title: How to analyze this constraint question

text: Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. Consider the system with potential energy $$U_N = Nq_2^2+U_0(q_1,q_2)$$ depending on the parameter $N$. And the initial conditions are $$ q_1(0) = q_1^0, \qquad \dot{q}_1(0)=\dot{q}_1^0,\qquad q_2(0) = 0, \qquad\dot{q}_2(0)=0 $$ Denote $q_2 = g(t,N)$ the evolution of the coordinate $q_2$ under a motion with these initial conditions in the field $U_N$, then the question is to prove $$\lim_{N\rightarrow \infty} g(t,N) = 0$$ I have tried as following: Assume the motion is with the whole energy $E = T + U_N$, then $$q_2^2 = \frac{E - T - U_0(q_1,q_2)}{N}$$ If $U_0(q_1,q_2) \geq U_{min}$, then we know $$q_2^2 \leq \frac{E - T - U_{min}}{N}$$ which tends to zero as $N\rightarrow \infty$ for any $t$. But what if there is no lower bound for $U_0(q_1,q_2)$?


title: Creating a 2d magnet simulator

text: I need to create a "simple" magnet simulator, unfortunetely I have little to no physics knowledge. The best way to describe my idea is, imagine a hockey table. There's a disk and I need to throw it around. So far no problems, I just have to calculate the disk acceleration and position along the time based on an initial force. Now, let's suppose it's a magnetic disk and I want some fixed magnetic disks around the table. Now I want to throw the disk around the table again, but it have to have it's route modified when it goes near another disk(either by being repelled or attracted). I have been looking for a few hours but I can't wrap my head around these formulas and how to put them together. Can someone please enlighten me on how would I go about solving this problem?


title: Quantum Temperature?

text: I was thinking if temperature is proportional to the rate at which energy is transferred then does it exist on a quantum as well as a classical scale? If so what is it proportional to exactly?


title: What does the latest $B_s^0\rightarrow \mu^+\mu^-$ results mean for SUSY?

text: A paper from the LHCb collaboration just came out last week, stating basically that the $B_s^0\rightarrow\mu^+\mu^-$ decay matches standard model predictions, and people are already shouting that SUSY is dead. Is SUSY actually in trouble or is it just hype again?


title: What is the P-parity, T-parity and C-parity of graviton? Are these conserved in general curved space-time?

text: I'm curious about the P,T,C-parity of graviton? 1)Are these graviton's parities even or odd? 2)Is the C,P,T-parity alternatively conserved in Einstein gravity? And does the CPT theorem still hold in Einstein gravity? 3)And do these conservations rely on the background space-time?


title: QED: Would atoms without electrons be visible?

text: I have been reading a lot of QED books lately, and understand (as well as possible anyway) the interaction between electrons and photons. But I can't seem to get a clear indication of the interaction between photons and **protons**. It seems that normal light (not talking about high-energy levels or anything exciting, just the stuff that comes out of a light bulb) would be insufficient to really produce a reflection, but, so far, it depends upon who I ask. That said, to boil down what I am really trying to determine: Would otherwise- normal atoms (or matter, really) with no electrons be visible? Would the protons take up the role normally provided by the electrons and cause a similar scattering of light, or would it really just mess things up?


title: What is the weakest magnetic field that can practically be measured?

text: I would like to be able to measure magnetic fields that are generated by nerve impulses in extremities like in fingers. I know that they are very weak: about 100 mV electric potential would give us about current 10 mA if resistance = 1/10 Ohm, but interested in measuring magnetic fields. **How many rounds should you have in a coil to measure 10 mA current?** The jumping of electrons from one level to another may become a problem in this case. However, I am not sure where and how. Assume that there is a steady current in the situation so Biot-Savart law. The radius of the nerve is r = 0.003m.


title: Restoring Force

text: Consider a pendulum hanging straight down. Say I pull it to the $\textbf{right}$ some angle $\phi$. Using Newton's laws in polar coordinates, I find that in the that force in the $\hat\phi$ direction is simply the weight in that direction pulling it back down to equilibrium. $$F_{\phi} = -mg\sin\phi$$ I get that it is negative because the force acts to pull the pendulum bob back down to $\phi = 0$. Also, usually when Newton's laws are derived in polar coordinates, $\hat\phi$ is make to point counterclockwise. My question is what if the pendulum was pulled to the $\textbf{left}$ and let go? Would $F_{\phi}$ still have that negative sign? In polar coordinates, $\phi$ is positive going counterclockwise, which is the way the pendulum will go now that I pulled it to the left. However, I feel like it would still be negative. So does that mean $\phi$ is positive only when the pendulum is swinging away from equilibrium (it swings away from equilibrium in two directions, one counterclockwise and the other clockwise) and negative when it is swinging towards it? How can this be?


title: Gravitational Potential Energy to Kinetic Energy

text: When a yo-yo is released from a height $h$, the gravitational potential energy is converted to kinetic energy. However, the yo-yo obviously has less acceleration than $g$, $9.8\frac{m}{s}$. This means that the final speed of the yo-yo is less than what it would be if it were in free fall. It seems that the kinetic energy at the end is less than the potential energy at the beginning. What happened to the rest of the energy?


title: How does pressure relate to cosmological expansion?

text: In my cosmology class, we've been talking about pressure in the Friedmann Equation for acceleration: $$ \frac{\ddot{a}}{a}=-\frac{4 \pi G}{3} \left(\rho+\frac{3p}{c^2} \right) $$ In particular, I've been a bit hung up on this passage in the textbook: > Notice that if the material has any pressure, this increases the > gravitational force, and so further decelerates the expansion. I remind you > that there are no forces associated with pressure in an isotropic Universe, > as there are no pressure gradients. How, physically, does pressure do work decelerating expansion if there is no pressure gradient or forces involved? How can pressure affect any of the properties of spacetime?


title: Why do Photons want to be together?

text: So I've heard that when a photon flies by a atom excited enough to release a photon there's a good chance it will. Because Photons want to be together and have the same direction etc? Is this true? And then, why is it like this?


title: A flammable gas in a energized conductive container

text: I place a flammable gas (natural gas) inside a metal container and energize the container (apply AC current). Do I have any concerns at this point. I understand for combustion to occur I have to have oxygen available. I also know that some gasses combust underwater (torch welding). I assume it's because oxygen is applied. To extend my example, would there be any concerns from this community with someone using a natural gas pipe in their home as a ground?


title: Can auxiliary fields be thought of as Lagrange multipliers?

text: In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + B^a\partial_\mu A^{a\,\mu}+\partial_\mu\bar\eta^a(D^\mu\eta)^a,$$ and in the superfield formalism of SUSY, the field $F(x)$ also appears as an auxiliary variable: $$\mathcal{L}_\text{SUSY}=\partial_\mu \phi\partial^\mu\phi+i\bar\psi^\dagger\bar\sigma^\mu\partial_\mu\psi+F^*F+\ldots\,.$$ It is very tempting to view $B^a(x)$ and $F$ as Lagrange multipliers since their equations of motion leads to constraints. But, these variables do not enter into the Lagrangian linearly, like a conventional Lagrange multiplier. Rather, they enter into the Lagrangian _quadratically_. However, in Kugo and Ojima's paper Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theores (1978), they refer to the $B^a(x)$ fields as the 'Lagrange Multiplier' fields (p.1882). **So my question is** : _Can_ these auxiliary fields be viewed as Lagrange multipliers? and in what ways do they behave differently/similar to the conventional Lagrange multipliers that enter into the function _linearly_?


title: Virtual particles contributing to universal expansion.

text: > **Possible Duplicate:**   >  Virtual particle production in space-time. Does virtual particle production contribute to observed universal expansion?


title: Find magnetic scalar potential for superconducting sphere

text: In regions where $J = 0$, the curl of the magnetic field $B$ is necessarily zero (since $\nabla \times B = \mu_0 J$). Therefore $B$ can be written as $B = -\nabla V_m$, where $V_m$ is a scalar function of position.   A superconducting ball of radius $a$ is placed in an otherwise uniform magnetic field $B = B_0 \hat {z}$. Since the $B$ field is zero inside the ball, it must hold that $B_{\perp} = 0$ just outside the surface of the ball (by magnetic boundary conditions).   (a) Find the magnetic _scalar_ potential $V_m (r, \theta)$ everywhere.   Far from the sphere the field is $B_0 \hat {z}$, and hence $V_m \rightarrow -B_0 z$. So in spherical coordinates, $V_m \rightarrow -B_0 r \cos \theta$ for $r \gg a$. Is the following true? $$ \frac {\partial V_{\text{above}}}{\partial n} = \mu_0 K $$ Since $V_{\text{below}} = 0$ due to the sphere being a superconductor. Using separation of variables, the first condition yields $$ \sum_{l=0}^{\infty} A_l r^l P_l (\cos \theta) = -B_0 r \cos \theta, $$ So, only one therm is present, $l = 1$. Then $A_1 = -B_0$ with all other $A_l$'s zero. From the second condition, $$ \frac {\partial V}{\partial n} = A_1 \cos \theta - \sum_{l =0}^{\infty} (l+1)\frac {B_l}{R^{l+2}} P_l (\cos \theta) = \mu_0 K. $$ Now I'm stuck as to how to proceed and remove some of the $B_l$ terms. Perhaps my boundary conditions are not correct.


title: Is flying really easier on smaller scales?

text: In the book Playing with Planets, the author makes the following argument, pertinent to flying robots of the future: > As it is, an important law of physics says that smaller organisms fly much > more easily than larger ones. This can be seen clearly in living organisms: > small animals have a lot less trouble getting off the ground than larger > ones. Therefore, once miniaturization sets in, we can expect to see lots of > small flying robots. I'm familiar with the upper-limit for size of birds, but I don't know if it can be expressed as a simple law of physics versus complications with the feather replacement intervals. On the other hand, insects have some of the highest mass-specific metabolisms of any animals on Earth. Most people's experience with RC helicopters is probably consistent with this - the flight time is typically a matter of minutes, severely limiting its usability. Large aircraft can fly clear across the world! For all of these reasons, the idea scaling laws would be favorable to small flying robotic drones seem to violate intuition. Is there any truth to the idea that smaller flying machines would fly easier? Is there an argument that can be made with Reynolds numbers? Tiny gnats can have Reynolds numbers in the 100s. Would laminar conditions set a lower limit on flight? How does scale affect the specific energy consumption needs of flying machines?


title: Will the plates of a parallel plates capacitor keep its charge after being charged then seperated from the non- conductor?

text: If i had 3 plates 2 metals and 1 glass .I put them together to form a basic parallel plates capacitor. After charging it and approaching it to an electroscope nothing happened that's because electric field outside a capacitor =0 . But what happens if i have separated the 3 plates , will the metal plates keep its charge?why? What happens when i approach them to the electroscope?


title: Wave function of IQH and FQH electrons

text: What are the wave functions of the ground state of Integer Quantum Hall (IQH) and Fractional Quantum Hall (FQH) electrons?


