title: Relationship between mass preserving four-fources and proper acceleration

text: **Background** I studied Rindler's book on Relativity. Relevant information from this book is available on-line here: > http://www.scholarpedia.org/article/Special_relativity:_mechanics Given that we work with four-vectors $\mathbf{R} = \left(\vec{r},ct\right)$ where the four-velocity is defined as $$ \tag{1} \mathbf{U}=\frac{d\mathbf{R}}{d\tau}=\gamma\frac{d\mathbf{R}}{dt}=\gamma\left(u\right)\left(\vec{u},c\right), $$ $\vec{u}$ is the spacial "three-velocity" and the four-acceleration is defined as $$ \tag{2} \mathbf{A}=\gamma\frac{d\mathbf{U}}{d\tau}=\gamma\frac{d}{dt}\left(\gamma\vec{u},\gamma c\right)=\gamma\left(\dot{\gamma}\vec{u}+\gamma\vec{a},\dot{\gamma}c\right), $$ we can define the four-force as $$ \tag{3} \mathbf{F}=\frac{d}{ {d\tau } } \mathbf{P}=\frac{d}{ {d\tau}}(m_{0}\mathbf{U})=m_{0}\mathbf{A}+\frac{ {dm}_{0} }{ {d\tau} }\mathbf{U}. $$ The four-force $\mathbf{F}$ is said to be **mass-preserving** if $dm_0/d\tau$ equals zero, i.e. the force leaves the resting mass of the particle unchanged. Moreover $$ \tag{4} \mathbf{F}=\frac{d}{d\tau}\mathbf{P}=\gamma(u)\frac{d}{ {dt} }( \vec{p},mc)=\gamma (u)\left(\vec{f},\frac{1}{c}\frac{dE}{dt}\right). $$ We note that $\mathbf{A}\cdot\mathbf{U}=0$ and therefore for a mass preserving four-force $\mathbf{F}\cdot\mathbf{U}=0$ from Eq. (3) or from Eq. (4) $$ \tag{5} \mathbf{F}\cdot\mathbf{U}=\gamma^2(u)\left(\frac{dE}{dt}-\vec{f}\cdot\vec{u}\right)=0 $$ which means that $$ \frac{dE}{dt}=\vec{f}\cdot\vec{u} $$ and $$ \tag{6} \mathbf{F}=\gamma(u)\left(\vec{f},\vec{f}\cdot\vec{u}/c\right). $$ We also have the relationship $$ \tag{7} \gamma(u)\vec{f}=m_0\frac{d^2\vec{r}}{d\tau^2}. $$ Moreover we have that the **proper acceleration** $\alpha$ is defined as $$ \mathbf{A}\cdot\mathbf{A}=\mathbf{A}^2=-\alpha^2. $$ I studied the questions formulated in > Why proper acceleration is $du/dt$ and not $du/d\tau$? > > relativistic factor between coordinate acceleration and proper acceleration and realized that the spatial vectors can be broken down into a parallel component and an orthogonal component, i.e. $\vec{v}=\vec{v}_\parallel+\vec{v}_\perp$ or $\vec{a}=\vec{a}_\parallel+\vec{a}_\perp$ where one makes a distinction between the component of the velocity that is parallel with the acceleration and vice versa. **Question** Now it can be shown that for a force $\vec{f}$ acting upon a particle traveling with a velocity vector $\vec{u}$ where there is an angle $\theta$ between $\vec{f}$ and $\vec{u}$ the relationship between the proper acceleration of the particle and the force $\vec{f}$ is $$ \tag{8} \vec{f}^2=m_0^2\frac{\gamma_\parallel^2}{\gamma^2}\alpha^2 $$ where $$ \tag{9} \gamma_\parallel=\frac{1}{\sqrt{1-u_\parallel^2/c^2}}\;\;\;\text{and}\;\;\;\gamma=\frac{1}{\sqrt{1-u^2/c^2}}. $$ Now how can I derive or prove equation (8)? I have tried every feasible combination of Eq. (1) to (7) without success. I've also tried the more basic variants of the Lorentz transformation > https://en.wikipedia.org/wiki/Lorentz_transformation but it gets very confusing when there are two components to consider when assessing the relativistic effects. Any ideas?


title: Detecting the presence of a delta potential

text: Suppose you have a particle in a box, and there may or may not be a Dirac delta potential somewhere in the box. How could one detect whether or not the potential is present? Furthermore: If there's more than one way to do this, what's the easiest or "cheapest"? How long will it take? Can it be done with certainty, or will there be some chance that you're wrong? The energy spectrum of the particle will depend on whether or not there's a potential, so if there's a good way to distinguish between two different energy spectra, that would answer my question.


title: Work done by complex field on complex plane

text: > A force field is given by $F = 3z+5$. Find the work done in moving an object > in this force field along the parabola $z = t^2 + it$ from $z = 0$ to $z = > 4+2i$. ![enter image description here](http://i.stack.imgur.com/HO4ot.png) I don't understand why conjugate is taken. How do I interpret it?


title: Determine orbital position after change in velocity

text: I am working on a satellite simulator for a project/game and I am stuck on this one bit of physics. So far I have a satellite that revolves around earth on a 2D plane following Keplerian motion using Kepler's equation. Everything is fine and the satellite will orbit many times without problem. I can also change the velocity of the satellite to manipulate apo/peri points as well as augment of periapsis. However, the problem occurs if I change the velocity once the satellite passes the `PI/2` mark, the entire orbit gets flipped (augment of periapsis is offset by `PI`) and the true anomaly resets to 0. Here is how it is currently implemented: With the assumption that the satellite starts at the periapsis at t0, I find the eccentric and true anomaly given a time since periapsis $w$. Then I find the radius at that angle to get the final position. $$t_0 = 0$$ $$E_0 = 0$$ $$M(0) = E_0 - e \sin E_0$$ $$n = \sqrt{ \frac{GM }{ a^3}}$$ $$E = {\rm kepler}(e, M)$$ $$v = 2 \tan^{-1}{(\sqrt{\frac{(1+e)}{(1-e)}}\tan\frac{E}{2})}$$ $$r = a\frac{1-e^2}{1 + e\cos{v}}$$ When I update the velocity of the satellite (flight path angle $\phi$ is recomputed after change in velocity), I recalculate a new true anomaly using $\phi$, I then find the difference between the old true anomaly and the new true anomaly, and add that to the augment of periapsis $w$. After that I recompute time since periapsis $t$ using the new true anomaly and recompute the orbital parameters as well. $$vel = \sqrt{GM\frac{2}{r} - \frac{1}{a}}$$ $$\phi = \tan^{-1} ( \frac{e\sin{v}}{1 + e \cos{v}})$$ $$v_2 = \tan^{-1}{\frac{r v g m \cos{\phi} * \sin{\phi}}{r v g m \cos^2{\phi} - 1}}$$ $$w \mathrel{{+}{=}} v_2 - v$$ $$E = \cos^{-1}{ \frac{e + \cos{v}}{1 + e \cos{v}} }$$ $$M = E - e \sin{E}$$ $$n = \sqrt{ \frac{GM}{a^3} }$$ $$t = \frac{M}{n}$$ $${\rm calculateOrbit}()$$ I noticed if I were to $v = v + \pi$ if ever $v_2 < 0$ and $v > 0$, then the weird orbit/position flip won't occur until the $v = \pi$ Otherwise I do not know what the problem is. I would really appreciate it if someone can point me the way, as I have been scratching my head over this for a long time now.


title: Coupling of open and closed channels in Feshbach resonance model

text: Feshbach resonance is described with coupling of 2 systems differing in the form of potentials :- one is said to produce a bound state (in 'closed' channel) and other is to produce scattering states (in 'open' channel). The two channels are said to be coupled by another potential term. The resonance occurs where a bound state energy from the closed channel matches a scattering state energy in the open channel. I understand that if we have a 2*2 potential matrix, we may have non-diagonal terms which can be said to bring the coupling between two systems. The calculation is done by taking the schrodinger equation for the whole system and projecting onto the two separate systems using projection operators onto the subspaces of states of the subsystems. Is there a way to describe the resonance (bound state energy = scattering state energy) without dividing the original system into two coupled systems of closed and open channels i.e. taking the schrodinger equation for the whole system with the 2*2 potential containing non-diagonal coupling terms ? Refer section 2.3 here :- http://www.physics.ox.ac.uk/Users/godun/FilingCabinet/FeshbachTheory.pdf


title: Matrix element of an operator

text: Suppose we want to calculate the following matrix element. $$ \langle \alpha |\hat{o}|\beta\rangle $$ Where $\alpha$ and $\beta$ are two arbitrary basis states in a many-particle basis set, and $\hat{o}$ is an arbitrary operator. If we now insert two identities, we can rewrite this as $$ \int dx \int dx' \langle\alpha|x\rangle \langle x|\hat{o}|x'\rangle\langle x'|\beta\rangle $$ Then, in the lecture notes, it says that in this context $$ \langle x|\hat{o}|x'\rangle = \hat{o} \delta (x-x') $$ My question is, why is $\hat{o}$ still an operator. I know that $\langle x | x' \rangle$ is $\delta(x-x')$, but it does not make sense to me that it remains an operator and does not become an eigenvalue.


title: If quantum gas goes below 0K, is calling 0K absolute zero irrelevant?

text: Lord Kelvin defined the absolute temperature scale in the mid-1800s in such a way that nothing could be colder than absolute zero. Physicists later realized that the absolute temperature of a gas is related to the average energy of its particles. Absolute zero corresponds to the theoretical state in which particles have no energy at all, and higher temperatures correspond to higher average energies. This and many other articles out there say that quantum gas can go below zero kelvin.They(the professors) teach that -273.15ºC is lowest temperature anything can attain and that's why it is called **absolute zero**. My question is not about whether materials can or can't attain sub-0K temperatures. I'm asking, **should we stop calling 0K as absolute zero?**


title: Compressible flow - subsonic to supersonic and the 2nd law of thermodynamics

text: I'm reading chapter 16 in Fluid Mechanics by Kundu. It is stated (figures 16.16 and 16.17) that in a constant area duct flow with heating or friction, to go from subsonic conditions to supersonic violates the 2nd law of thermodynamics. How can that be possible? How does a fluid attain supersonic speeds in the first place? Somehow, one must be able to get ambient quiescent air to supersonic velocities (it's done all the time). I know that you cannot generate a normal shock by going from subsonic to supersonic, only the other way around, but why can't you speed a fluid from subsonic conditions to supersonic in the duct? In that case, there just would not be any shock generated? Would that violate the 2nd law? I'm missing something here...


title: Myopia / Hypermetropia eye glasses inverting image on retina

text: As far as I'm aware, the eye acts like a pinhole camera in that it inverts the image on the rentina. This makes sense as the rays converge and form a focal point that is upside down. Myopia (shortsightedness) is described as the rays focusing before the retina, resulting in a blurred distance image. This would still be the right side up. However with Hypermetropia (farsightedness) the focal would be behind the retina, so the rays should project a blurry but right side up (but interpreted by the brain as upside down) image. I know this is not the case, by why?


