Parity of proton is 1?

I have found from Wikipedia that "a parity transformation is the flip in the sign of spatial coordinates". Now when we operate parity operator, does that mean we are taking any physical entity at ${\bf x}$ to $-{\bf x}$. Or we are just reverting axes of the co- ordinate system? However if we take parity transformation as active transformation then what does it mean that parity of proton is 1? Doesn't it anyway depend on the **origin** of the co-ordinate system? Please elucidate the meaning of the parity. I get too much confused whenever I hear "parity"!


PDE from dispersion relation?

Suppose I have knowledge of a system's dispersion relation $f(\omega,k)$. Is it possible to recover the underlying PDE describing the system? Can I simply use the replacement $k=-i\nabla$, $\omega=i\frac{d}{dt}$ to go back? I came across one source which claimed that the original PDE could only be recovered to a certain extent. An example given was the Dirac and Klein Gordon equations which both satisfy the same dispersion relation. But I didn't quite follow. Example: I have a polynomial dispersion relation of the form $\omega^2+c^2k^2=1$. Can I automatically say that the underlying PDE is $\frac{d^2u(x,t)}{dt^2}+c^2\nabla^2u(x,t)+u(x,t)=0$? Or is there a subtlety I'm missing? Thanks


How do objects heat up?

If every body emits radiation at a given frequency and temperature exactly as well as it absorbs the same radiation, how do objects heat up?


What is the density and energy of a photon?

As I understand, photons are considered mass-less, which is a necessary condition for moving at the speed of light. However, does that mean their density is 0, as they will occupy some volume. If their density is zero, that means there is no matter inside a photon. Thus, shouldn't a photon be able to pass through matter instead of colliding with it? As $E = mc^2$, shouldn't a photon have zero energy, as it has zero mass?


Power and magnetism

If we have a solenoid and a magnetic field passes through it, a DC voltage will be produced in the wire. If we want to calculate the power, we find out the current using Ohm's law. I know there is power loss due to the resistance (joule effect). But what about the magnetic field due to current (lenz's law), does it contribute in the power loss?


Deriving entanglement entropy from Renyi entropy

My questions are based on this paper - http://arxiv.org/abs/0905.4013   * Firstly I want to know as to whether some assumptions are needed about the relationship between the systems $A$ and $B$ for the Hilbert space to factor as tensor products as ("assumed"?) on page 3?  I mean assume the more common reverse scenario - if you are given a system C and you decide to call some part of it as $A$ and the rest as $B$ then does it automatically mean that the Hilbert space of C factors between A and B? (...that doesn't intuitively feel to be true...then what exactly is the assumption being made here?..)   * Secondly given the definition of $S_A$ and $S^{(n)}_A$ as in equations 2 and 3 how does this claimed equality follow that, $S_A = \lim _{n \rightarrow 1} S^{(n)}_A = - \lim_{n \rightarrow 1} \frac{\partial \rho^n_A }{\partial n }$ I am unable to see the proof of the above 2 equalities. It would be great if someone could help.


Alternate young double slit experiment

What will happen in young's double slit experiment, when instead of screen I put a black screen with a hole and a second screen behind the black one. Will it still form interference pattern albeit a faint one or no interference pattern will be observed?


Hydrogen extraction efficiency of newish nano particle AA battery

The internet is ablaze with the new nano particle based extraction method of converting water to hydrogen. In 2002/2003 when I was 16 there was a similar craze about a lawn mower motor which supposedly ran on water. Indeed there were even conversion kits on the internet for vehicles. They didn't work because of a fundamental inefficiency of the conversion. Will this new nanotechnology improve the process so that cars can run on converted water or does there still exist a limit?


Definition of Ampere

On Wikipedia it says: > This force is used in the formal definition of the ampere, which states that > it is "the constant current that will produce an attractive force of $2 × > 10–7$ newton per metre of length between two straight, parallel conductors > of infinite length and negligible circular cross section placed one metre > apart in a vacuum." In reference to the definition of an Ampere, why was $2 × 10–7$ chosen?


What is the reasoning behind hole carriers being able to carry heat?

In the Peltier effect, we consider charge carriers being able to carry heat. As for electrons or ions, this attitude makes sense, since external electric potential drives particles with mass in a direction and effectively transfers heat from one point of material to another. But for holes, this situation is only virtual, holes moving in a direction is only reformulation of the fact that the electron making up the hole environment are traveling opposite direction. I just cannot grasp the concept that the holes can indeed carry heat. I know one can argue that the holes have effective mass, but effective mass is only a measure of how much can you accelerate / decelerate the electron (or hole) in a crystal field. Can you help me out please?


Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for this statement and trying to verify it myself I did not succeed. If I make a general ansatz with a spherically symmetric Gaussian wave packet with time dependent width and time and space dependent phase $$ \psi(t,\vec x) = (\pi a(t)^2)^{-3/4} \exp\left(-\frac{x^2}{2 a(t)^2} + i \phi(t,\vec x)\right)$$ and insert it into the Schrödinger equation $$ i \dot{\psi}(t,\vec x) = -\frac{1}{2m} \Delta\psi(t,\vec x) + \frac{k}{2} x^2 \psi(t,\vec x) $$ I get relatively complicated differential equations involving first time derivatives of a and $\phi$ as well as first and second order spatial derivatives of $\phi$. I failed to solve those equations or even show that a solution exists. Is there an easy way to show this? Is there any reference where this is shown? Which are the solutions for $a(t)$ and $\phi(t,\vec x)$ for a Gaussian (given initial conditions $a(0) = \sigma$ and $\phi(0,\vec x)=0$)?


Faraday tensor, antisymmetric electromagnetic tensor

I want to write $F^{\mu \nu}F_{\mu \nu}$ in terms of $F_{\mu \nu}F^{\mu \nu}$. How to do it?


Why linear wave equation does not have solitonic solutions?

As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for solitons in nonlinear wave equation?


Relativity addition and signs

I have just covered a very brief module on special relativity as a part of my physics course. I have also done some extra reading mostly; Morrin's Classical Mechanics. While I found the book really illuminating in some aspects, I still feel that regardless of how hard I try there is something with relativity that prevents me form doing anything but the simplest questions. I was trying to pinpoint my problem and I think that a big part of it is velocity addition. I understand that the Galilean transformation would predict the $$V_{A}=V_{B}-V_{rel}$$ provided that A and B are two frames of reference. I also understand that we need to use the Lorentz transformation to get the velocity transformation in relativity; $$ \begin{pmatrix} c \Delta T_A \\\ \Delta x_A\\\ \end{pmatrix} \begin{pmatrix} \gamma & \gamma \beta \\\ \gamma \beta & \gamma\\\ \end{pmatrix} = \begin{pmatrix} c \Delta T_B \\\ \Delta x_B\\\ \end{pmatrix} $$ Transforming the velocity u measured in frame to frame B; $$u = \dfrac{\Delta x_A}{\Delta t_A} = \dfrac{v_B + u_{rel}}{1+\dfrac{v_B u_{rel}}{c^2}}$$ But as far as I understand we could equally reverse the frames A and B and simply transform the other way around which means we need the inverse of the transformation matrix; \begin{pmatrix} \gamma & \- \gamma \beta \\\ \- \gamma \beta & \gamma\\\ \end{pmatrix} This will yield the formula; $$u = \dfrac{\Delta x_B}{\Delta t_B} = \dfrac{v_A - u_{rel}}{1-\dfrac{v_A u_{rel}}{c^2}}$$. However since the naming of frames is arbitrary, how do I know which of the two formula to use, the one with the all plus and the all minus signs. I have tried to look on the internet for the explanation of this, but I could not find anything. Also provided that I know which equation to use how, do I use it what is the sign convention for the velocities? Thank you very much for all the help and sorry for the long post P.S. I would be also very grateful if someone could point me to some good and simple resources on relativistic dynamics especially collisions. Thanks again.


Formation of atoms question

Could you please, explain to me the logic of the folllowing process as you would do to your 8 y/o sister: > Ubiquitousness and stability of atoms relies on their binding energy, which > means that an atom has a lower energy than an unbound system of the nucleus > and electrons. Where the temperature is much higher than ionization > potential, the matter exists in the form of plasma – a gas of positively- > charged ions and electrons. When the temperature drops below the ionization > potential, atoms become statistically favorable. Source: wiki How that that binding energy can be explained in plain English? Is it somewhat intrinsic to atoms, with each kind having its own level of binding energy? I understand that this can be translated in the language of closed systems using terms as Gnatt energy (G), etc. - but how?


Chemical potentials for multicomponent solids

I have a problem understanding how to deal with the chemical potential when two thermodynamic systems are in contact. For simplicity, let us consider $T=0\ K$. We have a system that is composed of two parts - I and II - where I occupies a space of fixed volume and II occupies the remaining space (essentially going to infinity in all directions). Both parts are initially filled with a crystal that is made up of a finite number of atomic types $i$. This is an equilibrium configuration (1). Now, I apply a long-range displacement field that originates in part I. If the boundary between parts I and II is permeable for the matter, some atoms move across this boundary. Here, part II can be thought of as a reservoir for the matter. The system eventually settles into another equilibrium (2), where the number of atoms $n_i$ in the part I is different from the above, i.e. there is nonzero $$\Delta n_i=n_i^{(2)}-n_i^{(1)}\ .$$ The free energy difference between the two equilibrium states of part I is then $$\Delta F \equiv F^{(2)}-F^{(1)} = V\sum_{ij}\sigma_{ij}\Delta\epsilon_{ij}+\sum_i\mu_i\Delta n_i\ .$$ In each of the two configurations (1) and (2), the parts I and II are in contact equilibrium, which means that $\mu_i^{I(1)}=\mu_i^{II(1)}$ and $\mu_i^{I(2)}=\mu_i^{II(2)}$, where the number in the bracket refers to the configuration. So, we have two chemical potentials for each $i$, one corresponding to configuration (1) and the other (generally different) for the configuration (2). Moreover, the particles move into I from the immediate neighborhood of the boundary I-II, which implies that $\mu_i$ should correspond to the atoms in the part II that are close to this boundary. However, how am I to understand a single $\mu_i$ in the equation above?


what kind of transformation can be applied to qubits?

I had a doubt on what kind of transformations can be applied to qubits. I understand that the transformations need to be reversible , but they also have to preserve the norm thats why the transformations need to be unitary as $U^{\dagger}U=I$.      But then I was reading the paper 'Quantum Mechanics Near closed Timelike Lines , by D.Deutsch'.In it he has applied a transformation give by   $|x\rangle |y\rangle \Rightarrow |x\rangle |y \vee f(x)\rangle$ , where $\vee$ stands for XOR and $x,y \in \\{0,1\\}^{n}$ and $f:\\{0,1\\}^{n} \rightarrow \\{0,1\\}^{n}$      Now I can see that that the transformation is reversible ( apply twice and get the same thing ) but it is not unitary , am I missing something ?


