title: Compressing a fluid with arbitrarily large force in an infinitely strong chamber

text: Imagine I have an infinitely strong container of volume $v_1$, and I fill it with some monoatomic fluid like liquid hydrogen. I then proceed to compress the walls of the container to reduce its volume by some fraction to $v_2$. How much force/energy is required to reduce the volume of the container to ~99%, ~50%, then perhaps ~0.1% of its original volume? How might one characterize the various states of matter (presumably plasma) inside the container at different ratios of $\frac{v_2}{v_1}$?


title: Heat exchange depending on coolant flow direction

text: Consider a simplest case of a heat exchanger - two parallel pipes of flowing liquids (say, hot and cold) that have physical contact along some part of their length. Hot water of a certain temperature goes from A to B. Cold water can go either from C to D or from D to C. Assume that heat exchange between liquids occurs only where pipes contact (at XY part). What is the favorable direction (meaning "the most heat is transferred) of a coolant flow relative to hot flow - in the same direction (C->D) or in the reverse direction (D->C)? How the coolant and hot flows' temperatures are distributed along the pipe contact? ![enter image description here](http://i.stack.imgur.com/Yr9sq.png)


title: Explain the direction of waves on sea shore

text: Why do waves on the sea shore move towards the shore even when the tide is going out?


title: How does the internuclear repulsion vary in Hydrogenic atom collision?

text: Hydrogen fusion requires two hydrogen nuclei to get close enough (typically a few fm) to fuse. Much of the problem of creating a fusion reactor is overcoming the Coulomb repulsion between a pair of nuclei - the millions of degrees for Maxwellian distributions, the Bremstrahlung losses for inertial confinement. If we could align the paths of two neutral Hydrogen **atoms** (of whichever isotopes), what would the repulsion look like between them as they approach collision? Obviously at long range there is negligable force as both are neutral. But as they approach each other, what happens to the electron distribution? Intuitively, I expect a bonding cloud to form between the nuclei, and antibonding clouds beyond them. This would presumably attract at first until reaching the usual Hydrogen covalent bond length, after which the internuclear repulsion would increasingly dominate. But how does that compare to bare ionic collision? How much lower is the potential barrier? Obviously if it was significantly lower and we could somehow engineer the collision to achieve fusion, the cross section would be larger than ionic fusion, but how much? Or would the barrier be just as high over the final few femtometers?


title: The quantized energy level E depends on which power of n?

text: A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?


title: Depth of sea and point at which waves break

text: Following up on this answer, is the point at which waves break on the sea shore a guide to the depth of the sea at that point? Could it indicate eg hidden rocks? Explain the direction of waves on sea shore


title: What interpretive difference is there between defining a function with or without a differential as a postfactor?

text: I have thought about this and looked for answers for a long time now, but I do not have any name or label for this problem, which is the reason for the long title of this question. I have come across this many times, but I have never found any rationale for this, i.e. a clear interpretation or explanation, so here are a few examples of what I mean. I suspect that they have different answers. I) In e.g. Shapiro and Teukolsy's book _Black Holes, White Dwarfs and Neutron Stars_ , (1983), the Salpeter birthrate function for stars is defined as $\psi_s\mathrm{d}\left(\frac{M}{M_\odot}\right) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{d}\left(\frac{M}{M_\odot}\right)\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$ Questions: why does it matter that those differentials are included? Why make the statement look messier by multiplying both sides of the equation with something that obviously cancels out? If it is just to show what $\psi_s$ depends on, then it makes mre sense to me to define the function like so: $\psi_s(M) = 2\times10^{-12}\left(\frac{M}{M_\odot}\right)^{-2.35}\mathrm{\ stars\ pc}^{-3}\mathrm{\ yr}^{-1}$ What is (or may be) the answer? II) When defining the equation of continuity for brownian motion particles in 1D space, I have a lecture note that reasons in the following way: Given the setup                        |  n(x,t) |              |         |         J(x,t) ---->     ----> J(x + dx, t)              |         |              |         |         ---------------------->              x        x+dx      Where $n(x,t)$ is the number density of particles between positions $x$ and $x + \mathrm{d}x$ and $J(x, t)$ is the flux of these particles at given position $x$ and time $t$. The particle number is conserved, and they are assumed identical and non-interacting with each other. In the notes that I have, the following equation is readily displayed as though it was crystal clear why it was written inthis way: $n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = - (J(x + \mathrm{d}x, t)\mathrm{d}t - J(x, t)\mathrm{d}t)$ or (multiplying the minus into the parenthesis) $n(x, t + \mathrm{d}t)\mathrm{d}x - n(x, t)\mathrm{d}x = J(x, t)\mathrm{d}t - J(x + \mathrm{d}x, t)\mathrm{d}t$ Question: I can not myself justify why it makes sense to multiply with the diferentials as they are here. What is the significance, or rather the line of thought that lies behind the way this equation turns out? I am unsure of my own answer which is only to look at the dimensions of the two functions themselves and then conclude what unit the factor should have on each side in order for the dimensions to make sense. Just to complete the derivation, rewriting the equation by replacing the functions $n(x, t + \mathrm{d}t)$ and $J(x + \mathrm{d}x, t)$ with their Taylor expansion (to first order) gives $n(x, t + \mathrm{d}t) = n(x, t) + \frac{\partial n}{\partial t}\mathrm{d}t$, $J(x + \mathrm{d}x, t) = J(x, t) + \frac{\partial J}{\partial x}\mathrm{d}x$, which, when inserted into the original equation gives $\frac{\partial n}{\partial t}\mathrm{d}t\mathrm{d}x = -\frac{\partial J}{\partial x}\mathrm{d}x\mathrm{d}t$ so producing, again, a shared factor ($\mathrm{d}t\mathrm{d}x$) on both sides of the equation which can just be divided away (right?), and we end with the coninuity equation $\frac{\partial n}{\partial t} = -\frac{\partial J}{\partial x}$ I hope this question makes sense or at least allows you to suggest a label for this kind of problem that I can use to find more information, Or better, help by answering the wuestion here. Cheers.


title: Renormalization beta-function

text: I want to know the other forms of beta-function that make manifest certain properties of renormalization group, for instance dependence on poles/residue and more. If possible can you state a reference, or perhaps a link, or sketch the method.


title: Boltzmann distribution with interaction between particles?

text: First of all, I would like to apologize in advance if I make stupid mistakes. I am a mathematician and I am trying to apply the Boltzmann distribution to places where I am not sure if it is applicable (albeit I have no choice). The situation is: I have a system which consists in a discrete line of $M$ positions in which $N$ elements are distributed with a separation of at least $D$ positions between them. The state of each element should be its position in the line. Finally, (here's the fun part) each position in the line has an associated potential (so putting an element in the position $i$ means spending $\epsilon_{i}$ units of Energy). The usual approach to this problem as far as I have seen is just assigning $p_{i} = e^{-\epsilon_{i}/kT}$, where $p_{i}$ is the probability that there is an element in the position $i$. I don't understand this approach and I am trying to derive that new one, but I am stuck when trying to force the particles to be separated. Any insight or reference would be very much appreciated. Edit: If it is needed, we can also say that the particles might have a velocity (i.e. they can oscillate), but they should not be able to pass through each other.


title: fun computational physics projects

text: I want to learn how to code in Java and am in need of some cool computational physics projects that would provide motivation to do so. I was wondering if you guys could list some medium-level computational physics projects (from any field) that you think would provide a well-rounded insight into programming.


title: Why can you remove the gravitational constant from a computer game simulation?

text: I've seen in a few gravity simulation games (ie. bouncing balls) the equation:               force = G * m1 * m2 / distance^2      shortened to this by removing the gravitational constant:               force = m1 * m2 / distance^2      I accept that it works fine and saves some calculations, but I'm wondering _why_ it still works? Is the value just too small to matter? What's the physics behind this?


title: ODE envolving current

text: I have to do this exercise: > 9\. The current $I(t)$ at time $t$ flowing in an electric circuit obeys the > differential equation > > $$I''(t) + RI'(t) + I(t) = \sin \omega t$$ > > where $R$ and $\omega$ are positive constants. The solution can be expressed > in the form $I(t) = F(t) + A\sin(\omega t + a)$, where $F(t)\to 0$ as $t\to > +\infty$, and $A$ and $\alpha$ are constants depending on $R$ and $\omega$, > with $A > 0$. If there is a value of $\omega$ which makes $A$ as large as > possible, then $\omega/(2\pi)$ is called a _resonance frequency_ of the > circuit. > > (a) Find all resonance frequencies when $R = 1$.   >  (b) Find those values of $R$ for which the circuit will have a resonance > frequency. I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega t)+B\cos(\omega t)$. I got the first and second derivates $$I_{p}'=A\omega\cos(\omega t)-B\omega\sin(\omega t)$$ $$I_{p}''=-A\omega^{2}\sin(\omega t)-B\omega^{2}\cos(\omega t)$$ then, when I substitute in the equation $$I''(t)+I'(t)+I(t)=\sin(\omega t)$$ I got the system on $(A,B)$: $$\left\\{\begin{array}{l} -A\omega^{2}-B\omega+A=1 \\\ -B\omega^{2}+A\omega+B=0 \end{array}\right.$$ so $$A=\frac{(1-\omega^{2})}{(1-\omega^{2})^{2}+\omega^{2}}$$ I calculated $A$ because it follows $\sin(\omega t)$ and with this I can find all resonance frequencies. Well, to find them, I gotta find the values of $\omega$ which makes $A$ as large as possible. To $A\to+\infty$, so, its denominator must goes to 0: $$(1-\omega^{2})^{2}+\omega^{2}\to0$$ But that's not possible. The answer is $\displaystyle\frac{1}{2\pi\sqrt{2}}$. Comparing with the exercise, I know that $\omega=1/\sqrt{2}$. If the denominator was $(1-\omega^{2})^{2}-\omega^{2}$, I got the result, but I don't know what's wrong here. **EDIT:** I found after $$I_{p}(t)=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}\sin(\omega t-\alpha)$$ where $\alpha=\arctan(\omega/(1-\omega^{2})$. So, $$A=\frac{1}{\sqrt{(1-\omega^{2})^{2}+\omega^{2}}}$$ and it's large just if its denominator is small... but how?!


title: Determining the limits of an integral

text: In Griffiths' _Introduction to Electrodynamics_ , Problem 1.28 (the triangular prism question) is especially challenging for me. I do not know how the limits of x are **0** to ( **1-y** ). My concern is the latter limit, **1-y**. Can you explain systemically how this is arrived at? On the face of it, I'm inclined to say the limits of x are **0** and **1**. I can see it's the diagonal across the x- and y-axes that's presenting the extra challenge, but I do not know how to go about establishing that indeed the limit is **1-y**.


