title: Failure of a simple stat mech simulation

text: so I did a simple simulation that I thought would yield a Boltzmann distribution, but it failed to, and I was wondering if anyone has insight into why it failed. Ok, so I had a simple discrete system of $N$ energy levels, $E_j=j \epsilon, j=1...N$. Each of these energy levels was given a (initially random) occupation of $o_j$ "particles". Thus the total number of particles is $\sum_j o_j$, and the total energy is $\sum_j o_j E_j = \epsilon \sum_j o_j j$. Then, for a large number of iterations, I altered the occupancies in a way that preserved the energy and number of particles. I tried a number of different "alteration schemes"; for example, one scheme involved picking a random k, and if $o_k>=2$, I kicked one of the particles down to $o_{k-1}$ and one up to $o_{k+1}$. My hope was that these iterations would cause the state of the system to wander about on the "constant energy surface", and would, in great likelihood, wind up in a Boltzmann distribution, $\log o_k \propto -k$. But I wasn't seeing that at all. Does anyone know why these kinds of iterations don't lead me to the Boltzmann distribution? All I can think of is that there is something special about the iterations I'm doing that keeps the state of the system in some "subgroup" of the constant-energy surface, and I'm not properly "wandering" in phase space.


title: Can one of Newton's Laws of motion be derived from other Newton's Laws of motion?

text: Can one derive Newton's second and third laws from the first law **or** first and third laws from the second law **or** first and second laws from the third law I think Newton's laws of motions are independent to each other. They can not be derived from one another. Please share the idea.


title: Can energy be taken out of the QFT vacuum?

text: There have been recent questions about the vacuum. In my simplified knowledge the vacuum is like a ground state energy level, and also that there might even exist other lower energy levels than the vacuum we find ourselves in. The sea is a soup of created and annihilated pairs of virtual particles with virtual energies and momenta. In a normal sea on earth, which also represents a ground state of water, energy can be taken out of random waves by the clever construction of valves that allow only one way motion of water. Is it conceivable that a gadget of similar function could be found for the vacuum sea, or is it forbidden by conservation laws? My intuition tells me that it might be possible if GR is taken into account, but my physics knowledge does not stretch to support this. **Edit** : An explanation of why I am asking this question: Let me expand on the example of the sea. The energy from the waves comes from either tides, i.e. gravitational forces, or wind (temperature differentials). If these were missing the oceans would be like glass representing a unique ground state of the gravitational well of the earth. In an analogy, a gravitational wave going through the vacuum would be supplying energy to the vacuum sea. I have been thinking of this analogy ever since cold fusion surfaced and refuses to die out, the most recent one being discussed here too. Approached from nuclear physics orders of magnitude the claims seem preposterous. There are people though who believe they have results of extra energy over input energy, much more than chemical reactions could supply. This set me thinking on vacuum energy and the analogy with getting energy from the sea. A crystal is a prime candidate for any exploration of such concepts and in all cold fusion "successful" results crystals have been used. Now if the effect depended on the vacuum and how much distorted it was by a passage of a gravitational wave at the time of the experiment, or the exact orientation of the crystal, or the type of impurities in the crystal ( F centers etc) one would expect to get haphazard results, and non repeatable by other experimenters. Of course this would be the first experimental evidence of gravitational waves :). **Edit 20/7/12** Maybe I should clarify that an acceptable answer in the negative would be one based on conservation laws. I believe that data trumps theory, and next in line are conservation laws,because they are the distillation of an enormous amount of data. Some people seem to think that theoretical definitions can substitute for proof in physics, but physical theories change, solid data do not, and this is physics, not axiomatic mathematics.


title: How do you calculate the Milky Way’s galactic year?

text: > The Solar system moves at a speed of 220 km / s around the galaxy. It’s > about 27,000 light years from the Galactic Centre. How long does it take for > the solar system to orbit around the Milky Way? I first calculated the circumference = $\mathrm{2 \times 3.14 \times 27,000 \times 9.4605284 \times 10^{15}\ m}$ Then, I divided the circumference by 220 km/s. However, I get $7.2 \times 10^8$ Can someone give me idea what I did wrong?


The Earth moves at a high rate of speed around the Sun, and the solar system is moving quickly around the Milky Way. How is it that the Earth's atmosphere is not “blown away”?


title: What interesting physics problems can't be solved because mathematics is not developed enough?

text: I'm curious as to what sorts of physical problems to which we don't have an answer, because we haven't developed the right mathematics yet (or advanced- enough mathematics). Related to this question on the Math StackExchange: In what ways has physics spurred the invention of new mathematical tools?


title: Why don't the components of these forces add up correctly?

text: > In the arrangement as shown in the figure below, the ends P and Q of an > inextensible string move downwards with uniform speed $u$. Pulleys A and B > are fixed. Mass $M$ moves upward with a speed: > > a. $2u\cos\theta$   >  b. $u/\cos\theta$   >  c. $2u/\cos\theta$   >  d. $u\cos\theta$ > > ![figure](http://i.stack.imgur.com/2tRxY.png) In the preceding question , both the strings move with velocity $u$ in their respective directions , thus block will move with velocity $2u\cos\theta$. Since each of the two strings has velocity in vertical direction as $u\cos\theta$ . But the answer says it is $\large\frac{u}{\cos\theta}$ . Which seems right as when you take components of this total velocity of block $M$ in directions of strings, it gives components as $u$, but those 2 components don't add up to $u\cos\theta$. What is the mistake here ? Why are they not adding up to $\large\frac{u}{\cos\theta}$? And what if the options weren't available and it was a free-response question. Then how to deduce total velocity of block-mass $m$ is $\large\frac{u}{\cos\theta}$ by just knowing that velocity of both strings in their respective directions of motion is $u$ and the angle between them is $2\theta$? How to find the correct answer directly?


title: Objects in Physics as a mathematician would see them

text: I'm a mathematician with hardly any knowledge of physics. Before I start reading volumes of physics books, I have a few questions that have been bugging me and that will help me start reading physics. Let's forget Quantum Mechanics for the purpose of the discussion, and focus on Relativity. I always fail to understand what the objects are and how to relate them intuitively to perceivable space and time. What are the objects? Here's what I mean: we begin with spacetime being a pseudo-Riemannian manifold with some metric (is this metric assumed to be locally Lorentzian?). For each point in this manifold, there should be some parameters, right? Like whether there's a particle there, what particle is it (information that includes, for example, mass), is there a magnetic field there and so forth. Together, this pseudo-Riemannian manifold, with a pre-specified metric, and with a set of parameters for each point, is what described the universe in relativity, right? What _precisely_ are those parameters in General Relativity?


title: Can spheres leaking charge be assumed to be in equilibrium?

text: I am struggling with the following problem (Irodov 3.3): > Two small equally charged spheres, each of mass $m$, are suspended from the > same point by silk threads of length $l$. The distance between the spheres > $x \ll l$. Find the rate $\frac{dq}{dt}$ with which the charge leaks off > each sphere if their approach velocity varies as $v = \frac{a}{\sqrt{x}}$, > where $a$ is a constant. This is embarrassingly simple; we make an approximation for $x \ll l$ and get $$ \frac{1}{4 \pi \epsilon_0} \frac{q^2}{x^2} - \frac{mgx}{2l} = m \ddot{x}. $$ We can get $\ddot{x}$ from our relation for $v$, so we can solve for $q$ and then find $\frac{dq}{dt}$. However, in general, $\frac{dq}{dt}$ will depend on $x$ and hence on $t$. The answer in the back of the book and other solutions around the web have $\frac{dq}{dt}$ a constant. You can get this by assuming that at each moment the spheres are in equilibrium, so that you have $\ddot{x} = 0$ in the equation of motion above. Does the problem tacitly imply we should assume equilibrium and hence $\frac{dq}{dt}$ is constant, or am I missing something entirely? I.e. why is the assumption of equilibrium justified? I understand reasoning like "the process happens very gradually, so the acceleration is small compared to other quantities in the problem," but I don't understand how that is justified by the problem itself, where we are simply given that the spheres are small (so we can represent them as points) and $x \ll l$ (which we have used to approximate the gravity term in the equation of motion).


title: Surface integral of a line

text: Suppose I am calculating the (homogenous) electric flux through a straight line. How do I calculate the surface integral of a straight line? Is it simply a line integral, or is it more complex (I don't think the first is true because if the surface integral is a measure of area, but the line integral is a measure of length). I found this on wikipedia. > In mathematics, a surface integral is a definite integral taken over a > surface. It can be thought of as the double integral analog of the line > integral. Given a surface, one may integrate over its scalar fields (that > is, functions which return scalars as values), and vector fields (that is, > functions which return vectors as values). Basically, how do I calculate the surface integral of a line y= mx+c? What about more complex curves, for example the Archimedean Spiral ($r= a+ b \theta$ in polar form)?


title: Could it be a limit in our measuring the speed of photon, instead of photon having a speed?

text: Are we (not) limited by our own speed through time (1 sec / sec) to measure highest possible speeds(c)? I fire a laser which hits a mirror and comes back and I precisely measure the time, and knowing the distance I calculate the speed.. but when I measured time, wasn't I limited by my own speed in time? Here I'm imagining time in a 2-d graph with me just moving upwards (stationary) and light making a `|>`. Let's call bottom part where the light leaves `A` and top where it comes back `B`. I measured `AB`. Which is my own speed...


title: Conservation of Energy and Momentum Regarding Forces - clarification needed

text: The other day, my teacher stated something along the lines of, "Conservation of momentum is not violated by the actions of internal forces, but the conservation of energy is violated. Energy is conserved only when/if internal forces are conservative." Can anyone explain this is simpler terms or reiterate it in some fashion?


title: Photoelectric Effect - How are the electrons regained?

text: When the photons with enough energy impinge on a photocathode, it emits electrons. Does this mean that the solid will lose all its electron at one point? If not, how are electrons restored?


title: Saturn V specific impulse issue in velocity modelling with Tsiolkovsky's equation against actual values?

text: I've been trying to model the Saturn V's velocity using Tsiolkovsky's ideal rocket equation, and in the process, I think I may have made a mistake with regards to the specific impulse? **I've come up with the following equation** , taking the change in gravity into account: $$ \Delta v(t) = I_\mathrm{sp} g_0 \left(\frac{R_\oplus}{a(t)+R_\oplus}\right)^2 \ln\left(\frac{m_0}{m_0-c(t)}\right). $$ $a(t) = 0.0033 t^2 - 0.1462 t + 1.9487$ is the altitude function derived from actual values, with a very small error range, and $c(t) = 12579.9 t$ is the fuel consumption of the rocket per second. Also I have \begin{align} \Delta v & && \text{change in velocity} \\\ t & && \text{time elapsed} \\\ I_\mathrm{sp} & = 263\ \mathrm{s} && \text{specific impulse time} \\\ g_0 & = 9.81\ \mathrm{m/s^2} && \text{acceleration of gravity on Earth} \\\ R_\oplus & = 6371\ \mathrm{km} && \text{radius of Earth} \\\ m_0 & = 2{,}800{,}000\ \mathrm{kg} && \text{gross mass of Saturn V rocket.} \end{align} **And this a plot of the model** I created versus the actual values plotted against time: ![velocity versus time](http://i.stack.imgur.com/Ot7sP.png) The percentage error here between the two sets of values vary from ~80% to ~14%, and the graph shape is vastly different. I'd like to ask if there is a change in specific impulse? Or have I done anything else wrong in modelling the equation above? Also, I apologize if I've made a stupid mistake, I'm a HS student doing some fun research, quite new to this!


