question stringlengths 37 38.8k | group_id stringlengths 2 6 | sentence_embeddings listlengths 768 768 |
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<p>How do 3d <a href="http://en.wikipedia.org/wiki/Holography" rel="nofollow">holograms</a> work exactly? I read that there is a laser in use, but how are the multiple perspectives generated and how is the light <em>trapped?</em> in a certain area to create the effect?</p>
<p>Furthermore, how do you generate different colors? Is it still using multicolored projectors or is there something special? What kinds of lenses are used?</p> | g930 | [
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<p>How would a society go about either preventing our sun in its primary sequence from going into a Red Giant a billion years from now? Or perhaps, accelerating the process of going from main sequence of our start to a red giant prematurely. </p>
<p>What catalysts would have to happen, implemented by humans, so no natural phenomenon could play a part in it, that would have an impact, even slight, on the future processes of the sun's lifecycle. </p>
<p>Any thoughts or help would be great!!!!</p> | g8434 | [
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<p>I googled the question and found no explanation. It seems that dog ears are inevitable (for paperbacks, notably) even if you've always been careful. From my experience, they are about equally likely to appear on the top corners as on the bottom corners (for both the beginning pages and the ending ones). Dog ears for the middle pages of the book are less likely but they can also appear in frequently used old books. Can someone explain why? </p>
<p><img src="http://i.stack.imgur.com/rw85F.jpg" alt="dog ears on the bottom corners for the beginning and ending pages"></p>
<p><img src="http://i.stack.imgur.com/bbpa6.jpg" alt="dog ears for the beginning pages of a book"></p>
<p>I apologize if this is not the right kind of question to post here. I can find no other sites on SE for it. </p> | g8435 | [
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<p>I am interested in (typically topological) field theories arising from Lagrangians of the form.</p>
<p>$f(\Phi) \lambda$,</p>
<p>where $\lambda$ is a Lagrange multiplier field not appearing in $f(\Phi)$. Perturbatively, the partition function just counts the solutions to $f(\Phi)=0$. What about nonperturbatively? Are there exponentially suppressed kinks that affect the partition function?</p>
<p>Has anyone considered these?</p> | g8436 | [
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<p>This problem is puzzling me.</p>
<p>Scenario 1:
A certain sphere is having an innelastic collision with a plane of a very large mass. Let the angle between the sphere's velocity and the surface be 90 degrees. The escape angle will be the same as the incident angle, therefore the sphere velocity will be 90 degrees from it's origin.</p>
<p>Scenario 2:
The same as above, but now the ball is static, and the surface is moving towards the sphere with the same speed.</p>
<p>I was expecting exactly the same result, since on my mind (and obviously wrong), the speeds were relative. The sphere in scenario 2 leaves the collision plane on a trajectory normal to the surface.
Note that I mention speed, because I have a gut feeling that the problem I'm having is with summing up the vectors.</p>
<p>I'm having problems visualizing and coming up with an intuitive explanation for why.</p> | g8437 | [
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<p>This must be the first question everyone asks when starting to study field theory at finite density and zero temperature. </p>
<p>To introduce a finite density one adds a Lagrange multiplier which fixes the number of particles in a non-relativistic field theory or the number of particles minus the number of antiparticles in a relativistic field theories. To be concrete, in a non-relativistic field theory one has the Lagrangian density:
$$\mathcal L=i\psi ^*\partial _t \psi -{1\over 2m}\nabla\psi ^*\nabla\psi - V[\psi ^*\psi]$$
and then the same theory at finite density is described by:
$$\mathcal L=i\psi ^*\partial _t \psi -{1\over 2m}\nabla\psi ^*\nabla\psi - V[\psi ^*\psi]+\mu \psi ^*\psi \,\,\,\,\,\,\,\, [1]$$
where $\mu$ is the chemical potential and the number of particles is $N=\int \psi ^*\psi$.</p>
<p>What disturbs me is that one may redefine the field:
$$\chi\equiv e^{-i\mu t}\psi$$</p>
<p>with $t$ the time. And after replacing in [1]:</p>
<p>$$\mathcal L=i\chi ^*\partial _t \chi -{1\over 2m}\nabla\chi ^*\nabla\chi - V[\chi ^*\chi]$$
The "finite density term" disappears. Moreover, the path integral measure is invariant under this trivial field redefinition so that the finite density QFT seems equivalent to the zero density one. (Those who are familiarized with the non-relativistic limit of relativistic equation —such as Klein-Gordon or Dirac— will recognize this term and field redefinition.)</p>
<p>Something similar happens in the relativistic case. </p>
<p>What am I overlooking? </p>
<p>(Note that the theory is at zero temperature so I am in real time formalism and I don't have to impose periodic boundary conditions (I think). I am interested in time evolution, scattering, etc.)</p>
<p><strong>Reply to Ron's answer:</strong></p>
<p>(I here will overlook Thomas' comments, I will answer him tomorrow.)</p>
<p>Ron makes an interesting point, but I think his answer is not correct. The Hamiltonian density of the theory in the original fields is:$$\mathcal H={1\over 2m}\nabla\psi ^*\nabla\psi + V[\psi ^*\psi]-\mu \psi ^*\psi $$
What he correctly pointed out is that the Hamiltonian in the new fields is not just the previous Hamiltonian written in the new fields. This is right and in classical mechanics it is well-know that one has to add a piece to the Hamiltonian when one makes an explicit time dependent transformation. He pointed out the case of quantum mechanics, and the field theoretic version is:
$$i\partial _t \, \chi =i\partial _t \, (e^{-i\mu t}\psi )=i(-i) \mu \, e^{-i\mu t}\psi + e^{-i\mu t}[\psi , H]=[e^{-i\mu t}\psi ,\, \mu N + H] $$
And therefore the Hamiltonian for the field $\chi$ is:
$$H'=\mu N + H=\int {1\over 2m}\nabla\chi ^*\nabla\chi + V[\chi ^*\chi]$$
This is of course the Hamiltonian corresponding to the Lagrangian written in terms of $\chi$. In this way, one sees how the "standard" theory (without the $\mu$ term) is recovered in the Hamiltonian formalism as it was in path integral formalism. And this makes a lot of sense because the theory with $\mu = -m$ has to be equivalent to the theory with $\mu=0$ because we are just subtracting the rest energy.</p>
<p>Of course, this is true as long as one does not impose periodic boundary conditions. </p> | g8438 | [
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<p>Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways:</p>
<ol>
<li><p>First obtain the equation of motion in $\{q\}$ by the Euler Lagrange equation
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right)-\frac{\partial L}{\partial q}=0$$
and then rewrite the equation in terms of $\{q'\}$.</p></li>
<li><p>First transform $L(q,t)$ to $L'(q',t)=L(q(q',t),t)$ and then obtain the equation of motion
$$\frac{d}{dt}\left(\frac{\partial L'}{\partial \dot q'}\right)-\frac{\partial L'}{\partial q'}=0$$</p></li>
</ol>
<p>Are the two answers just the same?</p> | g8439 | [
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<p>I have seen on the internet that many times people assert that inside a cylindric <a href="http://en.wikipedia.org/wiki/Condenser" rel="nofollow">condenser</a> the electrostatic field is null due to the fact that the Gauss flux inside is null. But I wanted to make clear that this is not the reason. The reason lies on the symmetry of the condenser. Am I right?</p>
<p>If the electrostatic field was null inside the smaller radius conductor, it would mean that if we put a test particle there, it would not resist any force. However if I am right, this would not happen. The only thing that could happen so that the particle did not move, would be that the charge in the condenser could be reorganized in presence of that particle.
Do I have the concepts right? </p> | g8440 | [
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] |
<p>I have a few questions about 3D waves:</p>
<ol>
<li>Are all 3D waves types of electromagnetic radiation?</li>
<li>Are all 2D waves mechanical?</li>
</ol> | g8441 | [
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<p>I am approaching this from an intuitive perspective and I don't speak the language. However, I have been doing a lot of reading about Bell's Theorum and how invalidates either locality or counterfactual definiteness. </p>
<p>From my point of view, locality is the concept that there is no way that two particles can interact with each other "instantaneously" at a distance (i.e. faster than light.). Realism/counterfactual definitiveness is the concept that all particles have some true value for all variables at some instant regardless of whether or not that value was observed. Let's assume for the sake of this question that locality is true, which I believe is the current popular model in QM.</p>
<p>Assuming locality, why do Bell's Inequalities show that conterfactual definiveness must be false?</p>
<p><strong>Edit to respond to below post:</strong></p>
<p>This is crazy. Let me see if I am following you. I'll use your framework. </p>
<p>I'm not really sure what spin is but I'm seeing it as an observable phenomenon that exists as a function of a specific direction in a given particle at an instant of time, and the spin itself at any angle changes (randomly?) over time.</p>
<p>A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and experimentally we would see that they are the same spin This is crazy. Let me see if I am following you. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and they will be the same spin 99% of the time.</p>
<p>Similarly, if we measure the spin at θ in A and at 2θ in B, we will see that these two values are the same spin 99% of the time. Assumingly, this will work for any two angles that are exactly θ apart -- with enough measurements they will be correlated exactly 99% of the time.</p>
<p>However, if we measure the spin at 0 in A and at 2θ in B, they are only correlated 96% of the time. If V0 is at most 1% off from V1, and V1 is at most 1% off from V2, V0 should be anywhere from 0-2% off from V2. But, we find, according to QM and experimentation, it is not! It is less heavily correlated than that.</p>
<p>This is where I lose you. Why does saying no to realism resolve this issue?</p> | g8442 | [
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<p>When we study the interaction of the electromagnetic radiation with free electrons we can find two different approaches in the literature: for low frequency (RF, light...) a classical view is used and it is said that the electrons oscillates due to the electric field of the wave, but for high frequency radiation (X-rays, gamma), the type of interaction seems totally different (Compton effect and the like) and radiation is usually modeled as "particles". For an radiation with "intermediate" frequency such as UV or very low energy X-rays, would interactions be "electric-like" or "compton-like"?. How is the transition between both regimens? </p> | g8443 | [
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<p>If we could irradiate a metal or a plasma with a coherent beam of hard X-rays, would the main effect be an ordered oscillation of the electrons like with a classical EM wave, or "compton-style" collisions?</p> | g8444 | [
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<p>Like any set of massive objects separated by distance, the Earth and Sun have a center of mass. How far in meters from the Sun's center is the Earth-Sun center of mass? Just curious? Also what percent of the Sun's radius is this distance?</p> | g8445 | [
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<p>Some of you may be familiar with the <a href="http://en.wikipedia.org/wiki/European_Union_energy_label" rel="nofollow">EU energy consumption classes</a> for domestic appliances (or equivalents in other areas).</p>
<p>Let's say we have a domestic appliance which was made before the energy directives were made active.</p>
