title: Which qubit states are accessible with linear optics operations?

text: Given a quantum state of $n$ qubits, and being restricted to linear optics (that is, the output annihilation operators are linear combinations of the input annihilation operators):   * Which states are accessible, that is can be made from $|\psi \rangle$ with certainty? (Are there any simple criteria?)   * If we want to get (any) $|\phi \rangle$ from $|\psi \rangle$ and we allow post-selection, are there any known bounds on the success rate?  If it is going to simplify anything, one may assume that we consider states having exactly one photon in each rail: $\langle \psi | \hat{a}_{\updownarrow i}^{\dagger 2} \hat{a}_{\updownarrow i}^{2} | \psi \rangle = \langle \psi | \hat{a}_{\mathord{\leftrightarrow} i}^{\dagger 2} \hat{a}_{\mathord{\leftrightarrow}i}^{2} | \psi \rangle = \langle \psi | \hat{a}_{\mathord{\leftrightarrow} i}^{\dagger} \hat{a}_{\updownarrow i}^{\dagger}\hat{a}_{\mathord{\leftrightarrow}i} \hat{a}_{\updownarrow i} | \psi \rangle = 0$.


title: Modified Maxwell's equations

text: In 1985, Harmuth wrote that Maxwell's equations are incompatible with causality, and overcame the problem by adding a term for magnetic dipole currents, and as a consequence the problem of infinite zero-point energy and renormalization disappears. At least according to Harmuth's book: Calculus of finite differences in quantum electrodynamics By Henning F. Harmuth, Beate Meffert The foreword is readable at http://books.google.com/books?id=EQFENquY6fwC Are Harmuth's modifications generally accepted by the physics community as a more accurate description of reality than the unmodified equations ?


title: Decomposition of a vectorial field in free-curl and free-divergence fields

text: Is it always possible to do that decomposition? I'm asking it because Helmholtz theorem says a field on $\mathbb{R}^3$ that vanishes at infinity ($r\to \infty$) can be decomposed univocally into a gradient and a curl. But I also know, for example, that a constant field $\mathbf{E}$ on $\mathbb{R}^3$ is a gradient (not univocally definied): $\mathbf{E}(x+y+z+\mbox{constant})$. And the electric field is $-\nabla G+ d\mathbf{A}/dt$, where $\mathbf{A}$ can be (Coulomb Gauge) free-divergence. So, is it always possible to do the decomposition of a (regular, of course) field on $\mathbb{R}^3$ into two fields, free-curl and free-divergence? And on a limited domain?


title: Finding Distance an Object Travels Up an Incline After Launch

text: I've been doing a review for an introductory physics course final. I have a question on one problem though. Here is the problem: > A mass (M=2kg) is placed in front of a spring with k=900N/m, compressed > d=50cm. The spring is released, shooting the mass forward from A to the > spring’s equilibrium position at B (A to B is frictionless). The mass then > travels along a flat surface from B to C (L=20m), with μk=0.15. At C the > surface becomes frictionless and smoothly inclines upwards. The speed of the > mass at point D is 0m/s. ![Diagram illustrating > problem](http://i.stack.imgur.com/mZRnJ.png) What I've been asked so far: velocity at point B, then at point C. I calculated these out and got $10.61$ ${m}\over{s}$ and $7.33$ ${m}\over{s}$, respectively. However, now I'm being asked the height, $h$, of point $D$. **EDIT: Solution**   How could I have forgotten about gravatational potential energy!? I used $K_i$ = $U_f$, to retrieve the answer, knowing that $K_i$ = ${1}\over{2}$$mv^2$ and $U_f$ = $mgh$. This also made me notice (as I've been taught before) that mass is irrelevant in this part of the problem. **Attempt at solution:**   I know 3 equations that I assumed would help:   $x =$ ${1}\over{2}$$at^2$ + $v_{ox}t$ + $x_o$   $v = at$ + $v_{ox}$   $v^2 =$ $v_{ox}^2$ + $2a(x - x_o)$   I know that $v_o$ in any of these would be equivalent to the velocity I found at $C$, and that $v$ would be 0, since the object is at rest at point $D$. However, I don't know any times nor accelerations (as far as I'm concerned), so I'm stuck.   Other info: we've learned about conservation of energy and momentum, work, power, and a few others, but simple kinematics seem to be the only appropriate application here, unless I'm wrong on that too. The answer is supposedly $2.74m$, but once again, I'm unsure how to get here. Any pointers would help, as I'd like to be well prepared for tomorrow's final! Relevance of question (in my opinion): yes, this is a specific question, which isn't necessarily favored, but this problem seems rather common in introductory physics courses, so I'm sure many others that need help could see this and apply it appropriately as well.


title: About heat rate and dimensions convention

text: Suppose I have a ceiling and its dimension are given in $Z \times C \times Y$, and a it's got a thermal conductivity $k$. Now, I want to know H, or the heat rate. It's given by $ H =\frac{kA(T_h -T_c)}{L}$. Where $T_h \ and \ T_c$ are the hot and cold temperature difference. How do I use the given dimensions for $A$ and $L$? Can I just assume $Z \times C$ is the area $A$ and $Y$ is $L$?


title: Are quantum mechanics calculations useful for engineering?

text: I heard it's is pretty tough to get results for more than a few quantum particles. Are quantum mechanical calculations useful at all for any technology that is being sold? Or do they use quasi-classical results at most? Is there hope that advances in QM calculations would make any change to the technological world? I'm basically considering everything that could make a change to our life disregarding pure theoretical research results. Which path of QM advances could potentially be the most yielding in terms of practical results? EDIT: From what I experienced, experiments are already producing results, when theorists are still trying to fit their theories to the data. So why do you need the theoretical calculations then? Do they have predictive power that would be found easier and more precise with experiments?


title: What happens at the subatomic level when energy is added to an atom?

text: For the sake of simplicity I shall use the hydrogen atom. I know that adding energy to the hydrogen atom means bombarding it with photons. I am curious about what happens to the quarks and electron of the atom. Does the electron move to ever higher electron shells as energy is added? Does the velocity of the electron increase? What happens to the quarks as they continue to gain energy?


title: Coupled Spring System (3 mass 3 springs)

text: Hello I am having trouble trying to find the correct model for this coupled spring system. The scenario is the following we have: Ceiling - Spring - Mass(1) - Spring(2) - Mass(2) - Spring (3) - Mass(3) End. I came up with the following system of differential equations in the 2nd order to model this problem. $x_1^{''}=[-k_1x_1-k_2(x_2-x_1)-k_3(x_3-x_2)]/m_1$ $x_2^{''}=[-k_2(x_2-x_1)-k_3(x_3-x_2)]/m_2$ $x_3^{''}=-k_3(x_3-x_2)/m_3$ Is this the correct model? Afterwards I am trying to linearize these equations into 6 differential equations that I can input in matlab and plot the position of each spring. So I linearized them and obtained the following: $y_1^{'}=y_2$ $y_2^{'}=(-k_1y_1-k_2(y_3-y_1)-k_3(y_5-y_3)/m_1$ $y_3^{'}=y_4$ $y_4^{'}=(-k_2(y_3-y_1)-k_3(y_5-y_3)/m_2$ $y_5^{'}=y_6$ $y_6^{'}=(-k_3(y_5-y_3))/m_3$ I am not sure if this is correct or not. When I plot them in matlab I dont get a sinusoidal wave. A big plus if you guys can tell me how I could animate this system in matlab so that I can see the change in position in all three of the springs.


title: advantage of string theory over other theory-of-everything candidates

text: I am getting curious over why string theory, especially M-theory, is the most popular candidate for the theory of everything. It seems that all candidates of the theory of everything lack substantial experimental supports, so why is string theory so special? Thanks.


title: Lowest gravity on Earth's surface?

text: I am trying to determine which on Earth's surface has the lowest gravity. Googling is not finding anything concrete. My natural inclination would be to think of Mount Chimborazo in Ecuador, being on the equator (centripetal force) and also being the furthest point from the Earth's center. However, the GRACE gravity map has the area listed in red for high gravity, presumably due to local mineral anomalies or mantle structure. Is there any definitive source for information on the topic?


title: What "propagates" a force through the rest of a solid?

text: So, in typing the title of this question I was recommended this awesome one, which confirmed my guess that this effect "propagates" at the speed of sound (though I just had a feeling, I don't really understand why). I also only have high-school level physics experience. I feel like I don't know physics well enough to really know what to ask correctly, but I'll try to explain. So, I rotate my arm, and, even though tendons are just pulling it at a certain location, the rest of it follows suit. Why is that? What is happening at an atomic/molecular scale that ends up "conveying forces" over a distance? And, something else I don't understand- what is so special about the speed of sound that makes it this "fundamental unit of translation", or something? Maybe the better question is "what processes at an atomic/molecular scale lead to the speed of sound being associated with all of these behaviors", which ties both questions together.


title: Are planetary orbits measured from the Sun's surface or centre?

text: I would imagine planetary orbits are measured from the Sun's centre and not its surface. Is that true? I can't find anywhere that actually states this.


title: The path integral and Feynman diagrams

text: This question is somewhat of a historical one, but it also contains some physics. I am curious to find how exactly the concept of Feynman diagrams arose (I assume from Feynman's path integral)? The leap from path integrals to diagrammatic computations isn't obvious (to me, at least); I'd like to understand better how Feynman's thinking approximately developed. For instance, how did he come up with interpreting the propagator as the propagation of particles? Was there a particular analogy that can be made? Is there any understanding to be gained by learning how the technique was originally developed? I realize that my phrasing might be a quite vague. If the question is too broad as of right now, please let me know how I can improve on it!


title: Can neutrino detectors tell what direction the neutrinos came from?

text: I was reading this question and got to thinking. Can neutrino detectors give us any clue where the neutrinos came from or when a supernova may occur? I was unsure and decided to ask that here.


title: Polarization in Lasers and Liquid Crystal Displays

text: When we talk about Polarization in Lasers we mean placement of the crystal at Brewster angle to block the s-polarization and only the p-polarization will survive. How do we compare the term polarization in Lasers and in liquid crystal displays? So, for example, we know there are polarizers present in the case of LCD. Does that mean we are also using these polarizers to block the back-light s-polarization and want the p-polarization at the output?


title: Electrons Orbiting Sphere Magnet in Crooke's Tube

text: Were I to assemble a Crooke's Tube and insert a spherical magnet, with the poles of the magnet perpendicular to the electron stream, would the electrons begin to orbit the poles of the magnet as in the images of Terrella experiments? I am pretty sure yes. If so, are there equations to deduce the ideal distance from the electron emitter that the spherical magnet should sit? To determine how strong my magnet should be? What should I be reading to become more fluent in these sorts of experiments? Thank you for any assistance.


