Fock matrix elements for RHF formalism

Here I refer to a particular book _Molecular Quantum Mechanics_ by Peter W. Atkins and Ronald S. Friedman, but similar derivation could be found in many other texts. So, when obtaining the explicit form of the Fock matrix elements for RHF formalism (p. 295 in 4th edition), authors go from ![enter image description here](http://i.stack.imgur.com/GtcVE.png) to ![enter image description here](http://i.stack.imgur.com/kGKFJ.png) just by mentioning that $\psi_u$ is expanded as a linear combination of basis functions $\theta$. The only way I could go from the first equation to the second one is by expanding the same spatial orbital $\psi_u$ on the left and on the right sides of integrand expression, i.e. before and after $1/r_{12}$, _differently_ , $$ \psi_u = \sum_l c_{lu} \theta_{l} \quad \text{"on the left side"} \, , \\\ \psi_u = \sum_m c_{mu} \theta_{m} \quad \text{"on the right side"} \, . $$ Is it true? And if yes, why on earth this should be done this way?


Amount of 2 amperage

I have 2 equation describing the alternating amperage $I_1$ and $I_2$. I need to get amount of these amperages. My equations: $$I_1=10\sin(\omega t+30)$$ $$I_2=20\sin(\omega t-50)$$ How can i make it? I forgot. Thank you.


Exact summation of a sub-class of diagram: do we know the exact solved problem?

In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re- summation of diagrams is _exact_. I will keep italics to refer to this _exactness_. Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By "problem" I obviously mean the eigenstates/eigenvalues problem of a secular equation given by a Hamiltonian/Lagrangian. My question is the following: **do we know the problem we _exactly_ solved summing only partial class(es) of diagram ?** I mean: can we write out an effective Hamiltonian/Lagrangian that our _exact_ summation is the _exact_ solution of ? Can we at least know the _exact_ sub-space (eigenstates and/or eigenvalues), the solutions we found _exactly_ belong to ?


Is the Fock state a superposition of product states?

I know that in principle, the Fock state is not the same as a product state because if it were, the photons would be uncorrelated. What I don't understand is, can it be expanded in product states? For example, if two photons share the same mode, is the state $|2,0\rangle$ the same as the product $|1,0\rangle|1,0\rangle$? If they are in different modes, is the Fock state $|1,1\rangle$ the same as superposition of two Fock states, $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ (with $a$ and $b$ labeling the photons)? If yes, then is this superposition, in turn, the same as $\frac{1}{\sqrt{2}}\left(|1_a,1_b\rangle+|1_b,1_a\rangle\right)$ ?


Can all canonical transformation be obtained through generation function approaches?

The question can be formulated as following: Suppose $$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$ $$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = P(p,q,t), Q = Q(p,q,t)$$ is an invertible transformation. Can we prove that there must exist a $\lambda$ and function $G(p,q,t)$ (or $G(p,Q,t)$, $G(P,Q,t)$, $G(P,q,t)$), such that $$\lambda[p\cdot \dot{q} - H(p,q,t) ] = [P\cdot \dot{Q} - K(P,Q,t) ] + \frac{dG}{dt}~?$$


What are the strongest sources of collimated neutrons and protons?

I am imagining an unusual experiment which will require intense beams of either protons or neutrons. The experiment would work better with neutrons, but neutron sources are much ~~weaker~~ messier so I am considering both. I am interested in beams of almost any non-relativistic energy. I am not sure yet about how focused I will need the beam to be. I know this is a beginner question, but what are the basic options here for the strongest source (as measured by neucleons per second)? * * * Neutron apparently come from two sources: reactors and accelerators. (As Sir Patrick Stewert can attest.) In the latter case ("spallation"), a proton beam is just slammed into a heavy-metal target and the neutrons that fly out are caught. It appears that the highest intensity protons beams are often used to produce the highest intensity neutron beams in this manner. Saclay High Intensity Proton Injector Project (IPHI): $6 \times 10^{17}$ protons/second. Low-Energy Demonstration Accelerator (LEDA): $6 \times 10^{17}$ protons/second. (These agree with Andre's chart below showing 100 mA. Note also that the LHC current is about 500 mA, although the protons are grouped into bunches. And as Andre pointed out, the LHC circulates the same protons and so could not provide a continuous 500 mA beam into a target/experiment that consumed them.) Neutron sources, being the result of either fission or spallation, are messy. I still don't have a good handle on what kind of neutron intensities are possible for what momentum ranges. The Lujan Neutron Scattering Center at LANCE is a neutron source for many experiments. * * * In spallation, the number of neutrons produced per incident proton is usually on the order of 40, but the phase space distribution is much wider than the incident protons. (Protons beams can be directly focused using magnets.) According to a CERN report by Lengeler, reactors typically can produce thermal neutrons (~0.025 eV) fluxes of order $10^{14}$ neutrons/cm${}^2$s, whereas spallation sources can exceed $10^{17}$/cm${}^2$s. * * * It is probably best, then, to think of continuous neutron sources as phase- space-density limited, with the above numbers giving a rough estimate of the currently achievable densities. Apparently, spallation sources are limited by the need to dissipate the heat from the incident proton beam. If high fluxes are needed only briefly, laser pulses can produce $4 \times 10^{9}$/cm${}^2$ over a nanosecond, i.e. an instantaneous flux of $4 \times 10^{18}$/cm${}^2$s. As for protons, it's worth noting that the high-intensity 50 mA beam at the European Spallation Source is limited by "space charge effects at low energy, by the power that can be delivered to the beam in each cavity at medium and high energies, and by beam losses."


TKNN invariant changes due to continuous deformation of parameter space

Naively, I would assume that a topological invariant remains invariant under continuous deformations of whatever space the invariant belongs to. In the case of topological insulators, this space is the space of parameters characterizing the material and its environment. If we consider the magnitude of a magnetic field incident onto a material supporting metallic edge states a parameter, then why does the Chern number (TKNN invariant) change when $|\textbf{B}|$ is increased smoothly and slowly? Since the Hall conductivity experiences sharp increases due to slow increase of magnetic field, the Chern number must be increasing sharply (by integer jumps), signifying sharp changes in the topology of the parameter space. What parameter is changing so drastically in response to the increasing magnetic field?


Probability distribution of two particle types system

Suppose that particles of two different species, A and B, can be chosen with probability $p_A$ and $p_B$, respectively. What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? I'm trying to apply the Binomial distribution here but am bothered by the fact that it applies for N trials (whereas here we only do 1).


Why the free energy is called 'free'?

The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \begin{equation} e^{-\beta F} = \mathcal{Z} = \sum_{\\{configuration\\}} e^{-\beta E(configuration) } \end{equation} where $\beta = 1/k_BT$ and $E$ is the energy of a certain configuration. Now my question is: **Why the free energy is called 'free'? I mean it is free in which sense?**


Remaining Potential Experimental Particle Physics Discoveries at the TeV Scale?

With the discovery of the Higgs Boson, some have been calling it the end of experimental particle physics for our generation, due to the fact that all of the particles predicted by the standard model have been found (the W and Z Bosons in 1983 and the Top Quark in 1995) and no new physics is expected at the energy ranges we can probe. I have also generally heard from my friends who work in experimental particle physics (or, at the very least, accelerator physics) a similar story, that there isn't too much left to be discovered. Is this an accurate assessment? What about dark matter particle models, like the axion/axino? Can the LHC (or SLAC, etc) probe enough of a range to perhaps shed light on this cosmological problem? Can we hope for any more insights into beyond the standard model theories from current accelerators?


Efficiency of Metropolis algorithm

Context is 1D Ising model. Metropolis algorithm is used for simulate that model. Among all possible spins configurations (states) that algorithm generates only states with the desired Boltzmann probability. Algorithm chooses spin at random and makes a trial flip. If trial satisfies certain conditions related to Boltzmann probability, flip is accepted. Otherwise flip is rejected and system is unchanged. Define "acceptance ratio" as a percentage of accepted trials. Simulation shows that acceptance ratio is higher on higher temperature (behave as increasing function of temperature). Questions:   1. Why Metropolis algorithm is not efficient at low temperatures?    2. Is efficiency of algorithm related to acceptance ratio?


Why is moonlight less colourful than yellow light?

Just what the title states; and this isn't an original question ... On full-moon, moonlight is almost as bright as mid-dawn ; yet there is hardly any colour visible. This is not the case with sunlight. I'm curious to know why this is the case


Where did the idea of a neutrino come from?

Exactly what paper by W. Pauli introduced the idea for the existence of a neutrino and how was its existence confirmed experimentally (who did that and in what paper)?


Planetary Gravity and its effects

This is my first question on the Physics portion of Stack Exchange. I was hoping to get some light on the topic of gravity. I don't have much background knowledge of physics so I might as well start here. I have a few planet and gravity related questions: **1).** Does planetary gravity depend on size or density? Or both? **2).** Is it plausible for a large (very large), terrestrial planet to have tall mountains like Mount Everest or even like on Mars? **3).** Is there a way, given an average density per cubic meter along with the average radius of the planet, to calculate the pull of gravity? I've been scouring the internet for these things with no luck so far. The idea behind this is to make a planet rendering program which takes all of this into effect. If this forum doesn't answer this sort of thing, feel free to point me in the right direction. Thanks!


is this generalized Hawking radiation formula right?

Look at equation 11.2.17 in this page. The expression is: $$ T = 10^{-5} \text{K m} \frac{\xi}{\frac{GM}{c^2} \lbrace \frac{GM}{c^2} + \xi \rbrace - e^2 }$$ where $$ \xi = (r_s^2 - a^2 - e^2)^{1/2}$$ and the usual parameters $$ r_s = \frac{GM}{c^2} $$ $$ a^2 = \frac{L^2}{M^2 c^2}$$ $$ e^2 = \frac{Q^2 G}{4 \pi \epsilon_0 c^4}$$ This formula is supposed to describe the temperature of a black hole with angular momentum $L$, charge $Q$ and mass $M$ > **Question:** is the above formula correct? I'm trying to find the limit temperature for $a=0$ and $e=r_s$ The temperature expression could be simplified as $$ T = 10^{-5} \text{K m} \frac{\xi}{\xi^2 + a^2 + \frac{GM}{c^2} \xi }$$ if $a=0$, $$ T = 10^{-5} \text{K m} \frac{1}{\xi + \frac{GM}{c^2} } $$ so when the black hole has extremal charge, $\xi=0$ and the temperature looks like the normal black hole temperature for the Schwarzschild black hole, which looks very wrong Any idea where is the mistake? I was expecting the temperature of the charged extremal black hole with $a=0$ to be infinite


why does water do that?

