Tidal acceleration for a retrograde rotation?

Consider two nonelastic spherical bodies with uniformly distributed density, a small such body in a circular orbit around the bigger one. And consider the smaller body's rotation is matched (as if "tidally locked") to its orbit so the same hemisphere always faces the larger body. Now, here's my question: a particle on the daylight surface of the orbiting body is orbiting the "sun" with a smaller radius than a particle resting on the "night" side. If the two particles weren't attached to the body, the daylight particle would have a faster solar orbit and the nighttime particle a slower orbit. Would this not impart a retrograde rotational force on the orbiting body? If this is an already understood concept, what is it called? I read up on tidal locking and tidal acceleration on Wikipedia, and this dynamic wasn't mentioned. I think it's kind of interesting because I never thought about tidal forces doing anything with inelastic objects.


AdS Space Boundary and Geodesics

I'm new to working with AdS space and am primarily concerned with black holes. I'm just playing round with the metric for AdS$_4$ $$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\zeta^2$$ for $f(r)=r^2+m $, $\space\space\space\zeta=d\theta^2+\sin^2\theta d\phi^2$. My problem is trying to understand the boundary; specifically when considering particle trajectories:   1. For null geodeiscs, I've read that they reach the boundary of AdS space, which seems to commonly expressed as saying they are represented as straight lines. I don't understand how these two phrases are the same and how to show that this is the case starting from the metric I've stated. Using constants of motion etc, and assuming a radial path, I find the equation $\frac{dr}{d\lambda}=k$, for $k$ constant.   2. For timelike geodesics, I know they do not reach the boundary and equivalently I read that they are represented by the boundary of slices of the hyperboloid i.e. ellipses. Again, how do I show that this really represents timeline geodesics? As above (but $ds^2=1$ in this case) I find the equation $\Delta\tau=\log(r+\sqrt{k^2+m+r^2})\space \vert^b_{r_0}$ where $b$ is the boundary and $r_0$ the initial $r$. I've been reading (as much as I can using the fairly limited coherent literature on the topic) and I can only find discussions on this matter, with some diagrams. None seem to go about this question the way I have above and consequently I'm thinking there must be something wrong with what I've done.


How does one prove that the current of a spontaneously broken symmetry generates a particle?

I am having a hard time arguing that, after spontaneous breaking of a continuous symmetry of a field Lagrangian, local fluctuations around the vacuum can be interpreted as particles (without referring to analogies from condensed matter physics). I encountered a treatment which stated that, when the symmetry is unbroken, the corresponding current annihilates the vacuum $$J^{\mu} | 0 \rangle = 0$$ while after spontaneous symmetry breaking, the current creates a state out of the vacuum with some momentum $k^{\mu}$ $$J^{\mu} | 0 \rangle = k^{\mu} | k \rangle $$ My question is why this second equality holds only after spontaneous symmetry breaking.


formal framework for talking about 'minimal couplings'

usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields which will be used to derive some physical states and then, retroactively support the coupling from a match with the physics Usually one reads about 'minimal couplings' and for scalars with vectors on might see a factor like $\frac{ \partial{\phi}}{ \partial{ x_{\mu}} } A_{\mu}$ But my question goes on in what general or abstract sense such factors represent a minimal coupling? minimum of what? if is the total degree of powers of the fields? do we have some variational principle for lagrangian couplings that these terms can be called a stable point in such variations?


Certain material heating water in a recipient

I don't know how to resolve a problem, but I don't want the answer since I'm almost going to have it resolved. What the problem says is we have 85 liters of water at 7ºC in an iron pot of 29kg. We want the water to be at 86ºC. The temperature of the iron pot is 12ºC. The water is heated by fire wood (65% of the thermal energy is wasted in combustion), and has a heating value of 12 MJ/kg. We need to determine the amount of wood needed to heat the water. I would apply the normal formula of energy transfer: m1*c1*(t2-t1)=m2*c2*(t2-t1). But since we are using fire wood there, I don't know how to resolve this problem, probably because I'm missing something there... and I don't know what should I do. Which formula or principle would I need to use when having the material and the heating value of the material used?


Do all event horizons emit radiation?

So, the event horizon around a black hole emits radiation, and Rindler space is full of thermal energy. I guess I have two questions- does the Unruh effect have anything to do with radiation from the apparent horizon in Rindler space? And what about the cosmological horizon- any radiation there?


What is the energy functional for $\nu=5/2$ Moore-Read state?

I am trying to do some Monte Carlo simulations for Pfaffian state from Fractional Quantum Hall effect. I am wondering what is the energy functional for $\nu=5/2$ Moore-Read state?


How would one calculate the amount of water contained in a cloud?

So I was looking out the sky one day and I wondered how I would go about calculating how much water was contained in a cloud. I figured the following simple outline 1) We need to roughly know how big it is. We'll also be figuring out how high the cloud is. Use some sort of geometric method? Triangles? 2) We need to relate its color to its density. (Darker clouds more dense) (White fluffy clouds less dense) 3) Correction factors due to lighting conditions during different parts of the day. Can some body help me figure out the relevant calculations needed? Or how much water does a cloud contain based on its category?


What does the wavevector $\textbf{k}$ mean?

In Ashcroft, Mermin Solid State Physics, Eq. 17.43 is $$ \epsilon(\textbf{k}) = \frac{\hbar^2 k^2}{2m} - e\phi(\textbf{r}) $$ where $\textbf{k}$ is the wavevector and all other symbols have their usual meaning. What does the wavevector tell me and why do we use it instead of position $\textbf{r}$?


Bekenstein bound for electron?

Using the Wikipedia version of the Bekenstein bound, and substituting the Wikipedia values for electron mass and radius, one obtains 0.0662 bits. Does this really mean that a system, any system, placed inside a sphere the size of an electron, and weighing no more than an electron does, is almost determinate? How about an electron itself? Wouldn't one need at least a few bits to characterise the behavior of an electron in magnetic space? (I am a professional mathematician but I know very little about physics, I'm sure I'm missing something obvious here...)


Would half of electromagnetic wave be absorbed on first sensor?

If I would set up electromagnetic emitter that would emit one wave, but while it would be emitting it I would spin emitter 180 degrees at such speed that one half of wave would be emitted in that time (or it could be 1 degree, but the speed of turn would have to match the half of wave's oscillation) and then would set up a lot of sensors on the path of the wave that are able to measure strength of electromagnetic field (so that it would hit multiple sensors with that partial wave, sensors can be placed as far away as it needs be to allow the small turn in angle allow to have bigger affect). Would all the sensors show the value of the field going up to the peak and back down to zero, or would first sensor absorb the wave and all other sensors would show 0 values. **Update:** Reason I am asking is that I can't wrap my head around - for sensor to register something it has to be affected by electromagnetic field - to affect sensor some energy has to be used, if wave would affect all the sensors it would imply that it has infinite energy as increasing the amount of sensors would still 'do to the work' register a value, but there has to be more 'work' done with each additional sensor, so in my understanding 1st sensor (or some other) would have to take all the energy of the wave and rest would not get anything. Please forgive me if question is unclear or silly, I am not a physicist. If you need additional information please ask I will do my best to clarify.


Why does solar cell will create as a Multiple cells instead of one large cell area?

Why is a solar module assembled from multiple cells instead of one large cell area?


absorption length of thermal neutrons in a medium

My book states that $$N(x)=N_0\cdot e^{n\cdot \sigma \cdot x}$$ for $n$ absorbing particles per volume element with a cross section $\sigma$. Can somebody explain where this comes from?


Widom-Larsen Theory

Does the Widom-Larsen theory pose a credible underpinning for Low Energy Nuclear Reactions, often incorrectly referred to as "Cold Fusion"? http://arxiv.org/abs/cond-mat/0505026


What happens if a black hole sucks up a massive amount of oxygen and hydrogen?

Suppose a black hole were to suck up a massive amount of surrounding oxygen and hydrogen. Would this ignite inside the black hole and turn it back into a star?


Evaporation from a capillary tube

Consider a capillary tube (say from a liquid / capillary thermometer), that means a tube of small internal diameter which holds liquid by capillary action . The tube is fulfilled with water and closed at one end. How long it would take for water to completely evaporate out through the open end of the tube? To be specific, let the diameter of the tube be $d=1$mm, the length of the tube $L=1$m, the environmental temperature $T=300$ K and pressure $P=10^5$ Pa, the relative humidity $f=60$ %, the contact angle a) $\theta=0^{\circ}$ and b) $\theta=180^{\circ}$. Does it have any effect on the time when the tube is positioned horizontally or vertically(open end above)? ![A capillary tube with a liquid](http://i.stack.imgur.com/1akdv.gif)


Explanation of Cardy's "a theorem"

There seems to have been some discussion of Cardy's "a-theorem" recently: > “It is shown that, for d even, the one-point function of the trace of the > stress tensor on the sphere, Sd, when suitably regularized, defines a > c-function, which, at least to one loop order, is decreasing along RG > trajectories and is stationary at RG fixed points, where it is proportional > to the usual conformal anomaly.” said Cardy. “It is shown that the existence > of such a c-function, if it satisfies these properties to all orders, is > consistent with the expected behavior of QCD in four dimensions.” I would appreciate it if someone could give a reasonbly concise explanation of what the theorem states and implies, at a level for someone with a superficial understanding of quantum field theory. For example, what is the sphere in the above quote and how does the existence of the c-function lead to the main conclusion of the a-theorem which is purportedly: > "... a multitude of avenues in which quantum fields can be energetically > excited (a) is always greater at high energies than at low energies. " Edit: I think I found what I needed. Section 4.4 of David Tong's string theory notes gives a nice explanation of the trace anomaly and the "a" and "c" theorems. I assume the sphere they were talking about in the Nature article is just the $S^4$ of Euclidean compactified spacetime, in which case "a" is its Euler characteristic.


Can spacetime exist in the absence of matter and energy?

I'm pretty sure Ernst Mach would have said that spacetime cannot exist without matter in it. But I'm also pretty sure that a black hole can be described as a self- sustaining gravitational field, which is (I guess?) a case of spacetime sustaining itself without the matter meaning much anymore. So, what is the current theoretical view on this?


How many ways are there to distribute M excitations of N identical particles among K=3 quantum harmonic oscillators?

I'm trying to numerically calculate a partition function of N non-interacting but identical particles in a 3D SHO. To do this, I'd like to know the degeneracy of $M$ excitations, $N$ _indistinguishable_ particles, in $K=3$ _distinguishable_ oscillators. In other words, what's the degeneracy of $E = \hbar \omega (M + 1/2)$? I want to capture the Bose statistics and calculate something like a few-particle Bose-Einstein condensate. For example, if I have two excitations, two particles, and three oscillators, the degeneracy is nine (please correct me if I'm wrong!):   * Both excitations in one oscillator: 3 ways   * One excitation in each of two oscillators: 6 ways My current approach is to write an algorithm that counts all of these combinations, but I feel like there must be a simpler way. Alternatively, say we only have one oscillator. How many ways can I distribute M excitations among N identical particles? Edit: I originally miscounted the degeneracy of my example. I think it's 9, not 6.


