How should semiconductors be understood?

In the chemistry setting, a doped semiconductor is a crystal lattice with holes/extra electrons in it. In the band/quantum mechanical picture, these holes/extra electrons can be seen as the amount of missing/excess charges from a full band of energy states. Therefore, in a pure silicon crystal, with full covalent bonds and thus no holes or extra electrons, the valence band is completely full, and the conducting band is completely empty. Which means that Silicon is an insulator, since the Fermi level lies above the valence and below the conducting band. What are the holes in my reasoning?


Mathematical Physics SUSY QM Resource Recommendation

I want to study SUSY QM. I found some excellent physically motivated articles on Arxiv. Despite, I am especially interested in the mathematical structure behind SUSY QM. Does anybody know whether there is a particularly good reference to start studying SUSY QM from a rigorous mathematical point of view. The reference should cover Spectral analysis and a wide range of all the algebraic properties concerned with the topic.


What are the general solutions to a hard sphere collision?

Surely someone has found the solutions to the hard sphere collisions (in $n$ dimensions) of two bodies of mass $m_1$ and $m_2$, respectively--that is the resultant velocities (or momenta) of the two bodies after a hard sphere collision given $\vec{p}_1$ and $\vec{p}_2$. I have a solution (both for elastic and inelastic collisions), but I wonder whether or not it's correct. FYI, here are my solutions for elastic, hard-sphere collisions: $$ \vec{p}_{1f} = \vec{p}_1 + 2\frac{\left(m_1\vec{p}_2 - m_2\vec{p}_1\right)\circ \hat{r}}{m_1 + m_2}\hat{r} \\\ \vec{p}_{2f} = \vec{p}_2 - 2\frac{\left(m_1\vec{p}_2 - m_2\vec{p}_1\right)\circ \hat{r}}{m_1 + m_2}\hat{r} $$ wher $\hat{r}$ is in the direction of the "hard sphere" (i.e. the unit vector connecting the two radii of the spheres). You can see that my formulation trivially conserved momentum--the question is whether or not it conserves energy as well.


Symmetry breaking in Bose-Hubbard model

According to Landau's symmetry breaking theory, there is a symmetry breaking when phase transition occurs.   1. What is the symmetry breaking of superfluid-Mott insulator transition in Bose-Hubbard model?    2. Why metallic state to Mott insulator state transition in Fermi-Hubbard model is not a phase transition, but a crossover.


Why do physicists use LHC?

1. My question is why are we colliding particles in LHC to produce new ones?    2. And these particles that they sometimes say live for a fraction of a second, how in space they exists then?    3. In space all these particle exist without smashing to each other, why do we need to smash them to produce them?


If batteries are a source of energy, would not lower-valued resistors cause a violation of the conservation of energy?

Here's the conundrum I have been facing. Yesterday I asked the question: how can batteries constantly motivate electrons to complete a circuit? (What maintains constant voltage in a battery?) After some thought I realized that the concept is that of an energy source. The batteries are a source of energy, and regardless of the mechanism, they will constantly 'push' for a goal. My solution finally came when I realized that speaking about someone's 'motivation' was identical to the constant push of the battery. Here's the issue: My sister wanted to get me coffee yesterday for my birthday. She was attempting to give it to me before my class started. Unfortunately, she had difficulty with traffic, and so was able to deliver only after my class started. I realized when she told me that she tried as hard as she could, and that what really mattered was her motivation. The push she gave was constant regardless of the resistance she encountered. The parallel here to the constant push of a battery was strong; voltage didn't matter in respect to resistance for a given battery. Now I have a different issue. The voltage drop is described as the per-capita potential energy difference of electric charges at points a and b right? The total energy would then be the amount of 'capitas' multiplied by this per capita potential energy difference. More capitas mean more total energy. Here's the problem. If the motivation of a battery is described by voltage, then if we look at a circuit with a resistor and compare it to another circuit with an equal-length resistor but with a wider surface area, the potential difference is supposed to stay the same in both cases. However, in taking the charge from one surface to the other, more energy is required for the resistor with larger surface area! The amount of energy supplied is different in either case, yet from the same 'constant' energy source. This seems so wrong to me right now. Yet I know that Power=Voltage*(Current)^2, so this actually makes sense in terms of the equation. I wonder what is lacking in my fundamental understanding of the potential difference in comparison to the idea of motivation/constant energy source. I'm really fundamentally misunderstanding something complex here that I don't quite have my finger put on yet. Please help clear up my view of a battery as a constant pusher to line up with this changing energy ouput a battery can give.


How is it possible to move something without completely lifting it?

For example, let's assume a chair here: It can be "slid" across the force if we use minimal upwards force, but not enough to actually "lift" the chair. Why should it move? Here's a better example: I have seen some strong people lift and slide the back of cars, however, the back tires of the car never elevate completely off the ground. With regards to friction, though, how is it possible to actually "move" something without lifting it completely? How does this work is a better question. This image should illustrate the car example: ![enter image description here](http://i.stack.imgur.com/foRJg.jpg) Is there enough space within it to allow the angular force to make it "slid" a pinch, or what else would explain its ability to move without being completely suspended in the air that would, say, differ from this with a human?


Are there known turbulent nonlinear equations where the cascade is a thermal gradient?

In a recent answer (here: The equipartition theorem in momentum space ), I suggested that if you have an appropriate first order equation (in the answer I used a second order equation, but it is more likely to be true for a first order equation) of the form: $$ i\partial_t \phi = (\nabla^2)^N \phi + \lambda \phi^3 + f + D $$ (EDIT: I first had $|\phi|^2\phi$, which is the nonlinear Schrodinger equation, but this has the same frequency shifting property as the relativistic version--- the frequencies are not completely perturbatively decoupled even with a resonance condition. The only reason I went to first order was to fix this annoying problem, and there is no condition of being physical. Although this will not conserve the total $|\phi|^2$, it has a conserved energy.) Where f is a driving force with only long-wavlength components, and D is a damping force which only affects short wavelength components. The field $\phi$ is a complex scalar. Then if N is large and $\lambda$ is sufficiently small (by scaling, $\phi$'s magnitude can substitute for $\lambda$, so the limit of small $\lambda$ is really the limit of small $|\phi|$), the cascade due to the nonlinear term is a thermal gradient draining energy from the hot low frequency modes to the cold high-frequency modes. I don't know if this model is in the literature, of if the thermal analysis is accurate (I am pretty sure it is correct, because the resonance matching does lead to local flow of energy in |k|, but I am not sure if there isn't a block to thermalization due to some problem with mixing in each |k| modes separately). In the original answer, I used a second order equation, which has frequencies of both signs. This allows some extra mixing between distant modes, which shifts the frequency of low modes based on the occupation of high modes. While I don't think this changes the cascade significantly, I wanted a clean example, so I used a first order equation here. (EDIT: The clean example does not use the absolute value, so that the frequencies have to strictly additively combine.) Is this model analyzed in the literature? Does it thermalize into a thermal gradient cascade?


One more relation with spherical spinors

Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\\ \sqrt{l \mp m +\frac{1}{2}}Y_{l, m + \frac{1}{2}} \end{pmatrix}. $$ How to prove the relation $$ (\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -r\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}? $$ I tried to rewrite the $(\hat {\sigma} \cdot \mathbf r )$ as $$ (\hat {\sigma} \cdot \mathbf r ) = r\begin{pmatrix} \cos(\theta ) & \sin(\theta )\,e^{-i\varphi } \\\ \sin(\theta )\, e^{i \varphi } & -\cos(\theta ) \end{pmatrix}, $$ but it's hard to use recurrence relations for associated Legendre functions after that. Is there some hint which simplifies the proof? Maybe, I can first show that $(\hat {\sigma} \cdot \mathbf r )\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = -ra\mathbf {Y}_{j, m, l = j \mp \frac{1}{2}}$, because the parity of the left side is opposite to the parity of $\mathbf {Y}_{j, m, l = j \pm \frac{1}{2}}$, and $(\hat {\sigma} \cdot \mathbf r )$ commutes with rotation operator, because refers to the scalar pdoduct?


The Holstein-Primakoff Representation (approximation)

I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \sqrt{2S - n_j} a_j \\\ S_j^- = a^\dagger_j\sqrt{2S - n_j} \\\ S^z_j = S - n_j $$ Where $a_j$ and $n_j$ are operators. When we derive the magnon dispersion relation, we make the assumption that $$ \frac{\langle n_j \rangle}{\langle S^z \rangle}\ll 1 $$ which is fine. As far as I understand this only means that we assume that most of the spins are pointing along the z-direction. However, when we go further in the derivation, we do a series expansion in $n_j/S$, so that i.e. $$ S_j^+ = \sqrt{2S}\sqrt{1- \frac{n_j}{2S}+...} \approx \sqrt{2S}\sqrt{1- \frac{n_j}{2S}} $$ This is where I don't understand. Say that we are in a spin $1/2$-system. In that case we can for one site have at most 1 magnon excitation and $S=\pm 1/2$, so for each individual site $n_j/2S$ is not much smaller than one. My question is then, how can we justify that the series expansion makes sense in the operators at each individual site? Will the contributions from each site when summed up not contribute because $$ \frac{\langle n_j \rangle}{\langle S^z \rangle}\ll 1 $$ or have I misunderstood something? Any thoughts would be much appreciated.


Research on galaxy "stiffness"?

> **Possible Duplicate:**   >  Are galactic stars spiraling inwards? Has there been any research on the effects of interstellar gravitational attraction that would make collective star motion in a galaxy behave more like a solid disk? I understand rotational velocity has been observed to be higher than expected for stars farther from galaxy centers. This doesn't agree with Newtonian physics in which Pluto goes around the Sun more slowly than Mercury. An explanation for the faster star motion observed in galaxies was created and called dark matter. This seems to not comply with Occam's Razor, where the most simple possibility is probably right. I visualize the attraction between stars as stiffening the collection, or increasing the viscosity of the rotational behavior which could explain faster rotation towards the outside of a galaxy. The rotation would behave somewhere between atoms locked together in a frisbee, and a fluid circling the drain. Has any research been done to prove this wrong?


How would water drain out of a sealed pipe?

