title: Elliptical orbits passing 2 points

text: I would like to find **set of possible elliptical orbits which pass 2 points in plane**. I was searching for some solutions in orbital mechanics texts but I didn't found any. There are several possible approaches but I'm not sure which is the best - both looks quite difficult to solve algebraically.   1. using **polar equation relative to focus** with $(R_1,\phi_1),(R_2,\phi_2)$, being coordinates of points $$ R_1 = \frac{a(1-e^2)}{1-ecos(\phi_1-\theta)} $$ $$ R_2 = \frac{a(1-e^2)}{1-ecos(\phi_2-\theta)} $$ then for given $\theta$ solve for semimajor axis $a$ and eccentricity $e$   2. using deffinition of elipse as a **set of points of the same distance from both foci**. Given 2 points of cartesian coordinates $(x_1,y_1),(x_2,y_2)$ and one focus in origin $(0,0)$. For each given distance parameter $L$ solve for coordinates of second focus $(x_f,y_f)$, $$ L = \sqrt{x_1^2 + y_2^2} + \sqrt{(x_1 - x_f)^2 + (y_1 - y_f)^2} $$ $$ L = \sqrt{x_2^2 + y_2^2} + \sqrt{(x_2 - x_f)^2 + (y_2 - y_f)^2} $$   3. I can also first rotate the coordinate system (or my input points) by given angle (which is my arbitrary parameter) and then use some simplified equation of **ellipse which has major axis parallel to x-axis** which has just 2 degrees of freedom. But even after this rotation I don't see much simplification of algebraic solution.  * * * Nevertheless, the resulting equations are difficult to solve.I solved it using sympy, but the solution is very long expressing hard to simplyfy. I would like some more elegant solution if there is any. I would also like to implement this into computer as a part of orbital transfer optimization, so I would prefer some explicit expression which is fast to evaluate numerically ( for example goniometric functions are quite slow to evaluate )


title: Do primordial background neutrinos orbit in dark matter halos?

text: According to Wikipedia, neutrinos separated from other matter seconds after the Big Bang and formed a separate background radiation field which now fills space at a temperature ~2 K. Supposing neutrinos have a rest mass of 2 eV/c2, they should be traveling at an average speed $$v_\nu = c \sqrt{2 \cdot k \cdot \frac{2\mathrm{K}}{2 \mathrm{eV}}} \approx 4 \cdot 10^6 \mathrm{m/s}$$ Seems to me that as neutrinos slow down due to the metric expansion of space, they should be clustering into orbits around galaxies. Distant galaxies should show the effect less because the neutrinos were going faster at the time. Is this correct? According to the same Wikipedia article, the background neutrinos originally had an energy of 2.5 MeV and were in thermal equilibrium with electrons, positrons, and photons. So whereas they once accounted for a quarter of the mass-energy of the Universe (or half, because there are three flavors?), their current, essentially rest mass is less than a millionth of it. That's too little to put a dent in the dark matter mystery, or even observe the time-dependent effect I just mentioned, right? Also, galactic orbits aren't metrically expanding so this essentially traps the neutrinos around any galaxy into a spherical shell with the orbital radius and speed dependent on the galaxy mass. A neutrino which is going too fast will escape and pass a galaxy, but by the time it reaches the next galaxy, it will be further redshifted, and might be gravitationally trapped. Then it won't slow down any further because the neighborhood of a galaxy is geometrically stable. If there were something larger for neutrinos to orbit, they could stabilize at a higher speed, but it still couldn't be relativistic, right? Since nothing (besides lots of empty space) can slow down a neutrino, this sets a lower speed limit on all neutrinos in a galaxy or galaxy cluster, right?


title: Can we determine whether or not a particle is entangled?

text: Suppose Shaniqua and Tyrone have four pairs, a, b, c, and d, of entangled particles. They take their particles and go very far apart. If Tyrone can determine whether or not a particle is still entangled, Shaniqua could observe, for example, a and c, transmitting the binary number 1010 faster than light. So, can we determine whether or not a particle is entangled? If so, why doesn't it lead to faster-than-light information transfer?


title: How do I calculate speed of air on each side of airfoil?

text: I am doing a project on Bernoulli's principal for my High School physics course. I made an airfoil in accordance with NACA 2412 (same as a C-152). I am going to put it in a wind tunnel where the wind speed is going to be measured. How can I find the wind speed on top of and on the bottom of the airfoil, once I know what the outside wind speed (or velocity of the airfoil) is? My end goal is to find the pressure differences on the outside of the wing, and I can do that very easily once I know the speed on each side of the airfoil.


title: Is the momentum operator well-defined in the basis of standing waves?

text: Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely $$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot \sin \left(\frac{n\pi x}{L}\right)$$ In this basis, the operate $\hat{p}^2$ is diagonal by construction, so every state has a definite energy. But suppose I want to construct this operator from the matrix elements of $\hat{p}$? These are ($\hbar=1$) $$\left<m|\hat{p}|n \right>=-\frac{2imn((-1)^{m+n}-1)}{L\cdot(m^2-n^2)}$$ which vanishes for $m$ and $n$ with the same evenness, making this an off-diagonal "checkered" matrix. I tried taking a finite portion of this matrix and squaring it in order to obtain $\hat{p}^2$, but from what I could tell, this yields an on-diagonal checkered matrix, contrary to the construction that it would be strictly diagonal. I assume this has something to do with the fact that the wavefunctions obtained by applying $\hat{p}$ do not satisfy the same boundary conditions as the basis functions (do not vanish). Would this problem be solved I had taken the the infinite matrix in its entirety before squaring? Or is this entire problem (decomposing $\hat{p}$ in this basis) ill-defined?


title: Deriving combined gas law from Boyle's and Charles' laws

text: I know that the combined gas law, $$\frac{PV}{T}=k$$ should be derivable from Boyle's Law and Charles' Law. Since these are very basic equations, I presumed that it would be a simple matter, so I tried it myself. Charles' Law is $$\frac{V}{T}=k_1$$ and Boyle's Law is $$PV=k_2$$The subscripts are arbitrary. In the derivation on Wikipedia, they jump from this to $$PV=k_2T$$ I'm sure I'm just overlooking something silly, but I see no way of combining Charles' and Boyle's to achieve an equation in which we don't cancel at least one of $P$, $V$, or $T$. What am I missing? Thanks.


title: Why is there a line in the middle of the Planck full sky map?

text: Why there is a straight line (or perhaps a flat surface) in the middle of the Planck full sky map? ![Planck-FSM](http://sci.esa.int/science-e- media/img/e5/PLANCK_FSM_03_all_views.gif) and zoom ![Zoom surface collision of big bang](http://handshakemag.com/wp- content/uploads/2010/07/PLANCK_FSM_03_Black.jpg) (it makes it seem that the big bang was a collision between two unknown surfaces)


title: What is the physics behind a soap bubble?

text: A soap bubble is an extremely thin film of soapy water enclosing air that forms a hollow sphere with an iridescent surface. What fluid dynamical process occurs during the popping of a soap bubble?


title: What's the proper way to approximate the position uncertainty of a particle?

text: In this problem: ![image of the problem](http://i.imgur.com/oiF8i.jpg) shouldn't $\Delta x\sim\lambda/\sin\theta$ be $$\Delta x\sim \frac{\lambda}{\sin\theta} - \left(\frac{-\lambda}{\sin\theta}\right) = 2\frac{\lambda}{\sin\theta}$$ instead such that the final answer is $\Delta x \Delta p_x \sim 8\pi\hbar$?


title: Is 3+1 spacetime as privileged as is claimed?

text: I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia. For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two. Am I being overly skeptical in thinking that while the force of _gravity_ may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension? Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self- aware structures form through completely different processes. While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?


title: What is the critical mass of a planet to have an atmosphere like Earth's?

text: Small planets/orbits like Moon cannot have atmosphere because of their masses. They don't have enough gravity to hold an atmosphere. Then what is the critical mass that makes enough gravity to keep an atmosphere? Please explain by Mathematical equations with the data given below:   Density of the planet is uniform everywhere; 5 g/cm3.   The atmosphere consists of pure O2 gas.   Temperature of the the atmosphere is uniform everywhere and fixed to 300K.   What is the critical size of radius or mass of this planet that makes it possible to have an atmosphere?


title: Acceleration and relativity

text: Usually in physics books this equation used $a=\frac {dv}{dt}$ to calculate the relativistic acceleration. It is true that speed $v=\beta c$ don't have relativistic (Lorentz) factor, But time have a relativistic factor $t=\tau \gamma$, what about acceleration!?


title: Why is the magnetic field proportional to distance?

text: It's extremely difficult to create a magnetic field that can influence objects 1 m away, and somewhat impossible at 10 m away... Why is that? Why is the magnetic field heavily dependent on distance? A solenoid, has $I$ flowing and generates $B$. $B$ is only applied at small distances, by increasing $I$ to infinity(for example) would $B$ expand greatly over large distances? Or would that depend on $n$ of the solenoid?


title: Is cooling really exponential?

text: I did the following experiment: I took boiling water, put a thermomenter (from Vernier hooked up to a TInspire calculator) into it and pulled it out. Then we started measuring the temperature it read with time. We waved the thermometer so that the air aorund it was not heating up. I did an exponential regression on the data (well, appropriately shifted data). The results show a clear non- exponential heat loss. So what is this curve? Or what have I done wrong?![time v temp](http://i.stack.imgur.com/92MaS.jpg) From a comment I have added a further plot: where the y-axis is logged. Again, it is clear that the data is not linear. ![time v ln\(temp\)](http://i.stack.imgur.com/CMkEf.jpg)


title: What is $\hat{p}|x\rangle$?

text: Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\hat{p}|x\rangle$, where $\hat{p}$ is the momentum operator and $|x\rangle$ is an eigenstate of the position operator, with $\hat{x}|x\rangle = x|x\rangle$. It turns out that $\hat{p}|x\rangle$ should be equal to $-i\hbar\frac{ \partial |x\rangle }{\partial x}$. But what does this even mean? I'm not sure what to do with an expression like that. If I try to find $\langle x | \hat{p} | x_0 \rangle$, I end up with $-i\hbar \frac{\partial}{\partial x} \delta(x-x_0)$, which I'm pretty sure is not allowed even in physics, where we would scoff at phrases like "$\delta$ is really a distribution and not a function". Edit: Motivation. I have solved Heisenberg's equation and found, for a Hamiltonian $H = p^2/2m + m\omega^2x^2/2$, that $\hat{x}(t) = \hat{x}_0 \cos(\omega t) + \frac{\hat{p}_0}{m\omega} \sin(\omega t)$. I am given that the initial state is $|\psi(0)\rangle = |x_0\rangle$, i.e., an eigenstate of position, and I have a hunch that by finding $\hat{x}(t)|x_0\rangle$ I might find a way to get $|\psi(t)\rangle$. But this is not what I need help with; I am now curious as to what $\hat{p}|x\rangle$ might be, never mind my original problem.