title: Derivation for 7.14 in Atomic Physics by Foot

text: I was going thru Ch7 of Foot and trying to fill in the gaps. However I got stuck on (7.14). So Foot was working with a two level system with a small perturbation in the Hamiltonian resulted from an oscillating electric field $$H_I(t)=e\mathbf{r}\cdot \mathbf{E_0}cos(\omega t)$$. Now we assume that the new eigenstate for the perturbed Hamiltonian is $$|\Psi(\mathbf{r},t)\rangle= c_1 |1\rangle e^{-i\omega_1 t}+c_2 |2\rangle e^{-i\omega_2 t}$$ And we work out relationships for $c_1$ and $c_2$ and try to solve them. It's not hard to get (7.9) and (7.10) in his book. $$i\dot{c_1}=\Omega cos(\omega t)e^{-i\omega_0t}c_2$$ $$i\dot{c_2}=\Omega^* cos(\omega t)e^{i\omega_0t}c_1$$ where $\Omega$ is the Rabi frequency and $\omega_0=\omega_2-\omega_1$. Now Foot says "when all the population starts in the lower level, $c_1(0)=1$ and $c_2(0)=0$. Integration of eqns (7.9) and (7.10) leads to $$c_1(t)=1$$ $$c_2(t)=\frac{\Omega^*}{2}\\{\frac{1-e^{i(\omega_0+\omega)t}}{\omega_0+\omega}+\frac{1-e^{i(\omega_0-\omega)t}}{\omega_0-\omega}\\}"$$ I was wonder how we could integrate them and get those two eqns at the bottom? I tried two different ways but I still can't quite get what I want. Also I tried to simplify (7.9) and (7.10) a bit $$i\dot{c_1}=c_2(e^{i(\omega-\omega_0)t}+e^{-i(\omega+\omega_0)t})\frac{\Omega}{2}$$ But not sure if this helps with the integration. Any thoughts? **My apologies for previous ambiguity. In particular, I apologise to @neutrino for asking the wrong question. The question is: We all know that it would make our lives much easier if we apply rotating wave approximation before solving the ODEs. However Foot solved the ODEs first to get this result $$c_1(t)=1$$ $$c_2(t)=\frac{\Omega^*}{2}\\{\frac{1-e^{i(\omega_0+\omega)t}}{\omega_0+\omega}+\frac{1-e^{i(\omega_0-\omega)t}}{\omega_0-\omega}\\}"$$ He then applied the rotating wave approximation. I was wondering if anyone could derive the above equation in the way that Foot did?**


title: Magnification of an astronomical telescope not in normal adjustment?

text: I am stuck on this question: A telescope consists of two thin converging lenses of focal lengths 100cm and 10cm respectively. It is used to view an object 2000cm from the objective. What is the separation of the lenses if the final image is 25cm from the eye- lens (in front of I.e. A virtual image)? Determine the magnifying power for an observer whose eye is close to the eye lens. I am fine with the first part of which the answer is $\frac{14950}{133}$ but am really stuck on the second (finding the magnification). Here are my thoughts: The magnification is given by the angle subtended at the eye by the image (b) / the angle subtended at the unaided eye by the object (a). This using small angle approximations gives me $a=\frac{h}{2000/19}$ where 2000/19 is the distance of the intermediate image from the objective lens and h is the hight of the intermediate image and $b=\frac{h}{50/7}$ where 50/7 is the distance from the intermediate image to the eyepiece lens. This gives me a magnification of 14.73. Whilst my textbook gives the answer as 15.6. I have also tried finding the linear magnification of both lenes and timesing them together, but this to gives me the wrong answer. Please can someone explain where I am going wrong or direct me to any relevant formulas. thanks.


title: Is speed of light ( Sun-to-Earth ) related to the movement speed of solar system?

text: The speed of light has been measured to be 299 792 458 m/s. Now, the Solar System is traveling at an average speed of 828,000 km/h (230 000 m/s). Summing up the numbers we get close to 300 000 000 m/s Does it mean, that the speed of light can be actually a relative number based on the movement of the source? Does it mean that it IS actually 300 000 000 m/s but is calculated incorrectly due to sun moving round the galaxy?


title: A conductor passing a magnetic field?

text: A large thick copper wire(or bar if you like,) passes a magnetic field. What formula is best to calculate the EMF here? $$EMF = BA / Delta(t)$$ or, $$EMF = vBL$$ When the conductor passes the magnetic field **B** it is connected to a circuit to allow current flow. **EDIT**. Example: ![enter image description here](http://i.stack.imgur.com/h2MN6.png) This image was used to demonstrate Lorentz force, however, it can be used to illustrate my point, lets imagine the conductor in the middle is moving and its passing the magnetic field, when it reaches that point in the diagram it is connected to a circuit.


Do other particles besides scalars admit tachyonic solutions? For example fermions or gauge-boson tachyons? The picture in my head is that a tachyonic scalar simply rolls off some unstable potential until it finds a stable position in field space (so the higgs field is basically a tachyon until it condenses). But I don't see a similar picture for fermions or gauge bosons (they have higher dimensional operators, but not a potential in the same sense as a scalar particle). Furthermore for fermions I believe we can just perform a chiral phase transformation to get rid of any imaginary part of the mass term.


title: Feynman Green Function in D=2 for D'Alembertian

text: I'm trying to obtain the Feynman Green Function (i.e. I'm using the Feynman Causal prescription to compute the green function) for the D'Alembertian in D=2, I'm finding $$G^{(2)}_F (t; \vec x) = \frac{1}{4\pi} \frac{\Theta(t^2-\vec x^2)}{\sqrt{t^2-\vec x^2}} - \frac{i}{4\pi^2} \int_{-\infty}^\infty dz' \text{P.V.}\left(\frac{1}{t^2-\vec x^2 - z'^2}\right)$$ **This result is right ou wrong?** I don't have any reference with the answer. Supposing the result is right, there is some way to simplify it more? * * * _The way I find the result._ I've used the usual procedure (as in Eleftherios Economou and in Morse Feschbach) to find the Green function in D=2 as a potential generated by an infinite line of charge in D=3. $$G^{(2)}_F (t; \vec x) = \int dt' d^3 r' G^{(3)}_C (t-t'; \vec r-\vec r') J(t',\vec r')$$ The Feynman green function in D=3 is, $$G^{(3)}_F (t-t'; \vec r-\vec r') = \frac{1}{4\pi} \left[\delta((t-t')^2-||\vec r - \vec r'||) - \frac{i}{\pi} \text{P.V.}\left(\frac{1}{(t-t')^2-||\vec r-\vec r'||}\right)\right]$$ as can be checked on Bogoliubov-Shirkov (Appendix II, pag 605, A2b.6) And the source is, $$J(t',\vec r') = \delta(x') \delta(y') \delta(t')$$ So that, $$G^{(2)}_F (t; \vec x) = \int_{-\infty}^\infty dz'\frac{1}{4\pi} \left[\delta(t^2-\vec x^2 - z'^2) - \frac{i}{\pi} \text{P.V.}\left(\frac{1}{t^2-\vec x^2 - z'^2}\right)\right]$$ Using a basic Dirac Delta property, $$\delta(x^2-a^2) = \frac{1}{2|a|} \left(\delta(x+a) + \delta(x-a) \right)$$ we get for the first integral, $$\frac{1}{8\pi} \int_{-\infty}^\infty dz'\frac{1}{||\sqrt{t^2-\vec x^2}||} \left(\delta(z-\sqrt{t^2-\vec x^2}) + \delta(z+\sqrt{t^2-\vec x^2})\right)$$ For $T^2>\vec x^2$ (time-like interval) the points $\pm \sqrt{t^2-\vec x^2}$ are real and belong to interval $(-\infty,\infty)$. So we have (for the first integral), $$\frac{1}{4\pi} \frac{\Theta(t^2-\vec x^2)}{\sqrt{t^2-\vec x^2}}$$ And, finally, $$G^{(2)}_F (t; \vec x) = \frac{1}{4\pi} \frac{\Theta(t^2-\vec x^2)}{\sqrt{t^2-\vec x^2}} + \int_{-\infty}^\infty dz' \text{P.V.}\left(\frac{1}{t^2-\vec x^2 - z'^2}\right)$$ * * * Just putting here what happens with my solution if we are put of the light cone singularity ($t=\pm \vec x$). I think that we can forget about the principal value at this case. If I take the integral and solve it, I get $$\int_{-\infty}^\infty dz' \left(\frac{1}{t^2-\vec x^2 - z'^2}\right) = \frac{i\pi}{\sqrt{t^2-\vec x^2}}$$ So I get, $$G^{(2)}_F (t; \vec x) = \frac{1}{4\pi} \frac{1}{\sqrt{t^2-\vec x^2}} \left(\Theta(t^2-\vec x^2) + 1\right)$$


title: How to deal with conflicting "no-slip" Navier-Stokes boundary value constraints?

text: The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces. Now that my class is done, I've been playing with solving an idealized fluid flow problem: figuring out the rate of "spin-down" of stirred fluid in a non- bottomless cylindrical container (ie: my coffee after being given a good stir). For the non-time dependent portion of the problem, I'm not able to apply the no-slip condition as taught in class to my cup of coffee. Matching velocities to my stir stick (I've idealized that as a cylinder as in Acheson's bottomless tea example (pg 45) ), I've got a problem at the base of the cup: If I match velocities of the fluid to the "stir-stick", I can't also match velocities to the bottom of the cup (zero velocity) where the stir-stick touches the base. I've been pointed to some journal articles to read, but won't be able to do so until the fall when I enroll in my next course (and get library access again). In the interim, I was wondering if I could get some pointers on an approach to deal with conflicting 'no-slip' conditions like this one (ie: any problem with a moving interface in contact with a static one has this trouble.) EDIT: I believe that I can work around the inconsistency problem with no-slip constraints for this problem by "stirring without touching the bottom of the cup" ... ie: set the stir depth to 1cm (say). This makes the non-time dependent solution uglier, since I have to treat three different regions:   1. Below the stir stick (zero velocity no-slip condition on the inside of the cup in this region ; velocity matching the stir velocity along the circle at the stir radius at the stir depth).   2. Between the center of the cup and the stir stick. "no-slip" conditions: match velocities below the stir stick. match the stir velocity at the stir radius.   3. Between the stir stick and the edge of the cup. "no-slip" conditions: zero velocity on the cup edge. stir velocity at the stir radius. This makes the problem much messier ... I already have Bessel functions to deal with, and now have to apply them separately in three regions. I'm still interested to see if there's a better way to deal with the boundary value conditions ... ie: without forcing an artificial distance between a static and a dynamic interface so that they don't touch.


