title: Matrix operations on Quantum States in a composite quantum system

text: **Intro (you may skip this if you're an expert, I'm including this for completeness):** Say I have two bases for two systems, The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\\0 \end{array}\right),|-\rangle=\left(\begin{array}{c} 0\\\1 \end{array}\right)$ The second is a spin-1 system, with $|1_+\rangle=\left(\begin{array}{c} 1\\\0\\\0 \end{array}\right),|1_0\rangle=\left(\begin{array}{c} 0\\\1\\\0 \end{array}\right),|1_-\rangle=\left(\begin{array}{c} 0\\\0\\\1 \end{array}\right)$ Now for the first system, I can use the Pauli matrix $$\hat{S_z}=\frac{1}{2}\hbar\hat{\sigma}_z = \left( \begin{array}{cc} -\frac{\hbar }{2} & 0 \\\ 0 & \frac{\hbar }{2} \\\ \end{array} \right)$$ in order to get the projection of my state on the z-axis. Likewise, I could use the projection matrix $$\hat{J}_z=\hbar\left( \begin{array}{ccc} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & -1 \\\ \end{array} \right)$$ To project the other state on the z-axis. Those operators will act on my basis in the following way: $$\hat{S}_z|+\rangle=\frac{\hbar}{2}|+\rangle$$ $$\hat{J}_z|1_+\rangle=\hbar|1_+\rangle$$ **Problem: (here comes the question)** So far everything is good! Now the problem comes when I introduce a space for the composite system, so I'm getting the basis $$|S_z\rangle\otimes |J_z\rangle\rightarrow\left\\{ |+,1_+\rangle,|+,1_0\rangle,|+,1_-\rangle,|-,1_+\rangle,|-,1_0\rangle,|-,1_-\rangle\right\\}$$ Now the question is: how do I use the matrix formalism to have such operations just like I had them before in the single systems: $$S_z|+,1_+\rangle=\frac{\hbar}{2}|+,1_+\rangle$$ $$S_z|+,1_-\rangle=\frac{\hbar}{2}|+,1_-\rangle$$ $$J_z|+,1_0\rangle=0|+,1_0\rangle$$ $$J_z|+,1_-\rangle=-\hbar|+,1_-\rangle$$ In other words, how do I write the state-kets and the operators in the composite system in matrix formalism (just like I showed in the beginning) to give results compatible with what I would expect in the examples? Is this wrong in some way? Every time I try to do this with Kronecker Product (like $\hat{S}_z \otimes \hat{J}_z$) I arrive at a mess, and I get terms proportional to $\hbar^2$, and I don't get the eigen-values I expect, and I'm not sure what I'm doing wrong. Could you please show me how to do this? Thank you.


title: A relativistic meter stick and a thin disk

text: I have a question like the "pole in the barn" special relativity "paradox", but I'm not sure what to make of this: * * * ## Question A meter stick lies along the $x$-axis and approaches the origin, moving along its length, with velocity $v_x$. A very thin plate, parallel to the $xz$ plane in the laboratory, moves upward in the $y$ direction with velocity $v_y$. The plate has a circular hole with diameter $1$ m centered on the y-axis. The center of the meter stick arrives at the origin at the same time as the center of the hole in the plate. In the laboratory frame, the meter stick is Lorentz contracted, so it's in the hole in the plate, and plate and stick continue on their paths without a collision. In the rest frame of the meter stick, however, it is the plate, and the hole in it, that is contracted along the $x$-axis. This would seem to predict a collision, contradicting the requirement that physical predictions must be invariant. Which of these two predictions are correct? * * * My thinking is that simultaneity will play a role here, but I'm not sure how to write down the Lorentz contraction of the disk from the meter stick's point of view. It will be contracted in both the $x$ and the $y$ direction by a factor of $\gamma_y = {1\over \sqrt{1 - v_y^2}}$ but will this cause the disk to seem "tilted" from the stick's point of view? A hint would be great thanks.


title: Which particle rolls faster? Is the speed of the particles which are rolling differs with their shape?

text: A solid disk and a solid sphere each of them has the same mass and same kinetic energies, which one is faster?


title: Further explanation of the Penrose Conjecture

text: I'm currently a third year maths undergrad, writing a dissertation on the application of minimal surfaces in space. I have recently come across the Penrose Conjecture that the mass of a spacetime is: $$ m \geq \sqrt{\dfrac{A}{16 \pi}}.$$ What significance does this have in terms of the geometry of a black hole? Also how does this directly relate to that of minimal surfaces? In that, is it purely to do with the apparent horizons corresponding to a minimal surface?


title: Charged quantum particle in a magnetic field - choosing a different gauge leads to different wavefunctions

text: Consider a charged quantum particle confined to the $xy$ plane, subject to a magnetic field $\mathbf{B}=B\hat{z}$.   The Hamiltonian is: $$ H = \frac{1}{2m} \left( \mathbf{p} - \frac{e \mathbf{A}}{c}\right)^2~, $$ where $\mathbf{A}$ is the vector potential, for which we have a freedom of gauge in choosing. One possible choice is $\mathbf{A} = B x \hat{y}$, which leads to the Hamiltonian: $$ H = \frac{1}{2m} \left[ p_x^2 + \left(p_y- m \omega_c x\right)^2\right],$$ where $\omega_c = eB/mc$ is the cyclotron frequency. Following the usual derivation of Landau quantization, we get the wavefunctions: $$ \psi(x,y) = f_n ( x- k_y / m \omega_c ) e^{i k_y y},$$ where $f_n$ are the eigenfunctions of the simple harmonic oscillator ($n=0,1,2...$). However, we can also choose $\mathbf{A} = -B y \hat{x}$, which following the same lines would lead to the wavefunctions: $$ \psi(x,y) = f_n ( y+ k_x / m \omega_c ) e^{i k_x x}.$$ **The two outcomes seems different by more then just a global phase, yet all we did was use our freedom in picking a gauge.**   How does this work out?


title: Would not gravity negate entropy?

text: Back in high school, I asked my teacher gave us a quick explanation of relativity. Specifically, he told us what $E=mc^2$ meant. He explained that, at least as far as we needed to be concerned, matter is simply condensed energy. This, to me, was amazing. Obviously, it's not exactly that simple. Condensation isn't something that just happens with energy. However, it does mean that the forces which act on matter also act on energy. Now, I am aware that gravity is a relatively weak force. Years after taking physics in both high school and college (though never at any real "scholarly level", I'm afraid), I remember asking my teacher something that made him smile and admit "I cannot answer that, and that is why I am a high school physics teacher and not an applied physicist. Questions like that are going to get you very far in this field." That question was this: **If gravity acts on all matter and all energy in the universe, wouldn't the universe eventually condense into pockets of energy and matter, after entropy causes the "heat death" of the universe?** Edit: better wording Why wouldn't the same coalesce nice of matter and energy happen after the heat death of the universe? If not, why? Please note that I don't necessarily believe that the high-entropic state of the universe in billions and billions of years will coalesce into matter. I just want to know why entropy would continue.


title: Superposition of wavefunctions

text: Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then is the normalization factor for the superposed wavefunction? Perhaps an argument by the symmetry?


title: Relativistic effects

text: When are relativistic effects justifiably negligible? (I know that that is true for 'small velocities', but how small is 'small enough'?) 0.1c, 0.01c, etc.? And how does one properly justify that? I reckon by Taylor expansion of the relativistic formula, but still, how do you set the threshold speed? Thanks.


title: Line width of green DPSS laser

text: Could someone suggest what is the line width/mode stability/coherence length of generic cheapo green (532nm) DPSS laser? I've found lots of nice info here: http://redlum.xohp.pagesperso- orange.fr/laser/modeanalysis.html but there are no any DPSS lasers.


title: Can a particle have momentum without energy?

text: Can a particle have linear momentum if the total energy of the particle is zero? Even if a particle has a certain velocity, can its potential energy cancel out the kinetic energy as to add to zero ?


title: How are light and time related?

text: So, from what I understand: Special Relativity says that light is always observed moving at the speed of light (c). If some object had a velocity of (3/4)c, and the object had some sort of clock attached to it, it would measure differently from a still clock. 1) What would a clock moving the speed of light measure compared to a still clock? 2) Why does it seem that the speed of light is a focal point of time? EDIT: I may need to rephrase this slightly. What I want to know is, why is it that observed time changes at high velocities?


title: Determination of entangled states

text: The definition of an entangled state $|\Psi\rangle$ is that it CANNOT be factored into $$|\Psi\rangle=|\psi\rangle_1\otimes|\phi\rangle_2$$ I am kind of confused on what is meant by a quantum factoring. For example, my professor gave us an exercise to do ourselves: $$|\Psi\rangle_{12}=\frac12(|+_z\rangle_1|+_z\rangle_2+|+_z\rangle_1| -_z\rangle_2+|-_z\rangle_1|+_z\rangle_2+|-_z\rangle_1|-_z\rangle_2)$$ I believe that this is not entangled, because it would be the product of states with total j=1 and total j=0. Is that correct, and how could I show this mathematically?


title: Time for which a drop stays in the leidenfrost point

text: Is there any way to find out the time required for a drop of given dimensions to vaporize after attaining Leidenfrost point?


title: Pseudocubic unit cells: how to construct one?

text: I keep coming across the term `pseudocubic unit cell` while reading about `orthorhombic perovskite` structures. No clear explanation is given in papers. Can someone please tell me how these structures are generated and what exactly is meant by them?


title: Electric field in a sphere with a cylindrical hole drilled through it

text: Suppose that you have a sphere of radius $R$ and uniform charge density $\rho$; a cylindrical hole with radius $a$ ($a\ll R$) is drilled through the center of the sphere, leaving it like a "necklace bead". I would like to find a function for the electric field (1) very far away from the sphere ($r\gg R$) and (2) inside the hole, near the center of the bead $r\ll R$. In case (1), I simply treat it as a point charge and calculating the electric field is trivial. However, I am uncertain how to approach part (2) and would appreciate any assistance. The combination of spherical and cylindrical geometries seems to make this quite tricky. I am unsure what approximation or simplification to make from the knowledge that $r\ll R$. Would it perhaps be correct to find the electric field from (1) a complete, uniformly charged sphere and (2) a cylinder of charge density $-\rho$? Summed together, the charge densities would result in our original "bead" system, so then I can just add together the expressions for the electric field. Doing case (1) is quite easy, but (2) is nontrivial for positions that are not along the axis of the cylinder, but perhaps due to our condition that $r\ll R$ and $a\ll R$, we can assume that the field from the cylinder along the $z$-axis is a good enough approximation.


