Is inertia present in a body while in state of motion?

According to Newton's law of inertia (written in book: **The Mathematical Principles of Natural Philosophy**** ) it is said that in all objects having mass, have always inertia included in their property but if a body is in motion well then, would there be inertia in that body. For instance: A ball is hit by a bat and is in state of motion in air so would there be inertia present in that current state of ball or not? According to friction law of physics friction is any thing which exists when two or more bodies come in contact with one-another and it is against the direction of force (pushing a body) which is against inertia so would friction be helpful too in supporting inertia of a body consequently? If yes then could it be said that, inertia is directly proportional to the friction?


Are principle of Conservation of energy and principle of conservation of momentum consequences of Newton's laws?

It is known that principle of Conservation of momentum and principle of conservation of energy are two fundamental principles of physics.But in RP Feynman's _Lectures of physics,_ in the chapter of conservation of momentum, it is said that conservation of energy and conservation of momentum are consequences of Newtons laws of dynamics.Is it correct?


Photon temperature above Electroweak Symmetry Breaking (EWSB) transition

In discussions of the history of our Universe, photon temperature is substituted for time. As the Universe cools, phase transitions break symmetries, including electroweak symmetry. Why does it make sense to speak of photon temperature prior to Electroweak Symmetry Breaking (EWSB)? Is this a misnomer? Is it shorthand for "boson for unbroken $U(1)$" temperature? Or did I misunderstand something.


What will be the effect of placing a light source very close to a photodiode?

What will be the effect of placing a photodiode really close to a laser source and what should be the appropriate distance between a light source and photodiode to get maximum output current?


How do we show that no hidden variable theories can replace QM?

I've always hit two big stumbling blocks in conceiving of the proof or disproof of hidden variable theories as being even valid idea, let alone an answerable question... I feel I must be misunderstanding some very fundamental things.   1. Quantum mechanics is deterministic, ignoring the bit where we take our complex number and fudge it into a probability.   2. We use computers to make calculations to provide the predictions of quantum mechanics - how is this not itself a hidden variable theory? From what I understand the famous Bell theorem and work around it doesn't disprove hidden variable theories, just one specific flavour of them, and even then I am not convinced. I feel like I must be missing something again, because this result seems to me in no way profound... although it certainly fits into the picture of the EPR 'paradox' and is a good demonstration of the validity of QM. I guess the crux of my problem is that this 'area' really addresses wave function collapse and the problems it causes, but I don't see the need for some collapse mechanism to begin with... again I feel like I am missing something really fundamental. Apologies if this is a bad question. I have had a look around and there are similar questions, but I don't feel that either of my points are addressed...


Energy required to remove both electrons

The question is: **Q.** An energy of $24.6$eV is required to remove one of the electrons from a neutral Helium atom. The energy(in eV) required to remove both the electrons from a neutral Helium atom is: $a.$ 38.2 $b.$ 49.2 $c.$ 51.8 $d.$ 79.0 **My try:** Since after removing an electron the Helium atom becomes a single electron atom, Bohr's Model can be applied. The energy of the single electron is hence given by $-13.6\frac {z^{2}}{n^{2}}$ (where z is atomic no. and n is the no. of the shell in which the electron resides). Thus the energy comes out to be $-13.6\frac {4}{1}$, which is $-54.4$. Thus the total energy required to remove both the electrons is $24.6+54.4=79.0eV$, which should be option $d.$. But the answer is $a.$. Did I go wrong somewhere? Please correct..


Crash course how cosmologists develop mathematical model for the Big Bang theory

First of all, probably it might be useful to introduce me. I have sufficiently good enough mathematics and physics knowledge but only in bachelor level in engineering course. I am extremely interested in cosmology and right now I cannot wait to know how the cosmologists develop mathematical model to formulate the Big Bang theory chronologically from the earliest stage up to the conclusion that the space- time is expanding. The explanation should not be too detailed but please keep the mathematical aspects intact by providing me with the main, basic formulae. Ten paragraphs with 5 sentences each might be more than enough. Hopefuly I can have a good guidance to start learning cosmology from the given answer.


Given partial derivatives find the state function

I have to find the state function of a gas (which under very low pressures behaves as an ideal gas). So, first I tried writing it as part of an equation $ dV = \dfrac{\partial V}{\partial T}\ dT + \dfrac{\partial V}{\partial p}\ dp $ and then checking if the "switched" derivatives will be equal. And then I tried to find an integrating factor as a function of $p$ and then $T$. The result in both cases was overly complicated which suggests that it is not the correct answer. Could you please explain to me how to solve it? And when is it alright to change $V, p, T$ using the ideal gas equation to make calculations easier? Moreover, I have a conceptual difficulty - what would the integrating factor times $dV$ represent? I only used integrating factor to prove that $1/T$ was the integrating factor of heat (which gives entropy). ![the partial derivatives](http://i.stack.imgur.com/YWeR4.jpg)


Rotational velocity of face-on spiral galaxies using spectroscopy

I am doing my first steps in spectroscopy (IFS actually) and how we can learn more about galaxies by using it. I came up with a simple question which, unfortunately, I can not answer: How can we measure the rotational velocity of a spiral galaxy given that the galaxy is perfectly face-on? I understand how to do so if the galaxy was tilted or edge-on: By measuring the Doppler shift of the spectral lines for each spatial position. One side will be red, the other blue, and the rest somewhere in between. But what about if the galaxy is perfectly face-on? The Doppler shift will be zero right? Most of the resources, papers etc I find online are about tilted galaxies.


Is there a thought experiment which brings to light the contradiction between General Relativity and Quantum Mechanics?

I've been told that GR and QM are not compatible, is there an intuitive reason/thought experiment which demonstrates the issue? (Or one of the issues?)


Is there a strong force analog to magnetic fields?

In special relativity, magnetism can be re-interpreted as an aspect of how electric charges interact when viewed from different inertial frames. Color charge is more complex than electric charge, but also has deep similarities to it. This leads to three questions:   1. _Is it mathematically possible to re-interpret standard chromodynamics in terms of frame-dependent combinations of “chromostatic” and “chromomagnetic” interactions?_ The obvious analogy would be that moving color charges would create chromomagnetic fields that could in turn alter the paths of other nearby moving color charges. (Note: My just-created word “chromomagnetic” is analogical only, since the component of chromodynamics that it describes would be no more related to ordinary magnetism than color charge is to electric charge.)   2. _If such re-interpretation is possible, has there been prior work on this idea?_ That is, has the idea of a chromomagnetic analog to the magnetic fields proposed or assessed? If so, does anyone know the correct keywords for searching for such works, or have any specific references?   3. _If such a re-interpretation is possible, is there any real advantage to it?_ That is, even if chromodynamics _can_ be split into two frame-dependent components, that does mean it should be. It could be an idea that is plausible in terms of the some mathematical framework, but has no real conceptual or modeling merit. Conversely, if meaningful cases exist in which a sustained chromomagnetic field could be generated, the concept of chromomagnetics might prove useful for the same reason that we still have a concept of independent magnetic fields, more than a century after such fields were shown fundamentally to be only another aspect of moving electrical charges. **2012-09-09 Comments on Answer by David Bar Moshe** I am awarding the answer to David Bar Moshe specifically for the excellent reference he provided to a popular-style 1997 Edward Witten paper on the issue of quark confinement. In particular, on page 3 of that paper in the bottom left corner Witten says this: > "As a non-Abelian gauge theory, QCD has fields rather similar to ordinary > electric and magnetic fields but obeying a nonlinear version of Maxwell’s > equations. Quarks are particles that carry the QCD analog of electric > charge, and are confined in vacuum just as ordinary magnetic charges would > be in a superconductor. This analogy led in the 1970s to the idea that the > QCD vacuum is to a superconductor as electricity is to magnetism. This is an > important idea, but developing it concretely was out of reach in that > period." This answers all three of my questions: 1. = Yes, QCD analogies to EM were explored decades ago (!), back in the 1970s, albeit incompletely due to lack of sufficient mathematical support at that time (and possibly still today, according to David Bar Moshe's comments); 2. = Yes, there was prior work and continuing work, to which David Bar Moshe has provided some references; and 3. = Yes, the interpretation does appear to have have value, since it enables a useful analogy between monopole magnetic fields in a superconductor and QCD "magnetic-like" color force components that in a vacuum produce a similar effect. So, David, thanks! For anyone else reading this, I would note that I'm pretty sure we are still talking here about an _analogy_ of magnetic-like components within the color forces and the separate but mathematically similar description of true magnetic fields between (hypothetical) monopoles. Dave, if I am mistaken about that -- if you are suggesting that there is more than just an analogy going on between the component of QCD that produces confinement and the monopoles with conventional magnetic fields -- then you may want to add a note to that effect, since I'm still assuming (as Witten seemed to do) that this is an analogy or mathematical similarity only, versus an actual link between QCD and magnetic fields. I may add more comments after reading more from your other references. I appreciate your observations about unknown areas, also. So again: Interesting answer, and one that for me had a very unexpected outcome. Since I was only basing my question on an analogy to relativistic electromagnetics, I never would have guessed that it was used decades ago for something as important as quark confinement.


Electromagnetic fields and human behaviour

To what extent does electromagnetic fields and/or radiation from space have influence on human behaviour and/or physiology?


Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a plate on my palm to explain this to my fiancee! I have also looked at spinors as mathematical objects, such as the 2d subspace of complex 3-space such that $X·X = 0$, and feel that I understand this well also. I am confused by spinors in physics. Are they still vectors that are isotropic (inner product with themselves is 0)? In what vector space? Normally states are vectors in infinite dimensional spaces! Every attempt to find literature which pins down specifically what a spinor is (in physical terms) seems to assume that one is already well acquainted with the idea. Take, for example, the Dirac equation. I can see that the solutions are four- component wavefunctions, which then splits into two parts. Is this a spinor? Why? What vector space do these solutions live in? I believe I've heard that the answer has something to do with representation theory, perhaps of the Poincare group? I am also familiar with the basics there, so don't hesitate to explain in terms of representations.


Is there a term for the argument of the sine function outside of geometry?

Are there similar terms in other areas for the idea the "angle" conveys in geometry? I find that functions for abstract things such as pressure, electrical currents (nothing geometric there) on AC circuits and are described by the sine function (argument is an angle, therefore a geometric entity). Do we really have that abstract things are necessarily described by geometric entities? Maybe the concept of angle we know is a simply an example of a greater idea (like a a measure of difference between instances of the same thing) applied to geometry or am I tripping?


Does a lightning rod prevent lightning strikes?