Wave function interpretation $y(x,t) = (0.35m)\sin(10\pi t-3\pi x + \frac\pi{4})$ I used to deal with function with one variable And now theres are two, how can I interpret them?   Is $10\pi$ still equal to $\omega$ ? how about $-3\pi x$ and $ \frac\pi{4}$? I just want to know the form of the equation? Just like $y = mx$ I know $m$ is the slope.


title: How Chern-Simons gauge field transform fermion to scalar?

text: In A.Zee's "QFT in a nutshell"book Page 324, after he wrote down the general Lagrangian, he said "in previous chapter, we learned that by introducing a Chern-Simons gauge field we can transform $\psi$ to a scalar field" However, I didn't found any clarification about this argument in previous chapter. Any comment or references are greatly appreciated.


title: Why does the overhand knot jam but the figure-8 knot doesn't?

text: After tensioning a rope with an overhand knot in it, it is often very hard if not impossible to untie it; a figure-8 knot, on the other hand, still releases easily. Why is that so? Most "knot and physics" literature seems more concerned with the _topology_ of knots (see e.g. here), but I suppose friction and elasticity of the rope play a role in knowing whether a knot jams or not. I found this reference that explains why the overhand knot breaks earlier than the figure-8 — the former has a higher maximum curvature than the latter. Why doesn't this larger curvature make the overhand knot **easier** to untie? The knot would have more stored elastic energy that would aid overcoming friction when picking the knot apart.


title: Eigenstates of a shifted harmonic oscillator

text: Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume that we add some kind of shift so that we get the following Hamiltonian (up to a constant) $$H = \omega a^\dagger a + \omega (a + a^\dagger) = \omega (a+1)^\dagger (a+1)- \omega $$ Is it possible to express the eigenstates of this shifted harmonic oscillator with respect to the old eigenstates? It should be possible by using a coherent state I guess, because a coherent state can be seen as kind of a 'shifted' number state. Do you have any ideas/experiences on how to do this? The background of the question is that I want do diagonalize a similar hamiltonian of two coupled systems, where the coupling is of the order of $\omega$.


title: Find work done by force along a path - is parameterization the only way?

text: $F = x^3y^4 \hat i + x^4y^3 \hat j$ from $(0,0)$ to $(1,1)$. I am given different paths. For example, "ﬁrst along x axis and then along the y axis" is one of the paths. Is this problem solvable only using parameterization or is there another way to handle/integrate it?


title: Square of Laplace–Runge–Lenz vector in Hydrogen atom

text: I have a problem. I've tried this question, but I don't get the correct expression. Can someone give me some ideas? Thanks! Consider the Hydrogen Atom Hamiltonian:   $$ H = (\mathbf p^2/2 \mu)-(e^2/r) $$ Define a vector operator: $$ \mathbf M = (1/2 \mu)(\mathbf p\times\mathbf L - \mathbf L\times\mathbf p)-(e^2/r)\mathbf r $$ Show that: $$ \mathbf M^2 = (2H/\mu)(\mathbf L^2+ \hbar^2) + e^4 $$


title: Relation between the spin of a particle and the polarization of it's wave

text: Is there any intrinsic relation between the spin of a particle, and the degree of freedom of it's polarization? does it holds for any particle-wave couple? like EM-photon, gravitational_waves-graviton etc?


title: Find angle of force vector in equilibrium

text: So my question is * * * Rope BCA passes through a pulley at point C and supports a crate at point A. Rope segment CD supports the pulley and is attached to an eye anchor embedded in a wall. Rope segment BC creates an angle of ϕ = 62.0∘ with the floor and rope segment CD creates an angle θ with the horizontal. If both ropes BCA and CD can support a maximum tensile force Tmax = 700N , what is the maximum weight Wmax of the crate that the system can support? What is the angle θ required for equilibrium? !(http://i.imgur.com/S45QRZO.png) * * * My first thoughts were to make the sum of all forces acting along each axis = 0 as it is in equilibrium. (X axis) TCD cos θ - TBC cos 62 = 0 (We know TCD = 700) Therefore 700 cos θ = TCB cose 62 (Y axis) 700 sin θ = TCA + TCB sin 62 = 0 But I have no idea where to go from here.


title: How does one derive the 2 halo term in two-point correlation function

text: This question is in reference to the paper here. In Equation (86) on page 28, the authors have given the two point correlation function \begin{equation*} \xi(\mathbf{x}-\mathbf{x}^{\prime}) = \xi^{1h}(\mathbf{x}-\mathbf{x}^{\prime})+\xi^{2h}(\mathbf{x}-\mathbf{x}^{\prime}) \end{equation*} where the first term comes when we compute the density correlation using the same halo. The superscript $1h$ stands for "1 halo term" which is given by \begin{equation*} \xi^{1h}(\mathbf{x}-\mathbf{x}^{\prime}) =\int dm \frac{m^2 n(m)}{\bar{\rho}^{2}}\int d^3y\ u(\mathbf{y}|m)u(\mathbf{y}+\mathbf{x}-\mathbf{x}^{\prime}|m) \end{equation*} This term is easy to derive once we use formula (83). However, the issue is with the 2halo term which is: \begin{equation*} \begin{aligned} \xi^{2h}(\mathbf{x}-\mathbf{x}^{\prime}) &=\int dm_1 \frac{m_1 n(m_1)}{\bar{\rho}}\int dm_2\frac{m_2n(m_2)}{\bar{\rho}}\int d^3x_1 u(\mathbf{x}-\mathbf{x}_{1}|m_1)\\\ &\int d^3x_2 u(\mathbf{x}^{\prime}-\mathbf{x}_{2}|m_2)\xi_{hh}(\mathbf{x}_1 - \mathbf{x}_2|m_1,m_2) \end{aligned} \end{equation*} where $u$ is the normalized density profile i.e $\int d^2x\ u(\mathbf{x}-\mathbf{x}^{\prime})=1$. I have two questions regarding this:   1. How do I exactly derive the 2halo term (I tried using Equation for $\rho(\mathbf{x})$ as given in Equation (83) but, it does not give much).   2. Also, what does the term $\xi_{hh}$ physically imply?


title: Criticism of the Thomas precession literature

text: From an earlier version of the Wikipedia article on Thomas precession concerning TP and LP=Larmor precession, regarding the paper: G B Malykin, "Thomas precession: correct and incorrect solutions", Physics-Uspekhi 49 (8) 837-853 (2006) > In a 2006 survey of the literature Malykin notes that there are numerous > conflicting expressions for Thomas precession. This is partly explained by > the fact that different authors use "Thomas precession" to refer to > different things, often without saying what they are referring to and > subsequent authors then misinterpret the results and apply them to other > things, but, even taking this into account, some of the expressions in the > literature are just plain wrong. > > Malykin explains the source of some of these errors: _"We emphasize that > Thomas considered the rotation of the axes of the coordinate system > accompanying the electron in its motion rather than the electron spin > rotation. Subsequently, this led to a misunderstanding and the emergence of > incorrect work on the TP problem. **It is possible to introduce three > different reference frames accompanying the electron motion** around a > circular orbit and, in the most general case, along a curvilinear > trajectory: (i) a reference frame whose coordinate axes remain parallel or > retain their angular position relative to the axes of a laboratory IRF, (ii) > a reference frame one of whose coordinate axes is always coincident with the > electron velocity vector, and (iii) a reference frame in which the electron > spin vector retains its orientation relative to the coordinate axes. It is > evident that the electron spin vector precesses relative to the coordinate > axes of the two first systems, but the angular velocity of its precession is > different in these systems. ... In several papers concerned with the TP, > calculations are performed in the first approximation in v^2/c^2, where v is > the speed of an elementary particle in the laboratory IRF and c is the speed > of light. In this case, all authors arrive at the same expression first > derived by Thomas, this being so irrespective of whether they consider the > relativistic rotation of the particle spin or the relativistic rotation of > the axes of the coordinate system comoving with the particle. In the most > general case, however, the expressions for the TP obtained by different > authors are radically different. As noted above, the problem is complicated > by the fact that different authors assign different meaning to this > expression: some imply the relativistic rotation of the particle spin in the > laboratory IRF, some in the comoving reference frame (in this case, as noted > above, the rotation law for the axes of this system may be defined in three > ways), while others refer to the relativistic rotation of the axes of the > coordinate system accompanying the particle in motion. ... As noted above, > the expression for the TP in Thomas's first paper was obtained in the first > approximation in v^2/c^2 and is always correct when this condition is > fulfilled. In his subsequent work, on performing calculations for an > arbitrary electron velocity v, Thomas derived an expression that correctly > describes the relativistic rotation of the axes of the comoving coordinate > system relative to the rest-frame (laboratory) system. However, because the > majority of authors use the term TP in reference to the precession of the > spin of an elementary particle, this subsequently led to several errors and > misunderstandings. ... **In 1952, in his famous monograph [78], the Danish > scientist C Møller (1904-1980), an acknowledged expert on the theory of > relativity, derived an expression for the TP that coincides, up to a sign, > with the corresponding Thomas expression and is correct in the comoving > frame of reference. However, it was stated in Ref. [78] that this expression > was written for the laboratory IRF, which is incorrect. Møller's immense > scientific prestige played a negative role in this case: since then, the > majority of authors of papers, monographs, and lecture courses started using > the expression for the TP from Ref. [78] or, in the derivation of suchlike > expressions, tried to make them coincident with that given in Ref. [78].** > ... At the same time, because the experimentally observed particle spin > precession is caused by the sum of two effects, the TP and the LP, it is > possible to choose different expressions for each of these effects, only > provided that these expressions add up to correspond to expression (14). we > conclude that the preferred method is the above-discussed method of > recording the TP with the aid of mechanical gyroscopes in their orbital > motion because quantum mechanical effects in experiments on charged > elementary particles partly complicate the interpretation of experimental > data."_ Question: Is Malykin's criticism of the literature correct ?


title: Can a nuclear reactor meltdown be contained with molten lead?

text: If lead can absorb or block radiation, would it be possible to pump molten lead into a reactor core which is melting, so that it would eventually cool and contain the radiation? Is there something that can be dumped into the core that will both stop the reaction (extremely rapidly) AND will not combine with radioactive material and evaporate into the atmosphere, thus causing a radioactive cloud?


title: What do gravitons interact with?

text: The other three forces' mediating particles (photons etc.) are absorbed by their appropriate charge-carrying particles, but I can't seem to find a clear answer that applies to the gravitational force in a quantified scenario. The different answers to this most basis question seem to elude me in frameworks of String theory and LQG, though it becomes more intuitive in the latter.


title: Can the universe be described by a Markov chain?

