What is the probability of ice in boiling water?

Ice crystals are spatially ordered, and in every randomness there is a low possibility of temporarily order. If given enough boiling water, and sufficient time, could local clusters water molecules happen to be in a crystalized state? This may seem absurd, but I believe it must be possible, imagine dropping an ice cube in boiling water(and water vapor) in a perfect closed system, the ice cube melts while the water keeps boiling (because of the water vapor), then because of Poincare recurrence theorem, there will be an similar ice cube after a sufficiently long but finite time. EDIT: The temperature is 100 Celsius and 1 atm (not at triple point)


What are good mechanics experiments for 10 year olds?

I'm trying to explain elementary mechanics - without the benefits of calculus or even algebra - and struggling. I'd like to find reasonable ways to demonstrate Newton's laws, minimally, and possibly continuing though basic projectile motion, although that may be too much to hope for.


Distinction between Larmor frequency and Rabi frequency

Wikipedia said "In the context of a nuclear magnetic resonance experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radiofrequency field. (Note that this is distinct from the Larmor frequency, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)" However the two definitions seem equivalent to me... Any clarifications?


Are electrons on a conductor, in electrostatic equilibrium, stationary?

Electrostatics basically means dealing with time independent electric fields (which was produced by stationary charges) Now consider a neutral conductor. We know that putting a net negative charge on the conductor, the charge will very quickly spread over the surface till electrostatic equilibrium is reached. What does this exactly mean? does it mean that electrons on the surface are not "moving" anymore and they become stationary? So in this situation we have 2 kinds of electrons: 1-electrons inside the meat of the conductor whose dynamics is described by quantum mechanics 2-electron on the surface which are not moving in the classical sense?


Why the generators of boosts transform like a vector under rotation?

$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ is the generator of boost of Lorentz group In many textbook of QFT, they say that the second one implies that the generators of the boosts transform like a vector under rotations. But I can't see it explicitly. Can anyone give me the explanation.


Aren't all collisions elastic at some level?

The more I think about it, inelastic collisions produce heat and sound which imply motion at some scale, right? Are inelastic collisions macroscopic events that ignore motion at microscopic levels? Maybe I'm just misunderstanding the definition of an inelastic collision.


Anomalously broken conformal symmetry

I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential contains $\mu^2\phi^2$. Bardeen argues that   1. Setting $\mu^2=0$ would increase the symmetry of the Standard Model by a conformal symmetry. > _But the things we can measure - which are basically the Green's functions - > are no more symmetric with $\mu^2=0$ than with $\mu^2\neq0$! Because the > symmetry is anomalously broken._   2. The conformal symmetry is broken anomalously by non-zero beta-functions.   3. Quadratic divergences do not contribute to the beta-functions.  > _Whilst its true that they don't contribute to the running of the > renormalised parameters, they contribute to the running of the bare mass._ >   4. Quadratic divergences must therefore be a separate (explicit) source of > symmetry breaking. > _Not true if you consider that quadratic divergences contribute to the > running of the bare mass._   5. Quadratic divergences must therefore be an non-physical artefact of our re-normalisation procedure, and we must remove them with, for example, counter-terms, and this is not a fine-tuning. I have added my comments. My main question is, > If conformal symmetry is broken anomalously, why should our Lagrangian > respect conformal symmetry? The Green's function are not more symmetric with > $\mu^2=0$. I am interested in this for any symmetry, but especially > conformal symmetry. Also, I don't understand 3. and 4. Quadratic divergences would contribute to the running of the bare mass. Wouldn't that break conformal symmetry anomalously? or is it only the renormalised parameters that must have vanishing beta-functions? The distinction seems artificial. I can't understand why quadratic divergences are an explicit source of symmetry breaking, whereas logarithms etc are anomalous. This is key to solving the naturalness problem, and I can't follow the argument. My feeling is that these arguments are faulty (which makes me think I must be making mistakes because Bardeen is a real expert who has surely thought a lot about it!). I'm certainly unconvinced. Have they been confirmed/refuted at length in the literature?


Imagining to speed up the Earth's revolution around the Sun until we feel its motion

If the Earth were revolving around the Sun much faster than it does, let's say 100 or 1000 times faster or whatever big-enough multiple of its actual speed, would we feel it? What sensation would we feel? How could I estimate how big the revolution speed would be for us to feel it? Would it be possible to speed up the Earth motion around the Sun? How? What basic physics concepts can be used to explain what would happen?


Are Negative Eigen Values of a Hessian Matrix physically acceptable?

Suppose I have a Hessian Matrix of a System with 3N degrees of freedom, What are the physical significance of eigen values of the Hessian, Are negative Eigen Values physically acceptable?


What do we mean when we say the QM wave function is a section of the $U(1)$ bundle?

I have a couple questions here. To keep the discussion simple lets stick to the following case: what is the quantum mechanics of a single particle in the presence of a background EM field, such as that of a monopole, where multiple coordinate patches are required to define the vector field everywhere without singularity. I have learned about the geometric formulation of EM - where the field $B$ of a monopole is described as a real-valued two-form that is closed but not exact. I also know about using the vector potential as a connection for the complex wavefunctions, using the usual covariant derivative. The questions I have in mind are >   1. What (complex line) bundle is the wave function a section of? >   2. What are the bundle data (i.e. transition functions) that specify this? >   3. In what way can we see the monopole vector potential as a connection on > this bundle? >   4. People seem to talk about the monopole bundle only with the monopole > connection in mind - what would it look like to have a different connection > instead? > On a similar note, I've seen physicists refer to the classification of bundles, which there is one for every integer (based on the homology of $\mathbf{R}^3-\\{0\\}$), and call that integer the monopole number. But I've also seen people call the flux integral of the magnetic field two form over some closed surface, which relates instead to the curvature of some _connection_ on the bundle - which should be quantized by Gauss-Bonnet theorem - the monopole number. Is there some reason that these are the same? I was reading a great set of lecture notes around 10 pages which explained the need to use a section of a bundle for the wave function and the relation to the Dirac quantization condition, but I can no longer find them. Links to related resources would be very welcome.


Question on circular motion — car turning on road

Lets say a car is turning on a road that has friction. I made this small diagram ![enter image description here](http://i.stack.imgur.com/3fhVd.png) where the two parallel lines indicate the tires of the car. In order for the car to keep turning in a circle, there must be an force directed to the center of curvature — the centripetal force. I have read that this force comes as a result of friction. However, in order for there to be a frictional force towards the center, doesn't there have to be a force directly opposite pulling away from the center, as frictional forces cannot exist by themselves — they have to act in order to counter an opposite force. Thus, how can there be a frictional force towards the center without a corresponding force pulling away from the center? Does this have something to do with the centrifugal pseudo-force? Further, where exactly does this frictional force come from? Also, why is the frictional force directed towards the center? Shouldn't it be opposite the direction the car is traveling at the moment -- thus in a direction opposite the tangential?


Which physically acceptable quantum interpretations do not require the existence of any observer at all?

Some interpretations of quantum mechanics -- like the Copenhagen interpretation in particular -- require the existence of an observer. The role of the observer is a bit mysterious. After all, observers are not needed in classical mechanics. Which physically acceptable interpretations of quantum mechanics do not require the existence of any observer at all? By that, I mean they would continue to make sense even in the absence of any observer. The many worlds interpretations appears to require an observer to subjectively determine which branch is selected.


Line of action force

The task is to reduce the two forces into a single net force and then find the equation of its line of action. I chose to find the net force and momentum with respect to the origin. The magnitude of each force is $P$. ![asd](http://draw.to/static/d/3yezUM.png?v=ec211) We can therefore write $F_{net} = (P,0,0) + (0,0,-P) = (P,0,-P)$. The distance vector from origin to $F_1$ is $(0,b,c)$ and from origin to $F_2$ is $(a,0,c)$. The net moment is then $$M_{net} = (0,b,c)\times (P,0,0) + (a,0,c)\times (0,0,-P) = (0, aP + cP, -bP ).$$ What I tried to do was to assume the attacking point of the net force to be $(x,y,z)$. This force must cause the net moment with respect to the origin. So I set up the cross product $$ (x,y,z) \times (P,0,-P) = (0, aP + cP, -bP).$$ but I was unable to determine $x$, $y$ and $z$. So what can we do?


Joint probability function for the values of a free field at two different points

For a free real field $\phi$ in its ground state, is there a way to find the probability distribution $p(\phi_x,\phi_y)$ for joint measurement of $\phi(x)$ and $\phi(y)$ at two spacelike-separated points $x$, $y$? Presumably the information needed to construct $p(\phi_x,\phi_y)$ would be contained in the expectation values of products of powers of $\phi(x)$ and $\phi(y)$, since we could invert the formula $$ E[\phi(x)^n\phi(y)^m] = \int d\phi_xd\phi_y p(\phi_x,\phi_y)\phi_x^n\phi_y^m $$ but the expectation values are divergent for $n$ or $m$ greater than $1$.


Trying to make a realistic simulation of 'breaks' (Pool)

I'm working on a 2D pool simulation project right now, and I'm trying to create good looking physics that are also performant. After collisions are detected, I update the velocities of the balls. So far, I've assumed collisions are elastic, which gives pretty good results. The formulas for the final velocities of both balls after a 2D elastic collision are easy to find, and I can follow the derivations of them well enough. The problem is that 'breaks' are not realistic: momentum is often passed along the top and bottom of the triangle of balls, and only the top and bottom corners fly off. I was wondering if changing the equations to use partially inelastic collisions would make the break look more realistic. I would use a coefficient of restitution that is close to 1. However, I'm not able to find many examples of 2D partially inelastic collisions that give final velocities. I'm not sure if I can derive these easily myself, and I'm a bit short on time. Would this approach be worthwhile? Will using slightly inelastic collisions give me the results I'm looking for? If so, are there any resources that can help me solve this problem?


Aftermath of Plasma Arc

You have a chamber with constant amount of gas (say either helium or argon). You create the plasma with an arc and turn it off. How much of the gas used to create the plasma returns back to helium or argon as opposed to dissipating into the chamber or creating new elements?