I found recently that water fills a bottle quicker through only observation if I hold the tube away from the filling water than if I leave the tube inside the bottle. why does that happen? does potential energy have anything to do with this? ![enter image description here](http://i.stack.imgur.com/VUXJU.jpg)


Is traveling at the speed of light the same thing as teleportation?

If I were on one side of the room and moved at the speed of light to the other side of the room, to an observer it would appear that I teleported. If time stops at that speed, it would be instantaneous but we know that light is not instantaneous (see light years). So I'm curious what the difference between traveling at the speed of light and teleportation is. I think teleportation has to do with deconstructing objects and then reconstructing them which allows them to travel through things etc., but is there any other noticeable difference?


Question about exterior derivatives

I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge dx^{n-1}=\frac{1}{n!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_1}\ldots dx^{\mu_n}$$ Now, I have an expression that is given in spherical coordinate system, where I have $$\int_\Sigma f(\theta,\phi) d\theta\wedge d\phi$$ when I want to integrate this (the epsilon part is already computed), do I just have $\int\int d\theta d\phi$ to integrate, or do I need to put the part from integration in spherical coordinate system $\int\int\sin\theta d\theta d\phi$? I haven't done much integration that involved forms before, so any help is appreciated :) EDIT: From the article I have: $$k_\xi[h,\bar{g}]=k^{[\nu\mu]}_\xi[h,\bar{g}](d^{n-2}x)_{\nu\mu}$$ $$(d^{n-p}x)_{\mu_1\ldots\mu_p}:=\frac{1}{p!(n-p)!}\epsilon_{\mu_1\ldots\mu_n}dx^{\mu_p+1}\ldots dx^{\mu_n}$$ $$k_\xi^{[\nu\mu]}[h,\bar{g}]=-\frac{\sqrt{-\bar{g}}}{16\pi}\ldots$$ where $\ldots$ is an expression. Does this mean, since I have an $n-2$ form and I'm in 4 dimensional space, I need to include this $\sin\theta$ after all? EDIT2: I need to add the $\sin\theta$. I got it. Thanks :D


2 Event Horizons come into contact

Just a question out of curiosity, what would happen if the event horizons of 2 black holes of the same mass were to come into contact? Would both gravitational accelerations be canceled where the event horizons overlap?


Air insulation cavity depth: rule of thumb to avoid convection

I want to know more about optimising an air gap between two surfaces, such as window panes or a building's walls. I was taught that as a general rule, a 6cm air gap is the optimum. Any larger and convection counters the effectiveness. But something tells me this is hardly an accurate figure! Can anyone improve on this rule of thumb? What are the most significant variables which would alter the optimum separation distance? Examples of what I would want to apply this to are:   * A building's window panes of different sizes: glass, perspex or combinations   * Oven / furnace surfaces of different sizes: between two ceramic refractory walls, or refractory walls and steel. (Temperature differentials and absolutes of perhaps up to 1000 degrees C) Perhaps there are some approximated formulas to calculate optimum separation distance? And if the effective thermal conductivity of the air changes, a formula for that too. Thanks! For interest to other readers, this question is related.


Eigenfunction associated with the $\hat{x}$ operator

Consider the following operator $\hat{x}=i\hbar \frac{\partial}{\partial p}$. I am trying to show that the eigenfunctions of $\hat{x}$ are not square- normalizable. I am interested in doing so since theoretically, we notice that the eigenfunction of the momentum operator $\hat{p}=\frac{\hbar}{i}\frac{\partial}{\partial x}$ is not square- normalizable. How do we set up the equation for the $\hat{x}$ operator?


Fourier series of single tone modulated wave

When a single-tone continuous modulating signal modulates a sinusoidal carrier, isn't the modulated wave periodic? If so, can't we apply fourier series and determine the harmonic frequency components instead of finding the frequency components using fourier transform?


How does a speaker produce sound?

What I read is that a speaker produces sound by the movement of a coil attached to a cone which moves back and forth. So, If I try to move the coil by hand, would it produce sound? If not, why? or Why would it produce sound only when the current moves the coil?


What is the Jost function in scattering theory? Is it an operator or some kind of determinant? How is it obtained?


Zero-zero (lower indicies) term for affine connection ($\Gamma_{00}^\lambda$), why do some terms dissapear?

More simply a tensor algebra question, but in General relativity I have the following when I calculate $\Gamma_{00}^\lambda$:- $$ \Gamma_{00}^\lambda = \frac{1}{2}g^{\nu\lambda}\left( \frac{\partial g_{0\nu}}{\partial x^0} + \frac{\partial g_{0\nu}}{\partial x^0} - \frac{\partial g_{00}}{\partial x^\nu} \right) = -\frac{1}{2}g^{\nu\lambda} \frac{\partial g_{00}}{\partial x^\nu} $$ Why are the $\frac{\partial g_{0\nu}}{\partial x^0}$ terms zero?


Semiclassical description of EM waves reflection from metallic surfaces

Imagine an EM wave impinging on a metal. Fresnel's formulas tell us that no wave can propagate through the metal, or that the transmitted field is an evascent wave with some penetration depth dependent on the refraction index of the metal. If we now would like to zoom into the microscopic physics of the reflection, we would have to take into account the crystalline structure of the metal, which imposes a certain electronic band-structure, which in turn determine the way the electrons respond to the external EM perturbation. We should calculate the reflected field from the back action of the electron motion on the incident field. Do you know of a semiclassical way of solving this problem? Is it feasable to compute numerically a set of coupled Maxwell-Schroedinger equations?


What's The Minimum distance?

My Friend gave me a question today. The question was.:: > We have a point A. At a distance of $x_0$ from the point. There is a > particle$(P_1)$. Also, a particle is present at the point A $(P_2)$.The > Particle $P_1$ moves with a velocity of $u_1$ towards Point A. At the same > instant the particle $P_2$ moves making a angle $\theta$ with $P_1A$ with a > velocity $u_2$. So what will be the minimum distance between the two > particles when both of them start at the same time. So can anybody help me with this problem. I tried decomposing it into vectors, used $\tan\theta$, but it did not give me a correct answer.


Why can't helicopters reach mount everest?

Is there a reason why people can't just take the helicopter to mount Everest? Why is it that helicopters can't reach that high?


Wheatstone bridge galvanometer error

We had to measure the resistance of $R_x$, we balanced the Wheatstone bridge and did calculations. My question is: we didn't include galvanometer error into calculations. Why is that? I read that it's very precise, but that doesn't seem like a good enough explanation in exact sciences :/ Edit: The precision is not the case as I was told, I need to go into more detail. When I measured, I set a value on the potentiometer, then I adjusted the adjustable resistor, so that the galvanometer would show zero. Is the error somehow compensated? ![enter image description here](http://i.stack.imgur.com/yODpD.png) ɛ - electromotive force. K - switch. R - adjustable resistor. C - potentiometer.


When can photon field amplitudes be written as field operators?

Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as ${A}_1=\alpha f(A_1,A_2) \\\ {{A}_2}=\beta g(A_1,A_2) $ Under what conditions can I make the replacement $A\rightarrow \hat{A}$ ? I am uncomfortable with the Bohr correspondence principle because the correspondence principle seems to be a very weak argument. Edit: $z$ is position, $\alpha$,$\beta$ are real numbers. $f$ and $g$ are functions of the amplitudes.


How can I make a dry dilution refrigerator quiet?

The pulse tubes are really loud. We've wrapped the helium lines in sound absorbing foam, and we have a cage around the top of the cryostat also lined with foam, but it's still pretty loud. Are there any really good and practical ways to damp the sound?


Normal distribution of x, xdot

I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, the y-axis is xdot. ![x-xdot-histogram](http://i.stack.imgur.com/rQfWk.png) Is there some well known process which exhibit this property? What kind of dynamic equations on x(t) cause this?


How do we know that the nucleus isn't a quark-gluon plasma?

The standard picture of the nucleus of atom is that is several distinct nucleons, which themselves are composed of quarks. However, it seems to me like a much simpler picture is that the nucleus is directly made out of quarks, without having nucleons as substructures. That is, that the nucleus is the ground state of a quark-gluon plasma. I can't find any evidence that this is false. For instance, if the distances between the nucleons was significantly larger than the size of the nucleons, there would be a difference in size between hydrogen and other atoms which can't be explained by this new framework, but according to wikipedia the diameter of all of the nuclei is the same order of magnitude. What is the evidence that large nuclei have nucleons as substructures?


Small angle approximation for coupled pendulum

I am looking at a coupled plane pendulum problem where the pendula are identical with mass $m$ and length $r$, and connected by a spring with spring constant $k$ and an equilibrium seperation $d_0$. If $\theta_1$ and $\theta_2$ are the angular displacement with the vertical then in these coordinates then the Lagrangian (I believe) is $$L=\frac{1}{2}mr^2 \left( \dot{\theta_1^2} + \dot{\theta_2^2} \right) -mgr \left( \cos \theta_1 + \cos \theta_2 \right) -\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2.$$ I'm trying to rewrite the approximate Lagrangian for small oscillations about the bottom in terms of $\eta_1$ and $\eta_2$ where $\eta_{\alpha}$ represents the horizontal displacements. I'm not sure how to go about doing this. Am I supposed to assume that $\theta_1=\theta_2=0$ is an equilibrium point and then expand the potential to second degree? In order to do this, can I assume that my $-mgr\left(\cos \theta_1 - \cos \theta_2 \right)$ can be approximated as $$-mgr\left(\cos \theta_1 - \cos \theta_2 \right) \approx mgr\left(\frac{\theta_1^2 + \theta_2^2}{2} \right)$$ and then for the spring terms write $$\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2=P(\theta_1, \theta_2)$$ $$\frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2 \approx \frac{\partial^2 P}{\partial \theta_{\alpha} \partial \theta_{\beta}}\bigg|_{\theta_1=\theta_2=0} \theta_{\alpha} \theta_{\beta} $$ Then finally combine all like terms? Or before doing this can I simplify the root by writing $\sin \theta_{\alpha} = 0$ before doing the partial derivatives? I'm not sure which process is correct here. Any help would be appreciated. Thanks. **\------------------------EDIT----------------------------** If I write $$V(\theta_1, \theta_2) = mgr \cos \theta_1 + mgr \cos \theta_2 + \frac{1}{2}k \left[ \sqrt{\left(r \sin \theta_1 - r \sin \theta_2 \right)^2 + \left(r \cos \theta_1 - r \cos \theta_2 \right)^2} -d_0\right]^2$$ then according to my calculations $$\frac{\partial^2 V}{\partial \theta_1^2} = -\frac{1}{2}kr(2r-d)-mgr = \frac{\partial^2 V}{\partial \theta_2^2}$$ $$\frac{\partial^2 V}{\partial \theta_1 \partial \theta_2} = -\frac{1}{2}kr(2r-d_0)$$ Can I therefore write the Lagrangian as $$L \approx \frac{1}{2}mr^2 \left( \dot{\theta_1^2} + \dot{\theta_2^2} \right) + \frac{1}{2} \left[ \frac{1}{2}kr(2r-d_0) + mgr \right] \left(\theta_1^2 + \theta_2^2 \right)+\frac{1}{2}kr(2r-d_0) \theta_1 \theta_2$$ and then proceed to find the vibrational frequencies and normal modes using the K-M matrix procedure (after making the $\eta$ substitutions)? I'm wondering if there are any simplifications I can make to the potential (i.e. treat the approximation of the potential in multiple separate parts where I take derivatives of each individually) before I start taking partial derivatives to simplify the problem. Or does a multidimensional potential such as this always have to be treated as one massive beast?


