a free particle does not go to zero for $v=0$, but rather takes on > a finite value $$\mathscr E=mc^2.\tag{9.5}$$ This quantity is called the > _rest energy_ of the particle. But this is the direct result of choosing Lagrangian as $$L=-mc^2\sqrt{1-\frac{v^2}{c^2}}.$$ We could as easily subtract $-mc^2$ from the Lagrangian to have $L^\prime=L+mc^2$ and have nothing changed in equations of motion, but now having the energy $$E^\prime=mc^2\left(\frac1{\sqrt{1-\frac{v^2}{c^2}}}-1\right),$$ which would give rest energy $E^\prime_0=0$. So my question is what is the physical significance of using $(9.4)$ and corresponding Lagrangian to define rest energy? Why is such a seemingly special-case result makes the authors say "in relativistic mechanics the energy of a free particle does not go to zero" as if it's a general result?


title: Quantum superposition and fate

text: First of all, sorry for my knowledge of physics. Maybe my question is too obvious but I want to ask it. I am thinking about fate and if it exist or no. According to my assumption if I take any system, for example cube, throw into it some atoms and I will know all the laws which exist at this cube, I can predict position of this atoms at any moment of time. On the other hand I know about such experiment called "Schrödinger's cat" and quantum superposition. So, it brings uncertainty to my system. If this uncertainty will exist at cube than I can't predict anything. Taking into account this assumptions I can say that at our universe fate doesn't exist. But I know that during this choices parallel universe can be created. It's horrible for me to imagine numerous number of parallel universes. What you can say about it? It's naive or it makes sense?


title: Does large acceleration have to cause damage to the human body?

text: My whole life, I've heard that large accelerations cause damage to humans (e.g. g-forces in space movies). However, after reading about general relativity, it seems to me that a strong force which affected a body uniformly (such as a very massive object a good distance away) would cause no stresses whatsoever on the human body due to the lack of tidal forces. It seems to me that the usual stresses of acceleration (like those experienced in a spacecraft) are caused by the fact that the engines push the ship, which must then push the person, causing the part of the person touching the ship to be accelerated at a different rate than the rest of his body. Is this true? Or would a uniform acceleration on the body still cause stresses?


title: Intuitive description of what a "Fermi Gas" really is?

text: This question is based in the area of material equations of state. I am wanting to know what a Fermi Gas really is. I have searched in several places for a decent description, but I have not found one that makes sense to me. Wikipedia gives a definition of an ideal Fermi gas as > _It is the quantum mechanical version of an ideal gas, for the case of > fermionic particles._ I guess I am looking for a more tangible or relatable description of a Fermi Gas if there is one...


title: What is R-symmetry with supersymmetric theory?

text: What is R-charge and R-symmetry? In usual context, $N=2$ supersymmetry has $U(1)$ R-symmetry. I don't understand what this means. Could you explain to me with more examples? i.e R symmetry with relevant gauge groups will helpful. For example in $d=7$, $N=2$ supergravty we have R symmetry $R=sp(2)$. I wonder how this comes from.


title: Questions about Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper

text: I am reading the famous and concise Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper _Quantized Hall Conductance in a Two-Dimensional Periodic Potential,_ Phys. Rev. Lett. 49, 405–408 (1982), where I have several subtle questions about the details. At the beginning, it says with a weak periodic potential and flux $\varphi = p/q, p,q\in \mathbb Z$ per unit cell, "each Landu level is split into $p$ subbands..." But on the last page, it says with strong periodic potential, $p$ and $q$ are interchanged.   * Are there $q$ subbands in the latter case?   * Why $\varphi \mapsto 1/\varphi$ in the strong potential case? In the figure below,   * what do the arrows and dots all mean?    * How do they explain the quantized conductance?! ![enter image description here](http://i.stack.imgur.com/uqS0C.png) Any ideas? Many thanks! To be continued...


title: Showing the equivalence of lagrangians?

text: I have a lagrangian written as: $$\mathcal{L}_H = \text{Tr}\left[\,(D_\mu \Phi)^\dagger D^\mu \Phi\right] - \mu^2 \text{Tr}\left[\,\Phi^\dagger \Phi\right] - \lambda (\text{Tr}\left[\,\Phi^\dagger \Phi\right])^2 $$ Where the field is: $$\Phi \equiv \frac{1}{\sqrt{2}}(i \sigma_{2} \phi^*, \phi) = \frac{1}{\sqrt{2}} \begin{pmatrix}\phi^{0*} & \phi^+ \\\\-\phi^- & \phi^0\end{pmatrix} \;,$$ and $$D_\mu \Phi \equiv \partial_\mu \Phi + i g \frac{\sigma^{i}}{2} W^{i}_\mu \Phi - i \frac{g'}{2} B_\mu \Phi \sigma_3 \;.$$ And I need to show that that is the same as: $$\mathcal{L}_H = \left[(\partial_\mu+ig{\bf{T}\cdot W_\mu}+ig'\frac{Y}{2}B_\mu)\phi\right]^\dagger\left[(\partial_\mu+ig{\bf{T}\cdot W_\mu}+ig'\frac{Y}{2}B_\mu)\phi\right]-\mu^2(\phi^\dagger\phi)-\lambda(\phi^\dagger\phi)^2 $$. Now, I have easily shown that the potential part really is the same, by explicitly doing matrix multiplication, and showing that the expression with the trace is the same as the one without, but how to show the kinetic part? What to do here? Any help is welcome :)


title: If everything is relative to each other in this universe, why do we keep the Sun to be the reference point?

text: and study the solar system and universe relative to it and why not relative to the Earth?


title: At the atomic level, is heat conduction simply radiation?

text: Radiation and conduction are two ways that heat is transferred. Convection isn't really a mode of transfer as the actual heat transfer really occurs through radiation/conduction and not by some other process. What I was wondering was if this were more general; that at the atomic level, radiation is the _only_ mode of heat transfer. So during heat conduction, do the atoms transfer heat by applying forces on each other or do they emit radiation which gets absorbed by the neighbouring atoms or is it some other way? Thank you.


title: Integral of the product of three spherical harmonics

text: Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: $\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)Y_{l_3}^{m_3}(\theta,\phi)\sin(\theta)d\theta d\phi =$ $\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\\ 0 & 0 & 0 \\\ \end{array} } \right) \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\\ m_1 & m_2 & m_3 \\\ \end{array} } \right)$ Where the $Y_{l}^{m}(\theta,\phi)$ are spherical harmonics. Or does anyone know of a reference where the derivation is given?


title: What is the difference between north and south magnetically?

text: Is there a reason that magnetic field lines are drawn from north to south or is this a purely arbitrarily defined protocol (for convenience)? Essentially what I am asking is: without a reference magnet (like the earth) is there a way determining what pole of a magnet is north and which is south?