What is the basic difference between absolute pressure and gauge pressure? I have a great confusion regarding these two pressures. Please someone explain it to me


Why isn't there heterotic holographic QCD?

I think the question speaks for itself... Top-down holographic QCD, like Sakai-Sugimoto, always involves the Type II string. There are one or two papers on hQCD using the Type 0 string. But I can't see anyone making hQCD models using the heterotic string. Why not? I may as well explain how the question arose: I was thinking about two forms of conventional string phenomenology - M-theory on G2 manifolds, and E8xE8 heterotic models - and was looking at possible beyond-standard-model symmetries, like flavor symmetries, family symmetries, etc. Somehow I stumbled into a time warp and ended up in 1965, looking at "The Covariant Theory of Strong Interaction Symmetries" by Salam, Delbourgo, and Strathdee, in which "A classification of particles is suggested based on a U(12) symmetry scheme". They want to make hadrons out of a "12-component (Dirac) quark" (i.e. _u_ , _d_ , _s_ as a single "quark"), and propose representations for vector and pseudoscalar mesons, and spin 1/2 and spin 3/2 baryons. It occurred to me that it would be rather eccentric to think about ways to embed a variation of their scheme into E8xE8, but that someone working on holographic QCD might well be interested in forgotten approaches to the strong interactions; and yet, top- down hQCD employs Type II string theory, which is dual to the heterotic string. Why haven't insights from heterotic string theory played a role in holographic QCD so far?


Lepton Number Conservation

What is the global symmetry of the electroweak Lagrangian that gives rise to lepton number conservation? As I understand it, electric charge is some linear combination of the conserved quantities corresponding to the $SU(2)$ symmetry and the $U(1)$ symmetry of the electroweak theory. If something similar is true for lepton number, then perhaps it would not be easiest to give the global symmetry directly. This is perfectly fine of course.


Coulomb interaction and conservation laws

In many-body solid-state physics, the Coulomb interaction term in the Hamiltonian usually implies the momentum conservation law in indicies: $$H_c=\frac{1}{2} \sum_{\mathbf{k},\mathbf{k}',\mathbf{q} \neq 0} V_{\mathbf{q}} a^{\dagger}_{\mathbf{k}'-\mathbf{q}} a^{\dagger}_{\mathbf{k}+\mathbf{q}}a_{\mathbf{k}'} a_{\mathbf{k}},$$ where $\mathbf{k},\mathbf{k}',\mathbf{q}$ are quasi-momenta and quantum numbers for the continuum spectrum of electron gas simultaneously. In quantum chemistry textbooks, the Coulomb term usually looks like: $$H_c=\sum_{i,j,k,l} V_{i,j,k,l} a^{\dagger}_{i} a^{\dagger}_{j}a_{k} a_{l}$$ Numbers $i,j,k,$ and $l$ are running over some discrete energy spectrum. Is it possible to state any conservation laws for quantum numbers $i,j,k,$ and $l$ in the expression above? Should not they obey any conservation law, selection rules or any additional restrictions? I will appreciate any references to textbooks or papers.


A rod filled with water and charged particles is rotated in magnetic field

Assume we have a rod of diameter $d$ and length $l$. It is filled with a mixture of water and, say, sodium chloride. Thus there are positively charged sodium ions and negatively charged chloride ions in the water. Now, assume that the rod is rotated perpendicular to the ground. I should add that the rotation happens at the mid point, not at one of the ends. Earth's magnetic field $B$ thus asserts force to the moving particles according to $q (v \times B)$ (Lorentz force). This means (I think) that the particles should start to move to the ends of the rod. If the rotation is in counterclockwise direction, the right hand rule states that positive particles should move to the right side and opposite for negative. (I think) Question is, does this really happen or is the centrifugal force going to push the particles to the sides? If so, how strong magnetic field $B$ or the velocity $v$ should be in order for this to happen?


If a pendulum is on a rotating table, will a torque be generated?

Here is the set up. Very simple. A flat (i.e. horizontal table, there is no gravity) and rounded table that spins on its axis (through the center of the table). A spring mass system is now put on the table inside a slot along the radius of the table. One end of the spring is attached to the origin of the table, and the other end is the small bob mass. Lets assume the relaxed length of the spring is half way along the radius. So the above is the physical set up. Here is a picture, showing the spring with a little bit of tension in it. ![enter image description here](http://i.stack.imgur.com/u5ijO.png) So, the pendulum can only move along the radius, as it is inside a slot in the table. Now I spin the table, say clockwise with some initial angular speed, and at the same time release the bob so it also starts to vibrate back and forth on the radius as the table rotates. What I am not sure about is that whether a torque be generated due to the reaction of the bob on the edge of the slot as the disk spins or not? This reaction force has an arm length from the center, hence a torque will be present. In otherwords, will this system's angular momentum be constant or not? If there is no torque generated, then the angular momentum is ofcourse constant. Thanks **Edit** May be I should explain why I thought at first there might be a torque generated. When the bob is rotating along with the table, it will have 2 components of accelerations. One component is along the radius direction, which is `(x'' - x (theta')^2)` where `x` is the bob coordinate along the radius, and `theta(t)` is the angle of rotation. In the above, the term `x(*theta')^2` is the centripetal acceleration and `x''` is the acceleration of the bob along the radius. There is also an acceleration perpendicular to that, which is `(x theta'' + 2 x' theta')`. In the above, the second term is the coriolis acceleration and the first term is the standard Euler acceleration. I made this diagram ![enter image description here](http://i.stack.imgur.com/GVb2S.png) Now, whenever I see something accelerating, then there must be a force along that direction. Right? This is the force that I took as generating a torque on the disk (actually it is the reaction to this force, ie. back to the disk). This is why I thought there is a torque. But when I did the same derivation using Lagrangian method `L=T-V`, I got different equation of motion. (no torque). Hence I thought to ask the experts here. **update 4** I have finally I think sorted this whole thing out!. I have updated the derivation. Solved this problem using Newton and Lagrangian methods, and both give the same equation of motions, and verified that the angular momentum remain constant. I have a PDF and HTML page and the applet here http://12000.org/my_notes/mma_demos/slot_on_disk/index.htm This is not as easy as it first seemed. There is actually a torque involved. Due to Coriolis force. This torque however changes in value such that the angular momentum remain constant. The critical thing for me to see was that the spring in infinitly rigid against rotation. This was mentioned below by Shaktyai. This is critical, as it means a slot is not even needed to keep the spring stright, and hence there is no Euler acceleration term perpendicular to the radial direction. The only side acceleration is the Coriolis one. This does generate a torque. I am not sure what this torque is called ( Coriolis torque?). It all now fits nice. I removed the slot from the applet since it is not needed, and updated it. If someone can verify it for me it will be appericated. The applet runs in the browser. Written in Mathematica. Wiki has a nice illustration of the Coriolis effect here http://en.wikipedia.org/wiki/Coriolis_effect


Symmetries, Generators, Commutators and Observables

I'm learning about generators and conservation laws and have derived the equation (1) $$[Q,A]=-i\hbar f(A)$$ which is satisfied by the observable generator $Q$ for a transformation group with elements of form $$g_a(A)=af(A)+\textrm{O}(a^2)$$ The lecture notes I'm reading say that this equation (1) defines $Q$ provided we know $f(A)$ for all observables $A$. Why is this true mathematically? And what does all observables mean? The example in the notes applies it to transformations along the $k-$axis for a system of $r$ particles, obtaining $$[Q,\hat{x}_i^r]=-i\hbar\delta_{ik}$$ $$[Q,\hat{p}_i^r]=0$$ It then states that $Q = \hat{P}_k=\sum_r \hat{p}_k^r$. This is obviously a solution, but do I know that it's the only one? Many thanks in advance.


Would a Helium powered balloon vehicle work on Mars? Things that might need to be answered:   1. How much liquidHelium would you need to pack in order to fill a balloon with enough helium gas to travel the planet.   2. How much energy does it take to re-compress Helium per/litre at low temperatures on the surface of mars and would a solar cell be enough?   3. Would a "Balloot" (balloon/parachute) be enough to complete spacecraft re-entry on arrival to mars without really needing to immediately "touch down"?   4. Any other things to think about by anyone else?


How exactly does mass affect speed?

I konw that mass affects weight (force), so how does that relate to speed? F=ma. so how does all this affect speed?


Commutators with a density matrix

The equation describing the evolution of our system is as follows: $ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a a^\dagger) $ where $\rho$ is the density matrix and $a$ is the photon annihilation operator. I need to rearrange this equation to look something akin to: $\dot{\rho} = u_1(t) \hat{A} \rho + u_2(t) \hat{B}\rho $ How do I handle the commutators and the like in order to rearrange it?


What is the "foamy space" hypothesis that has been debunked recently?

In "Space-Time Is Smooth, Not Foamy", a Space.com article, it is stated: > In his general theory of relativity, Einstein described space-time as > fundamentally smooth, warping only under the strain of energy and matter. > Some quantum-theory interpretations disagree, however, viewing space-time as > being composed of a froth of minute particles that constantly pop into and > out of existence. What are these "quantum-theory interpretations" the article is talking about? Can you quickly summarize the physics behind this experiment?


What is the exact definition of center of gravity?

I've come across many definitions. Is it 1) The point from which the weight of the body acts, i.e., the point at which if the entire mass of the body is assumed to be concentrated, the **gravitational force** acting on the body remains the same 2)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational potential energy** of the body remains the same 3)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational torque** acting on the body about the origin remains the same 4) All/some of the above? I came across this while trying to answer this question.


Positivity for the level of Chern-Simons theory

In many classical papers about Chern-Simons theory (see, e.g. [1]), it is claimed that the Chern-Simons theories with gauge group $G$ are classified by an element of $k\in H^4(BG,\mathbb Z)$, the so-called "level". (I'm not assuming here that $G$ is simply connected, or even connected) Now, I have reasons to believe that this description is not quite accurate. Rather, I suspect that only the levels subject to a certain **positivity condition** yield actual quantum field theories. The exact positivity condition I have in mind can be formulated by saying that the bilinear form on $\mathfrak g$ which is the image of $k$ under the Chern-Weil homomorphism should be positivie definite . (Here, the Chern-Weil homomorphism goes from $H^*(BG)$ to $Sym(\mathfrak g^*)$) **Question:** Is there any reference on Chern-Simons theory in which the above positivity requirement on the level is noted? * * * **Reference:** [1] R. Dijkgraaf and E. Witten, Topological Gauge Theories And Group Cohomology, Commun. Math. Phys. 129, 393 (1990).


minimizing the Higgs potential equivalent to finding the minimum?