title: Dark matter and dark energy

text: > **Possible Duplicate:**   >  Dark matter references I have recently read about dark matter and dark energy, and why physicists think it must exist (dark matter: mass of galaxies are far bigger than expected, its gravitational effect on visible matter; dark energy: the force that causes the accelerated expansion of the universe). I however don't find the case for dark matter or dark energy particularly strong, and I have a couple of questions regarding this:   1. Why do so many physicists accept the existence of dark matter and dark energy?    2. Are there any good alternative theories to explain these phenomena without the use of dark energy and/or matter?    3. Is the field of study regarding dark matter and dark energy (and possible alternative explanations) 'alive' or not so much? I still am young, but I find this interesting and if it has good prospects (like for example string theory does) I might like to work in it in the future.


title: What actually Dark energy is ?

text: I recently read an article on dark matter and I could understand this things given as follows All the ordinary matter we can find accounts for only about 4 percent of the universe. We know this by calculating how much mass would be needed to hold galaxies together and cause them to move about the way they do when they gather in large clusters. Another way to weigh the unseen matter is to look at how gravity bends the light from distant objects. Every measure tells astronomers that most of the universe is invisible. I want to say that the universe must be full of dark clouds of dust or dead stars and be done with it, but there are persuasive arguments that this is not the case. First, although there are ways to spot even the darkest forms of matter, almost every attempt to find missing clouds and stars has failed. Second, and more convincing, cosmologists can make very precise calculations of the nuclear reactions that occurred right after the Big Bang and compare the expected results with the actual composition of the universe. Those calculations show that the total amount of ordinary matter, composed of familiar protons and neutrons, is much less than the total mass of the universe. Whatever the rest is, it isn't like the stuff of which we're made. But what actually is dark energy ? How should I connect dark energy with dark matter ?


title: References about rigorous thermodynamics

text: Can you suggest some references for rigorous treatment of thermodynamics? I want things like reversibility, equilibrium to be clearly defined in terms of the basic assumptions of the framework.


title: What would happen in a collision of an antimatter singularty and a matter singularity?

text: Would energy be released if a black hole made out of antimatter and another of matter were to collide?


If an anti-matter singularity and a normal matter singularity, of equal masses, collided would we (outside the event horizon) see an explosion?


title: No non-trivial UV asymptotically free and IR free

text: How it could be proven that a non-trivial theory cannot be both asymptotically free and IR free (g=0 both in the UV and IR with some interpolating function in between)? This is of course contrary to the behaviour of both QED and QCD in which we have monotonically RG flow.


title: Is it part of special relativity that mass possessing energy is more dense?

text: I was reading http://www.edge.org/3rd_culture/hillis/hillis_p2.html and it says that a charged battery weighs more than a dead one or a rotating object weighs more than a stationary one (i.e. mass containing energy weighs more than mass that doesn't). Is this true in special relativity? Because the energy would be confined to the area of the original mass, wouldn't it become more dense if it is true? Also, if the mass we are speaking of does become more gravity-sensitive on energy gain, does it actually gain physical mass (protons, electrons and neutrons)??


title: Drag force on a cone

text: I was asked to calculate the drag force on a cone with velocity 10 m/s , everything was okay until i needed to calculate the cross sectional area , the radius of the base was 0.5 m , radius of the top 0.0005 m , given that the cone falls top first, which one should i use ? should i get some kind of an average?


title: How to measure (missing) transverse energy

text: There is traditionally a bit of confusion between missing transverse energy, and missing transverse momentum. I've seen both used interchangably, and sometimes even things like "$\not E_T = -|\sum \vec p_T|$". Just to clarify before my question, if both quantities are used, then missing transverse momentum usually refers to the sum of all tracks' or reconstructed objects' $p_T$: $$ \not \vec p_T = - \\!\\!\\!\sum_{i \;\in\; \mathrm{tracks}}\\!\\!\\! \vec p_T(i) $$ and missing transverse energy refers to something calculated using mainly the calorimeter, right? So, how do you calculate (missing) transverse **energy** from calorimeter readings? Especially, how do you do a vectorial sum, even though calorimeter energies are scalar values? You don't have particle momentum vectors or similar, but only the readings of calorimeter cells. (Reconstructed objects come into play later, for corrections, but are not involved at first order.) Naively, I'd say you construct a vector from the position of each calorimeter cell, and give it a length proportional to its energy: $$\qquad\quad \vec E_\mathrm{cell} = \frac{\vec x_\mathrm{cell}}{|\vec x_\mathrm{cell}|} E_\mathrm{cell} \qquad (?)$$ and then just sum those vectors up. But you want to have _transverse_ energy. How do you do that? Just by using 2-d vectors (only $x$ and $y$ coordinates)? And finally, is the geometry of the cells a factor in the calculation? * * * (I'm looking for a general answer, but where it is experiment-specific, I'm interested in ATLAS and the way it was done at DZero. I can imagine the answer would be very different for CMS with their special calorimeter. And sorry, I tried to read the design documents, but I couldn't really understand how it is calculated. I'd be happy if someone would point me to some clear documentation though. There are a couple of similar questions, but they don't hit the point I'm wondering about. This question basically asks "why use transverse quantities", and the answer conflates transverse energy and momentum (I'm interested in the one determined by the calorimeter where you don't have $\vec p_T$ vectors!). And this question comes from the other side and asks how to calculate $\not E_T$ from four-vectors, not from experimental readings.)


title: Calculating torque applied on domain?

text: How is it possible to calculate torque that is applied on a domain due to the magnetization force[H]?


title: Calculate average speed with unknown variable accelaration

text: I am in the middle of a vehicle tracking project where I have to calculate the distance traveled by the vehicle in a given amount of time. Data I am getting:               Speed : 30.2 km/hr   12.7 km/hr    15 km/hr    21.8 km/hr      Time :  11:00:00     11:00:22      11:00:45    11:01:10      That is I am getting the speed of the vehicle every 20-25 seconds. So what is the best way to calculate the distance traveled by the vehicle during this whole duration? Is taking the median of two speeds the best way to calculate the average speed here?


title: Electric field lines density

text: When we're dealing with problems in electrostatics (especially when we use Gauss' law) we often refer to electric field lines density which is inversely proportional to radius in case of a single point charge (all field lines are directed radially). My question may sound dumb, but despite the fact that this concept is quite intuitive, if you think about it - there is an infinite amount of lines which we can draw everywhere, so we can't really think of an area with denser lines. Then why can we still use such a concept? Why it really works?


title: Two particles in infinite square potential

text: Consider two non-interacting distinguished particles in one-dimensional infinity square potential. Suppose the particles have the same mass $m$, and the potential is zero in the region $\frac{a}{2}>x>-\frac{a}{2}$. In the coordinate of $x_1,x_2$, the Schrodinger eq. $$-\frac{\hbar^2}{2m}\partial^2_{x_1}\psi(x_1,x_2)-\frac{\hbar^2}{2m}\partial^2_{x_2}\psi(x_1,x_2)=E\psi(x_1,x_2)$$ can be seperated and the solution is $$\psi(x_1,x_2)=\frac{2}{a} \sin [\frac{n_1\pi}{a}(x_1+\frac{a}{2})] \sin [\frac{n_2\pi}{a}(x_1+\frac{a}{2})]$$ $$E_{n_1n_2} =\frac{\pi^2\hbar^2}{2ma^2}(n_1^2+n^2_2)$$ For $n_1=n_2=1$, we get the ground statewave function and energies. **My question is how to sovle the above problem in center of mass frame?** In center of mass frame the schrodinger equation becomes: $$-\frac{\hbar^2}{2M}\partial^2_{R}\psi(R,r)-\frac{\hbar^2}{2\mu}\partial^2_{r}\psi(R,r)=E\psi(R,r)$$ Naively if we require wave function is nonvanshing in the region $$-\frac{a}{2} < R<\frac{a}{2}\\\ -a <r<a $$ Then we will find the ground energy is half of $E_{11} =\frac{\pi^2\hbar^2}{2ma^2}(1+1)$. What's wrong wth the center of mass frame calculation?


title: Is $k_B \rightarrow 0$ the classical limit of stat. mech., as $\hbar \rightarrow 0$ is in QM?

text: I hear very often among my peers and seniors that just as how $\hbar\rightarrow0$ takes me to classical mechanics from quantum mechanics, $k_B\rightarrow0$ will take me to classical thermodynamics from statistical mechanics. As nice as this sounds, my gut feeling and intuition tells me this is not the right analogy. I think $N\rightarrow\infty$, the ensemble size of statistical mechanical system, is closer to $\hbar\rightarrow0$ in that respect. Is this correct? If so, what role does $k_B$ play then?


title: In a non-degenerate plasma, why are e-e collision negligible compared to e-ion for thermal conduction?

text: I'm trying to make some order of magnitude estimates of heat transfer in stars - to better understand 1) why conduction is said to be negligible (for non- degenerate matter) and 2) when convection happens and is dominant. For thermal conduction, I have the expression that the flux of energy over a surface $f \approx - \rho \langle v \rangle \cdot \lambda \cdot a k_B \frac{dT}{dx}$ For a mean-free path $\lambda$ and degrees of freedom over two '$a$'. I.e. an expression for the thermal conductivity. Anyway, everything I read (e.g. See section 'Electron Conductivity'), says that the mean free path should be calculated for electron-ion collisions instead of electron-electron collisions. Can anyone explain this?


title: What is the mechanism by which magnetic fields do work?

text: I've seen some conflicted answers to this question in texts and on the web, in the case of a dipole, for example. Do magnetic fields do work directly, or is it their induced electric fields that do work?


title: How to find out the maximum radius of a hole that can keep water stay in a container by water viscosity?

text: Assume I have a inverse cone which holds 200ml water. I am going to cut the tip of the cone to create a small hole. How to calculate the maximum radius of the hole that the water will still stay in the container ?


title: Asymptotic Invariants in General Relativity

text: I was trying to understand Witten's proof of the Positive Energy Theorem in General Relativity by reading the original argument given by Witten. I am comfortable with the overall argument, but I would like to understand the following statement, made on the second paragraph of page 394 in the previous link: > "The only invariants that can be formed from the $1/r$ term in the metric > tensor are the total energy and the total momentum." These invariants (which I will call "$1/r$-built" for short) refer to an asymptotically flat initial value (spacelike smooth) 3-surface in spacetime. Is there an obvious reason why this should be true? Looking at the definition of the ADM-energy and momentum, it seems plausible because the only data we have are the first and second fundamental form, and it should be possible to write any invariant as a combination of these, and consequently, any $1/r$-built invariant as a function of ADM energy and momentum. However, this reasoning is too hand-wavy, so I wonder if there is a clear cut explanation. My interest in this fact is that although it is not really a logical step in the proof, if true it is probably one of the best ways to motivate a spinorial proof (I was thinking of something along the lines of "if we can construct a manifestly non-negative $1/r$-built invariant by means of the asymptotic behavior of spinors, then it should be possible to prove that energy is non- negative by writing this invariant as a function of ADM energy and momentum"). From a purely physical perspective, if it were true that ADM energy and momentum suffice to specify the system (to order $1/r$), then they should be the only independent invariants. However, this suggests that an asymptotic observer who knew the total energy and momentum could reconstruct the metric up to order $1/r$. I was thinking if it is not possible to construct a counterexample by defining an axisymmetric spacetime in which the Killing field of azimuthal symmetry is build up only with $1/r$ terms and showing that it is asymptotically distinguishable from its non-rotating counterpart. In this spirit, I think it is worth asking a broader version of my question: > What kinds of "physically interesting" boundary terms can appear in a > Hamiltonian formulation for an asymptotically flat spacetime manifold in > General Relativity? Thanks in advance.