title: Classifying all possible BPS configurations in string theory

text: Is there a classification of _all_ possible BPS configurations in string theory? It has to include Calabi-Yau orbifolds, intersecting D-branes, coincident D-branes, etc. . To simplify matters, take the limit of zero string coupling. Thanks.


title: Why build nuclear reactors on shorelines?

text: While not directly a physics question, I can't think of forum better capable of answering my question. In discussions over Japan's nuclear reactor situation the observation was made that reactors there and in the United States are built on shorelines or close to large bodies of water. Is this purely for the "backup backup backup" cases where immediate water is necessary or are there more reasons justifying these locations?


title: Relationship of the abundance of an isotope and the vapor pressure

text: Say I have a sample of some alkali metal - Rubidium for example. A sample of Rubidium consists of approximately 27% $^{87}$Rb and 73% $^{85}$Rb. Does the vapor pressure of each isotope individual _necessarily_ have to be in the same proportion? More specifically, assuming I have enough Rubidium around that I have enough atoms of each isotope to reach the saturated vapor pressure, is there any physical reason why the ratio of the vapor pressures would match their relative abundances? I guess this is coming from this perspective - the linear absorption spectrum for a sample of natural rubidium at a specific fine transition (D2 line - 5S1/2 ->5P3/2 if you are interested) looks like image below, the $^{87}$Rb peaks (outer) are clearly smaller than the $^{85}$Rb peaks (inner), and this amplitude ratio is directly related to their natural abundances. If the atoms are in a gas, what forces their vapor pressure to have the same ratio as their relative abundance? ![enter image description here](http://i.stack.imgur.com/0WgIG.jpg) Along a similar line of questioning. A sample of Rb will naturally have some trace amount of Cs in it. The saturated number density of Cs at room temperature is ~$10^9$ atoms per cm$^3$, and I'm looking at a volume of say 1cm$^3$. If I have a droplet of 1 gram of Rb in the volume, which is .001 moles, even at .001% impurity I have $>10^{17}$ atoms of cesium in the sample, clearly enough to reach saturated density. -Does this imply that you can do spectroscopy on a volume with Rb and obtain the same spectrum as a volume with a droplet of Cs in it? If not, what prevents the Cs impurity in Rb to reach full saturation. *Note: Perhaps I should make it clear : The linear absorption spectrum is proportional to the theoretical absorption (for a single atom) times the number density of atoms in the sample. The relative amplitudes of the peaks are therefore a proportional to the relative abundances of the two isotopes.


title: Several questions about spring chiller and LMTD

text: I'm designing a simple copper coil wort chiller. Digging formulas I'm having hard time to figure some things out, perhaps some simple guides may help me to go the right direction:   * LMTD need inlet and outlet temperatures differentials. On Time = 0 this can be for the inlet the initial refrigerant temperature and wort initial temperature. But what about then $T$ = 5 mins? And Outlet temperature? How do I even calculate it on $T = 0$? I bet this is related to the refrigerant flow inside the tubing, but can't find any paper to help me out.   * Wort Reynolds number. How the flow turbulence helps to improve the $U$-Number? Is there a way to calculate it (related to RPMs?) or I just use empirical numbers from tables?


title: Does local physics depend on global topology?

text: **Motivating Example** In standard treatments of AdS/CFT (MAGOO for example), one defines $\mathrm{AdS}_{p+2}$ as a particular embedded submanifold of $\mathbb R^{2,p+1}$ which gives it topology $S^1\times \mathbb R^{p+1}$ where the $S^1$ factor is "timelike." This leads to the property that there are closed timelike curves, so to obtain a causal spacetime, one "decompactifies" the timelike direction by unwrapping the $S^1$ factor to $\mathbb R^1$ while retaining the same expression for the metric (in other words one considers the universal cover of $\mathrm{AdS}_{p+2}$ It seems to me that since the metric has not changed, but the topology has, that at least at the classical level, one cannot locally distinguish between $\mathrm{AdS}_{p+2}$ and its universal cover. **The Question** _Can one locally distinguish between the two at the quantum level?_ Namely, if we were working with fully quantum AdS/CFT, would quantum effects due to global topology become apparent in local physics? I would think that the answer is yes in some sense based on intuition from a free particle moving in 1D on a circle, for example, versus on the entire real line. Apologies for the vagueness of parts of the question.


title: How can a satellite's speed decrease without its orbital angular momentum changing?

text: I have no idea what the answer is. I'm supposed to answer it within 3-4 sentences.


title: Is it possible for $\Delta x$ ($\sigma_x$) of any free particle wave packet to be decreasing at any time?

text: Consider _any_ wave packet describing a free particle (so no potential or other forces acting on it). Then it can be shown that $\Delta p$ does not change in time. However, my question is what happens with $\Delta x$ as we go forward in time? Does it have to increase at all times? Or is there a counter- example where the uncertainty in position is decreasing, if even for a short time period? My initial guess is $\Delta x$ must always increase, because $p \neq 0$, so that $\Delta p \neq0$ and hence $\Delta v = \frac{\Delta p}{m} \neq 0$. But if there's a spread in velocities, then the wave packet must also spread. Is this logic correct? Or could we have a wave packet where the back of it would move forward faster than the front, and for a certain period until that back end catches up with the front one, it would actually be narrower than at the outset, i.e. reducing $\Delta x$? If yes, how would one describe such a free particle (wave packet)? So it does seem to me that every wave packet describing a free particle will _eventually_ spread, but the question is whether there can be a time period in its evolution when it is actually becoming narrower. edit: In particular, if it does not have to increase at all times, can this be shown without appealing to time reversal?


title: Equivalence of simple formulations of qubit entanglement

text: I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a two-qubit system are entangled if the state cannot be expressed as the (tensor) product of two one-qubit states. Another "definition" states that (2) a two-qubit system is entangled if "we cannot determine the state of each qubit separately" or if (3) "measuring one qubit determines the distribution of the other". Perhaps my confusion is a simple matter of not fully understanding what the "independent" means and how it relates to the tensor product; but it's not clear to me how these statements are related. Does, for example (1) imply (2)? Are (2) and (3) equivalent?


title: How does paper make sound when it is torn?

text: We know that from our experience when we tear up a piece of paper, we can hear a characteristic sound. What is the underlying mechanism behind it? What do the dominant frequencies (edit: I don't mean there is one main frequency of the sound, it is more likely the noise produced contains mostly higher or lower frequencies) and intensity of the sound depend on? The explanation should agree with the real life cases -- here are a few that I observed (correct me if I'm wrong):   1. Dry paper makes louder sound and higher frequencies than wet paper   2. Paper produces louder sound and higher frequencies if we tear it faster   3. It is hard to tell which one produces louder sound, folded paper or single layer paper   4. Folded paper produces lower frequencies than single layer paper Also, is it the same mechanism that explains the sound produced when we are cutting paper using a scissor?