<p>How can one measure (approximately) the energy consumption using whatever is available in one's kitchen or home and check which class the appliance could be put under?</p>
<p><strong>Clarification:</strong></p>
<p>Of course one can use a power meter an extract some results. The question is more precisely about what can one do using things (and methods) available in one's kitchen. It would be best to describe a mehod to implement this task and furthermore the method should be able to classify (approximately not laboratory precision required) the appliance in terms of the EU energy classes.</p>
<p>Answers that mention the power company power supply meter are in the spirit of what is asked.</p> | g8446 | [
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<p><strong>1. The problem statement, all variables and given/known data</strong></p>
<blockquote>
<p>Beam of electrons with energy $10eV$ hits the potential step
($8eV$ high and $0.5nm$ wide). How much of the current is
transmitted?</p>
</blockquote>
<p><strong>2. Relevant equations</strong></p>
<p>I know that energy mentioned in the statement is kinetic energy so keep in mind when reading that $E\equiv E_k$.</p>
<p>In our case the kinetic energy is larger than the potential energy $\boxed{E>E_p}$ and this is why stationary states for the regions 1, 2, 3 (picture) are as folows: </p>
<p>\begin{align}
\psi(x)_1 &=Ae^{iLx}+B^{-iLx} && L=\sqrt{\tfrac{2mE}{\hbar^2}}=1.619\times 10^{10}\tfrac{1}{m}\\
\psi(x)_2 &=Ce^{iKx} +De^{-iKx} && K=\sqrt{\tfrac{2m(E-E_p)}{\hbar^2}}=0.724\times 10^{10}\tfrac{1}{m}\\
\psi(x)_3 &=Ee^{iLx}
\end{align}</p>
<p><img src="http://i.stack.imgur.com/RbNgm.png" alt="enter image description here"></p>
<p><strong>3. The attempt at a solution</strong></p>
<p>I first used the boundary conditions for the border 1-2 and got a system of equations:</p>
<p>\begin{align}
A+B &= C+D\\
iLA-iLB&=iKC-iKD\\
&\downarrow\\
A+B &= C+D\\
A-B &= \tfrac{K}{L}(C-D)\\
&\left\downarrow \substack{\text{I used the assumption that}\\\text{$E \gg E_p$ and got the simplified}\\ \text{relation between L and K:}\\ \tfrac{L}{K}=\sqrt{E/E_p}\gg 1 \Longrightarrow L\gg 1\\ \text{and hence}\\ \tfrac{K}{L}=\sqrt{E_p/E}\ll 1 \Longrightarrow K\ll 1} \right.\\
A + B &= C+ D\\
A-D &= 0\\
&\downarrow~\Sigma\\
2A &=C+D\\
&\downarrow\\
C&=2A-D
\end{align}</p>
<p>Now i used the boundary conditions for the border 2-3 and got a system of equations in which i inserted the above result:</p>
<p>\begin{align}
Ce^{iKd} + De^{-iKd} &= Ee^{iLd}\\
iKCe^{iKd} -iK De^{-iKd} &= iLEe^{iLd}\\
&\downarrow\\
Ce^{iKd} + De^{-iKd} &= Ee^{iLd}\\
Ce^{iKd} -De^{-iKd} &= \tfrac{L}{K}Ee^{iLd}\longleftarrow\substack{\text{Things get weird}\\\text{when i insert }C=2A-D}\\
&\downarrow\\
2Ae^{iKd} -De^{iKd} + De^{-iKd} &= Ee^{iLd}\\
2Ae^{iKd}-De^{iKd} -De^{-iKd} &= \tfrac{L}{K}Ee^{iLd}\\
&\downarrow \substack{ \text{$L$ is very big while $K$ is very small}}\\
2A &= Ee^{iLd}\\
2A -2D &= \tfrac{L}{K}Ee^{iLd}
\end{align}</p>
<p>Now this result is what i don't understand. First equation gives me a ratio $E/A=2\exp[-iLd]$ which i think i need to calculate the transmissivity, but it is a complex exponential. Is this possible? How do i continue? Is this the way to solve this kind of case or did i go completely wrong?</p>
<p>This is my first finite potential well problem and i have been searching the web for a loong time to even find 2 videos (<a href="http://www.youtube.com/watch?v=vMndTqowzqU&list=FLUuDEwpR43gvcFvCBovHBUA&index=25" rel="nofollow">video 1</a> and <a href="http://www.youtube.com/watch?v=gwSBFZYfSCc&list=FLUuDEwpR43gvcFvCBovHBUA&index=24" rel="nofollow">video 2</a>) for a <strong>similar problem</strong> but for the case when $E<E_p$. The professor in the video allso uses similar approximation - but in his case $E \ll E_p$.</p>
<p>On this hyperlink there is allso a scan of the problem solved on the paper just in case anyone likes it this way. It is in Slovenian language but i added some headings to help you reading.</p> | g8447 | [
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<p>Only because</p>
<ol>
<li><p>Rep is unitary, so saves positive-definite norm (for possibility density),</p></li>
<li><p>Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and</p></li>
<li><p>It is the representation of the global group of relativistic symmetry, </p></li>
</ol>
<p>yes?</p> | g8448 | [
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<p>Assume we had an elevator shaft long enough for a free-falling elevator to reach terminal velocity.</p>
<p>As I understand it, when the elevator begins to fall a person inside would experiences weightlessness because they would be accelerating downward at the same rate as the elevator. The rate of downward acceleration for the elevator should reach a constant when air resistance equals gravity (right?). However, the person in the elevator would not be exposed to air resistance and should continue to accelerate downward. </p>
<p>So would they gradually sink back to the floor and experience gravity as though the elevator were at rest... at least until the inevitable termination of the experiment? </p> | g8449 | [
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<p>I am trying to determine if a radiant barrier on the inside of a hot water tank would increase its insulating value. Normally radiant barriers are used in house construction to reduce heat loss by reflecting radiant heat back into the house. The reflective surface is placed against an air gap on the exterior wall facing inwards and the thermal radiation is reflected back across the air gap into the drywall and hence into the home. I am trying to determine if a radiant barrier placed directly against a liquid would have the effect of reducing heat loss from the liquid.</p>
<p>Lets say we have a tank designed to store hot water and keep it hot by reducing heat loss to the ambient surroundings. For example, the tank is metal and surrounded by a insulating layer of foam, fiber glass etc. with a high R value (A measure of resistance to heat transfer). Lets say there are two versions of this tank. The first tank has a grey or dark interior, similar to what would be found in a glass lined steel water tank commonly used a domestic water heater. The second tank is identical except that its interior is coated with a reflective silver surface facing towards the mass of hot water. Would the reflective surface increase the effective R value of the tank's walls and reduce heat loss from the tank? The possible effect is that thermal radiation emitted from the hot water molecules would reflect back from the silver surface and be reabsorbed by the hot water mass as it passes back into the body of the water.
This might seem similar to a thermos type vacuum flask, the main difference being that in a thermos flask there is a vacuum behind the radiant barrier so I don't know if their is difference in the radiant barrier effect between having a vacuum behind the radiant barrier and having solid conducting material behind it as we do in the water tank example. if their is any R value enhancement by virtue of the slivered lining of the water tank, how could we quantify it in terms of R value? Thanks.</p> | g8450 | [
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<p>The photon has a mass of 0, but it has energy because of its motion. When it is sucked into the black hole and becomes a singularity, it loses its energy because it is no longer moving. It is not possible for something to be sucked out of a black hole, so it has no potential energy that the kinetic energy could have been transformed into. If I haven't missed a point, then energy has just been destroyed. Everyone knows that it isn't possible for energy to be destroyed because of the first law of thermodynamics a.k.a. the conservation law of energy. Therefore I must have missed a point, because I am not ready to criticize famous scientists that have indefinitely more knowledge then I do. Please tell me what I did wrong :)</p> | g8451 | [
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<p>I am a layman, so take this with a grain of salt.</p>
<p>I saw a TV show the other day which showed a Russian Cosmonaut who had spent more time in space than any other human. The relativistic effects of the low gravity and extreme speeds at which he had spent a decent part of his life had pushed him a small, but surprisingly non-trivial fraction of a second into the "future" as compared to the rest of us observers here on earth. I want to say it was something like a 50th of a second.</p>
<p>What, if any, are the relativistic effects all of us experience in an average lifespan simply by being on the earth as it travels through space in orbit and as the galaxy rotates and the universe expands. </p>
<p>By effects I mean as compared to a hypothetical observer who is able to remain completely motionless in space? Is it even measurable? </p>
<p>I know gravity is not being taken into account, so is this question even answerable?</p>
<p>Thanks for your patience with my question.</p> | g8452 | [
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<p>What is a good way to block neutrons and what is the mechanism that allows this? It's my understanding that polyethylene is somewhat effective. Why?</p> | g8453 | [
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<p>Assume that we have access to a computer with unlimited computing power.</p>
<p>As with the Lattice QCD can try to prove that we live in a simulated universe?</p>
<p>Apparently, one may use the issue of cosmic rays.</p>
<p><a href="http://www.wired.co.uk/news/archive/2012-10/11/universe-computer-simulation" rel="nofollow">Here is full article</a></p>
<p>Quote:</p>
<blockquote>
<p>If we were living in a simulator, we'd expect to find evidence of that lattice if we looked close enough to the edges of the observable universe - and that's what Silas Beane from the University of Bonn and colleagues have calculated, in a paper published in arXiv. As cosmic particles fly through the universe, they lose energy and change direction and spread out across a spectrum of energy values. There's a known limit is how much energy Those particles have, though, and Beane and his colleagues have calculated That this seemingly arbitrary cliff in the spectrum is consistent with the kind of boundary that you ''d find if there was an underlying lattice governing the limits of a simulator. It should also, if present, scatter the particles in a Certain Way as they come up against it, and the should be able it investigate Whether that's the case.</p>
</blockquote>
<p>Do you have any idea?</p> | g8454 | [
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<p>I'm reading through Vol. 1 Chaper 18, and Feynman says that in the system:</p>
<p><img src="http://i.stack.imgur.com/G3AZG.png" alt="enter image description here"></p>
<p>(<a href="http://www.feynmanlectures.caltech.edu/img/FLP_I/f18-01/f18-01_tc_big.svg" rel="nofollow">See here</a> for a higher resolution copy - can't embed SVGs), the length of $PQ$ is equal to $r\Delta\theta$:</p>
<blockquote>
<p>If $OP$ is called $r$, then the length $PQ$ is $r\Delta\theta$, because of the way angles are defined.</p>
</blockquote>
<p>I'm probably being stupid, but surely this should be $r*\tan(\Delta\theta)$? Also, the diagram itself appears to depict an impossible figure. If the lengths of $OP$ and $OQ$ are equal, then the angle $OPQ$ cannot be 90 degrees. Like I said I'm probably being stupid, so hopefully someone can show me where I'm going wrong.</p> | g8455 | [
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<p>This is really a question to get a little more insight about a homework exercise.</p>
<p><strong><em>1.</em></strong> In my text book, the sum of multiplicities for all possible macrostates is given to be:</p>
<p>$$\sum \Omega_\mathrm{total}=\binom{N_A+N_B+q_\mathrm{total}-1}{q_\mathrm{total}}$$</p>
<p>For two Einstein solids $A$ and $B$ of $N_A$ oscillators in solid $A$, $N_B$ oscillators in solid $B$ and $q_\mathrm{total}$ units of energy in the system. I tried deriving this myself (which the text book doesn't require) but no dice there. How can I derive this myself?</p>
<hr>