title: Blackbody radiation

text: Looking at the radiation from the sun (T=5800K) I got a little surprise which I do not understand. I first calculated the energy density u and also the number of photons per unit volume ng. From this one gets the average energy per photon: $$ E_g = \frac{u}{n_g} $$ Converted to a wavelength one gets: $$\tag{1} \lambda = 920\:\mathrm{nm} $$ The average frequency using $$ \nu_{\text{av}} = \frac{\int \nu \cdot u_{\nu} \cdot \mathrm{d}\nu}{\int u_{\nu} \cdot \mathrm{d}\nu} $$ Converted to a wave length I get $\lambda_{\text{av}} = 650\:\mathrm{nm}$. Next I calculate the average wavelength using $$ \lambda_{\text{av}} = \frac{\int \lambda\cdot u_\lambda \cdot \mathrm{d}\lambda }{ \int u_\lambda \cdot \mathrm{d}\lambda} $$ One gets $\lambda_{\text{av}2} = 920\:\mathrm{nm}$ Questions: I was expecting some value which would be of course different from the value of the frequency distribution. This is actually the case. The same is of course true for the max of the distributions. However to my surprise this is also equal to the the value of eq. (1). Why? What is the physical meaning of the average frequency $\nu_{\text{av}}$? If there is no particular meaning of $\nu_{\text{av}}$ it looks like the wavelength distribution has more physical meaning which I think is nonesense.


title: Showing symmetry of the stress tensor by applying divergence theorem to $\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$

text: I'm currently working through the symmetry of the stress tensor, in relation to viscous flow. I am looking at this by examining the conservation of angular momentum equation for a material volume $V(t)$ with unit normal $\vec{n}=(n_1,n_2,n_3)$. I am having issue with applying the divergence theorem to this term $$\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$$ Where $\vec{x}=(x_1,x_2,x_3)$ and $\vec{t}$ is the stress vector where $\vec{t}=\vec{e}_i\sigma_{ij}n_j$, using the summation convenction, where $\sigma_{ij}$ is stress vector. If I can extract a normal from this expression I can use the divergence theorem to convert to a volume integral and combine with the other terms of the conservation of angular momentum equation, which are volume integrals, this will lead to showing $\sigma_{ij}=\sigma_{ji}$. Many thanks to anyone who could help.


title: What is the physical meaning of this simplification to calculate the effective coupling constants for a Gaussian model with quartic interactions?

text: To calculate the effective coupling constants $u'_2(q)$ and $u'_4(q)$ of the effective Hamiltinian eq (4.9) of this paper $$ H' = -\frac{1}{2}\int\limits_q u'_2(q)\sigma'_q\sigma'_{-q} \- \int\limits_{q_1}\int\limits_{q_2}\int\limits_{q_3}u'_4(q_1,q_2,q_3,-q_1-q_2-q_3) \sigma'_{q_1}\sigma'_{q_2}\sigma'_{q_3}\sigma'_{q_-q_1-q_2-q_3}$$ The following simplifications are introduced into eq (4.20) and (4.21) to calculate $u'_2(q)$ and $u'_4(q)$ respectively   1. $u'_2(q)$ is only evaluated to order u, which means only tree level diagrams are considered   2. Higher order than quartic interactions are neglected   3. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r} \rightarrow \frac{1}{1+r}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $   4. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r}\frac{1}{[(\frac{1}{2}q_1+\frac{1}{2}q_2-p)^2 +r]} \rightarrow \frac{1}{(1+r)^2}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $ What is the physical meaning of 3. and 4. ? Is there an "intuitive explanation for the physical meaning of these two simplifications? PS: here is an alternative link to the paper that maybe works better.


title: Finding Current and Voltage through resistors given overall voltage and resistor values

text: Given a setup like: ![enter image description here](http://i.stack.imgur.com/SEW8T.png) Is it possible to find current and voltage through each resistors given resistor values and overall voltage (battery voltage I suppose). I think $$I_0=I_A=\frac{\Delta V}{R_{total}}$$ I know $I_B+I_C=I_A$ but how do I find $I_B$ if I don't have $\Delta V_B$?


title: Total power of CMBR

text: Could you check me on this: if I'm interested in total power (summed over all frequency) of CMBR absorbed by a black body in empty space, would that be proportional to the surface of BB?


title: What is "code" in "toric code"?

text: When I first heard people talking about using Kitaev's toric code to do topological quantum computation, I was thinking how many lines does the toric code have. Then I was told that the "code" really represents quantum states. Later, I understood that the "code" is the degenerate ground subspace of a gapped Hamiltonian. But then I was corrected by people from quantum information who pointed out that code is a subspace of the Hilbert space of a quantum system, but code is also more than a subspace. So now I am wondering what is "code"?


title: What is spontaneous symmetry breaking in QUANTUM systems?

text: Most descriptions of spontaneous symmetry breaking, even for spontaneous symmetry breaking in quantum systems, actually only give a classical picture. According to the classical picture, spontaneous symmetry breaking can only happen for non-linear systems. Classical linear systems, such as Harmonic oscillators, can never have spontaneous symmetry breaking. (Here "linear" means that the equation of motion is linear.) But the real QUANTUM systems are always linear, since the Schroedinger equation is alway linear. But how can a linear quantum system have spontaneous symmetry breaking? Do we have a simple intuitive understanding spontaneous symmetry breaking WITHIN QUANTUM mechnics? (ie without using the classical picture, such a Mexican hat -- the logo of physics.stackexchange) The Mexican hat does give us a intuitive and pictorial understanding of spontaneous symmetry breaking in classical systems. Do we have a intuitive and pictorial understanding of spontaneous symmetry breaking in quantum systems.


title: How is energy produced in Alpha emission (radioactivity)?

text: ![enter image description here](http://i.stack.imgur.com/pW1vD.gif) In the above Alpha decay reaction, we get thorium by the alpha decay of uranium with an alpha particle. But we also get energy of amount Q by this process. And according to my course books, this Q energy comes from E = mc2 i.e. some of the mass is converted into energy. My question is that which mass converted into energy? I mean if you add the nucleons of alpha particle (helium nucleus), you will get the total nucleons present in Uranium. But, where did the energy come from then?


title: Coherence, Entanglement, Correlation

text: Let's deal specifically with a two-level system. I understand that ``coherence'' is due to the off-diagonal elements in the density matrix $\rho(t) = \sum_{i}|\psi_i(t)\rangle p_i\langle\psi_i(t)|$ for the system (generally written for mixed states). Entanglement can be measured by finding the eigenvalues $p_i$ of the density matrix at some time $t$ and adding up the terms $p_i\times \log(p_i)$. What are these values physically? I have seen problems where one would start off with a pure product state of one type and another, and observe as the two states become entangled and coherence is transferred from one to the other and back. What does this actually mean? I also find mention of correlations. How do those fit into this picture? Can one have entanglement without coherence, or coherence without entanglement? Edit: "Entanglement can be measured..." actually means that we are quantifying the entanglement spectrum via entropy.


title: Rubber Band Forces

text: I have a question regarding the force a band places on an object. Say I have a rubber band wrapped around 2 pegs at a certain distance, and at that distance I know the pounds of force per inch it is pulling at. For example, if the pegs are placed such that the band is stretched to 52 inches, the force the bands apply to the pegs are 57.6 lbs per inch. My question involves what happens with the addition of a third peg? If I make a triangle with the pegs such that it makes a triangle with the same height as before, namely 52 inches, but the base is now 24 inches. What would the force be pulling straight down (no longer along the band itself) from the high point of the triangle toward the 24 inch base? The free length of the band is 41 inches(peg to peg).


title: What is mass of free up and down Quark?

text: 1. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation.   2. Color confinement, often simply called confinement, is the physics phenomenon that color charged particles (such as quarks) cannot be isolated singularly, and therefore cannot be directly observed.   3. In theoretical physics, quantum chromodynamics (QCD) is a theory of the strong interaction (color force), a fundamental force describing the interactions between quarks and gluons which make up hadrons (such as the proton, neutron or pion).   4. Yang–Mills theory List of unsolved problems in physics Yang–Mills theory in the non- perturbative regime: The equations of Yang–Mills remain unsolved at energy scales relevant for describing atomic nuclei. How does Yang– Mills theory give rise to the physics of nuclei and nuclear constituents? When you read about quarks, All folders of Quark model have not been defined. I just want to know what is mass of free Quark? But in my calculations seem free quark have no mass. while Quark become free is like to melt ice.


title: Is it possible to trap smoke or a smoke like gas inside a glass bubble?

text: I am a glassblower, and I would like to trap smoke in a glass bubble, my hope is to create an infinitely lingering smoke inside. Is this possible?


title: Charging Glass Rods

text: When we rub two glass rods with their respective pieces of silk cloth, the two glass rods would repel each other. What if we rub the glass rod against the other glass rod? Will they repel each other? Or they only get charged when they are rubbed by silk cloth?


title: How to determine the amount of light energy (photons) being released from an incandescent light bulb?

text: I have got this all down pat: 1.Collision with a moving particle excites an atom. 2.This causes an electron to jump to a higher energy level. 3.The electron falls back to its original energy level, releasing the extra energy in the form of a photon. Here are a few questions: How does the quark, constituent component combining of the hadron, respond to the emitting particle, electron displacement, and photon disbursement on the scale of Planck length? I want to know the order and installment, arrangement, and implementation of the individual atoms on the subatomic scalar level when releasing photons, all on Planck scale measurements.


title: Computing the efficiency of a heat engine

text: With the heat engine ![enter image description here](http://i.stack.imgur.com/ulLxt.png) we have an ideal gas working the substance. I want to show that the efficiency of this heat engine is $$\eta = 1 - \frac{1}{\gamma}\frac{\bigg(\frac{V_3}{V_1}\bigg)^\gamma - \bigg(\frac{V_2}{V_1}\bigg)^\gamma}{\bigg(\frac{V_3}{V_1} - \frac{V_2}{V_1}\bigg)} $$ where $\gamma = \dfrac{C_p}{C_v}$ is the adiabetic index. So we have that $Q_1$ enters during the expansion $B \to C$ and $Q_2$ leaves during the compression $D \to A$. Therefore, $$\eta = \frac{Q_1 - Q_2}{Q_1} = 1 - \frac{Q_2}{Q_1}$$ I'm kind of stuck at this point. Any further help would be appreciated.


title: Can we observe diffraction even if the slit size is tending to zero?

text: According to Huygens principle, if we consider any point of the primary wave fronts, each point is the source of secondary disturbance. If the point size we consider tends to zero, the number of points on that wave front which enters the slit tends to infinity. Thus, if we consider the slit how much ever small, we must see the diffraction pattern. Because we can consider many points on the wave front that enter slit, whose size is very very small-thus producing interference of waves emanating from them which can give diffraction pattern. But in reality as I know we don't see any diffraction pattern if we have the slit of very small in width compared to wavelength. Is the point size that we consider in diffraction is restricted to some value? If not as explained above, we can have many points of very very small size on the wave front that enter the slit, and we can expect diffraction even if slit size is tending to zero. I don't know whether I have misunderstood anywhere, if yes pardon me and explain. If not, is it that (according to huygens principle) we can observe diffraction pattern even if slit size is tending to zero? Sometimes I might not be communicating with you better with my problem, if so please comment on where there is problem.