Aufbau principle in modern quantum theory

What is the rigorous definition of the Aufbau principle and the mathematical model used for its description? From Wikipedia, we have that the principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons. For modeling the postulate, we use the Madelung rule, Pauli exclusion and "if two same energy orbitals are available, choose the less occupied one)". So, $n+\ell$ ordering with constraints. And I also know that the Aufbau principle has its exceptions.


Physical, intuitive reason for divergence of dielectric constant at electronic percolation transition?

Several papers such as this (warning, PDF) and this (PDF again) talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (which, if you read those papers, gives rise to some neat effects). I'm having trouble understanding how, though, aside from just the math. The 2nd paper mentions a more qualitative interpretation: > Near the percolation threshold the metallic clusters are separated by thin > dielectric regions. Each pair of nearest clusters forms a condenser [i.e., > capacitor] whose effective surface tends to infinity near the percolation > threshold. Then the effective capacity of the system diverges, too. Ok, let's assume for simplicity the "dielectric" here is vacuum, and use the form of a parallel plate capacitor, $C=A\epsilon_0/d$. By "effective surface tends to infinity", I'm guessing they mean that $d$ is going to zero (because the distance between clusters is going to 0), and thus $C$ is increasing to infinity. From there, I'm a little confused. My best guess is that the impedance goes as $Z\propto 1/(i\omega C)$, and conductivity $\sigma \propto 1/Z$, and dielectric constant $\epsilon \propto i\omega \sigma$, so when the C increases, Z decreases, the imaginary part of $\sigma$ increases, and thus the real part of $\epsilon$ increases. But I'm not really sure about that, and I'm having trouble reconciling it with what must happen after the percolation transition. Does anyone have a better physical understanding of this? thank you!


For Faraday's law, why does the emf decrease as you increase the area of the loop?

I've only recently started learning basic electrodynamics, but I don't understand why a loop of coil with a small area and a magnet falling through will produce a larger emf than a loop of coil with a larger area with the same magnetic field falling through. To clarify, lets say you have a loop of coil and you drop a magnet through the coil. This is will produce an emf according to Faraday's law. Now if you have a loop of coil with a larger radius, and you drop the same magnet through at the same speed, it will produce a _smaller_ emf. Why is this?


Wave/particle duality

Apologies if this has been asked before (I did check and I believe it wasn't). I have a question about the particle/wave duality of photons (or other particles). Depending on what and how we measure the photon turns out to be either a wave or a particle. Recently I saw some web page (and I can't remember where) that maybe neither is true. Both the wave perception and the particle perception are just that: perceptions, strengthened by our perhaps limited ability to observe. What if the reality of the photon is something else, something "on top" of our two perceptions?? Could someone direct me to a site that could tell me more about it (or maybe debunk the whole idea?)?


Why aren't all photons virtual particles even in the "vacuum" of empty space?

I'm thoroughly confused about the nature of electromagnetic radiation. Light is supposed to exhibit both wave and particle characteristics. But does that mean that it is _both_ a wave and a particle or _neither_ a wave nor a particle? If the latter, then it is not a "real" particle at all, but only occasionally behaves as one (hence, a "virtual" particle). And if the former, then it is, depending on the experimental setup being used to detect it, a "real" particle, but one that only exists (as a particle) for a limited time, which again, corresponds to the definition of a _virtual_ particle ("a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time" - Wikipedia). So, if photons invariably appear to exhibit the characteristics of virtual particles, why are they not actually virtual particles (even in a vacuum)?


Particles acting like waves

Wave–particle duality is kinda bothering me... I read that electrons can act like waves, but I know that electrons are actually particles. The theory says that if you have not observed the particle it acts like a wave and can explore all classically available particle trajectories simultaneously and when you observe it it ends up in just one place. But doesn't that just mean that this particle is a particle the whole time, but you can't know it's location without observing it and the wave just presents a range of possible locations where it might be if you do observe it? And if so why would you call it "Wave–particle duality" if this particle is just a particle doing it's thing and a wave is just a best guess of where it is?


Does radio use virtual photons?

In radio communication each accelerated electron in the transmitter antenna interacts with an electron in the receiver antenna by exchanging a photon. Is that photon always a virtual photon as described in the diagram below rather than a "real" photon? ![enter image description here](http://i.stack.imgur.com/i4Zqd.gif) As I understand it, Feynman's definition of a virtual photon in his book QED is any exchanged photon that does not appear in the initial or final conditions. Therefore by his definition radio works by using virtual photons.


The role of dark matter in black holes and star formation

In my understanding, there exists a critical mass for which a star needs to be in order for it to collapse into a black hole. This also applies to a certain critical density of gas in order for stars to form in the first place. However, given a massive star or a massive amount of gas, it must have a relatively large gravitational field and therefore will interact with dark matter, attracting dark matter towards itself. This makes me question, doesn't the critical mass or critical density that we've figured out before our knowledge of dark matter now need to take that into account? OR Shouldn't the critical mass and critical density resemble the following pseudo equations? Critical mass (normal and dark) for a star to turn to black hole = A*normal matter mass + B*dark matter mass, where A and B are fractional amounts Critical density for a cloud of gas (normal and dark) for star formation = C*normal matter mass + D*dark matter mass, where C and D are fractional amounts So if the answer is yes, then what are the fractions, A, B, C, and D? Moreover, What kind of experiment or calculation would allow us to figure out these fractions?


Matter-wave interference from free falling cold atoms

and another exam question, this is about current research: > Interference of matter waves has been studied using ultra-cold atoms. The > phase of a matter wave for free-falling cold-atoms at time $t$ and height > $y$ is given as $$\phi(y,t) = \phi(y_0,t) + Et - \int_{y_0}^y dy' k(y')$$ > where $E$ is the total energy of individual coherent atoms and $\hbar k(y)$ > is the semi-classical momentum of falling atoms at height $y$. A recent > experiment observed coherent matter waves continuously emitted from two > vertical micro-traps (you can neglect the size of the traps) with height > $-\lambda/2$ and $\lambda/2$. The atoms are emitted at rest from the traps. > > A) For a given height $y$ below the two micro-traps, derive the condition of > constructive and destructive interference. Well, from basic wave mechanics I know that the phase difference of the two waves has to be an even (constructive) or odd (destructive) multiple of $\pi$, but apart from that I find the question poorly worded: What does $t$ refer to? Total time in laboratory frame, or time since leaving the trap? And what about the function $\phi(y_0,t)$? If I don't know its form, can I say anything meaningful about the interference pattern at all?


Non conducting charged planes

I have two parallel non conducting charged planes with opposite charges $6\mu C/m^2$, area $A = 3m^2$ and distance between the planes $d = 0.004 m$. I know the potential between these two planes is $$V = \frac{\sigma\times d}{\epsilon_0}$$ But if I put a conducting sheet of thickness $h = 0.001 m$ with the same area $A = 3m^2$ between these two planes, the new potential between the the planes would be $$V = (2\times\frac{\sigma}{2\epsilon_0} + 2\times\frac{2\sigma}{2\epsilon_0})\times d$$ Is this right? or Am I wrong somewhere? I mounted this schema. ![](http://i.stack.imgur.com/j1peM.jpg) Is that right?


Work done on gas by piston

In this problem there is a cylinder that is closed at both ends and has thermally insulated walls. It is divided into 2 parts by a movable, frictionless, thermally insulated piston. There is a heating coil in the left side of the cylinder and there are $N$ molecules of an ideal gas in each compartment. The original temperature and volume on each side are To and $V_o$. The left side is slowly warmed with the coil until its pressure doubles. How much work was done on the gas on the right? I calculated the final temperature and volume of each side after the left side is heated but I'm unsure about the work calculation. Would the work simply be $W = -P(V_f - V_o)$ where $P$ is 2 times the original pressure (so its $2NkT_o/V_o$) or do I have to do something with the pressure since it changes from $P$ to $2P$?


Confusing Classical Mechanics Question

I don't want this to be a "do my home work question" so please tell me how I can make this question helpful for other people. In my physics assignment I found the question below. I'd think that both FBD's and $F_{net}$s are the same for each scenario; since the track is friction- less the rider doesn't need to input anything to maintain a constant velocity. If it helps, the answers for 'c' were found to be: $ac = g\tan(\theta)$, $a = g\sin(\theta)$ for each respective scenario. Any help is much appreciated. ![Problem text](http://i.stack.imgur.com/7A93J.png)


How much faster would a Clock without gravity run?

Pardon the misleading title. It is to my understanding that moving/heavy clocks run slow. The Earth itself is under gravitational influence from many sources, and is moving. Is there a way to know how much 'faster' a clock would run if those influences were removed? I don't need a precise number, an order of magnitude is fine, just to get the idea. Any insight is appreciated.


Shot noise at high frequencies (can it really be ideal white noise?)

Quantum shot noise (either optical intensity noise or electrical current noise) described by the noise spectral density of $2 e I$ (electrically) or $2 h \nu P$ (optically). So it is white noise. I know this basically comes from the derivation, where we model electron or photon events as infinitely short (with delta impulses). See e.g. Eq 1 in http://123.physics.ucdavis.edu/shot_files/ShotNoise.pdf. $$I(t)=\sum_j q \;\delta(t-t_j)$$ So far so good, but can this really be true in an exact sense? A consequence would be that with growing bandwidth, the fluctuation (variance) grows indefinitely. That cant be physically correct. So things I would like to know specifically:   * Are there any experimental results where the spectral noise density of shot noise is measured? If e.g. there is a (nonzero) transit time of electrons, we should see some interesting things happen in the noise density at the inverse transit time. has something like this been observed?    * Is there a "more correct" (quantum physical) theory to derive shot noise? I.e. gives the correct spectral density? If yes, what is it and what is the idea behind the theory?   * Might there be a fundamental reason why it is impossible to measure shot noise at high frequencies? e.g. there is a fundamental limit for the gain bandwidth product of amplifiers (with which we could observe the shot noise at arbitrary frequencies in, for example, a diode)?


