title: potential energy of an object due to other two objects

text: consider object A with mass $m_{A}$ and positional vector $\overrightarrow{r_{A}}$ object B with mass $m_{B}$ and positional vector $\overrightarrow{r_{B}}$ object C with mass $m_{C}$ and positional vector $\overrightarrow{r_{C}}$ since reference frame is inertial (assumed) so $m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F}{}_{AB}+\overrightarrow{F}{}_{AC}$ $\Rightarrow$$m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})$ ....[1] for potential energy equation is to be integrated with a positional vector and i am trying hard to figure out what that vector could be, but still no success. please help * * * Potential energy of a object due to another object $m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{A}}-\overrightarrow{r_{B}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})$ assuming $\overrightarrow{r_{0}}=\overrightarrow{r_{B}}-\overrightarrow{r_{A}}$ $\Rightarrow m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}=-(1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})$ $\Rightarrow\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}\right).d\overrightarrow{r_{0}}=-\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left((1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})\right).d\overrightarrow{r_{0}}$ $\Rightarrow\left.\frac{1}{2}m_{A}v_{0}^{2}\right|_{v_{0}(\overrightarrow{r_{0i}})}^{v_{0}(\overrightarrow{r_{0f}})}=\left.G\frac{m_{A}(m_{A}+m_{B})}{r_{0}}\right|_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}$ So What positional vector should be integrated with equ [1] ? How it is decided ? (i mean what kind of property(s) that vector should possess ?) EDIT 1 $\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}\right).d\overrightarrow{r}=\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})\right).d\overrightarrow{r}$ .....[2] questions a) what is $\overrightarrow{r}$ in terms of $\overrightarrow{r_{A}}$,$\overrightarrow{r_{B}}$or $\overrightarrow{r_{C}}$ ? b) solve equ [2].


title: Polarization rotation: Jones Matrix that maps Horizontal to right circular

text: I am looking at the Poincaré sphere and I am trying to compute a Jones matrix for a particular rotation. Specifically, I would like it to perform the following maps: $O :|H \rangle \rightarrow |R \rangle$ $O :|V \rangle \rightarrow |L \rangle$ $O :|L \rangle \rightarrow |H \rangle$ $O :|R \rangle \rightarrow |V \rangle$ where $|R \rangle$, $|L \rangle$ are right circular and left circular light. Is this possible? I should mention that I would also accept the same equations with L,R replaced by linearly polarized light $|D \rangle$, $|A \rangle$. If it is possible, what would be common plates that could do it?


title: Relation between external magnetic field intensity H, magnetisation M and the entropy?

text: How are the external magnetic field intensity H, magnetisation M and the entropy related to each other? i.e. if I change the magnetic field intensity by dH what will be the change in entropy dS in terms of M.


title: Connection between entropy and energy

text: An isolated system $A$ has entropy $S_a>0$. Next, the isolation of $A$ is temporarily violated, and it has entropy reduced $$S_b ~=~ S_a - S,\space\space\space S\leq S_a.$$ Is it true to say: **the process of lowering entropy of a system requires work and energy?** I am not sure if energy of the system must be changed when entropy is reduced. However, energy certainly is required – changing entropy is work and uses energy?


title: How does gravitation propagate along curved spacetime?

text: In this wikipedia article it is described how a beam of light, with its locally constant speed, can travel "faster than light". That is to say it travels a distance, which, from a special relativistic point of view, is surprisingly big. I wonder if a gravitational wave on such a curved spacetime (of which the wave is actually part of) behaves equally. > Does a gravitational wave also ride on expanding spacetime, just as light > does? Do the nonlinearities of gravitation-gravitation interaction influence > the propagation of a wave (like e.g. a plasma) such that light and gravity > are effectively not equally fast? > > If I want to send a fast signal in this expanding universe scenario, in what > fashion do I decide to I send it?


title: Does Quantum Physics really suggests this universe as a computer simulation?

text: I was reading about interesting article here which suggests that our universe is a big computer simulation and the proof of it is a Quantum Physics. I know quantum physics tries to provide some explanation about unpredictable behavior of universe at atomic scale but could not believe that thing has anyway relates to pixels of the universe ? Can it be true that behavior of universe at atomic level coming from some super computer ?


title: Might quantum gravity be a barely inconsistent theory running as a quantum computer simulation?

text: > **Possible Duplicate:**   >  Does Quantum Physics really suggests this universe as a computer > simulation? Might our universe described by "quantum gravity" really be a finite but superastronomical quantum computer simulation which is barely inconsistent? The only way to describe the universe completely is to step outside the universe. Being barely inconsistent, we sentient beings can't find out the inconsistency for a very very long time, maybe ages, but as the looming inconsistency approaches, the computer simulation is stopped, and end information about the universe is saved and read to be used by another computer simulation at a higher level, and this process goes on ad infinitum. Might our world really be an infinite hierarchy of computer simulations?


title: A particlar normal ordering problem

text: Say we have an expression of the form: $$ \left<0\right|:\phi(x)^2: : \phi(y)^2:\left|0\right>, $$ where $\phi$ is some scalar field. I have heard the claim several times, that in evaluating this expression using Wick contractions, one only has to contract terms between groups of normal ordered terms. In this example this would mean we only have to contract $\phi(x)$ with $\phi(y)$ but not $\phi(x)$ with $\phi(x)$. I have no clue how to derive this. Has anyone got an idea? EDIT: And say we consider anticommuting operators, would we have: $$ \left<:\Psi^{\dagger}(x)\Psi(x)::\Psi^{\dagger}(0)\Psi(0):\right> = (-1)^3\left<\Psi^{\dagger}(x)\Psi(0)\right>\left<\Psi^{\dagger}(x)\Psi(0)\right>, $$ or with a plus sign?


title: Information bearing degrees of freedom of a quantum simple harmonic oscillator

text: I am trying to make sense of arXiv:physics/0210005. I am confused with the concept of information bearing degrees of freedom of a system mentioned at the very beginning. To verify the arguments of the paper, I would like to start with a 1-D quantum simple harmonic oscillator. We know that it has two degrees of freedom, momentum and position. You can also use generalized degrees of freedom if you want. My question is which of these are the information bearing degrees of freedom?


title: What is the difference between lattice vectors and basis vectors?

text: Google has not been very useful in this regard. It seems no one has clearly defined terms and Kittel has too little on this.


title: Transient radiation--heating a slab

text: Hey guys I really need help on this problem. > A ceramic slab of dimentions 5cm x 10 cm x .25 cm has to be heated to > $177\,^{\circ}{\rm C}$. The ceramic slab travels on a conveyor belt > traveling at $.1 \frac{m}{s}$. The slab is initially at a temperature of > $18\,^{\circ}{\rm C}$ when it enters a radiant heater on the conveyor belt. > The heater surrounds the whole slab and emits heat as a black body at > $1000\,^{\circ}{\rm C}$. Since the heat surrounds the slab, all the energy > that leaves the heater is absorbed by the slab. The slab has an emissivity > of .85. Find the amount of time it takes to heat the slab to > $177\,^{\circ}{\rm C}$. The slab has properties $c_{p}=1500\frac{J}{Kg K}$ > and $\rho=900\frac{kg}{m^{3}}$ and $k=1 \frac{W}{mK}$ I started off by creating an energy balance on the slab as it's in the heater. I assumed convection that is caused by the conveyor belt traveling at such a slow speed to be insignificant to all calculations. $$\dot{E}_{in}- \dot{E}_{out}+\dot{E}_{g}=\dot{E}_{st}$$ $$A\alpha\sigma T^{4}_{heater}-A\epsilon\sigma T^{4}=\rho c_{p}V\frac{dT}{dt}$$ $$A\sigma(T^{4}_{heater}-\epsilon T^{4})=\rho c_{p}V \frac{dT}{dt}$$ I then plugged in the values and got: $$44.185-1.428x10^{-11} T^{4}=\frac{dT}{dt}$$ I realized that this comes out to be a VERY nasty integral so I tried to use matlab and numerically integrate it using Euler's method. My graph for that came out to be linear which I knew was not right and I came here straight away to figure out what I was doing wrong


title: What is light localisation?

text: Reading about plasmonic nanoparticles I faced the term "localised light". How can one localise light? What are applications of it?


title: Do black holes have infinite areas and volumes?

text: How to calculate the area / volume of a black hole? Is there a corresponding mathematical function such as rotating $1/x$ around the $x$-axis or likewise to find the volume?


title: Computing the path of photons near a black hole

text: For a simulation, I want to compute the path that light follows near a black hole. Non-relativistically, a massive point particle in a central newtonian gravitational field follows either an ellipse, a parabola, or a hyperbola. Is the same true relativistically for light around a black hole? A problem I see with this, is that while particles gain velocity when approaching a black hole, a photon gains energy instead. So do photons behave differently?


title: How to calculate the amount of energy in joules/s that gas oven uses?

text: I know there may be missing information, but my question is really simple and yet I haven't found an answer to it. To make things easy, let's assume we have a $40×40×40$ cm oven that we want to keep its temperature at $200$ degrees Celsius using natural gas. How much energy per second is consumed to achieve this?


title: Is it ok to have two events $A$ and $B$ so that for one person $A$ occurs before $B$ but for another $B$ preceds $A$

text: Imagine two laser beams A and B are released at the same moment to bounce between two mirrors, A was moving and B was at rest, doing the calculations I found that for a person at rest B would reach the upper mirror before A because in his frame of reference A travels less distance. but for another person in the same reference frame of A, A would reach the upper mirror first. Is that OK in relativity!![the upper is moving and the lower is constant](http://i.stack.imgur.com/MJn2B.jpg)


title: Alternative derivations of vis-viva equation?

text: The proof of the vis-viva equation of orbital mechanics found on wikipedia looks, in my opinion, somewhat convoluted and unenlightening. Considering the simplicity and importance of the vis-viva equation, is there a shorter or more insightful derivation? The vis-viva equation states that for a Kepler orbit of a mass around a central mass $M$, the magnitude of the velocity at any distance $r$ from the center obeys $v^2 = GM(\frac{2}{r}-\frac{1}{a})$, where $a$ is the semi-major axis of the orbit. It is equivalent to saying that the energy of a Kepler orbit with semi-major axis $a$ is $-\frac{GM}{2a}$. This is a very natural generalization of the 'circular' case where the total energy is $-\frac{GM}{2r}$. Because of the simplicity of the result, I guessed that there could be a relatively straightforward derivation. The derivation on wikipedia involves quite some algebra and, importantly, only holds for elliptical orbits.


