title: What is the reason why anyons escape spin-statistic theorem?

text: I'm wondering about the exact reason why anyons escape the spin-statistic theorem (SST), see _e.g._ http://en.wikipedia.org/wiki/Spin–statistics_theorem. I've read somewhere (the wikipedia page is sufficient I believe to understand this point, to be honest I don't remember where I read this) that the reason is just that anyons do not belong to the relativistic sector. Despite Lorentz invariance is still the cornerstone of the SST, anyons can have fractional exchange statistic because they belong to the Galilean invariance. See also this SE question about the SST and its demonstration. On the other hand, it seems to me that this argument might well be wrong. Indeed, I've the feeling that the exact reason is that the permutation group is not enough to understand the exchange of two particles. If one discusses the rotation group instead of the permutation group, then one ends up with the Leinaas and Mirheim [please see the bottom of the question for the reference] argument, saying that in 2D, the homotopy group of SO(2) is $\mathbb{Z}$, no more $\mathbb{Z}_{2}$ as for SO($n$) with $n\geq 3$. **I'm wondering about the quantum field community perspective ?** Do they continue to discuss the Lorentz invariance as the key point to obtain SST, or do they generally accept the Leinaas and Mirheim point of view [see below]. The two arguments might be almost equivalent, except that I believe the permutation group does not care about the space dimension. Moreover, I think it has no one-dimensional-non-trivial-projective representation, as required for anyons to exist, am I correct ? NB: There are some alternative proof of the SST linked from the Wikipedia page about it. I gave the link at the beginning of this question.   * Leinaas, J. M., & Myrheim, J. _On the theory of identical particles._ Il Nuovo Cimento B, 37(1), 1–23 (1977). http://dx.doi.org/10.1007/BF02727953


title: If $SU(2)_{L} \times U(1)_{Y}$ breaks to $U(1)_{em}$ when a non-zero mass for the Higgs boson is chosen, why do we still have weak interactions?

text: As I understand it, when we say that the $SU(2)_{L} \times U(1)_{Y}$ is broken via the Higgs mechanism, this is because the symmetry acts on the Higgs mass in a way that would change it's value. If we want to pick a particular model we need to pick a fixed value of the Higgs mass, and this is only possible if we say $SU(2)_{L} \times U(1)_{Y}$ is broken to $U(1)_{em}$. The Lagrangian is always invariant to $SU(2)_{L} \times U(1)_{Y}$ (even if the Higgs mass changes under $SU(2)_{L} \times U(1)_{Y}$ the Lagrangian is such that it remains invariant). The symmetry breaking is just neccesary in choosing a theory with one particular value of the Higgs vev. Real life does appear to have a fixed Higgs mass so we require that $SU(2)_{L} \times U(1)_{Y}$ breaks to $U(1)_{em}$. But then aren't we saying that observable physics is described by $SU(3)_{C} \times U(1)_{em}$, why does the weak interaction still work in our universe that contains a fixed Higgs mass?


title: What's wrong with this application of Thomas Precession to circular motion velocity measurements?

text: If you happen to have the Third Edition of Classical Electrodynamics by John David Jackson, turn to section 11.8, as that's where I'm getting all this from. If not, you should still be able to follow along. In said section, Jackson gives us this equation that relates any physical vector **G** in a rotating vs. non-rotating reference frame: $\left(\frac{d\mathbf{G}}{dt}\right)_{nonrot} = \left(\frac{d\mathbf{G}}{dt}\right)_{rest frame} + \boldsymbol{\omega}_T \times \mathbf{G}$ where $\boldsymbol{\omega}_T = \frac{\gamma^2}{\gamma+1}\frac{\mathbf{a}\times\mathbf{v}}{c^2}$ "where **a** is the acceleration in the laboratory frame," according to the textbook. Also, gamma is defined using **v** , the velocity of the particle as measured in the lab frame. Ok. So I decided to check this by setting **G** = **x** , the position vector, for a particle that is undergoing circular motion in the laboratory frame. So we have $\left(\frac{d\mathbf{x}}{dt}\right)_{nonrot} = \mathbf{v}$ and $\left(\frac{d\mathbf{x}}{dt}\right)_{rest frame} = 0$ because the particle doesn't have any velocity in its own frame (right?). So far so good (I think). Now, this implies that $\boldsymbol{\omega}_T \times \mathbf{x} = \mathbf{v}$. So if we can verify this using the definition of $\boldsymbol{\omega}_T$, we're golden. However, if you use the fact that $|a| = \frac{v^2}{|x|}$ for circular motion as well as the fact that **a** is perpendicular to **v** , and that **a** is (anti)parallel to **x** , and carefully apply the right hand rule, you'll find, after the algebraic dust settles, that $\boldsymbol{\omega}_T \times \mathbf{G} = (1-\gamma)\mathbf{v}$ So this is definitely a contradiction. Because it implies that $\mathbf{v} = (1-\gamma)\mathbf{v}$. Can anyone tell me where this went horribly horribly wrong? I worked on this with my professor for two hours yesterday and we couldn't figure it out.


title: Is the static gauge pressure of a free jet always atmospheric?

text: Let's say I have a free jet of air leaving a pipe into the atmosphere. I know that the static gauge pressure at the pipe exit is equal to the atmospheric. But what about the static gauge pressure 10 meters away if the air is still traveling as a free jet? Is it still atmospheric?


title: Gravity, a weak force?

text: Why is gravity such a weak force? It becomes strong for particles only at the Planck scale, around $10^{19}$ $\text{GeV}$, much above the electroweak scale ($100$ $\text{GeV}$, the energy scale dominating physics at low energies). Why are these scales so different from each other? What prevents quantities at the electroweak scale, such as the Higgs boson mass, from getting quantum corrections on the order of the Planck scale? Is the solution super symmetry, extra dimensions, or just anthropic fine- tuning? Can we relate few problems of quantum mechanics with gravity ? Despite the fact that there is no experimental evidence that conflicts with the predictions of general relativity, physicists have found compelling reasons to suspect that general relativity may be only a good approximation to a more fundamental theory of gravity. The central issue is reconciling general relativity with the demands of quantum mechanics. Well tested by experiment, quantum mechanics is the theory that describes the microscopic behavior of particles. In the quantum world, particles are also waves, the results of measurements are probabilistic in nature, and an uncertainty principle forbids knowing certain pairs of measurable quantities, such as position and momentum, to arbitrary precision. The Standard Model is the unified picture of the strong, weak, and electromagnetic forces within the framework of quantum mechanics. Nonetheless, theoretical physicists have found it to be extremely difficult to construct a theory of quantum gravity that incorporates both general relativity and quantum mechanics. At the atomic scale, gravity is some $40$ orders of magnitude weaker than the other forces in nature. In both general relativity and Newtonian gravity, the strength of gravity grows at shorter and shorter distances, while quantum effects prevent the other forces from similarly increasing in strength. At a distance of approximately $10^{-35}$ $\text{m}$, called the Planck length, gravity becomes as strong as the other forces. At the Planck length, gravity is so strong and spacetime is so highly distorted that our common notions of space and time lose meaning. Quantum fluctuations at this length scale produce energies so large that microscopic black holes would pop into and out of existence. A theory of quantum gravity is needed to provide a description of nature at the Planck length. Yet, attempts by researchers to construct such a theory, analogous to the Standard Model of particle physics, have lead to serious inconsistencies.


title: What nonlinear deformations will a fast rotating planet exhibit?

text: It is common knowledge among the educated that the Earth is not exactly spherical, and some of this comes from tidal forces and inhomogeneities but some of it comes from the rotation of the planet itself. The deformation from the rotational effect makes it more oblate spheroid-like, or as I would prefer, "like a pancake". Here is one site illustrating the behavior, and image: ![Earth's shape from rotational distortion from Mathematical Imagery](http://i.stack.imgur.com/gV7Ay.jpg) Literature exists detailing the mathematical expectations for a rotating planet using just hydrostatic forces, for example, see Hydrostatic theory of the earth and its mechanical implications. I like to imagine a ball of water in space held together by its own gravity. I also don't want to deviate from consideration of only hydrostatic (and gravitational) forces because I think it is sufficient for this discussion. It would seem that the solution of the described problem is in terms of a small change in radius as a function of the azimuth angle, or z-coordinate if you take the axis of rotation to be the z-axis. This is using rotational symmetry. In other words, Earth's deformation due to rotation does not depend on longitude. I want to ask about the extreme case. Imagine a planet rotating so fast that it is a very thin pancake. What will occur in this case? I am curious:   * Will the center hollow out, creating a donut shape?    * Will it break up into a multi-body system? It seems to me that it would be logical for the high-rotation case to break up into 2 or more separate bodies. The reason is that a 2 body system is stable an can host a very large angular momentum. But would it be an instability that leads to this case? When would such an instability occur and could a rotating planetary body deform in a different kind of shape from the beginning, such as a dumbbell-like shape, which would transition into a 2-body system more logically than the pancake shape? ![Would THIS happen?](http://i.stack.imgur.com/aHZ9N.jpg) image link To sum up, how would a pancake shape transition into a dumbbell shape? Or would it? What are the possibilities for the described system?


title: scattering theory problem

text: I am studying scattering theory right now in my QM class, and I'm attempting the Griffiths problem 11.4 as an exercise (it's not for homework). The problem is: Consider the case of low-energy scattering from a spherical delta-function shell: $V(r) = \alpha \delta(r-a)$. Calculate the scattering amplitude $f(\theta)$, the differential cross-section, and the total cross-section. He outlines the following method in the section: in the exterior region where $V(r) = 0$, you get an expression for the wavefunction in terms of some complicated Hankel function expression $$ \psi(r,\theta) = A \sum i^l (2l+1)[j_l(kr) + ika_lh_l^{(1)}(kr) ] P_l(cos \theta). $$ You then match boundary conditions with the explicit solution to the wavefunction inside the region where $V(r) \neq 0$. I'm confused how to carry this out. It seems that the solution of the wavefunction inside the region $r<a$ is your typical plane wave $f(r) = \sin(kr)/r$. Then I'm thinking that you should match $f(a) = \psi(a)$ and $\psi'(a)-f'(a) = - \frac{2m\alpha}{\hbar}^2 f(a)$. However, this doesn't appear to give me the right answer.. So is this the correct approach (if not, how do you go about these kinds of problems)?


title: Why every galaxy is moving away from every other galaxy on astronomical scale? because of an explosion or because of creation of space?