text: This may be a fairly basic question as I don't have a strong background in physics. I intuitively thought that the universe must be able to be described by a Markov chain. That is, I thought you could feed the current state of the universe into a process and it would spit out the next state conditional on the laws of the universe. However, I have found no mention of the universe as a Markov chain outside of speculations on message boards. Can the universe be described as a Markov chain or is there some reason to suggest that the next state in the universe is dependent on more than just the current state and constant universal laws?


title: Linearized gravity and symmetries

text: I have naive question. When we analyzing weak gravity field we introduce expression for metric tensor as $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \quad \eta_{\mu \nu} = diag(1, -1, -1, -1), \quad h_{\mu \nu} = \varepsilon c_{\mu \nu}, $$ where $\varepsilon$ is small parameter. But $c_{\mu \nu}$ also can't be equal to the great value, so "linearization" condition somehow limits the functional relationship of $h_{\mu \nu}$ if it depends on coorditates. Also linearized gravity laws look like Maxwell's equations. So there is the next question: do linearized gravity laws act in space-time with some symmetries (like isotropy), which doesn't exist in a case when we don't linearize gravitational interaction?


title: How to find 'good' literature on a phenomenon?

text: With this being a potentially subjective question I know I'm threading on thin ice, but this is an actual "physics" problem I face from time to time: The situation is this: I hear about some phenomenon. For the sake of definiteness, let's say it is the Jahn-Teller-Theorem. I'd then like to find a good source that has a graduate level _introduction_ to this phenomenon. The problem is that all Google turns up is either a very simplified explanation of the thing at hand (think Wikipedia), or a research level text that assumes its readers already have a perfect understanding of what the thing is. And original papers often do not provide such a clear answer (Legend has it that Born and Oppenheimer's paper on the adiabatic approximation is quite inaccessible and the relevant thing, the BO-Expansion, appears hidden somewhere in the appendix...). I believe that I am not alone with this problem and I wonder if there is a straightforward approach to finding good introductory (but NOT superficial) texts on a given phenomenon.


title: Kinematics - moving body

text: If I have an inclined plane, which has acceleration $\alpha$, will a block placed on the inclined plane slip back with the same acceleration? (considering that the surface is frictionless)


title: Magnetic Force on a Loop and number of turns

text: Suppose we have a rectangular current-carrying loop with current $i$, then we know that the magnetic force on each side can be found as: $$F=iL\times B$$ Where $L$ is the vector in the direction of the current with the length of the corresponding side. On the other hand we know that if there are $N$ turns instead of just one this will affect the torque, since it'll also affect the magnetic dipole moment. However what about force? Do this also affect the force? My intuition tells that it'll affect the force and that it'll become $F=N(iL\times B)$, however I'm unsure if I'm correct and I also don't know how to prove it. Is this result correct? How do we prove it if it is?


title: Capillary action column height in a tube fitted inside another tube?

text: I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved). Height of liquid column: ![enter image description here](http://i.stack.imgur.com/P5s6S.png) where: $\gamma$ = liquid-air surface tension $\theta$ = contact angle $\rho$ = density of liquid $g$ = gravity acceleration $r$ = radius I tried my best to draw the examples I'm interested in order to help my explanation. I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that "a/2" in example #1 is close to "c" in example #2 and #3 (drawings not to scale). ![enter image description here](http://i.stack.imgur.com/TCZr6.png) Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if "c" in example #2 can be considered as "a/2" in example #1 to calculate column height using above formula. _Also I'd like to know how having beads of slightly smaller diameter than "c" between the two tubes (example #3) would affect the column height._ _If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness?_ _What'd be the column height of example #2 and #3 assuming "c" as 1mm ?_ I'm quite sure that given the same reached height "h" in example #2 and #3, "c" of #2 has to be smaller than "c" in #3, though it's apparently the opposite (beads lower the water column) and pointed at sciencedirect article by another user. Edit: I've been told that If $c\ll R$ the radius of your outer tube, the total curvature is approximately $\cos \theta/c$, so you will get $$h = \frac{\gamma}{\rho g} \frac{\cos \theta}{c}.$$ Beads will usually lower the apparent surface tension, so you'll get a lower column, although the amount of that depends on their wetting properties and of their arrangement (packing) _However by $c \ll R$ which orders of magnitude/fractions are we talking about?_ _I guess $c$ still has to be smaller than water'scapillary length (about 2.7mm), right?_ _If $c$ is slightly smaller than the outer tube radius (like 1/6), how'd total curvature and hence Jurin's law (above formula) be affected?_ Thank you very much


title: what is the static pressure in a yield stress fluid?

text: Suppose I have a tank filled and there is no slip at the walls. If the tank is filled with a Newtonian fluid and is in static equilibrium, we know that the pressure is defined as $p = \rho g z$. But what if the tank is filled with a yield stress fluid. For example, the Bingham plastic model defines viscous stress as: $$\tau = \tau_0 + \mu \gamma$$ (or if it is more clear, the definition of shear rate) $$ \gamma = \begin{cases}0 & |\tau|\leq\tau_0 \\\ \frac{1}{\mu}\left(\tau-\tau_0\right) & |\tau|>\tau_0\end{cases} $$ where $\tau_0$ is the yield stress of the fluid, $\mu$ is the plastic viscosity, and $\gamma$ is the shear rate. The material acts as a solid below the yield stress, and then as a Newtonian fluid above it. Intuitively, I feel as though the static pressure is reduced relative to the yield stress and that the pressure distribution is not a continuous function. Is my thinking correct? How do I calculate the static pressure in a yield stress fluid?


title: Verification of the Poincare Algebra

text: The generators of the Poincare group $P(1;3)$ are supposed to obey the following commutation relation to be verified: $$\left[ M^{\mu\nu}, P^{\rho} \right] = i \left(g^{\nu\rho} P^{\mu} - g^{\mu\rho} P^{\nu} \right)$$ where $M^{\mu\nu}$ are the 6 generators of the Lorentz group and $P^\mu$ are the 4 generators of the four-dimensional translation group $T(4)$. For $\mu = 3, \nu=1, \rho=0$ the LHS becomes: $ [M^{31},P^{0}] = M^{31}P^{0} - P^{0}M^{31}$. Here $M^{31} = J^2 = -J_2= \begin{pmatrix} 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & -i \\\ 0 & 0 & 0 & 0 \\\ 0 & i & 0 & 0 \end{pmatrix}$ and $ P^0 = P_0 = -i \begin{pmatrix} 0 & 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$. My question is that how can I multiply $M^{31}$ and $P^0$ when they are $4\times4$ and $5\times 5$ matrices respectively?


title: Sea surfer position displacement

text: Waves are means by which the _energy_ propagates through a medium (e.g., sea water). This is not associated with a net movement of water in the direction of wave propagation. If this is the case, then how does a surfer move with sea waves toward the seashore?


title: Why is the pressure inside a soap bubble higher than outside?

text: Apparently, the air inside a soap bubble is under higher pressure than the surrounding air. This is for instance apparent in the sound bubbles make when they burst. Why is the pressure inside the bubble higher in the first place?


title: conducting heat to infinity

text: I am modeling a situation in which an object that is a small object that is spherical in shape is receiving heat at a constant rate and then conducting it away into a medium that can be model as infinite in size relative to it (large glacier) such that the bulk temperature at infinity of the glacier is unchanged by the conduction. In a steady state, there should be some sort of temperature gradient set up in which all of the energy coming into the sphere is going out into the ice. in order to evaluate this Fourier's law of q = -k* dT/dr (assuming spherical symmetry such that all variations are only caused by variations in r and this becomes a 1D problem) should be integrated across the surface of the sphere, yielding dQ/dt = -kA* dT/dr. In order to evaluate dT/dr, the most common approach is to approximate it as (T1-T2)/L where L is the distance between the two different temperature sources. Unfortunately in the case where these two objects are "infinitely" far apart as L approaches a large number at which the bulk temp is unchanged relative to object, the quantity of (T1-T2)/L approaches 0, yielding that no heat would flow between the object and surrounding medium, which is unphysical.


title: Understanding units and the units of the derivative operator

text: Suppose that $f$ is a function from unit $A$ to $B$, then what is the unit of $f'(x)$?. We can do $f'(x)\Delta x$ to get an estimate of $f(x + \Delta x)$. Since the latter has unit $B$, so has the former. $\Delta x$ has unit $A$, so $f'(x)$ has unit $B/A$. So far so good, but what if $A$ is the unit of temperature? Intuitively it does not make sense to give $\Delta x$ the same unit as temperature, because a temperature difference cannot be used interchangeably with an absolute temperature. In particular it does not make sense to do $f'(x)\cdot y$ where $y$ is an absolute temperature. Are there extensions to the system of units that let you recognize that as an error just by looking at the units, i.e. a system of units where $\Delta x$ has a different unit than $x$? Similarly, adding two absolute temperatures is invalid. Is there a system of units that lets you only add two relative temperatures to get a new relative temperature or a relative plus an absolute tempurature to get a new absolute temperature? Now for a more general and more vague question: I was reading the page on extensions to dimensional analysis on Wikipedia. It describes a system of units that seems to capture the rotational invariance of physical laws. All physical laws should be rotationally invariant, and Siano's system of units makes sure that if a law is not rotationally invariant then it has a unit error (i.e. it does not matter what we define as our zero angle). Is this a correct intuition? Are there similar extensions to the unit system that capture other physical invariances? It seems like the standard units capture scale invariance (i.e. it does not matter what mass we call 1 kg as long as we do it consistently), and the system that I'm after in the first paragraph should capture translation invariance (it does not matter which point we define as zero). Of course there are other symmetries. Is there a system of units that lets you easily check from the units whether a law satisfies these symmetries? Another limitation of our current unit system is that some mathematical laws are not expressible. For example $e^{ab}=(e^a)^b$ is not expressible if $a$ and $b$ have units. Is there a way to solve this problem?


title: Moment of inertia of a football and its angular momentum

text: What are the ways to create a mathematical model for the moment of inertia of a football? Can the moment of inertia of the football be simplified to two cones stack against each other? I'm trying to find the angular momentum of the football by using $L=I \omega$ Is the angular momentum of football actually conserved? What are the ways one can obtain the angular velocity on a rotating football?