<p><strong><em>2.</em></strong> The problem here is that the followup question pretty much defines the long term entropy of this system to be:</p>
<p>$$S=k\ln\left(\sum\Omega_\mathrm{total}\right)$$</p>
<p>Now there's confusion. I simply have no idea what I'm doing. Why is the sum of all the multiplicities together the long term entropy? How should I reason that this is the correct answer? Some explanation would be very, <em>very</em> welcome.</p> | g8456 | [
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<p>Reading Landau's Statistical Physics Part (3rd Edition), I am trying to calculate the answer to Chapter 39, Problem 3.</p>
<p>You are supposed to calculate the total kinetic energy of the particles in an ideal gas hitting the wall of a vessel containing said gas.</p>
<p>The number of collisions per unit area (of the vessel) per unit time is easily calculated from the Maxwellian distribution of the number of particles with a given velocity $\vec{v}$ (we define a coordinate system with the z-axis perpendicular to a surface element of the vessel's wall; more on that in the above mentioned book):
$$
\mathrm{d}\nu_v = \mathrm{d}N_v \cdot v_z = \frac{N}{V}\left(\frac{m}{2\pi T}\right)^{3/2} \exp\left[-m(v_x^2 + v_y^2 + v_z^2)/2T \right] \cdot v_z \mathrm{d}v_x \mathrm{d}v_y \mathrm{d}v_z
$$</p>
<p>Integration of the velocity components in $x$ and $y$ direction from $-\infty$ to $\infty$, and of the $z$ component from $0$ to $\infty$ (because for $v_z<0$ a particle would move away from the vessel wall) gives for the total number of collisions with the wall per unit area per unit time:
$$
\nu = \frac{N}{V} \sqrt{\frac{T}{2\pi m}}
$$</p>
<p>Now it gets interesting:
I want to calculate the total kinetic energy of all particles hitting the wall, per unit area per unit time. I thought, this would just be:
$$
E_{\text{tot}} = \overline{E} \cdot \nu = \frac{1}{2} m \overline{v^2} \cdot \nu
$$
The solution in Landau is given as:
$$
E = \nu \cdot 2T
$$</p>
<p>That would mean that for the mean-square velocity of my particles I would need a result like:
$$
\overline{v^2} = 4\frac{T}{m}
$$
Now, I consider that for the distribution of $v_x$ and $v_z$ nothing has changed and I can still use a Maxwellian distribution. That would just give me a contribution of $\frac{T}{m}$ each. That leaves me with $2\frac{T}{m}$, which I have to obtain for the $v_z$, but this is where my trouble starts:</p>
<p>How do I calculate the correct velocity distribution of $v_z^2$?</p> | g8457 | [
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<p>Do we know exactly the difference between <a href="http://en.wikipedia.org/wiki/Particle" rel="nofollow">particles</a> and <a href="http://en.wikipedia.org/wiki/Quasiparticle" rel="nofollow">quasiparticles</a>? Is Higgs boson a particle or a quasiparticle? I ask this because if I understood well, Higgs boson created by a spontaneaous breaking symmetry and correspond to a Goldstone mode. Maybe I am wrong in my explication.</p> | g8458 | [
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<p><a href="http://en.wikipedia.org/wiki/Noether%27s_theorem">Noether's (first) theorem</a> states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.</p>
<p>Is the converse true: Any conservation law of a physical system has a differentiable symmetry of its action?</p> | g740 | [
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<p>Good afternoon everybody.</p>
<p>I am reading on a book about semiconductor mobility. I have fully understood the definition, but I also noticed that often one talks about high or low mobility. My question is: what are the <strong>advantages</strong> of having a <strong>high</strong> mobility? I mean, in the technological applications or experimental uses of the semiconductor/etherojunction.
Do you know any main <strong>methods</strong> to improve mobility (in Si and/or Ge semiconductors for example)? Any <strong>bibliography</strong> about them?</p> | g8459 | [
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<p>Recall the first law of BH thermodynamics</p>
<p>$ dM=\frac{\kappa}{8\pi} dA + \Omega dJ + \Phi dQ $</p>
<p>Now, let's consider the Reissner-Nordstrom solution $J=0$ such that $m>Q$ but only slightly greater. Suppose I have a small bit of charge $dQ$ which I bring in from infinity to the BH horizion. </p>
<p>1) Question: Is the extremal BH solution $m=Q$ possible? </p>
<p>I would think that if we consider the work to be done to bring $dQ$ from $\infty$ to BH horizon, this would blow up $\Phi$ and cause M to go to $\infty$ as well. Therefore, I would guess the extremal solution cannot be achieved in this scenario. </p>
<p>2) Now, this extremal solution DOES seem possible in the Kerr solution, but how would this affect $J$ and $\Omega$ ?</p> | g8460 | [
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<p><a href="http://en.wikipedia.org/wiki/Maxwell_equations" rel="nofollow">Maxwell's equations</a> allow for wave solutions via oscillations between electric and magnetic field content.</p>
<p>Couldn't we generate electric waves also if that solution didn't exists?
Imagine there was no magnetic field, only the electric field. If at one point in space we generated an electric field nicely varying in a sinus pattern of constant frequency, wouldn't this propagate with light speed, just like a plane EM-wave of that frequency, even though there was no magnetic field?</p> | g8461 | [
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<p>I'm pretty ignorant regarding neutron and nuclear transport modelling, but i'm interested in trying to pursue it for a particular pet project. It regards modelling of nuclear reactions like those required for NERVA-like rockets or <a href="http://path-2.narod.ru/design/base_e/nswr.pdf" rel="nofollow">this one</a>.</p>
<p>I would appreciate any references or some explanations of the required approximations for the models required to compute neutron transport that i would have to take in consideration for this exercise.</p> | g8462 | [
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<p>Water falling from a tap eventually breaks into droplets at a particular distance from the tap. The distance(from the tap) at which it breaks into droplets is observed to be an increasing function of the volume flux of the water flowing out of the tap. What is the valid explanation for this phenomenon?</p>
<p>My own line of reasoning is as follows : let $D$ be the critical distance measured from the tap, i.e. the distance at which the water breaks into droplets. At a distance $d<D$ the flow can be assumed to be streamlined to a good approximation; in which case the cross-sectional area of the stream goes on steadily decreasing down the flow. Due to this, the surface energy per unit volume associated with the surface of the water should go on increasing with the distance from the tap until it reaches a point where it becomes greater than/equal to the surface energy of the same volume of water as a water drop. At this distance, by minimisation of potential energy the water should break into droplets to minimise the surface energy. Let's say this happens when the cross section of the flow reaches a threshold value $A$. Now if the volume flux is low; it would mean that the cross sectional area at the tap is itself low. Due to that, the threshold area $A$ would be attained at a lesser distance from the tap. This probably explains the observation that the distance(from the tap) at which it breaks into droplets is observed to be an increasing function of the volume flux of the water flowing out of the tap. Am I right? Please correct/add to my explanation. </p> | g8463 | [
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<p>I'm learning quantum mechanics on my own. I've known that energy is quantized and I've started wondering about temperature. From thermodynamics we have:
$$U=\frac{3}{2}NkT $$
(for ideal gas, of course)<br>
Both U and N aren't continous, so i think T shouldn't be, too. Is that formula correct also for quantum mechanics?<br>
I'm really sorry because of my language, I'm still working on it.</p> | g8464 | [
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<p>I'm trying to program an $N$-body simulation and I'd like to be able to test it with a known solution to a simple, two-body problem. I've looked at multiple sources, but I just don't know how to apply it to my simple test case.</p>
<p>Two objects at rest placed 10 meters apart with mass of 1. The force between them is a modified gravitational force of F = 10 * m1 * m2 / r^2. How long will it take for each object to travel 4 meters?</p> | g134 | [
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<p>What is the Quantum Boltzmann equation and what does it describe? I think it describes the propagation of electrons and photons but I am not sure.</p> | g8465 | [
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<p>Why will the resulting force lines of two positive point charges be like this:</p>
<p><img src="http://i.stack.imgur.com/3jnkx.jpg" alt="enter image description here"></p>
<p>I would expect this:
<img src="http://i.stack.imgur.com/KIltz.jpg" alt="enter image description here"></p> | g8466 | [
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<p>Approaching the following question</p>
<blockquote>
<p>Imagine we have a collection of $N = 100$ simple harmonic oscillators.
The total internal energy is $U = q \epsilon$ where $q$ is the number of energy
quanta. Assume $q >> N >> 1$. If we double the internal energy, by how
much will the entropy of the collection change?</p>
</blockquote>
<p>The question was originally found <a href="http://online.physics.uiuc.edu/cgi/courses/shell/phys213/practice.pl?exam1/su11" rel="nofollow">here</a> under <em>Question 5</em>.</p>
<p>The answer is $\sigma_{final} - \sigma_{initial} = 69.3$</p>
<p>How is this determined? </p>
<p>I know $S = k_b ln \Omega$, where $S$ is entropy, $\Omega$ is the number of micro-states, $\sigma = \frac{ 1}{ \Omega}$, $k_b$ is the Boltzmann constant.</p>
<p>I know $<energy> = 1/2 k T$ per quadradic term.</p>
<p>I know $\frac{ 1}{ T} \equiv \frac{ \partial S}{\partial U}$ </p>
<p>But how does doubling energy effect entropy? Specifically, what equation am I looking for to determine this?</p> | g8467 | [
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<p>I would like to ask if there is any Nuclear process which can be used to produce a <a href="http://en.wikipedia.org/wiki/Wireless_power" rel="nofollow">wireless power</a> ? </p>
<p>Is there any nuclear process can be radiate to the environment with no harm to human beings?</p>
<hr>
<h1>How can I use the photo electric effect with some high Intensity radiation ?</h1> | g8468 | [
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<p>The Lagrangian density for QED is</p>
<p>$$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-m)\psi $$</p>
<p>with</p>
<p>$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} $$
$$ \bar{\psi}=\gamma^{0}\psi^{\dagger}$$
$$D_{\mu}=\partial_{\mu}+ieA_{\mu}$$</p>
<p>$U(1)$ local gauge transformations are</p>
<p>$$\psi\rightarrow\psi^{'}=e^{-i\alpha(x)}\psi$$
$$A_{\mu}^{'}=A_{\mu}+\frac{1}{e}\partial_{\mu}\alpha(x)$$</p>
<p>I'm trying to see that $\mathcal{L}$ it is invariant under those $U(1)$ transformations, but I finish my calculation with</p>
<p>$$\mathcal{L}^{'}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+e^{-i\alpha(x)}\bar{\psi}[i\gamma^{\mu}D_{\mu}-m-\gamma^{\mu}\partial_{\mu}\alpha(x)]e^{i\alpha(x)}\psi $$</p>
<p>Any hint will be appreciated thanks</p> | g8469 | [
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<p>Can someone kindly help me to know how can i get electron momentum density for one orbital like home?
what is the theory of electron momentum density? how can I derive electron momentum density from real electron density?
how can i use Fourier transform?</p> | g8470 | [
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<p><strong>Can a student with a heavy math background start learning physics with Goldstein's "Classical Mechanics"?</strong></p>
<p>Or is the book too obtuse with basic physics that I need to start elsewhere?</p> | g277 | [
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<p>I want to know about the importance of the stability,
generally,why is stability important in super-gravity?