Why can't the work done by a non-conservative force be zero? The displacement along a closed path is always zero. So, whatever be the type of force, variable or constant, the work has to be zero. Why do we need to calculate the work done for individual paths?![enter image description here](http://i.stack.imgur.com/jtrI2.jpg) This is a non-conservative force that starts from $A$ moves via Path 1 to $B$ and then back to $A$ via Path 2. Since the displacement is anyways going to be zero, why can't work done be zero?


the energy relations in oceanography description

I'm reading the following paper and do not entirely understand a point that the author is trying to make. In page 99 (2 of article) the author refers to the following equation: $$ Q^{*} = \beta S^{*}e^{-\beta z} + 2B^{*} \delta \left(z\right) $$ and in the following paragraph he states that "The first term on the right represents the convergence of the penetrating component S^{*} of solar radiation" My immediate question is does "first term" refer to the $2B^{*}\delta\left(z\right)$ or does it refer to "2B"? i.e. does rhs mean rhs of the equal sign or what? In addition, can anyone explain, in layman's terms, what this equation is referring to?


How can I calculate the relationship between power and acceleration for a beam driven photon sail?

I am writing a novel, and although I have a background in physics, I am unsure of the exact equations to use. Specifically: **For a photon-sail ship, how powerful would a driving laser need to be in order for the ship to reach an acceleration of ~0.5g?** Assuming we are only talking within the solar system here, and assuming that the ship is heavy enough to carry passengers. Would such a thing ever be feasible? I am trying to strike the right balance between fun and feasibility for some method of regular interplanetary transport (think the space equivalent of commercial jets).


Separation of perturbative and non-perturbative contributions in partition function computation

The following is defined, where $\epsilon \to 0^+$ is a cutoff: $$ \mathcal{F}(Z)=\int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: how do we see that $\mathcal{F}$ has a perturbative + non- perturbative expansion (in $x \gg 0$) given by $$ \mathcal{F}(Z) = \sum_g \mathcal{F}_g x^{2-2g} + O(e^{-x}),$$ for some coefficients $\mathcal{F}_g$ ? Reference: the issue is discussed in arXiv:hep-th/9809187 eq. (3.2) and below. The book _Quantum Field Theory_ by Itzykson and Zuber eq. (4-117) and below should help clarify. It should be possible to see this kind of expansion by rescaling the variable $s$ in the integral and making a Taylor expansion of the $\sinh^2$ in the denominator. The non-perturbative terms should arise from picking up poles at imaginary values of $s= 2\pi i n.$ Note that the $\epsilon$ parameter is supposed to ensure convergence of the integral. * * * I think I found a partial answer in the paper http://arxiv.org/abs/0907.4082 paragraphs 2.2 and 2.4, although I'm not 100% sure how one relates formula (2.20) here with (2.1) in http://arxiv.org/abs/hep-th/9812127


Spectral Decomposition Grover operator

I am studing the Grover operator. Let be $x_0$ the marked element. Let be $t_f = \pi/2\omega$. The measurement of the register in the computational basis return $x_0$ with probability $$p_{x_0}(t_f)=\langle x_0|U^{t_f}|D\rangle|=1-1/N$$ Where $|D\rangle$ is a diagonal state state, $U$ the grover operator, $\cos \omega = 1-2/N$ and $N=2^n$. My lecture say $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\lfloor t_f\rfloor).$ $\lfloor t_f\rfloor=\lfloor (\pi/4)\sqrt{N}\rfloor + O(1/\sqrt{N})$ was obtained taking the asymptotic expansion in $N$. My question is Why $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\lfloor t_f\rfloor).$?


Moon's pull causes tides on far side of Earth: why?

I have always wondered and once I even got it, but then completely forgot. I understand that gravity causes high and low tides in oceans, but why does it occur on the other side of Earth?


Infinite Potential Well Energy for Piece-wise Constant Wave Function

I'm trying to compute the expectation value of energy for a certain state in an infinite potential well but I'm getting contradictory answers. The well has potential \begin{align} V(x) = \left\\{ \begin{array}{lr} 0 & : 0 < x < L\\\ \infty & : \text{ elsewhere} \end{array} \right. \end{align} which has eigenstates $\phi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n \pi x}{L})$ with corresponding energies $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$. Now, consider the state \begin{align} \psi(x) = \left\\{ \begin{array}{lr} 0 & : 0 < x < L/2\\\ \sqrt{\frac{2}{L}} & : L/2 \leq x < L\\\ 0 &: \text{ elsewhere} \end{array} \right. \end{align} We want to compute $\langle H \rangle$ for this state. One way to do this is simply using the definition: $\langle H \rangle = \int_{0}^L \psi^*(x) H \psi(x) dx$. The problem with this though is that $\psi(x)$ is piecewise constant, and therefore this will give you $0$. The other option is to expand $\psi$ in terms of the energy eigenbasis by computing the coefficients $c_n = \langle \phi_n | \psi \rangle$, and getting $\langle H \rangle = \sum_{n=1}^\infty | c_n |^2 E_n$. As $E_n > 0$ for every $n$, this quantity will be strictly greater than $0$ and therefore will differ from the previous answer. What is the discrepancy? It surely has to do with the discontinuity at $x = L/2$, but I can't figure out how to deal with it.


Velocity of a Charged particle in a magnetic field according to Biot-Savart Law

According to Biot-Savart Law, if there is a charged particle in motion, there will be a magnetic field. My question is whether the counterpart of this law also holds true, i.e. if there is a magnetic field, whether there will be a charged particle be in motion. Let me explain my question a bit more clearly. First of all I don't know whether we can have a single charge in motion and another at rest in reality. If we can have then there may be two cases: Case 1: At time t=0 a charged particle is placed at rest at point P where B=0 and at time t>0 an external B field is created in a region R covering the point P. Case 2: At time t=0 a magnetic field B is created in a region R. At time t>0 a charged particle is placed at rest at point P in R. Note of Caution: The point P is chosen in such a way that there won't be any external electric field at P. Now come to the question: Will the charged particle move? If yes, under which case? If 'NO', whether the charged particle will absorb any energy or not?


Interpretation of a density matrix as an observable

In quantum mechanics, any density matrix (or density operator) is Hermitian. Observables are also represented by Hermitian operators. So it follows that a density matrix can also be interpreted as an observable. What is the physical meaning of this observable?


Please explain this circular movement problem

I was working over some problems from my physics textbook, and I came across this one, it involves circular movement: A car with the mass of 1t is moving over a hill with the velocity of 20 m/s. The radius of the hill (which can be viewed as a half-circle) is 100m. When it comes to the top of the hill, with what force does it act upon the hill? The correct answer is, apparently, 6kN. How do I solve this? The situation can be illustrated like this I guess: ![enter image description here](http://i.stack.imgur.com/zkukZ.png)


Optical Tunneling

Is there light tunneling taking place in optically transparent mediums like glass? wherein light travels a larger path in these mediums without interacting with atoms and without any change in velocity, frequency, wavelength or momentum? but just getting affected by the field of atom and following the least resistant path through the medium?


Software for simulating the motion of a test particle in a force field

> **Possible Duplicate:**   >  Software for simulating 3D Newtonian dynamics of simple geometric objects > (with force fields) Is there any software for linux, which can simulate the motion of a test particle in a force field. I.e. I want to define a force field and the initial definitions of the test particle and the software simulates the moving particle (based on the (numerical) solutions of the corresponding differential equations of motion). Would be great if the software could display the vector field and the field lines as well.


Non reflecting boundaries in waveguides

Can someone please explain the Sommerfeld radiation condition and what is the alternative non-reflecting boundary conditions for waveguides of general geometries?


In terms of physics, does the phrase "time slows down" mean the same thing as "things happen more slowly?"

The common definition of "time" is a type of measurement, like size. But the sentence "size gets bigger" doesn't make any sense. Is "time slows down" an odd phrasing of "events occur more slowly" or is there a deeper meaning to the phrase?


Einstein equation and scalar field stress-energy tensor

Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where $$ L_{m} = \frac{1}{2}g^{\mu \nu}\partial_{\mu}\varphi \partial_{\nu} \varphi - V(\varphi ) $$ So the variation $\delta S_{m}$ must give $\frac{1}{2}\int d^{4}x \sqrt{-g}\delta (g^{\mu \nu})T_{\mu \nu}$, where $T_{\mu \nu}$ refers to the stress-energy tensor of scalar real field. I tried to do it, but this leads me to the answer $$ \delta S_{m} = \int d^{4}x\delta (\sqrt{-g})L_{m} + $$ $$ +\int d^{4}x \sqrt{-g}\left(\frac{1}{2}\delta (g^{\mu \nu})\partial_{\mu} \varphi \partial_{\nu} \varphi - g^{\mu \nu} \partial_{\mu}\varphi \partial_{\nu} \delta \varphi - \frac{\partial V( \varphi )}{\partial \varphi }\delta \varphi\right)= $$ $$ =\frac{1}{2}\int d^{4}x\sqrt{-g}\delta (g^{\mu \nu})T_{\mu \nu} + \int d^{4}x\sqrt{-g}\left( g^{\mu \nu} \partial_{\mu}\varphi \partial_{\nu} \delta \varphi - \frac{\partial V( \varphi )}{\partial \varphi }\delta \varphi\right). $$ What to do with the second integral? Does it equal to zero according to Euler-Lagrange equation, or I can't say so?


Who is usually credited for the creation of QFT?

I read in a book just now that says: > [...] but it was not explained until the invention of quantum field theory > by Richard Feynmann [ _sic_] and others in the 1940's. I have been under the impression that it was Paul Dirac and Hermann Weyl in the early to mid thirties who developed the foundations of QFT.


I am trying to calculate how $<r>$ in the hydrogen atom evolves with time

I am working on the Hydrogen atom and I was trying to calculate $\frac{d<r>}{dt}$ using $$\frac{d<r>}{dt} = \frac{i}{\hbar} <[\hat{H} , \hat{r}]>.$$ Here $r = \sqrt(x^2 + y^2 + z^2)$ and $H = \frac{p^2}{2m} + V$ where $p^2 = -\hbar^2 \nabla^2 $. Now according to Ehrenfest's theorem should behave classically and give me some equivalent of velocity, and indeed I do get something but it does't resemble velocity: $\frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r)$ where $f$ is a test function. Steps: $[H ,r]f = [\frac{p^2}{2m} + V , r]f = \frac{p^2(rf)}{2m} + Vrf - \frac{rp^2(f)}{2m} - rVf = \frac{1}{2m}[p^2 ,r]f = \frac{1}{2m}[-\hbar^2\nabla^2 , r]f = \frac{-\hbar^2}{2m}[\nabla^2 , r]f $$= \frac{-\hbar^2}{2m} (\nabla^2(rf) - r\nabla^2(f)) = \frac{-\hbar^2}{2m} (\nabla r\nabla f + r\nabla^2f + \nabla f \nabla r + f\nabla^2 r - r\nabla^2f) = \frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r) $ Am I doing something wrong?