text: Why galaxies move away from each other in general on the astronomical scale? Which answer is correct of the following? It is because of the big bang theory, everything is just moving away from everything else on the astronomical scale, because of the explosion that happened in the past, just by inertia. It is because space is being created all the time between galaxies, that is why they look receding from each other. Like drawing dots on a rubber sheet, then start stretching that rubber sheet. Dots are not moving, just more and more space is created between them. Which answer is the right one? and if the 2nd answer is the right one, is it only space that is being created or spacetime?


title: Quantum Mechanics - Hidden Variables

text: In Steven Weinberg's Lecture on Quantum Mechanics (p. 342), he writes: > The correlation between the spins of the two particles can be expressed as > the average value of the product of the $\hat{a}$ component of the spin of > particle 1 and the $\hat{b}$ component of the spin of particle 2: > > $\langle (\textbf{s1} \cdot \hat{a}) (\textbf{s2} \cdot \hat{b}) \rangle = > -\frac{\hbar^2}{4} \int{d\lambda\ \rho(\lambda) S(\hat{a}, \lambda) > S(\hat{b}, \lambda)}$ > > where $\hat{a}$ and $\hat{b}$ are any two unit vectors. > > In quantum mechanics, we have: > > $\langle (\textbf{s1} \cdot \hat{a}) (\textbf{s2} \cdot \hat{b}) \rangle = > -\frac{\hbar^2}{4} \hat{a} \cdot \hat{b}$ > > There is no obstacle to constructing a function $S$ and a probability > density $\rho$ for which are equal for any single pair of directions > $\hat{a}$ and $\hat{b}$. Can someone explain to me precisely how this is done and what $S(\hat{a}, \lambda)$ and $\rho(\lambda)$, along with the hidden variables $\lambda$ would be? My guess is that since we want something that is rotationally invariant since it depends only on the dot product, we want some $S$ and hidden variables $\lambda$ that act nicely under rotations. On the other hand, since the first equation is a local hidden variable theory, I thought this was not possible by Bell's inequalities...


title: What exactly is temperature in the Gibbs free energy?

text: In equation of Gibbs free energy change $(\Delta G) = (\Delta H)-T(\Delta S)$, is $T$ the system's or surrounding's temperature? Edit:Oh sorry I am not clear earlier now I get clear question.I know we have to calculate gibbs free energy of system but criterion for spontaneity of ΔStotal should be greater than zero.When we relate it to gibbs free energy to show that gibbs free energy change should be always negative we keep both system and surrounding temperature same with pressure also constant.Then how process can occur?Wikipedia says that it is chemical potential that undergo changes there but what about gibbs free energy change. I think I made too hurry to ask question when I am studying but I am running out of time so I ask here. **NEW EDIT TO QUESTION** :I found that above equation can only be applied to open systems but the quoted statement at top of **_Free energy of reactions_** section taken from Wikipedia Article > To derive the Gibbs free energy equation for an **_isolated system_** , let > Stot be the total entropy of the isolated system, that is, a system that > cannot exchange heat or mass with its surroundings. According to the second > law of thermodynamics: the word isolated system means that the formula $\Delta G=\Delta H-T\Delta S$ is derived for isolated system.But I think exactly in opposite way that it is true for open systems.Read Above question before this New EDIT to understand my question.Please Help me.


title: Energy Functional

text: I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\int_M(R+|\nabla f|^2)e^{-f}d\mu$ is introduced as an energy functuional, where $M$ is a closed manifold, $g$ is Riemannian metric, $R$ is Ricci scalar, and $f$ is any function that in the physics literature is called dilaton. I do not know why these functionals are attributed to the energy concept and why does $f$ show dilaton concept? Can anyone help me?


title: Calculating water flow as function of the height of water above the inlet pipe

text: The Source of Water is Pond where the water level do change randomly. A pipeline of 200mm pipe diameter run for around 2500 m. The Height difference between the level of the inlet pipe and the level of the exit pipe is around 9m. In other word: Water need to run by gravity for 2500 m. Difference in height is 9m. My question: How I could calculate the water volume/flow received at the exit pipe as function of the height of the water above the inlet pipe. Ex: If the water in the Pond is covering the inlet pipe by say 10 cm, what is the expected flow at the exit pipe.


title: Emissivity and thermal equilibrium

text: Lets say I enter a closed room with the walls and everything in it (including me and my eyes) at thermal equilibrium. Its a very hot room, but my super-eyes still work at 5000 degrees Kelvin. I have two propositions which are in conflict. The first is that I should be effectively blind when I am in the room. For me to see anything at all, the pigments in my retina would have to absorb some of the thermal radiation and convert it into electrical impulses for my brain. (The same applies for anything which detects radiation). But we can't extract energy from the heat without a cooler object, and there is none, so we can't use the energy in the black-body radiation for vision. There is no available free heat energy; everything is at thermal equilibrium. What is happening is that the pigments in my retina are generating black body radiation at the same rate as they are absorbing black body radiation from the object being looked at, so no net chemical change can occur and hence no vision. However, I also know that objects have different emissivity, which means some objects in the room will be brighter than others. If I am looking at an object of lower emissivity than the pigments in my eyes, my eyes should radiate more power out than they absorb from the object, and vice versa for an object of higher emissivity. This means that if I look at different things, the pigments in my eyes will react differently to different objects, meaning vision is possible. But this contradicts the first proposition. Can somebody explain this "paradox" ?


title: Tunneling and transmission

text: Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step: ![enter image description here](http://i.stack.imgur.com/aFgQ4.png) If particle is comming from the left a general solutions to the Schrödinger equations for sepparate intervals I, II and II are: \begin{align} \text{I:}& & \psi_1 &= \overbrace{A e^{i\mathcal L x}}^{\psi_{in}} + \overbrace{Be^{-i \mathcal L x}}^{\psi_{re}}& \mathcal L &= \sqrt{\tfrac{2mW}{\hbar^2}}\\\ \text{II:}& & \psi_2 &= C e^{\mathcal K x} + De^{-\mathcal K x}& \mathcal K &= \sqrt{-\tfrac{2m(W-W_p)}{\hbar^2}}\\\ \text{III:}& & \psi_3 &= \underbrace{E e^{i \mathcal L x}}_{\psi_{tr}}& &\\\ \end{align} Where $\psi_{in}$ is an incomming wave, $\psi_{re}$ is a reflected wave and $\psi_{tr}$ is transmitted wave. I used the boundary conditions and got a system of 4 equations: \begin{align} {\tiny\text{boundary}}&{\tiny\text{conditions at x=0:}} & {\tiny\text{boundary conditions}}&{\tiny\text{at x=d:}}\\\ A + B &= C + D & Ce^{\mathcal K d} + De^{-\mathcal K d} &= E e^{i \mathcal L d}\\\ i \mathcal L A - i \mathcal L B &= \mathcal KC - \mathcal K D & \mathcal K C e^{\mathcal K d} - \mathcal K D e^{-\mathcal K d}&= i \mathcal L E e^{i \mathcal L d} \end{align} So now i decided to calculate coefficient of transmission $T$: \begin{align} T &= \dfrac{|j_{tr}|}{|j_{in}|} \\!=\\! \Bigg|\dfrac{\dfrac{\hbar }{2mi}\\! \left( \dfrac{d\overline{\psi}_{tr}}{dx}\, \psi_{tr} - \dfrac{d \psi_{tr}}{dx}\, \overline{\psi}_{tr} \right)}{\dfrac{\hbar}{2mi} \\!\left( \dfrac{d\overline{\psi}_{in}}{dx}\, \psi_{in} - \dfrac{d\psi_{in}}{dx}\, \overline{\psi}_{in} \right) }\Bigg| \\!=\\! \Bigg|\dfrac{\frac{d}{dx}\big(\overbrace{Ee^{-i\mathcal L x}}^{\text{konjug.}}\big) Ee^{i\mathcal L x} - \frac{d}{dx} \left( Ee^{i\mathcal L x}\right)\\! \overbrace{Ee^{-i\mathcal L x}}^{\text{konjug.}}}{ \frac{d}{dx}\big(\underbrace{Ae^{-i\mathcal L x}}_{\text{konjug.}}\big) Ae^{i\mathcal L x} - \frac{d}{dx} \left( Ae^{i\mathcal L x}\right)\\! \underbrace{Ae^{-i\mathcal L x}}_{\text{konjug.}}}\Bigg|\\! = \nonumber\\\ &=\Bigg|\dfrac{-i\mathcal L Ee^{-i\mathcal L x} E e^{i \mathcal L x} - i\mathcal L E e^{i \mathcal L x} Ee^{-i \mathcal L x}}{-i \mathcal L A e^{-i\mathcal L x} Ae^{i \mathcal L x} - i \mathcal L A e^{i \mathcal L x}Ae^{-i \mathcal L x} }\Bigg|=\Bigg|\dfrac{-i\mathcal L E^2 - i\mathcal L E^2}{-i \mathcal L A^2 - i \mathcal L A^2}\Bigg|=\Bigg|\dfrac{-2 i \mathcal L E^2}{-2i\mathcal L A^2}\Bigg| = \frac{|E|^2}{|A|^2} \end{align} It accured to me that if out of 4 system equations i can get amplitude ratio $E/A$, i can calculate $T$ quite easy. Could anyone show me how do i get this ratio?


title: Dynamic ferrofluid sculptures

text: http://www.youtube.com/watch?v=UJJuq_pcyIQ What exactly is going on in the video example? I understand the phenomena occurs because of magnetism but I am trying to figure out the mechanics behind this sculpture. There obviously is a magnet underneath but what is it doing? Is it moving? Is it getting some type of charge? Any insight is great!


title: Collision of 2 particles - calculating the mass and a speed after the collision

text: Lets say we have a particle of mass $m_1$ which has a kinetic energy $W_{k1}$. This particle collides with another same particle. How can i calculate mass $m_2$ and the speed $v_2$ of the particle which is formed out of the colliding two particles? I know that relativistic preservation of energy $W$ and momentum $p$ must hold and i know these equations: \begin{align} W^2 &= p^2c^2 + {W_0}^2\\\ W &= W_k + W_0\\\ W &= mc^2 \gamma(v)\\\ p &= mv\gamma(v)\\\ \end{align} * * * My attempt was something like this: \begin{align} p_2 &= p_1\\\ p_2 c&= p_1 c\\\ {p_2}^2 c^2&= {p_1}^2 c^2\\\ {W_1}^2-{W_{01}}^2 &= {W_2}^2-{W_{02}}^2 \longleftarrow \smash{\substack{\text{Doesnt it hold that $W_1 = W_2$?}\\\ \text{Can i cross them out to get}\\\\\text{the line below?}}}\\\ {W_{01}}^2 &= {W_{02}}^2\\\ \left(2m_1c^2\right)^2 &= \left(m_2 c^2\right)^2\\\ 4{m_1}^2c^4 &= {m_2}^2 c^4\\\ m_2 &= 2m_1 \end{align} Is this even correct? My professor said it is not... Why?