and why do we like these solutions?</p> | g8471 | [
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<p>We know that nuclear magnetic moment can be expressed in terms of the expected value for nuclear spin as:</p>
<p>$$\langle\mu\rangle =[g_lj+(g_s-g_l)\langle s_z\rangle]\frac{\mu_N}{\hbar}$$</p>
<p>(Cf. Krane), where $\vec{j}$ is the <em>total</em> angular momentum, $\vec{l}+\vec{s}$.</p>
<p>How does the expected $\langle s_z\rangle$ value relate to the $\vec{j}$-component of spin, $\langle s_j\rangle$? Krane mentions that only that value is needed, given that it remains constant. </p> | g8472 | [
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<p>How deep can I go if jumping from 50m?</p> | g8473 | [
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<p>The electric field of a point charge goes like $\displaystyle\frac{1}{r^2}$</p>
<p>The electric field of an infinite line goes like $\displaystyle\frac{1}{s}$</p>
<p>The electric field of an infinite plane is constant</p>
<p>Is there an intuitive way or some insight to understand such results and realize that they are expected?(One of my friends said that the electric field due to an infinite plane does not change at all because it looks infinite no matter how close or far from we are it. Is this argument correct? I am not convinced because we can say the same about the infinite line, it looks infinite no matter how close or far we are from it, yet it changes parametrically with the distance)</p> | g8474 | [
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<p>I'm very young so know no calculus whatsoever.</p>
<p>I want to be able to calculate uncertainties for a Physics course I'm doing online, but I believe you need to have knowledge of differentials to do this in some cases.</p>
<p>Do you need knowledge of calculus to be able to calculate with differentials?</p> | g8475 | [
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<p>I have earlier posted the same question <a href="http://math.stackexchange.com/questions/117876/conversion-of-motion-equation-from-cartesian-to-polar-coordinates-is-covariant">here</a> on <code>math stackexchange</code> but without any answer. As the question concerns tensors, I guess that I have come to the right place i.e. to physicists.</p>
<p>Say I have the following equation of motion in the <code>Cartesian</code> coordinate system for a typical mass spring damper system:</p>
<p>$$M \; \ddot{x} + C \; \dot{x} + K \; x = 0$$</p>
<p>where the dot $^\dot{}$ represents differentiation with respect to time.</p>
<p>Now I would like to convert this equation to <code>Polar</code> coordinates. So I introduce</p>
<p>$$x=r \; \cos{\theta}$$ to obtain</p>
<p>$$\dot{x}=\dot{r} \; \cos{\theta} - r \; \dot{\theta} \sin{\theta}$$</p>
<p>and
$$\ddot{x}=\ddot{r} \; \cos{\theta}-2 \; \dot{r} \; \dot{\theta} \; \sin{\theta}-r \; \dot{\theta}^2 \; \cos{\theta}- r \; \ddot{\theta} \; \sin{\theta}$$</p>
<p>I can insert $x, \; \dot{x} \; \text{and} \; \ddot{x}$ in my original equation in the <code>Cartesian</code> coordinate system to yield</p>
<p>$$M \; (\ddot{r} \; \cos{\theta}-2 \; \dot{r} \; \dot{\theta} \; \sin{\theta}-r \; \dot{\theta}^2 \; \cos{\theta}- r \; \ddot{\theta} \; \sin{\theta}) + C \; (\dot{r} \; \cos{\theta} - r \; \dot{\theta} \sin{\theta}) + K \; (r \; \cos{\theta}) = 0$$</p>
<p><strong>Note: I am just showing the equation and derivatives in the x-direction. But the full system has both $x$ and $y$ components.</strong></p>
<p>I wonder if the above way of thinking is right. I am <strong>very</strong> new to tensors and I after reading about <strong>covariant derivatives</strong>, I am now thinking that one should include consider the basis vectors of the <code>Polar</code>coordinate system (a <code>non-Cartesian</code>coordinate system) also since unlike the basis vectors of the <code>Cartesian</code> coordinate system which <em>do not</em> change direction in the 2D space, <code>Polar</code> coordinate basis vectors <em>change</em> direction depending on the angle $\theta$.</p>
<p>I am thinking about <code>covariant derivatives</code> as the conversion process includes differentiation with respect to the bases. For example if $x=r \; \cos{\theta}$, then</p>
<p>$$\dot{x}=\frac{dx}{dt}=\frac{\partial{x}}{\partial{r}} \cdot \frac{dr}{dt} + \frac{\partial{x}}{\partial{\theta}} \cdot \frac{d\theta}{dt}$$</p>
<p>So we have terms like $\frac{\partial{x}}{\partial{r}}$ and $\frac{\partial{x}}{\partial{\theta}}$ that concern basis vectors both in the <code>Cartesian</code> and in the <code>Polar</code> coordinate systems.</p>
<p>Hope that someone can shed some light on this.</p> | g8476 | [
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<p>When I first studied friction I faced f = $\mu$ N where f is force of friction, $\mu$ is coefficient of friction for the surface considered and N is the normal force for the body on surface. Now f is a force so it must be a vector which means it has a direction. N is normal force so it has direction too. But both f and N are mutually perpendicular to each other! And $\mu$ is a constant, so how can we write f and N in one equation? They do not have the same direction!</p>
<p>I think this is not clear to me because I have never read about the origin of this equation. From where do we get it?</p> | g8477 | [
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<p>Can you explain, please, <strong>step-by-step</strong> how an <a href="http://en.wikipedia.org/wiki/Electronic_filter" rel="nofollow">electronic filter</a> does work?
For example, high pass filter. I know It's a trivial things, but I can't get it completely. Don't bring me formula and etc. Just explanation in three words.</p> | g8478 | [
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] |
<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/2219/beginner-physics-resources">Beginner Physics Resources?</a><br>
<a href="http://physics.stackexchange.com/questions/12175/book-recommendations">Book recommendations</a> </p>
</blockquote>
<p>I'm wanting to learn physics. In fact i'm a software engineer and i find physics very fascinating. But everything I read things concerning physics, I don't understand many things. What are the books that can be helpful for someone who don't know physics but have a good foundation of mathematics??</p> | g98 | [
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<p>When considering the 'Heisenberg' picture of the harmonic oscillator, I've come across the step:
$$\begin{align}
\left\langle n\left|(\hat{q_H}\hat{H}-\hat{H}\hat{q_H})\right|k\right\rangle &= (E_k-E_n)q_{nk}
\end{align}$$</p>
<p>where $\hat{q_H}$ is a position operator. I realise that this must be a pretty simple step for someone who is familiar with Dirac notation, but I'm not sure how the author has <em>unlocked</em> the $\hat{H}$ from the $\hat{H}\hat{q_H}$?</p>
<p>Any comments greatly appreciated as always.</p> | g8479 | [
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<p>From Wikipedia (http://en.wikipedia.org/wiki/Water_electrolysis#Efficiency):</p>
<blockquote>
<p>The electrolysis of water requires a minimum of 237.13 kJ of electrical energy input to dissociate each mole.</p>
</blockquote>
<p>Each mole of water gives you 2 grams of Hydrogen and 16 grams of Oxygen (http://www.lenntech.com/calculators/molecular/molecular-weight-calculator.htm).</p>
<p>The energy density of the Hydrogen is 141.86 MJ/kg (http://en.wikipedia.org/wiki/Energy_density#Energy_densities_ignoring_external_components).</p>
<p>Calculation for 1 kg of water (55.55 moles):</p>
<blockquote>
<p>Energy for electrolysis: 237.13 kJ * 55.55 = 13.173 MJ</p>
<p>Energy released by Hydrogen combustion: 0.002 * 55.55 * 141.86 MJ = 15.76 MJ</p>
</blockquote>
<p>These calculations are not taking in account efficiency and energy loses, they are purely theoretical.</p>
<p>In various Wikipedia articles there are claims regarding to electrolysis similar to following:</p>
<blockquote>
<p>The energy required to generate the oxyhydrogen always exceeds the energy released by combusting it.</p>
<p>Electrolysis-based designs have repeatedly failed efficiency tests and <strong>contradict widely accepted laws of thermodynamics (i.e. conservation of energy)</strong></p>
</blockquote>
<p><strong>First Question:</strong> In theory (in practice we always have less efficiency and must take those in account), is there anything wrong with my calculation?</p>
<p><strong>Second Question:</strong> Can someone clarify to me the claims about laws of thermodynamics and conservation of energy - I do not see ANY CONNECTION between energy needed to electrolyze water and energy released by hydrogen combustion?</p> | g8480 | [
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<p>Static electric fields are supposed to be conservative in nature and therefore give $0J$ work if traversed over a loop. However in the following problem I got non zero work by a static electric field. How can I explain this work ?</p>
<p><img src="https://lh4.googleusercontent.com/-S3oWGGy_30Y/UuLvVIXZnFI/AAAAAAAAAhE/kRutOGK9Q88/s1668-Ut/2014+-+2" alt="enter image description here"><br>
<strong>Note :</strong> This is done in field of line charge, diagram shows a cuboid type structure but that is just for differentiating between traversed path and line charge.</p>
<p>I am writing some of the important parts/equations here, but I have written the whole derivation <a href="http://thinklo.blogspot.in/2014/01/non-conservative-behaviour-of-static.html?m=0" rel="nofollow">here</a>.
