title: Comparison: graph of radius vector and angular velocity

text: $$ r(t)=R \left(\begin{array}{c}\cos(\omega t)\\\ \sin(\omega t)\end{array}\right) , $$ $$ v(t)=\omega R \left(\begin{array}{c}-\sin(\omega t)\\\ \cos(\omega t)\end{array}\right) $$ where the symbols have their usual meaning. Angular velocity vector is tangential to radius vector at a point. But I want to draw the graph of the vectors separately, without thinking one will be tangential to other at the beginning.


title: Why can we simply absorb the positive coefficient of $i\epsilon$ in a propagator?

text: As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification. In Peskin page 286, he did this: $$k^0\rightarrow k^0(1+i\epsilon)$$ $$(k^2-m^2)\rightarrow (k^2-m^2+i\epsilon)$$ In M. Srednicki's Quantum Field Theory, page 51, > The factor in large parentheses is equal to > $E^2-\omega^2+i(E^2+\omega^2)\epsilon$, and **we can absorb** the positive > coefficient in to $\epsilon$ to get $E^2-\omega^2+i\epsilon$. Why and does this kind of manipulation affect the final result of calculation?   Although $\frac{1}{k^2-m^2+i\epsilon k^2}-\frac{1}{k^2-m^2+i\epsilon}$ is infinitesimal, but the integration of such terms may lead to divergences, and this is my worry. Also the presence of $k^0$ in the coefficient of $i\epsilon$ could potentially influence the poles of an integrand and consequently influence the validity of Wick Rotation.


title: Quantum mechanics formulations

text: There are about 9 different formulations of quantum mechanics. What are the merits and setbacks of each formulation? In the path integral formulation, is the sum over all possible paths idea purely mathematical, or is it an actual probabilistic sum over trajectories?


title: Non-Constant Acceleration due to Gravity

text: Recently, I had the first physics lab for my university physics course. This lab was fairly simple, as we were merely using a computer and a distance sensor to graph the position, velocity, and acceleration of a cart as it moved along a linear track. One of the situations we captured data for involved starting the cart at the bottom of an inclined ramp and giving it a push upwards. As expected, it rolled up, came to a stop, and then came back down the track to its starting position. The position-vs-time graph was essentially parabolic, the velocity- vs-time graph was essentially linear, and the acceleration-vs-time graph was essentially linear. So far, so good. At this point in the lab, the instructor pointed out that, if the data was examined closely, the acceleration of the cart was greater while the cart was traveling upwards than when the cart was traveling downwards (approximately $0.546\frac{m}{s^2}$ and $0.486\frac{m}{s^2}$, respectively), and asked us to determine _why_ in our lab report. Now, gravity was the only force acting upon the cart, and thus it's acceleration should be a constant, at least at the scale our experiment was conducted at, so these results are completely baffling my lab group. So far, we have proposed the following ideas, but none of them seem very plausible.   * Doppler effect on the ultrasonic distance sensor   * Friction   * Air resistance   * Human error The first seems highly improbable, and the last three are more obfuscation and hand waving than actual theories. **Why does our experimental data show the acceleration due to gravity to change based on the direction the object is moving?**


title: How can a generalised force be dependent on an angle i.e. not a vector?

text: I'm currently working through an example question in Patrick Hamill's 'A Student's Guide to Hamiltonians and Lagrangians'. The question I'm having conceptual difficulty with is: > A particle is acted upon with components $F_x$ and $F_y$. Determine the > generalized forces, $Q_i$, in polar coordinates. It's simple to find the generalised force dependence on $r$: ($Q_r = F_x\cos(\theta) + F_y\sin(\theta)$) however I can't seem to get my head around the answer for the dependence on $\theta$, (which is $-F_x r\sin(\theta) + F_y r\cos(\theta)$). I think that the difficulty I'm having stems from $\theta$ looking a lot like a vector here in component form. As far as I am aware, angles are never vectors.


title: What is the function of the top point of a bouncing ball?

text: ![enter image description here](http://i.stack.imgur.com/fqSMt.png) A ball is thrown away as parallel to x axis from M(0,h) point with speed V . After each jumping on x axis , it can reach half of previous height as shown in the figure.(Assume that no any air friction and the ball is very small.) Question-$1$: Find f(x) that passes from of top points ? Question-$2$: I have just a claim about the system but I need to prove that if we throw away the ball from a point of $f(x)$ as parallel to x axis with the speed of $V_{initial}=V\sqrt{\frac{f(x)}{h}}$, the maximum points of the ball will be on same $f(x)$. How can the claim be proved or disproved? * * * My attempt to find f(x) $f(0)=h$ $f(x_1+x_2)=h/2$ $f(x_1+2x_2+x_3)=h/4$ $f(x_1+2x_2+2x_3+x_4)=h/8$ . . . $x_1=vt$ $mgh=\frac{1}{2}mV^2$ $h=\frac{1}{2}g t^2$ After first bouncing , On first top point: N $(x_1+x_2,h/2)$: $mgh/2=\frac{1}{2}mV_1^2$ $V_1^2=V^2/2$ $V_1=\frac{V}{\sqrt{2}}$ $h/2=\frac{1}{2}g t_2^2$ $t_2^2=t^2/2$ $t_2=\frac{t}{\sqrt{2}}$ $f(x_1+x_2)=h/2$ $f(Vt+t_2\frac{V}{\sqrt{2}})=h/2$ $f(Vt+\frac{t}{\sqrt{2}}\frac{V}{\sqrt{2}})=h/2$ $f(Vt(1+1/2))=h/2$ After second bouncing, On second top point: O $(x1+2x2+x_3,h/4)$: $mgh/4=\frac{1}{2}mV_2^2$ $V_2^2=V^2/4$ $V_2=\frac{V}{2}$ $h/4=\frac{1}{2}g t_3^2$ $t_3^2=t^2/4$ $t_3=\frac{t}{2}$ $f(x_1+2x_2+x_3)=h/4$ $f(Vt+Vt+V_3t_3)=h/4$ $f(Vt+Vt+Vt/4)=h/4$ $f(Vt(1+1+1/4))=h/4$ On 3th top point: $f(Vt(1+1+1/2+1/8))=h/8$ On 4th top point: $f(Vt(1+1+1/2+1/4+1/16))=h/16$ . . On nth top point: $n--->\infty$ $f(3Vt)=0$ How can I find $y=f(x)$ Thanks for answers * * * EDIT: I have found $f(x)$. MrBrody is right. The function is a line. my Proof: On nth top point: $f(Vt.a(n))=h/2^n$ $a(0)=1$ $a(1)=1+1/2=1+1-1/2$ $a(2)=1+1+1/4=1+1+1/2-1/4$ $a(3)=1+1+1/2+1/8=1+1+1/2+1/4-1/8$ $a(4)=1+1+1/2+1/4+1/16=1+1+1/2+1/4+1/8-1/16$ . . $a(n)=1+1+1/2+1/4+....1/2^{n-1}-1/2^{n}$ $a(n)=1+\frac{1-(1/2)^{n})}{1-1/2}-1/2^{n}=1+2(1-(1/2)^{n})-1/2^{n}=3-\frac{3}{2^n}$ On nth top point: $$f(Vt[3-\frac{3}{2^n}])=\frac{h}{2^n}$$ where $t=\sqrt{\frac{2h}{g}}$ $Vt[3-\frac{3}{2^n}]=x$ $3-\frac{3}{2^n}=\frac{x}{Vt}$ $1-\frac{1}{2^n}=\frac{x}{3Vt}$ $1-\frac{x}{3Vt}=\frac{1}{2^n}$ $$f(x)=\frac{h}{2^n}=h(1-\frac{x}{3Vt})=h(1-\frac{x}{3V\sqrt{\frac{2h}{g}}})$$ Now I am trying to prove or disprove my claim in second question. If I find the solution I will post it.


title: Choices within the many-worlds interpretation

text: I should preface this question by pointing out that I am a completely amateur scientist and may be missing something very obvious. I've been learning about the many-worlds interpretation recently and I seem to be missing something that seems fairly basic. My question is about choices within the many-worlds interpretation. As I understand it, each choice made during your life results in a parallel universe where that choice was different. _For a really simple example: You walk into the street and in one universe a car hits you but in a parallel universe, you decide to walk down the sidewalk and you're not hit._ This is the way I've seen it described by other articles I've read about it and it makes sense so far to me. **But what I don't understand is what constitutes a "choice" that would result in the universe being split?** Would hypothetical choices that you had no intention on acting on count? For example: _If I'm sitting in a room with my friend and I vaguely think "Should I go get a gun and kill him?", but then I realize that's ridiculous and don't do it, wouldn't I still by definition be making a choice, just by the fact that I'm choosing not to act at all? In theory that would make an alternate universe where I really did kill my friend._ **I guess my main confusion stems from what type of choice constitutes making a split in the universe** , because wouldn't even the tiniest of decisions like what I decide to make for lunch that day result in infinite possibilities ranging from eating everything in my house, to going to every restaurant in my neighborhood?


title: How long does an object experience motionlessness at the beginning of its descent?

text: When a rubber ball bounces, it first accelerates in the positive direction (downward) then it bounces (hits the floor) .The velocity then becomes negative as the ball declerates while travelling upward - accelerates in the negative direction. After some time, the velocity is finally cancelled out by the downward force of gravity on the ball. At this point the same amount of force pushing the ball upward is the same pushing it downward - it is at zero velocity. My question is, how long does this period of zero velocity (motionlessness) last.


title: A graphical proof that the $SU(2)/\mathbb{Z}_2$ vortex is non-orientable

text: The text, see [1], compares the vortex solutions of a spontaneously broken symmetry $U(1) \rightarrow 1$ and $SU(2)\rightarrow U(1) \rightarrow \mathbb{Z}_2$. The vortices can be classified by elements in the first homotopy group which are respectively $\mathbb{Z}$ and $\mathbb{Z}_2$. **My questions**   1. What do they mean by an orientable vortex?    2. Now this picture should show that the $SU(2)/\mathbb{Z}_2$ vortex is non-orientable: > One could try to label the quantum numbers of the vortices by arrows. But as > the figure indicates, these arrows are unstable in the $SU(2)/\mathbb{Z}_2$ > case. ![Proof](http://i.stack.imgur.com/E56lp.png) I don't understand how the arrows are associated to the quantum numbers: a single arrow is $-1$, a looping arrow is $+1$?! What does they mean with 'unstable'/'inconsistent'? 3 . I neither understand the physical consequence of this. The total flux of two $SU(2)\rightarrow U(1) \rightarrow \mathbb{Z}_2$ vortices is again a vacuum state (with zero flux). They conclude from this that there have been created `a pair of something that carries magnetic charge.' I don't see how they come to this conclusion. This is corresponding pictorial representation: > ![snapping](http://i.stack.imgur.com/mhdaw.png) Source: arxiv.org/abs/hep-th/0010225, p 20.


title: Finding out the potential

text: ![enter image description here](http://i.stack.imgur.com/v5LA9.png) > According to me, if we want to find out the potential the the equation will > be, $$dV = \int \frac{dQ}{4 \pi \epsilon_0 x}$$. But the answer is given is on the basis of $$dV = \int \frac{dQ}{4 \pi \epsilon_0 \sqrt{(R^2+ x^2)}}$$. > Why the the distance x from the center of the radious is not allowed to get > the potential?