title: Why is GR renormalizable to one loop?

text: I have read in a few places that GR is renormalizable at one loop. (hep- th/9809169 for example, second sentence, although they don't seem to develop this point at all). Is this do to some hidden symmetry in the theory? Naively we need new counter terms at all orders, even one loop, in perturbation theory, right?


title: Coriolis Force in a Merry go Round

text: Suppose there is a person standing in a Merry go Round, which is rotating at a constant angular velocity $\vec \omega$. He experiences, of course, a centripetal acceleration $\vec a_{cen}$ and has some tangental velocity $\vec v_{tan}$. This person now starts to walk in a circular path against the direction of rotation with a velocity $\vec v$ with respect to the Merry go Round. From an inertial frame of reference outside the Merry go Round, an observer should notice a Coriolis acceleration pointing radially outwards, given by $\vec a_{coriolis}=2 (\vec \omega \times \vec v)$. If $2|\vec v|=|\vec v_{tan}|$ then the Coriolis acceleration would be equal and opposite to the centripetal acceleration and the observer in the inertial frame of reference would see this person standing still. If $2|\vec v|>|\vec v_{tan}|$ then the Coriolis acceleration is greater than the centripel acceleration and the person would move in a spiral with an increasing radius against the direction of rotation of the Merry go Round, as seen from the inertial frame of reference. Is this reasoning correct?


title: How to estimate size of nucleus from minimum scattering angle of electrons?

text: I have the following problem: > Electrons of energy 350MeV are scattered from a nucleus. The scatter pattern > shows a minimum at a scatter angle of 45°. Estimate the size of the nucleus. I feel like this should have a relatively straight forward solution, but I can't seem to figure out what equation(s) I should be using here.


title: Is Einstein-Hilbert action the unique action whose variation gives Einstein's field equations?

text: I know that   1. scaling the action with a non-zero multiplicative constant, or    2. adding a total divergence term to the Lagrangian density  do not change the Euler-Lagrange equations, cf. e.g. this Phys.SE post. Apart from such trivial modifications (1&2), is Einstein-Hilbert action the unique action whose variation gives Einstein's field equations? If not, is there any other action known which differs non-trivially from Einstein-Hilbert action and whose variation gives Einstein's equations?


title: Physics of a skateboard ollie

text: Does anyone have a good explanation of the physics and vectors of force involved in the skateboarding trick the ollie (where the skater jumps and causes the skateboard to rise off the ground with him)?


title: What is a good mathematical description of the Non-renormalizability of gravity?

text: By now everybody knows that gravity is non-renormalizable, what is often lacking is a simplified mathematical description of what that means. Can anybody provide such a description?


title: Why is lagrangian density correct?

text: The textbooks I have available explain that due to the infinite degrees of freedom of a field, the relevant object in QFT is the lagrangian density. A lagrangian is then obtained for the field by integrating over space. I find the justification for this procedure unclear. In classical mechanics, the lagrangians of two particles may be added only if the particules do not interract. Does it mean that the lagrangian density concept is only valid for a free field? What happens in the case of interracting particles?


title: Are double-slit patterns really due to wave-like interference?

text: According to various sources on the web, it seems like the general concensus is that there isn't actually any wave-particle duality with quantum particles. For example, this article implies that diffraction patterns in double-slit experiments were interpreted as wave interference due to _apparatus limitation_ at the time they were first performed. Does this mean that all those sources and animations showing two waves interfering are simply incorrect, classical conclusions which don't have anything to do with (quantum) reality? What's actually the most confusing is that most sites which state that it is now possible to pass individual photons through these slits, also claim that these individual photons somehow _interfere with themselves_ resulting in the observed patterns. That seems like a rather thin explanation, doesn't it? So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the "spooky" explanations?


title: (Polar jets) Is it possible that black holes radiate substantial amount of the mass/energy they accrete?

text: Is the amount of energy a black hole emit through polar jets comparable to the mass it accretes or is it negligible? Here I am mostly concerned about the observational findings as opposed to the theoretical ones. In other words, any mass in the accretion disk has two possible fate: ending up inside the event horizon or escaping through jets. What is the possible probability range of the second happening to each unit of mass?


title: Pulley problem with spring balance

text: ![enter image description here](http://i.stack.imgur.com/i73YD.jpg) Please excuse the crude drawings The question is, if I burn the thread t, how will the reading in the spring balance(S) change? Ie, increase, decrease, or remain the same? My approach: When we burn the thread, the objects will move, but the tension should increase a bit. ( I can't explain this, I only get this intuitively.) and, since the spring balance reads the double of the tension, I think this should make it a lot more than the initial reading. Since if the tension in the intact string changes by $\triangle T$, the reading should increase by $2 \triangle T$. That's all I can think of. So as you see, my approach is very intuitive, and hence I am asking this in order to get a logical and stronger reasoning


title: Why does static electricity not make a charged body reflective?

text: If mirrors work by deflecting photons by free electrons in surface layer of mirror, so it could be possible to take a glass pane and provide it with extra free electrons by giving it massive static electricity charge, so it will become reflective — but it seems it would not. Why?


title: Understanding units manipulation (speed of falling coconut after 20m)

text: When I was on holidays, I was told a story about how someone passing under a palmtree and almost got a coconut fall on his head. Given that these palmtrees where about $20m$ high, we wondered at what speed the coconut was when it reached the ground... I took a pen and a paper and tried to remember what I knew about velocity, acceleration etc. When I got home I found out that the formula to compute this is $v = \sqrt{2 * g * d}$, giving a final speed of about $71.3127 km/h$. However, the question is not about the real formula, it is about my failed attempt at solving the problem. I tried reasoning about units and ended up with this:   * I know that $g = 9.81 m/s^2$   * I have a distance ($20 m$)   * I want my result to be in $m/s$   * Hum, hey look if I multiply $g$ by the distance I end up with $m^2/s^2$   * And then I can simply do $\sqrt{m^2/s^2} = m/s$, giving the final formula for computing the speed to be $v = \sqrt{g * d}$ If I apply the math, I end up with a speed of $50.42 km/h$, which looked reasonable enough for me to believe I was right until I got home. Ok, after all this introduction here comes the questions... I know that the reasoning is wrong, that I should integrate the speed etc and eventually end up with the correct formula, however I can't help thinking that what I'm computing has to have a meaning in the real physical world:   1. What does the intermediate result in $m^2/s^2$ represent?   2. What does the final result in $m/s$ represent?   3. Should I avoid thinking about units when trying to solve a problem?   4. Does every units manipulation result into something that means something in the real world, or does some manipulation just end up in non sense?


title: 1D Motion under Gravity

text: I have attempted this question in so many different ways but I am getting nowhere. Could someone point me in the correct direction so I can work it out myself? Here is the question: > _An object $K$ is shot upwards with a velocity $v_i$, after $t$ seconds it > is 50 metres high and time $2t$ seconds it is 80 metres high. Solve for > $v_i$ and $t$._ I have tried breaking it into two parts but I always hit a wall.


title: Tensors in general relativity

text: This is a question on the nitty-gritty bits of general relativity. Would anybody mind teaching me how to work these indices? **Definitions** : Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form $X$: $$(*X)_{a_1...a_{n-p}}\equiv \frac{1}{p!}\epsilon_{a_1...a_{n-p}b_1...b_p}X^{b_1...b_p}$$ Exterior derivative of p-form $X$: $$(dX)_{a_1...a_{p+1}}\equiv (p+1) \nabla_{[a_1}X_{a_2...a_{p+1}]}$$ Given the relation $$\epsilon^{a_1...a_p c_{p+1}...c_n}\epsilon_{b_1...b_pc_{p+1}...c_n}\equiv p!(n-p)! \delta^{a_1}_{[b_1}...\delta^{a_p}_{b_p]}\,\,\,\,\,\,\,\,\,(\dagger)$$ where $\epsilon_{a_1...a_n}$ is an orientation of the manifold. Why then is $$(*d*X)_{a_1...a_{p-1}}=(-1)^{p(n-p)}\nabla^b X_{a_1...a_{p-1}b}$$? **How far I've got myself:** Firstly, I believe $(*d*X)$ means $*(d(*X))$? $$(d*X)_{c_1...c_{n-p+1}}=\frac{n-p+1}{p!}\nabla_{[c_1}\epsilon_{c_2...c_{n-p+1}]b_1...b_p}X^{b_1...b_p}$$ Then $$*(d*X)_{d_1...d_{p-1}}=\frac{n-p+1}{(n-p+1)!p!}\epsilon_{d_1...d_{p-1}c_1...c_{n-p+1}}\nabla^{[c_1}\epsilon^{c_2...c_{n-p+1}]b_1...b_p}X_{b_1...b_p}$$ $$=\frac{1}{(n-p)!p!}\epsilon_{d_1...d_{p-1}c_1...c_{n-p+1}}\nabla^{[c_1}\epsilon^{c_2...c_{n-p+1}]b_1...b_p}X_{b_1...b_p}$$ Now I know that I should apply $(\dagger)$ but I don't know how to given the antisymmetrisation brackets. Would someone mind explaining it to me please? Thank you!


title: Acceleration in special relativity

text: I am currently studying the motion of relativistic charged particles in electromagnetic fields. More exactly, we first derived the equation of motion in the 4-vector formalism. I was a bit confused when my teacher talked about acceleration in special relativity. In which cases are we allowed to talk about it specifically in special relativity? (I mean that I have not studied general relativity yet) What stay constant ? Or are we making an approximation? It may of course have a sense, though I have the feeling that in my courses, we only talked about constant speeds in special relativity. I hope my question has a sense, Isaac


title: Minimum required to lift a roller up a step

text: Consider a roller of diameter $0.82m$ which is to be raised up a step measuring $0.32m$. Find the minimum force required to do so. The roller weighs $500N$. ![Diagram](http://i41.tinypic.com/2qmlytu.png) The roller could be considered as a point mass $F$, and its body could be considered a lever with $F_{applied}$ acting at $G$ and $F_{perpendicular}$ acting perpendicular to diameter $GC$. Let $\angle FCH = \theta \implies \angle IFG = \theta$. $F_{perpendicular} = mg\sin\theta$. After that's its a simple application of torque and moments. I'm not quite sure if this resolution of vectors into components is quite the right approach. At least, the value given in my textbook doesn't correspond to the one I got. EDIT: In response to the comment below, I decided to post my workings: $\sin\theta = \frac{FH}{FC} = \frac{0.09m}{0.41m} = 0.2195 \implies F_{perpendicular} = 500N\cdot 0.2195 = 109.756N$ The anticlockwise torque equals $\tau_{anti-clockwise} = 109.756N\cdot 0.41m = 45Nm$ This has to equal the clockise torque $\tau_{clockwise} = F_{applied}\cdot 0.82m$ $\implies F_{applied} = \frac{45Nm}{0.82m} = \boxed{54.878N}$ According to the book, the answer is $110N$? Maybe the book is incorrect?