title: AdS/CFT and boundary translational invariance

text: I work in quantum information theory/condensed matter and have some very basic questions about AdS/CFT correspondence. For simplicity, I would like to restrict to 1+1 CFT <-> 2+1 AdS. I apologize in advance if the following questions are too basic or ill-posed, clearly owing to a lack of familiarity with the subject. Broadly, I am interested in the relation between boundary translations and bulk diffeomorphims, and what happens in the scenario where the boundary translational invariance is broken by introducing a conformal defect or boundary. QUESTIONS:   1. Are translations at the boundary identified with diffeomorphisms in the bulk? More specifically, given a translation at the boundary, what is the corresponding diffeomorphism?   2. When the boundary translational invariance is broken, say by introducing (A) a conformal defect in the CFT or (B) a boundary, what is the "residual" bulk diffeomorphism invariance (in cases where there is still a dual gravity theory)? Any pointers or references would be most helpful.


title: How do we know that these radio bursts are from billions of light-years away?

text: NASA just announced that they detected the first radio bursts from outside of our galaxy. > Astronomers, including a team member from NASA's Jet Propulsion Laboratory > in Pasadena, Calif., have detected the first population of radio bursts > known to originate from galaxies beyond our own Milky Way. **The sources of > the light bursts are unknown** , but cataclysmic events, such as merging or > exploding stars, are likely the triggers. > > The new radio-burst detections -- four in total -- **are from billions of > light-years away** , erasing any doubt that the phenomenon is real. If we don't know what the source is, then how do we know how far away the source is? How can we tell how far the light waves have traveled?


title: Gravitational Constant in Newtonian Gravity vs. General Relativity

text: From my understanding, the gravitational constant $G$ is a proportionality constant used by Newton in his law of universal gravitation (which was based around Kepler's Laws), namely in the equation $F = \frac{G\cdot M\cdot m}{r^2}$. Later, Einstein set forward a different theory for Gravity (based around the equivalence principle), namely General Relativity, which concluded that Newton's law was simply a (rather decent) approximation to a more complex reality. Mathematically speaking, Einstein's Theory was completely different from Newton's Theory and based around his Field equations, which also included $G$ in one of it's terms. How come two different theories that stemmed from completely different postulates end up having this same constant $G$ with the same numerical value show up in their equations? What exactly does $G$ represent?


title: probability of sequence for given rate constants

text: lets consider a copolymer, $C_{r,s}^A$ containing r number of A monomers and s number of B monomers with A at the reactive end of the polymer. The equilibrium constant for A-A, A-B, B-A, and B-B bonds are $K_{AA}$, $K_{AB}$, $K_{BA}$ and $K_{BB}$ respectively. $K_{AA} = exp(-\beta * \Delta g_{AA})$, Where $\Delta g_{AA}$ is free energy change of A-A bond formed. other equilibrium constants are also defined in similar way. $C_{r,s}^A$ = A------------- Hence i am interested in following quantities, P(AB)= probability of sequence AB--------- and P(AA)= probability of sequence AA--------- Thank you, Nitin


title: Rate of twinkling of stars

text: According to "Why do stars flicker?", the twinkling of stars (stellar scintillation) is caused by the thick layers of turbulent air in the Earth's atmosphere. While the explanation is convincing, it would be better to have some quantitative justification. Since the twinkling of stars is easily observed, it is reasonable to estimate the twinkling rate as ~ $1\textrm{s}^{-1}$. How can one obtain a similar estimate from the underlying theory?


title: Why does dark matter form walls and filaments

text: Related: How are galaxy filaments formed? And do they have any analogues in stellar formation? But I want to come at this from a different angle. Like the user asking that other question, I was a bit surprised by the walls, filaments and nodes of the large-scale structure of dark matter: ![Dark matter structure from the Millenium Simulation](http://i.stack.imgur.com/FAwhm.jpg) Intuitively, I might have expected more spherical shapes. I got a bit closer to a satisfactory explanation when I heard about an equivalent way of thinking about the situation. Instead of thinking of overdense regions collapsing (e.g. the Zel'dovich pancakes mentioned in the answer to the question linked above), one can think about underdense regions expanding to form voids/supervoids/etc. The voids are (roughly) spherical, and as they push out and collide, they compress the dark matter into walls and filaments. I'm picturing something like blowing bubbles in soapy water, giving a nice intuitive picture. ![Dark Soap Matter Bubbles](http://i.stack.imgur.com/alVQn.jpg) Now where I get hung up is that there is an obvious symmetry between thinking about overdensities collapsing and underdensities expanding, but there is an obvious _lack_ of symmetry between the structures resulting from collapse and expansion, for instance why wouldn't we get the opposite case, where the voids are filament shaped and the dark matter forms roughly spherical blobs? I have a feeling that the key is in the strictly attractive nature of gravity, but can't really put my thoughts together coherently. Would be interested to hear if anyone can elaborate a bit on this. Feel free to get technical in the answers, I have a solid math & physics background to help me interpret. Bonus points if you can draw a nice intuitive picture, though :)


title: How do I use the direction of a cross product?

text: The question is: > An electron is projected at a speed of $3.70×10^6\text{ m/s}$ in the > $u=\frac{(i+j+k)}{\sqrt3}$ direction into a uniform magnetic field $\vec {B} > = 6.43 i + 3.43 j −8.29 k$. Calculate the $x$ component of the vector force > on the electron. So I know that I'll need to use $F = q\vec{v}\times\vec{ B}$ and do the cross product between the velocity and the magnetic field but I'm still getting the hold of cross products and vectors and the direction of the velocity is really throwing me off so I don't know how to write it down and solve. If anyone can put me in the right direction that would be awesome. Thanks.