title: Power of viscous friction on a falling sphere

text: I have derived a simple model of a rotameter using an homogeneous solid ball in a rigid cone where a fluid flows. I consider 4 forces: Weight, Buyancy, Viscous Friction and Drag. I have written my forces balance, and I made no assumption about momentum balance: $\sum\limits_i\vec{F}_i = \vec{0},\quad \sum\limits_i\vec{M}_i\neq\vec{0}$ I also found useful dimensionless relations in Perry's Chemical Engineering Handbook. So I am able to estimate terminal velocity of the sphere in a fluid for a given flow domain (Stokes, Allen, Newton). My questions are the following: 1) I would like to estimate the power required to maintain the plunger at its position. Is it right to estimate it that way? $P_\mathrm{trans}=\vec{F}_W \bullet \vec{v}_s$ Where $\vec{F}_W$ is the only driving force (Weight or may I withdraw the Buyancy) and $\vec{v}_s$ the terminal velocity of the sphere. 2) I would like to estimate the power absorbed by the plunger rotation. Can I do it the same way? $P_\mathrm{rot} = \vec{\tau}\bullet\vec{\omega}$ If so, how can I estimate $\vec{\tau}$ if I have a good estimation of $\omega$ but I have no idea how $\vec{\omega}$ vary? It should help to underline if rotation is negligible in front of suspension. 3) Knowing those two powers, can I estimate the pressure drop $\Delta \xi$ for a given volumetric flow $\dot{V}$ with the following formula? $P_\mathrm{tot} = \dot{V}\cdot \Delta \xi$


title: Shock speed in air/vacuum shock tube

text: Some of you are probably aware of What If, xkcd's blog about interesting physics problems. One episode, Glass Half Empty, concerns itself with what would happen if a **glass of water is half water, half vacuum**. One scenario he looks at is water in bottom, vacuum on top, with air rushing in from the top end. He says that after a few hundred ns, the air has filled the vacuum completely. **I would very much like to know how he arrives at that estimate!** _What I thought about so far:_ He cites a max speed of the air molecules of ~1000m/s, but this can't be the only mechanism, cause after 500ns the fastest molecules would've only travelled about 0.5mm. I figured that the air rushing in would have to happen via a shock, and that the situation described is essentially a **shock tube** with air as driver and vacuum on the driven side (neglecting the vacuum/water interface for convenience). **However, I could not find a formula which would let me calculate the shock speed (in m/s) for the given configuration.** Specifically, I'm not sure the "standard" handbook formulae even apply here, because of vacuum as "driven gas". Can anybody shed light here? Conditions: Let's say it's dry air at 300K and 1013mbar, and perfect vacuum (or very low pressure if that's necessary). I'm not interested in what happens after the shock arrives at the end of the tube, so no need to open that can of worms. Thanks for any insight!


title: How do physicists justify using $c^2$ in equations?

text: Excuse my ignorance but I see physicists use $c^2$ in equations and I was wondering what reasoning they use to justify $c^2$ since, as far as I understand, $c^2$ must be an absurd quantity because there is a speed limit imposed by physicists themselves at $c$. If there is no motion faster than $c$, how do you justify using $c^2$ in an equation?


title: prove that flat shape minimizes a functional

text: The following functional arises in an information theoretic problem that I work on currently. $I(G(\omega)) = \int\limits_{-\kappa\pi}^{\kappa\pi}d\omega \frac{A}{G(\omega)+A}-\frac{| \int\limits_{-\kappa\pi}^{\kappa\pi}d\omega \frac{A}{G(\omega)+A}\exp(-i\omega)|^2}{ \int\limits_{-\kappa\pi}^{\kappa\pi}d\omega \frac{A}{G(\omega)+A}}$, where $\kappa<1$, $A>0$, and $G(\omega)\geq 0$. Now I would like to minimize $I(G(\omega))$ under the constraint of unit area of $G(\omega)$, i.e., $\int\limits_{-\kappa \pi}^{\kappa \pi}d\omega\, G(\omega)=1$. My hypothesis is that a flat $G(\omega)=\frac{1}{2}\kappa\pi$ is optimal, but I cannot prove that (Matlab hints towards it).


title: How should one combine the uncertainties from the ATLAS and CMS measurements?

text: First off, a naive theorist question - How are measurements divided between the different detectors at the LHC? I would imagine that for a short run time, say, the CMS detector is active and all the others are turned off (the beam just passes through) and the beam is directed so as to collide in the CMS detector. Then for another run period, ATLAS is activated and collisions happen only there, giving another set of data. This would mean that the measurements at ATLAS and CMS are completely independent. Is this correct? Given that, maybe I missed it, but I never saw anyone quote a combined average and uncertainty for both the ATLAS and CMS excesses. If, let's say, there's a 1 in 300 chance that the CMS excess is a statistical fluke and 1 in 200 that the ATLAS excess is a statistical fluke, as someone with a very limited understanding of probability and statistics, I would just multiply the probabilities to see how likely it isn't a Higgs signal and looking at the small value, would be very excited. How does one go about rigorously calculating the combined likelihood that the excess is not a Higgs signal?


title: Would a sea level rise (on the other side of the Earth) if you would dip your finger into the ocean?

text: (I'm not a physicist & I hope this question isn't too basic/lame) The question says it all & assuming:   * No atmosphere    * No Earth rotation   * Not a single object in the ocean(s)   * Ignore all other disturbances (earthquakes, etc.)   * The ocean is 'perfectly' still If (due to-me-unknown reasons) the level wouldn't rise, how big an object had to be, so the sea/ocean level rise would be measurable?


title: What routine tasks have to be performed on the outside of spaceships/stations?

text: What kind of routine tasks have to be performed on the outside of spaceships or space stations? Repairs, I presume. Anything else? Is there anything that needs to be cleaned off from the outside, like debris, or some other substance? Can anything on the surface freeze?


title: Why is the probability that Kip could go back and kill his grandfather  $10^{-10^{60}}$?

text: In _The Universe In A Nutshell_ , page 153, Hawking says that > _The probability that Kip could go back and kill his grandfather is > $10^{-10^{60}}$._ How do we get this result?


title: valence bands in graphene

text: In Graphene, each carbon use 3 electrons to form sp2 bonding with neighboring, and in a unit cell, there are 2 carbon atoms, so at least these 6 electrons contribute to 6 valence bands. Then my question is, why the two Pz orbital electrons in each unit cell consist one conduction and one valence band, but not both valence bands? thanks


title: What determines the charge of an electron?

text: So I am thinking of making a machine that turns electrons into positrons. So you put as many electrons as you want in one side and positrons come out the other side. This will solve all out energy problems! But before I start my project I need to know what determines the change of an electron. Right now, the only way to make antimatter is particle accelerators but these are super uneconomical and inefficient so I want to make a machine that is economical and efficient but I don't know how... if my machine is physically impossible to make, theoretically, what machine will allow me to do this? Will it ever be invented or is it just too impossible to make?


title: Dirac equation in curved space-time with Torsion

text: I am looking for pedagogical references in which Dirac equation in space-time with curvature and torsion were discussed.


title: Why are intensive variables easier to control than extensive?

text: In a IUPAC Technical Report ( ** _Alberty** , Use of Legender Transforms in Chemical Thermodynamics_, pdf) I have found the following assertion (p. 1359): > "Intensive variables are introduced because they are often easier to control > than extensible variables." Without any further explanation. I does not hurt my scientific feeling but I am looking for a good justification. I will assume that control means you are first able to measure it and then you may be able to regulate it. I would agree with this affirmation for system at equilibrium. In this case, measuring intensive variables (such as temperature, pressure or wind speed) is not a big deal because the scalar/vector field is homogeneous or easily modeled and you do not need to analyze the whole system in order to complete your measurement. Instead accurately measuring volume or amount of matter might be not trivial (eg. for atmospheric system). But how could this affirmation stand for more general case?


title: Weird consequence of Gauss's law

text: According to Gauss's Law, the electric field at a surface is the function of only the charge enclosed inside it. But that doesn't make sense. I mean, if I put the surface in an electric field, won't the resultant electric field at the surface change?


title: Is Hyperbolic Space $H3$ the best representation space for momenta, in momentum-scale invariant theories?

text: The motivation is the following: For each particle of mass m, we could write : $- (E/m)^2 + (p_x/m)^2 + (p_y/m)^2 + (p_z/m)^2 = -1$, which is nothing than a equation for a point in the hyperbolic space $\mathbb H3$, which can be seen, with Poincaré representation (Poincaré ball), as a (full) sphere (with hyperbolic distance) For photons, we may give us a very tiny mass, so photons could be considered, in the limit where this tiny mass is zero, as living at the boundary of $\mathbb H3$, which is $\mathbb CP1$ (Riemann sphere $S^2$) With these prescriptions, we cannot distinguish, photons with proportionnal momenta, but if the theory is momentum-scale invariant, it seems OK. So could we use Hyperbolic Space $H3$ as representation space for momenta, in momentum-scale invariant theories?


title: About turbulence modeling

text: I have some questions about this paper: > Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids. R. > J. Becker. _Phys. Rev. Lett._ **58** no. 14 (1987), pp. 1419-1422. After reading the paper, the question arises how far can we investigate turbulence with this approach? With all the mathematical machinery available to us in solving classical field theory and QFT, will it be of any help if we start with a Lagrangian density in taming the turbulence problem?


title: Bogolubov coefficient identities

text: Along the lines of Birrell and Davies, which contrary to Mukhanov-Winitzki (which is actually my level) gives a quite general but short account on Bogolubov transformations, I tried to follow the formula and obtained a little different result. Before thinking too much about it I rather ask here, but first a summary of the notation and logic:   1. Suppose there are two "complete" sets of solutions $\\{u_i\\}, \\{\bar{u}_i\\}$ to some eigenvalue problem that enable us to expand a general solution to the operator equation with periodic boundary conditions (see my question Scalar product for scalar field mode functions ) as $\phi(x) = \sum \left[ a_i u_i(x) + a^\dagger_i u^*_i(x)\right]$ and $\phi(x) = \sum \left[ \bar{a}_i \bar{u}_i(x) + \bar{a}^\dagger_i \bar{u}^*_i(x)\right]$ and there is further some scalar product and both sets are orthonormal, i.e. $(u_i,u_j)=\delta_{ij},(u_i^*,u_j^*)=-\delta_{ij}, (u_i,u_j^*)=0$ and similarly for the other set.    2. It is clear that we must have $\bar{u}_j = \sum_i (\alpha_{ji}u_i + \beta_{ji}u_i^*)$ for some coefficients.    3. It is also clear that these coefficients are given by $\alpha_{ij} = (\bar{u}_i,u_j), \beta_{ij} = -(\bar{u}_i,u_j^*) $   4. It then follows from the properties of the scalar product that the expansion of $u$ in terms of $\bar{u}$ has related coefficients, $u_i = \sum_j (\alpha_{ji}^*\bar{u}_j - \beta_{ji}\bar{u}_j^*)$   5. From equating both $\phi$'s and doing the and the other replacements it is then clear that $a_i = \sum_j(\alpha_{ji}\bar{a}_j+\beta_{ji}^*\bar{a}^\dagger_j)$ and $\bar{a}_j = \sum_i(\alpha_{ji}^*a_i-\beta_{ji}^*a^\dagger_i)$   6. From demanding that both $a$ and $\bar{a}$ have the right commutation relations we find a condition for the Bogolubov coefficients. There are two ways to do this (starting with the commutation relation of $a$ or $\bar{a}$) and what I find is:   1. $\delta_{ik}=\sum_n ( \alpha_{kn} \alpha_{in}^* -\beta_{in}\beta^*_{kn})$ and    2. $\delta_{ik}=\sum_n ( \alpha_{ni} \alpha_{nk}^* -\beta_{nk}\beta^*_{ni})$ This suggests that $\alpha_{ij} = \alpha_{ji}^*$. Question: is this true? How to see it? (Maybe it is very easy, but I am too tired now) Then, what Birrell-Davies says, is instead $\delta_{ik}=\sum_n ( \alpha_{in} \alpha_{kn}^* -\beta_{in}\beta^*_{kn})$ Now, this would only be true (modulo calculation mistakes I made) if $\sum_n \alpha_{in} \alpha_{kn}^*$ is real! Question: is this true? How to see it? Thank you!