Where I can find a torsional stiffness table for different types of stainless steels?

I am trying to comparing the torsional stiffness of different types of stainless steels. I googled the keywords but I could not find any useful information. Would anyone please give me some links which provide those information?


How does electron capture occurs?

Electron capture is a kind of decay by which a nuclear transformation takes place. Below is an example of it. $$ _{29} ^{64} \text{Cu} + e^- \rightarrow _{28}^{64} \text{Ni} $$ Of course, nucleus being positively charged and electron being negatively charged, there is an attraction between them, but electron generally don't fall in the nucleus. I am unable to understand that exactly how does nucleus is able to capture electron, using which interaction or force? Uncertainty principle won't allow electron in nucleus, though it is being captured. Exactly how?


How long did inflation take to happen?

If inflation happened, it ended about 13.7 billion years ago. But how long was inflation going on before that? Was it just a short burst of time so that it's really okay to say that the age of the universe is 13.7 billion years, or was it going on for a long time before it ended (maybe forever?)?


Poynting's Theorem simplified?

How can I apply Poynting's theorem to any system that has a magnetic field & electric field, to state conservation of energy? How does Poynting's theorem state conservation of energy in EM? I struggled to understand with the differential equations. Example: Work being done by the Lorentz force on a conductor by a supplied power source?


Matter - Antimatter Reactory Practicality

With **current technology** , would the energy released by a matter-antimatter annihilation be more than the energy needed to created the antimatter in the first place? Would it be worth it? Just curious, even if was on a small scale, Could one produce a machine that uses annihilation to produce a net energy gain?


Wen's question What is spontaneous symmetry breaking in QUANTUM systems? is cute, but here's an even cuter question. What is spontaneous symmetry breaking in QUANTUM GAUGE systems? There are some gauge models where the Higgs phase and the confinement phase are continuously connected, you know. Why are the Higgs phase and the confinement phase identical in Yang-Mills-Higgs systems? You mention cluster decomposition? Sorry, because bilinear operators have to be connected by a Wilson line.


Stablising the Higgs without SUSY

Should the Higgs be found at the LHC, but no supersymmetry (assuming for the sake of argument that the LHC be capable of eliminating all versions of SUSY that are motivated by solving the hierarchy problem), what then? Are there alternatives for stablising the Higgs scale that don't involve SUSY (other than Randall-Sundrum)?


Why will clay, when heated to high temperatures,  harden and become waterproof?

I have a question > Why will clay, when heated to high temperatures, harden and become > waterproof?


One-dimensional (1D) CdSe nanostructures

I'm not a physicist, but I'm really interested in Nanotechnology. I've a question, and I hope that I can be provided by an answer. I was reading some articles about the one-dimensional CdSe nanostructures. My question is why it's called one-dimensional? Is it because of their geometrical shapes or something else? I truly appreciate any hint or a resource could lead to the answer. Thanks.


Using Lagrange's Equations with Generalized forces

I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my Lagrangian is: $$\frac12m(\dot x^2+\dot y^2)-mgy$$ I can easily solve the Euler-Lagrange equations, to see that $\ddot y=-g$ but is there a way I could do this with generalized forces. I feel that this is an awful example. I vaguely remember my professor using generalized forces to find friction force. Any help with understanding generalized forces would be appreciated. The reason I want to know is that I am trying to model a curveball (baseball) and treating the lift and drag as generalized forces then using the E-L equations to solve for the motion.


Continuously feeding an evaporating micro-black hole?

What would happen if you created a micro-black hole and could continuously feed it as quickly as it evaporates? Is it possible that it would remain relatively stable? If so, how might such a thing be continuously fed?


What kind of cooling mechanism could be used in outer space?

This question arises out of this question on Quora - Apollo 11: 1969 Moon Landing: Did Neil Armstrong really land on the moon? I'm convinced with most of the explanations provided in the first answer to that question. None of these conspiracy theories have bothered me. However one thing that confuses me is the heat. On earth close to 47% of heat is absorbed or reflected by the atmosphere. On moon an object would get struck directly with thermal radiation from sun acquiring about 50% more heat then it would on earth. I am wondering what kind of cooling device could we create for such an environment? A system like an AC or a refrigerator won't work in a vacuum. Off the top of my head the only method I could think of is circulating some liquid with high specific heat in the suits and letting it evaporate. But is this feasible? How much heat do we need to loose? What rate of evaporation would be needed? **What volume of liquid would be needed** to support this rate? Given the high amount of electricity it takes to makes our refrigerators work **how much electricity would such a system need?** Because of very large amount of heat and large surface area (the lunar module, the suits, and the camera casings) I'd assume a large quantity of water (say a few thousand liters) would be needed. How could they carry such a large amount of fluid?


What can make signal triangulation tricky?

Usually, when you're trying to find a source of whatever signal, you move around the source and then calculate the position using triangulation. I'm specifically talking about low frequency signals, such as radio waves. I was wondering what can make this process harder or completely impossible (the former seems very unlikely).   If there were two transmitters, would it be easy to tell that there are two and not one? Could the signal be dispersed unevenly in the area, making it appear somewhere else than it is?


Witten's constrained S-matrix and Coleman-Mandula Theorem

I remember reading somewhere that Witten argued that if the Poincaré symmetry of spacetime were nontrivially combined with internal symmetries, then the S-matrix would be so constrained that the scattering amplitude would be non- vanishing at discrete angles only. I would like to know more about this:   1. At _which_ angles would the scattering amplitude be non-vanishing   2. By crossing symmetry, wouldn't this mean that in the crossed channel, the amplitude be non-vanishing at discrete _energies_?   3. Is there a reference that contains more information on this?


Thrust to Weight ratio in Space with an off set CoM

With regards to this thread, Thrust center in space My question is, if the thrust to weight ratio was increased so that it was much higher than the weighted mass of the sphere (ship), would the sphere then not start to follow a straight trajectory along the axis of the thruster.


In what applications are lasers necessary vs. just convenient?

I’m trying to understand, given the wide array of current laser applications:   1. Which applications actually require laser characteristics, as opposed to just use lasers because they are the most practical/efficient source of light with characteristics _not unique_ to lasers   2. What alternatives exist, or may exist, to lasers for applications that don’t require laser characteristics My key assumption here is that the essential characteristic of a laser is that it produces temporally coherent light. Often, but not necessarily, lasers produce light with two other useful characteristics:   1. Monochromaticity   2. Collimation But these convenient characteristics can be produced using non-lasers. For example, LEDs produce monochromatic light. And lenses can take diffuse light and produce a beam collimated to near the diffraction limit, right? If this is the case, then the only applications that require lasers are those that depend on interferometry, and that’s pretty much just holography and measurement, right? All of the other applications – directed energy, medical therapies, optical discs, optronics – could be accomplished with non-coherent collimated and/or monochromatic light, right? Presumably lasers are used in these cases only because they are currently the most efficient or practical way of producing light with those characteristics. But now I am wondering: What other technologies could supersede lasers in practice (and efficiency) for applications that require light with high energy, collimation, and/or monochromaticity?


Geometric/Visual Interpretation of Virasoro Algebra

I've been trying to gain some intuition about Virasoro Algebras, but have failed so far. The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I just can't seem to gain some intuition about it. As a central extension to Witt Algebras, I was hoping that there has to be some geometric interpretation, as I can imagine Witt Algebras rather well. If anyone has some nice Geometric or Visual Interpretation of Virasoro Algebra, I'd greatly appreciate it!


Effects of parallel superconducting plates

Assuming the existence of virtual particle field ( zero point energy field) Casimir force is produced by 2 parallel conducting plates excluding some of the frequencies between the plates, if these 2 plates are superconducting does that change the amount of frequencies being excluded? One theory I've read postulates that it is interaction of matter with the ZPF (=zero point field) that is the origin of mass and inertia, if the parallel plates are suppressing the appearance of some of these virtual particles, would that not in some manner be modifying inertia for matter passing between the plates?


Why is work in a constant electric field equal to the force times distance?

Let's say, for example, that an infinite plate generates a constant electric field that exerts a force of 6N on a 1C charge. To move the charge from a distance of 5m to 1m from the plate, the work done is given by 6N * 4m = 24Nm. The way this has always been explained is that we apply a 6N force over the 4m distance. However, this doesn't make intuitive sense to me:   * If we apply a constant 6N force to the charge in the opposite direction of the electric field, won't the charge simply remain stationary? The force generated by the field and our applied force in the opposite direction should net to zero.    * If the electric field exerts a force of 6N on the charge at every single infinitesimal point along the path from 5m to 1m, when we apply an opposite force of 6N over that same distance, shouldn't the charge simply stay still?   * I think what's troubling me is that I don't see how applying a 6N force can "overcome" the electric field's force and actually move the charge in the opposite direction. I think the intuition is the same for the force of gravity. A 1kg object that is 1m above the ground is experiencing a force downwards of 10N. If I apply a constant upwards force of 10N, the object won't fall. But how is that if I apply a 10N upwards force over 1m, I can actually move the object upwards? Gravity is exerting a downwards 10N force on the object the entire time I am exerting an upwards 10N force over that 1m. If the forces are equivalent, shouldn't the mass stay still?


Understanding Triplet And Singlet States

We know, $2\otimes 2=3\oplus 1$. Thus we have a spin triplet of states and a spin singlet. Can we regard these states as the spin part of wavefunction for the excited states and the ground state of the deuteron nucleus? What is the difference between the members of a triplet?


Connection between the Big Bang and Black Holes

So... this is my first question here after a little while of lurking; so be gentle. I've been thinking lately about trying to wrap my head around the concept of spacetime and how it can be affected by gravity and the density of matter in space (i.e. near a black hole). I've been wondering. Is it possible that matter - which is broken down (spaghettified?) into it's constituent parts and into energy - is drawn to a point in spacetime which has already got a massive density? Would it be possible that any black hole, which has been or will ever be, causes energy to be drawn to a single point in space or time? I.E. all black holes are "feeding" into a single, infinitesimal time and location; causing (possibly the wrong word - it seems to me that it would be less cause vs. effect and more that it just _is_ ) the big bang? I tried to find some papers that would touch on the topic but couldn't find anything specific - which likely means I'm being a total dolt and there are many good reasons why this isn't the case :) I would appreciate any information which would correct my thinking and counter my naivety.


