title: Faraday rotation effect in circularly polarized waves?

text: We all know farady effect is observed in linearly polarized wave when it passes through a dielectric medium and magnetic field is along the direction of propagation. Is this phenomenon observable in circularly polarized waves? ![enter image description here](http://i.stack.imgur.com/q0H18.png) Circularly polarized wave travel through the medium with different phases. But as soon as they are out of the medium they retain their polarization. So basically a circularly polarized wave 's polarization is not affected but only it's phase. Whether we split circularly polarized wave into two linearly polarized wave and take the equation of form: E(z)=E.e^[i{(2πz/λ)+ϕ}]----------- eq.1 (I am unable to add equations from MS word equation object) Frequency and wavelength remain same for the em wave when it is entering and when it is leaving the dielectric medium. Say the em wave is split into two linearly polarized waves- one is sin wave along yz plane and the other is cos wave along xz plane. Both waves rotated by same angle due to faraday rotation because rotation is not dependent on plane polarization or phase of the wave. When we combine them again only the phase(ϕ) of circularly polarized wave changes. First question is this theory correct? Second question is how do we calculate the change in phase. For example, a linearly polarized wave's change in polarization is given by experimental formula of becquerel equation(change in angle of polarization(β)=Verdet constant(V) X Magnetic field(B) X Length of dielectric(d), see the figure to get an idea ). Is there any experimental formula for circularly polarized waves? **Edit** : I guess the change in phase of circularly polarized wave follows different equation as seen in the becquerel equation. I am unable to find any experimental data on this.


title: Looking for solutions to problems in Feynman and Hibbs path integral and QM text

text: I've just started reading Feynman and Hibbs path integrals and Quantum mechanics after a decade hiatus from my undergraduate math degree (including a few semesters of physics for engineers). It would be tremendously helpful to see the step by step solutions to some of the first set of problems (p27-28, #2.1 and on) As many as folks are willing to post! Or if anyone knows of the solutions being available on-line that would also be helpful. The specific concept question is finding the extremum of functional that is the integral of the lagrangian of a system (classical action). The easiest example is 2.1, show that for L=m/2 (dx/dt)^2, the extremum is m/2*(Xb- xa)^2/(tb-ta). I have tried manipulating the integral by replacing 'partial of L with respect to x' with 'd/dt (partial L with respect to dx/dt) but haven't gotten there. I know it's the easiest problem but I think if I see an example, I will be able to apply what I learn to the harder subsequent problems I hope that's enough to come off "hold" Thanks!


title: Why does Hydrogen bubbles gather in anode instead of cathode during electrolysis?

text: Just wondering when water undergoes electrolysis why does oxygen goes towards cathode and hydrogen towards anode why not the other way round


title: Capacitors in series: Why is the equivalent charge the same as the individual charges

text: Consider two parallel-plate capacitors $C_1, C_2$ in series. For the "equivalent" circuit, clearly $$ Q_{equiv} = C_{equiv}V$$ should hold, where V is the total voltage drop between input and output. It is also obvious that $$ Q_1 = Q_2 $$ where $Q_1$ and $Q_2$ are the charges (plus and minus on opposite sides) accumulated on each of the two sets of plates. Obviously in general, $$ V_1 \neq V_2 \qquad \text{}$$ However, just because we have two capacitors with equal charge inside the circuit why is it true that for the equivalent circuit $$Q_{equiv} = Q_1 = Q_2$$ I see no obvious reason why this should be true, and this is _assumed_ , though not explained in the discusssion below: http://farside.ph.utexas.edu/teaching/302l/lectures/node46.html


title: Spike when wavefunction collapses

text: So, when wavefunction collapses, there is a spike occuring. Does this mean that there are parts with the continuous probability of 0? (For example, x position from -9 to -3 has probability of 0, while from -3 to 3, the probability of 100% (1) and so on)


title: Reconciling topological insulators and topological order

text: We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many- body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter. My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this? Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"? **Edit:** While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena. However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this. On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the _geometric_ properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?


title: How the boundary term in the variation of the action vanishes

text: In David Tong's QFT lecture notes ( _Quantum Field Theory: University of Cambridge Part III Mathematical Tripos,_ Lecture notes 2007, p.8), he states that > We can determine the equations of motion by the principle of least action. > We vary the path, keeping the end points fixed and require $\delta S=0$, $$ > \begin{align} \delta S &= \int d^4x\left[\frac{\partial \mathcal > L}{\partial\phi_a}\delta\phi_a+\frac{\partial \mathcal > L}{\partial(\partial_\mu\phi_a)}\delta(\partial_\mu\phi_a)\right] \\\&= \int > d^4x\left[\frac{\partial \mathcal L}{\partial\phi_a} > -\partial_\mu\left(\frac{\partial \mathcal > L}{\partial(\partial_\mu\phi_a)}\right) \right]\delta\phi_a > +\partial_\mu\left(\frac{\partial \mathcal > L}{\partial(\partial_\mu\phi_a)}\delta\phi_a\right) \tag{1.5} \end{align} $$ > The last term is a total derivative and vanishes for any $\delta\phi_a(\vec > x,t)$ that decays at spatial infinity and obeys $\delta\phi_a(\vec > x,t_1)=\delta\phi_a(\vec x,t_2)=0$. Requiring $\delta S=0$ for all such > paths yields the Euler-Lagrange equations of motion for the fields $\phi_a$, > $$ \partial_\mu\left(\frac{\partial \mathcal > L}{\partial(\partial_\mu\phi_a)}\right) -\frac{\partial \mathcal > L}{\partial\phi_a} =0 \tag{1.6} $$ Can someone explain a little more that why the last term in equation (1.5) vanishes?


title: Why do liquids separate in space?

text: I've seen videos of people in space (on ISS) who squeeze a bottle or something and liquid comes out, it then separates into smaller balls. Why is this surely it should stay pretty much together because theres no gravity from the Earth so the liquid should attract itself?


title: Redshift Mechanism

text: How does the stretching/expansion of space cause redshift in light from distant stars? What is the mechanism that causes the electromagnetic crests to be further apart?


title: What is the link between the metric signature of spacetime and fundamental field equations?

text: The signature of Minkowski spacetime is 2, as is explained here: metric signature explanation. The signature is related to the form the fundamental equations take, but I'm not totally clear on the link here. Furthermore I believe there is a link between second order PDEs and the metric of spacetime. I was wondering if anyone could clarify the links here, or could provide more details of how it goes. I have been reading from the following essay: http://www.fqxi.org/data/essay-contest-files/Callender_FQX.pdf


title: What can be the smallest chaotic system?

text: As I am talking about 'smallest' can I expect that it should be a quantum system? I understand that we use quantum chaos theory instead of perturbation theory when the perturbation is not small. For example when quantum numbers are very high for an EM field interacting with an atom, we cannot use perturbation theory. In this case we will use the theory of quantum chaos. My question is what is the lower boundary of the size of the system or more specifically of the quantum number in case of a EM-field atom interaction for which we must use quantum chaos theory?


title: How do you transform between theoretical Hertzsprung-Russell Diagrams and Colour-Magnitude Diagrams?

text: When discussing stars, theorists tend to use effective temperature $T_\text{eff}$ and luminosity $L$ (on logarithmic scales). Observers, on the other hand, usually talk about observed colours and magnitudes (e.g. $B$ vs. $B{-}V$). These two sets of parameters are obviously related. Temperature corresponds to colour; luminosity to magnitude. How does one transform between them? That is, given $L$ and $T_\text{eff}$ (from which one can derive $R$ through $L=4\pi R^2\sigma T_\text{eff}^4$), what are $B$ and $B{-}V$, or any other colour/magnitude? Alternatively, given $B$ and $B{-}V$, what are $L$ and $T_\text{eff}$? I figure $(L,T_\text{eff})\to(B,B{-}V)$ is easier. I'll start trying to answer it myself just by presuming the star is emitting a blackbody spectrum and integrating against the $B$ and $V$ filter functions. I realise this will be a major simplification but it's a start. I'm hoping it's been done (and published) before, maybe with better treatment of reddening etc.


title: In what situations do water levels not reach equilibrium?

text: I have been taught that water levels will always equal out. However, now I find that sumps and some other setups allow for water not to become equal. What other arrangements allow for the water levels of two containers not to be equal? Also, and more importantly, why does this occur? I hope someone is able to explain this for an aspiring physicist. Edit: Another example I have found is a phenomena called Heron's fountain, how does the water level rise instead of all of it pooling up in the lowest container?


title: What is Mathematical formulation of Holographic principle?

text: What is Mathematical formulation of Holographic principle > The holographic principle is a property of quantum gravity and string > theories which states that the description of a volume of space can be > thought of as encoded on a boundary to the region—preferably a light-like > boundary like a gravitational horizon.


title: What is the smallest amount of He you can use to cool a superconducting magnet?

text: This is an engineering question, but it is adressed to physicists who build accelerators. This question: An electromagnetic space elevator? notices that a NbSn superconducting ring around the equator will launch itself into orbit. The question is whether there is enough He to cool such a massive structure. What is the minimum amount of He required to maintain a stable 4-degree temperature safely throughout a long cylindrical wire? Is is necessary to use a fluid at all, or can one stably and safely maintain a stable He-range temperature in a very long wire without a liquid coolant?


title: Smallest minimum feature size possible for transistors printed by photolithography

text: I'm doing a project about Moore's Law, one of the subtopics I've come to is photolithography. The way I understand it is that the MOSFET transistors are currently printed on a silicon wafer by projection printing (mostly with 193nm UV light). With this technique your minimum feature size (CD) will be: $CD = k_1 \cdot \frac{\lambda}{NA}$ but you're limited by your depth of focus ($D_f$) $D_f = k_2 \cdot \frac{\lambda}{NA^2}$ From what I've read immersion lithography (increasing the NA) is the most advanced form of photolithography, but it's reaching its limit. Other lithography techniques are being researched; smaller wavelengths (EUV, X-ray lithography, electron beam lithography), each with their own issues and too expensive as of yet for large scale production. My question is, since I'm looking into the physical bariers Moore's law is going to face, is there a calculable limit to the minimum feature size possible with photolithography and/or next-generation lithography.


title: What is the relationship between the Higgs field and quarks?