title: How deep is the well, given this data?

text: A rock is dropped into a well. A timer starts when the rock is dropped and is stopped when the noise of the rock hitting the ground of the well is heard. How deep is the well? Here's what I have so far: $t = \left(\left(\frac{2\times h}{g}\right)^{\frac{1}{2}}+\left(\frac{h}{340}\right)\right)$ when I try to use the quadratic formula, I can express $h$, but when substitute different amounts for $t$, it doesn't seem right. I get a very small and a very large value for $h$. For example for $t=30\:\mathrm{s}$ I get $3\:\mathrm{m}$ and $42\:\mathrm{km}$, which is quite odd. I also tried it with the following set of equations: $$\frac{1}{2}\left(g\times t_1^2\right)=h$$ $$t_2\times 340 \:\mathrm{m/s}=h$$ $$t_1+t_2=t$$


title: What causes a force field to be "nonconservative?"

text: A conservative force field is one in which all that matters is that a particle goes from point A to point B. The time (or otherwise) path involved makes no difference. Most force fields in physics are conservative (conservation laws of mass, energy, etc.). But in many other applications, the time paths DO matter, meaning that the force field is not "conservative." What causes a force field to be "non-conservative?" Could you give some examples (probably outside of physics)?


title: What are the expressions for rotational and translational kinetic energies of a system of point particles?

text: Consider a system of point particles , where the mass of particle $i$ is $\mu_i$ and its position vector is $r_i$. What are the expressions for translational kinetic energy and rotational kinetic energy of the system of particles ? The expression for the total kinetic energy for the system is given as $$\frac{1}{2}\sum_i \mu_i(\dot{r}_i\cdot\dot{r}_i).$$ The sum of the rotational and translational kinetic energies should be equal to the total kinetic energy of the system. My guess is that the translational kinetic energy of the system is given as $$\frac{1}{2}(\sum_i \mu_i)(\dot{r}_{cm}\cdot\dot{r}_{cm}),$$ where $r_{cm}$ is the position vector of the center of mass of the system. But I am not able to get the expression for the rotational kinetic energy of the system and appreciate some help.


title: Really basic mechanics and Galilean Relativity question

text: Consider two solid objects: A and B. System 1:               A        <----- B               10m/s      System 2:               A ----->        B       10m/s      They are identical except for the reference frame. Common relativity tells us they are equivalent and will evolve in the same way, right? Now add masses: A weights 1kg and B weighs 10kg. In system 1, B will hit A with 500J of kinetic energy. In system 2, A will hit B with 50J of kinetic energy. So... the energy involved in the impact (and effects such as how far debris is flung) depends on what reference frame I'm in? This doesn't seem right. How do we resolve this?


title: What if the 126 GeV particle found at LHC that LOOKS like the Higgs is actually not the Higgs Boson?

text: I am suspicious of the Higgs announcement. Press mania aside, the scientists seem careful to say only that "we've found something which _looks_ like the Higgs Boson". The difference between Higgs-containing theories and non-Higgs-containing theories is a tiny bump on the energy curve, smaller than the resolution of the detectors, which is why we have to perform statistical gymnastics on large data sets to discover or refute the deviation. In other words, the energy distribution of the reactions is consistent with there being a short-lived particle weighing 126GeV, and that's all the announcements seem to really say. Is there anything else in the results which tells us that the extra element is Higgs and not something else? Is there any identifying feature other than lifespan and mass?


title: Laplace's Equation - under what circumstances does it hold?

text: I'm currently taking an EM course whereby we deal with systems that satisfy Laplace's equation $\nabla^2 \phi = 0$. Examples include permeable sphere in a magnetic field and metal sphere in electric field, where the problems are solved usually through boundary conditions. But firstly, under what circumstances does Laplace's equation hold? I'm thinking there needs to be a surface of constant potential? But is that all is required?


title: Which kinds of Physics laws do and don't comply with the principle of relativity?

text: In Physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. However, according to this and this paper, it seems Bernoulli's Equation is a frame-dependent example in both Newtonian and relativistic mechanics. **Which kinds of laws are frame-independent and which are not?**


title: Given two surfaces, one that absorbs light and the other that reflects it, will there be a signficant temperature difference?

text: My likely wrong understanding of light is this: By some way the molecules are structured, a photon hits an atom or a structure, is absorbed, then re-emitted and it's that photon wavelength that I see as light when my eyes absorb the photon/wave. So if say something is painted black, does it follow that it would absorb light? And if so, when compared to an object that was painted white or not naturally black in color would it have a cooler temperature than the surface painted black?


title: D'Alembertian and Laplacian Green's Fucntions

text: There is a way to obtain the Green's Function for the Laplacian as a limit of the Green's function of the D'Alembertian? For the Laplacian ($-\nabla^2$) we have $$ G_1(\vec X) = \frac{1}{4\pi X}$$ And for the D'Alembertian ($\Box$) using the retarded prescription we have $$ G_2(T,\vec X) = \frac{1}{4\pi X} \delta(c T-|\vec X|) $$ I suppose that a way to go from $G_2$ to $G_1$ would be take the limit $c\rightarrow \infty$. That supposition is based on: $$ \left. \Box \right|_{c\rightarrow \infty}= \left. \frac{1}{c^2} \partial_t^2\right|_{c\rightarrow \infty} - \nabla^2 = - \nabla^2$$ The problem is that I can't see a way to have something like $$ \lim_{c\rightarrow \infty} \delta(c T-|\vec X|) = 1 $$ What's going on?


title: What allows for more traction: Rubber or skin?

text: Suppose I am on a soccer field running, and the field is composed of grass and soil. I also own a pair of shoes in the shape of feet, so when I am put them on, it's really just rubber protection. I am wondering: What would offer better traction, so I slip less: rubber, or skin? What is the larger coefficient of friction between both? On dry soil, wet soil?


title: Popular depictions of electromagnetic wave: is there an error?

text: Here are some depictions of electromagnetic wave, similar to the depictions in other places: ![enter image description here](http://idol.union.edu/malekis/Phys%20&%20Pol%202010/Images/e&mWave.gif) ![enter image description here](http://micro.magnet.fsu.edu/primer/java/wavebasics/basicwavesjavafigure1.jpg) ![enter image description here](http://www.who.int/entity/mediacentre/factsheets/images/fs205.gif) Isn't there an error? It is logical to presume that the electric field should have maximum when magnetic field is at zero and vise versa, so that there is no moment when the both vectors are zero at the same time. Otherwise one comes to a conclusion that the total energy of the system becomes zero, then grows to maximum, then becomes zero again which contradicts the conservation law.


title: Where does light energy go when the electric and magnetic components equal zero?

text: > **Possible Duplicate:**   >  Popular depictions of electromagnetic wave: is there an error? The electric and magnetic components of light increase and decrease at the same time, pass through zero magnitude and then increase in the opposite direction. When |electric component|= 0 =|magnetic component|, where does the energy go? I asked my physics professor this and he said that it went into momentum, which allowed the fields to disappear into and emerge from seemingly nothing. I remain skeptical. Is there an equation somewhere that describes this momentum? If not, is there another explanation?


title: Simultaneously commuting set

text: How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these are all of them or that these particular operators should be included? Is it because such a set is sufficient to distinguish all the eigenstates, degeneracies notwithstanding? Also, why is $$L^2|\ell,m\rangle = \hbar \ell (\ell+1)|\ell,m\rangle$$ and $$L_z|\ell,m\rangle = \hbar m|\ell,m\rangle,$$ where $\ell,m$ are integers? Is it defined as such or is it the result of the definition of some more fundamental things?


title: What is the likelihood of ever discovering the graviton?

text: How would one look for and confirm existence of a graviton? Someone was speaking to me about perhaps one day discovering the graviton, but to me it seems unlikely, although I'm young and essentially quite naive, so am coming to you physicists to ask   1. What actually is the likelihood of finding it?    2. How would we find it?


title: Emission and absorption spectra of sun

text: I heard in a documentary that the elements of the sun can be read out from the absorption lines. But I have also heard that an elements absorption lines and emission lies coincide. That means the sun can emit only the colors that its elements absorb. So if that is the case where does all the other colors (which are not in the absorption spectra) of the sunlight come from? And why aren't they being absorbed? Thanks a lot.


title: Is it possible that only one hemisphere of a planet has an atmosphere?

text: Suppose there is a tidally locked planet orbiting a star. The planet's surface consists of a global ocean, that is, liquid water. At the inner hemisphere the temperature is so high that the water is constantly boiling, creating the atmosphere of water vapor. But the vapor does not reach the dark side of the planet, and precipitates around the terminator line. The further side has no atmosphere and an icy surface. Is this setup possible? Can it be such thing that one side of a planet has liquid (and boiling) water surface while the other side has no atmosphere? Is it possible at all that a part of an ocean was boiling while the other one was icy?


title: Omega-frequency or pulsation?

text: I'm reading a book about optics in English. However, I speak a different language and have a problem with distinction. What does $\omega$ mean? $\omega=1/T$ or $\omega=2\pi/T$ What is the convention?


title: Characteristic length of differential element of cylinder surface?