title: Is nature symmetric between particles and antiparticles?

text: Is nature symmetric with respect to presence of particles? Do we have an antiparticle for every particle thought of? Are there any proven examples where we don't have an antiparticle? And what about antiparticle of a photon (we know it can also behave as a particle)?


title: Microscopy types and techniques?

text: I didn't get the basic difference in between the different types of microscopy. for example there are several different microscopy techniques are available such as Bright field, Dark field, Confocal, STED, electron microscopy etc. Is only the illumination system varies in between different techniques or are there any other parameters?


title: Spinning bucket of water in zero gravity

text: Everyone knows how the surface of a spinning bucket of water would look like on earth - parabolic. But what if we turned off gravity (for instance by doing the experiment in a freely falling lift)? Would the surface be still parabolic? I'll explain my confusion in more detail. The velocity of the spinning bucket is transferred to the water by means of frictional forces arising in the boundary between the bucket and water. But these frictional forces exist no matter whether there is gravity or not. So if I consider the whole bulk of water inside the bucket as a single system, this frictional force would give it a positive torque. Thus the water has to rotate. For the sustained rotation of water, a centripetal force has to exist. In normal gravity, the water surface changes its shape into a paraboloid so that there is a net force on any particle directed inward. But in free fall, there is no pressure on a particle inside the liquid. Thus the only force that can supply the centripetal acceleration is inter-molecular force between the particles which is weak to sustain huge velocities. So what exactly happens?


title: Is Quantum Physics used in manufacturing CPUs?

text: If I remember correctly, I heard some people saying that the transistors on CPUs today are so small, that they have to use quantum physics to make CPUs. Is that correct?


title: Simple exercises about U-tubes

text: I want to solve some exercises involving U-tubes, like You pour two non-mixing fluids with densities that go like 1:2 into a U-tube. The fluid is 5cm higher in one arm than in the other. Where is the seperating line? Unfortunately, I don't know any of the formulas one can use for such exercises, all I know is that the pressures must be equal on both sides. Can you supply me with the most common formula(s) used in this situation?


title: Angular Momentum with Upper Index

text: I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i L^i,L_i] = L_i [L^i,L_i]$. So I could not understand this term with upper index. What is $L^i$ ?? Any clues?


title: When I stretch a rubber band, it breaks. When I hold the broken ends together, why doesn't it join again?

text: The question is simple. When we join the two broken surfaces, what is it that keeps the surfaces from connecting with each other, while earlier they were attached to each other? Also, would the two sides join again if I hold them together for an infinite amount of time?


title: Derivation of the general Lorentz transformation

text: The _standard Lorentz transformation_ or _boost_ with velocity $u$ is given by $$\left(\begin{matrix} ct \\\ x \\\ y \\\ z \end{matrix}\right) = \left(\begin{matrix} \gamma & \gamma u/c & 0 & 0 \\\ \gamma u/c & \gamma & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{matrix}\right) \, \left(\begin{matrix} ct^\prime \\\ x^\prime \\\ y^\prime \\\ z^\prime \end{matrix}\right) = L_u \,\left(\begin{matrix} ct^\prime \\\ x^\prime \\\ y^\prime \\\ z^\prime \end{matrix}\right)$$ where $\gamma = \gamma(u) = 1/\sqrt{1-u^2/c^2}$. In the standard Lorentz transformation, it is assumed that the $x$ and $x^\prime$ axes coincide, and that $O^\prime$ is moving directly away from $O$. If we drop the first condition, allowing the inertial frames to have arbitrary orientations, then "we must combine [the standard Lorentz transformation] with an orthogonal transformation of the $x$, $y$, $z$ coordinates and an orthogonal transformation of the $x^\prime$, $y^\prime$, $z^\prime$ coordinates. The result is $$\left(\begin{matrix} ct \\\ x \\\ y \\\ z \end{matrix}\right) = L \,\left(\begin{matrix} ct^\prime \\\ x^\prime \\\ y^\prime \\\ z^\prime \end{matrix}\right)$$ with $$L = \left(\begin{matrix} 1 & 0 \\\ 0 & H \end{matrix}\right)\, L_u \,\left(\begin{matrix} 1 & 0 \\\ 0 & K^\textrm{t} \end{matrix}\right)$$ where $H$ and $K$ are $3 \times 3$ proper orthogonal matrices, $L_u$ is the standard Lorentz transformation matrix with velocity $u$, for some $u < c$, [and 't' denotes matrix transpose]." I have two questions:   1. Why are two orthogonal transformations, for both the unprimed and primed spatial coordinates, necessary? That is, why isn't one orthogonal transformation sufficient to align the axes of the inertial frames?   2. Why does the first orthogonal transformation use the transposed orthogonal matrix $K^\textrm{t}$?


title: Mean stopping range of $\alpha$-particles in air

text: I did an experiment in university in which I determined how far $\alpha$-particles emmited from an $Am^{241}$ source penetrate into air. I want to compare my result to literature values but... I cant find any on google! Can somebody point me to a source for this? Cheers in advance!


title: Why does the voltage increase when capacitor plates are separated?

text: In lab, my TA charged a large circular parallel plate capacitor to some voltage. She then disconnected the power supply and used a electrometer to read the voltage (about 10V). She then pulled the plates apart and to my surprise, I saw that the voltage increased with distance. Her explanation was that the work she did increased the potential energy that consequently, increases the voltage between the plates but the electric field remains constant. Although I tried to get more of a physical explanation out of her, she was unable to give me one. Can someone help me here?


title: Force problem related to adhesive and bonding

text: I have two PCBs (printed circuit board), and they are glued by adhesives, as show in the pictures. And the location of the adhesives are indicated on the picture (please notice that NO adhesive is applied between the PCB boards). **50N** force is applied on the upper PCB (Z direction). I use adhesive to prevent the PCB from being separated. Here are the dimensions of the adhesives: **All the wedge shaped adhesives have height 3mm (Z-direction) (the same as the PCBs), and width is 1mm (x-direction),** **The long 80mm adhesive is 1mm thick (y-direction) and height is 6mm (z-direction).** I read from the adhesive specification sheet, the adhesive have the following properties: $\mathrm{tensile\: strength}: 22N/mm^{2}$ $\mathrm{shear\: strength}: 18N/mm^{2}$ How can i judge whether the adhesive is strong enough? What kind of formula should i use? ![enter image description here](http://i.stack.imgur.com/4zVqD.jpg) _**I attempted to solve the problem like this:_** **Step 1:** Total area of glue contact on the blue board $= 2(3(10) + 3(5.5)) + 80(3) = 333mm^{2} $ Force resisting detachment of the blue board (shear force only in this situation) $= 333(18) = 5994N $ **Step 2:** Total area of the 4 glue contacts on the grey (ground) board $= 2(1(10) + 1(5.5)) = 31mm^{2}$ Area of the glue contact at the edge of the grey board $= 80(3)=240mm^{2} $ So, Total force resisting the glue from being pull out from the grey board (tensile+shear force in this situation) $= 31(22) + 240(18) = 5002N $ **Therefore, in 5002N is required to pull the blue out?**


title: What is the mechanism for equilibration?

text: I read on page 5 of Matthew Schwartz' book QFT & the SM that if you heat a box with monochromatic light, then (later) all the frequencies will get excited. The author says that particles have to be created and destroyed to accomplish this. This must show the need for QFT. I'm just wondering if I will eventually see a mechanism by which equilibration is accomplished or if I will see only a calculation that gives the correct result without an underlying mechanism. This is not to say that a calculation would not be a great thing but I want to know if there's a mechanism.


title: Standard error for expected value

text: I have a computer simulation from which I get a sample of values of some microscopic quantity $X$, i.e. $\\{ x_1,\ldots,x_N \\}$. I'm interested in estimating the expected value of $X$, i.e. $E[ X]\approx\frac{1}{N}\sum x_i$, and the variance associated with this **estimate**. Now, do I understand correctly that:   1. taking the estimate of the variance of $X$, $Var[X]\approx\frac{1}{N}\sum (x_i-\bar{x})^2$, is NOT what I'm looking for (which in this notation would be $Var[E[X]]$)?   2. I need to make use of some statistical technique, such as bootstrapping?


title: Why do the $1/2$ factor appear in the Majorana mass Lagrangian?

text: In case of Dirac neutrino there is no $1/2$ factor in the mass Lagrangian but for Majorana type neutrino there is a half factor in the mass Lagrangian.


title: Zeroth Law of Thermodynamics, temperature, and ordering

text: In my thermodynamics course (and in other places on the internet) it is asserted that the Zeroth Law of Thermodynamics can be used to define the concept of temperature. One statement of the Zeroth Law I have seen states that the relation _thermal equilibrium_ on two closed systems brought into diathermal contact defined is in fact an equivalence relation. The argument continues by saying that if we call the equivalence classes so defined _isotherms_ , then we can assign arbitrary numbers to these isotherms and these numbers are what we call _temperature_. We now have a set of numbers assigned to a set of equivalence relations. But what I **don't** see is how this numerical assignment bears any relation to physical temperature. Where is order defined? For example, if one of the classes gets the number "1", another gets the number "2", and a third "3", what in the above derivation shows that the isotherm "2" comes between the isotherm "1" and the isotherm "3"? Maybe more is needed than just the Zeroth Law. If so, what is necessary in order to complete the argument?


title: Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

text: I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In particular I'm looking at the exercises on page 287 of Nakahara's book, these respectively ask you to compute the Riemann curvature tensors with the Levi-Civita connection explicitly for flat space with spherical coordinates (for which we obviously expect the Riemann tensor to vanish), the FRW metric and the Schwarzschild metric. The simplifications that I'm able to make so far are:   1. The anti-symmetry of the Riemann tensor leaves $m^2(m^2-1)/12$ independent components, where $m$ is the dimension, that are possibly non-vanishing, see for example page 290 of Nakahara's text. We can write down these combinations of indices explicitly and see which ones vanish.    2. Since the metrics in these particular examples are all diagonal, so the non-vanishing components of the Levi-Civita connection coefficients are either a. both lower coordinates are the same or b. the upper coordinate coincide with at least one lower coordinate. My question is whether there is any further simplification that can be made, or must one proceed with brute force computation after the simplification mentioned above have been made.


