When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen the usage of spherical BZ in these answers(1,2) by SE user Heidar. I can think of two possibilities: (1)Some physical system has a spherical BZ. This is hard to imagine, since it seems to me that all lattice systems with translational symmetries will have a torodal BZ, by the periodicity of Bloch wavefunctions. The closest scenario I can imagine is a continuous system having $\mathbf{R}^n$ as BZ, and somehow(in a way I cannot think of) acquires an one-point compactification. (2)A trick that makes certain questions easier to deal with, while the true BZ is still a torus. Can someone elaborate the idea behind a spherical BZ for me? **Update** : I recently came across these notes(pdf) by J.Moore. In the beginning of page 9 he mentioned > We need to use one somewhat deep fact: under some assumptions, if $π_1(M) = > 0$ for some target space $M$, then maps from the torus $T^ 2\to M$ are > contractible to maps from the sphere $S^2 \to M$ I think this is a special case of the general math theorem I want to know, but unfortunately Moore did not give any reference so I'm not sure where to look. **EDIT** : The above math theorem is intuitively acceptable to me although I'm not able to prove it. I can take this theorem as a working hypothesis for now, what I'm more interested in is, granted such theorem, what makes a $\pi_1(M)=0$ physical system candidate for strong topological insulators(robust under local perturbations), and why in $\pi_1(M)\neq 0$ case we can only have weak topological insulators. Crossposted: When can we take the Brillouin zone to be a sphere?


title: Stopping an isolated metal ion

text: If we obtain something like a single isolated hydrogen atom I.e. $H^+$ is it possible by keeping it in a system of charged rings to contain and stop it at the centre of the system? Any positively charged ring would have electric field pointing towards its centre, if the ion tries to move then it will be pushed towards center! Clearly this would violate the uncertainity principle because we would know that the atom is at the Centre of the ring and is stopped, I think therefore it must have atleast some difficulty but I can't figure out why it may not be possible! The question is that if it is possible, what does it say about the uncertainty priniciple and if it is not possible then why not?


I have a length of string and I want to know the maximum tension the string can support. I tie one end of the string to the ceiling and the other end to a glass of mass 100 g. The glass is cylindrical, with a cross-sectional radius of 4 cm and a height of 15 cm. I fill the glass with water (density of 1 g/cm3) and discover that the string breaks when the glass is 2/3 full. What is the maximum tension in N the string can support?


title: How can I find the frequency?

text: Grocery stores often have spring scales in their produce department to weigh fruits and vegetables. The pan of one particular scale has a mass of 0.5 kg, and when you place a 0.5 kg sack of potatoes on the scale the pan lowers by 6 cm. As we reach for the potatoes to toss in the cart, we accidentally knock the pan, making the pan and potatoes oscillate up and down. What is the frequency of their oscillation in Hz?               we may treat the spring as an ideal spring.     The acceleration due to gravity is −9.8 m/s2.


title: What are the conditions to be satisfied by a theory in order to be a quantum theory?

text: This is in continuation to my previous question. It is not a duplicate of the previous one. This question arises because of the answers and discussions in that question. Can we call a theory, quantum theory, if it is consistent with HUP? For example, suppose there is a finite and self consistent theory of gravity which incorporates the uncertainty principle. Can we at once call this theory a quantum theory of gravity or does it have to satisfy other conditions too? This question may be too basic but it is intriguing my mind.


title: What allows the Wave Disk Generator to be so efficient?

text: Researchers at Michigan State University recently invented the Wave Disk Generator that is supposed to get 60% fuel efficiency. What allows it to be so much more efficient than a traditional Internal Combustion Engine? I am aware that there is better mixing of fuel and air, but surely this alone does not produce the extreme efficiency.


title: Mobile "muonic hydrogen"

text: If we look at the atomic positions in a single crystal sample with a diamond like lattice, there exist directions along which there are long hexagonal "tubes" (I'm not sure if these have a proper term). ![hexagonal tubes in diamond like lattice](http://i.stack.imgur.com/a4Y2v.jpg) This happens in silicon for example. Now if a hydrogen atom was inserted in the material, it would be too large and just get trapped in the 'matrix' of the silicon and present a defect in the material. If instead I took a proton and muon in a bound "muonic hydrogen" state, the bohr radius which goes like 1/m would be ~ 200x's smaller. Due to being neutral, small, and tightly bound (a binding energy > 2.5 keV), I am tempted to view it almost like a large neutron. My question is:   Due to its much smaller size, could such a 'large neutron' pass freely through a crystal along one of these tubes?   Could one eliminate trapped hydrogen in a crystal by making it "mobile" in this fashion by putting the sample in a muon beam of low enough energy that it won't damage the crystal structure but still create 'muonic hydrogen'?


title: What is a virtual ground?

text: What is a virtual ground? I would like to know what it is.


title: Are not the field equations proof of the holographic principle?

text: The field equations (e.g. Schrödinger’s/Maxwell’s) describe the four- dimensional universe as the time evolution of a three-dimensional system. This implies that the universe contains only three dimensions of information: the fourth is derivable deterministically (albeit not practically) from the other three via these formulas. My understanding is that this is exactly what the holographic principle posits: that the universe contains only three dimensions of information (commonly presented as two spatial, one temporal). Yet the holographic principle remains unproven. Where is my misunderstanding?


title: Strong interacting v.s. Strong Coupling v.s. Strong Correlated

text: One of the active research areas in present is Strong interacting, Strong Coupling, Strong Correlated regime of the phases of matters. It seems to me that some physicists in the fields often mix the usages of these twos: **Strong Coupling, Strong Correlated**. However, in my viewpoint, they are NOT the exactly same, I regard that > $\bullet$ **Strong Coupling** : implies the large coupling of interactions > comparing to the free part of theory. Say, suppose there is a Lagrangian > description, then the action $S$ $$ S=S_{free} +g S_{interact} $$ the > **Strong Coupling** means $g >>1$. So this can be the confined phases of > QCD, where coupling $g$ of quarks and gluons runs large. \-- > $\bullet$ **Strong Correlated** : in my view, usually implies the > **fractionalization of the elementary particles into fractional quantum > numbers**. For example, this happens at 1+1D Luttinger liquids, where spin > and charge can separated their degree of freedom from the elementary > constituents(electrons), but the system needs NOT to be **Strong Coupling**. > i.e. this example is **Strong Correlated** but NOT **Strong Coupling**. This > is about the fields of **Strong Correlated Electron** on arXiv. \-- **My question** , so what are other **examples of systems** that are: > **1\. YES Strong Coupling and YES Strong Correlated** > > **2\. YES Strong Coupling but NOT Strong Correlated** > > **3\. NOT Strong Coupling but YES Strong Correlated** See also this relevant post.


title: How does a (near) elastic collision work at a microscopic level?

text: Suppose a large mass $M$ travelling at $v$ impacts a small stationary mass $m$, then following through the elastic collisions from high school, we get that the smaller mass can exit the collision FASTER than $v$. In the limit of huge $M$, we can change reference frame and make $M$ basically stationary, and $m$ simply bounces off, thereby having a velocity change of $2v$. How can I visualise this from a microscopic level? If the normal force between objects only exists while they are in contact, the moment contact was lost between two objects they no longer act on each other. So as soon as the target balls speed reaches the tiniest amount above the impact ball speed, the balls should separate and no longer interact. This would lead to all collisions being "sticky" and completely inelastic. So why are not all collisions sticky? How do collisions work that apparently allow acceleration past the point where contact is lost? And if this is something to do with Coulomb interaction, is there any historical commentary on how Newton and peers felt about this prior to the understanding of electrodynamics?


title: Parity on gamma matrices

text: I want to understand clearly why $ P \gamma^{\mu} P = \gamma^{\mu} $, where $ P $ is the parity operator. This result follow for example from pag. 66 of Peskin-Schroeder. The parity operator acts on Dirac fields in this way: $ P \psi (t, \textbf{x}) P = \gamma^0 \psi (t, -\textbf{x}) $, assuming no other phase factors. On the Dirac bilinear $ \bar{\psi} \gamma^{\mu} \psi (t, \textbf{x}) $, using the fact that $ P^2=1 $, you have $P \bar{\psi} \gamma^{\mu} \psi (t, \textbf{x})P = P \bar{\psi} P P \gamma^{\mu} P P \psi (t, \textbf{x}) P = \bar{\psi}\gamma^0 P \gamma^{\mu} P \gamma^0 \psi (t,- \textbf{x}) = \bar{\psi}\gamma^0 \gamma^{\mu} \gamma^0 \psi (t,- \textbf{x}) = (-1)^{\mu}\bar{\psi} \gamma^{\mu} \psi (t, -\textbf{x}) $ With $(-1)^{\mu} =1 $ if $ \mu=0 $ and $(-1)^{\mu} =-1 $ if $ \mu=1,2,3 $. So $ P \gamma^{\mu} P = \gamma^{\mu} $ follows because $ P$ and $\gamma^{\mu} $ act on different spaces? Or there are other explanations? (Please, do all the steps in the answer) To be clear: I use $ \gamma^0= \bigl(\begin{smallmatrix} 0&1\\\ 1&0 \end{smallmatrix} \bigr)$ and $ \gamma^{i}= \bigl(\begin{smallmatrix} 0&\sigma^i\\\ -\sigma^i&0 \end{smallmatrix} \bigr)$.


title: Current known *lower* bounds to the neutrino masses?

text: I started a little bit of Googling around the topic, and found very few information. There are mainly upper limits. I found lower limits only to the rest mass differences of the different neutrino flavors. Thus, what are the actually current known lower mass limits to the neutrino flavors?


title: How does one find the average relaxation time for a current through a conductor?

text: For determining the drift velocity of a flow of charge we need to find the **average relaxation time** for that flow at that specific temperature. My question is, how does one find that? Is there some experimental way to find that? Or it is derived by some Mathematical derivations and statistical calculations?


title: How Light or Water Intensity is equal to square modulus of wave function of Light or Water Waves $I=|\psi|^2 \,$?

text: I've seen the Wave Function as a psi $\Psi$ $\psi$. And always heard that the wave function is the Complex Number as Imaginary and real number. But I've never seen it I've never seen components of wave function But I have seen it everywhere in the book of Quantum Mechanics. i want to know how wave function could be intensity! in double slit experiment sometimes called Young's experiment, that is relation: $$I=I_1+I_2$$ $$I=|\psi_1+\psi_2|^2=|\psi_1|^2+|\psi_2|^2+2Re(\psi_1 * \psi_2)$$ i want to know how Intensity could be square modulus of wave function $|\psi|^2$ $$I=|\psi|^2$$ $$I=I_1+I_2$$ $$I=I_1+I_2+2\sqrt {l_1 l_2} cos\delta$$


title: Scattering of light by light: experimental status

text: Scattering of light by light does not occur in the solutions of Maxwell's equations (since they are linear and EM waves obey superposition), but it is a prediction of QED (the most significant Feynman diagrams have a closed loop of four electron propagators). Has scattering of light by light in vacuum ever been observed, for photons of any energy? If not, how close are we to such an experiment?