text: I am trying to find the Nusselt number for a small element of the outside of a cylinder that has a height of $Delta z$. I found the average Grashof number of a surface as $$Gr_{L}=\frac{\beta \rho (T_{s}-T_{\infty})L_{c}^{3} }{\nu^{2}}$$ If I wanted to find $Gr_{x}$ of a certain element, would I use a characteristic length of $\Delta z$ or could I find the z-location of the elements centroid and use that as the $L_{c}$. For example, If the cell was located at a z coordinate of 1 m, would that be the characteristic length or would the characteristic length be the height of the differential element? I was thinking that I could treat each element as a small isothermal cylinder and have the characteristic length of each element be $\Delta z$ Also, does anyone know the a Nusselt number correlation for a vertical cylinder with a Prandtl number of less than one? I have only found one for when it is greater than 1. Thank you


title: number of microstates associated with two-level quantum systems

text: this is a very simple question, but apparently one that has no simple answer, at least from standard quantum mechanics theory I'm trying to figure the number of simple quantum states (microstates) of the allowed configuration space of a two-level quantum system, ignoring for now mixed states. So if i have a state like: $$ \Psi = \psi_0 | 0 \rangle + \psi_1 | 1 \rangle $$ removing all redundancies from the representation, we are left $$ \psi_0 = \rho_0 $$ $$ \psi_1 = \rho_1 e^{-i \theta_1}$$ $$ \rho_0^2 + \rho_1^2 = 1$$ Now, this state space looks like $[0,1] \times S_1$. I've seen discussions that quantum states take some volume in phase space, but i want to understand that in detail: so in this case, > **Question:** how many distinct states are in this configuration space? is > this a finite set? infinite but numerable set? nonnumerable? **Following Steve suggestion** , lets write the Shannon entropy of this wave function: $$ S( \Psi ) = - \rho_0 \ln \rho_0 - \rho_1 \ln \rho_1 $$ There is some conflict when i see this entropy; it cannot derive from any counting state scheme, in particular because it completely ignores the relative phase factor between the basis states, and as we all we know by now, states with different relative phase factors are physically different; otherwise there would be no interference terms, and no quantum mechanics to speak of!


title: What cools off faster a square or a disc?

text: What cools off faster and why, a square (4 right angles, all sides same length)with an area of 100 inches or a disc with an area of 100 inches. The same material and thickness.


title: Gauss Law for Magnetism,Non Instantaneous Field Propagation

text: ![Fiels lines](http://i.stack.imgur.com/wZIuW.jpg)![Magnetic Flux**Field lines**](http://i.stack.imgur.com/1Nqsb.jpg) Is the magnetic force instantaneous? And, are all field lines established simultaneously? Otherwise, for example, the field line marked 'L' will take longer time to propagate than the ones above it, so there will be a net outward flux in the time when L is out and not yet in into the surface.What happens to Gauss Law then? Are all the field lines established simultaneously, like 'L',that stretches out so far, and the straight one? If not, then the force between the magnets must increase with time.


title: Benefit of using Matsubara Green function

text: Physicists often calculate Matsubara Green function and then perform an analytic continuation $i\omega_n \rightarrow \omega +i\eta$ to obtain the retarded Green function. Why is doing so better than directly computing the retarded Green function?


title: Rms value of rectified output from a half wave rectifier

text: The rms value of an alternating quantity which is the input to a half wave rectifier is $\frac{I_{max}}{\sqrt2}$. Then the rms value of output should be $\frac{I_{max}}{2} \sqrt 2$. But it is given every where that it is $\frac{I_{max}}{2}$. Please tell me where am I wrong.


title: There seems to be no definition of "stability" in axiomatic QFT. Is there? And, if not, is this a problem?

text: "stability" is invoked as the justification for the axiomatic requirement that the spectrum of the generators of the translation group must be confined to the forward light-cone. The _spectrum condition_ has pervasive, significant effects in axiomatic QFT. There seems to be no proof, however, that the spectrum condition actually ensures that a quantum field will be stable, partly because there is, AFAIK, no mathematical specification of what stability consists of in QFT. IF there were, I suppose the _stability axiom_ would be central to axiomatic QFT instead of the positive spectrum condition. Stability is intimately related with positive energy in classical physics, of course, but the concept of energy is rather different in classical relativistic field physics than in quantum mechanics, being the 00 component of the stress-energy tensor instead of being the 0 component of the 4-vector of generators of translations. The relationship between positive energy and stability in classical physics does not seem enough to justify an uncritical adoption of the spectrum condition in quantum theory _as an axiom_ , which is supposed to be obvious enough that it is almost beyond question. Negative frequencies are certainly not ruled out for classical field theories, because the energy is not a linear functional of the frequency of the Fourier components of the field. An axiomatic definition of stability would presumably have to specify what deformations would or should not affect the stability of a given construction. A building is only stable, for example, provided a strong enough earthquake does not occur, it is not stable _sine die_. Given that the deformations that are possible in quantum field theory are more varied than the deformations that are possible in classical field theory, the spectrum condition seems to require a more substantial justification. Less axiomatically, Feynman integrals include negative frequency/energy components in intermediate calculations, though _not_ in observables, which seems to bring the spectrum condition into at least some question. Haag discusses the relationship of stability with the spectrum condition only extremely perfunctorily (p.29 of the 2nd edition of _Local Quantum Physics_ ), and I am not aware of an elaborate discussion by other authors. Is there one? EDIT: Streater & Wightman, in _PCT, Spin & Statistics, and all that_, discuss collision states. Their discussion is entirely in terms of perturbation theory, which seems not adequate _enough_ for an axiomatic discussion. However, because of asking this question I'm starting to see slightly more clearly why a conventional Physicist might be entirely satisfied with what there is on this. EDIT(after acceptance of Tim van Beek's Answer): The other aspect of this is that the restriction to positive frequency is apparently not enough to ensure "confinement", at least not in an elementary way. That seems to me to be more what "stability" _ought_ to mean.


title: How to use specific heats and temperature change in a system to calculate mass?

text: In a system where multiple liquids and solids are mixed together with different specific heats at different initial temperatures, reaching an equilibrium temperature, how do all of these things relate to the masses? How could I use all the information given of the other objects to find the mass of one object in the system? I know Q=cm*deltaT is the formula that relates heat added to the mass, specific heat and temperature change, where normally I could just take m=Q/cdT (I think) and find the mass easily but I am just not sure how to relate this to an entire system where many things are working together. Any pointers would be appreciated, thanks.


title: Neutrino versus Anti-neutrino Detection

text: Is there a that detection method in use that can distinguish between a neutrino from its anti-neutrino?


title: What is a linear probability density function?

text: In the following question, what is meant by linear probability density function? Is it a uniformly distributed variable or triangularly distributed? Thanks in advance. > _The kinetic energy of any object in motion is given by the > $E(v)=\frac{1}{2}mv^2$, where $v$ is the velocity in m/s. Someone measures > the speed of students walking along Symonds St. to the Engineering > buildings, and **notices that the speed can be described by a linear > probability density function in the range 0-1m/s.**_ > > _(a) Write an equation for the probability distribution and sketch the > probability density function. Make sure to label values on both axes._


title: Two-Electron System

text: I'm reading the section "Two Electron System" in Sakurai's textbook and I'm stuck on the following reasoning: > Let us now consider specifically a two-electron system. The eigenvalue of > the permutation operator is necessarily $-1$. Suppose the base kets we use > may be specified by $\mathbf x_1, \mathbf x_2, m_{s1}$, and $m_{s2}$, where > $m_{s1}$ and $m_{s2}$ stand for the spin-magnetic quantum numbers of > electron 1 and electron 2, respectively. > > We can express the wave function for a two-electron system as a linear > combination of the state ket with eigenbras of $\mathbf x_1, \mathbf x_2, > m_{s1}$, and $m_{s2}$ as follows: > > $$ \psi = \sum_{m_{s1}}\sum_{m_{s2}} C(m_{s1}, m_{s2})\cdot \langle \mathbf > x_1, m_{s1}; ~\mathbf x_2, m_{s2} \,\lvert \, \alpha\rangle. $$ > > If the Hamiltonian commutes with $S_{\text{tot}}^2$, > > $$ \left[ S_{\text{tot}}^2, H \right] = 0,$$ > > then the energy eigenfunction is expected to be an eigenfunction of > $S_{\text{tot}}^2$, and if $\psi$ is written as > > $$ \psi = \phi(\mathbf x_1, \mathbf x_2)\chi(m_{s1},m_{s2}), $$ > > then the spin function $\chi$ is expected to be one of the following: > > $$ \chi(m_{s1},m_{s2}) = \begin{cases} \chi_{++} & \\\ > \frac{1}{\sqrt{2}}\cdot (\chi_{+-} + \chi_{-+}) & \\\ \chi_{--} & \\\ > \frac{1}{\sqrt{2}}\cdot (\chi_{+-} - \chi_{-+}) &\end{cases} $$ I have the following questions:   1. What's the role of the Hamiltonian in that section?   2. Why can we decouple position and spin, i.e. find independent function $\phi$ and $\chi$ yielding $\psi$ at all?   3. Why, in principle, couldn't $\chi$ be of form $a\cdot (\chi_{++} + \chi_{+-} + \chi_{-+})$? I understand that this has to do something with the symmetrization postulate, but I can't see why exactly this needs to be the case: If $P$ is the permutation operator, then $$ P(\chi_{++} + \chi_{+-} + \chi_{-+}) = (\chi_{++} + \chi_{-+} + \chi_{+-}),$$ which is exactly the same and thus should be symmetrical.


title: Why is the Faraday Tensor derived from the Lorentz force?

text: If we start from the Lorentz force, $$\textbf{F}=q\textbf{E} +q\textbf{v}\times\textbf{B}$$ and use the four velocity u$^{\mu}$ and the four momentum p$^{\nu}$, then we get to $$\frac{dp^{\mu}}{d\tau}=qF^{\mu}_{\nu}u^{\nu}$$ OR $$\frac{dp^{\mu}}{d\tau}=qF^{\mu\nu}u_{\nu}$$. Two questions: 1) Which one is the Faraday tensor? $F^{\mu}_{\nu}$ or $F^{\mu\nu}$? What's the difference between the two and what is the significance of the latter being antisymmetric? 2) What is the "physical meaning" of deriving the Faraday tensor from the Lorentz force? And why does it relate to the four velocity? Is it that there must be a momentum change in all frames?


title: Why does pitch in a helicopter take effect 90 degrees later?

text: In a helicopter if you want to give it a forward pitch, you change the angle of the blades when it is in this position ---- ![enter image description here](http://i.stack.imgur.com/6178t.jpg) So the two blades experience unequal lift and because o gyroscopic precession the helicopter pitches forward (instead of pitching sideways, which is intuitively expected) Up till this it's clear to me. But what I dont understand is, why does this motion have to take effect 90 degrees later when the orientation of the wings is like this --- ![enter image description here](http://i.stack.imgur.com/4e38c.jpg) I got these images from the helicopter physics videos of smarter every day. Please watch this video to understand my question better. It says that if the helicopter wants to pitch forward, it changes the angle of the blades 90 degrees before. And gyroscopic precession takes place. But it also says that the effect takes place 90 degrees later. That I do not understand. The effect should take place immediately.


title: What is the relation between renormalization in physics and divergent series in mathematics?

text: The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from divergent series. Or so. In physics, it is said that the pertubative expansion for the calculation of QFT scattering amplitudes is a divergent series and that its summation is solved via the renormalisation group. Is there some explicit connection between the mathematical theory and the physics formalism? Perhaps the question can have different answers for two different layers: the "renormalized series", which can be still divergent, and the structure of counter-terms doing the summation at a given order. If so, please make clear what layer are you addressing in the answer. Thanks!


