title: Capacitance of a conducting disk

text: I'm reading this(PDF) derivation of the capacitance of a thin conducting disk. The surface charge density of such a disk can be shown to be: $\sigma(r) = \frac{Q}{4\pi a\sqrt{a^2 - r^2}}$ (in Gaussian units) where r is a point on the disk, and a is the radius of the disk. The text at the end of the first page states: "If we let s measure the distance radially inward from the circumference of the disk, then the charge density (3) varies as $\frac{1}{\sqrt{s}}$ for small s. However, the charge density near the edge of a conductor whose surfaces intersect at an exterior angle of $\frac{3π}{2}$, as in the present problem, is known to vary as $\frac{1}{\sqrt[3]{s}}$, for s measured normal to the edge along either surface." I am having trouble visualizing what the author means by this. Could someone help me understand the geometry better? Thanks.


title: How can a point-like particle "feel" gravity, if locally the curvature of spacetime is always flat?

text: I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that > _Locally, as expressed in the equivalence principle, spacetime is > Minkowskian, and the laws of physics exhibit local Lorentz invariance._ > (Wikipedia:General relativity) But if this is the case how does a point-like particle know, which direction it should follow in a gravitational field? Can this be a seen as a hint that point-like particles cannot be used as underlying concept for a field theory like general relativity?


title: Video documentaries about physics?

text: Does anyone know about good video documentaries about physics, possibly on YouTube? I've seen a couple of them about string theory, but I could not find many others.


title: Why is there a scarcity of lithium?

text: One of the major impediments to the widespread adoption of electric cars is a shortage of lithium for the batteries. I read an article a while back that says that there is simply not enough lithium available on the entire planet to make enough batteries to replace every gasoline-powered car with one electric car. And that confuses the heck out of me. The Big Bang theory says that in the beginning, there was a whole bunch of hydrogen, and then lots of hydrogen started to clump together and form stars, and those stars produced lots of helium through fusion, and then after helium, all the rest of the elements. That's why hydrogen is the most common element in the universe by far, and helium is the second most common. Well, lithium is #3 on the periodic table. By extrapolation, there ought to be several times more lithium around than, say, iron or aluminum, which there is definitely enough of for us to build plenty of cars with. So why do we have a scarcity of lithium?


title: What near-future measurement technique improvements will extend our physical knowledge?

text: Inspired by the wiki-article on atom-lasers: > Another application, which might also benefit from atom lasers, is atom > interferometry. In an atom interferometer an atomic wave packet is > coherently split into two wave packets that follow different paths before > recombining. Atom interferometers, which can be more sensitive than optical > interferometers, could be used to test quantum theory, and have such high > precision that they may even be able to detect changes in space-time , i would ask you to list more examples of **CURRENT measurement techniques getting developed & tested, that will likely drastically enlarge our knowledge and falsification options of physical theories. But not Sci-Fi and Star Trek Physics ;)**. Mainly i can think of improvements that will allow   * more accurate measurement of physical constants/parameters (higher particle energies CERN, better atomic clocks/time definition by Pulsar periods)   * direct measurement of so far only indirect definable physical quantities (e.g. atom laser)   * at all new concepts of (real-time) measurement techniques due to improvements e.g. in computational physics (e.g. quantum computing) or better visualisation of phenomena (adaptive optics in telescopes) Please name the technique and give short example/link to the physics field profiting from it. Examples i mentioned can also be explained further, as im no expert in these fields


title: why do we Carbonate drinks not any other gas

text: Carbonation is done in all fizzy drinks. Also it forms a small amount of carbonic acid which we feel on our tongue. We can diffuse oxygen as well in the water at higher level. Except for acid formation is there any other reason we don't use any other gas other than $CO_2$?


title: Is there a four dimensional form of Born's Rule -redub

text: Generalizing Born's Rule for 4-dimensions $x_4$, write $$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$   1. Is this consistent with quantum mechanics?    2. Is this a generalized form of the Born's Rule for the phase space of the wave function? @DavidZas I was thinking of a four dimensional space where Time is not a parameter. There been recent work on Time Reversed light pulses. http://prl.aps.org/abstract/PRL/v106/i19/e193902 and the PHYSORG description http://www.physorg.com/news/2011-05-physicists-time-reversed-pulses.html And this leads me to wonder about abstract phase space, and I mean phase both in the epoch angle of a wave function and the phase space of a Hamiltonian (which yes, is very confusing). The way I see it, the wave function implictly has the Action of phase space built into it. $\psi = e^{i(\omega t - k x)} = e^{\frac{i}{\hbar }(\mathbb{E} t - \mathbf{p} x)}$ Where the phase and the action are connected by $\frac{\mathbb{E} t - \mathbf{p} x}{\hbar }$ Imagine dividing the phases by the action of the particle It's a bit like there's a blob (note the very precise language) of phase space wobbling about like the liquid drop model of the nuclear physics. Only this is a more abstract phase space that's oscillating. So, again, would Born's Rule apply to such a system? Actually after rereading what I just written, I'll be surprised if anyone replies this time! :)


title: Is this claim from historician true for physicist point of view?

text: "The original Naxos I had a vertically polarized antenna, with poor results as the British radars initially used horizontal polarisation. (This seems to have been a case of the German designers being smarter than the British ones, with unfortunate consequences.) Naxos Ia had a triple antenna, with elements crossing each other at 45 degrees to avoid this problem. Over 1000 sets of Naxos I were produced, for U-Boot installations but also for the Luftwaffe, the original customer." is a passage from http://uboat.net/technical/detectors.htm The question is: Why the British antenna seem to have advantages over the German ones?


title: This sentence makes no sense, electrostatics and electrons moving in a conductor - current

text: ![enter image description here](http://i.stack.imgur.com/QW7ff.jpg) I highlighted the part where the confusion is. The sentence said that the potential difference is 0, yet it then immediately talks about how electrons can have motion. What are they trying to say?


title: Where can I get latest full updated list of supernovae candidates from the SDSS-II Supernova Survey?

text: I am doing a project where I need the list of supernovae candidates from the Sloan Digital Sky Survey II (SDSS-II) supernova survey. **One list** is available as part of the data-release-7 (DR7) website. The list contains 902 confirmed candidates. On the other hand, **there is a list** on another website for the supernova- survey project, where the list is supposedly confirmed _and_ updated. This list has 1030 candidates. Not only is is the number of candidates on each list different, but each list has candidates that are not on the other list. **The question at first, then, was which list is correct one?** There is no information on the SDSS DR7 website on when the list was last updated, but on the main site for the external project site, the footer says that the page was last modified in 2009, which is after DR7. I am not sure if this date is the same for the page with the actual list, though. When I inspected the overlapping entries (those sharing the same internal SDSS id), I noticed that the larger list at least for the ones that I looked at, in some places had information in a field that was blank in the smaller list. This was usually the value of the redshift for the given supernova candidate. I took this as an indication that the larger list, i.e. the one that is not on sdss.org, is actually the newer of the two lists. **But what about the confirmed candidates that are still listed in the older (smaller) list? Are they no longer considered supernovae candidates?** I do not want to have to guess which list is better one to use, and I am not sure that merging the candidates from the older confirmed list that are not on the newer into the newer list, since there must be a reason why they are not included in the latter in the first place. But this rationale for this decision is not given alongside the list or anywhere else on the project webpage that I looked. _What I really want is the latest complete and updated list of supernovae candidates available._ **So the question is, where do I obtain such this list? And when was it made?**


title: How should I be thinking about tides?

text: I am working on a project for physics that involves tides. This is my current mind set when thinking about tides: The earth applies a gravitational force on some mass `M`. The moon & sun apply a gravitational forces on the mass `M` away from the earth. The bodies of water on the earth have some acceleration towards the moon & sun due to their forces (F=ma). The bodies of water accelerate till they reach a sort of equilibrium between forces. I would like you to poke holes in my understanding of tides and try and answer my naive questions bellow. How would you describe tides effected by the moon using forces? How can that be used to calculate the height of tides?


title: Careers in physics that are related to quantum mechanics or relativity?

text: Personally I learned Physics in high-school and found it very interesting, I read up a lot about physics in my free time. Personally I am also a programmer which I think is also good when it comes to Physics. Now I want to start studying in a university and my dilemma right is mainly between Math/Computer Science/Physics. It's possible to do a degree in just one or combine two of them. Currently, the subjects that I find most interesting in Physics are the most advanced ones: Relativity and most of all Quantum Mechanics. I've always been fascinated with materials, what they are formed of at the most elementary level. Now for the real question. From what I usually read it's very hard to get into physics research, of course it's not impossible but still I'd like to ask: Are there careers that are related to quantum mechanics or relativity that are not as a researcher? If so, what are they? can you give examples?


title: Could geothermal energy extraction in Yellowstone significantly diminish the threat of a supervolcano?

text: I recently read an article about a team in Iceland that had managed to drill directly into magma coming up from the earth's mantle, and then use the resulting bore-hole to generate steam to power turbines. This generated significant amounts of electricity, but would also presumably cool the magma slightly. If we did this sort of energy extraction on a much larger scale in active hotspots, could it prevent volcanic eruptions, or at least significantly diminish their potential danger? Might it possibly work as a prophylactic against a possible supervolcanic eruption in Yellowstone?


title: For 2 electrons in a simple harmonic oscillator (SHO) potential, is $\langle x^2\rangle$ the same as $\langle(x_2 - x_1)^2\rangle$?