While hiking through the Grand Canyon, I started wondering. Say we have a pipe for the purpose of transporting water across a canyon, with the bottom submerged in a pool of water. Like the blue line in this terribly drawn image: ![enter image description here](http://i.stack.imgur.com/7iVLE.png) Now let's seal the two ends of the pipe, which is full of water. What would happen if the pipe broke at the bottom of the U? Note: It shouldn't matter whether the pipe is a U or a vertical line, I simply thought about this with a U-shaped pipe. * * * If the ends of the pipe weren't sealed, the water inside the pipe would leak out until the height of the water outside the pipe was the same as the height of the water inside the pipe. But the ends are sealed. A vacuum can support 10m of water, so perhaps the water will drain out of the pipe until it is 10m above the height of the water outside of the pipe, leaving a big near-vacuum behind. This would explain why the Grand Canyon's pipe has such a large number of valves. On the other hand, we learn from http://what-if.xkcd.com/6/ that water boils into a vacuum. Does this leave a significant amount of water vapour behind?


Uniqueness of quasiclassical consistent histories

The current zeitgeist here is on interpretations of quantum mechanics, so let me add my own two cents here. As you may know, consistent histories is an alternative interpretation proposed in a series of papers 1990-1994 by Gell- Mann and Hartle. In a shocking article, Dowker and Kent proved there are many distinct consistent histories for most quantum systems. Even if we constrain ourselves to quasiclassical histories, i.e. those histories which are close to classical, being quasiclassical now does not guarantee being quasiclassical in the future, or in the past. This is disturbing. Do histories which are quasiclassical for all times exist? If they do, are they unique? Is the nightmare scenario of two everywhere quasiclassical consistent histories which are mutually incompatible possible?


Weyl Large Number Coincidence

I've just learned of the Weyl Large Number Coincidence on the Wikipedia. It looks interesting. Here is my interpretation of it. What is the smallest "quantum" of energy in the Universe? $$E_{min} = \hbar H$$ where $H$ is the Hubble parameter. This make sense as the Hubble parameter is roughly the inverse of the age of the Universe ($H=2\times10^{-18}s^{-1}$). You can't define a non-zero frequency smaller than that. Now consider the gravitational self-energy of a particle, $E_g$, given by $$E_g = \frac{Gm^2}{r}$$ where $m$, $r$ is the mass,size of the particle. The smallest possible particle must have a gravitational self-energy of the size of $E_{min}$ therefore we have $$\frac{Gm^2}{r} = \hbar H$$ We can link the mass $m$ and size $r$ of a particle by using the Compton wavelength quantum relation $$m c r = \hbar$$ Putting the two equations together and eliminating $r$ we find $$m^3 = \frac{\hbar^2H}{Gc}$$ If we plug values into this expression we get a mass given by $$m \approx 10^{-28}kg$$ This is nearish to the proton mass. Interesting or just a coincidence?


D-branes wrapping divisors and/or cycles

What is the difference between a divisor and a homology cycle? What is the difference between a D-brane wrapped around a divisor and a D-brane wrapped around a cycle? Thanks.


How is it possible for objects to travel faster than the speed of sound when particles interact at the speed of sound

First of all, I am sorry if this is a stupid question but: I've heard that atoms interact with each other at the speed of sound (when you for instance push a chair, the atoms collide with each other in a chain reaction at the speed of sound, making the chair move). How can then airplanes fly faster than the speed of sound without something going crazy?


Ballistic Pendulum Demo Problem

I have a question about the following problem: > ![enter image description here](http://i.stack.imgur.com/nOtw4.png) I got the solution $v=\frac{M+m}{m} \sqrt{2gh}$. But my real question is in the following picture: > ![enter image description here](http://i.stack.imgur.com/gVtk4.png) In the above slide, how can you derive the solution $v=(\frac{M+m}{m})d\sqrt{\frac{g}{L}}$ and thus, $v=3d$? Also, it seems that $v=3d$ is not correct since their unit is not matching... Can somebody explain this???


Schrödinger's equation, time reversal, negative energy and antimatter

You know how there are no antiparticles for the Schrödinger equation, I've been pushing around the equation and have found a solution that seems to indicate there are - I've probably missed something obvious, so please read on and tell me the error of my ways... Schrödinger's equation from Princeton Guide to Advanced Physics p200, write $\hbar$ = 1, then for free particle $$i \psi \frac{\partial T}{\partial t} = \frac{1}{2m}\frac{\partial ^2\psi }{\partial x^2}T$$ rearrange $$i \frac{1}{T} \frac{\partial T}{\partial t} = \frac{i^2}{2m}\frac{1}{\psi }\frac{\partial ^2\psi }{\partial x^2}$$ this is true iff both sides equal $\alpha$ it can be shown there is a general solution (1) $$\psi (x,t) \text{:=} \psi (x) e^{-i E t}$$ But if I break time into two sets, past -t and future +t and allow energy to have only negative values for -t, and positive values for +t, then the above general solution can be written as (2) $$\psi (x,t) \text{:=} \psi (x) e^{-i (-E) (-t)}$$ and it can be seen that (2) is the same as (1), diagrammatically ![energy time diagram](http://i.stack.imgur.com/0JBc2.jpg) And now if I describe the time as monotonically decreasing for t < 0, it appears as if matter(read antimatter) is moving backwards in time. Its as if matter and antimatter are created at time zero (read the rest frame) which matches an interpretation of the Dirac equation. This violates Hamilton's principle that energy can never be negative, however, I think I can get round that by suggesting we never see the negative states, only the consequences of antimatter scattering light which moves forward in time to our frame of reference. In other words the information from the four-vector of the antiparticle is rotated to our frame of reference. Now I've never seen this before, so I'm guessing I've missed something obvious - many apologies in advance, I'm not trying to prove something just confused.


Distribution of charge on a hollow metal sphere

A hollow metal sphere is electrically neutral (no excess charge). A small amount of negative charge is suddenly placed at one point P on this metal sphere. If we check on this excess negative charge a few seconds later we will find one of the following possibilities: (a) All of the excess charge remains right around P. (b) The excess charge has distributed itself evenly over the outside surface of the sphere. (c) The excess charge is evenly distributed over the inside and outside surface. (d) Most of the charge is still at point P, but some will have spread over the sphere. (e) There will be no excess charge left. Which one is correct and why? I guess it is some kind of electrostatic induction - phenomena going on. Am I right? I understand that excess charge is distributed over hollow sphere and that negative and positive charges are distributed opposite sides, but don't know which one positive or negative go to inside surface.


On the shape of magnetic and electric fields in an electromagnetic wave

Electromagnetic waves are generally depicted like this: ![enter image description here](http://i.stack.imgur.com/Gn4wv.png) Where the electric fields and magnetic fields exist in the planes perpendicular to the direction of propagation. I also realize that as the electric field changes while the wave is propagating, a magnetic field is induced and vice versa (by faraday's and maxwell's laws of induction). But, those laws predict that the fields will be circular. So, won't the electric and magnetic fields look different? Won't they be circles along arrows that are drawn in the figure? I haven't seen anything written about this anywhere.


Coulomb force in SI and cgs

Coulomb force in SI is $ F = \frac{Q1*Q2}{4\pi\varepsilon R^{2}} $ while in CGS $ F = \frac{Q1*Q2}{R^{2}} $ why is it? I mean doesn't it any make difference in dimension? since $ \varepsilon $ itself has dimention. And similarly for magnetic force also.


Why is there a factor of $4\pi$ in certain force equations?

I mean to ask why there is $4\pi$ present in force equations governing electricity? Though all objects in universe are not spherical and circular, the constant of proportionality in both equations contain $4\pi$. Why?


Uncertainty of permittivity of vacuum

**Question** : The value of permittivity of vacuum, $\epsilon_0$, is given with absolutely no uncertainty in NIST Why is this the case? * * * **More details** : The permeability of vacuum can be given by $$\mu_0=\frac{1}{\epsilon_0 c^2}$$ which comes from the definition of a magnetic field in special relativity, where we solve the problem of a wire with electrons flowing in, and we calculate the force exerted on an external charge moving with some velocity (for details, refer to the book Electricity and Magnetism, E. M. Purcell), and we define a new field called "magnetic field" with the form of Lorentz force, where the magnetic field is $$B=\frac{I}{2\pi \epsilon_0 c^2 r}$$ where $I$ is the current due to the flow of electrons in the wire, and $r$ is the distance of the external charge from the wire. And there we get the definition of $\mu_0$ that makes $B$: $$B=\frac{\mu_0 I}{2\pi r}$$ and there starts the concept "magnetism". Why am I giving this detailed example? Because I wouldn't like to get the answer that $\mu_0$ has no error, and that's why $\epsilon_0$ has no error, and then we fall into circular logic. So I expect a reason which is independent of $\mu_0$. Thank you in advance.


Why do we take the value of the constant in Coulomb's law as $\frac{1}{4\pi\varepsilon_0}$?


Coulomb's Law: why is $k = \dfrac{1}{4\pi\epsilon_0}$

This was supposed to be a long question but something went wrong and everything I typed was lost. Here goes.   1. Why is $k = \dfrac{1}{4\pi\epsilon_0}$ in Coulomb's law?    2. Is this an experimental fact?    3. If not, what is the significance of this definition?


Why do stickers curl?

When peeling a sticker off its base, the immediate reaction is that it curls; why is this? I am having trouble finding an answer to this. Could it be that the glued side expands upon contact with the air? And, on a somewhat related note, is this a similar reason for why a ribbon curls when glided over with a blade?


Topology needed for Differential Geometry

I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from the Internet on topological spaces, connectedness, compactness, metric, quotient Hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge. Could you please also suggest a good differential geometry books that covers diff. geom. needed in physics in sufficient detail, but not too mathematical? I heard some names such as Nakahara, Fecko, Spivak. How are these?


Presence Of an Another Universe

> **Possible Duplicate:**   >  Experimental evidence for parallel universes Is there an universe similar to ours somewhere else, I mean I had heard from Einsteins theories that,actually when I am typing this question here, there is an another world somewhere else where I would be typing this question simultaneously. Is this true, if so any evidences.


Why does a system try to minimize potential energy?

In mechanics problems, especially one-dimensional ones, we talk about how a particle goes in a direction to minimize potential energy. This is easy to see when we use cartesian coordinates: For example, $-\frac{dU}{dx}=F$ (or in the multidimensional case, the gradient), so the force will go in the direction of minimizing the potential energy. However, it becomes less clear in other cases. For example, I read a problem that involved a ball attached to a pivot, so it could only rotate. It was then claimed that the ball would rotate towards minimal potential energy, however $-\frac{dU}{d\theta} \neq F$! I think in this case it might be equal to torque, which would make their reasoning correct, but it seems like regardless of the degrees of freedom of the problem, it is always assumed that the forces act in a way such that the potential energy is minimized. Could someone give a good explanation for why this is? Edit: I should note that I typed this in google and found this page. where it states that minimizing potential energy and increasing heat increases entropy. For one, this isn't really an explanation because it doesn't state why it increases entropy. Also, if possible, I would like an explanation that doesn't involve entropy. But if it is impossible to make a rigorous argument that doesn't involve entropy then using entropy is fine. As a side note, how does this relate to Hamilton's Principle?


