title: How do Einstein's field equations come out of string theory?

text: The classical theory of spacetime geometry that we call gravity is described at its core by the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy in spacetime. For example: $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ is an important concept in General Relativity. Mathematically, how do the Einstein's equations come out of string theory?


title: How do I find the force on the charged sphere due to a dipole sitting inside?

text: I have a dipole sitting inside a charged sphere i.e. at the centre of the sphere. As far as I can see the force on the Dipole is zero because the external electric field on the dipole at $r=0$ is $0$. But what about the force due to the dipole on the sphere? We know that the electric field due to a dipole is given by $-\bar p\cdot \frac {\nabla \bar r}{r^3}$. I have come to the step (I don't know if I am right): $$\frac{-3Q}{4\pi R^3} \int_{\partial V}\frac{\bar p\cdot\bar r}{r^3}.$$ Another case, If the sphere was a hollow sphere with the thickness $t$, what would be the force on the sphere due to this dipole?


title: How to determine whether a large container is air-tight?

text: In constructing a kitchen-waste digester at home, I use a 50 Litre HDPE drum. The base of the drum is holed with a plug fitted to allow drainage when necessary. The top has two openings - one for inlet, the other to act as outlet for the generated fluid CO2/CH4. The drum is to be used filled to about 2/3rds lying on it's side. Therefore any leakage at the drain will be immediately visible. The same applies to the inlet pipe - it too shall be filled with water. A heated iron nail was used to make the outlet hole. Then a ball-point pen was inserted into it; a very snug fit. I then melted a silicon glue-stick to attempt to seal any leaks at this point. The problem at hand is to determine whether the outlet is air-tight. It is possible to dunk this container in water and test for bubbles; but the same test will not be possible with a larger container (300 litres) to be used if this experiment is successful. Suggestions, anybody?


title: Plate tectonics and water

text: In a couple of places already i've read the following assertion: > It is believed that plate tectonics might play an important part in > maintaining the presence of water on a planet's surface one source is this article. But the assertion is never argumented. What is the correlation between the two? what does the models says about what happens to the surface water when there is no plate tectonics?


title: The Role of Gravity among the Fundamental Forces of Nature

text: If we look at the standard model, we have 4 fundamental forces which include   1. Gravity,   2. Electromagnetism,   3. Nuclear weak force,   4. Nuclear strong force. I would like to look at Gravity for a minute. Scientists are still searching for what causes gravity. But is it possible that gravity is only a property of the space-time fabric? For example take a blanket that is stretched tight and put different weight balls on the blanket. As one would see, the balls sink into the blanket differently. If you look at the space-time fabric as the blanket and the different balls as objects in the space-time fabric when it comes to planets and more on the macro scale, thinking gravity as a property of space-time fabric seems to work. Since the standard model deals more at the quantum level though and we are looking at it here on earth where gravity is $9.8\,m/s^2$, it seems to me that the standard model would also need to stand up to say if we had gravity that was $11.4\, m/s^2$ or something more absurd like $20.6\,m/s^2$ . Also could we not say that gravity is really just a property of space-time fabric?


title: Vector Addition -- Direction

text: Say we have three forces $F_1, F_2, F_3$, such that $$ F_1 + F_2 - F_3 = 0\hspace 10mm (1) $$ And let us say that $F_1$ and $F_2$ have the same direction and magnitude, and that $F_3$ has double the magnitude of either, in the opposite direction. From this it would seem that $F_1$ and $F_2$ had the same direction (in highschool physics, at least!), but if we treat these vectors like numbers, we can make another statement: $$ F_1 - F_3 = - F_2\hspace 10mm (2) $$ And yet this seems absurd to me, since equality of vectors seems to imply equality of direction. From the statement in $(1)$ I can also state that: $$ F_1 = F_2\hspace 10mm (3) $$ But this contradicts $(2)$! Edit: So they don't contradict, but I guess what I was wondering was the _notation_ \-- that is, if we say that there is a force $F = ma$, $F - ma = 0$ follows. Does this mean that $ma$ and $F$ are in opposite directions? What does the negative sign really mean? I'm sure I've probably totally missed the point of vectors, but I can't seem to be able to contract this question into a Google search.


title: Why does a superconductor obey particle-hole symmetry?

text: We normally solve the Bogoliubov-de Gennes (BdG) equations in order to compute the energy spectrum of a superconductor. The Nambu spinor is a common object that is used in formulating these equations. In other words, the problem is formulated with the a priori assumption that the superconductors obeys particle-hole symmetry. But how do we know that it will obey particle-hole symmetry? I'm sure that by assuming particle-hole symmetry, computing the spectrum by solving the BdG equations, and comparing with experiments we get a good agreement. So we know it's true. I just don't understand why that is the case. Is there any physical reason behind it?


title: "Forgetting" the initial condition in conservative oscillations; What has been "forgotten" exactly?

text: I am training myself on oscillations. The topic is self-sustained oscillations. The claim is these oscillations are NOT forgetful about their initial condition as opposed to conservative oscillators which are. What I don't understand is what are our prior information when we want to recover the initial condition? In conservative oscillators, we know that the state of the system is a closed curve and in self-sustained ones it is a limit cycle, but according to uniqueness and existence theorem for ODEs, knowing the state of the system at time $t$, i.e. $x(t)$, will let you recover $x(0)$. So what is "it" that is not forgotten in the former and is forgotten in the latter? And what is the prior knowledge that helps the recovery? **Note:** If there are people out there who don't have the time to write the answer, I would be happy to be referred to a reference book. Thank you


title: Is the Hawking radiation of a charged black hole thermal?

text: Suppose you have a Schwarzschild black hole of mass $M$ and angular parameter $a = 0$ (no rotation). **Question:** is it possible to throw a charge $Q$ at a faster rate than it will be re-radiated? Will the radiation profile be still thermal? If it is thermal, it would mean that even big, cold black holes would emit a lot of energy in the form of electromagnetic radiation just to be able to get rid of the extra charge? A thermal spectra that starts at $511\: \mathrm{KeV}$ (the energy of the lowest charged particles and has very little emitted power at lower energies) would be a very weird thing to call 'thermal'. There is a differential expression for the increase in temperature as a small charge (compared to the black hole mass) is added to the black hole that one obtains by taking the formula 11.2.17 in this page (Modern Relativity, 2005, David Waite) and deriving against $e$ and $M$ and taking $$ \delta T = ( \frac{\partial T}{\partial e} + \frac{c^2}{G} \frac{\partial T}{\partial M} ) \delta e$$ so this gives a profile for the variation of black hole temperature with charge. **Question:** Is it correct to conclude that you can estimate the overall Hawking radiation of a relatively small black hole ($M \approx 10^{18} Kg$) by adding electric charge at a faster rate than it will be re-emitted by the thermal spectra? Or is the spectra totally non-thermal, and the radiation will favour throwing away charged particles, while being electromagnetically cold?


title: Why is my approach to the equation of time off by a constant?

text: I'm trying to better understand the causes for the equation of time by deriving an approximation from first principles. My naive approach, $EOT_{NAIVE}$, is to take the difference between the right ascension of the mean sun, $\alpha_M(t)$, and the right ascension of the "real" sun $$EOT_{NAIVE}(t)=\langle\dot{\alpha }\rangle\cdot(t-t_0)-\alpha(t)$$ where $\alpha(t)$ is simply the "actual" position of the sun at time $t$ from ephemeris data (e.g. either of the RA values reported by JPL HORIZONS, or a similar source) and where $\langle\dot{\alpha }\rangle = 24/365.242$ and $t_0=t:\alpha_M(t_0)=0$. To confirm that this is about right, I compare my result with what I get using the USNO definition of GMST $$EOT_{GMST}(t)=GMST(t)-(t-t_{noon})-\alpha(t)$$ and with $EOT_{POLY}$, a standard polynomial expression from Dershowitz & Reingold as described in The Clock of the Long Now documentation. But the values I get for $EOT_{NAIVE}$ (blue dashed line) are consistently about $7.4$ minutes greater than these reference methods (exactly $7.4537$ for $EOT_{GMST}$, red line; and within a couple seconds of that for $EOT_{POLY}$, gray outline) give: ![enter image description here](http://i.stack.imgur.com/cqWAE.png) Surprisingly, I can fix this by simply changing $t_0$ from the date, $t_E$, on which the most recent vernal equinox occurred (2012-03-20T05:14:33Z) — which I had assumed would give a "starting" RA of 0 — to 2012-03-22T02:20:32.41Z, which brings my approximation (blue line) exactly in register with the GMST approach (gray band), and within a couple seconds of the polynomial approach (red line), as can be seen by taking the difference between the GMST approach (gray band) and each of these approaches (note the change of scale, this figure essentially takes the difference between $EOT_{GMST}$ and each of the plots in the first figure above, and "zooms in" on the gray band): ![enter image description here](http://i.stack.imgur.com/njj15.png) Why should changing the date in this way "fix" my approximation? Why doesn't my approximation work with $t_0=t_E$? Why does it work perfectly with a different date? Are the approaches I'm using as references ($EOT_{GMST}$ and $EOT_{POLY}$) the right ones for this kind of exploration? * * * Note that the reported RA of the sun at 2012-03-22T02:20:32.41Z is $0.1146^h$, and that the fit between $EOT_{GMST}$ and $EOT_{NAIVE}$ using that date is quite good (this figure zooms in further on the gray band in the second figure above): ![enter image description here](http://i.stack.imgur.com/1Mtl5.png) Note also that despite the documentation for the ephemeris I use claiming that "light time delay is not included", the values I get match those from JPL's "airless apparent right ascension and declination of the target center with respect to the Earth's true-equator and the meridian containing the Earth's true equinox-of-date. Corrected for light-time, the gravitational deflection of light, stellar aberration, precession and nutation."


title: How to calculate radiative transition rate of exciton in a quantum dot with specific dimension?

text: I am writing rate equations for a nanophotonic system including three quantum dots. I need to calculate that radiative transition rates of exciton in ground state in those quantum dots. In the paper named "Dynamics of ONF-Based Three- QD Nanophotonic AND Gates at Finite Temperatures", table1, there are the exact numbers for radiative transition rates but it is not clear how it is calculated! Does anyone here know it?


title: Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?

text: For a particle in the potential: $$V(r) = \begin{cases} 0 & \text{a < r < b}\\\ \infty & \text{otherwise.} \end{cases}$$ Does this guy in the ground-state exert a force on the shells a and b? I don't think so because at ground-state both the centrifugal term of the radial equation and the angular momentum of the particle is zero. What say you brethren?


title: Why isn't temperature measured in units of energy?

text: Temperature is the average of the kinetic energies of all molecules of a body. Then, why do we consider it a different fundamental physical quantity altogether [K], and not an alternate form of energy, with a dimensional formula derived from three initial fundamental quantities, length [L], mass [M] and time [T]?


title: Why regularization?