$$W_{AB} = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(L)^2+(r_1)^2}+L)(r_1)}{(\sqrt{(\frac{L}{2})^2+(r_1)^2}+\frac{L}{2})^2}|]$$</p>
<p>$$W_{BC} = \frac{q\lambda}{4\pi\epsilon_0}[ln|\frac{(\sqrt{(\frac{L}{2})^2+(r_1)^2}+(\frac{L}{2}))(r_2)}{(\sqrt{(\frac{L}{2})^2+(r_2)^2}+(\frac{L}{2}))(r_1)}|]$$</p>
<p>$$W_{CD} = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(\frac{L}{2})^2+(r_2)^2}+\frac{L}{2})^2}{(\sqrt{(L)^2+(r_2)^2}+L)(r_2)}|]$$</p>
<p>$$W_{DA} = \frac{q\lambda}{4\pi\epsilon_0}[ln|\frac{(\sqrt{(L)^2+(r_2)^2}+(L))(r_1)}{(\sqrt{(L)^2+(r_1)^2}+(L))(r_2)}|]$$</p>
<p>$$W=W_{AB}+W_{BC}+W_{CD}+W_{DA}$$</p>
<p>$$W = \frac{q\lambda}{4\pi\epsilon_0} [ln|\frac{(\sqrt{(\frac{L}{2r_2})^2+1}+\frac{L}{2r_2})}{(\sqrt{(\frac{L}{2r_1})^2+1}+\frac{L}{2r_1})}|]$$</p>
<p>If field were to behave conservatively $W=0$ must have been satisfied, but unless $r_1 = r_2$, $W \neq 0$. Thus field is not behaving conservatively.</p>
<p>PS : This is not a homework question ! I am trying to explain the behaviour.</p> | g8481 | [
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<p>The question is to finde the initial horizontal velocity of the ball at end of the ramp, where it is released.</p>
<p>I know how to do this using gravitational potential energy and kinetic energy ($v=\sqrt{2gh}$), assuming all potential energy is converted into kinetic energy but the question is asking me to find the error in the experiment.</p>
<p>It gives me the values on $y$ as the ramp is lifted up (it is lifted up about 10 cm each time) and the corresponding values of $x^2$. I drew a graph of $x^2$ vs $y$ and found the gradient.</p>
<p>How can I use this to find the experimental initial horizontal velocity?</p>
<p>This sketch shows what the experiment looks like:
<img src="http://i.stack.imgur.com/RfGvy.png" alt="Experiment"></p> | g8482 | [
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<p>White light is always said to contain all the different wavelengths of light. Why, then, can we 'make' new colors simply by adding wavelengths? Is it just a matter of our perception, that, when two colors are added together, they appear to be the wavelength of another color. Or are the waves actually forming into another, differently 'waving' wave, with a different wavelength? </p> | g8483 | [
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<p>For the same speed and inclination, is the physics acting on the body equivalent? Lets assume the person keeps pace with something that moves perfectly uniformly.</p> | g8484 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/21851/prove-that-negative-absolute-temperatures-are-actually-hotter-than-positive-abso">Prove that negative absolute temperatures are actually hotter than positive absolute temperatures</a><br>
<a href="http://physics.stackexchange.com/questions/28601/proof-of-existence-of-lowest-temperature-0-k">Proof of existence of lowest temperature $0 K$</a> </p>
</blockquote>
<p>On the Kelvin temperature scale, zero is the lowest possible temperature. Why there is no defined negative temperature?</p> | g278 | [
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<p>I'm an aspiring physicist who wants to self study some Quantum Physics. My thirst for knowledge is unquenchable and I can not wait 2 more years until I get my first quantum physics class in university, so I want to start with a self study. I am enrolled in a grammar school and the most gifted among the gifted (not my description, mind you, I hate coming off as cocky, sorry) are enrolled in a special 'project'. We are allowed to take 3 school hours a week off in order to work on a project, which can be about anything you want, from music to mathematics. On the 4th of April we have to present our projects. Last year an acquaintance of mine did it about university level mathematics, so I thought, why not do it about university level physics? It is now the 3rd of October so I have half a year. My question is, where can I conduct a self study of quantum physics? Starting from scratch? And is it possible for me to be able to use and understand the Schrödinger equation by April? What are good books, sites, etc. that can help me? My goal is to have a working knowledge of BASIC quantum physics and I would like to understand and be able to use the Schrödinger equation. Is this possible? What is needed for these goals? </p> | g1004 | [
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<p>I'd like answers both in the more intuitive side an on the more precise side. </p>
<p>Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ?</p>
<p>I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?</p> | g8485 | [
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<p>I was trying to find out the cause behind the Explosion at petrol pump due to the Mobile phones but eventually it turns out it is not Radio Frequency which may cause the explosion rather it is totally the static electricity which may cause serious explosion and ignition of vapors,
The other reason I found is, due to the spark by the mobile phone battery in case the mobile has fallen out of your pocket and it seems more legit.
The second scenario, Its been said that Electromagnetic Waves absorb in liquid across a wide wavelength , is there any possibility that the absorption of radio Frequency in Petrol may cause any explosion ?</p>
<p>My question is,
anything can produce Static electricity then why mobile phones are prohibited ? Do they produce more Static Electricity than other equipments ? </p> | g8486 | [
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] |
<p>How does <a href="http://en.wikipedia.org/wiki/Annihilation" rel="nofollow">annihilation</a> work? I'm wondering why matter and antimatter actually annihilates if they come into contact. What exactly happens? Is that a known process? Is it just because of their different charges? Then what about neutrons and anti-neutrons? I guess that would be about their quarks, but wouldn't it be very unlikely what all of their individual quarks would actually touch the anti-quarks of the other (anti-)neutron at exactly the same time?</p> | g935 | [
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<p>I recently heard about the possibility of the creation of a "baby" universe or "pocket" universe from our space time,the theory says that concentrating enough mass into a tiny volume of space (like in black holes) a pocket of false vacuum would form e perhaps it'll continue to expand following it's own inflation period and go on forever completly disconnected from our space time without influencing it any way. Now my question is : supposing that we live in a metastable vacuum we know that true vacuum bubble could form via quantum tunnelling and begin to expand into our space time where these false vacuum bubble (or pocket/baby universe) are supposed to go and expand ?</p> | g8487 | [
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<p>It seems impossible for an ice cube to contain bubbles. the ice should freeze far too slowly for a bubble to get trapped within it. However, virtually all ice (that I have seen in my freezer) contains gas bubbles. How is this possible? </p> | g279 | [
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<p>In our quantum optics lecture we derived the formula for the <a href="http://en.wikipedia.org/wiki/Casimir_effect" rel="nofollow">Casimir force</a> between two plates (or at least the force per unit surface, since for two infinite plates we would have an infinite force). It was then said, that in experiments one often uses <a href="http://www.google.com/search?q=Casimir+plate+sphere" rel="nofollow">a sphere and a plate</a> to measure the force, where the small distance between the sphere and the plate is varied by a piezo.</p>
<p>I'm not particularly interested in how exactly the force is measured but rather how to theoretically calculate it in this case. Our professor said, the measured forces agree with the theory up to 1% so I guess, there is a way to calculate it.
Let's say I have a sphere of radius $R$ in a distance $d$ from a plate where $R\gg d$. How can I theoretically calculate the Casimir force between a plate and a sphere?</p> | g8488 | [
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<p>What what I understand, hydrostatic pressure is the "weight" of the water pushing against objects. But if this is true, why is hydrostatic pressure perpendicular to the surface it acts on instead of always going down?</p>
<p>For example, if you placed a book on a desk, the book's weight would push against the desk, but gravity is pulling it "down". But if you put another book beside this book the first book wouldn't apply any force on it. The weight exerted from the book always <em>goes down</em>.</p>
<p>Similarly, why doesn't hydrostatic pressure always go down? I understand that if you put a plate on the seabed, the weight of the water would push down on it, but if you just had a vertical plate standing on the bed, why would force push it from the sides?</p>
<p>To sum up, why does hydrostatic pressure act perpendicular to the object instead of always down?</p> | g8489 | [
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<p>Actually I am not a physics student but I have to give a lecture about Fluorescence lifetime imaging(FLIM). I am thinking of comparing this techniques to another imaging techniques.
Does anyone know what is the best imaging techniques that is comparable to FLIM?
Thank you</p>
<p>Soha</p> | g8490 | [
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] |
<ol>
<li><p>What is the precise definition of <a href="http://en.wikipedia.org/wiki/Unified_field_theory">unification of fields</a> (in classical and quantum mechanics)?</p></li>
<li><p>In general, does unification of a field mean that we can write both of them at both sides of an equation (like Maxwell's laws)? Or does it mean that one of them can produce the other (like $E$ and $B$)?</p></li>
<li><p>Is there any intuitive explanation of how <a href="http://en.wikipedia.org/wiki/Electroweak_interaction">electroweak</a> unification works? Like an electric charge will feel a weak field or that a flavoured particle will produce a weak field?</p></li>
</ol> | g8491 | [
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<p>I've spent some time reading wiki etc. What I get now is that apart from the normal light amplitude information, <a href="http://en.wikipedia.org/wiki/Holography" rel="nofollow">holograms</a> also record the phase information of light. But this is so difficult for me to understand. In my understanding to display the 3D objects you have to reconstruct the 3D object field; the light source and exact light field at that moment, what does the phase mean here? why phase alone makes all these happen?</p>
<p>Apart from the principles, what is restraining holography display from being popular at the moment? </p> | g8492 | [
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<p>I am a physics enthusiast and I have a question: Why is it meaningless to express the '$\pm$' (standard deviation) value as a percentage?</p> | g8493 | [
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<p>let $|\psi\rangle = a_0|0\rangle + a_1|1\rangle \in H$. Show that there are 3 real $r_x,r_y,r_z$ s.t. $$|\psi\rangle \langle\psi|= \frac{1}{2}(I + r_x\sigma_x + r_y\sigma_y + r_z\sigma_z.)$$</p>
<p>Any suggestions on where to start with this?</p> | g280 | [
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<p>I have an idea of making a tennis serving machine. I will briefly describe what it is:</p>
<ul>
<li>The machine is configured to serve the ball at a fixed speed to the center of the left (or right) service court</li>
<li>Based on the rotation in 3 directions (X,Y,Z) of the ball, the machine would let the ball land in various positions in the left service court.</li>
</ul>
<p>How can I determine the relationship between the angular velocity of the ball and the location where it lands?</p> | g8494 | [
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<p>What really is <a href="http://en.wikipedia.org/wiki/Reflection_%28physics%29" rel="nofollow">reflection</a>? Is it just the reemission of EMR? I asked my teacher, he said in quantum sense, it is true. But when I read something about emissivity in Stefan Boltzmann's equation, it states that if an object reflects most light, then it absorbs and reemits less. So what it said means reflection and absorption and reemission are different concepts.</p> | g92 | [
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<p>The tensor of inertia of a solid sphere is $I_{ii}=\frac{2}{5}MR^2$ about an axis passing through its CM. Why would the tensor of inertia of each hemisphere about that axis be $I_{ii}=\frac{2}{5}mR^2$, where $m=\frac{M}{2}$ is the mass of the hemisphere? Is it because the tensor of inertia of each hemisphere would be expected to be $I_{ii}=\frac{1}{5}MR^2$, but then $M = 2m$, hence $I_{ii}=\frac{2}{5}mR^2$?</p> | g8495 | [
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<p>The escape velocity of Earth is $v=\sqrt{\frac {2GM}{R}}$, where $M$ is the mass of the Earth and $R$ it's radius (approximating it as a sphere), and is much less than light speed $c$.</p>
<p>But I want to know the escape velocity of black holes. Is it much more than light speed?</p> | g281 | [
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<p>First, I would like to apology for my lousy English used here, as being a non-native english speaker, </p>
<p>I am currently using Classical Mechanics as a tool to learn how to construct a theory as an end. Therefore, I have a few questions, and a few reasonings behind them (I only consider reality, not imagined world), and I would like your suggestions or corrections :D </p>
<p>To make things simple, I first consider when mass is a constant, so F=dp/dt can be written as F=ma.</p>
<p>Why F=ma? Instead of, say, I define new constants or variables, and claim</p>
<p>1.) $I=N \frac{d^3x}{dt^3} $ ? </p>
<p>Or using the same F both literally and physically, and claim
2.) $F=N(x,\dot x,\ddot x,...x^{(n)};t) \frac{d^3x}{dt^3} $ or </p>
<p>3.) $F=Q \frac{dx}{dt} $?</p>
<p>I mean, force is not something that you can touch nor see, either is mass;
so, we pretty much define them via indirect observations, recording the data, and then comparing with other objects, in order to give operational definitions on mass and force (in case of "mass", we will have to define a unit mass first, and then use F= ma as operational definition for mass.). </p>
<p>Under this line of reasoning,
So why can't we work things in a simple way - $I=Q \dot x $? Yes, we can indeed. </p>
<p>for question 1.) given reality: A falling object a=g:</p>
<p>$I=Q \frac{dx}{dt}$</p>
<p>$\dot x=gt$ </p>
<p>$I=t$</p>
<p>$Q=1/g$</p>
<p>So if I define such way $I=Q \dot x $, it is just the law of falling.