title: Why is Uranus's axis of rotation tilted?

text: The poles of Uranus are 'in the wrong place', why is this?   * historically, do we have any evidence of its past?    * also, do we have an understanding of how its rotational axis might be evolving?


title: Basic Grassmann/Berezin Integral Question

text: Is there a reason why $\int\\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\\! d\theta = 0$ which I can follow, but not for the former one.


title: How can I judge a globular cluster on the Shapley & Sawyer scale?

text: Globular clusters were classified by Shapley and Sawyer as being on a scale from 1 to 12 where 1 is very loosely packed central area and 12 is highly packed central area. As a visual astronomer, it seems difficult to judge the concentration of a globular cluster. Has anyone tried to classify globular clusters concentration classes for something like the Astronomical League Club for Globular Clusters? http://www.astroleague.org/al/obsclubs/globular/globular1.html It seems that to judge concentration you first need to know what percentage of the diameter of the globular cluster corresponds to the packed core. Visually, knowing the official stated size seems to be required since without that, it is hard to say how concentrated the core is as a glow percentage. Can any human eye get anywhere near detecting all twelve steps of concentration? It is easy enough to distinguish 3 or 4 levels, but twelve seems daunting. How can I attempt to properly judge globular clusters on the Shapley and Sawyer scale?


title: Young's double slit

text: Am I right to think the (general) probability distribution of photon in a double slit experiment at the screen has the form $|\psi|^2 = c e^{\alpha x^2}\cos^2(\beta x)$? (Due to the superposition of the 2 Gaussian shaped wavefunctions of the independent slits) Thank you.


title: is it possible to have magnetic flux density B not in the same direction of magnetic field intensity H?

text: it is said that direction of magnetic flux density B in the same direction of magnetic field intensity H for isotropic media so what is isotropic media and is it possible to have B not in the same direction of H


title: How long does it take for a chemostat to reach equilibrium?

text: A chemostat is a device to grow a cell culture in equilibrium. If we denote by $x$ the cell density and by $s$ the density of nutrients, we can write down the following equations (1): $$\frac{ds}{dt} = D(s_R - s) - \mu x /Y$$ $$\frac{dx}{dt} = (\mu - D)x$$ where $t$ denotes time. Here $D$ and $Y$ are constants called _dilution rate_ and _yield constant_ respectively. $\mu$ is the specific growth rate and it can be a complicated function of $s$. However, two characteristics of $\mu(s)$ are generally accepted: **1.** it is monotonically increasing; and **2.** it saturates, that is, there exists an asymptote limit $\mu_\text{max} = \lim_{s\rightarrow\infty} \mu(s)$. Usually, a so-called Monod function: $$\mu(s) = \mu_\text{max}\frac{s}{k+s}$$ with suitable parameter values for $\mu_\text{max}$ and $k$ gives a good approximation. This dynamical system has a non-trivial point of equilibrium (the trivial equilibrium point being $x=0,s=s_R$): $$\tilde{s}=\mu^{-1}\left(D\right),\quad\tilde{x}=Y\left(s_{R}-\tilde{s}\right)$$ Here one assumes that $\mu(s_R)>D$. This equilibrium point is stable. Suppose the chemostat is initially in equilibrium, and suddenly the parameters $D$ and $s_R$ change. What's a good estimate of how long it takes the chemostat to reach equilibrium again? I accept answers in the form of references to the literature. (1) Szilard, L. (2001). Nonlinear population dynamics in the chemostat. Computing in Science & Engineering, 48–55.


title: A gentle introduction to CFT

text: 1. Which is the definition of a conformal field theory?   2. Which are the physical prerequisites one would need to start studying conformal field theories? (i.e Does one need to know supersymmetry? Does one need non-perturbative effects such as instantons etc?)   3. Which are the mathematical prerequisites one would need to start studying conformal field theories? (i.e how much complex analysis should one know? Does one need the theory of Riemann Surfaces? Does one need algebraic topology or algebraic geometry? And how much?)   4. Which are the best/most common books, or review articles, for a gentle introduction on the topic, at second/third year graduate level?   5. Do CFT models have an application in real world (already experimentally tested) physics? (Also outside the high energy framework, maybe in condensed matter, etc.)


title: What is a "measure equation" as mentioned by this TeX Users Group guide?

text: In this TeX Users Group (TUG) document, Typesetting mathematics for science and technology according to ISO 31/XI by Claudio Beccari, the author makes various typesetting recommendations including: > 9\. Measure equations should be absolutely avoided in professional > scientific texts; measure equations were somewhat popular before the SI was > universally adopted; now they should not be used any more. They survived in > those countries where the "English system of units" is being used, but, > since scientifically speaking this traditional system of units is "illegal", > measure equations have no reason to be used anymore. What are these "measure equations" he talks of? I take it that should he write say $7.25\,\text{cm}$, he would call $7.25$ the "measure" and $\text{cm}$ the "unit of measure". From this and the context of the quoted paragraph I would guess that a measure equation is a relationship between different units of the same quantity, such as " $1\,\text{in}\equiv2.54\,\text{cm}$ ". However I find such things handy, or at least not unnecessary or obsolete, especially when I'm working on atomic scales where SI isn't the most convenient. Is this what the author means and if so, are they really that bad?


title: Why are scattering matrices unitary?

text: In Griffith's QM book, he introduces scattering matrices as an end-of-the- chapter problem. For a Dirac-Delta potential $V(x) = \alpha \delta (x - x_0)$, I've derived the scattering matrix and observed that it is unitary $S^{-1} = S^{\dagger}$. I'm trying to explain why this is intuitively, but I don't really have an intuitive picture of what hermitian conjugation $S^{\dagger}$ is doing here. Thoughts?


title: Experimental evidence showing the kinetic energy of an electron changes in a static non-uniform magnetic field?

text: In a previous question, Does a magnetic field do work on an intrinsic magnetic dipole?, one highly rated answer suggested that static magnetic fields do work on intrinsic magnetic dipoles in a non-uniform magnetic field. I can visualise the change in kinetic energy of the nucleus of an atom coming from a change in the configuration of the electrons around the nucleus. But for an electron, since it's truly fundamental, I'm scratching my head over where the energy comes from to change its kinetic energy. If it does, then it really must come from the static magnetic field. So what is the experimental evidence that shows the kinetic energy of an electron changes in a static non-uniform magnetic field?


title: Example of application of creation/annihilation operators in matrix form

text: I was wondering how it would sound like the creation/annihilation of particles that we usually do in the context of Dirac formalism, with matrices and vectors. As a reminder we know that: \begin{equation} a^{-}\left|n\right\rangle =\sqrt{n}\left|n-1\right\rangle \end{equation} and \begin{equation} a^{+}\left|n\right\rangle =\sqrt{n+1}\left|n+1\right\rangle \end{equation} The operators are defined as: \begin{equation*}a^{-}=\left(\begin{array}{ccccc} 0 & 1 & 0 & 0 & ...\\\ 0 & 0 & 1 & 0 & ...\\\ 0 & 0 & 0 & 1 & ...\\\ 0 & 0 & 0 & 0 & ...\\\ ... & ... & ... & ... & ... \end{array}\right) \end{equation*} and \begin{equation*}a^{+}=\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & ...\\\ 1 & 0 & 0 & 0 & ...\\\ 0 & 1 & 0 & 0 & ...\\\ 0 & 0 & 1 & 0 & ...\\\ ... & ... & ... & ... & ... \end{array}\right) \end{equation*} So how is the form of the ket $\left|n\right\rangle$ in terms of a vector? It must a tensor product of vectors for sure. Moreover the matrix $a^{-}$ (or $a^{+}$) times $\left|0\right\rangle$ (probably a vector with all its entries equal to $0$?) must be something that gives a vector with an entry increased of $1$. But, to be rigorous, a prefix should be added to the matrix in order to indicate that it is acting on a precise factor of the tensor product which made up $\left|n\right\rangle$. An example of a multiplication between a creation/annihilation matrix for a vector would be highly appreciated. If the question is not clear let me know!


title: Where is spin in the Schroedinger equation of an electron in the hydrogen atom?

text: In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom. We are told that the Pauli Exclusion principle is a consequence of two electrons not being able to share the same wave equation. However, in our derived equation, we did not have anything including the spin. We defined $\psi (r,\theta,\phi)$ as $\psi_{n,l,m} (r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi)$, where $Y_{l,m}(\theta,\phi) = f_{l,m}(\theta) e^{i m \phi}$. We then were given well-defined $R_{n,l}$ and $f_{l,m}$ that satisfied the partial differential equations in the Schroedinger Equation. Nowhere in our final $\psi$ do we find anything that varies depending on a fourth degree of freedom, not to mention one that behaved as $m_s$ should. Am I missing the point of the Pauli Exclusion Principle? Is there a part of the solutions for $\psi$ that I am not understanding? EDIT: I am referring to an $H^-$ ion, where there are two electrons each with their own wave equation. If we imagine the case where both have the same quantum numbers n,l,m, but different spin $m_s$, would not their wave equations be exactly the same, and therefore not allowed?


title: Red/Blue Shift in EM Waves

text: > A Doppler weather radar operates at a frequency of 3.40 GHz. The wave from > this radar system reflects from an approaching weather system moving with a > speed of 39.0 m/s. What is the difference in frequency between the outgoing > and returning waves? Ans: 884 Hz How do I compute this? I did $$f = f_s (1 \pm u/c)$$ $$f = 3.4\times 10^9 (1 + \frac{39}{3 \times 10^8}) = 442 Hz$$ I suppose thats the incoming frequency? How do I know the outgoing frequency difference? Looks like its just times 2? Why?


title: Michelson rotating mirror experiment

text: Could someone explain the calculation required to answer this question. It is from a text book and the answer is recorded as 585Hz but I cannot replicate the answer. > In 1931 Michelson used a rotating prism with 32 faces to measure the time > for light to travel through a 1.6km evacuated tube to a distant mirror and > back. From his results he calculated that the speed of light is 299774000 > m/s. > > Assuming the prism is rotated by only one face during the time of flight of > light calculate the rotation frequency of the prism. Is the book incorrect or am I? My attempt: $(1/(1600/299774000))/32=5855\:\mathrm{Hz}$ which is suspiciously similar to the suggested answer 585Hz. But then I remembered about the round- trip (3200m) which threw all the numbers again.


title: I need some help trying to convert the specific fuel consumption from imperial to metric units

text: Ok the imperial unit for the specific fuel consumption for a jet aircraft is shown below. Its the amount of fuel (in lbs) per hour required to produce 1 pound of thrust. $$\frac{\frac{lbs}{h}}{lbs}$$ The metric unit is. $$\frac{kg}{Ns}$$ I tried this. $$\frac{\frac{lbs}{h}}{lbs}=\frac{lbs}{lbs}\cdot\frac{1}{h}=\frac{2.2\cdot kg}{2.2\cdot9.81\cdot N}\cdot\frac{1}{3600 \cdot s}=\frac{1}{9.81\cdot3600} \left[ \frac{kg}{Ns} \right]$$ But apparently it is. $$\frac{1}{2.2\cdot9.81\cdot3600} \left[ \frac{kg}{Ns} \right]$$


title: Where do planets get energy to revolve around sun?