text: We're dealing with 2 electrons in a simple harmonic oscillator (SHO) potential. We're given the creation and annihilation operators as well as the position operator and I have to find the expectation value $\langle(x_2 - x_1)^2\rangle$ where $x_2$ and $x_1$ correspond to the positions of the two electrons. My problem is that I don't know how to input those two different positions into the position operator, which by the way is defined as: $$\hat{x}_i=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}_i+\hat{a}_i^\dagger)$$ So there you have it, I am stuck because I'm unclear on notation. **EDIT** I am still stuck, I don't know what the $x_1$ or $x_2$ operators would look like. The definition above of the position operator, as I read it, is telling me that I should make the following substitution to get the $x_1$ and $x_2$ operators: $$\hat{x}_1=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}_1+\hat{a}_1^\dagger)$$ $$\hat{x}_2=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}_2+\hat{a}_2^\dagger)$$ But this doesn't make a lot of sense to me, because now thelowering operator would be this: $$\hat{a}_i\vert n \rangle_i=\sqrt{n} \vert n-1\rangle \rightarrow \hat{a}_1\vert n \rangle_1=\sqrt{n} \vert n-1\rangle$$ $$\hat{a}_i\vert n \rangle_i=\sqrt{n} \vert n-1\rangle \rightarrow \hat{a}_2\vert n \rangle_2=\sqrt{n} \vert n-1\rangle$$ And the raising operator would be this: $$ \hat{a}_i^\dagger\vert n \rangle_i=\sqrt{n+1} \vert n+1\rangle \rightarrow \hat{a}_1\vert n \rangle_1=\sqrt{n+1} \vert n+1\rangle$$ $$\hat{a}_i^\dagger\vert n \rangle_i=\sqrt{n+1} \vert n+1\rangle \rightarrow \hat{a}_2\vert n \rangle_2=\sqrt{n+1} \vert n+1\rangle$$ which to me looks like they do exactly the same. Now someone much more knowledgeable than me said that this would be the wavefunction on which we operate: $$\vert \Psi\rangle = \frac{1}{\sqrt{2}}\lbrace\vert \Psi_0(x_1)\rangle\vert\psi_1(x_2)\rangle + \vert\Psi_0(x_2)\rangle\vert\psi_1(x_1)\rangle\rbrace $$ where $\vert\Psi_0(x_1)\rangle$ means the ground state of the first particle and $\vert\Psi_1(x_1)\rangle$ is the first excited state of the first electron, you get the idea. So this is fine by me, I still don't know how I am to solve the problem because I don't know what $x_1$ or $x_2$ would look like nor what the difference between those operators is. Am I making sense here? I am being thrown off by subscripts, I can't see any difference between those two operators, and so I can't do the problem.


title: Speed of the Moon

text: Why the motion of the Moon looks very slow in the sky? Doesn't it need the high speed in order to escape the earth's gravity?


title: Potential energy and average force?

text: A body of mass $15\,kg$ originally at rest $5\,m$ above the ground falls and penetrates $15\,cm$ into soft earth.   * Determine the loss of potential energy    * Determine the average resistance of the earth Please help explain and solve this. I got 735 Joules as the potential energy $E_{pot}=m\cdot g\cdot h$. As for the other one I am stumped. Please help.


title: Radiated power from a 'source volume' outside which charge & current are zero (i.e. derive radiated power from Jefimenko's equations)

text: In classical electrodynamics, what is the radiated power from a generalized source (consisting of charge density $\rho$ and current density $\vec{J}$) in vacuum? Let us define $V_s$ to be the smallest sphere that completely encloses all source points -- that is, it is the smallest sphere outside which $\rho\left(\vec{r},t\right) = \vec{J}\left(\vec{r},t\right) = 0$ for all $t$. Let us say $\mathcal{R}_s$ is the radius of $V_s$. Now let us define another sphere $V$ concentric with $V_s$. Let us say the radius of this sphere is $\mathcal{R}$, and $\mathcal{R} > \mathcal{R}_s$. So what is power crossing the surface of $V$ at any instant? In other words, what is the value of the following? $$ P\left(\mathcal{R},t\right) = \oint_{\partial V} \vec{S}\left(\vec{r},t\right) \cdot \space d\vec{s}\left(\vec{r}\right) = \frac{1}{\mu_0} \oint_{\partial V} \left( \vec{E}\left(\vec{r},t\right) \times \vec{B}\left(\vec{r},t\right) \right) \cdot \space d\vec{s}\left(\vec{r}\right) $$ Below, I have provided an answer for the 'far field' approximation ($\mathcal{R} \gg \mathcal{R}_s$). Are there 'sanitized' expressions for cases where this approximation does not hold? Even if there isn't a 'clean' expression for the general case, what I am looking for is the following. Let us say that there are two instants $t_1$ and $t_2$ such that for all $\vec{r} \in V_s$ we have $\rho\left(\vec{r}, t_1\right) = \rho\left(\vec{r}, t_2\right)$ and $\vec{J}\left(\vec{r}, t_1\right) = \vec{J}\left(\vec{r}, t_2\right)$. If we now define: $$ \mathcal{P}_{\infty}\left(t\right) = \lim_{\mathcal{R}\to\infty} P\left(\mathcal{R},t + \frac {\mathcal{R}} {c}\right) \\\ \mathcal{P}_{\delta}\left(\mathcal{R},t\right) = P\left(\mathcal{R},t + \frac {\mathcal{R}} {c}\right) - \mathcal{P}_{\infty}\left(t\right) $$ ... can the following be proven? $$ \int_{t_1}^{t_2} \mathcal{P}_{\delta}\left(\mathcal{R},t\right)\space dt = 0 $$ Also, is it true (and can it be proven) that if $\hat{n}\left(\vec{r}\right)$ is the unit normal at any $\vec{r} \in V$, there will always be one axis along which $\vec{S}\left(\vec{r},t\right) \cdot \hat{n}\left(\vec{r}\right) = 0$?


title: Space of states in quantum mechanics

text: A state in quantum mechanics I think is just a vector in a complex Hilbert space. As the physical properties are defined up to a phase $e^{i\theta}$ then this Hilbert space is invariant under the action of $S^1$ and the quotient which what is physically relevant is just an infinite complex projective space. Is this right?


title: Why is the dark matter density profile within the solar radius (and local density) uncertain?

text: It seems that we know the rotation curve inside the sun's galactic orbit fairly accurately. Then wouldn't we be able to just take the derivative* of this to get the DM density profile at smaller radii? What then is the primary reason we can't distinguish between different profiles like pseudo-isothermal, NFW etc? a) Is it uncertainty in the rotation curve? If so what is the source of this uncertainty? b) Uncertainty in the amount of luminous matter? Source? Something else? I can see why the profile is uncertain at larger distances from the galactic center since the rotation curve is uncertain there, but why is it uncertain in the region where the rotation curve is known more precisely? *Well, square, multiply by r and take the derivative, but you get the idea


title: Can light waves cause beats?

text: My question is pretty brief. When two sound waves of nearly same frequencies interfere, we get beats. But, I have not observed something like that happening in the case of light. In fact, most of the light around us is a collection of continuous wavelength range which must be all nearly same frequencies. **Can we observe beats in light waves, as in case of sound waves?** _If yes, how to observe them?_


title: Hypercharge of the complex Higgs doublet

text: I was wondering how one can see that the hypercharge of the complex Higgs doublet is $Y=\frac{1}{2}$. Complex Higgs doublet. $$ \Phi(x) = \begin{pmatrix}\ \Phi^{+}(x)\\\ \Phi^{0}(x) \end{pmatrix} $$ Lagrangian density: $$ L = \frac{1}{2}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi + \frac{m^2}{2}\vert{\Phi(x)}\vert^2 + \frac{\lambda}{4!}\vert{\Phi(x)}\vert^4 $$ There is now a **local** $U(1)_Y$ symmetry called hypercharge, given by $\Phi'(x) = e^{-i\tfrac{1}{2}\varphi}\Phi(x)$. But how is the hypercharged deduced? I would really ask more specific questions, but I dont really get it and I am learning this stuff just for me, because I am interested how it works.


title: Boltzmann Distribution of Electrons in Confining Potential

text: I have a particle simulation wherein many non-interacting electrons are trapped in a electric potential well. I would expect, and therefore I initialize according to this, that the electrons would take the shape of a Boltzmann distribution. However, when the simulation is allowed to run for a long time, the charge density converges to a slightly different shape: ![enter image description here](http://i.stack.imgur.com/kAjFj.jpg) In this chart, the horizontal axis denotes position. The red line is the negative of the confining potential. I flipped it just for readability. This is on one vertical axis. On the other vertical axis is both the charge density as the system is initialized (green), and the charge density that the system settles into (blue). The colors are somewhat hard to see, so to clarify, the initialized (green) state is the one that is more negative in the region from around z=20cm to z=60cm Any ideas as to why the Boltzmann distribution is not the correct initialization? UPDATE: This is the graph with the electron-electron interaction turned on. It has pretty much settled into equilibrium by now ![enter image description here](http://i.stack.imgur.com/6icd6.jpg)


title: Reference Request: Fluid dynamics/Elasticity via Lagrangians

text: Would there be a book that does what Landau does in Fluid Mechanics and Theory of Elasticity using Lagrangian's/Action-principles, analogous to the presentation in Landau's mechanics? I have only found brief mentions of a Lagrangian in Fluid mechanics, e.g. Stone and Lanczos, and nothing useful for elasticity. Would sincerely appreciate a reference, thanks. References:   1. Stone - Mathematics for Physics: A Guided Tour for Graduate Students, p. 25.   2. Lanczos - The Variational Principles of Mechanics, p. 360.


title: What is $\epsilon_\infty$ in this equation and why can it be neglected in the IR?

text: I'm reading this paper (warning, PDF) and they mention that the complex permittivity $\epsilon$ and complex conductivity $\sigma$ are related through the equation $$\epsilon - \epsilon_\infty = (4\pi i \sigma)/\omega$$ Then they say that in the far-IR region, $\epsilon_\infty$ can be neglected, so it simplifies the equation. What does $\epsilon_\infty$ represent physically? From Jackson, he says that if some electrons in the material are "free", then the complex permittivity is $$\epsilon(\omega) = \epsilon_b(\omega) + i\frac{Ne^2f_0}{m\omega(\gamma_0-i\omega)}$$, which is pretty similar in form to the first equation. In Jackson, $ \epsilon_b(\omega)$ is the contribution from the electrons that are dipoles rather than free. So why can they be ignored in the FIR?


title: What happens to the free energy of the two-dimensional ising model with vortices?

text: The classical 2d Ising model has a Hamiltonian of the form: \begin{equation} H = -\sum_{m,n = 0}^{M,N} J_1 x_{m,n}x_{m+1,n} + J_2 x_{m,n}x_{m,n+1} \end{equation} The partition function for the model can be written as the sum over all configuration sof the spins $x_{ij}$ times the boltzman factor. Up to an over all multiplicative constant we can re-write this as: \begin{equation} Z = \sum_{x_{ij}}\prod_m (1+t_1 x_{mn} x_{m+1,n})(1+t_2 x_{mn} x_{m,n+1}) \end{equation} Then we can perform the sum by which ever means we like. My current favourite is by following this reference: http://link.springer.com/article/10.1007%2FBF01017042?LI=true, or see http://link.springer.com/article/10.1007/BF02896231. They re-write the partition sum as an integral over Grassmann variables. The computation is technical at points but to my understanding, they essentially sum over all loops in the plane with appropriate boltzmann weights by a convenient choice of integral. This can be seen by expanding the exponential and recognizing that the only terms left after performing the integration will be loops that describe domain walls. They successfully perform the integration on a finite lattice with the topology of a torus and obtain the free energy. When I have more time I will detail some of these calculations. But, for now I have a question: What if we want to consider vortices in the system? One way to do introduce two vortices in the model is to require that some finite number of neighbouring rows satisfy different boundary conditions from all others. Another may be to consider the system Hamiltonian on a cylinder and specify antiperiodic boundary conditions on the lower half and periodic on the upper half. In particular, I would be interested in knowing what happens to the free energy relative to the case of no vortices in the thermodynamic limit (this may be the only tractable part of the problem).