text: I have some difficulty considering the relative size of each and the meaning behind the shape of Higgs boson. I ask relating to the _structures_ of both the Higgs field and quarks. How is it that the structure of a Higgs boson flows into that of, for instance, a bottom-antibottom quark pair? Essentially I am asking (or at least think I am asking): If the interactions for the field to exist occurred at some point in the universe's past, the particle is expressing it's shape in relation to the field, etc, etc.. Does this mean {when viewing some of the type of symmetries seen in readouts of the possible Higgs boson decay} quarks themselves are further expressions of the same field's shape or instead some manner of deformation? ![Now well known shape of a Higgs Boson, computer generated from Wikipedia](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs- event.jpg/650px-CMS_Higgs-event.jpg) This now fairly well known image from Wikipedia is a computer generated Higgs boson demonstrating simulated decay trajectories. This has often given me some considerations and can hopefully serve to slightly illustrate the structures I'm inquiring. (Knowing this is neither the boson or the quarks themselves)


title: Why are there photographs of nuclear tests?

text: I was looking at old photographs of the nuclear tests on the bikini atoll. It dawned on me that you don't want to run film through airport x-rays, as it exposes the film. I've been told that a nuclear explosion emits all energy on the spectrum (IR to Gamma). If this includes x-rays, why wasn't the film ruined? Is it because: 1) I'm an amateur radio operator. One of the things I'm familiar with is that RF signal strength starts at a point, and changes by a sum of squares (correct me if I'm wrong PLEASE!) as you move away from the point. If all energy behaves similarly in this respect, is this just a, "not enough x-rays at this distance to effect the film discernibly?" 2) The body of the camera must be taken into account (maybe?). Most cameras of this time period were metal (I believe). Most of the film curtains/iris' we're made of metal too. The only point of entry might have been during exposure? 3) A combination of 1+2 4) Some other awesome physics thing that I can't wait to learn about? 5) Pics of nuclear blasts are all photoshopped productions (I'm kidding of course)?


title: Distinguishing between an entangled and non-entangled state (mainly $S(H_A) \otimes S(H_B)$ vs. $S(H_A \otimes H_B)$)

text: Say I have two quantum systems $A$ and $B$ I can look at the joint (composite) system $AB$ which is given by $H_{AB} \in H_A \otimes H_B$ Measuring a subsystem with respect to a collection of measurement matrices $\textbf{M} = \\{M_i\\}_{i \in I} \in Meas_{I}(H_A)$ acts as measuring $AB$ with respect to $\textbf{M} \otimes \mathbb{I}_B = \\{M_i \otimes \mathbb{I}_B\\}_{i \in I}$ Q: Can I do this with entangled states? I know that making a measurement causes the entangled state of say 2 qubits to decompose in to two states. If I understand correctly, they are not entangled anymore. Now because of this we can seperate the states in to a sum of the product states $|AB> = \sum \alpha_j|j> \otimes |\psi_j>$ over all basis states $|j> \in S(H_A)$. What this says to me is that we can some how distinguish between these qubit states (hence they are not entangled anymore). Do I have this right? This leads me in to the difference between $S(H_A) \otimes S(H_B)$ vs. $S(H_A \otimes H_B)$. Now the first case is not entangled and we have the "product states" of two wavefunctions/state vectors , but in the second case we have some sort of combination (composite?) of states? I would guess this is when the states are entangled and after measurement they decompose in to say $|\psi> \otimes |\phi> \in S(H_A) \otimes S(H_B)$. Now if I look at an entangled 2-qubit state that I know is entangled $|\Phi> = |\Phi^+> = \frac{1}{\sqrt{2}}(|0>|0> \+ |1>|1>) \in H_A \otimes H_B$ (as written out in some notes) I see that we can have an entangled state embedded somehow in the tensor of two state vectors in Hilbert space. I don't know if this is supposed to be $S(H_A \otimes H_B)$ or $S(H_A) \otimes S(H_B)$ or something else completely. Edit: I removed some tensor math that was incorrect. I had initially thought that if some mixture of states could be decomposed and have just a $\otimes$ and nothing such as an addition or subtraction operator it. I know that is very rudimentary but from my basic understanding of product states: http://en.wikipedia.org/wiki/Product_state I see that if a probability density can be written as the tensor product of two different probability densities, then we have a non-quantum correlation, although in the wiki article above, this is neither quantum or classical in nature. I however also see that a mixed state such as $\rho_{AB}=\frac{1}{2}(|0_A0_B⟩⟨0_A0_B|+|1_A1_B⟩⟨1_A1_B|)$ only has classical correlations. Furthermore I have found that Two states $\rho, \sigma$ are called $\textbf{perfectly distinguishable}$ if there exists a measurement $M \in Meas(H) \text{ with } I = {0,1,...}$ such that $p_0(M,\rho) = 1 = p_1(M,\sigma)$. Now from my reading, in the case of a von Neumann measurement $M = \\{M_i\\}, p_i$ simplifies to $p_i = tr(M_i \rho)$ and in the case of a complete von Neumann measurement $M = \\{ |i \rangle \langle i | \\}$, $p_i$ simplifies to $p_i = tr(|i \rangle \langle i | \rho) = \langle i | \rho |i \rangle$ So basically, I just have to take the traces of the respective measurements on probability densities, if they are equal (hence perfectly distinguishable) then the system of, say qubits, are not entangled, otherwise they are? Is this correct? Thank you, Brian


title: Does the human body have a resonant fequency? If so, how strong is it?

text: Inpired by http://music.stackexchange.com/questions/7064/physiological- responses-to-different-rhythms, I'm wondering if the human body has a strong resonant frequency. I guess the fact that that it's largely a bag of jelly would add a lot a damping to the system, but is that enough to dampen it entirely? What models for resonance might be used to model the human body? (eg. weight- on-a-spring, with legs as springs?). What about individual, semi-independent body parts, like legs, or lung cavity (acoustic resonance?).


title: From Euler-Lagrange equation to non affine geodesic equation

text: I have some problems to demonstrate the non affine geodesic equation from Euler-Lagrange's equations. I start defining the Lagrangian $L=\sqrt f$, but then I'm not able to find the Christoffel symbol's expression. Can anybody help me?


title: Bloch-sphere-like representation of two-qubit density operators

text: The Bloch sphere is an excellent way to visualize the state-space available to a single qubit, both for pure and mixed states. Aside from its connection to physical orientation of spin in a spin-1/2 particle (as given by the Bloch vector), it's also easy to derive from first principles mathematically, using the Pauli spin operators $$\mathcal S = \bigl\\{\mathbf 1, X,Y,Z \bigr\\} = \Bigl\\{\mathbf 1,\; \sigma^{(x)}\\!,\; \sigma^{(y)}\\!,\; \sigma^{(z)}\Bigr\\}$$ as an operator basis, and using only the facts that $\mathbf 1$ is the only element in $\mathcal S$ with non-zero trace, and that $\def\tr{\mathop{\mathrm{tr}}}\tr(\rho) = 1$ and $\tr(\rho^2) \leqslant 1$ for density operators $\rho$. **Question.** Using either the Pauli operators as a matrix basis, or some other decomposition of Hermitian operators / positive semidefinite operators / unit trace operators on $\mathbb C^4 \cong \mathbb C^2 \otimes \mathbb C^2$, is there a simple presentation of the state-space of a two-qubit system -- or more generally, a spin-3/2 system in which we do not recognise any tensor- product decomposition of the state-space (but might use some other decomposition of the state-space)? My interest here is that the representation be _simple_.   * The representation doesn't have to be presented visually in three dimensions; I mean that the constraints of the parameter space can be succinctly described, and specifically can be presented in algebraic terms which are easy to formulate and verify (as with the norm-squared of the Bloch vector being at most 1, with equality if and only if the state is pure).   * The representation should also be practical for describing/computing relationships between states. For instance, with the Bloch representation, I can easily tell when two pure states are orthogonal, when two bases are mutually unbiased, or when one state is a mixture of two or more others, because they can be presented in terms of colinearity/coplanarity or orthogonality relationships. Essentially, it should be a representation in which linear (super-)operations and relationships on states should correspond to very simple transformations or relationships of the representation; ideally linear ones.   * The representation should be _unable_ to represent some Hermitian operators which are not density operators. For instance, there is no way to represent operators whose trace is not 1 in the Bloch representation (though non-positive operators can be represented by Bloch vectors with norm greater than 1). In fact, the Bloch representation essentially is that of the affine space of unit trace in the space of Hermitian matrices, centered on $\mathbf 1/2$. A simple and concise geometric description of density operators as a subset of the affine plane of unit-trace Hermitian operators centered on $\mathbf 1/2 \otimes \mathbf 1/2$, _i.e._  a _generalization_ of the Bloch sphere representation (but not necessarily using the Pauli spin basis), would be ideal. If there is such a representation, does it generalize? How would one construct a similar representation, for instance, for qutrits (spin-1 systems) or three- qubit states (spin-5/2 systems)? However, these questions should be understood to be secondary to the question for two qubits.


title: Equation regarding the Riemann tensor in the Cartan formalism

text: I have a problem verifying the following equation (in three dimensions) $$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$ where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature Attempt at a solution: $$\epsilon_{abc} e^a\wedge R^{bc}=\epsilon_{abc} e_\mu^ae_\alpha^be_\beta^c R^{\alpha\beta}_{\nu\rho} dx^\mu\wedge dx^\nu\wedge dx^\rho$$ Now the idea is that the number of dimensions and the Levi-Civita tensor and the antisymmetry of the three-form forces the set $\\{\alpha,\beta\\}=\\{\nu,\rho\\}$. This will give the expression \begin{align}\epsilon_{abc} e^a\wedge R^{bc}&=\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{12} dx^0\wedge dx^1\wedge dx^2+\\\&\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{21} dx^0\wedge dx^2\wedge dx^1+\\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{21} dx^0\wedge dx^2\wedge dx^1+\\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{12} dx^0\wedge dx^1\wedge dx^2+({\rm cyclic\,permutations})\end{align} The problem now is that the Ricci scalar is $R^{12}_{12}+R^{21}_{21}+({\rm cyclic\,permutations})$, so when counting the number of terms I obtain $2\sqrt{|g|}Rd^3 x$ which is wrong by a factor of 2. Can anyone see where I made a mistake?


title: Does the nonlocality of the preferred basis mean QM is nonlocal?