title: Equation describing magnetic hysteresis

text: So when you're looking at B-H curves for ferromagnetic substances, you often see these magnetic hysteresis curves, which occur, I gather, largely because of domain formation which has some reversible and some irreversible components: ![A magnetic hysteresis curve.](http://i.stack.imgur.com/DNjCd.png) I've been reading through many papers, web sites and this book (Hysteresis in Magnetism), but I haven't seen any equations for how to generate these curves. I recognize that there may be no easy way to express the entire curve in a single equation, but clearly people are generating these plots, and I think they're doing it from some equations, presumably based on some characteristic parameters like the saturation remanance, the coercive force, etc. Is there anyone out there who can help me with this? Personally, my first choice would be an equation that actually describes the system given some set of material conditions (subject to some constraints is fine, too) but if that's quite complicated and for the most part, people just stitch together two sigmoid functions until it looks about right, then I'm fine with that answer as well, so long as there's some justification for why this is done in there somewhere.1 1Full disclosure: I am in thesis deadline mode and I'm currently probably a 5 on the Stanford Sleepiness Scale, so I apologize if some of the words in this post don't make sense - I'll try and edit it if people find that I haven't adequately conveyed my question.


title: calculating work done by friction

text: I want to calculate the work done by friction if the length $L$ of uniform rope on the table slides off. There is friction between the cord and the table with coefficient of kinetic friction $\mu_k$. $$ W = \int F \cdot d \vec{s}$$ I think it would be: $$ W_{fr} = \frac M L g \int_{0}^{L} dx$$ But the solutions (which could be mistaken) say: $$ dW_{fr} = \mu_k \frac M L g \, x \, dx$$ which is then integrated. Should there be an $x$ in the integral? I don't think there should be because you are summing up over an infinitesimal displacement $dx$ and the force of friction is not proportional to the displacement at any instant (I think). ![enter image description here](http://i.stack.imgur.com/lKzuO.png)


title: Do high/low pass lenses exist?

text: For an experiment I will hopefully be soon conducting at Johns Hopkins I need two different lenses. The first needs to allow all wavelengths above 500 nm to pass (thus a high pass filter) and cut off everything else.   The second needs to allow all wavelengths below 370 nm to pass (thus a low pass filter) and cut off everything else. My knowledge of optics is middling. I know that good old glass cuts of UV light, but I was hoping for something more specific. Does anyone know of the theory necessary to "tune" materials to make such filters? Truth be told, I'm an experimentalist, so simply giving me a retail source that has such lenses would get me to where I need to go! But learning the theory would be nice as well. Thanks, Sam


title: Thermodynamically possible to hide a Dyson sphere?

text: You build a Dyson sphere around a star to capture all its energy. The outer surface of the Dyson sphere still radiates heat at much higher temperature than the cold space background, so you're easy to detect. But you'd like to stay hidden. So you cool the outer surface of the Dyson sphere to near cold space background. Of course you still need to radiate your excess heat somewhere, so you plan to radiate it off in directed beams, away from the directions of the nearby solar systems, to stay hidden from your neighbors at least. **Questions** :   1. Is such directed radiation of excess heat allowed by known laws of physics?   2. Would the energy of the star be sufficient for running the cooling system?


title: A spring with two masses in an inelastic collision

text: An ideal spring is attached to a wall, and the other end is attached to a mass $m$. The spring is initially compressed a distance $x$. After it is released, the mass collides with another mass $2m$ at a distance $x/2$ to the right of the spring equilibrium. The collision is inelastic and they slide together. How far will they slide before coming to a momentary stop? **My work** Since the collision is inelastic then mechanical energy is not conserved. And since the two-mass system has an external force, momentum is not conserved. So one has to use only mechanical principles to solve this. We can compute the velocity of the mass at the moment of impact using simple harmonic motion, $r(t) = -x\cos\left( t\sqrt{k/m}\right)$ so that $x/2 = -x\cos \left(t^{*}\sqrt{k/m}\right)$. With this I could do an awkward solution for time and maybe continue from there. At this point, though, I'm not perfectly clear on how to move forward. Do I just assume that momentum is conserved for a very, very short time period during the collision and use that to find the velocity of the combined mass system, then put that into another simple harmonic motion problem?


title: Is the photon energy required to cause an atomic transition $\Delta E+\Delta KE$, where $\Delta E$ is the "transition energy"?

text: An atom "at rest" can absorb a photon, and while some of this energy goes into increasing the energy level of the electron, momentum must be conserved, and so some energy must also increase the kinetic energy of the atom. The amount of $KE$ delivered to the atom is easy to calculate. Therefore, is the energy to cause a transition $\Delta E+\Delta KE$, where $\Delta E$ is the "transition energy"? So that no commenters cover old ground, please see this question.


title: What is the real-world significance of the Bekenstein bound?

text: The Bekenstein bound sets the maximum amount of information that can be contained in a region of space/energy, and is usually referred to in the same way as computer storage density: > For example, a single hydrogen atom, if it were to code as much information > as permitted by the Bekenstein Bound, would code about 4×106 bits of > information, since the hydrogen atom is about one Ångström in radius, and > has a mass of about 1.67×10−27 kilograms. (source) > Nature permits a surprising amount of information to be encoded before the > Bekenstein bound is reached. For example, a hydrogen atom can encode about 1 > Mb of information — most of a floppy disk. (source) > The "Bekenstein bound" leaves room for a million bits in a hydrogen atom > (source) But what does this really mean? How could _any_ information be stored in a hydrogen atom?


title: Is time going backwards beyond the event horizon of a black hole?

text: For an outside observer the time seems to stop at the event horizon. My intuition suggests, that if it stops there, then it must go backwards inside. Is this the case? This question is a followup for the comment I made for this question: Are we inside a black hole? > Food for thought: if time stops at the event horizon (for an outside > observer), for inside, my intuition suggests, time should go backwards. So > for matter, that's already inside when the black hole forms, it won't fall > towards a singularity but would fall outwards towards the event horizon due > to this time reversal. So inside there would be an outward gravitational > force. It would be fascinating if it turns out that all this cosmological > redshift, and expansion we observe, is just the effect of an enormous event > horizon outside pulling the stuff outwards. So from outside: we see nothing fall in, and see nothing come out. And from inside: we see nothing fall out, and see nothing come in. Hopefully the answers make this clear, and I learn a bit more about the GR. :)


title: In solar cells, do photons break apart electron-hole pairs, or create them?

text: Some sources say that when a photon hits the PV cell, it breaks apart electron-hole pairs. Other sources say that photons _create_ electron-hole pairs. Can anyone explain which one is right? I've read several explanations of what goes on in the solar cell, but they don't seem very clear. To me, who has little prior knowledge, a few aspects of the many different explanations seem to contradict each other. I don't know if it's because some of the explanations said things a certain way for simplicity's sake, but this part seems too significant to ignore. For example, http://www.solarenergyexperts.co.uk/buyersguides/photovoltaic- glass-how-does-it-work/: "When photons (light particles) from the sun hit the cell, the energy breaks up the paired particles. The freed electrons go into the n-type layer, while the holes go down into the p-type layer." science.howstuffworks.com/environmental/energy/solar-cell3.htm : "When light, in the form of photons, hits our solar cell, its energy breaks apart electron- hole pairs." solarjourneyusa.com/bandgaps.php has a hole section called " _Generation_ of electron-hole pairs" This one basically seems to be talking about the same thing, but doesn't really mention electron-hole pairs, and the explanation seems more confusing to me, but anyway, http://www2.pv.unsw.edu.au/nsite- files/pdfs/UNSW_Understanding_the_p-n_Junction.pdf: "...broken bonds created by the light act as holes ... and these holes are also free to move throughout the material. Electrons and holes created in this way are physically near each other: for every electron excited by the light there is a corresponding hole generated. These electrons and holes can remain excited only for a short period of time. In a process called recombination, excited electrons stray too close to holes and the two fall back into bonded positions." This one seems to be saying that photons break apart the bonds between the electron and its atom, which creates a free electron and a hole, i.e. another one saying that an electron-hole pair is created. solarcellcentral.com/junction_page.html : "When photons hit the solar cell, free electrons attempt to unite with holes in the p-type layer." www.solarenergy.net/Articles/how-photovoltaic-cells-work.aspx : "When enough photons are absorbed by the negative layer of the photovoltaic cell, electrons are freed from the negative semiconductor material." So what actually happens when light hits a solar cell?


title: Will relative velocity with respect to electrons effect current in wire and magnetic field produced by wire?

text: **Question:** > consider a long, straight wire of cross-sectional area $A$ carrying a > current $i$. Let there be $n$ free electrons per unit volume. An observer > places himself on a trolley moving in the direction opposite to the current > with a speed $v = \frac{i}{nAe}$ and separated from the wire by a distance > $r$. The magnetic field seen by the observer is very nearly **My Answer:** > Zero. Because current is $neAv$ where $v$ is drift velocity of electrons. > Relative velocity between him and electrons is zero. So, no flow of charge > through any cross-section according to him. So no current. So no magnetic > field. **Actual answer:** > $\frac{\mu\ i}{2\cdot\pi\cdot r}$ where $\mu$ is the permeabilty of free > space.


title: What is it called when a fluid will "jump" to grab onto an object that comes very close?

text: I'm doing an experiment where I bring a probe very close to a well full of fluid and then very slowly lower it to obtain some force deformation values. The material behaves very much like a fluid and when I bring the probe close to the fluid, it sort of jumps up and grabs the probe and forms like a "column?" that actually pulls on the probe rather than pushing on it. Anyway, I'm trying to coat the probe with an oil or something that will prevent this, but I don't know how to look into it because I don't know what its called or how its described in the literature. Any ideas on what this phenomena is called?


title: What do we learn from gravity in three spacetime dimensions?

text: The last decades there has been a lot of research going on in the the area of three dimensional gravity. The motivation, I understand, is threefold:   1. Whereas gravity is not perturbatively renormalizable in four spacetime dimensions, in three dimensions it is. To make it even more interesting it has black hole solutions and it is exactly solvable. This opens the way to to study quantum black holes. This make three dimensional gravity a very interesting system on itself.   2. Through the AdS/CFT correspondence there is a connection between conformal field theories (CFT) in two dimensions and gravity in three dimensions. CFT's are important in condensed matter physics and one can use 3D gravity to learn more about them.   3. Gravity in three dimensions is simpler to deal with then gravity in four dimensions. Therefor it can be used as a toy model for gravity in four dimensions.  I am wondering what are the most important insights that 3d gravity brought in these respects? In particular I am interested in point three: did 3d gravity provide any new view on 4d gravity so far?