title: Is the neutral pion a singlet?

text: In Griffiths' _Introduction to Elementary Particles_, it is mentioned p. 179 that the $\pi^0$ is a singlet under $SU(2)$ isospin. But it is also part of the $\pi^-,\pi^0,\pi^+$ isospin triplet. How can it be both? Don't particles of a given $SU(2)$ multiplet mix under a corresponding transformation?


title: Is all matter made of virtual particles?

text: This article in New Scientist says that all matter is actually virtual particles popping in and out of existence and nothing more. is this correct?


title: Newton's Law of Universal Gravitation for extended objects

text: Carroll and Ostlie write, in their book _An Introduction to Modern Astrophysics_ (second edition), at the page 33, right after deriving Newton's Law of Universal Gravitation for two point masses using Kepler's third law and the three laws of motion : > Newton's law of gravity applies to any two objects having mass. In > particular, for an extended object (as opposed to a point mass), the force > exerted by that object on another extended object may be found by > integrating over each of their mass distributions. My question is : how is the integrating process a "particular" case of the formula for two point masses ? Shouldn't the integrating process really be a postulate on its own ?


title: Gravity from a singularity as distance approaches zero

text: If you had a singularity (that had mass but took up no space), what would happen to the acceleration of an object as it approached this singularity? I would assume that it would be infinite, since as $r$ approaches $0$, the force of gravity approaches infinity.


title: Suppose a hollow metal sphere filled with helium is dropped in a body of water

text: What are the conditions that would cause said sphere to sink or float? What if the sphere was full of ice instead?


title: Why both yellow and purple light could be made by a mix of red, green and blue?

text: We see the mix of red light and green light as yellow light (#FFFF00). The wavelength of yellow light lies **between** red and green. ![enter image description here](http://i.stack.imgur.com/MmUwY.jpg) But the wavelength of purple light lies **outside of** red and blue. Why can we also see the mix of red and blue as purple? Is it real purple?


title: Are new elements still being created?

text: Last I heard, there were 115 elements (natural and artificial, of course). Are there more still being made? If so, why? I thought this question would be more appropriate on the physics site rather than the chemistry site. But if I'm wrong, do let me know, so I can post it there instead.


title: Calculate water flow rate through orifice

text: I'm not very good with fluid physics, and need some help. Imagine the following setup with water contained in-front of a wall with an opening on the bottom: ![Fluid image](http://i.stack.imgur.com/GjPyZ.jpg) How do I calculate the water flow $Q$?. I have made some re-search and found I need to (partially) calculate the pressure across the opening (orifice). But I don't know the pressure on the back side of the orifice. Can this be solved in any way? **Note:** I'm not saying _"please give me the solution, I'm lazy"_. I want to figure it out myself. But since, in this case, I only found formulas involving calculating pressure drop, I canno't use them to solve the problem. Therefore I'm turning my face to you, to see if there's another way to solve this problem. **Update:** The "tank" holding the water is actually `a big lake`, and the opening is how much the water gate have opened. I need to very precisely calculate how much water flows through the opening.


title: Does heat increase the volume of a gas, and in turn its pressure?

text: Lets say you have 1 liter of hydrogen in a sealed container, at 100 psi. If 50 cm^2 of the containers surface area is heated to a 1000 degree Celsius, will the psi increase over time? What would be the formula to figure this out in terms of the amount of pressure added and the time frame it takes? No need to do the math for me, I'm capable of that, but I do need help with understanding which formula to use. Disregard the heat loss of the container for now.


title: Would a three wheeled vehicle be faster than a four wheeled vehicle of the same weight?

text: If I have a four wheeled vehicle (small wooden block with metal nail axles and plastic wheels) and apply a force X to it, would it be made faster by keeping one wheel off the ground in order to reduce friction? My thought is that the remaining three wheels would then have more weight on them, and thus more friction -- but is this added force more than offset by the loss of friction in the missing wheel? Update: After some back and forth with Ruben I think I have gathered the following -- The friction per wheel exists in both contact with ground and to a much greater degree contact of axle to wheel. There is a small and most likely negligible wind resistance component.                                     4 wheeling........3 wheeling        * Friction per wheel = F ...........................4/3 F   * Wind resistance = W ...................................W


title: How to calculate the velocity needed for a rocket to get to a L1 point (escape a body without orbiting)?

text: I'm looking to calculate parameters around launching, say, a model rocket STRAIGHT to the moon. This does not mean through orbital insertion into a stable ~17,000 MPH relative to earth's surface and then Hohmann Transfer maneuvers. What I mean is reaching an altitude that would correspond to the Earth-Moon L1 Lagrangian Point https://en.wikipedia.org/wiki/Lagrangian_points, with a minimal amount of velocity left to fall toward the Moon. For the purposes of the calculation, this could mean getting there with zero velocity (though it would need just a slight bit more, and have to be timed perfectly with the Moon's rotation). ![Sun Earth System](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Lagrange_points2.svg/500px- Lagrange_points2.svg.png) Tentative Values:   1. Where is Earth-Moon L1, on average? 345,000km from Earth's center?   2. Earth's Radius at the Equator? 6,384km?   3. What altitude is the Earth-Moon L1? 338,616km?   4. Without air resistance, how much energy would be required to get a given mass to this altitude? (straight up, no orbital insertion)   5. What does the formula for this look like (getting to a specified altitude of a given astronomical body)?   6. If you were to launch from, say, 100km up with a few weather balloons, how would the formula need to be adjusted?   7. What type and how many rocket engines would be required to get a tiny model rocket to this altitude, regardless of feasibility?   8. Other than the issues of landing safely, would this be a more efficient way to transport supplies to the moon if they could survive the crash landing?


title: 'Fermi-Dirac'-like occupation probability at high temperature

text: Consider an ensemble of $N\to\infty$ free particles, each of which can assume energy states $E_i\in\\{0,E\\}$. Using the canonical ensemble one can compute the occupation probability for a single of those particles to be in the excited state $E_i=E$ (or equivalently the expectation value for what fraction of all particles is in the excited state). The result is: $$n_T(E)=\frac{1}{e^{\frac{1}{k_B T}E}+1}$$ Now, if we check this expression in the limit $T\to 0$, we properly obtain $n_0(E)=0$, telling us that at low temperatures almost no particles will be in the excited energy state. But then, in the opposite limit $T\to\infty$ we get $n_\infty(E)=1/2$, so apparently at infinite temperature there will be equally many particles in the ground and the excited state! I kind of feel like all the particles should go into the excited state for $T\to\infty$, so that this goes against intuition. But maybe I am wrong? What should I expect to happen for $T\to\infty$?


title: About Poincare section for the double pendulum

text: I am reading Prof. Louis N. Hand's _Analytical Mechanics._ In the chapter about chaos, it introduces the concepts of Poincare section based on the example of double pendulum. Also, it plot the section when the initial angles for the upper and lower pendulum is small (about 1 degree for each). So in this case, the section contains some closed curves. I have few questions about the section.   1. Firstly, is it possible for some curves shown in the section crossing? Why and why not?    2. Secondly, in the plot it shows in the book (fig. 11.4), there are two fix points but they are not symmetric about the generalized momentum. Why is that? Following the post found in http://mathematica.stackexchange.com/q/40122/, I have the code modified such that ![enter image description here](http://i.stack.imgur.com/JOFzV.png) ![enter image description here](http://i.stack.imgur.com/iiQuu.jpg)


title: Determine KE of electron given momentum & mass

text: Some info:   * wavelength of electron: $2.78 \times 10^{-10}$   * momentum of electron: $2.38 \times 10^{-24}$ Determine KE of electron. In a provided hint: $KE = \frac{p^2}{2m}$. So I have: $$KE = \frac{2.38 \times 10^{-24}}{2 \times 9.11 \times 10^{-31}} = 1.31 \times 10^6$$. But provided answer is $3.10 \times 10^{-18}$. How do I get this?


title: I don't get band structure of solids

text: If the energy levels of bound electrons are discrete, why do band structures in solids arise?


title: Geophysics with earth science

text: I am a communications engineering student. I am interested in doing a project on geophysics. The concept is for a hole-digging robot with a camera and sensors and wireless transmitter that is made of high mp alloys. It is intended to make a hole down to 3000 km and perform a sample-return for any interesting things that could be found at that depth. This could help scientists learn more about the interior of Earth. Is this something that could be done?


title: Cameras in Voyager probes

text: I've always wondered about the cameras in the space probes, especially in the Voyagers. 1) What kind of cameras do they have? **Digital?** (What kind of sensor and megapixel count?) **Analog?** (Do they develop the film and then scan it, as Luna 3?) 2) How do they transmit information to NASA?


title: How would two equally massed stars orbit?

text: In an empty universe, except for two equally massed stars, how would they orbit? Or, for another example, if the earth suddenly grew to be the mass of the sun, how would they orbit, or interact? Would they simple pull together and combine? In the earth example, because the earth is already moving, would inertia keep it orbiting the sun?


title: Are there websites or programmes that permit a simulation of the night sky in the past and the future on an ordinary computer?

text: Are there websites or programs that permit a simulation of the night sky in the past and the future on an ordinary computer? (For the past, I would be content with objects visible to the naked eye.)


title: Wouldn't 3 or more particle entanglement allow passing classical information?

text: My (simple) understanding of entanglement is that by measuring the spin of one entangled particle, the other entangled particles' spin changes to the opposite of measured particle. This act of measurement doesn't allow any communication. (Obviously entanglement isn't just the spin but also other physical properties, but for simplicity of this question). However, since more than two particles can be entangled, what if a group of 3 particles are entangled, with Alice having 2 and Bob having 1. At predetermined time intervals (eg every 0.1ms), Bob measures his entangled particle if he wishes to communicate a 1 in binary times. This will cause Alice's two entangled particles (in the same entanglement group) to have the same, opposite spin direction of Bob's particle. Bob does not measure his entangled particle if he wishes to communicate a 0. This keep all of the 3 entangled particles still in a superposition. At every 0.1ms, Alice measures both of her entangled particles. If Bob has already measured the entangled particle, Alice will find that both of her entangled particles have the same spin. However, if Bob has not measured, then when Alice measures one of her entangled particles, the other entangled particle will change to the opposite spin state, and Alice will find her two entangled particles are different. This would mean that information would be passable. What's the problem with this?