text: Take the Mach-Zehnder interferometer as an example. A photon passes through a beam splitter, is reflected off mirrors, and interferes with itself at another half-silvered mirror. No measurements or disturbances are made in between. The preferred basis of the photon is given by which path it takes after passing the second beam splitter. The problem is, this preferred basis is nonlocal in between. Is there a problem with locality here, or is silly ol' me just plain confused?


title: Role of unit vectors in cylindrical coordinates

text: I know how the unit vectors are defined in cylindrical coords. If I have a point P, how do I express it as a combination of the unit vectors uρ, uφ and uz. In the case of Cartesian coordinates this combination is linear. But what about cylindrical coords? Does such a combination exist for them? And, BTW, could you suggest a simple Physics problem where the use of cylindrical coordinates is convenient or, in general, the reason to choose them?


title: In QFT, why do fermions have to anticommute in order to insure causality?

text: I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the _anticommutator_ $[\phi,\phi^\dagger]_+$ to turn zero. The answer given to the question above does not seem to address this.


title: dimensional analysis of Grassmann integration/differentiation

text: There is another paradox that I need to resolve: The Berezin integration rules for Grassmann odd variables give the same result as differentiation: If $f=x+\theta\psi$ is a superfunction, the integral $$\int d\theta(x+\theta\psi)=\psi$$ gives the same result as differentiation $$\frac{d}{d\theta}(x+\theta\psi)=\psi.$$ How is this supposed to work in supersymmetry, where the Grassmann coordinates carry mass dimension -1/2? If I integrate, I expect a result to drop mass dimension by 1/2, whereas differentiation would lead to a gain. In the above example, I end up with the same object. What is its mass dimension?


title: Tensor Product of a Bra and a Ket

text: What does one get if the take the tensor product of a bra and a ket, for instance, $\langle\uparrow \rvert \otimes \lvert \downarrow\rangle$? What I mean it, what is this object? What does it act on? Or does it get acted on? Is it meaningful, like the tensor product of two kets is, etc?


title: Copenhagen interpretation

text: Reading some science history, Werner Heisenberg and Bohr created the Copenhagen interpretation, but what I didn't get is how can we connect this interpretation to Schroedinger's cat and the double slit experience? Are they confirming Heisenberg and Bohr's work?


title: Proving Simple Harmonic Motion (direction of acceleration)

text: A particle of mass $m=0.5kg$, is attached to a spring of natural length $l=0.6m$ and modulus of elasticity $\lambda=60N$, and the setup is on a horizontal smooth table. The other end of the spring is attached to a fixed point $A$ on the table. The particle is then pulled so that the distance $AP=0.9m$, and is then released from rest. ($P$ is to the right of $A$, and the tension in the spring is $T$). If you take $x$ (displacement from the centre of oscillation) as increasing to the right, then you can prove SHM if you can get equation that is like this: $$a = -w^2$$ So $$-T =ma$$ (The $T$ is negative because $x$ increases to the right making right positive, and $T$ is to the left, and $a$ is positive because in SHM $a$ is always in the direction of $x$ increasing.) $$-\frac{\lambda x}{l} = ma$$ $$-\frac{60x}{0.6} = 0.5a$$ $$-200x = a$$ (which fits the equation at the top so its SHM.) However if you take $x$ to increase to the left: $$T = ma$$ ($T$ is positive because $x$ increases to the left making left positive and $T$ is to the left, $a$ is positive because, again, in SHM $a$ is always in the direction of $x$ increasing.) $$\frac{\lambda x}{l} = ma$$ $$\frac{60x}{0.6} = 0.5a$$ $$200x = a$$ (which doesn't fit the equation because there is no negative.) So I don't understand what I'm not getting right in the second part? how can changing the defining of the direction of $x$ increasing have such an effect? (Instead of changing the direction of $x$ increasing I could have said the particle $P$ is pushed so that the spring is compressed and $AP=0.3m$ making the $T$ act towards the right which is the same direction as $x$ increasing).


title: Maxwell's Equations using Differential Forms

text: Maxwell's Equations written with usual vector calculus are $$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$ $$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times B=\mu_0j+\dfrac{1}{c^2}\dfrac{\partial E}{\partial t}$$ now, if we are to translate into differential forms we notice something: from the first two equations, it seems that $E$ and $B$ should be $2$-forms. The reason is simple: we are taking divergence, and divergence of a vector field is equivalent to the exterior derivative of a $2$-form, so this is the first point. The second two equations, though, suggests $E$ and $B$ should be $1$-forms, because we are taking curl. Thinking of integrals, the first two we integrate over surfaces, so the integrands should be $2$-forms and the second two we integrate over paths and so the integrands should be $1$-forms. In that case, how do we represent $E$ and $B$ with differential forms, if in each equation they should be a different kind of form?


title: Cosmological metric with off-diagonal terms?

text: In the context of Cosmology models, What are examples of metrics with off- diagonal terms?


title: Applying kinematics issue

text: I've been having an unusually difficult time solving kinematics problems in comparison to all the other students in the classroom. It appears that I'm one of the unfortunate people with an anti-math mind (although I've been expecting as much for some time now). In short, I have no idea how to solve this problem: > A carpenter tosses a shingle off a $9.4m$ high roof, giving it an initial > horizontal velocity of $7.2 m/s$.   >  a. What is the final vertical velocity of the shingle?   >  b. How long does it take to reach the ground?   >  c. How far does it move horizontally in this time? I tried to organize the data I had and the data I needed to calculate: **Horizontal:**   1\. time is unknown and the same as vertical time   2\. initial position is $0m$   3\. final position is unknown   4\. initial velocity is $7.2m/s$   5\. final velocity is $7.2m/s$   6\. acceleration must be $0$ because velocity doesn't change **Vertical:**   1\. time is unknown and the same as vertical time   2\. initial position is $12m$   3\. final position is $0m$   4\. initial velocity is gravity, or $9.81m/s$   5\. final velocity is unknown   6\. acceleration is $9.81m/s^2$ However, according to my unknowns and my equations, I cannot ultimately solve any of these because I cannot solve for time, which requires knowing distance, velocity, and acceleration, like this:   $t = -v_i \pm \sqrt{v_i^2 + 2a(\pm d)} \over{a}$ Is this correct and is there another way to find time with less unknowns? And none of my equations can explicitly find the initial or final positions of anything: > $\Delta x = v_{ix} \Delta t + 0.5a_x \Delta t^2$ (also works with $y$)   >  $v_{fx}^2 - v_{ix}^2 = 2a_x \Delta x$ (also works with $y$)   >  $v_{fx} = v_{ix} + a_x \Delta t$ (also works with $y$)   >  $\Delta y = 0.5(v_{fy} + v_{iy}) \Delta t$   >  $y = v_{avg} * t$   >  $\Delta y = 0.5(V_{fy} + V_{iy}) \Delta t$ What I want to know is **how to correctly solve this (with equations listed)**. Please do not just provide the answer; thanks in advance.


title: Is it possible to produce gamma radiaton using radio emitter?

text: As in the title, I'm wondering is it possible. I think it is possible, because we have powerful enough radiotechniques and gamma radiation are just EM waves, not particles. However I think is useless, because it costs too much. Can anyone say something more about this?   * How much power would consume such emitter?   * Would it be helpful anywhere?   * How huge frequencies of current would it use in the antenna?   * How big device would it be?


title: Maxwell demon : how can the demon move the door without doing any work ?

text: Here is something that I never understood about Maxwell's demon paradox : in order to contradict the second law of thermodynamics, the demon must open and close the door without doing any work... But how is that possible ? I know it is just a thought experiment but still, as I see it, it just shows (by contraposition) that, if you assume the second principle to hold true, then there cannot exist a device that will open and close the door in this way without doing any actual work (for instance, there will necessarily be some friction or whatever which will creates some work..) Seeing things this way there is no paradox and it only states, as a corollary of the second law, that such a demon cannot exist... Obviously, I am missing something here since there seems to be much more to this thought experiment than that... I would be extremely grateful if someone would take the time to explain to me where I am mistaken ! :) Thanks !


title: The speed of light and fields

text: Does the speed of light apply to the speed of (waves?) in every known field, or does it only apply to the electromagnetic field?


title: Hilbert Space of (quantum) Gauge theory

text: Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down **the Hilbert Space of quantum Gauge theory with spin-S**. (Are there something more rich/subtle than just saying the Hilbert Space are composed by the state space of infinite many sets and infinite many modes of harmonic oscillators? - i.e. more rich/subtle than usual spin-0 scalar fields' Hilbert space?) Is the quantum Gauge theory's Hilbert Space written **in a tensor product form or not** (eg. thinking about put this gauge theory **on the lattice** )? Whether there are differences for this procedure construction of Hilbert space for these three cases: (1) spin-1 quantum Gauge theory with Abelian $U(1)$ symmetry (2) spin-1 quantum Gauge theory with non-Abelian (such as $SU(N)$) symmetry (3) spin-2 quantum Gauge theory (Gravity? or anything else) Also, whether gauge-redundancy plays any roles? Is there similar thing like Faddeev-Popov ghosts happened in the path integral formalism, when one dealing with gauge-redundancy?


title: Subtlety of analytic continuation - Euclidean / Minkowski path integral

text: I subconsciously feel not fully comfortable about **Wick rotating or analytic continuation from Euclidean to Minkowski space**. I simply wonder whether there is any subtlety here, and when we need to be conscious whether **(i) this continuation can be dangerous or the continuation cannot be done; or (ii) the two theories(Euclidean and Minkowski) may not be the same?** For example, if you read Polchinski's String theory, most of worldsheet and CFT business is done in Euclidean signature. But in Chap 3 of Polchinsk's at p.83, says ``The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, and make the same rotation. **This provides a formal justification for the equivalence of the Minkowski and Euclidean path integrals.**  It has been shown by explicit calculation that they define the same amplitudes, respectively in the light-cone and conformal gauges.'' But right at p.83 footnote, says '' **In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.** '' Is Polchinski providing an example here? **Can someone explain in what (generic) scenarios there will be subtlety about the analytic continuation?**


title: What is the origin of the Dirac delta term in the dipole electric field?

