title: The Bhatnagar-Gross-Krook (BGK) approximation of the collision integral

text: Bhatnagar, Gross and Krook (BGK) proposed a relaxation term for the collision integral $ Q$ as follows $$J = \frac{1}{\tau} (f^{eq} - f)$$ where $f^{eq}$ is the distribution at equilibrium. $Q$ has following property: $$\int Q \psi_k d\mathbf{v} = 0$$ where $\psi_0 = 1,\hspace{.5cm} (\psi_1, \psi_2,\psi_3) = \mathbf{v}, \hspace{.5cm} \psi_4 = \mathbf{v}^2 $ Now we say that this term should have the following constraints: a) `It should have a tendency towards Maxwellian distribution` What is meant by this expression? My understanding is that f should be close to Maxwell distribution at all times. Is this correct? if not what is the correct interpretation of this constraint? b) `J Should conserve collision invariants of Q` i.e. $\int J \psi_k d\mathbf{v}d\mathbf{x} = 0 $ where $\psi_k$ is one of the collision invariant. Note that now integral is performed over both velocity and spatial space instead of velocity alone as in case of $Q$. $\int J \psi_k d\mathbf{v} = 0 \implies \int J \psi_k d\mathbf{v}d\mathbf{x} = 0$ but $\int J \psi_k d\mathbf{v}d\mathbf{x} = 0 \not \implies \int J \psi_k d\mathbf{v} = 0$. So why don't we put condition on J as $\int J \psi_k d\mathbf{v} = 0$? By adding extra spatial space in the integral do we put less restrictions on J? if I substitute the first expression for J into the above constraint b), then it implies $\int f \psi_k d\mathbf{v}d\mathbf{x}$ = 0 but this implies that f should be already Maxwellian for this relation to be true! What am I missing here?


title: How is the current same on both the sides of a resistor?

text: I am wondering that how can current be equal on either side of a resistor as the work of a resistor is to resist the flow of electrons i.e. decrease the current. How is this possible?


title: Field of a parallel plate capacitor

text: Pretty much in all the diagrams I have seen the electric field of the parallel plates is depicted as in this image ![enter image description here](http://i.stack.imgur.com/YyLco.jpg) The field lines seems to start bending only after leaving the periphery of the plates, shouldn't the field lines be depicted as starting to bend as soon as we start to move away from the exact centre of the capacitor ?


title: Spilling a cup of coffee

text: I've observed this many times when I spill coffee, It gets attracted towards the cup or the vessel. why is this so? Which is the force acting? (Hope you understood my question).


title: Investigating damped Harmonic Motion in a Spring?

text: I'm going to conduct an investigation into the dampening of a spring. Essentially, what specific factors could be investigated? Currently I'm planning to investigate the effect of changing the mass on the end of the spring, the effect of using different spring constants, and the effect of changing the initial extension of the spring on the amount of dampening.


title: Complex conjugate of momentum operator

text: Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ $$\hat{p}e^{ipx}=pe^{ipx}$$ Taking the complex conjugate of the equation, $${\hat{p}}^* e^{-ipx}=p^* e^{-ipx}$$ Because momentum eigenvalue is real, $p^* =p$. Thus $${\hat{p}}^* e^{-ipx}=p e^{-ipx}$$ Now consider, $$-\hat{p}e^{-ipx}=pe^{-ipx}$$ From these two equations we see that ${\hat{p}}^* =-\hat{p}$.\ Now consider the matrix representation of the momentum operator. In the basis of the momentum eigenstates, the momentum operator matrix (infinite dimensional) is diagonal and the diagonal elements represent the eigenvalues of the momentum operator just as in the case of other finite dimensional operators. This means that the complex conjugate of the $p-$matrix is $p-$matrix itself. However we saw from the above logic that the complex conjugate should be negative of the $p-$matrix.\ I can't see where the problem is!!


title: What are you studying when you study a Harmonic Oscillator in QM?