I was always under the impression that a Lightning rod worked to protect a building by being the tallest part of the building and having a low path of resistance to the ground, thus being a good target for lightning to hit, reducing the chances of lightning strikes elsewhere on the building. However when reading an Ars technica article, one comment (from user "malor") said that theory is incorrect, and explained that a lightning rod actually _prevents_ lightning strikes in the first place: > Lightning happens when a really huge negative charge builds up in the > ground, corresponding to a positive charge overhead, and the differential > becomes sufficient to jump the gap. (and, as others are pointing out > repeatedly, air is an excellent insulator, so it takes a whacking huge > voltage differential to make the arc.) > > A lightning rod doesn't exist to provide a spot for the lightning to strike. > Rather, it exists to dissipate the charge so that the strike never happens > at all. This comes from early experiments with the Leyden jar; if a pointed > metal rod was attached to the jar, it wouldn't charge. The electrons are > able to leap off a pointed tip, and into the air, dissipating charge as fast > as it accumulates. The lightning rod does the same thing, on a larger scale; > it spits out electrons into the air like crazy, so that the charge won't > build up sufficiently, and the lightning never hits at all. > > Now, if the charge accumulates faster than the rod can dissipate it, there > can still be a lightning strike, and the rod typically IS the best route to > ground. But everything around the rod is going to take a hell of a jolt > anyway, including, most likely, your electronics. A lightning rod getting > hit means it failed to work adequately; ideally, it should never be hit at > all. How does a lightning rod actually work? Does a lightning rod actually prevent lightning strikes?


Causality and Quantum Field Theory

I have a problem with proof of causality in Peskin & Schroeder, _An Introduction to QFT,_ page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for $y=0$ as follows: $[\phi(x,t),\phi(0,0)]=\int \frac{d^3p}{(2\pi)^3}\frac{1}{2\sqrt{p^2+m^2}}\left(e^{-i\mathrm{p}.\mathrm{x}-it\sqrt{p^2+m^2}}-e^{i\mathrm{p}.\mathrm{x}+it\sqrt{p^2+m^2}}\right)$ The book goes on about how the integrand being Lorentz invariant makes this integral zero for the x out of the light cone. But I (not being a special relativity expert) want to see it more rigorously: after changing variables $p\to-p$ in the first term, the equation simplifies to: $[\phi(x,t),\phi(0,0)]=\int \frac{d^3p}{(2\pi)^3}\frac{-2i}{2\sqrt{p^2+m^2}}e^{i\mathrm{p}.\mathrm{x}}\sin\left(t\sqrt{p^2+m^2}\right)$ using spherical coordinates: $[\phi(x,t),\phi(0,0)]=\int \frac{dpd\phi d\theta p^2\sin\theta}{(2\pi)^3}\frac{-i}{\sqrt{p^2+m^2}}e^{ipx\cos\theta}\sin\left(t\sqrt{p^2+m^2}\right)\\\ [\phi(x,t),\phi(0,0)]=\int_0^{\infty}\frac{dpp}{(2\pi)^2}\frac{-2i}{x\sqrt{p^2+m^2}}\sin (px)\sin\left(t\sqrt{p^2+m^2}\right)$ again after another change of variables $u=\sqrt{p^2+m^2}$, $[\phi(x,t),\phi(0,0)]=\frac{-2i}{x}\int_m^{\infty}\frac{du}{(2\pi)^2}\sin (x\sqrt{u^2-m^2})\sin\left(tu\right)$ I cannot see how this integral should be zero for $x>t$ !!! Can somebody please explain this to me?


Violation of Lorentz invariance (Lagrangian for particle)

I'm trying to get the relativistic action (or Lagrangian) for a free particle in the case of violation of Lorenz invariance. Suppose we have the modified dispersion relation: $$ E^{2}=\Omega^{2}(p^{2}) $$ Here $E$ and $p$- energy and momentum of the particle, $\Omega^{2}$ is a function that takes the form $\Omega^{2}(p^{2})=m^{2}c^{4}+p^{2}c^{2}$ if we have Lorentz invariance. In the other word, we have equation $$ \left(v\frac{\partial L}{\partial v}-L\right)^{2}=\Omega^{2}\left(\left[\frac{\partial L}{\partial v}\right]^{2}\right) $$ **We want to define the Lagrangian**. For example, if $E^{2}=m^{2}+(1+\xi)p^2$, we can obtain(overriding $\tilde{v}=v(1+\xi)^{-1/2}$, we present the equation written above to the standard relativistic form, above which we know everything): $$ L=-m\sqrt{1-\frac{v^{2}}{1+\xi}} $$ **I want to find lagrangian and action for the following case** : $$ E^{2}=p^{2}c^{2}+m^{2}c^{4}+\frac{p^{4}}{M^{2}} $$ Here $M\gg m$. _So, we have equation_ : $$ \left(v\frac{\partial L}{\partial v}-L\right)^{2}=\left(\frac{\partial L}{\partial v}\right)^{2}c^{2}+\frac{1}{M^{2}}\left(\frac{\partial L}{\partial v}\right)^{4}+m^{2}c^{4} $$ -I tried to find the solution in the following way: $L(v)=L_{0}+L_{1}v$, where ${{ L_0}}^{2}={{ L_1}}^{2}{c}^{2}+{\dfrac {{{ L_1}}^{4}}{{M} ^{2}}}+{m}^{2}{c}^{4}$, and $|L_0| \ge mc^2$. I set it in the original equation, but it didn't give significant results.   * _Then I've found the following article_ : http://arxiv.org/abs/1209.0464 (arxiv: 1209.0464) Here (pages $6$-$7$) they obtaining the Lagrangian for the case I described above, and they say that in the general case Lagrangian can be defined as (equation $18$) $$ L=-m\sqrt{1-v^{2}}F\left(\frac{1}{\sqrt{1-v^{2}}}\right) $$ I tried to find a solution in this form (by substituting in the equation) but I didn't succeed. **Can you help me to obtain Lagrangian (action) for the relation I wrote? Also it would be great if you gave the usefull references devoted to the topic** (for example, where they obtain Lagrangians, etc)


Stokes' theorem and alike for fractal surfaces

Has anyone ever read an article dealing with the extension of integral theorems to fractal surfaces ?


Detecting absolute motion with Doppler (reconciling receiver frequencies)

Suppose a transmitter emits a pulse of light every $T$ seconds. The spacial distance between pulses is $cT$. Suppose the receiver is approaching it with speed $v$. ![enter image description here](http://i.stack.imgur.com/M2EGq.png) Because, after receiving the first pulse, the next pulse is at distance $cT$ and it will be covered by both light and receiver approaching each other at speed $(v+c)$ so that the next pulse will be detected after time $$t_1 = {cT \over v+c}$$. Particularly, when receiver is moving at v=c, we have $t_1 = T/2$ -- the receiver detects twice the frequency emitted. Suppose that light source is moving instead ![enter image description here](http://i.stack.imgur.com/a7bof.png) Now, at a time the first pulse reaches the receiver, the next pulse is closer, by distance $cT-vT$ because the transmitter has traveled $vT$ meters when light traveled $cT$. This distance, $cT-vT$, will be closed by light in $$t_2 = {(c-v)\times T \over c}$$ seconds. Particularly, when transmitter speed is equal to the speed of light, v=c, the observer detects infinite frequency ($t_2 = 0$). By measuring the frequency we can detect who is moving, receiver or transmitter. Do you know that this contradicts to the principle of relativity and Michelson-Morley experiment which both say that we must not be able to detect who is moving? Einstein has corrected this by time dilation. Can you show how Lorentz arises here naturally for the Doppler correction? I can guess that time dilation can be modeled by scaling the time period of moving body. That is, when receiver is moving, we can have the observed frequency of $t_1 = {cT \over (c+v) } \times {1 \over f_1(v)} $ and when transmitter is in motion then true observed period $t_2 = {(c-v) T\over c} \times f_2(v)$. I have placed $f_1(v)$ into denomitator because receiver slowdown can be considered as transmitter's speedup. Now, receiver must observe the same frequency regardless of who is moving, $${c \over (c+v) } \times {1 \over f_1(v)} = {(c-v) T\over c} \times f_2(v)$$ or $$f_1(v) \times f_2(v) = {c^2 \over c^2-v^2}$$ which looks suspiciously identical to the square of the Lorentz factor squared. Should I guess that $f(v) = f_1(v) = f_2(v) = \sqrt{1 \over 1 - v^2/c^2}$?


Gravitational binding energy and integrated potential energy not the same?

Before looking up the formula for the gravitational binding energy of a uniform sphere, I simply figured that the general formula for binding energy of an arbitrarily-shaped mass distribution would be $\left\langle V,\rho\right\rangle $, where $V$ is the potential as a function of space due to the distribution, $\rho$ is the density distribution as a function of space, and $\left\langle ,\right\rangle $ is the inner product (ie, integral over all space). Going ahead for the special case of a uniform sphere of density $\rho$ and radius $R$, I used the well-known result for the potential inside of a sphere of uniform density, $$V(r)=\frac{2}{3}\pi G\rho(r^{2}-3R^{2})\mbox{ for }r\leq R. $$ I then computed the inner product, $$\left\langle V,\rho\right\rangle =\int_{0}^{R}r^{2}dr\int_{0}^{\pi}\mbox{sin}(\theta)d\theta\int_{0}^{2\pi}d\phi V(r)\rho=-\frac{32}{15}G\pi^{2}R^{5}\rho^{2}=-\frac{6GM^{2}}{5R}, $$ which is twice the correct result, $\frac{-3GM^{2}}{5R}$. (The last equality follows from $\rho=\frac{M}{4/3\pi R^{3}}$). I fully understand the geometric proof that the binding energy of a sphere is $\frac{-3GM^{2}}{5R}$ that proceeds by successively moving shells in from infinity, but was a bit confused when the inner product approach gave an extra factor of 2. My question is as follows:   * Is the extra factor of two in $\left\langle V,\rho\right\rangle$ due to double-counting the interaction energies?   * If so, is the correct formula for the binding energy of an arbitrary mass distribution $U=\frac{1}{2}\left\langle V,\rho\right\rangle$? I am trying to estimate the total energy released when two planets which are just barely touching each other collapse to form a single large sphere, and in the course of the derivation this issue sprung up. Any help explaining the extra factor of 2 would be greatly appreciated!


What causes electron to orbit the nucleus in an atom?

What causes the electron to orbit the nucleus? Which is the force that causes it to do so? Is it related to the Electro - Magnetic force? .


Potential functions

Can someone please explain what a potential is? Example. velocity potential in ideal flows, acoustic potential (gradient of which gives the particle velocity in a sound wave). Whenever I see potential functions complex analysis is applied to compute integrals, complex functions for conformal mapping. I vaguely understand that potential functions are independent of path and in complex plane the integration around a contour is independent of path as well (the integration depends only on the end points, hence in a closed curve say region $Z$, if function, $f$ is analytic then integration of $f$ over $Z$ is 0). Can you please explain on this?


Water and electricity? Hot or Cold?

Is hot water or cold water more conductive for electricity or is it the same regardless?


What can be done to increase the strength of a given electromagnet?

There's a lot of information online about increasing the strength of an electromagnet with more turns, different cores, etc, but not much about factors in the power supply that affect the strength of the magnet. I've got a commercial electromagnet rated for 12VDC that's supposed to produce 200 lbs of pull. What factors in the power supply will maximize this magnet's strength? Thanks!


Why is plasma the highest "state" of matter?