text: I am a bit lost how one has deduced the formula for electric field with electric dipole because of some inconsistency between different sources. The Wikipedia article contains a delta function in the formula, but this paper gives only the formula without the delta function, $$\bar{E} = -\nabla \phi = \frac{-p}{4\pi \epsilon_{0} r^{3}} ( \hat{e}_{z}-3\hat{u}_{r}(\hat{u}_{r} \cdot \hat{u}_{z}) ) $$ Is it intentionally left out? My notes mention that the dipole is with a circular thing $P=Q d$ where $d$ is distance and $Q$ is charge.


title: If a Kerr-Newman black hole is like a charged, spinning, heavy magnet, what kind of magnet is it like?

text: I was reading up on De Sitter spaces, which states that the gravitational effects from a black hole is indistinguishable from any other spherically symmetric mass distribution. This makes a lot of sense to me. I'm now super curious, can we just formulate all the properties of a black hole beyond the limit at which GR is needed in a completely Maxwellian / Newtonian sense? The Wikipedia article on the Kerr-Newman metric seems to indicate so, but the equations are in terms of the GR metrics. This is not what I want, I want a simplification, a limit-case of that math. Ted Bunn answered a part of my question in Detection of the Electric Charge of a Black Hole. Let me repeat the stupidly simple form of the gravitational and electric field for a black hole beyond the point at which GR is needed. $$\vec{g} = \frac{GM}{r^2} \hat{r}$$ $$\vec{E} = \frac{Q}{4\pi\epsilon_0 r^2} \hat{r}$$ Correct me if I'm wrong, but these would be a meaningful and accurate approximation in many, in fact, most situations where we would plausibly interact with a black hole (if we were close enough that these are no longer representative, we'd be risking a date with eternity). **Question:** Fill in the blank; what would the magnetic field $\vec{B}$ around a black hole be? Here is why I find it non-trivial: Every magnet in "our" world has some significant waist to it. So here is a normal magnet. ![normal magnet](http://i.stack.imgur.com/l7sr2.png) What happens when this is a black hole? Would the approximation for $\vec{B}$ that I'm asking for have all the magnetic field lines pass through the singularity? Or would they all pass through the event horizon radius but not necessarily a single point? * * * I think most people who have understood this already, but ideally the answer would use the 3 fundamental metrics of a black hole. Mass $M$, charge $Q$, and angular momentum $L$. The prior equations for gravity and electric field already fit this criteria. So the answer I'm looking for should be doable in the following form. $$\vec{B} = f \left( M, Q, L, \vec{r} \right) $$


title: Selection Rules in electron spectroscopy

text: How to derive the selection rules $\Delta L= \pm 1$, $\Delta S=0$ for electron spectroscopy?


title: Finding the vacuum which breaks a symmetry

text: I will start with an example. Consider a symmetry breaking pattern like $SU(4)\rightarrow Sp(4)$. We know that in $SU(4)$ there is the Standard Model (SM) symmetry $SU(2)_L\times U(1)_Y$ but depending on which vacuum we use to break this symmetry, in a case you can totally break the SM symmetry, with the vacuum : $$\Sigma_1 = \begin{pmatrix} 0& I_2 \\\ -I_2 & 0 \end{pmatrix}$$ and in another case, these symmetry is preserved, with the vacuum $$\Sigma_2 = \begin{pmatrix} i\sigma_2& 0 \\\ 0 & i\sigma_2 \end{pmatrix}$$ In the first case (with $\Sigma_1$), the generators corresponding to the SM symmetry are part of the broken generators so the SM symmetry is totally broken. In the second case ($\Sigma_2$), the SM generators are part of the unbroken generators then the SM symmetry is preserved. As you can read, I know the answers but not how to find them ! So, my questions are :   1. How is it possible in general (not only for the $SU(4)\rightarrow Sp(4)$ breaking pattern) to construct the vacuum that breaks the symmetry ?   2. Is it possible, when constructing the vacuum, to ensure that the vacuum will (or not) break a sub-symmetry like the SM symmetry in the previous example ?


title: Gravity beneath sea level

text: This is a very simple question, but I can't find an answer anywhere. Earth's gravity at sea level is 1 (approximately 9.8m/s^2) What happens being under sea level in a cavern/chasm/ravine, how much does gravity increase normally here on Earth?


title: How large is the information collected from an inverse femtobarn of  collisions?

text: I ran into this while looking at measures of humongous amounts of data. How does the information (data) collected in an inverse femtobarn exposure compare to a gigabyte of data ?


title: Can someone identify this Landau reference?

text: I'm having some trouble identifying a reference Landau and Lifshitz make in volume 3 - non-relativistic quantum mechanics - of their Course of theoretical physics. This is the page from the Oxford, Pergamon, 1977 edition: ![enter image description here](http://i.stack.imgur.com/OqC2L.png) There isn't a bibliography either at the end of the chapter or the book, and other references are usually well identified in footnotes. I know it's a long shot to identify and obtain a copy of this article (in English), but I would definitely like to have a look at it. It appears to be the original publication of Landau's method of complex-contour integration to find WKB- approximation matrix elements.


title: Why photons are emitted because of changes to electron behavior

text: Explanations I have read of why photons are emitted from atoms mention electrons being 'excited' to another energy level, and then returning to their base level, releasing a photon. I have also seen the occasional mention of 'fields' and I vaguely expect from what I've read that interaction of the electrons with these fields has some effect. I've also seen, but not used, Feynmann diagrams that involve additional particles. Please can you (ideally) clarify what are the different theories and experimental evidence relating to electrons and photons because of changes to electron behavior?


title: Dark light filters letting through less light

text: I'm looking for an explanation on why darker light filters let through less light than lighter filters (secondary/high school level). Although this seems to the layman like the obvious outcome, our science teacher was unable to explain why a darker blue and darker green filter let through less light than a yellow or turquoise filter. Thanks


title: Maximum resolution per lens size

text: This question is more practical than theoretical, but I am interested in the theoretical considerations as well. My wife just bought a Samsung S3 phone with a 8 MP image sensor hiding behind a tiny lens. In daylight the pictures come out fine, but it suffers horribly in low-light conditions. Is there a theoretical limit as to how fine an image sensor can be behind a lens of a specific aperture, given a reasonable amount of ambient light and a reasonable shutter speed? Will increasing the sensor resolution beyond this limit decrease the actual resolution (the ability to resolve two points as individual points) of the final image? Thanks.


title: Vector Potential for Magnetic field when the field is not in simply-connected region

text: According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some $(k-1)$-form, say $\eta$ with $\omega = d\eta$. Now, translating to vector fields, if we consider $U$ a star-shaped set in $\mathbb{R}^3$ and if $B$ is a vector field inside $U$ such that $\left\langle \nabla, B \right \rangle= 0$, then there's some vector field $A$ defined in $U$ such that $B = \nabla \times A$. I've heard that Poincare's Lemma turns out to be true even if $U$ is not star- shaped, but rather, simply connected. Now, in the hypothesis of Poincare's Lemma, the fact that the magnetic field satisfies $\left\langle \nabla, B\right\rangle = 0$ implies the existance of the vector potential $A$, with $B = \nabla \times A$. But now, what happens if the magnetic field is defined in some region of space that is not simply connected? In that case, the vector potential could not exist according to Poincare's Lemma (it doesn't says that it doesn't exists, but it doesn't guarantees the existance). So, if the region where the field is defined has holes in it, what happens? Is really a chance that the vector potential doesn't exists? In that case, what are the physical consequences of that? Since I was always told that the vector potential was just a mathematical tool introduced to make life easier, I think that there wouldn't be so great impact on the point of view of conceptual explanation of the situation, however, I'm not sure.


title: An objective question from electromagnetic induction

text: > A bar magnet is released from rest along the axis of a very long, vertical > copper tube. After some the the magnet : > > (a) will stop in the tube.   >  (b) will move with almost constant speed.   >  (c) will move with an acceleration g.   >  (d) will oscillate. ## Attempt: Let the magnet be dropped with its north pole downward. We can consider long tube to be made of many circular loops joined together. At an instant $t=0$, the force mg acts on the magnet. $m$ is its mass. As it moves, by Lenz's law, each circular loop in the tube opposes its downward motion and hence upward force will act on it by many such loops. A time will come when the magnet experiences zero net force. Even eddy currents are produced in the magnet. The net force is $F-mg$ where at some instant will be nil, i.e. $F=mg$ as I already stated. Even if it has constant velocity at this instant magnetic flux by it linked with those coils is still varied and hence upward force still acts. Hence after this $F-mg<0$, and magnet accelerated upwards. Now again by Lenz' law coils oppose upward motion of magnet and this process continues till then again it move down.. and so on it just keep on oscillating... So answer should be (d). But its wrong !! How ? Correct answer given in (b). Can anyone point out the flaw in my attempt ?


title: Width of Gaussian Beam and Refractive Index

text: I know that in free space, the width of a Gaussian beam can be written as $W=W_0\sqrt{1+(\frac{z}{z_0})^{2}}$. However, I was wondering if it was possible to express this width as a function of refractive index instead (since I don't believe a Gaussian beam originating in say, glass, will diverge in the same manner as one in air). Anyone have any ideas?


title: Holographic principle "inside-out view"

text: From the perspective inside a black hole: Is information about everything outside a black hole - the rest of the cosmos - represented on the inside of the (event) horizon too? NB. I realize it is the notion of a so called stretched horizon that is being used about the 2d-like sheet that information is scrambled on, but I haven't heard about its view-from-the-inside-analogy, that naively being a horizon hovering just "below" the event horizon; therefore the simplified use of "event horizon" in the question. I also realize that it would be impossible to sample (all) this information as an inside job, which I would suppose could have some bearing on the answer to the question.


title: Why is NASA interested in Gamma ray bursts?

text: Gamma-ray bursts(GRBs) are flashes of gamma rays associated with extremely energetic explosions why nasa interested to Gamma ray burst?


title: Find distance between turtles

text: I've been thinking about the following problem for some time now and was wondering if anyone could shed any light on it. > At time $t = 0$, turtle `A` sits at $(0,0)$ and turtle `B` sits at $(d,0)$. > `B` begins moving straight upwards perpendicular to the $x$-axis with some > constant velocity $v$. Simultaneously, the `A` turtle begins moving with the > same constant velocity in such a way that its nose is always pointed at `B`. > What is the distance between these turtles as $t \rightarrow \infty$? * * * **Edit** So basically I've arrived at a nasty system of differential equations. Here's my reasoning: We know that turtle `A` will always point towards turtle `B`, so if $(x(t),y(t))$ is the curve traced out by `A`, then we can determine $dy/dx = \dot{y}/\dot{x}$ at each point: $\dot{y}/\dot{x} = (vt - y)/(d - x).$ The fact that `A` has a velocity $v$ gives rise to a second equation: $\dot{x}^2 + \dot{y}^2 = v^2.$ If we solve these, the problem would be reduced to taking a limit. Anyway, this is what I have so far - not much really. Would this be a good/feasible approach to pursue so as to solve the problem? Or is there a better way?