title: Why do people claim electrons are accelerating

text: A lot of text books mention that one of the reasons that classical mechanics failed to explain atomic and subatomic processes is that electrons which accelerate should release energy in the form of electromagnetic radiation, which would lower the atoms overall energy level, but this does not happen. One place where I discovered this, for example, is in the description for the Bohr model. What I don't understand is why everyone takes for granted the fact that the electron is accelerating. I thought the electron orbits the nucleus at a, more or less, constant velocity. Are people referring to specific situations when the atom is excited? Furthermore, I was under the impression that electrons already travel at the fastest allowable speed, the speed of light.


title: Reference on electrodynamics with tempered distributions

text: Back in my undergrad I had a course on classical electrodynamics where the fields had values in the space of tempered distributions. In this way one could correctly treat self-interaction and effectively solve the differential equations involved in the distribution sense. Unfortunately the few notes that I have are in Italian, and I am looking for some resources in English, but don't seem to find anything, could someone point me in the right direction?


title: Work done by a frictional force on a cylinder

text: > _A cylinder of mass $m$ and radius $R$, is rolled on surface with > coefficient of kinetic friction $\mu_{k}$ about the axis passing through the > center and parallel to the surface, with initial angular velocity > $\omega_{0}$; the work done by frictional force from start until it begins > to roll w/o slipping is to be found out._ It is easy to see that $a=\mu_{k}g,\alpha=-2\mu_{k}g/R,t=\frac{R\omega_{0}}{3\mu_{k}g},d = \frac{1}{2}at^2 $ are acceleration, angular acceleration, time from beginning to the time of rolling w/o slipping and distance covered. I am somehow missing why I am getting $$W=\int F.ds+\int\tau.d\theta=\mu_{k}mg*d+\mu_{k}mgR*\frac{d}{R}\neq\triangle K.E.=-\frac{1}{6}mR^{2}\omega_{0}^{2}.$$ Please shed some light.


title: Type-II SHG of linearly polarized light

text: It's possible to perform type-II frequency doubling of unpolarized light in suitable nonlinear crystals. But for a single beam of linearly polarized input light, is it still possible to do type-II doubling? I vaguely remember reading that it can be done if the crystal is oriented at a 45-degree angle to the polarization direction, since the crystal will see half the light as ordinary- polarized and the other half as extraordinary-polarized. If it's possible to do this with a single linear beam, is there an efficiency reduction as compared to type-I SHG?


title: How did one get the defining equation of probability current and conservation of probability current and density?

text: I'm reading the Wikipedia page for the Dirac equation: > $$\rho=\phi^*\phi$$ > > _and this density is convected according to the probability current vector_ > > $$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$$ > > _with the conservation of probability current and density following from the > Schrödinger equation:_ > > $$\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0$$ The question is, how did one get the defining equation of the probability current vector? It seems that in most texts, this was just given as a rule, yet I am thinking, there must be somehow reasons for writing the equation like that.. Also, why is the conservation equation - the last equation - is kept?


title: Finding current using EMF & internal resistance

text: **What exactly is the difference between internal resistance and resistance?** * * * This came up in the context of a homework problem I have been given: > The circuit shown in the figure contains two batteries, each with an emf and > an internal resistance, and two resistors. ![Circuit](http://i.stack.imgur.com/3XWIF.jpg) I need to find the magnitude of the current in this circuit. I believe I'm supposed to be using the equation: $I = \frac E{(R + r)}$ where $E = 24.0V, R = 17 \Omega$. So how do I identify the internal resistance.


title: Mathematical probabilistic interepretation of probability amplitude

text: As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question: I'm looking at the possibility of using probability amplitude functions to represent probability distributions on surfaces. From my perspective, a probability amplitude function is a function $\psi:\Sigma\rightarrow\mathbb{C}$ satisfying $\int_\Sigma |\psi|^2=1$ for some domain $\Sigma$ (e.g. a surface or part of $\mathbb{R}^n$)-- obviously these are some of the main objects manipulated in quantum physics! In other words, $\psi$ is a complex function such that $|\psi|^2$ is a probability density function on $\Sigma$. **From this purely probabilistic standpoint, is it possible to understand why multiple $\psi$'s can represent the same probability density $|\psi|^2$? What is the most generic physical interpretation?** That is, if I write down any function $\gamma:\Sigma\rightarrow\mathbb{C}$ with $|\gamma(x)|=1\ \forall x\in\Sigma$, then $|\psi\gamma|^2=|\psi|^2|\gamma|^2=|\psi|^2$, and thus $\psi$ and $\psi\gamma$ represent the same probability distribution on $\Sigma$. So why is this redundancy useful mathematically?


title: Why helicity is proportional to the spin of particle and has two values?

text: How can it be shown without using the little group formalism? Let's have the Wigner's classification for the irreducible represetation of the Poincare group. For the massless case the eigenvalues of two Casimir operators of the group, the squares of Pauli-Lubanski operator and momentum operator, $\hat {W}_{\alpha}W^{\alpha}, \hat {P}_{\alpha}\hat {P}^{\alpha}$, is equal to zero. Together with $\hat {W}_{\alpha}\hat {P}^{\alpha} = 0$ it leads to an expression $\hat {W}_{\alpha} = \hat {h}\hat {P}_{\alpha}$, where eigenvalues of $\hat {h}$ has dimension like angular momentum. It is called helicity. I want to get it "properties" without using small groups formalism (by the other words, not as Weinberg).


title: Why do circuits work so fast?

text: Drift velocity (explained to me as how fast the electrons are moving) is _really_ slow. My book says the electrons move at around 10 mm/ s. If electrons move so slowly how do circuits work so fast? If you make a basic circuit with just a light bulb, the bulb lights up almost immediately after you connect the wire to the positive terminal.


title: Is it possible to solve a 2D relative motion question using components of relative velocity?

text: This is a question based on concepts of two dimensional motion . Here's how the question is: > A plank A is floating in air, gravity has no effect on it (See its > coordinates in the figure attached). At $t=0$ plank starts moving along the > x-axis with an acceleration of $1.5 \ \mathrm{m/s^2}$. At the same instant a > projectile is fired (see figure) from origin with a velocity $u$. > > A stationary person on ground observes the stone (projected particle) > hitting the object during its downward motion at an angle of $45^\circ$ to > the horizontal. Find the velocity with which the stone was fired. Also find > the time after which stone hits the object. ($g=10\ \mathrm{m/s^2}$). > > ![diagram of problem](http://i.stack.imgur.com/XYBw7.png) **TEXTBOOK APPROACH:** $$\begin{gather} s_x=u_xt=3+\frac12(1.5)t^2\tag{1} \\\ s_y=u_yt-\frac12gt^2\quad\leftrightarrow 1.25=u_yt-\frac12gt^2\tag{2} \\\ \tan 45^{\circ} =\frac{-v_y}{v_x}=\frac{-(u-gt)}{u_x}\tag{3} \end{gather}$$   These are now three equations in three variables and easily solvable. I was however curious to know if we can solve this question using the concept of relative velocity seperately for x and y components like we do for 1-D Motion for the collision kind of questions?   I tried but went horribly wrong but I am sure relative velocity in two different axes will also give us the right answer.


title: Field due to current in a wire

text: Suppose a current flows in a straight cylindrical wire so that an electric field $\textbf{E}$ is maintained in the wire. Will there be an electric field just outside the wire..?


title: What determines the (minimum) angle at which a domino falls over?

text: Dominoes, when placed upright, remain that way. Sometimes, even if you tip them a little bit, they will go back to their upright position. However, if you tip them too far, they will fall over. After trying this with many different sized/shaped dominoes and some textbooks, I've noticed that this angle of "maximum tippage" varies for each one, depending on its dimensions. Taller dominoes seem to have a lower maximum tippage angle. Dominoes with wider bases seem to have a higher maximum tippage angle. What are other factors are involved? Is there a way to compute this maximum tippage angle, given the height of the domino, and the width of its base, and any other factors that might be involved? If so, what is this relationship (mathematically)? (Dominoes start acting weirdly when their weight/density is unevenly distributed, so for this question, assume that dominoes are of constant density)


title: Special relativity: circumventing velocity-addition formula

text: Two spaceships approach an observer from an equal distance and from an opposite direction with an equal speed $v$ in the observer's intertial reference frame $O$. The speed of a spaceship in the intertial reference frame of the other spaceship $S$ is $0.8c$, what is the speed of one of the spaceships in $O$? I proceeded as follows: Let $2l'$ be the distance between the two spaceships in $S$. In S the two spaceships will collide after a time $t' = \frac{2.5l'}{c}$. Let $\gamma$ be the squareroot of $1 - v²/c²$. In $O$ the two spaceships will collide when $vt = l$ or $v\gamma t' = \frac{l'}{\gamma}$ ($O$ has to correct for what he perceives as the time dilations and space contractions of the measurements made in $S$) Substituting we get the equation $x(1-x^2) = 0.4$ with $x = \frac{v}{c}$, if you solve the equation you conclude that this line of reasoning was wrong (but when we replace $0.4$ by $0.375$ we do get the right solution, which is $0.5c$). Where's the flaw?


title: Schrödinger equation for two particles in a 3D box?

text: This is not a homework question, just a question I have developed to get a better conceptual understanding of the results of the Schrödinger equation. If I had a 3D spherical container or radius R, containing 2 particles of opposite charge, say a proton and an electron, what does the solution to the resulting Schrödinger equation look like? How does the solution compare to the solution of the Schrödinger equation for a simple hydrogen atom? What happens as R approaches infinity?


title: Atomic Hamiltonian, non analytic solution

text: How can be proved that terms $$\frac{1}{|r_{i}-r_{j}|} $$ are the ones that avoid the existence of an analytical solution for the many electron atom problem


title: Maxwell's Demon bug, trapdoor space and time

text: What about the size of the door (space) and how long has it to be opened (time)? I think Maxwell's demon would have a problem with space, if the door is too wide (more than one particle size), then direct interaction between the two part is possible and would lead to lose isolation restriction, like there were no demon, (even no isolation wall). On the other hand, if the door is too narrow, then no particle can go into or it will let pass only those which are direct aimed, then Maxwell's demon would lose its freedom of choice because the narrow gate itself "choose" particles having a direction perpendicular to the wall, despite any module criterion (no temperature can be choosen) again it is like there were no demon. Finally assuming that there were an exact size door being useful for the demon work, then time would be the problem, when the door is open too long, both side particles can cross or collide, like there were no demon!, and if the door is opened too little time, then it only could "select" fast particles from both sides,then it couldn't separate temperatures. I've read very complex arguments against Maxwell's Demon, concerning information store but I think size and time constraint could be enough arguments to defeat Maxwell's Demon, **Do you know any related work about size and time constraint ?** Thanks


title: Find the acceleration based on V² x d graph

text: How could I find the acceleration of a body knowing only that its position and velocity satisfy $v^2 = f(x)$, where $f(x)$ is a known function of $x$ (position)?


