title: How many Onsager's solutions are there?

text: **Update:** I provided an answer of my own (reflecting the things I discovered since I asked the question). But there is still lot to be added. I'd love to hear about other people's opinions on the solutions and relations among them. In particular **short, intuitive descriptions of the methods used**. Come on, the bounty awaits ;-) * * * Now, this might look like a question into the history of Ising model but actually it's about physics. In particular I want to learn about all the approaches to Ising model that have in some way or other relation to Onsager's solution. Also, I am asking three questions at once but they are all very related so I thought it's better to put them under one umbrella. If you think it would be better to split please let me know. * * * When reading articles and listening to lectures one often encounters so called Onsager's solution. This is obviously very famous, a first case of a complete solution of a microscopic system that exhibits phase transition. So it is pretty surprising that each time I hear about it, the derivation is (at least ostensibly) completely different. To be more precise and give some examples:   1. The most common approach seems to be through computation of eigenvalues of some transfer matrix.   2. There are few approaches through Peierl's contour model. This can then be reformulated in terms of a model of cycles on edges and then one can either proceed by cluster expansion or again by some matrix calculations. The solutions differ in what type of matrix they use and also whether or not they employ Fourier transform. Now, my questions (or rather requests) are: >   1. Try to give another example of an approach that might be called > _Onsager's solution_ (you can also include variations of the ones I already > mentioned). >   2. Are all of these approaches really that different? Argue why or why not > some (or better yet, **all** ) of them could happen to be equivalent. >   3. What approach did Onsager actually take in his original paper. In other > words, which of the numerous _Onsager's solutions_ is actually **_the_ > Onsager's solution**. > * * * For 3.: I looked at the paper for a bit and I am a little perplexed. On the one hand it looks like it might be related to transfer matrix but on the other hand it talks about quaternion algebras. Now that might be just a quirk of Onsager's approach to 4x4 matrices that pop basically in every other solution but I'll need some time to understand it; so any help is appreciated.


title: Electric power for current density

text: The electric power produced by a current $I\in\mathbb{R}^+$ and a voltage $V\in\mathbb{R}^+$ is $$ P = IV. $$ Now the current is given as an (alternating) current density $J(\mathbf{x},t)=\Im(e^{i\omega t} j(\mathbf{x}))$, $j:\Omega\to\mathbb{C}$, across the cross-section $\Omega\subset\mathbb{R}^2$ of a conductor. What is the generalization of the power in this case?


title: Intereference rings by light transmission?

text: Recently I recorded UV-Vis spectra of liquids with a two-beam spectrometer (Perkin Elmer 900). Lack of proper racks for cells of 0.5 cm optical path length let me use a stack consisting of a 0.2 cm cell and a one of 0.5 cm. Such a stack was hold in both beams -- for convenience, the smaller ones filled with water, the later ones with either reference or later the solution of the analyte -- by the racks for cells of 1.0 cm. The set-up was a quick resort, aiming only to narrow the region to be analyzed more in detail (and then, with a correct set-up of only one cell per optical path). Magnification of the spectrum displays a fine "ondulated" finer structure on top, as shown in the picture below. Here, different curves represent repeated recordings of the same sample at low concentration of the analyte. The little shift along Abs between one curve and the next one may be due to decomposition of the sample. Yet because these little waves are seen across the whole range recorded (190--800 nm), regardless of the samples analyzed, I conclude it is due to the experimental design, likely the stacks of two cells each, placed back-to-back. This is the question: Are these due to interference as Newton/Fringe rings are, even if the experimental set-up is in transmission (rather than in reflection mode)? ![UV-Vis spectra](http://i.stack.imgur.com/g0AQx.png)


title: Calculation of a Gravity Resonance Keyhole

text: Can anyone describe the mathematics behind the calculation of a resonance keyhole (for a two-body model)? It seems like the size and position of the keyhole should be a function only of mass and relative velocity, but I'm having difficulty deriving a formulaic solution.


title: Radio waves and frequency of photon

text: How radio waves create the current in antenna in terms of photons? If it is Compton scattering then why is not changed the freuency of photons?


title: Predicting Decay Rates via the Standard Model

text: Question 7584 illustrated a procedure to forecast the decay rates of isotopes with known long average lifetimes. Lifetimes of the many U isotopes vary from micoseconds to gigayears. F has only one stable isotope while Sn has 10. Can Standard Model principles be used to predict the stability of isotopes and the average lifetimes for unstable isotopes, or can this only be done by measurement?


title: effect of atmospheric pressure on reading of a weighing scale

text: Let us consider a completely sealed weighing scale such that the air pressure above and below the pan of the scale are equal and is equal to 1 atm. pressure. The scale initially reads zero. Now if this weighing scale is placed in a completely evacuated chamber, will there be any change in the reading? I am guessing that as the force due to atmospheric pressure above the pan of the scale is now absent, but the air pressure below the pan is still present, the scale will show a negative reading as it experiences a net upward force which is equal in magnitude to the force exerted by the atmosphere on the pan initially. Is this reasoning correct? If not, what is the actual answer?


title: Mass frequency problem

text: For Dispersion relation , according to Gaussian profile, the author in the equation 3 wrote as   $\omega= \left(k^2+\omega_{mass}^2\right)^{1/2}$ My question is what is mass frequency and how it arose in the frequency term?


title: Upper equilibrium point of a pendulum with small perturbations

text: I remember the following fact about a rigid pendulum: The point when the pendulum weight is lowest is a stable equilibrium while the point where the pendulum weight is highest is an unstable one. But if you perturb the suspension point of the pendulum with small, quick oscillations, the upper point becomes a stable equilibrium. Now, my question: > Can someone give me a reference for the stability of the second equilibrium > point of a pendulum with small perturbation? > > Has this been actually replicated in an experiment (preferably on youtube :) > ) or are the necessary parameters impractical?


title: Is equivalent resistance always lower if we add a resistor to a passive electronic circuit?

text: How to prove that equivalent resistance of any passive network is always lower if we add a resistor between arbitrary two nodes? Note that this is not necessarily a parallel circuit, 2 nodes that we connect with a resistor are not the same 2 nodes between we want to measure equivalent resistance but completely arbitrary 2 nodes in passive network. So imagine a circuit with 4 access points: A, B, C and D. We want to measure equivalent resistance between A and B. How can you prove that Rab will be less if we add a resistor between C and D? I tried to search the web but didn't have any success.


title: Why Transmission of electrical energy without wires was not practical even after Tesla's proposal

text: Why we couldn't transmit Electricity wirelessly even after so many proposals available proposed by Tesla ? > Tesla tried to generate more interest in Wardenclyffe by revealing its > ability to transmit wireless electricity, but Morgan was not interested, and > the 1903 "rich man's panic" on Wall Street dried up any further > investment.[8][9][10] By July 1904, Morgan (and the other investors) finally > decided they would not provide any additional financing. In May 1905, > Tesla's patents on alternating current motors and other methods of power > transmission expired, halting royalty payments and causing a severe > reduction of funding to the Wardenclyffe Tower. In an attempt to find > alternative funding Tesla advertised the services of the Wardenclyffe > facility but he was met with little success. By this time Tesla had also > designed the Tesla turbine at Wardenclyffe and produced Tesla coils for sale > to various businesses. Source > > In 1891 and 1892, Tesla demonstrated the oscillatory transformer that bears > his name in lectures delivered before meetings of the American Institute of > Electrical Engineers (AIEE) in New York City" and the Institute of > Electrical Engineers (IEE) in London, and in a later presentation titled "On > Light and Other High Frequency Phenomena" (Philadelphia/St. Louis; Franklin > Institute in 1893), where he put forward his ideas on the wireless > transmission of electrical energy. Source ![enter image description here](http://i.stack.imgur.com/O4iFF.png) The Tesla coil wireless transmitter


title: Non-stationary capacitor

text: A common exercise in non-stationary electromagnetism is to find the electric and the magnetic field generated by a capacitor with round plates, if the potential difference between the plates varies in time (typically $V=V_0 sin( \omega t)$ or $V= \alpha t$). While finding the fields internal to the capacitor is pretty easy, I was struggling with those on the outside. $B$ is given instant per instant by Biot-Savart law and varies in time, so exists an electric field following the third Maxwell equation; However, I couldn't find an expression for $E$ without evident absurdities. I thought that maybe with some approximations (quasi-stationarity?) such fields could be ignored. Some hints?


title: Does space have to be filled with charged particles to carry electromagnetic waves?

text: I'm a newbie here so have mercy. I'm studying electromagnetic waves. This is the propagation of energy via the vibration of charged particles, as I understand it. A charged particle could be like a proton, which has a positive charge. So I'm imagining a simplified "universe" that just exists along a two dimensional line, a long string of protons. You can send energy from one end of this space to the other by vibrating this string of charged particles. Maybe this could be done by "lifting" one proton at one end of space, and that would effect the next proton along the string in a similar way, only less so, to create a wave. You can move your end of space up and down as fast as you'd like. If you move it slowly, you create long waves (from crest to crest). If you move your proton at your end quickly, you produce short waves. But no matter how quickly or slowly you lift your proton up and down, the wave you create (either longish or shortish) will always reach the other end of space at the same time. In other words, you can't control how fast your wave travels through space, you can only control the shape of the wave itself. The energy transmitted always travels the same speed (C, the speed of light). Lift your proton up and down slowly, and you produce a radio wave. Lift it up and down really quickly, and you'll produce an x-ray wave. Lift it up and down somewhere in the middle, and you get a light wave. Here's one problem I'm struggling with. If we can see light emitted from stars that are light years away from us, that must mean that every part of space along that line of sight is full of charged particles right? If that's so, what are these charged particles? Is space full of particles in every direction, in every place? They aren't protons and electrons right, maybe more elementary charged particles like quarks? So space isn't empty, it's absolutely jam packed with charged particles. The "fabric" of space time is "made of something," and that something must be charged either positive or negative. Is this a correct line of thinking, or am I missing some fundamental idea here? Thanks for your help.


title: Intuitively Understanding Work and Energy

text: It is easy to understand the concepts of momentum and impulse. The formula $mv$ is simple, and easy to reason about. It has an obvious symmetry to it. The same cannot be said for kinetic energy, work, and potential energy. I understand that a lightweight object moving at very high speed is going to do more damage than a heavy object moving at a slower speed (their momenta being equal) because $E_k=\frac{1}{2}mv^2$, but why is that? Most explanations I have read use circular logic to derive this equation, implementing the formula $W=Fd$. Even Samlan Khan's videos on energy and work use circular definitions to explain these two terms. I have three key questions:   * What is a definition of energy that doesn't use this circular logic?   * How is kinetic energy different from momentum?   * Why does energy change according to $Fd$ and not $Ft$?