Current flowing through a square loop - Magnetic Field

> _A square conducting loop of wire of side $2a$ carries a uniform current I. > Calculate the magnetic field at the centre of square due to the current in > the loop._


What is the meaning of $U''(x)=0$?

Most potentials with a minimum can be described approximately as a harmonic oscillator. So the procedure is to Taylor expand $U(x)$: $$U(x)=U(0)+U'(0)x+\frac{1}{2}U''(0)x^2 +...$$ If we suppose that the potential is cero at the origin an it has a minimum there, we get: $$U(x)=\frac{1}{2}U''(0)x^2$$ We take $U''(0)$ to be the _spring_ constant $k$. So the angular frecuency is given by: $\omega=\sqrt{\frac{k}{m}}$ But what if $U''(0)=0$ and there is still a minumum at zero, like a potential $U(x)=x^4$? In this case if you blindly apply the formula you get zero frecuency, which is false. Does it just mean that to a small approximation a body will not oscillate?


First, pardons to all because I don't have a physics background and I'm probably using laymans terms. What is the rebound angle of a circle striking a concave surface? Assume the simplest case:   * the circle (red) is not spinning   * the circle does not compress   * the arc (blue) does not deform   * there's no friction   * there's no energy lost in the impact Here's my attempt at an illustration with a red circle moving in a blue circle traveling towards the blue circumference along the gold path and rebounding on an unknown green angle. ![enter image description here](http://i.stack.imgur.com/tt3gB.png)


Why does increasing the temperature of a thermistor decrease it's resistance?

Surely, upon an increase in temperature, the atoms within the thermistor would vibrate with more energy and therefore more vigorously, hence making the electrons flowing through the electric circuit more likely to collide with one of the atoms, so increasing resistance. However, the effect of temperature on a thermistor is contrary to this. I can't understand how it can be. It's analogous to running across a playground: if everyone is still you're less likely to collide with someone, however if everyone is constantly moving from left to right then a collision is more likely. So why does an increase in temperature decrease the resistance of a thermistor?


Horizon problem in the standard cosmological model

The horizon problem is usually stated as follows: Causally disconnected parts of the universe that we see today have the same temperature, so how could that be so if no information travels faster than the speed of light. But (without assuming inflation) wasn't the universe causally connected right after the big bag? And that those parts that are disconnected today only became disconnected due to expansion? Also, if temperature irregularities existed but to a very minute level, then why do we expect that the universe without inflation must look different than the homogeneous universe (on large scales) we see today?


Clarity in Electric field Definition?

The electric field at a point is defined as q/$r^2$. How does this definition take the nature of source and test charge into consideration. If I bring any positive/negative charge around the source charge, it brings a change in the distances involved due to forces of attraction or repulsion.how can I use this definition for practical purposes?


antiferromagnetic spin wave

I have a hamiltonian that is derived from a spin wave energy dispersion calculation for a nearest neighbor interacting cubic antiferromagnet. After a Holstein-Primakoff transformation and making a substitution of -q for q in one place, I get the following hamiltonian (J is the exchange energy, S is the magntiude of the spin on each sublattice, z=6 is the number of nearest neighbors of each site, and $\mathop \gamma \nolimits_q = \mathop e\nolimits^{i*\mathop q\nolimits_x } + \mathop e\nolimits^{ - i*\mathop q\nolimits_x } + \mathop e\nolimits^{i*\mathop q\nolimits_y } + \mathop e\nolimits^{ - i*\mathop q\nolimits_y } + \mathop e\nolimits^{i*\mathop q\nolimits_z } + \mathop e\nolimits^{ - i*\mathop q\nolimits_z } $ : $$ \sum\limits_q {\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q^\dagger }&{\mathop b\nolimits_q^\dagger }&{\mathop a\nolimits_{ - q}^\dagger }&{\mathop b\nolimits_{ - q}^\dagger }&{\mathop a\nolimits_q }&{\mathop b\nolimits_q }&{\mathop a\nolimits_{ - q} }&{\mathop b\nolimits_{ - q} } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {JzS/2}&0&0&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}\\\ 0&{JzS/2}&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}&0\\\ 0&0&{JzS/2}&0&0&{JS\mathop \gamma \nolimits_q /2}&0&0\\\ 0&0&0&{JzS/2}&{JS\mathop \gamma \nolimits_q /2}&0&0&0\\\ 0&0&0&{JS\mathop \gamma \nolimits_q /2}&{JzS/2}&0&0&0\\\ 0&0&{JS\mathop \gamma \nolimits_q /2}&0&0&{JzS/2}&0&0\\\ 0&{JS\mathop \gamma \nolimits_q /2}&0&0&0&0&{JzS/2}&0\\\ {JS\mathop \gamma \nolimits_q /2}&0&0&0&0&0&0&{JzS/2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q }\\\ {\mathop b\nolimits_q }\\\ {\mathop a\nolimits_{ - q} }\\\ {\mathop b\nolimits_{ - q} }\\\ {\mathop a\nolimits_q^\dagger }\\\ {\mathop b\nolimits_q^\dagger }\\\ {\mathop a\nolimits_{ - q}^\dagger }\\\ {\mathop b\nolimits_{ - q}^\dagger } \end{array}} \right)} $$ I then, as outlined in this paper (http://jmp.aip.org/resource/1/jmapaq/v19/i1/p277_s1) solve the equation det(H-x*{{1,0},{0,-1}})=0 for x, where H is the 8x8 matrix above and 1 is the 4x4 identity matrix. This procedure gets the right answer for the energy x, as expected. However, if I transform sums over $ \mathop b\nolimits_{ - q}^\dagger \mathop b\nolimits_{ - q} $ to $ \mathop b\nolimits_q^\dagger \mathop b\nolimits_q $ and similarly for a_q, I get this hamiltonian: $$ \sum\limits_q {\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q^\dagger }&{\mathop b\nolimits_q^\dagger }&{\mathop a\nolimits_{ - q}^\dagger }&{\mathop b\nolimits_{ - q}^\dagger }&{\mathop a\nolimits_q }&{\mathop b\nolimits_q }&{\mathop a\nolimits_{ - q} }&{\mathop b\nolimits_{ - q} } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {JzS}&0&0&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}\\\ 0&{JzS}&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}&0\\\ 0&0&{0}&0&0&{JS\mathop \gamma \nolimits_q /2}&0&0\\\ 0&0&0&{0}&{JS\mathop \gamma \nolimits_q /2}&0&0&0\\\ 0&0&0&{JS\mathop \gamma \nolimits_q /2}&{JzS}&0&0&0\\\ 0&0&{JS\mathop \gamma \nolimits_q /2}&0&0&{JzS}&0&0\\\ 0&{JS\mathop \gamma \nolimits_q /2}&0&0&0&0&{0}&0\\\ {JS\mathop \gamma \nolimits_q /2}&0&0&0&0&0&0&{0} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q }\\\ {\mathop b\nolimits_q }\\\ {\mathop a\nolimits_{ - q} }\\\ {\mathop b\nolimits_{ - q} }\\\ {\mathop a\nolimits_q^\dagger }\\\ {\mathop b\nolimits_q^\dagger }\\\ {\mathop a\nolimits_{ - q}^\dagger }\\\ {\mathop b\nolimits_{ - q}^\dagger } \end{array}} \right)} $$ But solving the equation det(H-x*{{1,0},{0,-1}})=0 (where H is the new 8x8 matrix above and 1 represents a 4x4 identity matrix) does not give me the same answer. I'd really like to know what is going on. Thanks!


Escape velocity to intersection of two gravitational fields

> Find the minimal velocity needed for a meteorite of mass $m$ to get to earth > from the moon. Hint: the distance between the center of earth and the center > of moon is $\approx 60 R_E$, and the meteorite should reach a certain point > $O$ on that distance, where the gravitational forces of the moon and the > earth are canceling each other out. It is located $\approx 6 R_E$ from the > center of the moon. I thought that the best way to solve this question is to find the difference between the initial enegy of the body and the final one at $O$. Beyond that point, the gravitational pull of the earth will 'overcome' the gravitational pull of the moon, and the object will gather speed itself by reducing the GPE in the earth's gravitational field. Therefore: $U_{G,i}=-G \frac{M_{moon}m}{R_{moon}}$ $U_{G,f}=-G \frac{M_{moon}m}{6R_E}-G \frac{M_E m}{54R_E}$ $U_{G,f}-U_{G,i} \approx m \cdot 1.51 \cdot 10^6 \text{J}$ (there's no mistake in the calculation) $E_k=m \frac{v^2}{2}=m \cdot 1.51 \cdot 10^6$ Therefore $v \approx 1739.97 \text{m/s}$ However this answer is wrong. The correct magnitude of the speed should be somewhere around $2.26 \text{km/s}$ which is close to the escape velocity from the moon ($\approx 2.37 \text{km/s}$). Where am I wrong? Why the difference in GPE is not satisfying the problem?