title: Deriving the expectation of $[\hat X,\hat H]$

text: For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, \hat{H}] = \frac {i\hbar} {m} \hat{P}\tag{1}$$ In the common eigenstate of $\hat{H}$ and $\hat{P}$, $|e, p\rangle$, can we do the following? $$\langle e, p| [\hat{X}, \,\hat{H}] |e, p\rangle = \langle e, p|\hat{X} (\hat{H}|e, p\rangle) - (\langle e, p|\hat{H}) \hat{X}|e, p\rangle \\\ = \langle e, p|\hat{X} (e|e, p\rangle) - (\langle e, p|e) \hat{X}|e, p\rangle \\\ = e( \langle e, p|\hat{X}|e, p\rangle - \langle e, p|\hat{X}|e, p\rangle ) \\\ = 0 $$ Since the $\hat{H}$ is Hermitian, the above derivation doesn't seem to show any flaw. Given the commutative relation, Eq (1), we know the result is wrong. What's wrong with the above derivation? [ **EDIT** ] Following the comment by Luboš Motl, I have worked out the solution and would like to share it here. The link provided by Qmechanic had the solution closely related to this question. $$ \langle e', p'| [\hat{X}, \,\hat{H}] |e, p\rangle \\\ = \langle e', p'|\hat{X} (\hat{H}|e, p\rangle) - (\langle e', p'|\hat{H}) \hat{X}|e, p\rangle \\\ = (e - e') \langle e', p'|\hat{X}|e, p\rangle $$ Note that: $$ e - e' = \frac{p^2}{2m} - \frac{p'^2}{2m} = \frac{(p+p')(p-p')}{2m} $$ $$ \langle e', p'|\hat{X}|e, p\rangle = -i\hbar \delta'(p - p') $$ where $\delta'(\cdot)$ is the derivative of the Dirac function, with respect to $p$. Then we get $$ (e - e') \langle e', p'|\hat{X}|e, p\rangle \\\ = -i\hbar \frac{(p+p')}{2m} \cdot (p - p')\delta'(p - p') \\\ = - \frac{i\hbar (p+p')}{2m} \cdot (-\delta(p - p')) \\\ = \frac{i\hbar (p+p')}{2m} \delta(p - p') $$ As we take the limit $p \rightarrow p'$: $$ lim_{p \rightarrow p'} \frac{i\hbar(p+p')}{2m} \delta(p - p') \\\ \rightarrow \frac{i\hbar}{m} p \delta(p - p') $$


title: Accelerometer problem - difficulty understanding my readout

text: I am having great difficult understand my readout from an accelerometer that I placed on a rotation wheel. The wheel spins at various speeds and i measured the radius from the center. I think I mentioned before that I had done some calculations and I was expecting $z = 1$g (which I know many disputed). And then I was thinking that the displacement in the x and y direction would be $$ x(t) = r \cos(ω t) = r \cos(2 π f t) $$ $$ y(t) = r \sin(ω t) = r \sin(2 π f t) $$ and so acceleration would be this differentiated twice. When I looked at the recording on my accelerometer it was strange as $z$ was indeed 1g, $x$ remained at 0 and $y$ does vary but not in the way I was expecting. ie like a sin or cos wave over time in fact the signal oscillated about a particular value in the same was $x$ and $z$ does. however, this value is not quite as much as I would expect. Any help or advice to offer insight into this would be greatly appreciated you guys


title: Can you extract energy from "hot" things without a temperature differential?

text: I've been reading about extracting energy from heat, particularly Sterling engines. There's always a temperature gradient. Heat flows from the hot side to the cold side through the working fluid. As the temperature differential approaches zero the engine stops. My question is that why do we even need a temperature differential? Even more broadly why do we use energy to cool things? Let's take a gallon of water at 50 deg C. Water has a specific heat of 4.18JdegC/g. Ignoring the heat of fusion and following Q = CMdT, just by virtue of it being "warm" (323 deg C above absolute zero) wouldn't the water "contain" 3780 * 4.18 * 323 = 5104 kJ of energy (3780 grams per gallon of water). Why couldn't you extract any of this energy as work and as a result of extracting the energy the water cools itself (it gives up energy, for every 3780*4.18 J extracted it should lower by 1 deg C). I'm invisioning the heated water (or anything heated above abs zero) as an energy sink or battery. Theoretically you should be able to extract 5104 kJ of energy from this 50 deg C gallon of water. In the Sterling engine we could lower the cold side with liquid nitrogen but it takes energy to make this. In an AC system we spend energy to lower the temperature in the room. Since "cold" is the absence of "heat" or energy, shouldn't cooling things actually create energy for us (the hot things that earlier had energy imparted into them give up their energy to make work). It seems wasteful to just waste the heat on warming up something at a lower temperature differential such as the cool side of the Sterling engine or the ambient air by the condenser.


title: Meaning of boundary condition for steady current density?

text: Although I understand the derivation of boundary condition in case of steady electric current but I did not understand, that the electric field which is in direction of $J$ current density that is applied between two conducting medium so that if we analyse this electric field we should get that both tangent and normal component bet two medium is the same not only tangent component and the value of $J$ change due to different value of conductivity bet medium while the direction should still the same. ![enter image description here](http://i.stack.imgur.com/QAlyw.png) ok my question what if the field like i draw why the direction of current change although the same source should make the current,electric field in both conductors in the same direction ![enter image description here](http://i.stack.imgur.com/D6YhS.png)


title: Change in the Higgs mechanism

text: The most important thing about the Higgs mechanism is that the higgs mechanism is one of the simplest mechanisms ( _certainly the simplest way to break the electroweak symmetry_ ) of elementary particles. There is an unwritten law in particle physics: When the mechanism is simple, just light Weight particles arise. What should be change to Higgs mechanism for the production of heavier particle?


title: Faraday's Law and Variable Reluctance Sensor

text: My coworker's and I have been discussing the variable reluctance sensor and have had trouble establishing the relationship between the magnetic flux, voltage and tooth position (when the VR sensor is coupled with a toothed wheel). Generic VR sensor configuration: ![Generic VR sensor configuration](http://i.stack.imgur.com/6isBd.png) Our understanding is that the magnetic flux will vary with the wheel position with the maximum occurring when a tooth is directly centered underneath the pole piece. The resulting waveform is sinusoidal. What we can't figure out is relationship between Faraday's law and the time varying magnetic flux. We know Faraday's law to be: ![Faraday's law](http://i.stack.imgur.com/n4Ikq.png) But we've also seen this graph from allaboutcircuits.com: ![Time varying magnetic flux and voltage](http://i.stack.imgur.com/dwfB3.png) Which implies that the voltage leads the magnetic flux by 90 degrees. This is where we're confused and probably missing some simple piece of the puzzle - Faraday's law implies that voltage is the negative derivative of magnetic flux. So if the flux is characterized by sin(t) then voltage would be -cos(t) which actually lags the flux by 90 degrees. Thanks!


title: Why are magnets attracted to negative side in battery

text: I have been experimenting with magnets, and built an Homopolar motor however it works but the magnet only attracts the negative side of the AA battery (the flat side of a battery) which I thought was intriguing as its my first time I found this out myself, but my question is why is the negative side only attracted to the magnet? The battery is made of Zinc Chloride.