When my advisor tells me to "minimize the Higgs potential", is she asking me to find the minimum (take the derivative of the potential and set it equal to zero)?


A question about Virasoro algebra

(skip disclaimer) I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra. Introducing complex coordinates $$w=\sigma^1 + i \sigma^2 $$ $$z=\exp (-i \omega) =\exp( -i \sigma^1 + \sigma^2) $$ and $$T_m=L_m - \delta_{m0} \frac{c}{24}$$ $$ \tilde{T}_m=\tilde{L}_m - \delta_{m0} \frac{\tilde{c}}{24}$$ It is said > The Hamiltonian H of time translation in the $w=\sigma^1+i\sigma^2$ frame is > $$H=\int_0^{2 \pi} \frac{ d \sigma^1}{2 \pi} T_{22} = L_0 + \tilde{L}_0 - > \frac{ c +\tilde{c}}{24} (2.6.10) $$ My questions are (sorry ask two questions in a thread, since they are closely related) (i) Does $T_{22}$ mean $T_{zz}$ or $T_{\sigma^2 \sigma^2}$? (ii) How the anti holomorphic operator $\tilde{L}_0$ are obtained in Eq. (2.6.10)?


Do all impacts create a wave-like disturbance in the medium through which they travel?

There is a scene in the first Matrix movie, where a helicopter strikes a skyscraper. The most interesting part is the 'slow-motion' bit where, as the helicopter strikes the building, a wave first propogates from the impact zone, outward along the facade, before the glass finally shatters. I was wondering whether this same thing happens across the board. Is a wave produced _every time_ two surfaces come into contact (AKA collide)? This is the clip I'm referring to: https://www.youtube.com/watch?v=2E3JDtqHaVM&t=3m56s


Free neutrons pass through most materials but how do they damage materials?

Neutrons has no charge so they are hard to interact with materials. That's what I'm thinking. But they collide with atoms and it causes nuclear fission. And they can also be a dangerous radioactive which bring out biological damage. Why? You know that the most of the atom is empty space.


Does the Lennard-Jones force equation give its answer in Newtons?

I'm trying to do the dimensional analysis of the Lennard-Jones force to work out what units are being used in my MD simulation. The lennard Jones force is given as the negative derivative of the potential: $F = 24e/r*[2*(s/r)^{12}-(s/r)^6]$ This article: http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Atomic_Theory/Intermolecular_Forces/Lennard- Jones_Potential suggests that the units of r and sigma are in angstroms and epsilon is in kj/mol. Upon performing dimensional analysis of the bracket I get 0 implying that the force is dimension-less? or 0? This cannot be correct; could someone tell me where I am going wrong? Thanks


Geometric quantization AND nuclear physics

Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. Geometric quantization is one formalization of the notion of quantization of a classical mechanical system/classical field theory to a quantum mechanical system/quantum field theory. Geometric quantization play an important role in Nuclear Collective Model. see here or restricted symplectic model of collective nuclear motion. I am looking for recent applications of geometric quantization theory in nuclear physics.


'There shall be reheating' and other inflation-related questions

I've some basic questions about inflation:   1. What is reheating exactly? What triggered the reheating? Is it something that one pulls out of a hat? Through which force did the decay of the inflation occur ?    2. How can I picture the inflaton? It is a scalar field, like the Higgs, could we ever hope to observe the inflaton? Can we assign to the field excitation a mass?    3. On the blog Quantum Frontier I read the following: > "A possible solution the the super-Planckian value of the change of the > inflaton field during the inflation is that the field didn't go in a > straight line but somehow turned in circles." (Preskill's post on Quantum > Frontiers) What is meant by "going into circles" ? That it oscillated around a minimum of the potential (but isn't that the phenomenon associated to reheating ?) Or is it an identification, e.g. $\phi(x)= \phi (x+2\pi)$? (This seems even stranger :S.) Thanks in advance.


How does tempered glass work?

I've read that tempered glass work by heating and then rapidly cooling the glass so that the outer layer 'freezes' (stops expanding/contracting) but the core contracts further as it cools. I'm told that this causes compression in the outer layer and tension in the core but I can't see how this could be. Surely, if the still-contracting core were 'pulling' on it, the outer layer would be under tension. For the outer layer to be compressed, surely the core must continue to expand after the outer layer 'freezes', pushing on the outer layer. Any help would be greatly appreciated.


Experiment dropping electrons into glass of protons

So, when you drop dye into a glass of water the dye spreads out. Now I realize you cant simply replace the water in the glass with protons (or a pure concoction of electrons) but I am wondering... What is the closest real world experiment to doing just that: mixing a drop of electrons with a glass of protons, or vice versa.


How large would the electric field need to be to split the mj -/+1 626.6 nm line of neon?

I am an undergraduate considering an extension on a quantum beats experiment with neon. The premise is to excite neon atoms coherently from a metastable 'ground' state into a superposition of excited states (transistion wavelength 626.6 nm) which have been zeeman split by a magnetic field. The decay of the coherently excited states shows beats in a time domain analogue of Young's double slit experiment. I would like to do essentially the same experiment but with the Stark effect instead of the Zeeman effect but I have been unable to find any literature which I can understand that would help me get a feeling for the electric fields involved. If i can get away with something below the kV range then I can probably do it.


Energy spectrum of a tight-binding model

Consider the one-dimensional tight-binding Hamiltonian $$\mathcal{H}=t\sum_m\left(a^\dagger_m a_{m+1}+a^\dagger_{m+1} a_{m}\right).$$ With the lattice constant set to 1, the energy spectrum is given by $$\epsilon(k)=2t\cos{k},$$ for $k\in[-\pi,\pi]$ in the first Brillouin zone. However, if we repeat the calculation with each unit cell containing two sites, we would obtain an energy spectrum of $$\epsilon(k)=\pm2t\cos{k}$$ instead, with the size of the first Brillouin zone reduced by half. Since the energy spectrum of a system can be determined unambiguously by experiments such as neutron scattering, it seems that we have a contradiction. What am I missing here?


Bose-Einstein condensate for general interacting systems

There is Bose-Einstein condensate (BEC) for non-interacting boson systems. Can we prove the existence of BEC for interacting systems?


Would the arms of a rotating ice skater still move outwards if there was no other object in the universe?

If there is no other object in the universe apart from a rotating ice skater, then nothing can be used as a reference frame. Would it make any sense to say that the skater is rotating? If so, rotating with respect to what? Would she feel an outward pull on her arms? Does rotation makes sense if there is no background reference frame?


Conductivity of a crystalline solid

In a crystalline solid each atomic level 'splits' into _n_ levels ( _n_ = number of atoms in the system). When the number of atoms is large each level becomes replaced by a band of closely spaced levels. In a semi-conductor we have an empty "conduction band" and a fully occupied "valance band". Conductivity arises because electrons get excited to the conduction band. **Question:** Why can't electron in the valence band freely move around and therefore conduct electricity? My question also applies to metals where the conduction band is already half-filled. What's special about this conduction band that allows electron to move around freely?


Determining Orbital Velocity

Is there any way to determine the orbital velocity of a point around another stationary point, if I don't know the mass of either of the points but know the force that gravity exerts and the distance between them. I'd like to know at what velocity I should push the non-stationary point perpendicular to the stationary point, so that it will maintain a circular orbit.


Degree of Time Dilation At a Distance From the Sun where acceleration = g?

At higher altitudes above a body, clocks tick more slowly, and gravitational field is weaker. But what is the relationship? It is tempting for a GR newbie such as myself to think that anywhere that the gravity is equal to that of the earth's surface that time-dilation would be the same, i.e. dilation as a simple function of local gravitational acceleration. I have been assured that this is not the case. I wonder if someone can calculate the following situation, which strikes me as the simplest 2nd case to compare with earth's surface: At the surface of the Sun, gravity is roughly 28 times that at the Earth's surface. If we move out to a suitable distance, we can find a place where the gravitational acceleration is the same as the surface of the Earth - 1g - and we can hold a clock there with a rocket. What is the time dilation there with regard to a "Far away" observer? I understand to the dilation on the Earth's surface to be 0.0219 seconds per year compared with the distant observer.


Specific heat of aluminium as a function of temperature

I am trying to find the curve describing the specific heat of aluminum as a function of temperature between about 50K to 300K(or room temperature) for an experiment I am doing. Ultimately I need to integrate between those bounds. Google doesn't seem to provide any useful links. I was able to find some values at specific temperatures, but not enough data to produce a curve. Any references to places that might contain this sort of data are welcome.


Entropy before and after collapse of the wavefunction/ and interpretation?

Seems like it might be pretty rudimentary but I want to see if my thinking is on the right track as well as what the result means. The question is, is the entropy of the collapse of a wavefunction or entangled system more before the collapse? My reasoning is that as Entropy is a measure of the "unpredictability" of information content (in this context), the more we know the less entropy there is. So for example if we have N fair coin tosses in which we don't know the outcome, we would have N bits of unknown information as $ \sum_{i = 0;N} 2^N = N$ but the first we flip and once the answer is known we would have an entropy of N-1 as we don't know those N-1 states. Once all is known we have an entropy of 0. Of course we would deal with a continuous function for waves, but am I on the right track of thought? If so, then how does the second law of thermodynamics fit in in that the Universe is supposed to go to a higher state of entropy? Just an aside question, but is the wavefunction collapse the same as the collapse of an entangled system? Seems that the wavefunction itself is just a superposition of several different eigenstates, is it safe to say that these eigenstates are "entangled"? Thanks, Brian


Could dark energy be the effect of gravity at great distances?

This may be a silly question, but is it possible that dark energy and gravity are related to each other? Space-time is deformed everywhere in space by objects with mass. The more massive the object, the more deformation in the local area, and, as a result, the more concentrated space-time is within that deformation. However, where is the "extra" space-time coming from to be able to form these super-concentrated regions of space-time? Is that what we observe as dark energy? Is it possible that at great enough distances, gravity could be pulling space- time apart due to the lack of mass in the great voids of space? Not only that, but if there was enough mass in a part of space to have a little more gravity than the surrounding area, it could prevent that area of space from being "stretched out." Thus, might dark matter just be regions of space where space- time is denser than the surrounding regions? I'm sure these aren't new ideas, but I'd like to know what premises and assumptions are incorrect.


How could they tell things weigh a sixth of their "Earth weight" on the moon?