title: Do resistor-based fan regulators save no power at all?

text: I have heard that the traditional resistor-based fan speed regulators are inefficient. In fact, I have noticed that such regulators tend to get hot when the fan is set to low speed. However, does that mean that they save **_no power at all_** at low speeds, or is it just that power is saved but the amount of power saved is far less than ideal? I was expecting that since power P = V / R^2, the overall power consumption of the fan plus regulator system would be comparatively lower at lower speeds.


title: Relationship between Alcubierre drive space-time evolution and speed of gravity

text: The top rated answer to this question about the Alcubierre drive asserts, "spacetime can dynamically evolve in a way which apparently violates special relativity," but according to the Wikipedia article on the speed of gravity, changes in the gravitational field propagate at the speed of light. The second answer under this question puts the limitation in broader terms: "distortions of spacetime are as limited to travel to the speed of light as any other physical influence." These statements appear to contradict one another. My question is: Does the Alcubierre drive depend on space-time distortions propagating at superluminal speeds, and is that possible under general relativity? If Alcubierre drive distortions of space-time can propagate at superluminal speeds under GR then why can't gravitational waves (another traveling distortion of space-time) also propagate at superluminal speeds?


title: Derivative of a Position Eigenket

text: I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that $\langle\vec{r}|\hat{\vec{P}}|\psi\rangle = -i\hbar\vec{\nabla}\langle\vec{r}|\psi\rangle$ (I've omitted the full derivation). However, from this relationship he concluded that $\hat{\vec{P}} = -i\hbar\vec{\nabla}$. I'm sure this is very basic but why can you immediately conclude this? Surely this assumes that $\langle\vec{r}|(-i\hbar\vec{\nabla})|\psi\rangle = -i\hbar\vec{\nabla}\langle\vec{r}|\psi\rangle$. I'm not sure why this is necessarily true.


I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation. In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote: > Where did we get that (equation) from? Nowhere. It is not possible to derive > it from anything you know. It came out of the mind of Schrödinger. -- > Richard Feynman (from the Wikipedia entry on the Schrödinger equation) However, some places seem to _derive_ the Schrödinger equation: just search for "derivation of Schrödinger equation" in google. This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not? Thanks in advance, and please bear with my physical ignorance.


title: Seeking an account of fundamental particle physics sensitive to the issue of theory-laden observation

text: The hypothesis of confirmational holism ( http://en.wikipedia.org/wiki/Confirmation_holism ) speaks to all sciences, but seems especially trenchant in fundamental physics, the would-be end of the road for any reductionist (or modified reductionist) account of the world. Ironically, but naturally, the most fundamental physics comes at the top of any typical pedagogical climb -- knowledge at the smallest scales is the hardest won and most remote from ordinary experience. That makes typical textbooks useless for the philsophically-minded reader -- they all start with a raft of scientific assumptions that no sane physics major or graduate student would dare to challenge. I don't expect a complete account of the bootstrap from plain sense observation to fundamental laws because no one thinks as of yet that we know those fundamental laws. Nevertheless I seek to read a careful and skeptical analysis of the present situation. PS I'd like to add some tags -- at the least philosophy-of-physic, confirmation-holism -- but lack the necessary reputation.


title: How the spectrum of the hydrogen atom is checked/detected experimentally?

text: When solving the hydrogen atom, as a 2 body problem, we have the motion of the center of mass and the motion relative to the center of mass. The well known energy spectrum, $E_n$, that goes like $1/n^2$ is the one resulting from studying the motion relative to the center of mass. Now the energy of the center of mass of the atom, which is purely kinetic, should be added to $E_n$ to get the full energy. So unless the atom is at rest, there will be an extra (constant) term to $E_n$ My questions are, in practice when people study the hydrogen spectrum experimentally: 1-do people look at the spectrum from 1 atom only? if this is the case, then how can they guarantee that it is not moving (to properly identify the $1/n^2$ behavior)? (very cold atom technology was not known back in the day!) 2-if they are studying a gas of hydrogen, how come the spectrum to be detected is not washed out by the transitions coming from so many atoms in all directions? 3-again for looking at hydrogen gas, how the spectrum can be studied even to see if it goes like $1/n^2$ if the atoms are moving in all directions like crazy?


title: Atmospheric Escape of Gas Molecules

text: Most of the Bodies and Objects in space are _likely to have_ an **atmosphere**. Since the space is empty, the gases in atmosphere should have either _dissolved or emptied_ into space. But, Why doesn't this happen? Still those objects are revolving with their atmosphere...   * Taking into account of Gas Giants - Since their atmospheres are much **denser** , there should be some pressure difference (like more pressure at highly denser areas and vice-versa). There are possibilities for the molecules to escape at highly denser regions of the atmosphere.  Well, Is this RIGHT...? Also, Answers based on Atmospheric & Hydrodynamic escape would be appreciated...


title: How much does the global structure of a Lorentzian spacetime restrict the metric? And vice versa

text: E.g. if I know that my topology is that of a hyperboloid, how much freedom do I have left for my choice of the metric? And the other way around: if my metric is some conformal factor times the unit hyperboloid how much freedom do I have left in my choice for the topology? Am I forced to conclude that the topology of this (Lorentzian) spacetime needs to be that of a hyperboloid?


title: How to calculate the exchange constant of the Ising model?

text: The Ising model is a mathematical model of ferro-magnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually, a lattice, allowing each spin to interact with its neighbours. The model allows the identification of phase transitions, as a simplified model of reality [Source] The algorithm I am using to calculate it is: $$ E = -\sum\limits_{\langle i,j \rangle}^{n}J_{ij} s_i s_j - \mu H\sum\limits_{i}^{n} s_i $$ Where $E$ is energy, $J$ is the exchange constant, $s_i$ & $s_j$ represent nearest neighbour pairs (Like $s_{<1,0>} s_{<2,0>}$ or $ s_{<5,4>} s_{<5,5>} $). $H$ is the external magnetic field and $\mu$ is the magnetic moment. Usually H is given in units of $\frac{H}{\mu}$ and J is assumed to be isotropic, so the equation simplifies to: $$ E = -J\sum\limits_{\langle i,j \rangle}^{n} s_i s_j - m\sum\limits_{i}^{n} s_i $$ where $m= \frac{H}{\mu}$. This model can be used to predict the temperature at which a substance looses its ferromagnetism, which for cobalt is 1388K, for iron its 1043K and for nickel its 627K. This change from ferromagnetic to paramagnetic is found by observing a second order phase transition. (A continuous, dramatic change in the slope of the magnetism). If you use $J=1$, $T_C \approxeq 2.27K$. How could I determine the exchange constants for other substances, if I know their internal molecular lattice structure, or must I determine it numerically, knowing $T_C$?


title: Vanishing Ricci flow on a curved manifold

text: If I understand this right the Ricci flow on a compact manifold given by $\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$ tends to expand negatively curved regions and to shrink positively curved regions. Looking at the above definition I`m wondering if the parameter n can be used to achieve $\partial g_{\mu \nu} = 0 $ even if the Ricci tensor is not zero such that the validity of physics, that depends on the metric to be constant (as a precondition), could be extrapolated to curved manifolds to describe an expanding universe with a positive cosmological constant?


title: NASCAR drafting at Daytona

text: This year, the racetrack at Daytona has been repaved. The track was always faster than other tracks NASCAR raced at and several cars in a "train" were faster than single cars or smaller trains. This year is different, however, because two car groups are by far the faster than any other groupings. When a third car joins a tandem, all three slow down noticeably. Why is this? Why wouldn't it still be faster to have more cars pushing?


title: Merry go round physics problem?

text: So I have the following statement. "A merry-go-round is spinning with a fixed angular speed. As a person is walking towards the edge, the force of static friction must increase in order for the person not to slide off." I have no clue why is it true. I tried to figure out the problem with the following equations: rω = v (mv^2)/r = Tension but I can't seem to get past the tension. I guess what it causes the person to slide off would be the tangential acceleration (and therefore a friction force is needed), but I can't find a way of getting to this last variable with the previous mentioned equations... I need help at figuring out specifically what (if I'm not right about the tangential acceleration) causes the person to slide off and how is it affected by an increase in the radius (as the person walk towards the edge.)


title: What is the variation of Gauss-Bonnet term a total derivative of?

text: What is the variation of Gauss-Bonnet term total derivative of? i.e. Variation of Gauss-Bonnet combination $= \nabla_{\mu} C^{\mu}$. What's $C^{\mu}$ in 4-dimensions?


title: Dimensional analysis

text: I was recently introduced to dimensional analysis and I wanted good references for learning the ideas behind it and representation of the natural world. I'm a grad student in biology. I don't have much of a physics background.


title: Calculating the energy of an electron given the wavelength

text: Okay, so I know that the wavelength of an electron is `5e-7m` and I am asked to calculate its minimum velocity and hence minimum energy. Calculating the minimum velocity via $\lambda = h/p$ gives `1454.67 m/s`. Next step is to calculate the associated energy. The actual answer given to me is $E_{min} = 1/2 mv^2 = 9.71e-25J$, whereas I decided to calculate it using: $E_{min} = hf = hv/\lambda_{max} = 1.93e-24J$. Why does my answer differ?


title: 3d Ising Fixed point on general space manifold?