title: Can a positron beam cut through metal?

text: Can a positron beam be used to cut through metal?


title: Why does parity violation in weak decay imply decay asymmetry?

text: I googled the sentence in the title of this question and found the famous experiment by Wu _et al_ demonstrating that electrons in weak decay are emitted ``in the direction of motion of a left-handed screw rotating with the nuclear spin''. This show that: asymmetry in decay -> parity violation. How to prove the opposite? (parity violation -> asymmetry ?)


title: Deriving probability distributions from the Wigner distribution

text: I know that I can calculate the probability distributions of $x$ and $p$ from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by calculating their eigenstates, Wigner transforming them, and projecting them onto the Wigner distribution of interest. Is there some way to use the Wigner distribution to calculate probability distributions of operators without finding the eigenstates in the operator representation of QM? For example, how would I go about finding the probability that the energy is within a certain range?


title: Question on the stability of the solar system

text: One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful concept, it hardly explains the long stability of the solar system. Normally an N body problem with inverse square mutual interaction is an example of a chaotic system. Is there any real progress about this stability issue?


title: What is the difference between 'running' and 'current' quark mass?

text: When looking at the PDG, there is a difference between the 'running' and the 'current' quark masses. Does anyone know which is the difference between these two?


title: Exercise QFT and CFT

text: Consider the action functional $S[z;t_1,t_2]=\int_{t_1}^{t^2}[g(z,\bar{z})\dot{z}\dot{\bar{z}}]^{\frac{1}{2}}dt$ with $z(t)$ a complex path with end points $z_i=z(t_i),\; i=1,2$. $g(z,\bar{z})$ is a positive real function on $H=\\{z\in C: Im(z)>0\\}$. After some easy questions the exercise asks me to determine (up to a multiplicative constant) the function $g(z,\bar{z})$ such that the action is symmetric under the transformation $z\rightarrow \gamma(z)= \frac{az+b}{cz+d}$, with $a,b,c,d \in R$. Can somebody help me or give me some hints?


title: Question regarding Schrödinger's equation

text: Here is my understanding of Schrödinger's equation and wave functions: (I'm considering the time independent equation.) > The equation takes in values of energy as an input and, being a differential > equation, gives functions as outputs. These functions correspond to the > different orbitals. > > When we input coordinate values we get an answer that we can use to plot the > probability distribution of an electron around the nucleus. Can someone tell me if my understanding is correct, and correct me if I'm wrong?


title: Free Particle Propagator Using Path Integrals

text: I'm trying to recreate some work that a professor explained to me in his office, specifically deriving the free particle propagator going from $(y,0)$ to $(x,T)$ using the Feynman Path Integral. I'm trying to reproduce $$K(x,T;y,0) = \sqrt{\frac{m}{2\pi i\hbar T}}\mathrm{exp}[\frac{im(x-y)^2}{2\hbar T}]$$Here's what I've done so far: $K(x,T;y,0) = \int_y^x\mathscr{D}[x(t)]e^{iS[x(t)]/\hbar}.$ So first I compute the action: $S[x(t)] = \int_0^T \frac{1}{2}m\dot{x}^2 \mathrm{d}t$ We can always split the path $x(t)$ in the following way: $x(t) = x_{cl} (t) +q(t)$, where $x_{cl}(t)$ given by $$x_{cl}(t) = \frac{(x-y)t}{T} + y$$ is the classical path and $q(t)$ is a "quantum fluctuation". Because the endpoints of $x(t)$ and $x_{cl}(t)$ are the same, we get that $q(0)=q(T)=0$, and because any path should be piecewise differentiable, we can represent $q(t)$ in a Fourier Series: $$q(t) = \sum_{n=1}^{\infty} a_n sin(\frac{n\pi t}{T})$$. The action is then $$S[x(t)] = \frac{1}{2}m\int_0^T (\frac{x-y}{T})^2 + 2\frac{x-y}{T} \dot{q} + \dot{q}^2 \mathrm{d}t$$ The first term is trivial, the second term vanishes due to the fundamental theorem of calculus and the fact that $q(t)$ vanishes at the endpoints. Now for the last term we get $$\int_0^T \dot{q}^2 \mathrm{d}t = \int_0^T \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} a_n a_m (\frac{n\pi}{T}) (\frac{m\pi}{T})cos(\frac{n\pi t}{T})cos(\frac{m\pi t}{T})\mathrm{d}t$$ but due to orthogonality only the $n=m$ terms survive so we get $$ = \sum_{n=1}^{\infty}(\frac{n\pi }{T})^2\int_0^T a_{n}^2cos^2(\frac{n\pi t}{T})\mathrm{d}t = \sum_{n=1}^{\infty}\frac{(n\pi)^2}{2T}a_{n}^2$$. Now to do the actual path integral, "all possible paths" would correspond to "all possible $q(t)$'s" which would mean all possible $a_n$'s. Thus our path integral becomes: $$K(x,T;y) = \lim_{N\to\infty}\int_{-\infty}^{\infty}\mathrm{d}a_1\dotsi\int_{-\infty}^{\infty}\mathrm{d}a_N \mathrm{exp}\\{\frac{im}{2\hbar}[\frac{(x-y)^2}{T} + \sum_{n=1}^{\infty}\frac{(n\pi)^2}{2T}a_{n}^2]\\}$$ Now the first term in the exponential is clearly the same as the one in the original propagator however for the other integrals, i get an infinite amount of integrals which are infinite! Where does my reasoning or algebra go wrong? PS I know there's probably a simpler way to do it, but since we started out this way I wanna know how it can be done with this method.


title: Isotropy and Noise

text: If you have a field which value is just Gaussian noise plus a constant do you call it isotropic?   * there is no preferred direction   * however it is not "the same" in all directions if "the same" means "constant" The question is about a definition of a word, thus obviously conflicting answers can arise. However what definition do **you** use or have seen used: "no preferred direction" or "constant wrt translation"?


title: Why would a screen(with touch controls) in a vibrating environment be readable if its refresh rate is dynamically matched with vibrations?

text: So, I was reading about this Dragon V2 and it has got touch controlled screens, but screen might be unreadable for pilots because of engine vibrations and whatnot. I came across this comment on reddit. _"I read an article on the SLS/Orion. They had a problem with vibrations making screens unreadable. It would have cost millions of dollars to dampen the pod vibrations so the pilots could read their instruments. Then a scientist realized that if you changed the monitor refresh rate to match the vibrations, the screens looked perfectly clear. The million dollar fix was abandoned for a 10 cent accelerometer that adjusted the refresh rate automatically. I see no reason why spacex couldn't use this solution to make the screens useful in the rigor of flight._ " Yeah, so, why does matching refresh rate with vibrations solves this?


title: During an Ice Age, would someone at moderate latitude get a sunburn/suntan on a sunny day?

text: Let's say you live at a moderate (temperate, in today's terms) latitude during the last glacial maximum. You're probably in some kind of steppe or taiga biome, even though you're at 40 degrees north, and you've never known hot weather. If you're outside on a sunny summer day, would you still get a suntan/sunburn like you would in today's climate? Aside from temperature, which doesn't cause sunburn, would there be any relevant atmospheric differences that might affect the UV radiation?


title: Free fall from space

text: If you leave a ball weighing 10 kg at a height of 500 km above sea level (neglecting air friction). How can calculate how long the ball hits the ground and what will be its speed? ![enter image description here](http://i.stack.imgur.com/5Emdd.png) I know that: On earth I can calculate the time taken to the object to fall by this equivalent:               x = x0 + v0t + (at^2)/2     x = 0     x0 = 500000     v0 = 0     a = -g = 10 m/s^2     0 = 500000 - 5t^2     t = 316.227sec      Also I know how to calculate the force between objects:               g = GMm/r^2


title: Stability of a rigid body spinning around minor, intermediate and major principal inertia axes

text: > **Possible Duplicate:**   >  Instability of a thrown tennis racquet   >  Why does a cuboid spin stably around two axes but not the third? I have a problem with one of my study questions for an oral exam: "Describe the stability of as rigid body spinning around the minor, intermediate and major principal axes. Consider a case of spinning purely around the minor principal axis. What will happen with the angular momentum sphere and kinetic energy ellipsoid if we have energy dissipation in the system? Explain using the angular momentum sphere and the kinetic energy ellipsoid. Will the angular momentum stay constant in the presence of energy dissipation—why or why not? In which ways can energy be dissipated from a spinning body?"


title: Instability of a thrown tennis racquet

text: Someone once mentioned to me that it's impossible to throw a tennis racquet (or similarly shaped object) into the air, perpendicularly to the string plane, in such a way that it won't turn. What is this effect, or was he talking rubbish?


title: Can a single photon be polarized non-linearly?

text: I want to check if I correctly understand polarization. Considering a single photon travelling in vacuum, it can only be polarized linearly under the same direction at any time, right? When we talk about circular polarization, or unpolarized light, we are talking about sums of linearly polarized photons, with their electric fields ranging from $0$ to $\pi/2$ phase difference?


title: How to calculate average velocity

text: The position of an object moving along X-axis is given as $X=a+bt^2$ where $a=8.5\,\mathrm m$, $b=2.5\,\mathrm {m/s^2}$ and $t$ is measured in seconds. What is it's velocity at $t=0$ and $t=2.0$ s? What is the average velocity between $t=2.0s$ s and $t=4.0$ s? I have no idea what to do. I have seen this in numerical in one book where they have used velocity $v=dx/dt$, but what does $d$ stand for? Is it delta($\Delta$)? I am totally confused.


title: Charge is an intrinsic property

text: How does a body acquire charge or get charged up? I am not talking about the method through which it gets charged up, but does every atom contain things that make a body charge up? If we talked about the process by which a body gets charge, how does it happen?


title: Does perpetual motion machine really exist?

text: I recently saw a video on Youtube about a perpetual motion machine. Here is the link: http://www.youtube.com/watch?v=OS1KXMsE2qk It's on Robert Boyle's flask. Where does the energy come from?


title: Young's Double Slit Experiment : What would happen if the "first slit" was too wide?

text: ![A picture showing Young's Double Slit experiment](http://i.stack.imgur.com/QTGeQ.gif) I was wondering what would happen to the fringe pattern displayed on the screen if the first slit (as shown in the picture), which is also known as "single slit", was made a bit wider. I read it in my book. I don't understand it. Anyway, it quotes > _If the single slit is too wide, each part of it produces a fringe pattern > which is displaced slightly from the pattern due to adjacent parts of the > single slit. As a result, the dark fringes of double slit pattern become > narrower than the bright fringes, and contrast is lost between the dark and > the bright fringes._ Please answer the question in your own words and try to explain to me what the quote is trying to say. Also, when it says "parts", what does it mean?


title: Angular momentum of anyons

text: Why is it true that anyons can have angular momentum taking any real value? Why aren't they restricted to the $j(j+1)$ integer values most are familar with?


title: Can an asteroid/meteroid/meteor/meteorite be stopped in its way to Earth?

text: I was just studying meteorology and had a doubt about the motion of meteors towards earth. Can a meteor be stopped while it is still under under the gravitational influence of Earth? As far as I know some meteors are very bulky and their motion towards the Earth's surface is with a huge magnitude of momentum. Has any organisation made any equipment to stop these meteors?