text: In quantum field theory when dealing with divergent integrals, particularly in calculating corrections to scattering amplitudes, what is often done to render the integrals convergent is to add a regulator, which is some parameter $\Lambda$ which becomes the upper limit of the integrals instead of $\infty$. The physical explanation of this is that the quantum field theory is just an approximation of the "true" theory (if one exists), and it is only valid at low energies. We are ignorant of the processes that occur at high energies, and so it makes no sense to extend the theory to that regime. So we cut off the integral to include the low energy processes we are familiar with only. My problem with this explanation is, even if we don't know what happens at high energies, is it okay to just leave those phenomena out of our calculations? In effect we are rewriting our integral as $\int_{-\infty}^{\infty} = \int_{-\infty}^{\Lambda} + \int_{\Lambda}^{\infty}$, and then ignoring the second term. Is this really okay? And even if we were able to ascertain the "true" theory and calculated the contribution to the correction from high energy processes, I feel like it would come out to be large, since looking at it from our effective field theory viewpoint it already appears to be infinity. Do we have any right to say it is negligible?


title: boundary conditions

text: let be the operator $ -i\hbar x\frac{df(x)}{dx}-i\hbar \frac{f(x)}{2}=E_{n}f(x)$ what kind of bundary conditions can i put ?? i have tried to find a function so for every integer 'n' i get $ f(nx)=f(x) $ but i only get that $ f(x) $ must be a constant :( of course i know how to solve it to get the solution $ f(x)= \frac{C}{x^{1/2-iE_{n}}} $ or perhaps to set the conditions $ F(nx)=F(x) $ for every integer n with given $ F(x)= \sum_{p} \sum_{k=-\infty}^{\infty}f(p^{k}x) $ with $ f(x) $ solving the above equation and 'p' runs over the primes $ p=2,3,5,7,11,... $


title: Find generating function $F_1$ for canonical trasformation

text: I'd like to know the steps to follow to find the generating function $F_1(q,Q)$ given a canonical transformation. For example, considering the transformation $$q=Q^{1/2}e^{-P}$$ $$p=Q^{1/2}e^P$$ I have thought to find p and P: $$p=\frac{Q}{q}$$ $$P=\frac{1}{2}\ln Q-\ln q$$   And so $$\tag{1}\frac{\partial F_1}{q}=\frac{Q}{q}$$   $$\tag{2}\frac{\partial F_1}{Q}=\ln q-\frac{1}{2}\ln Q$$ Then I have thought to integer these relations and I have obtained: $$F_1=Q\ln q$$   $$F_1=Q\ln q-\frac{1}{2}(Q\ln Q-Q) $$ The result must be: $$F_1=Q\ln q-\frac{1}{2}(Q\ln Q-Q) $$ Now, I'm trying to find the general rule: At the beginning I have thought that I have to sum the two $F_1$, but then the result is wrong. And so I ask you: do I have to integer the two relations and then take as $F_1$ that relation that verifies both (1) and (2)?


title: Is Inflation modelled by a field?

text: If Inflation is modelled by a field - is this a classical field or a quantum field? If classical are there good reasons not to quantise it? What are the implications of such a quantisation?


title: Using Fourier Transforms to Solve Systems with springs of high frequency

text: I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine and cosine waves, it takes a very long time for a numerical solver to find the answer. However, the fourier transform of the solution is probably fairly simple - it's nearly a delta function peaked at the frequency (or frequencies) at which the system oscillates. Since Fourier transforms turn $\frac{df}{dx}$ into $2 \pi i \xi \hat{f}(\xi)$, it would seem that we could turn all of Newton's equations of the form $F_k=m\frac{d^2}{dt^2} x_k$ (representing, say, the $k$th particle or rigid body in a complicated system) into $\hat{F_k} = m (2 \pi i t)^2 \hat{x_k}$ (where the Fourier transform is taken with respect to time). Then, we could solve all of these simpler equations for the values of the $\hat{x_k}$, then take the Fourier transform (numerically). So, do people ever apply Fourier transforms to the case of many-body systems with some harmonic oscillators?


title: What is the relationship between kinetic energy and momentum?

text: I can't seem to figure out the relationship between $E_k$ and $p$ or $F$. I understand that the units are pretty different. But for example: A bullet with a mass of _10.0g_ is moving at the speed of _1000m/s_. A bull with a mass of _400kg_ is charging at the speed of _5.00m/s_ Which has the greater kinetic energy? _5000J_ for the bullet and _5000J_ for the bull That worked fine but now if you calculate momentum you get:   _$p = 10 kg\cdot m/s$_ for the bullet and _$p = 2000 kg\cdot m/s$_ for the bull.   Which does not make sense! Does the bull hit harder than the bullet or the same amount as the bullet?


title: Questions about the Dyson equation

text: I'm studying finite temperature many-body perturbation theory, and am trying to understand The Dyson equation. In particular, I'm using Mattuck - A guide to Feynman diagrams in the many body problem. There appear to be two types of Dyson equation: one which is valid in the special case of no external potential and with diagrams calculated in $(k,w)$-space, \begin{equation} G(k,w) = \frac{1}{G_0 (k,w)^{-1} - \Sigma (k,w)}. \end{equation} And one valid at other times which is an integral equation for $G$ which can be different depending on what space you're in and whether you have an external potential or not, but is something like \begin{equation} G(k,\tau-\tau') = G_0(k,\tau-\tau') + \int_0^\beta d\tau_1 d\tau_2 G_0(k,\tau- \tau_2) \Sigma(k,\tau_2-\tau_1) G(k,\tau_1-\tau'). \end{equation} So, my questions:   1. What is an external potential? I can't find an actual definition in the book. At first I assumed it meant the part of the system you want to treat perturbatively, I.e. the Hamiltonian is $H=H_0+H_1$ where $H_1$ is small and can there for be considered a perturbation. But obviously this can't be the case otherwise there's no expansion to use in a Dyson equation.   2. Why is the second Dyson equation useful? - How does it allow partial summations?    3. Every book I've looked at seems to give slightly different rules for calculating the Feynman diagrams. Any thoughts on which is correct? (for example, Negele - Quantum many-particle systems is different to Mattuck).


title: Galilean invariance of Lagrangian for non-relativistic free point particle?

text: In QFT, the Lagrangian density is explicitly constructed to be Lorentz- invariant from the beginning. However the Lagrangian $$L = \frac{1}{2} mv^2$$ for a non-relativistic free point particle is not invariant under Galilean transformation. This does not ultimately matter because the difference is a total time derivative. However, is it possible to exhibit a Galilean invariant Lagrangian for a non- relativistic free point particle?


title: Force Carrier in Fermion Generations?

text: Assuming that before Electroweak Symmetry Breaking all particles of a generation were identical, is it possible that they could be related to some kind of symmetry that is unbroken at higher energies, making them distinct from each other and leading to the ability to distinguish the otherwise identical Fermions? Could this hypothetical force be something like the strong force minus confinement, with each of the generations forming a triplet of "flavor charge" and there being "flavor gluons" that transform between the different types? And if this was the case, what would be the relationship between this "Flavor Force" and the Electroweak force, seeing as it recognized that the particles were different and gave them different masses to reflect that? I'd prefer if one can answer as many of the questions as it's really possible to, especially the last one, which with my limited knowledge has been bugging me lately. Please don't hold back in pointing out errors in my thinking, I'm here to learn why I'm wrong about this, not to push the idea.


title: Are there examples of "loss-less" Forster Resonance Energy Transfer (FRET)?

text: When I think of a Forster Resonance Energy Transfer (FRET) process, I typically imagine the initial excitation of an absorbing chromophore with a photon followed by the subsequent emission of a lower frequency photon from a secondary emitting chromophore, where this process occurs as a result of dipole-dipole coupling between electronic states of the two chromophores allowing for an InterSystem Crossing (ISC) event. However, isn't this just because Kasha's rule (http://en.wikipedia.org/wiki/Kasha's_rule) or some variant of this rule, tells us that ISC events are most likely to occur from the lowest excited eigenstate of the initially excited chromophore, where we shed energy non- radiatively to get to this state? Can't we use a laser and a band-pass filter to directly excite to absorbing chromophore to the lower regime of the first excited singlet state where ISC can occur with high probability (provided the right chromophore system)? Are there examples of "loss-less" FRET energy transfer where the excitation and emission wavelengths are identical or near-identical (e.g. where the wavelengths are less than a nanometer apart)?


title: Heat transfer calculated from the specific heat formula

text: Say I have 10g of silver, whose specific heat is known to be 0.235. I've heated it up from 50.0C to 60.0C. How much heat has been transferred? The equation is: $$ Q = C_pm\Delta t $$ where Cp is the specific heat value, m is the mass of the object, and Δt is the change in temperature in Kelvin. I got the answer 23.5j, which I derived from $$ Q = (0.235)(10)(60.0-50.0) = 23.5j $$ My teacher said that we have to get 10.0 as our delta T then add 273 to make the equation work with Kelvin, which translates to $$ Q = (0.235)(10)((60.0-50.0)+273) = 665.05j $$ Now I don't see her reasoning in this answer since the difference in Kelvin is the same as the difference in Celsius. It also doesn't help that the book had the 665.05j answer. This was a test too, so my wrong answer bugs me. If I did the equation in Kelvin, I would have arrived at the same answer. My argument was that translating in Kelvin beforehand should work. Apparently, the answer for delta T needs to be scaled. So, can someone provide a good argument as to which one of these is the correct way to calculate specific heat? Thanks. (I already know what the right answer is and my friends do too, so please give an argument.)


title: Infinite plane wave and ray vectors

text: I want to clear my mind on one basic optics thing: If I have a point source some distance away from a lens, the position of the image formed can be found using traditional geometrical optics ray approach: one ray passing through the center of the lens and another one parallely to the axis and through the focal point. If I think about this situation as if the rays are the plane waves: I decompose the light coming from the point source into infinitely many plane waves each reaching the lens at some angle. Now I can draw infinitely many rays for each plane wave which are perpendicular to the plane waves front and all these rays are going through the lens. And I get a lot of different ray directions after the lens, no image is formed. So is there the one exact path during which a ray must be drawn if a plain wave is infinite? Thank you for clearing this up for me.


title: What was the Law of Gravity better explained by?

text: In mechanics, our professor made the declaration that "all laws of physics" have been disproven. He mentioned several examples including the Law of Gravity, mentioning briefly that it is better explained by Einstein's theory of General Relativity. He also mentioned Bohr's model of the electron, and how it was better explained by other models, which I've since learned. Now, haven taken Modern Physics as part of a Physics 3 class, I'm still scratching my head wondering how the theory of general relativity explains attraction between large objects. Has has Einstein's theory of General Relativity truly explained it or did I misunderstand? If so, how?