I frequently keep hearing stuff about "gravity on the moon is only a sixth of that as Earth's" and "of course you'll weigh more on Jupiter". I do know the relative sizes of those bodies allow for gravitational acceleration that's different from the $9.8 \mathrm{\; m\cdot s}^{-2}$ we're accustomed to here, and I'm pretty sure nobody's ever bothered to bring a scale to Mars or something. How can one reckon how much you'll weigh on the moon if you don't know the "one-sixth" bit? How about on other planet(oid)s and satellites?


Yang-Mills instanton

How can instanton solution to Yang-Mills theory with gauge group $SU(3)$ or $SU(N)$ be obtained? For $SU(2)$ it is explained in textbooks but what about more general color gauge groups? EDIT: How does $A^\mu$ look like for SU(3) or SU(N). Wikipedia only gives for SU(2). It is given by for SU(2) $$A^\mu_a = 2/g \frac{\eta^a_{\mu\nu}(x-z)}{(x-z)^2+\rho^2}$$


Transmission Lines + DC Current

I have a problem where I'm given a circuit that looks like: ![enter image description here](http://i.stack.imgur.com/YxYR1.png) It contains a transmission line with a characteristic impedance $Z_0$, a source resistance of $R_s$, and a load impedance of $R_L$. If the voltage source provides a constant DC voltage source of $V_0$, what are the voltages at points A and B? I was considering using the voltage divider equation, but I don't think that's the right approach.


Super cooled steam and entropy change

I was thinking about a situation where I have some super cooled steam which suddenly freezes to water.What are the entropy changes(positive or negative) for the system and the universe? My thoughts:On transition from steam to liquid water the molecules gets more packed together which reduces their giggling around and hence, the entropy of the system would decrease. Since it is assumed to be a spontaneous process the entropy of the universe should increase.Is my thinking correct?


Electric Field in Circular waveguide in TE mode

I know that the magnetic field inside a circular waveguide is $B_z=B_0\dfrac{\rho}{R}\left(1-\dfrac{\rho}{2R}\right)\cos{\phi}\sin{\frac{\pi z}{d}}$ How can I find the two components of the electric field, $E_\rho, E_\phi$ that is?


Can a multipass x-ray absorption cell be constructed?

I've been trying to understand the various concepts behind x-ray optics compared to standard visible/IR optics like mirrors and such. However, the x-ray mirrors I've been finding typically have optimal angles of reflectance on the order of a few milliradians, particularly for hard x-rays. I think I mostly understand the reasons for this (having something to do with Bragg diffraction?), but, say, for absorption experiments with low sample density, it may be necessary to use something analogous to a multipass absorption cell. Since x-rays can't be reflected at anywhere near 90° with high efficiency, can a multipass absorption cell be constructed for x-rays (say, with fiber optics or something else)? Has anyone done so? Multiple Google searches for such a thing have turned up nothing, but maybe I'm just searching for the wrong thing.


Reflected and refracted wave sphased

When we derive refraction and reflection laws for a generical plane wave on a surface, we say that reflected and refracted must be in phase with the incident wave. Why a medium cannot do a sphased reaction beside incident wave? Sorry for my trivial question.


Free surface of inviscid fluid flow

The following problem seems like it should have a definite solution, but I've been thinking about it for months and haven't got anywhere. It might not be a well-posed problem, but if it isn't I'd like to understand why. An incompressible, inviscid fluid of density $\rho$ flows continuously (in a steady state) as shown in the following diagram: ![diagram](http://i.stack.imgur.com/Y1qJz.png) We know the height of the fluid (and hence its pressure) at points $x_1$ and $x_2$, but we don't know the velocity of the fluid or its height at any other value for $x$. The top of the fluid is a free surface, i.e. it's determined by the properties of the flow rather than being specified as part of the problem. I've drawn it as slightly concave but I've no idea if that's right. Let us assume that the velocity profile at $x_1$ is vertical (i.e. velocity does not vary with height above point $x_1$). Because the fluid is inviscid it seems to me that the constant vertical velocity profile should be maintained as the fluid travels to the right. So if we were to dye a vertical line of the fluid a particular colour, it would remain a vertical line as it travelled to the right, because the pressure differential across the line is constant with depth. If this is correct it means we can think of the velocity component in the $x$-direction, $v_x$, as a function of $x$ rather than $x$ and $y$. Because the flow is incompressible we know that $h(x)v_x(x)$ must constant over space, and this is the value I want to solve for (although it might not have a unique value - in that case I just want to know the function $h(x)$). If we need to we can also assume we know the initial and final velocities, $v_x(x_1)=v_1$ and $v_x(x_2)=v_2$. It seems like Bernoulli's equation should have some relevance here. That would certainly be the case if the fluid were confined to a pipe instead of having a free surface. (In this case the pressure difference would be independent of the difference in height, so we'd need to know that as well.) But every time I try to solve this problem using the Bernoulli equation I get into a terrible mess. I'm really not sure of the best way to approach this problem, so any insight anyone can offer would be much appreciated.


Are proton, antiproton, electron, positron the only observed subatomic particles that can freely exist and don't decay, i.e. are stable?

Are proton, antiproton, electron, positron the only subatomic particles that can freely exist (i.e. I don't want particles that only exist in bound state as constituents such as quarks) and don't decay, i.e. are stable? What about muons? Are there other particles/hadrons that can exist freely (i.e. not in some bound states) and don't decay?


How was this dot identified as a mystery planet originally (Pluto in this case)?

I was watching a documentary on a mission to study Pluto. At 21:11, an image is presented with an arrow mark pointing a small dot which is now identified as Pluto. As there are hundreds of dots on the image (I'm not sure what kind of an image this is as I'm not an astrophysicist), how can stars, planets, asteroids and other objects be distinguished in this image? And what kind of an image is this? ![Enter image description here](http://i.stack.imgur.com/KXHRF.png)


Are Normal Stress Differences in Newtonian Fluids Equal?

I was reading about normal stress differences in the context of Non-Newtonian fluids. There was a remark that the normal stress differences in Non-Newtonian fluids are not equal whereas the normal stress differences in Newtonian fluids are equal. I can't figure out why the normal stress differences in a Newtonian fluid should be equal. Can anyone highlight a proof?


Could the Higgs boson field be responsible for the dramatic mass increase when an object begins to travel close to the speed of light?

could the Higgs boson field be responsible for the dramatic mass increase incurred when an object nears the speed of light?


What allows new born suns to travel away from each other?

What allows massive new born suns to move away from each other, as they have been observed. I would think that their massive gravity would prevent this and cause them to slam into each other.


Number of possible entangled states

Suppose we have four single particle states: $|u,\pm\rangle$ and $|v,\pm\rangle$. How many lineary independent entangled states can we have for a pair of particles? I know that if A and B are entangled, the state AB cannot be written as a product of state A and state B, i.e. $|AB\rangle \neq |A,i\rangle |B,j\rangle$. Does this mean that the state of AB has to have at least two terms? Is $|\psi\rangle = |u,+\rangle|v,-\rangle + |v,-\rangle|u,+\rangle$ such a state?


Can anyone analyze this perpetual machine that a Norwegian built?

Take a look at this video: http://www.youtube.com/watch?v=tlx2PgESXhs It shows a machine that a Norwegian artist built. It seemingly keeps going forever. They filmed it over 3 days and it kept going at the same time. Can someone analyze the energy losses in this machine? I'm seeing several losses at the moment:   1. The ball is losing energy due to friction while going around the track.   2. The pendulums are losing energy due to air friction. What other sources of energy losses can you notice?


Could tidal effects make matter hot enough for nuclear fusion?

There was a recent question about cosmological firewalls. Putting aside questions of entanglement and information and looking at the problem from a pure mechanical viewpoint, we know that although there is little perceptible radiation emitted from the blackhole, the tidal forces approaching the event horizon are enormous leading to spaghettificaton. Although the tensile forces will eventually tear a person apart, there are also compressive forces for material located in the same plane. A human body contains a considerable amount of hydrogen and other lighter elements. Would the compressive effects of spaghettification be sufficiently high to cause any of the hydrogen or other elements to fuse prior to reaching the event horizon?


what does "medium-induced gluon radiation" mean

I've often run across phrases such as "medium-induced gluon radiation" from some papers,what is "medium-induced gluon radiation"?


What is the difference between induced current and eddy current?

I was reading today about electromagnetic induction and eddy currents. But while reading I was unable to differentiate between induced current and eddy current. What is the difference?


Measuring voltage drop from induced current

I'm having trouble connecting voltage drops and induced current. Imagine you have a triangular loop make up off three resistors. You place this loop in a constantly changing magnetic field. This changing magnetic field will cause an EMF and a resulting induced current in the loop. Now imagine I place a voltmeter across one of the resistors. What does the voltmeter read? $I\times R$ or $I\times 2R$? If you look at the situation one way, you are measuring the voltage drop across one resistor, so V would equal $I\times R$. But if you look at in another way, you're also measuring the voltage drop across the other half of the circuit which contains TWO resistors (hence a voltage drop of $I\times 2R$). I have tried to model this situation in my schools physics lab and I do see that there is a voltage drop across the resistor so the answer is not "there is no voltage drop". My model wasn't perfect, but it seemed to imply that the voltmeter read $I\times 3R$? How could that be?


Do free electrons really not interact with photons?

If free electrons don't interact with photons, why are free electrons accelerated by electromagnetic fields?


Convert latitude of lowest altitude to argument of perigee?

I am designing an orbit around Mercury. I know the values I want for the semi- major axis, eccentricity, inclination, and RAAN. I want the altitude of closest approach (periapse) to occur at $60.0^{\circ}$ N latitude above Mercury's surface. **My Question** : How do I convert this constraint for the _latitude of closest approach_ into an _argument of periapsis_ , $\omega$? Specifically, this is modeled after the orbit below: ![enter image description here](http://i.stack.imgur.com/pPllt.jpg)


Counting D0-D4 Bound States

I have a slightly technical combinatorics question. Consider the degeneracy $D_n$ of bound states of $n$ D0 branes and one D4 brane. This is given in Polchinski by (13.6.24), \begin{align} \sum_{n=0}^{\infty}q^n D_n=2^8\prod_{k=1}^{\infty}\left(\frac{1+q^k}{1-q^k}\right)^{8}. \end{align} I was able to verify this up to $n=3$, basically you have to count all ways to form bound states of the D0 branes and then bound them to the D4 brane. However, I wasn't able to verify it in general, since brute forcing it is a bit of a mess. Is there some clever mathematical formalism that allows one to deal with combinatorical problems like this?


What would the gravity be like if Earth and the moon were touching? ![enter image description here](http://i.stack.imgur.com/qLzZG.png) Assuming the moon keeps its form and doesn't collapse on its own weight, would the area between it and the Earth (point B) have less gravity? What about point A? Would gravity in that area stay the same?


Why does electricity need wires to flow?