title: Bernoulli's equation and reference frames

text: So I was thinking about this while driving home the other day. I've never been quite clear on why when you drive with the windows down air rushes into your car. I thought this might be explained by Bernoulli's equation for incompressible flow, but I ran into what seems to be a contradiction. If we consider the problem from the reference of the car, the air in the car is stationary and the air outside the car has a certain velocity. Then, Bernoulli's equation implies the pressure outside the car is lower than that inside the car. However, if we take the reference frame of the road, the air in the car is moving and then the pressure in side the car is lower. Intuitively, this second situation seems to be correct since air apparently flows into the car (from high pressure to low pressure). However there seems to be a contradiction, as the pressure gradient depends on reference frame. So my question is what has gone wrong here? Is this a situation in which bernoulli's principle simply isn't applicable? Did I make some sort of mistake in my application of the principle? Any help is appreciated. Thanks.


title: Can decoherence time be shorter than Planck time?

text: Decoherence times can be estimated and are inverse functions of mass. Since there are no upper bounds on mass, can decoherence time be shorter than Planck time?


title: How to handle Yukawa contractions in calculating SUSY beta functions?

text: I'm reading Chapter 6 of Martin's introduction to SUSY http://arxiv.org/abs/hepph/9709356, which is about RGEs in the MSSM. I tried to convince myself of some of the calculations, and I was particularly stuck with anything that has Yukawas in it. I don't know how to deal with them. Take for example the anomalous dimension formula, $$\gamma^i_j =\frac{1}{16 \pi^2} [\frac{1}{2} Y^{imn}Y^*_{jmn} - 2 g_a^2 C_a(i)\delta^i_j]$$ where, $g_a$ is a gauge coupling corresponding to a group $a$, and $C_a(i)$ is the Casimir invariant. And $Y^{imn}$ denotes Yukawas. For example, $$\gamma^{H_u}_{H_u} =\frac{1}{16 \pi^2} [3 Y^*_t Y_t - \dots]$$ $$\gamma^{\bar{d}_3}_{\bar{d}_3} =\frac{1}{16 \pi^2} [2 Y^*_b Y_b - \dots]$$ $$\gamma^{\bar{Q}_3}_{\bar{Q}_3} =\frac{1}{16 \pi^2} [Y^*_t Y_t + Y^*_b Y_b - \dots]$$ the dots are for the second term which I know how to calculate. But I don't know how the results for the Yukawas were obtained! My understanding is that the indices $i,j,k$ are family indices for chiral superfields. Explicitly, $$ Y^{ijk} = Y^{\Phi_i \Phi_j \Phi_k}$$ So if I wanted to calculate, the Yukawa part of $\gamma^{H_u}_{H_u}$, I would try: $$Y^{H_u mn}Y^*_{H_u mn} = Y^{H_u \bar{u}_m Q_n}Y^*_{H_u \bar{u}_m Q_n} + Y^{H_u Q_m \bar{u}_n}Y^*_{H_u Q_m \bar{u}_n} + \text{other permutations in superfields?} $$ If I chose third family approximation like in Martin's notes, then $m=n=3$ But I'm not sure this is correct or where it leads. Could someone help me with this. How should I handle $Y^{imn}Y^*_{jmn}$ to get those results?


title: Why do batteries work?

text: ![enter image description here](http://i.stack.imgur.com/w4whn.jpg) Consider diagram A. In diagram A a car is at the top of a cliff. Gravity is pulling it down, but it does't move since its on a flat surface. Even though it would be at a lower potential at point B, it can't get to point B because the force of gravity is perpendicular to the direction it can move. Now consider diagram B. In diagram B, an electron is at the negative terminal. The electrical field is pulling in towards the positive terminal, but it can't get there since the conductor is perpendicular to it. Even though it would be at a lower potential at point B, it can't get to point B because the field is perpendicular to the conductor. Yet, somehow the in the second diagram/scenario, the current _does_ flow. Why? Gravity and electromagnetism are similar and the situations above are similar yet there is movement in the first one but no movement in the second. Why?


title: Why did my frozen water bottle explode when I opened it after it defrosted a bit?

text: Last night I filled a 20 fl oz bottle (http://www.thenibble.com/reviews/main/beverages/soft-drinks/boylans-mash.asp) with lukewarm water from my tap. I filled the bottle pretty much to the brim, capped it, and put it in my freezer. Today, I took the bottle out, and left it on the table in my far-too-hot room for 10 minutes or so. When I unscrewed the cap, water flew everywhere, like a can of soda that's been shaken up really intensely. What happened? Can anyone explain this?


title: Why are there only derivatives to the first order in the Lagrangian?

text: Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded? Is there a good reason for this or is it simply "because it works".


title: Motivation for form of Lagrangian

text: This question (in lagrangian mechanics) might be silly, but why is the Lagrangian L defined as: $L = T - V$? I understand that the total mechanical energy of an isolated system is conserved, and that energy is: $E = T + V$. This makes good sense to me, but why would we _subtract_ V from T in the expression of the Lagrangian? What does the Lagrangian represent, after all? Every text or reference I consult just glosses over that and presents $L = T - V$ without explaining anything... Background: In classical mechanics class, there are often multiple ways to solve the problem at hand. For example, if the problem at hand is the motion of a physical pendulum, or some Atwood machine, you can solve by figuring out the forces, moments, and applying Newton's law. But you can also solve by considering that the total mechanical energy is conserved. Usually, it's faster that way. Then there is a third way, which would be to write the Lagrangian of the system and use Euler-Lagrange's equation. I am trying to understand how the third method should be interpreted and compared to the second one.


title: How Long is this Projectile in the Air?

text: A projectile is launched a horizontal distance of 200 meters, at an angle of 45 degrees. Launch and impact heights are the same. How long is the projectile in the air? I think the initial velocity is sqrt(200*9.8) = 44.3 m/s or 99 mph but I'm not sure that is the right formula


title: Examples where momentum is not equal to $mv$?

text: I am aware that momentum is the thing which is conserved due to symmetries in space (rotational symmetry, translaitonal symmetry, etc). I am aware that in some systems, the generalized momentum, $$p_j = {\partial \mathcal{L} \over \partial \dot{q}_j}$$ is not simply $mv$. One example of this might be a charged particle in an electric field, $$p_j = mv_i + {q \over c} A_i$$ Can anyone provide other examples where this is the case (simple, preferabily) and an intuition perhaps of why the "old" definition of momentum breaks down?


title: Ball rolling into a bowl - where is its maximum KE (speed)... given there is friction. See diagram

text: Please examine this diagram and answer the apparently trivial questions. I am particularly interested in reasoned answers for part (a)(ii) - where is the maximum Kinetic energy? ![img](http://i39.tinypic.com/14m6zuo.jpg) I say it is at B (as does the answer key), but others are less convinced and prefer the lowest point, C. I also assume that the diagram implies the ball comes to rest at the end of the arrow marked D. [Apologies if I've broken any forum etiquette, this is my first post.]


title: Paramagnetism and large N

text: In a paramagnetic system, we have: $$N = N_\uparrow + N_\downarrow$$. If we have a large system, with $N >> 1$, is it generally okay to assume $N_\uparrow \approx \frac{N}{2}$ and $N_\downarrow \approx \frac{N}{2}$? I would like to make this approximation, but I'm having a hard time justifying it and determining why it is correct.


title: Light cones and reference frames

text: I'd like to know what does it mean exactly to find a reference frame in which two events occur at the same time or in the same space coordinates. As I picture it if we have two events A and B in a (x, y, t) space we would need to find, for each case, a "tilted" frame x' y' t'in which: a) the two events are in the x'y' plane, so that they have the same t' coordinate and b) the two events are in the time axis, so that they have equal x' and y' coordinates. The problem with this picture is that I can't see why we can't find a frame like the one on case a) if the interval is spacelike nor one like in case b) if the events are timelike. I understand the physycal meaning of it (more or less) but I'd like to know if there is also any mathematical/geometrical explanation of why is no possible to find this frames.


title: Conversion of the Nambo-Goto action into the Polyakov action?

text: I`ve read that the Nambo-Goto action containing the induced metric $\gamma_{\alpha\beta}$ $$\tag{1} S_{NG} ~=~ -T\int_{\tau_i}^{\tau_f} d\tau \int_0^{\ell} d\sigma \sqrt{-\gamma}$$ can be converted into the Polyakov action using an intrinsic metric $h^{\alpha\beta}$ $$\tag{2} S_P ~=~ \frac{T}{2}\int d^2 \sigma \sqrt{-h}h^{\alpha\beta} \partial_{\alpha}X^{\mu} \partial_{\beta}X^{\nu} \eta_{\mu\nu}.$$ How this conversion is done is not further explained in my book so my question is if somebody can give or point me to some more explicit hints how this can be done? (This would probably help me to see the general relationship between the two metrics which confuses me too ...)


title: If an event happens 10 light years away and we observe it here, did it actually happen 10 years ago?

text: I'm sorry if this question has been answered a squillion times, I need someone with smarts to explain it to me. If an event happens 10 light years away, and we observe it here, did it actually happen 10 years ago? Are photons of light effected by relativity I guess is my question, if anyone gets the gist of what i'm saying. Its just someone made this comment > "If they're ten light years away, that means they are millions of years in > our past. We are not seeing the present that far away." Is that correct?


title: Electron Electric Field Mass?

text: I am confused of whether or not the expected electromagnetic field generated by the point-like electric charge of the electron distributed smoothly across space as a probability distribution creates the presence of an effective field mass. I know that the probability current of the electron is $\langle \gamma^{0k} \rangle$, which is conserved. Multiplying the probability current of the electron by its total charge $q$ gives the charge current density $J^k = q\langle \gamma^{0k} \rangle$. From the current I can calculate the expected four-potential field as \begin{align} A^k(r,t) = \int \dfrac{J^k(r',t-c/r')}{|r-r'|} d^3 r' . \end{align} From $A^k$ I can calculate the expected electromagnetic fields as $F^{jk} = \partial ^j A^k - \partial^k A^j$. Finally I can calculate the expected electromagnetic energy as \begin{align} U_{\text{eff}} = \dfrac{1}{8\pi} \int F^{jk}F^{jk} d^3 r . \end{align} Applying Einstein's relation energy is proportional to mass, I get the following effect field mass of the electron as \begin{align} M_{\text{eff}} = \dfrac{U_{\text{eff}}}{c^2} . \end{align} Is $M_{\text{eff}}$ a real observable? I was unable to find an analytic solution for $M_{\text{eff}}$, however I did compute $M_{\text{eff}}$ for Gaussian distributed probability functions for the electron with varying standard deviation (spacial localization), and I noticed that for standard deviations of the order of $10^{-10}$ meters (lattice size), $M_{\text{eff}}$ was very small in comparison to the electron's rest mass. When the standard deviation was $10^{-15}$ meters (nucleus size), the $M_{\text{eff}}$ was comparable in size to the rest mass of the electron.