title: Moment of Inertia Tensor about non-principal axis

text: I'm part of the Western Martial Arts/Historical European Martial Arts community, and a debate that often comes up is parrying with the edge vs parrying with the flat of a blade. I want to do some calculations myself to really convince myself one way or another. Specifically what I'm looking to do is to calculate the moment of inertia tensor of a highly idealised sword (basically just the blade, tang and pommel treated as two rectangular prisms and a sphere) about some pivot. My intention is to calculate the moment of inertia of each body individually, and find a way to sum them together; does the parallel axis theorem apply to the tensor quantity, or only to the scalar quantity? It is likely that the pivot point will _not_ be about the principal axes.


title: How was the Oh-My-God particle observed?

text: How exactly was the Oh-My-God particle (ultra-high energy cosmic ray) observed and its energy measured?


title: Effect of Background Radiation on a Transmitted Signal

text: I'm coding a basic simulation of using error correcting codes to transmit data from a satellite back to earth. I'm not sure what to set the "noise level" to. Let's say a satellite orbiting Mars transmits a signal which contains 128 bits, approximately how many of those bits would be changed by the time the signal reaches earth? This is just for fun so I don't need any references, just an approximation, however it would still be interesting if someone could point me to a reliable source :) Update: I'm basing this on Mariner 9, I managed to find that the transmissions were at 20 watts, using the S-Band, which I think is somewhere around 2.2 ghz


title: How do you find the speed of sound in the problem?

text: A stone is dropped into a 40m deep well. The sound of the splash is heard 2.95 seconds after the stone is dropped. Find the speed of sound.


title: Is there a delay in the effect of gravitational force?

text: Let's suppose there is a very massive object and a small object that are 1 lightyear apart. The massive object is large enough that the gravitational force pulling the small object is easily noticeable. Suppose (never mind how) that through some freaky event the massive object suddenly disappears or is suddenly transported to some part of the universe too far away to have noticeable gravitational effect on the small object. Since nothing travels faster than light, does it take 1 year or more for the small object to "figure out" that the massive object is gone and to stop accelerating towards where it used to be? Or does the magical disappearance of the massive object have immediately observable effects on the small object? To put it more concisely, is there a delay in the effect of gravitational force?


title: Atomic Physics - Bohr's model of atom

text: Well I'm learning about the models that have been proposed for the atom, and the Bohr model came up. My teacher told me that the one of the main postulate of the theory is that when an atom is in ground state, it can neither absorb nor emit energy. This is what gives the atom its stability. However, it can obviously absorb energy to jump to a higher energy level. I don't understand, how can it jump to a higher energy level, from the ground state obviously, without absorbing energy? Some problem here?


title: What does this question about entanglement and classical geometry mean?

text: Below is the question from Andy Strominger's presentation at the String 2014 conference. The question was asked by credible physicist Ashoke Sen as an important question. > "What is the precise relation between quantum entanglement and classical > geometry?" Could somebody describe the question and explain why it is an important question and its implications?


title: Why is the spectrum of the $\beta$-decay continuous?

text: the spectrum of the Gamma and Alpha decays are both discrete, i.e. the $\alpha$-particles and the $\gamma$-rays take on only discrete values when emitted from a decaying nucleus. Why is it then, that the $\beta^{\pm}$ can take on continuous values? The main thing that distinguishes the beta decay from the other two is, that it is a three body problem, i.e. the nucleus does not only decay into an electron/positron, but also into a electron-neutrino/antineutrino. I don't see yet how this immediately implies that the spectrum of the electrons is continuous though. The way I understand the two body decays is, that the initial nucleus spontaneously decays into the two bodies, i.e. a smaller nucleus and a gamma or alpha particle. As the energy levels in both the nuclei are quantized, only certain values for the energy of the photon and the Helium core are allowed, sine Energy and momentum need to stay conserved. Does the third particle change things in the way that there basically are two things (i.e. the electron and the neutrino in the beta decay) that are not restricted by an inner energy level hierarchy in the way the nuclei are, thus allowing the energy given by the nuclei to be split arbitrarily (and continuously) between the electron and the neutrino? If this is the wrong explanation, please correct me.


title: Darboux theorem and the canonical decomposition of a two-fermion wave function

text: It is a classical theorem in quantum mechanics or quantum chemistry or quantum information that a two-fermion wave function has a beautiful canonical expansion: $$f(x_1, x_2) = \sum_{j=1}^N \sqrt{p_j} \left( \phi_{2j-1}(x_1)\phi_{2j}(x_2) -\phi_{2j}(x_1)\phi_{2j-1}(x_2) \right).$$ Here $f$ is the wave function which is antisymmetric, i.e., $f(x_1, x_2 ) = - f(x_2, x_1) $. The orbitals $\phi_i$ are the natural orbitals, i.e., eigenstates of the single-particle density matrix. They are orthonormal. The weights $p_j $ are related to the population in the natural orbitals. This theorem was first obtained in quantum chemistry by lowdin in 1956 as far as I know. See http://journals.aps.org/archive/abstract/10.1103/PhysRev.101.1730 I discovered it myself when i was studying the best Slater approximation of a fermionic wave function. Its simplicity and beauty surprised me. It is so disappointing that I never found a quantum mechanics textbook mentioning it, although almost every quantum mechanics textbook would mention the two- electron wave function. Now reading differential geometry, I find the Darboux theorem: http://en.wikipedia.org/wiki/Darboux%27s_theorem It seems that Darboux theorem is basically the same stuff as the canonical decomposition above. Is that right? I am not familiar with differential geometry.


title: energy source for phase transformation-latent heat of vapourisation

text: As per my knowledge, phase transformation require energy gain or loss.if we are changing a liquid to gas we heat it and that is how molecules of liquid gains energy and lattice spacing increases and bonds becomes weaker and thus gets converted to gas. But we know that if we lower the pressure of water (a closed container and evacuate air and so lowering pressure) and it start boiling even at 20degree c.from where it is getting energy for phase transformation?please help me if I am completely wrong.


title: Are quantum decoherence and Everettian approaches to the measurement problem necessarily distinct?

text: As I understand it, there is a large contingent of physicists who believe that the measurement problem is "solved" by decoherence, without, for example, needing to postulate the existence of "many worlds." Yet at the same time my understanding is that in the decoherence picture there is only unitary evolution of the wave function, and that while the _appearance_ of collapse is explained, the global superposition of states still in fact exists, and whether or not _multiple_ states within the universal wave function observe the same appearance of collapse (but to different eigenvalues) is a question that is left completely unaddressed. Therefore my reading of the decoherence picture is that it is virtually identical to an Everettian approach, except that it purposefully ignores an obvious interpretational consequence of its description. Is this true, or do decoherence-based approaches somehow argue that there really is only a unique observer within the universal wave function that observes a collapse to unique eigenvalues, and that there is some form of symmetry breaking that allows this to happen at the expense of all the other potentially conscious components of the universal wave function?


title: current in series resistors and voltage drop in parallel resistors

text: when we have resistors in series, the current through all the resistors is same and the voltage drop (or simply voltage) at each resistor is different. **Question 1:** it is fine that voltage drop (potential drop) is different because each resistor offers different resistance (suppose). but how is the current through each resistor same? if we have resistors of different resistance, shouldn't the current be different through each resistor? similarly, when we have resistors in parallel, the current through each resistor is different but the voltage drop at each resistor is same. **Question 2:** current is through each resistor is different because resistance of each resistor is different (suppose). but how is the voltage drop at each resistor same here? shouldn't the voltage drop at each resistor be different because each resistor offers different resistor?


title: Resistance between two points on a conducting surface

text: Suppose we have a cylindrical resistor, with resistance given by $R=\rho\cdot l/(\pi r^2)$ Let $d$ be the distance between two points in the interior of the resistor and let $r\gg d\gg l$. Ie. it is approximately a 2D-surface (a rather thin disk). What is the resistance between these two points? Let $r,l\gg d$, (ie a 3D volume), is the resistance $0$ ? * * * Clarification: A voltage difference is applied between two points a distance $d$ apart, inside a material with resistivity $\rho$, and the current is measured, the proportionality constant $V/I$ is called $R$. The material is a cylinder of height $l$ and radius $r$, and the two points are situated close to the center, we can write $R$ as a function of $l$, $r$ and $d$, $R(l,r,d)$, for small $d$. The questions are then:   What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow \infty} R(l,r,d) $$   What is $$ \lim_{r \rightarrow \infty} \lim_{l \rightarrow 0} R(l,r,d) $$


title: How is the speed of light calculated?

text: How is the speed of light calculated? My knowledge of physics is limited to how much I studied till high school. One way that comes to my mind is: if we throw light from one point to another (of known distance) and measure the time taken, we could know the speed of light. but do we have such a precise time measuring tool?


title: Fluid to particles under newtonian gravity

text: How to start with a perfect fluid concept and reach (by approximations through certain mathematically well defined assumptions) to the concept of particle ? Here newtonian gravitation is being assumed. The state equation of matter can be assumed to be that of dust. **EDIT** Assume any initial mass distribution $\rho(r,t=0)$ and momentum distribution $p(r,t=o)$ and state equation of dust. **EDIT 2** I don't intend to throw away the fluid concept,but still want to arrive at a particle as some form of a mass distribution.One example could be $\rho_p(r,t)$ which has a compact support in r describing a particle. **EDIT 3** Assume there are a number of such particles as described in the earlier edit part of the question as $\rho_p(\vec{r}-\vec{r_i},t=0)$ for $i = 1,2,3,...$. distributed arbitrarily in space such that no two are overlapping. This could be considered as $\rho(r,t=0) = \displaystyle\sum\limits_{i=0}^n \rho_p(\vec{r}-\vec{r_i},t=0)$. Similarily for momentum distibution at $t=0$ as $\vec{p}(r,t=0) = \displaystyle\sum\limits_{i=0}^n \vec{p_p}(\vec{r}-\vec{r_i},t=0)$. What would eventually happen to such a system under its own newtonian gravity. Just out of curiosity what would happen if GR is assumed.


title: Historically, how do we know that Earth moves around Sun? And it does so in an elliptical orbit?

text: I know the basics of solar system like how Earth moves around Sun, and that we have so many planets, elliptical orbit of earth, and how far is sun from earth etc etc. I want to take a step back and understand how mankind has been able to construct this knowledge of solar system piece by piece. Questions: \-- Sun is something we see on daily basis, so it would not be strange if humankind got curious to know about it, and made theories,observations,maths to prove it. How did Copernicus and others came up with the theory that earth moves around sun in elliptical orbit. How did they prove it? Has there been any actual way by which we have actually observed sun being in center and earth moving around in elliptical orbit. or we made such theory and result of this theory have been proved by other observations like day,night,change in weather at a point and at different points on earth.