title: Statistical Mechanics - Distribution of Energies

text: Consider a state space $\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\pi_{\mathbb{X}}(x)=\frac{e^{-\beta E(x)}}{\mathbb{Z}(\beta)}$$ where $E(x)$ is the energy of state $x\in\mathbb{X}$, and $\mathbb{Z}(\beta)$ the partition function or _Zustandsumme_. I see the marginal density of energy given as $$\pi_{U}(E)=\frac{\Omega(E)e^{-\beta E}}{\mathbb{Z}(\beta)}$$ where $\Omega(E)$ is the density of states. My question is how do you derive the marginal density for energies from the density over the state space $\mathbb{X}$?. $E:\mathbb{X}\rightarrow\mathbb{R}_{+}$ is not a one-to-one function so one cannot apply the change of variables theorem.


title: Is there a differential equation that can represent a circuit with an arbitrary voltage source connected acrorss an antenna?

text: An RLC circuit with a voltage source can be characterized by the differential equation: $$ LC\;\ddot{I}\left(t\right) + RC\; \dot{I}\left(t\right) + I\left(t\right)-C \;\dot{V}\left(t\right) = 0 $$ ... but this will be true only as long as **radiation** is ignored. My **question** is, what will this equation become in the general case, when the radiation **cannot** be ignored and also has to be accounted for? I ask this because the radiation resistance on any antenna depends on the frequency of the (sinusoidal) current passing through it -- so I suppose it cannot simply add it on to the $R$ term. I expect $\dot{I}\left( t\right)$ to be involved -- as the far field $\vec{\mathrm{E}}$ depends on $\dot{\vec{\mathrm{J}}}$ which must derive from $\dot{I}$ somehow. But I don't seem to be able to go beyond that. Looking for pointers -- are there any textbooks that address this part? So far my Googling hasn't turned up anything, so I guess I don't know what I should be searching for or maybe this is something that has fallen through the cracks between circuit theory textbooks and antenna theory textbooks (most electromagnetism texts seem to jump to $\cos{\omega t}$ or $e^{j\omega t}$ and stay there). Another thing -- would the L & C values that can be computed from the antenna geometry be sufficient to address the _reactance_ in the high frequency case, or is there something else to it? I'm looking for an understanding within the bounds of classical electrodynamics (that is, leaving quantum mechanics out). Thanks...


title: Why isn't temperature frame dependent?

text: In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames? (I.e. In the lab frame $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u = {1 \over 2} m \overline {(v+u)^{2}}$).


title: Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

text: Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the following hexagonal lattice system: ![enter image description here](http://i.stack.imgur.com/zmdZP.png) The 271 purple dots represent point masses of mass $m=1$ and are spaced apart by a spacing $a=1$, and each point mass is connected to each of its six nearest neighbors via springs with rest length $L=1$ and spring constant $k$. The red dots are fixed in place, and essentially act as an immovable boundary condition, while the purple dots are free to move horizontally and vertically in the plane. The lattice has $D_{6}$ symmetry, and three of the elements (vertical reflection $\sigma_v$, horizontal reflection $\sigma_h$, and $\pi/3$ rotation $C_6$) are shown for reference. The entire system potential becomes $$U(\mathbf{r})=\sum_{i,j}\frac{1}{2}k\left(L-\left|(\mathbf{r}_i^{e}+\mathbf{r}_i)-(\mathbf{r}_j^{e}+\mathbf{r}_j)\right|\right)^2$$ where $$\mathbf{r}=(\mathbf{r}_1,\mathbf{r}_2,...,\mathbf{r}_{271})^\mathsf{T}=(x_1,y_1,x_2,y_2,...,x_{271},y_{271})^\mathsf{T}$$ is the vector which represents the state of deformation of the system away from the equilibrium configuration $\mathbf{r}^e$ in the image above ($\mathbf{r}_i$ represents the vector displacement of the $i$th point from equilibrium), and where the sum runs over nearest-neighbor pairs. As per the usual normal mode analysis, it's easy to show that $U(\mathbf{0})=0$ and $\nabla U(\mathbf{0})=\mathbf{0}$, and thus the system admits the second-order Taylor expansion $$U(\mathbf{r})=\mathbf{r}^\mathsf{T}\mbox{H}\mathbf{r}+O\left(\mathbf{r}^3\right)$$ where $$\mbox{H}=\nabla\nabla U(\mathbf{0})$$ is the $542 \times 542$ Hessian matrix of the potential at equilibrium. For small oscillations, the normal mode picture is applicable, and the eigenvibrations are the columns of the orthonormal eigenbasis $\Lambda$ of $\mbox{H}$. $\mbox{H}$ has a closed-form block matrix expression, although I omit it here for brevity. Note that $$U(\mathbf{r})=U(\Gamma_g\mathbf{r})$$ for all $\mathbf{r}$ and for all $g\in D_6$ where $\Gamma_g$ is the matrix representation of the symmetry operation $g$- for example, $\Gamma_{C_6}$ is the block matrix that maps every coordinate to the one 60 degrees CCW rotated from it, and also rotates each vector by 60 degrees. Since $U(\mathbf{r})=U(\Gamma_g\mathbf{r})$, we can take derivatives twice to obtain $$\nabla\nabla U(\mathbf{r})=\Gamma_g^\mathsf{T}\nabla\nabla U(\Gamma_g\mathbf{r})\Gamma_g.$$ By noting that $\Gamma_g^\mathsf{T}=\Gamma_g^{-1}$ for all $g\in D_6$ and setting $\mathbf{r}=0$, we obtain $$\Gamma_g\mbox{H}=\mbox{H}\Gamma_g$$ and hence $$[\mbox{H},\Gamma_{D_6}]=0.$$ Mathematica revealed that 180 of the 542 eigenstates $\Lambda_k$ of $H$ were singly-degenerate. As is well-known, a singly-degenerate eigenstate of a matrix $\mbox{H}$ is automatically an eigenstate of any $\mbox{A}$ for which $[\mbox{H},\mbox{A}]=0$, and thus all 180 singly-degenerate vibrations satisfy $$\Gamma_g\Lambda_k=\lambda_k^g\Lambda_k$$ and thus manifest in numerical outputs as strikingly beautiful patterns, which I'll show in a second. In particular, they are eigenstates of $\sigma_v,\sigma_h,C_6$ and the inversion operator $i$. Meanwhile, the other 362 eigenstates are doubly degenerate and so Mathematica simply spits out an arbitrary pair spanning each eigenspace, so they don't look very nice. However, they can be "symmetry-projected" via the usual means, which I experimentally found was best done by choosing the pair to be a simultaneous eigenbasis of $\mbox{H}$ and $\Gamma_{\sigma_v}$ (which, again, can be done since $[\mbox{H},\Gamma_G]=0$). Likewise, since $\sigma_v$ commutes with $\sigma_h$, they are also in a definite eigenstate of $\sigma_h$, and by extension, eigenstates of $i=\sigma_v\sigma_h$. However, they are not necessarily eigenstates of rotation $C_6$ since $C_6$ doesn't commute with reflections. As a cheap aesthetic surrogate, I define "generalized" eigenvalues as $$\lambda_k^{g}=\left<\Lambda_k,\Gamma_{g}\Lambda_k\right>,$$ which gives the correct result whenever $\Lambda_k$ is an eigenstate of $g$, and gives a sort of average value when it is not. The 542 symmetry-classified vibrations were visualized in Mathematica and exported as a gigantic 5343 x 8913 pixel image. Here is a considerably smaller preview image (it's far too small to read the words, click the full-size to see everything in detail): ![enter image description here](http://i.stack.imgur.com/cPlWE.jpg) They've been color-coded, with the yellow ones being singly-degenerate ("pure") and the orange ones being doubly-degenerate ("mixed"), their eigenvalue of $\mbox{H}$ is given ("$\lambda$"), and their symmetry numbers with respect to the four symmetry operations I've mentioned is also given. They're ordered by vibrational frequency, from highest to lowest. Many of them are reminiscent of snowflakes when viewed at lower resolution, which I thought was kind of pretty. At this point, all I've done is just compute a bunch of normal modes, which are valid for small oscillations. For large-amplitude vibrations, anharmonic coupling effects come into play, mixing the modes together and distorting their frequencies. To visualize this nonlinear mixing process, I numerically simulated the system starting out displaced in the direction of a single eigenmode, but with large vibrational amplitude. The time-varying vector output $\mathbf{r}(t)$ from NDSolve was then converted into the symmetry- adapted eigenbasis as $$\mathbf{r}'(t)=\Lambda^\mathsf{T}\mathbf{r}(t)$$ to yield the "population" in each eigenmode over time, and the result was plotted using ArrayPlot. For visual simplicity, I used a smaller 37-point lattice, rather than the 271-point lattice I've been using previously, although the same code can be used interchangeably for both. For example, starting in one of the $(C_6,\sigma_v,\sigma_h,i)=(-1,-1,1,-1)$ symmetry states with an amplitude such that the points traveled a distance of about 1/8 (ie, smaller than lattice displacement, but still reasonably large) yielded the following: ![enter image description here](http://i.stack.imgur.com/xtWcw.png) The $x$-axis is time (from left to right), and each of the 74 horizontal bars represents an eigenmode, which have been grouped together by which of the 8 symmetries it belongs to. White indicates no amplitude, black indicates large amplitude, and the color scale is somewhat logarithmic to make things easier to see. As can be seen, near the very beginning amplitude gets mixed into other states, but the mixing is confined to states with $(-1,-1,1,-1)$ symmetry or $(1,1,1,1)$ symmetry only. Similarly, repeating this with a $(\frac{1}{2},-1,1,-1)$ symmetry state yielded the following: ![enter image description here](http://i.stack.imgur.com/MSkYC.png) This time, the coupling is limited to four symmetry classes. I found that coupling only depended on symmetry, which I'm sure people more knowledgeable than myself will know well. Finally, I noted that such symmetry restrictions would spontaneously lift themselves at extremely large displacements without warning after some time; the time at which the sudden change occurred was strongly dependent on the precision of the numerical solver, and random noise added to the initial conditions caused an immediate breaking of symmetry. Here is an example of such a phenomenon: ![enter image description here](http://i.stack.imgur.com/YGAaz.png) Here's a question: _is this an classical-mechanical example of spontaneous symmetry breaking_? It seems that the system starts out in a state of definite symmetry, but that upon provocation by numerical noise, population 'tips over the edge' and spills into the rest of the symmetry classes, equalizing everything. Here's another question. I found that coupling was nonreciprocal, in the sense that population from $(1,-1,-1,1)$ modes could spill into modes of $(1,1,1,1)$ symmetry, but that the reverse was not possible (ie, the flow of amplitude seems to have sources and sinks). To be more specific, the coupling directions looked like this: ![enter image description here](http://i.stack.imgur.com/LhOHg.png) If the $i,j$ entry is black, then energy can couple from symmetry class $i$ to symmetry class $j$, and if it's white, it's forbidden (with the exception of when spontaneous breaks occur at extreme amplitudes). Note that not only is it upper triangular with the way I ordered the symmetry blocks here, but it's actually a $2\times 2$ block matrix of the form $$\left(\begin{array}{cc} A & A\\\ 0 & A \end{array}\right)$$ where $A$ is a $4\times 4$ matrix which is also upper-triangular. _Does anybody have any idea how or why this happens?_ I have to admit it's all a bit over my head. I did all of this intending to print out posters as a Christmas present for my parents, but I noticed all this wacky stuff in the process. I suspect it's some kind of weird "classical mechanics selection rule" but I'm not very good at algebra, so the reasons are all opaque to me, and I haven't formally taken classical mechanics. As a final example of pretty pictures, I should point out that my choice of symmetry projection for the doubly-degenerate modes was in a realistic one, as it matches the eigenmodes of the corresponding lattice with a unitary horizontal shear transformation applied to the equilibrium positions in the limit as the shear matrix approaches the identity matrix. To be specific, applying the shear transform $$\left(\begin{array}{cc} \frac{x}{1-x} & 0\\\ 0 & \frac{1-x}{x} \end{array}\right)$$ to the lattice and computing the eigenfrequencies as a function of $x$ yielded the following beautiful pattern: ![enter image description here](http://i.stack.imgur.com/N2xyg.jpg) It's available in higher resolution here. The $x$-axis is $x$, and each line represents an eigenfrequency. They have lots of crossings and avoided crossings, reminiscent of Stark effect perturbations in atoms. In the limit as $x\rightarrow\frac{1}{2}$, the eigenstates of $\mbox{H}$ approach the symmetry-adapted ones.