title: energy momentum tensor and covariant derivative

text: In field theory, the energy momentum defined as the functional derivative wrt the metric $T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (up to a sign depending on conventions)For a theory in flat space, this has the advantage to give you directly an improved energy momentum tensor as the metric is symmetric, but it hold also for dynamical metrics. Now, my problem is the following: If you have in your Lagrangian a term like $A^{\mu\nu}\nabla_\mu V^\mu$, where $V^\mu$ is a vector field (not needed to be a gauge field, for instance say it represents the velocity in hydrodynamics) and $A^{\mu\nu}$ an arbitrary tensor a priori depending on anything (metric, $V$, or any other field), you will have a Christoffel symbol. If you take a functional derivative, the expression will contain terms like $\frac{\delta \Gamma^\rho_{\mu\nu}(x)}{\delta g^{\alpha\beta}(y)}\supset g^{\rho\sigma} \partial_\mu(g_{\nu\alpha}g_{\sigma\beta}\delta(x-y))$+permutations those terms will not combine again to form a christoffel symbol. How can one take the functional derivative of something covariant ($\nabla V$) with respect to something covariant (the metric) and end up with something not covariant? For flat space cases, everything is well since those terms will vanish, but for curved space, we loose general covariance/diff invariance. What am I missing here?


title: How the lagrangian density is found?

text: In Classical Mechanics one usually considers the Lagrangian as $L = K - U$ where $K$ is the kinetic energy of the system and $U$ is the potential energy. One then gets the Euler-Lagrange equations and everything is fine: if we have a system we can plug in the kinetic energy and potential and find the Lagrangian for it. The point is that I've already seen a different object: the Lagrangian density $\mathcal{L}$ which is on the other hand a $4$-form on space-time. The main difference being that the action is the integral of $L$ over time and the integral of $\mathcal{L}$ over all space-time. The problem is that apart from that, no relation between $\mathcal{L}$ and other quantities is given at first. So for instance, in electrodynamics we have $$\mathcal{L} = -\dfrac{1}{4\mu_0}F^{\alpha \beta}F_{\alpha\beta}-A_\alpha J^\alpha$$ Where $A$ is the $4$-potential and $F=dA$ is the electromagnetic tensor. It is not clear at first, why _this_ is the right Lagrangian density in the sense that it becomes a little hard to see where it comes from. So, the Lagrangian itself is just $K-U$, but what about the Lagrangian density? How does one find it?


title: "windows of order" in the Bifurcation diagram

text: When looking at the bifurcation diagram of a chaotic system, one observes "windows of order", namely short intervals where the system briefly leaves its chaotic state and then rapidly returns to chaos. These intervals are quite periodical. Does anyone have a good explanation for this? Thanks!


title: Is gravitational force affected by intervening medium?

text: If we leave a iron ball and feather into the water, feather returns to the surface and floats or moves into the water slowly. On the other hand, iron ball (of certain mass greater than mass of feather) moves into the water directly. Here, both iron ball and feather are accelerated towards the earth to the same extent. Then why does feather moves slowly into the water than iron ball? Shouldn't they reach the earth at the same time. If we assume that water medium is making the feather to accelerate to lesser extent than iron ball, then doesn't it mean gravitational force is affected by intervening medium? Similarly if we consider a feather and iron ball to be left from certain height, I hope iron ball reaches the earth first and then feather. The reason I think is because of air resistance, doesn't this mean air medium or intervening medium is affecting gravitational force? We know that gravitational force is not affected by intervening medium, but it is contrary to the above situations. Is it that gravitational force is affected by intervening medium? I might have misunderstood sometimes, if so, pardon me and explain.


title: Polarisation of Light and Atomic Excitation

text: How does an atomic transition between ground and excited states depend upon the direction of polarisation of incident light?


title: Special Relativity: Transforming Maxwell's equations

text: I'm working through Einstein's original 1905 paper*, and I'm having trouble with the section on the transformation of Maxwell's equations from rest to moving frame. The paper proceeds as follows: > Let the Maxwell-Hertz equations for empty space hold good for the stationary > system K, so that we have > > IMAGE: Untransformed Maxwell equations > > where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that > of the magnetic force. > > If we apply to these equations the transformation developed in § 3, by > referring the electromagnetic processes to the system of co-ordinates there > introduced, moving with the velocity v, we obtain the equations > > IMAGE: Transformed Maxwell equations I do not understand just where this second set of equations comes from. The only derivations of this transformation that I have been able to find involve either potentials, four-vectors, or both. However since four-vectors had not been invented in 1905, and because the statement of the transformation is so blunt, it seems that Einstein is using or appealing to a simpler method for finding the transformation. So my question is: **What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity?** Is Einstein appealing to a pre-existing Lorentz method here, or is there a trick to accomplish all of this quickly? Or does the text here simply belie the true work involved in the derivation?   * I cannot give a proper link to the paper owing to link restrictions, but it's at hxxp://www.fourmilab.ch/etexts/einstein/specrel/www/#SECTION21


title: Orbital radius of Geo-stationary satellite

text: Could you please tell, Why all the geo-stationary satellites are to be dropped at same height from earth? Why can't it be closer or away from its regular orbit(ie, 35,000 km)? If all satellites are dropped in the same orbit, then will not those collide one another?


title: Information about newton's third law

text: Well I am still small student of physics, but as far as I know, Newton's Third Law states that, > "For every action there is equal and opposite reaction." Now let me tell you how I understood this, Suppose there are two bodies A & B initially at rest (w.r.t the earth), now if A makes collision with B, of course there will be some force acting on A->B (say $F_{ab}$) and according to the third law there would be some reacting force from B->A (say $F_{ba}$). And both would be numerically equal. But now just consider body A, it has some acting force on it i.e. $F_{ba}$ which is being given by body B, so won't reaction produce for that force? Is that reaction is nothing but $F_{ab}$? Now for both bodies, there should be some **static frictional force** (before movement) acting against the force which (tends to) move the body, so whenever two bodies act with each other, are there 4 force produced?


title: Problem with an electricity / thermodynamics assignment

text: I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some. The task states: > A heater made out of a wire with a diameter $R = 0.2\text{ mm}$, length > $4\pi\text{ m}$ and electrical resistivity of $0.5\times 10^{-6}\ > \Omega\;\mathrm{m}$ is connected to a voltage source of $220\text{ V}$, > sinked in the water. > > Which mass of water will it heat up from $20^{\circ}\mathrm{C}$ to > $50^{\circ}\mathrm{C}$ in the time of 10 minutes? (C of water = $4200\ > \mathrm{J\;kg}/\mathrm{K}$) I know I have the electrical properties of the wire and the thermodynamic properties of the water, but I don't know how to proceed from there. We've been studying electricity and I am not really aware how I can connect it with thermodynamics?


title: Aurora Effect from Galactic Wind?

text: The Solar Wind creates auroras due to interactions with Earth's Magnetic Field. Do we witness any similar effects from the Galactic Wind interaction with the Sun's Magnetic Field?


title: simplification of Green's intergral for diffusion

text: \begin{eqnarray} I_1 = \int^{\infty}_{0} \frac{1}{\tau}d\tau \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2\tau}(x-x')^2 + (z-z' + \tau)^2\right]dx' \end{eqnarray} where, $p$, $x$ $z$ and $\tau$ are dimensionless. Integral arises in determining shape of steady state surface in solidification process. Lecture notes says that if we substitute $z=-\frac{x^2}{2}$ in the above equation resulting integral is function of $p$ only and not $x$. When I put the substitution, \begin{eqnarray} I = \int^{\infty}_{0} \frac{1}{\tau}d\tau \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2\tau}(x-x')^2 + (-\frac{x^2}{2} + \frac{x'^2}{2} + \tau)^2\right]dx' \end{eqnarray} I do not see any way proceeding forward. I know that \begin{eqnarray} \int^{\infty}_{-\infty} \exp-(x-a)^2dx = \sqrt{\pi} \end{eqnarray} but expression in the integral is 4th order polynomial with apparently no simplification. How can I show that, $I_1$ is independent of $x$? **EDIT** : I found the solution (mentioned few pages later in the notes): along with $z= -x^2/2$, we substitute : $x' = x+y$, $y = Y\sqrt{\tau}$ and $\tau = T^2$ \begin{eqnarray} I_1 &=& \int^{\infty}_{0} \frac{1}{\tau}d\tau \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2\tau}(x-x')^2 + (z-z' + \tau)^2\right]dx' \\\ &=& 2\int^{\infty}_{0} dT \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2}\Phi(x,Y,T)\right]dY \\\ \end{eqnarray} where, \begin{eqnarray} \Phi(x,Y,T) = Y^2 + \left[(1+\frac{1}{2}Y^2)T + xY\right]^2 \end{eqnarray} note that, $\Phi(x,-Y,T) = \Phi(x,Y,-T)$, switching the limits of integration, \begin{eqnarray} I_1 &=& 2\int^{\infty}_{0} dY \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2}\Phi(x,Y,T)\right]dT \\\ &=& \frac{2}{\sqrt{p}}\int^{\infty}_{0} dY\frac{\exp(-p/2Y^2)}{1 + Y^2/2} \end{eqnarray} We finally obtain integration as function of $p$ alone and not $x$


title: How far can something travel in a straight line?

text: Suppose you have an object some distance from you and moving at a velocity different to the Hubble velocity you'd expect at that point. How does the motion of this object change with time? Does it travel in a straight line (i.e. geodesic), and if you wait long enough how far will it get?


title: Complex semi-definite programming

text: I'm doing some calculations and I want to simulate them in python or matlab (or whatever). However I use hermitian matrices and I don't really manage to find a library which enables me to calculate primal problems in complex form. Do you know of any obvious extension to the real problem to calculate the problem using real numbers? The extension of taking a matrix $A = A_{R} + i A_{I}$ and then construct $$\tilde A = \begin{pmatrix} A_R & -A_I \\\ -A_I & A_R \end{pmatrix} $$ will NOT work since in general (and unfortunately in my case too) $\tilde A \neq \tilde{A}^T$, which is needed for real semi definite programs. So the questions are:   1. Does anyone knows of some software allowing complex semidefinite programming?    2. Does anyone knows about some algorithm to implement in python to solve primal problems of semidefinite programs?  Let me know if the questions are unclear.