Velocity distribution in Plummer's models and others mass distributions

The Plummer's sphere is an model for the mass density in a globular cluster of stars. For an $N$-body simulation I have initialized the position of $N$ masses with a Monte-Carlo technique but cannot find a way of initializing the velocity initial conditions. Is there a simple function that given a position in a Plummers sphere assigns a velocity to a given mass? Lots of sites list the velocity for a circular orbit but is this a good approximation to a globular cluster and how should it be treated off the $x$-$y$ plane?


Why is work not related to velocity?

A very simple question; why does it cost me more energy to **very slowly** lift a mass $m$ over my head compared to very fast? The definition of work does not state anything about velocity, only the distance travelled, but I definitely feel more exhausted in the first case.


Numerical problem in solving the Bogoliubov de Gennes equations- methods to solve?

I am trying to solve an assignment on solving the Bogoliubov de Gennes equations self-consistently in Matlab. BdG equations in 1-Dimension are as follows:- $$\left(\begin{array}{cc} -\frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}-\mu+V\left(z\right) & \triangle(z)\\\ \triangle(z) & \frac{\hbar^{2}}{2m}\frac{\delta^{2}}{\delta z^{2}}+\mu-V(z) \end{array}\right)\left(\begin{array}{c} u_{n}(z)\\\ v_{n}(z) \end{array}\right)= \epsilon_{n}\left(\begin{array}{c} u_{n}(z)\\\ v_{n}(z) \end{array}\right)$$ along with the equations for gap function $\triangle(z)$ and number density $n(z)$. $$\triangle(z)=U\sum_{n}\left(1-2f_{n,}\right)u_{n}(z)v_{n}^{\star}(z)$$ and $$n(z)=2\sum_{n}|{u_{n}(z)}|^{2}f_{n}+|{v_{n}(z)}|^{2}\left(1-f_{n}\right).$$ For the case of solving the BdG equations in Fourier space in Matlab for the case of a periodic potential and periodic gap function (assumed), we can take $$u_{n}(z)=\sum_{k}\exp\left[ikz\right]U_{n,k}, $$ $$\triangle(z)=\sum_{K}\exp(iKz)T_{K},$$ and $$ V(z)=\sum_{K}\exp(iKz)P_{K} $$ where the sum is over the reciprocal lattice vectors $K$ leading to the number equation $$N=2\sum_{n,k}\left[f_{n}|{U_{n,k}}|^{2}+\left(1-f_{n}\right)|{V_{n,k}}|^{2}\right]$$ with $ f_{n}$ as the Fermi distribution function. Solving the set of equations self-consistently for a fixed $N$, I am trying to get a value of chemical potential from the number equation each time after solving the eigenvector components $U_{n,k}$ and $V_{n,k}$, but due to the form of the exponentials in the number equation and sum over large number of them, I am unable to get a correct value of chemical potential out of them using Matlab routines as the root of the equation to put it back into the equations for eigenvector components. In most cases, I get random values of chemical potential since the equation is more or less insoluble. How can I avoid this error ? Is there a better way to numerically solve the BdG equations self-consistently ? I also want to do this assignment in real space avoiding finite size effects but started with the Fourier space case to avoid errors associated with discretizing the differential. Please guide and ask for any details you might need. Following is my MATLAB code to solve the equations in real space but the code does not work as fsolve does not find the mu value. http://postimg.org/image/v7amx5vd9/full/ ## http://postimg.org/image/do9pyyk7z/full/ **If you have solved BdG equations numerically, please tell about the method and steps you used such that the above problems are eliminated.**


Thermal Conductivity Graph

This is our first time (as an engineer) seeing this type of graph that we can't interpret. This is a rough comparision of thermal conductivity from the Wiki page. http://en.wikipedia.org/wiki/Thermal_conductivity **What we know** The more we go in the log x axis, the more conductive the material is. Say, _silver_ vs _insulation fiber_. **What we don't know** What the length of each bar presents (in terms of y axis). Say, why _silver_ bar is 3 times the _copper_ and what does it mean. ![enter image description here](http://i.stack.imgur.com/HsyLk.jpg)


Isotropy of Space

Weinberg writes in his Cosmology text "Likewise,isotropy requires the mean value of any three-tensor $t_{ij}$ at $x=0$ to be proportional to $\delta_{ij}$ and hence to $g_{ij}$, which equals $a^2\delta_{ij}$ at $x = 0$" May someone please illuminate the point.


Force acting on a simple rigid body in space

so this is the question that's been bothering me: Say you have a simple rigid body in space that is at rest or travelling with only translational motion at a constant speed. Say that the body is something like a rod and it's not rotating. So, at some point, an external force is acted on the rod, just for an instance and it is not acted on some axes that goes through the center of mass. What will happen?? My guess is that it will start rotating about the center of mass (because of torque) + it will get some translational motion towards the direction of the force. Is that correct? and if so, how much of the force becomes rotation and how much translational movement and why?? please forgive my english and thnx in advance!!


Why are roofs blown away by wind?

Whenever there are high winds, such as in storms, thin metal roofs on sheds as well as concave roofs on huts are sometimes blown away. One explanation provided to me said that the higher velocity of the air outside causes the air pressure above the roof to decrease and when it has decreased to a certain extent such that the air pressure above the roof is lesser than the air pressure beneath the roof and due to _some kind of osmosis_ , the air particles move from the area of higher pressure (beneath the roof) to the area of low pressure. In this process, the roof is blown away. Another explanation, specifically about the thin metal roofs, said that it was blown away due to the lift caused by the air and this is the same kind of lift you get when you blow on paper. Both these explanations puzzle me. What really bothers me is the basis of the first one, **how can an increase in velocity cause pressure to drop?** I can't seem to correlate that with the Force per unit area definition of pressure. Please, oh great physicists of the internet, help me and every other ordinary person to understand how and why roofs get blown away.


Magnetic declination

i couldnt just figure out when i got to know that declination can be zero also. How can true and magnetic north ever align themselves in a straight line in any place? Also if a compass aligns in the direction of horizontal component of MF at that place does this mean that at any place horizontal component is directed towards the magnetic north ?? Maybe i am misinterpreting this idea of direction. here is what i think : If u look at a bar magnet's field lines not every tangent to the curve will pass through the north pole. So the compass placed at a point will align with the field but it wont point at the north pole always . please explain


Mathematical formalism to include wave and particle perspectives of light

Does the exist any mathematical formalism (model) describing the behavior of light and incorporating its particle character (divisibility, quantization) and wave character? (i.e. quantized wave model)


Formation of black holes

If stars start with a finite density and light can escape from them, how can they be compacted to form a mass with infinite density which light cannot escape? The black hole will have the same mass as the original star (correct?) and therefore will act on the photons with the same force of gravity, right?


Ice cream cone and loop-de-loop

Was siting in class thinking about this problem, did some rough sketches of a solution but never really managed to solve it. ![Ice cream cone and a loop-de-loop](http://i.imgur.com/D8XAh.png) > Assume a boy starts at the top of a circle with radius R as described in the > picture. It is a snowy day and the path can be considered without friction. > The boy enters a loop with radius r at the bottom of the hill. At the top of > the loop the boy loses his icecream cone in such a way that it starts > faling. The initial velocity of the icream is 0 m/s straight down. The problem is to find R expressed by r, such that the boy reaches the icrecream just as he reaches the bottom of the loop. The problem boiled down to finding out how much time the boy uses getting from the top of the loop to the bottom. Any help, solutions or inputs would be great. * * * My attempt, I know that this is most likely 90% wrong By using conservation of mechanical energy. The speed at the bottom of the hill equals $$ v_b^2 = Rmg $$ And the velocity at the top of the loop equals $$ v_t^2 = 2g\left( R - 2r \right) $$ Vi know that the aceleration is constant and equals $ g $ (Here is where I think I make my mistake, forgot to acount for the angular velocity) $$ s = \dfrac{v_1 - v_0}{2} t $$ We use this equation to find out how long it takes the boy to get from the top, to the bottom of the loop. $$ \large t \, = \, \dfrac{2s}{v_1 - v_0} \, = \, \dfrac{2\left( \dfrac{2\pi r}{2}\right)}{\sqrt{2gR} - \sqrt{2g(R - 2r)}} $$ Now we figure out how long it takes the icream to fall the distance of the diameter or $ 2r $ . $$ s = v_0 + \dfrac{1}{2}gt^2 $$ $$ t = \sqrt{\dfrac{2s}{g}} \, = \, \sqrt{\dfrac{4r}{g}} \, = \, 2 \sqrt{\dfrac{r}{g}} $$ By setting these two equations equal each other, and solving for $$ R $ , we obtain that $$ R = \dfrac{16+8 \pi^2+\pi^4) r}{8 \pi^2} \cdot r \approx 2.43 r $$


"Single-shot" Heat engine efficiency limits

The sun is 5778K and Earth is ~290K. Using the sun as the hot reservoir and earth as a cold reservoir we get 95% Carnot efficiency. However, the solar power efficiency limit is only 86%, see: http://www.energy.udel.edu/pdf/Honsberg_UDEI_Symposium.pdf This is not just because of our atmosphere or a non-ideal sun. There is a more fundamental constraint: Carnot heat engines don't care how quickly energy is transferred from the hot to the cold. In our case, however, we are force-fed energy and must accept it as fast as possible. If energy is lost back to the sun we lose efficiency. Thus the term "single-shot": we only have one chance to generate power. What is the single-shot efficiency limit as a function of temperature ratio? Assume you are surrounded with black body radiation from all directions at the cold temperature (otherwise you wouldn't even need a sun to get energy!) and an unlimited supply of coolant. The "sun" is a hot black body radiator that takes up a small angle of the sky. Efficiency is (useful power)/(intercepted radiation power). This is a theoretical calculation, there are no restrictions on the design (solar thermal, PV, hybrid, etc).


How does one calculate the polarization state of random light after total internal reflection

How does one calculate the polarization state of random light after having been totally reflected by a single dielectric interface? Please consider pure specular reflexions from a plane interface between two dielectric mediums of indexes $n_1,\,n_2$ when the angle of incidence $\theta_1$ is greater than the critical angle $\arcsin(n_2/n_1)$.