title: Is it possible to more cloud-layers go to different ways on the sky

text: Sometime when I see the sky I see that the cloud-layers go to different ways. But how is it possible?


title: How Non-abelian anyons arise in solid-state systems?

text: Recently it has been studied non-abelian anyons in some solid-state systems. These states are being studied for the creation and manipulation of qubits in quantum computing. **But, how these non-abelian anyons can arise if all particles involved are fermions?** In other words, how electronic states can have different statistics than the fermionic if all electrons are fermions?


title: Current to cancel Earth's magnetic field

text: I'm making a lab report where I've used a Helmholtz coil and a magnet to find Earth's magnetic field intensity in the area by relating the frequency of oscillation of the magnet when a certain current is going through the coil, using the equation: $$ f^2 = \frac{μ_o}{4π^2m_i} \left(\frac{8\mu_0N}{5^{3/2}R}I \pm B\right). $$ Everything went ok and I found the experimental value of the magnetic field to be $17\ \mu\mathrm{T}$; the teacher said around $2\times10^{-5}\ \mathrm{T}$ would be fine. One of the questions was, "What is the current needed to cancel Earth's magnetic field?" I did some research and found that if the field is cancelled then the period of oscillation goes to infinity and so concluded that the frequency of oscillation ($1/T$) must go to zero. By setting $f^2 = 0$ in the previous equation I found that $$ I = \frac{5^{5/3}BR}{8\mu_0N}$$ Are these assumptions correct? I know you're not supposed to do my homework; if someone could only answer if $f = 0$ when the field is cancelled is good enough. Notation: $B$ is Earth's field;   $I$ is the current;   $R$ is the coil's radius;   $N$ is the number of loops/turns;   $\mu_0$ is $4\pi\times10^{-7}$;   $m_i$ (not needed in the second equation) is the moment of inertia of the magnet.


title: Explanation for this type of (magic-trick) suspension?

text: Well let's start off with that I'm not a physicist but I'd like some thoughts on something I came across in my hometown. This guy: ![enter image description here](http://i.stack.imgur.com/G3U6R.jpg) Is it possible that due to the electrical charge of magnets this guy can make the illusion that he can float ? Or is this probably a cheap trick that fools the eye ? I was standing there for quite some time watching the guy and he keep moving his feet. The resistance that he appeared to have was from a magnet force keeping him afloat. So after I passed this guy I did some physics searches on the web and the first thing that caught my eye was the electrical charge of magnets. So the question is : Is this related to the electrical charge of a magnet or a cheap trick ?


title: Question about Lagrangian mechanics

text: I have a question about the Lagrangian-formalism; they state that the holonomic constraints are expressed as: $$f_\alpha(x_1,y_1,z_1,...,x_n,y_n,z_n,t) = 0 $$ where $\alpha = 1,...,L$ They state now that these constraints may not contain irrelevant information. They express this by saying: $$rank(\frac{\partial{f^{\alpha}}}{\partial{x_k}}\frac{\partial{f^{\alpha}}}{\partial{y_k}}\frac{\partial{f^{\alpha}}}{\partial{z_k}}) = L$$ With $\alpha = 1,..,L $ and $k = 1,...,N $. This looks like a Jacobi- matrix. I don't understand however how this expresses that the constraints may not contain irrelevant information. (I see that they may not be linear dependent, is there a relation between linear dependence and the Jacobi- matrix?) Further they state that under these premises, we can solve the constraints and we can say: $z_k = z_k(x_1,y_1,z_1,...,t) $ (where $z_k$ is a symbolical notation for a parameter (so it can also be for instance $x_5$)), and $z_k$ depends on $3n-L$ parameters and the time $t$. With a regular transformation these parameters can be transformed to other generalized parameters. This should follow out of the theorem of implicit functions, but these theorem only states that we can solve $f_\alpha$ **local** : so I don't understand why they assume that these generalized parameters should completely determine the configuration of the system. It seems that these coordinates only determine the configuration of the system in the surrounding of a point that satisfies the constraints.


title: Tree level and loop level

text: I'm trying to read through a paper which explains the following about Universal Extra Dimensions (UED) vs ADD models: > The new feature of the UED scenario compared to the brane world is that > since there is no brane to violate translation invariance along the extra > dimensions, momentum is conserved at tree level leading to degenerate KK > mode masses at each level and conservation of KK number in the interactions > of the four dimensional effective theory. This statement is broken at the > loop level, where the fact that the extra dimensions are compact leads to > (calculable) violations of the full Lorentz symmetry, and as a result shifts > the masses of the KK modes away from their tree level values. I'm unfamiliar with the concepts of tree-level and loop-level. Could someone perhaps explain this in context? If I would have to guess, I'm thinking that tree-level correspond to the scattering correlation functions of the classical theory and loop-level are a kind of correction?


title: Why is it natural to look for solutions involving dimensionless quantities?

text: While studying the Heat Equation, I got stuck in a statement in my book. It says: > We have seen that the combination of variables $\displaystyle > \frac{x}{\sqrt{Dt}}$ is not only invariant with respect to parabolic > dilations but also dimensionless. **It is then natural to check if there are > solutions of the Heat Equation involving such dimensionless group. (...) it > makes sense to look for solutions of the form:** > > $\displaystyle u^{*}(x,t)=\frac{q}{\sqrt{Dt}}U\left( \frac{x}{\sqrt{Dt}} > \right)$ > > **where $U$ is a (dimensionless) function of a single variable.** " Why is the bold text so? I don't see why it is intuitive to search for such solutions.


title: Back $emf$ and power?

text: The formula relating current to back emf is $V-E=IR$ where $V$ is the source pd and $E$ is the back Emf. Thus it can be seen that as the back emf increases the current decreases. But what is the implication of the power ($PV$) delivered across by the source and that across the compenent. I presume that both just decreases, otherwise energy would be lost, is this correct, and if so please can you explain?


title: Which is the most fundamental constant between the Planck constant $h$ and the reduced Planck constant $\hbar$?

text: This question is related to Planck units (also called natural units, absolute units or God's units). I'm wondering which constant is the most fundamental and should be normalized to 1. I would like as much explanation as possible please.


title: Behavior of the electric field on boundary surfaces

text: Consider this ![this](http://i.imgur.com/ko1D4aX.jpg) picture. Integrating over this infinitesimal box gives the following equivalencies: $$\int_{\Delta V} d^3r~{\rm div} \vec{E}(\vec{r}) = \int_{S(\Delta V)} d\vec{f} \cdot \vec{E}(\vec{r}),$$ which for $\Delta x \rightarrow 0$ gives $\Delta F \vec{n} \cdot (\vec{E_a} - \vec{E_i})$. Since this integral has to be equal to $\frac{1}{\epsilon_0}\sigma \Delta F$ we can deduce that $$\vec{n} \cdot (\vec{E_a}-\vec{E_i}) = \frac{\sigma}{\epsilon_0}.$$ My question is, how do I calculate this limit of $\Delta x \rightarrow 0$? I get that the integral basically computes all the scalar products of the electric field on the surface and adds them up, but how do I get the vector $ (\vec{E_a}-\vec{E_i})$? Why is the electric field given by this vector? It's probably really simple but I can't make sense of this right now, so some help would be appreciated.


title: A Satellite's Perspective

text: If a planet is spinning east to west and there is a satellite spinning from west to east... Can the satellite travel at a speed sufficient to make the planet appear, from the vantage point of the satellite, to be rotating from west to east? What kind of calculations could be made to determine that speed?


title: Flux if there were only one type of charge in the universe

text: There was this question that i saw in a book and it also had an answer given. The Question was: > If there were only one type of charge in the universe, then: > >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} \neq 0 $ on any > surface. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} = 0$ is the > charge is outside a surface. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} $ will not be > defined. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} = > \frac{q}{\epsilon_o}$ if the charge is inside the surface. > The answers were given as option second and the last one. (Gauss' law statements) How?


title: Do ~1THz oscillators output "light"?

text: Upon looking at some radiation levels higher than Microwaves, I have come across "Terahertz radiation". According to the article, there are numerous ways to generate such radiation, and that brings questions to me. My confusion lays in how it is different (if it is!) from a heating fan emitting infrared radiation (simple), or an LED emitting visible green light, or a flashlight emitting various spectral frequencies. Are those examples emitting Terahertz radiation? I would hypothesize that is, but can see no reference to light being called "Terahertz radiation" even though it is radiating. Although I am unsure if the technologies listed in the article can go above infrared in to visible light, but assuming so, would it emit "light" (in the respective frequencies)? How would this be different than an ordinary visible light emitter? At that point is it no different from an "ideal" colour changing LED?


title: Normal force in a compound pendulum (physical pundulum) system?

text: Consider a compound pendulum pivoted about a fixed horizontal axis, illustrated by the force diagram on the right: ![image was taken from hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c4 ](http://i.stack.imgur.com/aInsV.gif) ## # Okay, I can't figure out where the normal force on the pendlum should point (this force isn't indicated in the diagram). On the one hand, I think it should point in the same direction as the tension force on the left-side diagram, in order to produce the rotation. But on the other hand, shouldn't the normal force point towards the pendulum rather than away from it? In any case, I know that normal face is perpendicuar to the contact surface, but how does "contact surface" fit into the context of this scenario? Please help me understand the concepts involved.


title: Constant Pressure lines in evaporation diagram

text: Assume that there is the following $T-H$ phase diagram with the constant pressure lines. ![](http://i.stack.imgur.com/IsC28.gif) Which is greater preasure? The preasure drops as we move to lower temperatures (i.e. from top to bottom)? How constant pressure lines are on the water side of the diagram? How to compute the amount of heat that the steam can carry(or that the water can use to produce steam in a pressurised water reactor)? Any ideas are welcomed!


title: Partial derivative potential energy of 'free' vibration

text: I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial x_i\partial x_j} x_ix_j.$$ Or if we use $b_{ij}$ for the Hessian : $$U(x_i)=\frac{1}{2}\sum_{i,j}b_{ij}x_ix_j.$$ I want to calculate the force: $k_i=\frac{-\partial U}{\partial x_i}$. This should be $-\sum_{j} b_{ji}x_j$. My questions:   1. Why is the summation over the index $i$ gone?   2. Why are the indices of the hessian $b$ switched? I hope someone can give me an answer. I put this in the physics section because it's a physics problem, but my question is actually purely mathematical.


title: What is the friction between cylinder and wall (ground)?

text: A hollow cylinder (radius $R$) is rolling against the wall at angular speed $\omega$. The coefficient of friction between the cylinder and the wall(ground) is $\mu$. After how many rotations the cylinder will stop rotating? So I figured I need to find the time taken for cylinder to stop moving, and that would be $$ \beta=-\omega/t => t = -\omega/\beta $$ Where $\beta$ is angular acceleration, which is known from torques: $$ 2*F_f*R = I*\beta $$ That's where I got stuck... How do I know the friction? I'm familiar with such equation: $$ F_f = \mu*F_n $$ How do I find the normal force? Does it have anything to do with centripetal force?