title: Photons and perfect mirror

text: A perfect mirror means, that all the photons which collided with the mirror will be reflected in the same amount, with the same energy and with the same - except sign - angle. Will the mirror get an impulse from the photons?


title: Incompressiblity and speed of sound

text: I have read that usually if the speed of a fluid is much lower than the speed of sound (in that medium) then it can be treated as an incompressible fluid. Where does this condition come from? Is it possible to show it from the Navier-Stokes equation?


title: How Hamilton's Principle was found?

text: Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral $$S = \int_{t_1}^{t_2}L(q(t),\dot{q}(t),t)dt$$ is stationary. And then we have that $L = T - U$ is the lagrangian. Now, how this was found? I mean, how could someone find that picking the quantity $T-U$, considering the integral and extremizing it would give us the actual path on the configuration space? I know that it works, and the books show this very well. But historically how physicists found that this would give the path? How they found the quantity $L$ and thought on studying it's integral?


title: Why did it take a long time to discover top quark?

text: In high school physics, I learned that it took a long time before the top quark was discovered. One of the reasons that was given in my book was that the top quark has a large mass, much larger than the other 5 quarks, and there just wasn't a powerful enough particle accelerator to produce large energies. My doubts:   1. In such an experiment, do high speed particles just collide and energy is just given off that can be converted to mass? Is this how quarks are produced?   2. As far as I can understand, in a particle collision, large energy is produced and this is converted to mass i.e. a top quark in this case. However, I also learned that individual quarks cannot exist and hadrons form... so how did scientists get an individual quark?   3. If the top quark was formed from energy, it must mean that there was a "pair production" and an anti-top quark is also supposed to be produced. So, does the collision have to produce energy that is equivalent to the mass of two top quarks?


title: Categorizing solutions to Hierarchy problem

text: We know that no gauge symmetry can prevent a term $m_\phi^2|\phi|^2$ for a scalar field, and that, given the quadratic loop corrections, the natural scale is $m_\phi \sim M_P$. This is related to the hierarchy problem. Now, I was thinking, is it fair to categorize the solutions to the hierachy problem into two sets:   1. **Solutions which _transfer_ a symmetry to the scalar:**     * Supersymmetry, in which $\phi$ is protected via a symmetry transferred from a fermion.     * Little Higgs, in which $\phi$ is a Goldstone-boson protected via a shift symmetry linearly realized from the original scalar that precipitated SSB.     * Composite Higgs, in which the fundamental degrees of freedom at high energies are not scalar, but fermionic, and thus $m_\phi$ is protected.   2. **Solutions which lower the true $M_P$**. i.e. the true $M_P \sim M_W$, but on the brane $M_P^\textrm{eff}\sim 10^{19}$ GeV, e.g. ADD and extra-dimensions. Is this division of solutions accurate? Are there any solutions that don't fall into these categories? And are there any other ways to transfer symmetries to scalars?


title: QCD string breaking and glueballs

text: When one tries to pull two quarks appart, a flux tube is created. The tube eventually breaks, creating quark anti-quark pairs and eventually hadrons. Can there also be creation of gluons to form glueballs? Does this happen at a higher or lower energy then hadron creation?


title: Particle accelerators by focused laser beam?

text: My (layman) estimate is that such accelerators could relatively easily and cheap override the current high-end, multimilliard-$ accelerators. Some years ago I've read some interesting breakthrough in the news, but practically nothing since that. What is the current focus of the research/development? Is there any obstacle found? What is the current estimate to the practical energy limit of such accelerators?


title: A double-sided lever and multiple forces

text: Assume a double-sided lever where the left arm has a length of `a` and the right arm has a length of `b + c`. Please refer to this link for a picture. `a`, `b`, `c` and `F` are known and `F1`, `F2` and `F3` are not. How do I find the `F1`, `F2` and `F3` forces? **EDIT** : The lever is weightless and it is in balance. However, it is also in motion and `F` is a force acting upon the lever as a whole - it's supposed to be some kind of sum of the `F1`, `F2` and `F3` forces - I'm just not certain if I could assume that `F = F1 + F2 + F3`. Anyway, the picture could be misleading - it **is not** acting upon the lever in addition to the other forces. It feels kind of intuitive to me that I've got a sufficient amount of data. I just don't seem to be able to write down all the possible equations. In other words - how do you describe a double-sided lever with multiple forces applied using math?


title: Difference between velocity vectors, relative speed

text: Consider two particles moving in the same direction instantaneously but one of them follows a circular path and one follows a straight path. ![Situation of relative motion](http://i.stack.imgur.com/U2k52.png) If I adopt a rotating frame of reference in which the particle $A$ is stationary, math tells me that the relative speed between $A$ and $B$ is $V_{rel}=V_B-V_A-\frac{V_B h}{R}$, where $h$ is the distance between the particles and $R$ is the radius of the circular path. I understand that this is right, I can infer this from the formulas of rotating reference systems. But can someone please explain why it doesn't work to simply take $V_A-V_B$. The relative velocity should be the difference between two velocity vectors, I think that's very intuitive. But in this case it isn't? Why? When I see a physical situation like this (e.g. they could be airplanes, one doing a loop), how can I know if I can simply take the difference between the velocity vectors to find the relative velocity or not?


title: How does the volume change affect an adiabatic process?

text: > _A gas consisting of rigid diatomic molecules was initially under standard > conditions. Then the gas was compressed adiabatically, 5 times the original > volume. Find the mean KE of rotating molecules in the final state._ We know from the law of equipartition of energy that for a gas at equilibrium at a certain temperature, the average KE per particle associated with each degree of freedom is $\frac{1}{2}kT$. Since for a diatomic gas, degree of freedom due to rotation only is 2, I can say that the mean KE for rotation of one molecule is $\frac{1}{2}(2)kT = kT$. I have also thought that, since the process is adiabatic, temperature remains constant. So the rms velocities of the molecules should not be affected. Also, $v=\omega r$, so $v$ and $r$ being constant for the diatomic molecule, $\omega$ should not change either. So KE doesn’t change. The problem is that I can’t seem to involve the volume change in any way.


title: An application of Toeplitz operators

text: I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I need the reason why the Toeplitz operator becomes important in the solution.


title: Do we need to expand the potential in a power series to show $[x, V(x)] = 0$?

text: Today in class (Intro to QM) we went over a couple of commutators. Among them was $[x, V]$, where $V=V(x)$ is a potential. What the teacher said to prove this is zero was: let's assume $V$ is analytic and can be expanded in a power series. Then we can take the commutator of $x$ and each term in the series, and since $[x, x^n] = 0$ everything is zero. I immediately thought of something much simpler: In coordinate space, both $x$ and $V$ are simply multiplication. For any function $\psi$, $xV\psi = Vx\psi$ trivially, because we're just taking products. Therefore, they commute. Is there anything wrong with my reasoning?


title: Zero uncertainty constant and a unit change

text: So, we know the speed of light with zero uncertainty. We also know that values of $\epsilon_0$ (electric constant) and $\mu_0$ (magnetic constant) are known with zero uncertainty. My questions are quite simple, but I seek a sanity check.   1. Why are $\mu_0$ and $\epsilon_0$ known with zero uncertainty?   2. What might happen to the uncertainty of the present zero uncertainty constants if the definition of the units is changed in the future **SI** ? My guess is **:** **1.** $\Large \frac{1}{\sqrt{ \mu_0 \epsilon_0}}$ and **2.** Nothing. But I would appreciate hearing other's thoughts. Cheers.


title: Is this picture of the electron dipole moment correct?

text: ![enter image description here](http://i.stack.imgur.com/EGUec.png) It's the electron spinning on its axis, and a magnetic moment shoots up. The direction of the North-South is opposite what I thought it should be. Why is the vector arrow pointing from North to South and not opposite that? Shouldn't the magnetic moment arrow be pointing down?


title: What is the minimum pressure difference for your ears to pop?

text: I'm assuming the answer to this largely varies from person to person. Assuming you could instantly change the pressure around your head by amount $\Delta p$, what is the minimum $\Delta p$ for your ears to "pop" and adjust their inner pressure to the outside pressure? I think the minimum building height that I have noticed this in is around 10 stories. Assuming 10 feet per story and $\Delta p = \rho g h$, the pressure difference comes out to about 0.05 psi. What I'm asking is if there is some sort of "pressure release valve" in the ear that triggers for about the same pressure difference for most people?


title: Mechanical Equivalent of Heat

text: Recently I have been looking up James Joule's experiment regarding the mechanical equivalent of heat. After viewing some drawings of the apparatus, I assumed that the lines holding the weights would have to be attached to the drum. They would then be allowed to fall completely through a certain distance until the lines were at full length and the mechanical motion of the fluid would have to be allowed to continue to flow, causing the weights to be pulled up a bit and drop again, oscillating up and down until the motion stopped and all of the mechanical energy from the weights would be transferred into the fluid. However, upon watching some videos, (this video is about 14 minutes long; but, if you advance to about 5 min 30 sec, it will show the experiment in question.) I noticed that the weights were allowed to be stopped by a level surface (an external force) instead of doing what I have described above. To me, this means the weights were still in motion and, therefore, still had KE that was not transferred into the fluid. Of course, the weights are moving very slowly, much more slowly than if they had been allowed to free-fall through the distance; so, their kinetic energy is very small compared to the potential energy they lost, meaning the vast majority of the potential energy was transferred into the fluid and into other, small energy losses. Should I assume the speeds of the weights were small enough to ignore the KE in the weights? The assumption seems to be that the potential energy lost by the falling weights was all transferred into the fluid. However, in the experiments, the lines have to be wound back up and the weights dropped several times in order to make a measurable difference in temperature. Each time the weights fall, they are stopped by an external force, meaning they have kinetic energy that is not transferred into the fluid after falling through a measured distance. It seems to me that some of the potential energy from the weights is transferred into the fluid; but, some of it is transferred into the kinetic energy of the weights falling.


title: A corollary of Mermin-Wagner Theorem

text: ![enter image description here](http://i.stack.imgur.com/UcKiX.png) The picture above shows Mermin-Wagner Theorem and its corollary. How can the corollary be derived from Mermin-Wagner Theorem?


title: Commercial Infrared lights

text: I purchased an infrared light. It's a 100 W Philips infrared lightbulb. Says it's infrared, but I haven't done any spectrum analysis so I don't know for sure if it's just red or really infrared. As I recall glass is specially opaque to infrared light. Doesn't the light bulb have to be of special glass? (I may be wrong in my assumption, though.) Assuming a non absorbing material, I always think of longer wavelengths as having higher lattice penetration power as longer wavelengths collide with less particles as they travel. Although humans are not very lattice-like, I just want to know if what it says on the box is true or not. That it has more penetration power than a pure white light bulb, and it's really not just warm because of Joule effect (100 W is a lot, after all). Mostly, I think of low energy radiation as harmless to humans.Is infrared radiation so?