title: Spring compression and Momentum

text: I am asked to rate a series of elastic collisions in order of greatest time of max compression to least time of max compression for several vehicles with varying masses and velocities, which strike a spring with a spring constant k. I can determine the Momentum of each case, as I am given the masses and velocities. Additionally, I can determine each of their kinetic energy. I know that the kinetic energy of the car will be converted into potential energy in the spring: $$1/2mv^2 =1/2kx^2$$ Also, I know the impulse of the car's is going to be $$Ft= Δvm $$ $$ t=Δvm/F$$ I also know that the Force on spring will be $F=kx$, but I am not sure how the magnitude of the momentum of the cars' is going to relate to the time of maximum spring compression. I am asking for a little guidance in my reasoning.


title: Why can i replace a gauge field by the current it couples to in the calculation of a greens function?

text: I am reading about anomalies in QFT at the moment and have a related question. Often people calculate the time ordered expectation value of some fields (in QED for example) by replacing the field $A_\mu$ with the current $j_\mu = \bar{\psi}\gamma_\mu\psi$. For example in Peskin and Schröder (p. 311) they derive the Ward identities in QED and state that the object $$\langle T j^\mu(x)\psi(y)\overline{\psi}(z)\rangle$$ corresponds to the electron-photon vertex. But should the vertex not correspond to $$\langle T A^\mu(x)\psi(y)\overline{\psi}(z)\rangle$$ where $A^\mu$ is the EM potential? The same thing happens in the calculation of the triangle anomaly. Often you see people calculate something like $\langle j_1^\mu(x)j_2^\nu(y)j_3^\rho(z)\rangle$ where the $j$s are (axial or vector) currents of fermions coupled to some vector fields. This gives than the diagram with three external vector bosons and a circulating fermion. It is clear to me that $\langle T j^\mu \ldots \rangle$ and $\langle T A^\mu \ldots \rangle$ give me the same diagrams when I start to calculate. I am just wondering if I can prove in any way (from a generating functional perhaps) that these Green's functions are really the same. (I use Green's function in the sense of time ordered vacuum expectation value of fields.) Hopefully somebody can clarify this a little.


title: Thorium based nuclear reactor

text: My question is how thorium works as a radioactive nuclear fuel and what are the special properties of thorium based nuclear reactor.


title: spectral eigenvalue staircase and quantum system

text: in a d-dimensional system of Quantum physics , does the Eigenvalue staircase $ N(E)= \sum_{E_{n}\le E} 1 $ determine ALL the properties of Quantum System ?? for example, let us assume that the System is Chaotic and that Spectral eigenvalue staircase is given by $ N(E)= <N (E)>\+ \frac{1}{\pi}ArgZ(1/2+i \sqrt E )$ with $Z(s)= \prod_{n=0}^{\infty}\prod_{l_{p}}(1-l_{p}^{-s-n})$ the Gutzwiller zeta function


title: How to draw the Poincaré patch of ${AdS_3}$?

text: My main reference for this question are these notes (maximally symmetric spaces.pdf) by Kurt Hinterbichler. I'm using Global Coordinates: \begin{align} x^0&=\sec{R}\cos\tau\\\ x^1&=\sec{R}\sin\tau\\\ x^2&=\tan{R}\cos\theta\\\ x^3&=\tan{R}\sin\theta\\\ \end{align} with $\tau\in[0,2\pi)$, $R\in[0,\pi/2)$ and $\theta\in(-\pi,\pi)$, and Poincaré coordinates: \begin{align}x^0&=\frac{1}{2u}\left[1+u^2-t^2+(z^3)^2\right]\\\ x^1&=\frac{t}{2}\\\ x^2&=\frac{1}{2u}\left[1-u^2+t^2-(z^3)^2\right]\\\ x^3&=\frac{{z}^3}{u} \end{align} with ${u\in\mathbb{R}^+}$ and ${z^3,t\in\mathbb{R}}$. This way, relating the coordinates, I find that $$u=\frac{\cos{R}}{\cos\tau+\sin{R}\cos\theta},\hspace{0.25in}t=\frac{\sin\tau}{\cos\tau+\sin{R}\cos\theta},\hspace{0.25in}z^3=\frac{\sin{R}\sin\theta_1}{\cos\tau+\sin{R}\cos\theta_1}$$ and what I want to do is to reproduce Figure 25 in the notes. I somehow figured that since this patch cover only ${x^0+x^2>0}$, then maybe the plot is simply the region $$\cos\tau+\sin{R}\cos\theta>0$$ so I made the plot in Mathematica **(*)** and got this: ![Poincaré patch of $AdS_3$?](http://i.stack.imgur.com/qCTyz.png) which is a bit different from Figure 25 in the notes or the drawings usually found in the literature, like this: ![enter image description here](http://i.stack.imgur.com/51mDa.png) or like this in Wikipedia: ![enter image description here](http://i.stack.imgur.com/ZUYOO.png) So how should the Poincaré patch of ${AdS_3}$ be drawn? Might it be a software issue from Mathematica? I've noticed that sometimes the edges doesn't fill completely because of them being boundaries, like with Minkowski's diagram: ![enter image description here](http://i.stack.imgur.com/aHxWS.png) So what is the issue here and how should the plot be done? **(*)** These are the command lines I used in Mathematica: ![enter image description here](http://i.stack.imgur.com/nxSms.png)


title: Weight on carpet vs. hard floor

text: Will the scale say I weigh more, less, or the same on a carpet as compared to a hard floor? You can assume the scale works via spring mechanism. Free body diagrams encouraged!


title: Why plane stress condition is taken for thin plates

text: Why plane stress is taken for thin plates? It says in the books that the stress variation is small for thin components and is close to zero. Why is that so? Also why stress at free surface is zero? (talking with respect to a solid under simple loading e.g. a bar under uniaxial loading) and free surface is the boundary of a specimen right?


title: Would weighing scales show a negative weight if placed in a vacuum?

text: The weighing scales in my kitchen are currently showing 0 Kg. However, there is a column of air between the scales and the ceiling that is presumably exerting pressure on them. If I were to place those scales in a vacuum, would those scales show a negative weight?


title: Why an object travels a circular path?

text: It might sound stupid but seriously I couldn't find _why?_ We have a mass with a constant speed, which is acted on by a unit force which is always at right angles to its direction of motion. Why do the mass travels a circular path!?