As far as solid, liquid, gas, plasma go, why is plasma the highest state? Are there any other states of matter?


If gauge symmetries are fake, then why do we care if they are anomalous?

My understanding is that gauge symmetries are fake in that they are only redundancies of our description of the system that we put in (either knowingly or unknowingly) see Gauge symmetry is not a symmetry?. Moreover it seems like we could in principal work without the gauge redundancy, and things would just be a bit nastier at intermediate stages of a calculation. At the same time, people put a great deal of emphasis that gauge symmetries cannot be anomalous. I understand that if we work with the gauge redundancy then we do indeed need it to be anomaly free so that the unphysical polarizations will cancel. However, what if we work without the gauge redundancy, with just the physical polarizations, what goes wrong then if we don't make sure the would-be gauge symmetry is not anomalous?


What is a mass gap?

I read that Quantum Chromodynamics is a theory with a mass gap. What is a mass gap in laymen terms? Why some theories have it? Which theories does not have it? **Note: I searched for mass gap before asking, all topics about it assume that the reader knows what it is, or define it in an advanced and technical way. So I'm looking for a simple answer if possible.**


Filming light in motion?

Regarding this video http://www.youtube.com/watch?v=Y_9vd4HWlVA, which claims to "film" light in its motion. Is it not an absolute nonsense? Even if photons could even be "seen" (meaning, returning light them self), isn't this video look unrealistic? You can see that the area around the "light" is getting brighter, but that absurd.. light is not a flash light... light does not illuminate the area around it, because it does not "emit" more light to all the directions.. Am I missing something?


Is the second law of Thermodynamics an immense tautology?

Let's imagine for a second a system that is changing constantly from one microstate to another one. It could be a given volume of a gas with its atoms moving and bouncing around, or a deck of cards being constantly shuffled by a monkey. If the starting microstate belongs to a macrostate with very few microstates, chances are that in the next step the system will be in a macrostate with more microstates in it. There is nothing mysterious about this, it is simply a matter of probabilities and how we define them. Now we have the second law of thermodynamics, that says that entropy always increases. It could have been reformulated like: a system that is permanently visiting different states, will spend more time in those which have a higher probability of being visited. Things more probable occur more times. And, since we define probabilities in terms of frequency: Are not we simply saying that things more likely to occur, occur more times? **Isn't it true then, that the second law is simply an immense tautology?**


Relativistic Time Difference

I thought that it might be interesting to calculate the time difference between a clock placed on Earth when I was born and a hypothetical clock placed at the centre of the sun at the same time. I thought that I might use special relativity and the Lorentz factor. However, even if the speed of the Earth were constant (which it is not), the Earth still experiences acceleration as it traverses its (almost) elliptical orbit around the sun. Given there is acceleration, I suppose that I must use general relativity. How might one calculate the time difference experienced between me and my barbecued brother?


Mean field theory = large-N approximation?

Wikipedia entry of 1/N expansion (or 't Hooft large-N expansion) mentions that > _It (large-N) is also extensively used in condensed matter physics where it > can be used to provide a rigorous basis for mean field theory._ I would like reference(s) that reviews this connection between MFT and large-N.


In a double slit experiment are any particles lost because they hit the space between the two slits?

In its wave-form a particle should pass through every time because it propagates in all directions. So there shouldn't be any losses of particles landing in between the slits, right?


Is having full information about the resonances of a rigid body equivalent to having full information about its material parameters?

Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, including spatial variation, with perfect accuracy? My intuition says that the answer is yes, but I am not sure how to prove it. For concreteness, here is a specific example. Given a right circular cylinder such as the one shown below; will measurements of its resonant modes allow me to reconstruct its spatially non-uniform Young modulus and Poisson ratio? ![enter image description here](http://i.stack.imgur.com/fFUyi.jpg)


Simple three-body-problem?

Consider the problem of three bodies two of which having mass _M_ , one of them having mass _m_. Body _m_ is in the middle between the other two, coupled to them by two equal linear springs in rest. Now fix the two bodies _M_ and move body _m_ for a small amount perpendicular to the connection line. Now let loose the three of them. The solution is trivial when we assume _m_ to be finite and _M_ to be infinite (or vice versa), or when the spring constant is 0. But the problem is so symmetric, that there might be hope to get a closed or approximate solution even for finite _m_ >> _M_ , _M_ >> _m_ , or even _M_ = _m_?


The Opera Neutrino Experiment and the Supernova 1987

So this probably stems from my massive ignorance about post-Newtonian physics but the supernova results of 1987 which measured neutrinos arriving 3 hours before the light from the supernova have been presented as a non-trivial objection to the Opera result. However if the Opera result is confirmed, then doesn't that weaken the supernova objection since star mechanics relies heavily on relativity and such a result might imply that relativity is not the complete picture which in turn could imply different star mechanics than the one we understand? Thanks Edit: To clarify, I'm asking whether the supernova objection may not apply given that this experiment seems to challenge relativity which is what has been used to understand the behavior of stars, supernovae and such.


Newton's laws of motion

According to Newton's laws, if net force is zero on a particle then the particle is at rest or in uniform motion, and if it is not zero then it is accelerating. So when a car moves on the road, friction helps it to accelerate. If the acceleration is zero, (i.e) it is moving with constant velocity, the net external force must be zero, so friction is zero. Then with friction equal to zero how is the car moving on the road?


Best Materials Science Introduction Book?

Why is this here? > Sorry, I know that this isn't the best place to put this however there does > not seem to be any other stack exchange specific to this so I thought this > would be the best place since it seemed the most related. Question: > What would you suggest as a good introduction to materials science? (Prior > usage and examples would be wonderful, my particular focus is neutron > scattering) What I've Found So Far: >   * Materials Science and Engineering: An Introduction by Callister (Seems > to have good reviews) > Thank you for the help. Note: I didn't see anything related on the book request wiki.


Is there a closed form solution to the Esdale river problem?

This is probably not well known problem but it looks like open problem. What kind of methods there are to find a closed form solution to the physical situation? * * * Can you solve this problem? You're passively flowing downstream in the middle of a river that has its fastest flow in the middle. Most places along the riverbank are too steep or too vegetated, but suddenly you spot a perfect location for coming ashore a distance y away (measured along the river bank). Note that y can be zero so that you only spot it when you're already drifting past it; or if you're slower still, y can be negative. Now assume your swimming speed is a constant v, while the river's flow is V in the middle and linearly decreasing towards the shore. What angle should you take, as a function of your distance to the shore (x), to reach the desired location using minimum effort (i.e. shortest journey or, equivalently, shortest time)? ![enter image description here](http://i.stack.imgur.com/ha4VF.jpg) Source and image credit: Hanna Kokko.


How to make a black hole?

Many Physics discussions I have often conclude with: > Well you will then form a black hole... My questions are:   * Is there a general recipe for making a black hole?    * If not, then can you list the families of black hole causes? This means I can put a more rapid halt to many future Physics discussions.


When experimental search for exotic dark matter-like particles began exactly?

I am looking for information about :   * **what was the first experiment that claimed to look specifically for non baryonic dark matter particles ;**   * **when occurred the first serious(*) reporting of an experimental result regarding a non baryonic dark matter candidate.** (*: published in a peer reviewed journal)


![setting](http://i.stack.imgur.com/rcorp.jpg) Hi Sir/ Madam Want to achieve maximum magnetic force & find out the max point for this setting, want to know   1. is the strongest magnetic force must in the 'dot' area ( middle poin of setting )   2. would the length of D1, D2, D3 affect the 'dot' location?   3. if add a metal sheet ( material AISI1010 ) between the magnet, would the magnetic force increase? would the thicknes of the sheet D4 affect the force? Magnet material: NdFeB 52 5x10x18mm D1 would be <4mm


Relating theta_QCD to neutron EDM

How do I relate the topological $\theta_\text{QCD}$ parameter to the electric dipole moment (EDM) of the neutron? I am very familiar with chiral perturbation theory. I just need to know how to take $\theta_\text{QCD}$ into account. Any good sources?


Action and reaction pair problem

If I sit on a chair, the first action and reaction pair force is my weight and the force acting on the earth by me. And the 2nd pair is the force acting on the chair by me and the opposite force. According to newton third law, the action and reaction force must be equalI. 'm confused because how come the force acting on the chair by me is equal to normal force? F=gmm/r2


Construction of Angular Momentum eigenvectors

I have a problem that asks (verbatim) > Carryout the construction of the eigenvectors of total angular momentum and > its z component for $j_1$=3/2 and $j_2$=1/2 I am not completely sure where to start out with this. Our professor has not given the best explanation. The equations I have been trying to use are the following: $$\textbf{L}^2|l,m\rangle =\hbar^2l(l+1)|l,m\rangle$$ $$\textbf{L}_z|l,m\rangle=\hbar m|l,m\rangle$$ I figure I probably want to use the same equations, but for $\bf J$ instead of $\bf L$. I am also confused what the differences between $j_1$ and $j_2$ would represent. I would greatly appreciate any help!


Number of wave modes in a cavity

I'm trying to calculate the number of acoustic modes that can exist in a room in a certain range of frequencies. I thought of using the Rayleigh-Jeans formula for the electromagnetic standing wave modes possible in a cavity ($\frac{dN}{d\nu}=\frac{8\pi V}{c^3}\nu^2$), dividing by two to account for the fact that longitudinal waves such as sound do not have two different polarizations. I'm not sure that this formula is valid for a non-cubical cavity, though, because the development represents the modes as points in 3d "mode space", and counts them by looking how many of the points are included within a sphere. If the cavity is not a cube, wouldn't there be more modes in a direction than in another, stretching the sphere into an ellipsoid? My book (Eisberg-Resnick, "Quantum Physics") says that the formula is valid for an arbitrary cavity, but doesn't explain that statement. Can someone explain to me why I'm wrong?


What is the perturbative vacuum?

Will someone please supply me with a good definition of the perturbative vacuum vis a vis quantum cosmology?


computing wavefront deviations for off-axis mirrors

Suppose you have two off-axis aligned mirrors which are nearly planar, with small deformations such that a wavefront at plane $P_I$ at distance $-D$ from the mirrors, is refocused at distance $D$ at plane $P_O$. I only care about monochromatic waves of wavelength $\lambda$ I want to compute how much orders of Zernike mode expansion are required to account in the deformations of the mirrors, such that an input gaussian beam with intensity $I_I$ is projected to a gaussian beam with intensity $I_O$, such that $\frac{I_O}{I_I} > 1 - f$. That is, only a fraction $f$ of the beam is lost at most from the $T_{00}$ mode I've been looking into the literature but most of it does the wavefront calculations for on-axis elements. I wanted to find some proper literature for the off-axis case that would suit the particulars of my problem


experimental bounds on microcausality violation

In "The Great Soviet Encyclopedia", 3rd Edition from 1970-1979, (evidently an old book), some V. I. Grigor’ev has a well-written little note on microcausality. Towards the end he states an experimental bound on the vialotation of microcausality. According to that ancient encyclopedia article, if such violation occurs at all then it must be on scales smaller than $10^{-15} cm$. I don't see a citation for that number in his text (I don't have the original book in my hand though, to be frank) and I am wondering which experiment Grigor’ev might have been thinking of. And: if there are decent such experiments indeed, then probably there are more recent numbers on whatever that experiment actually measured. My question is if anyone might be able to help me with finding whatever useful there might be to say about experimental bounds of the kind Grigor’ev seems to be referring to, or anything similar. Maybe I should say that I am not trying to find such violations. Rather, I am in some discussion with some mathematicians interested in axiomatic QFT, and I'd find it entertaining to be able to accompany the statement of the microcausality axiom (or a similar axiom) with something substantial that has the word "experiment" in it.