Questions about voltage

For some reason, I feel like the concept of voltage is escaping my grasp. I've done much research on these forums and through texts, and come across answers that seem quite well thought out, but still cant grasp the fundamental concept of what voltage is. I could explain (regurgitate) the definition of voltage in many ways, and plug it into equations, but when it comes to testing my fundamental understanding of it as a natural concept, I fall short. Ill explain what I know, and what I don't, and I have some theoretical questions which I hope will help give me a better understanding. **Here is what I think I know:** 1\. Voltage is electrical potential energy. It represents an imbalance of charge distribution.   1. This means that given a sufficient conductor, the electrons will do work by moving to a state of equilibrium to correct this imbalance.    2. Voltage must be measured between two points, because it can only exist in relation to an imbalance between one point and another.   3. Measures of voltage determine the work which can be done as a result of the imbalance?   4. The quantity of voltage in a battery would not change if you increased the size of the battery. It would only change if you increased the average negativity charge density in the negative terminal? **Here is what I don't know or have questions about:** 1\. So, increasing voltage increases current because higher voltage means a higher disequilibrium from the source, meaning more "pressure" on the electrons to move? Given a fixed wire diameter, does that means higher voltage would make electrons move faster?   1. Why do resistances in series add together? I don't understand. I picture a resistor as a bottleneck, in that the entire circuit is limited by the resistance of the most resistant component. But this is clearly wrong. Is it more correct to imagine that every time electrons travel through a resistor, they have to do more work and lose some of their ability to do future work, and have lower and lower energy the more resistors they go through?   2. If a current/stream of electrons begins to flow and is traveling through a wire with 1 ohm resistance, and goes through a 5 ohm resistor, then back to a 1 ohm resistance wire, since resistance decreases, would it speed up again upon exiting the 5 ohm resistor? If so, I'm assuming its new speed (relative to its initial speed upon entering the circuit) would be proportional to the fraction of remaining voltage after the total voltage drop up to that point?   3. When a battery dies, I'm assuming that its voltage isn't at 0. Does a dead battery just no longer have the minimum voltage required to supply the circuit with enough power to power the device? In that case does that mean that dead batteries could still power a circuit with less resistance or lower power demands?   4. This quote from Wikipedia confuses me: "In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of **the voltages** across each component". What does it mean that components "have voltage". I'm assuming this means the drop in voltage from before the component compared to after, right? But what is the reasoning for saying the component "has" voltage? Is it trying to express the quantity of work done (i.e. heat emitted) from that component?   5. Last one, another wiki entry: "In a parallel circuit, the voltage across each of the components is the same". This confused me, but I think I just figured it out. If a current is flowing through multiple paths, it would take the path of least resistance to its destination, thus if a parallel circuit has a 1 ohm resistor and a 5 ohm resistor in parallel, both would have the same voltage drop, but the current flowing through the 5 ohm resistor would be 1/5th the value of the 1 ohm? Thanks for taking the time to read my post. I hope I didn't get everything terribly wrong.


The state of Indefinite metric in Quantum Electrodynamics

I faced difficulties to grasp why indefinite metric is introduced from no where in QED, after searching internet I found that this is a problem in QED, because one needs it to preserve theory's covariance, and to solve ultraviolet divergence problem, but in the same time it raise other different problems such as negative probabilities, and even from mathematical point of view that it's can't be introduced a proper topology for a Hilbert space with such metric. (those point I get from N. Nakanishi: Indefinite-Metric Quantum Field Theory. _Progress of Theoretical Physics Supplement_ No.51 (1972) pp. 1-95, doi:10.1143/PTPS.51.1) Thus those information may be old, I would like to know the official (main stream) position of this problem, and if it considered as solved, especially that there is some papers that refuses this kind of metric and some usual books on QED doesn't mention it all what makes me even more confused.


Modeling incoming solar radiation

I want to write a model for estimating incoming solar radiation for a specific latitude on earth but I am struggling to find an appropriate source which shows the required equations for doing so. Would anyone be able to provide me with a link to where I can find equations for estimating solar radiation (irradiance) given a specific cloud cover, latitude, time of day, and day of year?


estimating the rotational speed of dc motor

What is the best strategy to roughly estimate the RPM of simple DC motor as seen in this video. I want to estimate the max possible rotation of the coil that has $n$ number of turns and has moment of inertia $I$ for a uniform field $B$. The one side of the axle has been scraped so what current only flows along one half of the rotation.   ![enter image description here](http://i.stack.imgur.com/8cNCH.png) There are factors like friction that slows down motion and back emf that reduces current. How do I put those into one equation for estimation. Or is there any better way? Any helpful links will be appreciated. Thanks for your time.


Frequency of an Electron

My question is very simple. If frequency is defined as the cylces per unit time, Then what is meant by "Frequency of an Electron" ? If the rotation of electron around a nucleus is considered then, Which phenomena we consider for a free electron i.e: electron in a force field (unstable alone electron). EDIT: Is "Frequency of an Electron" a experimental quantity? **EDIT AFTER HAVING 2 ANSWERS:** My teacher told me how to calculate the frequency of an electron. We started from finding Energy of Electron then difference in energy, then we get this equation according to Bohr's Radius of Hydrogen Atom and his postulates: i.e: $$ f = \frac{z^2e^42\pi^2m}{h^3} (\frac{1}{n_1^2} - \frac{1}{n_2^2}) $$ Where:   * $z =$ Atomic number    * $e =$ Charge on Proton   * $m =$ mass of electron   * $h =$ Plank's constant    * $n =$ Orbit number From the last part of my equation, I am confused. The $n_1$ and $n_2$ shows that, that frequency will be the frequency of energy? or electrons?


Inertia matrix of a rod rotating about an axis

I'll provide a picture for clearer understanding. The problem is to calculate the angular momentum of the rod rotating about the z-axis. I have serious difficulties in deriving the inertia matrix, that's all I need help with. ![enter image description here](http://i.stack.imgur.com/WGzDa.jpg) **Progress** Since the rod is rotating about the z-axis $\Rightarrow \omega _{x}=\omega _{y}=0$ the angular momentum simplifies to $ \boldsymbol{L}_{O}=-I_{xz}\omega _{z}\hat{\mathbf{i}}-I_{yz}\omega _{z}\boldsymbol{\hat{j}}+I_{zz}\omega_{z}\boldsymbol{\hat{k}} $ Split the rod in three parts (SEE FIGURE) and calculate for each body. I have correct terms for body **A**. For **B** : $I_{xz}=\overline{I}_{xz}+md_{x}d_{z} = 0 +\rho b(0)(\frac{b}{2})=0$ $I_{yz}=\overline{I}_{yz}+md_{y}d_{z} = 0 +\rho b(b)(\frac{b}{2})=\frac{1}{2}\rho b^3$ **Wrong** $I_{zz}=\overline{I}_{zz}+md^2 = 0 + (\rho b)(b^2+(\frac{b}{2})^2)=\frac{5}{4}\rho b^3$ (d is distance from midpoint of [B]B[/B] to origin O) For **C** : $I_{xz}=\overline{I}_{xz}+md_{x}d_{z} = 0 +\rho b(\frac{b}{2})(b)=\frac{1}{2}\rho b^3$ $I_{yz}=\overline{I}_{yz}+md_{y}d_{z} = 0 +\rho b(b)(b)=\rho b^3$ **Wrong**$I_{zz}=\overline{I}_{zz}+md^2 = \frac{1}{3}(\rho b)(b^2)+(\rho b)((2b^2)^2+(\frac{b}{2})^2)=\frac{55}{12}\rho b^3$ (distance d is from midpoint of [B]C[/B] to origin O)


If Renormalization Scale is Arbitrary, Why Do We Care about Running Couplings?

**For the bounty please verify the following reasoning** [copied from comment below] Ah right, so the idea is that overall observable quantities must be independent of the renormalization scale. But at each order in perturbation theory the result can depend on renormalization scale, right? And they do so in exactly the right way to invalidate the use of a perturbation series when external momenta are of the wrong order. This is because the external momenta get caught up in loop integrals due to momenta conservation at vertices and the renormalization group equation comes along and means the right things cancel. [original question] **In Brief** People say that perturbation theory breaks down when the couplings run to high values. But given that this running depends on an arbitrary mass scale, how is this argument logical?! **A Longer Exposition (TL;DR)** It's obvious that Feynman diagram techniques work best when the coupling constant is small, since then you can neglect higher order terms. But various sources (e.g. Peskin and Schroeder) make claims about running couplings that seem incomplete to me. I often read sentences like   * if the renormalized coupling is small at low energy, then Feynman diagrams are good in that region   * for asymptotically free theories, Feynman diagrams are good for high energy calculations I understand exactly why these are true. But surely something much more general is true, namely   * if the coupling constant is small **at any renormalization scale** the Feynman diagrams are good for calculations My reasoning is as follows. Observable quantities are completely independent of the running of the coupling, so you can just choose your scale appropriately so that the coupling is small for the expansion. Hey presto, you can use Feynman diagrams. Is this true? I strongly expect the answer to be **no** but I can't convince myself why. Here's my attempt at a self-rebuttal.   * my argument above is incorrect, because I've assumed there's only a single coupling. In reality there's contributions from "irrelevant" terms at higher energies whose couplings can't be fixed from low energy observations.  This leads me to hypothesise that the following is true (would you agree?)   * if the coupling constant is small **at any renormalization scale above the scale of your observations** then Feynman diagrams are good This seems more plausible to me, but it would mean that Feynman diagrams would be good for low energy strong interaction processes for example. This feels wrong is a sense because the renormalized coupling is large there. But then again the renormalization scale is arbitrary, so maybe that doesn't matter. Many thanks in advance!


AdS to dS uplifting and its opposite

So as I understand it, localized structures in AdS can wick rotated to dS, the boundary has to be complixified as can be seen here. Also, uplifting is another technique that can be used to move from AdS to dS. (Thanks to Mitchell Porter for this suggestion) My question is that just as an exercise, is it possible to move from dS to AdS, I don't know the name for such a process, it is in some sense "moving down" from dS to AdS. Take the following example, suppose I have a mapping of a scalar field from boundary to bulk in AdS, now that can be uplifted to dS, it won't hold make much sense anymore since AdS allows for SUSY however dS doesn't but let's say for a moment that it can be done. Now consider the opposite case, we have a scalar field mapping in dS, can that be brought down to AdS? I think it can be, and this should be actually easier since AdS allows for SUSY which should make the mapping a little easier. The lack of a concrete example for dS/CFT makes it (about the mapping which we considered in our hypothetical case for dS) difficult to ask this question but does it seem plausible that if such a mapping were to be constructed, it can be brought down to AdS? Thanks.