title: Spectral Line Width and Uncertainty principle

text: so I've been at this for about 3 - 4 hours now. It is an homework assignment (well part of a question which i've already completed). We did not learn this in class. All work is shown below. > An atom in an excited state of $4.9 eV$ emits a photon and ends up in the > ground state. The lifetime of the excited state is $1.2 \times 10^{-13} s$. > > (b) What is the spectral line width (in wavelength) of the photon? So lets look at what I have done so far. I have done the following: $$\Delta E \Delta t = \frac{\hbar}{2} $$ but $$E = h f$$ so $$\Delta f = \frac{1}{4\pi \Delta t}$$ but if I take $\Delta f$ and convert it into wavelength using $\lambda f = c $ then it gives me the wrong answer. I've tried MANY variations of the above formulas. The correct answer is $0.142 nm $ Can anyone give me a hint?


title: How is distance between sun and earth calculated?

text: How has the distance between sun and earth been calculated by scientists? and size of sun? Thanks,


title: Viscosity and Density

text: So I'm reading my textbook and it says that that magnitude of air resistance $f(v)$ can be given as a taylor expansion: $$f(v) = bv + c^2v = f_{\text{linear}} + f_{\text{quadratic}}$$ The linear term, $bv$ is related to the viscosity of the medium while the quadratic term is related to the density of the medium. If something is more dense, shouldn't it be more viscous as well? Or are viscosity and density not related?


title: Searching for Big Bang Neutrinos

text: How is it possible to discover neutrinos from the big bang? Were the neutrinos emitted just before the big bang? Or at the same moment? If they moved outward in all directions, basically unhindered by matter or gravity, how can we hope to ever detect them? Unless something caused some of them to turn back and travel in a different direction?


title: Does the real part of the inverse dielectric function have to be negative at some point for Cooper pairs to form?

text: Electrons naturally repel one another. However, in a superconductor, a phonon- mediated interaction causes the electrons to have a weak attractive interaction. Suppose that the interaction between two electrons is $V(\textbf{q})$ (the bare Coulomb interaction, which is $>0$) and that the effective interaction between two electrons is $V(\textbf{q})/\epsilon(\textbf q, \omega)$. My question is threefold: (1) Is the only way to have an attractive interaction between electrons is for the $\Re[\frac{1}{\epsilon(\textbf{q},\omega)}] $ to become negative at a certain value of $\textbf{q}$ and $\omega$? (2) If so, does $\Re[\frac{1}{\epsilon(\textbf{q},\omega)}] $ have to become negative at the values of $\textbf{q}$ and $\omega$ that correspond to a phonon? (3) (If yes to Question (1)): Since high-temperature superconductors form Cooper pairs, does this mean that in the space of $\textbf{q}$ and $\omega$ such that $\Re[\frac{1}{\epsilon(\textbf{q},\omega)}] < 0 $, there are clues lurking about high-temperature superconductivity yet unbeknownst to us?


title: Application of Poisson summation Formula to Casimir Effect

text: I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to use a modified Poisson formula over positive integers only that appears to be: $$\frac{1}{2}F(0) + \sum_{n=1}^{\infty} F(n) = \pi \tilde{F}(0) + 2\pi \sum_{k=1}^{\infty} \tilde{F}(2\pi k)$$ where $$\tilde{F}(k):= \frac{1}{\pi} \int_0^\infty F(t) \cos(kt)dt.$$ I can't understand how to demonstrate the validity of this expression starting from the standard summation formula, that is: $$\sum_{n \in \mathbb{Z}} F(2\pi n) = \frac{1}{2\pi} \sum_{k \in \mathbb{Z}} \hat{F}(k)$$ where $$\hat{F}(k):= \int_{-\infty}^\infty F(x) e^{-i k x}dx$$ is the standard Fourier transform. In fact this formula is deduced using tempered distributions so $F$ (that is a "test" function) has to have good property of convergence: this mean I can't simply define a non-continuous function null for $n<0$. In my particular case $F(x) = (ax)^2 \log(1-e^{-ax})$. Sorry for my bad English ;)


title: Why do people look for a field formalism for String Theory

text: String theory was originally formulated from a perturbative description (using quantum mechanics (QM) and replacing points by strings and evaluating path integral). Still, although QM has an upgrade to a field formalism -- QFT (that helps to solve the problem with causality and negative energy particles), I don't really see why people want to look for a field formalism of string theory. What problem does String Theory share with QM? What are the advantages of a string field theory? Why the underlying degrees of freedom for string theory might be fields?


title: Quantum Field Theory with (2,2) metric

text: Does someone know some reference which treats QFT in a space with the following non-Lorentzian signature: $g_{\mu\nu}=\text{diag}(-1,-1,1,1)$. I'm interested in basic stuff like the shape of the scalar propagator and Feynman rules in such a space.


title: Different results to a basic question ( Newton's law and perservation of momentum)

text: > Trolley with mass of $m_0=1 \ kg$ is moving without friction on the railway > track. It is raining so there is a constant mass flow of water $\Phi_m=0.1\ > kg/s$. Constant force $F=0.1 \ N$ is accelerating the trolly horizontally. > > What is the velocity at time $t$ if the trolly is stationary initially ? I tried two different aproaches and got different results. I graphed the both functions and noticed that both were similar at $t=0$. ### **1\. Newton's law** $$F=m(t)a$$ $$F=(m+\Phi t) \frac {dv}{dt}$$ $$\int dv=F \int\frac{dt}{m+\Phi t}$$ ..integrated 0 to v; and 0 to t $$v=\frac F\Phi \ln(m+\Phi t)$$ ### **2\. Momentum** $$(m+\Phi t)v - 0 = \int Fdt$$ as $F=const.$ $$v=\frac{Ft}{m+\Phi t}$$ Am I missing some concept behind differential equations?


title: Two-state system problem

text: Given a 2-state system with (complete set) orthonormal eigenstates $u_1, u_2$ with eigenvalues $E_1, E_2$ respectively, where $E_2>E_1$, and there exists a linear operator $\hat{L}$ with eigenvalues $\pm1$,   1. would the normalized eigenfunctions (in terms of the given eigenstates) just be $$u_1\over \sqrt{\int u_1^*u_1}$$ ? Since I am not given an argument/coordinate system for the eigenstates, perhaps I should first project them into one? But I don't have any info on the nature of the quantum system...   2. A second question asks for the expectation values of the energy in the respective states. But isn't that just $E_1$ and $E_2$ respectively?? Grateful for any enlightenment.


title: Projection of states after measurement

text: Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the probability that +1 is measured again if the system is measured again at $t$. I would have thought that this probability is 1 since measurement projects the system into the first measured outcome such that successive measurements are consistent. But clearly I am mistaken, since it also asks to show that the probability = 0 for t = $\pi\hbar\over \Delta E$. Could anyone please explain to me what is going on? Thanks.


title: Why can't quantum teleportation be used to transport information?

text: Kaku Michio says in an interview that we've teleported photons, cesium atoms and beryllium atoms. Having watched a lot of Kaku as well as way too many astrophysics documentaries in general, I know he's exaggerating. From what I understand, some _state_ of those particles were teleported. Here's my question: what _state_ exactly is being teleported, and why can't it be used to transfer classical information? How can we know experimentally that a teleportation occured, if the transported information can't be read classically? Most of the news articles for the general public just throw around the word "state" without explaining the idea (or flat out tell us like Kaku that atoms were teleported), and Wikipedia uses math and terminology that's way over my head. Is there a layman's explanation by any chance?


title: Wave / particle duality question for sound and light

text: Sound is usually referred to as just "sound waves" - we do not talk about a "sound particle" and only as a wave or "matter wave." Could something similar apply to light i.e. that there really is no "light particle" (photon) and that the wave respresentation is what applies both to sound and light? We say light is an electromagnetic movement, could it be that light is an electromagnetic movement with just movement and similarly to sound like there is no "sound particle" (only a movement that makes some sound from vibration and movement of particles that are "ordinary" particles in the air and not in fact any particular "sound particle" so similarly the photon could be just a name or just a concept in fact used to describe what is similar to sound and just a wave and not a special particle? I know this seems controversial like saying that there is no photon and I say it might be not that controversial since there is no particular particle for sound and a waveform only. Thanks for any insight!


title: Quantum corrections to massless fermionic field

text: in QED the corrections to electron propagator change the bare electron mass from $m_0$ to $m=m_0+δm=m_0+∑(\not{p}=m)$ (Peskin, formula 7.27). This is the consequence of the fact, that the quantum corrections change the propagator from $i/(\not{p}−m0)$ to $i/(\not{p}−m_0−∑(\not{p})))$. This part is written very well in Peskin's book, formula 7.23. Then (after fomula 7.30) Peskin goes on to explain that if the fermion was massless, then the quantum corrections would never give a mass to it, in other words, δm is 0 when $m_0$ is 0. The question is the following. When $m_0$ is 0, the quantum corrections change the massless fermion propagator from $i/\not{p}$ to $i/(\not{p}−∑(\not{p}))$, and to me this 'corrected' propagator looks like massive. Seems like the quantum corrections gave mass to the massless fermion. To make the point clearer, the quantum corrections to photon propagator leave the denominator as $(1/q^2)$ and don't make it $1/(q^2-A)$, which is what happens to massless fermion propagator. My question is only from the point of view of diagrammatic approach (and not from the point of view of left and right spinors and the mass term connecting these two spinors). Also, when we'll get to renormalization, seems like the fermionic field renormalization via $Z_2^{-1}=1-(d\Sigma/d\not{p})|_{\not{p}=m}$ will not be enough, we also would have to introduce mass renormalization because of that $\Sigma(\not{p})$ term in the 'corrected' propagator. Am I wrong?


title: Why does the existence of a quadrupole moment of the deuteron implies that the nuclear force is not a central force?

text: Source: Krane and Weisskopf page 95 (the tensor force).


title: Can Parallel Transport always move a Vector Parallel to Itself?

text: Consider two tiny plane(flat) surfaces A and B meeting at a straight line L.We have a preassigned vector on A at some point P on it.Is it always possible to have a vector parallel to the first one and lying on B at some point on it(say Q)? [The first vector(that on A) is assumed as _not_ parallel to L] The two planes may be considered as the adjacent tangent planes on a curved surface over which we are working out parallel transport along some curve lying on it.


title: Using Audacity to generate power spectral density

text: Say I record something with a sound recording program like Audacity (e.g. plucking a guitar string). How could I use that recording to plot a dB vs Frequency graph?