text: This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The Hamiltonian is presumed. Creation and annihilation operators are introduced, and consequently, the HO transitions from energy state to energy state. My question arises due to a quote from the same text that I previously asked about in this Phys.SE post: > States of well defined energy are unphysical and never occur in Nature. They > are incapable of changing in any way, and hence it is impossible to get a > system into such a state. Of course the math is beautiful and rigorous, but are we studying a pedagogical model, or is there a context of reality? Also, in that the creation and annihilation operators are not Hermitian, is this a clue that we are not dealing with real observables. The source is notes by James Binney (Oxford) http://www-thphys.physics.ox.ac.uk/people/JamesBinney/qb.pdf which have subsequently been published as a text.


title: When the matter is heated does the photon absorbed by electrons of atoms or by atoms themself?

text: When the matter is heated does the photon absorbs by electrons of atoms or by atoms themself?


title: A question about the energy of turning on and off interaction adiabatically in QFT

text: I read a saying as follows: > _In a theory with no particles which decay and no bound states, the turning > on and off of the interactions merely serves to limit the effective range of > forces. In this case, turning the interactions on and off adiabatically will > not significantly affect the evolution of any state, so the initial energy > and the final energy as determined by the full Hamiltonian $H$ will equal to > that determined by the free Hamiltonian $H_0$, so we find that scattering > preserves the $H_0$-energy of the states:_ $$\left[H_0,S \right]=0.$$ I didn't prove this saying, but I think for infinite past the state is asymptotically free, for example $|p_1 p_2\rangle$, and because the full Hamiltonian in the infinte past is equal to free Hamiltonian in adiabatically switching off, in the infinte past the energy of full Hamiltonian is equal to free Hamiltonain $E_1+E_2$. While the whole process must be conservation of energy, the energy of full Hamiltonian is equal to free one, $E_1+E_2$, in any time. While according to Gell-Mann and Low theorem, the energy of free Hamiltonian $H_0$ is different from the full Hamiltonian $H$: $$E = E_0 + \langle\Psi_0 | H_\epsilon-H_0 | \Psi_\epsilon\rangle$$ This also seems to be right. Because in adiabatic process, energy crossing will not occur, that is, some energy level $E_n(t_1)$ of $H(t_1)=H_0+H_I(t_1)$ will become corresponding energy level $E_n(t_2)$ of $H(t_2)=H_0+H_I(t_2)$. Therefore the energy will different in different time. But it also seems to be weird, since energy is not conserved in interacting QFT. How to resolve this feud?