title: Are all fluorophores dipoles?

text: Lately I'm reading about surface enhanced fluorescence. In many articles I can see that fluorophores are called "dipoles". Is it because that they can be modelled by a vibrating electric dipole? Or maybe they are all chemical dipoles?


title: One question about derivation of Maxwell equations

text: I saw the following way of derivation of Maxwell equations: author starts from Lorentz transformations for the 3-vector of force, then he applies them for the Coulomb law, after that gets the Lorentz force law with expressions for $\mathbf E $ and for $\mathbf B$ fields, then he takes curl and divergence and, finally, he derives the Maxwell equation for the field of charge moving with constant speed. After that he uses a few axioms: superposition principle and independence of equations on acceleration, and generalizes equations on all electrodynamical processes. But I have some doubts about this method. The most problematic question is followIng: there are EM fields of electric and magnetic type. For the first one $\mathbf E^{2} - \mathbf B^{2} > 0$, for the second one - $\mathbf E^{2} - \mathbf B^{2} < 0$. But the author have started from the electric type, so he can't generalize final result on all EM processes. The question: can we violate this argument against this methot of derivation by introducing some other argument?


title: Choice of basis for Fujikawa method to derive chiral anomaly

text: I am studying the Fujikawa method of determining the chiral anomalies in a $U(1)$ theory. As we know the basis vectors selected are the eigenstates of the Dirac operator. One of the reasons given is that the eigenstates diagonalize the action which is needed for determining an exact quantity such as Ward-Takahashi identities. Anyone care to explain? I am referring to _Path Integrals and Quantum Anomalies_ by Kazuo Fujikawa and Hiroshi Suzuki.


title: Does frequency f = $\frac1T$ if there is non-periodic signal

text: Through what I learned from school, frequency $f=\frac1T$ while $T$ is the period. However, things get harder in college where there are non-periodic signal. That means they have no period or their period is extended to infinity but they still have frequency. How can they have frequencies even there is no period to apply the formula $f=\frac1T$? Is there any definition of frequency in those cases.


title: Does the kinetic theory of gases means gases mix almost instantaneously?

text: This theory has bugged me ever since my first physics class on the subject. If this (http://en.wikipedia.org/wiki/Kinetic_theory) is true, it leads me to a few weird conclusions. Opening the rear window in a pickup truck at 10 m/s doesn't empty the cabin of all its air. This means that gas molecules can catch up with us from behind so we can conclude that gases move much faster than 10 m/s at normal temperatures. Then, shouldn't gases mix almost instantaneously? For example, shouldn't a fart's smell reach us faster than a car on an highway and then dissapear almost immediately into such a large area that the concentration is too low for our noses to detect?


title: how can u split the Photon?

text: **If any equations are there ,for split the light particle, means that i have a 1m length of light just consider that in that who can i split the light particle into 1/2 m length Pls tell**


While watching a Schlieren video of a hand clapping, I noted a very distinct difference between a sound wave and a puff of air, which were both created by a hand clapping. What is the difference between a puff of air and a sound wave regarding creation and propagation? In the video, it appears that some of the energy goes into the sound wave and some of the energy goes into the puff. Is there a principle that governs the distribution of impact energy between oscillation (sound) and pushing (puff?)


title: Conservation of Energy and Birkhoff's theorem

text: I am reading the original paper by Bondi, van der Berg and Metzner (link) regarding gravitational waves in asymptotically flat axisymmetric spacetimes. In the introduction, he makes the following comment - > The conservation of mass effectively prohibits purely spherically symmetric > waves and similarly, conservation of momentum prohibits waves of dipole > symmetry. I know that Birkhoff's theorem tells us that spherically symmetric asymptotically flat solution to GR is necessarily static, and therefore contains a timelike Killing vector, which implies conservation of mass (energy). Bondi et. al. seem to be stating the converse of this theorem, whose validity I do not immediately see. How do we show this? Also, what is the corresponding proof of the second statement made above?


title: How to get distance when acceleration is not constant?

text: I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant acceleration is: $$d=ut +\frac{1}{2}at^2$$ which can also be expressed $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}$$ (where x(t) is the position of the object at time t) That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed? The acceleration is obviously not constant, but what about the change in acceleration? Is it constant? I suspect not. And then what about the change in the change of acceleration, etc. etc.? In other words, how does one know how many additional terms to add in the series? $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}+\frac{\mathrm{d^3}x}{\mathrm{d}t^3}\frac{t^3}{3}+\frac{\mathrm{d^4}x}{\mathrm{d}t^4}\frac{t^4}{4}\cdot etc. \cdot ?$$


title: how can u control the speed of light?

text: The light traveling a 3 lake km/sec, but Pls tell me how can a control the speed of light, means can limit the speed like 2 lake km/sec are 1 lake km/sec Pls tell me equation


title: Calculating the size of the "lensing sphere" of a black hole

text: Given a black hole of some size, say $10^8$ solar masses, how can the size of its sphere of influence of light be calculated? To clarify, ultimately I'd like to be able to calculate the apparent angle of a black hole's lensing effects, as viewed from some distance. (I'm no physicist, so I might be making incorrect assumptions about gravitational lensing. If this is the case, sorry!)


title: Decompression cooler

text: How would I figure the needed pressure differential to freeze water inside a cooler using ambient air? I am assuming that the combined gas law would apply here and that the formula P1•V1/T1=P2•V2/T2 would provide the answer. My understanding is that by lowering the pressure I could effectively lower the temp as well. By this calculation I would need to reduce the air pressure from 1 bar to 313 milibar. Does this seem right on the math side? In an open system lowering the pressure should also lower the volume and therefore keep the temperature from dropping. Is this correct as well. I would assume an ambient temp of 100 F.


title: Time scale when measuring total activity from radioactive sample

text: I have measured the total activity from a radioactive sample during an experiment by counting the number of decays happening in 3 seconds intervals. I got data as: Time bin: 0 1 2 3 4 5 6 7 ...   Count number: 307 294 257 246 250 216 186 169 ... Now I want to curve fit an exponential $ae^{bt}$ to the data to determine $b$ as the decay constant. But since I have measured during intervals of 3 seconds, how should I convert/scale this data to get a proper estimate of the decay constant that has units s$^{-1}$?


title: Getting pairs of angle and velocity for a projectile to a given destination

text: I'm trying to calculate the initial velocity $v_0$ and angle $\theta$ for a given destination $(x, y)$ with a launch height of $y_0$. Obviously there will be a set of pairs of velocity and angle that will pass through the destination point. This set is given by $$ \left\\{(v_0,\theta) \middle|y = - \frac{g}{2 v_0^2 \cdot \cos^2(\theta)} \cdot x^2 + \tan(\theta) \cdot x +y_0 \right\\} \ .$$ I have already looked at this questions: Solving for initial velocity required to launch a projectile to a given destination at a different height How to get the angle needed for a projectile to pass through a given point for trajectory plotting So for example I could rewrite the set to $$\left\\{ (v_0,\theta) \middle| v_0 = \frac{1}{\cos(\theta)}\sqrt{\frac{\frac{1}{2} g x^2}{x \tan(\theta)+y_0}} \right\\}\ .$$ Is there an easy way to compute this set for one given destination without iterating over all possible velocities or angles?


title: Very strange shadow phenomenon

text: I was laying on my bed, reading a book when the sun shone through the windows on my left. I happened to look at the wall on my right and noticed this very strange effect. The shadow of my elbow, when near the pages of the book, joined up with the shadow of the book even though I wasn't physically touching it. I have posted a video of the event here. _(Note: the video seems to be wrong way up but you still get the idea of what is happening)_ What is causing this? Some sort of optical illusion where the light get's bent? Coincidentally, I have been wondering about a similar effect recently where if you focus your eye on a nearby object, say, your finger, objects behind it in the distance seem to get curved/distorted around the edge of your finger. It seems awfully related... _**EDIT:_** I could see the bulge with my bare eyes to the same extent as in the video! The room was well light and the wall was indeed quite bright.


title: Proof that the momentum operator is Hermitian

text: I am trying to prove that the momentum $p_x$ operator is Hermitian, my approach is the following $$<p_x>~=~\int \Psi^*(\vec{r},t)[-ih\frac{\partial}{\partial x}]\Psi(\vec{r},t)\, d^3r.$$ I try to do integration by parts but I cannot resolve the differential, as the integration is in respect to $\vec{r}$ and the partial is in respect to $x$.


title: Parity of the surface state in a Topological Insulator (TI)?

text: Please bear with this experimentalist trying to understand the subtleties of TIs in what may well be imprecise language. I appreciate that one can deduce the topological trivial or non-trivial nature of a filled band for particular crystalline structures by virtue of $(-1)^{\nu_0} = \Pi \delta_i$ taken over the TR symmetry points of the BZ. I also appreciate that the gap must close at the interface between topologically distinct insulating phases. In my naive understanding, I would expect that the surface state must "fix" the overall parity difference between the valence bands of the TI and trivial insulator and believe that $N_k = \Delta \nu$ mod 2, where $N_k$ is the number of Kramers pair edge modes implies this fix. I would have expected however that their would be some implication about the parity of the edge mode itself (i.e. that is must be odd). Why is there no statement about the edge modes parity?


title: Why do pores make ceramic opaque?

text: I want to know how a ceramic transparency is mostly affected by the pores, grain boundary, second phases etc. present inside of it, but the major contribution is due to the pores. Let's consider the glass slab with an air gap between them. It is still transparent. Hence: how can simple pores make ceramics opaque? I am after a theoretical description.


title: Non-normalizable QM bound state in 4 spatial dimensions?

text: **Edit 26/Sept/13: Fixed Typo in potential** I'm solving the following (seemingly simple) quantum-mechanical problem in _four spatial dimensions_. In natural units ($\hbar^2/2m=1$), the Schrödinger equation reads: $$ \Big[-\nabla^2-\frac{24 R^2}{(\mathbf{x}^2+R^2)^2}\Big]\psi(\mathbf{x})=E\,\psi(\mathbf{x})\,, $$ where $R>0$ is a parameter simultaneously characterizing the depth and range of the potential. The potential depends only on the distance away from the origin, so I can separate variables $\psi=R_{nl}(r)\,Y_l(\vec{\theta})$ and the radial equation then reads: $$ \Big[-\frac{\partial^2}{\partial r^2}-\frac{3}{r}\frac{\partial}{\partial r}+\frac{l(l+2)}{r^2}-\frac{24 R^2}{(r^2+R^2)^2}\Big]\,R_{nl}(r)=E_{nl}\,R_{nl}(r)\,. $$ **Problem:** I seem to have found an _s_ -wave ($l=0$) non-scattering state with zero energy $E_{nl}=0$ that appears to be localized: $$R_{n,l=0}(r)=\mathcal{N}\frac{r^2-R^2}{(r^2+R^2)^2}\qquad E_{nl}=0.$$ But, I am unable to normalize this "bound" state since the integral $\int_0^\infty dr\, r^3 |R(r)|^2$ does not converge. What is the nature of this state? Or am I just totally screwing something up?