title: What determines how much light is reflected and refracted?

text: I'm currently working on a physics-based puzzle game with light refraction. I've read about the reflection/refraction incidence ratio, and how it's the proportion of the sine values of the angle of incidences. However, what formula, if there are any, determines the threshold for % of light reflected and % of light refracted? is the % of light reflected/refracted always either 0% or 100% (that is, they're always either reflected or refracted)? Or is there a curve of some sort for this?


title: why is it hard to push the empty upside down mug inside the water

text: It is hard to push the empty upside down mug inside the water as it has air present inside its cavity which prevents the water from entering inside the mug.But when we tilt that mug which is still completely immersed in the water by some angle then water enters inside it accompanying the formation of bubbles.....even thogh buoyant force acting would be the same in all orientations it happens.......why is it so..


title: How does one write the Einstein field equations in terms of Ricci tensor?

text: How can I go from the 'standard' Einstein equations $R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$ to these equations: $R_{\mu\nu} = \frac{8\pi G}{c^4}(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T)$?


title: Trinary System Arrangment?

text: I'm writing a science fiction story and wondered if someone could help me with the setup of a trinary star system I have in mind. Basically I just wanted to know if: a) a system with a central, Sun-sized star could have two smaller (orange dwarf?) stars orbiting it in a circular trojan orbit, and... b) how much disturbance would this type of orbit cause to any planets orbiting the inner star? This may all sound ridiculous, but any input would be greatly appreciated. Thank you.


title: Gravity in other dimensions than 3 and stable orbits

text: I have heard from here that stable orbits (ones that require a large amount of force to push it significantly out of it's elliptical path) can only exist in a three spatial dimensions because gravity would operate differently in a two or four dimensional space. Why is this?


title: Kähler and complex manifolds

text: I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simplest $\mathcal{N}=1$ supergravity we'll get these. Now in the course-notes they're quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite complete book) about the subject? To get a rigorous mathematician's point of view, I've also posted this topic in on the math-stackexchange.


title: Construction of the supersymmetric Faraday tensor

text: When I first learned gauge theories in my introductory quantum field theory course, I was taught that the Faraday (field-strength) tensor can be constructed by computing the commutator of the gauge-covariant derivative: $$[D_\mu,D_\nu]=-ieF_{\mu\nu}$$ Now, I am studying supersymmetry following Martin's SUSY primer, and in chapter 4.8, the author immediately writes down the super-symmetric field strength chiral superfield out of the vector superfield $V$: $$\mathcal{W}_\alpha=-\frac{1}{4}D^\dagger D^\dagger D_\alpha V.$$ I would have liked a more gentle introduction to this in terms of something I am already familiar with: is there a way for me to have constructed this using the commutator of some 'gauge super-covariant derivative'?


title: What is the electrical conductivity (S/m) of a carbon nanotube?

text: I have been searching around for a while for this but I am having trouble finding any actual figures, all I can seem to find is that it is "very high". So I am wondering, does anyone have any figures of what the electrical conductivity of a carbon nanotube is, a theoretical or estimated answer is fine. I am preferably looking for the answer in $Sm^{-1}$.


title: Interaction potential analysis from $\phi^4$ model

text: In this paper, the authors consider a real scalar field theory in $d$-dimensional flat Minkowski space-time, with the action given by $$S=\int d^d\\! x \left[\frac12(\partial_\mu\phi)^2-U(\phi)\right],$$ where $U(x)$ is a general self-interaction potential. Then, the authors proceed by saying that for the standard $\phi^4$ theory, the interaction potential can be written as $$U(\phi)= \frac{1}{8} \phi^2 (\phi -2)^2.$$ Why is this so? What is the significance of the cubic term present? In this question Willie Wong answered by setting $\psi = \phi - 1$, why is that? Or why is this a gauge transformation? Does anyone have better argument to understand the interection potential?


title: Is the "particle number" of "electrons" well defined in Wen's string-net theory of elementary particles?

text: According to professor Wen's string-net theory, electrons can be viewed as the elementary excitations of string-net objects. Just like the phonons and magnons are the elementary excitations of crystals and spin systems, respectively. But as we know, the "particle number" of phonons and magnons are not conserved since the statistical ensembles of crystal and spin-system are both canonical ensembles, so the"particle number" of phonons and magnons are not well defined. Of course there are no chemical potentials $\mu$ for them. So I want to know whether the "particle number" of "electrons" is well defined in Wen's string-net theory? And what kind of statistical ensemble does the string-net system belong to,a canonical ensemble or a grand canonical ensemble? Thank you very much.


title: Empirical bound on sum of electron and proton charge

text: Followup to "Why do electron and proton have the same but opposite electric charge?". It is argued that even a tiny residual charge would result in huge amounts of electricity in bulk matter, everything would be different, etc. I do not find that a convincing answer: suppose $n$ protons plus $n+1$ electrons are neutral. Why wouldn't we also expect there to be $n$ protons to every $n+1$ electrons? That is, there is no bulk matter problem if every $n$'th atom is a negative ion (for hydrogen). Now, what empirical lower bound can we give for $n$ in that kind of scenario?


title: Why Is a star a Pure state?

text: I am reading some papers about black hole complementarity (Samir D. Mathur. The information paradox: conﬂicts and resolutions. Proceedings for Lepton- Photon 2011 (expanded). arXiv:1201.2079 [hep-th].) and I found the following sentence; > Thus an initial pure state (the gas making up the collapsing star) evolves > to a black hole, and after evaporation, into something that can only be > described by a mixed state. How is a star a pure state? Isn't it a thermal state and as a consequence is mixed? Isnt that why the light coming from the sun is unpolarized?


title: Magdeburg Hemispheres

text: The Magdeburg Hemisphere experiment was the experiment that showed the effect of pressure differences on a vacuumed sphere. We know that the Force caused by pressure is $\Delta p A$ and so you can calculate the force by using the area of the base of one of the hemispheres of the the vacuumed ball. It is intuitive to me that the area we use is the area created by the diameter of the sphere, a line through the center of the sphere, but is there a clearer reason that we use this area? I'm not sure how to explain why this "important" area.


title: Emergent symmetries

text: As we know, spontaneous symmetry breaking(SSB) is a very important concept in physics. Loosely speaking, zero temprature SSB says that the Hamiltonian of a quantum system has some symmetry, but the ground state breaks the symmetry. But what about the **opposite case** of SSB? The ground state of a quantum system possesses some kind of symmetry while the Hamiltonian does not have this symmetry. For example, the exactly solvable Kitaev-type model Hamiltonians explicitly break the spin rotational symmetry, but the ground states are spin liquids, which possess the spin rotational symmetry. I wonder whether this **opposite case** of SSB plays an important role like SSB in physics? **Note:** Examples with **exact emergent symmetries:** A simple example with _exact emergent $SU(2)$ spin-rotation symmetry_ can be found here A simple model that exhibits emergent symmetry? Another example with _exact emergent $U(1)$ symmetry_ is presented in the Supplemental Material of this paper, where it is appeared on page 2 under Eq.(A7). Examples with **approximate emergent symmetries:** A chiral spin-liquid phase with _emergent $SU(2)$ spin-rotation symmetry_ in the framework of Schwinger-fermion mean-field study of the strongly correlated Kane-Mele-Hubbard model. The example with _approximate emergent lattice 3-fold rotation symmetry_ is the existence of Ferromagnetic(FM) ground state in the Kitaev-Heisenberg model, where the model Hamiltonian explicitly breaks the lattice 3-fold rotation symmetry but the FM phase possesses the lattice 3-fold rotation symmetry. Another example with _emergent chiral symmetry_ was proposed by X.G.Wen in his paper, as seen on page 18, title **C**. A third example with _emergent time-reversal symmetry_ can be found here. An example with _an emergent global topological U(1) symmetry_ is presented here. **More examples with emergent symmetries are welcome.** **Erratum:** The example of "Kitaev model" I gave in my 2nd paragraph is not correct, please see Why we call the ground state of Kitaev model a Spin Liquid? for the reason.


title: What does symplecticity imply?

text: Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context of dissipative systems, so I am no longer confident in my assumption. My question now is, why do authors emphasize symplecticity and what is the property they typically imply with that? Or in other more provocative terms: Why is it worth mentioning that something is symplectic?


title: How to determine the instantaneous acceleration of each box right after the connecting thread is cut?

text: Consider the following diagram. The system is in equilibrium. When the red thread is cut, determine the magnitude and direction of instantaneous acceleration of each box? Neglect the mass of the thread and spring. ![enter image description here](http://i.stack.imgur.com/Lx3ZN.png) In my mental model, the $M$ will be accelerated by $mg/M$ upward and $m$ will be accelerated by $g$ downward. Is it correct? ## My calculation For $M$, there are $k\Delta y=(M+m)g$ force upward and $Mg$ downward. The net force is $mg$ upward, so the acceleration is $mg/M$ upward. For $m$, there is only one force, $mg$ downward, so the acceleration is $mg/m=g$ downward.


title: Electric field outside a sphere with a cavity

text: I have a sphere of radius $2a$ centered at the origin and made of a nonconducting material that has a uniform volume charge density $\rho$. A spherical cavity of radius $a$ eccentric to the right side (touching) with its center aligned with the x-axis is now removed from the sphere. Find the electric field produced by this distribution along the x-axis for $x > 2a$. I started by establishing expressions for $E$ in both the uniform section and the cavity: $E_u = \rho 2a/(3\epsilon_0)$ and $E_c = -\rho a/(3\epsilon _0)$ and $E_t$ inside the cavity is $ E_t = \rho a/(3\epsilon_0)$ Is it correct that the $E$ along the x-axis would have the same magnitude as $E_t$?   I'm thinking it's not, but I'm not sure about the other idea I had which is that it would be $E = \rho r/\epsilon_0$. $r$ now being the distance from the center of the solid/enclosing sphere. This is the result of exchanging the $-\rho$ that is required to make the net charge inside the cavity with $\rho$ and adding the $E$-s together again.


title: Expectation Values in Quantum Mechanics

text: Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?


title: why the force of earth on a object is different from the force of object on earth

text: I have a doubt in gravitation lesson about masses and forces. ![page 1](http://i.stack.imgur.com/ZVL6x.jpg) ![page 2](http://i.stack.imgur.com/coZDH.jpg) ![page 3](http://i.stack.imgur.com/7qFIY.jpg) I have a doubt in gravitation lesson about masses and forces.


title: The moving capacitor

text: To what extent can a charged capacitor mounted on a moving platform (e.g. a rotating wheel) be considered an electric current generator? Electric current, after all, is nothing more than the transport of charge. Some extra detail: Because a capacitor is a charge separation device, do the charges on the two plates effectively cancel to produce no net current? If we ground one side of the capacitor, does the situation change? If we rotate one plate and counter-rotate the other, do we succeed in producing net current?