title: Alternative solar panels

text: I was doing a bit of thinking recently, when I stumbled on an interesting idea. In a simplified sense, solar cells function by absorbing light and converting it into electrical energy. Since visible light is just electromagnetic radiation, would it be possible to build a "solar" cell that absorbs different wavelengths of electromagnetic radiation? This question actually has two parts: The first is whether such a construction is actually possible. The second is whether such a construction is actually _useful_. Most of the electromagnetic spectrum is absorbed by the atmosphere, so I get the feeling targeting any of these absorbed frequencies wouldn't be useful on Earth, because there wouldn't be much energy left to absorb. However, they might be useful outside of Earth's atmosphere. Here's a diagram of which electromagnetic frequencies are absorbed by the atmosphere, for reference: http://upload.wikimedia.org/wikipedia/commons/3/34/Atmospheric_electromagnetic_opacity.svg I did a bit of thinking/research about what these alternate "solar" cells might look like. I would imagine that infrared/ultraviolet solar cells would be fairly similar to normal solar cells, and sure enough, some Googling revealed infrared/ultraviolet solar cells are actually something that's being worked on right now. Below infrared light are radio waves. A "radio" solar cell is a much more interesting idea. I know that "capturing" an electromagnetic wave has a lot to do with the wave's wavelength. But of course, we already have tools to capture radio waves: antennas. So is a "radio wave solar cell" just an antenna? Does this mean that the units of a normal solar cell are basically just tiny antennas for light? If so, it bodes poorly for the idea of x-ray or gamma-ray solar cells, due to the small wavelengths involved.


title: How do GPS devices work?

text: I have GPS in my phone but I have only a very rough idea of how it works. I have checked the Wikipedia article but that gone over my head (I can not understand that maths and other strange stuff there). Please explain its working in simple words. What I think is that it sends signal to satellites, satellite checks the location of sending device and reply with the location of that device.


title: Simple friction formula for a car

text: I am making a 2D driving video game, and I would like to know the "simple" formula for calculating the friction force between the car and the road. I have read lots of friction diagrams involving balls rolling down inclined planes, but I'm having trouble applying it to this problem. Consider a car driving on a horizontal road, with some coefficient of friction, $\mu$. Its velocity is $v$, mass is $m$, and a constant driving force is accelerating the car at $a\ \mathrm{m/sec}^2$ in the forwards direction. What is the friction force, $F$, that acts on the car in the backwards direction? I _guess_ that the friction force is proportional to the car's speed, but I can't explain this guess. The reason I think that is because of a handful of case studies. Say that the driver's foot is on the pedal such that the car will get up to $16\ \mathrm{m/sec}$:   * If the car's velocity is $0\ \mathrm{m/sec}$, there is no force accelerating it backwards (it is stationary).   * If the car's velocity is $8\ \mathrm{m/sec}$, the friction force accelerating it backwards must be less (in magnitude) than the driving force, and hence the car keeps accelerating.   * If the car's velocity is $16\ \mathrm{m/sec}$, the friction force accelerating it backwards is equal (in magnitude) and opposite to the driving force, and hence the car maintains a constant velocity.   * When the driver releases the pedal, the same friction force accelerates the car backwards, but now there is no driving force, so the car drifts slowly to a halt. It would therefore make sense that $F \propto -\mu v$, such that in the first case, $F = 0$; in the second case $F = -\frac{1}{2}ma$; in the third and fourth cases, $F = -ma$, where $a$ is whatever acceleration is required to maintain a constant velocity of $16\ \mathrm{m/sec}$. But I can't figure out the exact relationship between $F$ and $v$, or explain why they are proportional.


title: What is the logarithm of a kilometer? Is it a dimensionless number?

text: In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is. Suppose I have something with length $L = 1 km$. $\lg L = \lg km$ It seems that the unit of $\lg L$ is $\lg km$, but I can also say $L = 1000 m$ and now: $\lg L = 3 + \lg m$ This doesn't appear to have any units at all. This suggests that $\lg km$ and $\lg m$ are actually dimensionless numbers. But wait, I can do this with any unit! Does it actually make sense to talk about the logarithm of a unit, or some other function for that matter?


title: How to make a single crystal silicon wafer?

text: How can I make a single crystal silicon wafer? What are the tools needed? What is the most diffcult part? Can I make it at home or at school lab? I am now interested on Czochralski process where you melt silicon and let it crystallize and cut afterwords.


title: Change of variables in calculating the integral of multivariable differential entropy

text: I have already asked this question in math.SX but here might be more proper. So I decided to put a copy here and delete the one which is not the one that got an answer: I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int p(x)\log(p(x))dx=\int\limits_{0}^{1}\frac{d}{dp}[\log F^{-1}(p)]dp$$. Where $F$ is the cumulative distribution function of $p$. To extend the idea for multivariate density functions my idea was to try to integrate on level sets of $p(x)$ to simulate somehow what happens in one dimension. As a results I took a look at this question and topics relevant to pullback method in differential geometry and co-area formulation but couldn't come up with anything by myself. Does anybody know of any method to get a similar formula for differential entropy when $p$ is a multivariable density function? **EDIT** : As it has been clarified by "NowIGetToLearnWhatAHeadIs" CDF does not have a unique generalisation. What I thought to be the natural one is: $$F(x_1,...,x_n)=\int\limits_{-\infty}^{x_1}\int\limits_{-\infty}^{x_2}...\int\limits_{-\infty}^{x_n}p(x_1,...,x_n)dx_ndx_{n-1}...d_2d_1$$


title: Mean value of the force in a collision problem

text: I have the following problem: A point with mass $m$ and speed $v$ collides with a fixed obstacle and penetrates it, stopping in a space $\Delta x$. Calculate $\Delta t $. I understand the way my book solves it: let $F_m$ be the mean value of the force during the collision. We have: $$\frac{1}{2} mv^2 = F_m \Delta x \qquad mv =F_m \Delta t $$ which is easily solved for $\Delta t$. Although it's pretty obvious to intuition that $F_m$ must be the same in both equations, I was wondering if this is mathematically (and generally) correct. From the equations: $$F_m=\dfrac{W}{\Delta x}=\dfrac{\int_{x_0}^{x_1}Fdx}{x_1-x_0}$$ On the other hand: $$F_m=\dfrac {\Delta p}{\Delta t}=\dfrac {\int _{t_0}^{t_1}Fdt}{t_1-t_0}$$ So the mathematical statement behind this is: $$\dfrac {\int _{t_0}^{t_1}Fdt}{t_1-t_0}=\dfrac{\int _{x_0}^{x_1} Fdx}{x_1-x_0}$$ (that one may prove to be generally false, see accepted answer) and also, if $v_1 \neq v_0 $: $$\dfrac {W}{\Delta p}=\dfrac {\Delta x}{\Delta t}=v_m \iff v_m = \dfrac {v_0+v_1}{2} $$ Am I confused? any help is appreciated.


title: What do the names "E mode" and "B mode" mean? Where do they come from?

text: This has been bugging me a bit since the BICEP announcement, but if there are any resources that answer my question in a simple way, they've been buried in a slew of over-technical or over-popularized articles; Wikipedia isn't much help either. It is clear to me that when the CMB is described as having "E modes" and "B modes", some reference is being made to electric and magnetic fields. **What is the precise nature of this reference?** I suspect it is simply an appeal to the fact that the polarization can be split into a curl-free component, which is the gradient of something, and a divergence-free component, which is the curl of something else, and these are formally analogous to electric and magnetic fields. Calling them that way certainly brings to bear all our intuition from electrostatics and magnetostatics into how such modes can look. Is this suspicion correct? Or is there some actual electric or magnetic field involved (as, for example, in TE modes in a waveguide)? Secondly, how exactly does one split the polarization field into these components? It's not quite the sort of 3D vector field to which Helmholtz's theorem applies:   * It is a vector field over a sphere instead of all space. This sphere can be seen as the celestial sphere, or equivalently as the surface of last scattering.   * It does not automatically have a magnitude on top of its direction, though I suspect one can happily use some measure of the degree of polarization for this. (Is that correct? If so, exactly what measure is used?) How exactly is the polarization field defined, over what space, and exactly what mathematical machinery is used to split it into E modes and B modes? Are there analogues to the scalar and vector potential? If so, what do they physically represent?


title: What happens at the moment black hole is created?

text: Suppose a body has just enough matter in it that just 1 more electron could turn it into a blackhole. When this happens, where does all that matter go?


title: Why can't electromagnetic radiation from magnets be used for apparent perpetual motion?

text: If electromagnetic radiation represents a transfer of energy, then does a permanent magnet represent unlimited energy, and if so, why can't magnets be used for perpetual motion? Even if permanent magnets aren't permanent and rather just have a really long usable lifetime, don't they at least represent a large resource of energy?


title: Big Bang, What is wrong with such thinking?

text: Also if I think the Expanding of Universe like this: At the Start : The Amount of Matter and Anti-Matter is equal and they annihilate and all we had is Large Amount of energy. But some 'X' Thing, disturbed this symmetry. So matter start to be in excess so we have some part of universe. The Excess rate of creation of matter continued for many year as resulted in our day universe. And still the rate of Creation of Matter is greater than its anti-matter pair. Thus the Universe is Expanding. Does this thinking seem correct? If not what is wrong?


title: What is the difference between Lehmann-Kallen and Dispersion relation?

text: I know that the Lehmann-Kallen (LK) form of an operator concerns just that, an operator. But the LK is very similar in form to dispersion relations found in analytic S-matrix theory.


title: Diagonalizing Van der Waals Hamiltonian

text: In Kittel's Solid State Physics, he attempts to find the energy exchange due to the van der Waals interaction. He starts by writing the hamiltonian: two oscillators with coordinates $x_1$ and $x_2$ $$H_0 = \frac{1}{2m}p_1^2 + \frac{1}{2}C x_1^2 + \frac{1}{2m}p_2^2 + \frac{1}{2}C x_2^2$$ and an approximate coulomb interaction $$ H_1 \approx -\frac{2e^2 x_1 x_2}{R^3}.$$ He then "diagonalizes by the normal mode transformation" $$x_s = \frac{1}{\sqrt{2}}(x_1 + x_2)$$ $$x_a = \frac{1}{\sqrt{2}}(x_1 - x_2)$$ Intuitively, I understand what this transformation is. You construct a new, orthonormal basis. $x_s$ and $x_a$ oscillate independently of one another. I would like to repeat this but with differing spring constants, $K_1$ and $K_2$. I would like to diagonalize with matrices, but the hamiltonian cannot be represented as such (due to the coupling term $x_1 x_2$). I have tried writing an arbitrary basis $x_1 = c_1 x_a + c_2 x_b$ and $x_2 = c_3 x_a + c_b$ subject to the constraint that the coupling terms cancel: $$k_1 c_1 c_2 + k_2 c_3 c_4 - (c_1 c_4 + c_2 c_3) = 0$$ but this doesn't feel like the right way to do it. How would I diagonalize the above hamiltonian where the spring constants differ?