title: TISE for a triangular potential

text: I want to solve the TISE of a particle of charge $q$ and mass $m$ in a one dimensional triangular potential, with an infinitely high potential wall at $x = 0$, i.e. $$\hat{H}=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x),$$ with $$V\left(x\right)=\begin{cases}qEx&{\rm if}\,x>0 \\\ \infty &{\rm if}\,x\leq0\end{cases}$$ For $x>0$, this gives the TISE $$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi_n}{\partial x^2}+qEx\cdot\Psi_n = E_n\Psi_n.$$ I know that the solutions of $$\frac{d^2y}{dx^2} - xy = 0$$ are given by the Airy functions, however I don't know how I can transform the TISE to this form. I know it's possible, because I have the solution to the problem, but I don't know how I can myself come up with this, I don't see the aim which has to be reached in order for a substitution to be successful. Also, I don't know how the differential will transform under a substitution including $x$. Any help? The substitution used in the solution is $$u=\left(\frac{2mqE}{\hbar^2}\right)^\frac{1}{3}\left(x-\frac{E_n}{qE}\right).$$


title: Nonuniform acceleration due to rubber rope

text: What I want: I have a rubber rope which is $5m$ in length when not stressed and is able to stretch about $100\%$ (to $10m$ long). I want to accelerate a constant mass horizontally, which has negligible friction. I'd like to have a function that tells me the velocity of the mass dependent on time, so for instance velocity $1 s$ after releasing it. What I did: I've done some measurements of forces of the rope when pulling it to different lengths. Of course, when pulling $0cm$ (total length $5m$) I got a force of $0N$. Here is a graph of my results. ![http://i.imgur.com/vtEnACQ.png](http://i.stack.imgur.com/Mok8k.png) $x-axis$: displacement of one end of the rope $y-axis$: measured force I was also able to do a regression and found a function which describes how much force I get after I pull a given length. I name this function $F(s)$ for Force dependent on displacement. From this, it's easy to get the acceleration function, which is $a(s) = F(s)/m$ with $m = mass$ of the object I want to accelerate. But now I'm stuck. I somehow need to get $a(t)$ instead of $a(s)$, thus the acceleration by time, not by length, so I can then integrate that to get $v(t)$. How do I convert the dependency of the function?


title: Parametric equations of a hypersurface

text: In light-front QFT, in the Minkowski space, we define a hypersurface, $\Sigma_+ : x^3+ x^0 = 0 $. How can I write its parametric equations?


title: Energy Density of Hydrogen Fusion

text: How much energy is derived from the full process from hydrogen-1 fused into helium-4? Including all the sub-reactions (H1 into Deuterium, etc).


title: movement of photons

text: In a typical photon experiment the photon is depicted as moving across the page, say from right to left. Suppose we were actually able to witness such an experiment, from the side (to position of reader to a page). If the photon is actually moving from left to right can I, standing at 90 degrees to the motion, see the photon?


title: Quantum mechanics of thermal radiation

text: What are the quantum mechanisms behind the emission and absorption of thermal radiation at and below room temperature? If the relevant quantum state transitions are molecular (stretching, flexing and spin changes) how come the thermal spectrum is continuous? What about substances (such as noble gases) which don't form molecules, how do they emit or absorb thermal radiation? Is there a semi-classical mechanism (with the EM field treated classically) and also a deeper explanation using the full apparatus of QFT?


title: What would ACTUALLY happen to a person jettisoned into space?

text: _[insert obligatory statement of my lack of knowledge in physics]_ Alright, so we have all seen the movies where someone gets blasted out of the airlock on their starship, or their suit decompresses while on a space walk. The poor schmoe usually either decompresses so violently that blood is oozing out of every orifice in their body, or they freeze instantly. From this I have two questions:   1. **Would the decompression really be that violent?**     1. Clearly the drastic difference in pressure from a normal "earth" like environment to space would be bad, but would it be _that_ devastating.     2. I vaguely remember that standard atmospheric pressure was something like 15 psi, which doesn't seem like enough to mess you up that bad.   2. **Would you _actually_ freeze instantly in space?**     1. Heat, or lack thereof is a measure of internal energy, but in a vacuum there wouldn't be anything to have internal energy, so does space even have a temperature?     2. Wouldn't some form of matter have to be present in order to cool off? If there were no matter besides yourself and a few stray particles here and there, it seems like it would take a very long time to cool off.


title: Constructing a CP map with some decaying property

text: Given some observable $\mathcal O \in \mathcal H$ it is simple to construct a CP (completely positive) map $\Phi:\mathcal{H}\mapsto \mathcal{H}$ that conserves this quantity. All one has to observe is that $$ \text{Tr}(\mathcal O \, \Phi[\rho]) = \text{Tr}(\Phi^*[\mathcal O] \rho).$$ Therefore, if we impose $\Phi^*[\mathcal O] = \mathcal O$, then $\text{Tr}(\mathcal O \, \Phi[\rho])=\text{Tr}(\mathcal O \rho), \; \forall \rho\in \mathcal H$. That amounts to impose that the Kraus operators of $\Phi^*$ should commute with $\mathcal O$. I'd like, however, to construct a trace-preserving CP map for which the expectation value of $\mathcal O$ does not increase for any $\rho \in \mathcal H$. More explicitly, given $\mathcal O\in \mathcal H$, I want to construct $\Gamma:\mathcal H \mapsto \mathcal H$ such that $$ \text{Tr}(\mathcal O\, \Gamma[\rho]) \le \text{Tr}(\mathcal O \rho), \; \forall \rho \in \mathcal H .$$ How would you go about that? Any ideas?


title: Possible values of energies and impulses of virtual photon emitted by real electron

text: I need to find the possible values of energies and impulses of virtual photon emitted by real electron. I don't understand how to compute it (look to the text below). Let's introduce the free electron. Then let's use the frame in which it is at rest. So the laws of energy-momentum conservation take the form $$ E_{0} = E_{1} + E_{photon}, \quad 0 = p_{1} + p_{photon} . $$ According to this, electron after photon emission must be virtual. What to do after that? Formally I have only two equations for four unknown values. What to do next? To use energy- impulse relation $p_{\mu}p^{\mu} = m^2c^2$ for virtual particles?


title: Quantum gravity at D = 3

text: Quantization of gravity (general relativity) seems to be impossible for spacetime dimension D >= 4. Instead, quantum gravity is described by string theory which is something more than quantization (for example it introduces an infinite number of fields). More direct approaches to quantization of gravity have failed (my opinion; not everyone would agree on this). However, gravity in dimensions D < 4 is special, because it is topological (carries no dynamic degrees of freedom locally). It is possible to quantize gravity coupled to other fields at D = 2: in fact perturbative string theory is exactly that! What about D = 3? Are there approaches to quantization of gravity (coupled to other fields) which have been more successful at D = 3 than at D = 4?


title: Pauli-Villars (PV) regularisation breaks supersymmetry. How to see that?

text: Does the PV regulator breaks SUSY? Take for instance the 1-loop (top/stop loops) correction to the Higgs squared- mass parameter in the MSSM, and you'll get something like, $$\delta m^2_{h_u} = - 3Y_u^2/(4 \pi^2) m_{\tilde{t}}^2 ln (\frac{\Lambda_{PV}^2}{m_{\tilde{t}}^2})$$ Where, $\Lambda_{PV}$ is the PV regulator/cutoff, and $m_{\tilde{t}}$ is the stop-quark mass. In my mind, as the calculation is performed before ElectroWeak Symmetry Breaking (EWSB) (i.e. no mass for the top), but at the same time it's considering softly broken susy (i.e. there is mass for the stop-quark), therefore we don't get perfect cancelation. But I heard someone saying that there's no perfect cancelation because PV regulator breaks SUSY! I don't see where the PV breaking SUSY argument fits. Can anyone enlighten me?


title: Traditional Kirchoff voltage law in AC circuit?

text: The traditional (not taking into account phasor addition or complex addition) application of Kirchoff Voltage law, i.e. $\Sigma\Delta V=0$ along a loop, does not work for AC circuits. We can sum the voltage drops to zero if we take into account their phase differences. But at some particular time, no matter what the phase differences in the general equation for the voltages across various components, there will be only one numeric value of $\Delta V$. > **Why doesn't this sum to zero over the loop in case of AC circuits with > non-zeroreactance?** Is this because there is some contribution to the $\oint E.dl$ due to changing magnetic flux? If it is, even in absence of any inductance, the voltage drops still do not sum to zero. why?


title: Is there a material that changes local conductivity

text: I hope this is the right forum to ask this question. Is there a material (preferably thin, like a membrane) that changes its local conductivity (by that I mean the permeability for an electric field; I hope it's the right term) upon excitation with light or heat? I have no idea where to start my search for something like that. Thanks a lot.


title: Exploiting the Heisenberg Uncertainty Principle as a means to communicate

text: It seems as though I've come across a rather unusual conclusion that could either simply be a misinterpretation or a contradictory discovery. I seem to have found a way to utilize the Heisenberg Uncertainty Principle (HUP) to our benefit to communicate faster than the speed of light (FTL). I am aware of many proposals that try to utilize entangled spin particles and try to communicate, but this schemes fail because the measurement outcome can NOT be controlled. Therefore, even if two people share an entangled pair of particles, and Alice measures spin up, although she knows Bob has a spin down particle, Bob didn't measure yet and has no way of knowing that Alice has measured her particle. Hence, FTL is impossible based on the reason that the measurement outcome can not be controlled. This leads to my proposal of using position and momentum entangled pairs. Consider **Alice and Bob hold an ensemble of entangled particles in position and momenta** , where each particle is trapped in a separate harmonic potential. At some specified time agreed upon in the future, Alice measures all of her particles' position to high precision. **Alice and Bob have synchronized clocks** and Bob measures all of his entangled particle's momentum to whatever accuracy he chooses. Alice can calculate the average value and also the standard deviation of position. The standard deviation of position would be extremely narrow, i.e, the spread of her measurements would be very small. From the entanglement relation, **x1 = x2 and p1 = -p2** , we know that when Bob measures his particle's momentum, the spread of momentum will be very large. This must be true because position and momentum cannot be measured to arbitrary accuracy at the same time. This is very similar to Einstein's proposal to violate the HUP, however I am exploiting HUP. What this all means is that the if Bob measures a very large momentum spread, it must mean that Alice has made her measurements. If Bob measures a relatively moderate momentum spread, then he knows Alice did not measure her particles. **Since the position measurement can be made to arbitrary accuracy, we are "controlling the spread or the standard deviation as the means to communicate."** (MAIN RATIONALE) Say Alice and Bob have multiple ensembles of entangled particles. Alice can relay a message by simultaneously measuring her first ensemble, meaning its a "1" and not touch her second ensemble meaning its a "0", and perhaps she chose to measure the third ensemble, "1", etc. Hence generating the series 101..., where **each ensemble of entangled particles represents one bit of information.** What is flawed in this proposal? Entanglement in position and momenta is well established. We can also choose to measure the position of a particle to arbitrary precision. The HUP must hold for Bob and everyone involved.