How to know what materials are good conductors of electricity?

I'm not asking a question like "Is the wood conductive?". No. I'm asking what properties do they have to have to be good conductors. Theoretically I mean. Thanks.


Ising model for dummies

I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge. Can you direct me to some documentation on it that can be parsed by my puny undergrad brain? If the answer is negative, can you explain it right there, on the answer form?


Gravitational field has no curl? What about gas discs around stars, black holes, etc.?

So everybody says the gravitational field has no curl, and is not comparable to a liquid swirling around a drain. Observationally, of course, there are many examples of vector fields (which I think are gravitational fields) which look like they have some curl. A pair of stars, for example, one being slowly devoured by the other. The path of the swirling gas seems to trace out a field with a lot of curl in it. Just as if you were to pour dye into water that is swirling around a drain. How do you reconcile the observed vector field with the gravitational field which is not supposed to have curl? Is there some way to work them into a single model?


Electric potential vs potential difference

What is the difference between electric potential and potential difference? In our course book, they are given as separate topics but their definition is given the same.


Escape velocity from long ladder

The escape velocity of earth is roughly $11 kms^{-1}$. However, what if a long ladder was built extending out of Earth's atmosphere and considerably more. Then if something was to climb up at much less than the escape velocity, what would happen when it reached the end? And what if the object that climbed the "ladder" then fired some kind of thruster/rocket and was going fast enough so that it orbited the Earth. It would mean less energy required to get into orbit?


Theoretical reasons for charge quantization

I'm aware of Millikan's oil drop experiment and I've read that quarks have fractional eletric charge, but I was wondering if there's any theoretical argument that makes us believe charge is quantized. I've also read about Dirac's famous work in which he demonstrated that if a magnetic monopole exists then electric and magnetic charge have to be quantized. But if it doesn't exist? Is there any other theoretical reason for the charge to be quantized or is that the only way we know?


Local Mach number for an airfoil

I would like to know how to calculate the local Mach number on the upper surface of an airfoil given the ambient temperature, the local velocity on the airfoil surface, the freestream velocity, and the ambient pressure. I am assuming from my knowledge that the Mach number depends only the temperature, so I would need to calculate the local temperature on the upper airfoil surface, which is pretty trivial using the isentropic gas equations. So $$a ~=~ \sqrt{\gamma * R * T}.$$ And then just divide the local velocity by the local speed of sound. Is this the right way to do it?


Which direction does a pendulum swing in a car going through a ramp?

Suppose you have a car going clockwise around a ramp at constant speed. A pendulum is hanging from the rear view mirror. Does the pendulum swing outward or inward to the turn?


Considering the theory of special relativity: Is torque still a vector? In classical mechanics it is easy: You have 3 axes and thus 3 planes. Every plane has its own torque so torque has 3 dimensions and thus can be represented as a vector. How are torques represented in the theory of special relativity? You have 4 axes and thus 6 planes and each of them has its own torque.


Entropy of a black hole can be zero?

Black holes are one of highest density objects in the universe. Because of this high density(m/v) more matter is highly compressed in a very small volume( i am guessing that molecules cant freely move in black hole thus forming a perfect crystals). my question is whether entropy($\bigtriangleup S$) of the black hole can be considered as zero or not? Then what is the free energy of the blackhole, whether it will like this $\bigtriangleup G=\bigtriangleup H$.


Is Space-Time a special form of energy?

I know space-time can be influenced by matter and energy, so it must be somehow mingled in with the mix of it all, but does space-time have a fundamental particle? Can we make a little bit of space-time with enough energy? Might the Planck length & time quantize space-time?


QM with complex eigenvalues

What class of theories/physical systems own finite/infinite complex eigenvalues? I do know that e.g., quasinormal modes of BH do have complex eigenvalues, but are they finite or infinite in number? What about Conformal Field Theories with complex eigenvalues? What are the general spectral properties of CFT? By the other hand, what kind of non-hermitian operators could provide spectra with infinite complex eigenvalues in pairs ( complex- conjugated)? What about the problems in the "new"/"recent" quantum graph theory?Is there some "infinite" graph with complex eigenvalues? Remark: I am not an expert on spectral graph theory but I do know a little bit about its methods.


Motivation For Definitions

I noticed in my physics textbook that we define certain relationships to be true. I can see how this is considerably helpful in deriving other relationships from these definitions; for instance, take position: we define these quantities to be so, and from it we can define other quantities like velocity, acceleration, etc. Moreover, most of the time these definitions are well-grounded and intuitive. However, at other times they aren't. To serve as some examples: force, torque, and electric fields. How were these things defined? What was the reasoning used to define these quantities? What are the motivations for these definitions?


For an isolated system, can the entropy decrease or increase?

In any sizable system, the number of equilibrium states are much, much greater then the number of non-equilibrium states. Since each accessible micro state is equally probably, it is overwhelmingly likely that we will find the system in an equilibrium state. However, for a closed system, one that does not interact with any external system, the number of micro states is fixed. Therefore the entropy is fixed. At any given time, the system will be in one of its micro states. Even if the system is in a micro state that does not look like equilibrium from a macroscopic point of view, its entropy will remain the same, since entropy is a property of all the micro states, not just a given micro state. So, are the number of accessible micro states (and hence entropy) of an isolated system constant?


Can we feel our own weight (can we feel normal reaction)?

If we place something on our head, we **can** feel it's weight pushing us down. But we ourselves apply normal reaction which balances the weight of the object and stops it from actually pushing us down. Can we feel our own weight in a similar way ? For that I suppose we would have to feel normal reaction from something we stand on ! I tried 2 things:   1. Standing up, and I could not really experience any force either in upward or downward direction.    2. Pushing/Punching a wall and at the instant I felt a force towards the wall and **not** towards my fist. I found it a lot weird that you can feel the weight of something kept on your head, but not the normal reaction of the body underneath us! Both the forces are completely balanced and neither is a pseudo but still we can feel only one! Is there a scientific reason for this observation? Maybe something about normal reaction which I do not know of.


What would a rotating black-hole look like to a "geo-stationary" observer orbiting the black hole

A rotating black hole is believed to contain a ring singularity rather than a point. However, if an astronaut is orbiting the black hole at exactly the same angular velocity as the blackhole (in direction and magnitude), then it is no longer rotating to him. Would it appear as a point singularity? (if he could somehow see it)


What does BICEP2's results tell us about gravitation waves and quantum gravity?

The BICEP2 results, unless I am mistaken, are a measurement of CMB polarization, i.e. photon polarization. That is, taken at face value they say nothing about gravity directly. Now, we can start to make some assumptions in order to see what this result tells us about other physics like gravity. These assumptions fall into 2 very different categories: (i) very general statements about the nature of quantum mechanics, field theory, and quantum field theory and (ii) specific field-theoretic model dependent assumptions. Assuming this is true, (1) What does the BICEP2 results say about gravitational waves? And what assumptions did we need to make in order to make this statement(s) (general or model dependent assumptions)? (2) What do the BICEP2 results tell us about quantum gravity? And what assumptions did we need to make in order to make this statement(s) (general or model dependent assumptions)? (3) Lastly, are my questions (1) and (2) actually the same question?


Work done by Lorentz Force in case of motional emf

In the classical example of the slidewire generator where the rod slides on a U-shaped conductor in a magnetic field, we get a charge separation due to the Lorentz force. The way the induced emf is usually derived is the following: $\mathcal{E}=W_l/q=vBL$ where $W_l$ is the work done by the magnetic (Lorentz) force. But we know that work is defined as a **scalar product** of force and displacement. In case of Lorentz force, which is always perpendicular to the velocity vector, the scalar product should always be zero. So why then we allowed to use the above expression? It even confuses me more, because later on the book states: > The Joule heating produced in the slidewire generator is: $\Delta Q=R > I_{induced}^2 \Delta t=\frac{(vBL)^2}{R} \Delta t$ > > .... > > But **the Lorentz force does no work** , so where does this energy come > from? It turns out that there is another Lorentz force acting in the system > due to the relative motion of the charges... So, on the one hand it derives motional emf by assuming $F_l$ **does** work $W_l$ and on the other hand, it says later that it does **no** work (which can cause a confusion because heat is still being produced in the circuit). What's going on here? ![enter image description here](http://i.stack.imgur.com/FifpA.jpg)


Foucault pendulum

The equations of motions for a Foucault pendulum are given by: $$\ddot{x} = 2\omega \sin\lambda \dot{y} - \frac{g}{L}x,$$ $$\ddot{y} = -2\omega \sin\lambda \dot{x} - \frac{g}{L}y.$$ What are the equations describing $\dot{x}$ and $\dot{y}$?


How to evaluate commutators?

I need to evaluate $[1/x, p]$. Note: the $p$ is the momentum operator. So far this is what i have: $$= (1/x)(p) - (p)(1/x)$$ $$= (1/x)(-ih*d/dx)-(-ih*d/dx)(1/x)$$ Ii then factor out $-ih$ to get... $$=-ih(\frac{1}{x}\frac{d}{dx}-\frac{d}{dx}\frac{1}{x})$$ This is where im lost. Does the final answer just equal 0 because $(1/x)(d/dx) - (d/dx)(1/x) = 0$?


Does quantum mechanics allow faster than light (FTL) travel?

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\frac{(x-vt)^2}{a+v_st}}$$ Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed. If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$? ## A little background When talking about superluminal tunneling speeds I usually hear explanations like this: > That speed is apparent. In those cases only some frontal part of the wave > function gets through the barrier. Although it's center of mass seems to > have travelled FTL, that function would still stay under the initial wave > function if it continued it's way without the barrier. The center just > shifted because of dropping the rear part. I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from. **EDIT:** Sorry, the question was not about what some classic QM models allow, but actual quantum physics. I understand that Schrodinger's equation allows me any speed, I want to know if this situation is allowed in actual quantum physics. I guess the use of term QM was wrong. Sorry, my bad. ## Question reformulated Is it possible to send a particle that's slowly _spreads_ and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under $c$). I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by _train dropping cars_ ), but is the same possible without any barrier? Maybe I could equivalently ask, if quantum uncertainties allow occasional exceeding of $c$. [1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way. [2] If you need, we can define the _front_ of the wave, for example first of the points where second derivative is zero. It's position is $vt+\sqrt{\frac{a+v_st}{2}}$.