title: Degree of divergence of a Feynman diagram

text: I am studying the degrees of divergence of Feynman diagrams. I feel that I miss something but I don't really understand what. Please apologize if this question is silly. Anyway. As an introduction to systematics of renormalization, Peskin and Schroeder (chapter 10) introduce the counting of ultraviolet divergences. They say that an expression for any diagram typically looks like $$\sim \int \dfrac{d^4 k_1 d^4 k_2 \dots d^4 k_L}{(\bar{k}_i-m) \dots k^2_j \dots }$$ where $\bar{k}$ is $k$ slash (i.e. $\gamma^{\mu} k_{\mu}$). The number of $d^4 k$ will depend on the number of loops, while the number of $(\bar{k}-m)$ and $k^2$ in the denominator will depend on the number of fermion/boson propagators. Ok. They then say : > _For each loop there is a potentially divergent 4-momentum integral, but > each propagator aids the convergence of this integral by putting one or two > powers of momentum into the denominator._ I don't understand that. To be more clear, I'll develop a little bit more... They then introduce the _superficial degree of divergence_ $D$ as $$ D \equiv (\text{power of } k \text{ in numerator}) - (\text{power of } k \text{ in denominator}) $$ and they say that we can expect a diagram to have a divergence proportional to $\Lambda^D$ ($\Lambda$ is a cut-off) when $D>0$, proportional to $\log(\Lambda)$ when $D=0$ and no divergence when $D<0$. I know that it's a tool and not always exact and so on to evaluate the divergence of a Feynman diagram. That's not the point. What I don't get is why is a diagram with $D<0$ not divergent ? Integrals like $$ \int \dfrac{d^4 k}{k^7} $$ don't converge, so the associated diagram should be divergent too... Or is it because we are only dealing with UV divergences, and that diagrams with $D<0$ diverge because of IR divergences ?


title: Functions of time

text: Following are the functions of time:   $s(t)$ = distance a particle travels from time $0$ to $t$   $v(t)$ = velocity of a particle at time $t$   $a(t)$ = acceleration of a particle at time $t$. If we want to see how the position of a particle changes with respect to time only, then its velocity must remain constant with time. Likewise, if we want to see how velocity varies with time, then the distance between the former position of the particle and the current position should remain constant with time. Similarly, if we want to see how acceleration varies with time, then the difference between the initial velocity $U$ and final velocity $V$ should remain constant with time. Is this what the above functions of time tell us?


title: A Game Of The Number Of Space-Time Dimensions

text: Holger Bech Nielsen, one of the founders of string theory, has apparently just played some sort of game between different potential dimensions for space-time and reached the conclusion that D4 wins in a natural way: http://arxiv.org/abs/1304.6051 Could someone please try to explain the gist of this game and the broader reasoning behind it in laymans terms, and perhaps answer the following related questions: Does anyone pay notice to this developing research? Does the result have implications beyond the Standard Model, e.g. for string theory?


title: Energy Gain with capacitor?

text: I have a question about energy gain in capacitors. Assume the following system: ![illustration](http://i.stack.imgur.com/hdOo6.png) As the electron gets accelerated inside the capacitor, it will have more kinetic energy coming out than going in. But the capacitor has not lost any energy. Where is the energy coming from? **Update:** Sry, too hasty – forgot important infos: All this is (of course ^^) happening in vacuum and the plates are connected by a non-conducting material. So they can only move together. But there is no way for them to exchange (or in any other way gain/loose) electrons.


title: Increasing potential in a voltage doubler

text: When I use a sinusoidal input AC-voltage from -1 Volt to +1 V , the potential between the diode and the capacitor is increasing. But why do I reach a value of 2 Volt over the diode? I deliberately don't post this in the electronics forum. After some time the voltage on the right side of the capacitor is +1 Volt. When the input voltage goes from 0 V to +1 V the potential on the right side of the capacitor also increases, what I can't understand. If I want to increase the potential I need more charges and a current right? ![voltage doubler](http://i.stack.imgur.com/k61AA.png)


title: Neutron conversion to energy?

text: Numerical: If a neutron be entirely converted into energy how much energy would be produced. ($M_n = 1.6743\times 10^{-27}$kg.) Explain


title: Momentum And Energy Problem

text: The information to the question is, "A $59.0~kg$ boy and his $38.0~kg$ sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with a velocity $3.00~m/s$ toward the west. Ignore friction." The specific question I am working on is, "How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system?" Well, I said that $E_{mech,~i}=0$, because no one has kinetic energy (no one is moving), and neither of them have the potential to move, that is, until the girl commences with the push; but, $E_{mech,~f}=PE=KE$, when she uses her potential energy to apply a force over a distance, on her brother, and by doing so she changes the energy of the system, by changing the speed of both of them, meaning the potential energy will convert to kinetic energy. I've tried to plug every piece of data given to me, but I still couldn't find how much $PE $ was converted to mechanical energy. How do I find it? Is my analysis correct; is there anyway to improve it? Edit: I have another question: why can't internal forces of a system cause momentum to not be conserved?


title: Solution of a partial differential heat equation with derivative and boundary conditions

text: I want to solve the following partial different equation. Find $u(x, t)$, satisfying $u_t = u_{xx}$ , $u(x, 0) = x − x^2$ , $u(0, t) = T_0$ , $u_x (1, t) = 0$ and $|u|$ is bounded. Using separation of variables, and eliminating solutions that diverge with time, one gets $$u(x,t)~=~e^{ -\lambda^2t}(A \cos(\lambda x)+B\sin(\lambda x)).$$ How do I proceed after this? The condition $u_x (1, t) = 0$ gives $ A=-B \tan(\lambda )$ which is giving me a contradiction for the condition $u(0, t) = T_0$ , as you get $$-e^{ \lambda^2}B\tan(\lambda)~=~T_0.$$


title: Number of Grassmann generators for Dirac field?

text: How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? _i.e._ The Dirac field is a map to $\Lambda_N$, the space of supernumbers with $N$ real Grassmann generators. What is $N$? This is a follow up question to my previous question Grassmann Paradox Weirdness. I am following Prakash's book Mathemaical Perspectives on Theoretical Physics, where they say a supernumber $z\in\Lambda_N$ can be thought of the extension of complex numbers by the addition of $N$ Grassmann generators $\zeta^1,\, \zeta^2,\,\ldots \zeta^N$. The most general supernumber is written $$z = z_0+z_i\zeta^i+\textstyle\frac{1}{2!}z_{ij}\zeta^i\zeta^j+\ldots,$$ where $z_i$, $z_{ij}$, $\ldots$, are complex-valued and antisymmetric. The odd part of this is anticommuting and is used to describe Fermion fields. The book says that for finite $N$, it takes $2^{N-1}$ complex numbers to specify an anticommuting number. How do I figure out how many Grassmann generators $\zeta^i$ I need to specify a Dirac spinor in 4 dimensions.


title: Could some Red and Blue shifts be the result of light passing through "dark matter"?

text: As i see it, light behaves in certain ways, as the Double Slit experiement shows, So when light comes into contact with dark matter, it becomes both a wave and a particle, the wave is bent around the dark matter, which we see as gravitational lensing, but the particle will go through the dark matter, but will also leave our frame of reference, follow the particle through dark matter and its speed would not change staying at $c$, but only relative to the the space it travels through, it would eventually pass through dark matter, we would see this first as a red or blue shift as it slowly emerges (from our perspective), and continues on it way, Never knowing it had passed through any medium, where as the wave had passed around the dark matter thousands, millions, or billions of years before, depending upon the mass of the dark matter.


title: Are Quantum Physics and statistical theory always the same as semiclassical approximations?

text: Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ? In the form: replace $ \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint dxdpexp(-p^{2}\beta +\beta V(x)) $ and for the Eigenvalue staircase $ \sum_{n} \delta (E-E_{n})\sim \iint dxdp \delta (E-p^{2}-V(x))$ so it is easier to evaluate quantities :)


title: How long must escape velocity be maintained?

text: Escape velocity from Earth's surface is 11.2 Kilometres/second How long would one need to maintain this escape velocity to actually escape Earth's gravitational pull? Must this 11.2 km/s velocity need to be maintained all the way up to geosynchronous orbit?


title: Finding an efficient strategy for walking

text: Let's say you are already walking at a maximally efficient combination of pace and stride (or $\omega$ and $X_0$ I guess) but you need to reach your destination faster. Should you increase/decrease your pace and/or stride to achieve this goal (assuming that any adjustment will result in a net increase in speed)? How can I interpret this question as a question about kinetic energy of oscillators? I have tried solving the equation $KE'=0$ to find a minimum $KE$ but this seems like the wrong approach, for all that seems to tell me is the if you do not walk (ie $w$ or $X_0$ is $0$) then yes, of course you have kept your kinetic energy at a minimum. Where does this idea about "efficiency" come into play? Edit: Pace is the frequency of steps, stride is the length of steps, and average speed is the product of pace and stride.


title: Does a magnet's field become stronger or weaker in a vacuum?

text: This article seems to indicate that a magnet (atleast magnetite, it may not apply to rare earth magnets) weakens when subjected to extremes of pressure. This is purportedly the result of a decrease in electron spacing ... or so I understand from the article. Does then a magnet's influence change in a vacuum? Does the magnet appear to become stronger in a vacuum? Or is the converse true again?


title: How does gravity work?

text: The math, I read, is $F = G\frac{m_{1}m_{2}}{r^{2}}$ What I'm unclear about is how does it work in practice. Say there are two identical pebbles massing 1 kilogram each outside Sol's gravity well close enough to attract each other. What happens then? And, How long does each 'stage' take? Do the pebbles start to orbit around each other and eventually coalesce? Do they, without coalescing, begin to attract other particles around them?


title: Is there a 2D generalization of the coefficient of restitution?

text: The coefficient of restitution characterizes a collision in one dimension by relating the initial and final speeds of the particles involved, $$C_R = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$ In a 2D collision, the velocity can be split into components parallel and perpendicular to the "plane of collision" (the plane tangent to the two objects' surfaces at their contact point), and the equation above applies to the perpendicular components of the velocities only. One could write $$\begin{pmatrix}v_{2f\shortparallel} - v_{1f\shortparallel} \\\ v_{2f\perp} - v_{1f\perp}\end{pmatrix} = \begin{pmatrix}1 & 0 \\\ 0 & -C_R\end{pmatrix}\begin{pmatrix}v_{2i\shortparallel} - v_{1i\shortparallel} \\\ v_{2i\perp} - v_{1i\perp}\end{pmatrix}$$ My question is, is it useful (i.e. does it produce a more accurate description of realistic 2D collisions) to generalize that matrix? Perhaps by allowing the top left element to be unequal to 1, or allowing nonzero off-diagonal elements? Or is there some nonlinear relation that works better? As usual, if you know of any relevant published research, references would be much appreciated.


title: Particle wavefunction and gravity

text: Suppose a particle has 50% probability of being at location $A$, and 50% probability being at location $B$ (see double slit experiment). According to QM the particle is at both $A$ and $B$ at the same time, so is there a force of gravity between the two particle superpositions? Is there _self_ -gravity when a wave-function reaches over a finite distance? I cannot seem to wrap my head around this. Is the gravity a proportional fraction of the entire mass based on the probabilities. How do you combine a wavefunction with Gauss' law of gravity? I have being trying to think about self-gravity for a long time now.