title: Global warming and planetary thermodynamics

text: To a first approximation, the earth currently radiates out as low frequency thermal radiation the same amount of energy as it absorbs as high frequency solar radiation. (This ignores energy generated within the earth, which is also radiated away. But that amount is constant and is not relevant to my question. It also ignores energy stored or burned as fossil fuels.) Let's assume that global warming will not change the amount of energy received from the sun and absorbed by the earth. (I realize that's not true. Global warming melts the ice caps, which reflect solar radiation. With the ice caps melted, the earth absorbs more of the solar radiation it receives.) But if we ignore the melting of the ice caps, the earth must receive and radiate away a fixed amount of solar radiation, which is independent of its temperature. I would have thought that a warmer earth would radiate more thermal radiation than a cooler earth. But the argument above says that's not the case. How is this explained?


title: Particle colliders: why do they need an accelerator chain

text: Particle colliders like the LHC or the Tevatron use a complex accelerator chain to have particles at a given energy before being accelerated. For example:   * The CERN accelerator complex to inject in the LHC: protons are first accelerated by a LINAC, then by a "booster" synchrotron, then by a larger synchrotron, and finally by a very large synchrotron where they reach 450 GeV/c.   * As was discussed in link text , the chain for the Tevatron They both require "injector" accelerator(s). ![alt text](http://i.stack.imgur.com/PzikJ.gif) My question is simple: why ? Why can't we have a single accelerator with a source and the possibility to accelerate the particles to the top energy. * * * Note: this is also a seeding question, but I am curious how complete and clear an explanation to that can be provided.


title: Source term of the Einstein field equation

text: My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's "simplified account of the Einstein field equation of general relativity did need a qualification that he did not quite give." Feynman's intuitive discussion rests on relating the "radius excess" of a sphere to a constant times the enclosed gravitational mass $M$: for a sphere of measured radius $r_{\mathrm{meas}}$ and surface area $A$ enclosing matter with average mass density $\rho$ smoothly distributed throughout the sphere, $$\sqrt{\frac{A}{4π}}−r_{\mathrm{meas}}=\frac{G}{3c^2}\cdot M,$$ wherein $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $M=4π\rho r^3/3$. I don't know what $r$ is supposed to be, but it's presumably $\sqrt{\frac{A}{4\pi}}$. Feynman gratifyingly points out that $\frac{G}{3c^2}\approx 2.5\times 10^{−28}$ meters per kilogram (for Earth, this corresponds to a radius excess of about $1.5$ mm). Feynman is also careful to point out that this is a statement about average curvature. Penrose's criticism is: "the 'active' mass which is the source of gravity is not simply the same as the energy (according to Einstein's $E=mc^2$); instead, this source is the energy plus the sum of the pressures". Damned if I know what that means -- whose pressure on what? So, taking into account Penrose's criticism but maintaining Feynman's intuitive style, what is the active mass $M$?


title: Quality factor for a quantum oscillator?

text: I've been reading papers about nanomechanical oscillators, and the concept of quality factor often pops up. I understand to some extent about Q factor in classical sense, but since nanomechanic oscillators are often treated quantumly, what does Q factor mean then?


title: Implications of a phase shift in a 2-ray propagation model

text: So I am working with a 2-ray (actually it's 8 ray now, but that's merely an extension) wireless signal propagation model. The equation of the component of interest: $ P = \vert{1-\Gamma\exp\left(j2\pi/\lambda*(\sqrt{(d^2+(2h^2))} - d)\right)}\vert^2 $ This produces a sinusoid with progressively increasing period, which is what I want. However, adding a phase shift to the sinusoid (extra term in the exponential) provides a better fit to my data by simply shifting the sinusoid over (the phase shift I need is 1 rad or so). I've been thinking and I can't quite figure out what this means exactly in the physical realization. It changes the phase difference between the two signal paths. Is it just accounting for the fact that there is cluster of signals, not a single path? Is it accounting for the inaccuracies in the 2-ray model? Maybe someone can recommend a good paper that has dealt with this? I looked on Google Scholar, but the searches don't yield much - probably because I'm not using the right search terms.


title: Is there a timeline in urban area night temperature?

text: **Fact:** We all know that during the day **concrete** absorbs heat and **releases it during the night** , making urban areas hotter than rural areas. I observed that after sunset the ambient temperature is going down until late at night when it actually starts to feel hot. I would assume that soon after sunset the temperature should decrease slowly until sunrise, but that's not the case. So, I would like to understand which phenomena take place from sunset to sunrise in urban areas. Take into account that I live in Athens (Greece) where there is a huge urban area and the sea is nearby.


title: A Question on Hamilton's Principle

text: In some literatures, the Hamilton's principle for conservative systems is introduced by this equation: $$\delta \int_{t_1}^{t_2}(T-V) ~\mathrm{d}t~=~0$$ In some others, this principle is introduces as follow: $$\int_{t_1}^{t_2}\delta(T-V) ~\mathrm{d}t~=~0$$ What is the difference between two equations? Are those expressions the same?


title: capacitance between widely-separated parallel plates

text: There's a caveat, which is often ignored, to the "easy" equation for parallel plate capacitors C = epsilon * A / d, namely that d must be much smaller than the dimensions of the parallel plate. Is there an equation that works for large d? I tried finding one and could not. (These two papers talk about fringing fields for disc-shape plates but don't seem to have a valid equation for d -> infinity: http://www.santarosa.edu/~yataiiya/UNDER_GRAD_RESEARCH/Fringe%20Field%20of%20Parallel%20Plate%20Capacitor.pdf and http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.167.3361&rep=rep1&type=pdf) My hand-waving intuition is that as d -> infinity, C should decrease to a constant value (which is the case for two spheres separated by a very large distance, where C = 4*pi*e0/(1/R1 + 1/R2) ), because at large distances from each plate, the electric field goes as 1/R, so the voltage line integral from one plate to the other will be a fixed constant proportional to charge Q.


title: Equation of motion for explicit time dependent potential

text: What is the equation of motion for a single scalar field, which has a Lagrangian density in which the potential explicitly depends on time? For example: $$U(\phi,t)=\frac{1}{2}\phi^2 - \frac{1}{3} e^{t/T}\phi^3 + \frac{1}{8}\phi^4$$ where $T$ is a constant.


title: Is it possible that windmills can consume the wind?

text: Can windmills allow us to consume (and, eventually, over-consume) the wind as a natural resource somewhat in the same manner that we are over-consuming many other natural resources?


title: Why does light reflect more intensely when it hits a surface at a large angle?

text: I mean, what is happening at a microscopic level to cause this behavior? Here's what I got from Wikipedia:   1. On Reflection (physics)#Reflection of light it says that " _solving Maxwell's equations for a light ray striking a boundary allows the derivation of the Fresnel equations, which can be used to predict how much of the light reflected, and how much is refracted in a given situation._ "   2. On Specular reflection#Explanation it says that " _for most interfaces between materials, the fraction of the light that is reflected increases with increasing angle of incidence_ $\theta_i$" (but doesn't explain why)   3. Finally, on Reflection coefficient#Optics, it says basically nothing, redirecting the reader to the Fresnel equations article. What I'm trying to find, instead, is a basic level explanation that could provide an intuition on why this happens, rather than analytic formulations or equations to calculate these values. Is there a good analogy that explains this behavior?


title: Cross-section in relativistic limit: Fermi's golden rule still valid?

text: In order to calculate the cross-section of an interaction process the following formula is often used for first approximations: $$ \sigma = \frac {2\pi} {\hbar\,v_i} \left| M_{fi}\right|^2\varrho\left(E_f\right)\,V $$ $$ M_{fi} = \langle\psi_f|H_{int}|\psi_i\rangle $$ Very often plane waves are assumed for the final state and therefore the density of states is given by $$ \varrho\left(E_f\right) = \frac{\mathrm d n\left(E_f\right)}{\mathrm d E_f} = \frac{4\pi {p_f}^2}{\left(2\pi\hbar\right)^3}\frac V {v_f} $$ I understand the derivation of this equation in the context of the non relativistic Schrödinger equation. But why can I continue to use this formula in the relativistic limit: $v_i, v_f \to c\,,\quad p_f\approx E_f/c$. Very often books simply use this equation with matrix element derived from some relativistic theory, e.g. coupling factors and propagators from the Dirac equation or Electroweak interaction. How is this justified? ### Specific concerns:   * Is Fermi's golden rule still valid in the relativistic limit?   * Doesn't the density of final states has to be adapted in the relativistic limit?


title: What's inside a proton?

text: What constitutes protons? When I see pictures, I can't understand. Protons are made of quarks, but some say that they are made of 99% empty space. Also, in this illustration from Wikipedia, what's between the quarks? ![image](http://i.stack.imgur.com/CcjvK.png)


title: Setting of renormalization scale in field theory calculations

text: In dimensional regularization an arbitrary mass parameter $\mu$ must be introduced in going to $4-\epsilon$ dimensions. I am trying to understand to what extent this parameter can be eliminated from physical observables. Since $\mu$ is arbitrary, physical quantities such as pole masses and scattering amplitudes must be independent of it. Nevertheless at any fixed order in perturbation theory these quantities contain residual $\mu$-dependence. One expects this dependence to decrease at higher orders in perturbation theory. For concreteness, consider dimensional regularization with minimal subtraction of $\phi^4$ theory, which has bare Lagrangian $\mathcal{L}_B = \frac{1}{2}(\partial \phi_B)^2 - \frac{1}{2}m_B^2 \phi_B^2 - \frac{\lambda}{4!}\phi_B^4$ Here are the 1-loop expressions for the physical mass $m_P$ and 4-point coupling $\lambda_P \equiv (\sqrt{Z})^4 \Gamma^{(4)}$ after minimal subtraction of poles and taking $\epsilon \to 0$: \begin{equation} m_P^2 = m_R^2 \left\\{1 + \frac{\lambda_R}{2(4\pi)^2}\left[\log\left(\frac{m_R^2}{4\pi\mu^2}\right)\right] + \gamma - 1 \right\\} \end{equation} $$\lambda_P = \lambda_R + \frac{3\lambda_R^2}{2(4\pi)^2}\left[\log\left(\frac{m_R^2}{4\pi\mu^2}\right) + \gamma - 2+\frac{1}{3}A\left(\frac{m_R^2}{s_E},\frac{m_R^2}{t_E},\frac{m_R^2}{u_E}\right) \right] $$ where $A\left(\frac{m_R^2}{s_E},\frac{m_R^2}{t_E},\frac{m_R^2}{u_E}\right) = \sum_{z_E = s_E,t_E,u_E} A\left(\frac{m_R^2}{z_E}\right)$ and $A(x) \equiv \sqrt{1+4x}\log\left(\frac{\sqrt{1+4x}+1}{\sqrt{1+4x}-1}\right) $. Both of these quantities ($m_P$ and $\lambda_P$) are physically observable. Suppose we conduct an experiment at a reference momentum $p_{E0}\equiv(s_{E0},t_{E0},u_{E0})$ and make measurements of the pole mass and 4-point coupling with the result $\lambda_{P0}, m_{P0}$. We now have a system of two equations in the three unknowns ($\lambda_R,m_R,\mu$). This means that in principle I can solve for $\lambda_R = \lambda_R(\mu)$ and $m_R = m_R(\mu)$. I would now like to make a prediction for the 4-point amplitude at a different momentum $p_{E}' \neq p_{E0}$. Since I have two equations in three unknowns I need to guess a suitable value for $\mu$ (say $\mu' = \sqrt{s_{E}'}$) which allows me to fix $\lambda_R$ and $m_R$ and then calculate $\lambda_P (p_E')$. This procedure seems quite ad hoc to me because a different (arbitrary) choice of $\mu$ (e.g. $\mu'/2$ or $2\mu'$) will lead a different physical answer (albeit only logarithmically different). From what I can gather from the literature, the problem of determining the renormalization scale I described above is a genuine problem in actual calculations of QCD (e.g. http://arxiv.org/abs/1302.0599) which leads theorists to introduce so-called "systematic uncertainties". What concerns me is that I haven't been able to find any mention of this problem in any textbook on quantum field theory that deals with QED or QCD (anyone know of a reference?). Since this problem appears already in arguably the simplest QFT of $\lambda\phi^4$, I would expect it also to occur in e.g. 1-loop calculations of Bhabha scattering but I haven't been able to find any mention of it in this context. Does anyone know how this problem is dealt with in real loop calculations (e.g. at LEP or LHC?) Also, I would be interested to know if there is any analogue of this problem in condensed matter theory.