title: Where is the Atiyah-Singer index theorem used in physics?

text: I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my question is, what are some examples of these applications?


title: Critical point vs Gibbs phase rule

text: Why is the critical point for the phase diagram of pure water degrees of freedom equal to 0? Maybe, you know what is the mathematical explanation for the fact that the number of degrees of freedom at the critical point is 0? What else is affected by the lack of degrees of freedom in addition to the Gibbs phase rule?


title: Speed of gravity

text: Consider two objects presented in the figure below. Objects have equal masses and are separated by a distance of 60 light seconds. Assume that we move left object by 3 light seconds to the left in 30 seconds. This requires energy input, lets say it's equal to X. Change in potential energy is also equal to X. Now, the right object does not 'know' yet that the left object was moved. Left object was moved 30 seconds ago and this information requires 60 seconds to reach the right object. We have 2 options: **A:** We can move the right object by 3 light seconds to the right with the speed of 0.1c now. **B:** We can move it with the same speed and by the same distance later - let's say after 60 seconds. Clearly case A requires the same energy input as when moving left object = X. This is because right object still 'thinks' that the left mass in the same place. In the case B the right object 'knows' that the left mass was move so we will need less energy to move it. Energy needed < X. The end state in A and B is the same. Why it is possible to achieve it with 2 different energy inputs? If we choose method A the increase in potential energy is smaller than energy input X. Where is the missing energy? Maybe the 'speed of gravity' is infinite? ![2 objects](http://i.stack.imgur.com/dgCfA.png)


title: Navy's new Mach 7 Rail Gun Photo

text: There have been a few articles about the Navy's new Mach 7 33 Megajoule railgun. As a physics teacher, I have a couple of questions about this, and was hoping for some help.   1. Is the kinetic energy of the projectile 33 MJ, or is that the energy that was delivered from the capacitor bank that was used to fire the projectile, so that some of it would have been lost in resistive heating, etc.    2. The photos of the projectile show a cloud of hot gas behind the projectile? What is this? superheated air? Plasma from the electrical breakdown inside the railgun? What causes this?


title: Hierachies of AdS/CFT holographies

text: One of the most disturbing aspects of General Relativity is the 'Marble versus Wood' duality of the theory: Matter creates curvature, and curvature doesn't create curvature (at least not directly) The holographic principle suggests a reason for this division: matter fields and gravity are paired to structures that exist in different dimensions: so 4D gravity (if taken over a finite region $\Omega$ of spacetime) should be paired with some (abstract or actual) 3D (2 spatial+1 time?) boundary $\partial \Omega$ where a QFT of some sort exists, but the curvature in this 4D spacetime is dictated by 4D distributed matter fields, which themselves might be related to some 5D gravity (possibly actual, but maybe unobservable) that has the 4D spacetime as part of its boundary. Arguably one could extend this idea to lower and higher dimensional mappings. Besides the pragmatic but inherently hard question about what could be gained by such insight, I'm asking if > Has anyone tried to formulate physical theory in terms of a hierarchy of > such mappings? It seems that such formulation would have to deal somehow with the fact that the boundary operator is nilpotent ($\partial \partial \Omega = 0$) when defining the hierarchy So, if the set $HM(D)$ gives all the possible meaningful holographic mappings between compact manifolds of dimension $D$ and their boundaries, then one can imagine that a full description of physics could be achieved by experimentally finding the elements $hm_0, hm_1...hm_X$ with $hm_i \in HM(i)$ and $X$ some maximum dimension above which all different mappings become completely unobservable at 4D


title: Applying Poynting's theorem to this system?

text: A wire placed in a magnetic field and has current flowing, it will experience the Lorentz force, work is done by the electric field and energy is conserved. Being converted from the power source to KE, and the wire moves. How can we apply Poynting's theorem to state that this system's energy will always be conserved? To fully observe the conservation of energy from the power source is equal to the work done by the wire when it gains KE.


title: Explaining chirality for spin 1/2 particle

text: I found the following explanation for chirality for spin 1/2 particles here > ![enter image description here](http://i.stack.imgur.com/mUD0C.png) > > What happens when you rotate a left- vs right-chiral fermion 360 degree > about its direction of motion. Both particles pick up a -1, but the left- > chiral fermion goes one way around the complex plane, while the right-chiral > fermion goes the other way. The circle on the right represents the complex > phase of the particle's quantum state; as we rotate a particle, the value of > the phase moves along the circle. Rotating the particle 360 degrees only > brings you halfway around the circle in a direction that depends on the > chirality of the fermion. My question is: Isn't the way the phase of the wavefunction changes related to the direction of rotation of the particle, which is an external parameter depending on how you rotate it? Is there a better explanation for chirality?


title: Range of electromagnetic waves

text: I was reading this article. There is a statement "It is a well-known fact that the telecom towers mounted with antennas in the lower frequency bands can cover far greater areas than those using the 1800 frequency bands." Is this accurate/wrong/a simplified version. One point which made me think that it is wrong is the fact that Satellite TV uses high frequency bands for communication


title: Could observing unknowingly cause the wave function to collapse?

text: I've recently been reading (at a basic level) about the double slit experiment and how the mere act of observing can cause the "wave function to collapse", as they say. I find this mind-blowingly fascinating, and it naturally sparks some questions. In particular, consider the following scenario: I perform the double slit experiment and detect which slit was traversed, sending the data to some computer. This would cause the wave function to collapse, as I understand. Now, imagine that I delete the data as it is being read by the computer. If I am not mistaken, the wave function would not collapse in this case, since there would be no way for me to determine which slit was traversed. _Now,_ imagine that I am under the _belief_ that the data is being deleted when in reality it is being saved to some disk. What would happen in this case? Assuming I never come to know about the existence of the saved data, would the wave function collapse?


title: Question about de Broglie Waves?

text: Is photon interaction , electrostatic interaction outside the nucleus and gravitational interaction is all due to electromagnetic waves ? and CAN be identified as with the de Broglie waves ? I thought of a theory in which is assuming that photon interaction , electrostatic interaction outside the nucleus and gravitational interaction is all due to electromagnetic waves and can be identified as with the de Broglie waves in order to explain how the force of gravitation act between particles . So can this assumption stand a chance of being theoretically and experimentally correct . Please add your comments regarding this in your answer and please explain why or why not .


title: Divergence equations (Maxwell)

text: Let $\mathbf{E}(r,t),\mathbf{B}(r,t)$ be two vector fields (in $\mathbb{R}^3$), s.t. they satisfy fot $t=0$ the equations:   1. $\nabla \cdot \mathbf{B}(r,0)=0.$    2. $\nabla \cdot \mathbf{E}(r,0)=\frac{\rho(r,0)}{\epsilon_0}.$ The question now is: What properties have $\mathbf{E}$ and $\mathbf{B}$ got in order to satisfy equation 1 and 2 for all $t>0$? I think $\mathbf{B}$ must be independent of $t$, i.e. $\mathbf{B}(r,t)=\mathbf{B}(r,0)$ for all $t>0$. But what about $\mathbf{E}$? Doesn't the answer depend on the function $\rho(r,t)$?


title: Plasmon Based Transistor vs conventional one

text: Conventional wires become very lossy at high frequencies. **Is this because of the skin effect?** Plasmon based computer chips, since plasmons can support much higher frequencies (~100 THz range). Aren't free electron oscillations in a metal plasmons. What is the difference and why can they support such high optical frequencies?


title: Lagrangian Dynamics Question

text: Two equal masses of mass M are glued to a massless hoop of radius R is free to rotate about its center in a vertical plane. The angle between the masses is 2$\theta$. Find the frequency of oscillations. My attempt at a solution is ($\phi$ is the displaced angle): $$x_{m1}=-R\sin (\theta -\phi)$$ $$y_{m1}=-R\cos(\theta-\phi)$$ $$x_{m2}=R\sin(\theta+\phi)$$ $$y_{m2}=-R\cos(\theta +\phi)$$ I got my lagrangian and then found the Euler Lagrange equations in terms of $\phi$. I found that $$\ddot {\phi}=\frac{-g\cos(\theta)}{R}\sin\phi$$ So for small angles, $\sin(\phi)\approx\phi$ which means $\omega^2=\frac{g\cos(\theta)}{R}$ Can anyone let me know if this makes sense. I think I did it correctly, but I am not completely sure. Thanks! EDIT:: Lagrangian: $$L=T-U=mR^2\dot{\phi}^2+2mg\cos(\theta)\cos(\phi)$$


title: Can sand falling in a floating hourglass cause it to sink? (Follow up to hourglass question)

text: This is related to this question:Does the weight of an hourglass change when sands are falling inside? At Brigham Young University, there is a display consisting of a sealed off liter bottle with a sunken sealed off hourglass. When one turns over the bottle, the hourglass floats as the sands flow down. Eventually, the hourglass reaches some transition, and precipitately changes from floating to sinking. It has a sign next to it saying that it's for a fishing lure,and that you should not reveal the answer if you get it. What causes it to change from floating to sinking so quickly?


title: Gaussian Probability Distribution?

text: The uncertainty principle states that, $$\sigma _{{x}}\sigma _{{p}}\geq {\frac {\hbar }{2}}.$$ It is mentioned from many sources that the probability distribution of the particle position and momentum would follow a Gaussian distribution. Why is it a Gaussian distribution? is this the distribution that minimizes uncertainty? Is this distribution definitely the case for the uncertainty principle or can it be different under different conditions? Has this been proven? What are the formulas for position and momentum probability distributions of a free particle? How is this derived from the wave function? What would be the formulas of the probability distributions for the position and momentum for a system of 2 identical bosons separated by a distance $R$?


title: Relation between the period of rotation and the period of revolution of a satellite

text: I read somewhere that the tidal forces between the earth and the moon causes the equality between the 2 periods of the moon and that every planet-satellite system will evolve to this condition (like pluto-charon of you consider pluto a planet). Can somebody give me some references (books, articles) discussing this idea and why doesn't this happen in a star-planet system?