title: Examples of projection of angular velocity

text: I am looking for examples where the projection of angular velocity vector onto a different axis, has some interesting physical meaning in day-to-day contexts. For example, if a gramophone turntable is inclined with respect to the axis of rotation, we can project this angular velocity onto the geometric (normal) axis. The linear velocity within the plane of the record is then constant at a given distance from the center, so that the frequency of the picked up sound will NOT warble. (Can someone confirm that this reasoning is correct?) Are there other simple examples to get students thinking about projected components of angular velocity?


title: Time evolution of symmetrical mechanical systems

text: I asked this previously in an earlier question, although admittedly my question was hard to find in the slew of info I provided there, so hopefully this will clarify things. Suppose there's a typical $F=ma$ system of the form $$M\ddot{\mathbf{x}}=-\nabla U(\mathbf{x})$$ where $M$ is a mass tensor, $U$ is the potential and $\mathbf{x}$ is a vector describing the state. Suppose that there's a matrix representation of a finite group $G$, $\\{\Gamma_g|g\in G\\}$, such that $U(\Gamma_g\mathbf{x})=U(\mathbf{x})$ for all $\mathbf{x}\in \mathbb{R}^n$ and $g\in G$. Likewise, let's assume there's a basis $\Lambda=\\{\Lambda_1,...,\Lambda_n\\}$ for $\mathbb{R}^n$ such that each basis vector is in a definite eigenstate of each $\Gamma_g$. If we further assume that the system starts out with initial conditions which are in eigenstates of all the $\Gamma_g$, $$\Gamma_g\mathbf{x}(0)=\lambda_g\mathbf{x}(0)\mbox{ and }\Gamma_g\dot{\mathbf{x}}(0)=\lambda_g\dot{\mathbf{x}}(0)$$ then what, if anything, can we say about the expectation values of the symmetry numbers of the state, $$\left<g(t)\right>:=\left<\mathbf{x}(t),\Gamma_g\mathbf{x}(t)\right>$$ as time progresses? When are symmetry numbers conserved over time? If nothing can be said, are there additional constraints which would allow us to say something definite? Here's what I tried: Without loss of generality we can switch to mass-weighted coordinates to get rid of $M$. I vaguely suspected that if you happened to be able to show that $\ddot{\mathbf{x}}(t)$ had the same symmetry as the position and velocity, then the symmetries would be conserved. We have $$\Gamma_g\ddot{\mathbf{x}}(t)=\nabla U(\Gamma_g \mathbf{x}(t))$$ and since $G$ is finite and $\Gamma_g$ is real, the only possible eigenvalues of $\Gamma_g$ are $1$ or $-1$. In the case where $\mathbf{x}(t)$ is a gerade eigenstate of $\Gamma_g$, we have $$\Gamma_g\ddot{\mathbf{x}}(t)=\nabla U(\mathbf{x}(t))=\ddot{\mathbf{x}}(t)$$ which implies the force is of $g$-gerade symmetry, and I feel like we can conclude in this case that $\left<g(t)\right>=1$ for all $t$. Thus there is a "transition selection rule" which says that the system will evolve in such a way which forces the $g$-ungerade entries of $\Lambda^\mathsf{T}\mathbf{x}(t)$ to always be zero. Is that correct? What about $g$-ungerade states $\mathbf{x}(t)$? By the way, I never learned Noether's theorem, although I vaguely recall it has something to do with conserved continuous symmetries, so if you feel this can be answered as a special case of Noether's theorem with discrete symmetries, by all means say so :)


title: Radio telescope alternative. Does a high resolution radio camera exists?

text: I'm looking for an alternative to a radio telescope. Does a high resolution radio camera exists?


title: Why is thermodynamic equilibrium not reached by the system during a thermodynamic irreversible process?

text: In reversible process,due to infinitesimal steps,the system remains in thermal equilibrium(why?) . But in irreversible process, thermal equilibrium does not exist. What is the cause for this phenomenon? Why the former attains thermal equilibrium and the later cannot? Plz help me explaining this.


We already know that non-commutativity of observables leads to uncertainty in quantum mechanics cf. e.g. this and this Phys.SE post. What about the opposite: _Does uncertainty imply noncommutativity?_ If the statistical version of an uncertainty principle (i.e. in terms of standard deviation) is assumed to hold in an algebra of observables, can we ever deduce from this that the ambient operator algebra is necessarily noncommutative? (Say we give the notion of standard deviation as in the Robertson uncertainty principle.) EDIT: In light of the first answer, I should clarify: If there is a universal lower bound for the measurement of the standard deviations in question, must there be noncommutativity? Let me rephrase the question: > **Question:** Why should the presence of a statistical uncertainty relation > imply that we need a noncommutative quantum theory? I guess the point is that Heisenberg's "matrix" explanation for Rydberg-Ritz shockingly implies the uncertainty relation. Particularly, imagining that the classical phase space is a circle, the set of real-valued functions on the space are the observables, and we assume that these functions can be expanded in Fourier series...and then by the Fourier transform the algebra of observables is identified with the convolution algebra of the integers. In the quantum case, Rydberg-Ritz gives that we should replace the above abelian convolution algebra with the convolution algebra of a groupoid...hence a noncommutative matrix algebra. From this noncommutativity, the uncertainty relation can be deduced. Is it possible that a fundamental uncertainty relation exist between commuting observables? No reason exists experimentally to think so, except for the fact that the noncommutative theory predicts it for noncommuting observables...I should be a bit ashamed of this question, as there is no physical reason to think it would be true as of now. Yet I ask anyhow... I think that the first answer does the trick, and that the answer is no, but I'd be interested in any further thoughts on the degree to which uncertainty should force a noncommutative quantum theory.


title: Car accident, put in park or neutral?

text: I was waiting on a red light the other day and was wondering. If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral? I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed.


title: Blonde hair looking green at certain angle

text: I have a friend who has blonde hair. And when look at an oblique angle, there's a tinge of light green in her hair. I've heard that the copper ion in pool water can make blonde hair green, but her hair had always been like that. Besides, it's not really green hair, but more like a green glow. So is it possible that the cuticles on the hair strand acted like a diffraction grating, and somehow only the green portion of the spectrum can be seen? Additional information: I have to look almost horizontally facing the sun to see the green glow. And it was 1 o'clock in the afternoon, so I'd say the solar elevation angle is about 70 degrees.


title: What is the work done against a force?

text: Suppose a particle travels a path $\gamma : I\subset \mathbb{R}\to \mathbb{R}^3$ subject to a force $\mathbf{F}: \mathbb{R}^3\to T\mathbb{R}^3$, then we know that we define the work done _by the force_ as $$W=\int_\gamma \mathbf{F} = \int_I \langle \mathbf{F}\circ \gamma, \gamma'\rangle$$ Now usually I see the term "work done against a force" and I don't really understand what it means. The reason is that in my understanding, work is always done by a force upon a particle or system of particles. If we talk about work done against a force, it is work done by which force on which particle or system? Also, mathematically, how it is obtained? If we want to know not the work done by a force, but against it how we obtain it? I imagine is just the opposite, just changing the sign, but I'm unsure, and if it is really that I can't grasp why it should be.


title: Ratio of energy of normal modes of string

text: Suppose there's a wave on a string represented by fourier series: $$y(x,t) = \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \sin(nx)\cos(nct)$$ I know the energy is $\propto y^2 n^2$, where $y$ is the displacement and $n$ is the normal mode level. Is the energy ratio between the fundamental mode and third mode given by: $$(\frac{2}{1})^2 (\frac{1}{5})^2 = \frac{4}{25}$$? Why is the second excited mode less energetic than the ground mode?


title: Majorana wavefunction

text: I'm trying to compute the wavefunction for a Majorana state in an nanowire/superconductor hybrid system, like arXiv: Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures. I use the same ansatz for the wavefunction $\Psi\left(x\right) = Ae^{zx}$ and obtain the characteristic polynomial $$ z^{4} + 4\left(\mu + 1\right)z^{2} + 8\lambda\Delta z + 4\left(\mu^{2} + \Delta^{2} - V^{2}\right) = 0\text{,} $$ where $\mu$ the chemical potential, $V$ the Zeeman field, $\Delta$ the superconducting gap and I know that $u_{\sigma} = \lambda v_{\sigma}$ where $u_{\sigma}$ describes electron states and $v_{\sigma}$ describes hole states with $\lambda = \pm 1$.To solve the above equation I use the fundamental theorem of algebra: $$ z^{4} + 4\left(\mu + 1\right)z^{2} + 8\lambda\Delta z + 4\left(\mu^{2} + \Delta^{2} - V^{2}\right) = z^4 - \left(z_1 + z_2 + z_3 + z_4\right)z^3 + \left(z_1 z_2 + z_1 z_3 + z_1 z_4 + 2_2 z_3 + z_2 z_4 + z_3 z_4\right)z^2 - \left(z_1 z_2 z_3 + z_2 z_3 z_4 + z_1 z_3 z_4\right)z + z_1 z_2 z_3 z_4 = 0\text{,} $$ where $z_{i}$ are the roots. We see directly that $$ \sum_{i = 1}^{4} z_i = 0 \text{ and } \prod_{i = 1}^{4} z_i = 4\left(\mu^{2} + \Delta^{2} - V^{2}\right)\text{.} $$ Now we can study different cases: the most interested case is when $z_{1/2}$ are complex and $z_{3/4}$ real. In this case we obtain $$ z_{1/2} = a \pm ib $$ and $$ z_{3/4} = -a \pm \sqrt{a^{2} - \frac{4\left(\mu^{2} + \Delta^{2} - V^{2}\right)}{a^2 + b^2}}\text{.} $$ In the publication they write that they have 4 boundary condition and 1 condition from the normalization.Okay, two from the fact that the wavefunction must localized at ends, so that I can write $$ \Psi\left(0\right) = \Psi\left(L\right) = 0\text{,} $$ and the same fact for the derivative of the wavefunction $$ \Psi^\prime\left(0\right) = \Psi^\prime\left(L\right) = 0\text{.} $$ Edit: In the paper above they study two cases. In the first $\left(\mu^{2} + \Delta^{2} - V^{2}\right) > 0$ they say that they have 4 boundary condition and 1 condition from normalization but only 4 coefficients. I can explain that they neglect the two complex wavefunctions. However, in the opposite case $\left(\mu^{2} + \Delta^{2} - V^{2}\right) < 0$ they write that they have 6 coefficients and 6 Condition to solve the equation. But now I'm completly confused! Why now 6 coefficients and 6 conditions?


title: Do Quantum Inequalities forbids even submicroscopic wormholes?

text: Im perfectly aware of the fact that the same laws of quantum mechanics witch allow the existence of negative energy densities also constraints these a lot on their magnitude ,the duration and also any possible application for any space-time "shortcuts" like wormholes or warp drives.These are well known as Quantum Inequalities.However i want to ask if these restrictions theorically forbids also for example submicroscopic wormholes,witch of course can't be used to "transport" people or spaceships ,but at the same time don't require large negative energy densities.