Is there anything interesting to say about the fact that $\hbar$, the angular momentum and the action have the same units or is it a pure coincidence?


title: Gauge fields and strings: Loop equations

text: I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) = \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, {\rm d} s $$ where the non-abelian phase factor around a closed loop $C$ is defined as $$ \Psi(C) = {\rm P}\exp \left(\oint A_\mu dx^\mu \right) = {\rm P}\exp \left(\int_{0}^{2\pi} A_\mu \dot{x}_\mu\, {\rm d}s \right)$$ It seems that he is using the relation given on p. 116: $$ \delta \, {\rm P} \exp \int_{0}^{2\pi} M(\tau) {\rm d}\tau = \int_{0}^{2\pi}{\rm d}t\,{\rm P} \left(\delta M(t) \exp \int_{0}^{2\pi}M(\tau){\rm d}\tau\right)$$ Matching with (7.25) I find $\delta A_\nu = F_{\mu\nu} \delta x_\mu$. This relation seems to be saying that if I change the position of the loop at the parameter $s$ by $\delta x_\mu(s)$ then the vector potential changes by $\delta A_\nu(x(s)) = F_{\mu\nu}(x(s)) \delta x_\mu(s)$. I don't know how to derive this relation. Is it legitimate?


title: Will a spinning object come to rest?

text: Will a sphere spinning on its own axis come to rest given enough time, provided no other forces act upon it? I know that if you have _two_ spinning spheres in the depths of space and orbiting each other they'll eventually stop spinning -- they'll tidally lock each other. The internal friction of the shifting tidal bulge eventually overcomes the rotation. But I don't see what forces would stop a sphere from rotating around its own axis, in absence of other forces. And yet, each point on the sphere (except points along the axis) is experiencing accelerated motion, which implies a force. My intuition says that something must maintain the acceleration of each point, which implies that the acceleration causes internal stress. I know this to be true macroscopically (the sphere will have an equatorial bulge), but what forces maintain the acceleration of each particle, and will the stresses ever overcome the rotation of the sphere? **Clarification** : I am _not_ considering infinitely rigid spheres or point particles. I'm wondering about actual non-rigid macroscopic spheres, such as planets.


title: Could this suggest that there is a wavelength smaller than Planck's?

text: Suppose the earth receives a photon with a wavelength $\gamma_1$. Since spacetime is expanding, we know that this photon had an original wavelength $\gamma_2$, such that $\gamma_2\lt\gamma_1$. This is phenomenon is known as redshift. Nothing special. Now, here's the thing, what if the earth receives from far away a photon whose wavelength is equal to Planck's length $\ell_p$. This means that the photon before traveling all this distance a wavelength smaller than $\ell_p$. But this is impossible, since $\ell_p$ is the smallest possible wavelength. So would this suggest that there is a wavelength smaller than Planck's?


title: What about negative Pressure?

text: Here is something I see : ![enter image description here](http://i.stack.imgur.com/gqXZX.png) Let's say the ideal fluid(water here) of density $\rho$ is drawn from a source by a motor and thrown upwards with a velocity $v$. Now we take the power of motor be constant and fluid to flow in streamlines (same velocity all over because of same cross section area) ,then applying Bernoulli's Equation for two points shown we get the $$P_{down}-P_{up}=\rho g h$$ $P_{down}=P_{atm}$ because the down point is open to air. So, we get $$P_{up}=P_{atm}-\rho g h$$ which will go negative if $h>10m$. How is it possible?


title: Discreteness of set of energy eigenvalues

text: Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we wish to seek solutions for $$ E < \lim_{|x|\rightarrow \infty}V(x), $$ I find textbooks like Sakurai and others assert that (without proof) set of all such eigenvalues $E$ that admits nontivial solutions is discrete, which gives us liberty to index them with natural numbers (principal quantum numbers). Can anyone help me with a little elaboration of this mathematical fact ? As a student of mathematics, I know that compact operators have discrete spectrums, so for the operator $ T : L^2 \rightarrow L^2 $given as $$ T(\psi) = -\frac{\hbar^2}{2m}\Delta \psi + V\psi. $$ Are there methods to show that $T$ is compact? Or possibly other ways to establish discreteness of its eigenspectrum?


title: Getting started with Landau Physics course

text: I want to learn Physics in my own time. I am an electrical engineer, working in the same field and am tremendously interested in Physics. I have heard a lot about the 10 book series of Landau Theoretical physics. I want guidance to get started with Landau Physics. More specifically, I want help with the following..   1. What subjects/books/courses should I take as prerequisites (especially in Mathematics?)   2. What are the best practices while going through such a self-learning course.   3. Any good resources on Problem sets that I can solve while going through the course.


title: Why do clouds appear black?

text: I have noticed clouds appearing black during rain. But I don't know what makes clouds to acquire that colour. ![Black clouds over skyline](http://i.stack.imgur.com/4vJZq.jpg) This phenomenon doesn't appear every rainfall. There has to be distinction to white rain clouds.


title: Does ABS shorten stopping distance of a car?

text: ABS, from German _Antiblockiersystem_ , is a device put in almost every new automobile. The web has lots of explanations about the system, how it works, but I don't understand how it shortens the way of stopping. The system (a Wikipedia link) is intended to prevent locking of wheels and thus allows keeping control of the car (this I understand). When I push a brake pedal, there are metal parts that touch a wheel. Because of friction force between these metal parts and wheel, the latter stops rotating and its kinetic energy is dispersed as thermal energy and both parts are hot. It does not matter how strongly I press the pedal, if only it makes wheel stop rotating. When it happens, there is a friction force between a tyre and ground. This force depends on (among others) car's mass. The kinetic energy of moving car is (please confirm) dispersed as thermal energy. Sometimes stopping car leaves a black trace on the ground. If now ABS starts to work, it moves the metal parts away, so the wheel can rotate again for some angle, even if I am still pushing the brake. This is intended behavior, and again the ABS stops the wheel. I heard that such act can shorten the time to stop a car, but I don't know why. If the ABS allows the wheel to rotate for some angle, in this moment there is no friction between wheel and ground (well, actually there is, because the wheel rotates, but in this case it is useless friction). So the way to stop the car should be longer. Again, I understand how the ABS prevents wheels from blocking and allows maintaining control, but I don't how it shortens this distance. The only thing I can imagine is that this small angle of rotation prevents tyre from getting hot. If it would be hot too much, it becomes more liquid and the friction force is smaller (this is why these tracks come from). So the ABS changes "used" part (hot) of tyre to a "fresh" part (cold). This explanation is not confirmed by experiment which I performed myself on ice with the speed ca. 10 km/h, and it shortened the stopping distance, but with this speed we should not think about thermal destruction of tyres.


title: Deuterium density in seawater

text: I heard today that deuterium for nuclear fusion could be extracted from the Oceans, in which it is present in "heavy water." I heard this claim: > _"Deuterium is distributed uniformly with Ocean depth."_ This contradicted my naive expectation that deuterium density would _increase_ with depth, on account of it being in "heavy water." For example, salinity increases with depth. I asked, and received two contradictory explanations. The first was that because powerful waves and Ocean currents mix Ocean water, gravity is negligible. This surprised me; I thought that Ocean's were stratified, with predominantly tangential, rather than vertical, currents. The second explanation was that my view of seawater is naive - that I should think of seawater not as heavy and regular water molecules, but as a soup of weakly interacting deuterium, the normal hydrogen isotope and oxygen, forming and breaking water and heavy water molecules with a short time-scale. If so, apparently, deuterium density and the normal hydrogen isotope would be distributed evenly with depth. This did not help me, because, I then naturally wondered why _deuterium's_ density doesn't increase with depth, since it is heavier than hydrogen-1? Is the claim correct? and if so, what is really going on? I have the feeling that Ocean behaviour with depth might be non-trivial...


title: Vacuum polarization in QCD and gluon bubbles

text: In analogy to QED, the following Feynman diagram is a diagram contributing to the vacuum polarization effect, leading to anti-screening, asymptotic freedom and running of the strong coupling constant. It can also be interpreted as a correction to the propagator of the gluon. ![](http://s1.directupload.net/images/130703/u52v6r7m.png) (Of course, there's no true analogy to this diagram in QED because photons do not couple to themselves, and here we make use of the existence of a 3-gluon vertex. In QED we would have a fermion-antifermion pair instead of bosons in the loop, similar to a gluon going to fermion-antifermion and back in QCD.) My question now is about **the following diagram** which has a 4-gluon vertex: ![](http://s1.directupload.net/images/130703/j3rjiwb3.png) **Does this also contribute to vacuum polarization and the effects it entails? Or is it something conceptually different?** (It seems different because we don't have the splitting of one object into two and back.) _(PS: Feel free to add more meaningful tags and correct any statements in the above which are wrong.)_


title: What is happening to the electrons, and E & H fields, in an antenna with a standing wave inside?

text: Diagrams like the one shown below are often shown to explain antenna theory, but I have always had problem with the concept of voltage being a wave, and because of this the diagrams never make any sense to me. ![enter image description here](http://i.stack.imgur.com/ACkWT.png) If the voltage source starts to generate a sine wave at time 0, then the voltage at any point along the transmission line will be: $$V(x,t) = V_{max}\sin(kx-\omega t)$$ Once the voltage reflects the equation becomes: $$V(x,t) = V_{max}\sin(kx-\omega t) + V_{max}\sin(kx+\omega t)$$   * First question: **What is the mechanism explaining how this voltage propagates?** $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ - A voltage is not a "real" thing, it's measurable, but it is just a concept made up to quantify a complex interaction of coulomb forces between electrons. Voltage is the electrical potential energy between two places a unit charge would feel. Energy is just the potential to do work, which is the integral of force over distance moved in the direction of the force. If there is a 1 volt potential difference between 2 points ($a$ and $b$) within an electric field, that means that a unit charge placed at point $b$ would require 1 joule to be moved to point $a$. That is all voltage is, I don't understand how this can propagate as a wave.   * Second question: **Why does the voltage reflect?** $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ - Similar to the reason above, if voltage just describes the amount of work done needed to move 1 coulomb between two places (the work done can be manifested as distance or force, as long as: $\:\:W=\int f \cdot x $), then how can it reflect?   * Third question: **What is the voltage on this line relative to?** $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ - Voltage always has to be with respect to something, as it is a potential difference. In closed circuits with an oscillating voltage source you usually treat one side of the voltage source to always be at ground (a constant 0V) and the other side of the voltage source to be oscillating between the max and min values. Because voltage is relative this is exactly the same as treating both sides of the source to be oscillating and the difference between the two always equaling: $V_{max}\sin(\omega t)$. It is just easier to treat one side as ground. In the antenna, what is ground, and what are the voltages relative to?    * Final question: **Why is voltage and current at nodes/ antinodes respectively?** $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ - Okay, if I just accept that voltage can be a wave and hence a standing wave in an open transmission line, I still get confused. Why are the points of maximum voltage (antinodes) at the point of zero current (nodes)? At a point of maximum voltage the potential difference is oscillating between its max and min values, so why doesn't the maximum current occur here? * * * _Note: I'm not completely confident with the equations I gave for the value of the voltage at time $t$ and position $x$. Please just edit these if they are not correct as they are not the main concern of my question. I mainly want to know what is happening to the electrons and fields inside the antenna so I can understand why the voltage propagates and why it reflects._


title: Airplane example of special relativity

text: I'm struggling with an introductory example of special relativity. We haven't done the math yet so I would like an explanation based only on the fact that the speed of light is the same in every inertial frame. An airplane travels east with a certain speed. There is a "clock" at both ends of the airplane. If there is a flash in the middle of the airplane and we're in the airplane's inertial frame, both clocks will register the flash at the same time, say 3. But if we are an observer standing on the ground, in our inertial frame the light will reach the clock at the end of the airplane faster than the clock in the front. So the clock in the back may reads 3 when the clock in the front only reads 1, for example. Question: Let's say the airplane has a speed v. The light's speed as it travels in the inertial frame of the observer towards the back will c+v, and the speed as it travels towards the front will be c-v. Let's say the distance from the middle to the back and front respectively is L. Then the time it takes for the light to travel from the middle to the back is L/(c+v) and to the front L/(c-v). Shouldn't these be the same? If not, why can we say that the clocks will read 3 when the light hits them, no matter what frame is used? This is the argument the book uses to say that the clock must be less than 3, i.e. 1, in the front when the light hits the clock in the back in the observer's inertial frame.