title: 1-dimension radiation problem

text: > A positive charge $q$ is fired head-on at a distant positive charge $Q$ ( > which is held stationary ), with an initial velocity $v_{0}$. It comes in, > decelerates to $v=0$, and returns out to infinity. What fraction of its > initial energy ($\frac{1}{2}mv_{0}^2$) is radiated away? Assume $v_{0} << > c$, and that you can safely ignore the effect of radiative losses on the > motion of the particle. First, I have the Larmor formula : $P=\frac{\mu_{0}q^2a^2}{6\pi c}$. Since the total radiation is that $\int P dt$, it suffices to find the $P(t)$. We have the equation of the motion. $$F=\frac{1}{4\pi \epsilon}\frac{qQ}{r^2}=ma$$ If we can solve this equation, we get $a(t)$. But this equation is non- solvable. I need your help. the answer is that $\frac{16}{45}(q/Q)(v_{0}/c)^3$


title: How can a qubit superposition state be written to a quantum register?

text: If a 3 qubit register can simultaneously store all 8 possible values in superposition, then how it is achieved to write 8 values in to the register? And How these 8 values can be processed parallel to realize the benefits of parallel processing?"


title: What happens if up quarks are replaced by down quarks and down quarks are replaced by up quarks?

text: I believe the quarks would flip the proton's internal charge, reverse the spin of the proton, transform into another flavor, and then become unconfined and free.


title: Seeing light travelling at the speed of light

text: Imagine there are two cars travelling "straight" at the speed of light*, $A$, and $B$. $B$ is following directly behind $A$. Suddenly, $B$ switches on its headlights. Will $A$ be able to see this light? * * * My answer is no, since $A_v = B_v = c$ (the light will always stay stationary relative to $B$. This will probably lead to it gathering up, and intensifying. * * * *I realize this is impossible, but it's a question my Grade 9 [Honours] teacher asked, so we don't need to get into Relativity, $m = \frac{m_0}{\sqrt{1 – (v / c)^2}}$, _cough cough_. (I think.)


title: meteor explosion, steam or shock/vibration trigger

text: Is the explosion of a meteor due to forces imparted to the meteor by entry into atmosphere, or is it due to heating of a meteor that contains water with conversion to steam triggering the explosion? Do chondrites explode more readily because of some water content than nickel/iron meteorites?


title: energy of Russian meteor, does it matter whether it explodes or not?

text: The kinetic energy calculation for any meteor would be the traditional one half times the mass times the velocity squared. In terms of the events that occur on the ground, does it matter whether the meteor explodes in the atmosphere or doesn't explode? Is the energy released to the atmosphere and structures and people on the ground by the shock wave it creates the same whether it explodes in the air or not?


title: What is the meaning of phase difference?

text: What do you mean by phase of a wave? And phase difference? Waves have always confused me as it's too difficult to visualize them. I am no good at waves mechanics, so if anyone could explain in simpler term? Also if anyone could direct me to complete solution to my problem as to study waves from where? How do you visualize it? How did you study and understand? I have been meaning to ask this question for quite a time but always thought it would be too dumb. Thank you and I am eagerly waiting for an answer and your valuable suggestion


title: How can I calculate the amount of bubbles needed to lift a submerged body?

text: Our teacher had us come up with our own thermodynamics experiments, so I put a glass of water inside a vacuum-capsule. Inside the glass of water I have placed a non-floating object, a small plastic ball. If I enable the vacuum, the water will begin to develop bubbles of vapor. Those bubbles attach themselves to the plastic ball and it floats. What I want to figure out now, is how I can approximate the amount of bubbles that are necessary to lift the object. Also, does the vacuum make it easier to lift it, as the pressure on the water surface is lower?


title: Conservation of momentum in collision of two bodies

text: > Suppose we have some ramp on wheels of mass $M$, standing on a frictionless > surface. A cart of mass $m$ moves with a certain velocity $v$ towards the > ramp. The cart moves up the ramp seamlessly, and there's no friction between > them and no loss of energy. > >   * Find the maximal height $h$ that the cart will reach after the collision >   * Find the time (starting from the moment of the collision) it will take > for the cart to reach the maximal height (given the angle of the incline > $\alpha$) > ![enter image description here](http://i.stack.imgur.com/WaxYI.png) I have a few questions regarding this tricky question: 1) What happens after the collision (between the moment the cart is getting on the ramp and till the moment it returns to the ground)? Does the cart and the ramp are moving with a common velocity at each point in time (i.e. they both have the same velocity all the time, which is changing due to the deceleration of the cart)? 2) I suppose both the cart and the ramp are moving with the same velocity when the cart reaches the maximal height. Why can I use the conservation of momentum $mv+0=(m+M)V$ here? The conservation of mometum law states that if no external forces are acting on the system then the momentum is conserved. However, during the collision, there IS an external force acting on the cart and that is the gravity. 3) How can I possibly find the time it takes for the cart to get to the maximal height (even if we suppose that the incline is not curved)?


title: Why is the (nonrelativistic) stress tensor linear and symmetric?

text: From wikipedia: "...the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. ...The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric." What is the more detailed explanation for this, and/or where can I find it?


title: How to determine the mass of a quark?

text: As far as I know quarks are never found in isolation, so how can we determine their rest mass?


title: Diffraction pattern threshhold

text: What is the characteristic bump height of periodic grating below which diffraction effect cease to exist (let assume a threshold of peaks to valleys intensity of 20% as the minimum detectable by human eye). Is it significantly less than lambda?


title: What materials focus EM radiation in the 2.4GHz range

text: If glass and similar materials refract visible light effectively, what materials would be best for focusing lower frequencies of EM radiation, if any? If not, what other methods exist for focusing these ranges? The thought was inspired by wondering how you might build a camera that captures 'light' given off by Wifi routers


title: measure higher frequencies then half of Planck-frequency?

text: The maximum frequency is defined by the Planck frequency $\omega_P$. Also there is the Shannon theorem which tells us that to lossless capture a signal, you have to sample it with minimum of the double frequency. That states that you cannot measure frequencies higher than $\frac{\omega_P}{2}$. Is this correct? I saw sites telling the highest frequencies measured are about $10^{30}$Hz which is under $\frac{\omega_P}{2}=9.27435\cdot10^{42}$Hz. So obviously there are some borders in test equipment and other physical boundaries, but could it be that in theory this is not possible to measure such high frequencies?


title: D'Alembert's Principle and the term containing the reversed effective force

text: For our Classical Mechanics class, I'm reading Chapter 1 of Goldstein, et al. Now I come across Eq. (1.50). To put it in context: > $$\begin{align*} \sum_i{\dot{\mathbf{p}_i} \cdot > \delta\mathbf{r}_i}&=\sum_i{m_i\ddot{\mathbf{r}}_i \cdot > \delta{\mathbf{r}_i}}\\\ &=\sum_{i,j}{m_i\ddot{\mathbf{r}}_i} \cdot > \frac{\partial\mathbf{r}_i}{\partial q_j} \delta q_j \end{align*}$$ > > Consider now the relation Eq. (1.50): $$\begin{align*} > \sum_{i,j}{m_i\ddot{\mathbf{r}}_i} \cdot > \frac{\partial\mathbf{r}_i}{\partial q_j}&= \sum_i{\left[ \frac{d}{dt} > \left( m_i\dot{\mathbf{r}}_i \cdot \frac{\partial\mathbf{r}_i}{\partial q_j} > \right) - m_i\dot{\mathbf{r}}_i \frac{d}{dt} \left( \frac{\partial > \mathbf{r}_i}{\partial q_j} \right) \right]} \end{align*}$$ I'm at a loss for how he resolved it that way. He goes on to explain that we can interchange the differentiation with respect to $t$ and $q_j$. My question is: **Why is there a subtraction in Eq. (1.50)?**


title: Are black hole states completely mixed?

text: A completely mixed state is a statistical mixture with no interference terms, and (QMD, McMahon, pg 229): $$\rho = \dfrac{1}{n}I$$ $$Tr(\rho^2) = \dfrac{1}{n}$$ Are black hole quantum states completely mixed, meaning that they each have the same probability of being observed if we could in fact observe a black hole?


title: Home-made lattice calculation?

text: The topic of Lattice QCD or Lattice gauge theory or even Lattice field theory is quite old now. And the main reason for the interest in the topic is the ability to calculate nonperturbative stuff on a computer. It seems that to do research with lattice you need an access to some supercomputer. But now everyone can afford something that is as powerful as supercomputers 20 years ago. Maybe it possible to redo some of the results of that time? Is there any project that is:   1. Relatively simple.   2. Allows to calculate some real-world quantity (like mass of the proton).   3. Can be done with an average home computer.


title: How local is the stress tensor?

text: I am confused by the definition of the stress tensor in a crystal (let's say a semi-conductor), I don't see how it could be "more local" than over an unit cell. I know that in field theory the stress tensor, or more precisely the whole stress-energy tensor, is local function, a function of the point $\vec{x}$ but I don't understand the interpretation you can give to shearing a point. Plus, in the lattice case I stressed (pun semi-intended), you shouldn't be able to look at displacements or deformations on smallest scales than your elementary cell?


title: "Optically performed" Fourier Transform

text: This article says that they are only able to achieve such extremely high fiberoptic data rates because the multiplex light and then use a Fourier Transform to split it up again. But they say that they perform the Fourier Transform "optically". Now I understand the math behing a Fourier Transform, but I still can't wrap my mind around how this would be done in analog with light. Any ideas?