title: Why do we assume local conformal transformations are symmetries in 2D CFT

text: The global conformal group in 2D is $SL(2,\mathbb{C})$. It consists of the fractional linear transforms that map the Riemann sphere into itself bijectively and is finite dimensional. However, when studying $CFT_2$ people always use the full Virasoro algebra, not just the $L_{0,\pm1}$ which actually exponentiate to invertible transformations. I would like to know why people consider the other $L_n$'s to be symmetries of the theory. I am aware that Ward identities are local statements, and that I can consider a coordinate patch where the additional conformal transformations are bijective in order to derive relations between correlation functions in this patch. I am also familiar with the representations of the Virasoro algebra and how constraining the symmetry is. However, we are doing quantum mechanics, and a symmetry of the theory should take me from one physical state to another. In addition, the symmetry should have an inverse which undoes this transformation. This means that the physical Hilbert space should organize itself into representations of the symmetries of the theory. However, the local conformal transformations cannot have inverses, and so they do not form a group as far as I know. So why is it assumed that the states of a $CFT_2$ should organize themselves into representations of the Virasoro algebra? (I am aware $L_{n \leq -2}|0\rangle\neq 0$, $\langle 0|L_{n\geq 2} \neq 0$ so all but the $L_{0,\pm 1}$ are "spontaneously broken" on the in/out vacuum, but this is not relevant since the states of the theory are still assumed to assemble into Verma modules since it is assumed that the Virasoro was a symmetry of the theory which is just violated by the vacuum). My question basically boils down to: How can I have symmetries of a theory which are not invertible? I'd appreciate any comment's that clear up my confusion.


title: How general relativity gets to an inverse-square law

text: I understand that a general interpretation of the $1/r^2$ interactions is that virtual particles are exchanged, and to conserve their flux through spheres of different radii, one must assume the inverse-square law. This fundamentally relies on the 3D nature of space. General relativity does not suppose that zero-mass particles exchanged. What is the interpretation, in GR, of the $1/r^2$ law for gravity? Is it come sort of flux that is conserved as well? Is it a postulate? Note that I am not really interested in a complete derivation (I don't know GR enough). A physical interpretation would be better. Related question: Is Newton's Law of Gravity consistent with General Relativity?


title: Group theory notation used in physics (AdS/CFT)

text: This in the context of the AdS/CFT correspondence. I am reading this review on AdS/CFT Aharony et. al. (The MAGOO review) The abstract can be found here Equation (2.50) of the above paper lists the $R$-symmetry groups of extended supersymmetries in $AdS_{p+2}$. He uses the following notation for the group: > $AdS_5: SU(2,2 | {\cal N}/2),~{\cal N}=2,4,6,8$ Can anyone explain this notation to me?


title: 1/2 freeze a water bottle

text: Is it possible to quick freeze half of an unopened bottle of water or soda by putting half of the bottle in a subfreezing solution liquid nitrogen or other solution without freezing the whole bottle?


title: Electric potential on a uniformly charged tube

text: A hollow cylinder, with no top or bottom, of radius $R$ and length $L$ is uniformly charged with density $\sigma>0$. I have to find the point on space where a point charge $q>0$ has to be drawn from $\infty$ with speed $v\rightarrow0$ so that the work done is maximum, so I have to find the point with maximum potential. What I did was find the electric field, using Gauss's law, and then set $V(\infty)=0$: $0<\rho < R: E=0 \\\R<\rho < \infty: E=\frac{\sigma R}\rho$ so $V(\rho)=-\int\limits_\infty^\rho{E.dl}$ . For $R<\rho<\infty$ I get $V(\rho)=\sigma R \ln(\infty/\rho) $, so here I don't know what to do. Maybe I shouldn't have used Gauss's law, and calculate $E$ the 'classic' way, integrating over the cylinder, but the integral looks so complicated. This problem was given on an exam and trust me, if the integral is actually that complicated, I wasn't supposed to do it that way. So, what should I do?


title: References for Understanding Minahan's N=4 SCFT review

text: This is about the same paper as this thread: Some questions about chapter I.1 (by Minahan) of the "Review of AdS/CFT Integrability" but it was never answered. I have some different questions about it and I'll separate them into a couple posts if need be. I'd also be grateful if anyone can recommend other introductions or reviews for understanding N=4 generally and the Minahan review http://arxiv.org/pdf/1012.3983v2 in particular. Some of the algebra/group theory was particularly hard for me to follow (highest-weight reps, Cartan subalgebras...). Some questions I'm particularly intrigued/troubled by are:   1. After (3.1) he says an operator $O(x)$ having dimension $\Delta$ means that when $x\rightarrow \lambda x$, then "$O(x)$ scales as $O(x) \rightarrow \lambda^{-\Delta} O(\lambda x) $." Should this be $O(x) \rightarrow \lambda^{-\Delta} O(x) $? If we say that $O(x)$ is some polynomial of degree $n$ in $x$, then after the rescaling $O(x)$ will be a polynomial of degree $n$ in $\lambda x$. So we'd have $O(x) \rightarrow O(\lambda x) \sim \lambda^n O(x)$. Then if we identify $-\Delta = n$ we have $O(x) \rightarrow O(\lambda x) \sim \lambda^{-\Delta} O(x)$. Am I missing something?   2. How does he get eq. (3.2)? It apparently follows from $D$ being the generator of scalings, by which he says he means that $O(x) \rightarrow \lambda^{-iD} O(x) \lambda^{iD}$. I'm confused by this, too, as I expect to see the generator exponentiated by $e$, not $\lambda$. I'd expect something like $e^{-i\lambda D} O(X) e^{i\lambda D}$, with $D$ as the generator and $\lambda$ as the parameter.    3. Later, in eq. (3.9), he introduces the $R_{IJ}$ as the $SO(6)$ R-symmetry generators, as well as some matrices $\sigma^{IJ}$ that he only addresses later. Here $I, J = 1...6$. I don't understand the notation. Why are there two indices on these guys? And if it's an $SO(6) \sim SU(4)$ symmetry group, then there should only be 15 generators. So are some of these $R$ and $\sigma$ redundant? Because naively it would appear that we have $6\times 6=36$ of each. I suspect that I'm missing something about how to understand these indices.    4. Kind of the same thing as 3. In (3.14) he gives some of those $\sigma^{IJ}$ and states that they are the generators in the fundamental $SU(4)$ representation. Why? Where did these come from? I'll stop for now and post any other questions I have in another thread so as not to go overboard.


title: Potential step and its transmission / reflection

text: Lets say we have a potential step with regions 1 with zero potential $W_p\\!=\\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \begin{align} \psi_1&=Ae^{i\mathcal L x} + B e^{-i\mathcal L x} & \mathcal L &\equiv \sqrt{\frac{2mW}{\hbar^2}}\\\ \psi_2&=De^{-\mathcal K x} & \mathcal K &\equiv \sqrt{\frac{2m(W_p-W)}{\hbar^2}} \end{align} Where $A$ is an amplitude of an incomming wave, $B$ is an amplitude of an reflected wave and $D$ is an amplitude of an transmitted wave. I have sucessfuly derived a relations between amplitudes in potential step: \begin{align} \frac{A}{D} &= \frac{i\mathcal L-\mathcal K}{2i\mathcal L} & \frac{A}{B}&=-\frac{i \mathcal L - \mathcal K}{i \mathcal L + \mathcal K} \end{align} I know that if i want to calculate transmittivity coefficient $T$ or reflexifity coefficient $R$ i will have to use these two relations that i know from wave physics: \begin{align} T &= \frac{j_{trans.}}{j_{incom.}} & R &= \frac{j_{trans.}}{j_{incom.}} \end{align} * * * **Question 1:** I know that $j = \frac{dm}{dt} = \frac{d}{dt}\rho V \propto \rho v \propto \rho k$ But what is a density $\rho$ equal to? **Question 2:** I noticed that $\mathcal L$ and $\mathcal K$ are somehow (i dont know how) connected to the wavevector $k$ from the equation in 1st question but how? How can i make it obvious?


title: equation of motion for the scalar field via variational principle in general relativity

text: I would like to find the equation of motion for the scalar field $\phi$ by varying the following action in General Relativity. Special Relativity: $$ S = -\tfrac{1}{2}\int d^4\xi\, \eta^{ab} \partial_a \phi\partial_b\phi $$ General Relativity: $$ S = -\tfrac{1}{2}\int d^4x \sqrt{g}\, g^{\mu\nu} \partial_{\mu} \phi\partial_\nu \phi $$ I was able to get the correct equation using the covariant derivatives. Since they are constant with respect to the metric partial integration works and one obtains $ \Box\phi = 0$. But since for the scalar field the covariant and the partial derivates are the same I wanted to vary the action with the partial derivatives. Note: $\phi \rightarrow \phi + \delta\phi$ $$ \begin{align} \delta S &= -\tfrac{1}{2}\int d^4x \sqrt{g}\, (g^{\mu\nu} \partial_{\mu} \delta\phi\partial_\nu\phi + g^{\mu\nu} \partial_{\mu} \phi\partial_{\nu} \delta\phi ) \\\ % &=-\int d^4x \sqrt{g}\, g^{\mu\nu} \partial_{\mu} \delta\phi\partial_\nu\delta\phi \\\ % &=\int d^4x \, \partial_\nu(\sqrt{g}\,g^{\mu\nu}\partial_\mu\phi)\delta\phi \end{align} $$ The third line one obtains after partial integration. But at this point I'm stuck. I know the following definition: $$ \Box\phi = \tfrac{1}{\sqrt{g}}\partial_\mu(\sqrt{g}g^{\mu\nu}\partial_\nu\phi) $$ But I'm not able to convert my obtained solution into this. Thanks for your help!


title: Accelerating faster per second than the speed of sound?

text: What would happen if you were to accelerate one end of a material, say a steel rod for instance, at faster per second than the speed of sound in that material? For example, if the speed of sound in steel is 6100m/s, what would happen if you pushed the end of that piece of steel at more than 6100m/s/s? This only occurred to me because I heard that the speed of sound is the speed an impulse travels through a material, and I was wondering if there could be some sort of limit to acceleration which would arise from this. I appologize in advance if I'm comparing apples with irrelevant oranges. Thanks!


title: Why does a temperature increase on a fixed volume increase entropy?

text: I heard that this statement is correct. However, it seems odd to me. The number of possible microstates is still the same, so isn't the entropy constant?


title: Are all spin states orthogonal?

text: For a spin 1/2 particle you have two spin states, either up or down which are orthogonal. But what about a spin 1 particle which has 3 spin states, either up, down, not up/not down?


title: Why does the colour wheel wrap around?

text: Red colour bares 430-480THz and Violet bares 670-750THz according to Wikipedia. What frequencies are the colours between red and violet? Will they be higher than red? or lower than violet? What physically properties do opposite colours carry? Cyan is the opposite colour of red; yellow is the opposite colour of blue; green is the opposite colour of magenta. Can it be expressed mathematically?