If you drop a really heavy ball the ball's gravitational potential energy will turn into kinetic energy. If you place the same ball in the pool, the ball will still fall. A lot of kinetic energy will turn into thermal energy because of friction, but the gravitational potential energy will still be converted. Similarily, why doesn't electricity flow without a good conductor? Why won't Electrons won't flow from the negative terminal to the positive terminal without a wire attaching them? I know how electricity flows like a wave and that metals have free electrons floating around that allows the wave to propagate. But when these free electrons aren't available to propagate the wave, why don't the electrons just "move" like the ball? Why don't the electrons just "move" through the air to the positive terminal. A slow drift speed means that tell probably take a long time, but they should still get there. Shouldn't they?


If temperature is average KE per particle, and heat is total KE of all the particles, how can molar heat capacity vary?

If temperature is defined as the average kinetic energy per particle, and heat energy is defined as the total kinetic energy of all the particles (or more strictly, heat transferred is the total kinetic energy transferred to those particles), surely we can get heat by just multiplying temperature by the number of particles? If so, shouldn't molar heat capacity be constant across all substances? I'm fairly sure I'm missing something that should be obvious here...


Interpretation of an "interaction" term

In QFT a polynomial (of degree >2) in the fields is said to be an interaction term, Ex.: $\lambda\phi^4$. **Question** Is it possible to give an interpretation to terms like $\frac{1}{\phi^n}$? (for $n\in\mathbb{N}$) Cheers


Can the quartic oscillator Hamiltonian be made quadratic?

I'm interested in turning a quadratic + quartic oscillator Hamiltonian, $$H = \frac{p^{2}}{2m} +\frac{kx^{2}}{2} + \lambda x^{4},$$ into a purely quadratic problem. One of the simplest things I can think of is replace the quartic term by $$x^{4} \rightarrow \langle x^{2}\rangle x^{2}.$$ Has anyone come across such an approximation? If yes could you give me some reference to the work where this approximation has been used. Is there any other simple approximation which can turn the above Hamiltonian into a quadratic problem?


Pohlmeyer reduction of string theory for flat- and AdS- spaces

The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's  DDF and Pohlmeyer invariants of (super)string) is the following: $ Z^{\mu_1...\mu_N} (\mathcal{P}) = \frac{1}{N} \int\limits_0^{2\pi} d\sigma^1 \int\limits_{\sigma_1}^{\sigma_1+2\pi} d\sigma^2... \int\limits_{\sigma_{N-1}}^{\sigma_1} d\sigma^N \ \mathcal{P}^{\mu_1}(\sigma_1)\ \mathcal{P}^{\mu_2}(\sigma_2)...\mathcal{P}^{\mu_N}(\sigma_N) $ Another reference is eq-3.29 in Thomas Thiemann's  The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space. These formulations of the Pohlmeyer invariants is used to show its equivalence to DDF invariants for flat-space (cf. eq-2.53 in Schreiber's paper) Now, what is the equivalent invariant for the Pohlmeyer reduction in AdS case (as in Miramontes' Pohlmeyer reduction revisited)? I cant seem to find one that resembles the flat-space expression.


Frame dragging around a black hole

If you have one stationary nonrotating black hole, being orbited by a 2 or more black holes just outside each other's event horizons does this alter the size of the central black hole's event horizon? what's the answer to this thought experiment, if you put a spaceship into orbit around the central black hole. and it Falls into the event horizon just barely, and then introduce the multiple black holes orbiting just outside the central black hole's event horizon would it enable the spaceship to escape central black hole's event horizon via the pull of the exterior orbiting black holes. (I don't know if rotating or nonrotating would have any effect on the event horizon


How are particle simplices associated into complex particles?

Nonfundamental particles are seen as made up of fundamental particles (in whatever specific theory). consider the simple case of 2 simplex particles (subscript 1 and 2) which form a complex particle (no subscripts):   * Are $\phi(x_1), \phi(x_2)$ _enough_ to determine $\phi(x_1,x_2)$ or is _additional interaction information necessary_?   * How is $\phi(x)$ related to $\phi(x_1,x_2)$? i.e. how is the wavefunction of the complex particle as a function of the _complex particle position_ related to the wavefunction of the complex particle as a function of the _individual subparticle positions_? I would wildly guess to use barycentric centering: $\phi(x)=\int \phi(x_1,x_2)\delta(x - \frac{m_1 x_1+m_2 x_2}{m_1+m_2}) d x_1 d x_2$. But does that really make any sense and does it violate normalization of the resulting wavefunction etc? now consider a collection of N simplex particles   * In classical physics one could use the energy of the particle to say if it is bound or not, how does this translate to conditions on the wavefunctions?   * One can envision 2 seperate solar systems in classical mechanics, where solar system 1 is not bound to solar system 2, but within each solar system the planets are bound to their star. Given only the positions and velocities of the stars and planets, is boundedness association possible in classical physics? how does one recognice that they form 2 stellar systems which are not bound, but withing stellar systems planets are bound?   * How does one do a similar association given wavefunctions of subparticles to identify which subparticles form a complex particle? Or does none of the above make any sense?


Voltage drop over a cell membrane

Again, a problem from exam preparation: > [A] cell's membrane allows sodium ions to pass through it, but not chlorine > ions. The cell is placed in a salty solution with a ten times higher > concentration of salt outside than in the cell. As a result, sodium diffuses > into the cell. Calculate the voltage difference across the cell membrane as > a result. Well, in equilibrium, the chemical potentials of the solution and the cell should be the same. Let $\mu_A$ be that of the solution and $\mu_B$ that of the cell. Assuming that the salt is in a dilute solution, I can write its chemical potential as $\mu_A = f(T,P) + kT \ln(n_A)$ where $f(T,P)$ is some function independent of the number of particles, and $n_A$ is the number-concentration of sodium ions in the solution. I'd assume that for the cell, I have $\mu_B = f(T,P) + kT \ln(n_B) + U$ where now I include the voltage that exists across the cell's membrane. Problem is, now I have two unknowns: I don't know the final concentration of ions in the cell and I don't know the final voltage of ions in the cell. **Alternative idea** but not sure if that makes sense: Osmotic pressure is defined as the pressure you must bring up to prevent osmosis from occurring, and it is also the pressure that finally builds up when you allow osmosis from happening. Could I analogously obtain the final voltage as the voltage that I'd have to apply manually to prevent any diffusion from happening at all?


Where can get channel waveguide dimension data ny nx na ns

I need channel waveguide dimension to test my waveguide mode solver. Can someone help me?


A naive question on the Quantum Hall Effect(QHE) and the confinement in gauge theory?

The non-interacting 2D lattice QH system is described by the Hamiltonian $H=\sum t_{ij}e^{iA_{ij}}c_i^\dagger c_j+H.c$ My confusion is: Does this imply that the $2D$ lattice QHE is described by the $2+1$ compact lattice $U(1)$ gauge theory? If it is, then according to the general conclusion " the $2+1$ compact lattice $U(1)$ gauge theory is always confinement ", thus, does this mean that the low-energy quasiparticles of $2D$ lattice QHE are not the free electrons? I'm confused on this point, thank you very much.


Why photo cannot be captured while facing toward the Sun /

When the Camera is facing toward the Sun and a person is standing to and We're capturing his photo , all we get is a black shadow even when we capture with the flash light or anything. Why does it happen ?


Canonical transformation and Hamilton's equations

I was trying to prove, that for a transformation to be Canonical, one must have a relationship: $$ \left\\{ Q_a,P_i \right\\} = \delta_{ai} $$ Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$. Now to do the proof I started with $\dot{Q_a}$:   1. Chain rule and Hamilton's equation for initial coordinates $q_i,p_i$ $$ \dot{Q_a} = \frac{\partial Q_a}{\partial q_j} \dot{q_j} + \frac{\partial Q_a}{\partial p_j} \dot{p_j} = \frac{\partial Q_a}{\partial q_j} \frac{\partial H_a}{\partial p_j} - \frac{\partial Q_a}{\partial p_j} \frac{\partial H_a}{\partial q_j} $$   2. Then I apply chain rule for the Hamiltonian derivatives: $$ \dot{Q_a} = \frac{\partial Q_a}{\partial q_j} \left( \frac{\partial H}{\partial Q_i} \frac{\partial Q_i}{\partial p_j} + \frac{\partial H}{\partial P_i} \frac{\partial P_i}{\partial p_j} \right) - \frac{\partial Q_a}{\partial p_j} \left( \frac{\partial H}{\partial Q_i} \frac{\partial Q_i}{\partial q_j} + \frac{\partial H}{\partial P_i} \frac{\partial P_i}{\partial q_j} \right) $$   3. Now reordering the terms yields us: $$ \dot{Q_a} = \frac{\partial H}{\partial Q_i} \left\\{ Q_a,Q_i \right\\} + \frac{\partial H}{\partial P_i} \left\\{ Q_a,P_i \right\\} $$ Now here the problem, for the transformation from a coordinate system $(q_i,p_i)$ to a coordinate system $(Q_a(q_i,p_i), P_a(q_i,p_i))$ to be canonical we require: $$\left\\{ Q_a,P_i \right\\} = \delta_{ai}$$ But why we have as well the following requirement, or is it just too obvious or true because of some property of any coordinate transformation? $$\left\\{ Q_a,Q_i \right\\} = 0$$ * * * The problem I am having is as follows. I agree, that the following two are true (if I used the covariant notation correctly): $$ \left\\{ q^i,q_j \right\\}_{q,p} = \frac{\partial q^i}{\partial q^k} \frac{\partial q_j}{\partial p_k} \- \frac{\partial q^i}{\partial p^k} \frac{\partial q_j}{\partial q_k} = 0 $$ $$ \left\\{ Q^i,Q_j \right\\}_{Q,P} = 0 $$ But why its the case that the following is also true? $$ \left\\{ Q^i,Q_j \right\\}_{q,p} = 0 $$


Paramagnetism what about Paraweakism or Parastrongism?

Ok, I was just curious but the electromagnetic force can allow paramagnetism macroscopically in some objects. Can this be done microscopically to the subatomic level? Also, what about other forces (besides gravity I'm guessing) meaning the weak and strong force? Strong force being the strongest, would its para- also be and always be stronger?


Kinetic energy of a rotating rod

I have no idea where to even begin with this... Find the kinetic energy of a record of uniform density, mass 50 gm and radius 10 cm rotating at 33 and 1/3 revolutions per minute. Normally I would do: $$ KE = \frac{1}{2} m v^2, $$ but when I do that, I get a different answer then the one that is in the book, which is: 15231. I don't know how to use the fact that the radius is 10cm, but I'm thinking I need to find some type of integral and taking the piece into slices.


What's the real fundamental definition of energy?