The Gunn diode and the two valley theory

In today lecture of microwave and radar my teacher explained about the Gunn diode.   He said it is made up of only one type of matrial e.g. $GaAS$ as shown in (a) part of the image.   ![image 1](http://britneyspears.ac/physics/highfields/images/Image93.gif)   He said that there is no depletion layer and no gate so this diode conducts in reverse bias. He explained its working by using two valley theory. I have mainly two questions which emerged from this lecture: >   1. Why a Gunn diode is called a diode as it conducts in both the > directions. > My teacher said that there are two valleys in the conduction band as shown. And the electron in the lower valley has lower effective mass as compared to that of in upper valley. ![image 2](http://britneyspears.ac/physics/highfields/images/Image79.gif) So my question is : >   1. Since electron in the lower valley is more near to the nucleus so its > velocity should be more so its effective mass should be more because > $m^*=\dfrac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}}$in the lower valley >


Moving towards a clock at .866C

If you set up a clock that sends out a light pulse every second, and move towards it at a speed of .866c, will the clock appear to run faster?


Force experienced on two particles in a rotating system?

I've a system of two particles of the same mass who rotate in a circle about the centre of mass of the two particles. Is the force experienced by the particles $F=MV^{2}/r$ or should I use $Torque=$Moment of Inertia*angular acceleration?


Momentum acceleration in space?

I'm an engineering student and have a few ideas to bat at the scientific community. First premise is based upon common sense. I understand that if I am standing on a skateboard on a relatively smooth surface and a stone of significant mass is thrown to me and I catch it, the momentum of the projectile is transferred to me causing a forward acceleration. My question: Suppose that a system is designed to provided constant acceleration in space utilizing the aforementioned intuitive concept. In the simplest form I can idealize such a system as a basketball constantly striking a flat surface. Opposed to a ball consider an arbitrary mass striking a greater mass. I suppose that if creative means are applied to control the striking action (e.g. magnetism, mechanical means) then this action can be repeated to provide constant acceleration without the use of carrying fuel loads. Would it not be possible to reset the system to an initial state without canceling the resulting acceleration? Due to the length of this question I will post the next question later. Thx


Why doesn't light kill me?

I was attending my philosophy class and in the middle of student presentations, I found myself mentally wondering off and thinking about light. After a few minutes of trying to piece together how the sun works I came to ask a question I could not answer myself. Why does each individual photon have such a low amount of energy? I am hit by photons all day and I find it amazing that I am not vaporized. Am I simply too physically big for the photons to harm me much, or perhaps the Earth's magnetic field filters out enough harmful causes such as gammy rays?


Actual meaning of "Gravitational Potential"?

In a gravitational field, the gravitational force acting on a body (of mass m) at a point x metres away from the attracting body (of mass M) is $\frac{GMm}{x^2}$. Integrating this force from a point at infinity to x gives $-\frac{GMm}{x}$, or the work done by the gravitational field on the mass in moving the mass from a point at infinity to x. Then, the gravitational potential at point x is $-\frac{GM}{x}$. However, the definition that my school gives is : "Gravitational potential at a point in a gravitational field is the amount of work done by an **external agent** in moving one unit of mass from a point at infinity to the point in the gravitational field, without any acceleration". The lecturers explained that as a body moves closer to the attracting body, it accelerates and gains kinetic energy. In order to maintain a constant velocity, negative work must be done by an external agent to remove this extra kinetic energy. Now I am very confused, because both seem valid to me. What does "gravitational potential" actually mean?


Landau & Lifshitz - Euler's equation for one-dimensional flow

One page 5 in Landau & Lifshitz _Fluid Mechanics_ (2nd edition), the authors pose the following problem: > Write down the equations for one-dimensional motion of an ideal fluid in > terms > of the variables $a$, $t$, where $a$ (called a _Lagrangian variable_ > ) is the $x$ coordinate of a fluid particle at some instant $t=t_0$. The authors then go on to give their solutions and assumptions. Here are the important parts: > The coordinate $x$ of a fluid particle at an instant $t$ is regarded as a > function of $t$ and its coordinate $a$ at the initial instant: $x=x(a,t)$. For the condition of mass conversation the authors arrive at (where $\rho_0 = \rho(a)$ the given initial density distribution): > $$ \rho\,\mathrm{d}x = \rho_0 \mathrm{d}a $$ or alternatively: > $$ \rho\left(\frac{\partial x}{\partial a}\right)_t = \rho_0 $$ Now the authors go on to write out Euler's equation, where I start to miss something. With the velocity of the fluid particle $v=\left(\frac{\partial x}{\partial t}\right)_a$ and $\left(\frac{\partial v}{\partial t}\right)_a$ the rate of change of the velocity of the particle during its motion, they write: > $$ \left(\frac{\partial v}{\partial t}\right)_a = -\frac{1}{\rho_0} > \left(\frac{\partial p}{\partial a}\right)_t $$ **How are the authors arriving at that equation?** In particular, when looking at Euler's equation: $$ \frac{\partial\mathbb{v}}{\partial t} + \left( \mathbf{v} \cdot \textbf{grad} \right) \mathbf{v} = - \frac{1}{\rho} \textbf{grad}\, p $$ what happens with the second term on the LHS $\left( \mathbf{v} \cdot \textbf{grad} \right) \mathbf{v}$? Why does it not appear in the authors' solution?


How to turn water opaque by pouring the smallest quantity of matter into it?

Consider a glass of water and a glass of coffee. Their contents differ by no more than a few grams of particles coming from the roasted and ground coffee, yet the former lets almost all visible light pass through, and the latter blocks most of it. I am wondering **what are the most efficient ways to turn water opaque by pouring matter in it** , under normal temperature and pressure. By efficient I mean minimising the mass of added matter. By opaque, let's say 99% of daylight is blocked by two inches of liquid. There is no other reason for my asking than sheer curiosity from the many hours I spend gazing fixedly through the coffee pot.


How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum? I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and its spin current. I multiplied the currents by $mc$ to covert them into momentum and then crossed the momentum with position to obtain the orbital angular moment and the spin angular momentum. However the spin angular momentum was twice as large as the accepted value of spin angular momentum. I am stuck on how to get rid of the extra factor of two. The Gordon Decomposition splits the probability current into two terms \begin{align} mc\langle \gamma^{0n} \rangle = \langle \gamma^{0} i\hbar \partial^{n} \rangle \+ \tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j} , \end{align} where $\langle \gamma^{0} i\hbar \partial^{n} \rangle$ looks like a orbital momentum current and $\tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j}$ looks like a spin momentum current? The angular moments of $mc\langle \gamma^{0n} \rangle$ can be computed by crossing then with position to obtain the following: \begin{align} mc \left( x^m\langle \gamma^{0n} \rangle \- x^n\langle \gamma^{0m} \rangle \right) = \langle \gamma^{0} i\hbar (x^m\partial^{n}-x^n\partial^{m}) \rangle \+ \tfrac{\hbar}{2}\partial_{j} \left( x^m\langle i\gamma^{0nj} \rangle_{n\neq j} \- x^m\langle i\gamma^{0nj} \rangle_{n\neq j} \right) , \end{align}


Quantum Field Theory and Hilbert space dimensionality

Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis.   How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is countable, or does it “need” to use an uncountable, continuous space (e.g. rigged Hilbert space) for mathematical consistency, or some other reason?


Extrinsic Photoconductor Charge Carriers

I'm studying some materials on semiconductors, and a section on photoconductivity has left me somewhat confused. In an extrinsic material, is photoconductivity the result of electrons moving from the valence band to the conduction band, or is it the result of dopant atom ionization? If it is the latter, is the complete ionization assumption invalid for photoconducting devices?


Accelerating cavity in synchotron- how does it actually work?

I wanted to know how a radio frequency accelerating cavity actually works. I know that an electric field is used to accelerate the charged particles. However, is this done by using metal plates with opposite polarities or are there tubes (like in the linac) within the accelerating cavity which have fields in between the gaps?


Why can we not reduce the size of a system below the correlation length without qualitatively changing its properties?

This question is posed in the context of thermodynamics/statistical mechanics. Suppose we define the correlation length as the $\xi$ in the exponential factor $e^{-r/\xi}$ that appears in the correlation function $$C(r) = \langle m(0)m(r) \rangle - \langle m \rangle^2$$ (where $m$ is some relevant order parameter). Then my question is: how can we, from this mathematical definition, argue the following two physical properties. **Suppose we have our thermodynamic system and we take a chunk out of it with side-length L.**   1. If $L > \xi$, the sample has the **same** qualitative behavior as the original sample;   2. If $L < \xi$, the qualitative behavior is **different**. If one can add extra conditions for these statements to be true, that would be nice/welcome.


Conservation of Energy in a Capacitor

Consider a parallel-plate capacitor in free space. A negatively charged point particle with initial velocity $v$ passes through the space between the pair of parallel plates (with an initial path perpendicular to the normal vector of the plates). The point particle accelerates towards the positively charged plate but passes beyond the edge of the plate. How is energy conserved, given that the capacitor does work on the particle by accelerating it in the direction towards the negatively charged plate? EDIT: Was reminded by Art Brown that a negatively charged particle accelerates towards the positive plate.