Uncoupling a coupled oscillator Hamiltonian by change of variables

I'm working on the problem of two entangled harmonic oscillators with Hamiltonian: $$H = \frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2].$$ Introducing the variables $x_± = \frac{1}{\sqrt{2}} (x_1 ± x_2)$, and frequencies $\omega_+ = k_0^{1/2}$, $\omega_- = (k_0 + 2k_1)^{1/2}$ I've tried to write this Hamiltonian in an uncoupled way. I have found that $(x_+^2 + x_-^2) = (x_1^2 + x_2^2)$ and $2x_-^2 = (x_1 - x_2)^2$. Furthermore, since $ p_{1,2}^2 = \frac{\partial^2}{\partial x_{1,2}^2}$ when ignoring constants, I find $$p_±^2 = \frac{\partial^2}{\partial x_{1}^2} + \frac{\partial^2}{\partial x_{2}^2} ± 2 \frac{\partial}{\partial x_{1}} \frac{\partial}{\partial x_{2}}.$$ Then $\frac{1}{2} (p_+^2 + p_-^2) = p_1^2 + p_2^2$. And: $$H = \frac{1}{2} [ \frac{1}{2} (p_+^2 + p_-^2) + k_0 x_+^2 + (k_0 + 2k_1)x_-^2] = \frac{1}{2} [ \frac{1}{2} (p_+^2 + p_-^2) + \omega_+^2 x_+^2 + \omega_-^2 x_-^2]. $$ The paper I'm working from gives the normalised ground state as $$\psi_0(x_1, x_2) = \pi^{-1/2} (\omega_+ \omega_-)^{1/4} e^{-(\omega_+ x_+^2 + \omega_- x_-^2) / 2}.$$ However, when I try to check that $H \psi_0(x_1, x_2) = E \psi_0(x_1, x_2)$ I don't find anything sensible ($E$ is negative in the constant terms, and still dependent on $x_±$). I suppose then that my Hamiltonian is wrong, but I can't find my error.


What are anomalous threshold singularities

In the papers of the 1950s and 1960s, I see reference to anomalous threshold singularities. What are these? Is there a good reference that discusses this subject?


Thermodynamic relations from Gibbs-Duhem

Given the Gibbs-Duhem relation $V dp = S dT - N d \mu$, I am having trouble deriving the following identity: $\ (\frac{\partial N}{\partial \mu})_{V,T} = N (\frac{\partial \rho}{\partial p})_T$ The problem is that the variables $N$ and $\rho$ don't appear as infinitisemals in the equation. How can I proceed?


How can we describe the trajectory of an object projected into a viscous fluid?

Can we quantitatively describe the motion of a particle inside a fluid? Let us consider the case when an object (say a sphere, so that stoke's law holds true) is projected into a fluid of viscosity $\mu$, with a velocity $v$, at an angle $\theta$ with the horizontal. I myself approached it through Newtonian mechanics. First off the problem is when the object enters the fluid. The buoyant force as well as the viscous force are varying. There will be some threshold velocity for which it enters the fluid. Let's assume that $v$ given to the ball is enough to take it into the fluid. But how do we describe the motion of the ball while it is midway into the fluid? I tried writing the forces (Buoyant, Viscous and Gravitational) with respect to displacement and form some sort of a differential equation but it din't work for me. So i skipped that part and considered that the ball has entered the fluid and has a velocity $v'$ and makes an angle $\phi$ with the horizontal. This part i think is simpler. At an arbitrary time, i assumed the $x$ component of velocity to be $v_x$ and the $y$ component to be $v_y$. $$\vec{F_x}=-6 \pi \eta r \vec{v_x}$$ &$$\vec{F_y}=-6 \pi \eta r \vec{v_y}+m\vec{g}+\vec{F_{B}}$$ I do not know what to do next. Help will be appreciated, in both the cases- first, when the ball is entering the fluid, and the second after entering the fluid.


Is it correct to assume that a stretched rubber-band has negative entropy change?

If so, how could we express it in equations connecting S,T,Q? I was wondering if the net change is heat transfer was positive; Since we could feel the heat when it is stretched.


Quantum superposition and fate

First of all, sorry for my knowledge of physics. Maybe my question is too obvious but I want to ask it. I am thinking about fate and if it exist or no. According to my assumption if I take any system, for example cube, throw into it some atoms and I will know all the laws which exist at this cube, I can predict position of this atoms at any moment of time. On the other hand I know about such experiment called "Schrödinger's cat" and quantum superposition. So, it brings uncertainty to my system. If this uncertainty will exist at cube than I can't predict anything. Taking into account this assumptions I can say that at our universe fate doesn't exist. But I know that during this choices parallel universe can be created. It's horrible for me to imagine numerous number of parallel universes. What you can say about it? It's naive or it makes sense?


Relationship of the abundance of an isotope and the vapor pressure

Say I have a sample of some alkali metal - Rubidium for example. A sample of Rubidium consists of approximately 27% $^{87}$Rb and 73% $^{85}$Rb. Does the vapor pressure of each isotope individual _necessarily_ have to be in the same proportion? More specifically, assuming I have enough Rubidium around that I have enough atoms of each isotope to reach the saturated vapor pressure, is there any physical reason why the ratio of the vapor pressures would match their relative abundances? I guess this is coming from this perspective - the linear absorption spectrum for a sample of natural rubidium at a specific fine transition (D2 line - 5S1/2 ->5P3/2 if you are interested) looks like image below, the $^{87}$Rb peaks (outer) are clearly smaller than the $^{85}$Rb peaks (inner), and this amplitude ratio is directly related to their natural abundances. If the atoms are in a gas, what forces their vapor pressure to have the same ratio as their relative abundance? ![enter image description here](http://i.stack.imgur.com/0WgIG.jpg) Along a similar line of questioning. A sample of Rb will naturally have some trace amount of Cs in it. The saturated number density of Cs at room temperature is ~$10^9$ atoms per cm$^3$, and I'm looking at a volume of say 1cm$^3$. If I have a droplet of 1 gram of Rb in the volume, which is .001 moles, even at .001% impurity I have $>10^{17}$ atoms of cesium in the sample, clearly enough to reach saturated density. -Does this imply that you can do spectroscopy on a volume with Rb and obtain the same spectrum as a volume with a droplet of Cs in it? If not, what prevents the Cs impurity in Rb to reach full saturation. *Note: Perhaps I should make it clear : The linear absorption spectrum is proportional to the theoretical absorption (for a single atom) times the number density of atoms in the sample. The relative amplitudes of the peaks are therefore a proportional to the relative abundances of the two isotopes.


Rate of twinkling of stars

According to "Why do stars flicker?", the twinkling of stars (stellar scintillation) is caused by the thick layers of turbulent air in the Earth's atmosphere. While the explanation is convincing, it would be better to have some quantitative justification. Since the twinkling of stars is easily observed, it is reasonable to estimate the twinkling rate as ~ $1\textrm{s}^{-1}$. How can one obtain a similar estimate from the underlying theory?


How do I use the direction of a cross product?

The question is: > An electron is projected at a speed of $3.70×10^6\text{ m/s}$ in the > $u=\frac{(i+j+k)}{\sqrt3}$ direction into a uniform magnetic field $\vec {B} > = 6.43 i + 3.43 j −8.29 k$. Calculate the $x$ component of the vector force > on the electron. So I know that I'll need to use $F = q\vec{v}\times\vec{ B}$ and do the cross product between the velocity and the magnetic field but I'm still getting the hold of cross products and vectors and the direction of the velocity is really throwing me off so I don't know how to write it down and solve. If anyone can put me in the right direction that would be awesome. Thanks.


Change in the Higgs mechanism

The most important thing about the Higgs mechanism is that the higgs mechanism is one of the simplest mechanisms ( _certainly the simplest way to break the electroweak symmetry_ ) of elementary particles. There is an unwritten law in particle physics: When the mechanism is simple, just light Weight particles arise. What should be change to Higgs mechanism for the production of heavier particle?


Thorium based nuclear reactor

My question is how thorium works as a radioactive nuclear fuel and what are the special properties of thorium based nuclear reactor.


spectral eigenvalue staircase and quantum system

in a d-dimensional system of Quantum physics , does the Eigenvalue staircase $ N(E)= \sum_{E_{n}\le E} 1 $ determine ALL the properties of Quantum System ?? for example, let us assume that the System is Chaotic and that Spectral eigenvalue staircase is given by $ N(E)= <N (E)>\+ \frac{1}{\pi}ArgZ(1/2+i \sqrt E )$ with $Z(s)= \prod_{n=0}^{\infty}\prod_{l_{p}}(1-l_{p}^{-s-n})$ the Gutzwiller zeta function


An error in Gravitation by Misner Thorne and Wheeler?

I was studying on Gravitation the PPN formalism. Since in equation (39.41) pag. 1087, the term $1 + \dfrac{v^2}{2}+(2+\gamma)U = 1 + \dfrac{v^2}{2}+3U$ (the second in GR) looked odd, I tried (several times) to derive the formula, but I always find: $1 + \dfrac{v^2}{2}+U$ Is this one of the two famous errors metioned by Wheeler? I am really going crazy over this...


Are the cylindrical and spherical form of Jeans' equations equivalent?

The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy axisymmetric Jeans equations? (numerically rather than analytically, otherwise it would all get very messy...) More background: Providing them in sufficient detail to be helpful would be very longwinded (as presumably you'd need to compare all three of the first moment equations in both co-ordinate systems) but for a bit more background people can always look here http://en.wikipedia.org/wiki/Jeans_equations, or in for an in depth discussion look in Galactic Dynamics by James Binney and Scott Tremaine.


plane wave move out

I have a plane wave that is recorded by a set of receivers with $x$ spacing between them (see the sketch). in the sketch (plane wave in black slant line, receivers are the little circles in black and the line in blue is the surface). The place wave arrives with an angle to the normal (alpha) at the receivers. The wave was generated from 2000 cells to the left of my receivers line and 5000 cell deep below my line. Given I have the recording time of the arrival at each receiver (black circle) I was trying to correct for it and shift it into a normal incident rather than slanted as of the plane wave originated from deep below the receivers (5000 below). I do have the velocity of the medium below, arrival time, and angel of incident and I want to accomplish that. My final gaol is not to linear move out the arrive but rather shift it is if it was originated below the receivers. one approach I am thinking of is by multiplying the recording time at the first station by the cosine of the angel and then subtracting that from the first recording time $t_1-\cos(\alpha)\times t_1$.![enter image description here](http://i.stack.imgur.com/zoqTu.png)


WHY is a Refracting telescope image distance $S'=-\infty$?