title: Find state equation given cubic expansion and isothermal compressibility coefficients

text: **Context** : I'm studying basic thermodynamics. Since I'm not confident that I translate the terms correctly, let me define the following coefficients mathematically: \begin{equation} \beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P, k=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T \end{equation} The first one is the coefficient of cubic expansion (or coefficient of volume expansion) and the second one is the coefficient of isothermal compressibility. **Question** : The coefficients are given as: $\beta = \frac{V-a}{TV}, k = \frac{V-a}{4PV}$ and we are asked to 1. check whether they are "compatible" and 2. find the state equation of the system. $a$ is a constant. **Attempted answer** : The only constraints I found about $\beta, k$ in my book where for $k$ which "shall always decrease when pressure increases". This is compatible with the value of $k$, since $P$ is in the denominator. Regarding the state equation, I start from the known relation: \begin{equation} \frac{dV}{V} = \beta dT - kdP \end{equation} where I substitute the given expressions. At some point I end up with: \begin{equation} dV=(V-A) \frac{dT}{T} -(3/4)(V-a)\frac{dP}{P} \end{equation} By integrating I arrive at: \begin{equation} V = (V-a) \ln{T}-(3/4)(V-a)\ln{P}=(V-a)\ln{\frac{T}{P^{3/4}}} \end{equation} If $V=a$, then RHS becomes zero, therefore the LHS must also be zero, $V=0$. Therefore $a=0$. So the equation becomes: \begin{equation} V=V \ln{\frac{T}{P^{3/4}}} \Leftrightarrow \ln{\frac{T}{P^{3/4}}} = 1 \end{equation} Does that seem right to you? I have a strange feeling that something is wrong in my solution.


title: Converse of the Lagrangian form-invariance

text: The form-invariance of the Lagrange equations implies the existence of a function $\ A( q_k, t)$ so that $\ \begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \frac d {dt} A( q_k, t) \end{equation}$ Is there any way of proving the converse? That is, asuming the existence of the funcion $A$ prove the form-invariance of the Lagrangre equations.


title: How the tube of a tire get puncture?

text: How the tube of a tire(cycle) get puncture? I mean, the inner tube is protected by an outer tube, so how is it possible to make a hole in the inner tube? And in general the puncture hole is very small. Why?


title: Calculation of spherical Bessel functions - meaning of $\left(\frac{1}{x}\frac{d}{dx}\right)^{l}$

text: I'm trying to understand the calculation of spherical Bessel functions in chapter four of Griffiths' Introduction to Quantum Mechanics (2nd ed, p142). He gives $$j_{2}\left(x\right)=\left(-x\right)^{2}\left(\frac{1}{x}\frac{d}{dx}\right)^{2}\frac{\sin x}{x}=x^{2}\left(\frac{1}{x}\frac{d}{dx}\right)\frac{x\cos x-\sin x}{x^{3}}$$ $$=\frac{3\sin x-3x\cos x-x^{2}\sin x}{x^{3}}.$$ I can't see how he arrives at this answer. I think my problem is the $\left(\frac{1}{x}\frac{d}{dx}\right)^{2}$ bit (the general term for $j_{l}\left(x\right)$ is $\left(\frac{1}{x}\frac{d}{dx}\right)^{l}$ ). I'm assuming this means $1/x^{2}$ multiplied by the second derivative of $\frac{\sin x}{x}$ but I make that $$\left(\frac{1}{x^{2}}\right)\left(-\frac{\sin x}{x}+\frac{2\sin x}{x^{3}}-\frac{2\cos x}{x^{2}}\right).$$ Any idea what I'm doing wrong?


title: Equations for the two-plane model of the greenhouse effect

text: I'm trying to understand this "toy model" of the greenhouse effect. http://www.realclimate.org/index.php/archives/2007/04/learning-from-a-simple- model The model predicts the surface temperature of the Earth given the sunlight intensity and the emissivity of the atmosphere, $\lambda$. To do this, they use a two-energy model of light. There's shortwave radiation from the sun, and there's thermal radiation from terrestrial sources. They way they define $S$, the radiation intensity from the sun, allows for the atmosphere to reflect some shortwave light back into space, but the model doesn't allow for the atmosphere to absorb any shortwave light, meaning that all of $S$ makes it to the ground. Then the ground is treated as a blackbody. It logically follows that it emits radiation of intensity $G$ upwards, of which $\lambda G$ is absorbed by the atmosphere. Atmosphere emits $1/2 \lambda G$ back down to the surface because it is also a blackbody. What I don't understand is the complexity they seem to invoke to write equations describing this. They use equations for surface, atmosphere, and planet. $$S + \lambda A = G$$ $$\lambda G = 2 \lambda A$$ $$S = \lambda A + (1 - \lambda) G$$ These are 3 equations, but aren't there only 2 unknowns? My question: how do you draw the necessary system boundaries to get closure for this system? Also, is there a requirement on the temperature of the atmosphere? I think the source argued that it need to be cold for it to have insulation properties, but I don't think that's reflected in the math, since we require (radiation in) = (radiation out). That would seem to imply the temperature of the atmosphere is arbitrary.


title: What happens to the acceleration of a body undergoing constant force?

text: Here's the problem I have, specifically relating to a rocket in deep space: A rocket burns fuel at a constant rate. Assuming its mass remains constant, what happens to its acceleration? I seem to get two different answers depending on how I solve it (using conservation of energy or conservation of momentum): # Conservation of Energy $\frac12mv^2=E_{exhaust}$, where $E_{exhaust}$ varies with time. $v=\sqrt{\frac2mt}$, meaning that velocity varies with $\sqrt t$, meaning that acceleration **decreases** over time. This also means that depending on your frame of reference's velocity relative to the rocket, the ship's acceleration **must** change...which makes about 0 sense, # Conservation of Momentum $P_{exhaust}=mv$ Meaning that as more fuel is consumed, the velocity of the ship increases too (as $P_{exhaust}$ increases linearly with time, $v$ must also increase linearly with time). I am assuming that the mass of the spaceship is constant. While not quite true, the effect would be negligible and the paradox I'm running into would remain true anyways. Can anyone tell me what I'm doing wrong? The acceleration **cannot** possibly change depending on where you observe it from, but the acceleration also **cannot** remain constant as that violates conservation of energy. Thanks in advance!


title: Expectation of 2-form field $B_{MN}$ in string theory

text: In the context of string theory, in particular when we're dealing with a low energy effective action, if we have an effective action of the form: $$S_{eff} \sim S^{(0)} + \alpha S^{(1)} + (\alpha)^2 S^{(2)} + \ldots$$ In which $\alpha$ is the Regge slope: $$S^{(0)} \sim \int\limits d^Dx H_{MNP}H^{MNP}$$ and $H_{MNP} = \partial_M B_{NP} + \partial_N B_{PM} +\partial_P B_{MN}$. When we are asked to put $H_{MNP}=0$ to compute the list of terms corresponding to the tensor expectation relative to the field, what are we supposed to do exactly? One more question: How can we find the field equations of the tensor to any order?


title: Hamiltonian and non conservative force

text: I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$. I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is conjugated momentum, $\dot q_i$ is the velocity and $L$ is the Lagrangian. The result that I should obtain is $$H=\frac{1}{2} m(\dot x^2+ \dot y^2)= \frac {1}{2m}(p_x^2+ p_y^2)+ \frac{1}{2}\omega^2 (x^2+y^2)+ \omega (p_x y- p_y x)$$, where $\omega= \frac{eB}{2mc}$. I obtain this result only if I don't consider in the Hamiltonian the potential magnetic, and if I substitute only at the last step the values of velocities in function of conjugata momenta. Considering that the magnetic field isn't a field of conservative forces, I ask you: **if I have a system in a non conservative field, is it correct to not consider the potential of the non conservative force when I'm writing the Hamiltonian?** are my steps correct? thank you.


title: Variation of modified Einstein Hilbert Action

text: In general relativity one can derive the Einstein Field Equations by the principle of least action through variations with respect to the inverse of the metric tensor. In some modified theories of gravity, such as the Brans- Dicke Theory, a scalar field is added to the Einstein Hilbert Action and the gravitational constant is replaced by a function of the scalar field. I am not quite sure how to derive the field equations from this action, more specifically the part where the scalar field is attached to the Ricci scalar $\phi R$. The Brans-Dicke Action is $$S_{BD} = \int d^4x \sqrt{-g}\left[ \frac{1}{16\pi}\left(\phi R - \frac{\omega }{\phi}g^{ab} \partial _a\phi\partial _b \phi \right) +L_M \right].$$ The resulting field equation is $$G_{ab} = \frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2} (\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi).$$ I also want to derive a new field equation for practice. So my questions are:   1. How does one derive the equations of motion?   2. How to perform the variation of the following action? $$S=\int d^4x \sqrt{-g} \left[ \frac{1}{16\pi G} R - \phi( \nabla_{\mu} g_{ab} \nabla_{\nu}g_{ab}) - 2\Lambda+L_M) \right] $$ The Ricci scalar, the cosmological constant, and the matter Lagrangian will variate simply like the Einstein Hilbert Action to: $$\delta S = \int d^4x \sqrt{-g} \left[ \frac{1}{\kappa} \left( R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab} \right) -T_{ab} \right]\delta g^{ab}.$$ What about the extra term? Would one simply variate with respect to $\phi$, or is the variation of the covariant derivative of the metric tensor also required? If the latter is true, then would the variation of this extra term be $$\frac{\partial L}{\partial g_{ab}}-\partial _\mu\frac{\partial L}{\partial (\nabla_{\mu}g_{ab})}=0.$$ Any help would be appreciated. By the way, is $\nabla_{\mu}g_{ab}\nabla_{\nu}g_{ab}$ an expression that shows the rate of change (derivative) of the metric tensor with respect to a coordinate $(t, x, y, z)$?


title: How depletion region is developed in a Schottky diode?

text: In a P-N junction depletion region is developed when electrons from N-region diffuse to P-region and hole from P-region diffuse to N-region. Here diffusion thereby development of depletion is caused by the movement of majority carriers. Schottky diode has a metal and N-type semiconductors junction. In both metal and N-type semiconductor majority carriers are electrons. Then how a barrier is developed at the metal-semiconductor junction of a Schottky diode?


title: Lie group of Schrodinger Wave equation

text: In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 generators (3 rotations - $L_i$ , 3 translations - $P_i$, 3 boosts - $G_i$ and time translation - $H$). Apart from this the Schrodinger solution (the probability amplitude) is arbitrary upto to a phase factor ($e^{i\phi}$). Hence we include one more generator induced by the transformation of phase. With this the general Unitary transformation, $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + \delta\phi\mathbb{\hat1} = \sum_{i=1}^{10} \delta s_iK_i + \delta\phi\mathbb{\hat1}$$ The commutation relations of the group altogether can be given as, $$ [K_i,K_j] = i\sum_{n}C_{ij}^{\;\;n}K_n + ib_{ij}\mathbb{\hat1} $$. Now, this commutation relation does not have the structure of Lie algebra. Since with Lie Algebra the elements are closed under commutation. This one has an extra element sitting with $\delta\phi\mathbb{\hat1}$. What is really going on here ? Is this really a 11-parameter Lie group ? If so how do we convince about the algebra of generators ?


title: Why does smoke go out the window of the car - and what if there's wind blowing instead of the car moving?