Where does the wave function of the universe live? Please describe its home. I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.) Or maybe it is the Fock space of the universe, or some still bigger, yet more complicated stringy thingy. I will leave it to you whether you want to describe the observable universe, the total universe, or even the multiverse. Please give a reasonably accurate and succinct mathematical description, including at least dimensionality. Thank you.


title: Which ferromagnetic material has the lowest Curie temperature?

text: It is hard to search for materials by their properties in general and I am trying to find a material with a very low Curie temperature. At the moment I am browsing different sites but can only find a number of 'normal' ferromagnetic materials but few with a small Curie temperature. So which material has the lowest Curie of all ferromagnets? (By material I mean pure substance, e.g. NdFeB, but not a material under pressure, or nanocrystals or other special conditions which should all allow the suppression of ferromagnetism).


title: Why a glass rod when rubbed with silk cloth aquire positive charge and not negative charge?

text: I have read many times in the topic of induction that a glass rod when rubbed against a silk cloth acquires a positive charge. Why does it acquire positive charge only, why not negative charge? It is also said that glass rod attracts the small uncharged paper pieces when it is becomes positively charged..ok i understand that positively charged glass rod attracts the uncharged pieces of paper because some of the electrons present in the paper accumulate at the end near the rod,but can't we extend the same argument on attraction of negatively charged silk rod and the pieces of paper due to accumulation of positive charge near the end.


title: How to find that a molecule has zero spin?

text: I read that Ne has $S=0$. How can this be found, knowing the electron configuration? Electrons, protons and neutrons all have 1/2 spin. The Pauli exclusion principle implies that the even number of electrons are half up-spin and half down-spin, so that they sum to 0. Is that argument valid, and if so, can the same be applied to show that the total spin of neutrons and protons, which occupy the same space, is 0? I suppose that would imply that $S=0$ for all molecules with even numbers of electrons and protons+ńeutrons.


title: precise definition of "moduli space"

text: I'm curious what the precise definition of the moduli space of a QFT is. One often talks about the classical moduli space, which then can get quantum corrections. Does this mean the quantum moduli space is something like the set of minima of the effective potential in the 1PI action (or does the 1PI action already presume a choice of vacuum?)? Is there a non-perturbative definition, without referring to field configurations or effective actions (eg, how do we define it in string theory, where there are no fields)? Maybe it can be identified with some submanifold of the (projective) Hilbert space? Can the metric or other structure be defined in such an invariant way?


title: Canonical averages in a Fermi gas aka generalized Fermi-Dirac distribution

text: I am in the process of applying Beenakker's tunneling master equation theory of quantum dots (with some generalizations) to some problems of non-adiabatic charge pumping. As a part of this work I encounter thermal averages of single- particle quantities with a fixed total number of electrons. They are pretty straightforward to derive, as recently discussed on Physics.SE, see Combinatorial sum in a problem with a Fermi gas. I have trouble finding references to prior work for this basic quantum statistics problem. Can you suggest some relevant references in this context? **Progress report:** Writing this stuff up, I realized that I'm basically asking for finite electron number and finite level spacing generalization of the Fermi function: $\langle \nu_k \rangle = Z_n^{-1} \sum_{n=0}^{n-1} (-1)^{n-m} Z_m e^{-\beta \epsilon_k( n-m)}$ where $Z_n$ is the $n$-electron canonical partition function. For either $n \gg 1$ ~~or~~ and $\epsilon_{k+1}-\epsilon_k \ll \beta^{-1}$ this reduces to the standard Fermi-Dirac distribution: $\langle \nu_k \rangle= \frac{1}{1+z_0 e^{-\beta \epsilon_k}}$ The advantage of the new formula is that it only depends on $k$ via $e^{-\beta \epsilon_k}$ with all the combinatorics hidden in $Z_n$. This is unlike the textbook expression summarized in Wikipedia, or Beenakker's implicit function $F(E_p |n)$ (which is $\langle \nu_p \rangle$ in notation). The question still stands: I don't believe the first formula above is new, what is the right reference?


title: Krauss operators for random unitary

text: Suppose I have a density matrix $\rho$ and I act on it with a unitary matrix that is chosen randomly, and with even probability, from $S = \\{ H_1, H_2 \ldots H_N \\}$. I want to write the operation on the density matrix in Krauss form: $ \rho^{\prime} = \sum_i O_i \rho O^{\dagger}_i $ Since the operator is chosen evenly, the probability of choosing any $H_i$ is $\frac{1}{N}$. What would be my choices for $O_i$?


title: Boundary conditions / uniqueness of the propagators / Green's functions

text: My question(s) concern the interpretation and uniqueness of the propagators / Green's functions for both classical and quantum fields. It is well known that the Green's function for the Laplace equation $$ \Delta_x G(x,x') = \delta^{(3)}(x-x') $$ with _boundary conditions_ $$ G(x,x') = 0 \text{ for } x\in\partial D, x'\in D$$ is determined _uniquely_. This is because any harmonic function that vanishes on the boundary of domain must vanish inside as well. In contrast, I am confused about the uniqueness of the Green's function for the wave equation $$ (∂_t^2 - \Delta)G(x,t;x',t') = \delta^{(3)}(x-x')\delta(t-t') $$ Without specifying boundary conditions, there exist many homogenuous solutions to the wave equation and thus many different Green's functions. Physicists usually offer me several canonical choices, like the _retarded_ , the _advanced_ and also the _Feynman propagator_ , but I don't understand what makes each choice unique compared to the others, or which boundary conditions correspond to which choice. Hence, my question is > Which boundary conditions correspond to retarded, the advanced and the > Feynman propagator? What other possibilities for boundary conditions and > propagators are there? I am also confused by the situation in quantum field theory, where we have various conventions for propagators and time-ordering like $$ \langle0|T_t\lbrace a(x,t)a^\dagger(x',t')\rbrace |0\rangle $$ Apparently, the ground state is very important for selecting the right Green's function (see also my previous question), but I still don't understand why that is. > How does the vacuum state act as a boundary condition in the spirit of the > previous question? There is also the issue of imaginary time. The point is that imaginary time turns the wave equation $(\partial_t^2 -\Delta)\phi=0$ into the Laplacian $(\partial_\tau^2 + \Delta)\phi = 0$, but I don't understand how the usual analytic continuation $\tau \to it \pm i\eta$ for various propagators depends on the boundary conditions. For instance, Tsvelik writes in equation (22.4) that the imaginary time Green's function for the Laplacian in 2D is "obviously" the one that vanishes when $\tau$ and $x$ go to infinity, but I don't understand the reasoning behind this choice. > What are the right boundary conditions for Green's functions in imaginary > time? Is it clear that the path integral formalism selects the right ones?


title: Length of publication cycle for peer-reviewed journals

text: Is there any data of the expected length of publication cycle with respect to different peer-reviewed journals in theoretical physics? I know that the length depends on many things, including the referees, the author and the submitted paper itself. However, clearly some journals have it shorter or longer. While a long response time is not necessary a disadvantage (the referees may actually care to understand paper thoroughly (as it happens commonly in mathematics), while the preprint is on the arXiv anyway), it may influence decision of sending the paper to one journal or another (along with the Impact Factor, the Eigenfactor, etc). Of course always it is possible to perform a _reconnaissance in force_. However, with little or no prior experience with a particular journal (1 data point is not enough to estimate the variance) it could be difficult to estimate if it is going to take a few month or 2 years (with all its consequences). So is there any list of times (mean, variance, median... or the full data) for any of the steps in the following sequence $$\text{submitted} \rightarrow \text{reviews}\ (\rightarrow \text{response}) \rightarrow \text{accepted} \rightarrow \text{available}$$ for different peer-reviewed journals? EDIT: To clarify: I am not interested in mean(/variance of) acceptance time _averaging all_ journals in TP. Rather, I am asking for a list of _time vs journal_.


title: How to calculate the total angular momentum (J) values

text: How can I calculate the values of $J$ (total angular momentum) for a particular term, for instance, ${}^3P$?


title: What causes multiple colored patches on a wet road?

text: I was going to school (after a rainy hour) when I saw some patches of shiny colours lying on road. Some small children surrounded that area and thought that it's a rainbow falling on the Earth. (For some 5-6 year old children it is a serious thing.) I have a similar photo from Google:   ![enter image description here](http://i.stack.imgur.com/rKa9J.jpg) It is definitely not a rainbow falling on the Earth but what is it? The road is a busy road and hence I think that's because of some pollutants, oil, or grease.


title: The self-coupling constant for the Higgs mechanism

text: I have a few questions about the parameter $\lambda$ in the Higgs-Lagrangian density $$\mathcal{L}_H=(D_\mu \phi)^\dagger D_\mu \phi + \mu^2\phi^\dagger\phi-\lambda(\phi^\dagger\phi)^2$$   1. Is it possible to measure it _directly_?   2. Does it absolute value have a meaning? It relates to the self-interaction of the Higgs boson (?), but can one compare this number with something else?    3. Before the mass of the Higgs was measured, did one expect a precise value for $\lambda$?


title: where does the energy of water droplet go?

text: My experiment:   I took a string (3 mm diameter and wet) and a water source (1 drops per 2 seconds). Then I attached the string to the water source. The string was then tied to ground Observation : I saw some small droplets coming down. Question : where is the energy of droplet getting dissipated ? Reason for questioning : the droplet moves slowly so it means its loosing some energy somewhere it can be fluid friction but that friction is too small then what is making water droplet go too slow ![enter image description here](http://i.stack.imgur.com/SMU1H.jpg)   ![enter image description here](http://i.stack.imgur.com/AnmjK.png)


title: Modeling a bar magnet's magnetic flux inside a coil

text: A bar magnet and a coil arranged alongside their common axis is a common setting in teaching. But how does the magnetic flux H(x) (where x is the distance of the bar magnet's center relative to the coil's center) look like? Is there a good approximation? I guess it should look like a Gaussian function, but how close does that resemble reality?


title: Virtual images in (plane) mirrors?