title: What to do when you are sitting in a closed room without any fan or AC

text: I suffered this today, when the solar panels of our college stopped to work. Imagine a situation in which you are sitting in a botany lecture conducted by a 73 year old man, who cannot speak loud and hates the noise of running fans. As I said earlier, solar panels stopped to work, so ACs were non-functional now and more over, my college is situated on the road side, and you might be knowing about the Indian traffic, population & noisy horn's of vehicles, so door was also closed, and all we have to do is just sit inside a room without any ventilation. After this nearly whole class started making air for themselves with papers, after which the intensity of heat increased more, I think the reason for this was the heat generated by everyone's body while tilting papers before their faces. So the question is: What should one do in that situation, or what will make one most comfortable, sitting idle without any expenditure of energy (which generally cant happen :)) or start tilting the papers to overcome heat in the surroundings? I have done both things, but both of them failed, so what should I actually do to minimise the heat intensity. _I want an answer related to thermodynamical physics._


title: An error in Gravitation by Misner Thorne and Wheeler?

text: I was studying on Gravitation the PPN formalism. Since in equation (39.41) pag. 1087, the term $1 + \dfrac{v^2}{2}+(2+\gamma)U = 1 + \dfrac{v^2}{2}+3U$ (the second in GR) looked odd, I tried (several times) to derive the formula, but I always find: $1 + \dfrac{v^2}{2}+U$ Is this one of the two famous errors metioned by Wheeler? I am really going crazy over this...


title: How does quantization solve UV catastrophe in black body radiation? What would happen if there was no Planck constant $h$?

text: Planck's Law is $$I(\nu,T)=\frac{8\pi\nu^3}{c^2}\cdot\frac{1}{e^{h\nu/kT}−1}$$ This solves the UV catastrophe. For higher frequencies, intensity goes to zero. It does so because of $e^\nu$ not because there is a $h$ constant. If we remove $h$ parameter from the equation, it still goes to zero for higher frequencies. Right? So why is "UV catastrophe is solved by quantization" said? Note: I'm not saying if h = 0. I'm saying if there was no h constant. Like this: $$I(\nu,T)=\frac{8\pi\nu^3}{c^2}\cdot\frac{1}{e^{\nu/kT}−1}$$


title: Heat loss using alternating current

text: I am looking for somebody who can explain this to me. As I have read in physics books, the Joule-Lenz Law (Joule effect of heating) is represented by the formula: $$ W = I^2Rt $$ which can be transformed into: $$ W = VIt $$ (using V = IR). The when I have reached the lessons about alternating current, it was said that this formula can be used also for problems connected with it but it was said that in order to minimize the heat loss, the alternating current that is produced by the power stations is transformed into alternating current with high voltage. If we see the formula $W = I^2Rt$, that makes sense but as soon as we substitute with $V = IR$ and get $W = VIt$ this sounds strange (we already know that power maintains same when we use transformers). So where do I make a mistake and how does it work? P.S. Why does all the energy of electricity is transformed into heat?


title: Jacobian for Kronecker delta

text: I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, > this expression yields: > > $$ \left|\frac{\partial y^i}{\partial x^j}\right| \left|\frac{\partial > x^\alpha}{\partial y^\beta}\right|= 1 ,$$ > > With $ \left|\dfrac{\partial y^i}{\partial x^j}\right|$ being the Jacobian > for transformation $y^i=y^i(x^1.....x^n)$, and $\left|\dfrac{\partial > x^\alpha}{\partial y^\beta}\right|$ being the Jacobian of the _inverse_ > transformation. My question is, how do you get eq. 1 from the Kronecker delta, as they are merely jacobians of coordinate transformatios, and being inverse of each other, are 1. But how do they follow from $\delta^i_j$'s expansion? _(i.e. DERIVE eq. 1 using the expansion for kronecker delta)_? I am most probably making a conceptual error, but this is the first time I have seen such a representation of the kronecker delta.


title: Are the cylindrical and spherical form of Jeans' equations equivalent?

text: The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy axisymmetric Jeans equations? (numerically rather than analytically, otherwise it would all get very messy...) More background: Providing them in sufficient detail to be helpful would be very longwinded (as presumably you'd need to compare all three of the first moment equations in both co-ordinate systems) but for a bit more background people can always look here http://en.wikipedia.org/wiki/Jeans_equations, or in for an in depth discussion look in Galactic Dynamics by James Binney and Scott Tremaine.


title: Transformation of a time-dependent acceleration to another frame

text: How do I transform a time-dependent acceleration between to rest frames? I was given a particular problem and while I found possibilities [1] [2] for acceleration-transformations when the acceleration $a$ is uniform, I did not find anything about transforming an $a(t)$. I was thinking about using infinite Lorentz boosts after an infinitesimal time $\delta t$ to accomondate for the change in $\gamma=(1-v^2/c^2)^{-1}$, but didn't get anywhere. Another idea was to consider the Lagrangian of a free particle where the potential results in a time-dependent acceleration. In my special case, the acceleration is given in a train's proper time $\tau$ as $a(\tau) = (1-\alpha\tau)^{-1}$ m/s² with a constant $\alpha$. The question is to compute the train's speed as a function of the proper time in the train station's frame. The train left the station at $\tau = 0$. My take using the Lagrangian was the following: $ L = T - V$ -- where $V = 0 \Rightarrow L = -mc^2/\gamma$. Now consider the acceleration to result from an external potential (the train's engines for that matter) with a force $\vec F = -\vec \nabla V = ma \vec e_x$. This leads to $V(\tau) = -\frac{mx(\tau)}{1-\alpha \tau} + \mathcal C(\tau)$. Using the Euler-Lagrange equation $\frac{d}{dt}\frac{\partial L}{\partial \dot x} - \frac{dL}{dx} = 0$ I unfortunately get nowhere as $\partial L / \partial \dot x = 0$ and $dL/dx = - m/(1-\alpha\tau)$ which therefore should equal zero. So I think I didn't understand the concept on a fundamental level and hope you can enlighten me.


title: plane wave move out

text: I have a plane wave that is recorded by a set of receivers with $x$ spacing between them (see the sketch). in the sketch (plane wave in black slant line, receivers are the little circles in black and the line in blue is the surface). The place wave arrives with an angle to the normal (alpha) at the receivers. The wave was generated from 2000 cells to the left of my receivers line and 5000 cell deep below my line. Given I have the recording time of the arrival at each receiver (black circle) I was trying to correct for it and shift it into a normal incident rather than slanted as of the plane wave originated from deep below the receivers (5000 below). I do have the velocity of the medium below, arrival time, and angel of incident and I want to accomplish that. My final gaol is not to linear move out the arrive but rather shift it is if it was originated below the receivers. one approach I am thinking of is by multiplying the recording time at the first station by the cosine of the angel and then subtracting that from the first recording time $t_1-\cos(\alpha)\times t_1$.![enter image description here](http://i.stack.imgur.com/zoqTu.png)


title: Does fractional charge imply fractional statistics?

text: Does fractional charge imply fractional statistics (e.g. anyons)? If not, are there some relations between them?