title: What is the difference between diamagnetism and superconductivity?

text: Basically, What is the difference diamagnetism and superconductivity? As far as I understand, diamagnetism comes from the fact, that all electrons in a solid, when exerted by an external magnetic field, create so-called Eddy Currents which oppose the external magnetic field, right ? And in a superconductor, the supercurrents, are actually are "normal" current in the material, which opposes the magnetic field. But is that the whole story, or am I simplifying it too much ?


title: Towing of asteroid

text: I recently studied that NASA has planned to tow and place it in the orbit of the moon. My doubt is when asteroid is placed in the orbit near moon.since the gravitational field of earth is very high.what will it revolve around the moon or the earth. Can anyone clarify my doubt ??


title: Long range order in the BCS ground state

text: I am trying to prove the following equivalent form for long-range-order in superconductivity (Annett's book states something like this) : \begin{equation}\lim_{|R-R'|\to \infty}<\psi^{\dagger}(R')\psi(R)> = <\psi^{\dagger}(R)\psi(R)> \end{equation} where the brackets are evaluated on the BCS ground state : $|\psi_{BCS}>= ∏_{k}(u_k^{*}+v_k^* P_k^{\dagger})|0> $ ($|0> $ being the zero-temperature Fermi sea), $P_k^{\dagger}=c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}$ when : $\psi^{\dagger}(R) = \int d^3r~\psi(r) \phi_{\uparrow}^{\dagger} (R+r/2)\phi_{\downarrow}^{\dagger}(R-r/2)$ ($\psi(r)$ being some wavefunction) and : $\phi_{\sigma}(r) = \sum_{k}e^{ikr} c_{k\sigma}$ ($c$ being the commun fermion annihilator, and the $P$s commuting like bosons for this particular state).(Sorry for the lengthy definitions) The gap function being defined as : $\Delta \propto \sum_k <P_k>$, we can see relatively easily that the right side of the first equation will be proportional to $|\Delta|^2$. But what I've found quite puzzling is what to do with the unhomogeneous terms (especially in the left term) where the $c$ and $c^{\dagger}$ do not form a pairing operator and the Fermi sea not being ideal to deal with them... Could somebody help me on this calculation ? Or am I taking the problem on the wrong side ?


title: 4-velocity and 4-acceleration in instantaneous rest frames

text: I am trying to solve this problem: > _Consider a rocket moving relative to an inertial frame $\mathcal{F}$ , such > that its worldline is given by > $$x^{\mu}=c^2/g(\sinh(g\tau/c),\cosh(g\tau/c)-1,0,0).$$ What are the > components of four acceleration relative to the instantaneous rest frame of > the rocket, $\mathcal{F}'$?_ I (think I) understand how to do this using Lorentz transformation: $dt/d\tau=(1/c) \cdot dx^0/d\tau=\cosh(g\tau/c)$. This is equal to $\gamma$ and it is then straightforward to compute 3-velocity and then use the Lorentz matrix to get $$dx'^\mu(\tau)/d\tau=(c,0,0,0) \quad ; \quad d^2 x'^\mu(\tau)/d\tau^2=(0,g,0,0)$$ However, when I first saw this I immediately thought that by definition the 4-velocity in the instantaneous rest frame of the rocket would be (c,0,0,0) because in $\mathcal{F}'$ the 3-velocity is zero and $\gamma$ would be 1 and I was wondering if this is a valid reasoning. Even if it is, **why is the following wrong**? If $v'^\mu=dx'^\mu(\tau)/d\tau=(c,0,0,0)$, then $a'^\mu=dv'^\mu/d\tau=(0,0,0,0)$ since $c$ is a constant. But this contradicts the calculations given by the Lorentz transformation and I don't understand why and given this I have no idea of **how to interpret 4-acceleration**.


title: Why do clocks measure arc-length?

text: Apologies in advance for the long question. My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$ t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$. **Why is this so?** * * * Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$. Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$. If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$. If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers. I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$ T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$. Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$ I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align} I_{ba} &= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\\ &= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\\ &= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result. **How do others feel about this argument?** I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$. I imagine that in GR a similar argument could be made by invoking local flatness of the metric.


title: Buoyancy of a positively charged object inside positively charged fluid

text: An object with a density of $x$ inside a liquid with a density greater than $x$ would float. If we assume that both of these are positively charged and the object is in the middle of the liquid:   1. Would the liquid touch the object?   2. Would the object still float up because of buoyancy?


title: Relativistic equivalent of a spring-force?

text: Usually what helps me understand a concept better in physics is to write a simulation of it. I've got to the point where I'm competent in the basics of special relativity, but, I can't figure out how to write a relativistic simulation! My normal approach in Newtonian mechanics is just to attach two objects together with a hookean spring force, maybe add a damping effect, and then just to use the calculated force to get acceleration, and numerically integrate over that. The first issue is length contraction. The issue isn't as easy as a contraction of $1/\gamma$. If I have two point events simultaneous in some reference frame, corresponding to locations of masses connected by a spring, I do know their spacelike separation, which is an invariant. However, even though the proper length is invariant, if I change reference frames, the events are no longer simultaneous, so I can't really get a consistent definition from this... I figured that I could apply the spring force instantaneously, "faster than light", and show how when I do that that leads to violation of causality, but I can't even define force consistently! I understand that if I use a wave equation or some sort of electromagnetic force, then I can have a force field that transforms correctly, but I really don't want to do this, because I'm not great at electromagnetism, and this is really just for me to better my understanding of special relativity. Plus it would be difficult computationally. I haven't been able to make any headway on assuming that the masses are connected by a chain of springs, whose velocities relative to each other are $<<c$, but I think the solution may lie there.


title: Properties of event horizon for incoming matter

text: In one of his lecture,Prof. Susskind mentioned that the event horizon "bulges" forward to meet any incoming radiation or matter; and it is a property of Einstein field equations. I have not come across any such property, and if it exists, shouldn't it belong to the Schwarzchild metric rather? Please explain me that property in detail.


title: $2\pi$ and Feynman Rules

text: I notice a $2\pi$ term in the $\delta$-function when trying to construct an amplitude using the Feynman Rules. The $2\pi$ also appears as an integration measure to enforce normalisation in the phase space, what is the origin of this $2\pi$ term? Where does it come from?


title: Nature of gravity: gravitons, curvature of space-time or both?

text: General relativity tells us that what we perceive as gravity is curvature of space-time. On the other hand (as I understand it) gravity can be understood as a force between objects which are exchanging (hypothetical) virtual particles called gravitons, similar to the way electromagnetic forces are due to objects exchanging virtual photons? At least at first glance, the two concepts seem mutually exclusive. Is there a description of gravity which includes both, or is this contradiction one of the problems in combining GR with quantum mechanics?


title: How do gravitons impact on general relativity?

text: As I'm reading about GR a lot lately, I was wondering: how do gravitons (if they exist ofc) impact the general relativity? Since in GR, when we look at particles moving in space-time, we are only looking from geometric point of view so to say. Since gravity is represented with curvature of space-time via Einstein equations, we don't say gravity is a force that influences on bodies, gravity is just curvature affecting the bodies. So if there is a graviton (gravitons) which would be mediators of gravity as a force within or not within the Standard Model, how would this be reconciled with the view of gravity as a curvature of space and time? I would guess that this kinda question was asked by some scientists and answered, but I never really read anything on it. I don't even remember seeing gravitons mentioned in standard books about GR. Are there any explanations about it?


title: Can we fire a single femtosecond pulse of light without an associated train of additional pulses? What sort of equipment allows for this?

text: What sort of mode-locked laser systems allow for the production of single isolated pulse of light (as opposed to a train of more than one pulse), where the individual pulse duration is on order picoseconds ($\approx 10^{-12}$ seconds) to femtoseconds ($\approx 10^{-15}$ seconds)? Is there any reason I couldn't use a mode-locked Ti-sapphire laser to produce a single pulse of some desired / achievable duration in the femtosecond regime? Or is it generally challenging to drop the pulse repetition rate to a sufficiently small frequency to allow for shutters / etc. to filter out the rest of a pulse train? My guess is that if we can get the pulse repetition rate down to the MHz regime, we can start to think about diverting or blocking additional pulses using voltage or mechanically switchable filters? What do people actually use? I should also mention that I'm particularly interested in emission wavelengths in the range of something like $100 \space nm$ to $1.5 \space \mu m$ or so. ... In terms of the different types of femtosecond lasers, there are fiber lasers, bulk lasers, dye lasers, semiconductor lasers, even free-electron lasers that one can use. Supposing I need something like a few hundred milliwatts to a watt of output power, and I'm interested in the visible to near-visible wavelength range, are there any families of femtosecond lasers particularly suited to the above task?


title: How do I calculate stopping power?

text: Malcolm Gladwell made a claim in a recent talk that a sling with a stone going at 30m/s has the same stopping power as a .45 calibre handgun. How would I verify whether or not this claim is true - even given some assumptions, like the stone has a weight of 1lb or 2 lbs or whatever. Not quite sure how to work this out. Assume that the bullet is a 185grain 0.45 calibre bullet.


title: Time dilation in special relativity?

text: I'm trying to understand the concepts of time dilation and spatial compression. I've been using the classic example of firing a photon of light inside a ship (spaceship, boxcar, etc.) moving at a constant velocity to the second observer. But when I do the Lorentz, I get different values depending upon the direction of travel of the photon relative to the ship. My understanding is that time dilation for the two observers should be a constant, as should spatial compression, since their relative velocities are constant (at least for the duration of the experiment). When I fire the photon in the direction of travel of the ship, the measurements all work out as expected. Time on the ship is passing slower. Then when I fire the photon backward in the direction of travel of the ship, all the values that should be constant have different values. So I assumed I'd done the transforms wrong. I redid them and still got different values. hen I simplified the equations by picking values that make the transforms very basic. Ship length = 1 light second for the ship observer. Ship speed = 0.1 light year for the second observer. Even then I get variances in the values that I expect to be constants. My problem might be in measuring the distance the photon traveled for each observer. In firing forward, the second observer sees the photon travel the compressed length of the ship plus the distance the ship travels. The ship observer see it travel the length of the uncompressed ship. When firing backwards, the second observer sees the photon travel the compressed length of the ship minus the distance the ship travels. Since the ship observer sees the photon travel the same distance and amount of time regardless of direction, the second observer must also see it travel the same time in both directions. But the distance is shortened for the secondary observer. So either the time dilation or the spatial compression would have to change for the equations to work. But my original premise was those are constants because the relative velocity is assigned to be constant for the experiment! Am I still doing the transforms wrong, or do I have a bad assumption in which values become constants by forcing relative velocity to be constant? I've reviewed dozens of the time paradox explanations. But those arise from the two observers returning to one frame of reference. I've let my ship sail on beyond the end of the experiment indefinitely, purposely to avoid those problems from interfering in learning how to do the transforms correctly. Now I'm not sure I didn't step into a different paradox. Maybe this should be solved in GR instead?