text: We know that every planet in our solar system revolve's around the sun in a particular orbit. But were to they get the energy to revolve around the sun. And why do they not drop into the sun there is only gravitational force acting which is always attractive in nature?


title: Simple Experimental Laser Characterization Parameters Ideas?

text: I have 2 months to do a project on characterization of a laser parameter. We have typical optics lab equipment (DFB Lasers, oscilloscopes, random number generator, etc.). I was told to choose a laser parameter to characterize experimentally, but after hours of reading research papers, they all seem like they will take much longer to understand and test experimentally. Any ideas for which parameter is relatively simple to characterize experimentally with my short time window?


title: Why polarization vector $= (0,1,i,0)$?

text: I know from CED that one has e.g. polarization $$\vec{E}(z,t) = \begin{bmatrix} e_{x} \\\ e_{y} \\\ 0 \end{bmatrix} \; e^{i(kz - 2 \pi f t)}. $$ Why do Peskin&Schroeder define a polarization vector as (see page 7 PS) $$(0,1,i,0)?$$ Where is the exponential? Why does he use $i$ for the $y$-direction?


title: Inflation Scale via CMB Polarization

text: COBE, WMAP, and now PLANCK, have or are in the process of measuring anisotropies in the cosmic microwave background. WMAP has dectected E-mode polarization from electron Thomson scattering, but not B-mode polarization. One of PLANCK's objectives is to detect B-mode polarization as a means to determine the scale of inflation. The B-mode is expected to originate from large scale gravitational waves created by inflation, but gravtational lensing is also a B-mode source. My question is, how is B-mode polarization generated by gravity waves, and how is it used to determine the scale of inflation? Also, can B-mode detection be a verification of the inflation theory?


title: Why does angle deviation of a satellite not captured my Mars decreases as the Velocity of the satellite increases?

text: When a satellite is not captured by a planet, it is moving sufficiently fast. so why is it that the angle deviation of the satellite decreases as its initial velocity increases?


title: Non-commutative property of rotation

text: Addition of angles are non-commutative in three dimensions. Hence some other angular vector quantities like angular velocity, momentum become non- commutative. What is the physical significance of this non-commutative property?


title: Definition of Electromagnetic field?

text: 1. Definition of Electromagnetic field?   2. What is the Electromagnetic field at a given point in space due to a point charge being accelerated non-uniformly?    3. Is there a single equation that can give me the electromagnetic force experienced by a body of unit charge (and should I include unit velocity/momentum/acceleration)?    4. And how do I determine from which direction an electromagnetic wave is coming from?


title: Renormalizing with external momenta set to zero

text: I've often seen in textbooks that authors renormalize diagrams by setting external momentum to zero. Under what conditions is this justified? An example of this is done in Manohar and Wise's book on Heavy Quark Physics after they renormalize QED and then calculate the operator renormalization, $ Z _S $, of \begin{equation} S = \frac{1}{ Z _S }\bar{\psi} _b \psi _b = \frac{ Z _\psi }{Z _S } \bar{\psi} \psi \end{equation} where $ \psi_b $ is the bare field. They calulate this through the diagram, $\hspace{5cm}$ ![enter image description here](http://i.stack.imgur.com/wWqzl.png) where the cross indicates an operator insertion. The authors then say ``The operator, $ S $, contains no derivatives (and $Z _S$ is mass independent in the $\overline{MS}$ scheme), so $Z _S$ can be determined by evaluating (the diagram) at zero external momentum (and neglecting the (fermion) mass).'' Are these the two conditions necessary,   1. The operator doesn't have derivatives   2. The quantity of interest has no mass dependence and if so how do we know that the quantity you want to calculate (in this case, $ Z _S $) is mass independent, ahead of time?


title: When and why spontaneous nuclear fission is preferred to alpha emission?

text: I reasoned as follow: the probability of an $\alpha$ emission $^A_ZX\to^{A-4}_{Z-2}X^{'}+\alpha$ is given by: $T=e^{-G}$ where the Gamow factor is given by: $G\simeq \pi \sqrt{\frac{2\mu c^2}{E}}Z_1Z_2\frac{e^2}{\hbar c}$ so $G\propto \sqrt{\mu}$ where $\mu=\frac{m_{\alpha}m_{X^{'}}}{m_{X^{'}}+m_{\alpha}}\simeq m_{\alpha}$ because usually $m_{X^{'}}>>m_{\alpha}$ In the case of a spontaneous fission I have that $^A_ZX\to^{A_1}_{Z_1}X_1+^{A_2}_{Z_2}X_2$ The only thing I can say is that the probability of tunneling is less than in the previous case because now the reduced mass is higher. Then I have some qualitative knolwledges like "the fission happen when the energy associated to proton repulsion become higher than the binding energy, so because of the repulsion energy is $\propto Z^2$ the fission is a mechanism that involves heavy nucleus" What is the correct way to calculate the spontaneous fission probability and so compare it with the $\alpha$ emission probability?


title: Using supersymmetry outside high energy/particle physics

text: Are there applications of supersymmetry in other branches of physics other than high energy/particle physics?


title: Nuclear fission mechanism: neutron capture

text: Consider this nuclear fission reaction: $\mathrm{^{235}U+{}^1n\to{}^{236}U \text{ (excited)}\to{} ^{92}Kr+{}^{141}Ba+3{}^1n}$ I have not understand why a thermal neutron ($^1$n with $E\simeq 0.025$ eV) has enough energy to start the reaction. I mean, to start the reaction the $^1$n should be captured by the nuclei of $^{235}$U. What are the conditions under which this neutron is captured? My reasoning is the following: the energy that the neutron must have to be captured by the $^{235}$U should be equal to the $S_\mathrm n$ (separation energy for a neutron) of $^{236}$U which I found is 6.34 MeV, higher than the energy of the thermal neutron. So my reasoning must be wrong but I cannot understand where I make mistakes... I looked on the Povh Rith Particles and Nuclei but it does not explain the reaction well. Could anyone give me a reference or explain here the fission reaction ?


title: Combined Translation and Rotation of a disk possible? material and references?

text: I am considering building a robot that can rotate and move at the same time. Since it's just a theoretical idea at the time and I need read up material, I thought I would ask here. I am thinking of a two wheeled robot that can spin on the spot. Fairly straight forward. If this robot needs to move from A to B. Most people would suggest to stop spinning circularly, point the direction of motion and move there through linear translation. I am wondering if there is a way to do that without stopping the spin and pointing in a specific direction. Someway of combining both the translational and rotational motion of the device to move the whole system in a particular direction. I am guessing that rather than both wheels spinning at a constant speed, such as during rotation, there would be two sinusoidal wave with a phase shift of some sort that will cause the robot to "drift" in the direction of choice. I have some vague ideas, but would like more understanding about it or if it is even possible. I am not even sure if this sort of motion has a name. I tried google, but most of the hits I am getting are about a disk rolling down a plank kind of motion. any references, books, articles, videos or any other material would be of great use


title: Vanishing diagonal matrix elements of pertubation

text: In time-dependent pertubation theory we can denote the Schrödinger equation by a set of two equations $$\dot{c_a} = -\frac{i}{\hbar}\Big[c_aH'_{aa}+c_bH'_{ab}e^{-i(E_b-E_a)t/\hbar}\Big] \\\ \dot{c_b} =-\frac{i}{\hbar}\Big[c_aH'_{bb}+c_bH'_{ba}e^{-i(E_b- E_a)t/\hbar}\Big] $$ with matrix elements $H'_{ij} = \langle \psi_i|H'|\psi_j\rangle $ and a time-dependent pertubation $H'(t)$ for a two- level system where we denote our wave function by $$\Psi(t) = c_a(t)\psi_ae^{-iE_a/\hbar}+c_b(t)\psi_be^{iE_b/\hbar} $$as a linear combination of two orthonormal eigenstates $\psi_a,\psi_b$ of our unperturbed Hamiltonian $H^0$. Why do we (does Griffiths) assume that the diagnoal matrix elements of $H'$ typically vanish?


title: What will happen if we put balloon in water tank in space

text: Imagine this. You have some object (let say barrel) and fill it with water. The water remains rather calm so it doesnt' spills out. Then we take a balloon filled with air and push it in that tank. What will happen when i release the balloon? Which side it will go? (if its in the middle) Will it direct to the open area of the barrel or to the bottom? Same goes if somehow we form a perfect sphere of water and put the balloon inside of it. Where it will be pushed?


title: Sun's tidal force on earth

text: My question is regarding effect of sun's gravity on earth. I want to know that if sun's gravity can change earth's landscape in long duration (i.e. billion of years ) or not?? Means if earth is dead planet and all other planets are not present in the solar system and earth is as near as sun as mercury currently orbiting. And also suppose earth's surface is not smooth (because of hitting with other small bodies at regular interval) then can earth's surface can be smooth in long duration by gravity of sun or not??


title: In the formula for the mass of a drop of water on the end of a tube, is r internal or external radius?

text: In the formula for the mass $m$ of a drop of water forming on the end of a vertical capillary tube, $$m = \frac{2\pi r \gamma}{g},$$ does $r$ refer to the internal or external radius of the capillary tube? ![drop on end of capillary tube](http://i.stack.imgur.com/psOmM.png) In the picture, the black part is meant to be the capillary tube, filled with water with a drop forming at the bottom.


title: phase transition by sublimation

text: Why do sublime solids sublimate in the first place? Is it that their melting point and their liquid's boiling point are the same so that they manage to magically skip the liquid phase? (if so, why does it have to be so?)And is the reverse possible?


title: Spherical charge in two different dielectric materials

text: I am trying to freshen up my memory about electrical fields and I came across this exercise from school. A sphere with a constantly distributed charge is located in between two different dielectrics (see picture at the bottom) and the task is to calculate the electrical field. I have a sample solution that makes the following assumptions:   1. The electrical field at the boundary between the two dielectrics has to be constant. (I understand this)   2. The electrical field has only a radial component. (I accept this as well)   3. The electrical field has spherical symmetry i.e. it has the same magnitude regardless if it is in the upper or lower dielectric. I am sort of stumbling over this assumption, because I feel that the setup is not symmetric.  So, how come the electrical field is radial symmetric? This doesn't seem intuitive to me. Or is the sample solution incorrect. ![enter image description here](http://i.stack.imgur.com/zyjfy.png) **EDIT** : Okay, I think I figured it out. **The sphere is actually a conducting sphere**. Now since the whole sphere has the same potential and lets assume that at infinity the potential is zero. The static electrical field has to be symmetrical around the sphere, since the integral from infinity over the electrical field onto the sphere has to be consistent from all directions. This also means, that the charges inside the sphere will shift accordingly. Can somebody confirm this?


title: Lagrangian and Equilibrium Points

text: I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, but they are messy in complicated situations. Is there a conceptual shortcut I could take, or am I expecting too much? Many thanks in advance!


title: Why does absorption cause seismic pulses to increase in length over distance?