The Four-Clock Special Relativity Conundrum

Two open-car trains approach each other at fixed velocities. Each has a radar to see how quickly the other train is approaching, but apart from that the trains have no _a priori_ knowledge of each other. Each train has engineers on its first and last cars. Each engineer has an atomic clock and a laser for communicating with their partner on the same train. Long before the trains meet, the engineers on each train use their lasers to adjust their relative separations to exactly one kilometer. Using the synchronization procedure first defined by Einstein, the engineers also exchange time data and use it to adjust their atomic clocks until they are precisely synchronized within their shared frame. The trains meet. The clocks are positioned so that they almost touch as they pass. At the moment of nearest contact, they exchange and record each other's values (timestamps). Some time later, the clocks on the trailing cars meet and perform the same procedure. The delay between the leading and trailing data events can now be measured in two ways. If the two timestamps taken from the first train are compared, the result is the time between the events as measured from the first train. This value is meaningful because the engineers on that train previously synchronized their clocks and stayed within the same frame of reference at all times. If the timestamps from the second train are used, a similar but distinct measurement of the time between the two events can be obtained. Now, three questions:   1. Is there anything wrong or impossible with this experimental setup? If so, what is it?   2. If you accept the experiment as realistic and meaningful, will the delays calculated from the perspective of each of the two trains be the same, or different?   3. If you answered "different," what is the correct procedure for predicting in advance what the ratio of the two delays will be? * * * **2013-01-10 - The Answer(s!)** I awarded the bounty to @FrankH for pages of excellent and educational work worth of reading by anyone who wants to understand special relativity better. However, I've also taken the unusual step of re-allocating the _answer_ designation to @MatthewMcIrvin. There are two reasons: (1) Matthew McIrvin was the first one to spot the importance of the symmetric-velocities frame, which others ended up gravitating (heh!) to; and (2) Matthew's answer is _short_ , which is great for readers in a hurry. FrankH, sorry about the switch, but I didn't realize I could split them before. So: If you are not overly familiar with special relativity and want to understand the full range of issues, I definitely recommend FrankH's answer. But if you already know SR pretty well and want the key insights quickly, please look at Matthew McIrvin's answer. I will have more "proximate data exchange" SR questions sometime in the near future. Two frames turns out to be a very special case, but I had to start somewhere. * * * For folks with strong minds, stout hearts, and plenty of time to kill, a much more precise version of the above question is provided below. It requires new notations, alas. To reduce the growing size of this question, I have deleted all earlier versions of it. You can still find them in the change history. * * * **Precise version of the 4-clock conundrum** Please see the first three figures below. They define two new operators, the frame view and clock synchronization operators, that make the problem easier to state precisely: * * * ![Setup of the four-clock conundrum using the frame operators](http://i.stack.imgur.com/JTWkQ.png) * * * ![Event T1: Leading edge clocks A1 and B1 pass very closely](http://i.stack.imgur.com/347QJ.png) * * * ![Event T2: Trailing edge clocks A2 and B2 pass very closely](http://i.stack.imgur.com/EVZtI.png) * * * Note that the four timestamps $\\{T_{A1}, T_{B1}, T_{A2}, T_{B2}\\}:>\\{A,B\\}$. That is, the four timestamps are shared and agreed to (with an accuracy of about 1 nanosecond in this case) by observers from _both_ of the interacting frames A or B. Since T1 and T2 are historically recorded local events, this assertion can be further generalized to $\\{T1,T2\\}:>*$, where $*$ represents all possible frames. (And yes, $:>$ is two eyes looking left, while $<:$ is two eyes looking to the right.) Using the four shared time stamps, define: > $T_{\Delta{A}} = T_{A2} - T_{A1}$ > > $T_{\Delta{B}} = T_{B2} - T_{B1}$ My main question is this: * * * > Assuming that $T_{\Delta{A}} = f(T_{\Delta{B}})$ exists, what is $f(x)$? * * * **Analysis** (why this problem is difficult) Based purely on symmetry in the setup, the most obvious answer is $f(x)=x$, that is: > $T_{\Delta{A}} = T_{\Delta{B}}$ There are some interesting reasons to be troubled by that seemingly straightforward and even obvious conclusion, not the least of which is that it violates the whole concept of special relativity (the Dingle heresy). That is, if you assume $f(x)=x$ and follow that line of logic through to its logical conclusion, you quickly end up with time that flows the _same_ for all frames -- that is, no relativity. Such a conclusion is in flat violation of over a century of very detailed experimental evidence, and so is just not supportable. The following four figures show why it's so hard to assert that $T_{\Delta{A}} = T_{\Delta{B}}$ without violating special relativity. * * * ![View of Event T1 from Frame A](http://i.stack.imgur.com/G59Bl.png) * * * ![View of Event T2 from Frame A](http://i.stack.imgur.com/j2K87.png) * * * ![View of Event T1 from Frame B](http://i.stack.imgur.com/IFj5Q.png) * * * ![View of Event T2 from Frame B](http://i.stack.imgur.com/jmoYk.png) * * * While the above figures accurately capture the reality of relativistic contraction of both distance and time, the problem in this case is simple: How do you decide which frame to select? The experiment as described can only give one outcome. Which one will it be?


What were the immediate consequences Yang-Lee work on Weak Interaction?

I am studying the history of Modern Physics and Yang-Lee earned their Nobel the next year after the Cobalt experiments. I am familiar with the chronology, but am not clear what those findings meant to the physics community to warrant such recognition for two relatively' unknown physicists. Other household names had to wait decades for more seemingly significant work. So can someone clarify what their work meant to the eventual development of the Standard Model?


How is wavefunction probability redistributed after partial wavefunction collapse?

Suppose I set up the double-slit experiment using photons as my particle. Behind the left slit I place a beam splitter that points some of the light off in the direction of a camera (represented as `C` in the diagram below). We'll say that the beam splitter splits a light beam passing through it into two equal beams. The light that passes through the beam splitter unaffected and the light that passes through the right slit continue on to another camera (represented as `CCC` in the diagram below). We'll say that camera `C` is closer to the beam splitter than camera `CCC` is.                    CCC          | |          | |        C-\ |          | |     -----|-|-----           |           |         Laser      If I fire a single photon through this apparatus, there's about a $1/4$ chance that it will be detected by camera `C` and about a $3/4$ chance that it will be detected by camera `CCC`. When the wavefunction reaches camera `C`, the photon is forced to either be detected by camera `C` or have the part of the wavefunction leading to camera `C` disappear. **Question:** In the case that the photon is not detected by camera `C`, how is the $1/4$ probability from that branch of the wavefunction redistributed to the remaining two branches of the wavefunction, the branch went through the left slit and straight through the beam splitter and the branch that went through the right slit? The first starts with about $1/4$ probability and the second starts with about $1/2$ probability. Does the first get a new probability of $1/3$ and the second a new probability of $2/3$? Does the first get a new probability of $1/2$ while the second keeps its old probability of $1/2$?


Potential due to a spherical surface charge

The potential at the surface of an insulating sphere (radius R) is given by   $$V(R,\theta) = k \cos(3\theta)$$ where $k$ is a constant. Use separation of variables to find the potential inside the sphere (r $\leq$ R) and outside the sphere (r $\geq$ R), and then use your answer to determine the surface charge density $\sigma(\theta)$ on the sphere. Assume there is no charge inside or outside the sphere. My approach: I will set my reference point at infinity, therefore the potential at infinity is 0, to set that as a boundary condition. Essentially, the approach in finding the surface charge density would be to find the potential function outside the sphere, then integrate that with the surface area. I am following Griffiths for E&M, but he doesn't include a specific example I can refer to for the separation of variables, and how to apply it. I can follow his examples for following potentials pretty easily, but cannot understand the first steps to set up this problem with SOV.


What is the initial velocity of a projectile so that it passes through a target point in its trajectory?

Let's say I have a projectile being thrown by a player in my 2-D game. I want to work backwards and find the initial velocity to apply to the projectile such that it passes through a target point in space given that the launch point might not be at $y=0$. Some applications of this would be a basketball shot simulator or a pub darts simulation. Given:   * initial or launch angle: $\theta_o$   * initial or launch height: $y_o$   * target point: ($x_f$, $y_f$)   * constant acceleration due to gravity ($g$)   * air resistance ignored I need to find:   * initial launch velocity: $v_o$ Here's an image of what I'm talking about. ![enter image description here](http://i.stack.imgur.com/uRRkX.png) I am interested in learning the derivation of the answer not just a formula that I plug-and-chug. Thanks!


Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? * * * If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$. I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem. Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too. Do we know the limit for making up these objects? What is the fundamental justification for order two?


Why there is the requirement for derivatives no higher than second order in free quantum field equations?

Why there is the requirement for derivatives no higher than second order in free quantum fields equations? We can get the equations for the free fields of an arbitrary spin by using the requirements of linearity of the equations, derivatives no higher than second order. But why there is the second requirement? Is it connect with initial conditions $A_{\mu}(0), \partial^{\mu}A_{\nu}(0)$ for the field? For example, Schrodinger equation is first-order time-derivative equation, because we have the postulate that if we know the $\psi (\mathbf r , t)$ in the present, we know it in the future.


Why do fundamental physical laws involve the second derivative?

The title says it all. This is a question that has been nagging at me for some time. Mathematically, the first derivative is not really any different from the second derivative, or the $k$-th. So I ask, what is so special about the second derivative?


Fermi Walker vs. Fermi transport

A vector field $f^\mu$ is said to be Fermi-Walker transported along a curve $\gamma$ parametrized with $\tau$ if the following holds $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = -(a^\mu v^\nu - a^\nu v^\mu) f_\nu,$$ where $v^\mu$ is the tangent vector $\gamma$ and $a^\mu$ is its derivative. The uppercase D denotes the covariant derivative. This is the usual transport law for nonrotating tetrads. However, in "Introduction to General Relativity" Lewis Ryder mentions the Fermi transport for which $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = v^\mu a^\nu f_\nu.$$ What is the significance of this transport? Are there any applications of it?


Mass of a galaxy via Luminosity

Is there a way of calculating the mass of a galaxy, or even a nebula from the luminosity? EDIT I'm deleting this, and moving the question to Astronomy Stack Exchange - thanks david


Conservation of angular momentum in helicopter

I have a small RC-controlled toy helicopter with removable tail rotor. Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and then let it go. Suppose that the heli remains on the floor all the time. Then the body of the helicopter begins to turn into the opposite direction of the rotor. My problem is to explain this in terms of angular momentum. At first this seemed to contradict angular momentum conservation because I assumed that the helicopter is a closed system and there is no external torque, but the rotor keeps the same angular velocity while the body starts to turn into the other direction. My second thought was that I have to take the air into account. If I do that it seems plausible to me to explain it as follows: The rotor transfers angular momentum from to the surrounding air. However to keep it running at constant angular velocity, the motor has to transfer angular momentum to the rotor, which is only possible if the body of the helicopter gets the same amount of angular momentum as goes to the rotor, but into the opposite direction. But then I thought, the body should turn faster and faster to infinity which obviously doesn't happen. So I guessed I should also take the friction with the floor into account where the angular momentum of the body is transferred to. This leads me to the following three questions:   1. Is my reasoning described above correct or I am completely wrong?   2. If I would do the experiment on an almost frictionless turntable (which I don't have available at the moment), would the body of the helicopter indeed become faster and faster almost to infinity?   3. If I would do the same thing in a vacuum, what would happen? I guess (see my first thought above), that the body wouldn't turn at all. Is that correct? So it would be great if someone could clarify this in terms of angular momentum. Since the same problem should occur in real helicopters, I guess that there should be some detailed literature about it's physics, but I didn't find any resources about this. So if you have any references for more details or further reading on this problem, it would be also great if you could post it.


Why does $c_{-k,-\sigma}$ create a particle with momentum $k$?