title: Is the pressure at the outer hole on a pitot static tube equal to atmospheric pressure?

text: I'm looking at a pitot static tube question right now and I just need some clarification. There is the outer hole on the pitot static tube, not the hole that the air stream goes through, but the other tiny one near that. Is the pressure at that hole equal to atmospheric pressure?


title: Is energy conserved in  decay of hydrogen atom in superposed state?

text: This looks like a paradox. Let's say we have an hydrogen atom. Superposition of states could be possible for electrons. But if an electron is in a superposition, I guess it could decay into a lower state by emitting a photon. Different initial and final energy expectance would make possible photons of arbitrary energy levels. But that is not the case. So that, what is wrong? Is energy conservation violated?


title: Why does a ping pong ball change direction when I spin it on a table?

text: When I spin a ping pong ball on the table, it rolls forward in the opposite direction of the spin, and then eventually changes direction and rolls backward. Here's a video demonstrating the effect. Why does that happen?


title: Stat mech explanation for separation of one liquid from another in gravity?

text: If one mixes two distinct ideal gases above the Earth's surface, one with a higher molecular mass than the other, then at equilibrium, their number density gradients will be such that at low heights, the density of the heavier gas will dominate while higher up, the density of the lighter gas will dominate. This is something I know how to show using stat mech + chemical potential etc. I would think that for liquids, the situation is different in the sense that the liquid with higher molecular mass would completely separate from the liquid with lower molecular mass. I suspect this because I've seen this happen with liquids in the real world, but is this always the case? If not, then what assumptions must hold about the liquids? Does surface tension play a role in all of this? Ultimately, is there a stat mech explanation for this?


title: How do I determine where the maximum air velocity around an object is?

text: I have a freestream of air directed at an object. Is there a way to find where the maximum air velcoty occurs with only having initial air speed and static pressure at the object?


title: Proof the superstring action is Weyl invariant

text: The superstring action is: $$ S = k\int \mathrm d\sigma \sqrt{-h} \left [ h^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X_\mu + 2i {\bar{\psi}} ^\mu \rho^\alpha \partial_\alpha \psi_\mu - i {\bar \chi}_\alpha \rho^\beta \rho^\alpha \psi^\mu \left (\partial_\beta X_\mu -\frac{i}{4}{\bar \chi}_\beta \psi_\mu \right)\right ]$$ This is Weyl invariant, but the symmetry isn't manifest. All the books and lecture notes I've seen just state this outright, without proving it. Is there a reference where I can see the proof done in detail? Even when they give the action in the superconformal gauge ($ h_{\alpha \beta} = \eta_{\alpha \beta}$ and $\chi_\alpha = 0$), they don't bother to prove it.


title: Resistance of Light Dependent Resistor (LDR)

text: Is there a mathematical expression relating the resistance of Light Dependent Resistor (LDR) with light intensity?


title: Is the E-cat for real?

text: Does this thing really do what they say? http://www.youtube.com/watch?v=EhvD4KuAEmo If it does, it looks like this will probably be the biggest breaktrough in science ever :)


title: Difference between action-at-a-distance and a field according to Maxwell?

text: My question is more on a historical note that involves Maxwell’s equations. Besides the information that I have obtained from textbooks, I am mainly getting it from History of Maxwell's Equations and was not able to find or understand the question below. I first started thinking about this last semester while tutoring a mechanics student who believed that action-at-a-distance was a correct way of thinking about gravitational interactions between two objects. I explained to this student that the “correct” interpretation was the idea of a “field.” And it was here that I stated asking myself **how did Maxwell know he was on the right track** in developing the field idea from Faraday’s work? From my readings, I believe that I understand that he first developed his mechanical analog with an imaginary weightless and incompressible fluid flowing (where the stream lines are the electric and magnetic field lines) through a porous medium to reach his conclusions. It seems to me that at an electromagnetic boundary, the action-at-a-distance theory would not be able to account for boundary conditions (in post Maxwellian language) established by a field. Even though I typically think of boundary conditions when referring to electromagnetic wave propagation across a boundary, I feel that somehow Maxwell “knew” this before he developed the electromagnetic wave equation and its associated boundary conditions: $$ϵ_1E^{\bot}_1=ϵ_2E^{\bot}_2; {\;} {\;} E^{\parallel}_1=E^{\parallel}_2;{\;} {\;} B^{\bot}_1=B^{\bot}_2; {\;} {\;} \frac{B^{\parallel}_1}{{\mu}_1}=\frac{B^{\parallel}_2}{{\mu}_2}$$ I would appreciate someone giving me insight on this historical question or setting me straight on were I got it wrong.


title: How does a ducted fan suck in air?

text: I am thinking of a simple ducted fan with a single set of rotating blades, no stator vanes behind. If the effect of the fan is to raise the pressure of the air, how is there flow into the fan against this pressure gradient? I understand that the rotating fan blades have the effect of increasing the stagnation enthalpy of the flow. For an incompressible flow, this is equivalent to the stagnation pressure increasing via the fan blades imparting kinetic energy into the flow. I know that static pressure and velocity increase across the fan, so how can there be flow against this pressure gradient? I think the explanation will have something to do with the different frames of reference chosen to analyse the problem.


title: Visualising the magnetic field

text: How can we visualise the magnetic field?How to visualise magnetic field due to current carrying conductor having poles(like north and south for ordinary magnetic). _Can we determine the north and the south of the magnetic field produced by the current_.


title: How to Extend Relaxation Methods for 2D Laplace Equation given in Jackson E&M to 3D?

text: In Jackson (3 ed) chapter 1.13 an outline is given for using relaxation to solve laplace equation in 2D. The general procedure in 3D involves minimizing the quantity $$I[\Psi]=\frac{1}{2}\int_V \nabla\psi\cdot\nabla\psi d^3x $$ But in 2D they seem to use the surface integral over an area A. This leads to equation 1.76 where the integration is taken over the north east corner of a box of dimension h x h $$I_{NE}=\frac{1}{2}\int_0^{h/2}dx\int_0^{h/2}dy\bigg [\bigg (\frac{\partial \psi}{\partial x}\bigg )^2+(\frac{\partial \psi}{\partial y}\bigg )^2\bigg ] $$ The approximation to the derivatives in the north east corner is given by the potential of the neighboring grid points $$\bigg(\frac{\partial \psi}{\partial x}\bigg )_{NE}=\frac{1}{h}(\psi_o-\psi_E)$$ $$\bigg(\frac{\partial \psi}{\partial y}\bigg )_{NE}=\frac{1}{h}(\psi_o-\psi_N)$$ Where $\psi_N$ is the grid point north of the center grid point $\psi_o$ and so on. Putting these approximations into the integral and integrating gives $$I_{NE}=\frac{1}{8}[(\psi_N-\psi_o)^2+(\psi_E-\psi_o)^2]$$. Now I try to do the same thing in 3D starting with $$I_{NE+}=\frac{1}{2}\int_0^{h/2}dx\int_0^{h/2}dy\int_0^{h/2}dz\bigg [\bigg (\frac{\partial \psi}{\partial x}\bigg )^2+\bigg(\frac{\partial \psi}{\partial y}\bigg )^2+\bigg(\frac{\partial \psi}{\partial z}\bigg )^2\bigg ] $$ where the + denotes the point above $\psi_o$ in the z direction. I can make approximations similar to those above and I get $$I_{NE+}=\frac{1}{16}h[(\psi_N-\psi_o)^2+(\psi_E-\psi_o)^2+(\psi_+-\psi_o)^2]$$ Here I have units of length times potential squared so I am a little concerned that I may have missed something. Could someone verify that this approach is correct? Thanks!


title: Where and how exactly does string theory and Q.E.D. use zeta function regularization?

text: In the video they mention it being used in many fields of physics inclusing String and QED theory. https://www.youtube.com/watch?v=w-I6XTVZXww But I remember reading somewhere that 1+2+3..=-1/12 is obviously a "mathematical trick" (something about stupidly equating incompatible sets), and if so, how does this turn out to be true for things that are real (like QED)?


title: Does Coriolis effect affect snowing?

text: Right now, outside is snowing. No matter what window I look from, the snow constantly for last few hours tends to fall to the right close to the window, and to the left close to the building across the street, just like it would suggest the Coriolis effect on northern hemisphere. I understand the Coriolis effect applies for air flow and the reasons for this may be in something completely different, specific to the street I am in. But in theory - **could the Coriolis effect on snowing be so dramatic, it would be observable?**


title: Simulate a physical impact of objects made of finite, small elements

text: I want to simulate an impact between two bodies according to gravity, and eventually considering other forces to stick matter together. I'd like to use python to do this, but I am open to alternatives. What kind of tools or libraries can I use to perform this task ?


title: What makes a space a real space?

text: By "real space" I mean a space in which physical particles move. Consider a color sphere and let a bunch of objects "move" on its surface. "Move" means "change colors". Let there be some rules governing the changes of color, maybe giving rise to something like inertness or collisions. > Under which conditions will we be willing or forced to think of the > movements as real movements in real space? ![alt text](http://i.stack.imgur.com/zhVN1.jpg)


title: Ion-neutralization processes and its energies

text: Ionization energies/Electron affinities are well mapped. I wonder about opposite processes... I imagine for anion the necessary energy will be equal to the electron affinity (energy released when the anion was created). What about cations? If I had an cation and free electron (both with low kinetic energies) will it automatically recombine? Or if I had let's say grounded metal surface and near it a cation of some gas, will it do the same (and how would react an anion in this case)? Thank you for answers. If you could direct me to the proper literature I would be grateful :) (I know it requires QM for any decent description).


title: Closed circuit integral, magnetic vector potential for straight line segment

text: Looking at a worked example in Chengs Field and Wave Electromagnetics (newest ed, p 236) about Biot-Savarts law. 'A conducting wire of length $2L$ is carrying the current $I$. Find $\textbf{B}$ by first finding the magnetic vector potential.' The book then applies the following equation $\textbf{A} = \frac{\mu_0 I}{4\pi}\oint_c \frac{dl}{R}$ and integrates from $-L$ to $L$. But how can we do that? It's not a closed loop. I guess this is due to some sloppy notation somewhere?