Where to find information on the pulsar map written to Voyager's golden record?

Where might I find details about the pulsar map written on the Voyager's golden record? ![enter image description here](http://i.stack.imgur.com/bFopN.png) > This diagram defines the location of our Sun utilizing 14 > > pulsars of known directions from our sun. The binary > > code defines the frequency of the pulses.


Inflation and scalar spectral index

I've been reading that the results from the Planck satellite constrain a number called the "scalar spectral index" to be 0.96 rather than 1 at the 5-sigma level. This is supposed to be big news, but I don't understand why (my ignorance, of course). For instance, with an index of 0.96, can we say how many e-foldings occured during inflation? What else does it tell us?


Questions about the Jeans length

I have a couple of questions about the Jeans length. Suppose the universe has a homogeneous energy density, except there's a spherical region that is overdense.   * I understand that if the region is smaller than the Jeans length, then pressure, which travels at the speed of sound (compared to the speed of gravity $c$), can build up fast enough to counteract the gravitational collapse. But if the region is bigger than the Jeans length, it collapses. What exactly does it mean? Does the region get smaller and smaller until it vanishes? Wouldn't this create a black hole, since you're stuffing all the energy of the overdense region into a smaller and smaller place?   * Or, if the region is collapsing, wouldn't it eventually be smaller than the Jeans length, at which point it stops collapsing?   * And what happens if the region is underdense, rather than overdense?


Hydrogen atom in quantum mechanics

I have problems following the calculations in Griffiths' _Introduction to Quantum Mechanics_ (Chapter 4.2.1): If you apply the Schrödinger equation to the Coulomb potential you get the following differential equation: $$\tag{4.56} \frac{d^2u}{d\rho^2} ~=~ [1-\frac{\rho_0}{\rho} + \frac{\ell(\ell+1)}{\rho^2}]u. $$ A few steps later my books substitutes $u(\rho)$ without explaining why this is allowed: The next step is to peel of the asymptotic behavior, introducing a new function $v(\rho)$: $$\tag{4.60} u(\rho) ~=~ \rho^{\ell+1}e^{-\rho}v(\rho). $$ How is this function still related to the Hydrogen atom? To me it looks like something total random.


Could negative dimension ever make sense?

After some quick check I found that negative dimensions are not used. But we have negative probability, negative energy etc. So is it so likely that we won't ever use negative dimension(s) ? ## Update I understand there're also dimensions that are not integers e.g. dimension 1½ (?) for fractals or so. Could there also be a dimension such as dimension i (imaginary)?


Black holes and entropy

1)I'm told that for Black Holes, when they radiate (Hawking radiation) particles and anti-particles) split at the event horizon, one going to infinity, the other into the BH. They then lose mass. How is that possible? Wouldn't their masses increase, since they are absorbing particle or anti- particles? 2)Also, their entropy varies as their masses, and since entropy increases, wouldn't their masses also be increasing, along with their area? How does that reconcile with case 1? Thanks


What is the difference between the properties of Electron spin and Photon polarization/helicity?

What is the difference between a photon's polarization/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half? This question stems from wondering why there isn't a classical wave equation like maxwell's equation for the electron.


How is the direction of time determined in general relativity?

In special relativity every frame has its own unique time axis, represented in Minkowski diagrams by a fan-out of time vectors that grows infinitely dense as you approach the surface of the light cone for a given rest frame. Viewed from any other frame, these same time vectors shift about exactly the same fan-out topology, so no one frame perspective is any more "correct" than any other one. In general relativity this does not quite seem to be the case. If nothing else, all those worldlines extend over billions of years and so presumably end up having a quite precise average time orientation for any given local region of space, even if you select a complicated overall topology. Conversely (and I assume equivalently), if space is viewed interpreted as manifold, the idea of an average "flatness" that is orthogonal to this long-term time axis appears to be assumed, since it would look quite odd to see the entire universe slanted relative to its own time axis, yes? So:   1. Am I understanding this correctly? Does GR applied over eons necessarily end up defining and distinguishing a unique and well-defined overall time axis?   2. If so, how is this unique time axis ascertained in experimental practice? Is it taken to be some sort of very large center-of-momentum average, as the billions-of-years bundle analogy would seem to suggest?   3. If a distinguished time arrow does exist in GR, does the existence of that axis imply any experimentally meaningful implications for its associated SR frame?


Why does air circulate on an airfoil --- The Kutta Condition

![enter image description here](http://i.stack.imgur.com/flYBz.png) Why does the air circulate on a flowing airfoil, thus giving rise to increased velocity (circulation + relative airspeed) above the wing and hence decreased pressure.


Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can be assumed constraint force. But in this point of view, I couldn't understand why friction is constraint force. In lagrangian formulation, we divide forces into two part, F= F(applied) + F(constraint) If particle moves in one dimension, and assuming there exists sliding friction, that particle can move anywhere. The sliding friction never restrict the domain that particle can move. so I think the sliding force is applied force, rather than constraint force. can anyone clarify why friction is constraint force??


Milky way rotation data

Can anyone tell me where I can download sample rotation curve data for the Milky Way in csv format?


Magnetic hot air balloon

Normally, hot air balloons are limited to 60,000 ft (related question), the limits being the mass of the physical structure containing the air, and the heat source. If one constructed a magnetic bubble and filled it with plasma (AKA mini- magnetospheric plasma propulsion) which was kept hot with a really powerful ground based microwave source (~3MW for a 1 ton payload, from my back of the envelope calculations), could it reach space (Kármán line, atmospheric pressure 32 mPa)? Or would something stop it working at a particular altitude, e.g. the plasma would cool down too fast, or become detached from the magnetic field, or the electrical conductivity of the air at some height would neutralise the plasma, or some other reason I don't know yet? If it is possible, how accurate are my estimates (I don't know how efficiently microwaves would heat a plasma, nor how much power would really be lost by something like this).


Are fundamental forces conservative?

I wonder whether fundamental forces are conservative. First of all, I'm not sure if we can talk about conservative forces, since to study electromagnetism, weak and strong interactions we need QFT. For gravity, I'd that it's not conservative, because energy conservation doesn't even apply.


How to convert RL Circuits?

_**How do we convert a parallel RL circuit into its equivalent series RL circuit ?_**


How is this process not quasi-static yet reversible?

Consider a (adiabatic) canister with a piston containing some gas kept in a vacuum. There are two weights on the canister which equalize the pressure of the gas on the piston. Assume the system is at equilibrium. I remove one weight from the piston. The system will go out of equilibrium. After some flux, it will again settle into equilibrium. Clearly, this is not a quasi-static process. But its a reversible one, isn't it? If I put the weight back on the piston, the system will again achieve its initial state. How can a non-quasi-static process be reversible? **EDIT:** According to Wikipedia, a reversible process "is a process that can be 'reversed' by means of infinitesimal changes in some property of the system without entropy production". So, if a process can be reversed _without_ the means of infinitesimal changes without entropy production. Would that qualify as reversible? (as in the example given above)


How reflected objects are composed and who is responsible for that?

Please refer to this image. The scene contains an object close to a mirror in the wall and a window, note that the reflected object is receiving more light than the object itself. I read some articles about the Law of Reflection and the Ray Principle but did not get a straight answer on how the mirror(or my eyes) knows how to compose that virtual image in a way it appears to receive more light. According to the definition of the Plane Mirror, the light rays do not actually pass through the virtual image, then why the virtual image is being composed with more light and who is responsible for that? The mirror? My eyes?


What would be the basis vectors for this 2D crystal structure?

![enter image description here](http://i.stack.imgur.com/hvaL7.jpg) In the above image, I have a 2D crystal structure. The lattice vectors are described by:               a = {-1/2, -Sqrt[3]/2};     b = {1, 0};      and the location of atoms A and B are given by:               \[Tau][A] = {2/3, 1/3};     \[Tau][B] = {0, 0};      Now the problem I'm tasked with is to plot the locations of the atoms A and B, which I have done as seen above, where A is red and B is blue. I also have to plot the lattice vectors which I have done in blue arrows as seen above, and lastly I have to plot the basis vectors and this is where I don't know what to do. I don't know what the basis vectors are. Googling has led me to discover that a lot of people use lattice vectors and basis vectors interchangeably, and overall, I have no clear definition to work with. Sometimes people say basis vectors are orthonormal, sometimes they need to be linear combinations with integers, sometimes with any real numbers. I'm frustrated by the looseness with which all kinds of sources use these terms, and ultimately I don't have a clear reference to go with.


A lightning protection device physics

Here is a description of the principle of the operation of a new lightning device: _During a storm the ambient electric field may rise to between 10 to 20 kV/m. As soon as the field exceeds a thresold representing the minimum risk of a lightning strike, the OPR lightning terminal is activated. **It draw its energy from the ambient electric field the energy required to generate high voltage pulses, creating and propagating an upward leader.** No other power sources are required, and no radioactive components are used._ It remains unclear to me the physics of the drawing the energy from the electric field. Obviously, you can not draw the energy from the electrostatic field, no matter how strong it is. So i suppose that the electric field must oscillate more or less. The question is: Which is a minimal frequency of the oscillations so that such device can work properly? A full description of the device


Relative Motion

I read the following: http://www.feynmanlectures.caltech.edu/I_15.html#Ch15-S4 > "Now let us see what happens to the moving clock. Before the man took it > aboard, he agreed that it was a nice, standard clock, and when he goes along > in the space ship he will not see anything peculiar. If he did, he would > know he was moving—if anything at all changed because of the motion, he > could tell he was moving. But the _principle of relativity_ says this is > impossible in a uniformly moving system, so nothing has changed." In my point of view Feynman uses the principle of relativity as a kind of an _axiom_ for his derivation of time dilation. **Question:** How do we know that the _principle of relativity_ is true and can use it like an axiom for new phenomena? If we had treated Newton's laws of motion as axioms we would never have found out how nature actually works. In other words, what is the law behind the fact that you cannot know how fast you are moving? How can you use this assumption as axiom for e.g. the behavior of clocks as Feynman does further down in the link? P.S.: Feel free to edit since I'm not a native English speaker. Nevertheless I want to improve my English and it would be great if you pointed out my errors.