title: Combinatorial sum in a problem with a Fermi gas

text: I'm solving a problem involving a Fermi gas. There is a specific sum I cannot figure my way around. A set of equidistant levels, indexed by $m=0,1,2 \ldots$, is populated by spinless fermions with population numbers $\nu_m =0 $ or $1$. I need to compute the following sum over the set of all possible configurations $\\{ \nu_l \\}$: > $Q(\beta,\beta_c) = \sum_{\\{ \nu_l \\}} \sum_{l} \prod_m \exp({\beta_c \, l > \, \nu_l}-{ [ \beta \, m + i \phi] \, \nu_m} )$. Any hints on how to deal with this are appreciated. This is not homework, it is a research problem. It is known that $\beta >0$, $\beta_c>0$, and $\phi \in [0; 2 \pi ]$. **EDIT** : corrected with the complex phase (the sum is coming from a generating function)


title: If LHC is the most powerful microscope in the world, what does it magnify?

text: Some experts (and non experts) have called LHC the most powerful microscope in the world. I am wondering, what does the LHC magnify? Or is calling it a microscope just wrong and misleading?


title: Embrace Physics

text: Does anyone have any suggestions as to what is a good topic for a short talk on theoretical physics to a bunch of Math and Physics undergrads that might make them "embrace" theoretical physics? Thanks. (Brain storm: String theory, quantum tunneling, etc...?) Also it would be great if someone could suggest some resources.


title: Finding shortest possible rotation period

text: How is it possible to find the shortest possible rotation period of a pulsar from a mass and a radius?


title: How does water falling down a waterfall gain energy?

text: I'm kind of a noob in the physics area. But I know that energy cannot be created, it can only be transferred. So how and what "type" of energy does water get when free-falling?


title: Relation between representations of boson operators?

text: I have a simple (I think !) question about the representations of boson operators and how they are related. First of all let's define two conjugate observables $Q$ and $P$ (i.e. $\left[Q,P\right]=i$ and $Q^\dagger=Q$, $P^\dagger=P$). If we further define: \begin{equation} a=\sqrt{\frac{\alpha}{2}}\left(Q+\frac{i}{\alpha}P\right)~~~~~~~~~~~~ a^\dagger=\sqrt{\frac{\alpha}{2}}\left(Q-\frac{i}{\alpha}P\right)~~~~~~~\alpha\in \mathbb{C}, \end{equation} (as in the harmonic oscillator problem) we have that $\left[a,a^{\dagger}\right]=1$. We can there identify $a^{(\dagger)}$ as boson annihilation (creation) operators. However we can also define: \begin{equation} b=\sqrt{Q}e^{iP}~~~~~~~~~~~~~~~~b^\dagger=e^{-iP}\sqrt{Q} \end{equation} which will verify $\left[b,b^{\dagger}\right]=1$ (this requires a bit more algebra though). **Question :** Is there a relation between these two representations ? These are specific examples, but one could probably think of other representations. Since these representations implement the same commutation relations, does it mean that there are related by some transformation (a unitary transformation in particular) ? (I give here specific examples for bosonic operators, but I guess one may extend the discussion to any type of operator satisfying some commutation relation).


title: Why is the Copenhagen interpretation of quantum physics the most widely accepted one?

text: I've been digging a lot into quantum physics in the last few weeks. I didn't care much about the maths, just about what empirically happens to get a conceptual idea about quantum phenomena. The most widely accepted interpretation of quantum mechanics seems to be the Copenhagen one. If I got it right, it's heavily relaying on the two following principles (among others):   * Superposition: a quantum system is at the same time in all the states it could possibly be in. When it's measured, it instantaneously collapses in a single state.   * Entanglement (aka "spooky action at a distance"): if two or more quantum systems are entangled, it means that some of their properties are correlated. When measuring a system, all the entangled ones collapse in a state coherent to the measured one. Simultaneously. No matter how far away they are to each other. I'm not able to believe it. It allows some unrealistic paradoxes (e.g. Schrödinger's cat paradox), and I have the feeling that this interpretation (and its consequences) is what makes quantum mechanics look so weird, mysterious, unnatural and spooky to the public. Besides I've read from a few sources (like this Google Tech Talk) that this interpretation has proven to be broken: the math says everything is continuous and doesn't hint to anything like collapsing, and even more important, the quantum eraser experiment contradicts the Copenhagen interpretation. The second most popular interpretation, many-worlds sounds a lot more _natural_ to me, although it strongly smells like science fiction. I believe there must be many interpretations that would hold better and would be a lot less weird than the two mentioned ones. What I'm wondering is, then: why does the Copenhagen interpretation (and to a lesser degree the many-worlds one) remain the most accredited one?


title: Ising model for ferromagnetism is not intuitive

text: In the Ising model for ferromagnetism a lower energy is assumed when two spin magnetic dipoles are aligned parallel to each other and the energy is higher when they are antiparallel. If I take two simple bar magnets I see precisely the opposite behavior: if I force their axes to be parallel they prefer to be in the antiparallel configuration (so that the N of one magnet is near to the S of the other). What happens in ferromagnets that makes the dipole to prefer to be parallel to one another?


title: Revealing Electric field lines through Grass seeds in mineral oil

text: In one of Walter Lewin's famous lectures, he takes a small container almost completely filled with mineral oil and grass seeds (I'll keep looking for the video to link to it). He connects two electrodes to the container and the electric field that is produced polarizes the grass seeds, aligning them in the direction of the electric field lines. I had a few questions about this experiment as I am interested in reproducing it.   1. What material should be used as the container?   2. Does the type of oil matter?   3. What voltage is required?


title: Speed distribution in 1 dimension

text: In 3D, the maxwell velocity distribution is: $$f = \left(\frac{\alpha}{\pi} \right)^{\frac{3}{2}} e^{-\alpha v^2} d^3 \vec v$$ To get the speed distribution in 3D, we simply expand $d^3\vec v = 4\pi v^2 dv$ Thus in 3D, the maxwell speed distribution is: $$w = 4\pi^2 \left(\frac{\alpha}{\pi} \right)^{\frac{3}{2}}v^2 \space e^{-\alpha v^2} d v $$ In 2D, the maxwell velocity distribution is: $$f = \left(\frac{\alpha}{\pi} \right) e^{-\alpha v^2} d^2 \vec v$$ To get the speed distribution in 2D, we simply expand $d^2\vec v = 2\pi v\space dv$ Thus in 2D, the maxwell speed distribution is: $$f = 2\pi\left(\frac{\alpha}{\pi} \right) v \space e^{-\alpha v^2} d \vec v$$ In 1D, the maxwell velocity distribution is: $$f = \left(\frac{\alpha}{\pi} \right)^{\frac{1}{2}} e^{-\alpha v^2} d \vec v$$ Following the same line of thought, how do I get the 1D speed distribution?


title: Ideal gas concentration under temperature gradient

text: I'm trying to calculate the concentration of an ideal gas in an adiabatic container as a function of position where the top and bottom plates of the container are fixed at temperatures $T_1$ and $T_2$, respectively, with $T_2 > T_1$. Thus, there is a temperature gradient from bottom to top, i.e. $T(z) = T_2 - (T_2-T_1)\frac{z}{h}$, where $h$ is the height of the container. How do I calculate the concentration of the gas, $n(z)$, which I expect to be smaller at the bottom (near the hot plate) and larger at the top? EDIT: I started with a similar problem but which included the gravitational field. The correct insight was to realize that the difference in pressure between $z$ and $z+dz$ was equal to the gravitational force due to the mass in the volume between $z$ and $z+dz$. From this it's easy to follow the math. In the present case, however, is the correct insight simply that the pressure is uniform? I.e., since there is no mass to balance, there are no pressure differences? If so, I have found the solution: $n(z) = n(0)T_2(T_2 - (T_2-T_1)\frac{z}{h})^{-1}$, which is inversely proportional to the temperature, and which seems kind of nice, but how can I properly justify that the pressure is uniform?


title: How would you use the Euler-Lagrange equation to predict the motion of projectiles with linear (Stokes) drag (but no wind)?

text: My first instinct would be to use the force $$\vec{F} =- \alpha \vec{v}$$ and therefore $$V(\vec{r}) = \alpha \int_C \vec{v}\cdot d\vec{s} = \alpha \int_C \vec{v}\cdot \vec{v} dt = \alpha \int_C g_{ab} \frac{dx^a}{dt} \frac{dx^b}{dt} dt$$ (where $C$ is the projectile's path) and go from there. Would this work?


title: How local fields transform in the holographic boundary

text: Consider a holographic description of gravity $f:\Omega \rightarrow \partial \Omega$ such that gravitational fields and curvature in a neighbourhood $\Omega$ of 4D spacetime induce local fields on $\partial \Omega$. So if $G(\Omega)$ is a shorthand to describe all gravitational fields in the neighbourhood, then $g := f(G(\Omega))$ is a shorthand to describe all local fields in the holographic dual in $\partial \Omega$ I understand that we rarely have detailed information of the $f$ map. But usually one can ask questions about $f$ properties. In this case, I'm interested by the fact that for a given point $x$ in the spacetime, there are infinite neighbourhoods $\Omega$ such that $x \in \partial \Omega$. How we expect the holographic fields to transform on $x$ as we change the $\Omega$? > **Question:** what would you call the following property: Be $x$ a point in > spacetime, and be $\Omega(x)$ a family of compact, simply connected sets > such that if $\Omega \in \Omega(x)$ then $x \in \partial \Omega$. Assume > that gravitational fields $G(\Omega)$ are defined everywhere on spacetime. > Then the corresponding fields in the holographic dual > $g:=f(G(\Omega)):\partial \Omega$ are such that if we take quantum field > observable operators on $g$ and obtain vacuum expectation values of the > operators around a small neighbourhood around $x$ with $x \in X_h$, then > vacuum expectation values $\langle A_{g(\Omega)} \rangle_{X_h}$ of operators > $A_g$ in the dual $g$ satisfy the following invariance property: > > be $\Omega \in \Omega(x)$ and $\Omega' \in \Omega(x)$. Then $$ \langle > A_{g(\Omega)} \rangle_{X_h} = \langle A_{g(\Omega')} \rangle_{X_h} $$ > > for $X_h$ small enough This property seems to be intuitively some kind of condition that the holographic dual fields also behave as local fields in the embedding space. I'm not sure if it makes physical sense to demand it, but I'm trying to understand if it is an interesting property to demand from holographic duals. Naively I would be tempted to think that such invariance suggests that the holographic dual is a _real physical_ field, in some sense


title: Electric force in DNA molecule

text: Given that distance between O-H and N-H bonds are 0.11 nm and ![enter image description here](http://i.stack.imgur.com/KnBfr.png) How do I compute the net force exerted from Thymine and Adenine? A hint given is: > _Hint: To keep calculations fairly simple, yet reasonable, consider only the > forces due to the O–H–N and the N–H–N combinations, assuming that these two > combinations are parallel to each other._ Do I compute the force on O due to H and N like: $$F_{OH} = k \frac{e^2}{(0.17 \times 10^{-9})^2}$$ $$F_{ON} = k \frac{e^2}{(0.28 \times 10^{-9})^2}$$ $$F_{O} = k \frac{e^2}{(0.17 \times 10^{-9})^2} - k \frac{e^2}{(0.28 \times 10^{-9})^2}$$ Then about the HN part? Do I consider that 1 molecule or something?