title: Is kinetic energy a relative quantity? Will it make inconsistent equations when applying it to the conservation of energy equations?

text: If the velocity is a relative quantity, will it make inconsistent equations when applying it to the conservation of energy equations? For example: In the train moving at $V$ relative to ground, there is an object moving at $v$ relative to the frame in the same direction the frame moves. Observer on the ground calculates the object kinetic energy as $\frac{1}{2}m(v+V)^2$. However, another observer on the frame calculates the energy as $\frac{1}{2}mv^2$. When each of these equations is plugged into the conservation of energy, they will result in 2 distinct results (I think).


title: Human perception of distance

text: When we see things around us, distant objects look smaller to our eyes than nearby objects do. Is there any physics-related reason why our eyes or brain perceive things like this? Or if this is purely biology issues (<-but nothing is purely biology or something like that..), can anyone briefly show me the processes behind those things?


title: How would an earth-spanning ring around the equator behave?

text: If a ring/belt would be constructed   * of a sufficiently strong material   * around the entire equator   * at 100m height from the surface   * assuming the land/water beneath is flat (no mountains get in the way), How would that ring behave - would it remain still, or crash against one side of the Earth?   Would it rotate with the Earth (maybe slightly slower, like the athmosphere)?   Could Earth slip out of that ring?


title: Could there be more universes?

text: In the documentary: "Curiosity - Did God Create the Universe (on YouTube)", theoretical physicist and cosmologist Stephen Hawking states that time did not exist before the big bang. The first question that popped up in my mind was: "Could there be more universes with their own space and time?"


title: What are the calculations for Vacuum Energy?

text: In wiki the Vacuum Energy in a cubic meter of free space ranges from $10^{-9}$ from the cosmological constant to $10^{113}$ due to calculations in Quantum Electrodynamics (QED) and Stochastic Electrodynamics (SED). I've looked at Baez and references given on the wiki page but none of them give a clear working for how these values are derived. Can someone point me in the right direction, as to how values like $10^{-9}$ are derived from the cosmological constant; **OR** $10^{113}$ due to calculations in Quantum Electrodynamics?


title: Can I just ask what these pulleys-and-constant-lengths problems are called?

text: I am not sure if this question is appropriate for this section, but I just want to know what these type of questions are called and when do physics majors learn them? These problems have to do with pulleys and constant lengths. I found the problem in an engineering dynamics course. Example ![](http://img812.imageshack.us/img812/8305/25857449.jpg)!


title: faster-than-c photons

text: As far as I know, according to quantum field theory, there are some photons that go faster than _c_ , which is the speed of light in vacuum. However, there seems to be a paper and a corresponding experiment that show every photon obeys the speed limit of _c_. (http://physics.aps.org/synopsis- for/10.1103/PhysRevLett.106.243602) So, my question is:   1. Is this experiment accepted universally?   2. Regardless of the acceptance of the experiment, if every single photon is shown to obey the speed limit of _c_ , what does this mean for quantum field theory? Thanks.


title: Attractive higgs force and inflation

text: Inflation was the extreme accelerating expansion of the universe, see here: http://en.wikipedia.org/wiki/Inflation_(cosmology) It worked in a similar way to dark energy but was so strong it would easily tear atoms apart (if it wasn't far to hot for atoms to form in the first place). The weird thing about inflation and dark energy is that they are under _tension_. **[Optional reading]: Explanation of how pressure works backwards in general relativity** In general relativity, pressure _itself_ is attractive. This is not because it takes energy to compress materials, the energy put in to compressing something simply shows up as extra mass. This is an _extra_ effect. Suppose you dive deep into a (non-rotating) neutron star (pretend the center is liquid) and measure the central density (as a swimmer would gauge the inertial resistance of water). You then move 1mm away from the core, release a pellet of dark matter (which goes right through everything so it has no buoyancy), and measure the acceleration towards the center (because of the shell theorem and the slow maximum speed of the pellet (~10 m/s), you can assume newtonian gravity in the local vicinity). The measured acceleration will be greater than the calculated acceleration, up to almost twice as much. This is a consequence of light bending twice as much as "expected" and can be derived from special relativity using a reference frame with constant acceleration. Pressure is equivalent to an exchange of fast-moving particles. These "virtual" particles "bend more" toward a mass and conversely the mass is pulled back more toward it. The inflation field was under enormous _tension_ , which is the exchange of negative mass particles, and it created a _repulsion_. Another effect is that work must have been done on the inflation energy as space expanded, just as it energy is added to a rubber band when you pull on it. This work showed up as more inflation energy, and meant that it _does not dilute_ when space expands. Eventually, the pressure/density ratio, (equation of state) went above the critical value of -1 and the field began to dissipate. The -1 "critical value" also causes that the fluid to be Lorentz invariant, which means there is no preferred reference frame. **The question itself** The Higgs particle is a purely attractive force, see here: http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the- known-forces-of-nature/the-strength-of-the-known-forces/ Could the Higgs force create a strong enough tension (perhaps with help from other forces?) to overwhelm the positive pressure effects of quantum degeneracy and heat and leave enough tension to cause inflation? If so you would not need GUT or TOE speculations for the mechanism of inflation. Furthermore, you could quantify it's strength and the nature of it's eventual decay.


title: The quantum-mechanical description of an electron motion in a magnetic monopole field

text: The quantum-mechanical motion problem of an electron in electric field of the nucleus is well known. The quantum-mechanical description of electron motion in a magnetic field is also not difficult, since it needs to solve the Schrödinger equation of the form: $$\frac{(\hat p + eA)^2} {2m} \psi = E \psi $$ But if we want to consider the motion of an electron in a magnetic monopole field, the difficulty arises because the definition of the vector potential in the whole space. See, for example. Was this problem solved? What interesting consequences derived from this task? (for energy levels, angular momentum etc.)


title: What does a planet look like in the infrared?

text: I've been trying to find a good source for what the planets look like in the infrared, specifically if viewed as a point source. I've been working with a low resolution telescope that senses in the infrared. What will a planet look like?


What is the experience using the Sky-Watcher Skyhawk 1145P SynScan AZ GOTO telescope and can it be recommended for a new astronomer? Is it any good?


title: Did physical models of galaxies come before they were actually observed?

text: Black holes were first predicted by astrophysics, then observed. Was the existence of galaxies first predicted by astrophysics, or first observed by astronomers?


title: Is it true that our solar system belongs to another galaxy which collides with the milky way galaxy?

text: I have just read an article about it here(http://www.greatdreams.com/2012.htm)(Scroll to bottom). I'm just a website developer, and I really want to know if this is really true. I'm curious. I'm pasting the content from it: > It turns out that our solar system appears to belong to another galaxy that > is colliding with the Milky Way. This was recently discovered when > scientists were trying to figure sources for "dark matter" that would > account for forces we can measure but not see visibly. Using near-infrared > (wavelengths of light outside human eye and optical telescopes) a huge > sister galaxy circling the Milky Way was discovered. It's called the > Sagittarius dwarf galaxy (SGR for short). > > For those keen on the 2012 data, this is the reason our entry point to the > Rift, center, heart (HunabKu) of the Milky Way is thru Sagittarius. The two > collide at that point. This explains why our solar system is at an angle to > the plane of the galaxy and also why we dip above and below that center line > every 12,000 yrs or so.


title: Troubles with the Nucleon Bound Energies

text: I was reading my quantum mechanics text and I have a doubt. I have the energy levels well defined for the finite square well and the author suddenly compares (I believe) those levels with the levels of the nucleon with the following phrase: "is the spacing between levels on the order of MeV for a nucleon bound in a several-fm well?". So I have some troubles understanding the meaning of that particular pharse. Thanks.


title: Is it possible to split baryons and extract useable energy out of it?

text: Since baryons (e.g. protons, neutrons) are composite particles it should be possible to split them apart. If so, is it then possible to extract useable energy out of the splitting of baryons in analogy to nuclear fission?


title: What is the effect of an increase in pressure on latent heat of vaporization?

text: What is latent heat of vaporization ($L_v$) in the first place? Wikipedia seems to indicate that it is the energy used in overcoming intermolecular interactions, without taking into account at all any work done to push back the atmosphere to allow for an increase in volume when a liquid boils. If that is so, then would it be correct that $L_v$ decreases as boiling point rises, because at the higher boiling point, less energy is required to overcome the weaker intermolecular interactions? Otherwise, would it then be correct to say that $L_v$ _in_ creases as boiling point rises, assuming constant-volume? Let's say there is a beaker containing 1 kg of liquid water at the boiling point, and an identical beaker, containing an identical amount of water at the same temperature. However, this second beaker is perfectly sealed such that the volume of its contents, both liquid and gaseous, will not change. Since both liquids are at the boiling point, applying heat should cause boiling to occur. Compared to the first beaker, then, would the second beaker (in theory) require more or less heat for its 1 kg of liquid water to completely boil? Thanks!


title: Energy of a simple pendulum like device

text: If there is a device in which a ball is attached to a pivot by a rod of length R and it is left to oscillate by dragging it to one side so that it is completely horizontal, at any point on its decent the ball has some velocity and also has some angular velocity about the pivot. In this case is the rotational kinetic energy about pivot equal to translational kinetic energy of the ball ? Why/ Why not ? If they are same then what is better way to deal with the calculations of such problems, the rotational KE about the pivot or translation KE of the ball alone ?


title: Transformation from waves to matter

text: Let $|\psi \rangle$ represents a wavefunction through and $\langle \psi |$ represent the dual   vector. Now there are things such as matter waves, could $|\psi \rangle$ represents a matter wavefunction, (of course representation is a definition), which is analogous to that of the massless wavefunction $|\psi \rangle$? If so, what would the $\langle \psi |\psi \rangle = n \in \mathbb{R}$ represent, an object? and the outer product $|\psi \rangle \langle \psi | = \rho$ would that be similar to the probability of localization of matter? I guess what I am trying to do is understand mathematically how I would be able to represent matter and how, if any, way there is to represent this in an analogous matter with the bra-kets. Seems there should be a relation as if waves are representable and matter is a wave, then matter is representable. Thanks, Brian


title: Deviation from 2D trajectory

text: I need to find out how far a ball is from the predicted trajectory in 2 dimensional space and I know the start and end position of the ball in both dimensions. Along with that I know the initial velocity and the angle at which the ball was launched. **Every single variable is known in this picture.** ![enter image description here](http://i.stack.imgur.com/pfvQH.png) The problem is that I can predict where the ball is going to land based on the angle α and the initial velocity v0, but the actual landing is a bit off due to variables assumed to be zero (wind, friction, etc.) and I need a mathematical way of calculating this deviation. NOTE: The mathematical model may not include calculus as we haven't covered that part of our curriculum yet. _Any suggestions on how to do that?_


title: Generalized tight-binding model - how to solve it?