In Mudelung's book, Introduction to Solid-State Theory, I am confused by the following statement. > For many applications a further simplification is helpful. The concept of > the hole presents us with the confusing situationthat a hole in the state > $\mathbf{k},\sigma$ has a momentum $-ħ\mathbf{k}$, while an electron has a > momentum $ħ\mathbf{k}$. This asymmetry can be avoided by defining new > _quasi-particles_ which have momentum $ħ\mathbf{k}$ inside and outside the > Fermi sphere. When we note that the creation of a particle with > $+ħ\mathbf{k}$ is achieved by the operator $c_{\mathbf{k}\sigma}^+$ when > outside the Fermi sphere and by $c_{-\mathbf{k},-\sigma}$ when inside the > Fermi sphere, it is only a small step to define the following operators: $$ > \begin{align} > \alpha_{\mathbf{k}\sigma}^+=u_\mathbf{k}c_{\mathbf{k}\sigma}^++v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma} > \\\ > \alpha_{\mathbf{k}\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}^+ > \end{align} $$ with $$ \begin{align} u_\mathbf{k}=1,\; > v_\mathbf{k}=0\quad\text{ for }\quad k>k_\text{F}, \\\ u_\mathbf{k}=0,\; > v_\mathbf{k}=1\quad\text{ for }\quad k<k_\text{F}. \end{align} $$ He showed the motivation of introducing Bogoliubov tansformation. But I don't understand his statement. "Creation of a particle with $+\mathbf k$ is acvieved by the operator $c^{+}_{\mathbf k,\sigma}$ when outside the Fermi shere and by $c_{-\mathbf k,-\sigma}$ when inside the Fermi sphere". $c^{+}_{\mathbf k,\sigma}$ means create a particle with $\mathbf k$. So, why must we distinguish ouside and inside the Fermi sphere? And why does annihilation operator $c_{-\mathbf k,-\sigma}$ inside the Fermi sphere have the same effect as creation operator outside the Fermi sphere?


Calculate the center of mass of a semicircle

How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a homogeneous mass distribution.


Finite amount of matter and space

(Please bear with me.) If the universe included multi-universes, and black holes transferred matter between them, AND yet there was a large, but FINITE amount of matter and space, How would that max-limit change our understanding/perception or theories of the universe? Would it make any difference or change current concepts?


A 0.1kg ball of dough is thrown up with a velocity of 15m/s. What is the momentum halfway up?

I know that $p=mv$ and (0.1kg)(15m/s)=1.5 kg m/s and the momentum at the vertex is 0, but what is the momentum halfway up?


Pressure vs Volume Calculation for air

I have 1 litre of air in a sealed container at atmosphere pressure (approx 15 psi absolute). I want to reduce pressure in the container by 1 psi ( 14 psi absolute or -1 psig).   * If I understand correctly that 1 psi reduction is 6.66 % reduction in pressure compared to standard atmospheric pressure,    * Applying the same to volume would it be correct to say that if I increase the volume of the container by 6.66 % the pressure would drop by 1psi for air OR   * If i remove 6.66% volume of air in the container I would still be able to achieve the 1psi drop in pressure? Do some other factors come in to play for pressure / volume calculation...?


What is known about quantum electrodynamics at finite times?

I'm aware that we can describe the time evolution of states/operators (choose your favourite picture) of non interacting quantum fields and that perturbation theory is very effective in computing S matrix elements between free states in the remote past and free states in the remote future. Clearly the non-perturbative description of what's going on at finite times, where the interaction is active is intractible, but my question is - are there simplified toy models (scalar electrodynamics ? reduced numbers of dimensions ?) where we _can_ describe what's happening non perturbatively. Even if nothing like electrodynamics has been treatable like this, any results on the other "textbook" QFTs (like $\phi^4$) would be interesting.


Elastic vs Inelastic vs isospin violating scattering particle physics models

I'm looking for a nice paper that explains the difference between three particle physics models for spin-independent dark matter interaction with nuclei: elastic, inelastic and isospin violating scattering. I've found a nice paper that is giving a nice summation of the current results in direct dark matter searches ( http://arxiv.org/abs/1210.4011 ) and want to read up some more. Cheers, Adnan


Why is the charge naming convention wrong?

I recently came to know about the Conventional Current vs. Electron Flow issue.   Doing some search I found that the reason for this is that Benjamin Franklin made a mistake when naming positive and negative charges.   There is even this little comic about that http://xkcd.com/567/ My question is, how can a naming convention be wrong??   Perhaps I don't understand what is the actual mistake here, I don't know. But I don't see how a naming convention could be wrong or right. There is no right and wrong about that. It could've been any other name, like charge A and charge B. I'll appreciate anyone can help me understand what is wrong in all this.


Why are electrons defined to have negative charge?

We normally think of the "default" or "root" state of things as being on the positive side of the spectrum. For example, we don't normally use a `+` symbol to indicate the sign of positive numbers, whereas negative numbers are invariably preceded with `-` to indicate their negative sign. Similarly, positive temperatures (in °C or in F) are those that we tend to encounter more typically. Why, then, do we define the basic unit of _electricity_ as something with a negative charge?


Who (and Why) started the "electrons are negative, protons are positive" convention?

For some reason everyone labels electrons using a minus sign and protons using a positive sign, even though the opposite seems more intuitive: ![enter image description here](http://i.stack.imgur.com/Palaa.png) ![enter image description here](http://i.stack.imgur.com/RTgX7.png) Who started the convention that electrons should be "negative" and protons should be "positive"? What's the reason for choosing so?


positive charge , current and electron flow in a simple circuit …confusion

my question is that in a simple circuit one wire attached with battery cell ,and then electrons start flowing from lower potential to higher and as we know in metal wire only electron is the thing which is carrying charge ,,, then why we say there is current and it flow opposite to electron ??? even there is no other thing , please dont go in other stuff yeah i know positive charge also moves some where like alpha particals in some cases but here in simple circuit why they dont just say electron flow ? why current and why its opposite ???


Semileptonic decays of the $B_c$ meson

I am struggling with calculating the exclusive semileptonic $B_c^+\rightarrow J/\psi l^+\nu_l$ decay. I learnt that the amplitude is given by a product of the leptonic current $L^{\mu}$ and the hadronic current $H^{\mu}$ $$ \mathcal{M}(B_c\rightarrow J/\psi l^+\nu_l)=\frac{G_F}{\sqrt{2}}V_{cb}L^{\mu}H_{\mu} $$ where $V_{cb}$ is the CKM parameter, $L^{\mu}$ and $H^{\mu}$ are expressed as $$ L^{\mu}=\bar{u}_l\gamma^{\mu}(1-\gamma^5)v_{\nu},\quad H^{\mu}=\langle J/\psi|J^{\mu}(0)|B_c\rangle $$ where $J^{\mu}$ is the $V$-$A$ weak current. However, I did not know how this result can be derived. Could anyone provide some help? There is a second problem. On the tree-level, we have the following Feynman diagram ![enter image description here](http://i.stack.imgur.com/l84vp.png) If we calculate $\bar{b}\rightarrow\bar{c}l^+\nu_l$ as a three-body decay in the electroweak theory (not the four-fermion approximation adopted above), how does it relate to $B_c^+\rightarrow J/\psi l^+\nu_l$?


How does one determine black hole mass based on the curvature of a photon?

1. How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km/s at the horizon? (assume a non-spinning black hole, then contrast that with a photon traveling in either direction along the equator of a spinning black hole) Eg. [![Example](https://drive.google.com/uc?export=download&id=0B7P471Xpxoc1N2FoN1FtRmttV2c)]


Finding the tangential force experienced by a bob of mass m on a simple pendulum via the gradient/nabla operator)

The problem was posed as follows. Given a pendulum of length $L$ with a mass $m$ attached to it, which forms an angle $\theta$ from the y-axis to the direction of swinging. First we had to find the potential energy as a function of the angle, which was trivial enough. $$U = mgL(1 - cos(\theta))$$ But the next question was to develop the gradient for this potential energy function via the arc length $s = L\cdot\theta$, where $s$ is the arc length. I calculated the gradient by using the trigonometric definition of $\cos(\theta)$, but I developed the gradient by changing the variable of my function $U(\theta)$ to $U(x,y)$ after which I calculated it further, and found that $F_{tan} = -mg\sin(\theta)$, as desired. But is there an alternative way to calculate it, by using the gradient, and by utilizing the arc length formula above? The question explicitly asks to develop the gradient via the arc length rather than $x$ and $y$ components.


What is the largest diameter an optical fiber can have? - for nonimaging visible light transmission

Is there any theoretical limitation on the size of the diameter of an Plastic optical fiber? I would like to transmit visible sunlight through it. I see bundles of small cores available but wouldn't it be more efficient to make a very large single core?


Can't understand the principle of least action

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone please explain it to me easily? Comment: for example if I have in _énoncé_ , light travel between point $a$ and point $b$ in time $s$, we should formulate the equation of motion where the path is the shortest such that it takes $s$ time to travel from point $a$ and point $b$ by that time? Also how does this relate to Newtonian mechanics and how is it better?


Navier-Stokes equations: conservation of momentum

The first Navier-Stokes equation (conservation of mass) says: $\vec \nabla \cdot \vec v=0$ For a stationary flow, the l.h.s of the second equation is (conservation of momentum): $\rho \frac{D\vec v}{Dt}=\rho (\underbrace{\frac{\partial \vec v}{\partial t}}_{=0} + (\vec v\cdot \vec \nabla) \vec v)=\rho ( \underbrace{(\vec \nabla \cdot \vec v)}_{=0??} \vec v)\stackrel{??}{=}0$ I find that the l.h.s of the conservation of momentum equation is always equal to zero for a stationary field. I know this isn't true but where am I wrong in this reasoning ?