text: When driving a car while smoking with the window open (safety and legal issues aside), I've noticed that the smoke tends to go outside the window.   1. Why does the smoke go outside?   2. If the car is standing still and there is wind blowing at the same velocity the car was going - will the smoke behave the same?


title: Why don't the derivatives of the blackbody spectrum over frequency and wavelength match?

text: The question is: The radiation emitted by a black body can be represented either by the energy distribution over the wavelength or by that over the frequency. In the first case the maximum energy corresponds to $\lambda(m)$ , while in the second case there is a maximum corresponding to frequency $\nu(m)$. Is $\nu(m)$= $c$/$\lambda(m)$? Now I have basically solved the problem by showing that $dE(\nu)/d(\nu)$= $0$ does not correspond to $dE(\lambda)/d(\lambda)$=$0$. My question is: how do I account for this result physically?


title: When does the angular momentum point in a different direction from the angular velocity?

text: I read this somewhere: $$\mathbf{L} = \tilde{\mathbf{I}}\mathbf{\omega}$$ In general, the angular momentum vector, $\mathbf{L}$, obtained from Equation above, points in a different direction to the angular velocity vector, $\mathbf{\omega}$. In other words, $\mathbf{L}$ is generally not parallel to $\mathbf{\omega}$. I don't quite get it: when is the angular momentum not parallel to the velocity? Because judging from the above equation, it should be parallel, since moment of inertia is a scalar?


title: Utilizing maximum acceleration $a$ for displacement $d$ with initial velocity $v_0$ and final velocity $v_1$

text: # Problem My goal is to move an object from point a to b (displacement $d$) as fast as possible utilizing the maximum available acceleration $a_{max}$, taking into account the initial velocity $v_0$ and final velocity $v_1$. # Elaboration I'm working on spaceship physics which means there is no friction or drag. The direction of the object is not necessarily parallel to the direction of it's velocity, neither are the initial and final velocities necessarily parallel. * * * # Solving for one dimension I started off by solving this in one dimension using the formulas for constant acceleration, specifically: $$ d = \frac{v_1^2-v_0^2}{2a} $$ As my intuition tells me there should be a displacement $d_0$ where constant acceleration $a_{max}$ is performed (where the object reaches its maximum velocity $v_{max}$) followed by a displacement $d_1$ where constant deceleration $-a_{max}$ is performed. From the above equation, and from: $$ d = d_0 + d_1 $$ We find that: $$ d = \frac{v_{max}^2-v_0^2}{2a_{max}} + \frac{v_1^2-v_{max}^2}{-2a_{max}} $$ Solving for $v_{max}$ gives: $$ v_{max}^2 = \frac{2a_{max}d+v_1^2+v_0^2}2 $$ Using this information I can accelerate the object until $v_{max}$ is reached and then decelerate it. * * * # Solving for 3 dimensions Doing this in 3-dimensions is a bit more difficult. I'm assuming that $||\vec{a}||$ should be constant and equal the maximum available acceleration $a_{max}$. My first attempt here was to calculate $v_{max}$ (like above) for each axis seperately. The problem though is to find a proper acceleration distribution along each axis when the displacement changes (having a moving target), so that the object for example does not go into orbit. In another attempt I did the following spring/damper like calculation to find $\vec{a}$: $$ \vec{s} = \vec{v_1}-\vec{v_0}+\vec{d} $$ $$ \vec{a} = \frac{\vec{s}}{||\vec{s}||} * a_{max} $$ This works fine when accelerating but deceleration starts off too early. I have been trying to search the web for this information but to no avail, any suggestions?


title: Why the direction of dipole moment is from negative charge to positive charge?

text: An electric dipole moment is defined as $p = q\times 2d$. How to understand it physically? Why the direction of the electric dipole moment is from negative charge to positive charge?


title: Is there any example where electric and magnetic fields are not perpendicular?

text: Perpendicular electric and magnetic field creates light or other electromagnetic waves. Is it a necessary property to have a perpendicular fields? If not what would happen when the fields are not perpendicular?


title: Can virtual image be photographed

text: Is there any instrument that can photograph a virtal image? How would such a device work?


title: Can acceleration feel like constant gravity for indefinitely long?

text: So here's the setup: I'm in a spaceship, without windows as always, and the ship is accelerating upwards at a constant rate of $1\,\text{g}$. So inside the spaceship it feels like I'm being pulled down by gravity as normal. But this constant acceleration can't last forever: I have to approach $c$ asymptotically. I believe the equation for my speed is: $$v/c = \tanh(at/c).$$ That's what an outsider would observe. Correct? Not really my question though. What I want to know is, as my speed approaches $c$ and the outsider observes my acceleration slowing down, do I feel that? Or does time dilation and length contraction mean that I would feel the exact same acceleration/gravity inside, forever?


title: Simple harmonic oscillator system and changes in its total energy

text: Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to point $B$ and $CB=d$. Its motion is harmonic. The total energy of such a system is simply $\frac{1}{2} k \left(\frac{d}{2} \right)^2=\frac{1}{8} k d^2$ (because the equilibrium point is at $d/2$ and $d/2$ is the amplitude). Now, suppose we drop vertically some plasticine with the same mass $M$ from some height $h$. After it hits the oscillating object, it just sticks to it. My question is - why the total energy of the system doesn't change if the plasticine hits the object at point $C$ but it changes when it hits the object in the middle of $CB$ (it equals there to $\frac{1}{16} k d^2$)? Intuitively, I do understand that such a plastic collision contributes to the loss of energy, but I'm not sure how exactly it fits here and how the energy is lost. And even so, I do not understand how the position of the body influences the change in energy. * * * **Proposed solution:** The total mechanical energy at point $d/2$ is $E_{k,max}=P_{total}=\frac{1}{8} k d^2$: $E_k=\frac{M}{2}V^2\\\ V=\sqrt{\frac{2E_k}{M}}=\sqrt{\frac{2P_{total}}{M}} $ Due to the conservation of the horizontal momentum we get: $MV=2MV'\\\ \frac{2E_k}{M}=4V'^2\\\ V'=\sqrt{\frac{E_k}{2M}}=\sqrt{\frac{\frac{1}{8}kd^2}{2M}}=\sqrt{\frac{kd^2}{16M}}$ Therefore, the momentary kinetic energy (which is the total energy) after the hit is: $$E_{k, after}=\frac{2MV'^2}{2}=M \frac{kd^2}{16M}=\frac{kd^2}{16}$$


title: Coarse-graining on a second channel decreases mutual information?

text: Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$. This can be thought of as a bound on the capacity of a quantum channel called 2 (for example, you either perfectly correlate input $B_2$ with output $X_2$ (measurement result), or input $A_2$ with output $Y_2$, depending on the value of a bit $C_2$). Do we have: $I(X_1:B_1|C_1=0) \geq I(X_1\oplus X_2:B_1\oplus B_2|C_1=0, C_2=0)+ I(X_1\oplus Y_2:B_1\oplus A_2|C_1=0, C_2=1)$ ? $I$ is the mutual information and $\oplus$ the sum modulo 2. If this inequality is true, how would you show it?


title: Why are string theorists interested in entanglement entropy?

text: I have been reading some papers by Ryu-Takyanagi but I am not seeing a good explanation as to why entanglement entropy of the boundary CFT is a good observable to probe the possible bulk quantum gravity. Is there any sense of non-renormalizability about this observable? Any kind of deformation invariance? (like Witten index) Is it even defined for CFTs at finite temperature or at arbitrary coupling?


title: Quick question on convolution - Diffraction through a pair of slits

text: We know that the fourier transform of the amplitude function (in terms of $y$) gives you the amplitude function (in terms of $\theta$) Consider a pair of triangular slits: ![enter image description here](http://i.stack.imgur.com/RJ438.png) Fourier transform of G(y) is: $$FT[G_{(y)}] = G_{(\theta)} = e^{-i\beta a} sinc \left( \frac{\beta a}{2}\right) cos\left( \frac{\beta a}{2}\right)$$ Fourier transform of H(y) is: $$FT[H_{(y)}] = \frac{1}{2} e^{-\frac{i\beta a}{2}} sinc \left(\frac{\beta a}{2} \right) $$ Since $T = G \otimes H$, it implies that $FT[T_{(x)}] = FT[G_{(x)}] \cdot FT[H_{(x)}] $ Thus in terms of $\theta$: $$ G_{(\theta)} = \frac{1}{2} e^{-i\beta a} sinc \left( \frac{\beta a}{2}\right) cos\left( \frac{\beta a}{2}\right) e^{-\frac{i\beta a}{2}} sinc \left(\frac{\beta a}{2} \right) $$ Thus the intensity is: $$G \propto |G_{(\theta)}|^2 = I_0 sinc^4 \left( \frac{\beta a}{2}\right) cos ^2 \left( \frac{\beta a}{2}\right) $$ This makes sense because for a double rectangular slit, due to its non-zero width, the envelope falls as $\propto sinc^2 \left(\frac{\beta a}{2} \right)$. Now the triangular slit has **half the area** of the rectangular slit, so we would expect the envelope to fall even faster. In fourier space, factor of $\frac{1}{2}$ the area is brought up to the power, giving $\propto \left( sinc^2 \left(\frac{\beta a}{2} \right) \right)^2$ Does this make sense?


title: Does object sealed in a airless place weigh more?

text: In a case where you vacuum your items in those airless bags(Space bags), do they weigh more ? Or is it just a feeling because of increase in density due to decrease in volume ?


title: How to ‘correct’ reflected image on my telescope?

text: When viewed using my reflector telescope, the image, for example the moon, is flipped over and upside down (reflected). Therefore I need to do the ‘opposite’ movements when I want to align something. Is there something I can do to correct this, or is that part of using a reflector, or am I doing something wrong ? e.g. : The moon’s crescent is currently to my bottom right, however, when viewed through my 'scope, it’s on my top left. thanks


title: Why is it hard to write a computer program to detect patterns which planethunters is asking us to figure out?

text: I recently came across this site http://www.planethunters.org. It shows brightness observed for a star vs time. It asks questions like if star is exhibiting variable or fixed pattern. Later it asks if there are any transits which can indicate presence of planet. While I find this interesting, it made me wonder why can't a computer program detect these patterns. Is this really hard? Given the fact this field has been subject of research for past so much time. These days we have programs can detect face, read text on live phone camera and translate it. So what's stopping this? P.S. Also how can I ask same question across multiple stackexchange site? Maybe this is more SO / Physics question.


title: Balancing stick on hand - inverted pendulum motion

text: How do I calculate the velocity and acceleration of a stick that is vertically on my hand? How fast and far do I need to move my hand from side to side to keep balancing the stick? Assuming that we neglect friction force.


title: Contents page of Exoplanets book

text: I can't find an online contents page for the book _Exoplanets_ , edited by Sara Seager, University of Arizona Press, 2011. Can you point to such a page, or say what the contents page says?