title: Deducing Young Tableaux from symmetries

text: I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to either$a_i's $ or $ b_i's$ 2) Total interchange of $a_i$'s with $b_i$'s are symmetric, that is $T^{a_1 \dots a_p;b_1 \dots b_p} = T^{b_1 \dots b_p;a_1 \dots a_p}$ Is this an irreducible representation of the full $S_{2p}$ group? if not how to decompose it? And what are/is the associated young tableau. In general I would be grateful if anyone can provide me with a reference on how to go about finding the associated tableau given arbitrary symmetries, such as cyclic symmetry etc.. I am guessing the usual procedure of the Littlewood-Richardson rule can be constrained further when additional symmetries are present? Thanks for you help. \---------------------- ADDITIONAL DETAILS -------------------------------- $\textbf {Question 1}$ An arbitrary Tensor $T$ is decomposed into irrep's, ie, it is projected by a young operator. I call one of the projectors, $P_1$. So the tensor corresponding to this particular irrep is given by $T_{irrep_1} = P_1.T$. Now I have some additional information about T, that is a symmetrisation of certain indices vanish. I take this symmetrisation operator to be $\mathbf S$. So the additional information I have is $\mathbf S.T =0$. Q1) Does this in general imply $\mathbf S.T_{irrep_1} =0$. For this to be true the two operators should commute, $[P_1, S]=0$. Is this true in general? If this is true, $S.T = 0 \implies S.T_{irrep_1} =0$. Let me make my question clear by taking an example. HERE EVERYTHING IS DONE IN SYMMETRIC BASIS. Consider $T^{ab,cd}$ symmetric in the comma separated indices. My question is that How do I go about decomposing this one with the additional constraint that $(abc)=0$. So I think of T as the product of two tensors with the symmetry {(ab)X(cd)} and decompose this. That would give me i) {ab,cd} ii){abc,d} iii){abd,c} iv) {abcd} (HERE COMMA SEPARATES THE ROWS). At this point I want to apply the additional constraint $(abc) =0$. and this where my confusion lies. I DO understand that these components do not mix and I can treat them sepearately. Q) If I assume the additional symmetry is inherited by the each component I will just apply this to i,ii,iii, iv and I indeed see only the component {ab,cd} satisfies. So it somehow works? Does that mean the two projectors always commute? Or It works by someother reason? Another way to see {ab,cd} is the only one is to decompose using two different ways (symmetric basis) 1) Decomposing ${(ab) X (cd)}$ 2) Decomposing ${ab,c X d}$ the right component/s should be whatever that is the intersection of these two operations and I see only the {ab,cd} one is shared by 1) 2). This method- I see why it works. but the previous method is puzzling for me. **Question 2** Given $T^{a[p],b[p]} $computing $1^p \otimes 1^p$ gives ${a[p]b[i],b[p-i]}$ with the young condition this tensor corresponds to one of the irreducible representation. I two questions regarding this, 2.1) For the same reason I was confused in question 1, Is saying the tensor T, satisfies condition two, ie, total pairwise interchange of a's and b's is symmetric, imply, the pairwise interchange of an irreducible component symmetric. 2.2) You said that anti-symmetrising over all b's in ${a[p]b[i],b[p-i]}$ yields a tensor satisfying condition one and lies in the same invariant sub- space. I AGREE. but is this the only tensor in this invariant sub-space that satisfies condition one? If I take for example $a[p-1]b[i+1],ab[p-i-1]$ this tensor lies in the invariant subspace. If I antisymmetrise over all b's do I get another independent tensor that satisfied condition 1?


title: Exercise Lorentz Transformations

text: I have 2 inertial frames, both are in relative motion along the x axis moving with constant velocity $v_x$. It is called with Standard figuration where the Lorentz transformations relate the coordinates of one system with another. In non-primate evolution of a system is observed vector, obeying the equation $\dfrac{d\xi^\mu}{d\tau}=\Omega^\mu\,_\alpha\xi^\alpha$ In this equation, $\tau$ represents the proper time or arc length and $\Omega^{\mu\nu}=-\Omega^{\nu\mu}$ is an antisymmetric operator like the electromagnetic field tensor, for the rest, arbitrary. Find the explicit form, matrix, operator $\Omega_{(Th)}$ for the primacy system: $\dfrac{d\xi'^\mu}{d\tau}=\Omega^{\mu}_{(Th)}\,_\alpha\xi^\alpha$ I don't know how to get the matrix form!


title: How to decompose a divergence operator

text: I am reading a paper, and see someone decompose a divergence operator as follows, could someone judge and see if it is correct? $$\nabla \cdot {\bf{v}} = \left( {{\bf{n}} \cdot \nabla } \right){v_n} + {\nabla _\parallel } \cdot {{\bf{v}}_t}$$ $\bf{v}$ is velocity vector field, and $n$ means normal, $t$ is tangent component. And I do not understand why ${\bf{v}}_t$ is a vector and $v_n$ is not?


title: Perfectly centered break of a perfectly aligned pool ball rack

text: Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a triangle) has every ball equally spaced apart and all balls touching all other appropriate balls. All balls are perfectly round. Now, imagine that the cue ball was hit along a friction less surface at the center axis for this triangle O-------<| and hits the far left ball of the rack dead center on this axis. How would the rack react? I would imagine this would be an extension of newtons cradle and only the 5 balls on the far end would be the candidates to move at all. But in what way would they move? Which ball, for instance, would move the furthest? Thanks


title: Why does water pouring from a glass sometimes travel down the side of the glass?

text: If you have a glass of water, say, three quarters full and you pour it at an angle of say, $45^{\circ}$ with respect to the the table, the water comes out of the glass and goes directly down towards the floor. However, when the glass is more full, or even three quarters full and the 'angle of pouring' is far less with respect the table, when the water comes out of the glass, rather than going straight down it kind of stays stuck to the glass and travels down the outside of it. Why does this happen? (I'm sure many of you have observed it yourself by accident and ended up making a mess).