Feynman's sum over histories?

The concept requires all possible path's to be mapped out, and any remaining paths not cancelled out represent the most probable path of the object. Considering this:      i) If "infinite" paths are mapped out (since there are infinite paths for every movement) , isn't it guaranteed for every path to be canceled out? Every path has an opposite, and if infinite paths are mapped, the opposites are guaranteed to be present. ii) How do you map out infinite paths? Nothing infinite can be completely mapped, on the basis that it is infinite.


What's the basic premise of General Relativity?

What is the basic assumption(s) required to explore general relativity? For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the same for all observers, then it's possible using a mix of algebra and some calculus to derive every SR result from time dilation to length contraction, $E = mc^2$, and so on. What is the basic premise of general relativity? I assume that all of SR is included but is there anything else?


What is the effect of total internal reflection on polarized light

Total Internal Reflection causes phase change of a light beam. I searched for effect on a polarized light beam by Total Internal Reflection but could not find much. I am assuming polarization does not get altered. I am unable to understand the reason why atoms in glass are able to change phase but unable to change the orientation of electric field of an EM wave. I am re-studying optics after 20 years so please excuse me for being rather naive.


Optical absorption -- what are the common ranges and mechanisms?

So let's say you do some reflection/transmission spectroscopy of a material. It's clear that it's absorbing in some range. What would be your first step in identifying the source of the absorption? The two main clues I would look at would be   1. The range it occurs in (if your incident light is in the x-ray range, it's clearly not phonon absorption, for example)   2. The shape of the absorption (is it a peak? Or an absorption 'edge', where it isn't absorbing below some energy, and absorbs everything above, for example) But I'd like to know if there are other ways as well. Additionally, does anyone know of a 'map' of the ranges you'll typically find different types of absorption in? I saw one on wikipedia once, but it only had three types, and now I can't find it of course. For example, a few types of absorption I can think of off the top of my head are   * Interband electron transitions   * Intersubband electron transitions   * Impurities   * Phonons   * Vibrational modes   * Atomic transitions I understand that a lot of these will be material-dependent, but often the same mechanism in different materials will yield absorption in the same order of magnitude range. What ranges would one typically see these (and other) absorption mechanisms in? Thank you!


Stationary wave on a string

The problem says: > _A string lies on the x-axis, and is fixed at $ x=-a $ and $ x=a $. At $ t=0 > $ we impose a transversal velocity given by $ a^2-x^2 $. Assuming that the > velocity of the wave is $ c $, find the solution to the wave equation._ So here's what I did so far: I set up the solution in complex form as $$ \tilde{y}(x,t) = \tilde{A}e^{i(kx-\omega t)}+\tilde{B}e^{i(-kx-\omega t)} $$   Then imposing the condition of fixed string, I got the values for $ k_n =\frac{n \pi}{2a} $ and $\omega_n =\frac{n \pi}{2a}c $ In the process I got the relation $\tilde{A}=-\tilde{B}e^{2ka}$ Now my problem is when I plug back into the original equation, I get something like this: $$ \tilde{y}(x,t)=\tilde{B}e^{-i\omega_n t}(e^{ik_nx} -e^{i(k_nx + 2k_na)})$$ I know that $2k_na=n\pi$, but that gives me two possible results for $e^{i2k_na}$, 1 and -1. That confuses me a bit, and I don't know how to continue from here. I've seen the example of the plucked string that goes from 0 to $L$, and that's much simpler than this case. I guess I just don't know what to do with the constants introduced by the fact that the string goes from $-a$ to $a$. Any help is appreciated.


Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} \sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu,\gamma_\nu]~? \end{align*}


how dense fluid affect the buoyancy force?

I read a story regarding the Archimedes' principle in a magazine of popular science and I am thinking of the following question: how does the density of the fluid change the buoyancy force for the same object? As we know, the Archimedes' principle tell that for any object in a fluid, the buoyancy force equals to the weight of the fluid displaced by the object. It is pretty straightforward. Now, if I have an object partially floating on the surface of a liquid, so we have $$ F_b = \Delta V \rho g $$ where $\Delta V$ is the volume of the displaced liquid and $\rho$ is the density of the liquid. So what happen if we place the same object into a denser liquid? Physically or intuitively, since the liquid is denser, it is harder for the object to 'inject' into the liquid, so the buoyancy force should be bigger, so less part of the object submerge into the liquid. But if you look at the math, it seems not like this. Well, now $\rho$ is bigger, but the volume of displaced liquid will be smaller because it is harder to submerge the object into a denser liquid too. So how do we that for denser fluid, the same object will experience bigger buoyancy force instead of being the same? So my question is from intuition, the same object in the denser liquid should submerse less than the case in less dense liquid. But from the math, it buoyancy force could stay the same or more. So how to prove from the math that our intuition is correct?


How can I cheat an XRD sample (~10 mg) from a total 1g required? How can I use for the background.

How much can affect (in terms of hump) my X-ray diffraction pattern the "amorphous" glass or maybe someone can suggest me other material to compensate for the small amount of the sample. Note: my sample is a powder.


Why should the Standard Model be renormalizable?

Effective theories like _Little Higgs models_ or _Nambu-Jona-Lasinio model_ are non-renormalizable and there is no problem with it, since an effective theory does not need to be renormalizable. These theories are valid until a scale $\Lambda$ (ultraviolet cutoff), after this scale an effective theory need a UV completion, a more "general" theory. The Standard Model, as an effective theory of particle physics, needs a more general theory that addresses the phenomena beyond the Standard Model (an UV completion?). **So, why should the Standard Model be renormalizable?**


Could the acceleration of universe expansion be caused by gravity itself?

Dark energy is suggested to be a repulsive force in the universe causing an accelerated expansion. If the amount of mass outside our observable universe is greater than inside (higher mass density), would it not cause an accelerated expansion from our viewpoint?


How is time distributed through space?

I was greatly confused when I saw video saying that if two wormholes were separated by _time_ instead of _distance_ , it would allow time travel. My question is what did they mean by this statement? What do they mean by time instead distance? How is time distributed through space?


Conservation of Energy and Momentum Regarding Forces - clarification needed

The other day, my teacher stated something along the lines of, "Conservation of momentum is not violated by the actions of internal forces, but the conservation of energy is violated. Energy is conserved only when/if internal forces are conservative." Can anyone explain this is simpler terms or reiterate it in some fashion?


How to deal with conflicting "no-slip" Navier-Stokes boundary value constraints?

The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces. Now that my class is done, I've been playing with solving an idealized fluid flow problem: figuring out the rate of "spin-down" of stirred fluid in a non- bottomless cylindrical container (ie: my coffee after being given a good stir). For the non-time dependent portion of the problem, I'm not able to apply the no-slip condition as taught in class to my cup of coffee. Matching velocities to my stir stick (I've idealized that as a cylinder as in Acheson's bottomless tea example (pg 45) ), I've got a problem at the base of the cup: If I match velocities of the fluid to the "stir-stick", I can't also match velocities to the bottom of the cup (zero velocity) where the stir-stick touches the base. I've been pointed to some journal articles to read, but won't be able to do so until the fall when I enroll in my next course (and get library access again). In the interim, I was wondering if I could get some pointers on an approach to deal with conflicting 'no-slip' conditions like this one (ie: any problem with a moving interface in contact with a static one has this trouble.) EDIT: I believe that I can work around the inconsistency problem with no-slip constraints for this problem by "stirring without touching the bottom of the cup" ... ie: set the stir depth to 1cm (say). This makes the non-time dependent solution uglier, since I have to treat three different regions:   1. Below the stir stick (zero velocity no-slip condition on the inside of the cup in this region ; velocity matching the stir velocity along the circle at the stir radius at the stir depth).   2. Between the center of the cup and the stir stick. "no-slip" conditions: match velocities below the stir stick. match the stir velocity at the stir radius.   3. Between the stir stick and the edge of the cup. "no-slip" conditions: zero velocity on the cup edge. stir velocity at the stir radius. This makes the problem much messier ... I already have Bessel functions to deal with, and now have to apply them separately in three regions. I'm still interested to see if there's a better way to deal with the boundary value conditions ... ie: without forcing an artificial distance between a static and a dynamic interface so that they don't touch.


Are double-slit patterns really due to wave-like interference?

According to various sources on the web, it seems like the general concensus is that there isn't actually any wave-particle duality with quantum particles. For example, this article implies that diffraction patterns in double-slit experiments were interpreted as wave interference due to _apparatus limitation_ at the time they were first performed. Does this mean that all those sources and animations showing two waves interfering are simply incorrect, classical conclusions which don't have anything to do with (quantum) reality? What's actually the most confusing is that most sites which state that it is now possible to pass individual photons through these slits, also claim that these individual photons somehow _interfere with themselves_ resulting in the observed patterns. That seems like a rather thin explanation, doesn't it? So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the "spooky" explanations?