text: Consider a generalized 1D tight-binding model (without spin) with the following Hamiltonian \begin{equation} \mathcal{H}\left(\\{\chi_{r,r+1}\\}\right)=\sum_{r}(\chi_{r,r+1}c^\dagger_rc_{r+1}+h.c.)~, \end{equation} and suppose that the complex number $\chi_{r,r+1}$ is a variational parameter. The $r$ index runs over a lattice sites of a 1D chain (you can think of this as a periodic boundary condition problem with $N$ lattice sites). **Q** : Given an arbitrary set of $\chi_{r,r+1}$, is there a systematic procedure to find the ground-state of the model (numerically or analytically) ? I'm not looking for a detailed answer with the calculation here. Perhaps just a good reference or starting point. For instance, in the case where $\chi_{r,r+1}=\chi$ $\forall r$, the system can be diagonalized by a simple lattice Fourier transform.


title: What is the current status of string theory (2013)?

text: I've seen a bunch of articles talking about how new findings from the LHC seem to disprove (super)string theory and/or supersymmetry, or at least force physicists to reformulate them and change essential predictions. Some examples:   * Did the Large Hadron Collider Just Debunk Superstring Theory?   * Is Supersymmetry Dead?   * String Theory Now on Life Support   * Supersymmetry Fails Test, Forcing Physics to Seek New Ideas So I'd like to know: has string theory really been hit that hard? Is it losing ground in the scientific community? Do you think it can recover from it? Are there any viable or promising alternative theories? (I've seen Michio Kaku in some clips saying string theory is "the only game in town".) Note: a related question is What if the LHC doesn't see SUSY?, but I'm asking for more direct answers in light of the results found in the last 2 years.


title: What role has our Moon played in creating a persistent geomagnetic field?

text: The question comes from a comment by Mark Rovetta on my earlier question about the Earth's core going cold.


title: Can we excite a nucleus by means of very intense low energy gamma-photon irradiation?

text: The phenomenon of multi-photon ionization of atoms has been studied, both theoretically and experimentally, for several decades. Intense laser beam devices are the apparatuses used for the experimental study of this phenomenon. **QUESTION:** Would it be possible to use similar excitation processes with nuclei, using "low energy" $\gamma$-photons in order to manufacture nuclear isomers for industrial and medical applications?


title: Is it possible to use a powder-based firearm in space?

text: A firearm relies upon some kind of explosive powder to drive the slug out of the barrel. My guess however is that in space (at GEO, or higher) a firearm would be unusable due to the extremes of temperature/pressure. Secondly the powder probably would not ignite when the hammer fell. Are my assumptions correct? Can a firearm be used in space?


title: Why can't we store light in the form of light?

text: We can store cold (ice), heat (i.e. hot water bag) and electrical charge (batteries). We can even "store" a magnetic field in a magnet. We can convert light into energy and then, if we want, back to light. But we can't store light in form of light in significant amounts. What is the explanation of that in physics terms?


title: Boundary condition Hamiltonian with point tinteractions

text: I`m studying the Hamiltonian with point interaction centered in $y$ in three dimensions. I know that the elements in the domain of the Hamiltonian are of the form $$\psi=\phi+qG^z(\cdot-y)$$ where $G^z(x)=\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}$, $q\in\mathbb{C}$ and $\phi\in H^2$. The boundary condition is $$q=\frac{4\pi\phi(y)}{4\pi\alpha-i\sqrt{z}}$$ Many times it is written in the form $$\lim_{r\to 0}\bigg[\frac{\partial (r\psi)}{\partial r}-4\pi\alpha(r\psi)\bigg]=0$$. Are these two relations equivalent?


title: Can you prepare the superposition of two arbitrary light beams?

text: More specifically, how do you physically prepare the superposition of two beams of polarized light? Suppose beam A and beam B, the two input beams, each have unknown polarizations.   Beam C, the output beam, must be the superposition of beams A and B.   How do you do it? You can create an overlap of two beams with a half silvered mirror at 45%.   You can also creat a localized overlap by crossing the beams with a small angle.   In both cases you get a mixture, not a superposition. How do you get the superposition? Bonus Points: How do you prepare $ a|A \rangle + b|B\rangle $?


title: Is it possible that quantum mechanics is not truely fundamental?

text: That there is some form of generalized framework from which quantum mechanics follows as a certain limit? Is there any theory that has been discovered that can lead to the same predictions of QM but is not strictly QM?


title: When I ride my bike, does half of the energy go into the earth?

text: When I ride my bike, does half of the energy go into the earth because of Newton's third law? Does the energy in the earth transfer into heat upon my brakes when I utilize them? If the earth was 2000 times less massive than it is, and I applied the same energy to my pedal, would my bike move (relative to an observer not connected to earth) at the same speed as it would if the earth was normal? I think if you think about it these two questions are the same.


title: Whats the contact area for elastic contact between a cylinder and a plane surface?

text: I am trying to determine the contact area between a cylinder and a plane surface of two different materials so that the plane lies tangent to the cylinder. There is elastic contact between the two materials so the plane surface deforms to create a contact area around an area of the cylinder because there is a force pushing down on the cylinder (its weight). I have looked around the internet and have only been able to find the scenarios for a sphere on a plane surface, two spheres in contact, and a cylinder on a plane surface but the but the cylinder's flat top is touching the plane instead of the plane being tangent. It would be great if anyone could guide me to this equation.


title: A thought on definition of momentum

text: Well, this is a simple, basic and I think even silly doubt. The first time I saw the definition of momentum as $p = mv$ I started to think why this is a good definition. So I've read the beginning of Newton's Principia where he said that momentum is a measure of quantity of motion. Well, this started to make sense: if there's more mass, there's more matter and so there's more movement going on. If also there's more velocity, the movement is greater. So it makes real sense that quantity of motion should be proportional both to mass and velocity. The only one thing I've failed to grasp is: why the proportionality constant should be $1$? What's the reasoning behind setting $p = mv$ instead of $p = kmv$ for some constant $k$? Thanks in advance. And really sorry if this doubt is to silly and basic to be posted here.


title: Influence of charged particle's own electric field on itself

text: I read this in my textbook: _A charged particle or object is not affected by its own electric field._ Since I find this completely unintuitive and my mind is yelling "wrong! wrong! how could a particle even distinguish between its own field and the external fields?" I would really like to hear an explanation about why this is true, if it is. _(A corollary question that comes to my mind after thinking about this is... What about gravity and mass considering the analogous situation? And the other two forces?)_


title: Kolmogorov/Energy spectrum for turbulent boundary layer

text: Previously, I have calculated energy spectrum for 3D isotropic turbulent flow data which is equally spaced in all three directions and then to compute the energy spectrum, one performs Fourier transform and then accumulates energy located in different wavenumber bins and then gets the ($k^{−5/3}$) slope and it all works fine. Now if I want to extend the same for an anisotropic case like turbulent boundary layer case, how should I modify the technique? I cannot take a Fourier transform in the wall normal direction anymore and I am not sure how to compute the energy spectrum as such. There is no problem if I compute the spectrum for streamwise and spanwise plane since they are both defined using Fourier basis. So my question is two-fold:   1. How do we compute energy spectrum for a 3D turbulent boundary layer?   2. If we cannot compute it for a 3D case i.e. if the argument is valid only for a 2D planar cut at various wall normal locations, then how do we define a scale for turbulent boundary layer cases, particularly along wall normal direction?


title: Why do He-3 atoms repel each other much more strongly than electrons?

text: Is there a simple answer to this question ? see last line of this paragraph http://en.wikipedia.org/wiki/Fermionic_condensate#Fermionic_superfluids


title: Diver view of refraction

text: I'm studying the refraction in optics. If a red light monocromatic beam of red light (700 nm) passes from air to water it becomes with a wavelenght of aprox 526 nm. So, my question is: How is going to see this beam a diver? Red (700 nm) or something more like green (526 nm)? (Let's suppose that the diver isn't wearing glasses). I think that he is going to see the beam green? I'm a little confused...


title: photoelectric cell

text: In the photoelectric cell my teacher says that the electron emission from the cathode depends on the frequency of the incident photon and it doesn't depend on the light intensity (I = nhU/ta , Right?) so if the light intensity increased the photoelectric current won't change . but what if this increase in the light intensity is due to an increase in the frequency(energy) of each photon. to make a correct statement shouldn't the light intensity be replaced with rate of photons? so that if the number of photons increased even to millions and if the frequency of each was less than the threshold frequency , the photoelectric current won't be affected.


title: Galilean group time interval

text: I have a question that arised by reading _A Course in Modern Mathematical Physics_ by Peter Szekeres. He defines the galilean space $\mathbb{G}^{4}$ to be the space of events with a structure consisting of three elements:   1. Time intervals $\Delta t=t_2-t_1$   2. The spatial distance $\Delta s=|\mathbf{r_2}-\mathbf{r_1}|$   3. Motions of inertial free particles, i.e rectilinear motions, $\mathbf r (t)=\mathbf u (t) + \mathbf r_0$. This is all fine but then he proceeds by saying: > _Note that $\Delta t$ is a function on all $\mathbb G^4 \times \mathbb G^4$ > while $\Delta s $ is a function on the subset of $\mathbb G^4 \times \mathbb > G^4$ consisting of simultaneous pairs of events $\\{((\mathbf r, t),(\mathbf > r',t'))|\Delta t=t'-t=0\\}$ [...]_ I don't see how the time interval is a function on all $\mathbb G^4 \times \mathbb G^4$


title: Deriving formula for time dilation

text: Last week in class we derived the formula for time dilation using light clocks and got $$t=\gamma t_0\quad \gamma=\left(1+\left(\frac vc\right)^2\right)^{-1/2}$$ So far so good. However, after class I was thinking of an alternate proof, but for some reason I got another result and I can't tell why. Basically you have a train with an observer A inside who emits a beam of light to the left which is reflected off a wall at distance $d$ from A. The time it takes for the beam to get back to the observer is $t_0=\frac{2d}{c}$ which is the proper time. Now consider an observer B outside the train. The train is moving at a velocity $v$ to the right relative to B. Thus the time it take for the light to hit the wall is $\frac{d}{c+v}$ and the time it take for it to return to A is $\frac{d}{c-v}$. Thus the dilated time is $$t=\frac{d}{c+v}+\frac{d}{c-v}=\frac{2dc}{c^2-v^2}=\frac{2d/c}{1-\left(\frac vc\right)^2}=\frac{t_0}{1-\left(\frac vc\right)^2}=\gamma^2t_0$$ Where did I go wrong?