Finding Max Radius of Propeller (angular velocity)

I am looking at a question from University Physics ![enter image description here](http://i.stack.imgur.com/loQo7.png) The given answer ![enter image description here](http://i.stack.imgur.com/cgP7p.png) Whats the intuition behind using the below diagram ![enter image description here](http://i.stack.imgur.com/BX14V.png) of finding $v_{tip}$? I was looking the the $v_{tan}$ not knowing how to continue. Also $v_{tip}$ will rotate and turn positive and negative


Andrew's experiment

In Thomas Andrew's experiment, consider the dome shaped saturation region. If we increase the pressure at constant volume until we reach the critical point, why does the density of vapours rise and the density of liquid fall? Moreover, why will the meniscus or the separation boundary of two phase slowly become less distinct and finally at critical point disappear?  ![enter image description here](http://i.stack.imgur.com/LXo52.gif) Will vapour pressure increase or decrease as we move vertically up in the P-V plane in the saturation region?


Online magnetic field solver

Is there a site that allows you to give the approximate specs of a set of magnets and then it produces the graph of the magnetic field for you? Here is a situation I try to understand: I have a set of rectangular magnets about 1*0.75*.25 inches with north and south on large faces. The magnets are glued in two long lines (like dominoes) about 1 inch apart, all north sides are up. In the area between the magnets there is a strong magnetic field, as if a magnet with opposite polarity is in that places (as if a south pole is there). Why does that happen?


Gravity Question Can this be reopened?

Anyway what I am trying to figure out is something I have been thinking about for many years. The dinosaurs didn’t die from a big explosion or anything just that the earths gravity increased and made them less competitive to smaller creatures. I have heard bone mass decreases for astronauts so I am think it must increase if gravity increased. Also I am not looking for a number but an understanding of what a creature would weigh with the increase of matter that has been hitting the earth over time. The 37,000# number is what I found on the internet as a theory to what is being added to the earth ever year so over 60 million years there must by an increaseHope someone can help


Why does kinetic energy increase quadratically, not linearly, with speed?

As Wikipedia says: > _[...] the kinetic energy of a non-rotating object of mass $m$ traveling at > a speed $v$ is $mv^2/2$._ Why does this not increase linearly with speed? Why does it take so much more energy to go from 1 m/s to 2 m/s than it does to go from 0 m/s to 1 m/s? My intuition is wrong here, please help it out!


Is acceleration quadratic or linear outside of a gravity field?

So as I understand it: If you hit a wall at twice the speed you hit four times as hard? I often ask people this and everyone says "at twice the speed you hit twice as hard". $K=\frac{1}{2}mv^2$ is very counter intuitive. I have read an explanation that seems to say gravity is acceleration and that's where the squared comes from. Do we have anyway of knowing if the kinetic energy of an object outside of a gravity field is linear or quadratic? Could the gravity field somehow match the acceleration to give the $v^2$ Bonus question: If you get pulled over for speeding should you ever ask the policeman why kinetic energy is quadratic instead of linear? $K=\frac{1}{2}mv^2=$($300) dollar fine where I live.


How - The force of a 60 mph crash is not just twice as great as a 30 mph crash; it’s four times as great!

The DMV manual says that > _The faster you go, the less time you have to avoid a hazard or collision. > The force of a 60 mph crash is not just twice as great as a 30 mph crash; > it’s four times as great!_ My physics is quite rusty, so I could not figure it out. I guess the above statement is correct, but how do we prove it? **Edit** I figured this out myself, but alternative methods or new ways of understanding are still welcome.


Newton's Third Law of Motion

I want to start off by saying that I've looked around for other explanations, but I've not really found any satisfying ones. My question is basically the whole "Why can anything move at all?" question, with a little twist. Refer to this picture: ![](http://i.stack.imgur.com/bmhkv.png) Now I understand that opposite forces act on different objects, so in the top one, the rocket would have a net force of a 100 N to the right, and accelerate, as per Newton's Second Law. However, in the bottom example, wouldn't the box push back on the rocket, equal and opposite, as the rocket pushes on the box, therefore cancelling the forward push from the exhaust, making the rocket not move at all, whilst the box gets a net force of a 100 N? This isn't true obviously, but why not? Also, in this earlier Phys.SE post there is a great answer depicting a finger and a matchbox. Now this post is very related to mine. I'm wondering why in that picture, the force of the finger pushing on the matches doesn't equal the force of the muscles pushing forwards in the finger? Surely when I'm having a force pushing the finger forwards, that same force applies to the matches?


How to calculate the Gibbs free energy out of a given Entropy?

I'm stuck in a small calculation. I try to solve a small exercise. You should calculate for a given entropy the Gibbs free energy, where the entropy is given as $S\left(U,V\right) = \frac{4}{3}\left(\alpha V U^3\right)^{1/4} \qquad\dots\qquad U$ is the internal energy, $V$ is the volume and $\alpha > 0, \text{const}$ The exercise say something like "you get a surprisingly expression for the Gibbs free energy". * * * Basically I have to use the definition of the Gibbs free energy: $$ \begin{equation} G\left(T,p\right) = U\left(S,V\right) - T\cdot S + p \cdot V \label{eqn:gibbs} \end{equation} $$ The question is now, how can we exchange the volume $V$ and the entropy $S$ in the equation above. My idea was to calculate $p=p\left(T,V\right)$ and transpose it to $V=V\left(p,T\right)$. The thing is, that I get for the pressure $p$ something like: $p = \frac{\alpha}{3}T^4$, which is not $p = p\left(T,V\right)$ and it's not possible to find a relation for $V=V\left(p,T\right)$. I thing, I'm making some mistake. Can someone help me?


Calculating the force of a mass bearing down on an object

I need to calculate the force of a weight bearing down on a smaller object, constraining it from expansion. The weight bearing down has a much bigger surface area than the smaller object. I know the mass of the weight. I know the surface area of the smaller object. What else do I need to know in order to calculate the force bearing down on the small object? What I mean is, what do I need to know in order to calculate what the force bearing down on that smaller object is? I think I might need to know about the acceleration of gravity (9.8 m s^-2), but I'm not sure how that applies if I don't know the dimensions of the larger object. I know that F=ma, but I can't visualise how that applies to an object bearing down on an object with a smaller surface area.


Is Dirac's description of a photon in a split beam still seen as correct today?

This comes from the _Interference of Photons_ section in the book _The Principles of Quantum Mechanics_ by P Dirac: > We shall discuss the description which quantum mechanics provides of the > interference. Suppose we have a beam of light which is passed through some > kind of interferometer, so that it gets split up into two components and the > two components are subsequently made to interfere. We may, as in the > preceding section, take an incident beam consisting of only a single photon > and inquire what will happen to it as it goes through the apparatus. This > will present to us the difficulty of the conflict between the wave and > corpuscular theories of light in an acute form. > > Corresponding to the description that he had in the case of the > polarization, we must now describe the photon as going partly into each of > the two components into which the incident beam is split. The photon is > then, as we may say, in a translational state given by the superposition of > the two translational states associated with the two components. We are thus > lead to a generalization of the term "translational state" applied to a > photon. For a photon to be in a definite translational state it need not be > associated with one single beam of light, but may be associated with two or > more beams of light which are the components into which one original beam > has been split. In the accurate mathematical theory each translational state > is associated with one of the wave functions of ordinary wave optics, which > wave function may describe either a single beam or two or more beams into > which one original beam has been split. Translational states are thus > superposable in a similar way to wave functions.   1. Is this the view held by leading physicists today?   2. Can we really talk of the incident and exiting photon in the beam splitter being the same photon?   3. Is Dirac saying that the photon is partly in the two split beams, but no where else in space?


Is an electric lamp a transducer?

Silly thought. A transducer, by definition, is a device that converts variations in one form of energy to another. An electric lamp converts electricity into visible light - the brightness may vary depending upon the electric potential applied. Is it correct to state an Electric Lamp is a transducer?


what property of steam makes it the preferable motive fluid in jet ejectors? why not air or nitrogen?

I want to know what properties of steam make it the fluid of choice in   1. steam turbines   2. jet ejectors. I want to understand the mechanism of energy conversion in these equipment and hence understand why steam is the preferred choice. Please leave out economic factors for now and keep your responses rooted only in physics/thermodynamics.


Dual particle-wave behavior

If electrons and photons, and possibly more particles, exhibit dual character, why don't physicists create a new classification for them? Why describe them as both waves and particles. Why not rather create a new classification, which accurately describes their behavior?


Term for "atmospheric ricochet" due to wrong "angle-of-attack"

I watched "Apollo 13" yesterday, and they had the "angle-of-attack" problem that had to be manually solved, to prevent the ship from " _ricochet[ing] off the atmosphere like a rock skipping off a pond_ ". I found and read this, but couldn't manage the term to further learn about this "ricochet". **Is there a term that reflects this "ricochet"?**


Effect of Current on spring

When Current passes thru a spring , some books mention that it gets compressed. However, I think due to the heating effect of current, molecules will increase kinetic energy and the spring should get elongated. What is the physical mechanism for the compression instead of elongation?


What is the difference between negative acceleration and decreasing acceleration?

According to Reisnick and Halliday, retardation is the acceleration in the negative direction of time axis ie. opposite to the motion. Decreasing accleretion is the decrease in the magnitude of time rate of change of velocity. Am I correct or not? What is the $v$-$t$ graph with retardation and decreasing acceleration? [I don't think they are same.]


Accelerating an elevator

A person is standing on a weighing scale in an elevator in upward acceleration. Let $N$ be normal reaction force exerted by the weighing scale to the person (upward). It is known that the person will experience a normal reaction force ($N$) larger than his own weight (larger reading on the weighing scale). While $N$ is equal, but in opposite direction, to the force that the person exerts to the weighing scale due to action-reaction pair (the person and the weighing scale). That means other than the person's weight, there should be an extra force exerting to the weighing scale downward ($A$). So $-N$ (upward)$ = mg + A$ (downward). I can't figure out where does $A$ come from?