title: Spring Damper System for a Vibrating Motor

text: Good day people of SE I have a friend that has a final year project and is stuck. He has a motor with a small weight at the end of the shaft that causes vibrations. This motor is on a thin plate. Beneath the plate, in each corner, is a spring and damper. He has to formulate a model that describes the power absorbed by the spring damper system. The aim is to minimize or negate the vibrations to the platform under the spring-damper system. The weight, spring and damper can be adjusted. The level of the student is about 3rd year engineering. The standard of the mechanical engineering department is quite low compared to the other engineering practices, so a purely mathematical solution will be sufficient. I have no information regarding the position of the weight on the shaft. I assume it will be a rectangular weight positioned slightly to one side to allow some serious vibrations, but this is a guess. Where do you suggest we start researching this? Thank you EDIT : Thank you Tobias for the comment. I changed the question as instructed.


title: When a wire is stretched by a load, is there any change in its volume?

text: When a wire, or a rod(whose diameter is not negligible) is subjected to a tensile stress, is there any change in its volume? If yes, does the volume increase or decrease? Take into consideration both the longitudinal elongation and the lateral contraction. If no, then what is wrong with the following derivation? $V=\pi d^2 l/4$$\ \ \ \ \ \ \ \ \ \ \ (V$ is the volume of the wire) $\implies\Delta V/V=2\Delta d/d +\Delta l/l$ $\implies\Delta V/V=(1-2\sigma)\ \Delta l/l$ $\ \ \ \ \ \ \ (\Delta d/d=-\sigma \Delta l/l)$ where $\sigma$ is Poisson's ratio.


title: Why do lightbulbs continue to glow after the light is turned off?

text: I've noticed that whenever I turn the lamp off in my room at night, the lightbulb seems to continue to glow for a minute or so after that. It's not bright though; the only way I even notice it is if the room is dark. So why does it keep glowing?


title: quasiparticles resembling monopole energy generator

text: Is it possible using quasiparticles that superficially resemble monopoles to make a generator providing a sustainable source of power


title: Numerical: Work Power Energy

text: > _A particle moves along a straight line. A force acts on the particle which > produces a constant power. It starts with initial velocity 3 m/s and after > moving a distance 252 m its velocity is 6 m/s. Find the time taken._ Here are my attempts at solving it.


title: Derivation of a quantum dynamical map on open quantum system

text: Let us consider a initial total quantum system as $\rho(0) = \rho_S(0)\otimes\rho_B$ where $\rho_S(0)$ is initial open system and $\rho_B$ is density matrix for environment. We can use partial trace to get open system after some time, $\rho_S(t) = \mathrm{tr}_B \\{ U(t, 0) [\rho_S(0) \otimes \rho_B] U^{\dagger}(t, 0) \\}$ This will be a dynamical map $V(t): S(\mathcal{H}_S) \mapsto S(\mathcal{H}_S)$ where $\rho_S(t) =V(t)\rho_S(0)$. And given $\rho_B=\sum_\alpha \lambda_\alpha | \phi_\alpha > <\phi_\alpha|$ . The question is, I am not sure how in detail get the following $V(t)\rho_S = \sum_{\alpha, \beta} W_{\alpha, \beta}(t) \rho_S W_{\alpha, \beta}^{\dagger}(t)^{}$ where $W_{\alpha, \beta}(t) = \sqrt{\lambda_\beta} <\phi_\alpha|U(t, 0)|\phi_\beta>$. I tried to put $\rho_B$ into the $\rho_S(t)$ as $\rho_S(t) = \mathrm{tr}_B \\{ U(t, 0) [\rho_S(0) \otimes (\sum_\alpha \lambda_\alpha | \phi_\alpha > <\phi_\alpha|)] U^{\dagger}(t, 0) \\}$ But how to proceed from here ?


A spacecraft leaves the point of origin and has a maximum acceleration of 30 m/s. It can only maintain this rate for 10 minutes before it runs out of gas and then proceeds to drift at a constant velocity. Thirty minutes later, a second spacecraft (identical to the first) leaves the point of origin. But it has more gas and can maintain its rate for 15 minutes before it runs out of gas. How far will the first spacecraft travel before the second one catches up to it?


title: Quantum Mechanics: Show that the expectation value of angular momentum does not change with time

text: The potential is given by $V\left(\left\|(x,y,z)\right\|\right)$, so $[\hat{L}, \hat{H}] = 0$. Using the definition of $\langle \hat{L} \rangle$ and the time-dependent Schrödinger equation, show that the expectation value of angular momentum does not change with time. Expectation value of angular momentum, $\langle \hat{L} \rangle$ = $\langle \psi \mid \hat{L} \mid \psi \rangle $ where $\psi$ is a generic wavefunction.


title: The chiral ring of N=2 WZW models

text: Does anyone knows if the chiral ring of the Kazama-Suzuki models have been found? If yes, please send me the reference. Thanks


title: Rocket engine, perpendicular to the end of stick, in space

text: Rocket engine, perpendicular to the end of stick, located in cosmos, without atmosphere and gravitation. As in picture below: ![enter image description here](http://i.stack.imgur.com/DM0AM.gif) In what direction will the stick move? In _V_ or another direction?


title: QED photon propagator to one-loop order gets different answers

text: I'm a self-studying 14-year-old who has a passion for particle physics. I'm currently trying to calculate the QED photon propagator to one loop. However, in all the places I've looked, even with the same renormalization scheme, the results are different. I don't get it. Also, I need a step-by-step tutorial on how to calculate that trace in the numerator, specifically, the one with those gamma matrices.


title: Can a object with constant acceleration change its trajectory?

text: That's a question from previous Physics tests which I am using to study. I would say that no, if the acceleration of the object is constant, its velocity, which is directly related to the acceleration, will also remain the same and make the object always go on the same direction. Is my way of thinking correct? Please correct me if I am wrong and give me better explanations if possible.


title: Deceleration of Vehicle off pavement

text: How would I calculate the deceleration of a vehicle (ATV), wheels locked, on dirt? How long would it take to halt? How fast is it decelerating? I used this calculator, but it seems incorrect for a light vehicle.


title: Decibel equation, dyne or Watt

text: I know decibel is used as a ratio between two physical quantities with the same units on a logaritmic scale. Is also used for calculate the intensity of a sound, but I found it reported in two different ways: $$\text {db} = 20 \log \left(\frac{p}{p_0} \right) \qquad\text{with: }p_0 = 0,0002\,\text{dyne}/\text{cm}^2$$ or $$p_0 = 10^{-12}\,\text W/\text m^2$$ are those equivalent? If so, where does that 20 come from? is that because of this (quoting from wikipeida): > "(...) commonly used in acoustics to quantify sound levels relative to a 0 > dB reference which has been defined as a sound pressure level of > $.0002\,$microbar, or $20\,$micropascals." is the 20 micropascals? Then should that be equivalent to $0.0002\,\text{dyne}/\text{cm}^2$?


title: Would a considerably big asteroid be disintegrated by the Earth's Roche limit?

text: If there was a big asteroid with a diameter of say 50km+ in a _collision_ course with the Earth (not orbiting), would it disintegrate into smaller chunks due to the Earth's Roche limit, or the time it will spend in the Roche radius won't be enough for the tidal forces to have an effect? My simple calculations and assumptions of an asteroid with density as the moon will have ~9500km Roche radius with the Earth, so an asteroid with velocity of 20km/s will have about 8 minutes as soon as it enters the Roche radius until it collides with the surface of the Earth. My question here : is this time enough to disintegrate the asteroid?


title: wavefunction antisymmetry as a limit of a deeper geometric constraint

text: Recently there was an interesting reformulation of Pauli principle in terms of polytopes: http://physics.aps.org/articles/v6/8 My question is, can this suggest that fermionicity is not a fundamental property, but an emergent property of the quantum low-energy Hamiltonian? **Bonus question:** Can this imply that Coleman-Mandula theorem doesn't really apply to physics, and hence supersymmetry is not as useful as we thought?


title: Why does it take so long to make a nuclear bomb?

text: So as I know nuclear bombs are derived from fission reactions: By providing the nucleus with enough power to trigger a chain reaction. If uranium was present why does it take so much to make a nuclear bomb.


title: Is it possible to clearly receive weak signal from space?

text: Imagine there is communication satellite on geostationary orbite. It transmits signal in a defocused beam, "illuminating" whole Earth hemisphere it's floating over. For example, signal is amplitude-modulated with 10GHz carrier frequency. What if satellite transmitter power is very, very low, like 25mW, is it possible to receive this signal on Earth with temporal resolution high enough for high-speed data transfers by just using sufficiently large or huge satellite antenna dish; or will noise get amplified too and this approach will be fruitless? What if two identical satellites are transferring signals of their own, and floating close enough (say, 1° away) to each other to be visible at once from surface, is this possible to receive separate signals from both satellites, or interfence will cause total mess? Are there ways to overcome this?


title: penetration of a solid body in a liquid

text: A solid (for example a steel ball) is moving with a certain constant velocity U toward a liquid in a container; I can write the equations of motion of the solid when it has a little part of it in the liquid (the forces on the solid are its weight, the buoyancy force and the viscosity) but I want to know what happens at the instant the solid collides with the surface of the liquid and why at high velocity U, the liquid acts as a solid in the collision? (the usual answer "the liquid doesn't have time to move isn't very scientific")


title: Can we measure the "Tilt-A-Whirl" effect of the Earth+Sun orbiting the galactic hub?

text: You've probably ridden on the fairground ride called the "Tilt-A-Whirl", or-- as Disneyland calls it--the "Spinning Teacups", as well as other fairground rides that employ epicycles. You can really feel the centrifugal force strongly when your car spins complementary to the spin of the main rotor. Now, can that extra force be measured on Earth as our orbit around the Sun complements the Sun's orbit around the galactic hub? Has anyone done it?


title: Wheeler-Feynman theory, QED without fields, vacuum polarization

text: Initially Wheeler and Feynman postulated that, the electromagnetic field is just a set of bookkeeping variables required in a Hamiltonian description. This is very neat because makes the point of divergent vacuum energy a moot point (i.e: an example of asking the wrong question) However, a few years later (1951), Feynman wrote to Wheeler that this approach would not be able to explain vacuum polarization. Anyone knows what was the argument for saying so? I don't see how allowing both processes with entry and exit particles and processes that begin in pair- creation and end in pair-annihilation makes the existence of a field a requirement.


title: Good resources for learning/reviewing complex time propagator formalism

text: I studied this at the beginning of my graduate degree but have to review it for my graduate exam. If it's not clear I'm talking about the $\beta = \frac{it}{\hbar} $ turning the integral of your propagator into a "partition function" and the ensuing analysis of eigenvalues as poles in the complex plane. I remember this as being a particularly beautiful and powerful approach but one that was difficult to follow the details and keep an intuitive picture of what's going on. I'd like to find books (or articles or websites) that have a good qualitative and quantitative explanation of what's going on without getting too bogged down with the mathematical details. Shorter is better but not at the expense of clarity.