text: The specific question I'm trying to answer is "How does the progressive loss of higher frequencies in a propagating seismic pulse lead to an increase in pulse length?" I understand how the higher frequencies attenuate faster than lower frequencies (through the absorption constant), thus resulting in a change of shape but not so much how this increases the pulse length? I've been researching for a while and have come across multiple ideas including: 1) I've read that in non-porous media velocity = freq x wavelength, as the frequency components reduce in value, the wavelength would increase to balance this? Though I'm not sure I can assume the media is non-porous. 2) I've been reading about wave dispersion which discusses how higher frequency components move faster than lower frequency components in dispersive media, thus the wavelength increases. My issue with this though is that the question specifically refers to the loss of higher frequencies influencing the pulse length, and just talking about different frequencies having different velocities doesn't seem to cut it. 3) Geometrical spreading could be increasing the wavelength? Though again this doesn't really relate to attenuation as the question is asking. Any help letting me know what I'm missing here would be fantastic. This question has been bugging me for a couple of days now. Cheers


title: Calculating Quantum number from initial conditions

text: I have solved the particle in a box problem to get energy eigenstates and wave vectors: $$E_{n}=\frac{\hbar^{2} k^2}{2m} ,\hspace{1cm} k_{n}=\frac{\pi n}{L}$$ And now I am trying to figure out how we could determine the quantum number $n$ from the initial properties. For instance if we give the particle a mass of say $0.2 \,\text{kg}$ and initial velocity $v=1\text{ ms}^{-1}$ and set $L=1.5\text{ m}$. How could we determine the quantum number $n$. We note that we have associated de Broglie wavelength: $$\lambda = \frac{h}{p}=\frac{h}{0.2}=5h$$ We note that our wave number is given by: $k=\frac{2\pi}{\lambda}=\frac{2\pi}{5 h}$. However, this does not result in an integer $n$. What am I misunderstanding here?


title: How does Passivation in Mercury-arc-valve cause "diode" behavior

text: Does passivation of lead anodes cause them to become semiconductors? Or even become diodes (eg. a p-n passage)? How does this work on an atomar level? This question has been risen by trying to understand an old AC-DC converter alos known as Quecksilberdampfgleichrichter or Mercury-arc valve. Shortly: It turns rotating or alternating current into DC (for street cars traditionally) by that it ignites an arc above a mercury lake (the catode) in an containment. Above that lake are the anodes that hand the arc around as the current rotates. This requires some place in the anodes to be current conducing in only one direction (diode like behaviour). Otherwise anodes would exchange current between themselves instead of between catode and anodes. It is being said that "Passivation" on those lead anodes in the mercury environment is the reason for that. Therefore: How does passivation (on an atomar or subatomar level) cause the electrons to only move one way, not both?


title: Kinematics and Energy - Finding the work done in projectile motion

text: If you are asked to find the work done by gravity on a projectile, the force will obviously be the mass multiplied by gravitational acceleration. What will the displacement be, the horizontal or vertical one?


title: Dimension of Dirac $\gamma$ matrices

text: While studying the Dirac equation, I came across this enigmatic passage on p. 551 in _From Classical to Quantum Mechanics_ by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: > $$\tag{16.1.2} (\gamma^0)^2 = I , (\gamma^j)^2 = -I \ (j=1,2,3) $$ > $$\tag{16.1.3} \gamma^0\gamma^j + \gamma^j \gamma^0 = 0 $$ $$\tag{16.1.4} > \gamma^j \gamma^k + \gamma^k \gamma^j = 0, \ j\neq k$$ _In looking for > solutions of these equations in terms of matrices, one finds that they must > have as order a multiple of 4, and that there exists a solution of order 4._ Obviously the word _order_ here means dimension. In my QM classes the lecturer referenced chapter 5 from _Advanced Quantum Mechanics_ by F. Schwabl, especially as regards the dimension of Dirac $\gamma$ matrices. However there it is stated only that, since the number of positive and negative eigenvalues of $\alpha$ and $\beta^k$ must be equal, $n$ is even. Moreover, $n=2$ is not sufficient, so $n=4$ is the smallest possible dimension in which it is possible to realize the desired algebraic structure. While I got that the smallest dimension is 4, I fail to find any argument to reject the possibility that $n=6$ could be a solution. I also checked this Phys.SE post, but I didn't find it helpful at all. Can anyone help me?


title: How scientists could say that such meteorite comes from Mars

text: How could scientists affirm that a meteorite comes from Mars and not from another source ? This is a probability or an absolute certainty ? How much percent ?


title: Fluctuation-Dissipation theorems in an infinite quantum system

text: So for a quantum spin chain, one can easily prove via the partition function that you have a fluctuation-dissipation type relation between the magnetic susceptibility and the variance of the magnetization in cases where the Hamiltonian commutes with the external field. For example, if I have a Heisenberg Hamiltonian in a transverse field, I can show $$\frac1\beta\frac{d\langle M\rangle}{dh}=\langle M^2\rangle-\langle M\rangle^2$$ Where $M$ is the magnetization operator, and $\beta$ is the inverse temperature. My question is whether this relationship should be expected to hold in the thermodynamic limit, i.e. if I let $T\to0$ and the length of the chain go to infinity. The usual derivation would seems to break down.


title: Olympic weight lifting bar, plate location affect

text: I apologize in advance for the rudimentary nature of my question. > _A perfectly balanced weight lifting bar is loaded with three weight plates > on each side. The three plates are 25kg, 10kg and 5kg. On the left side of > the bar the 25kg plate is innermost, then the 10kg plate, then the 5kg > plate. On the right side of the bar the 10kg is innermost, then the 25kg > plate, then the 5kg plate._ I say the center of gravity will no longer be in the center of the bar but shift toward the right side since the 25kg plate on the right side has moved farther from the center of the bar. My two friends say the center of gravity will remain dead center on the bar.


title: How accurate are our calculations about distant stars keeping in mind their enormous distances?

text: Since many stars are hundreds of light years away from the Earth and therefore, what we observe of them today is really their distant past, how can we say anything with certainty about their composition, size and nature? Betelguese, for example, is said to be in the last stage of stellar evolution, but taking into consideration its approximately 640 light years of distance from the Earth, aren't we actually making assertions about the star that are 640 years old? If that is the case, how can we claim to know the actual present status of the star?


title: Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?

text: Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation that has a different covariance group? In other words: consider some solution to the linear Klein-Gordon equation, and model that solution by a mapping $\psi : M \rightarrow \mathbb{R}$, where $M = (\mathbb{R}^4, \eta)$ is Minkowski spacetime. Now consider a coordinate transformation---but not necessarily a Lorentz transformation. Instead, consider any coordinate transformation modeled by an autohomeomorphism $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$, where $\mathbb{R}^4$ has the standard induced topology. This gives $\psi'(x', t') : \mathbb{R}^4 \rightarrow \mathbb{R}$. The question is this: if one has free choice of metric on the new coordinates established by $\phi$, is there such a $\psi'$ that solves another equation with some different covariance group? Further, if this is possible for simple, plane-wave solutions, then what are the conditions under which it is no longer possible? What about linear combinations of plane-wave solutions? What about the case of multiple solutions, the values of one being coincident with the values of the other(s) in $M$? (Note: empirical adequacy is irrelevant; this is just a question about the conditions under which the values of a scalar-field solution to a partially- differential equation with some particular covariance group can also act as the solution to another equation with some different covariance group, while preserving their topological (or perhaps even differential) structure.)


title: First Order Phase Transition - Heat Capacity

text: ![\(a\) First Order \(b\) Second Order](http://i.stack.imgur.com/bO3T9.gif) At constant pressure : $\frac{dH}{dT}=C_p $ Hence, why doesn't the graph of $C_p$ against $T$ show two perfectly horizontal lines (with the obvious discontinuity) ? I mean the gradient of $H$ versus $T$ graph is constant so the $C_p$ value should be constant as well.


title: Minkowski Metric Signature

text: When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ x^0,x^1,x^2,x^3 $ come from $ x^{\mu} : \mu = 0,1,2,3 $, and $ c $ is the speed of light. The first resource I had access to--I have to do a bit of digging for the exact paper-- of course addressed that this invariant takes the tensor form: $$ ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu} $$ as well. This two things I've seen in all of my texts and online resources regarding resources. The element in question that varies between authors, is the time coordinate, $ x^0 $. When it was first explained to me, it was using the standard Cartesian representation for the spatial portion of the coordinates, and the time coordinate was labeled as $ x^0 = ict $. Squaring gives $ (x^0)^2 = -c^2t^2 $, and applying differential calculus, we get $ (dx^0)^2 = -c^2dt^2 $. Sensible and expected to have the tensor formula spit out the Minkowski Metric. The author then later explicitly states the coordinates are $ x^0 = ict, x^1=x,x^2=y,$ and $x^3 = z$. My questions is then, why is it most authors on the subject omit the imaginary unit on the time coordinate? For example, here. The only reason I can fathom omission is if the author is using metric signature $ [+,-,-,-] $, where I started off learning the theory with signature $ [-,+,+,+] $ which may be the reason seeing the time coord with no imaginary unit seems dissonant to me. All help appreciated! Edit: After reading the other answers, my questions is now one of _why_ and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign.


title: Attempts to explain Higgs coupling as a gauge transformation symmetry

text: As is (supposedly) well known, Electromagnetic coupling can be "explained" as a closure term to a langrangian comprising a free Dirac field and a free vector field that are required to be invariant under transformations that add arbitrary gradient quantities to the vector field and matching local phase $U(1)$ adjustments to the Dirac field, a construction better known as _gauge invariance_. There are similar gauge invariant arguments to obtain the electro-weak and chromodynamic couplings. None of such applies so far to gravity in a way that is entirely satisfactory (to me, yes), specially with all the non-sense that its straightforward application would imply (such as no local observables, etc.) This particular question is about the Higgs field; from my understanding it is entirely added by hand to all (or most?) fermion fields. It couples also to gauge vector bosons but unlike those, i've never seen any allegations that such coupling would arise naturally from a gauge invariance schema. Are there any attempts to produce any of such constructions that is worth mentioning?


title: What is the math showing that the time reversed version of an electron is a positron? (+general time reversal question)

text: As in Wheeler's One Electron Universe idea, how do you show that electrons and positrons are time-reversed versions of each other? Do you just apply time reversal to an electron and out pops a positron? Maybe a more accurate question would be "How do you describe a particle moving backwards in time?", out of which the transformation from electron -> positron should become apparent. Edited to add: Also, what is the difference between time reversal and moving backwards in time? Is time reversal the observer moving backwards in time (in which case we would see an electron as an electron) vs. the electron moving backwards in time (when we would see a positron)?


title: Why are overtones forbidden within the harmonic approximation?