What are relativistic and radiative effects (in quantum simulation)?

I'm reading about Quantum Monte Carlo, and I see that some people are trying to calculate hydrogen and helium energies as accurately as possible. QMC with Green's function or Diffusion QMC seem to be the best ways to converge on the "exact" solution to Schrodinger's equation. However, if one wants to be very exact, then the Born-Oppenheimer approximation must be removed. A lot of papers mention that the results are still not exact enough, and must be corrected for relativistic and radiative effects. I'm pretty sure I know what relativistic effects are -- the non-relative Schrodinger's equation cannot account for GR as particles approach the speed of light (or even small but measurable effects at lower speeds). But what are radiative effects? And I would think you would include these two things into your QMC calculation instead of applying a post-simulation correction factor if you wanted to be ultra-precise (e.g. use the Dirac equation instead for relativistic effects). So why don't most researchers do this? Does it raise the calculation time by orders of magnitude for an additional 4th decimal place of accuracy? Finally, is there anything at a "deeper" level than relativistic and radiative effects? In other words, if I left a supercomputer running for years to compute helium's energies without the BO approximation, and with relativistic and radiative effects included in the MC calculations, would this converge on the _exact_ experimental values? (Actually, I just thought of one such left-out factor -- gravity ... and might you have to simulate the quarks within the protons individually? Anything else?)


Find Angular Momentum about any point

How do I find the angular momentum of a body about any point?   We know that $L=I\omega$ for a body rotating in space, where $L$ denotes the angular momentum, $I$ denotes the moment of inertia and $\omega$ denotes the angular velocity. However, this is **only applicable** for **Fixed Axis of Rotation, Instantaneous Axis of Rotation and Center of Mass**.   Can somebody state and prove the value of the angular momentum of a body about any point? (If a formula for that exists).   Thanks!


Does an uneven and/or moving surface affect quality when using a solar filter?

I have chosen to make my own solar filter using Baader as opposed to Mylar or anything else based on these: http://physics.stackexchange.com/a/25204/6805 and http://irwincur.tripod.com/mylar_vs__baader.htm and others. My last solar filter was the product which is no longer available direct from Celestron (I'm not even sure what it was made with). **My questions** : Does "flapping in the wind" baader effect image quality? See the images below to understand what I mean... Is it possible to achieve a better taut fitting when making my own? As seen here ![enter image description here](http://i.stack.imgur.com/BcsMd.jpg) it is not flat or tight like this glass one is ![enter image description here](http://i.stack.imgur.com/EHj3t.jpg)


What happens when a lightning rod is not earthed?

A Lightning rod is fixed to the building and it is not given earthing then what happens? How does the lightning reach the ground through the building?


Moving conducting bar in (changing) magnetic field

In my exam today I've been given this problem, yet even with the results at hand I simply can't warp my head around it; Given the picture below, a bar is placed on two conducting rails with a resistor (in the form of a single thin wire) at the left. Now at t=0 the magnetic field changes which causes the bar to move. (and a current flows through the "circuit"). The current can easily be found by Lenz's law: $$ \mathscr{E} = -\frac{\mathrm{d}\Phi_b}{\mathrm{d}t} = -\ell\frac{\mathrm{d}\left( Bx \right )}{\mathrm{d}t} = -\ell x\frac{\mathrm{d} B }{\mathrm{d}t} -\ell B v $$ ![enter image description here](http://i.stack.imgur.com/TnAlF.png) Now this is follow up text for the problem > It may be assumed that the conducting bar **has a negligible mass and > friction** , so it will immediately move to a position $x$ in which there is > no force exerted on it. > > b) Explain why the relation between the actual position $x$ and the actual > value of the magnetic field $B$ is given by $Bx = B_0x_0$. What did the prof try to explain here? No force means constant speed, so not a "position" at all right?   Well the result sheet shows: > No force means $\frac{\mathrm{d}(Bx)}{\mathrm{d}t} = 0$ Hence $Bx$ is > constant ..... what???? Sorry but maybe it stems from the fact that I already didn't understand what "no force" meant in the text but I can't understand this.


Dispersion relation and Heat Capacity

I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation $\omega = c_sk$. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume $$C_V\sim T^3$$. Now I want to show that for bosons with dispersion relation $\omega\sim A\sqrt k$ has heat capacity $$C_V\sim T^4$$ for $T\ll T_{Debye}$. In the case studies I have read, I can't find where the dispersion relation comes into play. I have no idea how to see this. Please help! * * * Perhaps I could get a response if I change the question to: How does the dispersion relation come into the calculations for the Debye model? Links to notes or an answer would be very much appreciated!! * * * @MarkMitchison kindly pointed out that the density of state depends on the dispersion relation. What are the general definitions of $$E, p$$ in terms of $$\omega, k$$? E.g. for the first case $E=\hbar \omega$ and $p=\hbar k$. So the question is: what are the respective values for $E,p$ in general?


Is normalization consistent with Schrodinger's Equation?

Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's Equation.


Specific problems with the quadrupole formulation of gravitational radiation

the quadrupole formula has some counterintuitive consequences, when analysing the power output averaged over a period $$ P = \langle \frac{d^3 Q_{ij}}{dt^3} \frac{d^3 Q_{ij}}{dt^3} \rangle $$ Specifically, if two separated sources are considered inside the radiation zone, $\rho_1$ and $\rho_2$, and assume that each orbit around separate centers $C_1$ and $C_2$ by a length $R$, the quadrupole moment of the pair is going to be (around the common center $C_1 + \frac{R}{2} = C_2 - \frac{R}{2}$) $$ \int ( \rho_1(x) + \rho_2(x) ) x^j x^k d^3x = \int \rho_1(x) (x_1 - \frac{R}{2})^j (x_1 - \frac{R}{2})^k d^3x + \int \rho_2(x) (x_2 + \frac{R}{2})^j (x_2 + \frac{R}{2})^k d^3x = \int \rho_1(x) x_1^j x_1^k d^3x - \frac{R^j}{2} \int \rho_1(x) x_1^k d^3x - \frac{R^k}{2} \int \rho_1(x) x_1^j d^3x \+ \int \rho_2(x) x_2^j x_2^k d^3x + \frac{R^j}{2} \int \rho_2(x) x_2^k d^3x + \frac{R^k}{2} \int \rho_2(x) x_2^j d^3x$$ $$ = \int \rho_1(x) x_1^j x_1^k d^3x + \int \rho_2(x) x_2^j x_2^k d^3x + \frac{R^j}{2}( \int \rho_2(x) x_2^k d^3x - \int \rho_1(x) x_1^k d^3x ) + \frac{R^k}{2}( \int \rho_2(x) x_2^j d^3x - \int \rho_1(x) x_1^j d^3x )$$ plus some other terms that are constant over time that are not needed However, the terms $\frac{R^k}{2} \int \rho_c(x) x_c^j d^3x$ will have nonzero third time derivatives in general, so they will contribute to the overall power, even though they depend on an arbitrary choice of a center over which the quadrupole moments are calculated? What is wrong with this expression, and how to obtain a power output from the quadrupolar moment that doesn't depend over what origin of coordinates is chosen for the calculation?


Where is the particle during a tunneling event?

If, say, a particle with energy $E<V_0$, approaches a finite potential barrier with height $V_0$, and happens to tunnel through, where would the particle be during the time period when it is to the left of the potential barrier and to the right of the potential barrier? Surely there must be a finite amount of time for it to travel through to the other side, unless it simply teleports there? If it travels through with energy less than $V_0$, however, doesn't that mean it cannot enter in the region of the potential barrier?


Detection of W and Z bosons

What specific behaviour confirmed the existence of the W and Z bosons at the UA1 and UA2 experiments? Thanks!


How should I simulate the electric potential field from a wavefunction?

I was interested in making what I thought would be a simple simulation of an electron encountering a positron by numerically solving the Schrodinger equation over several time steps, but I've run into 1 problem: the potential used in most textbooks (to solve the hydrogen atom for example) are still of a classical point particle, rather than a wavefunction. How do I generate the potential field from two real, moving and interacting wavefunctions?


Experiments proving/disproving extra dimensions

I have been recently reading quite a little about Kaluza Klein theories. I am still far from mastering this but I am curious if any experiment that may disprove or give hints of the existence of extra spatial dimensions is to be expected to be performed relatively soon. This page of CERN, for example, says that the LHC might give some clues, but it doesn't really dive into much detail. So, in some detail, which experiments might say something about this, and when might they be expected?


Why does laser light not affect glowing materials?

I have this childrens rubber ball which glows in the dark after it's exposed to light. I "charge" it with a flash light then play with my dogs at night. I thought to try a very intense green laser, and see how the ball reacted. The laser light had no effect on the balls ability to glow. So I'm left wondering, why does laser light not allow luminescence (maybe not the right word) materials to glow? EDIT In Response to Answer. So I tried a little modification. I tried exciting the ball with three different light sources; a "super bright" Red LED, a very very "super bright" white LED and a blue LED of unknown specs (no package, bottom of my kit). I held the ball to each light source (driven with the same current) for the same approximate amount of time and compared the results. The red LED had no effect. The white had a bit of an effect, enough to see dimly in normal room lighting. The blue led had a significant effect, causing a bright glow. This was interesting as the blue LED was the least bright visually. Yay science!


Mathematically, how do we deduce that angular momentum is bounded?

So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$? I.e. that m is bounded by j. We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its norm-square?


Exotic coupling

I have encountered the minimal coupling between a field and charges before $$H = \frac{1}{2m}(p-qA)^2,$$ whereby I am considering the classical case. The description minimal leads me to ask if there exist exotic couplings (non- minimal)? Could some examples of such couplings be discussed?


How did gunsmiths create revolver cylinders in 1850's without the use of electrical drill?

How did gunsmiths create revolver cylinders holes in 1850's without the use of electrical drill? Which referces/ books specialize providing knowledge in similar molding method?


$E=mc^2$ and particles

This is a simple question. Does the Energy create both Fermion and Boson particles ? or just only the Fermion particles?