text: The following image is taken from teaching physics lecture Was man aus virtuellen Bildern lernen kann (in German): ![enter image description here](http://i.stack.imgur.com/Dzq9r.jpg) Now the cited paper claims that the left hand side is the correct picture to explain the virtual image if you look with your eyes, the right hand side is correct if you use a camera (but incorrect if the camera is replaced with the eye). In the case of a plane mirror both constructions lead to the same result. However in the case of curved mirrors they lead to different results (which is also discussed in the cited paper). Now I am a bit confused since the right picture (with an eye instead of the camera) is that one typically drawn in physics textbooks (for example Pedrotti "Introduction to optics" p.38 or Halliday Resnick "Fundamentals of Physics" p.1140). So my question is, which picture is the correct one? Do you know further references of papers or books which discuss this problem more in detail? (I have only found this cited paper in German).


title: A video conference between earth and a space shuttle

text: I have just started looking into special relativity and I have come up with an intriguing _gedanke_ , as Einstein himself called such theoretical thought experiments. Imagine a space shuttle traveling through space at a constant velocity close to $c$. As the shuttle passes earth, a previously set-up camera starts broadcasting from earth to the shuttle. Since radio waves travel at the speed of light, the shuttle is receiving a constant transmission feed, assuming the camera is broadcasting 24/7. Now, from what I have understood of special relativity so far, time will flow slower for the astronaut than for the earthlings. Hence, assuming $v=0.8c$, the astronaut will after 30 years have received a video transmission 50 years long! Is my reasoning correct, that even though the transmission is live, the astronaut would actually be watching things that happened many years ago, while still receiving the "live" feed, which would be stored/buffered in the shuttles memory, thus making it possible for the astronaut to fast-forward the clip to see what happened more than 30 years after passing the earth? My second question is, what happens when we consider the space shuttle to be at rest and the earth to be moving instead? If that would imply that it has been 50 years from the astronauts point of view, while only 30 years have passed on earth, then the astronaut would run out of video material after the first 30 years of watching the broadcast. Then what? I hope it makes sense. Thank you!


title: Deriving the Lorentz Transformation

text: I have been trying to understand a more or less geometric derivation of the Lorentz transformation, and I'm getting stuck at one spot. The wikipedia article for the Lorentz transformation for frames in standard configuration lists the following equations: $x^{\prime} = \frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}$ $y^{\prime} = y$ $z^{\prime} = z$ $t^{\prime} = \frac{t-(v/c^2)x}{\sqrt{1-\frac{v^2}{c^2}}}$ I've been able to work everything out except for $-(v/c^2)x$ in the $t^{\prime}$ equation. I haven't seen any explanations for this, which makes me feel like I'm missing something simple. Where does this part of the equation come from? Shouldn't $t^{\prime} = \gamma \cdot t$? **EDIT:** Ok, so I reviewed the idea I was using to derive the Lorentz factor and thus the transformation for $t^{\prime}$. Suppose you have the two frames I've described, and you have a light wave moving perpendicular to the X axis in the second ($\prime$) frame. Light Path Diagram Using basic trig with the diagram, you can derive: $t^{\prime}=t\cdot\sqrt{1 - \frac{v^2}{c^2}}$ Obviously this would contradict the transformation provided by wikipedia. What step am I missing here? I don't really want a proof that I'm wrong or that the equation I've derived is incorrect - I'm already pretty convinced of that. What I would really like is an intuitive explanation as to why mine is invalid and how I would go about deriving the correct equation through similar means.


title: How to measure the spin of a neutral particle?

text: If a charged particle with charge $q$ and mass $m$ has spin $s \neq 0$ we can measure an intrinsic magnetic moment $\mu = g \frac{q}{2m}\hbar \sqrt{s(s+1)}$. This is how spin was discovered in the first place in the Stern-Gerlach Experiment. But for a neutral particle $\mu = 0$, so we cannot measure the spin of the particle in the same manner. But it is said, that e.g. the Neutron or the Neutrino both have a spin $s=1/2$. How was or can this be measured?


title: Simple formula for liquid heat transfer

text: I'm trying to do a simple simulation of a solar panel coupled through some piping to a boiler, with the aid of a pump. My input constants could be something like: volume of liquid inside the panel, volume of liquid inside the boiler, pump transfer rate. For simplicity I can assume that connective piping does not absorb any heat, so panel and boiler are coupled next to each other through some hole and thermally insulated from another.   Possible constants: ambient temperature, heat loss (depends on ambient temperature). The variables: panel temperature, boiler temperature, whether the pump is running or not. What I'm doing right now is like linear estimation, applying some energy to the panel (raises the heat linearly), running the pump transfers linearly energy from one vessel to another (temperature increase/decrease is inversely proportional to volume of liquid), heat transfer to boiler raises its energy linearly. I would like to have some thing like: estimating the rise and fall of temperature in both vessels given the above constants and variables. It doesn't have to be scientifically exact, but I know that linear aproximation is wrong as the temperature increase is somehow inversely proportional to absolute temperature (i.e. it 'tapers').   The end result will be like: apply xxx units (Joules) of energy to solar panel for 1 minute > temperature rises by y degrees; start pump (open valve) for 1 minute -> temperature in boiler goes up by z degrees, temperature in panel goes down by w degrees.   Any graphs or formulas will be helpful, I can continue from there. This is not a homework or scientific study, just something for personal use.   Also some explanations about differences between water and oil energy absorbtion would be nice.


title: In Orbital Mechanics what is the quantity described below called?

text: I seem to recall that $r^2 \dot{\theta}$ is a conserved quantity in orbital mechanics, which I just proved using the Euler-Lagrange equations. Namely via: $ \mathcal{L} = \frac{m}{2} (\dot{r}^2+r^2 \dot{\theta}^2)+\frac{GMm}{r}$ $\frac{\partial{\mathcal{L}}}{\partial \theta}=\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial \dot{\theta}}$ $\therefore 0=\frac{d}{dt} (r^2 \dot{\theta}) $ $\implies r^2 \dot{\theta}=C$ My question is, what is this quantity called as I can't seem to recall its name, but I think it does have a special name.


title: Migration of an ion in a non-conductive medium

text: Someone asked me why, when performing DNA electrophoresis we need to put the gel inside a conducting buffer. Couldn't we just run it in distilled water? My answer was that if we had distilled water we would not be able to close the circuit between the anode and the cathode of the electrophoresis apparatus, no current (or very little) would flow and no movement would be possible. The next question then was: why do we need a current to pass through the system in order to have movement of the DNA (or, in general, any charged particle) towards the anode? What physical forces enter into play? In other words: if we had a bath of distilled water, we put two electrodes connected to a generator at each end and put a charged molecule in the middle of the bath, would it be able to migrate? Why? Which forces would be involved? I found this document which essentially confirms my first answer: > **Electrophoresis Buffer:** Several different buffers have been recommended > for electrophoresis of DNA. The most commonly used for duplex DNA are TAE > (Tris-acetate-EDTA) and TBE (Tris-borate-EDTA). DNA fragments will migrate > at somewhat different rates in these two buffers due to differences in ionic > strength. Buffers not only establish a pH, but provide ions to support > conductivity. If you mistakenly use water instead of buffer, there will be > essentially no migration of DNA in the gel! Conversely, if you use > concentrated buffer (e.g. a 10X stock solution), enough heat may be > generated in the gel to melt it. However, I have not been able to find a proper explanation of the physical forces involved in the process. PS: of course in a real DNA electrophoresis the charged molecule is trapped in a gel, which itself has been made using a conducting buffer, so the question becomes probably a tad more tricky...


title: Intuition behind mass corrections to massless fermions

text: I'm trying to understand the intuition behind the mass correction to massless fermions. To be concrete lets consider a theory with a massless Weyl fermion ($\psi $), as well as two massive particles, a complex scalar ($\phi $) and another Weyl fermion ($ \psi ' $), \begin{equation} {\cal L} = {\cal L} _{ kin} - \frac{1}{2} m ^2 \left| \phi \right| ^2 - M \left(\psi '\psi ' + h.c. \right) - \frac{ \lambda }{ 4!} \left| \phi \right| ^4 - g \phi \psi \psi ' + h.c. \end{equation} This Lagrangian would be valid if for example we impose a $ U(1) $ symmetry such that, \begin{align} & \psi \rightarrow e ^{ i \alpha } \psi \\\ & \phi \rightarrow e ^{ - i\alpha } \psi \\\ & \psi ' \rightarrow \psi ' \end{align} Now consider the lowest order mass correction the for massless fermion, $\hspace{4cm}$![enter image description here](http://i.stack.imgur.com/sOhQI.png) The mass correction can be calculated by setting external momenta to zero: \begin{align} i {\cal M} & \sim g ^2 \int \frac{ d^4 \ell }{ (2\pi)^4 } \frac{ 1 }{ \ell ^2 - m ^2 } \frac{ M }{ \ell ^2 - M ^2 } \\\ & \sim \frac{g ^2}{16 \pi^2}M \left( \frac{1}{ \epsilon } + \log \left( \frac{ \mu ^2 }{ \Delta ( m , M ) } \right) \right) \end{align} where $ \Delta $ is a function of $m$ and $ M $. In $\overline{MS}$ we toss the infinity and the first order correction is \begin{equation} m _{ \psi } \sim g ^2 M \log \frac{ \mu ^2 }{ \Delta } \end{equation} This has three interesting features which I am trying to understand:   1. It seems that the fermion doesn't decouple from the theory in the limit that its mass goes to infinity. Why are we not justified in integrating it out in this case?   2. The calculation is very insensitive to the boson mass. Is this an accident or has a deeper explanation?   3. The calculation has large logs unless $ \mu \sim m, M $. It seems if you renormalize at this scale the log goes away and the physical mass goes to zero! So if I am understanding correctly the particle is massless when doing experiments at the scale of the other particles in the theory (at least at lowest order in perturbation theory) but heavy when its much below this scale. This seems very strange!


title: what is the 'resolution' of the reality in pixel terms

text: 1.What resolution should a TV screen have so that its image were so faithful as reality as if the TV were a window ? 2.Also what would happen if Physics could reproduce a 'pixel' of the size $ l_{p}^{2} $ the square of the Planck length ? Would be then this the resolution of our reality like watching through a real window ?


title: Doesn't Warp theory violate causality?

text: I have heard many physicists (ex:- Michio Kaku) saying "Warp speed" from Star Trek doesn't violate any known physical laws. But doesn't it violate causality? Say, we make warp drive possible and travel towards Alpha Centauri (4.22 light years away) in warp speed and reach there, say, within a year (+/- few months) and blow it up. Now using Special Relativity, we could devise a frame of reference wherein the observer would see the Star blow up before we ever left planet Earth. Wouldn't that violate causality and make warp speeds unattainable?


title: Heat equation : boundary conditions?