title: How to test that a flat metric represents a global three-torus geometry

text: When introducing Robertson-Walker metrics, Carroll's suggests that we > consider our spacetime to be $R \times \Sigma$, where $R$ represents the > time direction and $\Sigma$ is a maximally symmetric three-manifold. He then goes on to discuss the curvature on $\Sigma$ which yields the metric on this three-surface > $d\sigma^2=\frac{d \bar{r}^2}{1-k\bar{r}^2}+\bar{r}^2d\Omega^2$ Case $k=0$ corresponds to no curvature and is called flat. So the metric, after introducing a new radial coordinate $\chi$ defined by $d\chi=\frac{d\bar{r}}{\sqrt{1-k\bar{r}^2}}$, the flat metric on $\Sigma$ becomes > $d\sigma^2=d\chi^2+\chi^2d\Omega^2$ > > $d\sigma^2=dx^2+dy^2+dz^2$ > > which is simply flat Euclidean space. Carroll then points out that > Globally, it could describe $R^3$ or **a more complicated manifold, such as > the three-torus $S^1 \times S^1 \times S^1$.** I see that the metric on $S^1 \times S^1 \times S^1$ is also given by $d\theta^2+d\phi^2+d\psi^2$ and therefore there could be a fourth spatial dimension in which $\Sigma$ is a submanifold. However, I am unsure how can we test by experiments or cosmological observations to know for sure whether the flat metric is indeed Euclidean or to conclude a more complicated global three-torus geometry.


title: In an Evolving Block Universe, does the growth rate of the universe give the value of C

text: In an Evolving Block Universe (http://arxiv.org/abs/0912.0808, http://arxiv.org/abs/1407.7243) the future does not exist. The present moment is the bounding edge of the universe in the time dimension. As the present moment advances it leaves an eternal block universe of the past behind it. GR tells us that photons do not experience time and that would suggest that photons only exist in the advancing present moment (a 3d surface precisely as we experience the universe). This would suggest that photons are moving at the same rate as the bounding edge of the universe (at least in the time dimension). Does this not suggest that C gives us the rate of growth of the universe, or more importantly, that the rate of growth of the universe gives us C? This seems like a really profound result. Edit I wonder if this also means that the value of C is not constant but has followed a sigmoid curve i.e. inflation. This would also seem to suggest that if you could visit the past (rather than just rewind time) that it would be cold and dark - the ashes of the present.


title: How does this series expand the expression?

text: How does $$\sqrt{R^2 + |x|^2} = R + \frac{|x|^2}{2R}+\dots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out.


title: Particle Acceleration in three dimensions

text: ![enter image description here](http://i.stack.imgur.com/1WzTY.jpg) So here is the problem I dont know how to approach it.


title: Energy not conserved with accleration by a constant force

text: Suppose I have an object with mass $m$ in vacuum that I propel by applying a constant force, $F$, on it, with a rocket engine that I supply a constant amount of energy, $\frac{\delta E_{supply}}{\delta t}$, to. Then the object's acceleration is given by $F = ma \implies a = \frac{F}{m}$. Thus, it will have a constant increase in velocity, $\frac{\delta v}{\delta t} = a$. But, $E = \frac{mv^2}{2}$, and so the objects increase in energy equals $\frac{\delta E_{object}}{\delta t} = \frac{m}{2} \frac{\delta}{\delta t} (v^2) = mv \frac{\delta v}{\delta t} = mva$. This energy increase, $\frac{\delta E_{object}}{\delta t}$, is not constant, but $\frac{\delta E_{supply}}{\delta t}$ is and so the principle of conservation of energy is violated. Where is my error? **EDIT:** Many of you point out that the problem lies in how I assume a rocket will convert my energy to a constant force (@BMS for example says that $F = \frac{dK}{dx}$ instead of $F = \frac{dK}{dt}$), but there is one thing about that I don't understand: Imagine that instead of a rocket, I propel my object with an electron accelerator. I accelerate my electrons over an electric potential of $V$ volts and aim the beam opposite to the direction of desired travel. This will consume a constant amount of power since $P = IV$ (power equals current times voltage and the rate of electron ejection (current) can be kept constant). This should mean that that all my electrons are ejected with the same velocity, $V_e$, relative to my object at the time of ejection, shouldn't it? (or does $F = \frac{dK}{dx}$ make this statement invalid?) And since the electrons should require a known force to be accelerated to $V_e$ and because of Newton's third law, every force has an equal and opposite force, shouldn't this apply a constant force on my object?


title: What is a measurement in MWI

text: I was reviewing the MWI, and couldn't figure what a measurement is in the MWI? Everett claims that split happens when you measure, but what is it for MWI (not for general QM)? Yes, I know about decoherence, but here I am talking specifically about the split, and how MWI considers the measurement. Also, decoherence is just a part of the problem. I am talking about actual collapse or a split. If an apparatus is of an intermediate size (neither a macroscopic nor a microscopic), how do we handle 'half-splits'? Since MWT is an interpretation and not just a postulate, this should be handled somehow. This question is not a duplicate of question about general measurements in QM like this one: What constitutes an observation/measurement in QM?


title: Conditions for hyperregularity

text: (Definitions from Marsden & Ratiu, ``Introduction to Mechanics and Symmetry''): A Lagrangian is _regular_ if the Hessian $\partial^2 L/(\partial \dot{q}^i \partial \dot{q}^j)$ is weakly non degenerate (i.e. - I think? - has zero determinant somewhere); this condition is equivalent to the condition that the Legendre transformation $\dot{q} \mapsto p := \partial L/\partial \dot{q}$ be locally invertible. A Lagrangian is _hyperregular_ if the Legendre transformation is a diffeomorphism; similarly, a Hamiltonian is hyperregular if the inverse Legendre transformation is a diffeomorphism. I have three (related) questions:   1. Is there a more "intrinsic" characterisation of the hyperregularity condition for Lagrangians (i.e. one that does not explicitly relate it to the Legendre transformation)? In particular, is hyperregularity equivalent to the Hessian being strongly non-degenerate (i.e. - again, I think? - its having zero determinant everywhere)?   2. Is there a natural counterpart to the regularity condition for Hamiltonians?   3. Again, is there an intrinsic characterisation of the hyperregularity condition for Hamiltonians?


title: Why is there a 'loophole' in Mermin Wagner for rotations?

text: I'm just starting out in my mathematics career by looking at some simple stuff on broken symmetries in statistical mechanics. Since 3D is 'hard' it would be very nice to look at 2D toy models of crystals. But by (what I assume is down to) the awkwardness of logarithms and two dimensions, this leads to some complications in the form of the Mermin-Wagner theorem, which most people state in words as 'it is impossible to break symmetry in two dimensions'. However, many people including Mermin himself in _Crystalline Order in Two Dimensions_ , point out that this doesn't apply to the action of rotations, and the gap between mathematical statements and physics starts to get difficult for me. There was an excellent question that gives a precise statement of the Mermin Wagner theorem. Starting from this and Mermin's paper I have two questions: Suppose we have a simple model where we sit an atom at each point in $\mathbb{Z}^2$, and we restrict their motion to some ball in $\mathbb{R}^2$ centered on each lattice site so we have somtheing like a crystal, and equip them with something like harmonic interactions for simplicity. Then as Mermin points out in his paper, we can show rotational symmetry is broken, but translational symmetry isn't. In this case-   1. Does the 'Mermin Wagner theorem' not apply in this case because the atoms actually take values in different compact spaces, and not all in the same one $\mathcal{S}$, so we're really looking at a different thing all together with rotations? But then it does apply to translational symmetry, so I guess this isn't why.    2. Can anyone explain to me exactly how Mermin concludes as he does in this remarks about rotational symmetry breaking, if you happen to have read the paper? I'm not quite sure what the reasoning is, which I guess is exactly why I'm asking on here! Thank you very much, and sorry the question is a little vague.


title: Could the universe have evolved WITHOUT the non-determinism of quantum mechanics?

text: (I'm going to make a few conjectures here - please answer the question in light of them as if they were true, even though of course they may be overly simplistic or wrong) Assuming that:   * the universe had a beginning, at a singular point in space    * the net energy of the universe is zero   * all conservation laws and symmetries hold perfectly at the highest energy scales   * all fields are perfectly unified at the highest energy scales Could the universe have evolved beyond this dimensionless singularity if the laws of physics were perfectly deterministic? Would such a system need an initial seed to break symmetries and begin evolving, and could it be given such a directed push in an entirely deterministic framework, if it started with no net energy and no particular bias (geometric or otherwise)? Is it meaningful to ask if the process had to be initiated by an element of randomness? Sorry if the question is overly vague and superficial; my knowledge of physics is somewhat limited (at least compared to most of you kind folks). [edit] I understand why the question was put on hold. Thanks all the same. :) If I think of a more appropriate way to recast the question I will submit a new one if it is sufficiently different to warrant it.


title: Reasoning transformer identity for currents

text: When looking for reasons for the Transformer Identity for Currents with a short secondary circuit $$\frac{I_2}{I_1}=\frac{N_1}{N_2}$$ I find two common answers:   1. Both coils create the common magnetic field together. Its magnetic flux density $B=\mu N\frac{I}{l}$ is proportional to $I$ and $N$ in each coil separately:      * $B\sim I_1\cdot N_1$      * $B\sim I_2\cdot N_2$   Consequently, it must be $$I_1\cdot N_1=I_2\cdot N_2$$ and the identiy follows by algebraic transformations.   2. No, it's not all like 1. The secondary current creates it's own, antiparallel magnetic field, which reduces the overall field lowering the induction voltage in the primary circuit, letting the primary current get bigger, …. If you want to solve the problem, you have to use Ampère's Law (or similar):   Take a rectangular path through the iron of the transformer and you get [blabla] $$I_1\cdot N_1=I_2\cdot N_2$$ My question is now: Why renders the second version the first version not only obsolete but also wrong? The formula for the magnetic flux density in a coil is also retrieved from Ampère's Law. The Length $l$ doesn't stem from the law itself, but from the model, that the magnetic field density is nearly zero outside the coil. So, the only grave difference between version 1 and 2 is the length and material of the coil. But material and effective length of the magnetic field is indeed equal for both coils, as the transformer core is homogenous and a complete circuit core.


title: ket vector with two "entries"

text: This is a very simple question. I am learning about angular momentum. In my lecture notes, the symbol $|\lambda,m_l \rangle$ was defined as a eigenfunction of a central potential. Two assumptions are introduced: $L^2 = \lambda \hbar^2$ and $L_z = m_l \hbar$. So, $$ \hat{L}^2 |\lambda,m_l \rangle = \lambda \hbar^2|\lambda,m_l \rangle \\\ \hat{L_z}|\lambda,m_l \rangle = m_l \hbar|\lambda,m_l \rangle $$ But what does $|\lambda,m_l \rangle$ mean exactly? I am comfortable with $|\psi \rangle$, but I do not understand what having two variables in the ket means.


title: Precise meaning of composition of ket and bra, e.g. $|\psi\rangle\langle\psi|$

text: I'm currently studying density matrices, and have been frequently coming across the construction $$|\psi\rangle\langle\psi| \,.$$ What is the formal meaning of this composition? I understand $\langle \psi|$ to be an element of the dual space (to that vector space for which $|\psi\rangle$ is a member) but I don't quite understand what it means to put them together. I have been treating this object as a linear operator on the space of ket vectors, and assuming a certain associativity to its composition, such that $$\bigg(|\psi\rangle \langle \psi|\bigg) |\phi\rangle = |\psi\rangle \bigg(\langle \psi|\phi\rangle \bigg) = \alpha |\psi\rangle \,,$$ but is there a way to make this more formal? Thank you.