So my answer to my 1st question is:
Yes, we can define things such way, but it has nothing to do with the subject we are dealing right now - inertia. Lesson learned: Know what you want to deal with.</p>
<p>For question 2.) $F=N(x,\dot x,\ddot x,...x^{(n)};t) \frac{d^3x}{dt^3} $</p>
<p>Since I only consider our reality, that means F=ma is truth.
So I will comparing F=Na' with F=ma with a few cases (constant a, and SHM)
And it turns out that F=Na' is far more complicated (often a variable) and N is going to be ill defined given constant a.</p>
<p>Reality: A falling object a=g:</p>
<p>$F=N\dot a=0
=>N\dot a=mg
=>N=mg/\dot a$.</p>
<p>as $\dot a=0$, N(x,t) is ill defined.</p>
<p>Reality: SHM:</p>
<p>$ m\ddot x=-Ax
=>\dddot x= -A\dot x/m
$.
Given
$ Na'=ma
=> N= ma/a'=-m^2\ddot x/\dot x A=(-m^2/A)(d\dot x/dx)
$</p>
<p>Too complicated. (comparing with the fact that m is just a constant.)</p>
<p>Lesson learned:
1.) best work for most of the cases. 2.)When a constant will do, don't use a function. 3.)Theory is about productivity.</p>
<p>For question 3.) $F=Q \frac{dx}{dt} $
or any other orders,
actually, I am just asking: </p>
<p>for $ F= \frac {d^nR(W,\dot W,...,W^{(n)},x,\dot x,...,x^{(n)};t)}{dt^n}= \frac {dP(m,\dot m,x,\dot x;t)}{dt}
$</p>
<p>4.) is there exist a n, such that there exist a R makes things simpler than $F=\dot P$ ? Define $R(W,x;t)=W(x,\dot x,\ddot x,...x^{(n)};t) x$, (R, W new variables, x is position function of time)</p>
<p>if there doesn't exist such an n, then
5.) why F= ma turns out to be the simplest way to work things out?</p>
<p>And thanks for your reading. Please give me some criticisms to my reasonings. </p>
<p>p.s.
I think I am making the reinvent wheel mistake.
I hope these fancy mathematical philosophy will end up some educational purpose on physics. </p> | g8496 | [
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<p>I always think that it is not possible to escape from within event horizon. However, some one recently told me with deep conviction that it <em>is</em> possible with sustained energy output. I countered with calculations in Schwarzschild metric showing that any objects below event horizon in the simplistic black hole will hit the singularity within a finite proper time, and was met with the response that Schwarzschild metric was a bad choice for coordinate system.</p>
<p>Now I understand that the singularity at $r=\frac{2GM}{c^2}$ in Schwarzschild metric is an artifact of the coordinate choice, and thus it seems that he could have a point. So my question is, can one prove that it is (or not) possible to escape from within the event horizon of a simple static black hole using a better metric?</p> | g8497 | [
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<p>I'm a little unclear as to how <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" rel="nofollow">QFT</a> differs from Elementary <a href="http://en.wikipedia.org/wiki/Particle_physics" rel="nofollow">particle physics</a>. They both use pictorials of Feynman graphs, is it that Elementary particle physics assumes the point particle perspective, while QFT treats them as fields?</p> | g8498 | [
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<p>I'm afraid this question is a little too open-ended, but bear with me while I find a better formulation.</p>
<p>carbon allotropes (like fullerenes and graphene) are regular patterned. Conduction bands of electrons in graphene in particular, have energy-momentum relationships that make them behave as massless carriers (sticking to what is called a Dirac cone). More interestingly the propagation dynamics seem to be at first order, direction independent.</p>
<p>Given the above facts, is possible that the interactions of carriers on such 2D sheet is describable by the limit of some Conformal Field Theory. If that were to be the case, and I have a big closed surface of graphene material, what does the holographic principle will say about the fields in the volume bounded by the surface? What is the equivalent of the Gravitational Theory in that volume?</p> | g8499 | [
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<p>What are some good lecture notes to learn quantum field theory from, available online?</p> | g282 | [
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<p>I have a top of unknown mass that has a moment of inertia $I=4\times 10^{-7} kg \cdot m^2$. A string is wrapped around the top and pulls it so that its tension is kept at 5.57 N for a distance of .8 m.</p>
<p>Could somebody help me derive some equations to help with this? Or to get me in the right direction? I have been trying to derive some sort of equations from $E=\frac{I \cdot \omega ^{2}}{2}$ but I cant get anywhere without ending up at radius = radius or mass = mass.</p>
<p>I need the final angular velocity.</p> | g894 | [
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<p>Where/what is emitting cosmic background radiation, and when did it come into existance, was immediately after the big bang?</p>
<p>I know that the universe isn't 3D in the traditional sense, and I don't pretend to understand it. However if the big bang originated from a single point, and the cosmic radiation was emitted from that point at the very start, then unless he explosion/expansion/inflation (whatever happened) wasn't faster the speed of light, then the CMB would have reached the edge of the universe and gotten out of it.</p> | g8500 | [
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<p>Recently I got the book "On the Shoulders of Giant" from Stephen Hawkings. It consists of more than 1000 pages of classical publications in physics. However 900 pages are given to the work of Copernicus, Galilei and Newton and 100 pages are given to the papers of Einstein. Therefore I think this books gives a wrong impression about what is really important for our knowledge in physics today. </p>
<p>If you could fill a book with the most important papers relevant for our knowledge in physics today, which ones would you select? </p> | g8501 | [
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<p>So I know that conduction is transfer of thermal energy by direct contact via molecular collisions. </p>
<p>For conduction, $Q/t$ is the rate of heat flow and is heat current ($I$). My textbook says that the rate of heat flow is constant across any number of slabs between 2 heat reservoirs, and I understand why this is so because if I use electric current as an example, the current in constant through resistors of different ohms in a series circuit too.</p>
<p>But then, the textbook goes on to show how conservation of energy also explains why. It says that "If the rate of energy transfer were not steady across all slabs, the slab that conducted heat the fastest would become cold. A cold slab could not conduct heat to warmer slabs because heat moves from hot to cold."</p>
<p>I don't understand their explanation that they offered from a conservation of energy standpoint. Could you please clarify this for me?
Why would the slab that conducted heat the fastest become cold if the rate of energy transfer were not steady?</p>
<p>For your reference, the equation for conduction is:
$$\frac{Q}{t} = kA\frac{T[\text{hot}] – T[\text{cold}]}{L}$$</p> | g8502 | [
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<p>Why sometimes we remember that "classical" lagrangians of fermions are constructed from grassmann numbers, while sometimes don't? </p>
<p>For example, for Majorana's field in terms of 2-component spinors (you can read about this representation <a href="http://www.mdpi.com/2073-8994/5/4/271" rel="nofollow">here</a> in details),
$$
L = \bar{\psi}(\alpha^{\mu}\partial_{\nu} - m)\psi , \quad \alpha^{\mu} = (\sigma_{2}K, \sigma_{1}, \sigma_{2}, \sigma_{3}), \quad K\psi = \psi^{*}, \quad \bar{\psi} = -\psi^{\dagger}\alpha_{0},
$$
and the mass term become
$$
\bar{\psi} \psi = -\psi^{\dagger}\sigma_{2}\psi^{*} = -\begin{pmatrix}\psi_{1}^{*} & \psi_{2}^{*}\end{pmatrix}\sigma_{2}\begin{pmatrix} \psi_{1}^{*} \\ \psi_{2}^{*}\end{pmatrix} = 0
$$
if the fields are "classical". So we must assume their grassmanian nature before making all calculations (there this expression won't be equal to zero).</p>
<p>But when we assume the mass matrix for see-saw type 1 model, we introduce the term
$$
-M_{ij}(\psi_{R}^{T})^{i}\hat{C}\psi_{R}^{j} + h.c., \quad M_{ij} = M_{ji}, \quad \hat{C} = i\gamma_{2}\gamma_{0}.
$$
If the fields are grassmann numbers, this expression is equal to zero (don't forget about symmetric mass matrix). So we forget about grassmannian nature of the fields even in "classical" level.</p>
<p>Here "classical" means that we set all of anticommutators to zero. If we set them not to zero, we will have new problems, but it's the other story.</p> | g8503 | [
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<p>What is the modern mainstream way to deal with the <a href="http://en.wikipedia.org/wiki/Measurement_problem" rel="nofollow">measurement problem</a>? For this question, the measurement problem is the problem of formalizing wave function collapse when the observer is part of the model (say I want to simulate an observer+observee quantum system on a computer). I am aware of proposed solutions: Everett, Bohm, and decoherence (though decoherence apparently does not explain collapse). How does mainstream physics deal with this issue?</p>
<p>For example: Given two electrons initially isolated from each other and the universe which then interact , can each electron be considered an observer? With no-collapse theories each electron sees the other election's wave function collapse relatively. If the mainstream view holds that a single electron cannot be an observer what is the criterion for an observer?</p> | g8504 | [
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<p>This might be a stupid question, but nonetheless, it has been bothering me.</p>
<p>If you take a photon, make it go through some atoms in a solid, liquid or whatever, then you have the chance of this photon being absorbed by an electron, and thereby exciting the electron. This requires the photon to have enough energy to actually excite the electron to another energy level.</p>
<p>My question is then: How does the photon know if it has enough energy or not? Do they interact very quickly to determine if it's okay or not, or is it just something it "knows?"</p> | g723 | [
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<p>I started studying the BMS symmetry in connection with the paper:
<a href="http://arxiv.org/abs/1312.2229" rel="nofollow">http://arxiv.org/abs/1312.2229</a>
and there are a few strange things I noticed.</p>
<p>First of all, from reading the original papers by Bondi, Metzner and Sachs, I know that the "BMS symmetry" is just an allowed subset of coordinate diffeomorphisms which leaves the asymptotic flatness of the space-time intact. However, when I read the paper above, the BMS symmetry is stated in form of a vector field eq. (2.10) and (2.14). Therefore, my first question:</p>
<blockquote>
<p>How do I obtain from a given subset of coordinate diffeomorphisms (which is being considered a symmetry) a corresponding vector field a'la Killing vector?</p>
</blockquote>
<p>Furthermore, later in the paper they go from the BMS vector fields to generators of BMS symmetry in eq. (3.3). They do not mention how to do it, so I'd like to know:</p>
<blockquote>
<p>How do I go from vector fields characterizing a symmetry to an actual generator of the symmetry?</p>
</blockquote>
<p>I realize that this is some pretty advanced stuff. I am grateful for any help or suggestion!</p> | g8505 | [
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<p>Consider the standard map (also known as <a href="http://www.scholarpedia.org/article/Chirikov_standard_map" rel="nofollow">Chirikov map</a>):
$$
p_{n+1} = p_n + K \sin(\theta_n) \\
\theta_{n+1} = \theta_n + p_{n+1}
$$</p>
<p>I know that the diffusion coefficient according to the Einstein relation is defined through the relation:</p>
<p>$$
\sigma^2 = 2Dt
$$</p>
<p>being $\sigma$ the variance of the position in the case of the $1$-D random walker. Can you figure out why the diffusion coefficient for the standard map is given by:</p>
<p>$$
D = \lim_{n \rightarrow \infty} \frac{<(p_n - p_0)^2>}{n} \quad ?