title: Malus' Law (Polarization)

text: Consider the image below: > ![enter image description here](http://i.stack.imgur.com/HoQET.png) Let $\theta_2$ be the angle between the axes of transmission of Polarizer 2 and Polarizer 3 and $\theta_3$ be the angle between the axes of transmission of Polarizer 3 and Polarizer 4. Derive $I_4$, in terms of $I_2$, $\theta_2$ and $\theta_3$. Any idea how to do this one?


title: Need help with this derivation (Sound intensity)

text: I will copy a little bit of what my textbook says. By the way, we're talking about sound here. $$Intensity = (0.5)BwkA^2$$ The textbook claims that by using the relationships $w=vk$ and $v^2=\frac{B}{\rho}$, we can transform the previous equation into $$I = (0.5)(\rho)(B)^{0.5} w^2 A^2$$ So I'm here manipulating these equations and can't see how they got to that derivation. Any ideas?


title: Why is the bispectrum not commonly used in experimental physics?

text: Power spectra, coherence spectra, and linear transfer functions are ubiquitous tools of experimental physics. However, our instruments often retain small nonlinear effects which can contaminate measurements. It appears that higher order spectra, in particular the bispectrum, would be ideal tools to investigate nonlinear interactions. Nonetheless, I've never actually seen them put to use in experimental physics. For example, consider the (frequency-domain) coherence: $C_{xy}(f) = \frac{\langle X(f)Y(f)^*\rangle}{\sqrt{\langle X(f)X(f)^*\rangle\langle Y(f)Y(f)^*\rangle}}$ The bicoherence considers not two but three signals, and looks for correlations between oscillations at frequencies $f_1$ and $f_2$ combining nonlinearly to produce a signal at $f_1+f_2$: $C_{xyz}(f_1,f_2) = \frac{\langle X(f_1) Y(f_2) Z^*(f_1 + f_2)\rangle}{ \sqrt{\langle X(f_1)X^*(f_1)\rangle\langle Y(f_2)Y^*(f_2)\rangle\langle Z(f_1+f_2)Z^*(f_1+f_2)\rangle} } $ ...which seems like a useful thing to do. Why are the bispectrum and bicoherence not used more frequently in experimental physics? I am specifically thinking about time domain, multi-input/multi-output systems where one is looking for nonlinear couplings between various signals. One of the top Google hits on the subject is for the Matlab Higher Order Spectral Analysis (HOSA) Toolbox, which seems like a nice resource (though it appears to be no longer maintained and now suffering from bit-rot).


title: What are the coefficients α, β, and κ in this equation for decompression-induced gas bubble growth?

text: I am reading a text about gas nuclei I encountered the following formula: $$r = \alpha + \beta \left( \frac{T}{P} \right)^\frac{1}{3} + \kappa \left( \frac{T}{P} \right)^\frac{2}{3}$$ $r$ is the critical radius of the nucleus, while $T$ and $P$ are the current temperature and pressure respectively. I have no idea what $\alpha$, $\beta$ or $\kappa$ are. Also, I don't even known the name of such formula and the principle behind it. Can anybody explain what $\alpha$, $\beta$ or $\kappa$ are and what the underlying principle is? The text is REDUCED GRADIENT BUBBLE MODEL, page 12 equation 61. **UPDATE:** The problem is that I'm not able to find the values of the three constants. If I use the values from the first three rows of the table below the equation:   * $T = 293 K, P = 13 fsw, r = 2.10$   * $T = 293 K, P = 33 fsw, r = 1.36$   * $T = 293 K, P = 53 fsw, r = 1.34$ I get (approximately) $\alpha = 4.395, \beta = -3.260, \kappa = 0.866$. However such values do not work for e.g. $P = 273 fsw$. If I use my constants, I get $r = 1.966$, while the table reports $r = 0.80$. Also, the table shows that $r$ decreases for $P \in [13, 273]$, my function increases approximately at $P = 73$ (where my $r$ is 1.403 against 1.32). I think $\alpha$, $\beta$ and $\kappa$ must be functions of pressure. The Wikipedia page about Equation of State shows $\alpha$ and $\kappa$ as functions of pressure and temperature, but there are no traces of $\beta$.


title: View of the Sun from Voyager 1

text: Today Voyager 1 is approximately 122 AU distance from the Sun. What does the sun look like from this distance? How much brighter does the sun appear to the naked eye compared to other stars in the same field of view? Are any of the other planets from our solar system visible to the naked eye at this distance? Would the sun produce enough light for a normal person to read a book, or would the light be too dark for reading (just trying to get a practical feel of the amount of light)?


title: Universal central charge in higher dimensional AdS/CFT?

text: In the $AdS_3/CFT_2$ correspondence, the central charge of the dual CFT2 is universally given by $$ c = \frac{3\ell}{2G} $$ This is independent of the matter in the bulk of AdS3. Is it also universal in the higher dimensional analogues $AdS_d/CFT_{d-1}$ or does it depend on details of the matter in the bulk, etc.? In particular, what is the central charge for $d=4$?


title: Question on kinematics

text: A particle is projected vertically upwards from point $A$ on the ground. It takes $T_1$ time to reach point $B$ but it still continues to move up. If it takes further $T_2$ time to reach ground from point $B$ then height of point $B$ would be?


title: Fluctuations in energy for macro and micro canonical ensembles

text: I was thinking about fluctuations in energy for a system in thermal equilibrium. I think that the Boltzmann distribution itself has an standard deviation approximately equal to the mean, as it is approximately an exponential distribution. From the microcanonical ensemble perspective, I think the reason the fluctuations in energy (the standard deviation) grow as $\sqrt{N}$ is because in statistics we know that for $N$ independent random variables, the variance of their sum scales with $N$. However, if I consider the whole gas as the system (this is the macrocanonical ensmeble, isn't it?) for which I know the average energy (the same as before) and I can apply Boltzmann statistics directly, will I not get a different answer for the size of energy fluctuations? Will I not get an standard deviation on the same order as the mean energy, as expected for a plain Boltzmann distribution?


title: Predicting Faraday's law, Changing Fields

text: Are there other equations that we can predict Faraday's law from? I know that each of Maxwell's equations are 'fundamental', but I feel like Gauss's law and Ampere's Law are very "nice", and for some reason or another I don't feel the same about Faraday's law. I know that, obviously, any other way of 'deriving' Faraday's law will in effect just be another way of restating Faraday's law, but is there _any_ other way besides noting the fact that changing electric fields and changing magnetic fields are related? If this is the only way, can someone provide a quick run through of the process that went into developing this law?


title: How to load Bose-Einstein Condensates into an optical lattice?

text: In cold atom experiments, what techniques are used to load Bose-Einstein Condensates into an optical lattice??


title: Cross-field diffusion from Smoluchowski approximation

text: I'm reading _An Introduction to Stochastic Processes in Physics_ by Don S Lemons. Problem 10.2 leads to a pair of equations: $dV_x = -\gamma V_xdt+V_y\Omega dt-V_y\sqrt{2\gamma dt}N_t(0,1)$ $dV_y = -\gamma V_ydt-V_x\Omega dt+V_x\sqrt{2\gamma dt}N_t(0,1)$ ($\gamma$, $\Omega$ constants, $X$, $Y$ position, $V_x$, $V_y$ velocity. The two noises are identical.) It then asks for the application of the Smoluchowski approximation to these equations. As the book is very hand-wavey in justifying the use of this approximation I find it tricky to reason with. I only know how to use the search-and-replace rules the book gives, eg. set $dV_x=0=dV_y$ and replace $V_xdt$ and $V_ydt$ with $dX$ and $dY$. So what do I do with the terms like $V_y\sqrt{2\gamma dt}N_t(0,1)$ for the Smoluchowksi approximation? Presumably all references to $V_x$ and $V_y$ need to be eliminated.


title: Strange behaviour of water drops on a heated pan

text: I did an experiment in my home and I was surprised by the results. I heated a pan without anything in it. After some time it became very hot and I took some water (2 drops) and put it into the pan. It started behaving very strangely. The water is forming droplets, and the droplets are jumping here and there and fizzing and rapidly moving across the pan. When I put more water, then it forms big drops. And the drops are floating here and there like mercury drops. You can see the video here. What is happening here? Why is the water behaving like this?


title: Twin Paradox in case of two twins that don't meet

text: If there are two twins. such that one of them goes on a space voyage maintaining a constant velocity, and that one never returns to earth, and the twins decide to never meet but send information about their age to each other by some light pulse or any other method. If then one knows that he is younger, won't he know he is the one travelling thus violating the postulate of relativity that all inertial frames are equal w.r.t. all phenomena and that one can never know if he is moving with a constant velocity.


title: can interaction between a massless fermion and external sourse exist?

text: For example, let's consider the electromagnetic interaction between a massless fermion and a electromagnetic externel sourse $A^\mu$, then the lagrangian is $$\mathcal{L}=\bar{\psi}\gamma^\mu\partial_\mu\psi+ieA_\mu\bar{\psi}\gamma^\mu\psi$$ and how can we calculate the Feynman diagram and $\mathcal{M}$ of it? But according to relativity, the equation of motion of a particle with charge is $$e F^a\ _bU^b=U^b\partial_bP^a$$ where $F_{ab}$ is the electromagnetic tensor, $U^a$ is the 4-velocity of the particle and $P^a$ is the 4-momentum of the particle. If the particle is massless, then the speed will be $1$ and this equation fails. So can a massless particle have charge and interaction with externel sourse? If could, for example, how can I calculate the force and equation of motion? If couldn't, why?


title: Using local U(1) Transformation to solve Problem in Path Integral

text: When we develop photon path integral, we assume that the current is always conserved. But if we consider interaction between electron/positron and photon, the Noether current is conserved only when the fields obeys the equations of motion. To solve this problem, a gauge transformation including a transformation on the Dirac field as well as the electromagnetic field is introduced. $A^{\mu}(x)\rightarrow A^{\mu}-\partial^{\mu}\Gamma(x)\\\\\Psi(x)\rightarrow exp[-ie\Gamma(x)]\Psi(x)\\\\\bar{\Psi}(x)\rightarrow exp[+ie\Gamma(x)]\bar{\Psi}(x)$ But how specifically does this transformation solve that problem?


title: Why does electromagnetic induction actually occur?

text: In my book, it is written that "An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing" (Faraday's law) I understand that whenever there is a change in magnetic flux, there will an emf induced in a loop and this in turn will induce a current. However, I really want to understand what exactly goes on to cause this. I just want to check if I am right:   1. There is a relative movement between the loop and the magnet (e.g. loop moving towards magnet)   2. Thus, electrons in the loop are "moving" relative to the field   3. A moving charge experiences a force perpendicular to the direction of the field   4. As a force is exerted on the electrons, they move in a particular direction   5. Current is induced I do not know much calculus and studying high school physics but could someone please help me get an intuition behind Faraday's law or tell me if my way of thinking about it is right?