text: Say we have a bar centered at $x=0$, that is heated. We have the 1D Heat equation which we can solve to find a parabolic temperature profile : $$\kappa \frac{d^2 T}{d x^2}=-\frac{Q}{L S}$$ with $$T(L/2)=T(-L/2)=T_{\infty}$$ $T_{\infty}$ is the ambient temperature at the end of the bar. This boundary condition alone is enough to solve the equation. However, now say we want to add a very tiny rod at the center of the bar that is connected to the environment to act as a path for heat flow. $R$ is the thermal resistance of the rod. $S$ is the cross sectional area of the bar. $\kappa$ is thermal conductivity of the system. $Q$ is the heat dissipated by joule heating. The corresponding boundary condition is: $$\frac{T(0)-T_{\infty}}{R_{\rm thermal \ rod}}=\kappa S \left(\frac{d T}{dx}_{x=0^+}-\frac{d T}{d x}_{x=0^-} \right)$$ ![enter image description here](http://i.stack.imgur.com/BHOqJ.png) I have fiddled around with this math for a while but cannot figure out how to implement it. In particular I am following this paper on the T-Bridge Method for thermal conductivity.


title: What makes matter travel along geodesics?

text: The relativistic explanation of gravity is geometric, the motion of a body in a field of space-time distortion can be described as being at rest and travelling along a geodesic of that field, but why does it travel along that geodesic? What causes it to move?


title: Eikonal approximation in QFT

text: Does the eikonal approximation for calculating a scattering amplitude in QFT provide the exact result in the limit of $s\rightarrow\infty$ at finite $t=0$ ($s$ and $t$ are the usual Mandelstam variables)? If so, does it match the answer obtained in the Born approximation in the same limit? See e.g. Eq. (15-16) of http://arxiv.org/abs/hep-ph/0112161 for an explicit expression of the eikonal approximation.


title: spontaneous disintegration of an unstable particle

text: Suppose one wants to describe an unstable particle that spontaneously disintegrates with a life time say "tau". In that case the total probability of finding the particle is not constant. But should decrease at some rate, say exponential rate. Now my question is that the textbook asks me calculate the rate of change of probability by inserting a complex potential in the schrodinger equation, I did that and indeed found that the probability itself decays with an exponential rate. I would like to know why does one get this by introducing a complex potential? And instead of spontaneous disintegration of the particle if I had decay, lets say that the particle decays exponentially. Then I know that the probability of finding a particle in some region changes as the particle decays, but how do I calculate the rate of change of probability, because now I have the fact that mass is a function of time, how does this fit into the schrodinger equation?


title: Deflection/diffraction of static electric field

text: I know the fenomenon of diffraction (one slit) in case of electromagnetic waves. http://en.wikipedia.org/wiki/Diffraction What happens if there is only a **static** electric field? Let' see a figure:                       C               ---------- B -------------                     /    \             /      \            A        A'      There is a point-like charge at the point C. The line is a wall that blocks the electric field. At the point B there is a hole in the wall. A and A' are test charges to measure the force, the field direction. What we will measure? Possible answers: 1: The picture is correct, at the point A the force directs to the hole, the point B. The test charge A will feel a force which is as big as it would come from the distance AB+BC. 2: This experiment is not possible, there is no way to block the static electric field by a wall. 3: The electric field "flows through" the hole, but the force on the test charge A will still direct to the point C. The measured force will as big as it would come from a distance AB+BC. 4: The electric field "flows through" the hole, but the force on the test charge A will still direct to the point C. The measured force will as big as it would come from a distance AC, just like as it would be without the wall. 5: There is no flowing or deflection or diffraction. The point A will feel no force at all. The test charge at A' have straight view to the C, so will feel the force, as it is on the picture, as big as it would feel without the wall. 6: There is no flowing or deflection or diffraction. The point A will feel no force at all. The test charge at A' have straight view to the C, so will feel the force, but a weaker one than it would be without the wall. 7: Other possibilities? (What I assume is that if the hole size approaches zero, the hole will behave like a point-like charge.) What do you think, which is the correct answer? Is it possible to solve this (or a simplest case similar to this) analytically and find a mathematical formula to count exactly the direction of the force, the diffraction of a **static** electric field? Thank you!


title: Superconducting magnet

text: This is a somewhat more detailed question related to this one. The problem I want to solve is problem 1 here. What I tried: a) From $V=-L\frac{\mathrm{d} i}{\mathrm{d} t}$, we can integrate and obtain t, right? b) Here start my doubts: in a normal coil, the energy inside the coil is given by $U=\frac{1}{2}LI^{2}$, right? Is it the same for a superconducting coil? In a superconductor, the magnetic field inside the superconductor is $B=0$, am I right? Does this answer the question? c) The energy released when the superconductor goes to normal state is $\Delta U=\frac{1}{2}L(I^{2}_{super} - I^{2}_{normal})$, right? And this is the energy that causes the helium to evaporate, right? Could anybody please tell me if I am on the right direction? Or am I missed something? Thank you in advance,


title: Rigorous definition of frame of reference

text: I'm looking for a mathematical definition of frame of reference. Most of the textbooks I have seen take it for granted and they just refer to some set of spacetime coordinates. A more mathematical definition I found defines spacetime $E$ as an affine space over $\mathbb{R}^4$ and then defines a frame of reference as a 5-tuple $(P,v_1,v_2,v_3,v_4)$ where $P\in E$ and $(v_i)$ is a basis of $\mathbb{R}^4$. But with this definition I cannot make sense of the phrase "a frame moving with respect to another".


title: Surface charge density relation with the radius of curvature at the surface of a conductor

text: In a text book it was given that the surface charge density of a conductor at a particular region on its surface is inversely proportional to the radius of curvature at that region. I didn't understand it. I want proof for that can anyone please post the proof.


title: What is the cause of Special Relativity

text: Why does special relativity exist? Via studying the outcome(s) of such relativity, one can end up with all the equations involved in special relativity. Thus one runs into things such as contraction, dilation, transformations, velocity addition, etc. But these are the outcomes. My question is what is the cause of special relativity, and is the knowledge of this cause currently being shared in the world of physics? Or, to put it another way, is it currently accepted despite there being no cause that is known of, thus an acceptance of an effect without a cause? If you give this question a negative vote, please present your reasoning and or evidence to support such a vote.


title: What are the statuses of Silicene and Graphene for real world circuit production?

text: A lot of hype is out there about both of them (especially the latter) and I was wondering if there is more concrete information about them other than the news IBM posted on a circuit 2 years ago and the patent war about Graphene transistors involving Samsung and IBM among others. Also, what's up with Silicene? Is it of any importance or is the hype there only because it is based on Si?


title: Spinor integration

text: I am learning on-shell methods for one loop integrals from this paper: Loop amplitudes in gauge theory: modern analytic approaches by Britto. Starting with formula (18) spinor integration is explained. One first writes the loop momentum as $$(l)_{a\dot{a}}=t\lambda_a\tilde{\lambda}_\dot{a},$$ where $t\in \mathbb{R}^+$ and $\lambda$, $\tilde{\lambda}$ are spinors with $\tilde{\lambda}=\bar{\lambda}$. Why can I write $l$ in this way? I know the decomposition $p=\lambda\tilde{\lambda}$ for momenta with $p^2=0$, but the loop momentum is not lightlike. I guess the Dirac-Delta in the integrand puts it on shell, but then where does the $t$ come from? Next the integral over the loop momentum is turned into an integral over the spinors: $$\int{d^4l\delta^{(+)}(l^2)f(l)}=-\int_0^\infty{\frac{t}{4}dt\int_{\tilde{\lambda}=\bar{\lambda}}{\langle\lambda \; d\lambda\rangle} [\tilde{\lambda} \; d\tilde{\lambda}] f(t,\lambda,\tilde{\lambda})}. $$ Can someone give the derivation of this formula or give me a reference where this is done? Britto refers to the paper MHV Vertices And Tree Amplitudes In Gauge Theory by Cachazo, Svrcek and Witten. But there I can't find an explanation that I understand.


title: Improvements on a makeshift dessicator

text: I am a first year undergrad working in a lab, and I have these Plexiglass containers that I tried to convert to dessicators, since the commercial bell jars are really expensive. I used these weather strips from a hardwarestore that are suppose to be moisture tight, and sealed the connections with silicone. The craftmanship is kind of poor, but I am pretty sure there are no leaks in the silicone seal. Then I used some vacuum grease between the strip and the glass. I have these containers connected and have N2 flowing through. However, even though there is gas flowing through, moisture indicators still showed (after a few of days) that the last couple boxes are not sealing properly. I was wondering if you guys have any ideas on improving the seals, or improving the system in general to help prolonging the period which the containers can seal properly. Couldn't post any images since I'm new here =(


title: Transition between 2D and 3D quantum systems

text: Quantum Hall effect and anyonic particles are examples that occur in a two- dimensional system. However, experiments for such systems can only be realized in a pseudo-2D environment, where the third spatial dimension is much smaller than the other two dimensions. How do we expect the results from such experiments to differ from a true 2D system? In particular, how/when does an anyon begin or cease to exist when we transit between a 2D and 3D system?


title: What does $\left|x,t\right>$ actually mean (Heisenberg picture)?

text: I am pretty much confused with this notation I believe. The Heisenberg states are denoted by $\left|x,t\right>$ and the Schrodinger states are given by $\left|x(t)\right>$. It seems like both of these are parameterised with time, but the Schrodinger state is parameterised through the position vector, But then in the case of Heisenberg picture, is the state parameterised using two variables $x,t$ ? What is this Linear Vector space ? Secondly, in the case of position operator in the Heisenberg picture $\hat X_H(t)$ the eigenvalue equation is given like this, $$ \hat X_H(t)\left|x,t\right> = x\left|x,t\right> $$ So does that mean that for every time $t$ of $\hat X_H(t)$ there is a eigenvalue $x$ with the eigenvector $\left|x,t\right>$. So does that mean $$ \hat X_H(t')\left|x,t\right> = 0 $$


title: Thermodynamics of a bar, specific heat

text: I have a problem and I don't understand how the book solves it. It says I have a bar such that $$F=aT^2(L-L_0)$$ where $F$ is the force or tension of the bar, $T$ is the temperature, and $L$ the longitude. $L_0$ is the longitude when $F=0$. It says that $C_{L_0}=bT$ and it asks me about the general function $C_L(L,T)$, being $C_L$ the specific heat at constant $L$. The book starts solving it with this expression: $$\left(\frac{\partial C_L}{\partial L} \right)_T=-T\left(\frac{\partial^2 F}{\partial T^2}\right)_L$$ From there, I know how to go on, but I have no idea where that comes from. I have tried to get that out of the generalised Mayer relation but if that's the way, I can't do it.


title: Why can't microwaves use molecules' dipole moments to cool?

text: My understanding is that microwaves use the dipole moments of water molecules to heat things. The microwaves resonate with the dipole moment, and add energy to the system. How exactly does the microwave-molecule interaction work, and why can't that same resonance be used to _reduce_ molecular movement? For analogy: If there was a bell with a resonant frequency, you could add energy to the bell (and ring it) by hitting it with sound at that frequency. If the bell was already ringing, you could slow it down by hitting it with the same frequency, but 90-degrees phase shifted. Why can't the same be done with water molecules?