title: A question about Haldane's conjecture

text: Haldane's conjecture states that the integer spin antiferromagnetic Heisenberg chains have a gap in the excitation spectrum. However, the dispersion relation of the antiferromagnetic spin wave is $\omega_k\sim k$ in the long wave length limit, meaning that the excitation energy could be zero. What is the matter?


title: Relativity of Simultaneity

text: Relativity of Simultaneity seems to be about OBSERVING two events simultaneously (please correct me if I am wrong). However, as long as the two events are separated by a distance (any distance) then two observers (in the same frame) cannot agree that they happen simultaneously unless they are equidistant from the two events within the frame. Consider the example of A and B at rest with respect to another observer C. A and B are stationary in space but separated by 1 light second of distance. An event occurs simultaneously at A and B, the time should be the same since they are at rest with respect to each other. However, A will OBSERVE event at A and exactly after 1 second, the event at B. B will OBSERVE the event at B and exactly after 1 second, the event at A. C may observe A first, B first or A and B simultaneously (which is fine). My questions are:   1. Are A and B in the same inertial frame? And if so, if simultaneity is about OBSERVING an event, then how do we account for the different observation times within the same frame? An event can never be simultaneous for A and B unless they are at the same location.   2. If A and B are not in the same frame, then they are not in inertial frames as there is no motion with respect to each other.   3. If simultaneity is not about OBSERVING two events, then how do we separate the visual aspect (time taken for light to travel) from the temporal aspects (time at which the two events actually occurred.


title: Phase transition water

text: The water-gas phase transition is said to be similar to the ferromagnetic- paramagnetic phase transition (same set of critical exponents = same universality class). In the former case the order parameter is the difference in the densities, while in the latter it is the magnetization density. In the magnetic case the $O(N)$ symmetry is broken spontaneously - hence we talk about two distinct phases as suggested by Landau, i.e. the ferromagnetic phase (which has $O(N-1)$ symmetry) and the paramagnetic phase (which has the full $O(N)$ symmetry). Now for the water-gas case, what symmetry is spontaneously broken? Both liquid water and gaseous water have translational and rotational symmetry.. Also, sometimes it is suggested that water and liquid should not be considered as distinct phases, since one can join the two phases by an excursion in the parameter space which does not encounter any singularities in the free energy. If so, then shouldn't we consider the paramagnetic and ferromagnetic phase to be just one phase as well? How do we reconcile the concept of phases being characterized by symmetry (and their breakings), or being characterized by excursions in the parameter space, since they seem to give contradictory results? Thanks.


title: Why is a flower shaped diffraction occluder the best solution for controlling diffraction?

text: According to this TED talk by Jeremy Kasdin, Nasa is planning to spend $1bn on a "Starshade" project, where a giant flower shaped metal eclipser 20 meters wide is placed 50k kilometers in front of a space telescope, to fit the telescope diameter. The idea is to occlude a star and photograph it's exoplanets. The above solution is surreal. Why can't they control the diffraction of the light around the occlusion circle with a refractive material, to direct it outwards? I suggest that they can design a round black occluder with soft edges overlaid with a refractive material that deflects the light away from the centre, similar to a lense. Why do the angles and shapes at the edge of the flower shaped occluder have to be very precise in order to control diffraction?


title: Why do $\beta$ particles not get attracted to the nucleus?

text: I currently have a very limited knowledge of how radiation works etc, but while sat in class the other day, one question occurred to me that even my teacher could not answer. We have been learning about alpha, beta and gamma radiation. I know that alpha is positive and that makes perfect sense. However, when the neutron is broken down to form a β particle and this particle is then sent out of the nucleus, why is this negatively charged β particle not attracted (presumably by electromagnetic force) back into the overall positive nucleus of the atom? To me, at least, it does not make sense that it would not be attracted back the nucleus. Sorry if this is simple stuff but even after some furious googling I haven't been able to find an answer.


title: Circular motion - is average velocity enough to tell?

text: If I knew the exact value of constant centripetal force and the **average velocity** of an object travelling around a circle for time $t$, would that be enough information to determine the final positon of an object? What I mean is what if I had another object, at the same initial position and with identical centripetal force applied to it, which would have a different velocity function $f(t)$, but identical average velocity - would it eventually be in the same place as the first object after time $t$? Deriving a formula answering this question was beyond my skills.


title: Thermal fluctuations in metals

text: My professor said that the $k_BT$ displacement in the energy levels of the band electrons is due to the space-thermal displacement of the potential of the ion host. I think that this displacement is due to the thermal fluctuations of the electromagnetic field coupled to the electron charge. Who is right?


title: The need for a 'particle description' of electrons

text: Is there any phenomenon where the 'wave description' of the electron's motion is not applicable? The reason for this question is to find out if there are any situations were quantum wave theories fail. If not, it seems that the Everett's multiworld analysis as described by Coleman, http://media.physics.harvard.edu/video/index.php?id=SidneyColeman_QMIYF.flv works.


title: How to find tension in the cantilever brakes?

text: ![enter image description here](http://i.stack.imgur.com/orT7X.png) In the left side example of cantilever brakes, how do I find the tension in strings DE and DC, in terms of T. According to me applying force balance equations, horizontal and vertical forces should be zero i.e. $T_{DE} \cos(45) + T_{DE} \cos(45) = T$... The answer given is $T/2$, which is against general understanding. **SOURCE:- Mechanics of Materials by JM Gere Ed2009- Problem 1.2.3**


title: Would the inside of a black hole be like a giant mirror?

text: As any light reflected or emitted from objects inside a black hole (if it is possible to be there) does not leave the event horizon and comes back inside, would it be like seeing yourself? What I mean is that would the light we might emit/reflect return to our eyes and make us sort of look at ourselves?


title: Meaning/picture of the statement: "Turbulent flow is chaotic. However, not all chaotic flows are turbulent"

text: Wikipedia states that > _Turbulent flow is chaotic. However, not all chaotic flows are turbulent._ Someone give a picture for that?


title: Behavior of water at temperatures above 100°C

text: Assuming you have container with heat insulation (something like boiler).   You can store water at any temperature **below 100°C**. At these conditions, you can drill a hole into container and **store water at air pressure**.   But when you heat water to tempertature **over 100°C** , it starts to boil, increasing pressure in contailner. If you drill a hole, water will start to evaporate and that'll cause temperature drop. If you try to maintain constant temperature, after some time **all water will evaporate**.   **Water with temperature over 100°C can be stored only at high pressure?**   If so, it'll reach some pressure (depending on temperature) and stop boiling?   Are there any other risks besides high pressure to be aware when trying to store water at these conditions?


title: force on a moving charge in magnetic field

text: Need help in understanding the direction of magnetic force in the magnetic field!Totally confused by directions. Why is it that magnetic force is perpendicular to the direction of magnetic field and velocity of charged particle. Why is it(force) not in the same direction as the magnetuc field


title: Two air bubbles inside a liquid

text: I know that if two air bubbles which are formed inside a liquid are somehow joined using something (say a small tube), then, as the bubble with the larger radius has less pressure and the one with the smaller radius has more air pressure, air will flow to the larger bubble from the smaller bubble as the excess pressure inside the smaller bubble is greater. But will we be able to quantitatively calculate the time taken for this to happen? If so then what is the relation connecting time and every other variable?


title: Electron distribution around atom when moving

text: I do not have much experience on this but if an atom has some electrons around nucleus and the atom itself it is moving at some speed does that affect the distribution of electrons around? I am presuming that the interaction between the nucleus and electrons has a constant speed $c$. Anything I found so far is a calculation for interactions that presume an infinite speed. As an argument I am thinking of relativistic Doppler effect that does not change proportional with $v/c$. So I am thinking that maybe the speed does affect the distribution and so that is why the difference in the emission energy.


title: Is the classical Doopler Effect, for light shift, $1-v/c$, exact? What is it an approximation of?

text: Is the classical doopler effect for light shift equal to $1-v/c$ exact or an approximation of a classical formula? I know that it is an approximation of the relativistic formula, but what was the corresponding classical formula? I ask this because in Einstein's On the Electrodynamics of moving bodies he derives $\sqrt\frac{1-v/c}{1+v/c}$ and notes that it is different from the classical case. I'm not exactly sure what formula he is comparing it to.


title: Say we're driving a bike and suddenly hold the brakes?

text: It's easy for me to imagine that if we brake the front wheel then there is a chance that I'll flip. On the other hand if I brake the back wheel, there is no way it'll happen no matter how fast I brake. But whether I brake the front wheel or back wheel, the force I've added is a backward force at the bottom of the wheel. So why the different outcome?


title: Regarding possibility of infinitely many spatial dimensions

text: Recently I was taking a look at a video explaining the existence of fourth spatial dimension and thereupon that infinitely many spatial dimensions are possible. Also it showed that what Einstein told about time being a dimension itself was also incorrect as time is always present, even in one dimension. So how's it all possible, is the statement by Einstein not correct?


title: Orbits for space missions

text: I am just wondering say if there is an expedition where some astronauts are sent to the moon, how do they choose the trajectory for the spaceshuttle (or whatnot)? I mean there are many possible trajectories depending on the initial velocity at which the vessel is launched. There must be some sort of optimal trajectory they choose right? What are the factors they take into account when choosing the trajectory? Things like costs, stability and so on? What does stability mean in this context?


title: Shor's Algorithm: Why throw away the f(x)?