title: Why the cavity radiation has the form of standing waves?

text: I'm reading about the cavity radiation in the context of blackbody theory. I'm asking myself: WHY do we describe this radiation by the use of standing waves? Why can't they be not-standing, maybe reflecting along some strange paths inside the cavity itself?


title: How does pressure factor into mechanics?

text: We just started learning about pressure in our school, and i am a bit confused. I tried to google extensively, but i didn't really find much. In mechanics, so far in our school we've been taught about Newton's laws of motion and some related stuff, and suddenly we started learning about pressure. I asked this same question to my physics teacher, but he failed to answer. How does pressure actually factor into mechanics? We studied so far that to cause motion, you require a force. Then we study about pressure, and the examples are ones like pressure of gas in a contained beaker and then lowering/raising it's volume. Suddenly in the applications section however there are things like Paper pins have a low surface area at the end to maximize pressure, and that you can't walk on sand easily because the sand depresses under pressure. I understand the math behind it, but what does the math actually mean? I mean, a force is being applied. It should cause motion, but apparantly if the surface area of the end is small then pressure goes up? The force here remained constant(?) so how does pressure factor into the mechanics? Why did that just happen? I realize i may be way off topic and asking a stupid question, but i am having some trouble comprehending how pressure works. I understand the examples of things like pressure in a container, atm. pressure e.t.c but how does pressure transmit in a case like this? My apoligies if this is a stupid question.


title: Is the potential energy of molecules related to its temperature?

text: Do ideal gases have a certain potential energy at a certain temperature?


title: What is the physical meaning of fact, that Reissner-Nordstrom black hole is thermodynamically unstable?

text: It is known, that Reissner-Nordstrom black hole is thermodynamically unstable [1].   * Does it mean, that there is no Reissner-Nordstrom black hole in physical world?   * Does it mean, that there may be phase transition?   * Does it mean, that it can be stable for enough long time? [1] For example, arxiv.org/pdf/0812.1767v2.pdf pp.19-20.


title: What's the criteria for black hole thermodynamically stability? (And dynamical?)

text: It looks like usual criteria (positivity of Hessian; what geometrically means a cancave of entropy) is no useful, becouse entropy is not additive and not extensive for black hole. Then what is the right criteria?


title: Dark and bright areas around atoms in a scanning tunnelling microscope image

text: Recently IBM created world’s smallest ever animation on an atomic scale video. Researchers made the animation using a scanning tunnelling microscope to move thousands of carbon monoxide molecules to show a boy dancing, throwing a ball and bouncing on a trampoline. My question is, why in this video we see a pattern of dark and bright circles around each molecule? What do they represent?


title: How we approach RLC circult from RLGC model?

text: In the text, it introduces a practical model to investigate a transmission line (like BNC cable), it considers the transmission line has resistive $R$, inductance $L$, conductance $G$ and capacitance $C$. The model is illustrated as follow ![enter image description here](http://i.stack.imgur.com/Iaxe3.jpg) It is easy to derive the (telegraph) equations and figure out the impedance Z to be $$Z = \sqrt{\frac{R+iX_L}{G+i/X_C}}$$ where $i$ is the imaginary unit, $\omega$ is the angular frequency, $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance. And in other section, it introduce a RC circuit and RLC circuit, in which, the impedance are $$Z_{RC} = \sqrt{R^2 + X_C^2}, \qquad Z_{RLC}=\sqrt{R^2 + (X_L - X_C)^2}$$ It is pretty confusing because from RLGC model, if we make the electrical conductance $G$ to zero and consider no inductance ($L=0$), so the circuit becomes RC circuit, but from the first equation for the impedance given by the RLGC model, the impedance should be $$Z = \sqrt{-iRX_C}$$ Why are they not the same? How to approach RC and RLC case from RLGC model?


title: Calculating the coefficient of thermal expansion in liquid

text: I am trying to write a matlab function that calculates the coefficient of thermal expansion of water from a given temperature. From what I understand the thermal expansion coefficient is calculated as the degree of expansion divided by the change in temperature, expressing the tendency of a fluid to change in volume with a change in temperature. The following code calculates this in matlab:               T = 20; % initial temperature     dT = 0.001; % change in temperature °C     V = 1; % volume m^3          rho1 = waterDensity(T); % density of temperature     rho2 = waterDensity(T+dT);% density of second temperature          V2 = rho1./rho2;     alpha = (V2-V)/dT; % coefficient of thermal expansion in deg C-1      where waterDensity is an external function that calculates the density of water in kgm-3 from a given temperature. Here, I'm having trouble making sense of this, mostly I don't understand why we need the line that starts with V2. I would have though that if mass is conserved the volume is given by the density so alpha should be calculated by the change in density divided by the change in temperature i.e.               alpha = (rho - rho1) ./ dT      Could anyone explain to me why this isn't the case. I know that the first method works because it gives the same result as:               alpha = 1.6e-5 + 9.6e-6 .* T;      which is commonly used in the literature.


title: What is the Andromeda Paradox?

text: I have been studying causality (specifically why there is no such thing as a simultaneous instant of time across all observers) recently and I keep hearing references to the Andromeda paradox. Can anyone tell me what it is and how it is resolved? I've tried reading what Wikipedia says about it, but I could really use someone's explanation. **Edit**   Ok, after reading the related question, now I'd like to know does this imply there is some sort of privileged reference frame, in which there exists absolute velocity and absolute time? I ask because the answer for the other question seemed to indicate that the actions seem to be at different points of simultaneity but that in fact the person not moving is getting it right. But this seems strange to me because if we accelerated our galaxy to a different speed, we would still perceive our stationary observer as being right even though they'd see something different than before the acceleration. So who is right? In what frame can we be sure that our present is Andromeda's present?


title: Parallax barrier to achieve "dual displays"

text: I am curious as to whether or not parallax barrier could be used to create the illusion that a screen is bigger than its physical dimensions. I assume this would only work in one dimension (width?). But could I potentially see a single screen as if it were a dual monitor setup? Right-eye sees the left screen, left eye sees the right screen? Sorry if this is the wrong exchange to post this on.


title: Thought experiment with entangled electrons

text: Suppose we start out by having two entangled electrons. We separate them by some distance and we put one electron inside a thin loop of wire connected to an extremely sensitive voltage measuring device at lab 1 and the second electron at lab 2. At this point, no measurement is made. Both electron's spin's are undetermined. Therefore... -The direction of the spin is undetermined. -We do not know if the electron's spin, $s=\pm\hbar$. -We do not know its magnetic moment,$ m = (-gu_bS)/\hbar$. $S = \frac{h}{2\pi}\sqrt{s(s+1)}$, $g$ is g factor, $u_b$ is Bohr Magneton. -We do not know the Magnetization, $M = (N/V)m$, where $N$ is number of magnetic moments and $V$ is the volume of the system in question. -We do not know the magnetic field, $B = \mu_0(H + M)$, where $\mu_0$ is vacuum permeability, $H = M/X$, where $X$ is the magnetic susceptibility. The magnetic field of the first electron at lab 1 is undetermined because no measurement has been made, therefore no magnetic field can possibly be present, $B = 0$. (Please correct if wrong) Now, we measure the second electron's spin by sending through a Stern-Gerlach device and having that electron hit a screen to record its spin value, $+$ or $-$, at lab 2. Regardless of whether or not the second electron's spin is up or down, we know that the first electron's spin is now determined. This means that the magnetic field at lab 1 has been determined and therefore a magnetic field must be present. Since there is a change in magnetic field from 0 to some non-zero value, there must be a change in voltage from the law of induction, $V = -\frac{d}{dt}\left(BNA\cos\theta\right)$, where $A$ is the area the magnetic flux is going through, and $N$ is the number of coiled wire. This suggest that there is a measurable effect, although extremely small, at lab 1 due to the entanglement breaking in lab 2. My question is, is this theoretically correct? If yes, then I would suggest this as a method of communication by the following. By creating a large ensemble of these entangled electrons, A-A', B-B', C-C', D-D', where A is an electron at lab 1, entangled to a second electron, A', at lab 2, etc. For example, by choosing to measure A' and C' and leaving B' and D' alone at lab 2, we create a measurable effect at lab 1 for the electrons' A and C, voltage is changed. Thus, this would constitute a sent message as (1 0 1 0). Where 1 would be a voltage change and 0 would be no voltage change. This would of course be a one time messaging system, but it still does not negate the fact that it would be able to send a message via entanglement by this specific scheme. This is true only if my scheme is logically and theoretically correct. I am open to scrutiny and correction. Please help me determine if my scheme is wrong. Thank you. :D


title: Angular momentum coupling-calculation of Clebsch–Gordan coefficients

text: I am facing problem in calculating the value of given Clebsch–Gordan coefficients representing the coupled angular momenta of two-particle system. For example $$\begin{pmatrix}2 & 1 & 2 \\\ 1 & -1 & 0\end{pmatrix}$$ In the book it is first expanded in three parts like $$\begin{pmatrix}2 & 1 & 2 \\\ 0 & 0 & 0\end{pmatrix},\begin{pmatrix}2 & 1 & 2 \\\ 1 & -1 & 0\end{pmatrix},\begin{pmatrix}2 & 1 & 2 \\\ -1 & 1 & 0\end{pmatrix}$$ I am really very much confused that which symmetry property should use? I know here orthonormality condition of coefficient applied here. But why here in the second row values are first $\begin{pmatrix}0 & 0 & 0\end{pmatrix}$ then $\begin{pmatrix}1 & -1 & 0\end{pmatrix}$ and then $\begin{pmatrix}-1 & 1 & 0\end{pmatrix}$? Please help! I will be thankful to you.