text: In vibrational spectroscopy only transitions between neighboring vibrational states ($\Delta \nu = \pm 1$, $\nu$ being the vibrational quantum number) are allowed within the harmonic approximation. Where does this selection rule come from? I have some ideas but haven't come to a satisfactory conclusion. I guess it has something to do with the transition dipole moment $P$ having to be nonzero. This is given by \begin{equation} P = \left\langle \psi^{\text{final}} \big| \hat{\mu} \big| \psi^{\text{initial}} \right\rangle \end{equation} where $\hat{\mu}$ is the dipole moment operator and $\psi^{\text{final}}$ and $\psi^{\text{initial}}$ are the wave functions of the final and the initial state, respectively. Since the wave functions of the harmonic oscillator are even functions for $\nu = 2n$ and odd functions for $\nu = 2n + 1$ (with $n = 0, 1, \dots$) and $\hat{\mu}$ transforms as a vector (and so can be considered an uneven function) the integral for $P$ vanishes if $\psi^{\text{final}}$ and $\psi^{\text{initial}}$ are both even (or uneven). So, if my ideas about the problem are correct (please correct me if they aren't) I would expect a selection rule $\Delta \nu = \pm 1, \pm 3, \pm 5, \dots$. But why is it only $\Delta \nu = \pm 1$?


title: Trying to understand Laplace's equation

text: I'm struggling here so please excuse if I'm writing nonsense. I understand that the gravitational potential field, a scalar field, is given by $$\phi=\frac{-Gm}{r}$$ where $\phi$ is the gravitational potential energy of a unit mass in a gravitational field $g$ . The gradient of this is (a vector field) $$g=-\nabla\phi=-\left(\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z}\right)$$ And the divergence of this vector field is $$\nabla\cdot\nabla\phi=\nabla^{2}\phi=4\pi G\rho$$ and is called Poisson's equation. If the point is outside of the mass, then $$\rho=0$$ and Poisson's equation becomes $$\nabla\cdot\nabla\phi=0$$ My question is, how do I express $\phi=\frac{-Gm}{r}$ as a function of $x,y,z$ so I can then end up with $\nabla\cdot\nabla\phi=0$ in empty space $r\neq 0$? I would have thought that I could write $$\phi=\frac{-Gm}{r}=\frac{-Gm}{\sqrt{x^{2}+y^{2}+z^{2}}}$$ but when I try to calculate $\nabla\cdot\nabla\phi$ from this, I don't get zero for $r\neq 0$. I'm confused.


title: How to solve Schrodinger Equation - Tunnelling

text: I have to solve analitically the Schrodinger equation in one-dimension with a barrier of potential (tunnel effect): $$ih \frac{d}{dt} U(x,t) = \left[ \left(-\hbar^2 \frac{d^2}{dx^2} \right) + q V(x) \right] U(x,t)$$ where: $i$ is the imaginary unit, ($d/dt$) is the time derivative, $\hbar$ is the Reduced Plank constant, ($d^2/dx^2$) is the second derivative in space, $V(x)$ is an external potential function of $x$, $U(x,t)$ is the wave function of time and place. The barrier of potential is: $$V(x) = \begin{cases} 0, & \mbox{if } -d<x<-L \\\ V_0, & \mbox{if } -L<x<L \\\ 0, & \mbox{if } L<x<d \end{cases} $$ with $d=10L$ and $V_0>0$; Also the boundary condition are: $U(-d,t)=U(d,t)=0$; and the initial condition is $$U(x,t_{0})=\frac{1}{\sqrt{Dx}} \exp \left(i P_0 \frac{x}{\hbar} \right)$$ if $-d<x<-d+Dx$ and $U(x,t_{0})=0$ if $-d+Dx<x<d$; where $Dx<<L$ and $P_0$ is the quantum moment at t0. Also I know that at time $t_0$, the Fourier Transform of $U(x,t_{0})$ is a sinc centered in $P_0$, the aspectated value of position is $-d+Dx/2$ and the aspectated value of velocity is $P_0/m$, where $m$ is the mass of the particle. Then I have to compare analytical results with results from FINITE ELEMENTS and FINITE DIFFERENCE method. I hope that someone can help me to solve this problem.


title: Discrete point particles stress energy tensor

text: I am trying to solve an exercise in Sean Carroll's GR book "Spacetime and Geometry". Basically we need to derive the stress-energy tensor of a perfect fluid (ie $T^{\mu\nu}=(\rho +p)U^{\mu}U^{\nu} + p\eta^{\mu\nu}$) from the stress-energy tensor of a discrete set of particles (ie $T^{\mu\nu}=\sum_a \frac{p^{\mu}_a p^{\nu}_a}{p^0_a}\delta^{(3)}(\mathbf x - \mathbf x^{(a)})$), under the hypothesis of isotropy. I managed to get in for the $T^{00}$ component and the $T^{0i}$ components, by replacing $p^{\mu}$ by $p^0$ a trivial sum appears: energy density for the 00-component and momentum density the 0i-components (vanishing by isotropy). But I am still struggling with the pure spatial part, I was thinking of substituting the sum by an integral, then the non diagonal part vanishes by isotropy again. Could it be something like: $\sum_a = \int d^3x \rho(x)$? Because then I need a relation that relates the density distribution $\rho$ and $p^{\mu}$ to the pressure (more exactly the definition of the pressure from these two concepts).


title: EM Fields in a Rotating Frame of Reference

text: I'm struggling on my approach to the problem of figuring out E and B fields in a non-relativistic way for a rotating frame of reference in the x-y plane around the z-axis. I am attempting to do this without the use of Tensors: Consider a particle at rest with respect to the rotating frame. Since there is no velocity there is no force felt from $\vec{B'}$. **Do I have to include the pseudo forces in this equation?** $$ \vec{F} = m \vec{a} = q \vec{E'} = q(\vec E+\vec v \times \vec B) \\\ \vec{E'} = \vec E+\vec v \times \vec B $$ Additionally, considering a stationary particle in the inertial frame of reference, with pseudo forces included: $$ \vec{F} = m \vec a + m \vec \omega \times \vec v \+ m \vec \omega \times \vec v + m\vec \omega \times \vec \omega \times \vec x \\\ = q(\vec{E'}+\vec v \times \vec{B'}) = q\vec E $$ However if I just plug in $\vec{E'}$ into the second equation, $\vec{B'}$ = $\vec{B}$ which seems incorrect. Looking at the non-relativistic portions of a Lorentz transformations has $\vec{B'}$ = $\vec{B} - \vec v \times \vec{E}$


title: Should the expression for energy conservation in alpha decay include the mass of electrons?

text: Let $M_P$, $M_d$, $m_\alpha$, $m_e$ and $Q$ the mass of the parent nucleus, daughter nucleus, alpha particle, electron and the disintegration energy, respectively. I understand that applying conservation of energy in alpha decay we get that $$M_{P}c^2=M_{d}c^2+m_{\alpha}c^2+Q \tag{1}$$ where $Q$ is the sum of the kinetic energy of the alpha particle plus the kinetic energy of the daughter nucleus. But reading a book written by my teacher, he says that the energy conservation law carry us to the equation $$M_{P}c^2=M_{d}c^2+m_{\alpha}c^2+2m_{e}c^2+Q \tag{2}$$ My question is, which equation is the correct? And if the answer is (2), how do I derive it?


title: Gravitational Redshift between spacecraft

text: I am trying to determine the frequency redshift when a signal is sent between two spacecraft orbiting a star at different radii. I have not been able to find a formula for this, I can only find formula for a signal emitted at a particular radius from a star and received at infinity. Both spacecraft are well outside the Schwartzchild radius, so I think I only need the Newtonian limit. I _think_ what I need is: Fr = Fs/(1+(Pr-Ps)/c^2) Fr = Frequency at receiver Fs = Frequency at source Pr = Gravitational Potential at receiver Ps = Gravitational Potential at source c = speed of light I determine Gravitational Potential as: P = -GM/r Is it correct to use the difference in Potential in this way? Is the rest of the equation correct? I suspect there may be something wrong because if the receiving ship immediately repeats the signal back to the sending ship, the above formula does **not** result in the final signal having the same frequency as the original transmission, i.e. the red shift in one direction does not cancel the blue shift in the other direction. I would think it should.


Did Hilbert publish general relativity field equation before Einstein?


title: Does a towel that's spread out cool faster than one that isn't?

text: I was thinking about how they say those sails on top of some dinosaurs helped regulate their body temperature. If a dinosaur didn't have that sail, would it _really_ make any difference? If you heated up two towels (large ones) to 50 degrees Celsius, and spread one out, and shaped the other one into a ball, which one would cool faster?


title: Finding final charge distribution on capacitors on grounding

text: I am trying to solve the following problem on capacitors: Here's the circuit > I have to find the total energy loss when _all_ three switches are closed, > and subsequent equilibrium attained. I had very little idea of how to proceed, but this is what I did. Even when the switches are closed, the _sum_ of charges on right plate of $C_1$ and the left plate of $C_2$ $(-30\ mC)$ will remain conserved. Let the final charge appearing on on the right plate of $C_1$ be $q\ mC$, then the charge on the left plate of $C_2$ will be $(-30-q)\ mC$. Now, will the charge on the left plate of $C_1$ and the right plate of $C_2$ be $-q\ mC$ and $(30+q)\ mC$ ? If this is so, then I think I can obtain the value of $q$ using the Voltage Law (and therefore the energy lost). If I am incorrect (which I suspect I am), I hope someone can point me in the right direction.


title: Give a description of Loop Quantum Gravity your grandmother could understand

text: Of course, assuming your grandmother is not a theoretical physicist. I'd like to hear the basics concepts that make LQG tick and the way it relates to the GR. I heard about spin-networks where one assigns Lie groups representations to the edges and intertwining operators to the nodes of the graph but at the moment I have no idea why this concept should be useful (except for a possible similarity with gauge theories and Wilson loops; but I guess this is purely accidental). I also heard that this spin-graph can evolve by means of a spin-foam which, I guess, should be a generalization of a graph to the simplicial complexes but that's where my knowledge ends. I have also read the wikipedia article but I don't find it very enlightening. It gives some motivation for quantizing gravity and lists some problems of LQG but (unless I am blind) it never says what LQG actually is. So, my questions: >   1. Try to give a simple description of fundamentals of Loop Quantum > Gravity. >   2. Give some basic results of the theory. Not necessary physical, I just > want to know what are implications of the fundamentals I ask for in 1. >   3. Why is this theory interesting physically? In particular, what does it > tell us about General Relativity (both about the way it is quantized and the > way it is recovered from LQG). >


title: Is there any metal having the properties of rubber?

text: I have seen tires of vehicles to be made of rubber with air filled tube inside.Why can't we have whole tire to be made of rubber? I hope there would be some use of the air filled tube.Then What is the importance of air filled tube? * * * As known rubber can get charge due to **friction** ,that would be hazardous.So, **is there any metal having the properties(greater coefficient of friction,high elasticity) of rubber?** so that charge produced due to friction would get discharged to the earth.Really it would be help full if any one can suggest such an metal.   ![enter image description here](http://i.stack.imgur.com/TD0xv.jpg)


title: How do I map color to temperature?

text: Following up to Do burning red coals and red-hot iron have the same temperature? and Are Colors Emitted at Specific Temperatures?, how do I determine the temperature of the object by the color of the light it is emitting? E.g., a _picture_ with different temperatures written in different colors would be nice. Note that this would have to take eye sensitivity to different temperatures into account as well as the emission spectrum.