title: Pressure inside a light bulb

text: It is known that nowadays most light bulbs are filled with gas in order to minimise the evaporation rate of tungsten. Is e.g. argon put into the bulb with reduced pressure, so that the glass doesn't explode while heating the wire?


title: Lifting house with balloons

text: I have an assignment about the movie Up. Here is the question: > _Critique the brief floating scene. Did the animators use enough balloons > during this scene? What important variables, (which affect gases) did the > animators seemingly fail to consider during the floating scenes as Carl's > house is slowly rising after initial takeoff? What important variables, > (which affect gases) did the animators fail to consider during the floating > scenes as Carl traveled from the US to Venezuela? Explain why these factors > should be considered._ For reference, we are supposed to assume that Carl's house was 125,000 pounds. Could someone please help me with this question? Thanks! I'm thinking for the important variables about pressure, volume, and number of moles.


title: What is sound in terms of acoustic sources?

text: Sound is nothing more than small amplitude, unsteady pressure perturbations that propagate as a longitudinal wave from a region in space which created it (called the source region) into a quiescent (still) region where it is observed by hearing. It is at this point where I get slightly confused in that we may distinguish between two types of pressure perturbations: **hydrodynamic pressure perturbations** and **acoustic pressure perturbations**. The acoustic perturbations are what we term sound, as they are characterized by their ability to _propagate_ into the hearing region. The hydrodynamic pressure perturbations could be a consequence of fluid flow simply changing in the source region, and this may not necessarily propagate and become what we term sound. Perhaps evanescent waves can serve as some insight to what I am trying to explain: with reference to this link the pressure fluctations created by the subsonically pulled wavy plate decay exponentially in the upward direction and so do not constitute a sound. Is my interpretation so far correct? If so are there any rigorous methods for determining if pressure fluctations will propagate (thus becoming sound), or is sound simply that which we can hear outside a source region? Are there other cases besides the linked example for which pressure fluctations are created, but do not propagate as sound?


title: Pendulum in an elevator

text: ![enter image description here](http://i.stack.imgur.com/p5vwp.png) Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\beta$. Now at some time when the bob of the pendulum was at the mean position(equilibrium position/center position), the elevator suddenly starts moving up with some acceleration $a$. I need to find out what is the new angular amplitude of the pendulum, in terms of $a$,$\beta$ and if needed $T$, and $l$(length of the string). I'm completely stuck at this concept. I absolutely do not know how to proceed. My ideas are that we could use some conservation, maybe the energy of the bob. Any hints? P.S. I'm sorry if anything is too elementary.


title: What stabilizes neutrons against beta decay in a neutron star?

text: Free neutrons are known to undergo beta decay with a half-life of slightly above 10 minutes. Binding with other nucleons stabilizes the neutrons in an atomic nucleus, but only if the fraction of protons is high enough (at least a third or so). **But what keeps a neutron star stable against beta decay?** Apparently, this is extra pressure due to gravity in contrast to "negative pressure" of proton Coulomb repulsion in a nucleus but how do we know that this is enough to stabilize the degenerate neutronic fluid? I am aware of a closely related question but not really happy with the answers there. There is lot of dazzling details here, but I am a looking for an answer suitable form a 8-year old with enhanced curiosity towards astrophysics.


title: Is the Green function a prescription for a connection?

text: I'm trying to learn connection on principal fibre bundle. As far as I can see, the connection is just a given prescription for the displaced field/function on the base space to remains on the horizontal subset of the global space. [The global space and the base space are related by gauge symmetry equivalence class, the base space identifying two points related by a gauge transformation from the global space] In other words, a connection is an other word for parallel displacement. So it relates the field/function at position $x$ to the nearby one $x+\epsilon$ infinitesimal close to the previous position. An interesting basic relation regarding connection is the covariant derivative, defined in Peskin and Schroeder (eq.15.4) [see the reference at the end] as $$n^{\mu}D_{\mu}\Psi=\underset{\epsilon\rightarrow0}{\lim}\dfrac{1}{\epsilon}\left[\Psi\left(x+\epsilon n\right)-U\left(x+\epsilon n,x\right)\Psi\left(x\right)\right]$$ where $D_{\mu}$ is the covariant derivative, $\Psi(x)$ the field, and $n$ the vector along which the field is displaced, and $U\left(y,x\right)$ describes the propagation of the field $\Psi$ along the trajectory $x\rightarrow y$. For two distant points, $U$ becomes the Wilson line (eq.15.53, after a path is chosen). If all what I said is correct up to now (please correct me otherwise), here is my question: I'm wondering about the Green function, which is also a kind of prescription for the displacement of a field along a trajectory. Does it have something to do with a (mathematically properly defined) connection on a fibre bundle ? I've the feeling it gives a peculiar path among all the allowed ones. Is this feeling justified ? _More details about how I came to this question:_ In the Peskin and Schroeder, I've been particularly interested in the definition of a gauge transformation applied to their $U$. It's defined like (eq.15.3) $$U\left(y,x\right)\rightarrow V\left(y\right)U\left(y,x\right)V^{-1}\left(x\right)$$ the $\rightarrow$ symbol meaning _gauge transformation_. This transformation law is the same as for a Green function defined as $$G\propto\left\langle \Psi\left(y\right)\Psi^{\dagger}\left(x\right)\right\rangle $$ so I was wondering about the infinitesimal value of $G\left(x\rightarrow y \right)$ which is not clear because of the discontinuity ()say I'm talking about fermion). But still appealing to me. Also, it is clear that having the same transformation law is not sufficient to make any proof.   * M.E. Peskin and D.V. Schroeder _An Introduction To Quantum Field Theory_ Frontiers in Physics (1995)


title: Materials with Zero Poisson Ratio

text: Poisson's ratio is defined as negative ratio between transverse and axial strain. So, a material with zero poisson ratio must necessarily exhibit no transverse strain. After checking the wikipedia, I was suprised to discover that a CORK has a near-zero poisson ratio. I haven't found any references yet to other materials that also have zero poisson ratio. What is so special about a cork that produces its zero poisson ratio? What other materials share this characteristic?


title: Example in the book: A simple accelerometer

text: # A simple accelerometer > _You tape one end of a piece of string to the ceiling light of your car and > hang a key with mass m to the other end (Figure 5.7). A protractor taped to > the light allows you to measure the angle the string makes with the > vertical. Your friend drives the car while you make measurements. When the > car has a constant acceleration with magnitude a toward the right, the > string hangs at rest (relative to the car), making an angle $B$ with the > vertical._ > >   * _(a) Derive an expression for the acceleration $a$ in terms of the mass > m and the measured angle $B$._ > >   * _(b) In particular, what is a when $B$ = 45? When $B$ = 0?_ > > I don't care about the answers, the important thing is the following:- The book says The string and the key are at rest with respect to the car, but car, string, and key are all accelerating in the +x direction. Thus, there must be a horizontal component of force acting on the key. That's the reason the book decided to consider a force in the $+x$ direction, but I'm looking for a better explanation: how would I find detect the force in the $+x$ direction in another way? To me, when I draw the free body diagram of the string, there looks to be no force acting on the $+x$ direction! I understand it starts with noticing that the string is attached to the ceiling of the car, and that the car has force causing acceleration in one direction, but I don't know how to go further than that.


title: How does one interpret the Dirac equation with a self-field potential?

text: EVERY QFT text I've ever examined states that if there is an external vector potential, $A_\mu$, then one writes the Dirac eq.(or Klein-Gordon eq.) using a covariant derivative to include this U(1) gauge field, $A_\mu$. No problem. Wikipedia states that one must also include the contribution due to the electron self-field, $A_\mu'$. Again, makes perfect sense. Thus if there is no external 4-vector potential, the Dirac equation should exhibit the internal one! However, all texts state the "free" Dirac equation, without an accompanying vector potential. If I now solve the free Dirac equation, the spinorial solutions can be used to construct the Dirac charged current. Using this current as the source, I can then solve the classical D'Alembertian eq. for the corresponding internal self-field $A_\mu'$. If I insert $A_\mu'$ back into the Dirac equation, the solution must be different than those obtained from the free Dirac equation. I am simply wondering what the interpretation of the solutions will be now?


title: In-flight damage to a supersonic jet

text: I'm doing some research for a book I'm writing, and, as the title implies, I'm looking to understand what happens when a supersonic jet sustains in-flight damage. Specifically, what happens to the occupants if the cockpit is breached? (I assume that this is why jet pilots wear masks?) Could a jet maintain super sonic speeds (for at least a short time) with such a breach? What would happen to a person in the cockpit without a mask if there was a cockpit breach at those speeds? Just how weird do things get at super-sonic speeds? Could a conventional airliner ever reach mach one in extreme circumstances? Many thanks!


title: What information is stored on gramaphones/tape recorders/CDs/DVDs

text: I'm a Software Developer by profession and my physics knowledge is limited what I had learned at high school level. Please excuse me if the question is trivial. Question: From what I know, a sound wave is set of different amplitudes spread across a time line. The amplitudes vary greatly. Hence, sound wave is largely aperiodic (amplitudes don't repeat often). Now, where does frequency come into picture here? If the wave is periodic like a sine wave or a deterministic mathematical function of time, then frequency can be measured as the number of cycles (wave reaching same amplitude) in one second. How can we define frequency of highly aperiodic sounds like human speech. If all that is recorded on a gramophone disc is varying amplitudes across timeline, where is frequency accounted for? Does the frequency remain constant in a typical human speech?