title: Rabbit and tortoise in a foot race

text: A rabbit (H) and a tortoise (t) are in a foot race. When the tortoise is 22 m from the finish line, he has a speed of 4.0 m/s but is 5.0 m behind the rabbit. The rabbit has a speed of 5.0 m/s. The rabbit is overconfident and assumes he has an easy win. So, during the remaining portion of the race, the rabbit decelerates at a constant rate of 0.46m/s2 to the finish line. I am not sure how to start it off.


title: Particle-hole symmetry and the sign of the superconducting gap

text: Time-reversal (TR) symmetry leads to topological insulator property. As expected, the topological invariant depends on TR (Pfaffian) operator. TR+particle-hole symmetry leads to topological superconductor. The topological invariant depends on the Chern number and the sign of the gap. Is it physically correct to say that the Chern number represents the TR symmetry of the band and the sign of the gap represents the particle-hole symmetry ? If yes, what is the physical intuition behind the representation of the particle-hole symmetry with sign of the gap ?


title: Invariant tensors of Symplectic and Exceptional groups.

text: We know that for special orthogonal groups $SO(N)$ there exists invariant tensors (invariant under the group action). These are $\delta_{ij}$ and the totally anti-symmetric $\epsilon_{m_1,m_2,...m_N}$ tensor. Similarly for $SU(N)$ the invariant tensors are $\delta_k^i$, $\epsilon_{m_1,m_2,...m_N}$ and $\epsilon^{m_1,m_2,...m_N}$ ($\delta_k^i$ is an invariant tensor of $U(N)$ too but not so for the $\epsilon$'s). These objects are very useful in constructing singlets out of objects transforming under representations of $SO(N)$ or $SU(N)$. Question 1: Are there such tensors for the Symplectic group and Exceptional groups? I am particularly interested in the groups $Sp(2N)$ and $E_7$. Is there a systematic method of obtaining the same? Question 2. This question is just for the sake of curiosity. Can we also find invariant tensors for supergroups like $OSp(4|\mathcal{N})$ or $SU(2,2|\mathcal{N}/2)$ that appears in numerous $\mathcal{N}$-extended supersymmetric field theories?


title: (Why) is dumping liquid nitrogen on your head dangerous?

text: A chemist who dumped liquid nitrogen on his head described the act as very dangerous; yet, at the same time, he mentioned that the Leidenfrost effect would protect him from the dangers of this act. Clearly, one can see from the video that nothing particularly dangerous happened to him. I'm therefore confused: why is this act dangerous, then, when the Leidenfrost effect clearly protects you?


title: Give a description of M-theory your grandmother can understand

text: Inspired by this question, let me ask a similar question. Is it possible to do the same (give a description of M-theory your grandmother could understand)for M theory? While I know even experts don't understand it properly, I still hope some basic ideas may be expressed. Is it just a hypothetical non-perturbative formulation of quantume gravity whose different limits for varisous coupling constants are five string theories and 11d supergravity or theorists know about it in more detail? **Edit:** As far as my grandmother is concerned let me inform that she has a Phd (HEP her specialization) and quite a smart woman. She has also a good knowledge of GR. So she won't mind technical staff. What she finds frustrating is that there is no rigorous definition of M-theory available to her. In her age (80) she doesn't hope to understand every details of M-theory but she certainly can find some definition of M-theory helpful.


title: What are the most realistic ways of high speed space propulsion?

text: Liquid and solid chemical fuels in rockets are very expensive and inefficient. I have heard of solar sails but what are the most realistic space travel fuels that will be used in the future to get close to the speed of light?


title: Noether's theorem vs. Heisenberg uncertainty principle

text: In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle. Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they? I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists. Also compare this question where again momentum and space are connected, this time through a Fourier transform. (*) i.e. electric potential and magnetic vector potential combined.


title: How do black holes in merging galaxies find each other to merge?

text: In many accounts of galaxy mergers, the prompt merging of their central black holes, if any, is stated seemingly as too obvious to need further explanation. While I don't dispute that this may indeed be the inevitable and prompt outcome, it doesn't seem self-evident, even if both black holes are near the centre of mass of their respective galaxies (and even if so, who is to say the centres of mass meet rather than in effect forming an orbit and the merged galaxy having a new resultant centre of mass?) After all, from a distance a black hole is no different gravitationally from any other mass distribution and, even if they end up fairly close, gravitational waves seem too feeble to degrade their mutual orbit in any reasonable time unless they are extremely close (by cosmic standards - a few diameters apart say).


title: Why does tachyon arise in bosonic string theory?

text: 1. I am looking for precise mathematical and physical reasons which cause the presence of tachyon in bosonic string theory(specially closed bosonic string theory). Has it to do with the specific form of the conformal field theory of free bosons?    2. In particular suppose one makes use of a conformal field theory of central charge 26 but different from the conformal field theory of 26 free bosons to formulate a string theory without local worldsheet supersymmetry. Will the tachyon problem still persist?


title: Time Dilation - what happens when you bring the observers back together?

text: I'm having trouble getting my head around the time dilation paradox. Observer A and B are at the same "depth" in a gravity well. Observer B then descends into the well. A will observe B's time as going slower than their own. B will observe A's time as going faster than their own. What happens if B were to ascend the well back to A's depth, would B's local time speed back up to the same rate as A's, but B would be younger (relative to A)? What about the paradox caused by relative motion (ignoring gravity)? If A is moving relative to B, A and B will both observe the other's time as going slower. If A and B were together initially, then B moves away and returns, do their clocks agree? they can't both be younger than each other :s (i get thats the paradox, but what explanation resolves it?) Thanks


title: Which big bang theory has more assumptions?

text: I'm trying to compare the two most popular big bang theories, the big freeze and the big crunch or big bounce theory, and apply Occam's razor, which says the more assumptions a solution has, the more likely it is wrong. First of all, if the universe had a beginning, then what was the cause of that beginning of time, if by definition, the lack of time means the lack of events? Is it plausible to assume that dark energy remains at constant density with the expansion, and why doesn't that break the first law of thermodynamics? Why do they assume that, without being able to even see it? Up until a couple of decades ago, the big crunch theory was most popular, what we would expect to happen, and do observations of an accelerating expansion really change that, given all the assumptions that need to be made? Dark energy could be decreasing in density, and in the future when all mass is in black holes, wouldn't this suck in dark energy with everything else, converting it's energy into a pure pulling force of gravity?


title: Why is everything floating in space so cold?

text: Of course there is pretty hot stuff too. For example Suns. But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space? Why aren't they hot from time to time? Is it because they loose their heat by radiant heat? And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors?


title: Commutation for constraints

text: Suppose from the Hamiltonian I got the Primary constraints $$(\Phi_m,\Phi)$$ And $\dot \Phi_m$ , $\dot \Phi$ leads to secondary constraints $$(\gamma_m,\gamma)$$ respectively. Now if the commutation of $\Phi_m$ with $\gamma_m$ is non zero (i.e they both are second class) does it terminate the commutation chain there?


title: In heterojunction problem, how to align the energy band in presence of bias voltage

text: For example, SiO$_2$ barrier embeded between Fe magnet and 2-dimensional- electron-gas such as Si. How to align the energy bands of the three materials when an electric field is perpendicular to the interfaces? Do they have the same Fermi level? Could somebody draw the band offset for me? Where can I find the data of band offset?


title: Is it true looking at an object from a great distance with a telescope will show the past version of the object?

text: Once again from my son's workbook. It discussed standing on a planet 65 million light years away from Earth, with an extremely powerful telescope pointed at Earth. It claimed that then you could see the dinosaurs if that was the case. Why?


title: What is the color of a mirror?

text: A mirror couldn't be white, as then you wouldn't be able to see your reflection so clearly. It wouldn't be transparent, as that then won't reflect. So what color is it?


title: Angular momentum

text: I'm given with the following problem - it's an easy one, but it's nearly 1AM, I'm tired and I need some push into the general direction to get the solution: A particle is assumed to be in the state $\left(-\sqrt{1 \over 3} Y_1^0(\theta, \phi), -\sqrt{2 \over 3} Y_1^1(\theta, \phi)\right)^T$ $Y$ are spherical harmonics. What are the   * total angular momentum $\vec{j}^2$   * total orbital angular momentum $\vec{l}^2$   * total spin angular momentum $\vec{s}^2$   * total angular momentum $j_z$ of the particle


title: converting force from spherical to Cartesian coords

text: So I am working on my assignment, and have a question about converting coordinates. I dont know whether I should ask here or the math SE, so lets give it a try here. The force in question is $$\vec{F} = -k\,r^{-n}\,\hat{r}$$ I know the conversion equations but I have no idea how to do it. I need to split the $\hat{r}$ into its corresponding i, j, k vectors. I wanna take the curl to make sure that its zero and hence the force is conservative. Note: the question asks to be converted. I know how to already do it in spherical coordinates.


title: Entropy change at varying temperatures?

text: Entropy change is defined as the amount of energy dispersed **reversibly** to or from the system at a specific temperature. Reversivility means that the temperature of the system must remain constant during the dispersal of energy to or from the system . But this criterion is only fulfilled during phase change & isothermal processes. But not all processes maintain constant temperature;temperature may change constantly during the dispersal of energy to or from the system . To measure entropy change ,say, from $300K$ & $310K$, the range is divided into infinitesimal ranges ,then entropy is measured in that ranges and then is integrated as said in this site . But I cannot understand how they have measured entropy change in that infinitesimal ranges as there will always be difference between the temperature however small the range might be . What is the intuition behind it? Change of entropy is measured at constant temperature,so how can it be measured in a range ? I know it is done by definite integration but can't getting the proper intuition . Also ,if by using definite integration to measure change, continuous graph must be there(like to measure change in velocity,area under the graph of acceleration is measured) . So what is the graph whose area gives change in entropy? Plz help me explaining these two questions.


title: Difference between sound and heat at particle level

text: If heat (or thermal energy) are vibrations of particles and sound is a wave that is propagated through medium e.g vibration of air particles, what indicates if vibration of particles will be perceived as sound or heat (what is the difference between these vibrations) ?


title: Blackbody radiation through quantum mechanics perspective

text: While explaining black body radiation, the body is assumed as a cavity radiator and the radiations are due to the oscillating electrons. But we know that the electromagnetic radiation emitted is quantized, energy of these radiation is in the form of photons which are emitted due to electrons going from higher energy level to lower energy level. But how does this picture fit into explaining black body radiation? Specifically, I cannot understand how, by increasing the temperature, the peak of the frequency shifts to higher energy density in terms of these photon emission?


title: Volume of Brillouin zone is the same as Fourier primitive cell?

text: In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and primitive parallelepiped in real space?