title: About Boltzmann H-theorem

text: What is the assumption for Boltzmann H-theorem? One can derive it just from the unitarity of quantum mechanics, so this should be generally true, does it imply a closed system will always thermalize eventually? Does it apply for many-body localized states?


title: Galilean, SE(3), Poincare groups - Central Extension

text: After having learnt that the Galilean (with its central extension) with an unitary operator $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + \delta\phi\mathbb{\hat1} = \sum_{i=1}^{10} \delta s_iK_i + \delta\phi\mathbb{\hat1} $$ This makes sure the commutation relations hold good in the group (especially for the boosts). However, in the case of Poincare group, the commutators still hold without a central extension. Similarly, is the case with SE(3) (there are no central extensions involved). My question is why is there a necessity for central extensions in the first case, but not later ?? PS: This answer is somewhat related to the question, but am not able to get to the bottom of this thing.


title: Tidal force of Sun?

text: As I understand, satellites and the Moon orbiting Earth are in free fall. Isn't the same true for Earth orbiting the Sun? My question is then: How can the Sun's gravity affect tides? Aren't the molecules in the oceans in free fall relative to the Sun?


title: Can we derive the relation between proper time and the spacetime interval?

text: In GR, it's usually taken for granted - or as a definition - that the time measured by an observer's clock is related to the geometry in a very simple way, $d\tau^2 = |ds^2|$. This is easy enough to see in some simple contexts, like special relativity, but it might not be obvious in general spacetimes. In principle I should be able to construct a model of a clock using the _matter sector_ alone, and then show that, assuming that matter is minimally coupled to the metric, the time this clock measures is related to the spacetime interval in the usual way. Is that possible, and if so does anyone know where I could find a derivation like that? EDIT: We can think about the question like this. I might consider having a second metric related to the first, e.g., by a conformal factor depending on a scalar field like in many modified gravity theories. So for which metric do you assume that $d\tau^2 = |ds^2|$ measures an observer's clock time? The standard answer is you use the metric to which matter is minimally coupled. But I would be very surprised if this were just an assumption; minimal coupling should be able to tell us the relation between proper time and geometry. But how?


title: Algorithm for identifying planes in a Bravais Lattice

text: I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general.   How can I identify the atoms/unit cells that belong to a plane - that is normal to a given direction.   I do recognise that the lattice might not be periodic in ANY direction - only specific ones. I worked out a way to calculate the periodicity of the lattice planes:   1\. Given the direction $\vec{t}$, construct the corresponding reciprocal lattice vector G.   2\. Project $\vec{G}$ in the direction of $\vec{t}$ and take the inverse of the length of the projected vector.   i.e. Distance between lattice planes normal to the direction $d = \vec{G}\cdot \frac{\vec{t}}{\vert\vec{t}\vert}$ My question, once again, is to find an algorithm that identifies the atoms in the crystal planes thus formed.


title: Maximum Shear on a Beam - beam with fixed support on one end and hinge on other end

text: A beam $\displaystyle 3m$ long with fixed support on one end and hinge on the other end is subjected to a uniform load of $10\ kN/m$. What is the maximum shear of this beam? The solution is this one: $\displaystyle Max.Shear = \frac{5\omega{L}}{8}=\frac{5\left(10\right)\left(3\right)}{8}=18.75\ kN$ where $\omega$ = distributed load $N/m$ $L=$ length My question is, how did he get this equation? All I know is that hinges have two forces acting on them. Can anyone derive this equation?


title: Future light cones inside black hole

text: In Caroll's Spacetime and Geometry, page 227, he says that from the Schwarzschild metric, you can see than from inside a black hole future events all lead to the singularity. He says you can see this because for $r<2GM$, t becomes spacelike and r becomes timelike. I don't understand his reasoning though. Why does r being timelike mean you can only travel towards the singularity?


title: How to show that in 1D scalar potential well there isn't pairs production (Dirac particle)

text: Let's have the potential $U = -V_{0}$ for $|x| \leqslant a$ and $U = 0$ for $|x| > a$. The stationary Dirac equation for bound states gives $$ tg(\frac{p_{2}a}{\hbar}) = \frac{2\Gamma}{1 - \Gamma^{2}}, \qquad (1) $$ where $$ \Gamma = \frac{p_{1}}{p_{2}}\frac{E + mc^{2} + V_{0}}{E + mc^{2}}, \quad p_{1}c = \sqrt{m^{2}c^{4} - E^{2}}, \quad p_{2}c = \sqrt{(E + V_{0})^{2} - m^{2}c^{4}}. $$ How to show that there isn't pair production independently on well's parameters $V_{0}, a$? My attemption. I assume that if scalar potential acts on particle and antiparticle by the same ways, there exists only one case when the production is possible. It is when $E_{1} = E_{2} = 0$, where these energies refer to the particle and antiparticle energies correspondingly. For case $E = 0 (1)$ gives me $$ \sqrt{V_{0}^{2} - m^{2}c^{4}} = -mc^{2}tg\left(\frac{\sqrt{V_{0}^{2} - m^{2}c^{4}}a}{c\hbar}\right), $$ and theoretically it doesn't have solutions for all possible values of $a, V_{0}$. But it seems that it has. Can you help me?


title: How can I determine the coefficient $k$ in $ \dfrac{dT}{dt} = -k(T - 100 \mathrm{^\circ C}) $?

text: I recently spend some time on cooking and I'm curious about the time evolution of the temperature of the water. I did some experiment and the temperature is of the form $$ T = 100 \mathrm{^\circ C} + (T(t_0) - 100 \mathrm{^\circ C}) e^{-k(t - t_0)} $$ for some positive constant $k$ somehow (for example, see my previous post and my answer). In other words, $$ \dfrac{dT}{dt} = -k(T - 100 \mathrm{^\circ C}). $$ My question: **How can I determine the coefficient $k$?** Of course, I can determine it by experiments as shown below. But what I want to do is understanding this as a function of volume of water $V$, power of heater $P$, and some other variables related to a pot. So my expected answer is something like "If the volume (power) is $n$ times larger then the coefficient is $n$ times ($1/n$ times) larger." Here is the experimental data and fitting graphs. ($V$ water is in the same pot with radius 9cm and heated by IH correspond to power $P$. I measure its temperature every 30 seconds.) **My aim is determine the time evolution of the temperature of the water a priori.** So if another approach is better, please tell me. I would appreciate if you help me. Thank you.


title: Does Light (EM Signal) undergo accelration during reflection?

text: Since Average acceleration is defined as, Average acceleration over a period of time is the change in Velocity divided by the Duration of the Period. Does Light (EM Wave) undergo acceleration during reflection from mirror or any other surface?


title: What happens when a photon hits a mirror?

text: When a photon of light hits a mirror does the exact same photon of light bounce back or is it absorbed then one with the same properties emitted? If the same one is bounced back does it's velocity take all values on $[-c,c]$ or does it just jump from $c$ to $-c$ when it hits the mirror? Or, is the phenomenon of a mirror better explained using a wave analogy? If so, what is this explanation?


title: A question on the existence of Dirac points in graphene?

text: As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ and $K^{'}$ in BZ. According to the von Neumann-Wigner theorem (no-crossing theorem): To make two eigenvalues of a Hermitian matrix (depending on some independent real parameters) cross, generally speaking, we need to change at least 3 parameters. But in the 2D graphene case, the variation of only 2 parameters $k_x,k_y$ can cause the energy levels cross. So I want to know whether there are some physical or mathematical reasons for the existence of Dirac points in graphene.


title: Bloch sphere representation

text: Suppose you know that a qubit is either is in state $|+\rangle$ with probability $p$ or in state $|-\rangle$ with probability $1-p$. If this is the best you know about the qubit's state, where in the Bloch sphere would you represent this qubit?


title: How long does it take for an electric car to go from 0 to 60 mph?

text: I found the freefall motion equation which describes terminal velocity of a falling body, but I can't find a similar equation for a vehicle subject to constant traction force, so I tried determining it by myself, but resulting equation is not plausible, as it shows dozens of seconds needed for a 1600 kg vehicle to go from 0 to 60 mph, so there must be something wrong. I'm using this equation: $$v(x) = v_f \cdot \tanh\left(\frac F {mv_f} \cdot x \right) = v_f \cdot\tanh\left(\frac {T/w } {m v_f} \cdot x \right)$$   * $v_f$ = terminal velocity = $\sqrt {\frac F c} = \sqrt {\frac {T} {wc}}$   * $c = \frac 1 2 \rho C_d A$   * $\rho$ = air density = 1.225 $\frac {kg} {m^3}$   * $C_d$ = air drag coefficient = 0.32   * A = frontal area = 2.19 m$^2$   * T = given torque = 220 Nm   * w = wheel radius = 0.25 m   * m = vehicle mass = 1762.5 kg Freefall motion equation is: $$ v(x) = v_f \tanh\left( {x\sqrt{\frac{gc}{m}}}\right)$$ with $v_f=\sqrt{\frac{mg}c}$ With above data for the car, I should get around 10s time for 0-60 mph, but I get 63 seconds! What am I doing wrong? With above data(1) for the car, I know (2) I should get around 10s time for 0-60 mph, but I get 63 seconds! What am I doing wrong? Other literature data:   * Fiat Stilo - 255 Nm, 1488 kg, 11.2 s   * BMW M3 - 400 Nm, 1885 kg, 5.3 s   * Citroen C3 - 133 Nm, 1126 kg, 14.5 s Literature data for electric cars:   * kg W Nm sec-to-60mph   * Chevrolet Volt 1715 63 130 9,0   * smart fortwo electric drive 900 55 130 12,9   * Mitsubishi i-MiEV 1185 47 180 13,5   * Citroen zEro 1185 49 180 13,5   * Peugeot iOn 1185 47 180 13,5   * Toyota Prius Plug-in 1500 60 207 10,7   * Renault Zoe 1392 65 220 8,0   * Renault Fluence Z.E. 1543 70 226 9,9   * Nissan leaf 1595 80 280 11,9   * Toyota RAV4 EV (US only) 1560 115 296 8,0 (1) "Evaluation of 20000 km driven with a battery electric vehicle" - I.J.M. Besselink, J.A.J. Hereijgers, P.F. van Oorschot, H. Nijmeijer (2) http://inhabitat.com/2015-volkswagen-e-golf-electric-car-arrives-in-the-u- s-next-fall/2015-vw-e-golf_0003-2/


title: What is $vdp$ work and when do I use it?

text: I am a little confused, from the first law of thermodynamics (energy conservation) $$\Delta E = \delta Q - \delta W $$ If the amount of work done is a volume expansion of a gas in, say a piston cylinder instrument at constant pressure, $$\Delta E = \delta Q - pdv$$ Here $p$ is the constant pressure and $dv$ is the change in (specific) volume. So, when do I take into account $$\delta W = d(pv) = pdv + vdp$$ I am assuming that for cases of boundary work, at constant pressure, the $vdp$ term is zero. So under what conditions should I consider the $vdp$ term?