title: Does the wave function of a particle completely describe the state of the particle?

text: In classical mechanics, if you know the position and momentum of a particle at time $t$ and the Hamiltonian, you can predict the particle's position and momentum at any time. In quantum mechanics, if you know the wave function of a particle at time $t$ and the Hamiltonian, can you predict the wave function at any time? Or do you also need to know $d\Psi/dt$ at time $t$? My understanding is that, from the Schrödinger equation, you only need to know $\Psi$ at time $t$ and $H$ to predict $\Psi$ at any time. That is, you don't need to know $d\Psi/dt$, because it can be obtained from calculation.


title: Has a human ever perished in space?

text: Apollo 13 returned safely. The Challenger was leaving when it exploded. The Columbia was coming back when it burned up, as was that Russian guy who was profiled on National Public Radio (NPR) and that recent book. Has any human ever died in space?


title: Mass of Milky Way Supermassive Black Hole

text: I am looking at data from Andrea Ghez (2005; ADS link) which states the mass of the supermassive black hole in the MW. Here's what I don't understand: She reports the value as $3.7 \times 10^6 [R_0/(8~\mathrm{kpc})]^3$ solar masses. The part I'm not clear on is the radius/8 kpc piece. Everything else I read (except for the journal article) leaves out the radius/8 kpc stuff and just reports it as $3.7 \times 10^6$ solar masses. (I know that other values have since been determined). What is that radius/8 kpc all about?


title: Where are exactly the charges of charged capacitor?

text: If a capacitor is connected to a battery and is charged, are there charges inside the wires or do they just accumulate on the surfaces of capacitor?


title: What if our Sun were located in the middle of a globular cluster?

text: Say you took our current solar system and relocated it deep in the heart of a globular cluster such as Omega Centauri. What would the night sky look like? Would the starshine of nearby stars be enough to turn the sky blue or cast shadows or are they still too far away to do that? What would the day sky look like? How would nearby starts affect the Earth's ecosystem? Is it even possible for a solar system such as ours to exist in a stable form in such surroundings?


title: Forces on a particle moving in a vertical circle

text: ![enter image description here](http://i.stack.imgur.com/pdFyz.jpg) In the diagrma, a particle (A, mass 0.6kg) is moving in a vertical circle. The question is: When it gets to the lowest point, what is the tension in the light rod that is between the center of the circle (B) and the mass A? At that point the particle is not moving vertically, so why doesn't the tension = the weight (0.6g)? The tension in the rod is providing the centripetal force so it's not like the tension and the centripetal force have to equal 0.6g?


title: Why can rolled up capacitors be modelled as flat parallel plates?

text: If you have two parallel plate of opposite charge with a dielectric in between, then Capacitance, $ C = \frac{A \epsilon_{0}}{d} K$, where $ K $ is the dielectric constant. This is used to calculate the capacitance of a capacitor. However, a real capacitor is actually two of those plates rolled up many times. Wouldn't the electric field $ \vec{E} $ be different? So why is the equation for the capacitance still valid?


title: What jobs can you get after Studying Physics at University?

text: What types of jobs can you get after a degree in Physics? My sister is choosing her course and thinking of doing physics but isn't sure what she can do after it.


title: What does it mean to expand a Hamiltonian using perturbation theory?

text: On UC Davis chemwiki website, the Hamiltonian for quadrupolar coupling in NMR is analyzed. (The details of this aren't important.) It is said in the analysis that: > _The expansion of the Hamiltonian is done usingperturbation theory._ It then proceeds to "expand the Hamiltonian" according to first and second order terms. I'm an undergraduate reading up on quantum physics for a summer research position, and I've spent the last day or so immersing myself in perturbation theory. It's my understanding that the basic premise of the theory is that there are two (and only two) parts to the Hamiltonian: $H^0$, the unperturbed Hamiltonian, and $H^1$, the perturbing one. What does it mean to expand the Hamiltonian itself? * * * EDIT: It appears the premise of this question is mistaken. As far as I can tell, perturbation theory itself **doesn't** result in first- and second-order Hamiltonians. Instead, NMR researchers have just happened to call a pair of Hamiltonians "first-order" and "second-order". Since both of these are perturbing Hamiltonians, I got confused at the overlap in terminology. Thanks for helping me figure this out!


title: Justification of discrete spectrum for V(x) unbounded at $\pm \infty$ in Pauling and Wilson

text: In Pauling and Wilson, Introduction to Quantum Mechanics, they offer the following intuitive reason for the discrete spectrum of a potential which is unbounded at $\pm \infty$: ![enter image description here](http://i.imgur.com/PpW1esP.jpg) ![enter image description here](http://i.stack.imgur.com/ziydP.jpg) ![enter image description here](http://i.stack.imgur.com/yCf7A.jpg) This is interesting but I'm not sure I "buy" it in it's current form. **Is there a way to make this argument mathematically precise?**


title: Why is stringless supergravity not considered by many to be a candidate theory of quantum gravity?

text: This paper seems to show that $d=4, N=8$ supergravity is finite. Yet the paper only has three citations in spires, and I certainly haven't heard talk of a new candidate theory of gravity. Why isn't perturbative supergravity with some supersymmetry breaking principle, coupled with the standard model considered a possible theory of the universe? Has someone checked the coupling to matter? Is that the problem?


title: Dimension analysis in Derrick theorem

text: The following image is taken from p. 85 in the textbook _Topological Solitons_ by N. Manton and P.M. Sutcliffe: ![enter image description here](http://i.stack.imgur.com/QVGeY.png) ![enter image description here](http://i.stack.imgur.com/EP93z.png) > What I don't understand from the above statement: > >   * why $e(\mu)$ has minimum for $d=2,3$, whereas when $d=4$, $e(\mu)$ is > scale independent and stationary points and vacuum solutions are possible? >   * How $e(\mu)$ is a continuous function bounded by zero? >


title: Necessary condition for square integrable functions?

text: I'm studying Quantum Mechanics and I came across this which I don't quite understand: For a vector space of functions $f(x)$ to be square integrable (i.e $\int{|f(x)|^2dx < \infty)}$, the necessary condition is $\lim_{|x| \to \infty}f(x) \to 0$. Can someone help me understand why that must be the case? I sort of understand that it must not blow up at the infinities but can a (continuous or piece-wise continuous) function have a non-zero finite value at the infinities and still be square integrable?


title: Is it really okay to treat the Earth as stationary when doing free fall experiments?

text: Typical treatments of free-fall experiments treat the the Earth as fixed. Using this model, object being dropped moves toward Earth, but the Earth doesn't move toward the object. A more detailed analysis would have both the Earth and object accelerating toward each other due to the mutual third-law interaction. Is it really safe to ignore the acceleration—and hence movement—of the Earth during such an interval?


title: Open problems in string theory

text: I'm thinking of applying to do a PhD in String Theory. I'm gradually learning more about the subject through external reading, but still most papers are impenetrable. Could anyone give me a description of the areas of current String Theory research that would be accessible for a doctoral student? Are there any particular open problems that might be worth getting to know? Also any general advice on how to choose a PhD topic would be most welcome.


title: The units of gain and number of atoms in population inversion in a laser

text: I am following my university course notes on amplification in laser media, and have come across expressions for the **gain** of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is: $$G = 10 \log_{10}(e^\gamma) \mathrm{dB} \mathrm{m}^{-1}$$ where $\gamma$ is _also_ the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\gamma = \Delta N \sigma$$ where $\Delta N$ is the "number of atoms in population inversion" and $\sigma = [\mathrm{m}]^{-2}$ is the "cross-section for stimulated emission". This clearly shows that $\Delta N = [\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\gamma$ is unitless, so I am mighty confused.


title: Vacuum energy extraction: What happens to the cosmological constant?

text: If the vacuum energy is extracted, then what will happen to the cosmological constant? Will it remain the same? .


title: Commutators of differential and field operators

text: In $p+ip$ superconductivity, a term in the Hamiltonian (in polar coordinates) is $$ H=\int \mathrm{d}^2\vec{r} \left[ \left( \partial_r -\frac{i}{r} \partial_{\theta} \right) \psi^\dagger(\vec{r}) \right] \psi^\dagger(\vec{r}) $$ where $\psi^\dagger$ is the field operator that creates a spinless fermion. To follow convention and arrange the term as a matrix element, it should be possible to put it on the form: $$ H=\int \mathrm{d}^2\vec{r} \; \psi^\dagger(\vec{r}) \left( -\partial_r +\frac{i}{r} \partial_{\theta} \right) \psi^\dagger(\vec{r}) $$ Which is to say the two factors anti-commute, or at least the commutator is something that becomes a constant when integrated? My question is how to show this is possible? I have tried expanding $\psi^\dagger = \sum \psi_k a_k^\dagger$, but I am not sure if the basis is supposed to depend on the boundary conditions or not. I am also not quite sure how to calculate the commutator between the differential operators and the creation operator, $a_k^\dagger$, either.


title: Convert units from cal cm^-1 day^-1. to Wm-2

text: I am trying to calculate the energy budget of a water body and the last term I'm trying to calculate is the energy advected from the water body ($Q_v$). This equation is given by: $Q_v = V ( (T_1 - T_0) / A)$ ![enter image description here](http://i.stack.imgur.com/9YcKV.png) where $V$ is the daily inflow (in $\mathrm{m}^3 \mathrm{day}^{-1}$); $T_1$ is the temperature of the inflow ($^\circ\mathrm{C}$); $T_0$ is the temperature of the outflow ($^\circ\mathrm{C}$); $A$ is the surface area of the water body ($1\times10^{10} cm^2$) and $Q_v$ is given in $\mathrm{cal}\cdot\mathrm{cm}^{-2} \mathrm{day}^{-1}$. How would I convert $Q_v$ into $\mathrm{W\cdot m}^{-2}$ i.e. Watts per meter squared? I've seen an example in page 881 in this paper http://ecologia.ugr.es/pages/publicaciones/publicaciones- pdfs/2004/thermalstructureandenergybudgetinasmall2004/! which shows that $1 \mathrm{cal\cdot cm}^{–2}\mathrm{h}^{–1} = 4.19\times10^{–4}\mathrm{J\cdot m}^{–2}\mathrm{h}^{–1} = 11.63 \mathrm{W}/\mathrm{m}^2$ Could anyone explain to me the process of calculating this and for the example I provided?


title: Derivation of the centrifugal and coriolis force

text: I was wondering how easily these two pseudo-forces can be derived mathematically in order to exhibit a clear physical meaning. How would you proceed?


Impedance is a concept that shows up in any area of physics concerning waves. In transmission lines, impedance is the ratio of voltage to current. In optics, index of refraction plays a role similar to impedance. Mechanical impedance is the ratio of force to velocity. **What is a general definition of impedance?** **What are some examples of "impedance matching" other than in electrical transmission lines?**