title: Moment of Inertia in SR/GR & Calculating it in General

text: In classical mechanics you want to calculate the moment of inertia for hollow & solid: lines, triangles, squares/rectangles, polygons, planes, pyramids, cubes/parallelepiped's, circles, ellipses, parabola's, hyperbola's, sphere's, ellipsoid's, paraboloid's, hyperboloid's, cones & cylinder's setting them up in either scalar notation (i.e. geometrically deriving it as in elementary physics books) or in tensor notation, which at least for me is an immense task with so many possibilities I still haven't fully carried it all out. My question is, how does all this translate over the special &/or (?) general relativity? Do you have to re-do every one of those calculations from a more general standpoint or is it just that the density in the integral is usually variable? As a side note, is there an easier & more way to do all of the above? For instance, in calculus books they sometimes put MoI into 3 different chapters, leading to single, double & triple integral modelling on top of physical modelling (in physics books) or tensor modelling (in advanced physics books) which is really 5 ways to set up about 30 geometric models... Thinking about it, you can apparently sometimes use Stokes theorem, e.g. slide 26, to show some of these models are exactly equivalent, but how do I deal with it all in general in a unified manner, where do I start to derive it all in as simple a manner as possible? Thanks


title: Hyperkahler manifolds and their use in theoretical physics

text: Just as the title says:   1. What is the easiest definition of a Hyperkahler Manifold?   2. Could you give some examples of Hyperkahler manifolds, and manifolds which fail to be hyperkahler?   3. Why are such manifolds considered to be interesting in physics, and how do they arise in the study of supersymmetric gauge theories?   4. Apart from SUSY, are there any other branches of physics in which Hyperkahler manifolds appear?


title: Could two colliding comets near Earth cause devastation?

text: Let's say two comets crashed into each other. If it was 0.1 AU away from the Earth, would the collision cause mass destruction here, or not affect us at all?


title: What is the analogy of $|x\rangle$ in quantum field theory?

text: Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ $|x_a \rangle$ is an eigenstate of position operator $\hat{x}$. In QFT we have $$ U(\phi_b,\phi_a;T) = \langle \phi_b | U(T) |\phi_a \rangle= \int \mathcal{D}\phi e^{iS} \tag{2} $$ $| \phi_a \rangle $ is an eigenstate of field operator $\hat{\phi}(x)$. By analogy with QM, it is tempting to relate $$|\phi \rangle \leftrightarrow |x \rangle \tag{3} $$ However, in Peskin and Schroeder's QFT, p24, by computing it is said > $$ \langle 0 | \phi(\mathbf{x}) | \mathbf{p} \rangle = e^{i \mathbf{p} \cdot > \mathbf{x}} \tag{2.42} $$ We can interpret this as the position-space > representation of the single-particle wavefunction of the state $| > \mathbf{p} \rangle$, just as in nonrelativistic quantum mechanics $\langle > \mathbf{x} | \mathbf{p} \rangle \propto e^{i \mathbf{p} \cdot \mathbf{x}} $ > is the wavefunction of the state $|\mathbf{p}\rangle$. Based on the quoted statement, seems $$\hat{\phi}(x) | 0 \rangle \leftrightarrow | x \rangle \tag{4} $$ If relations (3) and (4) are both correct, I should have $$\hat{\phi} ( \hat{\phi} | 0 \rangle ) = \phi(x) ( \hat{\phi}| 0 \rangle ) \tag{5}$$ seems Eq. (5) is not correct. At least I cannot derive Eq. (5). How to reconcile analogies (3) and (4)?


title: Normal force and reaction

text: It is said that the normal force comes into play any time two bodies are in direct contact with one another, and always acts perpendicular to the body that applies the force. This force is a reactionary force because its value depends on the force the body from which the Normal force is emanating experiences and changes with change in the force applied. When I apply force on a body at an angle to the normal, even then I am getting a reaction equal to my force. This, too, is emanating from the body that is reacting. So why is the normal special?? When we discuss the free body diagrams of forces applied on bodies, we do not always show this normal - harness attached to a horse, rope on a pulley, wire attached to picture etc.?


title: Which ball touches the ground first?

text: This is a very well known problem, but I can't find an answer in the specific case I'm looking for. Let's consider two balls :   * Ball 1 weighs 10 kg   * Ball 2 weighs 1 kg   * Balls have identical volumes (so Ball 1 is much more dense)   * Balls have identical shapes (perfect spheres) Let's drop them from a rather important height, on earth, WITH air. (That's the important thing, because all the proofs that I browse take place in a vaccum). I am arguing with a colleague. He thinks that ball 1 will fall faster in air, and that the two balls will fall at the same speed in a vacuum. I think that the identical shapes and volumes make air friction identical too and that the vaccum has no importance here. Could someone tell who's right and provide a small proof?


title: Peculiar Hamiltonian Phase space

text: I was solving an exercise of classical mechanics : Consider the following hamiltonian $H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$ Where $\lambda,m,\alpha$ are positive parameters. After having solved the equations of motion, I found something odd, and plotting the phase space (p,q) (which is y,x below) one can find that there are loops where momentum never changes sign, yet the system is localized (loop) ![enter image description here](http://i.stack.imgur.com/xBB2K.png) What sort of physical system does such a hamiltonian describe ? I cannot imagine something with an oscillating, never zero momentum, but still localized in space. I know this happens because of the cross-term, but I cannot find an interpretation for it


title: Is light's path a wave?

text: In a lot of textbooks I see a schematic of light drawn as a squiggly line. I have even heard that some things are too small to be seen because they are smaller than the wavelength of light (and therefore light wiggles around them.) But now I think that light actually travels in a perfectly straight line and that no particles will feel any force from this light when they are not on this line. My friend says I'm wrong, and that if you increase the amplitude of the light, the ray will wiggle into the particle. To back it up, he linked me to a video about the single photon double slit experiment, where a single photon seems to interfere with itself. I'm not sure that this experiment proves him right, but it does seem to say that light wiggles around.


title: Bunsen Burners and the Sun

text: 1. Why do Bunsen burners burn blue in the center? What element is being burned?    2. Why does the sun glow yellow, and not blue-a Bunsen burner is much cooler and yet it burns blue. Is it because the relative size, or are they two different phenomena? (i.e. black body radiation and something else?)


title: About deriving the multi-trace index in terms of the single-trace index

text: This question is in reference to this paper Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function, $Z(x) = \prod_{n=1}^{n =\infty}\left [ \int d\rho_n e^{-N^2\frac{\rho_n^2}{n}(1 - I_L(x,n))} \right ]$ where $I_L(x,n) = z_B(x^n) + (-1)^{n+1}z_F(x^n)$ ("x" is the fugacity, $z_B$ and $z_F$ are single particle Bosonic and Fermionic partition functions respectively and $\rho_n$ are Fourier components of the eigenvalue density on the circle of the gauge group $U(N \rightarrow \infty)$) -I would like to know as to how in the large N limit does this become their claimed equation 5.10, $Z(x) = \prod_{n=1}^{\infty} \frac{1}{1 - I_L(x,n)}$


title: Absorption of Water

text: 1. I was just watching the videos Chris Hadfield put on youtube from space, and was wondering why water doesn't get absorbed as well onto his toothbrush in space? And what characteristic of toothbrushes and wash clothes causes them to absorb water?   2. What exactly is water absorption?    3. Also, when he put water on his eye, it spread out over the surface. Why would water still want to spread out over his eye's surface without the force of gravity pulling on it?


title: Does quantum randomness predicate an infinite number of realities?

text: I am a layman when it comes to physics and especially quantum mechanics. I have seen many documentaries on the subject, and often in these productions there is a physicist featured explaining the "many worlds" interpretation of quantum randomness. This individual will give an example of a person making a split second decisions (e.g. step right or step left). The physicist explains that this decision stems from quantum fluctuations in the particles composing the hypothetical individual's mind, and, if assuming the many worlds interpretation is correct, this person walks left and right simultaneously, leading to the existence of two realities: one where the person chose left and the one where he chose right. At face value I have no qualms with this; however, this implies that all quantum fluctuations lead to multiple worlds where the position of the particle in question and every particle in existence assumes all points in its probability space simultaneously, branching into a mind-boggling infinity of realities in one "step" for lack of a better term. Assuming the many worlds interpretation is correct, the number of branching of realities is stupendous for any given moment. The implication of this is incomprehensible to me. Am I looking at this idea correctly? Or am I missing something that would make this idea a little more bearable?


title: Function to determine Earth-Moon distance vs. time

text: I had no luck trying to find a function that determines the distance between Earth and Moon versus time.   (Moon ground level to Earth sea level) _Accuracy:_   * the acceptable time range would be for years in [ 1900, 2100 ]   * 1 km error margin (or better)


title: How can new interpretations of QM help?

text: There is some current work on interpretations of quantum mechanics. How do you think can interesting results in that area help physics? Can it change quantum physics or make it easier? Which interpretation has to potential to change practical QM calculations? I mean if MWI turns out the best, then so what? It neither provides more intuition nor makes it calculations easier. If there are axioms and QM is derived from these, is there any practical value from this mathematical approach? I thought a statement like "it's the only mathematically consistent solution to the axioms", would provide no practical value? How is knowledge about QFT important to interpretations of QM or is QFT merely a handy mathematical framework?


title: Conserved quantum observables from symmetries *with density matrix*

text: I’ve read Ballentine where he derives the conserved observable operators (momentum, energy, ...) from symmetries of space-time. Can I read up such a derivation in more detail somewhere else or even one for the Poincaré group? And more importantly, the derivation uses gauge invariance of the wavefunction. Is there a derivation with the density matrix formalism where you wouldn’t need complex exponentials for gauge invariance?


title: What is the error in the measured value of the Hubble Constant?

text: I've found on the internet that the Hubble constant has been measured to be about $2.3\times 10^{-18} \, \mathrm{s}^{-1}$ Does anyone know what the current error bounds are on this value?


title: Moment of inertia of a cylinder

text: When I tried to calculate the moment of inertia ($I_C$) of a cylinder (mass M, height H, radius R) around the rotating axis going symmetrically through its middle, I came up with a different result than expected ($\frac{1}{2}MR^2$), but I do not spot my mistake, since my calculation makes perfect sense to me: $$ I_C := \int_V{ρr^2dV} = \int_0^H{\int_0^{2π}{\int_0^R{ρr^2 dr dφ dh}}} = ρ \int_0^H{\int_0^{2π}{\frac{R^3}{3} dφ dh}} = ρ \cdot 2πH \frac{R^3}{3} = Vρ\frac{R^2}{3} = \frac{1}{3}MR^2 $$ Can anyone spot what's wrong?


title: Peaks in binding energy per nucleon

text: Looking at the the binding energy per nucleon chart: ![enter image description here](http://i.stack.imgur.com/Essi0.jpg) I observe peaks for N=4,8,12,16,20,24 while I expected to observe peaks for 2, 8, 20, 28, 50, 82, and 126 because I have heard that in correspondence of these numbers there is the completion of a nuclear orbital, so I expect a local maximum of the stability an so a peak of binding energy. Is my reasoning wrong?