Polarization vector and bound charge

Why is it that the bound charge is $Q_b = - \oint_S{\mathbf{P} \cdot d\mathbf{S}}$? In particular, why is there a negative sign? Hayt's book on electromagnetism describes this as the "net increase in bound charge within the closed surface". Compared to Gauss' law $Q_{f} = \oint_S{\mathbf{D} \cdot d\mathbf{S}}$, the free charge is just the electric flux, so I don't see why the bound charge has a negative sign. Also, if the electric flux is independent of bound charge, how come the D-field changes across material boundaries? I thought the D-field was defined as $\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}$ to cancel out the effect of the bound charge (since the effect of polarization is already incorporated into the E-field). Is the D-field dependent on bound charge? Just to add to the second part of my question. I know that for a dielectric- dielectric boundary the tangential components $\frac{D_{t1}}{\epsilon_1} = \frac{D_{t2}}{\epsilon_2}$ and if $Q_{free} = 0$ on the boundary, then the normal components $D_{n1} = D_{n2}$. The tangential component will not contribute to the flux integral so for any $\epsilon_1$ and $\epsilon_2$ across a boundary the electric flux integral will evaluate to 0. It's clear that the flux is independent of the bound charge, but the the tangential component changes so the D-field is not?


What aspects of QFT do mathematicians find troublesome?

What aspects of "conventional" quantum field theory (i.e. what's used by most practicing physicists) are considered to be lacking mathematical rigor?


Argument for symmetry of potential of a spherical capacitor filled with dielectric

A spherical capacitor with inner radius $r_1$ and outer radius $r_2$ is filled with dielectric material with permittivity $\epsilon=\epsilon_0+\epsilon_1\cos^2\theta $.   $\theta$ is the polar angle. Furthermore there is a charge $Q$ on the inner surface at $r_1$ and a charge $-Q$ on the surface at $r_2$. Give reasons why the potential $\Phi(r,\theta,\varphi)$ is for all $r$ independent of $\theta$ and $\varphi$. The independence of $\varphi$ is clear as our problem is invariant under $\varphi$-rotations. I'm not able to find a valid argument why the potential is independent of $\theta$. Thanks in advance for help !


What does f/4.6 mean in a telescope and how important is this value?

In some specifications for telescopes, I saw a value marked as f/4.6. What does it mean exactly, and how important is when it comes to choosing a telescope?


What is the lenght of the rope?

The $E$ string of a violin has a linear density of $0.5 g / m$ and is subjected to a $80\text{N}$ of tension, tuned for a frequency $u = 660 \text{Hz}$:   1. What is the length of the rope? I know that in order to answer the question I have manipulate the equation: $$ \nu = \sqrt\frac{\tau}{\mu} $$ Getting this equation: $$ L = \frac{m v^2}{T} $$ I have tried to find the mass but my attempts was unsucessufull. I think that i could use the equation: $$ f_1 = \frac{\sqrt\frac{T}{m/L}}{2L} $$ In order to find the lenght, but I need help to find the mass of the rope.


What is the constant appearing in the low energy action?

Usually one finds this expression for the low energy action $$S = \frac{1}{2\kappa_0^2}\int d^D X\; \sqrt{-G}\; \mathrm{e}^{-2\Phi}\,(R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4 \partial_{\mu}\Phi\partial^{\mu}\Phi).$$ On dimensional grounds one can deduce $\kappa_0^2$ near $\ell_s^{24}$ (for $D=26$) or $\ell_s^8$ (for $D=10$). Some authors use (obvious for the superstring, not for $D=26$ string): $$2\kappa_0^2=(2\pi)^7\alpha'^4$$ with $\ell_s=2\pi\sqrt{\alpha'}$. Btw. this also confuses me, is $\ell_s=\sqrt{\alpha'}$? So my question is how is $2\kappa_0^2$ determined?


Amateur moon laser ranging

Questions first, then my rough estimations: 1) Is it possible to perform moon laser ranging with amateur motorized 114mm telescope? My calculations suggest that for 1mJ laser it should receive ~2 photons per source 1mJ laser pulse. 2) Given that we already talking about individual photons, how it was possible to perform moon laser ranging BEFORE retro-reflectors were deployed to the moon? Retroreflector sends back light in ~1arcsecond angle, while bare lunar sufrace - in ~6 archours, which means we supposed to receive signal ~(6*60*60)^2 = 4.5*10^8 weaker, i.e. even with 2.5 meter telescopes we are talking 1 photon per 250 1J pulses. **My rough estimations:** Given that atmosphere turbulence limits telescope resolution to ~1 arcsecond (adaptive optics was not available when laser ranging experiments started, nor it is available now for amateurs), if we use telescope with diameter larger than ~150mm (so that we are limited by atmosphere, not diffraction) to expand the laser beam we will get ~1939x1939 meter illuminated area on the moon surface (tan(1arcsec)*400'000km). Which means only 1/(1939*1939) part of our energy will reach reflector. Retroreflector is ~1x1 meter in size. It will reflect the light with same beam divergence - 1arcsecond. Too sad, as diffraction limit for retro-reflector of such size is ~0.2 arcsecond. So, if our receiving telescope has area of ~1 meter^2, we will receive again 1/(1939*1939) part of what reached the moon, so total attenuation is ~ 1.4*10^13. If we use 532nm pulse laser with 1mJ pulse energy, it will emit 2.67*10^15 photons, which means we are going to receive ~190 photons per pulse. Sounds realistic. These calculations suggests that 114mm amateur telescope should be able to detect 2 photon per pulse - again should be detectable statistically.


scattering of light

How does the path of light become visible due to scaterring of light? According to me it will only tell the presence of colloidal particle but the book give the example of cinema hall projector and says that you could tell path of light due to scattering of light. Nevertheless, without scaterring of light also we can tell its path.


Are galaxies, stars, planets, and ultimately life in violation of the second law of thermodynamics?

How can we think about entropy in these situations? To my knowledge all of these structures are born out of gravitational interaction. However, it would seem that the formation of these more organized structures conceptually violates the second law of thermodynamics. Is this dilemma arisen from my flawed understanding of thermodynamics or are these structures actually in a way defying an increase in entropy?


Zitterbewegung for massless particle

Is it possible for a massless particle to undergo zitterbewegung? In massive Dirac theory the Zitterbewegung frequency comes out to be $2mc^2/\hbar$. It looks like the effect will vanish for a massless particle. But the effect is due to the interference between particle and antiparticle state which is also there for a massless particle. So what would be the outcome?


Force inversely proportional to the squared distance

**Newton's law of universal gravitation:** "Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and **inversely proportional to the square of the distance** between them." **Coulomb's law:** "The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and **inversely proportional to the square of the distances** between them." How did Sir Isaac Newton and Sir Charles Augustine De Coulomb come to know that the force, gravitational or coulomb's, is _inversely proportional to the square of the distance between two point masses or charges_ , why didn't they just say that _the force is inversely proportional to the distance of two bodies from each other or two charges_? There must have been something that made them formulate these **inverse-square laws**.


Rotational Velocity and Rotational Frequency

What is the difference between rotational velocity & rotational frequency? Their units seem to be the same, and I've read that one is a 'scalar' and the other is a 'vector,' but how do they differ?


Bell Test statistics

I have some questions about the statistics of Bell Tests. For convenience I'm going to refer to the test described in this paper: http://arxiv.org/abs/quant-ph/0205171 They show a coincidence rate of 300 counts with polarizers aligned. I'm wondering how many counts there would be if you took out both polarizers altogether, and how many of those would be coincidence counts. I'm interested in the ideal case of 100% detector efficiency and 100% downcoversion efficiency, but I also wonder how close the ideal case can be approached in practise.


Ten-ping bowling: Can a ping pong ball knock over a bowling pin?

In this video we see a rather unorthodox bowling technique on display. The gentleman appears to knock over all of the pins using only a ping pong ball. It's a fake, of course: you could never knock over a bowling pin with a ping pong ball. Or could you? How fast would a ping pong ball need to travel in order to knock over a heavy object like a bowling pin? Naturally, you are free to make any simplifying assumptions and order-of-magnitude estimates you like.


"Vectors" (i.e. 1-tensors) their definition and motivation for relativity

I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, $V$, $p_V=(p^1,p^2)$ then when we transform it via a rotation $p_W = R(\theta)p_V$ then we do not violate any physical laws. To motivate the question further, I understand that individual terms after a transformation are not preserved but the law as a whole is: For example if $ma_V = F_V$ then $ma_W=F_W$ and although $F_W \neq F_V$ it is the case that the scalar (tensor) is preserved $$ma_W-F_w=0=ma_V-F_V$$ The question that motivated this post is the following: Prove that if $p_V$ is a vector then $p'=(ap^1,bp^2)$ cannot be a vector unless $a \equiv b$. This seems easy enough to show $$Rp'= (p_1a\cos(\theta)-p_2b\sin(\theta), p_1 a\sin(\theta) + p_2b\cos(\theta))^t$$ And clearly, we cannot just factor out the $a$ and $b$ unless $a\equiv b$. But what does this really mean? The converse must be true, namely if $p'$ is a vector then surely $p_V$ cannot be a vector. I suspect if I keep reading there will be ample examples and it will click better (for example the force-mass-acceleration equation makes sense in how it transforms). I suppose my confusion rests in the idea that "well of course a vector is just a tuple and if rotate it that's another tuple and that old-new tuple pair was prescribed by a rotation so of course all tuples are vectors!" EDIT: Here's another example to make the point (this is an actual exercise): Suppose we are given two vectors ${p}$ and $q$ in ordinary 3-dimensional space. Consider this array of three numbers: $(p^2q^3,p^3q^1,p^1q^2)^t$. Prove that is not a vector, even though it looks like a vector. (Check how it transforms under rotation!) In contrast, $(p^2q^3-p^3q^2, p^3q^1-p^1q^3,p^1q^2-p^2q^1)^t$ does transform like a vector (so it's a vector). It is in fact the vector cross product $p \times q$.


Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement > _In space of constant sectional curvature, $K$ is independent of position._ Here > _$$R_{abcd}\equiv K(x)(g_{bd}g_{ac}-g_{ad}g_{bc})$$ where $R_{abcd}$ is the > Riemann curvature tensor and $g_{ab}$ is the metric of the spacetime._ The proof goes like this: > _Contract the defining equation with $g^{ac}$, giving $$R_{bd}=3Kg_{bd}.$$ > and so on._ Problem is I don't understand why the contraction gives $$R_{bd}=3Kg_{bd}.$$ I can see the first term gives $$g^{ac}g_{bd}g_{ac}=4g_{bd}$$ since it's 4D spacetime. But as far as I can tell, the second term gives $g^{ac}g_{ad}g_{bc}=\delta_{bd}$ which is not necessarily $g_{bd}$. Where have I gone wrong?


Electric field of uniformly charged spherical cap?