$$</p>
<p>I mean, the $n$ makes sense, but what about the variance, why this is given just in terms of $p$, what happens with the $x$.</p> | g8506 | [
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<p>Imagine you have a sealed cylindrical vessel with a given radius with a compressed gas inside. Let's give some numbers, 5 cm radius and 100 atm pressure. You poke a hole in the vessel and the gas will flow out, propelling the vessel in the opposite direction with a given force. Intuition tells me that this force depends on the hole size. Suppose the pressure does not change as time goes by, what's the optimal hole size to deliver the best acceleration ? </p> | g8507 | [
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<p>Imagine there's a sphere of radius $R$ which has constant density $\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it.</p>
<p>Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\mathbb R^4$ so that we're on $S^4$.</p>
<p>What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?</p> | g8508 | [
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<p>could exist 2 different Hamiltonians $ H1 $ and $ H2$ so</p>
<p>a) The SEMICLASSICAL eigenvalue staircase is equal i mean the expression $$ \oint _{C} p.dq = n+\alpha $$</p>
<p>but the real eigenvalues $ E_{n} (1,n) $ and $ E_{n}(2,n) $ are completely different</p> | g8509 | [
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] |
<p>I am studying the transition from the second excited electronic state of molecular oxygen, $b^1\Sigma_g^+$ , to the ground state, $X^3\Sigma_g^-$. I know that the ground state has total angular momentum $J=1$, total spin $S=1$, and three spin sublevels $(m_s=-1,0,+1)$. The upper state has $J=0$, $S=0$, and one sublevel with $m_s=0$. I am specifically interested in the transition from the upper state to the ground state $m=\pm1$ level and I will refer to it as the $b-X,1$ transition. </p>
<p>I would like to understand the the selection rules that govern this transition and the language that is used to describe this transition. </p>
<p>Here is what I know so far about this transition.</p>
<ul>
<li><p><a href="http://campus.udayton.edu/~physics/rjb/MRSOxygenI.pdf" rel="nofollow">Brecha, Pedrotti, Krause, "Magnetic rotation spectroscopy of molecular oxygen with a diode laser" JOSA 1998</a>: They describe this transition as "doubly weak" and I quote, "First, it is a magnetic dipole transition and thus is roughly 5 orders of magnitude weaker than a normal electric dipole transition. Second, the transition is a singlet-triplet intercombination band, making it 3 orders of magnitude weaker still." </p></li>
<li><p><a href="http://pubs.rsc.org/en/content/articlelanding/1997/ft/a607263a" rel="nofollow">Minaev, Agren, 1997</a>: They describe this transition as "magnetic dipolar" and "spin-forbidden." They also say that it is spin-orbit coupling that accounts for this "doubly weak" transition being as large as it is. </p></li>
<li><p>Wikipedia-Selection Rules: Provides the rules for Magnetic Dipole (M1) transitions and discusses Spin-Obrit (LS) Coupling but I do not understand where this came from. </p></li>
<li><p>Sannigrahi, "Derivation of Selection Rules for Magnetic Dipole Transitions," 1982: This is pretty clear but just seems to say that $\Delta m_s=\pm1$. This is true for the $b-X,1$ transition but how do I tie in $L$ and $J$ into the selection rules?</p></li>
</ul>
<p>Most of what I have found on the internet and in textbooks relating to selection rules is directed to electric dipole transitions. Where I have found something discussing magnetic dipole transitions, it is either a summary of the rules or I do not have the required background to understand it. </p>
<ul>
<li><p>What does it mean for a transition to be spin-forbidden?</p></li>
<li><p>What is a singlet-triplet intercombination band? Does this change the selection rules? How is this related to being spin-forbidden?</p></li>
<li><p>What does spin-orbit coupling have with this?</p></li>
</ul>
<p>I look forward to any feedback on this topic. </p>
<h1>Follow up Questions</h1>
<h2>Meaning of Spin-Forbidden</h2>
<p>Does 'spin-forbidden' just mean that the transition from a $J=1$ to $J=0$ state is not allowed because the selection rules for magnetic dipole transitions say that $J$ cannot change? I expected 'spin-forbidden' to imply something about the change of the spin between the initial and final states. </p>
<p>For example, suppose I have a time-dependent perturbation like $V_{md}(t) = \frac{e}{m} \vec{S}\cdot \vec{B}(t)$ and I am interested in the transition rate between the initial state $\left| s m_s \right\rangle$ and the final state $\left| s' m_s' \right\rangle$ with the quantization axis in the z direction. As you pointed out, the transition rate will be proportional to $\left\langle s' m_s'\right| \vec{S} \cdot \vec{B}(t) \left| s m_s \right \rangle$. Now if $\vec{B}(t)$ is circularly polarized, the rules for an allowed transition will be $s'-s=0$ and $m_s'-m_s = \pm1$. For $\vec{B}(t)$ polarized in the z direction, $s'-s=0$ and $m_s'-m_s = 0$ so there is not a transition to other states. I would think that if $s'-s=0$ and $m_s'-m_s = \pm1$ are <strong>not</strong> true (like if $s'=1$ and $s=0$), then the magnetic dipole transition between $\left| s m_s \right\rangle$ and $\left| s' m_s' \right\rangle$ would be called 'spin-forbidden.' </p>
<p>The same arguments could be made for $\vec{L}$ or $\vec{J}$ as I did with $\vec{S}$. Would you also call a magnetic dipole transition between $\left| L=0 \right\rangle$ and $\left| L=1 \right\rangle$ 'spin-forbidden?'</p>
<h2>Spin-Orbit Coupling</h2>
<p>Now the difference in the energies of the singlet $(b)$ and triplet, ground state $(X)$ is $1.63\text{ eV}$. That seems too large to be due to spin-orbit coupling breaking a degeneracy. If I was pretending this was a hydrogen atom, I would say this was like a transition where the principle quantum number $n$ changed. I am not sure how to talk about this in a molecule.</p>
<p>This isn't key to the rest of your explanation but I did want to clarify to make sure we were on the page. </p>
<h2>Singlet-Triplet Intercombination Band</h2>
<p>Do you know what the expression 'singlet-triplet intercombination band' is referring to? After your explanation, it seems to refer to the mixing of the singlet and triplet states due to spin-orbit coupling (SOC). Is this true?</p>
<h2>Mixing of unperturbed states</h2>
<p>How did you know that the perturbed upper state could be written as a combination of the unperturbed upper state and the $M_s=0$ ground state? It makes sense that the states would get mixed up by SOC but I don't know how. If this is a messy explanation, don't worry about it. </p>
<p><strong>I would like to reiterate that your (George G's) explanation has been incredibly helpful. Thank you.</strong></p> | g8510 | [
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<ol>
<li><p>All the linearly independent eigenfunctions of the parity operator $\mathcal{P}$ form an infinite set and all the linearly independent eigenfunctions of the unit operator $\bf 1$ also form an infinite set. Is the cardinality of these two sets same or different? </p></li>
<li><p>Every function is an eigenfunction of the unit operator $\bf 1$. Right? Can any arbitrary function be written as the linear combination of the eigenfunctions of $\mathcal{P}$? If not, doesn't this imply that the set linearly independent eigenfunctions of unit operator $\bf 1$ forms a much bigger set than that of $\mathcal{P}$?</p></li>
</ol>
<p><strong>EDIT:</strong> I just want to know since $[\mathcal{P},\bf{1}]=0$, whether the total number of linearly independent eigenvectors of $\mathcal{P}$ and $\bf{1}$, forms a basis i.e., do they have same number of linearly independent eigenvectors and and whether both spans the space? </p> | g8511 | [
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<p>I am using <a href="http://chipmunk-physics.net/" rel="nofollow">Chipmunk</a>'s physics engine to simulate 2D classical mechanics of solid polygons (excellent by the way!). The only way you can alter the spontaneous behaviour of the the simulated objects is to apply <a href="http://chipmunk-physics.net/release/ChipmunkLatest-Docs/#cpBody-Forces" rel="nofollow">forces and impulses</a> to them.</p>
<p>I am given a certain amount $E$ of energy to spend (= to input into the system) over a certain period of time $\Delta T$, that I need to use to alter the motion of a given body. By the time I need to do so, it is rather difficult to access its mass, instantaneous speed, kinetic moment, energy, or other forces applied to it.. </p>
<p>(I can still track its shape, position and rotation.)</p>
<p>However, I feel like I don't need to access all these to actually spend this energy and alter its motion (with the only tools I am given) provided I decide I will do it by applying a single force $F$ to it over the period $\Delta T$, and provided I give myself an application point $P$ and a direction $d$ (both relative to the body's actual position and rotation).</p>
<hr>
<p>To be more precise about "my feeling": no matter the actual situation of the body, I will apply this force and spend this energy over $\Delta T$:</p>
<ul>
<li>if it turns out to be a very heavy object, then it won't move a lot</li>
<li>if it turns out to be in a super-fast race against $d$, then its speed won't change a lot</li>
<li>if it turns out to be blocked under a stone, then it won't move a lot</li>
<li>if it turns out to be free, light, resting.. then the change will be visible..</li>
</ul>
<p>(I am aware that the concept of <em>force</em> might be the wrong one to consider here, so I just want to make sure we'll agree about what I am talking about, whatever it might be. It is exactly like this body had a certain amount of energy to spend and used it to <em>move itself</em> without actually knowing anything about the world around.)</p>
<hr>
<p>If I was right about this feeling that I do have everything I need to perform this alteration without spending more than $E$, then how would I compute the actual intensity of $F$ to be applied?</p>
<p>If I was wrong, then what information am I missing?</p>
<hr>
<p>[EDIT]: It turns out from the discussion that I actually need to give myself a certain <em>cost of applying a force</em> $\beta\ (m.s^{-1})$, inspired from biophysics. Then the actual energy spent is simply $$E = \beta \int_{\Delta t} F(t) \mathrm{d}t,$$ and I just have to take care that it will never be lower than the actual work $W$ done by $F$ over $\Delta t$.</p>
<p>To achieve this, one solution seems to keep constantly track of the actual useful energy spent, which is difficult but.. not undoable.</p> | g8512 | [
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<p>I am trying to find the energy of a wave travelling through a solid material in a 2D Finite Element Method (FEM) - Simulation. </p>
<p>As a general approach I would try to use $E_{kin}=\frac{1}{2}mv^2$ at the highest particle velocity $v$, as this would be the point, where all the energy in the wave is kinetic and potential energy should be zero. So I would know the energy of the wave</p>