title: How does Newton's law apply to a person throwing snow onto their snowbank at the side of their driveway?

text: We know Newton's three laws:   1. A object at rest will remain at rest, and an object in motion will remain in motion unless a external force acts upon it.   2. If an unbalanced force acts on a object, the object will accelerate in the direction of the net force.   3. If an object $A$ exerts a force on object $B$, then object $B$ will exert a equal force to object $A$ in the opposite direction. $F_{a\text{ on }b} = -F_{b\text{ on }a}$ What I think is that as you throw the snow, it begins moving at a constant velocity. However, when the shovel stops moving, the snow will remain in motion, causing it to accelerate/fly toward the snow bank, according to Newton's first law. But is the snow **accelerating** when it leaves the shovel? How does this apply to Newton's second law? Thanks!


title: Confusion regarding photons?

text: I'm still in high school, and while I can't complain about the quality of my teachers (all of them have done at least a bachelor, some a masters) I usually am cautious to believe what they say straight away. Since I'm interested quite a bit in physics, I know more about it than other subjects and I spot things I disagree with more often, and this is the most recent thing: While discussing photons, my teacher made a couple of statements which might be true but sound foreign to me:   * He said that under certain conditions, photons have mass. I didn't think this was true at all. I think he said this to avoid confusion regarding $E=mc^2$, however, in my opinion it only adds to the confusion since objects with mass can't travel with the speed of light, and light does have a tendency to travel with the speed of light.. I myself understand how photons can have a momentum while having no mass because I lurk this site, but my classmates don't.   * He said photons don't actually exist, but are handy to envision. This dazzled my mind. Even more so since he followed this statement by explaining the photo-electric effect, which to me seems like a proof of the existence of photons as the quantum of light. He might have done this to avoid confusion regarding the wave-particle duality. This all seems very odd to me and I hope some of you can clarify.


title: Self-adjointness

text: I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by $i\partial_{t}\Psi[\Phi,t]~=~\underset{A}{\underbrace{\frac{1}{2}\int d^{3}x\left(-\frac{\delta^{2}}{\delta\Phi^{2}(\vec{x})}+|\nabla\Phi|^{2}+m^{2}\Phi^{2}\right)}}\Psi[\Phi,t]$. This is the Schrödinger representation of QFT. Now I want to know, whether the operator $A$ on the r.h.s is essentially self- adjoint? Any idea or advise? My problem is how to handle the functional derivative here.


title: What's this about kinetic energy increasing with the fifth power of length?

text: I don't quite understand this quote from Stephen J. Gould's _Ever since Darwin_ , where he talks about the compensating physical characteristics of organisms for their size. > Other essential features of organisms change even more rapidly with > increasing size than the ratio of surface to volume. Kinetic energy, in some > situations, increases as length raised to the ﬁfth power. If a child half > your height falls down, its head will hit with not half, but only 1/32 the > energy of yours in a similar fall. In return, we are protected from the > physical force of its tantrums, for the child can strike with, not half, but > only 1/32 of the energy we can muster. Mass increases with the third power of length, and even if we allow for a shorter fall (even though that is not clear from the wording) for the child (who is half as tall), I can only come up with the fourth power.


title: How to assign coordinates to the elements of a flat metric space

text: Consider the metric space $(M, d \,)$ where set $M$ contains sufficiently many (at least five) distinct elements,   and consider the assignment $c_f$ of coordinates to (the elements of) set $M$, $c_f \, : \, M \leftrightarrow {\mathbb{R}}^3; \, c_f[ P ] := \\{ x_P, y_P, z_P \\}$ such that distinct coordinates values are assigned to distinct elements of set $M$, and   such that for the function $f \, : \, ({\mathbb{R}}^3 \times {\mathbb{R}}^3) \rightarrow {\mathbb{R}};$   $f[ \\{ x_P, y_P, z_P \\}, \\{ x_Q, y_Q, z_Q \\} ] := $ ${\sqrt{ (x_Q - x_P)^2 + (y_Q - y_P)^2 + (z_Q - z_P)^2 }} \equiv {\sqrt{ \sum_{ k \in \\{ x \, y \, z \\} } (k_Q - k_P)^2 }}$ and for any three distinct elements $A$, $B$, and $J$ $\in M$ holds $f[ c_f[ A ], c_f[ J ] ] \, d[ B, J ] = f[ c_f[ B ], c_f[ J ] ] \, d[ A, J ]$. Is the metric space $(M, d \,)$ therefore flat? (i.e. in the sense of vanishing Cayley-Menger determinants of distance ratios between any five elements of set $M$.)


title: Strong interaction and the Lagrangian for electromagnetic interaction

text: The Lagrangian for electromagnetic field has the following expression: $$ L = -\frac{1}{c^{2}}A_{\alpha}j^{\alpha} - \frac{1}{8 \pi c}(\partial_{\alpha} A_{\beta})(\partial^{\alpha}A^{\beta}) $$ (I used Lorentz calibration $\partial_{\alpha} A^{\alpha} = 0 $). If I add the summand $\frac{\mu^{2}}{8 \pi c}A_{\alpha}A^{\alpha}$, I'll get an equations for field (which characterized by some 4-vector $A^{\alpha}$ (not electromagnetic (!!!))) of strong interaction and (for static case) the expression for Yukawa potential. So what is the physical meaning of summand written above? This term is somehow characteristic of the mass of the interaction carriers, but I don't understand the physical meaning of $A_{\alpha}A^{\alpha}$.


title: Proper notation for normalized scalar?

text: I have not been able to find a resource to tell me the standard notation for a normalized scalar value. Normalized vectors (i.e. unit vectors) are typically denoted by placing a hat over the variable, something like: $${\bf \hat{e} = \dfrac{e}{||e||} }$$ However, does the same apply to normalizing (and nondimensionalizing) a scalar? Would it be correct to write: $$ \hat{L} = \dfrac{L}{L_0} $$ This is assuming that $L$ and $L_0$ are just scalar lengths. If I am defining my own notation, is it _verboten_ to call this something like $\bar{L}$ (with a bar)? If it makes any difference, I am a mechanical engineer and this will be going in my thesis.


title: Quantum Circuit, example of the Bernstein-Vazirani problem

text: This question is regarding the quantum circuit in the picture below. Suppose we have the set up below, where U performs the operation $U:\mid x \rangle \mid y \rangle \rightarrow \mid x \rangle\mid y \oplus f(x) \rangle$. We define the function $f(x)$ on the 3-bit string $\mid x \rangle= \mid x_1 x_2 x_3 \rangle$ with $f(x)=a\cdot x=a_1 x_1 \oplus a_2 x_2 \oplus a_3 x_3$, where $\oplus$ is addition modulo 2 and $a$ is the fixed three bit string $\mid a \rangle=\mid a_1 a_2 a_3 \rangle$. The objective is to show how this circuit finds $a$. So far I have that the 3 zero qubits pass through 3 Hadamard gates, putting them in the bell state $\beta_{00}=\frac{1}{\sqrt2}(\mid 0 \rangle +\mid 1 \rangle)$. So at point 1 on the circuit the system has state: $$\frac14(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle +\mid 1 \rangle)(\mid 0 \rangle -\mid 1 \rangle)=\frac14 \sum_{x\in\\{0,1\\}^3} \mid x \rangle(\mid 0 \rangle -\mid 1 \rangle$$ Where we have found the RHS by expanding out and summing over all the possible 3 but qubits. So next we consider what happens when the system passes through the $U$ gate, we find: $$\frac14 \sum_{x\in\\{0,1\\}^3} \mid x \rangle(\mid f(x) \rangle -\mid 1 \oplus f(x) \rangle )$$ Here we notice that is $f(x)$ is $0$ nothing changes, but if it is $1$ it acts as a factor of $-1$, therefore the system is: $$\frac14 \sum_{x\in\\{0,1\\}^3} (-1)^{f(x)}\mid x \rangle(\mid 0 \rangle -\mid 1 \rangle )$$ Now $f(x)$ outputs sums of the elements of $a$ depending on the value of $x$, for example for $\mid x \rangle = \mid 110 \rangle$, $f(x)=f(110)=a_1 \oplus a_2$. Would anyone be able to help me understand how this helps us find the three values for $a$ at our measured gates? The main issue i'm finding is finding a useful of writing an expression for the system after it has passed through the last three Hadamard gates, one which we can use to find $a$, any help would be greatly appreciated! ![Figure 1](http://i.stack.imgur.com/nSGNe.jpg)


title: Measurement of Mass and Momentum of a particle simultaneously

text: In quantum mechanics can the mass and the linear momentum of a particle be measured precisely or do they commute ?


title: Does 1 volt cancel out 1 ohm? Can current be 0?

text: $V=IR$ Right? $100 (\rm{V}) = 0 (\rm{A}) \times 100 (\rm{\Omega})$ Lets say something has $100 \rm V$ potential But since this object is surrounded by air and current is not flowing therefore there has to be $100 \rm{\Omega}$. Right? But then the equation does not work as 100 does not equal 0? You get infinite resistance


title: Gravitional force

text: Does the gravitation of Earth have a limit? when a body projected vertically with $v=11km/s$ (escape velocity) from Earth's surface does this means that it does not return back to Earth?


title: Electromagnetic force interaction

text: As far as I know, the electromagnetic force only interacts on particles with electrical charge, but I was told that the electromagnetic force was involved in the following reaction: $$\Sigma^0\rightarrow\Lambda^0+\gamma.$$ How can this be, when the electric charge of both the $\Sigma^0$ particle and the $\Lambda^0$ particle is $0$? Or is it another fundamental interaction that is involved in the reaction? Thank you.


title: What happens to capacitor’s charge when the plates are moved further apart?

text: In my physics textbook there is an example of using capacitor switches in computer keyboard: > Pressing the key pushes two capacitor plates closer together, increasing > their capacitance. A larger capacitor can hold more charge, so a momentary > current carries charge from the battery (or power supply) to the capacitor. > This current is sensed, and the keystroke is then recorded. That makes perfect sense, and is kind of neat. What I am curious about, is what happens to that extra charge afterwards. Is there some sort of discharge mechanism? I suppose, that would be also necessary to differentiate between single keystrokes and continuous depression (register stroke current, then register the discharge). What would happen to the capacitor if there was no such discharge mechanism, but its capacitance was suddenly reduced? If capacitance is reduced, and the charge stays the same, then, according to $Q = C \Delta V_C$, the difference of potentials on plates of capacitor should increase and exceed that of a power supply thus reversing the current. Is that what is happening, and the keystrokes are recorded by sensing not only the existence of the current, but also its direction?