title: Entropy increase vs Conservation of information (QM)

text: Unitarity of quantum mechanics prohibits information destruction. On the other hand, the second law of thermodynamics claims entropy to be increasing. If entropy is to be thought of as a measure of information content, how can these two principles be compatible?


title: Relation between between linear momentum and translational kinetic energy

text: The momentum $m v$ of a particle is formally the same as the **derivative** its translational kinetic energy $\frac{1}{2} m v^2$ with respect to $v$. Similarly the angular momentum $I \omega$ is the **derivative** of its rotational energy $\frac{1}{2}I \omega^2$ with respect to $\omega$. Does this relation has any physical interpretation?


title: How can a metal heated at 600° emit thermal photons at 4000°+?

text: Suppose we have a cube of metal inside a room at temperature 27°. If we heat the metal up to 600° using _uniform radiation of that energy_ , **no part** of it should have higher T°, but nevertheless it will start emitting visible light, that is thermal photons with a temperature in the excess of 4000°. How is that possible? I am aware of thermal radiation, blackbody etc., the first question is: 1) - when the metal is at thermal equilibrium with the room (27°) are there inside it or inside the room any molecules/atoms with energy in the range of 4000°? if the answer is affirmative the question has been fully answered, if is negative we need a follow-up question


title: Change in Wavelength of a Photon Relation to Energy (specifically Compton Effect)

text: Given a photon dropping from $\lambda_1$ to $\lambda_2$, its energy will drop from $\frac{hc}{(\lambda_1)}$ to $\frac{hc}{(\lambda_2)}$. However, I was wondering if there is any significance in the energy of the change in wavelength itself. For $\Delta \lambda = \lambda_2 - \lambda_1$, this change in wavelength has an energy $\frac{hc}{(\Delta \lambda)}$, yet this value does not correspond at all (as far as I can see) with $\Delta E = \frac{hc}{(\lambda_1)} -\frac{hc}{(\lambda_2)}$. Is this just a subtlety in the math, or is there actually meaning behind the value $\frac{hc}{(\Delta \lambda)}$?


title: Diffraction at apertures with arbitrary angle to wavefront

text: Every physics book I know explains diffraction (e.g. at slits, gratings and circular apertures) in a way which assumes that the light wave impinges perpendicularly on the aperture. Would there be any difference if that was not the case (i.e. if I have a slit and adjust a laser such that it impinges with a certain angle instead of pointing directly towards the slit)? I know that diffraction can usually be explained using Huygens' principle but I am not sure what to get from that in this specific case. Especially, I am wondering if the time difference between the wave's arrival at the individual ends of the aperture would have any effect (I guess so but I could not tell how that would look like).


What will be the acceleration of an object $A$ when its released by another object $B$ moving with an upward acceleration $a$? Will the acceleration of the object $A$ be more than the acceleration due to gravity or less than it? Will the net acceleration reduce or increase?


title: Gerstner Wave Formula to Vertex information $(x,y,z)$

text: I have created a program, that draws a plane mesh, and allows to me edit or animate all the separate vertices along it. The idea behind it, is for me to be able to create a gridMesh based wave simulation. The common Gerstner Wave formula doesn't provide me with $x,y,z$ information for separate points on the waves. Is there any way to retrieve Vector information for certain points on the waves? Or any alternatives to calculating waves for Games Design?


title: Is it possible to harness energy from the exothermic reaction of changes of state of matter?

text: Since a substance emits heat as it transitions from a high energy state (gas) to a lower energy state (solid), is it possible to devise a method to capture this heat and convert it to usable energy? I'm not sure how economical this would be, but consider the exothermic reaction of water freezing. The heat of fusion for water is $3.33 *10^5 J/kg$ So if a 1 cubic meter of water was frozen, there would be: $3.33 * 10^5 J/kg * 1000kg = 3.33 * 10^8 J$ of heat energy produced. The heat value for the vaporization of water is $2.36 * 10^6 J/kg$ which would be even more efficient, though it is more difficult to capture 1000kg of water vapor. I imagine this might be akin to how an air conditioner works, but I cannot imagine how this may be done or if it is already being attempted. So I pose the question here, would it be possible to harness the heat energy from the air and water concentrate it into a usable form of energy--a clean form of energy at that. Why, or why not?


title: In a strong field, how long would it take for domains to align?

text: How fast is the rate of "switch" for the domains to align with an exterior field? Possibly in milliseconds? I assume it won't take much time since domains are very small, and they only turn from a certain degree to another. I know this would depend on a lot of factors, but if powerful large magnet can attract a ferromagnet with a force of nearly 4,000, brining it closer to it would take 0.005 seconds, then surely the magnetization was a lot faster than that. Since magnetization occurs before attraction.


title: String Vertex Operators in Light Cone Gauge

text: I have a basic confusion about how to compute string scattering amplitudes in light cone gauge, using the operator formalism reviewed in GSW. I am familiar with the covariant gauge formalism, where I know how to use the state-operator correspondence to find the vertex operators corresponding to a physical state, but I'm not sure how to carry this over to light cone gauge. To be specific, take the open string state $\alpha_{-n}^i|0,k\rangle$. A naive guess for the corresponding vertex operator is $\partial^n X^i e^{ik\cdot X}$. Is this right? If so, what is the relation between this operator and the DDF operator $A^i_n$? As far as I understand the DDF operator is supposed to be the covariant gauge realization of a light cone gauge state. Does this mean that the DDF operator $A^i_n$ reduces to $\partial^n X^i e^{ik\cdot X}$ if one fixes light cone gauge? The main quantity I am trying to compute is the scattering amplitude between three tachyons and a very massive state. If anyone knows a reference that does this in detail please let me know.


title: If fans increase the heat of the air and this DIY AC decreases it, why don't they neutralize eachother?

text: Please try to answer in layman terms, i am only starting to study thermodynamics (And physics) In my book one of the curiosity parts gives that fans don't actually cool the room but by hitting the air particles they increase their kinetic energy, these high speed particles hit us and hence evaporate our sweat, making us feel cool. I understand this part, but then recently my friend shown me an homemade air conditioner. It works by using a table fan and water is siphoned through a tube coiling through the fan. I understand some of the heat that the fan throws out is absorbed by the water which in turn heats up and cools down the air. But then, why does that actually work? Did the fan not just heat up the air, only for it to be cooled down? Or in other words, why do they not neutralize eachother? Is the cooling effect provided by this much higher than the fan's heating or is this due to some other concept?


title: What is a dual / cotangent space?

text: Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never really understood the physical significance of these "dual spaces," or why they should need to exist. What is a dual / cotangent space? Could someone explain why they are necessary, and what their physical significance is?


title: Eigenvalues of the Lagrangian?

text: It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent. We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about eigenvalues of Lagrangians. Why is this so? Is it not useful? is it not possible to do it?


title: Why are most accurate reflective surfaces not white?

text: Since white is our perception of reflection of light, why are mirrors and other metallic objects that are more grey or silvery capable more reflective than a white object? This is somewhat related: What is the difference between a white object and a mirror? but I'm wondering why more white objects aren't mirrors.


title: Why did nuclear testing not result in nuclear winter?

text: According to Wikipedia over 2000 nuclear tests have been performed since the Manhattan Project. If nuclear war would bring about a nuclear winter, why didn't testing do? Were they too much spread out in time to cause any real climate damage?


title: Ergodic Hypothesis; canonical ensemble

text: I'm currently studying for an exam in thermodynamics/classic statistical mechanics and 2 things came up which are confusing me. First the ergodic hypothesis states that it is equal to take the mean value of one micro state with respect to time or take the mean of the whole ensemble. In our textbook the state of the whole phase space is defined as a probability measure $d\mu (x)=\omega (x)dx$ on $\Gamma_N$ = $\mathbb{R}^6$. The ergodic hypothesis is formally stated as: \begin{align} &d\mu_{E}(x)=\Sigma(E)^{-1} \cdot \delta(H(x) - E)\ dx\\\ &\Sigma(E)=\int_\Gamma \delta(H(x)-E)dx \end{align} is the only invariant probability measure under $\phi_t$ on $\Gamma(E)$. Where $\phi_t$ is the hamiltonian flow and $\Gamma(E) = \\{x \in \Gamma_N|H(x) = E\\}$. Up to this point it's clear to me. But after this they say that also for a 'general state': $\omega(x)dx = \int dE(\delta(H(x)-E)\omega(x) dx)$ we can write the time average as an ensemble average: \begin{align} <f> = \int dE\ W_E \cdot \int_\Gamma f(x) d\mu_E(x)\\\ W_E = \int\omega(x)\delta(H(x)-E)dx \end{align} Where f is an arbitrary observable. They go on proving the statement: \begin{align} <f>=\int dx\ f(\phi_t(x))\ \omega(x) = \int dx\ dE\ f(\phi_t(x))\ \delta(H(x)-E)\ \omega(x)\\\ =\int dE\ W_E \int dx\ \frac{1}{W_E}\ f(\phi_t(x))\ \delta(H(x)-E)\ \omega(x)=\int dE\ W_E \int f(x) \ d\mu_E \end{align} The last step in this deviation is absolutely not clear to me. If just someone could point out what I'm missing this would be great. Second I got a problem with the definition of the canonical ensemble. We look at two systems in thermic contact with $V'>>V,\ N'>>N$ where V is volume and N is the number of particles. The state of the complete system "0" is the micro canonical ensemble to the energy E. The Hamiltonian is $H_0(x,x') = H(x) + H'(x')$ where we neglect the interaction Hamiltonian. The state of the small system is then given by: \begin{align} \omega(x) = \frac{1}{\Sigma_0(E)} \int_{\Gamma'}dx' \delta(H(x) + H'(x') - E) \end{align} Why is this the case? Again I would be very glad if someone could shortly give me a hint here why this holds... Thanks in advance. Edit: Maybe I just answered the sec ond question myself: Does this simply hold because the states of the complete system are the same as the states of the small system (because they are in thermal contact) and since we only integrate over $\Gamma'$ those are the states we are looking for?


title: Earthing of conductors

text: Can anyone please explain me the process of earthing of a spherical conductor in detail. I am aware about the basic idea of earthing but I am not able to understand and visualize it properly.