title: What determines the distribution of the energy between the beta-particle and the neutrino in beta decay?

text: When beta-decay occurs an electron or a positron is emitted along a neutrino or an antineutrino. The energy spectrum of the beta particles is continuous because, as I read, the energy is shared between the two emitted bodies. My question is: what determines the distribution of energy here? That is, what determines with how much energy each particle is emitted? And for that matter: why is this sharing of the energy continuous?


title: Finding out the Potential difference

text: > _Find the potential inside and outside a uniformly charged solid sphere > whose radius is $R$ and whose total charge is $q$. Use infinity as your > reference point. Compute the gradient of $V$ in each region, and check that > it yields the correct result._ The above one is a question cropped from Griffiths page no 82. When I try to solve the problem they used a potential equation inside the sphere ![enter image description here](http://i.stack.imgur.com/ctSoI.png) My problem is in the right part. Why do we need to consider the next part? we have been asked to solve potential inside a charged sphere. so can't I use only the limit $\infty $ to $ R$? I am saying this because, i have considered the infinity as the reference point and R as the point where I will find the potential. > Is it because we need to add the potential outside of the sphere and inside?


title: Conservation of angular momentum experiment

text: I've learned in that in this experiment: ![enter image description here](http://i.stack.imgur.com/hIlDX.jpg) ...the skater will start rotating faster when she brings her arms in and there is no net torque acting on her. But what would happen to her angular momentum and rotation if she only brings **one of her arms** inwards with the other arm sticking outwards? edit: will a net torque act on the body? What will this cause?


title: Is it easier to pump water up or down?

text: I'm trying to gauge the strength of aquarium air pumps, for use in a vertical hydroponic farm. A curious question arose: is it easier to pump water upwards vertically, or is it easier to pump it downwards (assuming a volume like an aquarium)? Upwards, you fight with gravity -- downwards you are aided by gravity but fight water pressure.


title: Conversion of energy to mass with radioactive decay

text: Consider the following problem: _The nuclei of Am-241 decay by the emission of $\alpha$-particles with a kinetic speed of $8.8 \cdot 10^{-19} J$. In a certain source of Am-241 there are $ 4.0 \cdot 10^3$ decaying nuclei per second. Caculate the mass this source loses per year._ So what the correction sheet tells me is that you basically just have to find the mass of an $\alpha$-particle in $kg$ and then just multiply that number until you get the amount of decaying $\alpha$-particles in a year. However, I think this is wrong, since the $8.8 \cdot 10^{-19} J$ isn't accounted for. I'd say you have to take the mass of the $\alpha$-particle, then add to that the $8.8 \cdot 10^{-19} J$ after you've converted it to $kg$ by using $E = mc^2$ and THEN multiplying that number in order to get the amount of decaying $\alpha$-particles in a year. Which is correct, or perhaps better: which is less wrong?


title: Principle of Reflection on atomic level

text: The well observed phenomenon has besides several others has always been a fascination to me, we are well aware of several theories, experiments and practical applications of the well known phenomenon. But is it established that what goes at the grass root of reflection ? My high school teacher once told me that whenever light is incident on any reflecting surface its electrons absorb the energy of photon and release back the same energy. If this is actually true then I have a plenty of questions to continue, but if it is not then what is reflection and how and why it takes place ?


title: Navier-Stokes system

text: I have to study this system which name is Navier-Stokes. Can you explain please what means that $p$, $u$ and $(u \cdot \nabla)u$. What represents in reality? Tell me please, how should I read the factor: $(u \cdot \nabla)u$? "$u$ multiplied with gradient applied to $u$ " ? $ (N-S)\begin{cases} -\mu \Delta u +(u \cdot \nabla)u+\nabla{p}=f &\mbox{in } \Omega, \\\ \mbox{div }u=0 & \mbox{in } \Omega,\\\ u_{\mid{\Gamma}}=0. \end{cases} $ one more question, what happens with with the system if $(u \cdot \nabla )u=0$ ? I found that the system describe the motion of a incompressible viscous fluid and it suppose the the motion is stationary but no slow, what means that **stationary** and that **slow**?


title: What are the coloured bands in this "Pale Blue Dot" picture?

text: You can see Earth in the right most band in this image: ![enter image description here](http://i.stack.imgur.com/UiTCm.jpg) What are these bands doing in space?! Can somebody please explain the coloured bands in this image?


title: Question about a finite time interval step in the derivation of the Feynman path integral in Sakurai

text: This may be a possible errata but Sakurai (pp 126 in the 2nd Edition) states that starting with $$S = \int dt \,\,\scr{L_{\mathrm{classical}}}$$ Looking at a finite-time-interval of the action: \begin{equation} S(n,n-1) = \int\limits_{t_{n-1}}^{t_n} dt \left( \frac{m \dot{x}^2}{2} - V(x)\right)\,\,\,\,\,\,(1) \end{equation} Now using a straight-line approximation for points $(x_{n-1},t_{n-1})$ and $(x_n,t_n)$, which implies $$\dot{x} = \frac{x_n-x_{n-1}}{t_n-t_{n-1}} = \frac{x_n-x_{n-1}}{\Delta t}$$ Sakurai then has Eq (1) become: $$S(n,n-1) = \Delta t\left( \left(\frac{m}{2}\right) \left(\frac{x_n-x_{n-1}}{\Delta t}\right)^2 - V\left(\frac{x_n+x_{n-1}}{2}\right)\right)$$ My question is why is the potential now dependent on $ V\left(\frac{x_n + x_{n-1}}{2}\right) \mathrm{\,\,instead \,\,\,of\,\,\,} V\left(x_n- x_{n-1}\right)$? I checked the most recent errata (4/5/13) posted by J. Napolitano ( http://homepages.rpi.edu/~napolj/ErrataMQM.pdf ) and there is none for this page. Could anyone clarify this step for me? Thanks.


title: Free particle in magnetic field / Landau quantization

text: I have a question concerning a possible derivation of the Landau quantization. In our lecture notes (and some other places as well), the following ansatz is used: $$ \Psi(x,y,z) = \exp{\left(-j(\beta y + k_z z)\right)}u(x), $$ which then leads to the known results ($u(x)$ is identified as a harmonical oscillator,...) But I'm unhappy with the ansatz itself: if I calculate the mean momentum in $y$ direction, I obtain $\hbar \beta$. From classical mechanics, I know the solution can be nothing but a spiral-like movement (should be compatible with quantum mechanics, as I could just use a particle with a huge momentum, which could be directly observed...). So, what am I missing?? I would also expect some symmetry around some path center?! Sorry for my maybe rough english, but I'm not too familiar with using it in physical context...


title: What is the reason of the semidiurnal pressure oscillation in the atmosphere?

text: It appears that in many (most) places on earth, a pressure oscillation of a 100-200 Pa takes place twice a day : that is, with maxima around 12AM and 12PM and minima around 6AM and 6PM. I've found almost no information about it, except highly technical (and not freely available) papers. I think it's called "semidiurnal oscillation", and it seems to be a vertical oscillation of the full atmosphere. But I'm curious : what is the basic mechanism of such an oscillation, and what makes its period a _half_ -day (instead of a full day)?


title: Capacitors in series?

text: > Capacitors in series have identical charges. We can explain how the > capacitors end up with identical charge by following a chain reaction of > events, in which the charging of each capacitor causes the charging of the > next capacitor. We start with capacitor 3 and work upward to capacitor 1. > When the battery is first connected to the series of capacitors, it produces > charge -q on the bottom plate of capacitor 3. That charge then repels > negative charge from the top plate of capacitor 3 (leaving it with charge > +q). The repelled negative charge moves to the bottom plate of capacitor 2 > (giving it charge -q). That charge on the bottom plate of capacitor 2 then > repels negative charge from the top plate of capacitor 2 (leaving it with > charge +q) to the bottom plate of capacitor 1 (giving it charge -q). > Finally, the charge on the bottom plate of capacitor 1 helps move negative > charge from the top plate of capacitor 1 to the battery, leaving that top > plate with charge +q. Why would inducing a charge of +q on one plate cause the other plate to acquire a charge of -q? I get that it would attract electrons from the other side, but the plates aren't the same distance from the electrons, so wouldn't the charge be less than q? For example, consider the following capacitor in series (the other capacitors are on its left). If you induce a charge on the right plate, electrons on the left side will build up on the left plate. But since the left plate is closer to the electrons on the left wire than the right plate, the electrons will stop moving when the field on the left plate is greater than -q. \-----||-----


title: Total angular momentum in a full shell

text: I do not understand why it's supposed to be vanishing. Rather than discussing the question in its full generality I prefer to consider the following scenario, which I think sums up anything that's actually important for my level and my purpose, but keep in mind the real point of the question. Consider a system of six identical particles with spin one-half, each one of them in a eigenstate with azimuthal quantum number $ l = 1 $ and identical quantum number $ n $ given by the following single-particle hamiltonian (no interaction then): $H_a := \frac{\vec{p_a}^2}{2m} + V(r_a) $ Due to the indistinguishability of the particles, the total wavefunction, orbital part plus spin part, must be antisymmetric with respect to the exchange between any two particles. Given six states, one for each particle, the Slater determinant identifies such a function. The six states must be different from one another, otherwise the Slater determinant vanishes and there's no wavefunction at all. More correctly, not only do they have to be different, they have to be linearly independent as well. If all particles are in an eigenstate of a projection of $ \vec{L_a} $ and $ \vec{S_a} $ then I can see clearly that the sum of these values add up to zero. Since $ l = 1 $, then $ m = -1, 0, 1. $ Each state with a definite $ m $ splits in two independent states, with spin up and down. I finally have six different state and the value of the projections clearly add up to zero. Yet by hypothesis I have only defined the azimuthal quantum number, meaning the above is only a particular case. The particles can actually be in any state, as long as they are independent. For example I could have some particles in an eigenstate of $ L_x $ and some particles in an eigenstate of $ L_z $ and in this situation I really can't see how the projections would add up to zero. Even by considering only the special case above I don't see how this implies that the total angular momentum, not only the the projection of the total angular momentum, is vanishing. A typical argument I have encountered says there's nothing special about a particular component, but I don't seem to understand it: if I say each particle is in an eigenstate of $ L_z $, the $ L_{ztot} = 0 $ and clearly this applies to $ L_x $ and $ L_y $ for instance but nothing says $ L_{ztot} $ , $ L_{xtot} $ and $ L_{ytot} $ are simultaneously zero. In fact it seems to me that I prepare the system in two different ways from the beginning because I would prepare eigenstates of $ L_z $ in the first case and eigenstates of $ L_x $ later. These are again particular cases and potentially different, I can't help but thinking a proper argument shouldn't rely on statements like 'Each particle is in an eigenstate of a projection' because all I know is that particles are in the eigenspace where $ l = 1 $, nothing else. I apologize if the post sounds too confusing.