title: How can you calculate air resistances at different speeds?

text: I've read that at 50mph air resistance to an average car is the equivalent of driving through water and at 80mph it's the equivalent of driving through oil. I can't find any references online to back up these figures. Is there a relatively simple way to explain the calculation to verify these figures, or are they incorrect?


title: Angular Momentum of Diatomic Molecules

text: I have run into a problem trying to calculate angular momenta and I wonder if someone can set me straight. I think I know that the lowest order rotational mode should have angular momentum of h, and that was the orbital momentum of the electron in Bohr's original hydrogen atom. The kinetic energy of the electron in its orbit was one Rydberg (~13 eV). For the rest of the calculations below it will be convenient to express this as an equivalent temperature of close to 150 000 K (see Wikipedia for the conversion). I wanted to see if I could understand the rotational modes of diatomic molecules by analogy with the hydrogen atom. So I tried the Hydrogen molecule. Using a bond length of approximately 1.4 Bohr radii (from this table), and mass of 2000 Me (electron masses) for each atom, I got a frequency factor of 2000x compared to the hydrogen atom, which equates to a reduced equivalent temperature of 75 K. This was not bad according to Table 5.1 from this article which gives an equivalent of 85 degrees for the characteristic rotational mode temperature. Similarly, I get reasonable answers for diatomic oxygen and diatomic Chlorine. But I am way off when I try to calculate HCl. My source gives a bond length of around 2.2 Bohr radii; because of the mismatch between H and Cl, I assume almost all the momentum is in the hydrogen atom and the calculation gives me a factor of 10 000 on frequency (it goes inversely as mass*r^2). This gives me an equivalent temperature of 15 K; but the table shows a value of 4K. I seem to be reasonably close on the other diatomic molecules. I wonder why I'm way off on HCl? Any ideas would be appreciated. EDIT: Okay, Fabian's answer is helpful. I probably shouldn't even have mentioned the hydrogen atom; since Fabian gives us the correct formula for the energy of the rigid rotor (first excited state), I can plug in the values for my diatomic molecules and see what happens. Here is what I get, using masses in AMU's and distances in Bohr radii, and letting h=1: ![Energy of Rigid Rotor, first mode](http://i.stack.imgur.com/zihzc.png) Hydrogen molecule: M1 = M2 = 1, d = 1.4, energy = 1.0 approx. Oxygen molecule: M1 = M2 = 16, d = 2.2, energy = 0.025 Chlorine molecule: M1 = M2 = 35, d = 3.8, energy = 0.0038 Taken in proportion, these numbers line up pretty well with the equivalent temperatures given in the table: ![Equivalent Temperatures of Rotational Energies](http://i.stack.imgur.com/VSI3I.png) But when I apply the formula to HCl, I'm way off: Hydrogen Chloride: M1=1, M2=35, d=2.2, energy = 0.2 Extrapolating from the other molecules, this would give me an equivalent temperature of 15 to 20 degrees K; but the table gives a value of 4.2 K. I wonder why the formula seems to give consistent values for the H2, O2, and Cl2, but is way out of whack for HCl?


title: Proof of the time-independent Schrödinger equation

text: I have a question regarding the proof of the time-independent Schrödinger equation. So if we have a time-Independent Hamiltonian, we can solve the Schrödinger equation by adopting separation of variables method: we write our general solution as $ \psi(r,t) = \psi(r)*f(t)$ and we get to the two equation : one for the f(t) and one for the $\psi(r)$. $$ f(t)=e^{-\frac{i}{\hbar}Ht} $$ and $$ H\psi(r) = E\psi(r) $$ Books in general refer to the second equation as the TISE and it is seen as an eigenvalue problem for the hamiltonian, in order to find the stationary states. Now what i don't understand is why we see that equation as an eigenvalue problem for the hamiltonian, since we have a wave-function $\psi(x)$ which is supposed to be eigenstate for H. So if $\psi(x)$ is eigenstate for H, it means that H is diagonal in coordinate basis, but i know it's not true since H has the term $\frac{P^2}{2m}$ in it, which is not diagonal in coordinate basis. Where am i wrong?? Thank you very much


title: Solving the Young-Laplace equation for arbitrary axisymmetric geometry

text: Say I have a non-ellipsoidal soap bubble and I want to numerically analyse the pressure in the inner lobe of this bubble here: ![](http://i.stack.imgur.com/d12UL.jpg) The Young Laplace equation gives the pressure difference across a fluid interface as a function of the curvatures. I have a set of points in 2D space (axisymmetry is assumed) for the inner lobe. How can I obtain the net force that acts over the entire inner lobe surface due to the Young-Laplace pressure gradient? This would be easy if the lobe itself was approximately ellipsoidal - then there are only two principal radii of curvature, and the pressure gradient follows from there. But what if I had a more complex shape for the inner lobe that wasn't ellipsoidal? Do I try to break the shape into many ellipses, however improbable that sounds?


title: Is the thermal time constant simply the heat capacity divided by thermal conductivity?

text: My professor gave me this temperature decay data to make an exponential fit and told me that the time constant tau should be heat capacity over thermal conductivity. I'm looking at the wikipedia page and that's not what I'm seeing. I don't know if there's some caveat to when we're doing 2D simulations but I just don't see how the time constant could be exactly heat capacity over thermal conductivity.


title: How is Laughlin's gauge argument explaining integer quantum hall effect(IQHE)?

text: It seems essential in Laughlin's gauge argument that the sample has to be cylindrical(or with similar toplogy), so that we can "thread" a thin solenoid through to control the gauge function on the surface of the cylinder. However, most IQHE experiments are done on flat samples, so how is Laughlin's argument really explaining IQHE? Certainly I can imagine some experiments being done on cylindrical samples, but the confusion comes from the fact that, in various sources that I've read, after an exposition of flat-sample IQHE experiment(and maybe a calculation of Landau levels), they would pose Laughlin's argument immediately as an explanation, so I guess somehow it does work for flat samples but there is something simple I missed?


title: Canonical partition sum for two fermions in harmonic potential

text: In an old exam, I found the following problem: > # Two Particles in a potential well > > We look at a onedimensional harmonic potential well that hold two spinless > particles that do not interact with each other. Each particle has the energy > eigenvalues $E_n = \hbar\omega(n + 1/2)$ with $n = 0, 1, \ldots$. > >   1. The particles are distinguishable. Show that the canonical partition > sum $Z_\text C$ factorizes, then calculate $Z_\text C$. > >   2. The two particles are now indistinguishable fermions. Calculate the > canonical partition sum $Z_\text C$ for this case. > > I got the first one done, that was pretty straightforward: ![My derivation](http://chaos.stw- bonn.de/users/mu/uploads/2014-02-12/Bildschirmfoto3.png) For the second part, I am not sure what my states are now. Since the particles are fermions, they will have Spin $s = 1/2$. That means that the wavefunction describing both particles will have to be totally antisymmetric regarding particle exchange. If they both occupy the same $n$ and their spins are opposing, this should be fine. What other states are allowed? If the spins are in the same direction, $n_1 \neq n_2$ for sure. Since the spin part of the wavefunction is symmetric, the other part has to be antisymmetric. The only other part that I could think of would be the spatial part of the wavefunction. If $n_1$ and $n_2$ are odd and even, respectively, each spatial wavefunction would change its sign when the particles are exchanged. However, since $n_1$ and $n_2$ are different, the spatial wavefunctions would look different as well, having a different amount of zeros (i. e. $|\psi|^2 = 0$). Which states are allowed?


title: Confusion with derivation of fictitious forces

text: I read the mathematical derivation of fictitious forces on Wikipedia and I'm having trouble understanding it. I went on a few other sites looking for a better derivation, but they're all basically the same. I've also seen this answer already. What I'm having trouble understanding is that, in every derivation, at some point, the expression $\sum_{i=1}^3 \hat{u}_i x_i$ (where $\hat{u}_i$ represents a unit vector of the non-inertial frame in the inertial frame, and $x_i$ is a coordinate of the particle in the non-inertial frame) is equated with "$\vec{x}_A$" or something similar (the position of the particle in the non-inertial frame). Likewise $\sum_{i=1}^3 \hat{u}_i \frac{dx_i}{dt}$ is equated with $\vec{v}_A$ and $\sum_{i=1}^3 \hat{u}_i \frac{d^2x_i}{dt^2}$ with $\vec{a}_A$. This doesn't make any sense to me. The position vector in the accelerating frame is just $(x_1, x_2, x_3)$ since $x_1, x_2, x_3$ were _defined_ to be the coordinates of the particle in the accelerating frame. It is _not_ $\sum_{i=1}^3 \hat{u}_i x_i$; that gives you the position vector in the _inertial_ frame. For example here's what Wikipedia says: > The interpretation of this equation [$\mathbf{x}_{\mathrm{B}} = \sum_{i=1}^3 > x_j \mathbf{u}_j$] is that $\mathbf{x}_{\mathrm{B}}$ is the vector > displacement of the particle as expressed in terms of the coordinates in > frame B at time _t_. Even though the derivation clearly gives the correct answers, I simply can't understand this step.


title: Physical interpretation of the change of diffusion term in navier stokes equations

text: In the Navier-Stokes Equations, there is one term accounting for convective flow and one term for diffusive flow. At high flow rates, the diffusive term becomes much smaller compared to convective term, therefore can be neglected, leading to Euler's equations. I can understand this in terms of mathematics: There is a multiplicative factor $\frac{1}{\text{Re}}$ in the diffusive term. With increasing flow rates, the inertial forces become larger than the viscous forces, so the Reynolds number increases and the term for diffusive flow becomes much smaller. But what is the physical interpretation of this? Does the influence of random motions decrease when the flow rate increases?


title: What can be known about the formulas for energy only from the fact that it is conserved?

text: The question is to figure out how the energy can be derived knowing just one thing: > **There is a quantity called Energy that is conserved over time.** The goal is to get an equation that somehow implies the basic formulas for kinetic energy and potential energy, $\frac{1}{2}mv^2$ and $mgh$. So how can we get this with just a rudimentary knowledge of physics and algebra, and the first principle (stated above)?


title: To which state of matter does the flame belong to?

text: I had this question from the day(9 years old, now 16) that i learned about states of matter. I have asked so many of my teachers some of them told me gas some as plasma etc. can anyone answer my question? Recently I've learnt that plasma state is obtained when all the electrons from the atom are removed. Obviously its present in sun due to high temperature.