![enter image description here](http://i.stack.imgur.com/vUi8g.png)![enter image description here](http://i.stack.imgur.com/kRqij.png)![enter image description here](http://i.stack.imgur.com/lOO0v.png) In the above slide, it says that "Eyepiece '2' makes the image distance $S'$ approach $-\infty$." However, What I don't understand is that if I place my eye right at the Eyepiece 2, then I would see the image as if the object (for example, moon) is right in front of me as if I'm observing a rock in my hand... Therefore, I feel like $S'$ has to be closer to zero or some finite number! not negative infinity... Also, eventhough $S'$ is actually negative infinity, I quite don't understand how the moon can be observed as if I am observing a rock right in my hand... If $S'=-\infty$, then my brain says that we can't really observe the cracks, rocks, or even sand in the moon since the object is so much farther away from me... Could somebody explain why I am confused???


Are right handed neutrinos actually antineutrinos and vice versa?

Is it experimentally ruled out that right-handed neutrinos are actually antineutrinos, and left-handed antineutrinos are neutrinos ?


Stronger than Newton's laws?

According to the Newton mechanics, the force is responsible for changing the body velocity, and the body mass is the body inertia - a property to resist to the applied force. These two things make a clear sense in the Newton equation (Second Law) $ma=F$. Both "parameters" (the body mass and the force strength) are observable and measurable. Now I am reading an **article** of G. 't Hooft where he states that "The interactions among particles have the effect of modifying masses and coupling strengths". So not only velocities are modified with forces but the particles masses and the forces themselves. I see here a contradiction with the Newton's definition of force and mass. As soon as the QFT is more fundamental than Classical Mechanics, does it mean that the 't Hooft's law is stronger than the Newton's ones? What is the impact (effect) of this stronger law on Classical Mechanics? How to define now the masses and forces if they are self-modifiable? **EDIT:** OK, is there a classical physics example to demonstrate that a potential interaction can modify mass and charge?


Cloud chamber video showing large particles - What are they?

Watching a video of a cloud chamber on wikipedia (http://en.wikipedia.org/wiki/File:Cloud_chamber.ogg), I cannot help noticing the large collisions that take place at 00:12 and 00:24. What are they? Alpha particles? They are huge compared to the small ones (i guess they are electrons)


kirchhoff's law.Simple question

So far I calculated total resistance and it is 4.66 ohms and there is 12 volt voltage across them, what is the best way or HOW to calculate current for each resistor ? ![enter image description here](http://i.stack.imgur.com/DZWKG.jpg)


Voltage drop along an idealized resistance-free wire in a circuit?

If you connected the positive terminal of a battery to the negative terminal to a battery with a wire with (hypothetically) no resistance, and are asked to give the voltage drop of a segment of wire in the circuit, how would you determine this?


How to solve Schrödinger's equation for this potential algebraically?

I want to solve Schrödinger's equation with the potential $$V(x)=\frac{1}{2}mx^2+\lambda x$$ algebraically? Is there any way to construct ladder operators that are similar to the one for the harmonic oscillator in order to get the full Hamiltonian written as $H=k_1(b^*b+k_2)$? A good reference may be suitable as well.


Lightest flying solid material

I need to use a number of lanterns, to write something with it! I would like to use a light material which will bind the lanterns together (separated by space) so it looks like dotted letter. I know that lanterns will lift up like helium balloon, but I don't know if I can think of a material which will be strong enough to not ruin/mis-shape the letter (one letter at a time), but not heavy enough to stop the lanterns from raising up or causing harm to life on earth (people mainly) when they fall back down to earth (gently). I hope I have given all the variables for the above.


Does "dark matter" explain how I can have -1 apples?

If I have 3 apples and you take 4 of them, that means I have -1 apples... is that apple made of dark matter?


What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?

What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & Schroder. And I have studied QM.


How to properly construct the electromagnetic tensor in curved space-time? (Part II)

In this question, I am testing what was previously discussed. I can't seem to get my results to match D'Inverno's electromagnetic tensor for a charged point (page 239 of his book - _Introducing Einstein's Relativity_ ). **Here are D'Inverno's steps:**   * The line element in spherical coordinates is ($\eta$ and $\lambda$ are functions of $r$ only) $$ \mathrm{d}s^2 = \mathrm{e}^\eta \mathrm{d}t^2 - \mathrm{e}^\lambda \mathrm{d}r^2 - r^2 (\mathrm{d}\theta^2 + \sin^2\\!\theta\ \mathrm{d}\phi^2). $$   * He defines this covariant electromagnetic field tensor: $$ F_{\mu\nu} = E(r) \begin{pmatrix} 0 & -1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{pmatrix}. $$   * He then proceeds to find the electric field, and consequently the electromagnetic field tensor by using the source-free Maxwell equations: \begin{align} \nabla_\nu F^{\mu\nu} & = 0 \\\ \partial_{[\lambda} F_{\mu\nu]} & = 0. \end{align}   * Solving the differential equation that appears from the equations above, he finds the electric field: $$ E(r) = \mathrm{e}^{(\eta+\lambda)/2} \varepsilon/r^2. $$   * He then notes that this field is that of a point charge at infinity ($\eta$ and $\lambda$ go to zero at infinity) where $\varepsilon$ is the electric charge. I managed to reproduce these steps. **Now, here are my steps, using the four-potential procedure (the line element is the same):**   * I define my contravariant four-potential (there is just the first element which is the electric potential of a point charge, just as D'Inverno found): $$A^\mu = (\varepsilon/r, 0, 0, 0). $$   * Then I lower the index of this four-potential to find the covariant one: $$A_\mu = (\mathrm{e}^\eta \varepsilon/r, 0, 0, 0). $$   * Finally I apply this equation to build the covariant electromagnetic tensor: $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$   * The result is $$ F_{\mu\nu} = \frac{\mathrm{e}^\eta \varepsilon}{r^2}\\! \left(r\frac{\partial\eta}{\partial r}-1\right) \begin{pmatrix} 0 & -1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{pmatrix}. $$   * Where: $$\frac{\mathrm{e}^\eta \varepsilon}{r^2}\\! \left(r\frac{\partial\eta}{\partial r}-1\right) = E(r)$$ And this is different from D'Inverno's electric field. I don't know what I am doing wrong. The calculations are not difficult for this simple case. The question is, due to these calculations: Do my contravariant four-potential needs to contain my metric funcions in some way? I was assuming it is just the four-potential for a electric charge in the flat space: $$A^\mu = (\varepsilon/r, 0, 0, 0). $$ If everything is right, the wrong assumption must be here.


Physical interpretation of the relation $\dot{x}(t) \rightarrow i \omega \tilde{x}(\omega)$

If $x(t)$ is some time-dependent real quantity i can interpret its Fourier Transform $\tilde{x}(t)$ as representing, in a generic sense, the frequency components of $x(t)$. What about the FT of $\dot{x}(t)$? Can this quantity be correlated in some intuitive way to the "motion" $x(t)$ itself? In other words, is there some intuitive picture explaining the fact that $\dot{x}(t) \longrightarrow i \omega \tilde{x}(\omega)$? This discussion should, of course, not be limited to functions that depend on time (only).


Properties of liquid and air bubbles

What is/are the properties of a liquid (e.g. viscosity, surface tension) which is conducive to formation of stable bubbles floating in air (not the bubble inside the liquid)? E.g., if soap dissolved water is bubbled through a straw, big-small air bubbles are formed which comes out of the liquid and keep floating in the air. Edit:   This Wikipedia page (http://en.wikipedia.org/wiki/Surface_tension#Surfactants) attributes this to low surface tension of the liquid, but I can enumerate several liquids like spirit, petrol which have quite low surface tension, still if you blow through a straw into it, you won't get bubbles.


Electric & Magnetic fields - clarification

If (changing electric field)=(magnetic field) and (changing magnet field)=(electric field), then why is there a constant magnetic field around a current carrying wire when the electric field does not change?


How can the Hall effect ever show positive charge carriers?

The Hall effect can be used to determine the sign of the charge carriers, as a positive particle drifting along the wire and a negative particle drifting the other direction get deflected the same (as $F = q \vec{v}\times\vec{B} = (-q) (-\vec{v})\times\vec{B}$). But I don't understand how positive charge carriers are ever possible. As I've understood, a positive hole is nothing more than the absence of an electron. As all the electrons in the valence band are still negatively charged, why would this hole behave in a magnetic field as if it were positive? Also, a hole is created if an electron is excited into the conductance band. If there is always the same numbers of holes as electrons, how can any Hall effect ever occur? Thank you very much


Why do birds, sitting on electric wires, not get shocked?

If we would touch electric wires, we would get a shock, even if we are not touching ground (so that no connection is complete form wire to ground through us). I always see that birds sit on electric wires. Why don't they get shocked?


Electric shock from a power line

A bird perches on a bare high power line and nothing happens to it. A man standing on ground touches the same line and gets a fatal shock. why?


How does a crow sitting on one of the electric lines attain the same potential as that of the line so as to prevent a charge flow through its body?

The crow doesn't get an electric shock while sitting on only one electric line because it has the same potential as that of the line. Since there is no potential difference the charge does not flow through the crow's body unless the crow touches the neutral line and creates a potential difference between one end of its body to the other. How could the crow attain this much electrical potential? The crow was at a zero potential when it came and sat on the line. Was there any charge flow to the crow's body from the line? If there was, how could the crow survive such a potential hike?


Experiment to find Ammeter Resistance

I've been studying experimental electrodynamics and I needed to describe an experiment to find the resistance of an ammeter if I just have the ammeter together with one voltmeter and a protection resistance. My idea was the following: since the ammeter has much less resistance than the voltmeter we connect them in paralel and connect this in series with a protection resistance connected to the battery. The idea is that the current will be reduced by the protection resistance avoiding damages to the ammeter. Then the current will divide and a much greater part will flow through the ammeter. Since the ammeter and the voltmeter are in parallel the voltage seen in the voltmeter is the same on both devices, and the ammeter will show the current that flows through it so that we can use Ohm's Law to find it's resistance: $R=V/I$. If we knew then the errors of the ammeter and of the voltmeter we could propagate to find the error in the resistance. Is this correct? Is this a valid way to measure the resistance of the ammeter? Thanks very much in advance for the help.