Some physical quantities like position, velocity, momentum and force, have precise definition even on basic textbooks, however energy is a little confusing for me. My point here is: using our intuition we know what momentum should be and also we know that defining it as $p = mv$ is a good definition. Also, based on Newton's law we can intuit and define what forces are. However, when it comes to energy many textbooks become a little "circular". They first try to define work, and after some arguments they just give a formula $W = F\cdot r$ without motivating or giving intuition about this definition. Then they say that work is variation of energy and they never give a formal definition of energy. I've heard that "energy is a number that remains unchanged after any process that a system undergoes", however I think that this is not so good for three reasons: first because momentum is also conserved, so it fits this definition and it's not energy, second because recently I've heard that on general relativity there's a loss of some conservation laws and third because conservation of energy can be derived as consequence of other definitions. So, how energy is defined formally in a way that fits both classical and modern physics without falling into circular arguments?


Questions on quantum mechanics

First Question I have problem with little alternative form of Dirac equation. This problem refer to a matrix $\alpha$ in this equation. I mean detailed general contravariant form of equation in which appeared matrices $\alpha$ and $\beta$. I need to change traditional place of matrix $\alpha$ and take it in new place in end of equation. Primary, original place of this matrix in this form of equation is instantly behind $\sigma$ sign of Hamiltonian. I want placed matrix $\alpha$ now beside matrix $\beta$ which is located beyond mass element of Dirac equation. The equation then takes the form where matrices $\alpha$ and $\beta$ are next to themselves. Will be correct this modification of equation? Second Question I have trouble with shape of equation element of massless particle in arbitrary equation of wave arbitrary of second rank. This equation will be similar to Klein-Gordon equation. In my opinion this massless element can be in expressed in this shape? Planck's constant on second power, angular frequency on second power, speed of light on second power. I understand this problem that forms of elements must be in coincidence with rank of equation. Then Planck's constant and circle frequency in numerator and speed of light in denominator will be all on (square) Will be this equation in right form?


What does Feynman mean when he says that $F=ma$ is not exact?

Chapter 12-2 in Feynman Lectures Vol. 1 states: > In fact the law, $F=ma$ is not exactly true; if it were a definition we > should have to say that it is _always_ true; but it is not ... First, > because Newton's Second Law is not exact, and second, because in order to > understand physical laws, you must understand that they are all some kind of > approximations.   1. What does he mean by _approximation_?    2. Also, how is $F=ma$ _not_ a definition of force?


CMB radiation and holography

If in principle someone have the exact data of the CMB of our universe and unlimited computability power, to what extent it is possible to determine how the universe should look like? Does it fully determines the later time structure of the universe or the same information could be matched to more than one universe?


Convective derivative of oscillating fluid with bulk motion?

I have a fluid $u$, which comprises a zeroth order constant motion $u_0$ as well as a first order oscillatory motion $u_1$. The convective derivative is $$\frac{\partial u}{\partial t} +u\frac{\partial u}{\partial x}$$ where $u=u_0+u_1$ We are assuming a linear solution, so that second order terms (any products of 1st order terms) are neglected. I know that the time derivative will be $\partial(u_1+ u_0)/\partial t$ but I have no idea how to linearized the flow part: $$(u_0+u_1)·\frac{\partial(u_0+u_1)}{\partial x}$$ Because we are assuming linear solution, we can move into frequency domain, and above expression is: $(u_0+u_1)·ik(u_0+u_1)$ - I'm afraid that I still don't get how to go about linearizing. My guess is FOILing it out and dropping $u_1u_1$ terms, but I want to be confident before moving forward.


Finding the total energy in centre of mass frame

I'm working through a problem in a special relativity textbook (Woodhouse) and I'm having some difficulty. I have to show that if I have a particle of rest mass $M$, total energy $E$ colliding with a stationery particle of rest mass $m$ then the total energy, $E'$ in the frame where their centre of mass is at rest is given by: $$E'^2 = (M^2 + m^2)c^4 + 2Emc^2$$. Now I understand that the 4 velocity of the centre of mass frame is $V$, and is given by $V = (c,0,0,0)$, I'll call the momenta $P$ and $Q$ (of M and m respectively). Now in the solutions it says that the centre of mass frame is defined so that it's 4 velocity is proportional to the sum of the momenta of the two particles (Say $V = k(P+Q)$) it then goes on to say that hence the total momentum in the centre of mass frame is $(E'/c,0)$ but how can this be, because the particles are still moving in the centre of mass frame? I can see that if this were the case then $g(V,P+Q) = E'$ and then if I could work out the form of $V$ I'm pretty sure I could calculate $E'$ okay. Any help is very much appreciated


Meaning and types of singularity in case of string or any cosmological model (Mathematical description)

What is actual meaning of singularity can we use this term for conclusion in any research paper( related to cosmological models ).what r the types .


Height of rebound of a projectile from a sphere

![enter image description here](http://i.stack.imgur.com/ss5ZM.png) My work here is: First, I use conservation of energy: (taking the plane as zero potential) $$\frac{1}{2}m(\frac{2ga}{5})+2amg=\frac{1}{2}mv^2+a(1+\cosθ)mg$$ $$\frac{1}{5}ga+2ga=\frac{1}{2}v^2+ga(1+\cosθ)$$ $$v^2=(\frac{12}{5}-2\cosθ)ga \tag1$$ Now resolving forces radially at the point of falling ($R=0$) $$mg\cosθ-\frac{mv^2}{r}-R=0$$ $$mg\cosθ=\frac{mv^2}{a}$$ $$v^2=ag\cosθ\tag2$$ Equating $(1)$ and $(2)$ $$\cosθ=\frac{4}{5}$$ As required. I am having problems in the second part. Let $v=v_1$, the velocity of arrival at the plane, $v_2$, and the velocity of rebound $v_3$. I considered the velocity of arrival equating energies. $$\frac{1}{2}mv_1^2 + mga(1+\cosθ)=\frac{1}{2}mv_2^2$$ $v_1=\sqrt{\frac{4ag}{5}}$ $$v_2^2=\frac{22ag}{5}$$ Using Newton's Law of Restitution: $$e=\frac{-v_3}{v_2}=\frac{5}{9}$$ $$v_3=\frac{5v_2}{9}=\frac{5\sqrt{\frac{22ag}5}}{9}$$ Using the dynamics equation $v^2=u^2+2as$ $$(v_3\sinθ)^2=2gh$$ From earlier, $\sinθ=\frac{3}{5}$ $$h=\frac{(\frac{3v_3}{5})^2}{2g}$$ But this gives a wrong result. I am unsure what I did wrong and, am I wrong in assuming that the particle will bounce of with the same angle $θ$ that it came out of the ball?


Chern-Simons Theory in 3D

For the CS theory on a three manifold $M$ with a gauge group $G$, it is said that the gauge field $A$ is a connection on the trivial bundle over $M$. Why the bundle should be trivial? I know that space of classical solutions is made of flat connections but why should we assume the bundle is trivial from the beginning? Another question: Can anyone explain the meaning of farming of a three manifold in simple terms? Thanks.


Isotropic harmonic oscillator in polar versus cartesian

I read another Phys.SE post here: 3D Quantum harmonic oscillator that I believe says the wave function in Cartesian coordinates for a 3D harmonic oscillator is the product of the 3 one dimensional wave functions. If so, are the angular and radial equations combined within it? Is it possible to separate the radial part from the angular part? The wavefunction for 1D in the ground state is $$\psi_{0}(x)=(\frac{m\omega}{\pi\hbar})^{\frac{1}{4}}e^{-\frac{m\omega x^{2}}{2\hbar}}.$$ I think this product would then be: $$\psi_{0}(r)=(\frac{m\omega}{\pi\hbar})^{\frac{3}{4}}e^{-\frac{m\omega r^{2}}{2\hbar}}.$$ Is this the radial equation for the ground state? Would I then need to multiply by the angular equation for the ground state to get the full wave equation?


Do the laws of physics that apply to all observers also apply to a non-observer?

The Timelessness of a photon https://www.youtube.com/watch?v=5ELA3ReWQJY An observer's laws of physics are time based. "When you're traveling at the speed of light, time does not exist" - Neil deGrasse Tyson. So a photon's laws of physics can not include time because it does not exist. How does the Photon exist in "space-TIME", if Time doesn't exist for the Photon? The Photon would not exist in "space time", it would exist only in "space". The physics of "space time", which is where we are, would not apply to "space without time", where the Photon is.   1. The rules of physics are the same for all observers in uniform motion relative to each other.   2. The Photon is never in uniform motion relative to any or all observer.   3. A Photon in an observer's frame is not the same as a Photon in a Photon's frame.   4. The Photon must have "a frame" (although it is not compatible with our own), because the Photon occupies coordinates in space as a particle.   5. The Photon's frame would not define Time.   6. The Photon does not exist in any observer's frame, because the Photon is never in uniform motion relative to any observer, but the Photon does exist.   7. If the Photon does not exist in any observer's frame, then the Photon must have a frame of its own that it does its existing in.   8. The laws of physics for all Observers includes a definition for time.   9. The Photon's frame has no time coordinates, so time coordinates from any observer's frame can not be meaningfully associated with the Photon or the Photon's frame. Time is relative to all "observers" but light is not an observer. Time can't be meaningfully applied to light because time doesn't exist to light. Time is not relative between an observer and light because they are not both "observers". If one physicist says that a particle is instantaneous And another physicist says the same particle takes 8 minutes to travel from the Sun to the Earth That's a contradiction 8 Minutes > Instant All physical laws should remain unchanged under a Lorentz transformation. An observer's physical laws include time laws. A Photon's physical laws do not include time laws. All physical laws can't remain unchanged when transforming an observer's physical laws into a photons physical laws. This would cause the application of time laws to the photon, but time laws and photons are incongruent. Assume that "space time" is a prison. In the prison there are guards and inmates. Everyone in the prison follows rules, but the guards follow one list of rules and the inmates follow a different list of rules. Some of the rules for the inmates and the guards are the same, some of the rules for the inmates and the guards are not the same. Among the inmates, all inmates must follow all of the inmate rules, no inmate receives special treatment. Also among the guards, all guards must follow all of the guard rules, no guard receives special treatment. The two lists are written in different languages. All of the inmates speak different languages, but one inmate speaks the language that the inmate rules are written in, and all of the languages that the inmates speak. All of the guards only speak the language that the guard rules are written in. It's possible for the one inmate who can read the inmate rules, to translate those rules into all of the languages that the inmates speak. And because all of the inmates follow exactly the same set of rules, the number of rules would be the same in each of the inmate languages. It's also possible for the inmate to translate the list of guard rules entirely into the language of the inmates, and the list of inmate rules into the language of the guards. This doesn't matter though, because the inmates and the guards still follow different rules. Writing the rules of the guard in the language of the inmate does not allow the inmate to follow the rules of the guard, and writing the rules of the inmate in the language of the guard does not subject the guard to the rules of the inmate. The laws of physics for any observer include laws that define the behavior of time. The entire system of physics for all observers then, is inherently based on the existence of time, and time is one of the defining features of the "existence" or "reality" that all observers occupy. It is the presence of time that allows an observer's laws of physics to establish a frame of reference. Because all observers follow exactly the same list of rules, like all inmates, all of the rules of "observer physics" can be translated (by the one inmate, "man") from one observer to the next, even though all observers speak a different language, or have differing personal frames of reference. However, the laws of physics for "light" or "the speed of light", do not recognize the existence of time. The entire system of physics for "light", is inherently without time, and this lack of time is one of the defining features of the "existence" or "reality" of "light" or, "anything 'traveling' at the 'speed' of light". It is the lack of "time", and therefore the lack of a definition between "at rest" and "moving", which is THE REASON why a photon can't be assigned a frame of reference when trying to translate the photon's physics to the observer's physics. It only APPEARS that photons exist by the same laws of physics as an observer because some of the laws cross over from one reality to the next. It appears that the physics of photons and the physics of observers are the same because the existence of time in our reality prevents you from observing the photon in it's "natural habitat" to see how we different we really are.