text: I'm having a little trouble understanding Shor's algorithm - namely, why do we throw away the result f(x) that we get after applying the F gate? Isn't that the answer we need? My notation: $\newcommand{\ket}[1]{\left|#1\right>}$ $F(\ket x \otimes \ket0) = \ket x \otimes \ket{f(x)}$ , $f(x) = a^x \mod r$


title: Representations of Pauli matrices involving outer product of qubit states

text: Let $| 0 \rangle$ and $| 1 \rangle $ be the states of qubit. Let $\hat{\sigma_x}$, $\hat{\sigma_y}$, $\hat{\sigma_z}$ be Pauli matrices: $$ \hat{\sigma}_{x} = \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right), \;\;\; \hat{\sigma}_{y} = \left( \begin{array}{cc} 0 & -i \\\ i & 0 \end{array} \right), \;\;\; \hat{\sigma}_{z} = \left( \begin{array}{cc} 1 & 0 \\\ 0 & -1 \end{array} \right). $$ Using the representation $\hat{I} = | 0 \rangle \langle 0 | + |1\rangle \langle 1| $ I tried to receive the similar representation for Pauli matrices: $$ \hat{\sigma}_x = | 1 \rangle \langle 0 | + |0\rangle \langle 1|, \\\ \hat{\sigma}_y = -i| 1 \rangle \langle 0 | + i|0\rangle \langle 1|, \\\ \hat{\sigma}_z = -| 0 \rangle \langle 0 | + |1\rangle \langle 1|, \\\ $$ Is it correct? I'm not sure if it is possible but I assumed that $|0\rangle = (0,1)^{T}$ and $|1\rangle = (1,0)^{T}$ and I looked at $|0\rangle \langle 1|$ as on outer product $(0,1)^{T}(1,0)$. I'm afraid that I use matrices instead of operators etc. Help me please to understand the topic.


title: Quantum tunneling and a football permeating a wall

text: I was wondering if I can say to a layman that **"upon throwing the ball on a wall an enormously large number of times, there is a small probability that the ball will go through the wall"** , while explaining quantum tunneling (alpha decay example is abstract and artificial for a layman). My doubt if whether the wall region can be modeled as a finite potential barrier (infinite potential barrier - which is not of Dirac delta form - will not allow tunneling). Also, the wall seems to have all the other characteristics of the artificial barrier potential we set up in quantum mechanics, am I missing anything?


title: Calculation of the spherical harmonic sum in the propagator of the particle on a sphere

text: I am calculating the propagator of the free particle on a sphere : $K(\theta_f \phi_f t_f; \theta_i \phi_i t_i)$. The wavefunctions in this case are the spherical harmonics $Y_{lm}(\theta, \phi)=\frac{e^{im \phi}}{\sqrt{2 \pi}}{\Omega(\theta)}$. So the kernel is $$K=\sum e^{\frac{-i E_n (t_f- t_i)}{\hbar}} \frac{e^{im (\phi_f - \phi_i)}}{{2 \pi}}{\Omega(\theta_f)}\Omega(\theta_i)$$ The sum is over $m=-l, -l+1, ....., l$, and $l=0,1,2.....$ How do I calculate this sum?


title: Gravity as a river

text: I understand that gravity is viewed as flowing as a river pushing objects down on the body of a planet. If that is the case and earth is a sphere, where does the gravity go when it hits the center of the planet? Does it get compressed? Is it destroyed?


title: Calculate heat loss of aquifer with unknown shape

text: I am trying to calculate the heat loss of an aquifer. An aquifer is basically a hot-cold storage $80\: \mathrm{meters}$ under the ground. It is an open system and utilizes groundwater. By pumping for example warm water ($70\: \mathrm{C}$) under the ground this can be stored for half a year and be used in the winter to heat a building. The problem is that I'm not quite sure how to calculate the amount of heat lost during those 6 months. I was thinking of using $\frac{dQ}{dt} = \mathop{UA}(T_2-T_1)$ Where $T_2-T_1$ is the temperature difference of the ground soil and water. However I am working with an undefined 3D shape. So if someone could point me in the right direction as to how to find out how much heat is lost during 6 months of storage, that would be great!


title: Change in Vapour/Liquid change point, at very low pressure

text: 1. In a previous question, I was given an answer: "A quick Google suggests that the triple point of Hydrogen is 13.8K and the triple point of Neon is 24.6K, so neither can exist as liquids at temperatures low enough to form BECs." CVB Note. It was inferred later in the answer, that a gas can never exist as a liquid, below its triple point.   2. From a separate source, I have a table, which shows that at 0.1atm press, the vap/liq point of Helium is 2.5066K. Being from, webbook.nist.gov (Note the ".gov" - See also comments below), I would assume that this is correct. The Vap/liq point of Helium at atmospheric pressure is 4.22K, so Helium's Vap/liq point has changed with a pressure change. I know that Helium is different in many ways, from other gases. One of these differences, is that it has no triple point.   3. I seem to remember, from somewhere, that although water's boiling and freezing points can change with pressure change, this only applies to water and not other elements. This appears to be untrue, in that Helium also seems to have the same property. My question is:   Which of these three are correct, and does any other element, other than Helium, have the property, of its Vap/liq point, changing, when it's pressure changes.?


title: What units are needed for the variables when I want the result to be in $erg$

text: $erg$ is an Energy unit, which means that it is equal to $\frac{mass{\cdot}distance^2}{time^2}$. If I want to calculate the kinetic energy of a body in erg units, in what units should the distance and mass be? By "what units" I mean $kg$ or $gram$ and $cm$, $m$ or $km$ not $inch$ vs. $m$.


title: S-operator lorentz invariance

text: How to show that $\hat {S}$-operator must be lorentz-invariant operator? $$ |\Psi (t)\rangle = \hat {S} | \Psi (0) \rangle , \quad \hat {S} = \hat {T}e^{-i\int \hat {H}_{I}d^{4}x}. $$ I have read that this result follows from the unitarity of the Poincare group operator $U_{0}(\Lambda , a)$ and the covariance of S-matrix $S_{out, in} = \langle out| \hat {S}|in \rangle$, but I don't understand how do we conclude that from this follows that $U_{0}(\Lambda , a)\hat {S}U^{-1}_{0}(\Lambda , a) = \hat {S}$. I'm not interesting in the derivation of lorentz-invariance of S-operator from causality principle at this moment.


title: Why does light travel in straight paths?

text: Why light travels in straight paths here on earth? What's the real cause that makes light photons to go in a straight line? and what are the factors that could change the path of light externally? (I'm excluding reflection, refraction and deviation here)


title: Maxwell equations: how to know the behavior of charge and current?

text: In school-level tasks, when (almost) all substances are linear, homogeneous and isotropic, we have $D=\epsilon E$, $H=B/\mu$ and thus Maxwell "in material" equations (1) say how $E$ and $B$ depend on time given known dependence of $\rho$ (free charge density) and $j$ (free current density). Here they are in CGS unit system: $$\left\\{\begin{aligned} \text{div} D=4\pi\rho\\\ \text{div} B=0\\\ \text{rot} E = -\frac{1}{c}\frac{\partial B}{\partial t}\\\ \text{rot} H = \frac{4\pi}{c}j+\frac{1}{c}\frac{\partial D}{\partial t} \end{aligned}\right.$$ Also we know continuity equation $\partial \rho/\partial t + \text{div} j=0$. But this is not enough to determine, how j will change over time or in statical case, how $j$ is distributed in the conductor. What are other equations for $j$? Are there any for some "ideal case"? For example, I don't know actually, is the following task correct or under determined: > Electric current I flows along infinite cylindrical conductor. Inner radius > is $r$, outer is $R$, magnetic constants of all substances are given > ($\mu_1,\mu_2, \mu_3$ from inside out). Find magnetic field ($B$ and $H$) > and current distribution in a conductor. **The question:** Is there any "standard" equations for $j$? Particularly, is the task above well-determined?


title: Nature of decoupling of matter and radiation

text: I have a question about cosmology. At popular level people explain the time of decoupling of matter and radiation as the moment when temperature falls enough for nuclei and electrons to recombine into atoms. People say that the Universe became "transparent". The photon cross section by an electro-neutral system is smaller than Thomson's one, i.e., technically it means that the mean free path became very large or even infinite. However, there is another mechanism for the mean free path to become infinite. It is because during its expansion the universe becomes less dense. Let me explain it by example. Imagine that you are in the forest where the diameter of tree is $a$, and mean distance between trees is $b$. What is the mean diameter of the observed area? It is proportional to $b^{2}/a$, I suppose. Now let's imagine that our forest is in the expanding universe, i.e., $b$ grows (linearly, for example) with time while $a$ remains constant. Then at some moment "the diameter of the observed area" starts to grow faster than the speed of light, i.e., becomes infinite. It implies that the recombination is not necessary for the decoupling. Did the recombination start earlier than the moment I described above? Or both mechanisms (recombination and density fall) are equally important for the decoupling?


title: What does the chromatic polynomial have to do with the Potts model?

text: Wikipedia writes: > In statistical mechanics, the Potts model, a generalization of the Ising > model, is a model of interacting spins on a crystalline lattice. From combinatorics conferences and seminars, I know that the Potts model has something to do with the chromatic polynomial, but it's not clear to me where it arises (and for which graph it's the chromatic polynomial of). > Question: What does the chromatic polynomial have to do with the Potts > model?


title: The two-body problem: What is force between them?

text: Two bodies with similar/different mass orbiting around a common barycenter.   * What is force between them, where $F_{12}$ is the force on mass 1 due to its interactions with mass 2 and $F_{21}$ is the force on mass 2 due to its interactions with mass 1?   * What is relation between $F_{12}$ and $F_{21}$?   * What is total force between them?


title: How do the different types of energy apply to voltage in a circuit

text: I was just wondering what happens in a circuit in terms of different types of energy transformations. If you apply a voltage to a circuit then electrons start moving (very slowly). Since the electrons want to flow to the positive terminal they will have electrical potential. But the electrons are also now moving once the voltage is applied as there is a current so would they have kinetic energy as well? Some information I have found online says that electrons collide with atoms in a bulb and this is why the filament heats up. But this would imply that kinetic energy is being transformed into heat and this can't be correct because surely any change in kinetic energy would alter the current flowing? Also, in a series circuit, if you measure the potential difference between any 2 points after all of the loads then you always get zero. I was just wondering why it is zero because don't the electrons keep moving even after passing through all the load in order to get back to the power source so surely they cannot do this without some form of energy?


title: Active and passive transformations and the change in potential energy

text: Under active transformation, the particle moves. On the other hand, for a passive one, the coordinate is just relabel. I've read that the passive one will not affect the potential energy and the active one will change it. This is the case for two massively different size objects, like earth and a ball. But basically there will be no difference between passive and active transformation if the two objects making the system are comparable in size. My question is how to prove that and which mathematical framework one should use?