Proton mass and Weizsäcker's formula

I'm _so_ confused in the use of nuclear masses and atomic masses. I have two questions. From the book "Outline of Modern Physics" by Ronald, I understand that the _semiempirical mass formula_ (Weizsäcker's formula) is $$M=Zm_{p}+(A-Z)m_{n}-b_{1}A+b_{2}A^{2/3}+b_{3}Z^{2}A^{-1/3}+b_{4}(A-2Z)^{2}A^{-1}+b_{5}A^{-3/4} \qquad (1)$$ and this formula is for the dependence of the mass of a nucleus on $A$ (mass number) and $Z$ (atomic number), i.e. is the formula that gives you (approximately) the mass of a _nucleus_. **1st question** $M$ is the nuclear mass or the atomic mass? (I understand that the difference between these two masses are the masses of the electrons. Obviously it's small, but is nonzero.) Then, he defines the _average binding energy per nucleon_ like $$BE=[Zm_{p}+(A-Z)m_{n}-M]c^{2}/A \qquad (2)$$ So far so good. Then, the book tries to do an example, and calculate the binding energy per nucleon for $_{42}^{98}Mo$ and writes $$BE=[Zm_{p}+(A-Z)m_{n}-M_{nuc}]c^{2}/A \qquad (3)$$ and says: "where the atomic masses are used for $m_p$ and $M_{nuc}$ (so that the electron masses cancel)." And insert the numbers. $$BE=[(42(1.007825u)+56(1.008665u)-97.905409u)/98] \times 931.5\frac{MeV}{u} \qquad (4)$$ But, according to Wikipedia the isotopic mass of $_{42}^{98}Mo$ ( _atomic_ mass) is 97.9054082 u, and the mass of the proton is 1.007276466812 u. So my **2nd question** is: why does the book uses the value 1.007825u for the mass of a proton instead of the value 1.007276466812 u, and why does it uses the atomic mass instead of the nuclear mass in $M_{nuc}$? **Note** The book uses the value 1.007825u for the mass of a proton and the atomic mass instead of the nuclear mass in $M$, in other examples. Why?


Alpha decay, why does it occur?

I was reading about alpha decay and why it happens. The strong force holds protons and neutrons together, but I don't get why does an atom emit helium nucleus when it has too many protons&neutrons? I mean why isn't the strong force holding on to it?


Changing Coordinate Systems in Two-Loop Integrals

Suppose we have the two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. In the first integration with respect to $k _ {1}$, I take $k _ {2}$ to be the z-axis. Then $k _ {2} = k _ {1} \cos \omega$ and the four-dimensional spherical volume element is $\mathrm{d} V = |k _ {1}| ^ {3} \sin ^ 2 \omega \sin \theta \, \mathrm{d} |k _ {1}| \, \mathrm{d} \omega \, \mathrm{d} \theta \, \mathrm{d} \phi$, where $|k _ {1}|$ is the Euclidean norm of $k _ {1}$. When we perform the second integration $\int \mathrm{d} ^ 4 k _ {2} \, g (k _ {2})$, where $g (k _ {2})$ is the result of the first integration, is it allowed to change coordinate systems and take $k _ {2}$ to be the radius in $S ^ 3$? That is, can we write the spherical volume element for the second integration as $2 \pi ^ 2 |k _ {2}| ^ {3}$, where $|k _ {2}|$ is the Euclidean norm of $k _ {2}$?


At an instant, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges?

**Question:** In principle, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges? _(At a single instance in time?)_ **Additional Notes:** I am aware that the evolution of this system would not behave the same because orbits cannot manifest in a simple system of like charges due to reasons regarding entropy; Just because it is entropically desirable to evolve from the big-bang state to the current universal state, it is not entropically desirable to evolve from the current universal state to the big-bang state just because we reversed the flow of time. (I intend to reverse the flow of time so that the fields reverse but not the global increase of entropy, this is why I specify _At a single instance in time?_ ). I would ask that SE users to consider that the argument of playing a tape backwards is an ill conceived method for generating an answer since it always admits an unnatural evolution of state of the system (One that would never play out if entropy was increasing, I only reverse time for the sake of reversing the fields). For example if I tape an ink drop falling in water and watch it in reverse, it becomes immediately become apparent to me that it is being played in reverse because the system evolves in a way which violates the second law of thermodynamics. Even though this is true, what I can say with confidence is that every electric field of every particle will reverse in sign and the same can be said about the gravitational field. That is the true purpose of the question. With that being said, I specify " _(At a single instance in time?)_ " because I am more concerned with this idea on a fundamental level _(i.e. The physical properties of the fields)_.


"Dynamic Albedo of Neutrons" and all these Farnsworth fusors

I've checked specs of Russian "Dynamic Albedo of Neutrons" module on Mars MSL rover - it can do 10^8 neutron pulses at 10 pulses per second. It works on D-T reaction in tiny linear accelerator, and power consumption is quite low (14W). If you check specs for all these fusors (Farnsworth, Polywell, e.t.c) - they are happy to have 10^6 neutrons (best ones barely scratch 2*10^8) per second while having huge power consumption and short construction lifetime (<1hour). Does that mean that fusors are just a waste of paper from the science prospective, and real scientists routinely make neutron sources based on nuclear fusion with much better parameters/efficiency? **Update:** Fusor: http://en.wikipedia.org/wiki/Fusor ![enter image description here](http://i.stack.imgur.com/LyXIY.jpg) DAN: http://mars.jpl.nasa.gov/msl/mission/instruments/radiationdetectors/dan/ More info on Russian: http://www.federalspace.ru/main.php?id=58


Does tapping at the side of a bottle prevent shaken soda from bubbling over?

Anecdotal evidence has it that a bottle of soda that was heavily shaken will not bubble over if tapped at the side multiple times. Yet I wonder: Has the tapping really any effect? Or could it be that the mere time that passes while tapping at the side of the bottle has the carbon dioxide not mixed with the soda anymore?


Why is energy conserved?

From basic principals, how does one prove that energy is conserved? Or a little more specifically - Why does this hold: $$\Delta \mbox{ PotentialEnergy} + \Delta \mbox{ KineticEnergy} = 0 $$ Or, for extra credit, why does this hold: $$\Delta \mbox{ PotentialEnergy } + \Delta \mbox{ KineticEnergy} + \Delta \mbox{ ThermalEnergy } = 0 $$


Is energy really conserved?

In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?


Why can't energy be created or destroyed?

My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. _Why_ is that? Is there a theory or scientific evidence that proves his statement true or false? I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :)


Conservation of Energy in the Universe

> **Possible Duplicate:**   >  Is energy really conserved?   >  Why can’t energy be created or destroyed? One of the laws of the universe that dazzles me the most is the law of conservation of energy. I however have a couple of questions regarding this law. Since Einstein's equivalence tells us that matter and energy are different manifestations of the same thing:   * Does this mean that the amount of energy + matter has been the same since the beginning of the universe? If this is true, then another question pops up. Since the universe is expanding at an accelerating rate, and the law of conservation of energy tells us that the amount of energy can't increase:   * does this mean that the universe is getting emptier?  Also, an intertwined question:   * Is there some sort of average 'energy density' in the universe? If so, can we notice the effects of the accelerating expansion of the universe by looking at this 'energy density'? (this might sound like/ be a stupid question).


Fluids in thermodynamic equlibrium

I am reading about the Euler Equations of Fluid dynamics from Leveque's numerical methods for conservation laws. After introducing the mass, momentum and energy equations, some thermodynamic concepts are discussed, to introduce an equation of state. He says               In the euler equations we assume that the gas is in chemical and thermodynamic     equilibrium and that the internal energy is a known function of pressure and density.      After this , the usual thermodynamics-related EOS discussions are carried out. Now chemical equilibrium I understand (number of moles of the chemical constituents do not change), however I don't understand how the assumption of thermodynamic equilibrium can be imposed. From what baby thermodynamics I know, any thermodynamic analysis is always calculated for quasi-static processes, like 'slowly' pushing a piston in a cylinder of gas. But in fluid _dynamics_ fluids are **flowing** and that too rapidly and from intuition there will not be any thermodynamic equlibrium during fluid flow. Where is my understanding going wrong?


How does one get the value of acceleration of gravitation on earth accurately by experiment without electronic device?

How does one get the value of acceleration of gravitation on earth accurately to 5 significant digits by experiment without electronic device?


Is every material able to exist in every state of matter?

I was shocked while reading Kittel's "Introduction to Solid State Physics", that the solid state of noble gases is a well described and makes one of the fundamental achievements of solid state physics. So I wonder: is every possible element and every possible compound able to exist in any of the 3 states of matter?


How do draw real path which represents the solution of the Landau-Lifshitz equation

![enter image description here](http://i.stack.imgur.com/Zux0C.png) how to draw two images look like images below with gnuplot.


Good algorithm for in-experiment 1-D optimization?

I'm running an experiment -- for the question, it doesn't matter which one, but I'm measuring an optical intensity $I$ as a function of two parameters: reflection angle $\theta$ and wavelength $\lambda$. I have motion control in place to move the setup to an angle $\theta_0$, and then I measure $I(\theta_0, \lambda)$ all at once using a spectrometer. I then move to the next angle $\theta_1$ and repeat. Due to the beam travelling through different media at different angles, I have to move the position of the detector $p$ slightly for each angle, which is also automated. I should move the detector so that $M(p) = \sum_\lambda I(\theta_n, \lambda, p)$ is maximized. (M stands for "figure of Merit".) $M(p)$ approximately has the form of a Gaussian plus noise, but I should be able to maximize it without caring what form it has. I have an amateurish algorithm in place to search for the proper position $p$ in order to maximize $M(p)$. The quick-n-dirty algorithm steps $p$ in one direction by a step size $\Delta p$, until the value of $M(p)$ becomes smaller than a previous value. Then it goes back one step and tries a smaller step size $\Delta p$ in the other direction. As you can see, it doesn't account for measurement noise. My thoughts on how to improve it were in the direction of measuring a small number of points spaced $\Delta p$ apart and then fitting a parabola through them. My question is, before I sit down and design a better algorithm, can anyone suggest an already-existing algorithm? I don't think a feedback algorithm (such as PID control) is appropriate, since I'm not trying to maintain a certain setpoint under perturbation of the system -- I just need to optimize to one value for each measurement. For bonus points, can someone point me to some papers on this subject?


What does it mean that particles are the quanta of fields?

I saw the question What are field quanta? but it's a bit advanced for me and probably for some people who will search for this question. I learned QM but not QFT, but I still hear all the time that "particles are the quanta of fields" and I don't really understand what it means. Is there a simple explanation for people who know QM but not QFT?


Derivation of the supergravity action in 11D

The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similiar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?


From knowing just the change in kinetic energy, can we find the friction force and engine power?

I understand this topic well enough to get all the task done because they aren't very creative. But for my exam I think I should have this clear. During the acceleration the force from the engine is of course bigger than air resistance and friction. This force, can we find it? And then the entire force the engine applies for the acceleration. Not just the stub you after subtracting for friction and air resistance. $W = F \times s$   $F = m \times a$ We have all the work done by forces at work, and the stretch of road is easy to calculate. If I now do this $\frac{F}{m} =a $ will that output be the correct acceleration? And this force that we found, is that a sum force? Because if that's the force sum on the car I can't find the engines power output which is what I want. And what about the friction force at work, I think we can't find it when we just have the change in kinetic energy. Is that right? Primarily I would like to know if the change in kinetic energy can be tied somehow to the engines output during the acceleration. The book has this nice equation too: $P = F \times v $   But that's just constant speed. Since we know the time maybe this can be used: $P = \frac{W}{T}$   That just seems a little too easy. Edit: I got a B on the exam, which means I'll be at again this fall. Not due to this question. Haha.