title: What are the lines visible between two cards held edge-to-edge?

text: Hold two cards (say credit cards) edge to edge, anything from a very slight touch to about 1/3 mm separation, in front of any ordinary light source. When I do this I see several fine dark parallel lines in the gap. What are those? Are they discussed in elementary texts?


title: How fast can toy helicopters change the turning direction of their propellers?

text: I saw someone do some tricks with a toy helicopter where he would turn it upside down for a while and it would still stay in the air. I thought it should have crash or at least not fly for very long in that position, but I was wrong. Is it possible for the helicopter to have change its propellers turing direction so fast (in less than a second) so it could still fly upside down?


title: What is the relationship between Cyclic Coordinates and Killing Vector Fields?

text: My question is related to this question. There are three or four other questions on Killing Vector Fields here, however none of them that I have seen address my question. $\\\$ I've been studying some Differential Geometry, and have been thinking about the Killing Vector Fields. $\\\$ In Stan Liou's answer he mentions cyclic coordinates. In the geodesic equation $$ \ddot{y} + \Gamma^y_{xx}\dot{x}\dot{x} + \Gamma^y_{xy}\dot{x}\dot{y} + \Gamma^y_{yx}\dot{y}\dot{x} +\Gamma^y_{yy}\dot{y}\dot{y} = 0 $$ we see that x is a cyclic cooordinate. Moreover, he mentions Killing Vector Fields. $\\\$ I am fimilar with the concept of cyclic coordinates giving an integral of the motion, as discussed in Landau Vol. 1, for example. Here, for $q_i$ a cyclic coordinate the Euler-Lagrange equation for $q_i$ $$ \frac{d}{dt} \left( \frac{ \partial L}{ \partial \dot{q}_i} \right) - \frac{ \partial L}{ \partial q_i} = 0 $$ reduces to $$ \frac{d}{dt} \left( \frac{ \partial L}{ \partial \dot{q}_i} \right) = 0 $$ whence $$ \frac{ \partial L}{ \partial \dot{q}_i} = E_i $$ say, is an integral of the motion. $\\\$ Moreover, we know that a Killing Vector Field $K$ is an isometry of the metric tensor $g$ such that $$ \mathcal{L}_{\small K} g = 0 $$ That is, $K$ is a symmetry of the metric tensor $g$. So for diffeomorphisms $\phi : M \rightarrow M$ which 'move us along the integral curves of $K$' (I don't know the best way to phrase this!) the metric tensor $g$ will remain the same. $\\\$ However, when we write out the geodesic equations, are these two things going to be the same in some way, such that we we find $$ K \sim q $$ for $K$ a Killing Vector and $q$ a cyclic coordinate? It seems like in the cyclic coordinate case we have a hypersurface $\Sigma \subset \mathbb{R}^4$ with Cartesian coordinates $x^{i} = (x,y,z)$ with $t$ our 'extra' coordinate that we 'move along' and the integral of the motion stays the same, where $(M,g)$ here is our Lorentzian manifold. $\\\$ Furthermore, I am quite fimilar with the Killing Form in the Theory of Lie Groups, as a symmetric bilinear form give by $$ K(X,Y) = \mbox{tr}(\mbox{ad}_X \mbox{ad}_Y) $$ where $X, Y$ $\in \mathfrak{g}$ for some Lie algebra $g$, with $\mbox{ad}_X$ the adjoint representation of $X \in \mathfrak{g}$. Then this Killing form is bi-invariant under the action of the Lie Group G, and has many other nice properties like non-singularity and being negative-definite for semi-simple compact groups. $\\\$ We can also use the Killing Form to define a metric on the underlying manifold of our Lie Group $G$, so part of me feels that these ideas are connected, but thus far I cannot merge them together in my head. $\\\$ So in brief, my question is, are Killing Vector Fields 'simply' cyclic coordinates? (I use simply here loosely) If not, what exactly is the difference? $\\\$ Thanks


title: Loopless voltage measurement

text: I think we are all very well familiarized with the classical voltmeter. Classical voltmeter has two conducting wires that bring two potentials into the box. In the box we have **well controlled conditions** , in which potential difference (voltage) starts current, and current through the galvanometer deflects the pointer. Anyways, conducting wires are absolutely necessary by this method in order to bring potentials to the box. However, a person posted the question, where it was necessary to measure voltage on the ends of the bar in the magnetic field. Of course, as soon as you do that with classical voltmeter, you create a loop and loop can generate additional voltage you simply do not want. So my half-way question is: **Is it possible to make a loopless voltage measurement?** Of course, in principle I could imagine such a measurement: I would make an electric field probe and put it into the bar, measuring electric field along the bar. After I've obtained electric field in the every point of the bar, I could calculate voltage by integrating electric field between bar's ends. However, this seems to be very difficult measurement and conditions are not really **controlled**. My final question is: **Is there any other loopless conventional method of measurement available?**


title: Application of trigonometry to the slingshot effect/gravity assist

text: I have been trying to understand the formula $$v_f^{2}=v_i^{2}+2V(V(1-\cos\beta)+v_i(\cos(\alpha-\beta)-\cos\alpha))$$ as it relates to Fig. 2 on page 5 of this exposition: http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf The angles between the positive directions of $V$ and $(v_i,v_f)$ are denoted by $(\alpha,\alpha^{\prime})$, respectively. $\beta$ is the positive rotation angle of $v_i$ arrowed between the dashed lines. I surmise that the law of cosines is at work, but I fail to see precisely how. Can someone provide hints as to how the formula relates to Fig. 2? (One way of answering my question is to partially/wholly derive the equation.)


title: Gravitational field strength and Horizon in Rindler coordinates

text: I came across the following statements in 't Hooft's black holes notes, but not being able to justify them. The metric in the Rindler coordinates $x=\tilde{x}, y=\tilde{y}, z= \rho \cosh{\tau}, t= \rho \sinh{\tau}$ is $$ds^2 = -\rho^2 dt^2 + d\rho^2 + d\tilde{x}^2+d\tilde{y}^2$$   1. Gravitational Field Strength > The actual gravitational Field strength felt by the (Rindler)observer is > inversely proportional to the distance from the origin. **How can I see this from the metric?** How do I mathematically and physically justify this claim? 2\. Horizon Why does the surface $x=t$, act like a Horizon i.e. anything from outside the shaded area shown in the picture can't enter into the shaded region? I think, this can be seen by showing that all time-like curves starting from a point outside the shaded area don't enter the shaded region. **But what are all the time-like curves in the Minkowski space-time?** and would this assumption still hold for any arbitrary geometry which is locally like the Rindler metric(e.g. the schwarzchild metric near the horizon?) ![enter image description here](http://i.stack.imgur.com/yIQMK.png)


title: Please correct my understanding of Clausius-Clapeyron and latent heat

text: As I work a problem set, I'm being asked to calculate the latent heat associated with a particular phase transition of $NH_3$ (solid -> liquid, liquid -> gas, solid -> gas). I'm being told to perform the calculation at the triple point (P, T), which I've already calculated based upon two phase specific formulas for vapor pressure: $$\ln{P} = 23.03 - \frac{3754}{T} \text{ (solid) }$$ $$\ln{P} = 19.49 - \frac{3063}{T} \text{ (liquid) }$$ What I'm not getting is how to use Clausius-Clapeyron to come up with a particular latent heat. Don't I need more information than this? It appears to me that if I can't make the simplification that when you transition to vapor you treat the vapor as an ideal gas, that I explicitly need more parameters ($\triangle V$) to make the calculation: $$L = \frac{dP}{dT}T\triangle V$$ Have I missed something simple about this conceptual exercise?


title: Clarification on the Seebeck Effect

text: Alright, I've been interested in the Seebeck effect lately, so I've been trying to learn it. From what I understand, this is measured with the Seebeck Coefficient, which gives you the $\mu\textrm{V}$ (Millionth of a volt) per $\textrm{K}$ (Kelvin). For example (according to this), if I take Molybdenum and Nickel, with 1 Kelvin of difference, I will produce 25 $\mu\textrm{V}$. This is where I need clarification, is this per contact (of any size)? I'd assume that size DOES matter, at which point I'd ask, what unit of surface area is this in? (ex: $\mu\textrm{V}/\textrm{K}/\text{cm}^2$) The only reason why I'd think that it is per contact, is that I can't find any unit of surface area. Thanks in advance for your time.


title: Spin Transition Energies

text: I am reading a paper: http://arxiv.org/ftp/arxiv/papers/1305/1305.2445.pdf On p. 22, the following Hamiltonian is given: $$ H = \mu_B g \mathbf{B} \cdot \mathbf{S} + D(S_Z^2+\frac{1}{3}S(S+1)) + E(S_X^2 - S_Y^2) $$ When $B = 0$, the Hamiltonian becomes: $$ H = D(S_Z^2+\frac{1}{3}S(S+1)) + E(S_X^2 - S_Y^2) $$. We are told that neutral divacancies correspond to $S = 1$, and $D$ and $E$ are the axially symmetric and anisotropic components of the crystal field interaction. They claim the spin transition energies are $D - E$ and $D + E$ when $B = 0$. Can someone explain how you get that? I solved for the eigenvalues of the above Hamiltonian after plugging in the spin-1 matrices into the above equation, but I did not obtain this result. Thanks!


title: Do modern Formula One cars produce enough down-force to drive upside-down?

text: For example, if they were driving at top speed through a long tunnel, could they transition to and stay on the ceiling?


title: Do all the forces become one?

text: 1. Were the forces of nature combined in one unifying force at the time of the Big Bang?   2. By which symmetry is this unification governed?   3. Are there any evidence for such unification of forces?   4. Has ever been published Theory or experiment in this issue? (Even original researches or unpublished theories. Anything that you can start with.)


title: What is the experimental status of AdS/CFT, AdS/QCD, AdS/CMT, etc?

text: What experiments have challenged or supported AdS/QCD, AdS/CMT, etc? What experiments should we look forward to do this?


title: How to understand the transmission coefficient from the following question?