Pulley problem with spring balance

![enter image description here](http://i.stack.imgur.com/i73YD.jpg) Please excuse the crude drawings The question is, if I burn the thread t, how will the reading in the spring balance(S) change? Ie, increase, decrease, or remain the same? My approach: When we burn the thread, the objects will move, but the tension should increase a bit. ( I can't explain this, I only get this intuitively.) and, since the spring balance reads the double of the tension, I think this should make it a lot more than the initial reading. Since if the tension in the intact string changes by $\triangle T$, the reading should increase by $2 \triangle T$. That's all I can think of. So as you see, my approach is very intuitive, and hence I am asking this in order to get a logical and stronger reasoning


Minimum required to lift a roller up a step

Consider a roller of diameter $0.82m$ which is to be raised up a step measuring $0.32m$. Find the minimum force required to do so. The roller weighs $500N$. ![Diagram](http://i41.tinypic.com/2qmlytu.png) The roller could be considered as a point mass $F$, and its body could be considered a lever with $F_{applied}$ acting at $G$ and $F_{perpendicular}$ acting perpendicular to diameter $GC$. Let $\angle FCH = \theta \implies \angle IFG = \theta$. $F_{perpendicular} = mg\sin\theta$. After that's its a simple application of torque and moments. I'm not quite sure if this resolution of vectors into components is quite the right approach. At least, the value given in my textbook doesn't correspond to the one I got. EDIT: In response to the comment below, I decided to post my workings: $\sin\theta = \frac{FH}{FC} = \frac{0.09m}{0.41m} = 0.2195 \implies F_{perpendicular} = 500N\cdot 0.2195 = 109.756N$ The anticlockwise torque equals $\tau_{anti-clockwise} = 109.756N\cdot 0.41m = 45Nm$ This has to equal the clockise torque $\tau_{clockwise} = F_{applied}\cdot 0.82m$ $\implies F_{applied} = \frac{45Nm}{0.82m} = \boxed{54.878N}$ According to the book, the answer is $110N$? Maybe the book is incorrect?


Trinary System Arrangment?

I'm writing a science fiction story and wondered if someone could help me with the setup of a trinary star system I have in mind. Basically I just wanted to know if: a) a system with a central, Sun-sized star could have two smaller (orange dwarf?) stars orbiting it in a circular trojan orbit, and... b) how much disturbance would this type of orbit cause to any planets orbiting the inner star? This may all sound ridiculous, but any input would be greatly appreciated. Thank you.


Emergent symmetries

As we know, spontaneous symmetry breaking(SSB) is a very important concept in physics. Loosely speaking, zero temprature SSB says that the Hamiltonian of a quantum system has some symmetry, but the ground state breaks the symmetry. But what about the **opposite case** of SSB? The ground state of a quantum system possesses some kind of symmetry while the Hamiltonian does not have this symmetry. For example, the exactly solvable Kitaev-type model Hamiltonians explicitly break the spin rotational symmetry, but the ground states are spin liquids, which possess the spin rotational symmetry. I wonder whether this **opposite case** of SSB plays an important role like SSB in physics? **Note:** Examples with **exact emergent symmetries:** A simple example with _exact emergent $SU(2)$ spin-rotation symmetry_ can be found here A simple model that exhibits emergent symmetry? Another example with _exact emergent $U(1)$ symmetry_ is presented in the Supplemental Material of this paper, where it is appeared on page 2 under Eq.(A7). Examples with **approximate emergent symmetries:** A chiral spin-liquid phase with _emergent $SU(2)$ spin-rotation symmetry_ in the framework of Schwinger-fermion mean-field study of the strongly correlated Kane-Mele-Hubbard model. The example with _approximate emergent lattice 3-fold rotation symmetry_ is the existence of Ferromagnetic(FM) ground state in the Kitaev-Heisenberg model, where the model Hamiltonian explicitly breaks the lattice 3-fold rotation symmetry but the FM phase possesses the lattice 3-fold rotation symmetry. Another example with _emergent chiral symmetry_ was proposed by X.G.Wen in his paper, as seen on page 18, title **C**. A third example with _emergent time-reversal symmetry_ can be found here. An example with _an emergent global topological U(1) symmetry_ is presented here. **More examples with emergent symmetries are welcome.** **Erratum:** The example of "Kitaev model" I gave in my 2nd paragraph is not correct, please see Why we call the ground state of Kitaev model a Spin Liquid? for the reason.


What does symplecticity imply?

Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context of dissipative systems, so I am no longer confident in my assumption. My question now is, why do authors emphasize symplecticity and what is the property they typically imply with that? Or in other more provocative terms: Why is it worth mentioning that something is symplectic?


Electric field outside a sphere with a cavity

I have a sphere of radius $2a$ centered at the origin and made of a nonconducting material that has a uniform volume charge density $\rho$. A spherical cavity of radius $a$ eccentric to the right side (touching) with its center aligned with the x-axis is now removed from the sphere. Find the electric field produced by this distribution along the x-axis for $x > 2a$. I started by establishing expressions for $E$ in both the uniform section and the cavity: $E_u = \rho 2a/(3\epsilon_0)$ and $E_c = -\rho a/(3\epsilon _0)$ and $E_t$ inside the cavity is $ E_t = \rho a/(3\epsilon_0)$ Is it correct that the $E$ along the x-axis would have the same magnitude as $E_t$?   I'm thinking it's not, but I'm not sure about the other idea I had which is that it would be $E = \rho r/\epsilon_0$. $r$ now being the distance from the center of the solid/enclosing sphere. This is the result of exchanging the $-\rho$ that is required to make the net charge inside the cavity with $\rho$ and adding the $E$-s together again.


Expectation Values in Quantum Mechanics

Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?


Car accident, put in park or neutral?

I was waiting on a red light the other day and was wondering. If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral? I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed.


Noether's theorem vs. Heisenberg uncertainty principle

In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle. Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they? I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists. Also compare this question where again momentum and space are connected, this time through a Fourier transform. (*) i.e. electric potential and magnetic vector potential combined.


Why is everything floating in space so cold?

Of course there is pretty hot stuff too. For example Suns. But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space? Why aren't they hot from time to time? Is it because they loose their heat by radiant heat? And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors?


In heterojunction problem, how to align the energy band in presence of bias voltage

For example, SiO$_2$ barrier embeded between Fe magnet and 2-dimensional- electron-gas such as Si. How to align the energy bands of the three materials when an electric field is perpendicular to the interfaces? Do they have the same Fermi level? Could somebody draw the band offset for me? Where can I find the data of band offset?


Entropy change at varying temperatures?

Entropy change is defined as the amount of energy dispersed **reversibly** to or from the system at a specific temperature. Reversivility means that the temperature of the system must remain constant during the dispersal of energy to or from the system . But this criterion is only fulfilled during phase change & isothermal processes. But not all processes maintain constant temperature;temperature may change constantly during the dispersal of energy to or from the system . To measure entropy change ,say, from $300K$ & $310K$, the range is divided into infinitesimal ranges ,then entropy is measured in that ranges and then is integrated as said in this site . But I cannot understand how they have measured entropy change in that infinitesimal ranges as there will always be difference between the temperature however small the range might be . What is the intuition behind it? Change of entropy is measured at constant temperature,so how can it be measured in a range ? I know it is done by definite integration but can't getting the proper intuition . Also ,if by using definite integration to measure change, continuous graph must be there(like to measure change in velocity,area under the graph of acceleration is measured) . So what is the graph whose area gives change in entropy? Plz help me explaining these two questions.


Volume of Brillouin zone is the same as Fourier primitive cell?

In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and primitive parallelepiped in real space?


The dimensional formula of distance travelled in $n$th second

I read that the dimensional formula of distance traveled in $n^{th}$ second is same as that of velocity. Okay, the formula for the distance traveled in $n^{th}$ second is $s_t= u+\frac{a}{2}(2t-1)$ where $u$ is initial velocity, $a$ is uniform acceleration and $t$ is the time. If we proceed and expand the aforementioned formula we will get $s_t = u + at-\frac{a}{2}$. Now, the last term i.e. $-\frac{a}{2}$ is not a velocity and isn't the principle of homogeneity violated?


Chaos and quantum physics: How many ways can a bonfire burn?

I'm interested in the extent to which quantum physical effects are seen at a macroscopic level. I might get some of the physics wrong, but I think I'll get it close enough that I can ask the question: Let's sat that we create a bonfire and let it burn until it burns out. As the smoke rises from the fire, turbulence takes over and the smoke particles and steam and hot air all mixed together. By the end of the night when the fire has burned out, the collection of molecules in the system are in some position/velocity X. My question: Let's assume the multiverse interpretation of quantum physics. How many possible end state superpositions can there be in this situation? Ok, that's imprecise and incorrect because it would actually be an uncountable infinitude of possible end states. How about this: Given the end state that we observed, what percentage of the end state superposition would be "visually" indiscernable from the end state that we observed so that each molecule would be in nearly the same end state across that portion of the multiverse? Or put another way: Do quantum effects sneak into everyday life fast enough that we can observe them? If we are effected by quantum physics at all, I imagine this is roughly a function of the timescale of the chaos effects.


Band offset and strain-induced valence band splitting in semiconductor compounds

My aim is to draw a plot of $GaAs_{1-x}Sb_{x}$/InP (GaAsSb on InP substrate) band offset as a function of As/Sb composition. The first part is easy if I'm doing this correctly: I found VBO (valence band offset) parameters for GaAs and GaSb, then the bowing parameter for $GaAs_{1-x}Sb_{x}$ VBO and finally the bowing parameter for $GaAs_{1-x}Sb_{x}$ band gap. This is what I got: ![Unstrained GaAsSb](http://i.stack.imgur.com/XOAPX.png) However, when I put GaAs or GaSb on InP, strains appear and the valence band is split into heavy hole and light hole subbands. In some articles I found the way to calculate the splitting: $\Delta E_{HH}=\partial E_H^V + \partial E_S$ $\Delta E_{LH}=\partial E_H^V - \partial E_S$ $\partial E_S=b(1-2\frac{C_{12}}{C_{11}})\epsilon_{||}$ $\partial E_H^V=2 a^V (1-\frac{C_{12}}{C_{11}})\epsilon_{||}$ a, b and Cs are material properties and can be found in this article: http://jap.aip.org/resource/1/japiau/v89/i11/p5815_s1 (Vurgraftman, Meyer, Ram-Mohan) However, I'm not sure what the $\epsilon_{||}$ is. Is it the direction of growth? Whatever it is, I can't see where I should consider the substrate, as I assume the strain to be substrate-dependent. Is it hidden in the $\epsilon$?