title: The orthogonalized plane waves

text: An orthogonal plane wave with wave number $k$ is written as $$ OPW_k=e^{ ik\cdot r}-\sum_\alpha \psi_\alpha(r) \int \psi^*_\alpha (r'') e^{ik\cdot r''} d\tau'',$$ where index $\alpha$ and $k$ stands for core states and conduction band states. In the above equation first term is plan wave term and in second term we use two wave functions $\psi_\alpha(r)$ and $\psi^*_\alpha (r'')$ for core states. My question is why do we use two wave functions and why do subtract second term from the first term ?


title: Why is the Gibbs Free Energy $F-HM$?

text: With magnetism, the Gibbs Free Energy is $F-HM$, where $F$ is the Helmholtz Free Energy, $H$ is the auxiliary magnetic field, and $M$ is magnetization. Why is this? Normally, in thermodynamics, we Legendre Transform the various free energies into each other to maximize the global entropy. In these cases, we subtract $TS$ when we are imagining a system exchanging heat with a thermal reservoir (i.e. heat bath at constant temperature $T$), add $PV$ when we exchange volume $V$ with a constant pressure reservoir at pressure $P$, and subtract $\mu N$ when we exchange particles with a chemical reservoir at constant chemical potential $\mu$. In every other case, we exchange heat, volume, and particles with the reservoir. How do we justify writing $G=F-HM$. Though it is true that $H$ is maintained constant, we don't exchange magnetization with a "magnetic reservoir".


title: Radiation Pressure

text: How much radiation pressure would be required from the sun for a carbon atom on the surface of the Earth to reach an acceleration of 1G? Can you show all the math required for the answer...thanks


title: Conservation, Maxwell tensor

text: Can someone please explain to me how the conservation of the energy momentum tensor $$\nabla_\beta T^{\alpha \beta}=0$$ imply the conservation of the Maxwell tensor $$\nabla_\beta F^{\alpha \beta}=0$$? Additional info: Note that the $\nabla_\beta$ denotes the covariant derivative. Energy momentum tensor Maxwell tensor = electromagnetic tensor


title: Faraday's law question: wire falling over two rails

text: I'm having trouble with this problem. The picture shows a wire of mass m and length l which slides on two rails, with certain angle with respect to the horizontal.The whole circuit has a resistance R and there is a B field which points upwards. I'm asked to find B so that the wire falls at a constant speed. (If you know spanish, you'll get a better undertanding of the situation. I'm not so good at technical english :D) ![enter image description here](http://i.stack.imgur.com/YSS6I.png) So here's what I did:   * From simple mechanics, we have that the force on the wire, parallel to the rail is $F=mg\sin{\alpha}$.   * The flux of B through the surface of the circuit is $\phi =\int \vec B \vec ds $, and the induced EMF is $\varepsilon =-\frac{\partial \phi}{\partial t}=Bgl\sin{\alpha}\cos{\alpha}.t$, and by Ohm's law is also $\varepsilon =iR$, so $i=\frac{Bgl}R \sin {\alpha}\cos{\alpha}.t$   * The magnetic force on the wire is $\vec {F_m}=\int i\vec{dl}\times \vec{B}=\frac{B^2l^2}R g \sin{\alpha}\cos{\alpha}.t$ So $F$ must be the same as $F_m$ for speed to be constant. Then $B=\sqrt{\frac{mR}{l^2\cos{\alpha}.t}}$. Here's the thing, this gives a time-depending $\vec B$, but in step 2 I supossed B was constant, so I don't know what's wrong, any help? Thanks


title: Perturbative Quantum Mechanics

text: I am, in full generality, confused about perturbation theory in quantum mechanics. My textbook and Wikipedia have the same general approach to explaining it: given some Hamiltonian $H=H^{(0)} + H^\prime$, we can break down each eigenfunction $\left\vert n \right\rangle$ into a power series in an invented constant $\lambda$ and the eigenenergies likewise: $\left\vert n \right\rangle = \sum\lambda^i\left\vert n^{(i)}\right\rangle$ $E_n = \sum \lambda^i E_n^{(i)}$ $\left(H^{(0)} + H^\prime\right) \left(\left\vert n^{(0)}\right\rangle + \lambda \left\vert n^{(0)}\right\rangle + \cdots \right) = \left(E^{(0)}+ \lambda E^{(1)} + \cdots\right) \left(\left\vert n^{(0)}\right\rangle + \lambda \left\vert n^{(1)}\right\rangle + \cdots \right)$ ... and then they take $\lambda\to1$. My question is - what's the logic here? Where did this come from? What purpose does $\lambda$ serve, given that the actual size of each contribution will be determined by the $E^{(i)}$'s and $\left\vert n^{(i)}\right\rangle$'s?


title: Symmetry Breaking And Phase transition

text: 1. Is every phase transition associated with a symmetry breaking? If yes, what is the symmetry that a gaseous phase have but the liquid phase does not?   2. What is the extra symmetry that normal $\bf He$ has but superfluid $\bf He$ does not? Is the symmetry breaking in this case a gauge symmetry breaking?


title: how to determine the direction of a wave propagation?

text: In the textbook, it said a wave in the form $y(x, t) = A\cos(\omega t + \beta x + \varphi)$ propagates along negative $x$ direction and $y(x, t) = A\cos(\omega t - \beta x + \varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\pm$ x direction, to my understand, we can drop the time term and ignore the initial phase $\varphi$ while analyzing the direction, i.e.$y(x, 0) = A\cos(\pm\beta x)$, however, because of the symmetry of the cosine function, $\cos(\beta x)\equiv \cos(-\beta x)$, so how come we determine the direction of propagation from that? I know my reasoning must be something incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\beta x$ corresponding to the propagation on positive x direction but not the opposite? Thanks.


title: why define relative permittivity and relative permeability?

text: I am reading something about electomagnetic field and the first introduce the free space permittivity and permeability for the electric field and magnetic field. And later when discussing the field in the material, the introducing the permittivity and permeability as the multiplication of the relative quantity and the free space one, i.e. $\varepsilon = \varepsilon_r\varepsilon_0$ and $\mu=\mu_r\mu_0$. For vacuum, we have $\varepsilon_r=1, \mu_r=1$. From the physical point of view, why we have to do this way and why we have to define the relative permittivity and relative permeability instead of just giving the actual permittivity and permeability for the material? Also, what that 'relative' refer to?


title: What is $p$ in this candidate solution for the motion of N masses?

text: In this problem: > Consider the system of $N$ masses shown in the figure (see picture 1). > > a) Using the approximation for small angles, write the equation of > transverse movement for the n-th particle. > > b) Propose a solution of the form: > ${ψ_n}^{(p)}(t)=A^{(p)}\cos(nk^{(p)}+α^{(p)})\cos(w^{(p)}t+\phi^{(p)})$. > Find the dispersion relation and decide whether it depends on the contour > conditions. > > **Figure 1:** > > ![picture 1](http://i.stack.imgur.com/MuQu5.jpg) > > **Figure 2:** > > ![picture 2](http://i.stack.imgur.com/NLmaK.jpg) With regard to part b), I don't understand the notation of the proposed solution, to be more specific, I don't understand what $p$ is, is it an index? Could someone explain me what $\ddot{y_n}$ would be just to see what $p$ stands by?


title: Finding Green function using eigenfunction expansion method

text: Given the Dirichlet boundary condition, I am to show that the functions that satisfy $(\nabla ^2 + k_{lmn}^2) \psi_{lmn} (x,y,z) = 0$ are given by $\psi_{lmn} = (\frac{\pi}{2x})^{1/2} J_{l+1/2}(x) Y_{lm} (\theta, \phi)$ for a hollow sphere or radius $a$, where the green function $G$ can be expanded as $G(\vec{x},\vec{x'}) = \sum_n a_n(\vec{x'}) \psi_n (\vec{x})$. Note that I can use the solution of the Helmhotz equation, which is given to me as ![enter image description here](http://i.stack.imgur.com/Ycew7.jpg) where the brackets express a linear combination of their arguments. I am confused as to how to proceed from there and how to use the BC's to make it look like the desired result. Any help would be much appreciated.


title: How to model the dynamics of human movement?

text: I am trying to derive a state space equation for human movement (i.e. walking, running, standing still) in some normal distribution sense. The following equations should not be so hard to follow: If the state $\bar{x} = (x, \dot{x})$ The state equation will be $\bar{x}(t) = A\bar{x}(t-1)$ With the covariance matrix $C(t) = AC(t-1)A^{-1} + Q$ Where $A = \begin{bmatrix} 1 & T \\\ 0 & 1 \end{bmatrix}$ where $T$ is the time period, and where $Q$ a "process noise" covariance matrix. $Q = \begin{pmatrix} \sigma_x^2 & \sigma_x \sigma_{\dot{x}} \\\ \sigma_x \sigma_{\dot{x}} & \sigma_{\dot{x}}^2 \end{pmatrix}$ In my view, the key of a correct model lies in getting correct values of $\sigma_x$ and $\sigma_{\dot{x}}$. According to wiki, the fastest human speed recorded is 44.72 km/h. What do you think?


title: A quantitative explanation of EM coherence domains in liquid with DNA

text: I've been looking with interest at a recent biology paper claiming that DNA molecules give off electromagnetic signals which can cause the same types of molecules to be reconstructed at a remote location. The Slashdot crowd seems to think the idea is pretty ridiculous, and I'm inclined to agree, but I still think it's worth digging into some of the physics they cite. A large part of the theoretical argument in the paper relies on what they call coherence domains. As far as I can tell, they're talking about a relatively large region of liquid (in this case water) in which all the molecules are in resonance with an oscillating electromagnetic field which is confined by the boundary of the region itself. I don't think I really grok that description, and a Google search for further information turned up nothing relevant. I'd like to see some of the math that backs this up. So can anyone provide a more mathematical explanation of how these coherence domains are able to exist? (I'm assuming the whole idea is not totally bogus :-P)


title: Measuring the Electromotive Force and Internal Resistance of a Battery?

text: Here is a problem that emerged some time ago: We needed to measure the emf(electromotive force) and internal resistance of a car battery. The only instruments were voltmeter and ammeter. Someone suggested: 1) the first time connect voltmeter and ammeter **in series** to the battery and write up the results of the measurements. 2)the second time connect voltmeter and ammeter **in parallel** to the battery and write up the results of the measurements. Based on these measurements only, calculate emf $E$ and internal resistance $r$ of the battery. To be specific, 1)for the first case $U_1=10V$ and $I_1=0.1A$ 2)for the second case $U_2=1V$ and $I_2=1A$ How do you solve this problem? Arguably, this method is advanced in the sense that the result do not depend on internal resistances(unknown) of voltmeter and ammeter. This last claim does not seem obvious.   Maybe there is an easy way prove(disprove) this claim!


title: What formulae for calculating fluid pressure change involves these parameters?