Gravitational force and time dilation

Suppose the radius of the earth is reduced by half but the mass is same, then how long will it take to complete one rotation, 24, 48, 12 or 6 h.? please give the mathematical relations and solution.


Why the initial spectrum of perturbation of inflation must be Gaussian? I heared it was because of Central Limit theorem. But I don't know how use this theorem explicitly in this question to get this result.


Bernoulli's principle on a curve ball

I've seen a few excellent answers here on the Magnus force, which explains why balls with a spin will curve. However, my intuition is still telling me that the Bernoulli's principle would push it the opposite way and I need help understanding why my reasoning is flawed. Imagine that you kick a soccer ball on the left side so that it's spinning clockwise (viewing the ball from above) and the ball will curve to the right. Since the left side of the ball is spinning against the air, wouldn't this mean faster relative speeds and thus a lower pressure than the right side? And wouldn't this lower pressure on the left side cause it to curve _left_ instead of right?


How to measure the mass of the electron?

I've done a little bit of research and it seems Millikan was able to measure the ratio between the charge of the electron and its mass. But how can one measure one of the two constants to get the value of the other?


Question about creation operator and its Hermitian adjoint

The definition of creation operator for bosonic system is $$a^{\dagger}|... n_i ...\rangle = \sqrt{n_i + 1} |... n_i + 1 ...\rangle $$ If I take Hermitian adjoint of this I will get $$(a^{\dagger}|... n_i ...\rangle)^{\dagger} = |... n_i ...\rangle^{\dagger} (a^{\dagger})^{\dagger} = \langle... n_i ...| a = \sqrt{n_i + 1} \langle... n_i + 1 ...|$$ But this can't be correct since $$\langle... n_i ...| a = \sqrt{n_i} \langle... n_i - 1 ...|$$ Can someone explain to me why my Hermitian adjoint equation is wrong. From the answer I got from JeffDror and the comment left by user26431. I infer that in dual space $A_1 = a$ acts as creation operator and $A_2 = a^{\dagger}$ as annihilation operator. I am a bit confused by what JeffDror wrote "if you are acting backwards with the operator then you need to use the Hermitian adjoint". In my mind it sounds like that to act with operator A on some bra-vector I must first take Hermitian adjoint of it and than act on bra. It make sense to me if imagine that bra-vector is row vector that I get for transposing ket-vector and non hermitian operator A in ket space is not represented by square matrix. Then in order to multiply matrix by column vector from the left I must transpose matrix that represents A. Is this correct way of thinking about operator how operator A acts in dual space?


Why are oxygen and hydrogen compressible, but water is barely compressible

What makes the two gases compressible, but does not apply to a liquid like water?


Isotropic systems and homogeneity

In isotropic systems, the atomic arrangement is homogeneous in all directions. In the case of glass, which has the atomic structure of a liquid and, therefore, a random atomic structure that is definitely not homogeneous, is it that the atomic arrangements in each direction are equally disordered and therefore homogeneous in that sense?


Lattice model completely constrained by boundary data

I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole volume) is fixed. This trait is true for all $N\times N \times N$ lattices with this model. I am interested if in the thermodynamic limit this behavior holds and if this fixing prohibits phase transitions. Any ideas on these two matters? Edit: These are things I think I can add. Since the internal variables are fixed by the boundary configurations, then for the partition function, there are bulk volume configurations where boundary spins are different but the internal spins are the same (compared to another configuration). Thus, I can define an equivalence between boundary configurations for two boundary configurations that yield the same interior. Then I only need to sum over equivalence class of surface configurations, and the model is somewhat 2D.


Easy question about magnetism?

I have to build a simple electric motor by attaching a magnet to a battery, extending the terminals of the battery (with stiff wires so they could act as supports), and placing a coil of wire on top of it. I know that the more loops you have in the coil the stronger the magnetic field and therefore the faster the coil will rotate. However, how does the diameter of the magnetic coil affect speed of the wire coil rotating? A smaller diameter might make the magnetic field stronger, but the magnetic force which produces torque would also be be applied closer to the center of rotation, therefore creating less torque.


Can the hot combustion products from a large flame be in "non-local thermal equilibrium"

**Question:** Does it take some time for the hot combustion products from a flame to reach local thermodynamical equilibrium (i.e. for the energy state populations to follow the Boltzmann distribution)? If so, on what type of time scales might this departure from local thermodynamical equilibrium occur? **Background:** I am trying to model infrared emission spectra in the mid-infrared region to reproduce emission spectra of hot exhausts from the flame of an industrial flare stack measured with an FTIR-spectrometer. The emitted spectral radiance is calculated using the Planck function and the absorption crossections of different molecular species. The absorption crossections are calculated for different species using the HITRAN and HITEMP line databases. This seems to work fine for some species and some regimes. However, I seem to have problems especially when the exhaust is really hot. For instance, in the region 900-1000 cm$^{-1}$ I can qualitativly see that I have strong water emission lines, but with only varying the temperature parameter I cannot get the relative intensities of the different lines right. I keep thinking that one possible problem might be that the exhaust product from the combustion are not in local thermodynamical equilibrium. In that case the state populations for the species would not follow the Boltzmann distribution which is assumed for calculating the line intensities and their temperature dependence.


Number density of LO and LA phonons as a function of temperature?

I'd like to know the how the number density of longitudinal optical (LO) and longitudinal acoustic (LA) phonons varies as a function of temperature of the material. Is there a simple expression for these two cases? I'm guessing that this would work, $N_{LO} = \int g_{LO}(E) f(E, T) dE$ $N_{LA} = \int g_{LA}(E) f(E, T) dE$ where $g(E)$ is the density of states for LO and LA phonons and $f$ is the Bose-Einstein distribution. What would be appropriate limits for the integrals. Does anybody know a reference where the density of states for these two modes is given? **EDIT** To improve the question, I'm interested in semiconductor 3D crystals. But maybe I left this too long, sorry. Best regards,


Cosmic Inflation: Lower Expansion Rate with than without?

When I read the Cosmic Inflation diagram at https://en.wikipedia.org/wiki/File:Horizonte_inflacionario.svg which I rotated to make it readable: ![Plot 1](http://yukterez.enabled.io/inflation.png) where the Hubbleradius, and therefore also the Hubbleparameter, which is c/Hubbleradius, is constant over inflation, then it seems to me, that with inflation the Hubbleparameter would have been much lower than without: ![Plot 2](http://yukterez.enabled.io/Inflation_Plot.gif) If the Hubble Parameter would go with Ω-Radiation (to the 4th Power), it would reach Infinity at 0, and about 10^103 m/sek/m at Planck size, which is way more than the expansion rate postulated for inflation: ![Radiation dominated Expansion Rate near 0](http://yukterez.enabled.io/Hp.PNG) Did I get this right, or do I have the wrong data? And if I have it right, is it more like the green curve with a hard inflation to radiation edge, or the blue one with a smooth blending; or is there all another way to get a plottable Hubble Parameter including inflation as a function of the scale factor? PS: the y Axis is the Hubble Parameter (c/Hubbleradius) in m/sek/m and the x Axis is the Scale Factor in m/m (dimensionless).


Can wind currents form in the mesosphere?

Probably not the right forum for this question. How high in the atmosphere may an air-current form? Can a wind current form up in the mesosphere?


What's the explanation for the Giant Magnetoresistance effect?

I've been reading this review on Giant Magnetoresistance, and something about it is bothering me. The basic effect is that, using a special "stack" of layers (alternating between ferromagnetic and not magnetic) you can cause great variations on the electrical resistance of the material by applying a small magnetic field.   ![alt text](http://i.stack.imgur.com/jJQRA.png)   The explanation given both on the review and on Wikipedia is the following. The configuration on the left will cause scattering for all spin down electrons that cross it, but will present very low resistance for the spin up electrons (thus a small resistance overall). The one on the right, will scatter all electrons and thus have very large resistance overall. **Here's my problem** (quoting the review) > These experimental results were all obtained for the ’Current In the Plane > of the layers’ (CIP) geometry. The current is then parallel to the plane of > the layers. How can this be? CIP in the above image would mean the current is horizontal, with probes positioned on the left and right of the middle layer. But, if you don't force the current to cross the ferromagnetic layers, why would there be an increase in resistance? I checked with a professor, and he confirmed that the standard method is indeed to apply the probes horizontally, but he couldn't tell why the effect still worked. One option that came up is that electrons don't follow a straight line from one probe to the other, and thus they would still "flow" through the other layers, but I know that's not the case. Using the resistors analogy seen in the image above, applying a CIP is equivalent to setting the resistors in parallel for each of the spin orientations, and parallel resistors _always_ have less (or equal) resistance than the weakest resistor. To be clear, my question: How can the above explanation be compatible with the CIP case? If it's not, then why does it still work?


Spin-orbit coupling constant for rubidium

I have come across the following question in my course notes: The $5s\to 5p$ transition in rubidium is split into two components with wavelengths of 780nm and 795nm respectively. For the $5p$ state, calculate the spin-orbit coupling constant $a$. I have calculated the spin orbit splitting energy to be $E=\frac{3}{2} a$, calculated the energy of the 795nm component to be $E=\dfrac{hc}{\lambda}=2.5\times 10^{-19} J$, suggesting that $a=1.67\times 10^{-19}J$. I'm not convinced by this answer as it gives a very large value for the magnetic field of the electron due to its orbital motion. Is the method I'm using correct, or does something seem to be out of place?