I was wondering what the electric field of a uniformly charged spherical cap is? Thereby I am referring to a spherical shell that was sliced into two pieces and we are only looking at one part of it. So in spherical coordinates it would mean that you would have a shell with radius $R$, it contains a full revolution of $2\pi$, but the polar angle does not go from $0$ to $\pi$ but rather from $0$ to some $\theta \in (0,\pi)$. And now I am looking for an equation that gives me the electric field for a given charge $Q$ on the shell. It would be sufficient if somebody could give me the integral, that I have to evaluate!


Velocity required for orbit of satellite

I am very confused about orbital velocity for satellite launched from earth. It is states that: $$ v=\sqrt{\frac{GM}{r}} $$ Some says that it is equal to 8000 m/s and some say 3.1 km/s. Please remove my doubt.


Heisenberg's uncertainty and $0 K$ temperature

when a body is subjected to $0 K$ temperature, it becomes rigid. hence if we see in terms of quantum the lattice vibration decreases, resulting in no change in the direction of the Random velocity, then can we increase the degree of probability of finding of an electron in an given sys


Are particles solid?

My intuitive understanding is that although atoms are mostly empty space, the nucleus is solid. Thus a neutron or a neutrino can collide with it and do things like deflect or cause a nuclear reaction. But does this intuition reflect the underlying reality? Since we usually think of a solid in reference to substances comprised of atoms, how does the concept of solid extend into the subatomic realm? Thanks


Why do comets have tails?

There is no atmosphere for comets to get hot and burn and show tails but they still have tails. Why do they? Edit: Isn't the answer "acceleration"? ![enter image description here](http://i.stack.imgur.com/bdR0F.gif)


What does the discovery of the Higgs Boson mean for physics?

Will this unite some theories, or cause some other change in physics, and perhaps our undertanding of the universe?


The relationship between the steady state theory and the notion of Higgs boson is not clear to me. What does the discovery of Higgs boson mean for steady state theory? Or are the two ideas purely orthogonal?


What's the problem with solar energy?

What "breakthrough" from a theoretical point of view is needed for solar energy to become feasible energy alternative?


Photon absorption probability for a given molecule in gas phase

So I'm pretty sure I'm approaching this problem in the wrong way and I need some guidance (my first hint is that I think I'm thinking about a quantum mechanical problem too classically) Suppose there is an isolated molecule in the gas phase with an average cross- sectional area to be exposed to radiation of $A$. (For my specific problem, the molecule is trapped in a superfluid Helium droplet, but I think the calculation should be roughly the same). If the radiation source has a flux $f$ (in units of photons/second/square area/0.1% BW) at energy $E$, what is the probability of the molecule absorbing a photon within a given interaction time $t$ if the absorption probability at a given energy is $P(E)$? This is pretty easy to calculate if I treat the whole problem classically, i.e. like a ball and a target model. For some reason, though, I get numbers that seem to be way too low if I do this. I know it has to be more complicated than that, since light is a wave also. What am I missing? I understand that transition probabilities are related to wavefunction overlap, etc. Also, I should note that the radiation in my specific problem is in the hard x-ray region, though I don't think that should change the answer. Thanks in advance for your help.


Quantum computing and ambiguity

I do a bit of hobby programming and I often search the internet for little oddities that are fun to ponder over. I have read a few passages that try to explain quantum computing to the layman like myself. I have read of the Qubit, the more 'power' version of the bit, and its bad habit of being in superposition. This, to me, sounds as if it sits halfway between 1 and 0. So, I reason that one can create a qu-binary number with these; something resembling a ternary number, made from 0's, 1's and 1/2's (or Q's). I have read that a quantum computer has more 'power' when it comes to computation because one qu-value is a possibility between at most n^2 regular values in n bits. I have constructed a little problem with this value when you try to store a specific set of regular values in a qu-value. Imagine a value is a superposition between 2 and 3. In qu-binary, I would write "10 or 11 -> 1Q", as the last bit is "both". OK, so this works. But what about real values 2, 3 and 4 in superposition? in my ternary notation "QQQ" is potentially any of the possibilities 0 through 7, and so actually represents a whole lot more values than I want!? My question is, how does it really work? Am I thinking about it all wrong? Because this is how the whole subject of Quantum computing looks like from the outside. Or is this an example of quantum computing's non-determinism? I assume all bits are completely isolated from one another and have no qu- knowledge of any other. Maybe something obscure like quantum gates sharing information between bits could explain the problem; or if the bits represent continuous probabilities. I don't know. Could someone explain it for me?


Ice cube on a scale

Lets put an ice cube on a fairly precise scale and take readings over a period of several hours. The ice cube is situated in a saucer so that the water is retained after melting. **What will the recorded curve look like?** Suppose ambients conditions: 25°C, RH40% and 1013 mbar.


Field At Magnetic Dipole

Suppose I have a pure magnetic dipole $\mathbf{\vec m} = m\hat z$ located at the origin. What is the magnitude of the field $|\vec B|$ as $r\to 0$? In other words, what is $\lim_{r\to 0}\frac{\hat{r}\cdot \vec{p}}{4\pi\varepsilon_0r^2}$? Is it just zero? $\infty$? Do I have to use some sort of quadripole term?


What is the physical or mathematical meaning of the Gibbs-Duhem equation?

The Gibbs-Duhem equation states $$0~=~SdT-VdP+\sum(N_i d\mu_i),$$ where $\mu$ is the chemical potential. Does it have any mathematical (about intensive parameters) or physical meaning?


Why is the Higgs boson spin 0?

Why is the Higgs boson spin 0? Detailed equation-form answers would be great, but if possible, some explanation of the original logic behind this feature of the Higgs mechanism (e.g., "to provide particles with mass without altering their spin states") would also be appreciated.


Moment of inertia of a coin

I have a a coin infinitely thin, rotating along the diameter. How to derive the formula for it's moment of inertia passing through the diameter. I was suggested to use the surface density and infinitely small part of the surface area, equidistant from the axis of rotation (marked as $dS$ on the picture). ![](http://i.stack.imgur.com/q6KEX.jpg) I've already figured out that: $$I~=~\int r^2dm~=~\int \rho r^2 dS. $$ And now I'm stuck. Any help would be much appreciated. Greg edit: My coin is infinitely thin, so it's two-dimensional object (only $X$ any $Y$ axis). So let's assume that my rotating axis is equal to Y axis. So I have to integrate from $R$ to $-R$ on the $X$ axis. And every $dS$ will have different surface area. But I know, that total area is $S=\pi r^2$. From Pythagorean theorem I know, that $ r^2 + h^2 = R^2 $. And my integrate is $$I ~=~ \int\limits_{-R}^{R} 2\rho r^2 \sqrt {R^2 - r^2} dr $$ ~~But now I'm confused how to solve that - every time I get different solution than inWolfram Alpha calculator ~~ Ok, I've solved that - final answer is $ I = \frac {1}{4}MR^2 $


Could an orbiting mass generate power?

If an heavy object (e.g. 10 tons) orbiting around Earth at 370 miles high, is connected with a cable back to Earth, we assumed either Earth is going to pull the mass or vice versa (or it will fall back to Earth). Assuming correct pressure/release/pull can be applied from the ground to prevent the fall, or it naturally starts spinning around in a circular motion - what do you think about its power generation capability? This was a fun chat between friends, so just wanted to run it by.


Discrepancy ( or Confusion ) in the mass of photon

$E$ = $mc^2$ And also $E$ = $hf$ (f - frequency) And hence Einstein said $m$ = $hf\over c^2$ And so photons have mass But later he also said $M$ = $M_0\over \sqrt {1-v^2/c^2}$ Where if we put $v = c$ we get $M = M_0/0 \leftrightarrow M=\infty$ And so photon travelling at the speed of light ($c$) have undefined mass (or $\infty$) And so in one equation it is said photons have mass and in one it is said they have $\infty$ mass. So do i have a problem in understanding or is there really any discrepancy ?


Stellar evolution simulation engine or software

Is there any general purpose stellar evolution simulation engine or software? Something to throw in properties of the star and to watch how (and why) they change along the timeline - with or without visualization (but preferably with). The best of what I have found on the net is this site: http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_links.html however, there is no good simulation software behind the links there (or at least I don't see one).


Bose gas with $T = 0$ and $\mu < 0$

Is it possible to have a Bose gas with $T = 0$ and $\mu < 0$ ? I think that there is a problem, because all the states $k$ are such as $$\langle n_k \rangle = \dfrac{1}{e^{\beta \\{\epsilon_k - \mu\\}}-1} = 0$$


Why aren't there any arrays of visible light telescope?

There have been many radio telescope arrays which are capable of resolving stars significantly fainter than can be resolved with a single "small" radio telescope. Why don't they do the same with light telescopes?


Pion production in proton-proton collision

Why is $\pi^0$ created in the high-energy collision $p+p\to p+p+\pi^0$?


Why does electromagnetic waves travel slower in well isolated conductor?

Wikipedia writes, that > Propagation speed is affected by insulation, so that in an unshielded copper > conductor ranges 95 to 97% that of the speed of light, while in a typical > coaxial cable it is about 66% of the speed of light Why does insulation affect this speed?


Are the solutions in radicals of cubic and quartic of any use in physics?

We all know that there are analytic formulae to solve quadratic, cubic and quartic polynomial equations. But it seems to me that the only solution that widely used is physics is the solution of quadratic we all learnt at school. Is it true that when physicist faces quartic or even cubic equation -- he turns to numerical methods to solve them? Have anyone seen the analytic analysis of some real problems in physics, where these formulae were used?


Deformation of light-cone

In the paper _The geometry of free fall and light propagation_ by Ehlers and his colleagues ( _Gen. Relativ. Gravit._ **44** no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable structure on page 70, they claim: > It should be realized that the representation of light in special relativity > by means of ordinary cones rather than by hypersurfaces of hour-glass shape > depends partly on a particular choice of differential structure. The question is: What do they mean by saying **particular choice of differential structure**? In what condition may the light cone be deformed and become like hour-glass as shown in this figure? Thank you in advance for any help. ![enter image description here](http://i.stack.imgur.com/XoQWY.jpg)


What are differences of different telecommunication wavebands?

I'm just wondering what are the differences (advantage and disadvantage) of different telecommunication bands ? For example:   * Why does satellite TV use Ku band?    * Is it true the rule is that greater band frequencies (10 ghz vs 1 ghz vs 800 mhz) can transport more data?   * Why mobile phone bands use (for example) 800 MHz or 1800 MHz?    * Is there any other "rules" related to band type and distance covered ? Yes, i know, i'm a little confused. I just would like to understand the differences between "radio frequencies", particularly their advantages and disadvantages. Are there references that explain those types of arguments?


Ampere's Law and Circulation?