title: Nature of tetragonal distortion in Jahn-Teller effect

text: I am wondering: If I have a regular octahedron as my starting point, oriented along the x-y-z axis, and now Jahn-Teller suggest I elongate or compress along the $z$-axis, what happens along the other axis? I expect that these move in the opposite direction, but by how much? Say my displacement in $z$-direction is $\delta$. Is the displacement in $x$- and $y$-direction then $-\delta$, or do I have to find $\delta_x = \delta_y$ so that the total Volume is conserved?


title: Internal Energy and entropy in a open system

text: Here http://en.wikipedia.org/wiki/Internal_energy we can read $$U= TS-PV+\sum_i \mu_i N_i$$ Let's suppose i=1 and a ideal gas. We know: $$U=N/N_A c_v T$$; $$PV=NKT$$; $$\mu=\frac{\partial U}{\partial N}= c_v T/N_A$$; so $S=\frac{U+PV-\mu N}{T}$ would be $ NK $ but it's not true. Where is the mistake?


title: Extracting water out of silica gel

text: From my preliminary research I've learned that desiccants are a good (best?) method to absorb water from air. Assuming I'm using silica gel (or kitchen salt, or rice) then what is the best way to extract the water out of the desiccant?


title: "Artificial gravity" on a rotating space stations

text: There have been a lot of sci-fi shows recently using the "rotating space station" explanation for gravity on space stations. After watching these videos: http://www.youtube.com/watch?v=49JwbrXcPjc&NR=1   http://www.youtube.com/watch?v=_36MiCUS1ro I was wondering what would happen, if you were facing the direction of rotation, while standing on the "floor" of a rotating space station and tossed a ball "up". From the video it looks like the ball should land in front of you. Is this in fact what would happen and if you dropped a ball would it land behind you?


title: why not just pumping to the 3rd level directly in a 4 level laser system？

text: why pumping to the 3rd level in a 4 level laser system will decrease the output PL intensity?


title: Diagonalization of Hamiltonian

text: Typically, one way of understanding the physics of an interacting quantum system is by diagonalizing the Hamiltonian. In principle, can we always diagonalize a Hamiltonian, such that it is expressed in terms of non- interacting particle states? If so, is the diagonalization unique?


title: Irreducible representation for the massless particle with helicity 2 and the Weyl tensor

text: As it can be shown, the equations for the irrep with zero mass and helicity 2, -2 respectively can be given in a form $$ \tag 1 \partial^{\dot {b}a}C_{abcd} = 0, \quad \partial^{\dot{b}a}C_{\dot{a}\dot{b}\dot{c}\dot{d}} = 0. $$ Here $C_{abcd} = C_{(abcd)}, C_{\dot{a}\dot{b}\dot{c}\dot{d}} = C_{(\dot{a}\dot{b}\dot{c}\dot{d})}$ are symmetrical spinor tensors or the reps $\left( 2, 0 \right), \left( 0, 2\right)$ of the Lorentz group respectively. From $C_{abcd}, C_{\dot{a}\dot{b}\dot{c}\dot{d}}$ we may construct 4-tensor $$ C_{\mu \nu \alpha \beta} = (\tilde{\sigma}_{\mu})^{\dot{a}a}(\tilde{\sigma}_{\nu})^{\dot{b}b}(\tilde{\sigma}_{\alpha})^{\dot{c}c}(\tilde{\sigma}_{\beta})^{\dot{d}d}(\varepsilon_{ab}\varepsilon_{cd}C_{\dot{a}\dot{b}\dot{c}\dot{d}} + \varepsilon_{\dot{a}\dot{b}}\varepsilon_{\dot{c}\dot{d}}C_{abcd}) $$ with properties $$ \tag 2 C_{\mu \nu \alpha \beta} = -C_{\nu \mu \alpha \beta} =-C_{\mu \nu \beta \alpha} = C_{\nu \mu \beta \alpha}, \quad C^{\alpha}_{\ \beta \alpha \delta} = 0, \quad \varepsilon^{\alpha \beta \gamma \delta}C_{\alpha \beta \gamma \delta} = 0. $$ In the formal language it is a direct sum of representations with helicity 2 and helicity -2. From $(1)$ we can also get $$ \tag 3 \varepsilon^{\mu \nu \alpha \beta}\partial_{\nu}C_{\varepsilon \delta \alpha \beta} = 0. $$ $(2), (3)$ gives us the Weyl tensor of the linearized GR. My question: why we have got the Weyl tensor for massless helicity-2 particles, not the curvature tensor? I.e., why the free gravitons are described by the Weyl tensor, not by the curvature?


title: Voltage and potential

text: I just need a confirmation I've understood this concept and help in a particular problem. So, what I understand under "voltage" is basically a difference between two electrical potentials (positions) which are actually just ratios of electrical energy per charge; if we knew how strong the probe charge is (how much coulombs does it have), and knowing the electrical potential - we would know the "exact" potential electrical energy at that point for that probe charge, measured from the middle of the bigger charge that emits the electrical field, to the probe charge. I understand the concept of potential energy and all. So voltage is proportional difference between two positions from a charge. Ok, now if I were to calculate the voltage ( $U$)between 2 charges ($q1$, $q2$; $r$), where $φ$ is electrical potential, and $k$ a constant ($9 \times 10^9 \frac{N\times m^2}{C^2})$: $$U=φ_1 - φ_2$$ $$φ=k\frac{q}{r}$$ I'm stuck. I can't think of it... I only understand voltage according to one charge... The best thing I could do is to make the starting electrical potential $0$ (if we don't care about the size of a charge... since it's very small, otherwise it would be big according to its radius), and finishing one with the distance $r$, then the voltage would be $U=φ_2$. All my textbook examples either give me electrical potentials, or the size of a charge which emits the electrical field, where the probe charge is moving, but none give me this kind of problem. Ty in advance. Addition: When we look at voltage in a battery, which has some excess of + and - charge at "both sides", so how can you say a battery has a voltage at all, do you fixate the + charge and calculate voltage (electrical potential of - charge at the distance from +)?


title: A block falling from a height on a block suspended by spring

text: The block suspended by the spring is hanging freely and its mass is M. The small block of mass m is dropped on the bigger block from height h. ![enter image description here](http://i.stack.imgur.com/XuQaO.png) After the small block is dropped 》》》 ![enter image description here](http://i.stack.imgur.com/lYZ6Q.png) I want help in analysis using simple harmonic motion. Find frequency, amplitude and shm equation. I want help in analysing this problem using simple harmonic motion. Thank u for giving me a -1 post when im being specific with my problem. Ive been trying to do this for a long time. Instead of help I get a negative vote thank u so much.......


title: What's the point of a cobasis?

text: I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term is more common). Basis vectors are simple enough, it's just a carry over from vector spaces in linear algebra. Cobasis vectors, on the other hand, don't seem well motivated to me. I understand that they're dual to the basis vectors, but why exactly do we need them? When should they be used over the normal basis vectors? I'm familiar with the treatment of vectors as elements of a tangent space of a manifold and dual vectors as linear operators on vectors, if that's the sort of explanation that motivates the cobasis most effectively, although I don't have a good intuition with that sort of thing quite yet.


title: How is UT1 being computed?

text: I've recently read up about time standards and now understand that UTC is a second-corrected version of international atomic time so that it is kept within 0.9s from UT1. And then UT1 is defined as the mean solar time at 0° longitude. However, it is not realized by measurements of the sun's apparent position, but via a sidereal time or earth rotation angle. Extensive googling hasn't helped me to understand how this observed earth rotation angle is used to compute UT1. This Phys.SE answer hints that the equation is quite complex. I'd like to understand why it needs to be complex, considering that the "mean sun" that mean solar time refers to is defined to follow a uniform movement (with respect to the stars) along the celestial equator. Therefore, shouldn't the difference between sidereal time and mean solar time be simply a linear function of the time of year?


title: Three body force

text: Reading on wikipedia about Faddeev equations, I've come across the notion of three-body force. The article which describes this type of force doesn't give me any understanding what it is about. My questions are: does every three-body problem have such force? Or are e.g. classical celestial three-body problems free of it? If it's not in every three-body problem, then what classes of problems do have it? Also, does such a force violate principle of superposition for forces?


title: Is there any inherent danger in operating a Thermonuclear Reactor?

text: Nuclear fusion research appears to promise sustainable, practically unlimited energy, and according to ITER it will be a perfectly safe, non-polluting source of energy. My question is if there are any plausible scenarios under which operating a thermonuclear reactor could cause a major environmental disaster? "Products of the fusion process are Helium, which is inert and harmless, and neutrons, which will lodge in the vessel walls and produce heat and activation of materials." This obviously assumes normal operating conditions, but what would happen if some man-made or natural catastrophe was to impact the reactor and create abnormal operating conditions?


title: Spectroscopy and the current state of our ability to determine the composition of extra-solar planets

text: I was interested in learning more about the status of our scientific understanding and technological instruments regarding extra solar spectroscopy. I am motivated by this almost 3 year old question/answer here at physics SE. Specifically: > The composition of extra solar planet should be taken with a pinch of salt, > the science is very new and uses instruments not designed for the task. Keep > an eye out for new missions such as ECHO which might fly in the next 5 to 10 > years... Since it has been almost 3 years, can someone give an update on our current understanding on this subject? I am curious as to how accurate our measurements of the composition of planets and stars outside our solar system are. Also, what is the impact of our atmosphere? Are we able to tease out the effects of absorption by our own atmosphere in order to get a clean signal?


title: What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

text: What is the difference between: $\nabla _{\sigma} $ and $ \nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and am still a bit unsure of the notation.


title: Mass-Energy Equivalence theory energy or momentum is not conserved?

text: The famous equation for mass energy equivalence: $E=mc^2$ It cannot conserve energy or momentum without some loss in one way or another. To elaborate further if I take $1kg$ of mass and I also take electromagnetic radiation with same energy as the $1kg$ mass and ensuring its accuracy using the above equation. Then use this radiation and I use it to smash it into a "crash" mat I would get this: $$E=mc^2$$ $$E=1.c^2$$ $$E=89,401,000,000,000,000$$ **Approx.** That being the energy we then use this to calculate the momentum of the electromagnetic radiation transferred onto the "crash" mat which would be: $$p=mv$$ $$p=mc$$ $$p=e/c^2 . c $$ $$p=e/c$$ $$p=89,401,000,000,000,000/c$$ $$p=299,000,000$$ **Approx.** where as if I now take $1kg$ of matter and accelerate it to high velocity (say, 10,000 kilometers per hour ) to measure its momentum we get roughly $10,000 kg.m/sec$ That being said, momentum cannot be conserved, why is that? Next, Energy cannot be conserved. For example if I used the same $1kg$ and I shoot it into space at a $x$ velocity and after say 1000 years I recieve it and measure mass of the object it would weigh exactly the same as 1000 years before therefore I can conclude it has same energy in its mass however if I use the mass-equivalent of electromagnetic radiation and do the same but my conclusion would be vastly different as first due to travelling in long fabric of space it would get slightly or even highly redshifted due to doppler effect. That being said the observer would get different energy reading to that of 1000 years ago. That in mind where did that energy go?