title: Example of a wavefunction that cannot be represented by a single Slater determinant

text: I know that in general, interacting fermions cannot necessarily be described by a single Slater determinant. Can anyone provide a simple example of a state that has no such representation?


title: Calculating the electro magnetic field strength needed to repel a permanent magnet

text: I have a design where a permanent magnet is placed on top of an electromagnet. When the electromagnet is powered off, the permanent magnet is attracted to the iron core of the electromagnet. When the electromagnet is powered on, it generates a magnetic field which repels the permanent magnet, pushing it away. Both objects are contained in a cylinder which controls the movement of the permanent magnet. My question is - how can I calculate the minimum magnetic field strength that the electromagnet needs to produce in order to repel the permanent magnet? I can calculate the electromagnet's strength in Newtons using these online calculators - http://www.daycounter.com/Calculators/Magnets/Solenoid-Force- Calculator.phtml http://onlinecalculators.brainmeasures.com/Electric/SolenoidCoil.aspx and I also know the weight of the permanent magnet, and I know the permanent magnet's pull force. Is the electromagnet's field strength force required to repel the permanent magnet equivalent to the permanent magnet's weight plus the permanent magnet's pull force?


title: Inverted Harmonic oscillator

text: what are the energies of the inverted Harmonic oscillator? $$ H=p^{2}-\omega^{2}x^{2} $$ since the eigenfunctions of this operator do not belong to any $ L^{2}(R)$ space I believe that the spectrum will be continuous, anyway in spite of the inverted oscillator having a continuum spectrum are there discrete 'gaps' inside it? Also if I use the notation of 'complex frequency' the energies of this operator should be $$ E_{n}= \hbar (n+1/2)i\omega $$ by analytic continuation of the frequency to imaginary values.


title: Is decoherence due to coarse graining or coupling with the environment?

text: In the literature, sometimes one reads that decoherence is due to the coupling of the system to the external environment, and sometimes one reads that it is due to coarse graining over the microscopic degrees of freedom. Are these two different cases of decoherence, or is one more fundamental than the other?


title: Chance of "macro tunneling"?

text: We know that subatomic particles can and do tunnel through barriers, so it is theoretically "possible" somewhat that a grain of sand could tunnel through a paper, but Id like to get some perspective on it. Can anybody give any sort of estimate of how long one would have to wait to expect to see a grain of sand tunnel through a sheet of paper? (for instance2 10 times the life of the universe")


title: Rotating a reference system on a vector

text: Assume you have a vector $\vec{x}=(\sin(\vartheta)\cos(\varphi),\sin(\vartheta)\sin(\varphi),\cos(\vartheta))$ given in spherical coordinates in a reference System "R". I want to rotate the reference system so that it's z-direction points along $\vec{x}$. All I have to do is a first rotation about the z-axis with the angle $\varphi$ and afterwards a rotation about the new y-axis with the angle $\vartheta$. Eulers angles could do the job (look here, german). So I define my transformation matrix like that, except that in the basic rotation matrices every $\sin$ has to be changed to $-\sin$ (since I am rotating the reference system and not the vector). The resulting matrix is (since I do not rotate again about the new z-axis, I can set the last angle to zero): $M=\left( \begin{array}{ccc} \cos(\vartheta)\cos(\varphi) & -\sin(\varphi) & -\sin(\vartheta)\cos(\varphi) \\\ -\cos(\vartheta)\sin(\varphi) & \cos(\varphi) & \sin(\vartheta)\sin(\varphi) \\\ -\sin(\vartheta) & 0 & \cos(\vartheta) \end{array} \right)$ So if I want to know the coordinates of a vector $\vec{y}$ in the new coordniate system R' I need to compute $\vec{y}'=M\vec{y}$. Now I want to do some tests. First, what is $\vec{e}_z'$ the old unit vector in the new coordinate system. $\vec{e}_z'=(-\sin(\vartheta)\cos(\varphi),\sin(\vartheta)\sin(\varphi),\cos(\vartheta))=\vec{x}(x,-\vartheta,-\varphi)/x$ which is correct. But if I compute $M\vec{x}$ it should yield $(0,0,1)$, which it doesn't. What am I doing wrong?


title: Arnold´s Math Methods of Classical Mechs - a question on Newtonian Mechanics

text: There is the following question/answer on Arnold´s book Mathematical Methods of Classical page 10. **Arnold's Question** A mechanical system consists of two points. At the initial moment their velocities (in some inertial coordinate system) are equal to zero. Show that the points will stay on the line which connected them in the initial moment. **Proof** A solution to this problem would be: Any rotation of the system around the line connecting the initial configuration is a Galilean transformation and so sends a solution to the differential equation of motion to another solution. Since these rotations fix the initial conditions, by the uniqueness of solutions of differential equations, it is easy to see that the motion of the points must be constrained to the afore mentioned line. qed. My qualms with this solution is that it assumes the equation of motion is nice enough and hence has unique solutions. My question is then: **Question** Is there a force field with a solution to the equation of motion that contradicts the above exercise question of Arnold? This force field will have to be "pathological" enough so that the solutions to the equations of motion are not unique - is there a known physical configuration (i.e. an existing real physical configuration) with this property? I came up with the given solution after trying, and not suceeding, to solve this problem via the creation of some conserved quantity (inner product) that would restrict the motion to the given line. Perhaps such an argument would rule out the above pathologies, and hence would be a stronger/better argument.


title: How can laser communication be faster than microwave?

text: My textbook states that "Laser communication is much faster than microwave communication." But, how can that be? Both are electromagnetic waves with different frequencies but, how can the speed be different? Or Is the statement referring to any kind of processing speeds involved?


title: About Dirac equation in curved spacetime (spherical)

text: I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta d\phi^{2},$$ where $\alpha$ is a positive constant. The Dirac equation(a part of) is given by $$i \gamma^{\mu}(x)(\partial_\mu+\Gamma_{\mu}(x))\psi(x)=0$$ where $\Gamma_{\mu}(x)$ is spinor affine connections and $\gamma^{\mu}$ are the general Dirac matrices. If we define the tetrad base as $$e^{\mu}_{(a)}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\\ 0 & \sin\theta \cos\phi &\sin\theta \sin\phi & \cos\theta \\\ 0 & \frac{\cos\theta \cos\phi }{r}&\frac{ \cos\theta \sin\phi}{r} & -\frac{\sin \theta}{r} \\\ 0 & -\frac{\sin\phi}{\alpha r \sin\theta} & \frac{\cos\phi}{\alpha r \sin\theta}& 0\\\ \end{array} \right) ,$$ using the tetrad base we obtain(But I cannot!): > Blockquote $\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2r}(1-\frac{1}{\alpha})\csc\theta(\gamma^{(1)}\cos\phi+\gamma^{(2)}\sin\phi)$ > Blockquote . Here, $\gamma^{(a)}$ are flat Dirac matrices. Moreover, $$\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2}\gamma^{(a)}[(\partial_{\mu}e_{(a)}^{\mu}(x)+e_{(a)}^{\rho}(x)\Gamma^{\mu}_{\rho \mu})+i\gamma^{(5)}\varepsilon_{(a)(b)(c)(d)}e^{(b)\mu}e^{(c)\nu}\partial_{\mu}e^{(d)}_{\nu}]$$ Now my question is: how one can obtain $\gamma^{\mu}(x)\Gamma^{\mu}(x)$ expansion above, which is $$\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2r}(1-\frac{1}{\alpha})\csc\theta(\gamma^{(1)}\cos\phi+\gamma^{(2)}\sin\phi),$$ because I couldn't obtain it. I'm not used to this notation :(


title: Magnetic muscles

text: Consider two flexible sheets glued at the ends, with electromagnets attached to both of them, so that depending on the directions and strength of currents passing through them, the system could be either                 / \               ||     m|   |m     or     m||w       \ /               ||      I could imagine multiple such systems combined to control a robotic limb. The up side of such construction would be few moving parts, possibly cheap construction. Is this ever done? What are the downsides?


title: Why does noise affect FM radio less than AM?

text: Frequency modulated waves are less susceptible to noise compared to amplitude modulated signal. This is because the information in an FM signal is transmitted through varying the frequency, and not the amplitude. But why does noise affect changing amplitude and not frequency or other characteristics?


title: Heat balance: how to obtain T(t) with energy in the form u(T(t))

text: I have the heat balance equation for a cooling case in the form: $$\frac{dU(T(t))}{dt}=-J$$ with   * $U$: energy (J)   * $T$: temperature (K)   * $t$: time (s)   * $J$: heat flux leaving the system (W) I have $U$ from experimental data; $U$ is a function of sample temperature, $T$, measured in the experiment, both are functions of the experiment time $t$. Now I want to solve the differential equation for $T(t)$; but the problem is that $U(T)$ is not definite because the sample releases crystallization heat and causes $U(T)$ to increase after initial decrease. I was thinking of using only $U(t)$ because this function would be definite. But I need $T(t)$ at the end. Do I need to go in 3D or are there other suggestions?


title: Bubbling metal vapor through liquid Helium?

text: When a metal cools from its liquid state, part of the nucleation happens from pre-existing clusters (a group of unit cells of the crystal lattice) already present in the liquid. In the case of a phase change from vapour directly to solid, on rapid cooling, is it possible for Aluminium (or some other metal) to exhibit a different crystal structure or worse, be completely amorphous? As I understand it, Aluminum atoms won't have the time to arrange themselves in FCC right? When BMGs (Bulk Metallic Glasses) are formed by rapid solidification, there is no long range order, only short range order, which is because this is already present at the liquid phase. But in this scenario of rapid gas-to-solid transformation, would there be any order, long or short? **What sort of a microstructure would you get on bubbling Aluminium vapors through a large tank of Helium? Would it be a glassy microstructure and/or would you get nano-powders and/or a new crystal structure formation because of the excess helium which would be present in the Aluminium matrix?**


title: To what degree does guitar construction affects the vibration of the strings?