![enter image description here](http://i.stack.imgur.com/V2GdM.png) The net circulation should be zero if the net current is zero. However, I don't see how the circulation in the Amperian loop above can be zero. All the lines interact the loop only once and all their circulation contributions are positive or negative depending on the assumed direction of integration. Could someone explain how the net circulation would be zero? Thanks!


Electrode potential

If I connect a zinc half cell (Left hand side) and a copper half cell (right hand side), I get an E cell value of 1.1V taken from the voltmeter. If I replace the voltmeter with a light bulb, I will get light and heat energy of the bulb. My question: Will the voltage across the cell remain the same? I mean, the potential difference arises due to the difference in the amount of charge build up on the electrodes... so an electron flows from the zinc electrode to the copper electrode, will the voltage across the bulb decrease?


I have the following questions.   1. Why can't matter pass the speed of the light?   2. What happens to an object when it comes close to the speed of the light?   3. What makes the object unable to surpass this speed limit? And finally, I need to know whether this speed limit has been experimentally proven?


Whats the anti-torque mechanism in horizontal take-off aircraft?

In most helicopters there is the anti-torque tail rotor to prevent the body from spinning in the opposite direction to the main rotor. What's the equivalent mechanism in horizontal takeoff single engine propeller, and jet aircrafts, where the air or the jet coming out back from the turbine or propeller is spinning and will cause such aircraft to roll?


An equivalent for the Bernoulli equation for viscous liquids

I was wondering whether there is any equation for viscous liquids (probably derived from energy and mass conservation principles) relating the pressure, rate of volume flow, area of cross section and height of the liquid (or any other parameters if required). NOTE : I am aware of Poiseuille's law, but that is not what I am looking for as it relates the pressure _gradient_ to the rate of flow.


Why are there 4 Dimensions and 4 Fundamental Forces?

Is it a coincidence that there are four fundamental forces and four spacetime dimensions ? Does a universe with three spacetime dimension contain four fundamental forces? Can magnetism be realized in three dimensional spacetime? Edit 1: Thanks alot for the answer. I'm a first year undergrad student so this may seem naive. If there is a symmetry that mixes spacetime symmetries and internal symmetries of the theory (e.g. gauge symmetry) then there might be ways that spacetime geometry have effects on the particle spectrum and interactions of the theory so you can describe things in geometrical manner . The mechanisms that lead to the structure of spacetime may explain why there are four forces.


geometry inside the event horizon

I'm trying to understand intuitively the geometry as it would look to an observer entering the event horizon of a schwarszchild black hole. I would appreciate any insights or corrections to the above. inmediately after your enter the event horizon, if you look back and try to reach again for the horizon, it will seem to be expanding away quicker than the speed of light. Near this region, the apparent shape of the horizon is a sphere expanding away, and we are inside the sphere Near the singularity, we really don't know what happens. I've heard that spaghettification is not a necessarily occurrence, since the metric field diagonal components are shrinking as the curvature grows, so it could very well be the case that a infinite length hyper-cylinder $S^3 \times R^+$ of constant _physical_ radius is being conformally mapped to the $S^3 - \\{0\\}$ region around the singularity, or that in general a region around the singularity can be mapped to anything in the other end, which is basically because the degrees of freedom of curvature and stress-energy in our end of the spacetime cannot really predict what sort of topology endpoint will connect to the matter in the other end. Since the metric components are tending to zero at the singularity, this argument sounds pretty interesting, since it would seem to imply that observers will "shrink" relative to kruskal coordinates, because the local physics would always be that physical observers will stay fixed relative to their local metric, since the metric is covariantly constant!. However, i'm not expert on how to describe the asymptotic physics in the neighbourhood of the schwarszchild singularity. (which is why i'm asking on this site, after all!). **Question:** _does this argument hold any water?_


Waveguides and Maxwell's Equations

**Background** In many textbooks on electromagnetism, the sequence in which topics are introduced are generally as follows -- electrostatics, magnetostatics, electrodynamics, wave propagation in free space, and wave propagation in confined space, in that order. As an example, consider Griffiths' 3rd edition of Introduction to Electrodynamics. Similarly, consider Orfanidis' Electromagnetic Waves and Antennas. By using the discussion on wave propagation in free space as a launchpad, the discussion on wave propagation in a confined space usually introduces some additional boundary conditions that the solutions to Maxwell's equations for wave propagation in free space did not have. If a proposed solution can be plugged into Maxwell's equations and not produce any contradictions, then they are valid. The goal is then to maintain the monochromatic wave solutions that worked for free space wave propagation with some "patches" to account for the fact that the propagating EM energy will be confined by the hollow waveguide. The reason monochromatic (one-wavelength) wave solutions are desirable is that, by superposition, any signal can be expressed as a sum of one or more monochromatic waves, and so any signal's behavior in a hollow waveguide can be understood. One of these "patches" is the boundary condition that $E_{\parallel}=0$. This is because the net electric field in an ideal conductor, such as the material that makes up the confining hollow waveguide, must be $0$ because any externally applied electric field will influence electrons to rush to counter the externally applied electric field. If there is no net electric field (meaning $\vec{E}=\vec{0}$), then there is no net electric field in the tangential direction ($E_{\parallel}=0$) along and inside the metal. If there is no net electric field along the metal, then, by Faraday's law, there must be no net changing magnetic field $\frac{\partial \vec{H}}{\partial t}$. This means that each of the components $\frac{\partial H_x}{\partial t}$, $\frac{\partial H_y}{\partial t}$, and $\frac{\partial H_z}{\partial t}$ must all be $0$ too. If there is no initial magnetic field (meaning $\vec{H}=\vec{0}$, then there must be no magnetic field at any later point. Therefore $\vec{H}=\vec{0}$ and $H_{\perp}=0$. So these are the two additional boundary conditions that must be considered. An implicit assumption is that the material on the interior of the waveguide is isotropic, homogeneous, and linear. This means that $\vec{B}=\mu\vec{H}$ and therefore $\vec{B}=\vec{0}$ and $B_{\perp}=0$ at the boundary of the metal + hollow interior interface as well. **Problem** Frequently, the monochromatic wave solution to Maxwell's equations given the boundary conditions of a hollow waveguide are presented as $$ \vec{F}\left(x,y,z,t\right)=\left[\text{spatial component}\right]\left[\text{temporal component}\right] $$ For example from Orfanidis, $$ \vec{E}\left(x,y,z,t\right)=\vec{E}\left(x,y\right)e^{j\omega t - jkz} $$ and from Griffiths, $$ \vec{E}\left(x,y,z,t\right)=\widetilde{\vec{E}}_0\left(x,y\right)e^{i\left(kz - \omega t\right)} $$ Further, $\vec{E}$ is a 3-dimensional vector with components $E_x\left(x,y\right)$, $E_y\left(x,y\right)$, and $E_z\left(x,y\right)$. **Bottom line** : Why is the spatial component only a function of $x$ and $y$? Further, why does spatial component have $x$, $y$, $z$ components [e.g. $E_x$, $E_y$, and $E_z$] (where $E_x$, $E_y$, and $E_z$ are each functions of only $x$ and $y$)?


Quantization surface in QFT

![enter image description here](http://i.stack.imgur.com/RKL4O.png) What does the Quantization Surface mean here? Reference: H. Latal W. Schweiger (Eds.) - Methods of Quantization


Please explain me how the Higgs boson gives mass to other particles, more detail?

As we know, the Higgs boson gives mass to other particles. But here is onething which is not clear for me. I mean, I do not understand how the Higgs boson gives mass to other particles? Does anyone could explain me more explicitly how procedure goes that the Higgs boson gives mass to other particles.


Neutrino mass difference and mixing angles

What are the most recent values of the mixing angles associated with the 3 neutrinos flavors and what are their mass differences? Have they been measured in the first place?


References requested on departments with recitation sessions

What physics departments offer recitation sessions for their physics courses on the undergraduate level? If there is a site which has such info it will be greatly appreciated.


View from a helicopter rotor: why is the horizon distorted?

This video ("rotor panorama") was captured by a camera attached to the rotor head of a radio-controlled helicopter, with the frame rate set to the rotor's frequency. During a long segment of the video, the horizon looks distorted: ![enter image description here](http://i.stack.imgur.com/Bl6c0.jpg) What causes this distortion?


Physical Interpretation of the Bloch vector

In the expression of the density matrix of a (Electron-Spin) Qubit $$ \rho=\frac{1}{2}(I + x \sigma_x + y \sigma_y + z \sigma_z) $$ where $\tau=(x,y,z)$ is unit vector in the Bloch sphere, which is the physical Interpretation of $\tau$? Can it be interpreted as the direction of spin axis of the electron?


What are the main algorithms the LHC particle detectors use to reconstruct decay pathways?

I am just starting to look into how we understand the data from particle collisions. My question is, what are the algorithms or ways that these detectors interpret the data? Are there standard approaches? Or if not, what are some good papers or places to look to get started in learning more about the implementation and/or details of how this works? So far I haven't dug into any textbooks, but many articles on the web and this was somewhat helpful in pointing to where to look: http://arstechnica.com/science/2010/03/all-about-particle-smashers-part-ii/ So from my understanding so far, there are a few different LCH "experiments", which are physical structures that are optimized to focus on specific aspects of data from a collision event. The detector measures all kinds of particle emissions and changes in electrical fields, and then seems to try to backtrack and figure out all the emission/decay events that might have taken place in that split second. From my understanding so far, basically the computer programs used to compute these possible "decay pathways" must be using some standard algorithms or something, and must have built into them all possible particle emission pathways (like all possible Feynman diagrams if there is such a thing). Are there any good resources or standard algorithms/approaches to understanding how particle detectors analyze their data?


Gravity change as a function of time only

Is it possible to find how gravity changes with time? That is, height is not part of the equation. For example, $g$ can be expressed as a function of height that does not involve time with this relation: $g_i=g_0 (r /(r+h))^2$ where $g_i$ is $g$ at any height and $g_0$ is on the Earth's surface. I also found through the above relation that $g_i = g_{i-1} ((r+h_{i-1})/(r+h_i))^2$ By the same token, there should be a relation between $g_0, g_i, t$ that does not involve distances. An object in free fall will be going through changes in $g$ as time progresses. Is it possible to create such an expression?


How to measure distances to stars by means of spectroscopic parallaxes?

How to measure distances to stars by means of spectroscopic parallaxes on practice? What is the accuracy of measuring distances using this method compared with distances based on HIPPARCOS trigonometric parallaxes?


Why don't bubble lights explode?