title: Home experiment question

text: I want to make a home experiment where I roughly explain the phenomenon of water remaining in a straw if you close one end of the straw. So I'm thinking that the weight of the water is pulling it down, but the pressure underneath the straw is keeping it up. If there was no finger on the top, the pressure would also be pushing it down so it would fall. Would this be a valid way to show this? Mass of water: volume*density: $\pi r^2*H*1000$ Weight of water: $\pi r^2*H*1000*9.8$ Pressure*area = force force= $101325*\pi r^2$ (weight of water) And I somehow want to combine this with the upward force of pressure in the room. Am I on the right track?


title: How do we know that earth is not perfect sphere?

text: I've just read here that: > Equatorial radius = 6378.16 kilometers. Polar radius = 6356.78 kilometers, > so the difference in circumference is 71.1 kilometers. It is not a perfect > sphere, but kind of pear-shaped. How correct is that information and what exactly are Equatorial radius and Polar radius with diagram if possible?


title: What is Fermi energy and Fermi level?

text: What is meant by Fermi level and Fermi energy? And what is the difference between the two?


title: Confused about charge seperation in solar cells

text: I'm a bit confused about how solar cells work. My understanding is that there is a p-n junction. A photon is absorbed which creates an electron-hole pair, and the idea is to separate the electron and hole before they recombine in order to produce current. n-type materials are doped with extra electrons and p-type materials are doped with holes. So why is that in all the diagrams I see, when charge separation occurs, electrons are conducted through the n-type material and holes are conducted through the p-type material? Shouldn't it be the other way round?


title: Weinberg's proof that Levi-Civita Symbol is a tensor

text: In _Gravitation and Cosmology_ , S.Weinberg states the following: $$\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} \propto \epsilon^{\alpha \beta \gamma \delta} $$ and the argument is that the left-hand side must be odd under a single permutation of the indices. I don't see why this is true. Say I interchange $\alpha\leftrightarrow \beta$: $$\Lambda_{\epsilon}^{\beta}\Lambda_{\zeta}^{\alpha}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} $$ I don't see why the above expression must satisfy $$\Lambda_{\epsilon}^{\beta}\Lambda_{\zeta}^{\alpha}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda}=-\Lambda_{\epsilon}^{\alpha}\Lambda_{\zeta}^{\beta}\Lambda_{\kappa}^{\gamma}\Lambda_{\lambda}^{\delta}\epsilon^{\epsilon \zeta \kappa \lambda} $$ Any hint will be appreciated.


title: How should one interpret the de Sitter slicings?

text: When 'constructing' the usual de Sitter space in $\mathcal{M^5}$ by invoking the contraint $-X^{2}_{0} +X^{2}_{1} +X^{2}_{2} +X^{2}_{3} + X^{2}_{4} = \alpha^2$ we quickly see that we end up with a hyperboloid suspended in Minkowski space. Most literature continues by invoking different coordinates on this embedding, varying from flat/open/closed slices corresponding to different solutions of the Friedmann equations of a universe with solely $\Lambda$ and curvature. I am not quite sure how to interpret these different coordinate systems physically; I can see what effect invoking them might have, but I am a tad confused due to the fact that we first map $X^{2}_{0}$ along the (arbitrary) $z$-axis and then continue to map a new variable $\tau$ for time on another axis. What allows us to create an embedding and invoke arbitrary coordinates on it within it's 'larger' Minkowski space?


title: Superconducting nanowire in a parallel magnetic field

text: Consider a nanoscopic wire (with radius $R$) of superconducting material. The wire lies along the $z$-axis and a magnetic field $\mathbf{H}_a = H\mathbf{e}_z$ is applied. The magnetic field is too weak to destroy superconductivity or to induce vortices and we assume that the superconducting parameter $\psi$ can be put equal to the constant value $\psi_{\infty}$ in the material (and of course, zero outside). Now, I want to use the second Ginzburg-Landau equation to find the vector potential $\mathbf{A}$ (and the magnetic induction field $\mathbf{B} = \nabla\times\mathbf{A}$ from this potential). With the assumptions mentioned above the second Ginzburg-Landau equation simplifies to   $\nabla\times\left(\nabla\times\mathbf{A}\right) = \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2\mathbf{A}$   where   $\psi(\mathbf{r}) = \left\\{\begin{array}{lr} \psi_{\infty} & \mathrm{in\,nanowire} \\\ 0 & \mathrm{outside} \end{array} \right.$   This expression is simplified even further by using the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$, leading to the following vector laplace equation in the case outside the wire:   $\nabla^2\mathbf{A} = 0$   Since the system has cylindrical symmetry, I'm working in cylindrical coordinates. The scalar equations associated with the general vector equation are then:   $\left\\{\begin{array}{rcl} \dfrac{\partial^2A_{\rho}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\rho}}{\partial\phi^2} + \dfrac{\partial^2A_{\rho}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\rho}}{\partial\rho} - \dfrac{2}{\rho^2}\dfrac{\partial A_{\phi}}{\partial\phi} - \dfrac{A_{\rho}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\rho} \\\ \dfrac{\partial^2A_{\phi}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\phi}}{\partial\phi^2} + \dfrac{\partial^2A_{\phi}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\phi}}{\partial\rho} + \dfrac{2}{\rho^2}\dfrac{\partial A_{\rho}}{\partial\phi} - \dfrac{A_{\phi}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\phi} \\\ \dfrac{\partial^2A_z}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_z}{\partial\phi^2} + \dfrac{\partial^2A_z}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_z}{\partial\rho} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_z \end{array} \right.$   while for the case outside the wire the right hand sides are zero. These equations are coupled and I'm not sure how to efficiently solve them. So my question is: could anyone point me in the right direction as to how I should try to solve this vector laplace equation in cylindrical coordinates? Or are there perhaps any possible further assumptions I failed to make? I'm not looking for a complete answer by the way, just a push down the right path, a method. Note: I've asked for a method to solve the vector laplace equation in cylindrical coordinates with less context on math.se yesterday (Solving the vector Laplace equation in cylindrical coordinates) but I thought it might be better to put this here, so I can describe the full physical problem. Perhaps the question on math.se can/should be deleted then?


title: polarization - quantum point of view

text: polarization could be easily imagined in classical model: direction of E vector. is there any simple image for polarization of single photon?


title: What is the escape velocity of a Black Hole?

text: The escape velocity of Earth is $v=\sqrt{\frac {2GM}{R}}$, where $M$ is the mass of the Earth and $R$ it's radius (approximating it as a sphere), and is much less than light speed $c$. But I want to know the escape velocity of black holes. Is it much more than light speed?


title: What is the relation between position and momentum wavefunctions in quantum physics?

text: I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or integral of other functions? Whereas the position and momentum wavefunctions are essentially different but related. They must preserve expectation values like the relationship of classical mechanics, $<p>=m~\frac{d<q>}{dt}$ (where $<p>$ and $<q>$ are now expectation values). For example, a momentum wave packet that has a positive expectation value constant over time implies a position wave packet that moves over time in some direction. Simply saying there is Fourier transform seems to obscure this important relation.


title: Calculating the height objects fall from

text: * Imagine an object of a given mass.   * The object falls from a certain height.   * On contact with the ground, the object has a certain force. If the force of the object and the mass of the object are know, what formula would be used to find the height fallen from?


title: Multiple classical paths from Hamilton's principle

text: Previous posts such as this ask about types of stationary point in Hamilton's Principle. There is, however, another aspect to discuss: the question as to whether the extremal path is unique. One geometric way to envisage this is to assume that multiple paths are simultaneously extremal. I believe that this is an explanation for lenses, but I have not seen lenses explained as multiple classical solutions to Hamilton's Principle. (The multiple paths being the 360 degrees of rays between source and focus, etc also demonstrable through Fermat's principle.) One can generalise lenses, but also consider a simpler case. Let the surface of a sphere be the action (phase space) surface which is minimized in classical paths. Thus (ignore antipodals here) between two points A and B the geodesic is the unique classical path. In quantum form the WKB approximation would no doubt have constructive maxima on this path. However if the sphere has a disk (containing that geodesic) cut out, the shortest path now has exactly two choices: around one or the other rim from A to B. Presumably WKB would maximize the quantum paths on these two (although I havent proved this). If so then classically we have a quantum-like phenomenon: a particle has a choice in going from A to B. Experimentalists might see this and wonder whether the particle went from A to B via the LHS, the RHS or both....


title: Your Mass is NOT from Higgs Boson

text: Your Mass is NOT from Higgs Boson? http://www.youtube.com/watch?v=Ztc6QPNUqls This guy can't be correct, right? He argues that because mostly of a nucleus' mass is made out of the space between quarks (the quark-gluon plasma) then this means that mostly all the mass we are made of doesn't come to be because of the interaction with the Higgs field. http://www.youtube.com/watch?v=Ztc6QPNUqls If this guy is correct then there really needs to be an easy to understand explanation for the masses because all popularizers of science make us lay people think that the Higgs mechanism is responsible for all the mass in the atoms and that would be of course v. misleading.


title: Mass and Higgs Field and Inertia

text: I'm starting to wonder if the description of Mass given to us when we hear physicists talk about Mass created by the Higgs field is the same type of mass we ordinarily think of, like inertial mass for example. I am wondering if they are talking about a field that stops the energy in "mass" from just flying apart and traveling at the speed of light, a field that in effect traps the energy and keeps it bundled up so that the energy stays as a particle?


title: Entropy of an empty universe

text: After watching the first episode of wonders of the solar system, one question came up which is not explained. Bryan Cox says that ultimately the universe will be devoid of matter, so not even a single particle will actually exist. This however, seems to be a state where entropy is zero. There is no entropy since there's no disorder. On the other hand the 2nd law of thermodynamics says that the disorder increases. So, I am pretty sure that I am thinking wrong about this in someway. I can imagine that the answer might be that an empty universe has an infinite amount of entropy but it feels pretty counter-intuitive for me.


title: Physics behind Water drops during falling from a tap

text: what is physics behind Water drops during falling from a tap. water drop animation A drop or droplet is a small column of liquid, bounded completely or almost completely by free surfaces.   * Why Water drops falling into spheres (during falling from a tap in vacuum )? please note that a drop of water made of $H_2O$ atoms. Drop of water


title: If the nucleation bubble radius is greater than the deSitter radius, does that make the de Sitter space stable?

text: In our de Sitter phase, the cosmological constant is tiny. $10^{-123}M_P^4$. Suppose there is another phase with a lower vacuum energy. Is de Sitter phase still stable? The tunneling bubble radius has to exceed the de Sitter radius. Suppose a metastable decay to such a bubble happened. Take that final state, and evolve back in time. It's unlikely to tunnel back because of exponential suppression factors. Light cones are dragged outward in expanding de Sitter at such radii, so, by causality, the bubble radius has to keep shrinking back in time until at least the de Sitter radius. This contradicts our earlier assumption. What about engineering a phase transition? Form a small bubble and stuff it with enough matter in the new phase with sufficient interior pressure to keep the bubble from shrinking. It collapses to a black hole if the radius R is much greater than $M_P^2/T$, which is much less than the de Sitter phase. The black hole then evaporates. Even if the cosmological constant in the new phase is large and negative, the tunneling radius still has to be larger than the de Sitter radius because of the hyperbolic geometry of AdS means the volume of the new phase is only proportional to the domain wall area? If we assume there is a large matter density in the de Sitter phase, it has to be very very large to make the tunneling radius smaller than the de Sitter radius. Is this deSitter phase stable and not metastable?