title: What's the efficency of a steam jet pump?

text: Jet pumps or venturi pumps are often stated as having a "terribly low" efficiency, steam jet pumps specifically are usually describes as "only justifiable when there's an abundant steam supply anyway" - however, I hardly find any numbers, and no formulas. My question: Is there a general form to describe the work done by a (non- steam) jet pump an the pumped medium as a function of the work used to pump the working fluid (gas or liquid) Is there a general form to describe the work done by a steam pumpe as a function of the thermal energyinput into the workng fluid? While real-world numbers are interesting, I'm mostly after the limits posed to jet pumps by thermodynamics, not engineering. EDIT: Can a jet pump be describes meaningfully in a p - V diagram? Put another way, can the changes of impulse in the media be expressed solely as functions of the pressures (before/after)? Thus, can a (steam-)jet pump be described as a thermal (steam-)engine?


title: Foliation of hypersurfaces in General Relativity

text: As I understand it, it is usual in GR to define a foliation of Spacetime by a family of spacelike hypersurfaces "indexed" by the time variable. Then, in the context of Schwarzschild metric in spherical coordinates, is it correct to define the same way a foliation of these spacelike hypersurfaces by a family of 2-spheres indexed by the radial variable? Can we find a correspondence with a topological ball of r radius? And does this second foliation possess the same properties as the first one? Is it correct to think of the spacelike hypersurfaces and of the 2-spheres as families each consisting of infinite elements empiled on each other along the t and r direction respectively ?


title: How to calculate ground state wave function?

text: I have seen many ground state wave functions. From where are they derived? How can one calculate them? Where can one find a list of all ground state wavefunctions discovered?


title: Transformers rate equation

text: Going back to transformers after some years I realize they hide a possible missconception I'm not sure I really understand. It is not easy to find a full derivation of the well known relation $\frac{\Delta V_2}{\Delta V_1}\\!=\\!\frac{n_2}{n_1}$, as if this question were being circumvented, and that's why in an aim to give an intuitive idea of it I fall in a nonsense. ![Typical transformer arrangements...](http://i.stack.imgur.com/UfNmA.png) I start thinking in the secondary winding: attending to Maxwell-Faraday-Lenz's Law, $\varepsilon=\Delta V_2=-\frac{d\phi}{dt}$, and given a certain primary winding signal frequency, the higher the maximum flux $\phi$ is, the higher the induced e.m.f. $\varepsilon$ will be. Then we want a secondary coil plenty of loops (high $n_2$), as this increases its effective surface, and it is _pierced_ by the same field $\vec{B}$ lines $n_1$ times. Ok, so $\Delta V_2\varpropto n_2$. On the other hand, to increase that flux, we need a strong field due to primary solenoid. And this, **for a given intensity $I_1$** (let $R_1$ be $n_i$-independent), is fulfilled if we have high $n_2$. Then we would say that, at the end, $\Delta V_2\varpropto n_1 n_2 \Delta V_1$, which is not true, and which would produce some symmetry problems. What physics am I missing? Something related to induction in the primary too? Is due to not considering that $I_i=\frac{\Delta V_i}{n_i R_o}$? What makes, physically, that by decreasing $n_1$ one increases $\Delta V_2$? How to understand that? Usual derivations, as in Wikipedia, apply that $\Delta V_1=-\frac{d\phi}{dt}$ should be verified, but I find it a bit misleading, as the _cause_ and _effect_ roles symmetry is not clear. Given the true relation, in fact, what is the reason not to set always just $n_1=1$? Just to have a more precise ratio control and/or to have a more efficient system with $\vec{B}$ lines better guided inside the iron or is there any other reason?


title: When can I use semiclassical approximation?

text: I know that I can use semiclassical approximation for path integral approach (in quantum mechanics) $\int d[q]e^{iA}$ when action $A >>1 $. But how shall I use such condition? For example, assume that I have a particle with mass M in potential $V(q)=\frac{\mu^{2}}{2}q^{2}-\frac{\lambda}{3}q^{3}$. Then action $A=\int dt \left(\frac{M\dot{q}^{2}}{2} -\frac{\mu^{2}}{2}q^{2} + \frac{\lambda}{3}q^{3}\right)$. **Suppose I want to determine $\mu$ and $\lambda$ for which we can use semiclassical approximation**. How to do this? I do not understand how to compare this action with $1$.


title: Heisenberg Uncertainity Principle

text: If any senior member of the group has access to the book, _The Physical Principles of Quantum Theory_ by W. Heisenberg, then please help me in understanding the first section of chapter 2 where he gives a mathematical proof of the uncertainty principle. In particular, I don't understand: > "Suppose the wave packet made up by superposition of plane sinusoidal waves, > all with wave lengths near $\lambda_0$. Then roughly speaking $n= \Delta > x/\lambda_0$ crests or troughs fall within the boundary of the packet. > Outside the boundary the component plane waves must cancel by interference, > this is possible if and only if the set of component waves contains some for > which at least $n+1$ waves fall in critical range. Now suppose I have $\lambda_0$ = 400 nm, then for varying $n$ I will have different $\Delta x$, and at only certain values of $n$ will the equation having $n+1$ will be satisfied but how does this happen is not clear to me.


title: Anchor of a boat

text: hope you can help I've been teaching my kids about pirates and when we were talking about pirate ships this morning I confused myself greatly. When carrying an anchor a pirate ship (or any vessel with an anchor) is free to float about where ever the crew pleases, however once the anchor is in the water the ship is held in place. This seems unusual as (according to Archimedes principal) the weight of the anchor when it's in the water is less than when it's on the ship (immersed weight = weight - weight of displaced fluid) so how come the boat can carry the anchor but as soon as the anchor hits the water the boat cannot move? As you can see below the ship on the left can move freely, the one in the right cannot. ![](http://i.stack.imgur.com/4mHp0.png) I thought of a few possible solutions but maybe I'm shooting in the dark here. Initially I thought maybe the part of the anchor that curls out at the bottom is holding a large volume of water (much more than that which the anchor displaced) and so holds the ship in place, like a scoop. The other idea I had was that maybe the anchor is on the bottom of the ocean but this seems absurd, I've read that the average depth of the ocean is around 14,000ft (obviously parts are much deeper) and a 30mm chain of that depth would weight almost 100,000kg, which not only seems like a lot for the boat to carry but also essentially adds to the weight of the achor itself, perplexing the question further still Thanks for your time


title: Unusual observation in photoelectric effect simulation

text: I was studying a photoelectric simulation (http://phet.colorado.edu/en/simulation/photoelectric) and I observed a really unusual thing. When I held intensity and potential at a constant value and then varied the frequency I observed that there was peak in photocurrent. That is, it first increased when moving towards ultraviolet and then decreased. Please try it yourself and explain.


title: How does one exert greater force on the ground by jumping?

text: When one jumps, how does he/she manage to exert greater force on their ground than their weight? Also, what is normal force and the reaction force (are they the same thing?) and by newton's third law, shouldn't the reaction(weight) when we are standing on the ground that the ground exerts on us send us flying above the ground- why doesn't the law apply here? Finally, when we drop a hard stone on the ground why doesn't it bounce? Plus, why is the force exerted by the stone on the ground greater than its weight?


title: Precession of angular velocity about the body-fixed axis

text: My textbook mentions that under force-free motion of a symmetric top, its angular velocity vector $\overrightarrow \omega$ precesses about the $z$-axis of the body-fixed coordinate system. This seems impossible to me. Assuming that the axis of symmetry of the top is the $z$-axis, how can $\overrightarrow \omega$ point in any direction other than the $z$-axis? It's got to rotate about the $z$-axis and hence point along it. What am I missing? ![enter image description here](http://i.stack.imgur.com/1lwZC.jpg)


title: E and H fields created by fiber optics?

text: When an EM wave travels down a conductor, it creates and electric and magnetic field around (H) the wire and normal to (E) the wire. My question is, when light travels down an optical material such as fiber optics, is there an similar magnetic and electric field created around the fiber? My gut tells me there should be, but I cant reason it out. I'm looking for some help with this thought experiment.


title: Misaligned Mirror on Michelson Inferometer

text: If one of the outer mirrors on a Michelson interferometer was to be misaligned by a small angle of theta, what would be the shape of the interference pattern in the detector plane? What would happen to this pattern as the other mirror moves?


title: Single-site superfluid wave function and coherent state

text: The Superfluid wavefunction in Bose Hubbard model can be written as $|\Psi_{SF}\rangle=\frac{1}{\sqrt{N!}}(b_{k=0}^{\dagger})^N|0\rangle$ in the U=0 limit, whereU is the onsite interaction, N is the total particle number, k=0 denotes the lowest energy state and $|0\rangle$is the vacuum state. By Fourier transform, $b_{k=0}^{\dagger}=\frac{1}{\sqrt{M}}\sum\limits_{i=1}^Mb_i^{\dagger}$, where M is the lattice sites number. So $|\Psi_{SF}\rangle=\frac{1}{\sqrt{N!}}(\frac{1}{\sqrt{M}}\sum\limits_{i=1}^Mb_i^{\dagger})^N|0\rangle$. How to understand in the U=0 limit, the single site many-body wave function of is equivalent to a local coherent state, or say $\vert\Psi_{SF}\rangle \propto \prod\limits_i \exp(\sqrt{\frac{N}{M}}b_i^{\dagger})|0\rangle_i $?


title: thermoelectric effect

text: The Absolute Seebeck effect states that an electric potential (voltage) is produced to any isolated conducting when subject to a temperature gradient.But why? My view of this is that when you apply heat to one end you create a net transfer of energy through phonons and electrons' collisions towards the colder end, however there should be no net drift of electrons in one direction, because electrons are just moving randomly and colliding with other particles to transfer their thermal energy. So how would that create a voltage?


title: Is a QFT in a classical curved spacetime background a self-consistent theory?

text: EDIT: Better rewording by Chris White: Is it possible to have a theory that treats both GR and QFT (e.g. QFT on a curved spacetime dynamically influenced by the standard QFT fields)? Is such a theory at least self-consistent (even if it does not apply to nature)? Or is there some fundamental incompatibility we run into without even trying to quantize GR (or perhaps we are somehow forced to quantize GR for consistency)?