title: Refraction seismology - travel time for wave

text: I am taking an introductory class in seismology, but have some difficulties understanding the logic behind the formula used to calculate the time it takes for a refracted wave to return to the surface from which it was sent out originally. So, say we have a seismic wave sent out from a horizontal layer. We also assume that the interfaces below are horizontal, and parallel to the top layer. If the wave hits the first interface at the so-called critical angle, it will be reflected back to the surface with the same angle at which it hit the interface. Now, the formula given in the book states that in order to calculate travel time, we use: $$t = \frac{2 h_1 cos(i_c)}{v_1}$$ where $h_1$ is the vertical height of the layer, $i_c$ is the critical angle, and $v_1$ is the velocity of the wave as it travels through the layer. As mentioned, I don't quite see the logic of this formula. I know that time = distance/velocity. Surely the shortest possible distance to travel back and forth between two horizontal layers, separated with distance $h_1$ is simply $2 h_1$. So by that logic, the shortest possible time a wave could possibly use, would be: $$t = \frac{2 h_1}{v_1}$$ But in the given formula we multiply the numerator with $cos(i_c)$, which would yield an even shorter value for distance. And I just can not see how this makes sense at all. Now, if we were to use the cosine-term in the _denominator_ , then the formula would make perfect sense to me. But why isn't it so? Any help here would be greatly appreciated!


title: How do you calculate the equivalent absorbed radiation dose from activity, type of emission, and the energy of the emission?

text: I have a sample of U-238 of which my Geiger counter detects beta activity at 700 events per second. Based on the counter's efficiency of 98.6% for U-238, the activity would be about 710 becquerels, I think. Here I am going off of my understanding based on explanations from many different people. From here I have figured I would just convert the beta energy in MeV to joules and then multiply it by the corrected activity (710Bq) to obtain the absolute absorbed energy per second. This is great and all and when considering the mass of the object absorbing the energy, we can get the number of Gy absorbed by an object. What I want is the equivalent dose in sieverts per hour. I understand there are many weight factors for different types of tissues and all that. But, there is a lot of misinformation floating around on the Internet about the calculations. Can someone point me in the right direction on how to go about calculating this?


title: What maintains constant voltage in a battery?

text: I know there's lots of questions that address similar situations, (Batteries connected in Parallel, Batteries and fields?, Naive Question About Batteries, and the oft-viewed I don't understand what we really mean by voltage drop). However, I have a question, and after examining just the battery structure, I have been wondering, exactly what structure/process maintains the constant voltage drop within batteries? I mean, certain chemical reactions are occurring in each half-cell, and electrolytes maintain charge conservation, I get that there's some motivation for the electron to move from one cell to the other. But why is this motivation so constant? I really want to get this.


title: Are solar physics images of use to the night-time community?

text: I'm actually interested in cases of cross-discipline data re-use. I know that the SOHO/LASCO coronographs are used for comet finding, that solar telescopes were used to get information about Venus's atmosphere during its solar transit in 2004, and STEREO were rolled to try to look at L4 and L5. But do the night-time or planetary communities use the images for anything else? For example, does STEREO's separation from Earth make the coronographs or heliospheric imagers useful for any triangulation, or are the exposure times or spatial resolution problematic, or is the relative separation insignificant in the grand scheme of things? **Update** : I'm not interested in space weather or for warnings about when to put satellites into 'safe mode'... I'm more interested if the low-level telescope data is useful.


title: Turbofan: Efficiency wrt. the bypass ratio: "a lot of slow air > a little faster air"?

text: Reading / viewing up on how jet engines work, this video explains at the 9:02 mark that, for turbofan engines, ".. it is more aerodynamically efficient to have a lot of air moving relatively slowly rather than a little air moving relatively quickly." I.e. a lot of air bypasses the combustion chamber and is only accelerated by the main frontal fan. This claim has me puzzled, given that kinetic energy is dependent on the square of the velocity, i.e. $E_k = \frac12 m v^2$, I would have guessed the inverse. Does "aerodynamically efficient" mean something very specific here?


title: Are all identical fermions in orthogonal states as opposed to different general states?

text: A professor told me that most physicists assume that all identical fermions are in completely orthogonal states. If that is true, then does that mean that that the total wave function is highly localized for a closed box of a huge number of electrons? I get confused between when your supposed multiply wavefunctions and when your supposed to add wavefunctions. I know that for one electron, adding all orthogonal states of that one electron in a closed box would yield a highly localized spatial wavefunction.


title: Why are orbits elliptical?

text: Just wanted to know why planetary orbits get to be elliptical? I'm not debating how stable these orbits are (obviously almost completely stable), but why don't planets revolve in the special ellipse known as a circle? Wouldn't the centrifugal/-pedal forces between the planet and Sun make the planet revolve around the Sun in a perfect circle? Is there a 'third force' that attracts planets at a specific point (a hypothetical reasoning for elliptical orbits)? I'm not trying to disprove Kepler over here, but I don't exactly get it.


title: Is turquoise closer to blue or green?

text: I am having a discussion with my coworkers. Does anyone know what is the wavelength of turquoise color and whether it is closer to green or blue when comparing their wavelengths ?


title: Interpretation of the WIMP annihilation cross section graph

text: I have some trouble in the interpretation of the WIMP cross-section annihilation versus their mass. ![Graph](http://www.quantumdiaries.org/wp- content/uploads/2012/07/XENON100_2012.png) I understand that the lines represent a upper bound on the cross section from the observation. But what do the contours from, e.g., DAMA mean?


title: Majorana Superfields

text: so apologies if this is a silly question... In the type 1 see saw model we add extra Majorana fermions to our model. These fermions have to be total gauge singlets in order to have a Majorana mass term and thus trigger the see saw mechanism. In a supersymmetric model we add gauge singlet superfields whose fermionic components are majorana. My question is regarding the scalar component of this superfield - is it real or complex? I think it should be real as the superfield itself has to be a gauge singlet and thus the scalar field has no charge associated with it. Yes? No? Thanks in advance


title: Why does a ball bounce lower?

text: If a ball hits the floor after an acceleration then why does it bounces lower? I mean the Energy is passed to the floor then why does the floor give back less Energy?


title: Will the black hole size increase?

text: I was thinking about the following thought experiment, but wasn't sure about its outcome. Suppose there is a black-hole and I enter it with a partitioned box containing two different gases on either side. After passing, the event horizon the radius of a black-hole increases as I added mass to the system (?). Now after I pass the event horizon, I open the partitioned box thereby increasing the entropy of the system. Will the area of the black hole also increase? If so, will the radius also increase? (Isn't the radius proportional to the mass of the black-hole?)


title: How to calculate the radius of a rain drop with variable mass?

text: I need help with the following problem, please help me get started as I do not know where to begin with One spherical raindrop is falling in the atmosphere. Mass of the raindrop increases proportional with it's area. There's no counterpoise in it's move. I should show that radius of a rain drop is increasing linearly with time.


title: What does the term 'a uniform RVB spin-liquid state' mean?

text: I encountered this term **a uniform RVB spin-liquid state** in some articles, for example, see the paragraph under Eq.(29) on page 9 in this paper. What does the word ' _uniform_ ' mean? Simply from the mean-field anzats(see Eq.(28)), how to see that the projected spin wave-function is a RVB state which is a superposition of spin-singlet states? And is the low-energy gauge structure of **a uniform RVB spin-liquid state** always $SU(2)$? Thank you very much.


title: What field of physics play a major role in water pipe function

text: Background: I'm a legal MMJ patient and I've been looking into water pipers/bongs recently. I understand that this question can be a little controversial but where I live this is all legal as long as you have the proper prescription. Last few years there has been quite a lot of "scientific" bongs that has been hitting the market and in my to medicate as healthy as possible I would like to understand how they work. I know about vaporizers and I have a few of them, this is a physics question. So to my question. What are the major fields that come into play when talking drag/resistance, smoke cooling and smoke filtration? My theory is that it is fluid dynamics. Are there any good books to read to fully understand the mechanics behind the water bong? Thank you


title: Boundary conditions on current carrying wire

text: I'm trying to simulate by finite elements method Maxwell equations for a current carrying wire. My 3d geometry consists of a cylinder and a box containing it. I will use a mixed formulation and Nedelec's elements introducing a vector potential. I'm in a magnetostatic regime. At the end of the simulation I would like to plot the lines of the magnetic field around the wire (the cylinder) and probably compute forces and see that the numerics method agrees with classical results. My troubles concern the boundary conditions I have to impose on the surfaces of the cylinder (side, top and bottom) and on the box. I think I will use a value of permittivity of $\mu_0$ for the box and a copper relative one for the wire. I hope someone of you can help me seeing that my physics background is not so good.


title: Does GR imply a fundamental difference between gravitational and non-gravitational acceleration?

text: 1. Does the equivalence principle imply that there is some fundamental difference between acceleration due to gravity and acceleration by other means (because there is no way to 'feel' free fall acceleration for a uniform gravitational field)?    2. Does General Relativity allow you to describe the acceleration due to gravity without Newton's second law (because every other source of 'push or pull' outside the nucleus involves the electromagnetic field)?    3. Is the acceleration due to gravity a result of changes in time dilation/length contraction as opposed to an actual push or pull?


title: Classical EM neglects electron recoil?

text: Imagine two electrons $A$ and $B$ at rest. Electron $B$ is at a vertical distance $r$ above electron $A$. Let us assume that the electrons are constrained to move on horizontal rails. At time $t=0$ I give electron $A$ a horizontal acceleration $a$ for a time interval $\Delta t$. At time $t=r/c$, due to the impulse provided by the classical Lienard-Weichert radiation field of electron $A$, electron $B$ gains horizontal momentum: $$\Delta p = F\Delta t=-\frac{e^2a\Delta t}{4\pi\epsilon_0c^2r}$$ In terms of quantum electrodynamics a photon is exchanged between electron $A$ and electron $B$. Therefore if electron $B$ receives a momentum $\Delta p$ from the photon does that mean that electron $A$ must recoil with momentum $-\Delta p$ as it emits the photon? If this is true then the classical (retarded) Lienard-Weichert interaction theory describes (later) momentum transfer to the static charge $B$ due to the (earlier) accelerating charge $A$ but it neglects the fact that charge $A$ receives (earlier) recoil momentum during the process. Maybe this recoil momentum is described by a time-reversed advanced Lienard- Weichert interaction in which the earlier momentum transfer to the accelerating charge $A$ is due to the later interaction with the static charge $B$?