title: Temperature measurement, precision

text: I am having problems with one of the problems![enter image description here](http://i.stack.imgur.com/HeDqP.jpg) I have done (a) which is basically $\theta = 273.16K (19/15) = 346.00K$ For (b) I am given that the length is 15.00cm when the thermometer is incontact with water at its triple point. But I have no idea what the length is when it is in contact with the freezing point. I am also confused with the "precision" thin and how it relates to $\theta = 273.16(X/X_{tp})$


title: Laplacian of a delta function as an interaction potential for Laughlin state

text: I am reading Xiao-Gang Wen's paper "Pattern-of-zeros approach to Fractional quantum Hall states and a classification of symmetric polynomial of infinite variables", on page 8, he gives three interaction potentials for various Laughlin state and claim that these Laughlin state are exactly the zero-energy ground state of the given interaction potential. I do have some questions about these interaction potentials.   1. I do believe that the second term in the interaction potential for $\nu=1/4$ is from Kivelson-Trugman's delta function expansion ("Exact results for the fractional quantum Hall effect with general interactions"), but I think in their paper, the Laplacian is with respect to the relative coordinate which is $z_{rel}=z_1-z_2$, but why in Xiao-Gang's paper that Laplacian is only with respect to $z_1$?   2. Why the delta function only involves $z$, but no conjugate $z^*$? Isn't a two dimensional delta function usually defined as $\delta(z,z^*)$?


title: How can I prove this inequality?

text: Prove that $$ \lambda _{1}\lambda _{2}^{*}\varphi _{1}\varphi _{2}^{*}+\lambda _{1}^{*}\lambda _{2}\varphi _{1}^{*}\varphi _{2} \leq \left | \lambda _{1} \right |\left | \lambda _{2} \right |\left \\{ \left | \varphi _{1} \right |^{2}+\left | \varphi _{2} \right |^{2} \right \\} $$ where all symbols are complex numbers. I encountered this while trying to prove that the set of all square integrable functions form a vector space.


title: The dimensional formula of distance travelled in $n$th second

text: I read that the dimensional formula of distance traveled in $n^{th}$ second is same as that of velocity. Okay, the formula for the distance traveled in $n^{th}$ second is $s_t= u+\frac{a}{2}(2t-1)$ where $u$ is initial velocity, $a$ is uniform acceleration and $t$ is the time. If we proceed and expand the aforementioned formula we will get $s_t = u + at-\frac{a}{2}$. Now, the last term i.e. $-\frac{a}{2}$ is not a velocity and isn't the principle of homogeneity violated?


title: $U(1)$ abelian/axial/chiral anomaly in 4D

text: I am reading $U(1)$ abelian/axial/chiral anomaly in 3+1 dimensions using the path integral method (Fujikawa). Am I wrong in assuming that the anomaly can be cancelled by introducing a counter term in the Lagrangian that exactly cancels the anomalous divergence of the $U(1)$ axial current? Literature sources beg to differ but I cannot understand why it is so. Any reference would be very much useful.


title: How does Dirac show that $\langle B|\bar{\bar{\alpha}}|P\rangle\;=\; \overline{\langle P|{\bar{\alpha}}|B\rangle}\;=\; \langle B|{\alpha}|P\rangle$?

text: Dirac shows that the conjugate imaginary of $\langle \\!P|\alpha$ is $\bar{\alpha} |P\\!\rangle$ and then starts with the identity on page 27 in his book: $$\langle B|\bar{{\alpha}}|P\rangle\;=\; \overline{\langle P|{{\alpha}}|B\rangle}\tag {4}$$ He then says this expression is true for any linear operator $\alpha$ and ket vectors$|P\\!\rangle$ and $|B\\!\rangle$; so replacing $\alpha$ with $\bar\alpha$ we get $$\langle B|\bar{\bar{\alpha}}|P\rangle\;=\; \overline{\langle P|{\bar{\alpha}}|B\rangle}\;=\; \langle B|{\alpha}|P\rangle$$ by using (4) again with $|P\\!\rangle$ and $|B\\!\rangle$ interchanged. **Why should this give the second equality?** If (4) is applied again, I would expect ${\bar\alpha}\rightarrow \bar{\bar\alpha}$ getting back to the LHS expression, yet Dirac has ${\bar\alpha}\rightarrow \alpha$


title: Horizontal velocity of a projectile

text: I was trying to prove that horizontal velocity of a projectile remains constant mathematically, but can't fully come close to proving it. Is it possible to do it, or do we have to live with this assumption which is proven correct only experimentally.


title: Question about Metropolis Monte Carlo in the case of equal energies

text: If configuration A is equal to configuration B in a Metropolis Monte Carlo method, do you still do the attempted update?


title: Why are some variables summed infinitesimally and others aren't?

text: This is something that has been bothering me and I hope the title kind of makes sense. It may be a stupid question but please be gentle. My question is, let's say we have current: $$I=\frac{dq}{dt}$$ And I understand that if we want to find the total charge over some time, for example, you do: $$q=\int I \; dt$$ And what this does, by the definition of an integral, is it sums every product of the infinitesimally small $dt$ and the value of $I$ at that $t$. However, how come you can't have an infinitesimally small current? Meaning, why can't this exist: $$dI=\frac{dq}{dt}$$ Is this a calculus question? Or is this just by the definition of current? I'm really confused. Thanks in advance.


title: What is the operator for the edge current of a fracional quantum Hall state?

text: The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson, $$ S = \frac{1}{4\pi} \int dt dx \left[\partial_t\phi\partial_x\phi - v (\partial_x\phi)^2\right]$$ Here the field $\phi$ is identified with the charge density operator: $$ \rho(x) = \frac{\sqrt{\nu}}{2\pi} \partial_x\phi$$ Now, you expect this operator to be the zero'th component of a two-vector, $J^\mu = (\rho, j)$ with $j$ the edge current. So how does $j$ relate to $\phi$ ? The reason I'm asking is that the literature gives different answers for this.   1. This paper states at the beginning of Chapter III to use the continuity equation to obtain $j = -v\frac{\sqrt{\nu}}{2\pi} \partial_x\phi$.   2. This paper (pdf warning) has equation (2.12) relating $j = -\frac{\sqrt{\nu}}{2\pi} \partial_t\phi$. No motivation though.   3. This paper first couples the theory to an electromagnetic potential with components $(a_0, a_x)$ at the boundary ($D_\mu = \partial_\mu + \sqrt{\nu}a_\mu$), $$ S = \int dt dx \frac{1}{4\pi}\left[D_t\phi D_x\phi - v_c (D_x\phi)^2\right]+\frac{\sqrt{\nu}}{2\pi}\epsilon^{\mu\lambda}a_{\mu}\partial_\lambda\phi$$ and defines the current through $J^\mu = \frac{\delta S}{\delta a_{\mu}}$ giving $J^\mu=(\frac{\sqrt{\nu}}{2\pi} D_x\phi, -v\frac{\sqrt{\nu}}{2\pi} D_x\phi)$. So I notice that case 3 reduces to case 1 when $a_x = 0$. Furthermore, case 3 is gauge invariant, so this operator seems like a logical choice. My problem with this choice is as follows: Consider a system with two edges (infininte quantum Hall bar) and suppose you have a non-zero potential along one edge, $a_t = U$ and $a_x = 0$ along this edge. You therefore expect a current to run through the system, because the edges are held at different potentials (and the current runs perpendicular to the potential difference, because of the quantum Hall relation). But: $\langle \partial_x \phi\rangle = 0$, so for case 1 and 3 there's no current.... Does that make case 2 the correct choice? So perhaps the question comes down to: What operator represents the edge current? What operator is 'measured' in an experiment where the current is probed?


title: Best Research Documentation Habit for Computational Physics Research

text: I am currently doing research in a subset of the field of gravitational lensing called weak lensing. I am simulating blended galaxy profiles and creating an algorithm to extract the true parameters I created my image with. I am then assessing the accuracy of my algorithm. Since this is my first venture in creating code modules, specifically in python, I am wondering how other researchers have used resources to create the best kind of documentation for work that is fundamentally computational, not experimental. For example, do I put all my work in a google logbook? Or do I use Ipython notebook to generate PDF's? I've had a tremendously hard time finding suitable materials. Thanks for the help!


title: Two expressions for potential energy in the gravitational field of the earth

text: Let $M$ be the mass of the earth, considered as a point mass, then the potential energy of a point with distance $r$ away from the center (assume $r > \textrm{radius of earth})$ is $$ U(r) = -\frac{G M m} r = -\frac{gm}{r} $$ with $g := G\cdot M$. Now, as it is written in the textbooks, the potential energy of a body of height $h$ on earth is approximately $$ U(h) = mgh. $$ Now for me both expression contradict each other, one is negative, the other positive, one decreases as $r \to \infty$, the other increases as $h \to \infty$ (and the height is directly related to the distance/radius $r$ I think). So whats my misconception, what have I missed that I came to conclusion that both expression contradict each other? Thanks for any help! _EDIT_ : Changed $$ U(r) = -\frac{GM}{r} = -\frac{g}{r} $$ to $$ U(r) = -\frac{GMm}{r} = \frac{gm}{r}. $$


title: Is there perfect vacuum 10000 billion lightyears away?

text: Is there perfect vacuum about _some_ 10,000 billion light years away? With perfect vacuum, I mean that there are no particles, not even virtual photons! I had this idea by assuming that all particle filled space was created by the big bang and the max speed of light combined with the age of the universe. Although probably impossible to measure, I wonder what the consensus is (if any) and what the theory says about it (if there is a theory about perfect vacuum at all)? By the way, I'm aware of the Cosmological constant problem, but that's not what I require as an answer. I haven't heard much about vacuum during education is that normal?


title: Why do we use Hermitian operators in QM?

text: Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') operators. The eigenvalues of the operator are the allowed values of the observable. Since Hermitian operators have a real spectrum, all is well. However, there are non-Hermitian operators with real eigenvalues, too. Consider the real triangular matrix: $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\\ 8 & 4 & 0 \\\ 5 & 9 & 3 \end{array} \right) $$ Obviously this matrix isn't Hermitian, but it does have real eigenvalues, as can be easily verified. Why can't this matrix represent an observable in QM? What other properties do Hermitian matrices have, which (for example) triangular matrices lack, that makes them desirable for this purpose?


What is the definition of a quantum integrable model? To be specific: given a quantum Hamiltonian, what makes it integrable?


title: Maximise magnetic force at one point while minimizing at another

text: Is it possible to have a magnet that attracts one object strongly, but not another object behind that first object? Given an (electro-) magnet M above two thin parallel steel sheets P1 and P2 in the following configuration:                     M          (magnet)     ----------- P1   (steel sheet)     ----------- P2   (steel sheet)      Is there a configuration that would maximize the attractive force between M and P1 while minimizing the attractive force between M and P2? I.e. a magnet that would grip P1 very strongly, but not P2? I know that generally the strength is proportional to the square root of the distance, but is there a particular configuration of multiple magnetic poles such as a Halbach array, that would change this? Can the gradient of the magnetic force, or the strength with which it permeates the first steel sheet be manipulated this way? Or is the force working on these objects simply a function of the overall strength of the magnet and the distances involved?


title: Is a total solar eclipse a common phenomenon?

text: Is our moon "special" for creating a total solar eclipse? Is it common or rather rare for a moon to create a total solar eclipse the way our moon does, with corona and such?