title: Almost identical fermions fighting for the same state

text: In quantum 101, we all learned that _identical_ particles behave strangely, even in the absence of interactions: no two fermions can be in the same state, but bosons love to be in the same state. But what if you have some fermions, all with the same spin. However, they have other difference(s) that make them distinguishable (i.e. mirror image molecules), but don't affect the overall shape of the wave-function very much. You put these molecules in a potential well in various states. You have a loose potential well so that the wavelengths of the different energy states are much larger than the size of the molecules; they are effectively point particles. There are too many fermions to all go in the ground state if they were identical, but they all can fit if they "know" that they are different. My reasoning is that they would initially refuse to drop into the ground state, like fermions, but eventually the tiny differences in mass/binding energy would distinguish them and allow them to settle in. Even if for mirror- image molecules with the same mass, asymmetrical multi-polar moments would introduce interaction energy terms that are different. Thus they would all end up in the ground state, but much more slowly than a single molecule would. Is this what would actually happen? Is there a way to estimate the "identicalness" (i.e. a difference of 0.001 eV in energy is slower to reach ground by a factor of x compared to a single molecule)?


title: What potential between charges leads to confinement

text: In (2+1)-d, instanton effect leads to a linear potential between charges. If we have two particles with opposite charges in this case, since linear potential diverges when the distance between the two charges goes to infinity, it is energetically favorable to produce two more charged particles to neutralize the original two particles, instead of letting them separate as far as possible. This is basically the argument on confinement of (2+1)-d EM field. It looks the essential element in this argument is the presence of a potential between charged particles which diverges when the distance goes to infinity, so does logarithmic potential lead to confinement? If so, we know that in (2+1)-d charges interact via a logarithmic potential, then why cannot we conclude that (2+1)-d EM field is confined without considering instanton effect? I think the answer is no, so how strong should the potential be in order to have confinement?


title: Why correlation functions?

text: While this concept is widely used in physics, it is really puzzling (at least for beginners) that you just have to multiply two functions (or the function by itself) at different values of the parameter and then average over the domain of the function keeping the difference between those parameters:   $C(x)=\langle f(x'+x)g(x')\rangle$ Is there any relatively simple illustrative examples that gives one the intuition about correlation functions in physics?


title: Christoffel symbol

text: For two nearby points in General Theory of Relativity. The change in the vector components when parallel transported is given by ![enter image description here](http://i.stack.imgur.com/VMaQx.png) Now, since the parallel transport change must depend on the path taken between the two points. What path does the christoffel symbol equate to being taken between the two points for parallel transporting a vector ?   * Geometric/intuitive Pictures/arguments will be appreciated alongside rigour as I am from a physics background.


title: How is a spherical electromagnetic wave emitted from an antenna described in terms of photons?

text: When an atenna transmits radiowaves isn't it true that the electromagnetic pulse is radiated away from the accelerating electron as a spherical wave in all directions simultaneously, and if so how can the associated photon be "everywhere" on this rapidly expanding sphere?


title: Are string theories extra dimensions required to be "higher" than the four we know of?

text: Does the mathematics of string theory require extra dimensions to be "higher" than our own? Is it possible that extra dimensions are lower that that the four we currently know? Could our four be placed somewhere in the middle of a proposed scheme of dimensions?


title: Why batteries in serial don't discharge each other?

text: A question that I've asked myself for long time about why batteries in series work as they do. I'll try to explain: If you take two batteries, let's say A and B, and you put A's anode touching the B's cathode (only that, without closing the circuit). It turns out that not only A and B poles don't discharge each other but the resulting device sums the voltages of both batteries. When I think of it my idea is that A and B poles should get "lost" (they have cancelled) but the poles that remain unconnected still have a different potential so it shoud work at that voltage, not the sum. After all you are connecting a + pole to a - one and it's supposed that one pole has a different charge that goes to the other until they are balanced. So, why batteries discharge when closing the circuit but not when it's open if poles of different batteries are in touch? Why the charge from one pole doesn't flow to the other? And why the voltages sum? I'm afraid that the explanation is quite simple but...which is? Thanks in advance! **NOTE: I edit my own question.** As Murphy states, once published I've found there can be a possible explanation here: Why two connected 1.5 volt battery become 3 volt I'll see if it tells me what I want. Nevertheless, feel free to give me your own answer. **NOTE 2:** Ok, guys, thank you very much for your answers. Finally I think I got it! The point is in the electrons that cause the potential difference. The accumulation of electrons in one pole makes impossible that the electrons of the other battery enter the possitive pole. They are rejected electrostatically. Allowing it you would have the two poles charged but the battery doesn't work that way. So, when you close the circuit, both accumulations of electrons sum because one battery would help expelling the other one's charge pushing with it's electrons. Am I right?


title: Velocity vector for constant acceleration in newtonian mechanics

text: If the acceleration vector of an object remains constant, is there no change in the direction of the velocity vector? The question applies to any movements in classical physics.


title: Didn't Newton's Corpuscular theory fail to explain reflection?

text: In 1637 Descartes gave the corpuscular model of light and derived Snell's law. This Corpuscular model of light was further developed by Isaac Newton in his famous book entitled _OPTICKS_ and because of the tremendous popularity of this book, the corpuscular model is very often attributed to him and is called Newton's Corpuscular theory (click here to see the video). Corpuscular theory predicted that if the ray of light (on refraction) bends towards the normal then the speed of light would be greater in the second medium. But the experiment of Foucault proved that, on refraction if the light bent toward the normal, then the speed of light will be lesser in the second medium. Thus, Corpuscular Theory didn't satisfactorily explain refraction. But, reflection is said to be explained by this theory. In case of reflection practically, when the light ray is incident at an angle zero with respect to the normal drawn at the interface, ray bounce back in the same direction with the angle of reflection zero. According to Corpuscular model, if corpuscle bounce back in the same direction (when angle of incidence is zero) it came from, there will be collisions with the corpuscles which are going to be incident later. Then, there will be random displacement of corpuscles. But, this is not what practically happens. So, _didn't corpusuclar theory also fail to explain reflection? does wave theory explain this case?_


title: General parameters of the stress energy tensor in local inertial frame

text: A general 4x4 symmetric tensor has 10 independent components. How many components are we free to prescribe in the local inertial frame? For example, relativistic dust is $\mbox{diag}(\rho c^2, 0, 0, 0)$ in local inertial frame (thus 1 parameter) which gives $$T_{\mu\nu} = \rho v_\mu v_\nu$$ that has 4 parameters (one is $\rho$ and then 3 parameters in $v_\mu$ due to the condition $v_\mu v^\mu = c^2$). Another example is perfect fluid with $\mbox{diag}(\rho c^2, p, p, p)$ (thus 2 parameters) which gives $$ T_{\mu\nu} = \left(\rho+{p\over c^2}\right) v_\mu v_\nu + p g_{\mu\nu} $$ That has 5 parameters ($\rho$, $p$ and the spatial components $v_i$). As such, it seems to me that there are only 7 independent parameters in the local inertial frame, as the other 3 degrees of freedom are given by the velocity (which is zero in the inertial frame). Is that correct?


title: How do I express the interaction energy between two charged spheres?

text: Consider two identical insulating spheres each with radius $R$ and uniform charge $Q$ through their volume. They are separated from their centers by a distance of $d>2R$. Here is my general equation for potential energy $$U_E = \frac{1}{2}\int_V \rho(r)\phi(r)\mathrm{d}\tau$$ How would I express the interaction energy between the two spheres? With this, would I use $\rho$ from sphere 2 and $\phi$ from sphere 1, evaluated at 2? Is this going in the right direction? I found the potential energy inside the sphere using ![enter image description here](http://i.stack.imgur.com/JF20y.gif) and ![enter image description here](http://i.stack.imgur.com/dk0QW.gif) to come up with ![enter image description here](http://i.stack.imgur.com/xyN2W.gif) but I believe this is for a single sphere, and am unsure how the second sphere would affect my results. Do I ignore the second sphere altogether?


title: How to calculate velocities after collision?

text: I'm currently writing a program for a particle simulator. One of the requirements is that the particles collide in a realistic way. However, I don't know how to calculate the final velocities. For each collision, I have the $x$-component and $y$-component of each velocity, as well as the displacement and mass of each particle. Is it possible to calculate the direction and magnitude of their velocities after the collision? If so, how?


title: Thermodynamic limit "vs" the method of steepest descent

text: Let me use this lecture note as the reference.   * I would like to know how in the above the expression (14) was obtained from expression (12).  In some sense it makes intuitive sense but I would like to know of the details of what happened in between those two equations. The point being that if there were no overall factor of "$\sqrt{N}$" in equation (12) then it would be a "textbook" case of doing the "method of steepest descent" in the asymptotic limit of "N".   * I am wondering if in between there is an unwritten argument that in the "large N" limit one is absorbing the $\sqrt{N}$ into a re(un?)defined measure and then doing the steepest descent on only the exponential part of the integrand.  I don't know how is the "method of steepest descent" supposed to be done on the entire integrand if the measure were not to be redefined.   * But again if something like that is being done then why is there an approximation symbol in equation (14)?  After taking the thermodynamic limit and doing the steepest descent shouldn't the equation (14) become an equality with a sum over all the $\mu_s$ which solve equation (15)? Though naively to me the expression (12) looks more amenable to a Dirac delta function interpretation since in the "large N" limit it seems to be looking like the standard representation of the Dirac delta function, $\frac{n}{\sqrt{\pi}} e^{-n^2x}$   * I would like to know of some comments/explanation about this general philosophy/proof by which one wants to say that the "method of steepest descent" is exact in the "thermodynamic limit".


title: Why is the objective function used in Nudged Elastic Band method reasonable?

text: In Nudged Elastic Band (NEB) method, which is used to find reaction pathways when both initial and final states are know, an objective function is first constructed and then minimized to find reaction pathways. NEB creates a string of replicas (or 'images') of the system between initial and final states, and connects them with springs to represent a path connecting initial and final states. Initially, the images may be interpolated between reactant and product linearly. Then an optimization algorithm is applied to relax the images down towards the Minimum Energy Path (MEP). The objective function is constructed as a combination of potential energy of each image and "spring energy" between each image as equation below shows. $$ S(\vec R_0,...,\vec R_N)=\sum_{i=1}^{N-1}E(\vec R_i)+\sum_{i=1}^N\frac{k}{2}(\vec R_i-\vec R_{i-1})^2 $$ Where $\vec R_i$ is the coordinates of i-th image. NEB solved "corner-cutting" and "sliding-down" problems in this model by considering vertical and parallel components of potential force and string force. I will not introduce here, you can look at this article for more details. My question is why the objective function above is reasonable. Does minimization of this objective function sufficiently leads to the MEP? If yes, how to prove it? If no, why is this "chain-of-states" model widely used? Thank you very much for answering.