title: Gravity on a doughnut-shaped/Möbius planet

text: How different would the effects of gravity be if the planet we're on is in the shape of a torus (doughnut-shaped)? For an (approximately) spherical planet, it's slightly clear that objects would tend to be gravitationally attracted towards the center. However a torus would have a hole in its center, and I'm not sure if the attraction towards the center still applies. In particular, could a person on such a planet walk in the vicinity of the hole without falling off? * * * Additional: Similar question, but now consider a planet in the shape of a Möbius strip. Not only do you have to contend with the hole, but with the "kink" as well. Can a person stand up on the kink?


title: What is the purpose of a "protect from light" warning?

text: Some food or medical items, like rasberry juice concentrate I just bought, have a "protect from light" warning on the label. How can light influence products like milk and what does this warning mean in practice?


title: Going Beyond Introductory Physics

text: Due to certain reasons, I don't have the chance for going to college for under-graduation, but that doesn't mean I can't choose to walk the hard path and learn. Over the past month or so a lot of people have asked about resources for physics online or offline, but what I want isn't just the resources, but advice on what to study. I have taught myself the basic undergrad stuff (essentially classical mechanics and electromagnetism with a bit of modern physics with things like the one dimension time in/dependent Schrödinger wave equation) through a combination of MIT OCW, Giancoli and asking around, but what boggles me is how to proceed ahead. I want to understand things instead of just knowing their names and there is so much to physics beyond this that I don't know where to start. I'm reaching out like this because I'm hoping that someone will give me a basic idea on where to start tackling these things. For example, I have a deep interest in thermodynamics to learn it what I would love is an answer that actually talks about; what's the next step (or book)? What do you recommend to get a solid grinding for any special math behind it? Any good experiments to actually see what I'm learning? And so on... I know this might be a tall order, but I just had to ask. I'm sorry if this detracts from the main purpose of this site. Oh and as far as where my interest lies. Well I just like to create things and what I'm fascinated about is creating systems no matter what form they take. Thank you. **Update:** I know that this question may seem broad and vague, but it's just that I see things in a sort of connected way in my mind. So, although I want to create things and it might appear to be an engineering question it really isn't. I think that traditionally what comprises of engineering is just one view at the same problem. Physics is another. So, if you're wondering what topic to answer on then pick anyone that you know well and you think is connected to the world and helps to expose some of our day to day realities. Things I can use to gain perspective while creating designs. (Particle physics for example, doesn't really make the cut, but fluid dynamics does and so on.) **Update 2** : I'm sorry for all of the confusion, but I'm essentially asking how to learn applied physics.


title: Why doesn't phase space contain acceleration/forces?

text: I'm watching some Physics lectures on the internet by Leonard Susskind:   http://www.youtube.com/watch?v=pyX8kQ- JzHI&feature=BFa&list=PL189C0DCE90CB6D81&lf=plpp_video In this lecture, and also in Wikipedia and other places phase space is described as the space of all the states we need to know to determine the configuration of the system infinitely into the future. But I don't understand how is position and velocity enough to determine this, what about forces? Let's say we have a particle and we want to know where it will be in 10 seconds, we obviously need it's starting position, it's starting velocity **and also** all the forces acting on it, or it's acceleration. Where is my misunderstanding?


title: How should the implicit sum $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted?

text: $C$ is a 3x3x3x3 tensor. How should the expression $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted? This is my guess: $$ \sum_{i=1}^3\sum_{j=1}^3 \sum_{k=1}^3\sum_{l=1}^3 C_{ijkl}u_{i,j}u_{k,l} $$


title: Photons and proper time

text: **Why is there no proper time without inertial frame?** In question n°95054 I learned that there is no proper time zero and no proper distance zero for photons because they are no inertial frames. That means, in my observer's spacetime Minkowski diagram with movements of objects (incl. photons) on it, I can assign to each observed straight worldline a proper time by applying the reciprocal Lorentz factor, except for particles moving at light speed. The result is the following: ![enter image description here](http://i.stack.imgur.com/766wf.jpg) Are these charts correct? What explanations, and which remedies are provided by physics for this apparent leap at v=c? Remark: This question is not meaningless because the answer is needed for question n°95054.


title: Have experiments ever suggested two different values to the same divergent series?

text: I believe to have understood that some physical experiments suggest finite values to divergent series (please correct me if I'm wrong, my understanding of these matters is limited). I heard, for example, that the "equality" $$ \sum_{n=1}^{ \infty} n = - \frac{1}{12} $$ was suggested by some experiment conducted by physicists. I was wondering if there are experiments in physics that seem to suggest two or more different values to the same divergent series. If not, why is this the case?


title: Can one raise indices on covariant derivative and products thereof?

text: Can the following be true?   1. $g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$   2. $g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$   3. $g^{\sigma\rho}\nabla_{\nu}\nabla_{\mu}T_{\sigma\rho} = \nabla_{\nu}\nabla_{\mu}T$


title: Will the water added to an ice piece freeze?

text: Water, at room temperature is poured into a hole made of a block of melting ice(kept at room temperature).I was wondering if the water will ever freeze? Thank you.


title: How to calculate pressure at altitude when there is 0 temperature lapse?

text: To calculate pressure at a given altitude, the following formula is used: $p = p_0 \cdot \left(1 - \frac{L \cdot h}{T_0} \right)^\frac{g \cdot M}{R \cdot L}$ _(the values can be foundhere)_ This formula is great, but only works completely with the US Standard Atmosphere. I really want to use ISA (International Standard Atmosphere) more, because it accounts for the dynamics of the atmosphere much better. The problem is that the Tropopause has a Lapse rate ($L$) of 0. I don't know how to get that to work in this formula because that would involve dividing by zero. Any ideas?


title: Band Structure and Carrier Recombination/Generation

text: So i've been a bit confused, looking at PN junction, semiconductors and the like (trying to nail down _how exactly_ semiconductors work, transistors and such). I've read the wiki on band structure (“holes” and free electrons). Apparently the big part is the band theory, and how the energy levels work. This is what I have so far (correct me if I’m wrong):   1. Materials/atoms have energy levels and bands based on these energy levels.   2. The forbidden band (I’m assuming closest to the atom): no free electrons are allowed in or out.   3. Above that is the valence band, which can force free electrons out with some energy leaving behind a “hole”. The valence band is usually pretty well occupied (Does this mean it might have some “unoccupied holes” in it?).   4. Electrons in the conduction band, I’m assuming, take very little energy to move out of the atom (or are already leaving the atom).   5. The band gap determines how much energy it takes to remove an electron from the valence band to the conduction band (Insulators have a _big_ gap, semiconductors’ gap is small. But does it, maybe, grow larger with heat?). So, I have a few questions:   * When an electron leaves a valence band, does it leave behind an actual hole, or is that just conceptual? What determines how “filled” a band is? How come electrons cannot just slide on into a band?   * When an electron reaches the conduction band, is it “free” from the atom? Or just takes very little energy to get discharged?   * When we refer to charge carriers, what is this? A Femi Gas? Are there holes in this gas that can move? Or do just electrons move in and out through the gas? I’m guessing the concept of “holes” has me slightly confused as I don’t understand how a hole can “move”.


title: Transformators (primary and secondary windings)

text: Consider the following problem: _Robby wants to put a light in the shed so he puts a cable between his house and the shed. In the cable there are 2 wires with a combined resistance of $0,92 \space \Omega$. Robby connects his transformator at his house on an electricity grid of $230 V$. He connects the secondary inductor($14.3V$) to the cable. A $12 \space V$ / $30 \space W$ lamp in the shed works perfectly._ There are a couple of questions about this piece of text, which have been bothering me:   * **Calculate the heat per second which is created in the cable.** My problem with questions like this is that I have an overwhelming amount of data and it is hard for me to distinguish which is needed and which is not. So my initial thought was that I'd calculate $I$ like this: $ I = 230 : 0,92 = 250 \space A$, and then use the formula $P = I^2 \times R$ which would give me a ridiculous amount of $57500 \space J/s$. I knew I did something wrong when I saw this, so I looked at the correction model and found out that they found the $I$ of the lamp, and afterwards used that in he formula I gave in combination with $R= 0,92 \Omega$. Why is it done like this? I first thought that you could you $I_{lamp}$ because this is in series and $I$ is the same everywhere in series, but I have 2 problems with that idea:   * How do we know it is in series?   * Also, if we do $ I = 230 : 0,92 = 250 \space A$ this isn't equal to $I_{lamp}$, so I dismissed that idea and now I am back at square one.. Then we get the following passage which confused me even more: _Robby decides the put the transformator in the shed. The lamp functions perfectly when he puts the loop ratio to 958:50._ What?! Does this mean that he didn't use the transformator in the first passage? But if he didn't why is the secondary inductor mentioned in the first passage then?! The following 2 questions are easier (if you made the first one correctly):   * **Calculate the primary voltage on the transformator** I just did $(958/50) \times 12 \approx 230V$   * **Explain without using calculations that he loses much less now** $I_{secondary}$ is much smaller now, and because of the formula $P=I^2 \times R$ this leads to less loss of energy. So the bulk of my problem lies with 'analyzing' the first text. How can you conclude from that test that the transformator isn't used?


title: What's the big deal with Nd:YAG rods?

text: I see the prices of Nd:YAG rods for lasers, and could someone explain why they are so expensive? Do they need to be mono-crystalline or something? Will Nd:YAG laser work with amorphous/poly-crystalline rod?


title: How come the universe is made of matter and not antimatter?

text: Antimatter is like matter on opposite day: it has the same properties as the stuff that makes up planets, stars and galaxies, but one vital piece is different—its charge. The universe supposedly started off with equal parts matter and antimatter, but somehow, matter won out. Most of both substances annihilated each other shortly after the big bang, and left a small surplus of matter remaining.


title: Does the uncertainty principle make simulation of systems impossible?

text: Is it possible to fully define a system, then be incapable of simulating or calculating its future states due to the Uncertainty Principle? If it can be done, how?


title: What is matter within the context of string theory?

text: I've been reading about string theory, and as I understand it, it proposes that matter is nothing more than vibrating membranes of space-time. Is this correct?