Why do we still need to think of gravity as a force?

Firstly I think shades of this question have appeared elsewhere (like here, or here). Hopefully mine is a slightly different take on it. If I'm just being thick please correct me. We always hear about the force of gravity being the odd-one-out of the four forces. And this argument, whenever it's presented in popular science at least, always hinges on the relative strength of the forces. Or for a more in depth picture this excellent thread. But, having had a single, brief semester studying general relativity, I'm struggling to see how it is viewed as a force at all. A force, as I understand it, involves the interaction of matter particles with each other via a field. An energy quantisation of the field is the force carrying particle of the field. In the case of gravity though, particles don't interact with one another in this way. General relativity describes how space-time is distorted by energy. So what looked to everyone before Einstein like two orbiting celestial bodies, bound by some long distance force was actually two lumps of energy distorting space-time enough to make their paths through 3D space elliptical. Yet theorists are still very concerned with "uniting the 4 forces". Even though that pesky 4th force has been well described by distortions in space time. Is there a reason for this that is understandable to a recent physics graduate like myself? My main points of confusion:   * Why is gravity still viewed as a force?   * Is the interaction of particles with space time the force-like interaction?   * Is space-time _the_ force field?   * If particles not experiencing EM/weak/strong forces merely follow straight lines in higher-dimensional space (what I understand geodesics to be) then how can there be a 4th force acting on them?  Thanks to anyone who can help shed some light on this for me!


Eigenvectors of a 4D rotation, and their interpretation

Let us define a 4D rotation by using two unit quaternions: $$\mathring{q}_l=\frac{a+ib+jc+kd}{\left|a+ib+jc+kd\right|}$$ and $$\mathring{q}_r=\frac{e+ib+jc+kd}{\left|e+ib+jc+kd\right|}.$$ They differ only by the real term (the denominators are only for normalization). And let us consider two cases: the $a,b,c,d,e \in \mathbb{R}$ in the first case and $a,b,c,d,e \in \mathbb{C}$ in the second case - that would be biquaternions. Now let us consider a rotation of an arbitrary (non-unit) quaternion $\mathring{q}$ so that: $$\mathring{q}_{rotated}=\mathring{q}_l\,\mathring{q}\,\mathring{q}_r^* $$ (note the conjugate of $\mathring{q}_r$). I can represent this 4D rotation using a $4 \times 4$ rotation matrix, and thus I can find eigenvalues and eigenvectors of this matrix. The question is - what is the interpretation of eigenvectors of such rotation? In 3D rotations the invariant vector (a vector that is not being rotated) is the rotation axis, and the eigenvalue has to be 1 (because it is a rotation, so no particular interpretation here). In 4D rotations I know that there are two planes around which the rotation occurs, but are those planes the invariant of the rotation? If yes, then how can I describe this plane as an eigenvector quaternion - would such quaternion be a "normal vector" of this plane, analogous to 3D space where normal vector of a plane is just a 3D vector? How is that different between quaternions and biquaternions? Please note that since $e \neq a$ the normalized quaternion $\mathring{q}_r$ will be very different from $\mathring{q}_l$ - all the imaginary components of unit quaternion (those created by $b,c,d$) will be in fact different, thanks to normalization.


Lightning and Charge Displacement

There is something I don't really understand about flashes of lightning. When a flash occurs, how come electricity be transferred at the speed of light since electricity's displacement is very slow ? It is not that I don't understand anything about that. I've read wikipedia and the others. But on that particular point there's something I really don't get ... But the question could be for lighting as for any charge strike between two points of course.


Layered CMOS structure question

I am trying to understand the workings of a CMOS image sensor. I understand that increasing wavelength results in an increased penetration depth in the silicon often used in CMOS image sensors. What I am in need of assistance in understanding is how the wavelengths (corresponding to the primary colours, red green and blue) relate to the vertical structure of the CMOS sensor. After reading several articles and performing some experiments, I am aware that CMOS sensors are indeed sensitive to UV wavelengths, but I am trying to understand how I was able to get a reddish-pink response using an incident UV source.


Why does a glass window reflect white objects white from an atomic scatterers’ viewpoint?

Related: Is a white object always white? If you are standing in front of a glass window during the day, you can see your dim white t-shirt’s reflection in the window. The reflection is dim because only 4% of the light is reflected (assuming you’re perpendicular to the window) while the rest is transmitted through the glass. If a white object is placed in front of this glass pane, it’s obvious that white light is reflected as white and not colored. Why is this? I vaguely remember that visible light that is reflected penetrates glass about λ/2, so that means that blue light penetrates the least while red light penetrates more. If I shine white light onto a glass window (assume RGB color model), I have to account for all three colored scatterers in roughly equal amounts of intensity to produce white. Here is what I think. • Blue light penetrates shorter distances into the glass window’s atomic layers and recruits less atomic scatterers that back scatter more intensity blue light. I assume Rayleigh scattering is valid since wavelengths of light are much greater than the atomic spacing’s in glass, therefore $I ∝ 1/λ^4$. • Green light penetrates deeper into the glass window’s atomic layers and recruits more atomic scatterers that back scatter more “less intensity” green light according to $I ∝ 1/λ^4$?. So the larger number of green atomic scatterers scattered less intense green light but there are more green atomic scatterers. • In a similar fashion, red light would recruit an even larger number of red atomic scatterers (but less intense than green or blue), however, the increased number of red atomic scatterers compensates for a more overall intense red intensity. In summary, there are roughly equal amounts of blue, green and red light reflected such that a glass window reflects white objects white. Here is what I am asking: is this correct way to view this? I was unable to find this anywhere online or on . I would greatly appreciate feedback, especially if this is not correct.


Why does density decide whether something floats or sinks?

> **Possible Duplicate:**   >  Balloons and lifting gases I understand that density of an object dictates whether or not it will float, but what I don't get and can't get any satisfactory explanation... What causes an object of lower density to float? Can someone clarify? Differences in pressure ( a little more on that) ? Thus far, I've rationalized it in this way ( I sense it's wrong, but still): Object with a lower density has a spread out mass distribution along the volume and therefore, the contact surface. Since water's density is 1 g/cm$^3$ ( our definition of mass says that 1 g is equal exactly to a 1 cm$^3$ volume of water)... When you have an object with a lower than water density, it exerts a force on the water due to gravity, but because at every specific point the weight of the object is always less than the one of water, it is unable to clear its volume of water out. I know, doesn't make any sense, please help! Every explanation on the Web just says it's dependent on the density of the object and that's that.


How does the entropy of an isolated system increase?

The change of entropy is defined $$\Delta S = \int \frac{dQ_\mathrm{rev}}{T}.$$ If a system is isolated the heat transfer between the system and the surroundings is zero ($dQ = 0$), thus $\Delta S = 0$. However, it is commonly stated that the entropy of an isolated system can increase. How is this possible, given the above definition of entropy?


List of immiscible liquids

I'm looking for a list of immiscible (and miscible) liquids. I currently developing a game that will use the property of differents liquid, some of them will mix, other will not (positioning in the recipient is important so density is). I'm searching but not found anything very relevant. Of course I know water, oil (and mercury). But I'm looking for more than two (at least 8 differents elements in three or four groups). I can always invent them but I'd like to keep things real as far as I can. The elements must be at the liquid state in normal conditions.


centrifugal force in static frame of reference

The other day we derived Kepler's third law. $$ \left( \frac{T_1}{T_2} \right)^2 = \left( \frac{r_1}{r_2} \right) ^3 $$ In order to derive this, you can look at a given planet that revolves around the sun. If you assume that the sun is way heavier than the planet and that the planet moves on a circle, you can simply state the force on the Planet by the sun: $$ F_G = G \frac{mM}{r^2} $$ Where $F_G$ is the force on the planet, $G$ is the gravitational constant, $m$ is the mass of the planet, $M$ the mass of the sun and $r$ the distance of the planet to the sun. Our tutor said that you now have to have a centrifugal force $F_c$ that is equal in absolute value to $F_G$, but opposite in direction so that the sum of the forces would be zero. The absolute value of this $F_c$ would then be equal to $F_G$: ![](http://wstaw.org/m/2011/11/15/m9.png) Then you plug in the formulas for the respective forces: $$ |\vec{F_c}| = |\vec{F_G}| $$ $$ m \frac{v^2}{r} = G \frac{mM}{r^2} $$ $$ \frac{v^2}{r} = G \frac{M}{r^2} $$ $$ v^2 = G \frac{M}{r} $$ $$ v = \sqrt{G \frac{M}{r}} $$ Where $v$ is the tangential velocity of the planet. Together with $v=\frac{2\pi r}{T}$, where $T$ is the revolution period, you get: $$ \frac{2\pi r}{T} = \sqrt{G \frac{M}{r}} $$ $$ \frac{4\pi^2 r^2}{T^2} = G \frac{M}{r} $$ $$ \frac{r^3}{T^2} = G \frac{M}{4\pi^2} $$ Since the right part is constant for each solar system, you can form a ratio and optain Kepler's law. So this approach does work. * * * I think that this view is only valid if you construct it from a rotating observation point, i. e. standing on the planet and facing the sun at all times. Our tutor said that this is from a static (with respect to the sun) observation point. In the rotating system, you do have the centrifugal force, but it is not a real force in this sense. So by saying that you have a centrifugal force, I think that you are in this rotating reference frame. And in the rotating frame, the planet is not moving at all. And by not moving in this rotating frame, it does revolve around the sun in the static frame. Since the planet is not moving, there must not be any force on it. Constructing a force equilibrium is the way to achieve this. But our tutor told me that there is a force equilibrium in the static reference frame as well. I say that if you have an equilibrium there, there is no net force, therefore the planet does only move along a straight line in a static frame. I was then told that although there is no net force, the forces are still there. At that point, I think that this is converting dynamics into statics by adding opposing forces so that everything cancels out. Say we _do_ have a net force (gravity) on the planet. ![](http://wstaw.org/m/2011/11/15/m10.png) This force is something like this, if the planet has rotated $\theta$ around the sun: $$ \vec{F_G} \propto \left( \begin{matrix} -\cos(\theta) \\\ -\sin(\theta) \end{matrix} \right) $$ If you integrate this, you will get the velocity and the position of the planet. $$ \int \vec{F_G} \, \mathrm d\theta \propto \left( \begin{matrix} -\sin(\theta) \\\ \cos(\theta) \end{matrix} \right) $$ This seems very good, since it goes along the circle, so this is the tangential velocity. $$ \iint \vec{F_G} \, \mathrm d\theta \propto \left( \begin{matrix} \cos(\theta) \\\ \sin(\theta) \end{matrix} \right) $$ And that is simply the position around the unit circle. You can substitute $\theta$ with $\omega t$ if you wish to express the angle with time. So I claim that you need a net force in order for the circular trajectory to appear at all. My tutor told me that if the force is too high (too weak), the planet will spiral into (away from) the sun, and therefore the equilibrium has to be there. That makes no sense to me. What makes sense to me is that if the trajectory is deflected too much (not enough) to form a circle, there is no circle. * * * If you have no force, integration turns out quite different: $$ F = 0 $$ $$ \int F \, \mathrm dt = \underbrace{0t}_{a=0} + \underbrace{c_1}_{v_0} $$ $$ \iint F \, \mathrm dt = \underbrace{0t^2}_{a=0} + \underbrace{c_1t}_{v_0} + \underbrace{c_2}_{d_0} $$ The latter is your $d = v_0 t + d_0$, which models a linear, unaccelerated movement. In the rotating system, $v_0$ is just zero, and it works out. In the static system, that would mean that the planet either remains still or moves on a straight line -- which is not what is really happening. * * * So I claim that you must _not_ have a force equilibrium for the earth to rotate -- that is the basic disagreement we have. Who is right? Or are we both right, just looking differently at the same problem?