text: Okay... this is a bit desperate... I am reverse engineering an excel program that is used for calculating outputs for a pressure vessel used for industrial process fluid heating. There is a part in the excel sheet where a pressure drop is being calculated. But there are too many unit conversions and other magic constants in intermediate steps i cannot regonise. I have some physics background but not enough in fluid mechanics to decipher. For starters, I have intermediate step formulas like:               (fluid flow rate[lbs/h]]) * (1545) * ((inlet plus outlet temp[deg c] / 2)*9/5+460) /          ((molecular weight[??])*((pressure[psig])+14.5)*144)      ## The formula looks like this: $\frac{\dot{m} \times 1545 \underbrace{\left[\frac{(T_\text{in} + T_\text{out})}{2}\times(\frac{9}{5}) + 460 \right]}_\text{Degrees C to Deg F to Rankine}}{M_w \times (P_g + 14.5) \times 144}$ I am hoping someone who deals with fluid mechanics will be familiar with many formulae to tell what it might be. The inputs to the entire calculation steps consists of flow rate, inlet and outlet temperatures, molecular weight, viscosity and thermal conductivity; and also dimensions of the vessel. Greatly appreciate everyones help! Thanks. **EDIT** : The actual problem is here http://www.jakesee.com/se/physics_se_16403.png and the birthplace of the image is here http://stackoverflow.com/questions/7647478/generate-a-flat-list-of- all-excel-cell-formulas


title: Interpretation of Wigner function in optics

text: I work in the field of synchrotron radiation sources where radiation (often x-rays) is produced from an electron beam going through magnetic fields. The quality of the resulting x-ray beam is determined by a parameter called the brightness, which is formally computed by the Wigner function for the radiation. The Wigner function is claimed to be a representation of the photon flux density in the radiation. It can sometimes be negative, which would be unphysical, but people say this is somehow due to Quantum Mechanics. There is this analogy between light optics and quantum mechanics, where $\hbar$ is replaced by the wavelength of the light. But I don't think that there is actual quantum mechanics involved, though I could be wrong. I think it is just an analogy. So my question is: When one represents radiation via a Wigner function, is this really quantum mechanics? (A kind of semi-classical approximation?) Can someone point me to good references on understanding this from a slightly deeper perspective? I'm interested if there may be some real mystery here, or if its actually well-understood. A reference by Kim: http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6202594


title: Entropic force in rubber bands

text: The formulation for entropic force of stretching a rubber band is now known. Are there any other such daily life examples that also have mathematical formulations of entropic forces associated with them? Or can the idea and theory be extrapolated in such a way that more physical phenomena can be explained by it?


title: Why is the angle of the wake of a duck constant?

text: Why is the angle of a wake of a duck constant? And why are some conditions on the water depth necessary? I realize that this question turns up in google searches, but I did not see a good discussion. I will be quite happy with a link. Edited to add: Could anyone tell me how the two up-voted answers are related?


title: Distance increase between Sun and Venus

text: Is the distance between the Sun and Venus increasing?   What is the measurement?   May I request for a formula?


title: Satellites and gravitation

text: A satellite with mass $m$ orbits a planet of mass $M$ in a circular path with radius $r$ and velocity $v$. Due to some internal technical failure, the satellite breaks into two, similar parts with mass $m/2$ each. In the satellite's frame of reference, it appears the two parts move radially, in opposite directions, along the line connecting the original satellite and the planet's center, each with velocity $v_0/2$. I am expected to show that right after the technical failure, each of the two parts has a total energy equal to $-3GM/(16r)$ and angular momentum equal to $(m/2)√(GMr)$, wrt the planet's center. The total energy of each of the two parts should be, I believe: $E_{tot} = mv_0^2/16 - GmM/(2r)$. Now, isn't angular momentum preserved despite the failure? However, why isn't the angular momentum zero if the two parts are moving in opposite directions?


title: Can you use a laser to measure the speed of light with a rotating mirror?

text: I have learned that the classical measurement of the speed of light with a rotating mirror does not work with a laser (as opposed to, say, a mercury- vapor lamp). Can you tell me if and how coherency of light is necessary/contraindicated for the success of this experiment?


title: Why wasn't the moon visible during the day a few decades ago?

text: I was born in 1949. When I was young we played outside and watched the clouds and the sky a lot, and I don't remember ever seeing the moon during the day. Is the sun closer to us now so we see it more often or has earth shifted enough to change how we see the moon?


title: $\mathcal{N}=2$ susy hypermultiplet self-CPT?

text: Is the multiplet given by $$\left( -\frac12,0,0,\frac12 \right)$$ self-CPT conjugate? There seems to be no common agreement upon that:   * Weinberg (QFT 3, page 47) and many others claim it is not, basing on $SU(2)$ $R$-symmetry reasoning    * Terning (modern susy, page 12) and some others claim it is, provided the fermions transform in a real representation of the gauge group, regardless of $R$-symmetry considerations.


title: Two-body problem questions

text: I am self studying the two body problem and I'm stuck on the following: I have given $$\ddot{\vec{x}}_1= - G m_2 \frac{\vec{x}_1-\vec{x}_2}{|\vec{x}_1-\vec{x}_2|^3}$$ and $$\ddot{\vec{x}}_2= - G m_1 \frac{\vec{x}_2-\vec{x}_1}{|\vec{x}_1-\vec{x}_2|^3}$$ with masses $m_1,m_2$ at the position $x_1,x_2$ and gravitational constant $G$. How can you show only with these two equations that total momentum $m_1\dot{\vec x_1}+m_2\dot{\vec x_2}$ is constant in time? And why is the total mechanical energy $$\frac12m_1|\dot{\vec x_1}|^2+\frac12m_2|\dot{\vec x_2}|^2-\frac{Gm_1m_2}{|\vec{x}_1-\vec{x}_2|}$$ conserved?


title: Equivalence between a charged rotating cylinder and a solenoid

text: Suppose we have a cylindrical shell of radius $r$ with surface charge density $\sigma$. Then we start rotating the cylinder at an angular speed $\Omega$. You can show that in this case the surface current density on the cylinder is $\sigma r \Omega$. Similarly, in a solenoid with loop density $n$ and current $I$, the surface current density can be thought to be $nI$, so you can use this to "convert" between a rotating cylinder and a solenoid. For example, the magnetic field inside the cylindrical shell must be $\mu_0 \sigma r \Omega$, because for the solenoid it's $\mu_0 n I$. I was wondering if you could establish another relation between the solenoid and the cylinder if the solenoid has a resistance $R$. Based on the analogy, I imagine the resistance must be connected to the cylinder's moment of inertia, but I can't quite figure it out.


title: Finding the work required to move an object under pressure without knowing volume

text: "What is the least amount work you must issue onto a silver ingot with a mass of 38 kg at 26 meters depth to bring it to the surface?" I'm not sure how to do this one. Surely it depends on the volume of the ingot, right? Archimedes' principle says that the buoyancy of an object is its volume times the density of the fluid times the gravitational constant. We only have two of those, so I can't find the lifting force. This is how my formula looks like (Since the force is constant across all depths): $F_L = $Lifting force Force required $= mg - F_L$ Force required $= 38kg * g - (1000kg/m^3 * g * V)$ Work $=$ Force required * distance Work $= 38kg * g - (1000kg/m^3 * g * V) * 26m$ Volume is _still_ unknown, so how do I solve this one?


title: What's my weight when jumping on a bicycle from a higher to a lower ground?

text: Before I start defining the situation and asking a question, I'd like to make a few things clear:   * this is **not** a homework, merely a matter of personal interest and enthusiasm   * I am neither a physicist, nor a native English speaker, therefore I encourage anyone with proper knowledge to go ahead and edit anything in my post to correct/improve any terminology, thread title including I'd like to start with an image I created for this purpose, I hope I didn't make any huge mistakes in it. So, here's the situation: ![The situation](http://i.stack.imgur.com/Ommgm.png) We have a biker weighing _m1_ riding a bicycle weighing _m2_ , therefore the total weight _m = m1 + m2_.   The biker is driving at a constant speed of _v_ and is about to jump down from a higher ground to a lower ground, while the total height he's jumping, the height difference of those two grounds is _h_.   According to how fast he's going and how high he's jumping from, he'll jump a distance _d_. **The question:** What's the biker's weight at the moment of landing with which he's "pushing" the bicycle down? As I'm not a physicist and I'm not sure what's the proper term, I'll try to form it like this:   I know what the maximum weight the bicycle can hold is according to the manufacturer and I need to calculate whether I will exceed this limit when performing such a jump. **Things to consider:** I believe there are many factors which can't be counted with properly, therefore   I _think_ , and now correct me if I'm wrong:   * that we can assume that the forward speed when landing will be the same as the forward speed before jumping. Of course it won't, but I believe this speed will not drop so much, so possibly we can ignore it   * I believe there's a huge difference between locking one's joints and dropping with a rigid body versus using the proper technique, keeping joints free and flexing one's arms and legs to absorb the mass of landing. I don't know how it'd be possible to include this factor to the calculation, I'll let people decide. Feel free to propose anything I forgot to include in my question, and again, if anyone who can edit posts can improve anything I wrote, please, don't hesitate and do so.


title: Does de Sitter space admit an asymptotic S-matrix?

text: From the Penrose diagram of de Sitter space, we see it has a future and past conformal boundary, and they are both spacelike. So, does de Sitter space admit an asymptotic S-matrix? Sure, in the usual coordinates which only cover half of the full space, we don't have an S-matrix because it's not causally complete, but in a coordinate system which covers the entire space, why not? Just because we don't have a globally timelike Killing vector field doesn't mean we can't have an S-matrix.


title: Why does a laser beam diverge?

text: I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on photons rather than wave physics.


title: Direct optical observation of the Arago spot: what would one see?

text: The Arago/Poisson spot is an excellent example of a constructive interference of diffracted light. In the traditional experimental layout, you observe _projected_ light on the screen. The theory in this case is completely straight-forward, just add together phase-delayed contributions (Fresnel integral). However, I'm not sure what would happen, if instead of a screen, you put an optical apparatus (eye, pinhole camera, lensed camera) on the symmetry axis? What confuses me is that all these optical imaging systems form an image _by angle_ , not by absolute spatial position! Adjacent pixels focus light from slightly different _angles_. Because the spot actually collects light coming _around_ the circular obstacle, you shouldn't see a bright spot in the middle. I'm almost certain that you wouldn't see anything special at all (nothing different from what you see if you are looking from an off-center position - you're in the geometric shadow after all). My reasoning is that because the eye projects light from different directions to different pixels, the contributions can't constructively interfere. I'm worried about the following things:   * The angular imaging description is ray-optics... the phenomenon requires wave-optics. Is the reasoning still valid if you pretend wavefront normals are rays?   * Constructive interference should form a head-on wavefront even before it gets to your eye. This indicates that you would indeed see a spot. If the front is wide enough to be considered a geometric ray and not get refracted by the lens.   * Would the optical apparatus just introduce slight variations in phase and ruing the phenomenon? Would you see "speckle" pattern?   * In a pinhole camera, the screen around the pinhole blocks precisely the path from the parts that contribute to the interference. If the wavefront forms in front of the screen, does it diverge back into a wide disperse wave after "cutting off" the income of new waves? Short of doing full wave optical calculation of a camera, can we reconstruct what would happen? Would a pinhole camera give a fundamentally different image from a lensed camera? Please elaborate. We can treat this as a discussion-type question.