text: I want to understand the transmission coefficient and construct a time- independent Schrodinger equation where $$ V(x)=\left\\{ \begin{array}{c c} \delta(x), & |x| < 1 \\\ \+ \infty, & |x| \geq 1. \\\ \end{array} \right. $$ $$ \Psi(x,0)=\left\\{ \begin{array}{c c} 1, & 0< x < 1 \\\ 0, & others. \\\ \end{array} \right. $$ I find out $\Psi(x,t)$ but it is too complict. And I find the transmission coefficient is zero since $\Psi(x,0)=0$ at $|x|=1$. In another hand, if $\Psi(x,t) \neq 0, (-1<x<0, t>0)$, there must be some 'probability' transimit to $x<0$. Thus, there will be a transimission coefficient. > Can you help me check out $\Psi(x,t)$ and the transimission coefficient? Any > help or suggestion about how to understand transimission coefficient from > this dynamic process will be appreciated!


title: Angular Momentum Operators Non-Degenerate

text: Typically one writes simultaneous eigenstates of the angular momentum operators $J_3$ and $J^2$ as $|j,m\rangle$, where $$J^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle$$ $$J_3 |j,m\rangle = \hbar m|j,m\rangle$$ There seems to be an implicit assumption that the eigenvalues of these operators are non-degenerate. I can't immediately see how this is obvious. Could someone point me in the direction of a reference, or clarify it in an answer? Apologies if I've missed something trivial!


title: Landauer's principle vs Wien's displacement law

text: Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of kTln2? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Wien's displacement law which says the total energy should be in the order of $k\frac{T^4}{h^4}$ (as integrated using Mathematica 8.0)?


title: What information about a meteor's trajectory, size, or height can be derived from a single location?

text: If one sees a meteor, is there any way to get even a rough approximation of its height, entry angle, size, or other characteristic _without triangulation_ from another position? If it appeared as a point source and got uniformly brighter, you'd know to take a step aside. And if it appeared on one horizon, traveled overhead, and disappeared over the other, you'd be able to say "Well, _that_ was a shallow angle of attack." But short of those scenarios (which are, probably for the best, rare), is there anything?


title: Are EM waves scattered the most when the wavelength and the obstacle have a similar size?

text: I heard that when the wavelength and obstacle are similar in size, the scattering is the greatest. Is this true?


title: About Vanishing of BRST commutator in path integral

text: In Witten's paper Topological Quantum Field Theory, about formula (3.2), the property $<\\{Q,\mathcal{O}\\}>=0$ depends on the assertion that $Z_{\varepsilon}(\mathcal{O})= \int \mathcal{D}X \exp(\varepsilon Q) [\exp(-\frac{\mathcal{L'}}{e^2})\mathcal{O} ]$ is independent on $\varepsilon$. Where does the assertion come from? Thanks!


title: Landauer's principle vs Rayleigh–Jeans law

text: Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of $kTln2$? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Rayleigh–Jeans law which says the total energy should be infinite?


title: Reference Request: Classical Mechanics with Symplectic Reduction

text: I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie algebra) to prove Symplectic reduction theorem (on locally free proper G-action), Arnold- Liouville Theorem (on completely integrable systems) and some more. For instance, both Arnold's mechanics book and Spivak's physics for mathematician does not explain these concepts. I think supplements will help me understand that book's appendix (where it explains reduction theorem with lots of machinery, Ehresmann connection, and so on). Any suggestions on this?


title: fitting van der waals law on real data

text: For an examples class on thermodynamics, I would like to fit Van der Waals law on data of real gasses, say $CO_2, H_2O_2, CH_4$. I want to set out measurements of pressure and volume at constant temperature in a P-V diagram, and make a fit to determine the van der Waals parameters a and b. Only I have no idea where to look for experimental data. Is such data available in databases or articles? Would I have to extract them by hand from plots or are there (electronic) tables available? Any suggestion is welcome.


title: Direction of friction on rollers with plank resting on top

text: In a question, its given that a plank is resting on 2 rollers like: ![enter image description here](http://i.stack.imgur.com/afMB1.png) Then the explanation, as part of computing the forces, for the direction of friction on the rollers look like: ![enter image description here](http://i.stack.imgur.com/KcJHQ.png) Why are the direction this way? Since the rollers are turning clockwise (plank is moving right), I expected the opposite? Explanation on the directions look like: (see line 5 counting from bottom) ![enter image description here](http://i.stack.imgur.com/K5lpx.png) Although, after the computation, the direction of $f_b$ is pointing to the right as its negative.


title: Why is the "canonical momentum" for the Dirac equation not defined in terms of the "gauge covariant derivative"?

text: The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor `i` in front of the vector potential. I know I'm combining two seemingly unrelated things, but they seem very similar, an the covariant form seems much "better" with respect to the inherent gauge freedom in the EM field. I can also see that with the canonical momentum form, the equations remain unchanged after an EM and a QM (phase) gauge transformation. Suffice to say my field theory knowledge is not that impressive.


title: Are there any good reading materials for variational approach in many-body theory?

text: I need something like a summary of existing results, including the treatment of BCS Hamiltonian and Hubbard model. Auerbach's book is a good one but I still hope to get more comprehensive review. My purpose is to compare the methods and give an outline of the mathematical framework behind this. Currently I am working on a small problem, where I am asked to obtain some effective action from variational method. To really understand what I am doing, I need to understand other applications of the variational methods and hopefully to draw some analogies.


title: Question about moments?

text: Take a look at this diagram: http://www.chegg.com/homework-help/questions-and- answers/7-pipe-assembly-supports-vertical-loads-shown-determine-components- reaction-ball-socket-jo-q2260091 If you calculate moment around point B, you can ignore the forces of the tension and only the forces at the ball and socket and the weights come into play. Therefore for moment at B to be zero, the Y and X components of the ball and socket have to be zero. However, if the x and y components at the ball and socket are at 0, theres nothing to cancel out the X and Y components of the tension forces, meaning the forces are at unbalanced. What am I missing? **Please answer MY question not the question at the link**


title: How would Kohn-Sham orbitals differ from 'true' elecron wavefunctions?

text: How would the non-interacting electron orbitals from a perfect DFT solution for a given potential shape differ from the 'true' electron wavefunctions? Or can you only really talk about the total wavefunction? Would they be less localised as they do not include interaction effects? Since Hohenberg-Kohn showed that a functional of the wavefunction could be transformed into a functional of the density, any 'measurements' on the total density would have to yield the same results as measurements on the total wavefunction, I think. Intuitively, if we imagine a 3-electron system, the ground state consists of spin-up and -down electrons in the lowest state, and a singlet electron in the next state. Since there is correlation and exchange, the total wavefunction would be a superposition of the various possible combinations of these.


title: Normal modes of a flexible rod clamped at only one point

text: I am interested in the vibrations of a thin, flexible rod that would only be clamped at one point, properly I'd like to calculate its eigenvalue. But the way I learned it in wave mechanics doesn't seem to apply here. The equation is: $$ \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}$$ with $u = u(x,t)$ defined has the micro-displacement in one transverse direction, $x$ the longitudinal direction of the string, $c$ the speed of sound. Or applying $$u (0,t) = \partial_x u(x,t)|_{x=0}=0$$ has no non trivial solutions, and so no spectrum. My real interest is to calculate the vibrational spectrum of a cantilever clamped at one of it extremities, which also obeys a second degree wave equation. (and I know that software can calculate these spectra, with the same boundary conditions, using the same equations)


title: Nanotechnologies: current problems and general limitations

text: I have some questions about Nanotechnologies. I'm not physicist and my knowledges about the topic are limited by the "Engines of Creation" book by Eric Drexler. So I think that my questions are related mostely to popular science.   1. As far as I know the molecular mechanisms like nanoassemblers and nanofactories exist in nature(I mean alive cells). Why have they not been created artificially by now?   2. In case that they have been artificially created yet, do they have any limitations in developing of macro objects? For instance, macromechanisms which exist in nature like animals are grow very slowly(months and years) and the initial program(i.e. DNA encryption) doesn't allow to make accurate copies. I mean that the twins who are growing from the same genetic material have a significant difference in adulthood(i.e. different phenotypes). Is this phenomena the result of some fundamental low of nature like Heisenberg uncertainty principle or any other that can't be resolved by modern technological methods?   3. What are the top well known laboratories and organisations in the world which are engaged to resolve such kind of problems? Thanks in advance.


title: Tracing out an observable vs integrating over unitaries

text: Let $O$ be an observable on a Hilbert space $\mathcal{H}$, and let $B$ be a subset of the spins composing $\mathcal{H}$, and let $\bar{B}$ be its complement. Now define $\displaystyle O_B = \frac{1}{\operatorname{Tr}_{\bar{B}}\mathbf{1}_{\bar{B}}} \operatorname{Tr}_{\bar{B}}(O) \otimes \mathbf{1}_{\bar{B}}$. Is this quantity equal to $\displaystyle \int d\mu(U) U O U^\dagger$? The integral is taken over the set of unitary operators acting on $\bar{B}$ and $\mu$ is the Haar measure of $U$. If so, why is this the case? What physics course/book/reference introduces these sorts of integrals? Note: this question came up from trying to understand the following paper: http://arxiv.org/abs/quant-ph/0603121


title: Does the Kelvin have a rigorous definition?

text: From Wikipedia: > _The kelvin is defined as the fraction 1⁄273.16 of the thermodynamic > temperature of the triple point of water._ That presupposes that we can take a fraction of temperature. Now, taking a fraction of mass, distance, or duration is well defined. Is taking a fraction of temperature well defined? Doesn't this depend on operational definitions of temperature - such as change in length? This is similar to the objection many physicists had with measuring human perceived loudness: What does it mean to say that humans perceive loudness as proportional to log of pressure? We can order perceived loudness, but how do you add two perceptions of loudness, so that any type of proportionality makes sense? Related question: If $K$ cannot be defined in terms of basic units, why do we say $mgs$ and not $mgsK$?


At what depth in the water atmospheric pressure is 100 times greater than on the ground? This question comes from the fact that average pressure in Earth( 1000 mbar) is 100 times greater than in Mars( 7 mbar).


title: Numerical aperture (NA) of an optical fiber

text: Numerical Aperture (NA) (for fiber optics) is usually used to denote the acceptance cone for a multi-mode fiber. **Does NA also describe the expansion of light emitted from the end of a fiber?** I have a 1mm core 0.22 NA PMMA fiber, I would like to collimate the light once it's reached a 1.5" diameter. $$ n\sin(\theta)=NA \\\ \theta = \arcsin(\frac{.22}{1.4914}) \\\ \frac{.5 \cdot 38 [mm]}{ \tan(\theta)}=\left \\{ \text{focal length} \right \\} $$ So, I should need a ~130 mm focusing lens? **edited in response to answer bellow** is n the index of refraction of the fiber core (1.4914), or free space (1)?


title: Why is the speed of light considered as a fundamental constant if its speed changes with medium resulting in refraction?

text: I know that the speed of light, the universal constant of gravitation and the Planck's constant are considered to be the three fundamental constants of the universe. But, why is speed of light considered as a fundamental constant? The speed of light changes from medium to medium for satisfying Fermat's principle then, how can it be a constant?