As a child, I loved to watch the bubble lights at Christmas time. The colorful bubbling was hypnotic. However, there was always something about them that bothered me. With all of that boiling happening within the glass vial, why didn't the entire thing explode? I knew how explosive steam boiling could be, why was the bubble light able to withstand the constant boiling, and pressure changes that were going on inside it? Also, how was the fluid able to boil within an enclosed space (where the volume could not change?)


How does one measure the mass of a galaxy? And other such large quantities

Trained as a pure mathematician, I see claims about the mass of a galaxy and other such huge measurements that are arrived at experimentally, and I just have to scratch my head. I know this is a bit of a vague question--but does anyone have a good resource for something like "measurement in astrophysics" or an introductory history of how astrophysicists figured out how to make measurements of this kind?


The Observance of Electromagnetic Phenomena

As far as I've seen, E&M waves are sinusoidal. Our eyes observe these patterns and give us colors, but in a situation like such: Why do electrons emit radiation? There is still a wave, or at least a pulse, but it's not necessarily sinusoidal, and I'm not sure what 'wavelength' would be used. Nonetheless, radiation produced in this way does have a wavelength-it is used in synchrotrons. So what am I missing?


What is the energy of interaction between a point charge and an infinite cylinder?

I don't remember enough from my electromagnetism course and I can't find any simple, full example on this subject. I know, that I can consider the cylinder as a wire with the same charge density (when outside of the cylinder), so I consider the wire case with the same charge density, $\lambda$. I set the $y$ direction to pass through the wire, and the $x$ direction to pass through the point charge and perpendicular to the wire. Everything is in the $z=0$ plane. Hence, the particle is at $x=r$ position. From Gauss law, I learn that the field at a point at distance $r$ from the wire I have a field which scales like $\lambda/r$. To find the potential, I need to solve the integral $V = -\int_b^r E dl = - k \int_b^r \lambda/x dx$ Where $k$ holds the constants and $b$ being a point where the potential is 0. The solution of this integral is $V = - k \lambda \log (1/r) + C$ When $b=1$ the potential is 0 so the potential is $V = k \lambda \log (1/r)$ and the energy is $U = k q \lambda \log (1/r)$ Here are my questions:   1. Is any of this true?   2. I tried to derive this by starting with coulomb law and calculating the energy for a segment of the wire, I get an answer which scales like $1/r$, can you derive the answer for this?


UCM unidentified force

> _A train rounds a curve of radius 235 m (turning right). The train track is > flat, and the train is traveling at a constant speed. A lamp in the train > makes an angle to the vertical of 17.5 degrees. How do you calculate the > speed of the train?_ The first thing I need is to draw a free-body diagram of the lamp.![free-body diagram of lamp](http://i.stack.imgur.com/SQTWH.png) Sorry about the bad drawing. What confuses me is what force drives the lamp to the left. It seems to have something to do with inertia, I think it is probably equal to the centripetal force that the train exerts on anything in it. Can someone identify this force and explain the problem?


How does the Alcubierre drive warp space in the vicinity of it's destination?

![enter image description here](http://i.stack.imgur.com/kWreF.gif) I don't understand how this could work. So lets say the drive is set to go 30,000,000 meters in a straight line to get to a distant planet. Now lets say the ship size is 20 meters. Now let's make the assumption that the warp field in the photo is 30 meters (10 meters bigger than the ship). So the warp bubble is applied. Now the 30 meters of space the ship occupied is cut. But it only could apply the warp bubble to this confined region which means 30 meters are cut instead of 30 million and now 29,999,970 meters are left and the ship is not even close to the planet it wants to go to (barely moves). Are there any solutions to this problem?


What provides the centrifugal force for planets orbiting a star?

this is a question I had when I was a kid. I'm a bit ashamed because I think I am missing out on something very obvious since I have the same question despite almost being an engineer now! From Newtonian physics, I understand how although the gravitational force pulls the planet towards a star, and the planet 'falls' towards the star, due the angular momentum, it also moves laterally. In case of a planetary orbit, this is just enough to keep it moving around it in an elliptical orbit. (With speed and radius being such that angular momentum and energy is conserved) But let's consider a single planet and star system where the planet moves around the star in a circular path for simplicity. If gravitational force provides centripetal force, what can account for the centrifugal force? I have a feeling it is related to some inaccurate view I have, of centrifugal force. The xkcd comic and wikipedia article talk about two concepts: fictitious and reactive centrifugal force. But it would be safe to assume the star(EDIT: sorry, I meant to choose a frame of reference such that the centripetal force on the planet has to be cancelled by a centrifugal force on the planet in the direction: star to planet) as a stationary frame of reference for the purpose of this question, right? Which means that it is not a fictitious force, right? Or is it not considered a centrifugal force if you take the star as the reference frame?


Why do books write $X(f)$ when they mean actually mean $\lvert X(f)\rvert$?

All books write $X(f)$ in plots - the Fourier transform of $x(t)$ - when they actually mean $\lvert X(f)\rvert$, without even mentioning in passing that they are dropping the mod sign. And also they consider only the mod of FT in the analysis, completely ignoring the phase term - especially communications analysis like AM and FM. Why? It is well known that in image processing phase also place an important role? Is there any proof or valid reason that phase is unimportant in communication engineering?


Simple applications of group theory which can be understood by a senior undergrad

I am looking for references (books or web links) which have "simple" examples on the use of group theory in physics or science in general. I have looked at many books on the subject unfortunately they usually require extensive technical coverage of the basics, i.e. the 1st 100 pages or something, to be able to start discussing applications. I believe that there is an easy way to explain anything (it's just hard to find it).


What will happen if we use a speed greater than light speed and find a body'motion and energy relative to it?

In Einstein's papers, he used light speed as a reference speed. What if we use a greater finite speed and do the same calculations. Won't this greater speed then be the limit.


what is the 2D gravity potential?

In 3D, I can calculate the total force due to gravity acting on a point on the surface of the unit sphere of constant density, where I choose units so that all physical constants (as well as the density of the sphere) is 1: $$F = 4\int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} \tfrac{x+1}{\left[(x+1)^2+y^2+z^2\right]^{3/2}} dz\, dy\, dx = \frac{16\pi}{3}.$$ This force agrees with what we get if we treat the sphere as a point particle at the sphere's center of gravity with lumped mass $\frac{4\pi}{3}$. So far so good. But now if I try to calculate the force of gravity of a point on the boundary of the unit disk in 2D, I get infinity: $$F = 2\int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \tfrac{x+1}{\left[(x+1)^2+y^2\right]^{3/2}} dy\, dx = \infty?$$ Intuitively, if I think of the total force as the sum of contributions from nested circular rings around the point of interest, the circumference of the rings scales like $r$, the force density like $\frac{1}{r^2}$, so the contribution of each ring scales like $\frac{1}{r}$ which diverges as $r\to 0$. What is going on here? How is the gravity potential derived? Is it an accident that in 3D, $\frac{G m_1 m_2}{r}$ is harmonic on the punctured space $\mathbb{R}^3 \backslash\\{0\\}$? Is the "right" gravity potential in 2D something like $G m_1 m_2 \log r$? If so, why, and isn't it a paradox if point masses in 2D orbit according to a different law than coplanar point masses in 3D?


Which mathematical operation does the right hand rule for current come from?

I am currently wondering about this famous rule: ![picture](http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Daniel_Lenord/righthand.jpg) Where does it come from mathematically that when you point with your thumb in the direction of the current, your curved fingers will point in the direction of the $B$-field? In other words, which mathematical operation does it come from? Probably it has to do with some vector product or Stokes theorem application, but I am not quite sure about it.


Is wearing 3D glasses from the cinema as sunglasses hurtful to the eyes?

I heard a few times that using them as sunglasses is hurting the eye since UV light is not filtered, but the pupil is wider than it would be w/o wearing them because the visible light is dimmed. I have always assumed since atmospheric (i.e. not reflected from flat surfaces) sunlight is unpolarised and since the glasses used in 3D cinema only allow circular polarised light to pass they do indeed filter out a large portion of the intensity of the incoming (unpolarised) light. Does this not work for the UV part of the spectrum or is my initial assumption wrong?


Expectation value of the stress-energy tensor in 2-D CFT

Due to a previous question, I am confused with the expectation value of the stress-energy tensor in a 2-D conformal field theory. Let's take the example of string theory, to sketch the problem. Defining the action by $S = \frac{1}{2 \pi \alpha'}\int d^2z \partial X^\mu \bar \partial X_\mu$, the (euclidean) expectation value of an operator is : $$\langle\mathcal F[X]\rangle=\int [dX]~ exp(-S)~F[X]\tag{0}$$ My questions are : 1) In string theory, what is : $$\langle \partial X^\mu \partial X_\mu\rangle \tag{1}$$ 2) In string theory, what is : $$\langle :\partial X^\mu \partial X_\mu:\rangle \tag{2}$$ 3) In a general 2-D conformal field theory, what is : $$\langle T^{\alpha \beta}\rangle \tag{3}$$


Brachistochrone problem in general relativity

This question Brachistochrone Problem for Inhomogeneous Potential has the obvious extension. Namely the same question, when gravity is treated according to general relativity. To make it specific let's consider the case of Schwarzschild metric. Given to points, outside the event horizon, what is the trajectory (or the worldline) of a test body, which goes from the first point to the second for least time. All considered from the point of view of a distant observer at rest (with respect to the star/back hole). Of course other metrics or the general case are also interesting. I couldn't find much by searching on the net. There is one article with the tittle suggesting relevance (actually more but this one seems very relevant), but it is over 60 pages of technicalities. So it would be nice to see a shorter answer.


Glauber's Formula

In the Cohen-Tannoudji Quantum Physics book, Complement BII, says: [...] two operators $A$ and $B$ with both commute with their commutator. An argument modeled on the preceding one shows that, if we have: \begin{align} [A,C]=[B,C]=0 \end{align} with $C=[A,B]$, then: \begin{align} [A,F(B)]=[A,B]F'(B) \end{align} Then this last property, is used to proof Glauber's Formula. \begin{align*} e^Ae^B=e^{A+B}e^{\frac{1}{2}[A,B]} \end{align*} I understand this proof, but I couldn't find a way to demonstrate, \begin{align} [A,C]=[B,C]=0 \end{align} with $C=[A,B]$, then: \begin{align} [A,F(B)]=[A,B]F'(B) \end{align} I would like to know to do this, so I can understand better the Glauber's Formula proof.


How to demonstrate that there are just 14 types of Bravais lattice?

Is it possible to demonstrate that there are just 14 types of Bravais lattice without the knowledge of group theory?


Expansion of $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$

How can I find the expansion of the Lagragian (it it only dependent on $v^2$) $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$ in powers of $\vec{\epsilon}$ ? (From L.Landau, E. Lifshitz, Mechanics , page 7)