title: Minimal Kinetic energy for particle in a box

text: This is driving me crazy! The question goes as follows: A proton is enclosed in a zone of length 2pm along the x-axis. The minimal kinetic energy of the proton lies closest to:   * 5000eV   * 0.5eV   * 50eV   * 500eV   * **5eV** (this is supposed to be the correct answer) So first I calculate the energy level for a particle in a box which is given by the equation $E_n = p_n^2/(2 m) = (n^2 h^2)/(8 m L^2) $ Here $L$ is the length of the zone. So I find $E_1=94007.53 eV$.. So.. How do I relate this to the minimal kinetic energy? \-- EDIT 1 -- After the first hint from someone, I changed my calculations (I used the mass of an electron instead of a proton..) NOW I get $E_1 = 51eV$, which is still wrong..


title: Optical chirality and its possible hierarchy of generalizations

text: Optical chirality refers to a constant of motion of the electromagnetic field, which measures in some sense how chiral a light field is. Specifically, the pseudoscalar quantity $$ C=\frac{\epsilon_0}{2}\mathbf{E}\cdot \nabla\times\mathbf{E}+\frac{1}{2\mu_0}\mathbf{B}\cdot \nabla\times\mathbf{B} \tag1 $$ obeys the continuity equation $$ \frac{\partial C}{\partial t}+\frac{1}{2\mu_0}\nabla\cdot\left(\mathbf{E}\times\nabla\times\mathbf{B}-\mathbf{B}\times\nabla\times\mathbf{E}\right)=0 $$ in free space. It was re-discovered by Yiqiao Tang and Adam E. Cohen, in > Yiqiao Tang and Adam E. Cohen. Optical Chirality and Its Interaction with > Matter. _Phys. Rev. Lett._ **104** , 163901 (2010); Harvard eprint. after having been discovered, puzzled over, called 'zilch' for lack of a better name, and forgotten in the 1960s. This quantity is useful because it is a direct measure of how strongly many chiral biological molecules will interact with a chiral electromagnetic wave, which is an important tool of biochemistry. This rediscovery is a huge step forward, but as Tang and Cohen note, it cannot be the whole story: > Similarly, there cannot be a single measure of electromagnetic chirality > appropriate to all EM fields. There exist chiral fields for which $C$ as > defined in Eq. (1) is zero. Indeed, the field of any static, chiral > configuration of point charges is chiral, yet by Eq. (1), $C=0$. (This is trivially seen as both curls vanish in the static case.) In response to this, Tang and Cohen offer some conjectures: > The optical chirality of Eq. (1) may be part of a hierarchy of bilinear > chiral measures that involve higher spatial derivatives of the electric and > magnetic fields [22]. We speculate that all linear chiral light-matter > interactions can be described by sums of products of material chiralities > and time-even pseudoscalar optical chiralities. In addition to this, there is the possibility of non-linear chiral interactions, which involve the product of three or more force fields. * * * To come, finally, to my question: **what is the current status of these conjectures?** Are there descriptions of higher-order tensors (involving higher spatial derivatives) or nonlinear terms (involving more than two force fields), which are also chirally sensitive? Is there some ordered hierarchy which contains them? Is there some sort of completeness result that guarantees said hierarchy contains 'all' the relevant quantities?


title: Special relativity, acceleration and being in two places at once (How to resolve this paradox?):

text: Imagine I am on Earth I have a clock which measures time $t$. When my Earth clock reads time $t$, then when I look at a moving spacecraft's clock I see the time $t'$. Let us assume time is measured in years. Let us suppose the clocks were initially synchronized so that when $t=0$, $t'= 0$. Now let us suppose that the speed of the rocket continues to increase in the following way (Note that I have increased the speed in discrete time steps for convenience, but I'm sure a continuous speed function could be obtained but just mathematically tedious): At $t = 0$, $v = c\sqrt{3/4}$. So when $t = 1$, $\delta t'= \frac{\delta t}{\gamma} = 0.5.$   At $t = 1$, $v = c\sqrt{15/16}$. So when $t = 2$, $\delta t'= \frac{\delta t}{\gamma} = 0.25.$   At $t = 2$, $v = c\sqrt{63/64}$. So when $t = 2$, $\delta t'= \frac{\delta t}{\gamma} = 0.125.$ and so on... If $v$ continues to increase over time in this way, then $t'$ will approach 1, but will never reach 1. It appears from the perspective of the Earthling, that the person in the rocket never ages by over 1 year. From the person in the rocket however, he will experience time as normal. Now let us assume that after the person in the rocket has aged 10 years, he leaves his rocket ship, and travels back to Earth, leaving his rocket to continue accelerating in the same pattern. Now when the man reaches Earth, will he not see himself in the rocket, where he sees himself not having aged 1 year? (Ignore optical effects for the purpose of this question.)


title: Why wouldn't Hubble's Law be directly in units of frequency?

text: Maybe using this as an example: > The energy of a particular color of yellow light is $3.44 \times 10^{-22}$ > $kJ$ So if I want to find frequency of that light, I take that number, divide by $h$ and I get $5.19 \times 10^{-14}$ (or something like that) which is in what units? Hz? What if I wanted to know the frequency of the light after travelling say 1 billion light years? Is there a version of Hubble's Law (and parameter I'd assume) in the form of $f_\text{new} = f_\text{original} \times H_u$ where $H_u$ is a version of Hubble's Parameter in units appropriate for the units of frequency?


title: Air velocity in a double-skin facade

text: I was wondering how the air flows in a double skin facade? Say I have a glass facade with a depth of 1m, a height of 200m and a width of 10m. It is a single vertical shaft all the way up with openings at the bottom to let air in and at the top to let air out: ![](http://i.stack.imgur.com/w4Xrc.png) There is a solar radiation of 100W/m2 hitting the outside of the facade and the air velocity at ground level is 1m/s. Can I expect the velocity of the air to increase as it travels through the shaft? and to what degree?


title: Why is Hawking's No Boundary condition described in terms of an instanton if there's no tunneling?

text: Regarding his take about combining No Boundary proposal with inflation theory (Hawking and Turok), he talks about the "pea instanton". To my very limited understanding I thought instantons were only for describing quantum tunneling phenomena. The No Boundary proposal doesn't subscribe to this, it just says that the Euclideanized 4-spatial-dimensioned pre-Planck-era universe was always there and there was no t=0? Or does it actually involve a tunneling process like Vilenkin's?


title: size and distance of mirror used in projected "smart" whiteboard

text: Apologies in advance for complete ignorance of optics (beyond conic sections) How is the size, distance, and angle of the mirror used here calculated? The mirror is I believe the open flap in front of the projector. ![enter image description here](http://i.stack.imgur.com/BNEiv.jpg)


title: Can surface dipoles/charges change the work function of a metal?

text: As typically drawn in simplified band diagrams (see picture below), the metal Fermi Level is shown as the top of the conduction band, with the entire band filled. In many situations, including joining a metal and semiconductor together to form a contact, charge is transferred between the two materials. For a metal/n-SC contact this forms a very thin layer of negative charge on the surface of the metal. However, the final band diagram is still drawn with the metal band unchanged. Is there simply a negligibly small change of the work function at the surface in this case that is not drawn? If so, what is the statistical meaning of the Fermi Level being above the top of the conduction band at the surface/interface (in the case of a thin sheet of negative charge)? ![enter image description here](http://i.stack.imgur.com/g0S9o.png)


title: Is it necessary to use all solutions when calculating an expectation value in a spin state?

text: I'm given an spinor $\Psi$ which is solution of the Free Dirac equation, such that is an eigenfunction of $\hat{\vec{p}}$ and has positive energy. Then I'm asked to calculate the expectation value of the operator $\beta$. I normalize the functions inside a box of volume $V$ So, is this correct? $$\langle \hat{\beta} \rangle = \displaystyle\int d^3r (\Psi^{\dagger}\hat{\beta}\Psi )$$ For instance, let me begin: $$\Psi(\vec{r},t)=\displaystyle\frac{1}{(2\pi\hbar)^{3/2}}\exp \left( \frac{\vec{p}\cdot\vec{r}-Et}{\hbar} \right) \psi(\vec{p)}$$ where $$\psi(\vec{p})=\left(\begin{matrix}1 \\\ 0 \\\ c\frac{\vec{\sigma}\cdot\vec{p}}{E_p+mc^2} \\\ 0 \\\ \end{matrix}\right)$$ In the matrix $\psi(\vec{p})$, did I miss half of the solutions? I mean, there is a pair of two element linearly independent matrices that satisfies the equation with the given constraints (one for each value of spin) but can I take, for simplicity the above matrix instead of a more general $$\psi(\vec{p})=\left(\begin{matrix}c_1\phi_1+c_2\phi_2 \\\ (c_1\phi_1+c_2\phi_2)c\frac{\vec{\sigma}\cdot\vec{p}}{E_p+mc^2} \\\ \end{matrix}\right)$$ with $c_1,c_2$ constants and $\phi_1,\phi_2$ arbitrary two elements matrices (the case I choose is a particular case of this one) Now, in the actual calculation, is this correct? $$\langle \hat{\beta} \rangle = \frac{1}{(2\pi \hbar)^3 V}\displaystyle\int d^3r \left(1-\frac{c^2(\vec{\sigma}\cdot\vec{p})^2}{(E_p+mc^2)^2}\right) $$ $$=\frac{1}{(2\pi\hbar)^3}\left(1-\frac{c^2(\vec{\sigma}\cdot\vec{p})^2}{(E_p+mc^2)^2}\right)$$ The solution shold be $\langle \hat{\beta} \rangle = \gamma^{-1}$ but first of all I don't know how to cancel the $\hbar$ and the rest of terms in the denominator, and the calculation is more than probably wrong for other reasons apart of that. I think the $\hbar$ stuff comes as an arbitrary multiplicative constant for a plane wave over the entire space time, wich is not normalizable. Now, for normalizable wave functions, I think it should dissapear, but then the other factor $$\left(1-\frac{c^2(\vec{\sigma}\cdot\vec{p})^2}{(E_p+mc^2)^2}\right) (1)$$ does not seem $\gamma^{-1}$. I know that $(\vec{\sigma}\cdot{\vec{p}})^2=p^2$ but I can't find the final answer. Thanks for your time. Note: I'm using Pauli representation of the $\gamma$ matrices. * * * As I said in the comment, the above is wrong, but I'm going to leave it unaltered and continue with I think it's better but not correct. The wave function should be $$ \Psi(\vec{r},t)=\frac{\exp(\frac{i(\vec{p}\cdot\vec{r}-Et)}{\hbar})}{\sqrt{V}}\psi(\vec{p}) $$ choosing the solution as $c_2=0, c_1=1, \phi_1=\left(\begin{matrix} 1 \\\ 0\end{matrix}\right)$ First, the action of $\hat{\beta}$ on $\Psi$ is $$\left( \begin{matrix} \mathbb{I} & 0 \\\ 0 & -\mathbb{I} \end{matrix} \right)\frac{\exp(\frac{i(\vec{p}\cdot\vec{r}-Et)}{\hbar})}{\sqrt{V}}\left(\begin{matrix}1 \\\ 0 \\\ c\frac{\vec{\sigma}\cdot\vec{p}}{E+mc^2} \\\ 0 \\\ \end{matrix}\right) = \frac{\exp(\frac{i(\vec{p}\cdot\vec{r}-Et)}{\hbar})}{\sqrt{V}}\left(\begin{matrix}1 \\\ 0 \\\ -c\frac{\vec{\sigma}\cdot\vec{p}}{E+mc^2} \\\ 0 \\\ \end{matrix}\right) $$ So the integrand $\Psi^{\dagger}\hat{\beta}\Psi$ is $$ \frac{1}{V}\left(1-c^2\frac{(\vec{\sigma}\cdot\vec{p})^2}{(E+mc^2)^2}\right) $$ The second factor is constant and the integral is $V$ which cancels with the $V$ in the denominator and then $$\langle \hat{\beta} \rangle = 1-c^2\frac{(\vec{\sigma}\cdot\vec{p})^2}{(E+mc^2)^2}=1-\left(\frac{cp}{E+mc^2}\right)^2$$ is my result, which is not $\gamma^{-1}$.