What happens when a supersonic airplane flies through a cloud? Will it punch a hole or is it more like a bullet through water (= hole closes immediately after the aircraft has passed)? Is there some special effect because of the supersonic speed? Or maybe the question should be: Does the airflow around an airplane change when the sound barrier is broken?


title: Boundary conditions for fields in Kerr/CFT

text: I am reading a paper by Guica et al. on Kerr/CFT correspondence (arXiv:0809.4266) and I'm not sure if I got this. They choose the boundary conditions, like a deviation of the full metric from the background Near- Horizon Extremal Kerr (NHEK) metric. Let's say that we can write that deviation like $$\delta_\xi g_{\mu \nu}=\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$$ And the most general diffeomorphism which preserve the boundary conditions given in the text is: $$\xi=[-r\epsilon'(\varphi)+\mathcal{O}(1)]\partial_r+\left[C+\mathcal{O}\left(\frac{1}{r^3}\right)\right]\partial_\tau+\left[\epsilon(\varphi)+\mathcal{O}\left(\frac{1}{r^2}\right)\right]\partial_\varphi+\mathcal{O}\left(\frac{1}{r}\right)\partial_\theta$$ What my mentor told me, while briefly explaining this, is that we basically need to find the most general $\xi$ such that $\mathcal{L}_\xi g_{\mu\nu}$ is within the class of the boundary conditions. But how do I find these $\xi$? How can you find these boundary conditions and diffeomorphisms? Or better jet, how do I find diffeomorphism using those boundary conditions? :\


title: If magnets exert greater force on heavier objects, why is it possible to separate magnets from very heavy objects?

text: I was thinking of the question "Do magnets exert more force on heavier (more massive) objects?" I would think the answer is "yes", because for example if you have a magnet and 2 paperclips, they will both be attracted by a force F to the magnet, and therefore if you tape them together the force will be 2F. But if this is true, then if you have a very heavy metallic object (e.g. a steel safe), why wouldn't a small magnet get stuck to it with a very high force? If a safe weighs 100kg and a paperclip weighs 1g, that means the force should be 100,000 times as strong. Given that the magnetic attraction force is enough to lift the paperclip, that means you should be able to lift the 100g safe with the same magnet, right? Well you can't, so I'm wrong. But why am I wrong? I was thinking maybe there is a dropoff in the force somewhere, but then isn't steel a good "conductor" of magnetic force?


title: Why is temperature constant during melting?

text: This is an elementary question but I do not know the answer to it. During a phase transition such as melting a solid to a liquid the temperature remains constant. At any lower temperature the heat provided went to kinetic energy and intermolecular potential energy. Why is it that at the melting point no energy goes into kinetic (that would increase the temperature)?


title: Frequency-averaged (gray) radiative transfer

text: The equation for radiative transfer is $$ \omega \cdot \nabla I = \kappa(B - I) $$ where $I$ is the intensity of radiation, $\omega$ is the ray direction, $\kappa$ the absorption coefficient, $B$ the Planck function. Here, $\kappa=\kappa(\nu)$ (i.e., it depends on the radiation frequency $\nu$). Often, though, the gray model, where $\kappa$ doesn't depend on $\nu$, is used. How can this be justified? What assumptions are used to obtain the gray model from the non-gray model?


title: Non-additivity of energy (thermodynamics)

text: In page 15, Chapter 1 (Laws of Thermodynamics) of the book "An Introduction to the Study of Stellar Structure" by Nobel laureate S. Chandrasekhar, it is mentioned (in context of total energy of two subsystems brought in contact): > In general when two bodies are brought in contact, the energy is not > additive; it is easy to see however, that the deviation is directly > proportional to common surface area of the bodies, hence, for the large > volumes the deviations from the additive law can be neglected I always thought that energy is additive as a fundamental property (or by definition). I did not find anything further to understand the statement. Please, help me understand this.


title: Numeric value of the electrons drift velocity in superconductors

text: Somebody knows the numeric value of electrons drift velocity in superconductors? How this value depends from the used superconductor material? What's about the current? Since the electrical resistance tends to zero, how many electrons can pass an area (a layer) that is occupied e.g. by 100*100 atoms?


title: How units were defined?

text: I was wondering how we humans can be sure that one meter is one meter and that one second is one second. Nowadays, except for the Kilogram, all other units are well defined using highly accurate techniques (frequency of atoms vibrations or stuff like that). But at the end all units are kind of related to each other and the definition of each unit is based on a combination of other units. There must be some viable sources that have constant measurable values that we used to define the basic units. What is those sources? To explain more let's start with the meter. From wikipedia, the definition is: The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. So here it is clear that the definition of a meter relies on the accuracy of how we define a second. Now let's look at the second: the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. How is this period calculated? The sensor has probably some equations that imply transformations using other units like Kg etc... Where does this loop stops? EDIT: I think I was a little bit mistaken. Not all units are directly related and there is 3 totally independent units which are : Time (second), Temperature (kelvin) and Mass (kilogram). Time and Temperature are well defined but Kilogram is still unclear. Every existing unit can be transformed into a combination of those three. It means that all units based on Kilogram are not absolute.


title: Perturbation method & eigenvalues

text: I have a problem but I don't understand the question. It says: "Show that, to first order in energy, the eigenvalues are unchanged." What does it mean? It means that if the Hamiltonian has the form $$H=H^{(0)}+\lambda H^{(1)}$$ Where $H^{(0)}$ is the Hamiltonian of the unperturbed system, $H^{(1)}$ is the perturbation and $\lambda$ is a small parameter, then if $$E_{n}=E_{n}^{(0)}+\lambda E_{n}^{(1)}$$ Where $$E_{n}^{(1)}=\left\langle \psi_{m}^{(0)}|H^{(1)}|\psi_{m}^{(0)}\right\rangle $$ I have to show that $$E_{n}^{(1)}=0$$ ? I'm confused. Thanks for your answers.


title: What is force? How does a constant force output a nonconstant power?

text: For a constant force, P=Fv. I understand the mathematical derivation of this, but this seems to me, intuitively, to be nonsense. I feel that my discomfort with this comes from a fundamental misunderstanding of force and Newton's Second Law, so I'm not really looking for any mathematical explanation. So, to begin: How is it that a constant force does not add energy to a system at a fixed rate? Consider a rocket burning a fuel at a constant rate. The chemical potential energy should be converted to kinetic energy at a constant rate, that is, 1/2mv^2 should be increase linearly. The magnitude of the velocity of the rocket would then increase at a less than linear rate, implying a nonconstant acceleration and therefore, a nonconstant force/thrust (F=ma). If force is indeed a "push or a pull," shouldn't that constant rate of burning of fuel yield a constant "push or pull" as well? Clearly not, so I would have to think that, somehow, a given force applied to a certain object at rest would in some way be different than that a force of the same magnitude being applied to that same object in motion. In this sense, is force merely a mathematical construct? What does it tangibly mean, in physical terms? Would a given force acting upon me "feel" differently to me (in terms of tug) as I am moving at differing velocities? Force being defined as a "push or pull," which is how it has been taught in my high school class, seems rather "handwavy," and maybe that's the issue. It's been troubling me for a couple of weeks and my teacher hasn't really been able to help, so thanks!


title: Why can't dark matter be black holes?

text: Since 90 % of matter is what we cannot see, why can't it be black-holes from early on? Is is possible to figure out that there are no black holes in the line of sight of various stars/galaxies we observe?


title: Scattering of phonons and electrons within solids

text: I got a question concerning the scattering of phonons and electrons. I read an introductory explanation to this process that is somehow not very satisfactory. It goes like this: Let $\psi_{k}$ and $\psi_{k'}$ be Bloch-waves within a solid. We denote the probability of transmission between these two states by $P_{k,k'}$. This probability is according to quantum mechanical perturbation theory proportional to $|<k'|H'|k>|^{2}$ where H' denotes the perturbed Hamiltonian or the perturbed potential (caused by either phonons or impurities). Now we assume that our two wave function are of the standard Bloch-form. Hence we obtain: $|<k'|H'|k>| = \int{ d^{3}r u_{k'}(r)^{*} H'(r,t) u_{k}(r) e^{i(k-k')r}}$ (Eq. 1) where $u_{k}$ and $u_{k'}$ have lattice-periodicity. Now in an inelastic collision with of an electron and a phonon we have by energy conservation: $E(k') - E(k) = \pm \hbar \omega(q)$ where $E(k)$ and $E(k')$ denote the electron energies and on the rights hand side there is the energy of a phonon with a wavevector $q$. Now comes the tricky part of the analysis: Now they say that the disturbed potential must include a dependency on $e^{iqr}$ Hence (Eq. 1) (which is the scattering probability) must include a matrix element of the form $<k'|e^{iqr} |k>=\int{d^{3}r u_{k'}(r)^{*} u_{k}(r) e^{i(k-k'+q)r}}$ . (Eq. 2) Why is that? I don't see their point here. I want to remark here that I do have heard an introductory quantum mechanics class and class in linear algebra. However I neither understand this from a mathematical point of view nor from a physical point of view. Can anyone give me a better explanation of this? Please note that the course I'm taking is not a course in theoretical solid state physics. Now the derivation goes on: The say that since $u_{k}u_{k'}$ can be expanded into a Fourier series of reciprocal lattice vectors. Fine - I agree that's legitimate. So assume: $(u_{k}u_{k'})(r) = \sum_{G}f_{g} e^{-iGr}$. Plugging this into this equation they claim the matrix element above (Eq. 2) only does NOT vanish is $k'-k+q=G$ Well this doesn't seem plausible. I mean then $e^{i(k-k'+q-G)r}=1$. Integrating over all space leads to infinity. Well I may agree that the integral over this term vanishes if the equality $k'-k+q=G$ does not hold because of periodicity/symmetry. Maybe one has to restrict the integral to the solid itself to keep it finite. Can anyone give me a more detailed explanation?? I'm looking forward to your responses. Thanks in advance.


title: RA/dec to Alt/Az program or method

text: I've been looking for a program to convert from RA/dec into Alt/Az; having used a couple of online versions I haven't seemed to find one yet that works reliably. I've tried to do it myself and half the numbers end up spot on, but the other half are way off. It's not something I can plug in completely by hand, I need to convert about 300 images. Is there good code, a program, etc. that will honestly always give back the right coordinates? Failing that, is it a common slip-up messing up the values?


title: Statistics in physics

text: What are the uses of statistics in physics? I am about to embark upon a study of statistics and I would like to know what the particular benefits I gain in physics.


title: What's the reason for the seasons?

text: ![Diagram of Sun and Earth](http://i.stack.imgur.com/fNtef.png) In the diagram, it shows that the fundamental reason for different seasons is when the northern hemisphere is titled towards the sun there's summer in northern hemisphere and winter in southern hemisphere and vice-versa after the six months. But because the earth spins around its axis, why doesn't the seasons change during day and night? is there something missing in the diagram to prove that?


title: Why can I never see any stars in the night sky?

text: I have always lived near a large city. There is a stark contrast between the picture linked below for example, and what I see with the naked eye. Sometimes I can see a few stars here and there, but usually they can be counted on my fingers. I'm looking up at the night sky right now, and there isn't a single visible star! Question: What creates this large gap in what I can see? Here's the photograph I was talking about: http://apod.nasa.gov/apod/ap120123.html


title: Why does the Gallum-Arsenite bandgap narrow for higher temperatures and is this because of unavoidable impurities?

text: I found this sentence: > GaAs at 300K contains 10^15 acceptor atoms per cubic centimeter. The bandgaps for Gallium Arsenite are: 0 Kelvin : 1.518 eV 300 Kelvin: 1.424 eV Now I am wondering:   1. Why does Gallium Arsenite have this impurity?   2. Is the bandgap at 300 Kelvin is stated for a _pure_ GaAs semiconductor or does it already incorporate the influence of the impurity?   3. If it doesn't what else does then cause the narrowing of the bandgap?