title: Confocal Microscopy

text: In the context of Confocal Microscopy literature state, "spatial rejection of out of phase light".Is that mean only light which is pass through the pinhole is used and the rest is blocked ?


title: Negative mass dirac equation -> Propagator?

text: There are two types of dirac equations: $(p_\mu\gamma^\mu - m)\Psi(x) = 0$ and $(p_\mu\gamma^\mu + m)\Psi(x) = 0$. Here $p$ are the momentum operators. The fermion propagator is defined in the equation $(p_\mu\gamma^\mu - m)S_F(x-x') = \delta(x-x')$. This equation can be easily solved in momentum space: $$S_F(p) = \frac{p_\mu\gamma^\mu + m}{p^2-m^2+i\epsilon}$$. As far as I can see this contains only the first dirac equation and not the second equation. Nevertheless we use this to calculate all processes in QED, so is the seconed dirac equation not included in QED?


title: What are the units of the Bekenstein bound?

text: Working with the Wikipedia definition of the Bekenstein bound: $S \leq \frac{2 \pi R k_bE}{\hbar c}$ $2\pi R \ $ is $m^2$ $k_b$ is $\frac{J}{K}$ $E$ is $J$ $\hbar$ is $J*s$ $c$ is $\frac{m}{s}$ $\frac{m^2 \frac{J}{K} J}{(Js \frac{m}{s})} = m \frac{J}{K}$ Am I overlooking something?


title: Optical coherence tomography

text: In Optical Coherence Tomography, Broader bandwidth/low coherence light sources are used. Is it only because to increase the resolution or are there any other reasons? What will happen if we use a high coherence source ?


title: Potential energy of the dipole-dipole interaction for two parallel dipole moments

text: I am looking for an equation that gives me the potential energy of the interaction between two parallel dipoles.


title: Converting indices of refraction

text: If I know that the index of refraction of a given substance is 1.4 for the average wavelength (say 550 nm), and I would like to know what is the index of refraction with a wavelength of 832 nm, how would I go about computing this? I'm not looking for an exact solution, just a rough estimate that is better than 1.4


title: Which experiments prove atomic theory?

text: Which experiments prove atomic theory? Sub-atomic theories:   1. atoms have: nuclei; electrons; protons; and neutrons.   2. That the number of electrons atoms have determines their relationship with other atoms.    3. That the atom is the smallest elemental unit of matter - that we can't continue to divide atoms into anything smaller and have them retain the characteristics of the parent element.   4. That everything is made of atoms.  These sub-theories might spur more thoughts of individual experiments that prove individual sub-atomic theories (my guess is more was able to be proven after more experiments followed).


title: How would a very nearby supernova shockwave and remnants affect the earth?

text: I've been reading about supernovae for a while, and I noticed how incredibly fast their shockwave and remnants travel shortly after the explosion. So I thought about how this would affect the earth if one happened nearby. I know that the earth's atmosphere already protects us from cosmic rays which I think have energy comparable to those particles from supernovae, so would a near-earth supernova be any different ? Of course I know it depends on how far the supernova is and I know that currently there are not too close stars that are about to go supernova. So for the sake of the question, let's assume it's a few light years away. Please notice that I am not asking about the possibility of something like this happening, I am just curious what it would do to earth if it does happen that close. So what do you think the effects of such close supernova can be ?


title: Do derivatives anticommute with Grassmann variables and complex numbers in a many-body path integral?

text: I'm trying to learn how to do a many-body path integral for **both fermions and bosons** , and I'm stuck. I'm following Altland and Simons - Condensed Matter Field Theory, chapter 4. On page 167, equation 4.27 is \begin{equation} Z = \int \prod_{n=1}^N d(\bar{\psi}^n,\psi^n) e^{-\delta \sum_{n=0}^{N-1}[\delta^{-1}(\bar{\psi}^n - \bar{\psi}^{n+1}).\psi^n + H(\bar{\psi}^{n+1},\psi^n)]} \end{equation} (I've set $\mu=0$ from the equation in the book). The limit $N \rightarrow \infty$ is then taken which involves various things, but the part I don't understand is this: \begin{equation} \lim_{N \rightarrow \infty} \delta^{-1}(\bar{\psi}^n - \bar{\psi}^{n+1})) \rightarrow -\partial_\tau \bar{\psi} \end{equation} which is fine, but the next is \begin{equation} Z = \int D(\bar{\psi},\psi) e^{-S[\bar{\psi},\psi]}, \hspace{4mm} S[\bar{\psi},\psi] = \int_0^\beta [\bar{\psi} \partial_\tau \psi + H(\bar{\psi},\psi)] \end{equation} My question is how do you get from $-\partial_\tau \bar{\psi}$, which is $-\partial_\tau \bar{\psi} \psi$ in $Z$, to $+\bar{\psi} \partial_\tau \psi$? If this was only for fermions, I would guess that the Grassmann variable $\psi$ and the derivative $\partial_\tau$ anticommute which is where the minus sign comes from. But the book says it's valid for bosons as well as fermions; for bosons, the $\psi$ is a complex number, and so I wouldn't expect the minus sign. Any help would be much appreciated!


title: Transformation of $t=0$ line in moving frame of reference

text: How does $t=0$ transform into $t - vx/c^2 = 0$ if a frame of reference is moving as given in here? It seems that the relativistic transformation is given by $$ \begin{bmatrix} x' \\\ ct' \end{bmatrix}= A\begin{bmatrix} x \\\ ct \end{bmatrix} = \left( \begin{array}{cc} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & -\frac{v}{c \sqrt{1-\frac{v^2}{c^2}}} \\\ -\frac{v}{c \sqrt{1-\frac{v^2}{c^2}}} & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\\ \end{array} \right).\begin{bmatrix} x \\\ ct \end{bmatrix} = \begin{bmatrix} \frac{x-t v}{\sqrt{1-\frac{v^2}{c^2}}} \\\ \frac{c^2 t-v x}{c \sqrt{1-\frac{v^2}{c^2}}} \end{bmatrix} $$ I was wondering if the transformation is equivalent to measuring in new basis $A\begin{bmatrix} 1 \\\ 0 \end{bmatrix}$ and $A\begin{bmatrix} 0 \\\ 1 \end{bmatrix}$ for an event?


title: Could there be a correlation between the Island of Stability and dark matter?

text: I can only boast of my passion for science and not of any credentials. I ask because I am curious, and it is not the type of question one can easily Google! What if the key to discovering and unlocking the secrets of dark matter and energy lie in the Island of Stability? Am I missing a vital reason that outright contradicts an attempted connection between the two?


title: Definition: Coupling

text: What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?


title: BEC in a rotating disc

text: Goodmorning everybody, I have to run a numerical simulation of a Bose-Einstein condensate on a rotating disc. Now, my problem is that I became suspicious about the equation I'm using, since the final result I obtain is not what I expected. First of all, the Gross-Pitaevskii equation (in a z-axis symmetrical trap) for the amplitude of the wave function of a condensate with a vortex is $$ \mu f(\rho,z) = -\frac{\hbar^2}{2m}\left( \frac{1}{\rho}\frac{\partial}{\partial \rho}\left( \rho \frac{\partial f(\rho,z)}{\partial \rho} \right) - \frac{f(\rho)l^2}{\rho^2}+\frac{\partial^2 f(\rho,z)}{\partial z^2} \right) + V(\rho,z)f(\rho,z) + U_0f^2(\rho,z)\cdot f(\rho,z) $$ where $l$ is the quantum of circulation, $\mu$ the chemical potential and $U_0 = 4\pi\hbar^2a/m $ ($a$ is the scattering length). Now, I want to consider the equation on a disk, so I put $f=f(\rho)$. To solve it numerically, I need to perform the substitution $$ f(\rho) = \frac{u(\rho)}{\sqrt \rho } $$ (while in a spherical confinment I would have put $f(\rho) = u(\rho)/\rho)$). The kinetic term is therefore given by $$ \frac{1}{\rho}\frac{\partial}{\partial \rho} \left( \rho \frac{\partial f(\rho)}{\partial \rho} \right) = \frac{1}{\rho}\frac{\partial f(\rho)}{\partial \rho}+\frac{\partial^2f(\rho)}{\partial\rho^2} = \frac{u''(\rho)}{\sqrt \rho}+\frac{u(\rho)}{4\rho^{5/2}} $$ so that the first derivative is absent (I need that because I'm solving it using the Numerov's algorithm). Putting all together $$ \mu u(\rho) = -\frac{\hbar^2}{2m}\frac{d^2u(\rho)}{d \rho^2} + \frac{\hbar^2}{2m\rho^2}\left( l^2-\frac{1}{4} \right)u(\rho) + V(\rho,z)u(\rho) + \frac{U_0}{\rho}u^2(\rho)\cdot u(\rho) $$ where, as external potential I use $V(\rho) = 1/2 m \omega^2\rho^2$. It seemed a good equation, until I noticed that if I try to test it in the case of the non- rotating disc ($l=0$, in this case the equation above must become exactly the GP equation in cylindrical coordinates) and non-interacting gas ($a=0$, that means $U_0=0$) the equation mantains a centrifugal term, and so the ground state solution vanishes as $\rho$ approches 0, that I don't think it's correct. What do you think? What did I do wrong? I assume that my program works, because the simulation in the spherical trap ended well (and I paid attention to modify the normalization condition and other stuff), but if you don't find any errors in my calculations, then I will go check again the algorithms! Thank you!!


title: Curves satisfying this functional

text: This is a problem in Hartle's "GRAVITY": Consider the functional $$S[x(t)]= \int_{0}^{T} \left[\left(\frac{dx(t)}{dt}\right)^2 + x^2(t)\right]\text{ }dt$$ Find the curve $x(t)$ satisfying the conditions: $x(0)=0$, and $x(T)=1$, i.e make $x(t)$ an extremum, and find its extreme value: whether it is maximum or minimum. Here's what I did: I evaluated the Euler lagrange equation : $\frac {\partial {L}} {\partial x}$=$\frac{d}{dt}$ $(\frac {\partial L} {\partial \dot x})$, where $L=\dot x^2 + x^2$, and obtained: $$\ddot x=x$$. Now, how do I solve this equation to do the rest of the problem?


title: Maximal kinetic energy due to gravitational attraction

text: Two related questions:   1. Small object of mass $m$ is falling into the supermassive black hole of mass $M$. What is the maximal kinetic energy can be acquired by the small object, from the point of view of an observer whose frame of reference is at rest near event horizon? Can it exceed $(M+m)c^2$?   2. Two small black holes of equal mass $M$ are attracted each other. What is the maximal kinetic energy can be acquired by each black hole due to gravitational attraction, from the point of view of an observer between them?


title: Cooling power for cloud chamber

text: I'd like to build a small Peltier cell cooled cloud chamber. The cold surface would be a copper sheet with a surface of 200x200 millimeters. Is there a way to even guess the cooling power needed to bring the copper sheet below -30°C ?