title: Primer on Liquid/Ink Splatter Physics

text: I'm looking for primer material on the modelling the physics of liquids. In particular I want to make a small simulation (I'm a programmer by profession) of throwing ink at a board, much like this. http://www.pond5.com/stock-footage/670934/ink-splatter.html I'm researching ways to approach the problem before I start banging on the keyboard. But I'm finding it hard to find an article about procedural generation of splatters and pools of liquid. There are a lot of search results for After Effects plugins, but this is of limited use. It could be possible that an open source AE plugin exists out there that I can analyze, but I would prefer to have a good maths based article to draw from. My math skills are quite basic so I'm fully prepared for a hard slog on this one. I would like to make some sort of road map for myself to get to a level where I can make a simulation such as this. Guidance is much appreciated.


title: What is the physical meaning of the connection and the curvature tensor?

text: Regarding general relativity:   * What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?   * What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ jkl}$), Ricci tensor ($R_{ij}$) and Ricci scalar ($R$)? For example why do the Einstein equations include the Ricci tensor and scalar, but not the Riemann tensor? To be clear, by "physical meaning" I mean something like - what physical effect do these components generate? Or, they make the GR solutions deviate from Newton because of xxx factor... or something similarly physically intuitive.


title: Frequency of kinetic energy in shm

text: I am currently learning about simple harmonic motion. In a book I am reading it says frequency of kinetic energy is twice the frequency of velocity for a harmonic oscillator by showing velocity vs time graph and KE vs time graph What I think it means to say is that kinetic energy reaches its maximum value 2 times in the same time it takes for velocity to reach its maximum value. Am I right?


title: pressure required for displacing a single electron off a crystal

text: I need to know this for my project- "power generation using the pressure applied on a keypad of a mobile electronic device". How much pressure does it take to displace a single electron off its crystal structure so that it is suitable for generating piezo-electricity?


title: Practical matter of the Higgs-Mechanism

text: My maybe very naive question is, of what practical importance will the discovery of the Higgs-Mechanism be for our technological advance in the near future?


title: What good is extraordinary matter to the consumer?

text: > **Possible Duplicate:**   >  Practical matter of the Higgs-Mechanism This video claims that for everyday matter you only need electrons, upquarks, and downquarks. So are all these other fermions useless? Has anyone ever made a shovel out of strange-charm-muon atoms or something?


title: What's next after Higgs Boson discovery?

text: > **Possible Duplicate:**   >  Practical matter of the Higgs-Mechanism As everybody knows that the Higgs Boson was discovered on July 4th,2012, I am so curious about it.   1. What are the possible outcome applications of Higgs Boson?   2. How Higgs Boson is related to Quantum Mechanics?   3. Does Higgs Boson fits in General Theory of Relativity?   4. Does Higgs Boson fits in String Theory? Please share the knowledge. I am very curious about it.


title: Not-so-hot black shirt

text: As it is getting warmer here by the minute I was asking myself: **Are there materials, that are black (in th visible range) but reflect (most) invisible light?** Furthermore, I asked myself **what fraction of heat could be kept** away using this material in daylight. For bonus points, it would be great if it could be used as some kind of fabric. I am not looking for a specific material but am curious, as e.g. metals do reflect all light up to a certain frequency and thus would not suffy. Therefore it would be nice to understand the underlying mechanism. In the winter of course it would be nice to reverse this, so to have white shirts, still getting hot in the sun.


title: What does it mean for an electromagnetic structure to be resonant?

text: There are many electromagnetic structures used in microwave engineering and EM devices. For example, patch antennas, metamaterials made from unit cells, etc. When they design structures like patch antennas, split-ring resonators in metamaterials, and many other similar microwave structures, **they want the device to resonate at a required frequency** , and somehow, the operation of the device depends on it being resonant at a given frequency. What is the meaning of this _resonance_ exactly? Why, for example a patch antenna or a split-ring resonator needs to resonate to function properly?


title: Attraction and repulsion of electron spin ups and electron spin downs

text: Alright, we know that copper is a diamagnetic material, which has paired electrons. These paired electrons have different spin. I'm specifically interested in what is going on with the electrons in a copper coil, when creating an induced current with a magnet. 1 Question: "Can you break the bond between an electron spin-up and electron spin-down in a diamagnetic material such as copper - using an external magnetic field?" 2 Question: "If yes, then could you say that a bar magnet with a north-pole would attract all the spin-ups and repulse all the spin-downs, due to the fact that spin-ups has south pole at the top, and spin-downs has north at the top. This would then create an opposing magnetic field, so that there in the copper would be a north-pole at the top, and a south-pole at the bottom." 3 Question: "If further yes, could you say that the movement of the electron spin-ups and electron spin-downs is the so called induced current." 4 Question: "And when the bar magnet is pulled out of the coil, the spin-ups and spin-downs will naturally find each other again, and bind and create a zero-magnetic field." Statement: If I make experiments with the galvanometer, I can see that the current is going different ways, depending on which pole I use, which in my mind can prove to be a valid proof that the bar magnet is attracting and repulsing at the same time. Either spin ups og spin downs, regarding which pole being used. 5 Question: "If all this is wrong, and you cannot break the bond between spin- ups and spin-downs, then what exactly is the bar-magnet doing to the electrons?" Many Regards


title: Equal and opposite electron flows in a magent?

text: If the electron flow in a magnet is say from South pole to North pole, then I can understand why you can't put the two north ends of magnets together, as the two flows of electrons repel each other. But then why can't I put the two south pole ends of a magnet together? If the electrons flow out the North pole and into the south pole, then what force is repelling the two south poles?? If the answer is to say there are two electron flow, in opposite directions, then why don't they just cancel each other? There must be "something" else flowing in the opposite direction of the electrons that causes the two south poles to be repelled. Edit: Perhaps electron flow was bad choice of words. Every diagram of a magnet I see shows the lines of flux flowing from out of North pole and into South pole. Given this flow of "whatever", it is easy to understand why the North poles would repel each other. It is also easy to understand why the North pole of one magnet is attracted to the South pole of another. My question really is why do the South poles also repel each other? If the magnetic field is really flowing "into" the south pole, then what force causes the two South poles to Repel each other.


title: Hyper surface orthogonal vector in Boyer-Lindquist coordinate

text: The Boyer-Lindquist coordinate coordinate of the Kerr Solution is $$ ds^2=\left(1-\frac{2Mr}{\Sigma}\right)dt^2+\frac{4Mar\sin^2\theta}{\Sigma}dtd\phi - \frac{\Sigma}{\Delta}dr^2-\Sigma d\theta^2-\left(r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2 $$ Let $$ K=\frac{\partial}{\partial t}=(1,0,0,0) $$ I am asking if it is possible for the vector field $K$ to be hypersurface orthogonal. The answer is that it is possible if and only if $J=Ma=0$, i.e. the cross term in the metric, $g_{t\phi}$, vanishes. I can figure it out only by the following (incorrect) argument. **Assuming we want $K$ to be orthogonal to the surface $t=$ constant** , the let $(t_0,r,\theta,\phi)$ be a curve in that surface, thus $V=(0,\dot r,\dot\theta,\dot\phi)$ is a tangent vector in that space. To make $K$ orthogonal, we must have $$ \langle K,V\rangle=0 $$ which is $$ \dot\phi g_{t\phi}=0 $$ so we must have $g_{t\phi}=0$. But I think the above argument is incorrect, since we only proved that $K$ cannot be orthogonal to the surface $t=$ constant, there maybe other possibilities. Then I tried to use the Frobenius Theorem, saying that $K$ is HSO if and only if $$ K_{[a}\nabla_bK_{c]}=0 $$ We have $$ K_a=K^bg_{ab}=g_{at} $$ Then equivalently we must show $$ g_{at}\nabla_bg_{ct}+g_{bt}\nabla_cg_{at}+g_{ct}\nabla_ag_{bt}=0 $$ But I don't know how to move on. Furthermore, if $\nabla$ is Levi-Civita, i.e. it is metric compatible, the above equation holds naturally, no matter whether $g_{\phi t}=0$. Could anyone point out to me how to deduce that $g_{t\phi}=0$ from this approach?


title: efficiency of reversible carnot engine

text: ![enter image description here](http://i.stack.imgur.com/IkDmz.jpg) here two reversible engines working between same temperature(1-a-b-c-a ;2- a-b-c-d-a),by carnot's theorem they will have same efficiency, but i could not understand from figures as the out put (closed area) of 1st one is less than second one , but they took same energy from the heat source, so where i am misleading? please explain from the diagram


title: Motional EMF and EMF?

text: What is the difference between motional EMF = $-vBL$ , and Faraday's law of induction $\displaystyle\mathcal{E} = \left|\frac{d\Phi_B}{dt}\right|$? Aren't they the same? What is the relation of Lorentz force to motional EMF?


title: If quantum computing requires hundreds of digits of accuracy, how will it be possible?

text: Leonid Levin said, "Exponential summations used in QC require hundreds if not millions of decimal places accuracy. I wonder who would expect any physical theory to make sense in this realm." See https://groups.google.com/forum/m/#!msg/sci.physics.research/GE5cz3xefCc/e0eh34MZGdwJ Given that no machine has ever been designed to be sensitive to physical quantities to hundreds of digits of accuracy, how will quantum computing ever be possible in the real world? To explain what I mean, in a QC model, the state vector has exponential size dimension in which the squares of the entries add to one. So if all of the entries are equal, when they are rounded to say the billionth digit, they will all be zero on a 100 qubit machine, contradicting the fact that they all must add to one. This is a big problem. EDIT: I think the question is this: How can we possibly perform sensible measurements on quantum computers, given their extreme sensitivity?


title: The relation between the movement of electrons and energy

text: So, I've been enjoying reading a lot of helpful posts, but now, I found myself in the need of asking something. I have a hard time grasping the general concept of electricity / how the relation between the movement of electrons and energy corresponds. But so far I've reach this conclusion / guess: 1) The movement of electrons, should rather be called mechanical energy, than calling it electric energy. Why? Electrons move by pushing to each other, typical mechanical energy example. That electrons has a electric charge, doesn't make the movement of them electric energy. 2) Every single electron has their own electric field. When there is a potential difference between minus and plus (electroncurrent), we say that there will be created an uniform field, because of the field lines between two parallel conducting plates with oppositely charges. Meanwhile, when electrons move in the same direction, they create a magnetic field. **And now comes my guess:** My statement: It is the effects of the magnetic and electric field in the wire, that do the work on electronic devices. We know that fields of all kinds can exert a force on any given object within its reach. So my guess is that this electromagnetic field the wire creates, can effect: a smaller wire in a computer, the wires that goes to the screen, the way the screen lights up. So you could say that the energy / what I would believe people in general thinks is 'electricity' exists not only inside the wire, but also outside. So when a device is 'using' / 'transforming' the energy to other energyforms, you could think of it this way. 1) Remembering that a current exists in a closed circuit. 2) When you exert a force from a battery or whatever, the electrons move very slowly themselves, but all in all as a system, they move at the speed of light. When they do this, the electromagnetic field will be a natural follower of this event. So when we transform energy/ use energy, it is the constant need for making the electrons move around, in order to create the electromagnetic field. And when there is no force to push the electrons around, the electromagnetic field disappears - and no work is done, and no money spent. This would also mean / or be the reasoning behind why devices / light or what so ever, starts in splitseconds. Because the electromagnetic field comes at the speed of light, because the electrons move the way they do. And using the expressions as potential energy, is just a way of saying that you create a strong difference between minus and plus (electroncurrent.) And the electromagneticfield will be "tapped" when there is devices nearby, but the current logically remains the same, buut, the electromagnetic field will be weaker near plus, because it has already done a lot of work. But we might be taking about fractions of seconds, so we probably wouldn't register it.. (is this right or just totally wrong reasoning??) So saying electricity, could in daily life relate to: the movement of electrons or the effects of electromagnetic field exertion on electronic devices. Many thanks in advance!