By what mechanism do quantum effects become observable in normal life at the macroscopic level? For instance, when two molecules "collide" is the momentum a probabilistic event wherein the end state is not unique? Another example, during a chemical reaction, it is a probabilistic event at the quantum level whether or not any particular molecule within the solution interacts with another molecule?


title: Chaos and quantum physics: How many ways can a bonfire burn?

text: I'm interested in the extent to which quantum physical effects are seen at a macroscopic level. I might get some of the physics wrong, but I think I'll get it close enough that I can ask the question: Let's sat that we create a bonfire and let it burn until it burns out. As the smoke rises from the fire, turbulence takes over and the smoke particles and steam and hot air all mixed together. By the end of the night when the fire has burned out, the collection of molecules in the system are in some position/velocity X. My question: Let's assume the multiverse interpretation of quantum physics. How many possible end state superpositions can there be in this situation? Ok, that's imprecise and incorrect because it would actually be an uncountable infinitude of possible end states. How about this: Given the end state that we observed, what percentage of the end state superposition would be "visually" indiscernable from the end state that we observed so that each molecule would be in nearly the same end state across that portion of the multiverse? Or put another way: Do quantum effects sneak into everyday life fast enough that we can observe them? If we are effected by quantum physics at all, I imagine this is roughly a function of the timescale of the chaos effects.


title: Band offset and strain-induced valence band splitting in semiconductor compounds

text: My aim is to draw a plot of $GaAs_{1-x}Sb_{x}$/InP (GaAsSb on InP substrate) band offset as a function of As/Sb composition. The first part is easy if I'm doing this correctly: I found VBO (valence band offset) parameters for GaAs and GaSb, then the bowing parameter for $GaAs_{1-x}Sb_{x}$ VBO and finally the bowing parameter for $GaAs_{1-x}Sb_{x}$ band gap. This is what I got: ![Unstrained GaAsSb](http://i.stack.imgur.com/XOAPX.png) However, when I put GaAs or GaSb on InP, strains appear and the valence band is split into heavy hole and light hole subbands. In some articles I found the way to calculate the splitting: $\Delta E_{HH}=\partial E_H^V + \partial E_S$ $\Delta E_{LH}=\partial E_H^V - \partial E_S$ $\partial E_S=b(1-2\frac{C_{12}}{C_{11}})\epsilon_{||}$ $\partial E_H^V=2 a^V (1-\frac{C_{12}}{C_{11}})\epsilon_{||}$ a, b and Cs are material properties and can be found in this article: http://jap.aip.org/resource/1/japiau/v89/i11/p5815_s1 (Vurgraftman, Meyer, Ram-Mohan) However, I'm not sure what the $\epsilon_{||}$ is. Is it the direction of growth? Whatever it is, I can't see where I should consider the substrate, as I assume the strain to be substrate-dependent. Is it hidden in the $\epsilon$?


title: Polarisation directions in standing waves in cubical cavity

text: I was studying Rayleigh-Jean's formula. The author has assumed a cubical cavity of each side $L$ with perfectly reflecting surfaces. According to author, there are two perpendicular directions of polarization for a standing wave taken in arbitrary direction. I am stuck up here, my question is why two directions of polarization?


title: Why are clouds wavy?

text: Say you're in an ascending airplane as your 10 year-old son asks you: "Dad, why are these clouds wavy?" Now, say you know a little about gravity waves and the formation of wavy clouds(Maybe I should be linking to As instead? Are these wiki pages basically talking about the same thing?). How would you go about explaining it to your son? PS. I'm looking for a simple and fluid explanation that would be able to give some intuition for the phenomena.


title: Why we can only perceive 3 dimensions

text: What is the reason that we cannot perceive more than 3 dimensions with our senses?


title: What is the attenuation spectrum of radio waves in pure water?

text: Here is a plot of microwave attenuation in atmosphere: ![enter image description here](http://i.stack.imgur.com/0NSM6.png) (source) I'm looking for a similar plot, but for pure water, and a frequency range covering at least 100 MHz to 2 GHz, hopefully more.


title: Time-dependence in LCAO

text: I would like to study time-dependence (TD) in linear combinations of atomic orbitals (LCAO). The Hückel method enables quick and dirty determination of MOs for suitable systems (view link for assumptions). The basis function $\phi$ is the 2pz orbital. Any molecular orbital (MO) is described as $$\psi_n(r) = \sum_k \phi_k(r)\ c_{k,n}$$ where $c_{k,n}$ is the coefficient on the $k$th atom at energy level (MO number) $n$. For benzene, the first MO is (calculation omitted): $$\psi_1(r) = \frac{1}{\sqrt{6}}\sum_{k=1}^6 \phi_k(r)$$ Pretty straightforward. Notice that the functions are all time-independent. I want to construct the TD wavefunction describing benzene in its ground state. $$\psi_{tot}(r,t) = \sum_j \psi_j \mathbf{C}_j\ e^{-itE_j}$$ Benzene has 6 $\pi$ electrons, thus the first three MOs will be occupied in its ground state, each containing 2 electrons. From the previous formula, the total wavefunction should be: $$\psi_{tot}(r,t) = 2\sum_j^3 \psi_j \mathbf{C}_j\ e^{-itE_j}$$ **How do I determine the second set of coefficients $\mathbf{C}_j$?**


title: Physical interpretation of infinite total cross section

text: What does it tell us about a process, say A+B->C+D, if the calculated total cross section is infinite?


title: If kinetic energy is a function of the square of velocity, how can gravitiational acceleration be constant?

text: Kinetic energy is defined as: \begin{equation} E_k = \frac{mv^2}{2} \end{equation} Because of the squared velocity term in there, the faster you're going, the more energy it takes to further accelerate. (This is why it takes longer to get your car from 30 to 60 than from 0 to 30, for example.) However, unlike normal methods of acceleration that add (more or less) constant energy over time to an object, the force of gravity is defined as imparting a constant _acceleration_ over time to an object. How does that work? Does that imply that the faster an object is already moving (in the direction of the mass exerting gravity on the object), the stronger gravity pulls at it?


title: What is the difference between Ricci curvature tensor and the Laplacian

text: Can someone please explain the difference between the information that the Ricci curvature tensor gives us and that which the Laplacian offers? Here is my shot at it: My concept of the Ricci curvature tensor is that it gives us information about the local curvature (how much it diverges from a flat plane) which can then be compared to any other local curvature. Universal comparisons of curvature can be made based on it. The Laplacian also gives us information about curvature, but it does not do so in the "absolute" way that the Ricci tensor does. Universal comparisons of curvature cannot be made based on it. Thanks


title: How does temperature affect photovoltaics (PV) efficiency?

text: I know that photovoltaic panels are more efficient at lower temperatures: As the temperature increases, the output voltage decreases. I am looking for an explanation of the mechanism behind this effect. What is going on in terms of electron energy level transitions, the band gap, etc?


title: Equations of circular motion

text: I have studied that a finite angular displacement $\triangle \theta$ is a scalar. But, $\delta \vec \theta$ is a vector. Now, when it is a uniformly accelerated motion we are dealing with, we use equation: $$\theta~=~\vec{\omega}_i t + \frac{1}{2} \vec{\alpha} t^2, $$ where $\vec{\theta}_0$ is initial angle and $\vec{\omega}_0$ is intial Now my question is, What is $\vec{\theta}$ here? Vector? If yes, then how can we get a finite value for it? (Since as it is a vector only when it is infinitesimally small.) If no, then how is this equation valid?


title: What is the consequence of "infinite" gravitational force?

text: ## Introduction I am a mathematically minded individual. I do not intuitively comprehend physics, and as a sophomore in high school who has only taken Intro to Physics in his freshman year, I may very well have a completely erroneous view of many concepts within physics. Recently, Einstein's model of gravity has been on my mind. I was watching the following lecture here wherein Professor Sera Cremonini explains that gravity is due to the curvature of 4 dimensional spacetime, and that this curvature is caused by the mass of objects within the universe. Furthermore, the higher the curvature (equivalently, the higher the mass of an object) the higher the force of gravity near that object. To illustrate this, she uses a stretched rubber disk and puts a large ball on it whilst smaller balls are on the disk. The large ball causes the rubber to stretch, and this curvature causes the smaller balls to be pulled toward the large ball just as gravity causes orbits of the solar system. This made me think. ## Question Theoretically, is it possible for the curvature of the universe at one particular point to be so strong that the gravitational force is infinite? If infinite is ridiculous in this context, consider the question reformulated: Is there, or can there be, a point in the universe where the curvature is greater than any other point in the universe? If so, why doesn't the strength of gravity at this point cause the entire universe to be contracted to this point? That is, isn't this just like the Big Crunch? ## P.S. Once again, I don't know if I've conveyed this using the proper terminology and I'm aware it's a very bizarre idea. I hope you all can entertain my thoughts and attempt to answer my question as best as possible, and correct my understanding and terminology as warranted. Thank you.


title: Compressibility and specific heat: interconnectedness and independence?

text: Consider a Mott insulator (insulator arising from strong correlations). This is an incompressible phase i.e. it costs energy to add a single particle to it; the ease of compression (compressibility) may be defined as $K = \frac{\partial \rho}{\partial \mu}$, where $\rho$ is the particle density and $\mu$ is the chemical potential. For a Mott insulator, the compressibility is zero. The Mott insulator's constant volume specific heat $C_v = T\frac{\partial S}{\partial T}$, with entropy $S$ at finite low temperatures $T$, will be exponentially suppressed due to the presence of such a particle gap. This simply indicates the absence of states close to the ground state "reachable'' by the energy provided by the low temperatures. I was wondering if there are any phases, exotic perhaps, where (a) the specific heat is suppressed but the compressibility is large, or (b) the specific heat is large but the compressibility is diminished.


title: Bound states in QCD: Why only bound states of 2 or 3 quarks and not more?

text: Why when people/textbooks talk about strong interaction, they talk only about bound states of 2 or 3 quarks to form baryons and mesons? Does the strong interaction allow bound states of more than 3 quarks? If so, how is the stability of a bound state of more than 3 quarks studied?


title: Which formula for entropy is correct? (OR Is the fundamental thermodynamic equality always right?)

text: Quick question, in the lecture notes to a thermodynamic course I'm taking, $$d\underline{S}=\frac{d\underline{U}}{T}+\Big(\frac{\partial P}{\partial T}\Big)_{\underline{V}} d\underline{V}$$ But everywhere else I've looked I've found (by rearranging the fundamental thermodynamic relation: $$d\underline{S}=\frac{d\underline{U}}{T}+\frac{P}{T}d\underline{V}$$ The two are obviously extremely similar, however I don't believe they're equivalent in general (I believe you'd have to make further assumptions about the system, particularly $P=kT$ at constant specific volume, to go from one to the other.) Right now I'm just asking if anyone knows which relation is the more general. I was of the opinion that the fundamental thermodynamic relation was always correct, but now I'm doubting myself. Thanks.


title: Is String Theory a Field Theory?

text: Is String Theory a Field or Quantum Mechanical Theory of the String rather than a Particle? I should know this having studied this for a term, but we jumped into the deep end, without really covering the basics of the theory.