title: A question to the theory of multiverse

text: Guys I couldn't catch a point of multiverse theory.. Theory: If space-time goes on forever, then it must start repeating at some point, because there are a finite number of ways particles can be arranged in space and time.. Question: It seems like as if multiverse theory stands on a optical argument. We call there is a next universe because we are not seeing that since there is a speed limit of light and our optical universe is 14 billions light yrs in diameter(or whatever it be..since we can not assume our space time a sphere or 2dimensional plane).Doesn't it sound weird? If there is a space between two universe then what is the value of calling it a system of two universe, Isn't it just a single universe? Question:Is it a good idea to call there is a next universe because it is exactly alike? Question: Another question is How does the multiverse theory provides solution to the grandfather paradox? Isn't it like we fold the two dimensional spacetime graph where universe exactly alike to our universe exits??If my above argument is correct then how is it valid?? Note that I am not questioning multiverse theory this time..So please Don't give answer to this question proving my above arguments wrong..Let this be independent question.. I know I am wrong But please Make me clear.


title: Acceleration, velocity and speed

text: It is given that acceleration is constant, so can we infer that average speed and velocity are the same?? Moreover, circular motion is out of the question, as the function of x(t) where x=displacement, suggests, that for any t>=0, displacement can not be zero... This is the conceptual problem I am facing in a question: My teacher was reading out the question, and it was asked only to find the avg velocity from the acceleration. She, on her own, added a part to it, asking us to also find avg speed, and then, while discussing solutions, said that a graph must be made in order to solve this...so do you think that it is absolutely necessary? Moreover, if my premise is flawed, then how can the graph even help?? Thanks in advance!


title: Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?

text: Lingua Franca links relativity and quantum theories with spectral geometry Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics community? Is this work ground breaking or just the start of something that might be? From what I can understand this physicist related two types of maths that we use to model the world around us. Which has been hard to do, because the types of maths are incompatible with each other. But what is this spectral geometry? How does it relate the two types of maths in relativity and quantum theory? What might those maths be? What about the two has made it hard to unite them?


title: What is the difference between Cramer and Vaidman?

text: Two very interesting new papers on arXiv last night by Lev Vaidman and friends lead me to ask about the differences between Cramer's transactional interpretation of quantum mechanics (TIQM) and the two state vector formalism (TSVF) advanced by Aharonov, Vaidman and others. At a first look, they both seem very similar to me.   The two papers:   http://arxiv.org/abs/1304.7474 “The past of a quantum particle”, and   http://arxiv.org/abs/1304.7469 “Asking a particle where it has been”.   From these two papers, you can find further references for TSVF, including its origin.   For TIQM, you can start with Wikipedia, and also trace back to the origin. http://en.wikipedia.org/wiki/Transactional_interpretation


title: Theorem of inclusion in the disordered Bose-Hubbard model

text: In a paper by V. Gurarie et al. , the theorem of inclusion is used to prove that there is no direct phase transition between Mott insulator and spuerfluid in presence of disorder. In Fig. 2 of that paper, why if $\Delta > \Delta_c $, there exist domains in phase B which locally look like phase A? Similarly, why for $\Delta < \Delta_c $, there exist domains in phase A which locally look like phase B?


title: Are atoms unique?

text: Do atoms have any uniquely identifying characteristic besides their history? For example, if we had detailed information about a specific carbon atom from one of Planck's fingerprints, and could time-travel to the cosmic event in which the atom formed, would it contain information with which we could positively identify that they two are the same?


title: How do we know that photon entanglement isn't the result of the photons's states being predetermined?

text: I know there is evidence that it is not predetermined and I tried reading articles on it but most of them either don't explain the intuition behind the experiment or they speak in a foreign language (That language being science). If you could explain the intuition behind the experiment and also give an analogy that would be great.


title: Why don't electrons crash into the nuclei they "orbit"?

text: I'm having trouble understanding the simple "planetary" model of the atom that I'm being taught in my basic chemistry course. In particular,   1. I can't see how a negatively charged electron can stay in "orbit" around a positively charged nucleus. Even if the electron actually orbits the nucleus, wouldn't that orbit eventually decay?    2. I can't reconcile the rapidly moving electrons required by the planetary model with the way atoms are described as forming bonds. If electrons are zooming around in orbits, how do they suddenly "stop" to form bonds. * * * I understand that certain aspects of quantum mechanics were created to address these problems, and that there are other models of atoms. My question here is whether the planetary model itself addresses these concerns in some way (that I'm missing) and whether I'm right to be uncomfortable with it.


title: motion of electrons

text: Do electrons move randomly, with no preference of directions? And why electrons don't fall into the nucleus? About this question, I read the article on Chemistry wiki, which says that when electron falls towards nucleus, its kinetic energy becomes extremely large so there is a battle between electrons and nucleus, and no one is gonna win the battle, there is a compromise state instead. My question is why can't we say that electrons with great kinetic energy are moving towards the nucleus? Why should it escape？


title: Why isn't Hydrogen's electron pulled into the nucleus?

text: > **Possible Duplicate:**   >  Why do electrons occupy the space around nuclei, and not collide with them?   >  Why don’t electrons crash into the nuclei they “orbit”? From what I learned in chemistry, the protons in the nucleus pull the electrons in and push on each other through electromagnetic forces, but are held in place by the strong nuclear forces in its gluons. Not much was said, however, about what keeps the electrons orbiting. I've always just assumed it was other electrons that prevented an electron from becoming part of the nucleus. In the form of Hydrogen that only has one electron, what keeps that electron from being pulled completely into the nucleus?


What is it that makes an electron maintain a distance from the positively charged nucleus? Why aren't electrons merely pulled into and absorbed by the nucleus ?


title: What prevents an atom's electrons from "collapsing" onto its protons?

text: Forgive me if the answer to this is obvious. I have no formal physics training, and I remember that when I asked my physics teacher this, she just frowned and said "Good question." An electron is negatively charged. A proton is positively charged. Based on basic principles, it seems like it would be logical for the electron cloud of an atom to "collapse" into the nucleus and become a part of it (especially since the electrons are so much lower mass). Why doesn't this happen? How do electrons maintain separation from the protons in the nucleus, when the opposite chargers ought to draw them together? I considered that perhaps the charge in the electron was too minuscule in comparison to the proton (like having a negatively charged magnet on Saturn while all of Earth was positively charged; obviously the magnet wouldn't just be drawn to the Earth because the forces weren't strong enough to act over that distance). But if that was or is the case, I would expect some other chemical behaviors to not exist. For example, the whole phenomenon of water being a "dipole". If the charge of the electron is too weak to interact with the proton, how could the oxygen in water more strongly attract them than hydrogen? I get that the oxygen has _more_ protons, and thus more positive charge in the nucleus, but that still would seem to support that the oxygen atom's _own_ electrons should be attracted to it... Can anyone explain the phenomena occurring here or simply point out the flaw in my thinking?


title: Why do electrons occupy the space around nuclei, and not collide with them?

text: We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However... How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons _do_ collide, and, if so, what occurs?


title: classical understanding of an atom

text: A problem in Bohr's day was understanding why an orbiting electron does not continuously radiate an EM field. An orbiting electron is a moving charge and according to Maxwell, this should generate an electric and magnetic field. Bohr said that photons (packets of EM energy) are only emitted when an electron jumps from a high quantum state to a lower one. Does that mean that an electron in a stable atom really isnt moving? Conversely, does Bohr's model mean that all motion requires energy transitions from one qunatum sate to another?


title: How to promote algebraic expressions to operators in quantum mechanics?

text: Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$ in the position basis. Now my question is, what if $x$ or $p$ appears in the denominator in a classical expression? How to promote this to a quantum expression? What would be the meaning of division by an operator? Edit: Thank you for your responses. My expression likely contains a mixture of x and p. For eg., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}$$. How to resolve products of non-commuting operators like x,p in a satisfactory way?


title: Degrees of Freedom for an Asymmetric top

text: How many degrees of freedom does an asymmetric top have if it is rotating about a fixed point?What are the generalised coordinates used then?


title: Is there a heuristic explanation for the derivation of Heisenbergs Uncertainty Principle from String Theory?

text: Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system. Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles. How does String Theory account for it? Heuristically, and mathematically?


title: What was a second in the early universe?

text: I have read some popsci articles and documentaries about the early universe and they often explain how various features of the universe came about and at what time. For example hydrogen atoms came about after hundreds of thousands of years. Now, the official SI time unit, the second, is based on the caesium atom's properties. How can we talk about time lengths in seconds or years if there are no atoms yet in the universe? (Or going back further where there aren't even electrons.) Clearly, there must be a way to do it. Then the next question is: why don't we use those field properties or whatever that existed before atoms to define the SI second? If atoms are not needed to get "the beat of time" in the early universe because something else can provide the tick then we could also use those things today instead of relying on the caesium atom for the definition of the second. I hope it's clear what I mean. Thanks!


title: geodesic conjugate points

text: I was reading "Nature of space and time" by Penrose and Hawking, pg.13, > If $\rho=\rho_0$ at $\nu=\nu_0$, then the RNP equation > > $\frac{d\rho}{d\nu} = \rho^2 + \sigma^{ij}\sigma_{ij} + \frac{1}{n} > R_{\mu\nu} l^\mu l^\nu$ implies that the convergence $\rho$ will become > infinite at a point $q$ within an affine parameter > distance$\frac{1}{\rho_0}$ if the null geodesic can be extended that far. > > _if $\rho=\rho_0$ at $\nu=\nu_0$ then $\rho$ is greater than or equal to > $\frac{1}{\rho^{-1} + \nu_0-\nu}$. Thus there is a conjugate point before > $\nu=\nu_0 + \rho^{-1}$._ I dont understand many terms here. Firstly, what is affine parameter distance? And I am at loss as to how does one get the 2nd relation between $\rho$ and $\frac{1}{\rho^{-1} + \nu_0-\nu}$. How can you derive it? Frankly, I dont understand ANYTHING about how does this equation come, though I suspect it just the Frobenius theorem. Because that is how you get conjugate points in spacetime. Please give me DETAILED asnwers, as I have mentioned before, I am not too comfortable with it. I dont understand anything in blockquotes other than the RNP equation. Thanks in advance!!!