title: What are the rainbow and ladder approximations in a solid state physics context?

text: All references I find talk about quarks and gluons, where I have only very limited knowledge about. From it's name (rainbow) I guess it applies to fermions coupled to bosons and we're interested in the single particle Green function for the fermion, and we approximate the sum over all diagrams by only summing those diagrams where no phonon lines cross (so that Feynman diagrams look like rainbows). Is that correct? I also noted that all search results always talk simultaneously about the rainbow and the ladder approximation, as was also pointed out in a comment below.


title: How would gravity change on a planet rotating around itself very fast?

text: Let's take a planet identical to Earth, but with rotation speed multiplied by ten thousand. What would happen with the gravity if it was spinning madly around itself? Would the centrifugal force make objects seem lighter than on normal Earth? Would the people on that planet be able to jump higher?


title: How do objects change their axis of rotation?

text: If I hold a pencil at its end and spin it, throwing it upwards, it will spin about its end, but will soon start spinning around its center. How is this? I would draw the following torque diagram for while it's in the air:   * Object: uniform thin rod with length $\ell$ and moments $I_{center}=\frac 1 {12} m\ell^2$ and $I_{end}=\frac 1 3 m\ell^2$)   * Center of rotation some small distance $d$ from the end   * Torque $m\vec g$ downward, at center of mass, with $\theta = 90°$ and $r = \frac \ell 2 - d$   * Possibly wind resistance $\vec D$ upward, at center of mass, with $\theta = -90°$ and $r = \frac \ell 2 - d$ Thus, $\vec \tau = \sum {\vec r \times \vec F} = \sum {rF~sin\theta} = \left (\frac \ell 2 - d\right)(m\vec g - \vec D)$. I could see how this might cause it to spin, but how does the center of rotation to move? **EDIT:** Here's a picture because apparently I wasn't clear. It definitely is spinning around the end (or close to it, anyway) _before_ I release the pencil. ![Pencil image](http://i.stack.imgur.com/7qUFu.png)


title: (Level: Undergrad) Continuity Conditions on the Wavefunction and Initial Values

text: I know that a physically meaningful $\Psi$ needs to be continuous. However, recently I came across a problem in which they were considering a wavefunction for the **infinite square well** potential and the initial condition was: $$\Psi(x,0)=\begin{cases}\sqrt{\frac2a} & 0\leq x\leq a/2 \\\ 0 & \text{otherwise}\end{cases}$$ Isn't this discontinuous? So can we even have such initial conditions? Also, for this case it makes you ask, why is the wavefunction not equal to zero at $x=0$? Is this even possible for the infinite square well?


title: Superfluid momentum distribution

text: Bosons confined in a optical lattice under some conditions can enter a superfluid phase. Momentum distribution can be measured using time of flight method. But why is there a sharp peak in momentum distribution?


title: The required energy to accelerate a particle from one velocity to another

text: Say I have a charged molecule running along a linear evacuated tube (so no wind resistance). In the laboratory frame, we can measure that the particle is moving at some speed $v_1$. Provided that kinetic energy scales as the square of the velocity, how much energy do we need to impart on the particle to accelerate it to some velocity $v_2$? Why should this depend on $v_1$ and not strictly the difference $v_2 - v_1$?


title: Flow rate of a syringe

text: Suppose a syringe (placed horizontally) contains a liquid with the density of water, composed of a barrel and a needle component. The barrel of the syringe has a cross-sectional area of $\alpha~m^2$, and the pressure everywhere is $\beta$ atm, when no force is applied. The needle has a pressure which remains equal to $\beta$ atm (regardless of force applied). If we push on the needle, applying a force of magnitude $\mu~N$, is it possible to determine the medicine's flow speed through the needle?


title: Hole in the disc shaped magnet

text: Let's suppose we have a magnetic disc. One half is magnetic north pole and the other half is magnetic south pole. Now suppose we drill a hole in the disc. Does anybody has an idea how the magnetic poles are arranged on the edge of the hole? Is the edge north, south or on the upper half is south and lower half is north ? (Th hole must be in the middle to make a ring) I placed an image of the disc, the hole and polarities of the parts. I want to find out if we place some sensors in positions 1,2,3,4,5,6 and 7 what polarites the sensor will detect ? An if we cut the yellow line of this material at positions 5-2 and 2-5 what will be the polarities ? **I drew an image of the disc but I have no enough reputations to place images so for now I will try to explain the details on my question.** Postions: 1 is center of the disc 2 are inner edge (cricle) at 3 and 9 o'clock 3 is at 12 o'clock 4 is at 6 o'clock 5 are outer edge of the disc at 3 and 9 o'clock 6 is on the outer edge between 7 and 8 o'clock 7 is on the outer edge between 1 and 2 o'clock. In the future when I earned at least 10 reputation I will post the image.


title: Optical Theorem ,how can experiment distinguish the unscattered wave from the forward scattered wave?

text: How can experiment distinguish the unscattered wave from the forward scattered wave? The Optical Theorem says the imaginary part of the forward wave determines the cross section for an initial two-particle state $|i\rangle$ in the center of mass frame through the equation $$2E_Tp_{in}\sigma=Im{\cal M}_{ii}$$


title: Simularity transformation of Heisenberg XXZ Hamiltonian

text: I am considering the Heisenberg XXZ Hamiltonian: $$ H(\Delta, J) = J\sum_{i=1}^L\left(\sigma^x_i\sigma^x_{i+1} + \sigma^y_i\sigma^y_{i+1} + \Delta \sigma^z_i\sigma^z_{i+1} \right) $$ Apparently, one can show: $$-H(-\Delta) = UH(\Delta)U^{-1}$$, where $U=\Pi_{m=1}^{M/2}\sigma_z^{2m}$ is a unitary transformation. I am having some trouble showing this. Of course the transformation commutes with the z-Pauli matrices and using $(\sigma^z)^{\dagger} = \sigma_z$ and $(\sigma_z)^2 = \textrm{id}$ this term is unchanged. However, I am left with mix terms for the first and second term. Furthermore, my reference claims that because of this simularity transformation, $J$ can be omitted if $-\infty<\Delta<\infty$. I don't really see how this follows from the simularity transform. Any help is much appreciated!


title: Do plasmons depend on the ambient EM field?

text: Imagine a situation: There's an illuminated metal slab in vacuum. Normally, there are some plasmons created running all over the slab. What would happen if we had turned a giant magnet near the slab? Or put the slab between two electrically biased plates? How would plasmons react?


title: What is the interaction with Higgs field(s) that give the quarks so much different masses?

text: The masses of quarks are:                mu 2∼3 MeV    md 4∼6 MeV      mc 1.3 GeV    ms 80∼130 MeV      mt 173 GeV    mb 4∼5 GeV


title: Modeling the trajectory of a particle in an Electric field

text: [This problem has been resolved, sorry for posting in the wrong forum!] (I was trying to model the trajectory of a particle in an electric field.)


title: Electron behavior changes when observed?

text: I saw this video of the double slit experiment by Dr. Quantum on youtube. Later in the video he says, the behavior of the electrons changes to produce double bars effect _as if it knows that it is being watched or observed_. What does that mean? How is that even possible? An atom knows if it is being watched? Seriously? Probably, likely are the chances that I dint understand the video?


title: Asymptotic Symmetry Group of General Relativity

text: This question is a little vague and I hope I can put across what I am looking for without too much confusion. > What is the motivation behind studying asymptotic symmetry groups in the > context of general relativity? How is it important in the AdS/CFT > correspondence? I have also read that the study of the ASG plays a role in > understanding scattering as well. Why? NOTE: I know what the ASG is. At least I think I do. I do not want to know how one goes about deriving it, etc. I am asking a more general question about the motivation behind its study. Indeed I am more interested in the motivations to study outside of the AdS/CFT correspondences and the likes. However, if you think the derivation of the ASG gives one some insight into this question, then please also mention that as well. PS - I am not just interested in the ASG of AdS space. I simply mentioned an example here.


title: does a charged particle revolving around another charged particle radiate energy?

text: The main drawback in Rutherford's model of the atom as pointed out by Niels Bohr was that according to Maxwell's equations, a revolving charge experiences a centripetal acceleration hence it must radiate energy continuously as a result of which its kinetic energy is reduced and it eventually should fall into the nucleus. Bohr then proposed his own model in which he explained this through quantum theory stating that the general laws of electrodynamics and classical mechanincs do not apply in the case of electrons. Does this imply that if a body is revolving around another, then it will tend to spiral in towards the centre? If so, then how is the stability of the solar system explained?


title: Relativistic corrections to classical physics formulae

text: How are classical formulas in physics (such as p = mv, or kinetic energy, or maxwell distribution of speeds) treated with the appropriate relativistic correction/modification? Is it done by using the Lorentz transformation equations? Could anyone give me a few examples of relativistic corrections to classical formulae? Thanks.


title: Is it possible to build a mechanical version of the basic logic gates AND, OR, XOR and NOT?

text: Is it possible to physically build a the basic logic gates with nuts ans bolts? I'm wondering if it's possible to build a mechanical adder unit.


title: Quantum XOR: How do you generalize it?

text: Consider the classical XOR Gate: Given a 2 bit system: $G = [u_1, u_2]$ $$XOR(G) = (u_1 + u_2) \ mod \ 2$$ Is the following a good generalizaiton of a Quantum XOR Gate: Given a 2-qubit system: $Q = a_1|00] + a_2|01] + a_3|10] + a_4|11]$ We now write: $$XOR(Q) = a_1|0] + a_2|1] + a_3|1] + a_4|0] = (a_1 + a_4)|0] + (a_2 + a_3)|1]$$ Notice that if we set $a_i = 1$ and the rest to $0$ that the quantum XOR behaves exactly like a classical XOR but it also has the ability to generalize to fractional and complex $a_i$


title: Synchronized rotation of the moon

text: Does the moon rotates? Yes. The rotation matches exactly the orbit of the Earth. Which means in 28 days the moon makes one rotation. Shouldn't this be also happening with the Earth rotation around the orbit of the sun? Video