Field operator eigenvalues

For an harmonic oscillator we can write the Hamiltonian eigenvalues in the basis of the amplitude eigenvalues : for example the ground state is a gaussian : $⟨x|0⟩=a.e^{-b.x^{2}}$. I was wondering if we can do that for the energy states of a field : can we calculate $⟨a|n⟩$ with |a⟩ an eigenvalue of the field operator $\phi$ (let's say a Klein-Gordon field) ? I failed to carry out the ladder operator method, but I think we should be able to calculate (or at least have implicit equation of) any operator eigenvalue in any other operator eigenvalue basis, or is there something obvious that I missed ? Because we always play to write state in all the basis we can : spin on position, energy on spin etc ..., but I have never seen in classes or in books energy states of a field written in the field operator basis. EDIT : I'll look in the reference given by Peter : "Weinberg I" and try to explain how to do. EDIT : I didn't find the answer in the given reference :(


How photons move along with EM wave?

So the wave moves like. a wave, it moves up and down, up and down. But how do photons move? Do they follow the same path or do they just go straight forward without oscillating?


Software for calculating Feynman Diagrams

Is there a software (open source preferred) where I would input something like "Ingoing: a fermion $(p1, s1)$ and a photon $(p2, s2)$. Output: A fermion $(k1, r1)$ and a photon $(k2, r2)$" and I would could then get each diagram (that is, each term in the Dyson series) up to whichever order I choose, also showing terms that evaluate to zero (just to see that they are there)? I have tried to compute the above diagram to 2nd order and I'm just baffled with how many details there are to collect, this seems like the perfect job for a computer, but I can't find anything that would do such a symbolic calculation (hopefully showing the steps along the way) and not just the final result.


Understanding that nothing can go beyond the speed of light

> **Possible Duplicate:**   >  Is the “How to break the speed of light” minute physics video wrong? Radius of the moon - 1737 km. Thus circumference of the side that can been seen from earth - pi * 1737 = 5457 km. Speed of light - 300 kmps So if I point a lazer at one edge of the moon from earth and then move it to the other side in one second I will ultimately make the lazer spot travel 5457km in 1 sec. Thus the speed of the light spot is 5457 kmps! What is it that I am missing interms of concepts here? When we say nothing can go beyond the speed of light then what does "nothing" mean?


Is it possible for a charged, fast-moving object to slow down and enter geo-stationary orbit?

I've had a wild idea which I can not discuss at length in this forum, but it comes down to the following problem: A sphere of radius R=~10μm and mass m=~10-16Kgr is travelling towards the earth at v = ~10^8m/sec. The sphere carries a charge Q and intersects the earth's magnetic field perpendicularly, at a distance r, such that it enters an elliptic orbit. Given an electron wave function of ~5eV, is there any possible configuration of Q and r that would allow the sphere to end up in a stable geo-stationary orbit? Would releasing charges at particular points of each rotation make any difference? Even though I did get a physics BS 13 years ago, I took a different direction and the calculations required are way beyond my current capabilities. If anyone finds the problem interesting, I'll be glad to hear from you.


Energy transfers in boiling a liquid?

When you boil a liquid the KE stays the same (as temp is the same), so where does the extra energy go? I know the extra energy put in breaks bonds between the molecules/ atoms, but once it has broken the bonds where does the energy go? And where does the potential energy in a liquid go- as (ideal) gases don't have PE? I've heard it's turned to latent heat, but what is this, how is it stored?


Transforming a sum into an integral

I posted this in the mathematical forums. Maybe you will help me. I found an hard article http://prola.aps.org/abstract/PR/v105/i3/p776_1 of yang huang and luttinger. The authors begins with the sum: $$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi ^2 a^2 \lambda ^2 }}{{V^2 }}\sum\limits_{} {'\frac{{\langle n_\alpha \rangle \langle n_\gamma \rangle \langle n_\lambda \rangle }}{{\frac{1}{2}\left( {k_\alpha ^2 + k_\beta ^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}}\delta \left( {\vec k{}_\alpha + \vec k{}_\beta - \vec k{}_\gamma - \vec k{}_\lambda } \right)} $$ which represents the second order perturbation term of the energy of a bose gas. ${\langle n_\alpha \rangle }$ is: $$\langle n_\alpha \rangle = \sum\limits_{n = 0}^\infty {n\left( {ze^{ - \beta \varepsilon _\alpha } } \right)} ^n /\sum\limits_{n = 0}^\infty {\left( {ze^{ - \beta \varepsilon _\alpha } } \right)} ^n = \frac{{ze^{ - \beta \varepsilon _\alpha } }}{{1 - ze^{ - \beta \varepsilon _\alpha } }} $$ In the sum the terms with a vanishing denominator are omitted. There is also the restriction ${\vec k{}_\alpha \ne \vec k{}_\beta }\ \ $ and ${\vec k{}_\gamma \ne \vec k{}_\lambda }\ \ $. Now the passage that is unclear to me is the passage to the integral: $$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi a^2 \lambda ^2 }}{{V^2 }}\left( {\frac{{4V^3 }}{{\pi ^3 \lambda ^5 }}} \right)\sum\limits_{i,j,k}^\infty {\frac{{z^{j + k + l} }}{{\left( {j + k + l} \right)^{1/2} \left( {j - k} \right)l}}} \frac{{\partial J}}{{\partial u}} $$ where: $$J = \int\limits_0^\infty {\int\limits_0^\infty {dqdq\frac{{\cosh \left( {upq} \right)}}{{q^2 - p^2 }}} } e^{ - vq^2 - wp^2 } $$ and: $$u = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{2\left( {j - k} \right)l}}{{j + k + l}}} \right) $$ $$v = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l}}{{j + k + l}}} \right) $$ $$w = \frac{{\hbar ^2 \beta }}{{2m}}\left( {\frac{{\left( {j + k} \right)l + 4jk}}{{j + k + l}}} \right) $$ $\frac{{\partial J}}{{\partial u}}$ is calculated here: http://math.stackexchange.com/questions/63534/difficult-integral Maybe we don't have to do thermal average of n and we need : $$n = \frac{1}{{z^{ - 1} e^{\varepsilon /kT} - 1}} = \sum\limits_{n = 1}^\infty {\left( {ze^{ - \varepsilon /kT} } \right)^l } $$ so: $$\sum\limits_{} {'... = \sum\limits_{i,j,k = 1}^\infty {\sum\limits_{} ' } } z^{ijk} \frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2 + k_b^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}} $$ This explain the factor $z^{ijk}$ and the sum. How can we continue?Now we have to transform the sum' in a integral. The density of state depends of $kdk$ so i think we have to evaluate: $$\int\limits_0^\infty {\frac{{e^{ - \frac{{\hbar ^2 k_a^2j }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\gamma ^2k }}{{2m}}} e^{ - \frac{{\hbar ^2 k_\lambda ^2l }}{{2m}}} }}{{\frac{1}{2}\left( {k_a^2 + k_b^2 - k_\gamma ^2 - k_\lambda ^2 } \right)}}k_a k_b k_\gamma k_\lambda } dk_a dk_b dk_\gamma dk_\lambda $$ We have to evaluate two integrals, i think. Do you have any suggestion?


Why isn't perpetual motion possible, even though we are so technologically advanced?

Why perpetual motion wouldn't be possible if we are so technological advanced? It is just a thing that I was wondering for too long. I mean, we are able to create so powerful permanent magnets, like neodymium magnets which can store an massive quantity of energy for a long period of time, but we are not able to use that energy at all. Why?


Black Body and Electron

My questions are:   1. How does a black body absorb photons?    2. Why does a black body absorb the most photons of all objects (e.g. those with another color)?    3. Are there any relationship between the electron and a black body?


Fourier transform and commutation of functions

The book I am reading takes the unjustified step $$e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}f(\vec{p}) = f(\frac{\hbar}{i}\vec{\nabla})e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}$$ and similarly, he uses elsewhere $$e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}g(\vec{r}) = g(i\hbar\vec{\nabla}_p)e^{-\frac{i}{\hbar}\vec{p}\cdot\vec{r}}$$ I have a gut feeling this follows somehow from the commutation relation $[r^i,p_j] = i\hbar\delta^i_j$ but I cannot justify this. Anyhow, the book has not introduced the standard commutation relations, and is working only with the fact that $\phi(\vec{p},t)$ and $\psi(\vec{r},t)$ are Fourier transforms of each other, and $f(\vec{p})$ and $g(\vec{r})$ are analytic. Could someone give me a hint?


Experiment demonstrating interference patterns of neutrons

The question is about **experiments** (and references) that demonstrate interference patterns of neutral (chargless) particles (fermions), especially neutrons. Like **double-slit experiments for neutrons** (or equivalent). Searching around did not yield sth (or maybe i did not notice it). My understading is that similar experiments have only been done for charged particles (i.e electrons, protons) or bosons (i.e photons). Is this correct?


How do traveling waves pass through a standing wave node, if the node doesn't move?

I'm having trouble with the explanation that a standing wave in a string is the superposition of traveling waves. ![standing wave](http://www.physicsclassroom.com/Class/waves/u10l4c5.gif) The nodes in the diagram above are points where the particles of the string's medium undergo zero displacement, i.e. they do not move at all. But if they do not move, _how is the disturbance of (any internal traveling wave) propagated past the node_? The usual explanation for how a wave is propagated is that when one particle is disturbed (say, moved up), it exerts a pull on another, which in turn exerts a pull on the next one, and so on. In other words, to exert a pull or push on the next particle there must be some movement/disturbance of the previous one. But the particle(s) at the node point do not move at all, so how does the disturbance of a traveling wave propagation pass through them? (I'm trying to understand the picture in terms of the mechanical forces between particles).