title: Energy conservation limited by uncertainty principle

text: The way I learned it from practicing Fourier analysis and signal processing besides quantum mechanics, is that Energy conservation cannot be achieved in short time scales, and that limits energy conservation in Quantum mechanics. In other words: _Energy conservation is limited by the Heisenberg uncertainty principle in our universe._ $$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$ As I posted this answer somewhere, someone said this is wrong. So I posted this issue here, for maybe I don't understand it properly and you guys could tell me why this is wrong. Example: A beta decay produces a W-boson, which is ~85 times larger in mass than the initial particles (which decays eventually to neutrino and some beta particle), and this is possible due to the very short time scale, during which the W-boson is produced. I.e, energy is not conserved in very short time scales. Do I understand this correctly? Thank you.


title: Mass loss rate of planetary nebulae

text: The “interacting wind” model of planetary nebulae is based on the idea that the white dwarf phase of stellar evolution is preceded by a red giant phase. A fast wind from the hot white dwarf overtakes the more slowly moving red giant wind and the region of interaction forms a shell of material which is driven outwards by the fast wind and photoionised by the white dwarf. The emission from this shell produces the observed planetary nebula. Assume that the fast wind switches on instantaneously at the end of the red giant phase at $t = 0$. The mass of swept up material is given by: $$ \frac{dM_s}{dt}=A(V_s-V_{RG}) $$ where $V_s=dR_s/dt$ is the velocity of the shell and $A=\dot{M_{RG}}/V_{RG}$ Assuming that the evolution of the interior of the mass-loss bubble is adiabatic, I have to find out the radius of the swept up shell and the pressure are given by $$ R_s(t) = \left ( 2L_w/3A \right )^{1/3} t $$ # Attempted solution I have started with \begin{eqnarray} \frac{d}{dt} \int_{shell} \rho d^3x = -\int_{S_2} \rho (v_i -u_i) n_i ds + \int_{S_1} \rho (v_i -u_i)n_i ds \end{eqnarray} Then finally I got the answer of The mass of swept up material. But I have started with Energy driven and momentum $$ \frac{d}{dt}\left (M \frac{dR_s}{dt} \right )= \frac{3}{2} \frac{L_w}{v_w} \frac{R_s^2}{R_1^2}= 4\pi p_bR_s^2 $$ I think from here I need find out the required equation. Could some one help ?


title: Spin polarization of decay products

text: A relativistic moving particle, e.g. muon $\mu^+$, described by its four- momentum vector $p_\mu$, charge $e$ and with a given spin polarization, ${\bf S}=(S_x,S_y,S_z)$, decays into three particles, e.g. $\mu^+\to\nu_\mu e^+\bar{\nu}_e$.   * How does one determine the spin polarizations of decay products? (what literature to search)   * Is the treatment different if particle decays in a (homogeneous) magnetic field or for non-relativistic velocities?


title: Why did Einstein get credit for formulating the theory of special relativity?

text: See _The Principle of Relativity_ here: _The Principles of Mathematical Physics_. This was written by Poincaré in 1904, a year before Einstein published his theory of relativity. It appears from this and other writings of Poincaré that Poincaré discovered the theory of special relativity before Einstein. So why does Einstein get the credit?


title: Time traveling and Time dilation within a multiplayer videogame

text: First thing first, I'm looking to implement this within a multiplayer videogame as a "special" skill or powerup, but, I'm willing to build it upon a physics based explanation/theory. The question may be misleading due to my language gap and lack of knowledge in physics, in that case, I'd be thankful if you'd correct me. Derived from: http://gamedev.stackexchange.com/questions/19819/how-to- implement-bullet-time-in-a-multiplayer-game What I think is associated to this question:   1. http://en.wikipedia.org/wiki/Time_travel   2. http://en.wikipedia.org/wiki/Time_dilation   3. http://en.wikipedia.org/wiki/Spacetime   4. General and Special relativity. The question is, what do I have to follow in order to implement a slow motion/bullet time effect for a player as a skill, while keeping these conditions in mind:   1. The player under the effects of slow motion has an advantage to make more precise actions (shooting, movement etc.), due to seeing everything in a much slower pace.   2. His actions should affect the other players in real-time.   3. The players looking upon our affected player, do not see any difference, except they can experience the actions he have done (like, player gets shot/affected dodges a rocket).   4. When the player comes out of this effect, he simply continues in the same pace as usual. (If this is not possible, then I guess he'll have to suffer until the time gets synced back) In case my imagined scenario is impossible, what would be closest possible solution for such an effect? I hope I've made myself clear and thanks in advance!


title: How to find out the compress happen when some force act on water/oil/air?

text: I need to do some calculation to find out whether my design works. I may use oil/water/air in my pneumatic cylinder (or you can call it hydraulic cylinder). Assume I have just a cylinder and I put oil into the chamber to raise a weight. My question is how much of volume will reduce when the weight is x kg (Let assumes x = 20). Also what if I change the oil to water or air. From my knowledge air is easier to calculate, because I can use the ideal gas law. Am I right? Thanks


title: sigma model on $S^1 \times S^3$

text: In arXiv:1207.3497 \- _4D partition function on $S^1 \times S^3$ and 2D Yang- Mills with nonzero area_ , Yuji Tachikawa explains the partition function for an 4d $\mathcal{N}=2$ sigma model on $S^3 \times S^1$ with target $T^*G_{\mathbb{C}}=G \times \mathfrak{g}\times \mathfrak{g}\times \mathfrak{g}$ is that of a q-deformed 2D Yang-Mills theory. What do the modes of this σ-model look like? How does the geometry/topology of $S^3 \times S^1$ play a role here?


title: Locally every force admits a potential?

text: I have a little doubt about a force being or not conservative. Well, as I understood, some forces cannot be expressed as exterior derivative of some scalar potential because the work done by the force is path dependent, in other words, it's not possible to determine the work based only on some value at the end points. But my question is: locally every force admits some potential? In other words, if $F$ is a force field defined on some subset $U\subset\mathbb{R}^n$, and if $a\in U$, is there always some $\epsilon$-neighborhood of $a$, $B_\epsilon(a)$ such that the restriction of $F$ to that $B_\epsilon(a)$ admits potential? Because for me it seems that for any force, it's possible to find some small region in which the "path-dependance" won't interfere that much. Is this correct? And in this case, I'm assuming that space is flat. If we're working with some curved manifold $M$, this idea yet applies? Sorry if that's silly, it's just a thought that I wanted to confirm. Thanks in advance for your aid.


title: The rebound height of a mass on a trampoline

text: When a mass(metal ball) lands on the end of a trampolne bed, the mass is displaced towards the centre of the trampoline. I have read about the energy changes and forces involved in a trampoline but can't find any information about this aspect. I'm guessing it has something to do with a spring always trying to return to equilibrium but can't work out what forces are involved. In my Advanced Higher Physics project I am changing the mass landing on the edge of the trampoline and measuring how this affects the amount it is displaced towards the centre. Obviously the displacement increases as mass increases. My other investigation is how the rebound height is affeced when I change the distance from the centre a mass dropped. The rebound height increases as I increase the distance. Why is this? Any suggestions about the physics behind these movements would be much appreciated and very interesting.


title: Wormholes and the illusion of time?

text: I was watching a video on Youtube by Brian Greene, "the illusion of time."http://www.youtube.com/watch?v=j-u1aaltiq4 In that video, he introduces to me the idea of a "brane," or a slice of the universe at specific points in time. And then, he talks about an alien that is tens of billions of lightyears away from Earth. At that distance, he says that even the slowest movements can angle his brane, or slice of time, such that he is angling it towards the Earth's past (if the alien is moving away from us) or future (if the alien moves towards us). So I got to wondering. What if you had a wormhole here on Earth, and walked through it to appear somewhere tens of billions of lightyears away from Earth. Depending on your speed of crossing through it, would you appear further into their past/future (depending on which direction you cross)? And, if you had another wormhole there connecting back to the Earth, wouldn't you be able to travel back into Earth's past? Which would violate causality? I know lots of people don't like the idea of wormholes, but it doesn't seem to have been scientifically ruled out. So, my question is, despite the fact that wormholes seem to possibly violate causality, why can't it be ruled out? Why is it still a theoretical possibility? As a side question, if the past, present, and future exist at the same time -- what does that even mean? (If anyone could edit my post, that would be great. My internet is giving me a horrible screen to work with. I have totally no ability to see my own tags -- look at this interface: https://dl.dropbox.com/u/106018191/phystackex.png) (The image above is going to take a while to upload because of the slow internet, possibly a few hours at worst.)


title: About space-time and its four dimensions

text: I explained to someone I know about General Relativity (as much as I know). He said that he didn't see how it could be correct. He argued: How is 4-dimensional space-time space different to 3-dimensional space? he doesn't agree that as that because the 4-dimensional space-time is only different to 3-dimensions because of the added time dimension. he doesn't think that the added dimension of time would change the space of the system (so 3-dimensional space therefore wouldn't be different to 4-dimensional) because they both still have 3 spacial dimensions, just space-time has an added dimension of time. I said there will be an explanation to how an added dimension of time would change the spacial structure. is there?


title: A certain regularization and renormalization scheme

text: In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove, (1) $\int ^\Lambda \frac{d^2 k}{(2\pi)^2}\frac{1}{k^2 + q_i ^2 \vert \sigma \vert ^2} = - \frac{1}{2\pi} ln \vert q _ i \vert - \frac{1}{2\pi}ln \frac{\vert \sigma\vert}{\mu}$ (..there was an overall $\sum _i q_i$ in the above but I don't think that is germane to the point..) (2) $\int ^\Lambda \frac{d^2 k}{(2\pi)^2}\frac{1}{k^2 + \vert \sigma \vert ^2} = \frac{1}{2\pi} (ln \frac{\Lambda}{\mu} - ln \frac{\vert \sigma \vert }{\mu} )$ I tried doing dimensional regularization and Pauli-Villar's (motivated by seeing that $\mu$ which looks like an IR cut-off) but nothing helped me reproduce the above equations. I would glad if someone can help prove these above two equations.


title: CY moduli fields

text: When one does string compactification on a Calabi-Yau 3-fold. The parameters in Kähler moduli and complex moduli gives the scalar fields in 4-dimensions. It is claimed that the Kähler potentials of the CY moduli space gives the kinetic terms of the scalar fields in 4d. Could anyone let me know why? I know that a consistent coupling of a SUSY multiplet containing scalars with supergravity requires the scalar kinetic term comes from a Kähler potential. But I am not sure why precisely this Kähler potential coincides with the one for CY moduli spaces in the case of string compactification. Could anyone explain it to me? Thanks!