title: The most stable way of standing in a bus

text: Here's what's bugging me for quite a long time. Imagine the every day situation, that you are standing in a bus with your back on wall having only limited space on the floor and no handle to hold. You are quite ok for acceleration effects in 3 of 4 directions. You will not fall behind, the wall is there. Right and left direction is relatively easily handled by a wide stance, but you are troubled about falling forward. Now how would you stand to be able to withstand the acceleration effects the most effectively? As it is probably a bit too vague (and maybe not completely understandable), let me elaborate it and put some numbers just to demonstrate the situation. Consider a normal human (average height and normal vertical weight distribution, if a notion like this exists) and let's state that the limitation on the floor space is defined as 50 cm from the wall. You cannot put your feet further, because someone is standing there. If we skip all exotic and unnatural stances, we are basically left with two extremes and a continuum between them. One extreme is that you are standing with your legs straight, second extreme is that you squat (let's consider here you will not go below the point, when your legs are in parallel with the floor) (and let's ignore how inconvenient this stance is). My physical imagination is too weak to at least guess, which of these stances is more stable, I just don't suspect seeing optimum somewhere between (but not sure about that either). I also don't have any idea, how it should be calculated and if it's something rather trivial, or something very hard. Here's a picture of what I was speaking about. Figure on the left is what I eventually do in the bus, figure on the right is what comes intuitively to me as better way (center of mass is lower and maybe the angles are more favorable). So.. is it better to bend the knees (even a bit)? ![two extreme stances](http://i.stack.imgur.com/gxlrW.png)


title: Is this double "double slit experiment" involving entanglement possible?

text: The experiment goes as follows: Put a particle emitter (photon, electron etc.) between a pair of double slits. The emitter launches pairs of particles that are entangled in such a way that if one goes through slit A the other goes through slit 2, if one goes through slit B the other goes through slit 1. ![enter image description here](http://i.stack.imgur.com/2TCUw.png) My prediction for this experiment is that if we put a detector on slit A (or any other slit) that can detect through which slit one of the particles went through then no interference patter will form on ether side of the emitter, if we don't place a detector we should see interference patterns emerge on both sides. Is this experiment possible? If so, was this or an equivalent experiment ever done?


title: GPS Satellite - Special Relativity

text: I'm going through an old relativity assignment, and I've been asked to calculate the time dilation for a satellite which orbits the earth in 12 hours at 26000km from the surface, and travels at a constant speed. The given radius of the Earth is 6400km. In the solutions, the author has calculated the velocity traveled by the satellite to be $\dfrac{3.24\times 10^7\times 2\pi}{12\times 3600} \text{ms}^{-1}\simeq 4500 \text{ms}^{-1} $. I'm fine with the calculation (if I've gone wrong with the numbers somewhere, I don't mind). What I'm not so OK with is that they proceeded to use $v=4500$ in the formula $\gamma=\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, which is needed to calculate time dilation. As far as I am aware, this formula only holds for observers whose relative velocity is $v$. However, in this case where there is an observer on the Earth's surface and a satellite moving in circular motion around the center of the earth, their relative velocity isn't constant!


title: Changing from potential to kinetic energy

text: During a conversation with a friend, I began to wonder if there's is a term for the transformation of potential energy to kinetic energy, and vice versa.Is there a term for the process of converting potential energy to kinetic?


title: How to calculate a collision which is partly elastic and partly inelastic?

text: _(For the purpose of this question, "calculating a collision" means: given the velocities and masses of two objects in a collision, figuring out the new velocities of both objects after the collision)._ I know how to calculate a totally elastic collision, and how to calculate a totally inelastic collision. But I don't know how to calculate a collision which is part elastic and part inelastic. I don't know where to start. Guidance will be appreciated. (Go easy on the math please).


title: What is the role of battery across a Parallel plate capacitor having equal charge?