Faraday's law for a current loop being deformed

I'm working through Jackson's Classical Electrodynamics text (3rd version, chapter 5.15) about Faraday's law: Faraday's law is pretty familiar: $\int_c E \cdot dl = -\frac{d}{dt}(\int_s B \cdot n da)$ which states that the contour integral of the electric field around a bounded surface is equal to the negative time rate of change of the flux bounded by that surface. No problem. Here we have a total derivative and Jackson addresses 2 cases. "The flux though the circuit may change because (a) the flux changes with time at a point, or (b) the translation of the circuit changes the location of the boundary" In either of these cases, it is easy to show that: $\frac{d}{dt}(\int_s B \cdot n da) = \int_s(\frac{\partial B}{\partial t} \cdot n da) + \int_c(B \times v)\cdot dl$ by using $\frac{d}{dt} = \frac{\partial}{\partial t} + v \cdot \nabla$ and then applying a "product rule", $\nabla \cdot B = 0$ and stokes theorem. i.e.: $\frac{d}{dt}\int_s B \cdot n da = \frac{\partial}{\partial t}\int_s B \cdot n da + (v \cdot \nabla) \int_s B \cdot n da$ **In the case where the surface is constant** , we can exchange the order of differentiation and integration: $\frac{d}{dt}\int_s B \cdot n da = \int_s \frac{\partial}{\partial t} B \cdot n da + \int_s (v \cdot \nabla) B \cdot n da$ I still have no problem with this. Now we can work on the second term on the right (apply a product rule, drop terms with $\nabla \cdot B$ and then apply stokes theorem) to get the $\int_c(B \times v)\cdot dl$ term. which, when applied to faraday's law gives the nice result: $\int_c [ E' - (v \times B)] \cdot dl = - \int_s \frac{\partial B}{\partial t} \cdot n da$ The problem arises when you allow the shape of the bounded surface to change. (Consider a circular wire loop in a magnetic field that you then hit with a hammer causing a dent in one side). How does changing the surface influence the result that we derived above? It seems to me that this would prevent us from being able to exchange the order of differentiation and integration which seems like it leaves us in a pretty tight spot ...


How does cryptochrome produce quantum entanglement?

I found this video on the net, which explains how a bird called "European robin" uses a trick based on quantum entanglement to see the earth's magnetic field. I am just asking if some of you know any research paper or reference on this subject. What I want to know is how exactly the molecule Cryptochrome in the bird's eye produces quantum entanglement after being hit by a photon.


Magnetic flux and transformers

I have two linked quesitons, 1. what is magnetic flux linkage's formal definitaiton and 2. when people say the flux flows better thru a soft iron core then the air what do they mean by flux flowing (in a transformer)???


Exciton energy splitting in indirect semiconductor

Let's take an indirect bulk semiconductor and imagine the exciton that build up at the indirect gap. Since we have a bulk semiconductor we should find three p-like orbitals: $p_x,p_y,p_z$. Would their energy levels be degenerate due to the mass anisotropy which we have at the indirect gap? My intuition screams yes. But if I try to bring it on the paper I actually find the opposite. Any suggestions, ideas or helpful tips?


Lamb-Chaplygin dipole

There is an interesting issue of hydrodynamics: whirlpools. I want to learn about the Lamb-Chaplygin dipole. Lamb-Chaplygin dipole corresponds to a steady solution of the two dimensional Euler equations. Can someone know where I can find a full development of Lamb-Chaplygin dipole?


Could there be massive gravitons and in our spacetime?

My question is: Could there in principle be sub-groups of gravitons (if they existed) that has mass (and in our spacetime dimensions) such as those affected by electric or by strong force or other forces which could be above the energy threshold of our current particle accelerators. (Perhaps they could be created, say, near pulsars due to high-energy collisions)?


One Loop Higgs Mass Correction

I am attempting to compute the one loop correction to the Higgs mass, which requires the evaluation of a scattering amplitude, namely $$\require{cancel} \mathcal{M} = (-)N_f \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \mathrm{Tr} \, \left[ \left( \frac{i\lambda_f}{\sqrt{2}}\right) \frac{i}{\cancel{k}-m_f} \left( \frac{i\lambda_f}{\sqrt{2}} \right) \frac{i}{\cancel{k} + \cancel{p}-m_f}\right]$$ which corresponds to the Feynman diagram: ![enter image description here](http://i.stack.imgur.com/pX1RH.png) After combining constants, and rationalizing the denominators, I obtain, $$-\frac{N_f \lambda_f^2}{2} \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \frac{\mathrm{Tr}\left[ \cancel{k}\cancel{k} + \cancel{k}\cancel{p} +2m_f \cancel{k} + m_f \cancel{p} + m_f^2\right]}{\left(k^2-m_f^2\right)\left((k+p)^2 -m_f^2 \right)}$$ Computing traces, via the relation $\mathrm{Tr}[\cancel{a}\cancel{b}] = 4(a\cdot b)$ yields, $$-2N_f \lambda_f^2 \int \frac{\mathrm{d}^4 k}{(2\pi)^4} \frac{k^2 +k\cdot p + m_f^2}{\left(k^2-m_f^2\right)\left((k+p)^2 -m_f^2 \right)}$$ At this point, I employed dimensional regularization, followed by Feynman reparametrization to combine the denominators, and then completed the square, yielding $$-\frac{2^{2-d}\pi^{-d/2}}{\Gamma (d/2)}N_f \lambda_f^2 \int_{0}^1 \mathrm{d}x \int_0^\infty \mathrm{d}k \frac{k^{d-1}(k^2 +kp + m_f^2)}{\left[ \left(k-(x-1)p\right)^2 +p^2(x-x^2 -1)\right]^2}$$ **Additional Calculations (Edit)** I attempted to further simplify the integrand using a substitution in only the first integral, namely $\ell = k-(1-x)p$ which implies $\mathrm{d}\ell = \mathrm{d}k$, yielding (after several manipulations), $$-\frac{2^{2-d}\pi^{-d/2}}{\Gamma(d/2)}N_f \lambda_f^2 \int_0^1 \mathrm{d}x \, \int_{(x-1)p}^{\infty} \mathrm{d}\ell \frac{(\ell + (1-x)p)^{d-1}[(\ell + \frac{1}{2}p(3-2x))^2 - \frac{1}{4}p^2 + m_f^2]}{[\ell^2 + p^2(x-x^2-1)]^2}$$ **N.B.** Mathematica evaluated the original integral over $k$, and outputted a combination of the first Appell hypergeometric series, which possess the integral representation, $$F_1(a,b_1,b_2,c;x,y) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 \mathrm{d}t \, t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_1}(1-yt)^{-b_2}$$ with $\Re c >\Re a >0$, which has a structure similar to the beta function. If I can express the loop integral in a similar form, I may be able to express it in terms of these functions. At the end of the calculation, I will take $d \to 4-\epsilon$ to obtain poles in $\epsilon$, using the usual expansion $$\Gamma(x) = \frac{1}{x} -\gamma + \mathcal{O}(x)$$ and a similar expansion should the final answer indeed contain the Appell hypergeometric series. **Passarino-Veltmann Reduction (Edit):** Based on my understanding of Veltmann-Passarino reduction, it is not applicable as the numerator contains an arbitrary power of loop momentum. I could plug in $d=4$, and impose a high momentum cut off, but this has already been done in many texts. As aforementioned, I would like a dimensionally regularized amplitude. I am stuck at this point, can anyone give some details as to how to proceed? In addition, I have a query regarding the hierarchy problem. If using a simple cut-off regularization, the one loop correction can be shown to be quadratically divergent. But why is this an issue that needs to be remedied, by for example, the minimally supersymmetric standard model? Can't the divergence be eliminated by a regular renormalization procedure?


What is the term for heat generation by a flowing fluid?

I would like to know more about the heat distribution over time in a flowing liquid. To this end, I consider the Navier-Stokes equation (where the coefficients may be temperature dependent) and the heat equation $$ \rho c_p \left(\frac{dT}{dt} + u \cdot \nabla T\right) = \sigma + S(u) $$ where $T$ is the temperature and $u$ the flow velocity. The term $\sigma$ may be some artificial heat source. The term $S(u)$ is now of interest to me: It corresponds to the heat generated by the friction of particles (I think), and I believe it must be proportional to the dynamic viscosity of the medium. I don't recall the specifics though. Can anyone help out on what $S(u)$ should be?


Finding a discontinuity and jump (time independent Schrodinger eqn)

A particle of mass $m$ is confined to move in a one-dimensional and Dirac delta-function attractive potential $$V(x)=-\frac{\hbar}{m}\alpha\delta(x)\text{ $,\alpha>0$ }.$$ Show that the function $\psi(x)$ is discontinuous at $x=0$, and determine its jump at the point. Showing its discontinuous at zero is easy. It follow from the following: $\frac{-\hbar^2}{2m}\int_{-\epsilon}^{\epsilon}\frac{d^2\psi}{dx^2}+\frac{-\hbar^2\alpha}{m}\int_{-\epsilon}^{\epsilon}\delta(x)\psi(x)=0\implies \lim_{\epsilon\to 0} \frac{d\psi}{dx}\bigg|_-+\frac{d\psi}{dx}\bigg|_+=2\alpha\psi(0)$ As for the jump, that's where I'm stuck. I assumed it was at $y=\alpha$. But I believe I'm wrong.


Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 -V(\phi)\right] \tag{1}$$ Where $V$ can be written as $$V = \frac{1}{8} \phi^2 (\phi-2)^2$$ But the author wrote in his article in the equation (1) including dimensions. $$V= \frac{1}{2} m^2 \phi^2+ \frac{\lambda_3}{3!}m^\frac{5-d}{2} \phi^3+ \frac{\lambda_4}{4!}m^{3-d} \phi^4$$ My question is how the dimensions incorporated with the potential?


Are there functions of the metric that are scalars under spatial diffs up to total derivatives?

Let $g_{\mu\nu}$ be a metric on a manifold with a time direction $x^0$ singled out. I'm wondering if there exists a function $F(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots)$ that transforms under spatial diffeomorphisms as \begin{align*} F(g_{\mu\nu}'(x'),\partial_\rho g_{\mu\nu}'(x'),\ldots)=F(g_{\mu\nu}(x),\partial_\rho g_{\mu\nu}(x),\ldots)+ \nabla_\mu \Lambda^{\mu}(g_{\mu\nu},\partial_\rho g_{\mu\nu},\ldots,x'), \end{align*} where $\Lambda$ is some functional of the metric and $x'$. This would imply that the integral \begin{align*} \int d^dx\, \sqrt{-g}F \end{align*} is invariant under spatial diffs. Any ideas?


Home experiments using wireless LAN or mobile phones about electromagnetism?

Are there any nice experiments using wireless LAN access points or routers or mobile phones to demonstrate physical features of electromagnetic fields, especially em-waves? More precisely I am looking for experiments which use one of the devices above as the main tool and some materials that can be purchased at a reasonable cost from, say, a hardware store or electronic supply store only. I do not impose any restrictions on the cognitive level the experiments aim at; undergraduate level is fine as well as graduate level in em-theory. Please use separate answers for different experiments.


Calculate pressure of air trapped underwater

Seal top end of straw, partially submerge straw in water, how do I calculate the air pressure inside the straw? Bonus points for compensating for any secondary factors like temp, humidity or flex of the straw. Application: measuring water level in 5 gallon aquarium to accuracy of about 5mm.


What process could produce two neutral pions only?