title: Basic question about the S-Matrix, Unitarity and Effective Field Theory

text: Consider scattering some particles in a state collectively denoted by $i$ to a final state denote by $f$. The scattering amplitude, S-matrix is then defined by: $S_{fi}\equiv \langle f|e^{-iHt}|i\rangle$. We then separate the S-matrix into the identity and another part as $S_{fi}=\delta_{fi}+iT_{fi}$. The statement of unitarity is that $S^\dagger S=1$ which implies that $2{\rm Im}T=T^\dagger T$ which leads to the optical theorem and all that. In field theory, what we calculate is the amplitude where we stick just $T$ between two states. That is, we only usually calculate the amplitudes where something interesting is happening. In the study of effective field theories, I often see statements about the violation of unitarity which confuse me. For example, if we took a simple scalar field theory with a derivative interaction $\mathcal L=\frac{1}{2}(\partial\phi)^2+\lambda(\partial\phi)^4/\Lambda^4$ then we could calculate $2\to 2$ scattering and we'd find something like $\mathcal{M}_{2\to 2}\sim \lambda k^4/\Lambda^4$. I've read and heard people say that for $k\gg \Lambda$, this leads to a violation of unitarity. I assume this means a violation of $2{\rm Im}T=T^\dagger T$. Why is this the case? Certainly the perturbative expansion breaks down in this regime but why is this connected to unitarity? If the above example does violate unitarity, then what's the difference between the above and the case of a normal, non-derivative $\lambda\phi^4$ interaction and with $\lambda\gg 1$ the case above? The key thing seems to be that $\mathcal{M}$ gets really big in the derivative example, but this would also occur in $\lambda\phi^4$ and I would doubt that this latter theory has any violations of unitarity.


title: Does pressure drop across pipe affect flow rate?

text: Let's say there is a main pipe containing $100\: \mathrm{m^3/hr}$ a fluid of density $750\: \mathrm{kg/m^3}$ and it's gonna be branched into 2 pipes (Pipe **A** and Pipe **B** ) of the same diameter. If Pipe **A** is much shorter than Pipe **B** , then the pressure drop across Pipe **B** will definitely be higher right? Will this condition affect the volumetric flow rate of the fluid in Pipes **A** and **B**? If it doesn't, will the volumetric flow rate of both pipes be the same, that is $50\: \mathrm{m^3/hr}$?


title: Is the sign in the Schrodinger equation physical?

text: I always have trouble remembering the sign in factors like $\exp(\pm ik\cdot x)$ (I'll use mostly minus signature here) that arise in field theory. My mnemonic is to remember that the Schrodinger equation is $i\partial_t|\Psi\rangle=H|\Psi\rangle$ and so the wavefunction goes something like $|\Psi\rangle\sim \exp(-i\omega t)$ where $\omega$ is the energy of the state and then I just remember that this is the same sign assigned to incoming particles in Feynman diagrams. Similarly, this is the same sign that comes with the annihilation operator when we expand out a field in terms of creation and annihilation operators. The sign in $|\Psi\rangle\sim \exp(-i\omega t)$ is then just determined by the sign appearing in the Schrodinger equation above. My question is whether the sign appearing in the Schrodinger equation is physical? Relatedly, are the signs appearing in Feynman diagrams or more generally the overall sign in the path integral physical? It would seem to me that all calculations would remain the same if we switched signs everywhere since physical quantities just depend on complex inner products and this would appear to be insensitive to flipping the signs everywhere..


title: Cosmology with a negative cosmological constant

text: Based on the Friedmann equation for a universe with only cosmological constant, $$\left(\frac{\dot{a}}{a}\right)^2 \sim \Lambda$$ I would expect the scale factor $a(t) \sim e^{-it}$ if $\Lambda < 0$. Is this an anti de Sitter universe? How do we interpret the scale factor?


title: Lenz's Law and Eddy Currents

text: You can determine the direction of eddy currents according to Lenz's law. E.g. If a metal sheet is losing flux into the page, it will experience induced eddy currents in a clockwise direction to replace it. However, this doesn't make sense to me logically. :S Consider a straight current-carrying wire. It has equal flux on either side of it's length. Now warp it into a circle (to simulate eddy currents). It should have equal flux on both outside and inside the the circle. Thus, in the above example, the eddy currents ARE NOT replacing the lost flux into the page, as it is also producing EQUAL flux out of the page (which "cancels" the other flux into the page). Am I misunderstanding something about Lenz's Law/Eddy currents? :S


title: What is the Sun's core made of?

text: The obvious answer is hydrogen and helium plasma but the nuclear fusion can also create heavier elements. Are these heavier elements a significant portion of the core? Do the heavier elements "sink" to the "bottom" of the core, like iron has during planetary formation? Presumably, during the Sun's formation it would have accreted heavy elements made by previous generations of stars - does this just get added to the mix?


title: If the current is increased, is there more charge flowing or is it moving quicker?

text: **Problem** Current is the amount of charge that is flowing through a component per unit of time. For a given voltage, Ohm's law tells us that if we increase the resistance, then the current must decrease. But _what's actually happening to decrease the current?_ **My reasoning so far - is it correct?** More resistance (if we're talking about something of the same size for simplicity) is more 'stuff' in the way (higher resistivity), so more collisions. More collisions means it takes longer for the charge to 'get through' the component. The charge is moving slower and so the current is lower. Or, is it that the speed is always the same and that somehow if we have more resistance then it means there is just less charge able to flow, hence a lower current. If so, why?


title: A few questions about interacting quantum field?

text: In interacting quantum field, we think that interaction is adiabatic switch on/off. So in the infinite past, we can think there is no interaction, so we can have particle interpretion. There are four questions: 1.$|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$ This state is in which picture? I think it is in Heisenberg picture. 2.Is $|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$ the eigenstate of the full Hamiltonian? 3.$|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$ This state has the definite energy and momentum and I think it is the eigenstate of the full Hamiltonian. Now that it is the eigenstate of the full Hamiltonian, why it can evolve to a different state ? According to Quantum mechanics, if a state $|n>$ is the eigenstate of the Hamiltonian, then the evolution of it is $e^{iE_nt}|n>$. 4.Another question is that according to Adiabatic Thoerm, if in the infinite past the state is in $|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$, then in the infinite furture the state is also $|p_1p_2>_{out}$ because of the adiabatic switching on/off the interaction. So $|p_1p_2>_{out} = U(\infty,-\infty)|p_1p_2>_{in}$ Apparently, there are huge faults in my arguments. So where is my fault? Thanks!


title: How to derive the commutation relationship between $\hat{L}^2$ and $\hat{\textbf{p}}$

text: How to prove that $$[\hat{L}^2,\hat{\textbf{p}}] = i\hbar(\hat{\textbf{p}}\times\hat{\textbf{L}} - \hat{\textbf{L}} \times \hat{\textbf{p}})$$ I tried to expand $\hat{L}^2$: $$[\hat{L}^2,\hat{\textbf{p}}] = [\hat{L}^2_x+\hat{L}^2_y+\hat{L}^2_z,\hat{\textbf{p}}]$$ And \begin{align*} [\hat{L}^2_x,\hat{\textbf{p}}] &= [\hat{L}_x\hat{L}_x,\hat{\textbf{p}}] \\\ &= \hat{L}_x[\hat{L}_x,\hat{\textbf{p}}] + [\hat{L}_x,\hat{\textbf{p}}]\hat{L}_x \\\ \end{align*} With $[\hat{L}_x,\hat{p}_y]=[\hat{p}_x,\hat{L}_y]=i\hbar \hat{p}_z$ and $[\hat{L}_x,\hat{p}_x]=0$, $$ [\hat{L}_x,\hat{\textbf{p}}] = i\hbar(\hat{p}_z\vec{e}_y-\hat{p}_y\vec{e}_z) $$ So the LHS of the equation in the question is simply $\textbf{0}$, which is wrong. Where did I make a mistake?


title: A pearl that moves in a smooth vertical hoop

text: I wanted to ask about the situation of a pearl that moves in a smooth vertical hoop in circular motion as described in the following sketch. ![enter image description here](http://i.stack.imgur.com/XleCl.png) According to a simulation found in the internet , a moment after this situation , when the pearl reached the point C , the pearl keeps moving in circular motion. What causes the pearl to keep moving the in circular motion after reaching point C? (The reference frame is Inertial frame) We already know that a(r) = V ^ 2 / Radius , at point C , v = 0 , therefore a(r) = 0. We also know that a(r) is the only acceleration of the pearl at point C. So what makes the pearl keep moving in circular motion ? From the equations we can conclude that the pearl should stop moving at that point but according to the simulation , that's not true.


title: Polarization of sound

text: Sound can't be polarized because the vibration of such type can't be polarized i.e, it can't be limited or controlled by any barriers and so polarization is not possible in them. This is what my teacher answered me when i asked the question. But i didn't understand what did he mean by "the vibration can't be controlled or limited." Does the word cant be limited or controlled make sense here? Moreover can anybody explain in details and more clearly to me?


title: Rolling sound of thunder

text: > _When sound of thunder travels to the listener from its location,different > parts of the same wave have to travel through air of different moisture > levels,densities and molar masses._ This is what I found when i wanted to know about the reason for the rolling sound(the sound which can be heard many times and for longer times) of thunder.But as I read it i found it unclear and failed to visualize the overall process when sound travels thereby giving the rolling sound of thunder.I also failed to understand how there will be different moisture levels in the air or what it meant! I also went through Extended sound of thunder this question but it didnt serve to answer my question and made me understandd the overall process.


title: Atmospheric heating and the reduction in viscosity

text: The oceans are becoming less viscous as they are heated. I'd imagine a similar effect is likely occurring in the atmosphere as well. What, if any, effect would this reduction of viscosity have on aeroplanes? Would they be more efficient and use less fuel?


title: Damped oscilator - logarithmic decrement of damping

text: Could you please tell me, where is the mistake? What is the **logarithmic decrement of damping** $Λ$ of damped harmonic oscillator, if its **mechanical energy decreases to the 50%** of its initial value **during first 10 seconds**? The **period of oscillations is T=2s**. [Result: 0.0693]. Formula from wikipedia: $$ Λ = \frac{1}{n} \cdot ln\left( \frac{x(t)}{x(t + nT)} \right) $$ My solution:   Since I dont have amplitude and mass of particle, I have to work with ratio $ \frac{1}{\frac{1}{2}} $. Since T=2s and elapsed time is 10seconds, $ n = 5 $. That gives me: $$ Λ = \frac{1}{\frac{t}{T}} \cdot ln\left( \frac{1}{\frac{1}{2}} \right) $$ $$ Λ = \frac{1}{5} \cdot ln\left( \frac{1}{\frac{1}{2}} \right) $$ $$ Λ = 0.1386 $$ Which is twice as much as was expected as result. So where is a mistake?   It's clear that it should be: $$ Λ = \frac{1}{10} \cdot ln\left( \frac{1}{\frac{1}{2}} \right) $$ but why?


title: General procedure for solving fluid flow problems

text: Could someone help me devise a short series of steps for solving an arbitrary fluid flow problem? Often the most difficult part of these problems is just figuring out what path to take in solving them, and I am trying to coalesce in my mind the best general procedure in preparation for my final exam. To give you an objective example of what I mean: to analogize with classical physics, the general procedure is to   1. Identify the forces (body forces, contact forces, normal forces)   2. Break the system into dimensional components   3. Solve the resulting system of equations. This works for "most" systems. I'm looking for a similar fluid flow procedure. This is my best attempt:   1. Write down the Navier Stokes equations for either cartesian, cylinder, or spherical coordinates. This gives you a momentum balance.   2. Cross out all terms based on assumptions (e.g. cross out viscosity if flow is inviscid. Cross out angular terms if the flow is symmetric within a cylinder, etc.)   3. Write down the continuity equation. This gives you a mass balance, so now you have mass and momentum taken care of.   4. Write down a constitutive equation relating any further unknowns to the properties of the material. Any suggestions for modification or elaboration? I haven't included anything in there about boundary layers or thermal boundary layers or turbulent/laminar flow, etc. What would be the best spot to put these considerations?