title: Reading list in topological QFT

text: I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm looking for pedagogical reviews rather than original articles. Though these things sit at the interface of mathematics and physics I'm interested in them more as a physics student. I remember someone asking for a suggested reading list for topological QFT in mathoverflow. The suggested papers were almost uniformly rigorous mathematics written by mathematicians. I am not looking for something like that.


title: References for Chern-Simons action $ \int d^3 x ~\epsilon^{\mu \nu \rho} A_{\mu} \partial_{\nu} A_{\rho}$

text: Can you recommend to me good references on this action? From basics to more advanced stuff.


title: Weightlessness on Earth

text: Would the following work? Imagine one would 1) create a straight 4000 km long tube (at ground level), following the curvature of the Earth, wide enough to hold a pod the size of an aircraft cabin; 2) remove the air in front of the pod and make sure it accelerates to a speed of 28000 kph. Would one achieve weightlessness inside the pod for ten minutes? If not, what are the obstacles?


title: Why isn't jumping from a high altitude fatal?

text: After seeing this answer claiming that displacing matter "In a very short time", "no matter whether the matter is solid, liquid, or gas" (even though he concludes that falling from a high altitude is fatal, independent of this). I wondered why then is the jump itself not fatal, considering that there is a significant amount of "gas", that does need to be displaced before even hitting water. Is it because there isn't enough mass per square inch to be fatal? And if so, at what speed would it be fatal? Or is there something else I or the guy who answered that question is missing?


title: Precession period matching solar year

text: Consider the case of a tidally locked planet: its axis of rotation must be perpendicular to the plane of its revolution around the parent star. Therefore, no precession. **It is possible for a planets to have a precession period that is the exact same time period as the planet's solar year** , such that **the planet rotates however its seasons never change?** One pole would be in perpetual sunlight (starlight) and the other in perpetual darkness. I am aware of the physical mechanism which causes stable tidally-locked planets, however could there be a physical mechanism which would cause a **stable precession-year ratio of 1:1**? I am interested in the subject for the dual purpose of better understanding precession, and additionally as a plot ploy in a short story that I am authoring. Therefore answers which tend towards either or both of those goals are most appreciated.


title: How to calculate new air pressure with temperature change?

text: > The question: _The gauge pressure in your car tires is $2.60 * 10^5 N/m^2$ > at a temperature of $35.0°C$ when you drive it onto a ferryboat to Alaska. > What is the gauge pressure later, when the temperature has dropped to −28°C? > Assume that the volume has not changed. Answer needed in atm._ Okay, Ive been working on this problem for about two hours already and I just cant seem to make it work, I know Im missing something with the equations but I cant seem to figure it out. I know that PV=NRT and that in the problem I am given initial P, and T and R is the gas constant, leaving N and V as unknown. I also know that I can use V=RT/P to find the volume (I got .009886, and I dont think that is right) and then plug that back in to calculate N (I got 1.659795*10^-32, not sure about that) at the given temperature (35C=309.15K) but when I do all this I get an outrageous answer that cant possibly be right (P=5.703*10^-59? no way on earth). I dont know if I am maybe using the wrong formula, maybe the wrong constant, messing up on algebra somewhere or if my calculator is just fudging my answer (I do that sometimes). Someone please give me some pointers on how to set this problem up to get the right P at the end! Thank you so much for the help guys, I think I actually understand what is going on with the algebraic organization (for once). When I do it myself I get an answer of 207,016 Pa which is 2.0431 atm, but the automated system I am using keeps telling me that is a wrong answer.


title: Three-body correlation function in kinetic theory

text: In Kinetic Theory, one studies the evolution of a system of $N$ particles interacting with each other. We use the notation $\boldsymbol{w}_{i}$ to describe the coordinates in phase-space of each particle. Liouville's Equation gives the incompressible evolution of the $N-$Body distribution function $$f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N})$$ One can then define the reduced distribution function $f_{i}$ , for $1 \leq i \leq N$ as $$f_{i} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{i}) = \int d \boldsymbol{w}_{i+1} ... \boldsymbol{w}_{N} \; f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N}) $$ A system is said to be separable, if we have $f^{(N)} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{N}) = \prod\limits_{i = 1}^{N} f_{1} (\boldsymbol{w}_{i})$. In order to measure the $\textit{distance}$ to measurability, one first introduces the two-body correlation function $g_{2} (\boldsymbol{w}_{1} , \boldsymbol{w}_{2})$ defined as $$f_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) = f_{1} (\boldsymbol{w}_{1}) \, f_{1} (\boldsymbol{w}_{2}) + g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) $$ This two-body correlation function plays a crucial role in the collision term appearing in the first equation of the BBGKY hierarchy. My question is : What is the definition of the three-body correlation function $g_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3})$ ? My feeling would be that it is defined as $$ \begin{aligned} f_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3}) = &f_{1} (\boldsymbol{w}_{1}) \, f_{1} (\boldsymbol{w}_{2}) \, f_{1} (\boldsymbol{w}_{3}) \\\& \+ \; f_{1}(\boldsymbol{w}_{1}) \, g_{2} (\boldsymbol{w}_{2},\boldsymbol{w}_{3}) + f_{1} (\boldsymbol{w}_{2}) \, g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{3}) + f_{1} (\boldsymbol{w}_{3}) \, g_{2} (\boldsymbol{w}_{1},\boldsymbol{w}_{2}) \\\& \+ \; g_{3} (\boldsymbol{w}_{1},\boldsymbol{w}_{2},\boldsymbol{w}_{3}) \end{aligned}$$ Is this definition correct ? Bonus question : Would you have a nice expression for the general $n-$body correlation function $g_{n} (\boldsymbol{w}_{1},...,\boldsymbol{w}_{n})$ ?


title: Correlation between outstanding hints in experimental particle physics

text: The 115 GeV ATLAS Higgs with enhanced diphoton decays has gone away but there are several other recent tantalizing hints relevant for particle physics, namely   * CoGeNT's 7-8 GeV dark matter particle that seems to support DAMA's and other signals   * D0's fourth-generation top prime quark that may be there at 325 GeV if a 3-sigma signal is right   * CDF's "new" force Z prime boson at 144 GeV which could be there if another 3-sigma bump is real   * Tevatron's top-antitop asymmetry above 400 GeV which seems very large   * A related CDF top-quark anomaly and maybe others I missed. My question is simple: > Is there some sensible theoretical basis (e.g. paper) that would > simultaneously explain at least two of the observations above if they were > real?


title: 7 GeV dark matter particle: how particle accelerators missed it?

text: This posting is regarding the recent confirmation of the DAMA results that might be due to underlying differences in proton and neutron cross section with the dark matter particles, which reflect on the differences between Xenon and Germanium detectors in the early days of particle physics, neutrinos were discovered as missing momentum in decay events my question is the following: **How did the current accelerators missed a dark matter product of 7GeV?** possible answers i think: 1) does by any chance theory predict that these relatively low energy particles will not be created with normal particle collisions in our current multi-TeV range? 2) does the current complexity of analysis of TeV-magnitude collision products just cannot detect such missing momenta signals? 3) there are no dark matter particles in such low energy range, otherwise we would have already spotted them in particle accelerators references: http://motls.blogspot.com/2011/05/cogent-sees-seasons-and-maybe-dark.html http://sixwoffers.blogspot.com/2011/05/second-experiment-hints-at- seasonal.html


title: Absorption/Extinction formula of nanoparticles

text: I know the absorption/extinction equations in nanoparticle physics should be: $$Q_{abs}=\frac{1}{2}\mathbf{Re}\int \mathbf{J}_{tot}\cdot\mathbf{E}_{tot}^\ast dV=\frac{\omega}{2}\mathbf{Im}(\epsilon)\int|\mathbf{E}_{tot}|^2dV$$ also, for the extinction, it reads: $$Q_{ext}=\frac{1}{2}\mathbf{Re}\int \mathbf{J}_{tot}\cdot\mathbf{E}_0^\ast dV$$ But I see in some papers people use the following equations: $$Q_{abs}=\frac{\omega}{2}\mathbf{Im}(\mathbf{d}\cdot\mathbf{E}_{inside}^\ast)$$ where $\mathbf{d}$ is the total dipole moment of nanoparticles. Also, for the extinction, it reads: $$Q_{ext}=\frac{\omega}{2}\mathbf{Im}(\mathbf{d}\cdot\mathbf{E}_0^\ast)$$ I failed to derive the two equations. Can anyone give some help? Or, some reference papers would be also very helpful. Thanks a lot for the help.


title: Only gravitation and Newton's $2^{\mathrm{nd}}$ law needed to derive Kepler's laws?

text: It is known that Kepler's laws of planetary motion can be derived from Newton's laws of motion and his law of universal gravitation. However, are all of Newton's laws of motion necessary? According to this website, all of Kepler's laws can be derived from gravitation and the Newton's second law ($\mathbf{F} = m\mathbf{a}$). Does this mean that Newton's first and third laws are irrelevant for planetary motion? Perhaps Newton's third law (action-reaction) is necessary to take into account the motion of the Sun as induced by gravitational pull from the orbiting planets, but when deriving Kepler's laws the Sun is assumed to be fixed...


title: Self-dual Maxwell equations, the second homology group, and topological invariants of a four manifold

text: In Witten's paper _Quantum Field Theory and the Jones Polynomial_ , he mentioned that: > Geometers have long known that (via de Rham theory) the self-dual and anti- > self-dual Maxwell equations are related to natural topological invariants of > a four manifold, namely the second homology group and its intersection form. **Q: can someone illuminate this more explicitly?** I suppose that self-dual and anti-self-dual Maxwell equations is source-free vacuum Equation of motions of $U(1)$ E&M in 3+1 dimensional spacetime. How do the **topological invariants of a four manifold, namely the second homology group and its intersection form** come into the story? How does **de Rham theory** help to understand this? Thanks for any comment or answer!