<p>But what would I use for mass? Since it is a 2D simulation, I image that, as the third dimension is infinitely small, there is no mass. Now I remember seeing in a paper once (sorry, I don't know the source anymore), that people used density $\rho$ instead of mass in their 2D FEM simulations.</p>
<p>My questions have 2 parts:</p>
<ul>
<li>Is using density instead of mass justifiable? If not, what other approach could someone point me to?</li>
<li>The units would not match up anymore. So if it was ok to use density how would I use it.</li>
</ul> | g8513 | [
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<p>If I stick a metal and insulator together, will the Fermi level of the metal align with the insulator?</p>
<p>When people draw a band diagram for a metal-oxide-semiconductor heterostructure, I never see them label the Fermi level of the oxide. What is the Fermi level of the oxide relative to the metal and insulator's?</p>
<p>I'm particularly interested in graphene/SiO2/Si field effect transistors. When we change the gate voltage between the graphene and Si, what happens to the SiO2 conduction band, valence band, and Fermi level? If the SiO2 Fermi level aligns with the graphene Fermi level, what is the mechanism that keeps graphene and SiO2 in thermal equilibrium so quickly?</p> | g8514 | [
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<p>The standard lore in relativistic QFT is that poles appearing on the real-axis in momentum-space Green's functions correspond to particles, with the position of the pole yielding the invariant mass of that particle. <em>(Here, I disregard complications tied to gauge-fixing and unphysical ghosts)</em></p>
<p>Schematically, I take this to mean:</p>
<p>$$\text{one particle state} \Leftrightarrow \text{pole on real axis}$$</p>
<p>It is easy to show the correspondence going $\text{particle} \Rightarrow \text{pole}
$. This is done in many textbooks, <em>e.g.</em> Peskin and Schroeder's text, where a complete set of states diagonalizing the field theoretic Hamiltonian is inserted into a $n$-point correlator, and a pole is shown to emerge.</p>
<p>However, how does one make the argument going the other way $\text{particle} \Leftarrow \text{pole}
$? That is, the appearance of a pole in a Green's function indicates an eigenstate of the Hamiltonian, with the eigenvalue corresponding to the particle energy?</p> | g8515 | [
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<p>Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group:</p>
<ul>
<li><strong>Local gauge group $G$</strong>. Corresponds to the fibers of the $G$-bundle. Local gauge transformations correspond to the change of coordinates in which fields and form are written in trivializations. In other words, as far as I understand, local gauge transformations = transition functions for trivializations of the principal bundle and the associated bundles.</li>
<li><strong>Global gauge group $\mathfrak G=Aut(P)$</strong>. Is the group of principal bundle diffeomorphisms, and is claimed to be much bigger then $G$.</li>
</ul>
<p>I would like to understand which notions in physics correspond to these two, apparently very distinct, notions.</p>
<p>First, in physics, there is the notion of <em>global symmetry</em>, as far as I understand this has nothing to do with gauge/bundles. But on the other hand, the limit of a gauge symmetry where the parameter becomes constant is a global symmetry. Typical $U(1)$ example: $\exp(i\alpha(x))$ = $U(1)$ gauge transformation, but if $\alpha(x)=\alpha=const.$, then it is a global symmetry. Is there any deeper connection between global symmetry and gauge "symmetry"/bundles?</p>
<p>Second, I would appreciate some examples which clarify the distinction between local and global gauge groups mentioned above. If, for example we take a prototype $SU(2)$ gauge theory of the form
$$
\mathcal L = -\frac{1}{4}(F_{\mu\nu}^a)^2+|(D_\mu\phi)^a|^2-V(\phi)
$$
then we can put it in the bundle picture as a $SU(2)$ principal bundle, the fields $A_\mu^a$ living in the associated bundle $$(P\times\mathfrak su(2))/\{(p,A)\sim(pg,Ad_{g^{-1}}A)\}$$ and the doublet field $\phi$ in fundamental representation of $SU(2)$ living in the associated bundle $$(P\times\mathbb C^2)/\{(p,v)\sim(pg,\rho(g^{-1})\phi)\}.$$ So, the $SU(2)$ group which we are referring to here is what was the local gauge group $G$ above, is this correct? The gauge transformations apply to coordinate versions of the fields in the trivializations of the associated bundles to change between different coordinates.</p>
<p>Now if, this is the case, what is the global gauge group $\mathfrak G$ then? In particular I am seeking the answer for the following questions:</p>
<ul>
<li>Do global gauge transformations apply to the model $SU(2)$ theory above?</li>
<li>In what sense is the global gauge group much bigger then the local gauge group?</li>
<li>Is there a model example where one can clearly write down the local gauge group and the global gauge group?</li>
<li>Are there cases where the two groups are the same?</li>
<li>How does this discussion relate to quantization of classical fields (if it does at all)?</li>
</ul>
<hr>
<h1>Edit</h1>
<p>I'll post here answers to various questions in the comments/answers.</p>
<p>About the definitions I used in my text. I took them from my lecture notes from a course on mathematical gauge theories taught by a mathematician. Although there is no on-line version of this course, I can provide more detailed definitions if needed.</p>
<p>Specifically, concerning the distinction global/local gauge, I have found this piece of information on nLab, which seems to be agreement with my definition:
<a href="http://ncatlab.org/nlab/show/gauge+group" rel="nofollow">http://ncatlab.org/nlab/show/gauge+group</a>. Although this page provides some explanations, it uses some mathematical terminology which I am not entirely familiar with. The problem both with the nLab page and with the lecture I attended is to make the connection to physics (I am a physicist).</p>
<p>Ok, I'll just write down the definitions from my lecture notes.</p>
<h2>Local gauge</h2>
<p>Let $P\to M$ a principal $G$-bundle, $\omega$ a connection 1-form on $P$. $U_i, U_j\subset M$ such that $U_i\cap U_j\neq\emptyset$ over which P is trivial:
$$
\psi_i:\pi^{-1}(U_i)\to U_i\times G\\
\psi_j:\pi^{-1}(U_j)\to U_j\times G
$$
with transition functions
$$
\psi_i\circ\psi_j^{-1}:(U_i\cap U_j)\times G\to (U_i\cap U_j)\times G\\
\phantom{\psi_i\circ\psi_j^{-1}:XXX}(m,g)\mapsto (m,\psi_{ij}(m)g)
$$
where $\psi_{ij}:U_i\cap U_j\to G$.</p>
<p>Using these trivializations we can construct preferred sections as follows:
$$
\sigma_i:U_i\to\pi^{-1}(U_i)\\
\phantom{\sigma_i:x}m\mapsto \psi_i^{-1}(m,e)\,.
$$
Define
$$
\omega_i:=\sigma_i^*\omega\,,
$$
which is a $\mathfrak g$-valued 1-form on $U_i$. Changing between coordinates on $U_i$ and $U_j$ is what is called choice of local gauge. We would like to write down a transformation prescription for $\omega_{i/j}$.</p>
<p>On G we have the canonical 1-form $\theta$ with values in $\mathfrak g$ defined as follows:
$$
\theta_g(X_g)=A\in\mathfrak g\quad\text{if }(A^*)_g=X_g
$$
where by $A^*$ we mean the fundamental vector field on $G$ corresponding to $A$. One can show that the 1-forms transform in the following way:
$$
\omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^*\theta\,.
$$
This looks to me like a gauge transformation of guage fields $A=A_\mu dx^\mu$ in physics where we write
$$
A'=gAg^{-1}+gdg^{-1}
$$
and I even believe that the above transformation for $\omega_j$ can be rewritten as something like this
$$
\omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^{-1}d\psi_{ij}\,,
$$
Then it looks like the transformation for $A$ with $g=\psi_{ij}^{-1}$.</p>
<p>So much about the local gauge transformation.</p>
<h2>Global gauge group</h2>
<p>The group of global gauge transformation $\mathfrak G$ is the set of automorphisms of the principal bundle P:
$$
\mathfrak G = Aut(P)\,.
$$
This must not be confused with the structure group $G$, which is sometimes called gauge group in leterature, but is much smaller.</p>
<p>The (global) gauge group can be described in three ways:</p>
<ol>
<li>$\mathfrak G=\{\phi:P\to P|\phi\text{ is a diffeo., }\pi\circ\phi=\pi, \phi(pg)=\phi(p)g\,\forall g\in G\}$</li>
<li>$\mathfrak G=\{u:P\to G|u\text{ smooth s.t. }u(pg)=g^{-1}u(p)g\,\forall g\in G\}$. Note that $\phi(p)=p\cdot u(p)$.</li>
<li>$\mathfrak G=\{\text{sections }s:M\to F\}$, where $F=(P\times G)/\sim$ with $(p,h)\sim(pg,g^{-1}pg)\forall g\in G$.</li>
</ol>
<p>If $\omega$ is a connection 1-form which corresponds to a choice to the horizontal tangent space $H$ on $P$, then $\phi^*\omega$ is also a connection 1-form, defining the pull-back connection
$$
(\phi^*H)_p:=(D_p\phi)^{-1}H_{\phi(p)}\quad\Leftrightarrow\quad D_p\phi((\phi^*H)_p):=H_{\phi(p)}
$$
We conclude that $\mathfrak G$ acts on connections, but the explicit form of this action if fairly complicated. In contrast, the action of $\mathfrak G$ on the curvature is easy to understand.</p>
<p>From the last equation above we conclude that $\phi$ maps $\phi^*H$ to $H$, and so it maps $\tilde\Omega$, the curvature of $\phi^*H$, to $\Omega$, the curvature of $H$:
$$
\phi(\tilde\Omega(X,Y))=\Omega(X,Y)\quad\forall X,Y\in T_mM\,.
$$
One can show that the following transformation relation holds:
$$
\tilde\Omega=Ad_{u^{-1}}\Omega\,.
$$
where $u:P\to G$ is the map from the second definition of $\mathfrak G$ above corresponding to $\phi$.</p>
<p>Two connections $H_1$ and $H_2$ on P are called gauge equivalent if there is a $\phi\in\mathfrak G$ with $\phi^*H_2=H_1$.</p>
<hr>
<p>One of the things which I am curious about is that from the local gauge transformations the transformation law for the 1-form $\omega$ seems to be what we call the gauge transformation of gauge fields $A$ (see above), but from the global gauge group transformation, the transformation of the curvature is what we see in physics, namely if $F$ is the field strength corresponding to $A$, and $g$ an element of the gauge group then
$$
F\to F'=gFg^{-1}\,.
$$
Also, as I said at the very beginning, I would like to pin down what these mathematical definitions correspond to in what we learn in physics. In physics, when we discuss a certain gauge theory, there is THE gauge group, such as $U(1)$, $SU(2)$, etc. under which fields transform in certain representations, as well as the gauge fields themselves, which transform in the adjoint representation. Now in mathematics I see the distinction local/global gauge group. The essence of my question is do understand this distinction and to relate it to phyics.</p> | g8516 | [
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<p>What happens when the explosive energy of a grenade or a bomb is contained in a container, if you will, and no energy can be released through the container in the form of heat or sound etc...? Just theoretically asking?</p>
<p>I just want to know what happens when we take a huge explosion or any explosion and shrink it to a finite point. Theoretically, what happens?</p> | g8517 | [
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<p>Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" potentials which have only a single discrete eigenvalue. I was wondering, what kind of potential do we need for producing an exponential $e^n$ set of discrete eigenvalues?</p> | g8518 | [
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