text: Each plate of a parallel plate capacitor has a charge $q$ on it. The capacitor is now connected to a battery. Now, which of the following options are correct? > a) The facing surfaces of the capacitor have equal and opposite charges.   >  b) The two plates of the capacitor have equal and opposite charges.   >  c) The battery supplies equal and opposite charges to the two plates.   >  d) The outer surfaces of the plates have equal charges. Before the battery is connected, Applying Gauss Law, the inner surfaces of both the plates will have no charge on them, the outer surfaces will have charge $q$. After the battery is connected, The potential difference across the capacitor will gradually become equal to that of the battery .The inner surfaces will have equal and opposite charges according to $Q=CV$ where $V$ is the potential difference across the battery. But, what will be the charges on the outer surfaces of the two plates ? I am not clear about the role of battery in affecting the charges on the surfaces of the plates. Option a) Looks right - Not sure about the other options. Kindly explain what happens to the plates of the capacitor after the battery is connected ? Does a battery always produce equal amount of negative and positive charges or it may produce additional charges ?


title: How does particles gain electrical charges and repel each others? (electrostatic stabilization)

text: When I study electrostatic stabilization, I understand that the particles have same charge and thus repel others, this is how colloid is stabilize. But how does particles gain electrical charges and repel each other in the first place? I think that there are weak electrostatic force on the particle and when electrolytes were added, it gain more charges. Thank you.


title: How does a hinge affect the amount of a submerged material?

text: Suppose I have a rod that has a density of $X <1$. If I were to submerge that rod in water (density 1), I would expect $X$ of the rod to be below water and $1-X$ of it to be above water (simple buoyancy). But here's the catch. Suppose I attach one corner of the rod to a frictionless hinge, suspended close to the water. How will that affect what percent of the rod is above water in equilibrium now? My teacher told me it is NOT simply $1-X$. He also didn't specify how high above the water the hinge was, so I suppose that doesn't matter. Please tell me if more info is needed.


title: How would a realist interpretation of the Mermin-Peres square look like?

text: How would a realist interpretation of the Mermin-Peres square with counterfactual definiteness and the existence of states prior to measurements look like?


title: Have we found a Higgsino?

text: In supersymmetry, for each particle (boson/fermion), there is a symmetric particle which is a fermion/boson. The MSSM predicts five Higgs bosons: two neutral scalar ones (H and h), a pseudo-scalar (A) and two charged scalars. **Does this mean that there are five higgsinos?** My question arises because I read: > What is the difference between the SM Higgs and the supersymmetric H and h? > > Nothing. So I'm confused, **is it saying that H and h are the supersymmetrical parters of the Higss boson?**


title: Rotation matrix for a coupled spin system

text: For an angular momentum basis with magnitude $F$ and magnetic numbers $m_F\in [-F,F]$, the unitary matrix that will perform the Euler rotations is the Wigner-D matrix of order $F$. I have applied the Hyperfine splitting to a system for an alkali atom, and now my basis of the angular momentum is both $F=3,4$ (For Cs atom, basically). What is the rotation matrix in a combination of two bases? Thank you for any efforts.


title: Can quantum entanglement be a proof of a 4th spatial dimension?

text: What I know about quantum entanglement is almost nothing. What I know can be resumed to: "two particles created together have a correlated behaviour no matter the distance between them". This is often interpreted as a faster-than-light phenomenon because it is instantaneous. But let's consider a 4th spatial dimension (we then have x,y,z,w)   * 2 particles are created in the point x=0,y=0,z=0,w=0   * Those 2 particles move apart in the 3D space such as: Particle 1 is @ x1,y1,z1,w and Particle 2 is @ x2,y2,z2,w Note that they are at the same w coordinate. Can this be an explanation for the fact that the correlation is instantaneous? I am not considering that this is a revolutionary thing. I am sure that a lot o people already thought about this, but I want to understand it better.