title: Equation of state of cosmic strings and branes

text: I'm sure these are basic ideas covered in string cosmology or advanced GR, but I've done very little string theory, so I hope you will forgive some elementary questions. I'm just trying to fit some ideas together here. Following my answer to this question I began to wonder what sort of fluid would have a cosmological equation of state $ w = -\frac{1}{3} $. Note the minus sign. I was interested because it seem like a critical case: the scale factor evolves as $$ a \propto t^{\frac{2}{3(1+w)}} $$ so for $ w = -\frac{1}{3} $, $$ a \propto t $$ which is neither accelerating nor deccelerating. I was thinking that a fluid of cosmic strings fits the bill since the energy momentum tensor for a string directed along the 3-axis is (effective on scales $\gg$ than the transverse dimension of the string) $$ T^{\mu\nu}=T_{\text{str}}\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0\\\ 0 & 0 & 0 & -1 \end{array}\right)\delta^{\left(2\right)}\left(x_{\perp}\right) $$ (c.f. Shifman Chapter 3) and $w$ is defined through the trace $$ T^\mu_\mu = \rho - 3 p = (1-3w) \rho = 2 \rho $$ the last equality using the explicit form of $T^{\mu\nu}$. This gives $w = -1/3$ as desired.   * Question 1: Does this continue to hold for a gas of non-interacting strings? I would expect so since the general case is described by the $T^{\mu\nu}$ above, suitably Lorentz transformed and convoluted with a distribution function. All the operations just mentioned being linear nothing should break, but is there a rigorous result?   * Question 2: Is the lack of acceleration/decceleration in any way related to the well known fact that cosmic strings do not gravitate? (There is a conical singularity on the string, yes, but no propagating curvature.) This is where I would really like some elaboration, because it seems intuitive. But I know that the gravitational effect of a slow rolling scalar is counter-intuitive so I don't want to jump to conclusions here. Finally, the obvious extension of these ideas are to branes. Take 2-branes embedded in our ordinary 4D universe. The energy momentum tensor is $$ T^{\mu\nu}=T_{\text{w}}\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\\ 0 & 0 & 0 & 0\\\ 0 & 0 & -1 & 0\\\ 0 & 0 & 0 & -1 \end{array}\right)\delta^{\left(1\right)}\left(x_{\perp}\right) $$ for a brane (domain wall) oriented in the 2-3 plane (see Shifman again, chapter 2). This gives $w = -\frac{2}{3}$ and an expansion history $a \propto t^{2}$, an accelerating expansion. Is this connected to the fact that domain walls antigravitate? * * * EDIT: Partial answer: Kolb & Turner do the calculation I outlined above for a non-interacting gas of strings and domain walls. The result is a little more involved than I had envisioned. For strings: $$ w = \frac{2}{3} v^2 - \frac{1}{3}, (7.57)$$ where $v$ is the average velocity of the strings. For branes they find $$ w = v^2 - \frac{2}{3}, (7.45)$$ where again $v$ is the average velocity. The equation numbers refer to Kolb & Turner, _The Early Universe_ , 1990 ed. So my speculations really only hold for the static case.


title: Why are AC quantities represented by sine waves always?

text: Usually we use a sinusoidal wave form to represent a alternating quantity. Why not a cosinusoidal wave or a ramp wave form? In sine wave forms we can indicate the maximum and minimum amplitude and its variation with respect to time. Ramp waves have the same properties; then why not a ramp wave form?


title: Why is gravity weak at the quantum level?

text: Why is gravity stronger than other forces at the macroscopic level, yet weaker than other forces at the quantum level? Is there an explanation?


title: In the known universe, would an atom not present in our periodic table exist?

text: I have watched this movie Battleship. In it the researchers say this piece of metal is alien because we cant find this metal on earth.   So that would mean somewhere else in the universe any of the following should be true?   1. Atoms' composition is not similar to that as on earth (nucleus, electrons, anything else)   2. Elements with atomic numbers above 120 or 130 are stable (highly impossible without point 1)   3. The realm itself is observed by different binding forces (but then, once that elements realm has changed, it should become unstable and collapse)


title: Space-time in String Theory

text: I would like to understand how Physicists think of space-time in the context of String Theory. I understand that there are $3$ large space dimensions, a time dimension, and $6$ or $7$ (or $22$) extra dimensions, and all these dimensions need to fit together in a way such that the extra dimensions are compactified (with a Calabi-Yau or $G_2$ structure). My question, however is not about the possible $10$, $11$ or $26$ dimensional manifolds that may be possible, but about whether string theorists consider space-time as somehow quantized (or discrete), or rather as a continuous manifold, or are both options possible? In other words, can strings move continuously through space, or are there a discrete set of locations where strings can be, and does string theory rule out one of the options? How about the same question in loop quantum gravity (LQG)? Should I think of the spin networks in LQG as describing a discrete space-time? Thanks for your insight, or any references you may be able to provide!


title: Is the thought on this problem on LC circuits correct?

text: I have the following problem and I want to know if my thought is correct: In one $LC$ circuit, on which $C=4,0\mu F$, the maximum potential difference across the capacitor during the oscilations is $V_m=1,25V$ and the maxium current across the inductor is $I_m=50mA$. $(a)$ Find the inductance of the inductor. My solution was: first we note that the charge on the capacitor satisfies the relation $q(t) = Q \cos (\omega_0 t + \phi)$ where $Q$ is the maximum charge. In that case, the current across the capacitor must be $q'(t)=-\omega_0Q \sin (\omega_0 t + \phi)$. The negative sign evidences that the charge is leaving the capacitor and flowing in the direction of the inductor. So this must be the same rate at which the charge flows through the inductor, so the current through the inductor must be $i(t) = \omega_0 Q \sin (\omega_0 t + \phi)$. In that case the maximum current must be $\omega_0 Q$. Since $Q = CV$, the maximum charge is obtained with the maximum potential, so $Q = CV_m$, in that case the maximum current is $I_m = \omega_0 CV_m$. Now I've used $\omega_0 = 1/\sqrt{LC}$, so that we have: $$I_m = \frac{CV_m}{\sqrt{LC}}\Longrightarrow \sqrt{LC}=\frac{CV_m}{I_m}\Longrightarrow L=C\left(\frac{V_m}{I_m}\right)^2$$ Putting the values there I've found $L = 2500 \mu H$ $(b)$ Find the frequency of oscilations: My solution was to use $\omega_0 = 2\pi f$, so that we have $f = \omega_0 / 2\pi$, in that case we have: $$f=\frac{1}{2\pi\sqrt{LC}}$$ And using $C=4,0\mu F$ and $L=2500 \mu H$ I've got $f = 10^4/(2\pi) \operatorname{Hz}$. $(c)$ Find the time on which the charge on the capacitor goes from zero to the maximum. I've analyzed graphically, the time it takes for the charge to go from the maximum to the maximum again is the period $T$. The time it takes to reach $0$ is one fourth of the period. So, since the time for the carge to get from maximum to zero must be the same as getting from zero to maximum, the desired time is: $$t = \frac{1}{4}T=\frac{1}{4f}=\frac{\pi\sqrt{LC}}{2}$$ And for this problem I've found $ t = (\pi \cdot 10^{-4}/2 )\operatorname{s}$ . Now, is my thought correct in all three items? I'm preocupated because I didn't even thought about what happened to the phase. My thought was: "well, it's arbitrary, it will just have to do with where we start measuring $t$", but I'm not sure. Also, the manipulations in general, are they all correct? Thanks very much in advance!


title: How to derive following thermodynamic relation?

text: I'd would like to derive the following $$\bigg(\frac{\partial U}{\partial V}\bigg)_T = T \bigg(\frac{\partial p}{\partial T}\bigg)_V - p$$ What I know is that the internal energy $U$ is a function of temperature and volume. Hence, a small change in $U$ can be related to changes in $T$ and $V$ by $$dU =\bigg(\frac{\partial U}{\partial T}\bigg)_V dT + \bigg(\frac{\partial U}{\partial V}\bigg)_T dV$$ But I'm not sure where to go from here.


title: Feynman diagram for $\overline{K}\,\!^0$ antimeson production on the quark-level

text: I've recently stumbled upon a physics problem concerning $\overline{K}\,\\!^0$ antimeson production. In this particular example, colliding a $\pi^-$ meson with a stationary proton yields a $K^0$ meson and a $\Lambda^0$ hyperon: $$\pi^-\,[\overline{u}d] + p\,[uud]\rightarrow K^0\,[d\overline{s}] + \Lambda^0\,[uds]$$ This can be expressed in a Feynman diagram by letting the $u$ and $\overline{u}$ quarks annihilate to a gluon, out of which a pair of $s$-$\overline{s}$-quarks is generated. However, if a $\overline{K}\,\\!^0$ particle would be generated by the same method, in order to conserve the baryon number and the strangeness, more than just a particle must be produced. For example, the following reaction could take place, so that every quantum number is conserved: $$\pi^-\,[\overline{u}d] + p\,[uud]\rightarrow \overline{K}\,\\!^0\,[s\overline{d}] + K^0\,[d\overline{s}] + n\,[udd]$$ However, I can't seem to find a corresponding Feynman diagram for the reaction. I am guessing that the $\Lambda^0$ hyperon decays weakly and somehow yields the antikaon and the neutron, but I can't figure out how... Does anyone have a clue what the Feynman diagram could be?


title: Is dark matter expanding the universe, and creating the structures within it?

text: If dark matter repelled other dark matter, as well as regular matter(atomic or celestial), and regular matter were repelled by other regular matter "less" than by the dark matter, would not this result in the coalescence of regular matter into spheres? And if so, and given the repulsive force of dark matter on regular matter is equal to the attractive strength of gravity. Could the influence of that dark matter on those spheres, be equivalent to, but opposite, the pull of gravity, thus eliminating gravity altogether from the equation? Just a hypothetical I've been pondering. Any input would be appreciated.


title: If the multiverse theory is true, can there be a Universe where there are different laws of physics?

text: This is probably a very difficult question. But my question is essentially this, if there are other Universes can different laws of physics exist in those Universes and if so, can't there be a Universe where the laws of physics are so different that the multiverse theory has to come out as false?


title: is really an atom stable?

text: Half filled and fulfilled atomic orbitals are stable because of : high exchange energy. The problem is with exchange energy. We have learnt that the half and fulfilled orbitals have maximum no. of electrons with the same spin so they can exchange their positions and thus release energy. BUT, we have also learnt that the electrons of the same orbitals have the same energy. Furthermore, the electrons of the same sub shell have the same energy in the absence of magnetic field. so, how on earth do they lose energy?


title: Helicity for Zero Rest Mass Field Equations

text: I'm trying to reconcile the usual definition of the helicity operator, namely $$ h = \hat{p}.S$$ with the definition of a massless helicity $n$ field as a symmetric spinor field $\phi^{A\dots B}$ with $2n$ indices, satisfying the z.r.m. equation $$\nabla_{CC'}\phi^{A\dots B} = 0$$ I can't find anywhere a proof that these concepts are equivalent, and I can't seem to prove it myself. I feel I need to Fourier transform somewhere and/or use a quantum argument, but I can't see what! Does anyone know how this works, or could point me towards a good reference? Cheers!