title: Where I can find information about how helium gas diffuses through different polymer materials

text: I need more experimental information about helium gas diffusion in solid materials such as different plastics, metals, ceramics etc. For example I am **scientifically curious** about what is the polymer material with lowest diffusion coefficient all around? Please help! Thanks!


title: WHY is a Refracting telescope image distance $S'=-\infty$?

text: ![enter image description here](http://i.stack.imgur.com/vUi8g.png)![enter image description here](http://i.stack.imgur.com/kRqij.png)![enter image description here](http://i.stack.imgur.com/lOO0v.png) In the above slide, it says that "Eyepiece '2' makes the image distance $S'$ approach $-\infty$." However, What I don't understand is that if I place my eye right at the Eyepiece 2, then I would see the image as if the object (for example, moon) is right in front of me as if I'm observing a rock in my hand... Therefore, I feel like $S'$ has to be closer to zero or some finite number! not negative infinity... Also, eventhough $S'$ is actually negative infinity, I quite don't understand how the moon can be observed as if I am observing a rock right in my hand... If $S'=-\infty$, then my brain says that we can't really observe the cracks, rocks, or even sand in the moon since the object is so much farther away from me... Could somebody explain why I am confused???


title: Fluctuation interaction between two uncharged spheres

text: I'm trying to figure out quantitatively what is the force, acting between two uncharged conducting spheres and I've got stuck. It is not a kind of homework - it is just a simple act of curiosity. I'd like to begin with several clarifications.   Firstly, let's consider where the interaction comes from. Imagine one uncharged metal sphere. Despite being uncharged, it has free electrons inside. So, there are thermal fluctuations of charge density in the sphere. One can think of them in terms of multipole moments: thermal fluctuations induce random dipole moment, quadrupole moment and so forth.   For simplification I'll consider only the first term in this row: fluctuations of dipole moment. Using formula: $$ U=\frac{1}{2}\int \rho \varphi dV $$ one can show that self energy of such a sphere in the first approximation is: $$ U_{self}=\frac{1}{2}\frac{\mathbf{d}^2}{R_0^3} $$ where $\mathbf{d}$ is dipole moment, $R_0$ is radius of the sphere.   Therefore, fluctuations are described statistically as follows: $$ W(\mathbf{d})\sim e^{-\frac{U_{self}}{kT}} $$ where $W(\mathbf{d})$ is probability of having a thermal induced dipole moment $\mathbf{d}$.   Now consider two uncharged spheres. The fluctuations in them are, of course, correlated. For example, imagine two spheres on the axis $Ox$ with the distance $R$ between them. Let it be so that due to fluctuations the induced dipole moment of the first sphere is directed upwards, along $Oy$ axis, and is equal to $\mathbf{d_1}$. The field this sphere would create is: $$ \mathbf{E}=\frac{3(\mathbf{d_1}\cdot\mathbf{R})\mathbf{R}-\mathbf{d_1}R^2}{R^5} $$ and in the position of the second sphere it would be directed downward, against the $Oy$ axis. Thus preferred direction for thermal fluctuations of the charge of the second spehere is created: the most probable state is that the dipole moment $\mathbf{d_2}$ of the second sphere is directed against $\mathbf{d_1}$. Therefore attraction must appear.   It seems clear that the dipole moment of one sphere would always be directed (most probably) so that attraction would exist.   However, when I try to describe this situation quantitatively, I face problems I don't know how to resolve. Please, take a look at my calculations and say, what's wrong. Let's begin.   The full energy in system is given as follows: $$ U(\mathbf{d_1},\mathbf{d_2})=\frac{1}{2}\frac{\mathbf{d_1}^2}{R_0^3}+\frac{1}{2}\frac{\mathbf{d_2}^2}{R_0^3}-\frac{3(\mathbf{d_1}\cdot\mathbf{R})(\mathbf{d_2}\cdot\mathbf{R})-\mathbf{d_1}\cdot\mathbf{d_2}R^2}{R^5} $$ (for simplicity let's take the radiuses of these spheres equal).   The probability: $$ W(\mathbf{d_1},\mathbf{d_2})\sim e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}} $$ To be more exact, if I understand it right, one can say, that: $$ dW(\mathbf{d_1},\mathbf{d_2})=A\cdot e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}}d\mathbf{d_1}d\mathbf{d_2} $$ where $d\mathbf{d}=d\mathrm{d_x} \cdot d\mathrm{d_y} \cdot d\mathrm{d_z}$ and $A$ is normalization constant. Using $\int dW=1$ we can find $A$ as: $$ A=\frac{1}{\int e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}}d\mathbf{d_1}d\mathbf{d_2}} $$ and then find average energy as follows: $$ \bar{U}=\frac{\int U(\mathbf{d_1},\mathbf{d_2})\cdot e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}}d\mathbf{d_1}d\mathbf{d_2}}{\int e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}}d\mathbf{d_1}d\mathbf{d_2}}=-\frac{d\mathrm{ln}\int e^{-\frac{U(\mathbf{d_1},\mathbf{d_2})}{kT}}d\mathbf{d_1}d\mathbf{d_2}}{d(\frac{1}{kT})} $$ Using the fact that energy is quadratic form of $\mathbf{d_1}$ and $\mathbf{d_2}$ one gets: $$ \bar{U}=\frac{6kT}{2} $$ as in any cases when energy is quadratic to "coordinates" (we have 6 degrees of freedom, and energy, equal to $kT/2$, corresponds to each one). So $\mathrm{grad}\:\bar {U}=\mathbf{0}$ and it looks like the interaction is absent.   **My questions are:**   What do I do wrong? Is there a lack in my qualitative understanding? Or there is a mathematical mistake in my calculations? Is this a well-known problem and if it is, I wonder if you can give me some links to the literature, where I can read about it.   I have several ideas on how to cope with difficulies.   It may be worth getting average of force between dipoles $\mathbf{F}=(\mathbf{d}\cdot \bigtriangledown )\mathbf{E}$, which looks the same as getting average only of the part of full energy in the system (skipping self-energy terms). I'm sorry for my grammar mistakes, I hope they didn't bother you much. Thank you for your attention and your advice.


title: Classical/Quantum Coin Toss

text: I am having a brainfreeze moment and have confused myself, help appreciated! Classical Coin: Heads OR tails. Quantum Coin: Superposition Heads AND Tails. Classical Mechanics: Deterministic (in principle, if not in practice) if I repeat the same experiment I get the same result. Quantum Mechanics: Non-deterministic no way that I can predict if I get heads or tails. Now think of some physical implementation of a quantum coin perhaps I send some electron to mirror, afterwards it is on a superposition on both sides of the mirror. Perhaps reflected (heads) with probability 0.9 and transmitted (tails) with probability 0.1. My question is does a classical analogy exist here? It can't be both deterministic and agree with the probabilities predicted by quantum mechanics right? Is the problem just that I should not be applying classical physics at all here? Does this question even make any sense?