title: Director field fluctuations in a liquid crystal in a magnetic field

text: If we consider the Frank elastic free energy in the equal constants limit (for more details, see here Chapter 4.1.) with an external magnetic field, we have $$ F = \frac{1}{2}\int \text{d}^3 r\left ( K|\nabla \cdot n|^2 + K|\nabla\times n|^2 -\Delta\chi |n\cdot H|^2 \right), $$ where $K$ is the elastic constant, $n$ is the unit director field ($n\cdot n =1$), $\Delta \chi$ is the magnetic susceptibility anisotropy and $H$ denotes the magnetic field. Suppose we take the ground state director field in de z-direction: $n_0 = (0,0,1)$. A fluctuation on his field will be of the form $\delta n \approx (\delta n_x, \delta n_y,0)$ for small $\delta n_x(r)$ and $\delta n_y(r)$. In the case of zero magnetic field, I understand the system. We introduce the Fourier transform $$ \delta n_{\alpha}(r) = \frac{1}{V}\sum_q \delta\tilde{n}_{\alpha}(q)e^{iqr} $$ and its inverse $$ \delta \tilde{n}(q)=\int \text{d} r \delta n_{\alpha}(r)e^{-iqr}, $$ to obtain $$ F \approx \frac{K}{2V}\sum_q \left( \delta \tilde{n}^2_x(q) + \delta \tilde{n}^2_y(q) \right)q^2. $$ Now one can impose equipartition of free energy: every mode in the Fourier decomposition carries $k_B T/2$ of energy. This yields the correlators $$ \langle \delta \tilde{n}_{\alpha}^2(q) \rangle = \frac{k_B T V}{Kq^2} \Rightarrow \langle \delta n_{\alpha}(0)\delta n_{\alpha}(r) \rangle = \frac{k_B T}{4\pi K |r|}. $$ However, I do not understand how this works in the case of a magnetic field. Suppose we take the initial magnetic field also in the z-direction. The free energy then becomes $$ F \approx \frac{K}{2}\sum_q \left( \delta \tilde{n}^2_x(q) + \delta \tilde{n}^2_y(q) \right)q^2 - \frac{1}{2}\Delta \chi V H_z^2. $$ I do not understand how I can invoke the equipartition theorem to find the correlation functions in this new situation. Can anyone help? EDIT: the form of the fluctuations is derived as follows. Say we have $\delta n = (\delta n_x, \delta n_y, \delta n_z)$, then we must have $|n+\delta n|^2 = 1$, yielding a conditions $1+\delta n_z = \sqrt{1-\delta n_x^2 - \delta n_y^2} \approx 1-\frac{1}{2}(\delta n_x^2+\delta n_y^2) \approx 1$ for small $\delta n_x$ and $\delta n_y$.


title: The most distant point from the earth that a space shuttle has reached

text: What is the most distant point from the Earth that a space shuttle has reached? When did this happen?


title: How would the night sky appear at the edge of the galaxy?

text: In Isaac Asimov's Foundation Series, there is a planet named Terminus which is believed to be the planet farthest from the galactic center. > There are almost no visible stars in its sky, only the huge lens of the > Milky Way galaxy But what would we really see if we look at the opposite side from the galactic center (without a telescope or binoculars)? Would we see a black sky or a sky full of galaxies?


title: Telescopes to avoid as a beginner?

text: I've heard people talk about "department store scopes" or "trash scopes". How do I know what to avoid in a beginner scope? How can I know that I'm not getting something we will be more frustrated with than excited about?


title: What criteria were used to set the "useme" flag in the NOMAD astrometric catalog?

text: How was the value of the "useme" flag in the NOMAD astrometric catalog set? The readme just describes the field as "Recommended astrometric standard," but says nothing about the basis on which that recommendation was made.