title: Question on energies obtained via WKB approximation

text: Suppose we are given an ODE problem $$ y''(x)+V(x)f(x)=E_{n} y(x) $$ with boundary conditions $ y(0)=y(\infty)=0$. Here $V(x)$ is a potential function. Then is it always true that (for $n \gg 1$) $$ E_{n}^{\rm WKB} \sim E_{n} $$ where WKB means the eigenvalues evaluated via the WKB approach?


title: Question concerning Isospin symmetry

text: I'm currently taking an introductory course to particle physics and I'm now trying to understand the concept of isospin. However I do have some trouble. So let's write the up- and down Quark as a Flavour-Doublet (u,d). We then obtain $\mid u \rangle = \mid \frac{1}{2} \frac{1}{2}\rangle$ and $\mid d \rangle = \mid \frac{1}{2} \frac{-1}{2}\rangle$. Now I want to know what particles I can get by combining these two quarks. So let's write $\frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1$. So I have a singlett and a triplet. Most books now tell me that the triplet corresponds to the pion-triplet and the singlet to the $\eta$-particle. Now I can for example write by using Clebsch Gordon coefficients that $\mid 1,1 \rangle = \mid \frac{1}{2} \frac{1}{2}\rangle \mid \frac{1}{2} \frac{1}{2}\rangle = \mid u u \rangle$. But this doens't correspond to any pion (They are mostly products of ups and anti-downs etc). However some books interpret this as the product of two proton wave function. This is quite confusing? Can someone help me? I'd me more than happy. Thanks in advance.


title: Lorentz invariance of a frequency- and wavelength- dependent dielectric tensor

text: Suppose we have a material described by a dielectric tensor $\bar{\epsilon}$. In frequency domain, this tensor depends on the wave frequency $\omega$ and the wave vector $\vec{k}$. Clearly not all $\bar{\epsilon}=f(\omega,\vec{k})$ are physically possible. Without considering the physics behind any particular $\bar{\epsilon}=f(\omega,\vec{k})$, is it possible to find general conditions that all $\bar{\epsilon}=f(\omega,\vec{k})$ must obey? I am mostly interested in Lorenz invariance : surely the most general description of any material must be parametrized by the material's velocity according to the observer ($\vec{v}_{mat}$), and then there must be some sort of symmetry between $\omega$ and $\vec{k}$ in $\bar{\epsilon}=f(\omega,\vec{k},\vec{v}_{mat})$ because the Lorentz transformation mixes up distances ($1/k$) and time intervals ($1/\omega$).


title: Does superluminal travel imply travelling back in time?

text: Is this statement true? > You can not travel back in time. If I travel fast enough the clock will > start to go backwards, but that does not mean I am traveling back in time. > It would only mean that the time reference is producing a negative count.


title: What can quantum adiabatic computation provably accomplish?

text: Let's say I have a quantum adiabatic computer in a black-box that works perfectly, doesn't suffer from decoherence/noise problems, etc. Are there any proven bounds for an adiabatic algorithm that guarantees an answer within some constant factor of an optimum solution? Say, for the NP-hard bin-packing problem (http://en.wikipedia.org/wiki/Bin_packing), for which a variety of classical algorithms have proven approximation factors?


title: Inertia in an empty universe

text: I was reading a recent article on Mach's Principle. In it, the author talks about inertia in an empty universe. I'll quote some lines from the article: > Imagine a single body in an otherwise empty universe. In the absence of any > forces, (Newton's second law gives): $$m\mathbf{a} = 0$$ What does this > equation imply? Following Newton we would conclude from that $\mathbf{a} = > 0$, that is, the body moves with uniform velocity. But we now no longer have > a background against which to measure velocities. Thus $\mathbf{a} = 0$ has > no operational significance. Rather, the lack of any tangible background for > measuring motion suggests that $\mathbf{a}$ should be completely > indeterminate. And it is not difficult to see that such a conclusion follows > naturally, provided we come to the remarkable conclusion that $$m = 0$$ In > other words, the measure of inertia depends on the existence of the > background in such a way that in the absence of the background the measure > vanishes! I don't see how the argument is complete. For example, in an empty universe, how is it possible to assign a value of 0 to a force? And wouldn't the existence of mathematics in such an empty universe be questionable?


title: Nature as a calculator

text: I wondered whether physical experiments can find the outcome of otherwise intractable mathematical problems. For example, solitons or other non-perturbative effects in QFT cannot be seen at any order in perturbation theory, and are difficult to calculate, but could in principle be measured. Is nature doing the whole path integral/the whole perturbative expansion at once? Does that mean a computer could, in principle do it, in a finite time? or can nature solve math problems that are otherwise impossible?


title: How does quantum jitter "fix" the masses of strings?

text: So the masses of strings should be huge because of the high energy levels. Which means, in other words, the strings are vibrating with enough energy to have large masses as point particles. But quantum jitter offsets the high energy via negative energy, and makes the masses just right (what we observe). How does this occur? Where does quantum jitter come from? What is negative energy? Is quantum jitter changing the vibration/resonance frequency of the string? If no, how is it acting upon the string?


title: Is velocity quantized?

text: If velocity is not quantized, then do moving objects have 'infinitely decimal place' velocities which we just can't measure to infinite decimal places? From my understanding the quantization of velocity in Bohr's model is just a consequence of the quantization of angular momentum.


title: Relation between multipole moment and angular scale of CMB

text: What is the relation between multipole moment $l$ and angular scale $\theta$ of the Cosmic Microwave Background? Somewhere on the web I found that $\theta\propto\frac{180^{\circ}}{l}$ but I need exact relation.


title: Radiation exposure to a child who was briefly in the presence of an adult who had received a 18FDG PET scan

text: I am a physician who thought she was good at math, but apparently am not as I cannot figure out this mathematical/physics question. (My background is obviously NOT nuclear medicine!) A family friend recently asked for my professional opinion on the possible radiation exposure to the child in the following encounter. This family friend had had a PET scan with 18FDG (18-fluorodeoxyglucose). She received a total dose of 11.6 mCi of tracer. Approximately 5 hours after the injection she had a brief contact with a 2-3 year-old child (a girl). She was within 10 feet of the child for approximately 5-10 minutes. (She had direct contact with the child for less time.) She had voided perhaps once (or twice) after the injection and before the above contact, so a fair amount of the 18FDG had presumably been removed from her body via her kidneys. From the literature I have read online, I don't believe there was any significant radiation exposure to be concerned about. But I am unable to arrive at an approximation of the dose equivalent (rems) the child may have received. (I am not sophisticated enough to convert radiation activity (mCi) to dose absorbed (rads).) I would like to know if such a calculation is, indeed, possible, as a raw number (even if only an approximation) would be much more reassuring. The child can be assumed to weigh 16 kg and the adult weighs 72 kg. Thank you for your help!


title: Can the expansion space time reverse itself and contract the same way?

text: If there's a mechanism for space-time expanding faster than the speed of light, is there an example of It contracting in the same manner? If whatever mechanism is causing it to expand, can the underlying mechanism reverse itself?


title: Are there any theories or suggestions for how the multiverse came into existence?

text: I've just seen a documentary about the multiverse. This provides an explanation for where the big bang came from. But it leaves me wondering: how did the multiverse come into existence? Because this is a step further than something we don't even surely know the existence of, I'm only wondering if there are any theories or suggestions.


title: How are the Pauli matrices for the electron spin derived?

text: Could you explain how to derive the Pauli matrices? $$\sigma_1 = \sigma_x = \begin{pmatrix} 0&1 \\\ 1&0 \end{pmatrix}\,, \qquad \sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\\ i&0 \end{pmatrix}\,, \qquad \sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\\0&-1 \end{pmatrix} $$ Maybe you can also link to an easy to follow tutorial ?


title: Perfect fluids in cosmology?

text: In cosmology, it is often assumed that the equation of state of a cosmological fluid is of the form $p=w\rho$. Why is this? Is it the equation of a _perfect_ fluid? Why does $w=0$ for matter $1/3$ for radiation and $-1$ for a cosmological constant?


title: Lenses and wave optics?

text: We all have studied lenses in the framework of geometrical optics, but how do they work within wave optics? I figure that the topic is quite broad, but I would appreciate any hints, like which equations can be borrowed etc


title: A question from Weinberg QFT

text: I'm self-studying Weinberg QFT. I'm confused by his treatment of scattering theory . I have the following question: He introduces the free particle states $\Phi_{\alpha}$ but I'm not sure what is it good for? It has the same shape of In and out states so why introduce it? So this equation $$\tag{3.1.12} \int d\alpha e^{-iE_{\alpha}t}g(\alpha)\Psi^{\pm}_{\alpha}~\longrightarrow~\int d\alpha e^{-iE_{\alpha}t}g(\alpha)\Phi_{\alpha}$$ at the asymptotic time (eq 3.1.12) is not clear.


title: Why is there no current flow when i connect two opposite terminals of two separate batteries

text: I don't understand this..maybe there is something wrong with my understanding of battery or electricity. I am learning about electricity more deeply this time. So please spare with me... So lets begin with what i know:   * follow of current occurs when there is voltage drop.   * voltage drop happens when there is imbalance in charge.   * a battery is basically two separated chemicals. One side with electrons to spare and other side that needs electron to balance itself. which is imbalance in charge.   * When both sides are connected **\+ side** snatches electrons from **\- side**. (And if there is a medium, conductor, then first electrons are taken from conductor that in turn snatches from **\- side** , which not being able to keep electrons lose them and give up those electrons, we label it as excessive.) The analogy i was taught with used marbles in pipe... When one side has excessive marbles And this pipe is connected to other side that has no marbles the marbles will instantaneously start falling out so even though marbles from container with excessive marbles haven't traveled there yet, they still start falling because filled side marbles are pushing marbles in pipe and which in turn push the last marbles in pipe out. 1st, Aren't electrons **pulled** by atoms that can use more electrons, and not move because one side is overfilled with electrons so they can't **wait to push** some out. Questions:   1. if i connect a conductor to positive side of a battery, shouldn't it make all electrons/marbles fall into empty container as empty container will pull marbles as atoms need electrons?   2. if i connect two batteries' **only opposite sides** to each other shouldn't they drain (balance the each other out, cancelling all **-** & **+** charges)?


title: What is opposite of electric field?

text: A negatively charged molecule will generate a field, an electric field, and it will repel all negative ions. So what does positively charged molecule generates that causes other positive molecules to be repelled?


title: Why four velocity under covariant differential is considered to be zero?

text: In Einstein's general theory of relativity the elements of four velocity $U^{\mu} (\gamma c, \gamma v)$ under covariant differential is considered to be zero, why? $$\mathcal{D} U^{\mu}=0$$ in other word: > Why the four velocity of a geodesic has zero directional covariant > derivative? > > * * * wikipedia four acceleration


title: Is broken supersymmetry compatible with a small cosmological constant?