title: How much merit is there in the heuristic argument of bulk-edge relation for topological insulators?

text: Take 2D quantum hall insulator for example. The typical argument goes like this: We have a Hamiltonian that has translation symmetry in both directions on a infinite lattice, and we assign a integer number(Chern number) to each of its band. The Chern number is a topological invariant(against homotopy, I suppose), hence remains constant if the Hamiltonian is varied continuously and as long as it is well defined. **Attach** a the quantum hall insulator(which has non-zero Chern-numbered bands) to a trivial insulator(of which all bands have 0 Chern number), and the band of the former must be concatenated with the latter. However this means the well-definedness of Chern number has to break down somewhere, or else one band could have two different Chern numbers. The place where the well-definedness breaks down must be a band gap closing, hence we have conducting (edge) states. My question is, the moment we decide to **attach** two insulators together, we must have broken the translation symmetry along at least one direction, so Chern number is already not well defined, how can we proceed with the argument that follows after? I am aware that there exists more serious arguments about bulk-edge relation like the ones by Yasuhiro Hatsugai, but I haven't got around to read it and I will in the near future. But can someone comment on the previous heuristic argument I phrased? Just how hard is it to make it a rigorous argument, if possible at all? Of course if I'm told Hatsugai's paper is exactly formalizing this heuristic argument I'll get to read it as soon as possible.


title: Matrix elements of linear operators - orthonormal basis required?

text: In an early linear algebra class of mine, I learnt that a linear map $\mathcal{A}$ acting on a vector space could be represented by a matrix $A_{ij}$ according to the rule: $$\mathcal{A}({e_j}) = A_{ij} e_i \,,$$ where $e_i$ was some chosen basis. In quantum mechanics, the rule for getting the matrix elements of a particular linear operator is seemingly very different: $$ A_{ij} = \langle i | \mathcal{A} | j \rangle $$ I was quite satisfied when I convinced myself that for _orthonormal bases_ , these two definitions coincide. My question is then essentially this: **is the above rule for quantum operators valid for orthonormal bases only?** The problem I see, if it's valid for all bases, is that the two definitions do not coincide. So when I proved results in my linear algebra class like "the trace of a linear map is a meaningful object, since the trace of its matrix representation is basis-independent", this was using the first definition, and so then surely it _won't_ be true if we use the second definition? According to this Wikipedia page on density matrices, in the third equation on the page, it suggests that we can write the trace of an operator $\rho A$ like so $$ \mathrm{tr}(\rho A) = \sum_n \langle u_n | \rho A |u_n \rangle = \sum_n (\rho A)_{nn} \qquad (\mathrm{second\ rule})$$ even if the basis $\\{|u_n\rangle\\}$ is not orthonormal. Will this not give a different trace to that evaluated using the first rule for the matrix elements of a linear map? Thank you.


title: Where can I find the datasets from LHC?

text: I know that most of the datasets from ATLAS or other particle physics experiments are terabytes, but I was just curious is there any place where one can find them to download? And one more question, which software they are using to analyze data and do the simulations? Plus if there is any guide available where they show setting up the software and analyzing some smaller particle physics datasets, I would be glad. I kept searching the Internet for similar things, but unfortunately couldn't find anything.


title: What circle aids/tools are there where angles are doubled?

text: I was thinking back to my undergraduate degree and remembered Mohr's circle, where the angles on the circle are double what they are physically (only thing I remember from that course because of the funny way the professor said it... all the time). In the electrical world, there is also the Smith chart, where 180 degrees electrically makes 360 degrees on the chart. I was wondering if there are any other such circle tools where angles are doubled? And if there are (many) more, why do circular tools work like that (angles doubled)? Offhand, I can't think of a circular physics/engineering analysis tool I've ever used where angles were actually 1:1 (except, perhaps, for the protractor).


title: Acceleration during collision with ground

text: For my physics class, I have to calculate an objects acceleration while it came to a stop. It was traveling at $6.26\: \mathrm{\frac{m}{s}}$ at impact, and travels a distance of $0.025\: \mathrm{m}$ while stopping. I assumed I would use a simple kinematics equation to solve. $$V_f^2 = V_i^2 + 2a\varDelta d$$ $$0 = 6.26^2 + 2*a*.025$$ $$-39.2 = .05a$$ $$a = -784\: \mathrm{\frac{m}{s^2}}$$ Is that right? It seems extremely high.


title: Definition of short range entanglement

text: When studying Symmetry Protected Topological phases, one needs to define what a short range entangled (SRE) states means. But there appears to be different definitions that are not equivalent to each other. In http://arxiv.org/abs/1106.4772, Xiao-Gang Wen defined SRE states to be a state that can be transformed into the unentangled state (direct-product state) through a local unitary evolution. This implies in particular, that there cannot be SPT phases with trivial symmetry, because states with trivial symmetry can always be unitarily evolved to a product state. This is apparently contradicted to Kitaev's notation of SRE. In http://arxiv.org/abs/1008.4138, Kitaev said that there can be non-trivial SPT phases for a Majorana chain with trivial symmetry in 1+1d characterized by dangling Majorana modes at the two ends. My question is, what is Kitaev's definition of SRE (I cannot find a reference where Kitaev explicitly defined this), and how is it differed from Wen's definition. Apparently, If a state is SRE in Wen's definition, then it is SRE in Kitaev's definition.


title: What is the spectral energy density of virtual photons around a unit charge at rest?

text: Given that my previous question, namely "What is the number density of virtual photons around a unit charge?" has no precise answer, here is a more precise wording: What is the virtual photon spectral density due to the electric field of a unit charge at rest (or, better, its difference with respect to the no-field case)? Or: How many virtual photons per volume and per energy are found around a unit charge, at a distance r?


title: Is Shor's algorithm a demonstration of the many worlds interpretation?

text: David Deutsch is very fond of pointing out Shor's integer factorization algorithm is a demonstration of the many worlds interpretation. As he often asked, where else did all the exponentially many combinations happen? Are there any other alternative interpretations of quantum mechanics which can explain Shor's algorithm, and the Deutsch-Jozsa and Simon's algorithm?


title: Inflation factor and doubling time

text: I get the general idea about cosmic inflation, but the numbers associated with it seem to be pulled out at random. For example, in The Elegant Universe, Brian Greene says that the universe doubled every 10^(-34) seconds about a hundred times to an overall factor of 10^30. Another astronomy text I read recently gives the doubling time as 10^(-35) secs and the overall expansion factor as 10^50. Where are all these numbers coming from ? For example, why couldn't the universe have doubled every 10^(-40) seconds many billions of times during inflation ?


title: Is there any simple proof of the no-ghost theorem?

text: Is there any simple proof of the no-ghost theorem in string theory?


title: Nuclear physics Radioactivity

text: 90Sr has a half life of 28.5y. It is chemically similar to Ca and enters the body through the food chain and collects in bones. It is a serious health hazard. How long in (years) will it take for 99.99% of Sr released in a nuclear reactor accident to disappear?


title: On the nature of the collapse of the wave function

text: The collapse of the wave function by measurements is one of the most mysterious properties of quantum mechanics.   1. At what scale does the wave function collapse? What are the conditions for a collapse?    2. Can it be made precise?    3. Does the collapse happen instantaneously, or is it spread out in time?    4. What is the precise equation for the collapse?    5. Why is the collapse time-asymmetric, and can this explain the arrow of time?


title: critical electric field that spontaneously generates real pairs

text: With the current QED framework, If an electric field is strong enough (say, near a nucleus with $Z > 140$) , pair production will occur spontaneously? Is this a real effect or an artifact before renormalization is carried out? how can energy be conserved in such scenario?


title: Slowing of time under gravity

text: I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle. Now imagine two light clocks (a clock that ticks due to light) are placed above the surface of the earth. One is placed at sea level and one very high. You might be beginning to think that I am copying Einstein but please bear with me. Consider the following image: ![enter image description here](http://i.stack.imgur.com/07OVX.jpg) The left part of the image shows the light clock placed closer to the earth's surface and the right part shows the clock placed very high. We do not need exact height as we are not doing any calculations. One light ray is emitted from the left end of the clock on the left of the image. Now since it is closer to the earth, it will bend more than the light ray in the other clock and take a longer time to reach the receiving end. The above clock (which is the clock on the right part of the image) is away from the earth and experiences less gravity. Therefore the light ray will bend less and reach the receiving end faster. Just imagine that the clocks are fixed in position and are not moving. They are only ticking when the light ray reaches the other end. Even GR predicts that a clock will run slower in high gravity but doesn't actually give a reason. Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)? Can this be an actual reason for gravitational time dilation or not? **NOTE** The clocks are not falling. The light ray inside them will still bend as it is not being held in position by anything. Only the clocks are fixed. How can a light ray be fixed? So do not think that the clocks are falling.


title: How classical chaos can be described quantum mechanically?

text: How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that basically underlies on such a linear equation to describe highly nonlinear chaotic properties of classical systems?


title: When matter and anti-matter collide

text: Do they create energy? Or do they just disappear with zero energy? If they create energy when disappearing, that means it takes energy to create them, right? If they disappear into zero energy, could that be a clue about how to create matter from nothing?


title: Nomenclature of hadronic resonances

text: I have the Particles Physics Booklet and I noticed that the resonances that decay into a nucleon and pion are indicated by an abbreviation. For example $P_{33}$ is associated to the $\Delta (1232)$ resonance. What is the meaning of this symbol?


title: Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

text: Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, the mathematical definition of the path integral is no longer obvious. Intuitively, I still expect the construction of this path integral to be much easier than the construction of path integrals for continuum QFT. This is because on the lattice we have an explicit UV cutoff and in particular the bare parameters of the action are finite so the action is an honest function on the field space. Naturally, I'm interested in interacting theories (e.g. lattice Yang-Mills) since free theories lead to Gaussian integrals which are relatively easy to define even in the infinite-dimensional case. *Usually periodic boundary conditions are used so the finite lattice is a product of cyclic finite groups


title: Why isn't the Gear predictor-corrector algorithm for integration of the equations of motion symplectic?

text: Okumura et al., _J. Chem. Phys._ 2007 states that the Gear predictor-corrector integration scheme, used in particular in some molecular dynamics packages for the dynamics of rigid bodies using quaternions to represent molecular orientations, is not a symplectic scheme. My question is: how can one prove that? Does it follow from the fact that the Gear integrator is not time- reversible (and if so, how can one show that)? If not, how do you prove that an integration scheme is not symplectic?


title: Why are different frequency bands used in different countries?

text: Why are different frequency bands used in different countries despite ITU's effort for a common frequency band use? There's got to be a reason behind this. For instance, U.S.-based Verizon Wireless uses the 700 MHz frequency band for its LTE service while European TeliaSonera and South Korean SKT uses 1800/2600 MHz frequency bands and 850/1800 MHz frequency band, respectively.