Phonon-phonon-interaction as higher order terms in the potential

Is there a simple way to understand why phonon-phonon-interaction is described by higher order terms in the potential? I mean: Having a quadratic potential is essential for the definition of the phonons as excitations of an harmonic oscillator. How can you suddenly talk about (interacting) phonons when you don't have a simple quadratic potential anymore...


Solving the Schrodinger equation with appropriate symmetry

In the paper _Markov Fields_ by Edward Nelson the introduction section claims that analytically continuing a Markov process with appropriate symmetry properties yields the solution of the Schrodinger equation. What is the appropriate symmetry required by a Markov process to solve the Schrodinger equation? Is the symmetry that of the Euclidean group? Furthermore, how would you show a Markov process with appropriate symmetry is a solution to the Schrodinger equation? Link to the paper: http://www.mathunion.org/ICM/ICM1974.2/Main/icm1974.2.0395.0398.ocr.pdf


Experimental results regarding non-extensivity in small systems

From a statistical mechanics point of view, the _extensivity_ of thermodynamics arises from the fact that we can neglect interaction terms in the Hamiltonian. This allows us to infer that when we put two systems in contact, the total energy is approximately equal to the sum of the individual systems' energy. If the systems are in thermal equilibrium it follows that the entropy is extensive as well. However, it is well known that if the interaction terms cannot be neglected then these extensivity results do not follow. In theory, this should typically be the case for very small systems. (It should also be the case for systems with very long range interactions, such as gravitational systems, but for this question I'm more interested in systems that are made of atoms and interact electromagnetically.) I'm interested in whether there has been any experimental work designed to measure this effect experimentally. In principle it could be done with calorimetry, but I'm not sure how sensitive the measurements would need to be. There might be ways to multiply the effect into a larger one, such as letting a very large number of tiny particles come into contact, rather than just two. Indeed, it might be a well-known effect that just isn't usually explained in these terms. I'd be particularly interested in work that measures the non-extensivity of _both_ internal energy _and_ entropy, in the same system, using independent measurements. However, any empirical work that addresses this issue is of interest. From searching Google I did come across a few experimental papers that mention non-extensivity. However, they seem to be concerned with testing the predictions of Tsallis entropy maximisation, which isn't what I'm interested in here. I'm really looking for calorimetric measurements of the non- additivity of the Hamiltonian in small systems, and the resulting changes in thermodynamic properties.


Why do moving particles emit thermal radiation?

While answering another question about heat in an atom, the discussion in the comments led to the question of how heat is related to thermal radiation picked up by infrared cameras. The answer is that molecules undergoing translational motion (which therefore have a temperature) emit energy in the IR range and the intensity of the radiation in that range is related to the temperature. What is the actual mechanism for the emission of this energy? What is it about translation that causes the emission? Does it require collisions with other atoms/molecules to cause a change in momentum and the emission of thermal energy (thereby slowing down/cooling the bodies involved in the collision)? Does that mean in rarefied conditions where the mean-free-path is relatively large, the rate of IR emissions decreases (while the intensity is still only dependent on the temperature)?


Does the energy of a magnetic field decrease when it moves a conductor carrying a current?

When a charged particle moves in an electric field, the field performs work on the particle. Thus, the energy of the field decreases, turning into kinetic energy of the particle. Does the magnetic field of a permanent magnetic similarly lose energy and perform work when moving a conductor with a current?


Is matter a continuous part of the field of space-time?

I recently found this quote by Einstein (in _On the Generalized Theory of Gravitation_ , 1950), and it seems to me like he is saying that matter is a part of the field of space-time, and is nothing more than an area of the field with very high energy. Is this correct? > According to general relativity, the concept of space detached from any > physical content does not exist. The physical reality of space is > represented by a field whose components are continuous functions of four > independent variables-the coordinates of space and time. It is just this > particular kind of dependence that expresses the spatial character of > physical reality. Since the theory of general relativity implies the > representation of physical reality by a continuous field, the concept of > particles or material points cannot play a fundamental part, nor can the > concept of motion. The particle can only appear as a limited region in space > in which the field strength or the energy density are particularly high.


The electron and classical dynamics

Can I use Newtonian laws of dynamics and kinematics applied to electrons (like $F = ma$ and $s = s_0 + v_0 t + at^2/2$)? If not, why not? I know that everything in principle behaves in a quantum way, even macroscopic bodies, but is the level of precision of using classical mechanical on electrons acceptable?


Can a static magnetic field turned into a static electric field? or vice versa?

Consider some positive charge that is distributed uniformly over a very long line along the z-axis. If I am stationary with respect to the line then there is only static electric field which has cylindrical symmetry. Assume now that I am moving with some constant velocity which has only a (positive) component along the z axis. With respect to me there is now current running down the (negative) z-axis, hence I expect to find a static magnetic field in my moving inertial frame. My question is, do I still find (a static) electric field as well in this case? [Because on one hand I find the answer to be NO, since static electric fields should come only from static charges, and the charges here will not be static. On the other hand, although the line carries current, the line is still charged (which is different from currents running in conducting wires where the wires are neutral at all times). That means the answer is YES and one expects to see static electric field because the line is continuously charged.] So is the answer yes or no (and why)?


Ropes and Pulleys - Really unintuitive answer

I usually don't want to do this, but please go to this link, the solution is too big to post it here http://engineering.union.edu/~curreyj/MER-201_files/Exam2_2_26_09_Solution.pdf And go to page 5 of the pdf. Briefly, the problem say > Determine the velocity of the 60-lb block A if the two blocks are released > from rest and the 40-lb block B moves 2 ft up the incline. The coefficient > of kinetic friction between both blocks and the inclined planes is $\mu_k$ = > 0.10. **Things I am confused with the solution** 1.First of all, I seriously thought lb was mass not force. after some googling, it turns out they are used interchangeably... 2.Where did they even get $2s_a + s_b = 0$ from? Why did they determine the change in distance this way? My first assumption was that if block B moved up 2ft, then block A should move down 2ft (the rope must "move" 2ft too right?). Then I wasn't sure, so I did a few triangles and found that the angle made a difference 3.Where did $2v_A = -v_B$ come from? 4 The FBD for block A is confusing, why is the friction force $F_A$ in the direction of the ropes? I thought it was block B that is going down? Am I the only one who had trouble deducting that the pulley and block A are the same object? 5.Look at the final answer, how could $v_b$ be negative? The problem says block B goes UP. If you are wondering, this is not homework. I am just interested in this problem and it is out of curiosity and very confused with the concepts. I've had at most introductory physics experience, but I think I should have been able to solve this still. Thank you for reading


How to derive Schrödinger equation?

How is the Schrödinger equation $$\frac {\partial }{\partial t}\psi=-\frac {i }{\hbar }H{\psi }$$ being derived?


Why the Hamiltonian near degeneracies should be proportional to Pauli matrices?

When a quantum system have a double degenerescence at one point, the Hamiltonian should be proportional to Pauli matrices near this point (also known as _diabolic point_ ) [Ref.]. But, why the Hamiltonian have this form?


$\gamma$ in Newton's Second Law of Motion in Differential Form

I am teaching myself Differential Equations from a website. In the website I am up to Direction Fields and an example of a differential equation is Newton's Second Law of Motion. It is written on the website like this: $$m\frac{dv}{dt}=mg-\gamma v$$ I know that $m$ is the mass, $g$ is the gravitational acceleration, $v$ is the velocity, and $t$ is the time, but what does $\gamma$ stand for?


I would like to know, what would happen if a primordial black hole mass of Earth, would impact with the Earth

Sorry for any mistakes, I'm Italian Hello everyone, I'm writing a novel, in which the Earth is devastated by enormous geological cataclysms (something like the end of the world), and to implement my cataclysm, I thought of a primordial black hole with a mass similar to that of the Earth. At first I thought to stabilize the black hole inside the Earth, but it would have destroyed the planet, and I do not want to destroy it; or so thought to pass away from one side to the Earth from the black hole, causing very powerful shock waves, which first pierce the mantle, causing the eruption of every volcano on earth, and then the crust collide, greatly destabilizing the tectonic plates , causing massive earthquakes on a global scale, and triggering the formation of giant tsunami impacting against every continent. such a scenario is likely, or not. ps: I hope that the question is connection to physics.


Is a quantum system mandatory for generating true random sequence?

Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of quantum mechanics involves randomness by definition. Can we argue based on these information that a true random number generator must use quantum mechanics? If anyone claims that s/he has a true random number generator and fails to prove that s/he is exploiting quantum mechanics can I discard the claim on the basis that s/he is not using quantum mechanics?


Cosmic Microwave Background and heat flow

Can the fact that energy is distributed homogeneously in the universe be explained through heat flow and not the Cosmic Microwave Background(or in other words, can we say CMB is a result of heat flow and not the universe expanding from a singularity)?


Couldn't we measure electrons with good enough technology?

I am a bit confused about the Heisenberg's Uncertainty Principle - just read about it in _How to Teach Physics to Your Dog_ , by Chad Orzel. He states that the reason electrons can't be measured is because the photons used to measure electrons, collide with them causing changes in momentum and velocity. Couldn't this simply be fixed by creating a vacuum and using a different method of observation? Maybe something that doesn't exist today? **Edit:** A full disclaimer: I never took a physics course higher than a basic intro to physics. All the answers below are still referring to light (photons) as the method for measuring the momentum/location of the particle. My question is: is it not possible to create a vacuum without any possible light source from entering; using this vacuum we can device a more advanced method which does not affect the momentum or position of the electrons.


Magnetic force on capacitor

I have this problem: I have a capacitor formed by two parallel square-shaped plates, of side $a$, and with distance $d$ between them. They're asking me out of other things, the magnetic force as a function of some things. I calculate de magnetic force by calculating the gradient of the magnetic energy, given by: $$E=\frac{1}{2}\iiint_VB\cdot H\;dV$$ I have calculated $B$ and $H$, that form, during the charge of the capacitor, loops around the center proportional to $r$, being $r$ the distance from the central axis. The problem is that I have to integrate that to a cubic volume: $a^2d$, one of those variables is independent, but the other two are not, and this problems never involve complicated integrals, such that one: $$\int_0^a\int_0^a rdxdy$$ If the plates were circles it would be trivial, but being squares, I'm sure I'm missing something.