title: When is the force null between parallel conducting wires?

text: Consider two long wire with negligible resistance closed at one end of the resistance R (say a light bulb), and the other end connected to a battery (DC voltage). Cross-sectional radius of each wire is in $x = 100$ times smaller than the distance between the axes of the wires. Question: At which value of resistance R (in ohms) the resultant force of the interaction of parallel conductors disappears? ![alt text](http://i.stack.imgur.com/Vxja0.png)


title: Reorienting a sensor axes according to particle displacement directions

text: Consider a sensor which is located inside the solid substance. This sensor is capable of detecting the substance oscillations along each of the three axes (usually orthogonal, but generally, any non-degenerated (non-coplanar) basis). This sensor produces a recording of the detected oscillations, called a trace, containing the displacement sampled at high-frequency, capable to capture any oscillation frequency existing in the substance. There are three traces, one for each axis. Consider that at some period of time the sensor registers an interference of an event, consisting of the compression wave, and two shear waves. Traces now contain a recordings of the event, effectively, a projections of the oscillations on the sensor's axes. How do I now virtually re-orient the axes, that is, perform the linear transform of the traces, so that they will be oriented each along a corresponding wave displacement vector, and thus will contain only single wave event recording in each of the traces after the transformation? Or should I ask this on `math` ?


title: Why is honeycomb lattice the ideal shape for solar cells?

text: Quoting this web site (which I haven't thoroughly verified for credibility yet): > The material is graphene, also known as graphite, a naturally-occurring > mineral that forms a one-atom-thick sheet of honeycomb crystal lattice > material (the ideal shape for solar cells) that has surprisingly robust > electric characteristics. In what context is this ideal shape? In the nano or macro scale? Why?


title: Observing a growing black hole

text: Suppose you are at rest relative to a black hole (so you are maintaining some constant acceleration in order to oppose that gravitational attraction), outside the event horizon. Now a continuous stream of energy starts falling into the black hole. This will cause the event horizon to grow and the gravitational potential of the black hole to increase, meaning that you would begin to fall toward the event horizon (because your acceleration is no longer sufficient to oppose the gravitational potential). Now suppose that rather than the black hole growing, it remains at a constant size (nothing is falling in), and the observer's acceleration (which is opposing the gravitational force) decreases (his rocket is losing power, but he is unaware and unable to measure this power - I'm thinking along the lines of Einstein's equivalence principle). Again he begins to fall toward the black hole which will seem to him to be increasing in size. Is there any way for the observer to tell the difference between these two cases (i.e. is there a way for the observer to know that the decrease in distance between him and the black hole is due to an increase of mass in the black hole or a decrease in his own acceleration)?


title: Counter rotating propellers: better in same column or staggered?

text: In an octo-copter design, is it better to put four pairs of counter-rotating props over each other so that each pair works in a column together? Or will you get more power or efficiency from staggering all eight props? It seems it would be more convenient to position them in pairs with a CW and CCW prop on the same axis, but I am wondering if this reduces the efficiency of the thrust to have them producing a single column of forced air rather than two separate columns. What are the physics involved?


title: How could I determine the temperature at which an egg explodes

text: A while back, some friends and I pondered "what will happen to an egg if placed very near a campfire". So we placed an egg on a grill, approximately 12 inches above a real campfire. Something like 7-10 minutes later, thankfully right as we walked away, the egg exploded in every direction in a gooey, golden glory. So I ask you, **how can I determine the internal temperature of an egg that will cause it to explode**?


title: How the gravitational constant $G$ is determined?

text: We know the gravitational constant is $G=6.67545 \times 10^{-11} m^3 kg^{-1} s^{-2}$. But how this value is determined at first?


title: Did a racer on a "restless" course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)

text: Consider two participants, a starting gate ($A$) and a finish line ($B$), in a (sufficiently) flat region, both undergoing uniformly accelerated motion, such that   * $A$ finds constant ping duration wrt. $B$: for any two (distinct) indications $A_J$ and $A_K$   $\mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }B}}} \\! \\! \\! \\! \tau_A[ A_J ] = \\! \\! \mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }B}}} \\! \\! \\! \\! \tau_A[ A_K ] = \\! \\! ({\small{\text{abbreviated as}}}) \\! \\! = \mathop{\Delta}\limits_{\small ABA} \\! \tau_A,$   * $B$ finds constant ping duration wrt. $A$: for any two (distinct) indications $B_P$ and $B_Q$   $\mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }A}}} \\! \\! \\! \\! \tau_B[ B_P ] = \\! \\! \mathop{\Delta}\limits_{\stackrel{\large{\text{until seeing}}}{\small{\text{the echo from }A}}} \\! \\! \\! \\! \tau_B[ B_Q ] = \\! \\! ({\small{\text{abbreviated as}}}) \\! \\! = \mathop{\Delta}\limits_{\small BAB} \\! \tau_B,$ and   * $A$'s ping duration wrt. $B$ is strictly less than $B$'s ping duration wrt. $A$: $\mathop{\Delta}\limits_{\small ABA} \\! \tau_A < \mathop{\Delta}\limits_{\small BAB} \\! \tau_B.$  (Evidently, therefore, $A$ and $B$ had not been at rest to each other; but they are rigid to each other.) Let there also be a racer ($R$) who passed the starting gate $A$ and (afterwards) passed the finish line $B$. For definiteness, suppose that $B$'s duration from having seen that $A$ and $R$ had passed each other until having been passed by $R$ equals $B$'s ping duration wrt. $A$.   (Consequently, $A$'s duration from having been passed by $R$ until having seen that $B$ and $R$ had passed each other equals twice $A$'s ping duration wrt. $B$.) **Question:** Can an "average velocity" of racer $R$ with respect to the race course, as far as it is specified by starting gate $A$ and finish line $B$ be defined, such that its magnitude could be expressed in terms of the given ping durations $\mathop{\Delta}\limits_{\small ABA} \\! \tau_A$, $\mathop{\Delta}\limits_{\small BAB} \\! \tau_B$, and the "speed of light, $c$"   (and where its direction is obviously "from $A$ to $B$") ?


title: Relationship between diffused light and intensity

text: I was wondering if the relationship of the intensity of diffused light is a linear correlation to the diffusing material. Basically will the intensity of the diffused light change in a linear fashion as the diffusing factor increases/decreases?


title: Non-abelian behavior of vortices in p-wave superconductors

text: I am trying to understand why vortices in p-wave superconductors are actually non-abelian anyons and how this relates to Majorana modes. However I am having a hard time finding proper resources (in particular since my background in superconductivity is not too strong)? These lecture notes that I found: http://www.phys.s.u-tokyo.ac.jp/curriculum/documents/2013lecture6_ver2.pdf are great in spirit but they lack some detail. Can someone provide me with recommendations on what to read and where to start?


title: If an atom is positively ionized, can is gain electrons if you emit photons at it?

text: I read somewhere that electrons and light are just electromagnetic radiation and are basically the same thing, does this mean that if you emit photons at an atom it will gain electrons?


title: Measuring the spin of a particle from a singlet state

text: Imagine that I have a singlet state: $|s\rangle = {1 \over \sqrt2}(| \uparrow_1\downarrow_2 \rangle - |\downarrow_1\uparrow_2\rangle)$ I want to measure the spin along the z axis of the first particle. So I should apply the operator $S_{z1}$ to the singlet state: $S_{z1}|s\rangle$ = ? The possible answer is 50% $\uparrow_1$ and 50% $\downarrow_1$. How should I write in the bra-ket notation the result of this measurement? Thank you!


title: Does the universal expansion affect the gravitational constant?

text: If the universe was not expanding, would gravity be stronger? since the expansion happens everywhere and can overcome gravity at long distances? If not, then are there any changes that would happen to the universe if there was no expansion?


title: Heat transfer and temperature difference of 2 liquids

text: We know that heat transfer is calculated by equation $Q$=$m$×$C_p$×$\Delta$$T$. Let's perform 2 experiments- Imagine I have 2 cups with water with same masses(volume). 1 is at 100 degree C and other at .5 degree C. If we pour these 2 cups of water in another bigger cup.how fast it's temperature reaches equilibrium. Same test I will perform with temperature of second cup at 30 degrees. Equation says heat transfer will be higher if temperature difference is high. That means 1st experiment will reach equilibrium faster than second. Is it right? I am completely confused. It really takes less time to cool a cup of tea at 70 degree C than 100 degree C.


title: Graded Index fiber

text: In the following picture of a graded index fiber two different modes are depicted. One is a higher mode which is propagating towards the core-cladding interface and one is through the higher refractive index region. What will happen at the core-cladding interface with the higher mode is there any small amount of evanescent field which will pass the core cladding interface or all of it will reflect back ? Second in case of multiple modes at what point they will intersect with each other ? in the case of single mode fiber there will be no ray directed to core cladding interface but still we get evanescent field what are the origin of this in single mode fiber ? ![enter image description here](http://i.stack.imgur.com/fbEWh.gif) ![enter image description here](http://i.stack.imgur.com/KYLBG.gif)


title: Unitary transformation between complete + orthonormal bases

text: Suppose the complete orthonormal bases $\\{|\psi_n\rangle\\}$ and $\\{|\psi{'}_n\rangle\\}$ are related by the transformation matrix $U$: $$ |\psi{'}_n\rangle = U|\psi_n\rangle \\\ \langle\psi{'}_n| = \langle\psi_n|U^{\dagger} $$ We want to prove that $U$ is a unitary operator, i.e. it satisfies the relationship $U^{\dagger}U = I$. By operating the LHS of the second on the LHS of the first, and similarly for the RHS: $$ \langle\psi{'}_n|\psi{'}_m\rangle = \langle\psi_n|U^{\dagger}U|\psi_m\rangle $$ Due to orthonormality $$ \langle\psi{'}_n|\psi{'}_m\rangle = \delta_{nm} = \langle\psi_n|\psi_m\rangle $$ Hence $$ \langle\psi_n|U^{\dagger}U|\psi_m\rangle = \langle\psi_n|\psi_m\rangle $$ My question is that is there any other relationship _other_ than $U^{\dagger}U = I$ that satisfies this equation? i.e. is this a sufficient proof that $U$ is unitary?


title: Is Kerr effect in glass observable?

text: If I apply high electric field to a glass piece can I observe Kerr effect to some extent or will it be too small for glass to observe with eyes? I want to perform simple experiment without any costly material e.g non-linear crystals .


title: Classical point particles to classical fields

text: I often hear that in the continuum limit we can study large numbers of particles as fields. I always imagined that by removing all bounds on the number of particles (while keeping total energy, momentum, charge etc. conserved), we can generate a field. Of course I have no idea how this is formally accomplished or whether it is even meaningful to do such a thing. Without appealing to any sort of quantum theory, how can one show that in this 'continuum limit' a large collection of indistinguishable classical point particles gives rise to a classical field? If not, why? I think this question might have been repeated, but all related questions I found involved taking limits of quantum systems.