text: I understand that we can find the energy of a bosonic field in its vacuum state via $E_{vac}^{(B)} = \sum_{\vec{k},s} \frac{1}{2}\hbar\omega_{\vec{k},s}^{(B)}$ and a fermionic one similarly, $E_{vac}^{(F)} = \sum_{\vec{k},s} -\frac{1}{2}\hbar\omega_{\vec{k},s}^{(F)}$ since anticommutators and commutators are as a rule swapped for fermionic fields. If these fields have masses $m_B$ and $m_F$ respectively, we can estimate the total energy density of both fields in their vacuum state from dimensional arguments as $\Lambda \propto m_B^4 - m_F^4$ As I understand it, part of the appeal of unbroken supersymmetry is that if each excitation has a partner of opposite statistics/spin ("commutativity"?) but identical mass, then $\Lambda = 0$. But experimentally, we don't observe equal-mass superpartners, or even similar mass ones, so I see no reason $\Lambda$ is required to be "small". The current lower bound on the selectron mass is on the order of $10$ GeV, so it seems to me that this contribution is still required to be some $10^{51}$ orders of magnitude too larger than the value known value from cosmology, which is admittedly better than $10^{120}$ but still not very convincing. How is spontaneously broken supersymmetry salvaged?


title: Oscillations in forces other than the Weak

text: As I understand it neutrino oscillations arise due to the neutrino mass eigenstates being distinct from the neutrino flavour eigenstates. Flavour eigenstates are the states in which neutrinos interact via the weak force, and so are the eigenstates in which they are created and detected, and consist of a superposition of the the mass eigenstates. The mass eigenstates correspond to how the neutrino propagates through spacetime. The finite neutrino mass causes the relative phase of each mass eigenstate to change as the neutrino propagates, changing the probability that it will be detected in a given flavour eigenstate (hence an electron neutrino can oscillate into a muon neutrino and so on). My question is, can other forces exhibit similar oscillations? For example, in principle is it possible for an electron to oscillate it's charge state as it travels, to be detected as a positron some distance later? Or are charge eigenstates not the similarly related to mass eigenstates?


title: Cosmic ray noise, strategic cancellation

text: With the radio telescopes, we measure the radio waves produced by every solar fluctuation in the near universe, which as I understand is heard similar to radio static we experience in tuning to a non local radio wavelength.. Anyhow, I'm wondering if we can plot our receiving position from these stars, and reciprocate only the waveforms we know are coming from these stars to effectively cancel their 'static' effect and potentially hear the alien communications we have been listening for? Assuming they use radio communication over galactic distances. This could have secondary use as a selective measurement device. We might measure the flux of distant sun with little interference by cancelling outside waveforms.


title: Ground State Functional and Vacuum-Vacuum Transition Amplitude

text: In Path Integral formalism, the vacuum-vacuum transition amplitude is defined to be (the functional integration is over all field configurations in the whole spacetime; $\Phi_{\vec{x}}(\tau)$ is the field configuration at a given time slice $\tau$): $<0,+\infty|0,-\infty> = \int D[\Phi_\vec{x}(\tau)] e^{i S [\Phi(x)]}$ $(1)$ It doesn't make sense to me. Or, at the very least, I expect the answer to be different (I use the Heisenberg picture, where the operator is evolved with time, so the field $|\Phi_{\vec{x}},\tau>$ eigenstates of eigenconfiguration $\Phi_\vec{x}$ is also time-dependence). Indeed, adding the whole $\hat{1}$ ($|\Phi_\vec{x},\tau><\Phi_\vec{x},\tau|$) at every time slice ($\tau$): $<0,+\infty|0,-\infty> = \int d\Phi_{f,\vec{x}} d \Phi_{i,\vec{x}} <0,+\infty|\Phi_{f,\vec{x}},+\infty><\Phi_{i,\vec{x}},-\infty|0,-\infty>\int D[\Phi(\tau \neq \pm \infty)] e^{i S[\Phi]}$ Note that $<A,t|B,t>=<A,t=0|B,t=0>=<A|B>$, hence: $= \int d\Phi_{f,\vec{x}} d \Phi_{i,\vec{x}} <0|\Phi_{f,\vec{x}}><\Phi_{i,\vec{x}}|0>\int D[\Phi(\tau \neq \pm \infty)] e^{i S[\Phi]}$ $= \int d\Phi_{f,\vec{x}} d \Phi_{i,\vec{x}} \Psi_0[\Phi_{f,\vec{x}}] \Psi_0[\Phi_{i,\vec{x}}] ^*\int D[\Phi(\tau \neq \pm \infty)] e^{i S[\Phi]}$ $(2)$ If only $\Psi_0[\Phi_{f,\vec{x}}] \Psi_0[\Phi_{i,\vec{x}}]$ is the same for any boundary of integration, then the answer of $(2)$ is physically the same as $(1)$, however in general it's not the case, I believe. Indeed, the ground state wavefunctional is complicated, and it's hard to consider $\Psi_0[\Phi_{f,\vec{x}}] \Psi_0[\Phi_{i,\vec{x}}]$ as a constant or something like that. So, how one can find the wavefunctional $\Psi_0[\Phi_\vec{x}]$ ($<\Phi_\vec{x}|0>$)? Solving the functional Schordinger equation for the lowest value $E_0$ of the energy spectrum $E$, as far as I know (the answer for some special cases, such as non-interacting theories, can be found in literatures): $\hat{H}(\hat{\Phi},\nabla \hat{\Phi},\hat{\Pi}) |\Psi> = E |\Psi> \Rightarrow H(\Phi,\nabla \Phi, -i\frac{\delta}{\delta \Phi}) \Psi[\Phi] = E \Psi[\Phi]$ **My question is: How $(1)$ is derived in QFT as vacuum-vacuum transition amplitude?** I don't see how this equation can be right. There's a trick to derive it in Euclidean QFT, but seriously I don't buy it. However, I do agree that the equation can be used for lattice QFT, since the calculations must be done on Euclidean time.


title: Question regarding inverse relation of resistance with area of cross section

text: It is said that resistance is inversely proportional to area of cross section. But greater area will have greater electric flux, and greater electric flux will have greater magnetic flux, and greater the magnetic flux will have greater eddy current, which is opposite to the current flowing. So this will oppose the current hence resistance increases. Am I right?


title: Voltage in a circuit

text: Suppose I have a following circuit: ![enter image description here](http://i.stack.imgur.com/kVQie.png) I do not understand, why the potential difference between the points $c$ and $d$ is equal to the potential difference between the points $b$ and $a$? That is, why $\Delta V _{cd}=\Delta V_{ba}$? As I understand, potential is the potential energy per unit charge, and the potential difference tells us how much work shall we do in order to bring a test charge from one point to another. If so, why the potential differences mentioned above are the same? The distances aren't the same, so it seems obvious for me that the work should also be different.


title: Given a potential energy function, find the natural frequency of small oscillations about stable equilibrium

text: This was a problem given to me on a quiz and I am having trouble with mainly part b, I know how to do part c once I get my solution to b. A particle of mass $m$ is constrained to move along the positive x-axis under the influence of a single force whose potential energy function is $$ U(x) = \frac A{x^2} - \frac Bx + C $$ where $A, B, C$ are positive constants. (a) Find the equilibrium point(s). (b) For each stable equilibrium, calculate the natural frequency of small oscillations. Write your answer in the form $ \omega_0 = am^bA^cB^d $ and explicitly determine the numbers $a,b,c,d$. (c) Suppose that initially the particle is at position $x = 1$ meter. For what range of initial velocities (sign and magnitude) will the particle escape to infinity? For this part, take the specific values $A=1,B=3,C=7,m=5$ all in SI units.


title: What happens to 5 electrons on a sphere?

text: Let's suppose we put 5 electrons on a perfectly conducting (no resistance at all) sphere. There's no equilibrium configuration with 5 (though there is with 2, 3, 4 or 6). So will they keep moving on the sphere for ever ?


title: Hamiltonian or not?

text: Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example $$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$ where $\omega_i$ are constants. How can I know if there exist suitable momenta and a Hamiltonian?


title: With ideal gases, varying quantity of moles, and having a constant volume how do temperature and pressure behave?

text: I'm trying to build a simulation of gases so I ended-up trying to use law of ideal gases ($PV = nRT$). In my scenario:   * volume is constant ($V=1\rm{m}^3$);   * a known quantity of moles are being added to the system ($n_{added}$);   * both pressure ($P$) and temperature ($T$) are unconstrained. Imagine it is a closed container where you pump gas into it. In other problems I've seen that either temperature or pressure would be considerer constant, so you can use one to find the other. In my problem there is no restriction on them, so theoretically both of them could fluctuate. I am left with $$\frac{P}{T}=(n_{orig}+n_{added})R$$ where both $P$ and $T$ are unknown and I don't know how to solve this without making one of those a constant. So how does one work out the value of pressure and temperature when you add molecules to the system given a constant volume? In reality does one of them remain constant so I could treat it as constant in my equation? Maybe I'm on the wrong track using those equations and I apologise for my blatant ignorance. Thanks in advance.


title: Calculating projectile range from known maximum height and time traveled

text: I've been stuck on this problem for many hours and I think I'm onto the right solution but I'm uncertain about my math. I've got a projectile that I know its maximum height and it's hang time and I need to figure out it's range. Is it possible to calculate this with so few variables? At the moment I'm ignoring air resistance and trying to work with a perfect parabola. I can approximate the launch velocity from the time the ball spends in the air, with $9.8m/s^2 * dt/2$ calculating gravity's effect to reach the time the ball starts falling. I'm still pretty confused with calculating the angle only knowing these variables. $v_0$ is what I calculated as the launch velocity from the known time to peak height. $g = 9.8 m/s^2$, $h$ is the known height Using the maximum height formula, I've worked out the following. $h=(v\sin\theta)^2/2g$ $2gh=(v_0\sin\theta)^2$ $\sqrt{gh}=v_0\sin\theta$ $v_0 = g * t/2$ $\sqrt{2gh}=(4.9 m/s^2 * t)\sin(\theta)$ $\sin\theta = \sqrt{(2gh}/(g*t/2)$ I used wolframalpha to solve for $\theta$. $\sin(\theta) = 2\sqrt{2}*\sqrt{gh}/gt$ $\theta=\sin^{-1}(2\sqrt{2}*\sqrt{gh}/gt)$ Am I on the right path with this? Any ways I can factor in wind resistance to this? Any help would be greatly appreciated.


title: Can light be canceled by merging with an inverted wave?

text: Can light waves be canceled by merging them with their inverted waves? Seems like it would violate conservation of energy but waves are added together when they overlap, right? Where is the flaw in this logic? I'm thinking polarized laser light, added to its opposite, might becomes dark again. \-- Appreciating your collective wisdom.


title: What would happen to electronic circuits when traveling near the speed of light?

text: Imagine a space ship, loaded with all sorts of computer systems, traveling near the speed of light. Electricity itself is very fast, and can reach speeds close the speed of light. (up to 99% according to wikipedia). So, what would happen to the electronic circuits in this spaceship? Will the computers shut down, because electricity can't reach the components? Or are they just not related to each other and will the computers keep working perfectly?