title: Euler Equations, Sod shock tube & conservation

text: Conservation of momentum? I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be conserved wrt time. Density and total energy are indeed conserved, however, momentum is not. The Euler equations are given by $$ \frac{\partial}{\partial t}\begin{bmatrix} \rho \\\ \rho u \\\ E \end{bmatrix} \+ \frac{\partial}{\partial x}\begin{bmatrix} \rho u \\\ \rho u^2 + p \\\ u(E +p) \end{bmatrix} = 0 $$ where pressure is related to the conserved quantities by $ p = (\gamma - 1)(E - 0.5 \rho u^2) $ The Sod shock problem splits the domain into two regions separated by a density and pressure discontinuity with initial velocity zero. That is, $\rho_L = 1, \rho_R = .125; p_L=1, p_R =.1; u_L=u_R =0.$ These initial conditions imply that ρu=0, therefore, momentum should be zero throughout the simulation. The solution profiles are well known and can be found here. We see that ρ >0 and u > 0, therefore, there is no way for momentum ρu = 0 (which it should be from the intial conditions). As a result, I do not even see why it is reasonable to expect that momentum would be conserved. A paper by Sod which surveys some methods for solution, on page 20, list a table which shows momentum to be increasing linearly. I generally do not work in this area, so maybe I am missing something basic. Can anyone shed some light on this? Thanks!


title: Are right handed neutrinos actually antineutrinos and vice versa?

text: Is it experimentally ruled out that right-handed neutrinos are actually antineutrinos, and left-handed antineutrinos are neutrinos ?


title: Stronger than Newton's laws?

text: According to the Newton mechanics, the force is responsible for changing the body velocity, and the body mass is the body inertia - a property to resist to the applied force. These two things make a clear sense in the Newton equation (Second Law) $ma=F$. Both "parameters" (the body mass and the force strength) are observable and measurable. Now I am reading an **article** of G. 't Hooft where he states that "The interactions among particles have the effect of modifying masses and coupling strengths". So not only velocities are modified with forces but the particles masses and the forces themselves. I see here a contradiction with the Newton's definition of force and mass. As soon as the QFT is more fundamental than Classical Mechanics, does it mean that the 't Hooft's law is stronger than the Newton's ones? What is the impact (effect) of this stronger law on Classical Mechanics? How to define now the masses and forces if they are self-modifiable? **EDIT:** OK, is there a classical physics example to demonstrate that a potential interaction can modify mass and charge?


title: Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?

text: In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22) $$ \Theta^{\mu} = x_{\nu} \Theta^{\mu\nu} + \Sigma^\mu.$$ $\Theta^{\mu\nu}$ is the energy-momentum tensor and $\Sigma^\mu = d_{\phi}\pi_i^{\mu}\phi^i$ is the internal part. This fact is just mentioned and I dont understand why this has to be the case, so can somebody explain it to me?


title: Dielectric in a parallel plate capacitor

text: Uniform charge: each atom has charge $q$. Magnitude of dipole moment is $q s$, where $s$ is the distance the nucleus is shifted. According to my notes, the charge on the surface of a dielectric in between the plates is $N q s S$, where $N$ is the number of dipoles and $S$ is the surface area of the plate. But surely this should be $N q s$, because the charge should on the surface should be the same irrespective of the surface area (because we are using the number of charges on the surface $N$).


title: Proving that the weak hypercharge gauge field is not A

text: Under the electroweak gauge group $SU(2)_LU(1)_Y$ one identifies the 4 gauge fields $W^+, W^-, W^0, B$. After symmetry breaking $W^0$ and $B$ mix to give the observed fields $Z^0$ and $A$. Is there an intuitive argument showing immediatly that $A$ can not be identified with $B$?


title: Cloud chamber video showing large particles - What are they?

text: Watching a video of a cloud chamber on wikipedia (http://en.wikipedia.org/wiki/File:Cloud_chamber.ogg), I cannot help noticing the large collisions that take place at 00:12 and 00:24. What are they? Alpha particles? They are huge compared to the small ones (i guess they are electrons)


title: How is energy conserved when a moving charge has false ideas about positions of other charges

text: An electron is shot towards a target that is negatively charged. While the electron is traveling, the target makes an abrupt move towards the electron. While the information that the target moved is traveling from the target to the electron, the electron behaves like an electron that is moving towards a target that is in the original position. How can energy be conserved when an electron that is moving towards a nearby charge behaves like it was moving towards a far away charge? Seems we end up with electron being at 2 meters distance from the target, while the electron had enough energy to travel to at most 4 meters distance from the target. It also seems to me that "moving the target requires energy" is not a solution to this problem.


title: Is special relativity an exact description of reality?

text: In discussing relativity with a (somewhat mathematical) friend the other day, I ran into a problem showing why special and/or general relativity could be considered as exact descriptions of reality rather than just approximations that are working okay so far. Here's his argument: _Special relativity applies to a situation where there is no curvature of spacetime. Since both matter and energy cause curvature, the only situation where special relativity would apply would be one which has no matter and energy, and therefore one which is not achievable in any experiment. And even as a thought experiment, a clock has to have mass and energy and so there can be no exact predictions in such a theory._ He went on to say that therefore the general theory of relativity rests on a very shaky foundation. This is as compared, for example, to thermodynamics which rests comfortably on statistical mechanics and quantum mechanics. A related question: Special Relativistic approximation to GR


title: kirchhoff's law.Simple question

text: So far I calculated total resistance and it is 4.66 ohms and there is 12 volt voltage across them, what is the best way or HOW to calculate current for each resistor ? ![enter image description here](http://i.stack.imgur.com/DZWKG.jpg)


title: Making a "heavier-than-air" craft float

text: How big would a hollow rigid object need to be to float, (not in water but in air) if all of the air was vacuumed out and the container sealed?


title: Voltage drop along an idealized resistance-free wire in a circuit?

text: If you connected the positive terminal of a battery to the negative terminal to a battery with a wire with (hypothetically) no resistance, and are asked to give the voltage drop of a segment of wire in the circuit, how would you determine this?


title: Poincare Symmetry of Nambu-Goto action

text: How do I show invariance under the Poincare transformations of the action for a relativistic string, $$S=-\frac{1}{2 \pi \alpha'} \int{\text{d}^2 \zeta}\sqrt{-\det(\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu \nu})}$$ This is specifically taken from West's "Introduction to strings and branes" and the reparameterisation invariance is explained; however this one isn't.