title: How to calculate the specific heat capacity of gases

text: Could someone explain why the Specific Heat Capacity (SHC) of dry air is 1.005kJ/kg.K whereas the accepted SHC for ventilation air is quoted 1.300kJ/kg.K? Assuming the air for the ventilation calc’s is humid say 60RH @20°C it is estimated that there would be about 8.8g/kg of moisture in the air. [Using the psychrometric chart from the British Standards Institute] The SHC of water is 4.200kJ/kg.K = 4.2J/g The SHC of water vapour is 1.86kJ/kg.K = 1.86J/g If there are about 9g of moisture (water vapour) in the air; 9 x 1.86 = 16.74J of energy will be required. The water molecules in vapour form will displace the dry air molecules therefore the dry air SHC will be less than 1.005kJ/kg.K. If dry air is about 1kg/m3 and the water vapour has displaced some of the dry air molecules then the SHC of the humid air (dry air + water vapour) is going to be less than 1.300kJ/kg.K. To convert H2O from liquid to gas (water to water vapour) requires 2.3kJ/g of latent heat which is more energy than is required to raise 1kg of water 1K. Also is there any relationship between atomic mass and SHC? As air is mainly Nitrogen (14 atomic number) @ 78%, and Oxygen (16) 21% [+ 1% of various other gases]. Both the main gases are diatomic: Dry Air = (0.78 x (2x14)) +( 0.21 x (2x16)) = 28.56 + Argon and CO2 = 28.946 atomic mass per mole. Water = Hydrogen (2x1) + Oxygen (16) = 18.00 or there about due to impurities per mole. The argument that humid air is less dense than dry air can be deduced from the above. If a mole of a gas at standard conditions has a volume of 22.4ltrs and 1000 ltrs = 1m3 therefore 1000/22.4 = 44.64. (moles in 1m3) 44.64 x 28.95 = 1292 which is close to the 1.300kJ/kg.K [ventilation air however it would be dry and not 60RH]. Deduct 1 mole from dry air (1000 – 22.4 = 978ltrs) 978/22.4 = 43.66 moles. (43.66 x 28.95) + (1 x 18.00) = 1.281kJ/kg.K. The SHC of humid air approx’ 1m3? Virtually the same as dry air. Or is it coincidental? Any help would be appreciated - thanks


title: EPR paradox: instantaneous vs very fast?

text: An EPR quantum experiment can be explained by instantaneous collapse of the wave function regardless of the distance separating a pair of entangled particles. But do we have the certainty that the process is really instantaneous? If not, what is the current experimental limit on how fast is the process?


Where and how is the entropy of a black hole stored? Is it around the horizon? Most of the entanglement entropy across the event horizon lies within Planck distances of it and are short lived. Is it stored near the singularity? How can you pack so much information in such a small region? Can you please help me?


title: Shouldn't Charge Conjugation be known as "positive/negative frequency symmetry"?

text: I know that charge conjugation exchanges the creation (or annihilation) operators of the particles with those of the anti-particles and therefore merits the name charge conjugation. However, if operated on the single electron Dirac plane wave $u(p)$ it results in v(p) and vice-versa. For me, however, $v(p)$ is not the single positron plane wave. For me it is the negative frequency solution. So for the single particles solutions of the Dirac equation it is more like a symmetry between positive and negative solutions. For a charge conjugation operator I would expect that it changes a in-going single electron plane wave to a in-going single positron wave. But $v(p)$ represents a out-going plane wave in Feynman diagrams. It is also said that $C$ changes the negative frequency wave $v(p)$ to a positive frequency wave solution $u(p)$ which finally represents the positron. Okay, but again then $C$ should not be called a charge conjugation, but symmetry between positive and negative frequency solutions. I would be grateful to get an explanation on that.


title: Overlap integral and probability

text: I have a question regarding how to extract probability from an overlap integral. Specifically, I am calculating the probability of a particle in a bound state in a delta potential $V=-\alpha \delta(x)$ ($\alpha$ is positive) remaining in a bound state after $V$ changes strength to $V=-\beta \delta(x)$. I am able to calculate the overlap integral of the bound state wavefunctions of these two $V$'s just fine; I am just wondering if I need to square the value of the overlap integral to find the probability of remaining in a bound state or if we just keep the value of the overlap integral as our probability. And if we do have to square the value of the overlap integral, why is that?


title: What's the relation between the Euler $\psi$ function, the digamma function, and the hypergeometric function?

text: Can somebody help me out with the intermediate details of eqn. (2.5) in this paper? > Generalized gravitational entropy. Aitor Lewkowycz and Juan Maldacena. > arXiv:1304.4926. Is the Euler $\psi$ function appearing in the above equation the same as the digamma function? I can't seem to figure out the relation between this Euler $\psi$ function and the hypergeometric function ${}_2F_1(a,b;c;z)$. It must be related to the derivative of the hypergeometric function but I can't figure out exactly how.


title: What is the importance of state functions in physics?

text: I'm currently reading about the Carnot cycle and its significance on the formulation of entropy (because I want to try to understand the concept better), but I can't seem to answer the following question:   * Why are state functions important in physics? What does it mean, physically, that a certain variable is a state function? I know the definition of state function, but I can't figure out why it would be important at all. In the aforementioned Carnot cycle, some variables that are state are internal energy and entropy. Apparently Rudolf Clausius felt the need to define a certain state function he discovered to be 'entropy', so it must have a reason.


title: Density operator time evolution in the path integral approach

text: I want to know how the density operator of a system evovles when we use path integral approach.


title: In Copenhagen, can this idea preserve locality for Bell inequalities?

text: Generate an entangled pair of qubits. Send to Alice and Bob far away from each other. Both measure along basis in one of two possible orientations. The result is sent to Charlie at some later time, who compares the corellations and concludes the Bell inequalities have been violated. But in Copenhagen, Charlie can say, the pair of results didn't become real until I collected both observations from Alice and Bob. They weren't real when either of them measured them. They only became real once I observed them. Then, both results, when they materialized, materialized at the same place. So, no nonlocality?


title: Methods for determination of crystal lattice's homogeneity

text: I know of X-Ray diffraction, which produces a pattern corresponding to the inverse Fourier Transform of the lattice (reciprocal lattice). While this method is widely employed, it provides more information than I need - I don't want to find out what Bravais lattice I'm examining, but just how homogeneous this lattice is. Which methods may be used to determine the homogeneity of the lattice? Can electron microscopes be used for this (SEM, TEM)? What spatial resolutions the additional methods posses? EDIT: I'm trying to figure out what techniques one can use in order to measure the lattice's deformation in vicinity of interfaces with another latices or amorphous materials (the picture is just a demonstration of such an interface between Si and Si02): ![enter image description here](http://i.stack.imgur.com/yu64s.jpg) To my best knowledge, XRD does not provide high enough spatial resolution for this kind of measurements.


title: On the theoretical aspects of the development of the first nuclear bombs

text: I've just read that 68 years ago _Little Boy_ was dropped on Hiroshima, which made me wonder about some rather historical facts about the development of the first nuclear bombs; they seem to be several questions, but they boil down to the same thing: the theoretical aspects of the development of the bombs. As far as I know, nuclear fission was already understood at the time, so was all the _competition_ between _the american team_ and _the german team_ to develop the bomb first, a mere matter of engineering? Or what was the role of theoretical physicists such as Richard Feynman? Also, I've heard things like that the _team_ led by Werner Heisenberg had _misconceptions_ about nuclear fission, so that his _team_ could not develop the bomb first. Can anyone put in perspective the theoretical aspects of the development of the bombs taking care of this historical issues?


title: Drawing the wave function for a wave packet

text: I have the following infotmation: Amplitude-Function: $U(k) = Ae^{-a|k-k_0|}$   Wave Function: $u(x,t) = \frac{A}{\sqrt{2\pi}} \frac{2a}{(x-vt)^2+a^2}e^{ik_0(x-vt)}$   Uncertainty in x: $\Delta x = 1$ (Linear dispersion-relation).   Uncertainty in k: $\Delta k = \frac{1}{\sqrt{2a}}$   And finally $a > 0$. The task is the following:   Sketch the Real-Part of $u(x,t)$ for $t = 0$ with 1) $k_0a = 0$ and 2) $k_0 a >> 1$ as well as for $t > 0$ for 1) $k_0a = 0$ and 2) $k_0 a >> 1$. Note that you should have $\frac{x}{a}$ on the abscissa. All I have done so far is calculating $u(0,0)$ for $k_0a = 0 \Rightarrow k_0 = 0$ and I got $u(x,t)_{k_0a = 0} = \sqrt{\frac{2}{\pi}} \frac{A}{a}$. And now I have absolutely no clue wjat to do. I am sitting on this for like ages... I hope anyone can help me out. Thank you very much.   FunkyPeanut Well I h


title: What the difference between "orbital" and "orbit"?

text: What's the difference between "ortibal" and "orbit"? Which one should be used in physics? In quantum mechanics, is "atomic orbital" or "atomic orbit" used? And what about in classical mechanics? A particle's orbit or orbital?


title: Doing a spring experiment, and need to find the stretch required

text: I'm doing an experiment where I have to calculate the stretch of a spring needed for it to launch itself into a trashcan located some distance away. The spring is located on the edge of a ramp that is 1.13 m high. The angle of the ramp is 30 degrees above horizontal. The spring constant of the spring is 175 N/m. The trashcan is located a horizontal distance 5.85 m away, height of the trashcan can be ignored. I must find the stretch (x) of the spring that will accomplish this. At first I thought I must solve for time of flight but there are too many variables involved. For instance, I don't know the initial velocity of the spring. If I could solve for time, then I could use it to find Vx (horizontal component of velocity), and then use the angle to find V. Then I would do Ee = Ek (elastic potential was converted to kinetic the instant the spring was released) and finally solve for the stretch (x). I tried to use the range equation: R = (v^2/g)sin2θ, but that is only for projectiles that land at the same height. Any guidance on how to approach this?


title: How to deduce equations 3.6.15 in Polchinski's string theory book?

text: In polchinski's first course on string, with ![enter image description here](http://i.stack.imgur.com/M9Cs0.png) how to deduce![enter image description here](http://i.stack.imgur.com/I6iW2.png)


title: Energy Levels of 3D Isotropic Harmonic Oscillator (Nuclear Shell Model)

text: One simple way of detailing the very basic structure of the nuclear shell model involves placing the nucleons in a 3D isotropic oscillator. It's easy to show that the energy eigenvalues are $E = \hbar\omega\left(n+\frac{3}{2}\right)$ with a degeneracy of $D(n) = \frac{1}{2}(n+1)(n+2)$, where $n = n_1 + n_2 + n_3 = a^\dagger_1 a_1 + a^\dagger_2 a_2 + a^\dagger_3 a_3$. However, in the book I am using presently (Ring, The Nuclear Many-Body Problem) and in several other texts, the authors write the quantum numbers as $n=2(n_r-1)+\ell$ (or, some places, $n=2n_r+\ell$). This seems odd, since we're trading 3 quantum numbers ($n_1, n_2, n_3$) for 2 ($n_r, \ell$). I suppose this is fine if we're only using $n_r$ and $\ell$ to enumerate our hypothesized nuclear energy levels, but in any case it's not clear to me _why_ you would make the choice $n = 2n_r + \ell$ to do this (besides, obviously, that it does an okay job of reproducing the lower magic numbers). These authors trot the expression out without any explanation, and I'd like to know why this expression is used. I'm still not clear on the significance of $n_r$ and $\ell$ in nuclei, so it's puzzling.


title: Can entanglement be explained as a consequence of conservation laws?

text: This article at NewScientist magazine (subscription required) describes entangling photons by passing them through a half silvered mirror. http://www.newscientist.com/article/mg21929282.100-quantum-weirdness-the- battle-for-the-basis-of-reality.html It talks about the "weirdness" of entanglement and how it means that either information is traveling faster than the speed of light or other fundamental assumptions about reality are wrong because the spins of the emerging photons are always correlated when measured. My question is just what's so strange about it? I don't get it... It seems to me to be just another variation of conservation laws. It would be like saying that it's "weird" for a photon in pair production to always produce particles of opposite charge. That's a given... Is there something to the timing of it that ruffles everybody's feathers? I think I'm missing something.