text: There's an old debate going on in the guitar community about how much does wood choice and body shape affect the sound of an electric guitar. No one denies that there's a difference acoustically (how the guitar sounds unplugged) because in this situation it's the wood and body shape that amplify the sound made by string vibrations, but when we're talking about the sound as it comes from the pickups, things get much more uncertain because there are a lot of variations even in supposedly identical guitar parts, and accounting for all of them for the purpose of doing an experiment where the only difference between two guitars is the wood is difficult (I certainly haven't heard of an experiment that was satisfying enough, but feel free to prove me wrong). I'm interested in a way to circumvent these practical difficulties using a theoretical explanation: Since the pickup only sees the string's vibrations, the question basically becomes "does body shape and wood make enough difference in the way a string vibrates that it changes the sound in a noticeable way", and this sounds like something that may be possible to figure out mathematically, or at the very least should be much easier to test because there are less variables involved. My question is: Is what I'm describing possible to calculate/test, and has this been done before?


title: What does $\nu$ mean in relativity?

text: I decided to teach myself relativity over the Christmas holiday, and I've gotten a bit stuck. Coordinates in space time can be defined by a collection of coordinates, $$ x^0 = ct \\\ x^1 = x \\\ x^2 = y \\\ x^3 = z $$ This collection is denoted $x^{\mu}$. My linear algebra kicks in here, and I see that it is a column vector: $$x^{\mu} =\begin{pmatrix}ct\\\x\\\y\\\z\end{pmatrix}$$ Next they define the metric tensor by a matrix, $$\eta_{\mu\nu} =\begin{pmatrix}-1&0&0&0\\\0&1&0&0\\\0&0&1&0\\\0&0&0&1\end{pmatrix}$$ And then say that the definition of the dot product is: $$A\cdot B = \eta_{\mu\nu}A^{\mu}B^{\nu}$$ This leads be to believe that $\nu$ denotes another column vector, but of what ? It is then used in the definition of the space time interval, $$ds^2 = \eta_{\mu\nu}dx^{\mu}dx^{\nu}$$ What is this $x^{\nu}$ ?


title: Could velocity be taken as fundamental instead of time?

text: In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as fundamental and then understand time by dimensional analysis in terms of l/v (length divided by velocity)? If not, why not?


title: How does spectroscopy determine what depth of the remote atmosphere a particular gas is at?

text: This article mentions "...there is less methane than expected in the upper atmosphere" of an exoplanet. How does spectroscopy determine what depth of the atmosphere a particular gas is at? Is it a matter of evaluating the pressure at which the gas would emit/absorb a given wavelength?


title: Why do quasicrystals have well-defined Fourier transforms?

text: I was recently reading about quasicrystals, and I was really surprised to learn that even though they do not have a periodic structure, and only have long range order in a very different sense to the usual one, they can still be detected via crystallographic techniques that involve Bragg diffraction patterns. More specifically, quasicrystals are like Penrose tilings in that > It is self-similar, so the same patterns occur at larger and larger scales. > Thus, the tiling can be obtained through "inflation" (or "deflation") and > **any finite patch from the tiling occurs infinitely many times.** The emphasized text also means that if I have a finite patch of quasicrystal to which I want to add atoms to make the full pattern, then there will be an infinite number of different ways to do this. Therefore, the process of adding atoms to a finite patch is not deterministic: it is constrained by certain rules but there is always some choice. To be more precise about what weirds me out, I feel this 'skips' an intermediate step. I can imagine there being a tiling which is not periodic but which is nevertheless deterministic in that given a starting 'seed' patch the whole pattern is determined. In such a pattern there is no translation invariance but there is nevertheless a very rigid sense of long-range order. * * * Edit: I was recently shown a construction that falls in this case. Consider the discrete one-dimensional point set $\mathbb Z\cup r\mathbb Z=\\{\ldots,-1,0,1,\ldots,\ldots,-r,0,r,\ldots\\}$ where $r$ is irrational. This set is not periodic (although any finite patch has infinite other patches that are arbitrarily similar to it), but it does have a well-defined Fourier transform: it is simply the sum of the transforms of $\mathbb Z$ and $r\mathbb Z$, which are single peaks. However, given an initial patch of length bigger than $r$ and $1$, the rest of the pattern is completely determined, and the long-range order is "rigid" without the pattern being periodic. * * * For quasicrystals, on the other hand, the local orders of two distant patches are definitely correlated but only loosely so. This being the case, I'm having some trouble visualizing how it is possible to obtain diffraction patterns from them, and understanding whether they have well-defined Fourier transforms. To bring this question to a more, precise footing then, let me ask this: **given a starting patch of quasicrystal and a (non-deterministic) rule for adding atoms to it, is the Fourier transform of the full, infinite pattern well defined? If so, what's the intuition that allows this to happen?** If this is actually way more complicated than I realize I would also be OK with a reference to an entry-level resource on the subject, but I would really like a nice explanation of this.


title: Do you feel gravity?

text: I have been reading a few articles about the question why we don't feel/notice gravity in everyday life, but I couldn't understand why exactly we don't feel/notice it, that is, why we don't feel a strong force pulling on us at every moment.


title: about wavefunction and vector entries

text: I am beginer of physics and I am studying some very fundamental idea of quantum mechanics by myself. In the introducing book I am reading, there is an example to show a particle diffraced by a slit or something small then there show diffraction pattern. So if the wavefunction of the particle is represented in momentum space such that it has n entries in the wavevector so integer n stands for the nth diffraction order. So what's the significance of the square of the nth entry? Does it mean the probabilty of finding the particle with momentum diffracted into the nth order of the diffraction pattern? Let's say I have a source of particles, each particle emitted at different speed towards the slit so they will be diffracted independently. How does the intial speed of each atom affect the diffraction pattern? If we look at the nth diffraction order, how will the speed of the particle change the probablity of finding the particle in the nth order? Thanks a lot


title: Retrieving Maxwell's equations from the minimum action principle

text: I'm currently working at the start of Alexei Tsvelik's book _Quantum Field Theory in Condensed Matter Physics._ I'm kinda stumped on a few essential steps. Starting with the action: $$S = \int dt \int d^3 x \mathcal{L}_{\mathrm{EM}} = \int dt \int d^3 x \left( -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^{\mu}\right).$$ In the absence of any sources $j_\mu = 0$ and $c=1$, which after pulling in the negative sign implies $$S = \frac{1}{2}\int dt \int d^3 x \,\,\left(E^2 - B^2\right)$$ The next step is to take $\vec{E} = -\vec{\nabla} \phi + \partial_t \vec{A}$ and $\vec{B} = \vec{\nabla} \times \vec{A}$. Letting $\phi = 0$ and directly inserting these into the above yields: $$S = \frac{1}{2} \int dt \int d^3 x \,\left( (\partial_t \vec{A})^2 - (\vec{\nabla} \times \vec{A})^2 \right) $$ Now considering $\vec{A} = \vec{A}_0 + \delta \vec{A}$ as some minimal change in our vector potential, with $\vec{A}_0$ gives a minimum for the action and putting this into the above expression we have: $$\delta S = \int dt \int d^3 x \left(\partial_0 \vec{A}_0 \partial_0 \delta \vec{A} + \left(\vec{\nabla} \times \vec{A}_0\right)\cdot\left(\vec{\nabla} \times \delta \vec{A} \right) \right)+ \mathcal{O}(A^2)\,\,\,\,\,\,\, (1)$$ where I have the term $$\mathcal{O}(A^2) = \int dt \int d^3 x \left( (\partial_0 \vec{A}_0)^2 + (\partial_0 \delta \vec{A})^2 - (\vec{\nabla}\times\vec{A}_0)^2-(\vec{\nabla}\times \delta\vec{A})^2\right)$$ The next step is where I lose our author and go from "Oh ok I get this" to "completely clueless". He rewrites $\delta S$ as: $$\delta S = \int dt \int d^3 (\delta x) \vec{A} \mathcal{F(\vec{A}_0)}+ \mathcal{O}(A^2)$$ where $\mathcal{F} = \frac{\delta S}{\delta A}$. What is the purpose of $x\to \delta x$ in the above expression? Is the functional derivative $\mathcal{F}$ a vector quantity? He has it bolded with a bolded $\vec{A}$ in the above expression. One last question takes the above expression further. We assume $\delta \vec{A}$ vanishes at infinity, as it should, and integration by parts yields: $$\delta S = -\int dt \int d^3 x \left( \partial^2_0 \vec{A}_0 - (\vec{\nabla} \times \vec{\nabla})\times \vec{A}_0 \right)\delta \vec{A}\,\,\,\,\,\,\,\, (2)$$ How exactly did we go from (1) to (2)? Thanks I appreciate your time!


title: Nuclear fusion: what causes this "resonance" peak?

text: Can anyone explain why the $^{11}\mathrm{B}\mathrm{H}$ fusion cross-section has a peak near 150 keV, and why $\mathrm{D}\mathrm{D}$ and $\mathrm{D}\mathrm{T}$ have no such sharp peaks? ![Fusion cross-section](http://i.stack.imgur.com/Xf20f.jpg)


title: Row of pivoted magnets and energy scale

text: This question is about a system involving a horizontal row of length **L** of equally spaced pivotable magnets, each with a pole at either end. These magnets will often be referred to as units. So each unit when rotated causes it's neighbours to rotate in the opposite direction to itself. When the first unit is quickly rotated 1/4 of a turn and fixed in place rotationally, each will likely turn less than the previous at first, taking longer to complete the full 1/4 of a turn. They will all eventually rotate the full 1/4 as the far end of the line is still not fixed. . > Turning the first magnet will require energy. For a line of 1000, while this > will be more energy than that for a line of for example 2, it will be less > than 1000 times the energy required for a line of 1. This is because: > >>   1. Momentum means that every unit doesn't need to be rotated the full distance to turn the first unit, it is not as if they are connected together by rods, the first can be turned all the way before the others have had time to rotate much at all. >> >>   2. More distant magnets contribute less force, so most units are contributing very little force to the first unit directly. Another way to think about this is that the line of separated magnets forms a compound magnet that will result in more force than each component magnet has individually, but not as much as 1000 times more, for example if metal becomes attached to an end it will take less than 1000 times the force of an individual magnet to remove. >> >> . Coils around each unit slow rotation but produce **N** electricity after 1/4 of a turn, however long the 1/4 turn takes. The amount of electricity produced appears to be L * N, for an input power that is increment by a diminishing amount for the growth of L, an example of this sort of growth could be sqrt(L). . > **Question:** This has the potential not to add up energy wise for long > lines, what's going on?