text: The headline question: Is it known how to construct an equivalent of the 3-D Ising Fixed point theory on an arbitrary 3-D manifold? Or any non-trivial d > 2 fixed point? The answer is maybe as simple as a careful conformal mapping of the action . I don't even know whether this is a hard question not. Let me try to explain what I mean by a fixed point theory on an arbitrary manifold and why I think such a thing should exist. By a fixed point theory I mean a QFT without a lattice scale, or put differently a theory with an actual continuum limit, with correlation function at all distances defined. If we have such a fixed point theory we can say it is the "Ising" if on small scales correlation functions look like the usual Ising correlation functions. We know how to use RG to construct fixed point theories on the $d$ dimensional plane. But the usual technique relies on the Fourier transform, and is therefore specific to the plane (or maybe other symmetric spaces). Does a fixed point theory exist for every manifold? It's true in $d = 2$, where I can use analytic maps to construct geometrically non-trivial theories. We can do it with topological QFTs. These are obviously special. It feels like scaling ideas should work - most crudely put a lattice on your manifold and then repeatedly refine the lattice while _locally_ scaling the parameters, so that large scale correlation functions remain constant. But I don't exactly see how geometric information should be encoded in this process and one has a tremendous amount of freedom in local scaling. I haven't figured it out. Maybe the process forces you somehow to a trivial theory or a topological blow up. Lastly, I haven't specified exactly what geometric information should go on this manifold. The naive assumption is that a conformal geometry is sufficient. But I honestly don't know.


title: Uniform Circular Motion

text: ![enter image description here](http://i.stack.imgur.com/6EXpa.png) Why does the object not go inward, into the circle if the acceleration is inward? I think its because the velocity to outward? So they sort of cancel each other out? But if the speed is kept constant and the acceleration is inward, won't the object eventially move inward? Also, what keeps the direction of acceleration always point inward?


title: Phase Plot for Harmonic Oscillator

text: This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\omega t)$: ![enter image description here](http://i.stack.imgur.com/v8EOE.png) However now that I am thinking about it, the phase is actually only the $\phi=2\pi f t$ part and therefore the real plot should look like this: ![enter image description here](http://i.stack.imgur.com/syt8f.png) What is the problem with the second plot? Shouldn't the second one be the phase plot? Note: Both of the plots are for the oscillator with $f=0.25$ and $t\in[0,10]$.


title: The radius of the universe and time

text: Ok - fair warning - Non-physicist asking dumb-assed questions here again. I've been reading a lot of Einstein, Feynmann, Ferris etc. I'm just loving this stuff. But I suddenly found myself looking at the 4-D sphere concept of our expanding universe, as well as the distortion of spacetime caused by objects with mass and so on, and suddenly thought - this 4th spacial dimension (that we cannot comprehend) through which our universe is expanding - is this actually the dimension of time? Is the time dilation experienced closer to bodies with mass not a literal curving of the skin of our 4-D sphere in toward its center? So at that point in spacetime the radius is increasing more slowly compared to the radius further from the mass? I'm afraid my brain just keeps failing when I try to incorporate other concepts from GR into this view - such as time dilation with increasing speed. However the shortening of length of an object in the direction of travel kind of makes sense there as well but only if there was a similar unilateral distortion near bodies of mass. And then I got to thinking about black holes and their extreme curvature - are they literally stopping universal expansion at their point of singularity such that deltaR=0 and therefore time stops? Or do they even stretch right back to R=0? Ok, enough of my sillyness - can someone put this into perspective for me?


title: The orthonormality conditions for plane waves

text: The notes I'm using for QFT give an orthonormality condition for the following plane wave: \begin{align*} u_{r,k}(x) = \frac{1}{\sqrt{V}} \varepsilon_r(\mathbf{k})e^{ik \cdot x} \end{align*} (where $\varepsilon_r(\mathbf{k})$ is a linear polarisation vector satisfying $\varepsilon_r(\mathbf{k}) \cdot \varepsilon_s(\mathbf{k}) = \delta_{rs}$ and $\varepsilon_r(\mathbf{k}) \cdot \mathbf{k} = 0$.) This condition is: \begin{align*} \int u_{r,k}(x) u_{s,k'}^*(x) d^3x = \delta_{k,k'}\delta_{r,s} \end{align*} So far so reasonable. * * * Now, later on in the the notes I meet a similar plane wave: \begin{align*} u_k(x) = \frac{1}{\sqrt{V}} e^{i(k \cdot x - \omega_k t)} = \frac{1}{\sqrt{V}} e^{-ik_\mu x^\mu} \end{align*} I am trying to construct an othornomality condition, but this time with two positions, $x$ and $x'$: \begin{align*} \int u_k(x) u_{k'}(x')^* d^3x = \frac{1}{V}\int e^{i(k \cdot x - k' \cdot x')}e^{-i(\omega_k - \omega_{k'})t} d^3x \end{align*} Now I'm a little unsure about this. My gut says it becomes $\delta_{k,k'}\delta_{x,x'}$ but I am worried about the legitimacy of this, and overall I am confused about standard procedures for these kinds of integrals.


title: Physical significance of Taylor and Maclaurin series - What is the significance of defining a Maclaurin series in Mathematical Physics?

text: In physics, usually Taylor series is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential energy as a Taylor expansion with respect to the coordinate about the equilibrium configuration. I am confusing this with the Maclaurin series in mathematics. If the equilibrium position is zero, it is Maclaurin series. We have the freedom to fix the equilibrium position at zero. In short, what is the physical significance of Maclaurin series? or else, what is the need of defining a Maclaurin series? Any physical interests on Maclaurin series? One more question I would like to add. If a quantity is changing with respect to its coordinate and doesn't have an equilibrium at any of its configuration, then how we can express it? Taylor series can be applied only if the quantity is changing about an equilibrium configuration?


title: If particles can find themselves spontaneously arranged, isn't entropy actually decreasing?

text: Take a box of gas particles. At $t = 0$, the distribution of particles is homogeneous. There is a small probability that at $t = 1$, all particles go to the left side of the box. In this case, entropy is _decreasing_. However, it is a general principle is that entropy always _increases_. So, where is the problem please?


title: Does entropy really always increase (or stay the same)?

text: ![enter image description here](http://i.stack.imgur.com/FVjLL.jpg) Consider this image. If the big (grey) molecules were all to spontaneously move to the left, and the small ones were to move to the right, there would be an increase in order. While unlikely, wouldn't this mean that entropy decreases?


title: Second law of Thermodynamics: Why is it only "almost" always true that entropy is non-decreasing?

text: Wikipedia - Second law of thermodynamics: > ...the entropy of any closed system not in thermal equilibrium **almost** > always increases. I understand that the second law of thermodynamics is based on the statistical unlikelihood of fast-moving molecules to aggregate. However, if it is only "almost always" then why is the phenomenon stated as a law? Is it because we have not yet observed the unlikely aggregation?


title: How light causes increase in temperature

text: Temperature is the measure of movements of atoms. So if something is said to have high temperature it means that its atoms are moving fast or have high KE energy. There are basically two ways heat can be travels from on object to another. 1) conduction. 2) Radiation. My question is how does radiation causes increase in KE energy of atoms, when photons only interact with electrons? Or how does electrons going up in energy state translates to atom having more KE energy.


title: Pair production at high laser intensity?

text: 1. Using high laser intensity to produce electron-positron pair, is it still required interaction with nucleus as is the case when gamma rays are used?   2. What causes the pair creation ?


title: Can the coefficient of static friction be less than that of kinetic friction?

text: I was recently wondering what would happen if the force sliding two surfaces against each other were somehow weaker than kinetic friction but stronger than static friction. Since the sliding force is greater than the maximum force of static friction, $F > f_s = \mu_s F_N$, it seems that the surfaces should slide. But on the other hand, if the force of kinetic friction is greater than the applied force, there'll be a net force $\mu_k F_N - F$ acting _against_ the motion, suggesting that the surfaces should move opposite to the direction they're being pushed! That doesn't make sense. The only logical resolution I can think of is that the coefficient of static friction can never be less than the coefficient of kinetic friction. Am I missing something?


title: On-shellness of photons

text: In principle, we could describe all physics without EM fields (or photons), as they are mainly a useful tool to describe "action at distance" (which does not mean instantaneous) between charged particle. In some sense, I could always integrate out photons, and describe only electrons and not have any changes in the observations (as detecting photons is done by observing how the motion of charged particles changes). With this picture, all photons are "virtual" (in the QFT sense), and we could expect them to always be off-shell. My question is : why do we expect that photons that are "really emitted" (in a sense that should be made clearer, but that might mean "travelling very far before being absorbed") are always on-shell (i.e. having $E=p$)? Is it because the propagator of an off-shell photon decays very fast and therefore these photons can not interact with long distance charges ? We could imagine that two very distant electrons (say, in two different galaxies) "scatter" each other (what we usually call "seeing a distant star") with off-shell photons. Why is it not so ? Is it just because the probability of this event is very small ?


title: Scattering from a box potential of width $L$ doesn't reproduce a step potential in the limit $L \rightarrow \infty$

text: Consider the scattering of a quantum particle in one dimension, caused by a step in the potential (this appears in many undergrad level QM books): $$ V(x) = \begin{cases} V_1 & x<0 \\\ V_2 & x>0\end{cases}. $$ The particle is incident from the left, so it's wavefunction is: $$ \psi(x) = \begin{cases} e^{i k_1 x} + r e^{-i k_1 x} & x<0 \\\ t e^{i k_2 x} & x>0\end{cases}, $$ where $k_i =\sqrt{2m(E-V_i)}/\hbar$.   Matching the wavefunction and its derivative at $x=0$ gives: $$ r = \frac{k_1-k_2}{k_1+k_2} ~~~;~~~ t = \frac{2 k_1}{k_1+k_2}.$$ Now we put another step in the potential at some distance $L$, which makes it a box potential: $$ V(x) = \begin{cases} V_1 & x<0 \\\ V_2 & 0<x<L \\\ V_1 & L<x\end{cases}. $$ We solve this in a similar manner as before, with the wavefunction: $$ \psi(x) = \begin{cases} e^{i k_1 x} + r e^{-i k_1 x} & x<0 \\\ a e^{i k_2 x} + b e^{-i k_2 x} & 0<x<L \\\ t e^{i k_1 x} & L<x \end{cases}. $$ Matching the wavefunction and its derivative at $x=0,L$ gives: $$ r = \frac{k_1^2-k_2^2}{k_1^2+k_2^2+2 i k_1 k_2 \cot{(k_2 L)} } ~~~;~~~ t = \text{(something)}.$$ **How come the second scattering problem doesn't reproduce the first scattering problem in the limit $L \rightarrow \infty$?**   I'm looking only at the value of $r$. I send a particle in, it scatters, and I get something back with an amplitude $r$. It seems unphysical that if the potential changed at $x=L$, it changes the scattering at $x=0$, no matter how far $L$ is.