title: Gauss’s Law inside the hollow of charged spherical shell

text: Use Gauss’s Law to prove that the electric field anywhere inside the hollow of a charged spherical shell must be zero. My attempt: $$\int \mathbf{E}\cdot \mathbf{dA} = \frac{q_{net}}{e}$$ $$\int E \ dAcos\theta = \frac{q_{net}}{e}$$ $$E \int dA = \frac{q_{net}}{e}$$ $E\ 4\pi r^2 = \frac{q_{net}}{e}$ and since it is a hollow of a charged spherical shell the $q_{net}$ or $q_{in}$ is $0$ so: $E = 0$. Is my reasoning on this problem correct? Essentially $E$ is $0$ because there is no charge enclosed.


title: Electric Field due to a disk of charge. (Problem in derivation)

text: This might be a really silly question, but I don't understand it. In finding the electric field due to a thin disk of charge, we use the known result of the field due to a ring of charge and then integrate the relation over the complete radius. But I had a problem in the derivation, as follows: We assume a ring at a distance $r$ , and of an infinitesimal thickness $dr$ from the center of the disk. Then the very next step in every book I've referred is $dA = 2 \pi rdr$. This is what I don't understand. Shouldn't the area be $$dA = \pi [ (r+dr)^2 - r^2] ~ ?$$ Please help me out here.


title: Lorentz Transformations and Aberration

text: Why does the azimuthal angle, $\phi$, remain unchanged between reference frames in special relativity? Edit: Here is a link from "Radiative Processes in Astrophysics": http://i937.photobucket.com/albums/ad216/Kyanise/Capture-5.jpg. I think this comes from the aberration formula, showing dependence only on the polar angle, $\theta$. The aberration formula is: $\tan \theta = \frac{u_\perp}{u_{||}} =\frac{ u' \sin \theta '}{\gamma (u' \cos \theta ' + v)}$ where ' indicates a property of the moving frame. But I'm not even sure why this is true. I know it comes from the transforms of the velocities, but why should $\tan \theta = \frac{u_\perp}{u_{||}}$ anyways? I mean $v$ can be in any direction, right? How come there is no dependence on $\phi$?


title: Minimum connectivity required for mean field to be a good approximation?

text: In spin models, it is known that mean field becomes a better approximation as the connectivity increases. My question is: Is there an estimate for the threshold connectivity (as a function of the system size, $N$), such that for connectivites greater than the threshold, mean field is a good approximation? The values I am mainly interested in are the free energy and the magnetization, so the mean field approximation should be good in the sense that these values are accurate.


title: Question about the vacuum bundle on A- and B-model

text: Let us consider the topological string A- and B-model (twisted SUSY non-linear sigma model on CY 3-manifold $X$). They are realization of $N=2$ SCFT and there are ground-states vector bundle $\mathcal{H}$ and vacuum line bundle $\mathcal{L}$ over the moduli space of the theory. In the A-model, $\mathcal{H}=H^{even}(X,\mathbb{C})$ and $\mathcal{L}=H^0(X,\mathbb{C})$. In the B-model, $\mathcal{H}=H^{3}(X,\mathbb{C})$ and $\mathcal{L}=H^{3,0}(X,\mathbb{C})$. The genus $g$ string amplitude is given as a section of $\mathcal{L}$ in either theory. Mirror symmetry is an identification of the geometry of A- and B-model ground-state geometry on distinct CY 3manifolds. My questions is following. In the A-model, it seems the splitting of the bundle $$ \mathcal{H}_A=H^{even}(X,\mathbb{C})=\oplus_{i=0}^3 H^{2i}(X,\mathbb{C}) $$ does not vary over the moduli space of the theory (Kahler moduli space). On the other hand, in the B-model, the splitting $$ \mathcal{H}=H^{3}(X,\mathbb{C})=\oplus_{p+q=3}H^{p,q}(X,\mathbb{C}) $$ varies over the moduli space (variation of Hodge structure). Moreover, $\mathcal{L}$ is the trivial line bundle in the A-model, while it is not in the B-model. Isn't this contradiction?


title: Atwood machine problem

text: ![enter image description here](http://i.stack.imgur.com/iQ5Zg.jpg) Sorry for the bad drawing, but I hope that this will help you get a hold of the problem. Consider an Atwood Machine with a total of two blocks, a mass less pulley, ideal string. One block rests on the floor, while the other one is at a height (H). Now, the string near the block that rests on the surface is slack. So, the other block falls freely, and later induces a jerk in the other block. How do I calculate the initial velocity of the two block just after the string is taut. My approach was: 1) Calculate the velocity of the block in motion (initially) at the point when the is ALMOST taut. 2) Now conserve mechanical energy b/w the point where string was almost taut but impulse wasnt generated, and the point where impulse was generated, and the second block had JUST started motion. But, my book says, I should conserve Linear Momentum between the same two points. I think this is wrong, because, The string that holds the pulley in place, will have an impulsive tension as soon as the impulse is generated in the string that joins the two masses. According to you, what is right, and why?


title: Symmetry breaking of the hidden sector

text: Lately, I have some read some papers about the hidden sector of particle physics which combines with the Standard Model through the so-called Higgs portal. Let the Lagrangian for this be composed of two simple scalar fields like this: $L=\partial_\mu \phi_{SM} \partial_\nu \phi_{SM} +\partial_\mu \phi_H \partial_\nu \phi_H -V(\phi_{SM},\phi_H)$ where $\phi_{SM}$ relates to the Standard Model and $\phi_H$ relates to the hidden sector. Assuming that the potential is: $V(\phi_{SM},\phi_H)=-1/2 \mu^2 {\phi_H} ^2 + 1/4 \lambda {\phi_H}^4 - 1/2 \mu^2 {\phi_{SM}} ^2 + 1/4 \lambda {\phi_{SM}}^4+ 1/4 \lambda_{mix} \phi_H^2 \phi_{SM}^2$ And since both of the fields gain a nonzero VEV like this: $\phi_H$=$v_H+h(x)/2^{1/2}$ $\phi_{SM}$=$v_{SM}+h(x)/2^{1/2}$ How would the spontaneous symmetry breaking mechanism then work? I am interested in how exactly the Higgs mechanism would work mathematically when two different mimimas "v" are involved.


title: What is the extent of the Galactic Magnetic Field?

text: What is the extent of the `Galactic Magnetic Field`? What does the `Galactic Magnetic Field` look like from afar (such as half-way between the `Milky Way Galaxy` and `Andromeda Galaxy`)? Does the `Galactic Magnetic Field` interact with other galaxies's magnetic field? Also, is there an `Inter-Galactic Wind` interacting with all of the above (in similar nature to how the `Solar Magnetic Field` interacts with the `Galactic Wind`)?


title: Inductance of a solenoid?

text: My textbook is using Faraday's law to explain the self inductance that happens in a solenoid with changing current. According to Wikipedia, Faraday's Law is: > The induced electromotive force in any closed circuit is equal to the > negative of the time rate of change of the magnetic flux through the > circuit. The loops in a solenoid aren't closed - it's loops are "almost closed", but the beginning and end don't touch each other. So how can Faraday's Law be used?


title: Simple Vector Problem

text: I'm a bit stuck on a homework question that I've been assigned. The question is as follows: > _You are paddling a canoe at a speed of 4 km/h directly across a river that > flows at 3 km/h. (a) What is your resultant speed relative to the shore? (b) > In approximately what direction should you paddle the canoe so that it > reaches a destination directly across the river?_ Below the question is a little diagram of a canoe in a river, perpendicular to the shore, with an arrow pointing to the right, labelled 4 km/h, and an arrow pointed downwards, labelled 3 km/h. I understand the first bit of the question, and have established that the resultant speed relative to the shore will be 5 km/h. Can you please help me with the second half?


title: Coupling constraint in massless Thirring Model in (1+1) Dimensions

text: In Coleman's paper, "Quantum sine-Gordon equation as the massive Thirring Model" (Link to the PRD paper http://prd.aps.org/abstract/PRD/v11/i8/p2088_1), he pointed out that the massless Thirring Model is sensible only for coupling $g>-\pi$. I do not understand why this is true. Can any one offer me an explanation to this? Is there relevant paper discussing this issue? Note the Lagrangian is $\cal L = \bar{\psi}i\partial_{\mu}\gamma^{\mu}\psi-\frac{g}{2}j_{\mu}j^{\mu}$, here $j_{\mu}$ is the usual current.


title: Does time exist in a vacuum?

text: Imagine a perfect vacuum devoid of all matter and radiation. Irrespective of whether such a space can even exist, my intuition tells me that time would not exist within such a space. What this thought experiment tells me is that the human concept of time is basically that it's a measure of motion. Whether its a car speeding down a highway, skin on your face as the years go by, or photons, it's really all about motion. Now what about something like a gravitational field? Could that also be excluded from acting within a perfect vacuum?


title: Expanding the Magnetic field?

text: A magnetic field $B$ can apply a force $X$ on a ferromagnet when there is no gap to separate them, $r$ = 0. When that gap ($r$) is increased to 3 cm, the force drops to zero. How can that be solved? How can I expand the magnetic field so that it can be applied over larger distances? What methods are known to solve such a problem?


title: Caculating thermal stresses in a composite rod

text: **The Question** > Between two rigid walls separated by a distance $l_1+l_2$, two rods of equal > cross-section area $A$, and length, coefficient of linear expansion and > Young's modulus $l_1, \alpha_1, Y_1$ and $l_2,\alpha_2,Y_2$. Suppose the > temperature of the entire system is raised by $T$ kelvins. What is the force > the two rods exert on each other?   >  ![enter image description here](http://i.stack.imgur.com/XhGMT.png) **Concepts Involved**   1. Tendency to expand of the two rods   2. Stress generated due to the constraint on total length which limits their expansion   3. Different increases in length of both the rods owing to different young's modulus and original lengths  **My efforts** Let at the increased temperature, the length of Rod 1 be $l_1+x$ and that of rod two be $l_2-x$. The "natural length" of both the rods at that temperature is $l_1(1+\alpha_1T)$ and $l_2(1+\alpha_2T)$. Therefore the longitudnal stress developed in each is:- $\delta l_1=l_1\alpha_1T-x$ and $\delta l_2=l_2\alpha_2T+x$. For the rods in equilibrium, the forces they exert on each other need to be equal. Therefore, using $F=\delta l/l\times AY$, and equating their forces we get:- $$\frac{Y_1l_2(l_1\alpha_1T-x)}{Y_2l_1(l_2\alpha_2T+x)}=1$$ This can be solved for $x$ which on substituting in $F=(l_1\alpha_1T-x)/l_1\times AY_1$, we should get the force. But no matter how much I rearrange this equation, it does not match with the given answer. **Is there a fault in my arguments/equations or calculations?** Sorry if there is trivial mistake.


title: Quantising the Electromagnetic Field in QED

text: ![enter image description here](http://i.stack.imgur.com/LW8GR.png) How exactly do you derive this result?