What is the difference between parallel universe and multiverse? Is it parallel universe or universes?


title: States and observables in quantum mechanics

text: I'm beginning learn quantum mechanics. As I understand, _state_ is a map $\phi$ from $L^2(\mathbb R)$ such that $|\phi|^2$ describes probability density of a particle's position. By integrating $|\phi|^2$ we can get the probability of a particle's position in a subset of $\mathbb R$. **My question is**. Why we need $\phi$? Why just not speaking about _probabilities_ itself, i.e. probability measures $\mathsf \Phi$ on $\mathbb R$ such that $\Phi(A)$ is the probability of particle's position in $A\subset \mathbb R$. Moreover, why we speaking about _observables_ as _operators_. Why not just speaking about _random variables_ on $\mathbb R$. (I know that observable isn't a arbitrary operator; but may be the restrictions can be expressed in terms of random variables?) Many expressions in quantum mechanics via integrals with observables and states, as I understand, just express expectation, variation and standard deviation of such random variables w.r.t. probability that define the states. For example, Heisenberg's inequality just inequality for product of standard deviations of position and momentum.


title: Relativistic Doppler Effect compared with Classical Doppler Effect

text: Looking at the classical Doppler effect there is one generalized equation, and they have a velocity of source, and observer. In the relativistic version, there is only one velocity taken into account, and, in my book, there are two separate equations for receding and approaching sources. These two formulas are leading me to be confused about the sign of the velocity, among other things. Is there a version with only one, generalized formula? And why is only one velocity taken into account, as opposed to the two from the classical version?


title: What are the characteristics of a quadripole circuit element?

text: I'm reading _Music, Physics and Engineering_ (H. Olson 1967), and contains models of musical instruments as electric circuits, i.e. uses the analogies between electrical circuits and mechanical (mass-spring-dashpot) systems. (The idea is that the reader will be more familiar with the electrical circuit representation.) In some of these systems he uses (without defining) a "quadripole" circuit element. A specific instance is to represent the sound board of a guitar (or other string instrument). I've seen some vague references that it performs some sort of amplification, but I would like to known a reasonable approach for mathematically representing it's effect within an electrical circuit. Another possibility is that the term "quadripole component" has fallen out of favor, and there is some other more modern name for this type of thing.


title: Applications of delay differential equations

text: Being interested in the mathematical theory, I was wondering if there are up- to-date, nontrivial models/theories where delay differential equations play a role (PDE-s, or more general functional differential equations). It is clear that   * in biological (population) models usually pregnancy introduces a delay term, or   * in disease transition in networks the latent period introduces a delay, or   * in engineering in feedback problems signal processing introduces the time delay. I would like to see a list of answers where each answer contains one reference/example.


title: Can I applicate the law of action and reaction on energy?

text: It is clear that if you push on some object, there is reaction of the same force. But is it the same energy? Thanks a lot.


title: Container in a Vacuum in a Container Problem

text: My friend puzzled me with this: Say I have a container that contains a vacuum and another container inside. And this inner container contains air. Also the outer container material is just barely strong enough to not implode due to the vacuum. If I remove the air from the inner container to outside of the outer container (for example through a tube that can be sealed), would the outer container implode? My friend says it will implode because the total volume of the whole thing is decreased which leads to decreased inside air pressure. Mathematically, it sounds right. But I thought it was weird because there was really no interaction between the air in the inner container and the outer container in the first place. So my answer is that it will not implode. But I'm not sure myself. What would actually happen and why? EDIT: I actually think it will implode if the inner container was a balloon or something due to having less area for the remaining air molecules in the "vacuum" to bounce around. My question is for an inner container that doesn't change shape like a balloon.


title: Higgs Mechanism

text: In Higgs mechanism, we take the combination of LH $SU(2)$ doublet and RH singlet along with Higgs doublet so that the overall weak hypercharge and weak isospin is zero to be $SU(2) \times U(1)$ invariant. I am pretty confident about my understanding on weak hypercharge calculation. But it seems weak isospin calculation is not quite clear. Specially when the terms comes from Weinberg's dimension five operator for leptons such as $\overline{L_{iL}^\mathcal{C}}$, Higgs $SU(2)$triplet or fermion $SU(2)$ triplet. To arrange them in $SU(2) \times U(1)$ invariant way how should I calculate weak isospin?


title: Entropy Change During Reversible Processes

text: I'm confused about the Second Law of Thermodynamics. The Second Law of Thermodynamics prohibits a decrease in the entropy of a closed system and states that the entropy is unchanged during a reversible process. Then why do we say that $\Delta S = \int_a^b{\frac{dQ}{T}}$ for a reversible process? Doesn't the second law simply state that $\Delta S = 0$? (I'm a high school student teaching himself the principles of thermodynamics, but I am struggling with more challenging material due to my poor understanding of these basics)


title: Ask procedure to obtain reduced density matrix

text: I have a mutipartite system; the subsysyems having interacted. Now, to find reduced density matrix for any single subsystem, is the order of partial traces(over remaining subsystems) important?


title: How do I calculate the experimental and theoretical rotational inertia of a point mass?

text: I'm getting some weird results from a calculation I'm doing and quite honestly, I'm pretty sure it's my fault. I do have an apparatus involved for the experimental process for my lab but I don't think it's that. I've come to the conclusion that my notes do not contain the right equations for these calculations. For the theoretical masses I know the total mass of the point masses and the distance from the axis to the masses. I'm pretty sure the equation of this is either 1/2mr^2 or just mr^2. The 1/2Mr^2 is from my notes but I think the correct answer would be mr^2. However the hanging mass lies on an apparatus that is a cylinder so 1/2Mr^2 could be correct. The experimental part of the lab involved an apparatus that looked like a wheel which lowered a hanging mass on a string by turning the wheel (also by gravity). For the point mass and apparatus combined, I know the hanging mass, slope, and radius. I also know this data for the apparatus. I started out by finding the Force (which I was told is also equal to the torque) using the equation F=m(g-a). I know a is acceleration, but I'm not exactly sure how that factors out. From there I'd probably use τ = I α but I know the angular acceleration nor how to calculate it. I'm pretty sure it involves integrals. Please help me with this, it's been three days and I still can't figure this out. The next part of my lab involves calculating the experimental rotational inertia of the ring and disk, and I know the hanging mass, slop and radius of each one. Would that be where 1/2Mr^2 comes in?


title: Relation of fraction of binary stars with spectral class (mass)

text: What is relation of fraction of binary stars with spectral class (mass)? For example, how many binary stars are among O,B,A,F,G,K,M stars separately?


title: Why is the orbit of an electron circular?

text: Why do the electrons revolve around the nucleus in circular motion not like the earth which revolves around the sun in an eliptical motion?


title: Crazy Dirac Deltas

text: I'm not expecting any rigor in the following and the answers...since we're dealing with Dirac deltas in the context of QFT. Consider the integral $$ \int d^4q\ \Theta(q_0)\Theta(p_{3,0}+q_0)\ \delta((p_3+q)^2)\delta((q^2))\frac{1}{(q+p_2)^2-m^2+i\epsilon} $$ The $p$'s and $q$'s are all Minkowski 4-vectors and $\ p_3=p_1+p_2$. For simplicity one can work in the $p_3$ CM-Frame so that $\ p_3 =(M, \vec{0})$. I have problem with the deltas (since they are coupled), **I was thinking about the** $q_0$ **integration first** , how would you guys proceed doing the $q_0$-integration?. I use the composition formula for the Dirac delta Dirac Delta Comp. Wikipedia, but when I replace $q_0$ in the argument of the other delta I get something like $\delta(M)$ where $M$ is the mass of $p_3$. This delta makes no sense to me since we're not even integrating over $M$? Anyone know how to do this integral from the begining? Any hint appreciated, thanks.


title: How the nonlinear equation can be written like this?

text: We consider a scalar theory in a $1+D$ dimensional flat Minkowski space-time, with a general self-interaction potential, whose action can be written as \begin{equation} A=\int dt\, d^D\\! x \left[\frac12(\partial_t\phi)^2-\frac12(\partial_i\phi)^2 -U(\phi)\right] \,, \end{equation} where $\phi$ is a real scalar field, $\partial_t=\partial/\partial t$, $\partial_i=\partial/\partial x^i$ and $i=1,2,\ldots,D$. The equation of motion following from \eqref{action} is a non-linear wave equation (NLWE) which is given as $$ -\phi_{,tt} + \Delta \phi = U'(\phi)=\phi +\sum\limits_{k=2}^{\infty}g_k\phi^k \tag{1}$$ where $$\quad{\Delta}=\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}\,. $$ I just want to know the logic, that how they transformed the equation (1)


title: Two blocks connected by spring on frictionless surface and Newton's third law

text: Suppose there is the following situation: > Blocks $A$ and $B$, with masses $m_A$ and $m_B$, are connected by a light > spring on a horizontal, frictionless table. When block $A$ has acceleration > $a_A$, then block $B$ has, by Newton’s second and third laws, acceleration > $-a_A\frac{m_A}{m_B}$. What does it mean for block $B$ to have a negative acceleration? Since the blocks are connected by a spring, does not block $B$ move in the same direction as block $A$?


title: What if the earth's core goes cold?

text: What effects would occur if the earth's core goes cold? Would the planet stay liveable after this happens?