title: Maxwell relations, confused on how the solution got to this answer

text: I have a problem where, basically, in part (a) I correctly found a fundamental equation $$TdS = dE - 2\sigma l dx$$ Then the problem goes on to say that the only parameters of interest are $x$ and $T$, so I would assume we can write the relation $$dS = \frac{\partial S}{\partial x} dx + \frac{\partial S}{\partial T}dT$$ by taking the total derivative. From this information, the solution goes on to say **"We can read off the Maxwell relation** $$\left(\frac{\partial S}{\partial x}\right)_T = \left(\frac{\partial(-2\sigma l)}{\partial T}\right)_x = -2l \frac{d \sigma}{dT} \qquad \sigma = \sigma_0 - \alpha T$$ Where $\sigma$ is given in the problem statement. I'm just not seeing where they find this maxwell relation....been stuck for quite some time. Any help would be appreciated, thanks.


title: Acceleration due to gravity?

text: I was looking into orbitals and found something I haven't been able to understand. http://www.math.ubc.ca/~cass/courses/m309-01a/hunter/satelliteOrbits.html There is a part on the page which states the following: $$\vec a = \frac{G(M+m)\vec{r}}{r^3}\approx GM\vec{r} / r^3$$ since $m << M$ I'm not sure why this is the case and why we do not use $\vec a = \vec F/m$ here. The cubed radius is really throwing me off. Can anyone further explain this or point me to a reference with a proof? Thanks


title: Why the magnetic flux is not zero?

text: If $\vec{\mathbf B}=B\vec{\mathbf a}_z$, compute the magnetic flux passing through a hemisphere of radius $R$ centered at the origin and bounded by the plane $z=0$. ![Figure](http://i.stack.imgur.com/WavK2.png) * * * **Solution** The hemisphere and the circular disc of radius $R$ form a closed surface, as illustrated in the figure; therefore, the flux passing through the hemisphere must be exactly equal to the flux passing through the disc. The flux passing through the disc is $$\Phi=\int_S\vec{\mathbf B}\cdot\mathrm d\vec{\mathbf s}= \int\limits_0^R\int\limits_0^{2\pi}B\rho\,\mathrm d\rho\,\mathrm d\phi =\pi R^2B$$ The reader is encouraged to verify this result by integrating over the surface of the hemisphere. * * * According to Maxwell's equations the magnetic flux over a closed surface must be zero, why in this case does not happen?


title: Some Dirac notation unclarities

text: **Q1:** Ok so i have come to a point where i know that $\Psi(r,t)$ which we denote only by $\Psi$ can be represented in a Hilbert space by a vector which we denote $\left|\Psi\right\rangle$. Does this mean $\left| \Psi(r,t) \right\rangle$? **Q2:** I know that equation $\Psi = \psi e^{- iWt/\hbar}$ represents a link between a stationary Schrödinger equation and a time dependant Schrödinger equation (we denote $\Psi(r)$ as $\psi$). I want to know now if we denote $\psi$ in a Dirac notation as $\left|\Psi(r)\right\rangle$ or any differently? **Q3:** How do we write a Schrödinger equation and a time independant Schrödinger equation using a Dirac notation?


title: Entanglement and Black holes

text: If you have two entangled quantum states, One state falls into a black hole and you measure the other state, What can you say about the state that has fallen into the black hole? If you have billions of these entangled pairs and one partner of each falls into the black hole, If you don't measure the states outside the BH , Can we say that the BH has a very large entropy? If we add more and more entanglements, Can the entropy increase without a limit? If we measure all the partners outside the event horizon, Will the entropy of the BH decrease to zero?


title: Astronaut on a rotating asteroid

text: This might sound like the silliest question ever, but can someone shed some light on what is actually happening in the following situation: **There is an asteroid in outer space what is rotating with high speed. There is an astronaut standing on its surface.**   1. Does the astronaut feel lighter compared to if the asteroid wasn't rotating?   2. Can he jump higher compared to if the asteroid wasn't rotating?   3. What happens if the astronaut makes a big jump vertically? Is he going to land on the exact same place, or the asteroid could "rotate under" the jumping astronaut?


title: Energy produced by a swing?

text: Please can someone help me with a equation for how much energy a person generates when swinging on a swing?


title: Duality in arbitrary finite dimension using the Levi-Civita tensor

text: In 4-D flat metric E&M context, given a rank $p$ tensor, one can construct dual of $4-p$ rank tensor by Levi-Civita tensor. Here dual is not in the same sense of mathematical dual. I do not know where it comes from. The followings are my questions:   1. Is it necessarily true for arbitrary dimension that $n$, given a rank $p$ tensor, one always can have a rank $n-p$ tensor constructed by Levi-Civita tensor dual to the rank $p$ tensor?    2. If it is true, in any dimensional space+1 dimension time, any tensor and its dual contain the same information. Shouldn't the bigger the space, the more information be thrown in? Why am I expect this duality is true up to any dimension?    3. It seems somehow there is some sort of one to one correspondence between dual and double dual itself where dual is still not the familiar mathematical dual. I knew in math double dual space isomorphic to dual space which is different from what we are talking here. What is the correspondence in the E&M dual context?


title: Trying to understand mixed states

text: I took a basic quantum chemistry course (McQuarrie's "Quantum Chemistry"), but it never dealt with mixed states \-- only pure states (or if it did, we never got to it in class). So I'm trying to understand them on my own. Consider a situation where Bob is in the lab and flips a coin. If it is heads, he prepares the system into pure state $|\psi_1\rangle$. If the coin is tails, he prepares the system into pure state $|\psi_2\rangle$. Now he invites Dave into the room. Bob knows which way the coin landed, but Dave doesn't. All Dave knows is that the system is either in the pure state $|\psi_1\rangle$ or $|\psi_2\rangle$, each with 50% probability. ...so could you say the system is in a pure state to Bob and a mixed state to Dave? Or am I way off base here?


title: Heisenberg uncertainty principle - question

text: > _A beam of particles each having mass $m$ and velocity $v$ in the incident > on a circular hole of radius $b$ located on a screen. If another screen is > placed at a distance $D$ from the hole, determine the minimum size of the > pattern formed on the screen by the beam._


title: How does optical phase modulation produce photons with different frequencies?

text: The classical description of electro-optic modulators is an index of refraction that depends on the applied voltage. For example, for a sine modulation $\sin(\Omega t)$, a monochromatic laser of frequency $\omega$ would get an additionnal phase $\varphi\propto\sin(\Omega t)$. This results in sidebands in the spectrum at $\omega-\Omega$ and $\omega+\Omega$. Now, what is the interpretation of this phenomenon in terms of photons? A photon with initial frequency $\omega$ will end up at $\omega-\Omega$ or $\omega+\Omega$. How can the time-variation of the refractive index create new photon frequencies? Is it a non-linear effect similar to second-harmonic generation? If yes, it could be explained by an interaction such as $\hbar\omega + \hbar\omega \rightarrow \hbar (\omega-\Omega ) + \hbar(\omega+\Omega)$? EDIT: A corollary to the original question. I shake my hand very fast in front of a laser beam, what happens to the photons? Do they get chopped in shorter photons? Instead of my hands, I could use a super-fast chopper. I would see photons with new frequencies (the sidebands) because of this modulation. How come the incident photons get a different energy?


title: how many qubit do we need to store "16"?

text: classical computers store information in bits which can either be 0 or 1, but in quantum computer the qubit can store 0 , 1 or a state that is the superposition of these two states so how this superposition help to use less qubit to stor information for example: how many qubit we are need to store "16"? I know a little about h(x) but i don't know how we store! And is there any fear about losing information because of probability?


title: Is there a depth at which a human body, with lungs full of air, will no longer float?

text: I was watching a special on divers who dove down hundreds of feet without tanks or gear. The show referred to a depth at which the diver was no longer buoyant and would actually sink, even with lungs full of air. Is there such a point and if so, why? What is happening there?


title: Precise statement of Mermin–Wagner theorem

text: Roughly speaking, Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions $d\leq 2$. What is the precise statement of this theorem? In particular:   1. What are the types of systems the theorem applies to?    2. How short-ranged must the interactions be?   3. Do the interactions have to satisfy other conditions, such as being isotropic?


title: What would happen if two entangled particles collided?

text: Does that is even possible? I have almost zero knowledge in quantum physics, it is just a curiosity that popped in my mind.


title: Non-zero net charge on cosmological scale

text: It seems very clear, that because atoms and ion compounds are electrically neutral, our world seems to consist of an equal amount of positive and negative charge resulting in a zero net charge. What would be the consequence of (small) non-zero net charges on cosmological scales? ## Planetary What would happen, if we gave an otherwise electrically neutral planet (e.g. Earth) 1000 (1 million, $10^{23}$, $10^{30}$,$10^{50}$) addtional electrons? Sure, these addtional electrons would somehow diffuse. Could we even measure this diffused charge or at least proof, that there is an additional charge? ## Galactical Could a non-zero net charge in (parts of) galaxies explain their behaviour (e.g. rotational velocity, expansion) better than _dark matter_ and _dark energy_? Suppose one galaxy had a positive net charge and a neighbour galaxy a negative net charge. How long would it take for these charges to neutralize itself?


title: What to work on?

text: (I am not exactly sure if this question is supposed to go here, but it seems like the appropriate place for it.) Hello Scientific Community, I am soon graduating from high school and, as everyone does, I am searching for something to study at college. I have always wanted to study straight forward physics, but recently I have been having problems finding the application (possible jobs) of a physicist. My parents are trying to push me towards mechanical engineering, but I am trying to convince them I can get a decent job studying physics. They hate the idea of having me become a teacher, so that is out of question. I know many of the members in the community have studied physics, and I would like all of you out there to please help me making a short description of your job and it`s availability. If you know of any other jobs apart from the one you are on, feel free to list them. Thanks a lot for the help P.S: I live in a country were not many resources are spent on scientific investigation. If you work in some kind of job on applied physics, please answer, at least a short response. Still make an answer if you are some kind of investigator for anyone else that reads the thread in the future.


I was wondering why water freezes from up to bottom and not the other way around. The reason I question this is because in a Video I came across, I discovered water freezes from top but surely with the current knowledge I would say it should be opposite because we know cool air drifts down while hot air drifts up so similarly the heat should fall down and this that the freezing should start from bottom and not up? Unless my knowledge applied is wrong and potentially stupid so I want to ask: Why does water freeze from top to bottom?


title: Absorption line detected with a significance of $2.2\sigma$

text: What do we mean by the following statement? > The SV $\lambda$ 786.46 line is detected with a equivalent width of $W = > 22.7\pm10.2$ corresponding to a significance of $2.2\sigma$ How does one calculate $\sigma$?


title: Exoplanet surface detail: Limitations on size of space telescope array

text: How big could an array of space telescopes acting as an interferometer be ? How big would it have to be to resolve exoplanet surface detail the size of Iceland at a distance of 100 light years ?