Is the proper interpretation of temperature missing in this book?

In Randall T. Knight’s textbook “Physics for Scientists and Engineers” in the first chapter on thermodynamics (Ch. 16: A Macroscopic Description of Matter) one of the first conceptual questions is stated like so: > The sample in an experiment is initially at $t_1$°C. If the sample’s > temperature is doubled, what is the new temperature in °C? The answer to that implies the reference to absolute temperature (for simplicity, to one significant figure, as the book required), i. e.: $t_2°C = (T_1 × 2)K - 273 = (t_1°C + 273) × 2 - 273$. I understand that the whole idea of this question to reiterate to the pupil that we ought to think of temperature as a specific physical property, and not in an everyday sense, but I am confused about the wording. In the section on temperature author writes: > We will begin with the commonsense idea that temperature is a measure of how > “hot” or “cold” a system is. These are properties that we can judge without > needing an elaborate theory. As we develop these ieas, we’ll find that > **temperature** _T_ is related to a system’s _thermal energy_. […] We’ll > study temperature more carefully in Chapter 18 and replace these vague > notions of hot and cold with a precise relationship between temperature and > thermal energy. That is, as yet we are still at the stage of “vague notions”. Few paragraphs later Knight introduces Kelvin scale but nowhere there or in between does he specifically insists on that idea I mentioned after the formula. At least not that I can see. Am I missing something that should be obvious to me, and I fail to grasp it instantly because my brain is depleted by 50+ hours a week of studies? Am I just plain stupid, because to my eternal shame it took me way too long to figure it out? Or was there a better way to home onto that idea by an author: what do other textbooks say on that topic, how do they word this concept? Is the question logically flawed or confusing, or is it the common way of describing the situation in scientific literature?


Delta Dirac Charge Density question

I have to write an expression for the charge density $\rho(\vec{r})$ of a point charge $q$ at $\vec{r}^{\prime}$, ensuring that the volume integral equals $q$. The only place any charge exists is at $\vec{r}^{\prime}$. The charge density $\rho$ is uniform: $$\rho(\vec{r}) = \delta(\vec{r} - \vec{r}^{\prime})\rho$$ But if I evaluate the total charge, I get $$ q = \int dq = \int^{\infty}_{-{\infty}}\delta(\vec{r} - \vec{r}^{\prime})\rho ~dV $$$$= \rho\int^{\infty}_{-{\infty}}\delta({x} -{x}')dx\int^{\infty}_{-{\infty}}\delta({y} -{y}')dy\int^{\infty}_{-{\infty}}\delta({z} -{z}')dz$$ The Dirac delta functions integrate to one each, but what becomes of the charge density $\rho$? For that matter, how does one integrate a zero dimensional point over 3 dimensinal space? Any help greatly appreciated. EDIT: So it seems that the charge density is just the charge itself ($\rho = q)$?


Single photon and double-slit experiment

Laser fires single particles of light, called photons, through the slits. Even though only single photons of light are being fired through the slits and They create three pattern again. How single particles of light can create this wave pattern?


Is there an explanation for the 3:2:1 ratio between the electron, up and down quark electric charges?

I understand that the NNG formula relates $Q$, $I_3$, and $Y$ and can be derived in QCD; does this unambiguously predict the electric charge ratios without making assumptions about the definitions of isospin and hypercharge? If so, this is unintuitive to me! Why does a particle carrying $SU(3)$ color charge care what charge it has under the totally separate electroweak $U(1)\times SU(2)~$ symmetries? If not, is there a name for the "problem" of explaining the charge ratios?


Why would the reflection of a lamp by a bowl look like a heart?

I have this bowl roughly the shape of a circular truncated cone, and the lamp shines on the bowl at an angle. The reflected lamp light hit the bottom of the bowl and created a light ring that looks just like a cardioid. Could someone help me, or give me a hint on how to really solve the equation for the light ring?


Energy of particle in electric field

I'm taking a physics class and the professor teaches us really basic things in lecture and then gives homework way beyond what he taught in lecture. Obviously I need to find some resource other than Stack Exchange, but in the mean time, perhaps you could explain some of the terms in this problem and give me a starting point for how to solve it. Regarding the problem, given below, I'm not sure what the significance of "exactly one mean free path through the air before hitting a molecule" is. I know eV is an electron volt, which is a unit of energy (from high school) but I'm not sure how this is related to the strength of a field. I would assume this energy is related to the force on the particle, which I can relate to the strength of the field and the force on the particle, as well as the distance the particle travels, but that doesn't seem right because no figures are given for distance or charge. I haven't got a clue what cross-sectional area might have to do with this. I do, however, know what a Maxwell distribution is from chemistry class. > (a) Assume that an ion moves, on average, exactly one mean free path through > the air before hitting a molecule. If the ion needs to acquire approximately > $1.11\ \mathrm{eV}$ of kinetic energy in order to ionize a molecule, > estimate the minimum strength of the electric field required at standard > room pressure and temperature ($300\ \mathrm{K}$). Assume that the cross- > sectional area of an air molecule is about $0.095\ \mathrm{nm^2}$. (Assume > the velocities of the particles have a Maxwell distribution of velocities.) > > (b) How does the strength of the electric field in Part (a) depend on > temperature? (Use the following variable as necessary: $T$.) > > (c) How does the strength of the electric field in Part (a) depend on > pressure? (Use the following variable as necessary: $P$.)


Is it ever necessary to extend an analysis of Grover's algorithm beyond $k/N = 1/2$?

Is it ever necessary to extend an analysis of Grover's algorithm beyond $k/N = 1/2$, where $k$ is the number of "hits" in a total of $N$ possible values for $|\,x\rangle$? If we know $k$, and know that it is greater than $1/2$, can't we just redefine hits as misses by modifying the oracle to produce $$-(-1)^{f(x)}\,|\,x\rangle$$ (note the additional negative sign) and then measure according to whatever strategy for $k<1/2$ produces satisfactory hit probabilities, $p_h$, since we can then say with confidence $p_h$ that one of the _unobserved_ outcomes is a hit?


Formulation of the Three-Body Newtonian problem

I am trying to understand three body problem in Newtonian space. I want to make formulation of differential equations for known initial conditions for the case with:   1. Identical three masses    2. Periodic solution    3. Zero Angular momentum The equations in vector form are: $$\frac{d^2 \overset{\rightharpoonup }{r}_1}{\text{dt}^2}=\frac{k^2 m_2 \left(\overset{\rightharpoonup }{r}_2-\overset{\rightharpoonup }{r}_1\right)}{r_{12}^3}+\frac{k^2 m_3 \left(\overset{\rightharpoonup }{r}_3-\overset{\rightharpoonup }{r}_1\right)}{r_{13}^3}$$ $$\frac{d^2 \overset{\rightharpoonup }{r}_2}{\text{dt}^2}=\frac{k^2 m_1 \left(\overset{\rightharpoonup }{r}_1-\overset{\rightharpoonup }{r}_2\right)}{r_{12}^3}+\frac{k^2 m_3 \left(\overset{\rightharpoonup }{r}_3-\overset{\rightharpoonup }{r}_2\right)}{r_{23}^3}$$ $$\frac{d^2 \overset{\rightharpoonup }{r}_3}{\text{dt}^2}=\frac{k^2 m_1 \left(\overset{\rightharpoonup }{r}_1-\overset{\rightharpoonup }{r}_3\right)}{r_{13}^3}+\frac{k^2 m_2 \left(\overset{\rightharpoonup }{r}_2-\overset{\rightharpoonup }{r}_3\right)}{r_{23}^3}$$ where $k$ denotes the gravity constant. I want to formulate (not solve) this problem properly. I've found this reference can help understand zero angular momentum. I don't understand it well.


Is it possible to detect fake Tungsten aka Wolfram gold bars with a strong magnet?

Tungsten aka Wolfram is paramagnetic so it is weakly attracted to magnets. A guy devised the following to test for Tungsten in gold bars: http://www.youtube.com/watch?v=foELQ7T8_90 But he is using a paperclip and not real Tungsten. Question: How strong a magnet do I need in order to achieve a significant attraction to Tungsten. By significant I mean that it should exert a force of several (1/100)*grams so that it can be measured with a scale as in the YouTube video. Will a Neodymium magnet do? If yes, what should be the strength of the magnet in kg?


What "I" should use in Rotational Energy formula $(I \omega^2)/2$

$\text{Rotational Energy} = \frac{1}{2} I \omega^2$. What $I$ should be used? $I$ as a inertia tensor matrix = stepRotation * inverse moment of inertia * inverse stepRotation; Or I as moment of inertia? Inertia tensor matrix gives changing energy due to rotation changes. So I don't believe its the right answer, but I'm not sure. Moment of inertia remains the same in simulation because it only represents the mass resistance to rotation in a fixed axis calculated by integration of massparticles * radius from object center.


Majorana-like representation for mixed symmetric states?

Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)? By Majorana representation I mean the decomposition of a state $$|\psi\rangle = \text{normalization} \times \sum_{perm} |\eta_1\rangle |\eta_2\rangle \cdots |\eta_n\rangle,$$ where $|\eta_k\rangle$ are uniquely determined (up to a global phase in each and the permutation) qubit states. Typically, they are presented as points on the Bloch sphere. When it comes to a desired generalization - by it I mean a unique set of invariants and covariants of $\text{SU}(2)$ assigned to each density matrix. The uniqueness (up to permutations) is crucial, otherwise one can just spectrally decompose the density matrix $$\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$$ and apply the standard Majorana representation to each of its eigenvectors (obtaining invariants $\\{ p_i \\}_{i\in\\{1,\ldots,n\\}} $ and covariants $\\{ |\eta_{i,j}\rangle \\}_{i,j\in\\{1,\ldots,n\\},}$ ). However, for the case of the eigenvalue degeneracy it is no longer unique. A naive dimension counting $(n+1)^2-1=2n + n^2$ gives a hint that there may be $n$ covariant points plus an invariant $n\times n$ matrix. An auxiliary question is if there is a unique representation of $k$-dimensional subspaces of pure symmetric $n$-qubit states? (If there is one, then it is possible to 'fix' the mentioned approach with the spectral decomposition.)


Coupling constant problem

In the scalar $φ^4$ theory we write Lagrangian as $$\mathcal{L}=\frac{1}{2}(\partial_t\phi)^2 -\frac{1}{2}\delta^{ij}\partial_i\phi\partial_j\phi - \frac{1}{2}m^2\phi^2-\frac{g}{4!}\phi^4. $$ I want to know details about the coupling strength g. How will I start? I studied some technique from Feynman diagram, that it represents the strength of a scattering. But I need to find out the value of this constant and need guideline how will I study so I can understand clearly. Is there any relationship which gives us the enough evidence for putting the coupling constant into Lagrangian.