Superpositions and expectation values in quantum mechanics

When the wavefunction of a particle is not an eigenfunction of an operator, the property to which the operator corresponds does not have a definite value, Why? Also consider a linear combination of basis functions: $$\psi=c_1\psi_1+c_2\psi_2+...=\sum_kc_k\psi_k$$ When the momentum is measured, in a single observation one of the eigenvalues corresponding to the $\psi_k$ that contribute to the superposition will be found, why?


Why is boiling water loud, then quiet?

Water in my electric kettle makes the most noise sixty to ninety seconds before the water comes to a full boil. I have been fooled many times by the noisy kettle, only to discover that the water was not yet hot enough for tea. The kettle is only at a full boil after the noise has _subsided_. I have noticed the same phenomenon with many other kettles, including conventional kettles on kitchen ranges; it is not a peculiarity of this electric kettle. Why does the boiling become quieter as the water reaches full boil?


When water is about to boil

Have ever noticed? When water is about to boil, no matters the kettle, there is some **sound** I have no idea where it comes from, sometimes long before it boils. Is there any explanation for this phenomena?


Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?

The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a _normal subgroup_ of Projective Symmetry Group(PSG), but it may not be a _normal subgroup_ of $SU(2)$, like $IGG=U(1)$. But this may results in a trouble: By definition, we can calculate the $IGG$ and $IGG'$ of the $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\psi_i)$ and $H(\widetilde{\psi_i})$, respectively. And it's easy to see that for each site $i$, we have $U_i'=G_iU_iG_i^\dagger$, where $U_i'\in IGG'$ and $U_i\in IGG$, which means that $IGG'=G_i\text{ }IGG \text{ }G_i^\dagger$. Now the trouble is explicit, if $IGG$(like $U(1)$) is not a _normal subgroup_ of $SU(2)$, then $IGG'$ may **not** equal to $IGG$, so does this mean that _two $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\psi_i)$ and $H(\widetilde{\psi_i})$ may have different IGGs_ ? Or in other words, does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom? Thank you very much.


Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega \times r.$ This does not make sense to me, since this centrifugal term is negative( I mean, compared to the energy of the rotating system the energy in the inertial system should be HIGHER, since there is an additional kinetic energy of the rotation) as I had exected something like $ E = \frac{m(v + \omega \times r)^2}{2} + U$ in the inertial frame. Then they claim that in the rotating system(in that case they use the notation $v_0$ instead of $v'$ we have $ E = \frac{mv'^2}{2} - m v' \omega \times r+ U,$ ( This looks reasonable to me, as kinetic energy is less than the inertial one), but still,in that case I had expected something like $ E = \frac{m(v' - \omega \times r)^2}{2} + U,$ so where am I wrong(especially about my reasoning in the first place) and why is it wrong, just to plug in the velocity of the transformation.


realization of: CFT generating fuction = AdS partition function

An important aspect of the AdS/CFT correspondence is the recipe to compute correlation functions of a boundary operator $\mathcal{O} $ in terms of the supergravity fields in the interior of the $AdS_{n+1}$ (as we approach the boundary). Namely, $\big< \exp \int_{\mathbb{S}^n} \mathcal{O} \phi_{0}\big > = \mathbb{Z}_{s} (\phi \big|_{\partial(AdS)} = \phi_0)$, where $\mathbb{Z}_s$ is the supergravity partition function. The review papers I have found (and Witten's original paper as well) explain how to use the above formula but fail to provide a satisfactory explanation why the formula ought to work, or even how it came about. Can anyone explain if there is a logical (and/or insightful) path that would lead to the above correspondence between the generating function of the $n$-points correlators and supergravity/string theory?


Why don't more rocky planets/moons have appreciable atmospheres?

It seems obvious why mercury has no atmosphere, given its proximity to the sun --but yet Venus is also fairly close, and has an extremely dense atmosphere. Titan is a large moon with an atmosphere thicker than Earth's, and then Triton, a more distant large moon, has only a tenuous Nitrogen atmosphere. Why aren't more rocky bodies able to maintain an atmosphere? What factors affect whether a planet or moon is able to hold on to its atmosphere? What factors determine whether a planet or moon develops an atmosphere to begin with?


Is our universe the only universe out there or are there also other possible universes?

There is the specific universe we live in, the "actual" universe with planet Earth, humans evolving, and us discussing metaphysical debates on physics.se in 2012, etc. There might or might not also be other possible/potential worlds out there, some differing majorly, like in the laws of physics, or minorly, almost like our current world, but maybe I didn't pose this question on physics.se. According to many worlds advocates, the other possible worlds also exist out there in Platonic space. Is this world we find ourselves the only universe out there, or do other possible worlds also exist?


Quantum Chemistry Kronecker Delta formality

In semi-empirical quantum chemistry, one frequently encounters the so called _zero differential overlap_ approximation $$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \mu | \lambda \lambda \rangle .$$ Why is it rather not written as $$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \nu | \lambda \sigma \rangle = \langle \mu \mu | \lambda \lambda \rangle $$ since on the right hand side of the first equation there are no $\nu$ nor $\sigma$ contained anymore. So either all four variables **plus** the Kronecker Deltas (middle expression of second equation), or only the "remaining" variables after evaluation of the Kronecker deltas (last expression of second equation).


Why does an airplane need to climb during a takeoff even if it is in emergency situation?

Right after take-off (which means an airplane already exceeded V1) it is recommended that an airplane keeps climbing even when emergency occurs. Beside worries of crashing into houses and buildings by flying low altitude, why should an airplane not stop climbing up? Why should it climb up and then, lowers to some altitude and then land?


Why doesn't diamond glow when hot?

In an answer to this SE question, the respondent explains that heating a perfect diamond will not cause it to glow with thermal blackbody radiation. I don't quite follow his explanation. I think it comes down to: there is no mechanism for diamond to generate light in the visible region of the spectrum. He mentions that interband transitions are well out of the visual range, so there will be no contribution from that. He mentions that the Debye temperature for diamond is > 2000 K. I presume that the argument here is that optical phonons will be frozen out, too. (But diamond doesn't have infrared-active phonons, does it?) So is that why hot diamond doesn't glow? I suppose that if one considers real (not ideal) crystals, imperfections, impurities, and the existence of surfaces lead to the possibility of emission mechanisms, and thus glow. In fact it might be the case that a finite but otherwise perfect crystal might have an extremely faint glow. Is this basically the reason that hot diamond does not glow? Further elucidation welcome.


What materials have an electroluminescent threshold in the millivolt or microvolt range?

Does anyone know any materials that have an electroluminescent threshold in the millivolt or microvolt range? Specifically, a stable solid state material. I hope to deposited the material as a thin film. Thank you for the help!


Deriving the curve of a cantilever

Essentially, there is a beam of length L and negligible mass sticking out of a wall with a mass Mg hanging at the end of it. We are given an equation for elastic energy (which I don't think needs to be modified), and we have to come up with the potential energy contribution from the mass, and then hopefully be able to minimize the energy using variational techniques. The elastic energy per unit length can be approximated to be $U[y]=\int_0^L \frac{1}{2} EI (y'')^2 dz$ If I can properly define the potential energy of the mass, I think I can do the rest, but I can't figure out the right term. Thanks for reading, and please comment if there is more I should add, including my work involved getting the (incorrect) gravity terms.


What are the implications of the possibility that the BICEP2 results are caused by a self ordering scalar field transition?

I've found this interesting paper that mentions another possible way to interpret the recent BICEP2 results, and that hadn't been ruled out yet 1. As interesting as the possibility that the BICEP2 team has observed CMB polarization due to gravitational waves resulting from a self ordering scalar field transition sounds, I would like to know what its implications/significance would be!


What is the composition of the universe's population of neutrinos?

I believe earth-based detectors measure mainly solar neutrinos, which have energies on the MeV scale of nuclear physics, are directed from the sun, and have flavors determined by the sun's nuclear reactions plus neutrino oscillations. In addition to this, I recently learned from a physics.SE answer that the universe contains neutrinos that are thermalized with respect to the 3 K background temperature of the universe. I assume these latter can't be detected with present technology...? What is our best present knowledge of the universe's population of neutrinos? Energy spectrum? Flavors? Directions of motion? (Are they isotropic?) Spatial distribution? Are there antineutrinos in the mix? Is the neutrino contribution to the cosmological stress-energy tensor primarily from one component of this population?


"Blue Bumper" Stars

I was recently overviewing various massive compact halo object studies (the Anglo-Australian MACHO collaboration and the French I/II EROS collaboration), and they frequently reference "blue bumper stars," irregular variable stars which produce light curves very similar to gravitational microlensing event. Further searches for more information about them was mostly fruitless, producing a few conference proceedings:   * The MACHO Project LMC variable star inventory: Aperiodic blue variables (code _1995llnl.reptR....P_ )   * The MACHO Project LMC Variable Star Inventory: Aperiodic Blue Variables (code _1995AAS...18710202P_ ),  but little else. Is there any more information on this type of variable star, especially a more rigorous and targeted study of them? Also, how is it possible to get the full text for either of those articles?


Question about electromagnetic Spectrum

I have question related to electromagnetic Spectrum. If energy of photon is $E=m_e c^2$, to which part of the electromagnetic spectrum does it belong ?


How do I place point charges when using the method of images with a sphere?

> We have a point charge $q>0$ in vacuum at a distance $a$ from the center $M$ > of an isolated metal sphere that has a total charge $-q$ and radius $R$. > ($a>R$) > > A) Use the method of images to find a charge configuration that has the same > boundary conditions. (hint: you need 3 charges) For question A I know that my point charge $q$ should stay in the same place and that I need to place 2 other point charges inside the sphere. But I can't figure out what the charge and place of these two charges should be. My book (introduction to Electrodynamics, Griffiths) has an example where the sphere isn't charged but they seem to magically find the answer. They place a new charge $q'=-\frac{R}{a}q$ at a distance $b=\frac{R^2}{a}$ but they don't explain how this is a logical configuration. Can someone please explain this to me, as I would like to know how to place my point charges for different situations and not just one.


Derivation of the oil drop experiment

The oil drop experiment performed by Robert Millikan in 1909 enables us to calculate the elementary charge. The following forces have effect on the droplet:   1. $F_\mathrm{G} = m g = \frac{4}{3} \, \pi r^3 \rho_\mathrm{Oil} g$   2. $F_\mathrm{A} = \frac{4}{3} \, \pi r^3 \rho_\mathrm{Air} g$   3. $F_\mathrm{E} = q E = \frac{qU}{d}$   4. $F_\mathrm{R} = 6\,\pi\eta r v$ However, which steps of calculation are necessary to finally calculate the elementary electric charge (charge of the electron) or at least a multiple of it? $$q = \frac{9 \pi d}{2 U} \sqrt{\frac{\eta^3 (v_1 - v_2)}{\rho^* g}} (v_1 + v_2)$$


How to mathematically formulate the Two Slit Experiment in Special Relativity?