title: How is a photon emission energy balance maintained?

text: How is a photon emission energy balance maintained ? If an electron is in motion across space at 260,000 kilometers per second, and it releases a photon in that same direction of travel, the photon will be in motion across space at the speed of 300,000 kilometers per second, thus there will be a difference of 40,000 kilometers per second. If another electron that is traveling along side the first electron at 260,000 kilometers per second, and it releases a photon in the reverse direction, the photon will be in motion across space at the speed of 300,000 kilometers per second, thus there will be a difference of 560,000 kilometers per second. In both cases each electron will lose energy when releasing a photon. Is the loss of energy equal in both of these cases ? ( assuming each photon is is identical in its properties of course )


title: Rotational behavior of objects in zero-g with forces

text: I would like to know how this object would rotate in free space out of curiosity. Below is a diagram of the object. It is of uniform density and has a center of mass in the center of the object, denoted by the green cross. On each long end of the bar, on opposing sides, are thrusters. They are placed at exactly the same distance from the ends and from the center of mass. They provide the same amount of force. They are labeled "A" and "B" in the diagram, and the red arrows show which direction the thruster fires. Edit: Assume that the thrusters are ideal and do not lose fuel/mass over time. ![Bar with thrusters](http://i.stack.imgur.com/yZBqF.png) If both thrusters are fired simultaneously, I believe the bar will spin about its center of mass, and there will be no translation. My questions are about what happens when you fire only one of the thrusters. # Question 1 What happens if you fire only thruster B? I think the bar will rotate, but not about it's center of mass, and not it's endpoints. I think it will also translate, but not sure how. # Question 2 What equation(s) govern the motion of the body in this scenario? I am almost certain that center of mass is a variable in this, because if you put a thruster directly in line with the center of mass, you would have no rotation at all, only translation. # Question 3 If left to spin to high velocities, will the object be "locked in" on it's translation vector? Because it is spinning so fast, the thrust wouldn't have a significant net force in any particular direction, so the object would forever travel at some particular velocity and heading. # Question 4 What happens if you fire thruster B at 100% force, and thruster B at 50% force? This question may already be answered by one of the other questions above. Edit 2: I wanted to see what this motion looked like, so I coded something up in an attempt to model it. I only implemented angular and translational acceleration/velocity from the equations Physics Llama provided.


title: Torque Required For a Motor to Move an Object on Wheels?

text: I've been attempting to calculate how much torque a motor needs to produce in order to start a stationary object on wheels moving. (The torque is being applied to the rear 2 wheels, the front 2 are on bearings.) I keep seeing Torque = Force * Radius (of the torqued wheel). I can't figure out how to calculate the Force in this equation though, so that I may find the Torque. The radius of the wheel is 3cm. The weight of the entire object (including wheels and everything else) is 5kg. I don't need an incredibly accurate result, so I haven't even been trying to factor in friction from the non-powered front wheels' bearings. I tried something with a friction coefficient and gravity, but the more I read the less I believe that my calculation was correct. Can anyone point me in the right direction?


title: Glass properties

text: This is a question about the properties and strength of glass. I know that glass is an amorphous solid but I am not sure if that is relevant to my question. My question is that if there's no physical cracks or blemishes, is it possible for glass to be damaged? For example, metal can be damaged by a sharp hit or force to it, as it is evident in a dent. For glass however, if a sharp force hits it and there is no evident crack or scratch, is the glass still in the same condition as it was before it got hit. For instance if I had something in contact with the glass that repeatedly struck it, would it over time weaken (ignore scratching)? (I know that with metal, each successive blow will slightly deform it)


title: Angular Momentum Operator

text: I'm looking at a review question I was given and it quite frankly has me stumped. "Using matrix representations find $L^{3}_{x},L^{3}_{y},L^{3}_{z}$ and from these show that $L_{x}, L_{y},L_{z}$ satisfy the same algebraic equations." What has me stumped is the $L^{3}_{x},L^{3}_{y},L^{3}_{z}$, I'm not even sure how it makes sense to talk of the cube of L? Any pointers in the right direction here would be incredibly helpful!


title: Number of Components of a Spinor

text: I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral representation of the Lorentz group. Now I've been reading that we can derive spinor representations by looking at the universal covering group of the proper orthochronous Lorentz group, which is $SL(2,\mathbb{C})$. Now $SL(2,\mathbb{C})$ acts on $\mathbb{C}^2$ by the fundamental representation. My book (Costa and Fogli) then calls elements of $\mathbb{C}^2$ spinors. But the second type of spinors have a different number of components to the first! What is going on here? Could someone clearly explain the link between these two concepts in a mathematically rigorous way? I come from a maths background of group theory and topology, but don't know much representation theory at present. Many thanks in advance!


title: When can a natural satellite be in polar orbit around it's primary?

text: A circular/inclined to the equator/non-polar orbit is typical for natural satellites; apparently a consequence of the energy/velocity requirements. This is applicable to natural satellites around planets, planets around Sol ... and perhaps upwards on the scale.   * When can a natural satellite possibly be in polar orbit around it's primary?    * Are there any such observations on record?


title: Sinusoidaly Driven Two-Level System (TLS)

text: I'm trying to solve the driven Two-Level System (TLS or qubit) question using a Fourier transform of the Schrodinger equation (SHE), but I'm getting stuck on solving the equation. Given Hamiltonian $$H=\left( \begin{array}{cc} -\frac{\omega _0}{2} & \frac{1}{2} V e^{i t \omega _D} \\\ \frac{1}{2} V e^{-i t \omega _D} & \frac{\omega _0}{2} \\\ \end{array} \right)$$ and plugging into SHE: $$i\frac{ d }{\text{dt}}\left( \begin{array}{c} c_a(t) \\\ c_b(t) \\\ \end{array} \right)=H \left( \begin{array}{c} c_a(t) \\\ c_b(t) \\\ \end{array} \right)$$ I get a set of two coupled 1st Order differential equations which i then Fourier transform and use the derivative rule and the shift rule to get: \left( \begin{array}{c} -\omega c_a(\omega )=\frac{1}{2} V c_b\left(\omega -\omega _D\right)-\frac{1}{2} \omega _0 c_a(\omega ) \\\ -\omega c_b(\omega )=\frac{1}{2} V c_a\left(\omega +\omega _D\right)+\frac{1}{2} \omega _0 c_b(\omega ) \\\ \end{array} \right) If I shift the $c_b$ equation: $$-\omega c_b\left(\omega -\omega _D\right)=\frac{1}{2} V c_a\left(\omega _D-\omega _D+\omega \right)++\frac{1}{2} \omega _0 c_b\left(\omega -\omega _D\right)$$ and solve for term i can plug back into the first equation: $$c_b\left(\omega -\omega _D\right)=\frac{V c_a(\omega )}{2 \omega +\omega _0}$$ and solve for $c_a$: $$c_a(\omega )=\frac{V^2 c_a(\omega )}{(2 \omega )^2+\omega _0^2} $$ So it looks like a lorentzian function with a width of the resonance frequency and centered at 0 multiplies $c_a$, but $c_a$ cancels unless it is 0? or some delta function? This is where I do not understand how to proceed to solve the problem further? Is part of the challenge that I am solving the problem without boundary conditions, just trying to find the steady state?


title: Finding interplanetary flight trajectory using calculus of variations?

text: Consider two orbits $x(t),\space y(t)$ representing the origin and destination for some spaceflight of interest. These could be, for example, cycloids describing LEO and another orbit circling, say, the moon or Mars. Suppose I want to leave at $x(t_i)$ and arrive at $y(t_f)$, departing and joining those orbits 'smoothly', so their derivatives $\dot{x}(t_t)$ and $\dot{y}(t_i)$ are also imposing boundary conditions on my trajectory. To accomplish this, I allow for rocket engines to be fired, most notably while leaving one orbit and decelerating into the other, but also at any time mid- flight. On the other hand, I restrict the total spent fuel along the trip, or the total "delta-v" changes, or some equivalent. What would be the best way to solve for this trajectory, including the firing pattern of the engine? What I thought about was devising some adjusted Lagrangian that includes the impulse caused by the engines and adding to it a Lagrange multiplier imposing fuel consumption. This is then used to derive new equations of motion. For example, I would consider an action such as: $$S=\int_{t_i}^{t_f}L[q,\dot{q},F_e,t]dt+\lambda\left(J-\int_{t_i}^{t_f}|F_e(t)|dt \right)$$ Where $q(t)$ is the desired orbit, $F_e(t)$ is the force exerted by the engines and unknown _a priori_ , and $J$ the total impulse supplied by them along the journey$^\dagger$. Now derive EOMs and solve for $q,F_e$. Eventually, I'd like to explore if this 'formalism' produces solutions (trajectories) that can be inverted, solving either for $J$ (allowing minimization of fuel) or $t_f,t_i$ (minimizing flight time, picking launch window). Where I am struggling is how to modify said Lagrangian. Should $F_e$ be considered as another generalized coordinate? If so, then a kinetic term for it is needed. Should it be treated as another term in the potential energy? Maybe it should be restated as a force originating from enforcing some constrained orbit? Maybe it's actually another 'Lagrangianesque' function of $q, \dot{q}$ and perhaps $\ddot{q}$ as well? $^\dagger$Admittedly, the instantaneous thrust is not constrained, but it's a start. **EDIT** Following the reference supplied in the answer below, I managed to make some progress with the problem. This was done by "inverting roles", so now the "Lagrangian" is the engine force and a Lagrange multiplier enforces the Newtonian equation of motion: $$S=\int_{t_0}^{t_f}dt\left( F^2 + 2\lambda(t)[m\ddot{x}+m \nabla \phi-F] \right)$$ Now, taking the differential of $S$ with regards to $x, F$ and $\lambda$, and doing the routine integration by parts on the variation $\delta\ddot{x}$, we obtain the following equations of motion: $$(1) \space F=\lambda$$ $$(2) \space m\ddot{x} = -m\nabla\phi +F$$ $$(3) \space \ddot{\lambda} = - H_{\phi}*\lambda$$ Where $H_{\phi}$ denotes the Hessian matrix of $\phi$, i.e. $(H_{\phi})_{ij}=\partial_i\partial_j\phi$. These equations can know be solved with given boundary conditions $x(t_0),x(t_f),\dot{x}(t_0),\dot{x}(t_f)$, which proved to be pretty straight- forward to do in MATLAB. The next thing on the agenda is making $t_0, t_f$ free variable subject to optimization as well, so we can find the best launch window. This is proving to be a bit more difficult. Introducing these extra degrees of freedom requires more Lagrange multipliers and produces algebraic equations for each of them and the multipliers. The major difficulty is that the equations of motion above can't be easily integrated and then solved for the time $t$, so they can be evaluated at the launch and landing times. Any suggestions?