Any examples? $$? \rightarrow \pi^0 \pi^0$$ If such a process exist, could there be nonzero total orbital angular momentum in the final states of the two neutral pions? But then how to understand the interaction between the two neutral pions that make them rotate with each other to contribute the non-zero L?


Online physics diagram tool

What is a suitable tool for drawing physics diagrams online? I would love to know if there is a program like the excellent schematic drawing tool that is integrated into the electrical engineering stack exchange, only for physics. This would really help in answering questions more graphically and in an intuitive fashion, without having to rely on formulas. I am aware of the offline possibilities, but for most problems, I just need to draw simple shapes such as lines, circles, boxes, and squiggly lines. I work from several machines, so an online tool would be a lot more flexible.


Is there a derivation of all possible Bravais lattices in 2D and 3D?

Do any of you know a (possibly elegant) derivation of all possible Bravais Lattice and the Crystallographic Group of Lattices in 2 and 3 dimensions? Group theory /abstract nonsense approaches are welcome.


Why don't we normally see the Higgs boson?

I am a physics student and my dad just asked me about the Higgs Boson. I've told him the little I know, that the Higgs field is a field that is supposed to give mass to elementary particles, and that finding the boson was crucial to see if this mechanism actually did exist. After telling that, a question has come into my mind. I have heard that it is very difficult to create the circumstances where we would have Higgs Bosons. I have been told also that this boson is the carrier of the field responsible of giving mass to other particles. So the question is quite natural to me, how come something which "is not there" in "normal" conditions happens to do its work? I mean, the carrier of the electromagnetic force are the photons and we have photons in "normal conditions"(by this I have meant not the conditions we have inside the LCH, for example) so it is natural to see their effects, but how is this possible with the Higgs?


Theoretical power limit of nuclear bomb

Is there any limits of power or power to mass (or any other) limits for nuclear bombs? I found this wiki article: Nuclear weapon yield: Yield limits. Is the information provided is correct? If yes, from where this limits come from?


Spontaneous conversion of heat into work at negative temperatures

Consider a heavy macroscopic object moving in a gas. Friction causes its kinetic energy to be converted into heat. Thermodynamically, there is (effectively) no entropy associated with the kinetic energy because all the energy is concentrated in a single degree of freedom. Therfore, if an amount $J$ of energy is converted from kinetic energy into heat, the total entropy change is $J/T$, so we can see that this is a spontaneous process. But now consider an object moving relative to a gas with negative temperature. Such a thing has been created in the laboratory, so this is not just idle theoretical speculation. If an amount $J$ of kinetic energy gets converted into heat, the total entropy change is still $J/T$, but now this is negative. This seems to mean that the opposite process - conversion of heat into kinetic energy, accelerating the object - would be spontaneous. This generalises to all other processes that convert work into heat. For example, performing the Joule heating experiment with a negative-temperature gas should cause the paddle to turn, and negative-temperature gas flowing through a pipe should experience an accelerating force rather than a decelerating one. Just as superfluids have zero viscosity, it seems that negative-temperature fluids must have negative viscosity. I realise that this does not lead to perpetual motion. As heat is converted into work the inverse temperature ($1/T$) will increase until it reaches zero. But what does look odd is that in some ways the arrow of time appears to be reversed. I realise that experimentally we're very far from being able to produce the macroscopic quantities of negative-temperature fluids that would be required in order to observe these things. But is it possible in principle? And if it is, would we actually see the phenomena I described, or is there some fundamental reason why they wouldn't happen after all? And has such a connection between negative temperatures and the arrow of time been discussed or debated in the literature?


Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD)

Consider a 2D _gapped_ CSL with a _nonzero_ Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$? For example, see this article, in the last paragraph on page 7, the authors give the **4-fold GSD** from the Chern number $m=\pm2$ for a CSL. I can not understand the explanation, can anybody present an intuitive illustration or a simple mathematical proof ? I will be very appreciated, thank you very much.


Is There a List of Physics Formulas?

I am looking for a list of _simple_ physics formulas \- like $F=ma$. Is there any website/book that is a list of such equations?


Does evaporation decrease entropy?

I have a container of water in thermal equilibrium (there are no temperature fluctuations inside the container). Some molecules will evaporate out of the container of water thus decreasing its temperature. This is clearly a spontaneous process. But doesn't this mean that the entropy of the system is decreasing? The molecules that evaporated were clearly in thermal equilibrium with the other water molecules before. But now they gained some energy and the other water molecules lost energy. How can this be?


Time travel and entropy

I saw a post on reddit regarding immortality and how it would never be possible due to entropy. That said, assuming time travel was possible, would it not be possible to never reach this state of entropy if you could keep traveling back in time? Say for instance that you couldn't travel back in time within your own time curve, but you could travel to other time curves. You would merely jump to another curve when your current universe was about to reach its end. Is it too big of an assumption to think that your own initial entropy (by this I mean your entropy from your first timeline) would not 'follow you' to other time curves? I may be totally off-base in my assumptions, as I know next to nothing about physics. Any feedback/criticism is welcomed. Here is the link to the post in reddit.


Weird resistor patterns

I came up with the following question: > The vertices of tetrahedron ABCD are all connected by 1-ohm resistors. What > is the total resistance between A and B? I haven't the faintest idea how to solve it. Does it involve infinite path summations at some point? [edit: how about from A to C?]


What's the electric field with a point charge not in the center of the sphere?

That's a hollow conducting sphere link My charge is at P (10uC). R = 0.15m. PS = 0.05m. DS = 0.35m. What's the electric field at point D? ![enter image description here](http://i.stack.imgur.com/UkYgu.png)


Non-vanishing commutator of potential and mass matrices for Majorana fermions interaction theory

Consider 2 different Majorana fermions $\Psi_{L}, \Psi_{R}$ (physically, neutrinos). In general case I can write the massive part of lagrangian of these fermions in the form $$ L_{m} = (\bar {\Psi}_{1} \bar {\Psi}_{2})\begin{pmatrix} m_{11} & m_{12} \\\ m_{21} & m_{22} \end{pmatrix}\begin{pmatrix} \Psi_{1} \\\ \Psi_{2} \end{pmatrix} = \bar {\Psi}^{a}\hat {m}_{ab}\Psi^{b}. \qquad (1) $$ Then let's introduce matrix of some gauge interaction $$ \hat {A}_{\mu}^{ab} = A_{\mu} = \begin{pmatrix} A^{11}_{\mu} & A^{12}_{\mu} \\\ A^{21}_{\mu} & A^{22}_{\mu} \end{pmatrix}_{ab}. \qquad (2) $$ and corresponding fermion's charge $g$. The full lagrangian with interaction and mass term takes the form $$ L = \bar {\Psi}(i\gamma^{\mu}\partial_{\mu} + g\gamma^{\mu}\hat {A}_{\mu})\Psi - \bar {\Psi}\hat {m}\Psi . $$ I want to represent the lagrangian in terms of states with definite masses. Let's call them $\Psi_{L}, \Psi_{R}$. So by introducing mixing unitary matrix $$ \begin{pmatrix} \Psi_{L} \\\ \Psi_{R} \end{pmatrix} = \begin{pmatrix} cos(\theta ) & -sin (\theta ) \\\ sin(\theta ) & cos(\theta ) \end{pmatrix}\begin{pmatrix} \Psi_{1} \\\ \Psi_{2} \end{pmatrix} $$ I can rewrite the mass term in a form of $$ L_{m} = (\bar {\Psi}_{L} \bar {\Psi}_{R})\begin{pmatrix} m_{1} & 0 \\\ 0 & m_{2} \end{pmatrix}\begin{pmatrix} \Psi_{L} \\\ \Psi_{R} \end{pmatrix}. $$ But I can't diagonalize the $A$-matrix in general case in this basis. So the $A$-matrix and $m$-matrix don't commute with each other. So there is the question: what is the physical meaning of this result in the case of Majorana fermions?


emergent phenomena from microscopic equations of motion

Is an emergent phenomenon like "superfluidity" or "quantum Hall effect" derivable from microscopic equations of motion? If No, why?


The reason(s) of seasons on earth

This maybe simple and usual question, although there is a lot of confusion about it over internet and even in some books, so I want from an astronomer / astrophysicist to fill the gaps for me accurately. Fixed axis tilt is the reason, this is the most common explanation, but with some research, you will find that it is not the only reason! Actually, earth distance from the sun playing its role: considering that the received energy from sun is inverse proportional to the square of earth distance, one can calculate that it's contribute to a change by 5-7% of the received energy. Moreover, it's effect is opposite, in senses that because we have summer when we are at the furthest point from sun, this actually reduces summer temperature due to axis tilt. Also I read a claim that mentions a third reason, that most of the texts ignores, it is the angle of earth orbit to the suns equator, anyway, I found no quantitative analysis. Your comments please.


Why is it so hard to accelerate macroscopic objects?

It seems all we're capable of accelerating currently are atomic particles. Why can't we, say, accelerate a clock to relativistic speeds?


Numerical Tools to find Braiding Statistics of Quasiparticles

While certain classes of systems that exhibit topological order can be solved exactly (such as the Toric Code, Abelian FQH Edges, etc.) there also exist systems (think of perturbed versions of the Toric Code or Abelian FQH Edges) that cannot be solved exactly. What are the common numerical tools that are used to extract the (not necessarily abelian) braiding statistics of quasiparticles from these models? What is the main idea of these techniques? For which systems do they work well and for which systems do they fail? Can one access the braiding statistics also from numerical exact diagonalization of a small system? (For say a perturbed Toric code I would guess so because at least in the unperturbed case the abelian braiding statistics do not depend on the system size) I would be very happy if someone could provide a brief overview. I am looking forward to your responses!


Where do atoms go after collision?

[I edited the question according to Mark's and Grisha's answers.] Consider two point-like particles of equal mass colliding centrally in 2D. The final directions of the momenta of the two particles are not determined by conservation of energy and momentum: they can go anywhere. > Under which circumstances the direction of the two particles _is_ > determined?


A charged sphere with pulsing radius

![enter image description here](http://i.stack.imgur.com/D0854.png) Radius increases and decreases periodically (as a pulse).And so does the charges on the surface of sphere. ![enter image description here](http://i.stack.imgur.com/SqSv2.gif) I can't get what is gonna happen.the EM waves are produced perpendicularly to motion of the charges,but here where is the perpendicular ? Will the EM waves get produced into the space of sphere? Also, can two EM waves intersect each other or not? [as Electric fields lines can't intersect.]


Standing sound waves phase difference

This is a basic question but I don't fully understand what is going on. When we have a pipe in resonance we get standing waves, where displacement antinodes are at pressure nodes and vise versa. So we have a situation where pressure and displacement are 90 degrees out of phase, this means you can differentiate the displacement wave to recover the pressure wave. Does anyone have a nice explanation of this 90 degree phase difference?