Is this really a golden ratio spiral?

In this blog post, I found this picture: ![enter image description here](http://i.stack.imgur.com/Qi3C2.jpg) Does the water really form golden ratio spiral in such cases? Or is the photo just a provocative example, without physics grounds for claims about "goldness" of the spiral?


Normalization for QFT single particle destruction operator

I don't understand a particular statement in the QFT book by Klauber. The particular page I'm having difficulty on is page 67 of chapter 3 (PDF link). The big picture is that the author wishes to investigate what the (operator) solutions to the Klein-Gordon equation, $\phi(x)$ and $\phi^\dagger(x)$, do when acting on the vacuum state $|0\rangle$. As prep for this, he creates a "general single particle state" ("general" meaning non $\mathbf{k}$-eigenstate) by operating on the vacuum with the operator $$C\equiv\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger,\tag{3-108}$$ $$C|0\rangle=\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger|0\rangle=A_1|\phi_1\rangle+A_2|\phi_2\rangle+\cdots\equiv|\phi\rangle\tag{3-109}$$ Each $A_\mathbf{k}$ is just a number, the absolute value square of which represents the probability of finding the $\mathbf{k}$ eigenstate for the single particle. The new state $C|0\rangle=|\phi\rangle$ is interpreted as a single particle state in a superposition of $\mathbf{k}$-eigenstates $|\phi_k\rangle$. The subscript $\mathbf{k}$ represents different momenta. For probability/normalization arguments, the numbers $A_\mathbf{k}$ should obey $$\sum_\mathbf{k}\left|A_\mathbf{k}\right|^2=1.\tag{3-110}$$ I feel like I understand the above statements. The author then introduces the "general single particle destruction operator" $$D\equiv\sum_\mathbf{k}a_k,\tag{3-111}$$ and shows that when applied to our general single particle state $|\phi\rangle$ above, the vacuum is (re)produced: $$\begin{eqnarray} D|\phi\rangle&=&\left(\sum_\mathbf{k}a_k\right)A_1|\phi_1\rangle+\left(\sum_\mathbf{k}a_k\right)A_2|\phi_2\rangle+\cdots\\\ &=&A_1\underbrace{a_1|\phi_1\rangle}_{=|0\rangle}+A_1\underbrace{a_2|\phi_1\rangle}_{=0}+A_1\underbrace{a_3|\phi_1\rangle}_{=0}+\cdots+\\\ &\ &+A_2\underbrace{a_1|\phi_2\rangle}_{=0}+A_2\underbrace{a_2|\phi_2\rangle}_{=|0\rangle}+ A_2\underbrace{a_3|\phi_2\rangle}_{=0}+\cdots+\\\ &\ &+\cdots\\\ &=&\underbrace{\left(A_1 + A_2 + \cdots\right)}_\text{can normalize = 1}|0\rangle. \end{eqnarray}\tag{3-112}$$ (Note the subtle but important differences in the underbraces; some are $0$ while others are $|0\rangle$.) **The part I am struggling with** is understanding how the underbrace "can normalize = 1" at the end of $\text{(3-112)}$ can be true given $\text{(3-110)}$. It seems to me that the $A$ terms appearing at the end of $\text{(3-112)}$ are the same ones defined in the construction operator $C$ and normalized so that their absolute values squared sum to $1$. How can their just-plain sum also be of magnitude $1$? I know that one would *like * the underbraced term to sum to zero, but I don't see how that can be. * * * It was suggested I consider the quantity $\langle\phi|D^\dagger D|\phi\rangle$. Here is my attempt to calculate it. $$ \begin{eqnarray} \langle\phi|D^\dagger D|\phi\rangle&=&\langle0|(A_1^\dagger+A_2^\dagger+\cdots)(A_1+A_2+\cdots)|0\rangle=\langle0|\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}|0\rangle\\\ &=&\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}\underbrace{\langle0|0\rangle}_{=1}=\underbrace{\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}}_\text{Can't simplify}\ne1 \end{eqnarray} $$


Solving systems of equations in dynamics

I have an exam in two days for first year university physics. Often for dynamics problems, I am required to solve algebraic systems of equations by hand, and this can be very daunting. When I see the solutions, however, the steps that the solver took to seem very clean and almost obvious. Are there some rules of thumb that physicists use to solve small systems of equations, either by elimination or substitution? Here is an example. Find $\frac{m_1}{m_2}$ in terms of only $\theta$ where the known quantities are $\theta$, $m_1$, $m_2$. $$F_T \sin \theta = m_1 g$$ $$F_T \cos \theta = m_1 a$$ $$F_T \sin \theta + F_N \cos \theta = m_2 g$$ $$F_N \sin \theta - F_T \cos \theta = m_2 a$$


Gauge invariance and diffeomorphism invariance in Chern-Simons theory

I have studied Chern-Simons (CS) theory somewhat and I am puzzled by the question of how diff. and gauge invariance in CS theory are related, e.g. in $SU(2)$ CS theory. In particular, I would like to know about the relation between large gauge transformations and large diffeos. If you know any good sources, I would be really grateful. Thank you!


Simulation of an one dimensional driven diffusive system

I'm currently writing a simulation in python with scipy and matplotlib to reproduce an one dimensional driven diffusive system described in this paper from M.R. Evans et al. The system consists of positive, negative and hole particles. On the left side of the system the positive (negative) particles are produced on the left (right) side with a possibility a and destroyed at the right (left) side with the possibility b. In some cases the system should show a flip between positive and negative high density states and I'm trying to reproduce this behaviour with my simulation. But all I can see is an increasing current within my simulation and I can't find any problems in my code explaining such behaviour. Does anyone here have some experience simulation such or similar systems? Cheers, Florian


Why can't light escape from a classical black hole?

Photons do not have (rest) mass (that's why they can move at speed of "light"). So, my question is how the gravity of classical$^1$ black hole can stop light from escaping? \-- $^1$ We ignore quantum mechanical effects, such as, Hawking radiation.


Kepler problem: flows generated by constants of motion

This is part of an admission exam problem, found at http://www.sissa.it/mp/admission/tests//2008_common.pdf Consider the Hamiltonian of Kepler problem $$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\boldsymbol{p}^2|}{2\mu} +\frac{\alpha}{|\boldsymbol{r}|}, \qquad \mu>0>\alpha,$$ where $\boldsymbol{r}\in M=\mathbb{R}^3\setminus\\{ 0 \\}, \ (\boldsymbol{r},\boldsymbol{p})\in T^*M$ and $|\boldsymbol{r}|=\sqrt{r_1^2+r_2^2+r_3^2}$. The quantities $$\boldsymbol{m}=\boldsymbol{r}\times\boldsymbol{p}, \qquad \boldsymbol{W}=\boldsymbol{p}\times\boldsymbol{m}+ \mu\alpha\frac{\boldsymbol{r}}{|\boldsymbol{r}|}$$ are constants of motion, as is well known. It is stated then that the flows generated by the functions $m_i$ and $W_i,\ i=1,2,3$ are canonical transformations. **I don't understand** is what is meant by this statement: I mean, I know what a canonical transformation is, but I would appreciate some explanation or reference about this precise statement. Thanks in advance for the help, and I hope this is formulated in compliance with the rules of this community.


Is Connes model a composite Higgs in disguise?

Most of the 5-dimensional Higgs models can be seen, if I understand correctly, as models where the Higgs is a composite. Now, is this true for Connes models? It is a model of extra dimensions too, in some sense. And when you look at the papers, at some moment a product of two symbols is substituted by a single $\phi$, so it seems that some composition is at work.


Wilson loops and gauge invariant operators (Part 1)

I guess the Hilbert space of the theory is precisely the space of all gauge invariant operators (mod equations of motion..as pointed out in the answers)   * Is it possible that in a gauge theory the Wilson loops are the only observables?  (...I would vaguely think that if a set of Wilson loops one for every cohomology class of the space-time is the complete set of observables then this is what would be "a" way of defining a Topological Field Theory but may be this is also possible for pure gauge theories in some peculiar limit or on some special space-time geometries..)   * When the above is not true then what are all the pure gauge theory observables?..I guess its only the local observables that is missed by the Wilson loops..   * In general is it always true that all gauge invariant observables are precisely all the polynomials in the fields which are invariant under the action of the gauge group? (..and this is a well studied question in algebraic geometry under the name of Geometric Invariant Theory?..)    * If one has matter in the theory then I guess the baryons and the mesons are the only matter observables? I guess there is no gauge group dependence on their existence?  (..though baryons can always be defined for any anti-symmetric combination of the flavour indices I guess mesons can be defined only if equal amount of matter exists in the conjugate representation of the gauge group also..right?..)   * Why are gauge traces of arbitrary products of matter fields neither baryons nor mesons? (...in arbitrary gauge theories is it legitimate to identify these states as ``chiral primaries" in any sense?..)


Coulomb potential energy functional derivative

I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional: $$ J[\rho] = \frac 12\int \frac{\rho(r)\rho(r')}{|r - r'|} drdr' $$ According to the functional derivative formula I should do something on these lines: $$ \frac {\delta J}{\delta \rho (r)} = \frac{\partial}{\partial \rho(r)} [ \frac 12 \int \frac{\rho(r) \rho(r')}{|r - r'|}dr' ]$$ In my wrong reasoning I would simply take $\rho(r) $ out of the integral and apply the derivative: $$\frac {\partial \rho(r)}{\partial \rho(r)} \frac 12 \int \frac{\rho(r')}{|r - r'|} dr' = \frac 12 \int \frac{\rho(r')}{|r - r'|} dr'$$ Which is wrong because the correct result should be: $$\int \frac{\rho(r')}{|r - r'|} dr'$$ 1) I'm quite confused by the notation and how to treat a partial derivative by $\rho(x)$ 2) What's the correct way to handle and compute functional derivatives in these cases? I'm actually in a similar situation with a much complex derivative, such as the same thing in the density matrix formalism: $$ J[\gamma_1] = \frac 12 \int \frac{\gamma_1 (x_1', x_1) \gamma_1 (x_2', x_2) \delta(x_1' \- x_1) \delta(x_2' \- x_2) dx_1 dx_1' dx_2 dx_2'}{|x_1 - x_2|}$$ $$\frac {\delta J[\gamma_1]}{ \delta \gamma_1 (x_1'. x_1)}$$