title: Physical interpretation of y-intercept in a Current vs Voltage graph

text: So I collected current and voltage data from a simple circuit with a power source and a resistor, using a multimeter. I created a graph for this data using excel and got the y-intercept (which is basically the value of the current when the voltage was zero) and found it to be a small negative number: ![enter image description here](http://i.stack.imgur.com/jecTE.jpg) What I'm wondering about is what is the physical explanation of this value, which by ohm's law should have been zero. Is it simply experimental error in the data collected by the multimeter? Or does it have any deeper physical meaning?


title: What Cat States of light have been experimentally produced?

text: I'm specifically looking for Schrödinger's Cat states involving superpositions of two, or if it's been done more, coherent states, i.e. monomodal states of the form $$|\psi\rangle=a|\alpha\rangle+b|\beta\rangle.$$ What states of this form have been produced in experiment? How even can the weights be? What regions of the $\hat{a}$-eigenvalue $\alpha$ and $\beta$ are accessible? If more than two coherent states can be superposed, how many? What phase-space geometries are possible so far? I'm also interested in what techniques are currently used to generate these states.


title: Confused about indices of the Ricci tensor

text: In an intro to GR book the Ricci tensor is given as: $$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu \lambda}^{\lambda}+\Gamma_{\nu \sigma}^{\lambda}\Gamma_{\mu \lambda}^{\sigma}]$$ I have gotten to the point where I can work out a given Christoffel symbol, but I am still having trouble working out the above tensor as a whole (just algebraically speaking). If I'm not mistaken, $R_{\mu\nu}$ should end up a $\mu$x$\nu$ (i.e. 4x4) matrix just like the energy-momentum tensor on the other side of the field equations. In the above rendering $\sigma$ is clearly a dummy index to be summed over, and I can see how $\lambda$ is also a dummy index in the first term. But the $\lambda$s in the other terms seem to be free indices, which would then introduce incompatible dimensions in the matrix operations. I appreciate it if someone can point out the error of my ways.


title: If microwave ovens and WiFi both operate on the same frequency, why doesn't WiFi cook things?

text: If we ignore 5GHz WiFi, then both microwaves and WiFi create photons at ~2.4GHz but one of them will boil water in a few seconds but the other doesn't have any effect. So what's the difference? Is it simply the number of photons created? Is that what the wattage of a microwave measures? If so, what would be the wattage of a wireless router? Does the enclosed space have anything to do with it? If it all has to do with power output could I put enough WiFi routers together in a room to cook a turkey (from microwaves and not waste heat)?


title: 2.45 GHz Microwave oven and Wi-Fi

text: I noticed that both microwave oven and Wi-Fi use the same frequency band 2.4-2.5 GHz. Yet microwave oven will certainly fry a living being the Wi-Fi has not done yet I suppose. So why is Wi-Fi harmless and microwave oven harmful to us? Why won't our bodies absorb the Wi-Fi energy through dielectric heating like it does if put in a microwave oven?


title: Reynolds number, turbulence regime, and drag force

text: I am trying to model a system in which cubes of about 2 cm in size are floating in a circular water thank of about 30 cm in diameter. The cubes move around under the influence of the fluid flow induced by four inlets that point toward the center of the tank, and are located at the positions $0$, $\pi/2$, $\pi$, and $3\pi/2$. The flow velocity ranges from 0 to 10 cm/s, with an average velocity around 6 cm/s. My questions are the following:   * What would be the Reynolds number of the system? In particular, should I take as characteristic length the size of the cubes, or that of the tank?   * For such a system, what is the limit Reynolds number for the turbulent regime?   * What would be the correct form of the drag force, and do you intuitively think that the orientation of the blocks is negligible from a drag coefficient point of view? Thanks for your help!


title: Similarity in two events from Dynamics point of view

text: What is similar from dynamic point of view between book lying stationary on horizontal table and rain drop falling down with constant speed? * * * I can find two similarities   1. Force acting on them is equal   2. Both have horizontal velocities zero Is there any other similarities possible? It is not mentioned how many similarities we have to give that means we should give all possible similarities. Answer should contain related basic physic like acceleration, velocity, displacement, force,etc. We can say stuff which can be understand by high school student.


title: Why does optical pumping of Rubidium require presence of magnetic field?

text: The optical pumping experiment of Rubidium requires the presence of magnetic field, but I don't understand why. The basic principle of pumping is that the selection rule forbids transition from $m_F=2$ of the ground state of ${}^{87} \mathrm{Rb}$ to excited states, but not the other way around ($\vec{F}$ is the total angular momentum of electron and nucleus). After several round of absorption and spontaneous emission, all atoms will reach the state of $m_F=2$, hence the optical pumping effect. But what does the Zeeman splitting have anything to do with optical pumping? Granted, the ground state, even after fine structure and hyperfine structure considered, is degenerate without Zeeman splitting, but the states with different $m_F$ still exists. In addition, how is the strength of optical pumping related to the intensity of magnetic field applied?


title: Differentiating inside an integral sign

text: I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation. Here is the part of the derivation that I didn't follow: ![enter image description here](http://i.stack.imgur.com/7B0wm.jpg) I don't get how he goes from 6.9 to 6.10 by partial-differentiating the term inside the integral. If this is allowed, I was probably missed my calculus class the day it was covered. Can someone tell me more about this? Which part of calculus is this from?


title: Why are D'Alembert's Principle and the Principle of Least Action Related?

text: Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( \mathbf F _i - \dot{\mathbf p}_i) \cdot \delta \mathbf r _i = 0$$ $$S[q(t)] = \int ^{t_2} _{t_1} \mathcal L (q,\dot{q},t)dt$$ After playing with d'Alembert's principle we find that we can rewrite the whole thing as $$\sum _i \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i}-Q_i\right] \delta q_i$$ This can further be rewritten under certain conditions so we get the exact form of the E-L equation. It seems to me that both ways of arriving at the result are fundamentally different. A function must obey the E-L equations in order to minimize the action over a path, but when we look at the virtual work, it appears that they come from the fact that (quoting Goldstein) "particles in the system will be in equilibrium under a force equal to the actual force plus a 'reversed effective force'." I think I understand the principle of stationary action, I can see how it leads to the E-L equations, but d'Alembert's Principle seems so arbitrary, I can't see any motivation for it.


title: Arguments for and against Many Worlds?

text: I would like to hear the best arguments for and against the Many Worlds interpretation of QM.


title: Newton third law. Is there a limit on how much force you can apply to an object?

text: Recently in class we went over Newton's Third Law. In the book they put an example of hitting a punching-bag with your fist and hitting a piece of paper, or an object with much less mass. It's clear that you cannot exert more force on the paper than what the paper can exert on you, otherwise it might stop your fist in the process, just like the punching-bag stops your fist. Is there a limit, however, on much force you can exert on a paper? This might sound silly but it got me thinking for the past couple of days. In order to make it a little more clear consider a Car with a paper in front of it. The car-system can continuously increase the acceleration, but this means that the paper will increase the acceleration as well. Ergo, by Newton 2nd Law, the force exerted on the paper has to increase. Right? That same force back to the car. Therefore, the car can continuously increase the force it exerts on the paper.


title: Expression of electrostatic field

text: An electrostatic field is characterized by the fact that it depends only by $r$, isn't it? If it is true, I don't understand why this expression, given in cylindrical coordinates, $${\bf E(r)}=\frac{\alpha}{z^2}{\bf u_r}-2 \frac{\alpha r}{z^3}{\bf u_z} $$ rapresents an electrostatic field.


title: What structural or chemical mechanism results in Saturn having a density less than that of water?

text: According to NASA’s Saturn fact sheet, Saturn has a density considerably less than water and the lowest density of the major planets. What compositional and/or structural mechanisms are theorised to be the cause of this low density? Edited to clarify: I am after not only calculations of densities, but from this, a model of what the structure of Saturn would be like to accommodate this.


title: Physical difference between two different attenuation coefficient functions

text: The attenuation of a wave through a medium can be modeled by the Beer-Lambert Law using an attenuation coefficient. If $I$ is the intensity, and $I_r$ is a reference intensity, then what is the physical difference between modelling attenuation using the two functions below? I am looking for a conceptual understanding of the difference between $I_1$ and $I_2$. What does the integral in $\beta_1(\tau,\omega)$ imply? In the equations below, $\tau$ is the time, $\omega$ is angular frequency, and $Q(\tau)$ is a real-valued function of time $\tau$. I have seen both $\beta_1$ and $\beta_2$ functions used in the context of the attenuation of seismic waves through _layered media_ , and I am wondering if one model is perhaps better than the other. ${I_1} = {I_r}{\beta _1}(\tau ,\omega )$ ${I_2} = {I_r}{\beta _2}(\tau ,\omega )$ ${\beta _1}(\tau ,\omega ) = \exp \left[- {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$ ${\beta _2}(\tau ,\omega ) = \exp \left[- {\frac{{\omega \tau }}{{2Q(\tau )}}} \right]$ A reference for these two equations is Y. Wang, Seismic Inverse Q Filtering (link). See for example, pg. 121.


title: Total Mechanical Advantage

text: How do you find the net Mechanical Advantage (MA) of two joint machines. Do you add or multiply the individual MA? Suppose I have two sets of wheel and axle connected by a fixed pulley. Each of the wheel has a radius of 100cm and each of the axle has a radius of 10cm. What will be their combined MA? I am quite sure that both of them has a MA of 10. But what will be their total MA? Is it 20 or 100? Should I add them or multiply them?


title: Is redshift the only way by which we can tell that space is expanding?

text: There's another question on physics.SE whose answer, if I have understood it correctly, explains that the farther the points are in space the faster they are moving away from each other. > Actually, the apparent speed with which two of the points on the circle in a > distance $D$ of each other would move relative to each other will be $v = > H_0 D$ where $H_0$ is the speed the balloon itself is expanding. This means that even the ruler sitting on my desk is expanding, just that the expansion is very very slow owing to the ruler's small size relative to the cosmic scales. However, if everything is scaling then everything should apparently be of the same size to an observer in such an expanding space. The _size million pants_ quote from the show "The Big Bang Theory" comes to mind. To be able to counter that, there're some phenomena that we can observe which are scale invariant. Red-shift would be one such phenomenon. But is red-shift the only way we can tell that the space is expanding? Is there a logical flaw in my thinking? I know that red-shift is a result of the Doppler effect - there's bound to be a decrease in the frequency of a signal, as perceived by an observer, emitted by a source which is moving away relative to the observer.


title: Can we develop a new kind of principles?

text: We have a basic equation $F=ma$. Now can we change this equation to another variable and start a new era in physics? If so, then how? If not, then why?