Frequency and wavelength of photons

I try to better understand how electromagnetic radiation works. So I have some questions. If an antenna emits at 100MHz (the charges on the antenna oscillate at 100MHz) what frequency will have the emitted photons? Will they have the same 100 MHz? In the case of the photons what will this frequency represent in fact? Also what does the wavelength of the photons represent? I mean in the case of an electromagnetic wave , the wave represents the variation in intensity of the magnetic and the electric fields, and the wave-length is the distance between two crests of this variations (correct me if I am wrong). How about in the case of the photon wave-length?


Do materials cool down in the vacuum of space? If yes, how does it really work?


Moons of the planets

What is the phenomena behind gas giants like Jupiter and Saturn having a large number of moons as compared to other planets. Were the present state moons a part of their parent planet long time ago ?


Random Walk Randomly Reflected

Hi I am not specialist in probability so I will not be surprised if the answer for this question is just a simple consequence of well known results from the random walk theory. In this case, I will be happy if you can tell me the "magic words" to find the material related to this problem. Let $\sum_{i=1}^n X_i$ a random walk in $\mathbb{Z}^d$ defined in the following way: In each step, the walker look at your current position and as long as its distance to the origin does not belong to the set $\\{d_1,d_2,\ldots\\}$ ( we assume that $d_n\in\mathbb{N}$ and $d_i<d_{i+1}$ ) it jumps for one of its nearest neighbor with probability $1/2d$. For the other hand if $\|\sum_{i=1}^n X_i\|\in \\{d_1,d_2,\ldots\\}$ then the walker toss a biased coin if the result is head, which occurs with probability $p$, it goes back to the previous position, that is, $\sum_{i=1}^{n-1} X_i$. If the result is tails it chose randomly one of its neighbors with probability $1/2d$. I would like to know if there exist a non-trivial $p_c$ such that this random walk is transient for $p<p_c$ in dimension $d=3$. Of course, for this question to be well posed we have to define the sequence $(d_n)$. I would like to chose it randomly but I have no idea for what kind of distribution this problem is tractable. So at this time we can assume that $d_n$ is deterministic sequence and, for example, has a polynomial growth as $d_n=n^2$. Any references or comments are very welcome. PS: Could an user with more rep. place random walk tag for this question ?


What is the shielding in nuclear reactors mainly against?

I have a little knowledge about ionizing radiation and I have been confused over why nuclear reactors need these massive shields. So, if I am not mistaken, Alpha and Beta radiation are not that dangerous since they can be shielded with relatively light materials, the main problem is because Gamma and Neutron radiation. So now, which one requires that heavy shielding ? Also, if the reactor released either Gamma rays only or Neutrons only, which one of them would need less massive material to be reduced to a non-harmful level ? Any numbers on the density of radiation this shield has to stop will greatly help here too.


kinetic and potential energy

I have 2 cubes where one mass is greater than the other. $M$ and $m$, where $M>m$, and there is a hill that is symmetrical on both sides, and has a friction factor of $k$ between the object and the surface. If we place both objects on one side of the hill, point 1, that has no slope, so it is flat, and give both of the masses the same velocities in the direction of the top of the hill so they finally land on the other side, on the flat area, point 2 (symmetrical to the point 1). How can I prove that the speed of the heavier cube is smaller than the lighter one in the point 2? (If I'm correct) I tried equating $E_{kinetic}$ and $E_{potential}$... I see there is work from the friction, but whenever I try anything I cross out the mass. Heres a diagram: http://www.wolframalpha.com/input/?i=sin%5E2(x)%2C+0%3Cx%3Cpi&x=0&y=0 Where point 1 is at (0, 0), and point 2 at (pi, 0). What I have so far: where $g$ is constant of gravity and $k$ friction factor, and $d$ distance. $E_{kinetic.start} - W_{friction} = E_{kinetic.end}$ $M(\frac{v_0^2}{2}-gkd)=M(\frac{v_1^2}{2})$ I could change $M$ with $m$ and nothing would change, they'd both have same speeds... So it's obvious that the masses cancel out and the energy is proportionally lower at both masses, the distance for friction is the same and starting velocities are the same, therefore the velocity should be the same for both bodies at the end. This confuses me as I believe the smaller body should be faster. (Cant noone help me with this? Im now doubting the solution i've provided first is right. I cannot prove otherwise.)


How to recognize broken candies from whole ones

Let's say I have a bag full of sugar candy. Some will be whole, some will be dent, some will be broken (in part, or half, etc). Let's say I have a device with an input box where I empty the bag, and the device has two output boxes where it start placing the whole candies in one box, and all the dented/broken one in the other box. What principle of the physic could use the device, to be able to tell the whole candies from the dented/broken ones?


Is it best to look at light as a particle when trying to understand special relativity?

So my course about special relativity explains time dilation using a moving train, where one sends up (i.e. perpendicular to the direction of movement) a light pulse which gets reflected etc. (a similar example is to be found here so you guys know what I'm talking about) Let's say the source of the light pulse is a flash light. In the moving frame of reference, it is pointed upwards, and thus only has an upward component. Everything fine so far. The thing is, that when you look at it from the outside ("the train station"), I picture the flash light to still be pointed directly upward, but somehow when the pulse leaves the flash light, it gets a horizontal component. This does not make sense if you think of it in terms of some stationary ether (which is also wrong as Michelson & Morley showed). The problem is that I can't seem to get my mind around the fact that it somehow seems that the flash light is pointed at an angle in the outside frame of reference. Except if I think of light as photons (particles), then it seems intuitively correct that they get a horizontal component (compare it to a skater throwing up a tennis ball). So, when trying to understand SR, do you have to think of light as particles? My course states this nowhere explicitly so I'm not sure. _I might have answered my own question again by formulating it in a more or less ordered fashion, as happens so often when I post it here :P_


Help finding Delta T for a known position displacement with known accel/decel, etc

I'm looking for an equation to find the time it will take for a position displacement to happen, given known $V_{max}$, a (known) constant acceleration, deceleration, jerk, and displacement... and an initial velocity of 0. I am trying to estimate/calculate the time it will take for a servo motor axis to travel a certain distance. Once I know the estimated time of that move, I can solve for jerk for the other axis. That way, I choose an optimally "slow" speed for whichever axis has a shorter move distance to make the maximum use of my time. FYI, this is a pick and place XY robot gantry.


Does the conversion of crude oil to greenhouse gases have any measurable effect on earth's gravitational pull?

Oil underground is much denser than greenhouse gas in the atmosphere. Does the conversion in anyway effect the gravitational force from earth.


Making fire using a filled water bottle

I have seen videos where a fire was started using a water bottle filled with water. I guess the bottle acts like a convex lens that converges light to a focal point to start a fire. But why does water have to be added?


What is the origin of the many-body expansion?

I'm looking for the original introduction of the many-body expansion (MBE) in the scientific literature. More specifically, I'm interested in a theoretical justification of the rapid convergence of the expansion, especially in the context of molecular physics. The MBE is often introduced in scientific papers without referring to foregoing work that introduces the concept. Instead, one often states something like: "Following the well-known many-body expansion, one writes ...". See for example http://pubs.acs.org/doi/abs/10.1021/ct600253j. (freely available at http://t1.chem.umn.edu/Truhlar/docs/758FAV.pdf) The many-body expansion is a scheme to decompose the energy of a general system of $N$ particles as follows: $$ V = \sum_{i = 1}^N V_i $$ with $$ V_1 = \sum_{i=1}^N E_i $$ $$ V_2 = \sum_{i=1}^N \sum_{j=i+1}^N E_{ij} - E_i - E_j $$ $$ V_3 = \sum_{i=1}^N \sum_{j=i+1}^N \sum_{k=j+1}^N \biggl( (E_{ijk} - E_i - E_j - E_k) - (E_{ij} - E_i - E_j)\\\ - (E_{jk} - E_j - E_k) - (E_{ki} - E_k - E_i)\biggr) $$ and so on. In these equations, $E_i$ is the energy consisting only of particle $i$, $E_{ij}$ is the energy of a system containing only particles $i$ and $j$, $E_{ijk}$ is the energy of a system with three partilces, $i$, $j$ and $k$, and so on.


Does electric field cause changing magnetic field also?

According to Faraday's law, changing magnetic field causes eletric field (imprecise wording, but generally accurate). My question is, can Faraday's law be interpreted in opposite way - that is, electric field causing changing magnetic field? Equation itself doesn't speak for the direction, so I ask.


Breathing Air Underwater From A Car Tyre

I've seen in many movies and programs the event where a car plummets off the road and in to a body of water, sinking rapidly. The car slowly fills with water as it sinks and the occupant(s) don't break free until well under the water (50m say). At this point, when outside of the sinking vehicle, they then breathe air from the tyres and swim to the surface. Obviously it's not that easy to breath air from a tyre due to the valve set up (having to push the pin in to draw air under pressure out). But what I really want to know is, if a tyre is at 30psi and there is no pressure regulator used, and your under 50m of water pressure... Could you actually do this? Surely there'd be issues with the pressure at that depth, or even at atmospheric!


Free spin (Curie) Paramagnetism

I'm working through a derivation for Curie paramagnetism and hope someone could help clarify a couple of steps. The way that makes sense to me (although now I have seen the wikipedia derivation below I realise this way is pretty long) is to not take any high temperature approximations until near the end of the derivation where I have: $M=g_J\mu_B[(J+1/2)coth[g_J\mu_B\beta(J+1/2)]-\frac{1}{2}coth(g_J\mu_B\beta/2)]$ now to get to the curie susceptibility it seems that when taking the high T limit of the above expression the leading $\frac{1}{x}$ term of the coth expansion is ignored and the second $\frac{1}{3}x$ term is considered (this pops out the correct answer $\chi_{curie}=\frac{n(g_J\mu_B)^2}{3}\frac{J(J+1)}{k_BT}$). I can't find or think or any sensible reason for this apart from when we take the limit of zero B the infinity is just a constant which has no temperature dependence (which is what we're interested in) so we happily ignore it. The wikipedia method below almost makes sense aside from the last equality where I can't follow how they've simplified the sums (I can see that you could drop the first parts of the sums as they have no H dependence so won't matter when it comes to finding $\chi_{curie}$ but this still doesn't seem to work) $\bar{m}=\frac{\sum\limits_{M_{J}=-J}^{J}{M_{J}g_{J}\mu _{B}e^{{M_{J}g_{J}\mu _{B}H}/{k_{B}T}\;}}}{\sum\limits_{M_{J}=-J}^{J}{e^{{M_{J}g_{J}\mu _{B}H}/{k_{B}T}\;}}}\simeq g_{J}\mu _{B}\frac{\sum\limits_{M_{J}=-J}^{J}{M_{J}\left( 1+{M_{J}g_{J}\mu _{B}H}/{k_{B}T}\; \right)}}{\sum\limits_{M_{J}=-J}^{J}{\left( 1+{M_{J}g_{J}\mu _{B}H}/{k_{B}T}\; \right)}}=\frac{g_{J}^{2}\mu _{B}^{2}H}{k_{B}T}\frac{\sum\limits_{-J}^{J}{M_{J}^{2}}}{\sum\limits_{M_{J}=-J}^{J}{\left( 1 \right)}}$ (Copied from Wikipedia paramagnetism article)