How to mathematically formulate the Two Slit Experiment in a Lorentz invariant framework. Is there a paper about this?


Number operator and Dirac field (with anticommutation relations)

Before using anticommutation relatives the energy, momentum, charge and number operators of the Dirac field have following expressions: $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {Q} = \int \left( \hat {a}^{+}_{s}(\mathbf p )\hat{a}_{s}(\mathbf p ) + \hat{b}_{s}(\mathbf p )\hat{b}^{+}_{s}( \mathbf p )\right)d^{3}\mathbf p, $$ $$ \hat {N} = \int \left( \hat {a}^{+}_{s}(\mathbf p )\hat{a}_{s}(\mathbf p ) + \hat{b}_{s}(\mathbf p )\hat{b}^{+}_{s}( \mathbf p )\right)d^{3}\mathbf p. $$ After using anticommutation relations and "neglecting" of infinite constant $\delta (0)$ (as we can do in a case of free field) in an expressions of energy and momentum operators, they transformed into $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ) \right)d^{3}\mathbf p . $$ But I have a couple of questions in connection with number and charge operators.   1. What can I do with infinite constant into espressions of charge and number operators? For example, $$ \hat {N} = \int (\hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \delta (0) - \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ))d^{3} \mathbf p . $$ Can I neglect summand with infinite constant in a reason that it is connected with infinite "vacuum" energy of field?   2. (Under the condition of neglecting infinite constant). When I acted on one-antiparticle state $|E_{\mathbf k }\rangle = \hat {b}^{+}_{s'}(\mathbf k )| \rangle$ by number operator, I got minus: $$ \hat {N}{b}^{+}_{s'}(\mathbf k )| \rangle = -\int \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p )\hat {b}^{+}_{s'}(\mathbf k)d^{3}\mathbf p | \rangle = -| E_{\mathbf k } \rangle . $$ But $\hat {b}^{+}_{s}(\mathbf p )$ is creation operator (i.e. it was the creation operator before using anticommutation relations; the relations changed this (?)). Did I make the mistake?


How do they measure the rotational speed of cricket balls?

The 2013 Ashes series (a cricket thing between England and Australia) are underway as of today and it seems they have a new (to me) gimmick. It appears that they are now able to measure almost instantaneously (within a few seconds) the rotational speed of balls bowled by spin bowlers. ![enter image description here](http://i.stack.imgur.com/lSivF.jpg) On the (golf) driving range, I was able to get similar readings, but they were from a machine only a few feet away from the tee. I don't quite understand that either, but it seems at least easier. Although there, they were even able to determine the rotational axis as well. ![enter image description here](http://i.stack.imgur.com/tpu8i.jpg) But in cricket they also have a rotational axis along the direction of the ball, which is even more difficult to measure (unless they have a third radar somewhere perpendicular- _ish_ to the pitch). _I think_ I now see that they placed very small black boxes between two of the stumps on either end of the pitch. I guess that must be the first two radars. (One of which will be useless because the batsman is standing in front of it.) ![enter image description here](http://i.stack.imgur.com/KaMpw.jpg) So, essentially: How do they do it? And, how accurate can it be, given that the reflection of the "sides" of the ball may be poor? Or are they using (visible-light) camera images, as suggested by the comments of udiboy and Deer Hunter below? (I'm fairly confident the golf system doesn't do that, but I don't _know_. Anyways, to make sure, the question is primarily about the cricket.) * * * _Added later._ I found that they measure spin rates in baseball as well: http://sportsillustrated.cnn.com/2011/writers/tom_verducci/04/12/fastballs.trackman/index.html. The http://trackman.dk site refers to these technologies and (patent/patent application) numbers: >   * Measuring spin rate of sports balls by radar using multiple harmonic > spectrum traces. (US2009/0075744, EP1698380, DE602006009719.0, > GB/EP/1698380, ZL200680006869.0, JP2008/538085A and KR10/0947898) > >   * Measuring spin axis orientation of sport balls from trajectory > measurements by radar. (US2009/0075744, DE602006015036.9, GB/EP/1853362, > ZL200680006869.0, JP2008/538085A and KR10/0947898) > >


state for a classical particle

> **Possible Duplicate:**   >  Why are coordinates and velocities sufficient to completely determine the > state and determine the subsequent motion of a mechanical system? In classical mechanics, if all the generalized coordinates and the generalized velocities are simultaneously specified, the state of the system is completely determined and one can calculate its subsequent motion. That also means if we know generalized coordinate $q$ and generalized velocity $\dot{q}$, then we can calculate acceleration $\ddot{q}$. How to prove this?


The electron jumps and lets loose photons

Where is the source of the photon. If the photon propagates from within the electrons transit does this point to some sort of field? Does the energy come from a boundary being broken in laymens terms a sound barrier type effect the electron performs on its shell as it changes orbit?


What is the Zero-dispersion wavelength?

There are a decent number of complex mathematical descriptions of the zero- dispersion wavelength online, but I can't seem to find any basic explanations. So to be redundant, what is it?


Is quantum perturbation theory taught in college?

Is perturbation theory usually taught in undergraduate physics, and how much of it is taught in quantum mechanics courses? Also, how much of quantum field theory would be taught in undergraduate physics?


Gravity waves detectors; are they all similar?

Are the gravity waves detectors all working on the same principle/effect ?


Elastic collisions with neutrinos

1: Do neutrinos undergo **elastic** collisions with fermions? 2: Would this imply a **variable** speed for neutrinos? 3: Can neutrinos transfer **momenta** in interactions?


Scale invariance symmetry as a simple argument in an electrostatics problem

In the comments to this post, it was hinted that proving that the force acting on a charge at a vertical distance from a uniformly charged plane is independent of that distance can be done by recalling the scale invariance symmetry. Can someone explain that to me?


How can a rapid change in the volume of a gas cause changes in its temperature?

We were learning about Boyle's law (pressure is inversely proportional to volume of a gas) and in the experiment to prove the law, we were told that we cannot change the volume of a gas too rapidly without affecting its temperature. I have two questions about this:   1. Why does temperature of a gas change when its volume changes very rapidly?   2. This process (rapid changes in volume) is used in the liquefaction of gases. How is this done?


How are physics and computer science getting united?

How is theoretical computer science getting united with physics? Phenomena like Quantum Computing uses Quantum Mechanics to be able to compute things, how are computers helping not just to model our equations but actually predict new equations, helping us to see the computational aspect of nature and how various things are being looked at from new perspectives using Computer Science?


Stability of nucleii and $A=5$

_Why there is no stable nuclei with $$A=5$$ in nuclide the chart and so in nature like we know it?_


Would oxygen in a container condense automatically at night in the exosphere?

If a container filled with pure oxygen and fitted with leak-proof valves were to be dumped into the exosphere at night, would the contained oxygen condense?


Thermal resistance of thermal interface materials?

Thermal conductivity are often used for surfaces between the computer chip and the heat sink to increase heat transfer and they want high thermal conductivity to decrease the thermal resistance. By $$\Delta T=RQ$$ and $Q$ is constant by the chip. When we decrease R, and keep Q constant from the chip, we decrease $\Delta T$ between the chip and the heat sink. This is where I get confused because I assumed we would want the temperature difference between the chip and heat sink to be as high as possible. If the two temperatures are close to each other then wouldn't that make the chip heat up and exceed the operating temperature? I thought we would want the heat sink to be much colder than the chip so the chip will cool. Can someone clarify?


How is the Earth heated by a Full Moon?

While the moon is certainly not a good reflector of solar radiation, surely the radiation it reflects back heats the Earth (even if it is a terribly small amount). How would one go about calculating (or estimating) this heating contribution on a night with a Full Moon?


Death by neutrinos - polonium go home

In Randall Munroe's _What If?_ He is calculating the Lethal Neutrinos dose. > If you observed a supernova from 1 AU away—and you somehow avoided being > being incinerated, vaporized, and converted to some type of exotic > plasma—even the flood of ghostly neutrinos would be dense enough to kill > you. **How do I stay alive to be killed by neutrinos?** Can I pick a large supernova or some other cosmic event, and hide behind a nearby neutron star?


Why are Hermitian operators linked to observables?

In Quantum Mechanics, why is it that a self-adjoint operator is linked to an observable? What makes it measureable? And why isn't a non-Hermitian operator linked to an observable? Also, what type of observables are we talking about here? Particles?


Neutrino path bent by gravity?

> **Possible Duplicate:**   >  Neutrino unaffected by gravity Other probably closely related questions here: Link Neutrinos unaffected by gravity   Link Superluminal Neutrinos My question is subtly different (I hope). Gravitational lensing demonstrates that light is bent by massive bodies. If neutrinos and light are bent by the earth's gravity would we be able to measure that? I suppose we would need to take measurements of the difference between traveling into/out of our gravity well and across the gravity well. A good deal of discussion is going on about the Gran Sasso experiment and one of the points that keeps being mentioned is the time it took for neutrinos to arrive from a supernova explosion. The light/neutrinos from the supernova are traveling **into** our gravity well and will therefore not be bent by the earth's gravity while the Gran Sasso neutrinos are traveling **across** and will be. What _should_ the difference be?


Rotation of a slipping ladder

Imagine a ladder leaning against a wall. All surfaces are smooth. Hence the ladder will slip and fall. While falling it rotates because there are external torques acting on it. My question is about which axis does the ladder rotate?


Do nanoscopes exist?

We are mostly all familiar with a microscope, and know that it helps to see MICRO components, like stuff that is photolithographically etched on silicon semiconductor die. (The latter can also be nano, but let's bide that.) Anyways, we rarely hear of nanoscopes, i.e. a lens that can digitally reflect traces of nanometer objects, like quarks, hadrons, and valence electrons. I want to purchase a good microscope, but that will only let me see cells, substrate micro-components, etc. I want to see the actual atoms that make up the silicon, the atoms that make up the doped logic gates in my tablet, the atoms that make up my arm, and the atoms in the air that surrounds our very demanding atmosphere of carbon dioxide! So I ask again, with endless respect for science, do nanoscopes exist, and if they don't, how close can I get to the atoms, quarks, and hadrons, and how do I tell them all apart!


Work done by introducing a spin in supersposition into a Magnetic Field

A spin is created in a superposition of up and down states. A magnet is moved very slowly, towards the spin. What is the work done by the magnet. It may be helpful to imagine that the magnet is connected to a spring that expands or contracts depending on work done by the magnet.