title: Dirac Delta Potential and bound/scattered states

text: Why does the attractive Dirac Delta distribution (function) potential $V = \alpha\delta$(x) (for negative $\alpha$) yield both bound AND scattered states? Is this due to the definition of the Dirac delta distribution? Or is it for a different reason?


title: Can the equivalence principle be shown to follow from special relativity?

text: Einstein showed in various papers that the energy content of a body was a measure of its inertial mass. If you increase the internal energy of a system while keeping the center of energy at rest, its effective inertial mass increases because of the increase of energy required to move the center of energy from rest to a velocity $v\ll c$. Is it possible to come up with a similar argument, just using special relativity, to show the equivalence of inertial and gravitational mass?


title: Bounded and Unbounded (Scattering) States in Quantum Mechanics

text: I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state implies that $E > V_0$. But I do not understand why for bounded states, the complete set of wave- function solutions to the time-dependent Schrodinger equation is constructed from a **discrete** linear combination of solutions, whereas for scattering states, the complete set, or general solution, is instead written as an integral over the **continuous** value of $k$ (or more appropriately, as a Fourier-type integral). Could anyone clarify why this difference is true for bound and scattering (unbound) states?


title: Are there reasons for the discrepancies in absolute temp units - Kelvin vs. kelvins vs. degrees Kelvin?

text: Before 1968, the units for absolute temperature were described as "degrees Kelvin" or "degrees absolute." After that, the SI system got rid of the idea of "degree" for absolute temperature, so the new unit should apparently be expressed as a "kelvin" (with lowercase k) and abbreviated simply "K" (without the degree sign). Also, official SI conventions suggest that not only should the unit name be lowercase, but it should be pluralized as other units would be: "Il en résulte que la température thermodynamique du point triple de l’eau est égale à 273,16 kelvins exactement, Ttpw = 273,16 K." Or, in English: "It follows that the thermodynamic temperature of the triple point of water is exactly 273.16 kelvins, Ttpw = 273.16 K." Despite the official SI usage, however, it seems that there are still a variety of conventions in use. Many of the questions on this forum, for example, use Kelvin (with a capital) instead of kelvin in referencing the unit. Also, it appears that the plural usage is somewhat mixed in the physics literature: something like "200 kelvins" occurs, but more rarely than "200 kelvin" or even "200 Kelvin." The NIST guidelines do not list the kelvin as an exception to the normal pluralization rules: "the following plurals are irregular: Singular — lux, hertz, siemens; Plural — lux, hertz, siemens." On the other hand CERN's writing guidelines suggest that there is an exception: "And note that it is always kelvin, even when plural (not kelvins or degrees kelvin)." Given all of this, here is my question: Is the SI standard actually to pluralize kelvins, as would be suggested from the quotations from the official SI guides above? Is this officially stated anywhere in some standards organization's guidelines? Or, is there some rationale given somewhere for the continued use of plural "kelvin" (as in the CERN guidelines) or even "Kelvin" (with an apparently anomalous capital)? Or, is it -- as I suspect -- just a failure to treat the kelvin as an actual SI unit, despite the redefinition from "degrees Kelvin" to "kelvins" that happened decades ago? (Perhaps we just dropped the "degree" but effectively still treat it the same way as Celsius or Fahrenheit?) **EDIT:** Just to be clear -- I did NOT intend for this to be only a question about linguistic convention. The SI redefinition of the absolute temperature scale and units was apparently meant to refine or change some physical conception about temperature. Resistance to this change may indicate some other elements about the underlying physics (perhaps including the fact that temperature is an intensive property, as suggested in one answer below). While I'd certainly be interested in other official standards and usage recommendations, I also wanted to know if there were other PHYSICAL rationales for the inconsistent units.


title: Rationale behind the 'joint cavity distribution'?

text: I have a question about equation (17) of this paper: http://arxiv.org/pdf/1009.1635v1.pdf First, I was hoping that someone could explain how it is arrived at. Second, I find the notation to be a bit perplexing--I do not quite understand the notation, particularly this "product of sums": $$\prod_{j\not\in\\{0,1,2,3\\}}\sum_{\sigma_j}e^{-\beta H_{G\backslash \\{0\\}}}$$ Suppose that we have some term of the product, say $j=n$ for some $n$ not in $\\{0,1,2,3\\}$. Then the summation is over $\sigma_n$, but what does that mean? $\sigma_n$ corresponds to a particular node in the graph (see Fig. 6), so I do not entirely understand how we can sum over it, or indeed why it would even be desirable to do so since the summand $e^{-\beta H_{G\backslash\\{0\\}}}$ certainly does not depend on which $\sigma_n$ we choose--it is just a function of the entire graph with one node 'carved out' as specified earlier in the paper.


title: Pulse Reflection

text: I wanted to ask why a pulse that reaches to a free end will reflect off and return with the same direction of displacement that it had before reflection. I am looking for explanation with forces.


title: Potential energy of a dipole in a uniform electric field convention?

text: When finding the potential energy of a dipole in a uniform electric field, I was told by my lecturer that the convention is that the potential energy is 0 when the dipole moment and electric field vectors are perpendicular. I don't understand why this is so, when it would make sense if the potential was zero when the dipole moment and electric field are parallel by agreeing with the minimum total potential energy principle (that is, a body will displace to minimise its total potential energy). The only reason I could possibly think for this convention is that the expression is simplified from $U=pE(1-cos\theta)$ to $U=-pEcos\theta$. If anyone can give me a better reason for this convention, it would really clear things up for me. Thanks.


title: what does "an average over noise" mean in Zwanzig's book

text: This is a very specific question about Robert Zwanzig's book _Nonequilibrium Statistical Mechanics_. Specifically, what is he talking about in equation 1.25 on page 10 that he calls "an average over noise", and how is it calculated? $$\left<v\left(t\right)\right>_\text{noise}=e^{-\frac{\xi t}{m}}v\left(0\right)$$ Is he just saying that the noise averages out if you average long enough? If so, how exactly do you show it. I realize this question is hard to answer without having the book, but I'm not sure how else to phrase it. I don't think he previously defined that term, and I'm not sure how to provide background without copying all the previous pages in the book. FWIW, here are the two preceding paragraphs: > The first example is to obtain the velocity correlation function of a > Brownian particle. In this example, it is instructive to calculate both the > equilibrium ensemble average and the long-term average. > > Calculating the equilibrium ensemble average involves both an average over > noise and an average over the initial velocity. The noise average leads to > > $$\left<v\left(t\right)\right>_\text{noise}=e^{-\frac{\xi > t}{m}}v\left(0\right).$$


title: Plastic bottle floating and sinking

text: I was doing this experiment: i take a plastic cup, put in a container of water, it floats – then when I put a coin in the water, it sinks and rests at the bottom. So to make the cup sink, i fill it with water – and the cup sinks, but it sinks until it's totally submerged with water and stops ( doesn't sink to the bottom ) Why did that happen? Why didn't it sink like the coin? * * * PS: I am aware of the presence of this thread, and it doesn't answer my question , so please do not mark duplicate.


title: Expansion in Quantum Fluctuations of the Path Integral

text: In this post: Dimensionless Constants in Physics there is a discussion about dimensionful vs. dimensionless constants in physics. In the context of this discussion, I'm wondering about the significance of setting $\hbar=1$ in calculations, specifically, in calculations where we are expanding the classical action around the classical solution to the equations of motion + quantum fluctuations. If I understand correctly, then, the quantum fluctuations are expanded in powers of $\hbar$, which measures just how much quantum effects you're taking into account in your approximation. I was wondering then, since $\hbar$ is dimensionful, then it must not be a fundamental measure of the amount of quantum effect in a calculation, is there another perspective on this? I have seen an ad-hoc coupling constant introduced, but I also remember specifically reading in books that $\hbar$ is an answer to the question "Is the action big?", a question which has no meaning in classical physics (e.g. in S. Coleman's report on instantons).


title: Form of the Classical EM Lagrangian

text: So I know that for an electromagnetic field in a vacuum the Lagrangian is $\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu}$, the standard model tells me this. What I want to know is if there is an elementary argument (based on symmetry perhaps) as to why it has this form. I have done some searching/reading on this, but have only ever found authors jumping straight to the expression and sometimes saying something to the effect that it is the "simplest possible". Thanks for the great answers, I accepted the one that was exactly what I was looking for, but the other two long ones were equally interesting and useful.


title: Fictitious forces confusion

text: I have a hard time understanding the subject of fictitious forces. Let's discuss a few examples: 1) I'm sitting inside a vehicle which is accelerating in a straight line. I feel like someone is pushing me to the seat. So, on the one hand, I'm told that this happens according to the third Newton's law: this pressure is the result of me pushing the seat as a reaction to the seat pushing me (because it is accelerating with the same acceleration as the car). On the other hand, these forces are acting on different objects and I'm told that there is another fictitious force acting on me in an opposite direction to the acceleration. So what is right? And if there is a fictitious force, then why some call it a "math trick" when they are real and I can feel them? 2) I do not understand why some call centrifugal force a fictitious force. The earth is pulling the earth with its "invisible" string called gravitation. That's why the moon is still there. And this is the centripetal force. However the moon is also pulling the earth according to the Newton's third law, and that's why we have tides. This is the centrifugal force. So why it is fictitious? What it has to do with frame of reference? When we observe this in non-inertial frame of reference (such as the moon), does it simply mean that we can't call it anymore a reaction force according to the Newton's third law? But why if it is virtually the reaction force? 3) Accelerating elevator - similar to the first example - let's say the elevator is accelerating upwards. So we get that $\vec{N}=m(g+a)$, and that's the same as me pushing the floor. That is why I feel heavier. Then why some add to here the fictitious force? I will appreciate any answer.


title: Charge propagation in a long circuit

text: Imagine a long circuit that goes around the Earth, or just spans a large enough distance to measure the effect, say 100'000 km. The circuit has efficient light bulbs installed at each 10km mark. There is a power source connected which powers all the light bulbs. Time N = all light bulbs are lit. > (+.................-) Time N+1 = the circuit is cut in the middle point, equidistant from the power supply terminals. > (+......./ /.......-) Question is, how do the light bulbs shut down along the timeline, starting with N+1.   1. Do they turn off one by one starting from the middle point, or does it start from the terminals moving towards the middle point?   2. Is the speed of propagation the same for both halves of the circuit?   3. What is the propagation speed? Thanks and sorry for stupidity if this question is trivial :)


title: Degeneracy Pressure, What is it?

text: There has been numerous question, some violent even in physics@SE regarding PEP and EM forces. But what baffles me is what is degeneracy pressure? I know there are 4 fundamental forces- EM, gravity, weak and strong. But then degeneracy comes along in everywhere, from neutron star to the electronic configuration. What causes this degeneracy in fermions?? Questions in SE suggests EM and degeneracy pressure are entirely different. But are they really? I mean what's the origin of degeneracy pressure among the fundamental forces?