title: Why, for a spin-½ particle, are the possible outcomes of measuring spin projection along any direction the same?

text: If one measures the projection of spin of a spin half particle along the $x$ axis one will always get $\pm\tfrac12\hbar$. Measuring it along the $y$ axis one will always get $\pm\tfrac12\hbar$. Measuring it along the $z$ axis one will always get $\pm\tfrac12\hbar$. Measuring it along "any" axis one will always get $\pm\tfrac12\hbar$. Why are the possible outcomes the same in all directions?


title: Question about infinite sum in quantum field

text: I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } s=1,$$or $$\sum_{n=1}^{\infty}\frac{1}{n^s} = c \pi \text{,when } s=1 \text{,where } c \text{ is a rational number ?}$$ I get a similar result in mathematics by analogue, I suspect the result may have some interpretation in physics.


title: How can I stand on the ground? EM or/and Pauli?

text: There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground. But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle. So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both?


title: Where does the Pauli Repulsive Force come from that counteracts the attraction between atoms and ions?

text: I'm learning about such things as ionic and covalent bonds, and the reason given for the ionic bonds is electrostatic attraction. However, if that were true, then the two ions would accelerate toward each other and shortly collide with each other. There must be a force opposite the electrostatic force that is _repelling_ the ions, with exactly the same magnitude so as to cancel out the electrostatic force. _(Edit: I'm actually referring to the ionic bond between cations and anions in this first paragraph, like sodium and chloride ions. Below is what I found by researching this topic, dealing mainly with covalent/molecular bonding.)_ The following quote is from _The Mechanical Universe_ : > For example, the potential energy of a pair of Hydrogen atoms has a position > of stable equilibrium. The potential energies of interaction between atoms > are due to the electrical forces between them, causing attraction when > they're far apart. At smaller distances, they resist being squeezed > together. The result is an equilibrium position where attraction and > repulsion are in perfect balance. At that position, they bind in a Hydrogen > molecule. I understand that on a larger (planetary) scale, there is a centrifugal potential that is inversely proportional to the distance $r$ that adds up with the gravitational potential to create this graph: ![enter image description here](http://i.stack.imgur.com/6TG03.jpg) I'm looking for an analogue to this on the molecular scale. I've recently found out about the Lennard-Jones Potential, which gives this equation and curve:$$V_{LJ}=\epsilon \left[ \left(\frac{r_m}{r}\right)^{12} - 2\left(\frac{r_m}{r}\right)^6 \right]$$ ![enter image description here](http://i.stack.imgur.com/T8eJF.png) In the wikipedia article, it says > ...the repulsive term describes Pauli repulsion at short ranges due to > overlapping electron orbitals Later on it is noted that > The repulsive term has no theoretical justification. So then this is my question: **What is the nature of the "Pauli repulsive force"?** If it's possible, I would appreciate an answer with cited sources. Edit: After some more research, I've found that attraction between two hydrogen atoms (which are both neutral) comes from Induced Dipole-Induced Dipole interaction AKA London Dispersion Forces. Collectively, the attraction and repulsive forces at the molecular scale are dubbed the Van der Waals forces. The wiki on Van der Waals forces mentions _A repulsive component resulting from the Pauli exclusion principle that prevents the collapse of molecules._ I would like to learn more about how the Pauli exclusion principle manifests itself as a repulsive force between any two atoms/ions.


title: What does it mean for two objects to "touch"?

text: If you've ever been annoyingly poked by a geek, you might be familiar with the semi-nerdy obnoxious response of > "I'm not _actually_ touching you! The electrons in the atoms of my skin are > just getting really _close_ to yours!" Expanding on this a little bit, it seems the obnoxious geek is right. After all, consider Zeno's paradox. Every time you try to touch two objects together, you have to get them halfway there, then quarter-way, etc. In other words, there's always a infinitesimal distance in between the two objects. Atoms _don't_ "touch" each other; even the protons and neutrons in the nucleus of an atom aren't "touching" each other. So what does it mean for two objects to touch each other?   1. Are atoms that join to form a molecule "touching"? I suppose the _atoms_ are touching, because their is some overlap, but the subatomic particles are just whizzing around avoiding each other. If this is the case, should "touching" just be defined relative to some context? I.e, if I touch your hand, our hands are touching, but unless you pick up some of my DNA, the molecules in our hands aren't touching? And since the molecules aren't changing, the atoms aren't touching either?    2. Is there really no such thing as "touching"?


title: Falling through the ground

text: I do not know much about physics but I know that according to Newtons third law of motion when we walk we are pushing the ground down but the ground is pushing us up. What force is making the ground push us up. How come gravity doesn't pull us and the ground down.


title: Proof that the ground pushes you up

text: It has been said that when you try to jump you are exerting your force on the ground and the ground _pushes you_ up. I was wondering if this was an over- simplification or not. If so, what's the full explanation? Otherwise, I would like to know the derivation of this proof. For clarification: If I jump and the ground is "pushing me up" (from a normal force), then assuming that this is not a simplification and is truly what happens, _why_ is this true? My current hypothesis is that it has something to deal with electromagnetism (pushing two repelling magnets together; how atoms swerve to avoid collisions, for example).


Do particles ever touch each other during a collision? My understanding is that they can get really close to each other but never actually touch, is that true? Another thing I've read about is, the closer the particles get to each other the greater the density of virtual particles until they reflect or particle pair creation occures due to the high energy density. If true, how does that work?


title: What kind of force is it when you push a door shut?

text: > **Possible Duplicate:**   >  How can I stand on the ground? EM or/and Pauli? It probably seems like a silly question to a physicist, but I'm from a pure maths background and don't know an awful lot about the real world. So what kind of force is it? It can't be gravitational because that attracts matter together. And it's not like the atoms in your hand are physically colliding with the atoms in the door, because there's mostly empty space and your hand might as well go straight through the door. I guess it's fallen out of fashion to think of things as made of 'particles' these days anyway, at these small scales I hear more about fields and coupling between fields. I'm doubtful it's anything related to strong or weak nuclear force. I've heard of van der Waals force, but didn't understand what it meant. So I guess that leaves.. an electric force or magnetic force? But my hand and the door aren't magnets. And we don't have much of a net charge, otherwise maybe I could push the door from a distance. What does it actually mean to 'touch' the door, push, and exert a force on it?


title: Is the electromagnetic force responsible for contact forces?

text: It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests? Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on contact forces seems to agree: > Molecular and quantum physics show that the electromagnetic force is the > fundamental interaction responsible for contact forces. An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what _force_ is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such. I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual "source needed" objections, and that this topic is either misunderstood by or poorly covered in the standard texts. (The argument can be found here). Which way is it?


title: The relation between the speed of light and the Big Bang Theory

text: I would like to know how much of the Big Bang theory is dependent on the constancy of the speed of light. **P.S.** : It might be guessed that I am asking this because of the recent CERN news. Yes, of course, I am. I was having some chat over this topic and I hit a wall before I could come to an answer for this because of my not-more- than-school-level-Physics knowledge. Thank you.


title: Why is the crystallography restriction obeyed?

text: The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of $rot_\theta(x)-x$ is less than the magnitude of $x$ if $\theta$ is less than $\pi / 3$. (So if there's a rotation by less than $\pi/3$, it will fail to be a lattice, since we can always get smaller and smaller elements.) That sounds fine to me, and I get why real crystals would have the requirement that all atoms be some epsilon away from each other. I don't understand why nature requires the lattice to be closed though - obviously real crystals aren't completely closed under addition, or else they would be infinitely large. So why does the crystallography restriction limit the symmetries of real crystals? (Or does it not, and this application of the theorem is made up by math teachers to convince their students that math is useful?)


title: weight function and the metric

text: The weight function comes from Dirac's book, PRINCIPLES OF QUANTUM MECHANICS. On page 66,he says that sometimes it is more convenient not to normalise the eigenvectors, i.e. $$\langle\xi_1'...\xi_u'|\xi_1''...\xi_u''\rangle=\rho'^{-1}\delta(\xi_1'-\xi_1'')...\delta(\xi_u'-\xi_u'')$$ Thus we have $$\iiint|\xi_1'...\xi_u'\rangle\langle\xi_1'...\xi_u'|\rho'd\xi_1'...d\xi_u'=1$$ Here Dirac says that $\rho'd\xi_1'...d\xi_u'$ is the weight attached to the volume element of the space of the variables $\xi_1',...,\xi_u'$. If we consider the space of the variables $\xi_1',...,\xi_u'$ as a Riemann manifold $M$ with metric $g$, then the volume element of $(M,g)$ is $\sqrt{\det(g)}d\xi_1'...d\xi_u'$. Does the two volume element equal? i.e. do we have that $\rho'=\sqrt{\det(g)}$? In his book,he then gives an example where the $\xi$'s are the polar and azimuthal angles $\theta,\phi$ giving a direction in three-dimensional space. In this case we have $\rho'=\sin{\theta'}$. And if we calculate the $\sqrt{\det(g)}$ of the unit sphere(in coordinates $\theta,\phi$),it is $\sin{\theta'}$,too! Is it just coincidence? Or is there an explanation? Would someone be kind enough to give me some hints on this? Thank you very much!


title: Graphene space elevator possible?

text: I just read this story on MIT working on industrial scale, km^2 sheet production of graphene. A quick check of Wikipedia on graphene and Wikipedia on space elevator tells me > Measurements have shown that graphene has a breaking strength 200 times > greater than steel, with a tensile strength of **130 GPa** (19,000,000 psi) and > The largest holdup to Edwards' proposed design is the technological limit of > the tether material. His calculations call for a fiber composed of epoxy- > bonded carbon nanotubes with a minimal tensile strength of **130 GPa** (19 > million psi) (including a safety factor of 2) Does this mean we may soon actually have the material for a space elevator?