title: X-ray production rate

text: A 100 MeV proton beam of $10^{14}$ proton/s is perpendicularly incident on a rhodium foil 25 $\mu$m in thickness. Estimate the production rate of K and L x-rays(use the figure below). * * * ![enter image description here](http://i.stack.imgur.com/CMncX.jpg) * * * So far I figured out from above the cross section for L x-rays to be $8\times 10^{2}$ barns and for K x-rays to be $1\times 10^{2}$ barns I have the right answers for this one but I can't seem to figure out the right formula to use in order to arrive the given answer. L $1.3\times10^{13}$ per second K $1.6\times10^{12}$ per second


title: Time evolution of quantum states

text: Time evolution of a quantum state is fully described by a one parameter family of unitary operators. What I can't seem to understand is, given some unitary operator acting on some Hilbert space, can it always be realized as a corresponding quantum gate. So far, I've understood that from the Stone Theorem, any such one-parametric (strongly continuous) family is fully understood in terms of a self-adjoint operator. So my question could be rephrased as: does every self-adjoint operator on a finite-dimensional Hilbert Space describes some physical Hamiltonian?


title: The Mysterious Reverb in a Jar of Hair Gel

text: I have a small jar filled with hair styling gel (or, as it calls itself: Ultra Gel-Wax). The jar is cylindrical(with the height being less than the width), has an unscreweable lid and is made of plastic. The jar is nearly fully filled with gel. The jar however seems to have a strange property: When hit(with a mallet or simply with your finger), it does not produce a simple short 'tok' sound like one would expect from a plastic jar filled with cream or gel. Instead, it seems like the sound inside is somehow reverberating: the sound continues on for a significant longer period of time than one would expect. It is much longer than when hitting an empty jar, a jar filled with cream or other substance, or even one filled with water. Possibly even odder, this effect still happens without the lid on. Also, it does not matter where you hit the jar: the bottom, the lid or on the curved side. (Thus, it is not vibrating like a drum). It also does not matter if you touch the surface you hit (or even grip the whole thing in your fist), it still works equally well. (This in contrast to empty jars or jars filled with water, which might resonate a little, but only if not touching the struck surface). I am trying to find out what the cause of this reverberation is. According to the research I did so far, reverberation in a room is caused by:   1. The **speed of sound** of the medium inside: lower sound speeds make the waves travel for a longer amount of time before hitting a wall or arriving at the listener. For instance, in water, sound travels much faster( _+- 1497 m/s at room temperature_ ) than in air(+- _343 m/s at room temperature_ ).   2. **Reflection** /absorption of the walls: The more sound gets reflected each time a wall is hit, the longer there will be sound inside of the room -> the reverb will be longer. In other words: It's better to coat the inside of the room with metal than with cloth, if you want a long reverb.   3. The **attenuation** of the sound: A sound gets quieter and quieter, the more distance it has traveled. In one medium sound might travel much further than in another. This also has a huge effect on the length of the Reverb: This makes Rubber, for instance, a very bad medium to use, because sound won't travel very far. However, I did some qualitative research, and I have trouble to use these three rules to understand why the effect in my jar with gel is going on:   * Speed of sound: It is very hard to find out what the speed of sound might be in gel, since it's such a complex mixture of different substances. However, it should be much faster than air because the gel is something between a liquid and a solid while air is a gas: Sound speeds are higher in solids than in liquids, and higher in liquids than in gases _(as a general rule: this is always true if looking at one element in its different forms, but also true in general since the steps in speed between gases, liquids and solids are significantly big)._   * Reflection: Plastic is a material that does not seem to work well at reflecting: It is much easier bendable and less dense than metal, so it should not work as good as it seems to do here. Testing the response of empty plastic jars vs empty metal jars at least showed a huge difference: Plastic does not nearly keep vibrating as long as metal.   * Attenuation: Could this be the reason that there is so much reverb in my jar of gel? I know that waves can travel much farther in water(attenuation coefficient of _0.253_ ) than in air( _137.0_ ) _(according to this source)_ Water is a mayor component of most gels, including this one. However, since the effect works about as good when I remove the lid of the jar, I am not sure if this alone can cause the long reverberation effect. **So, I must be missing something. I hope someone can explain why sound in the the jar of gel is behaving this way.** Listen to a sound capture of the jar here You first hear a jar filled with cream, then the jar filled with gel. Then you hear both clips in the same order again, but this time slowed down to 1/10th of the original speed.


title: Choice of X-ray scatterer in Compton effect

text: I am going to perform an experiment on Compton Scattering, and I am going to use an X-ray scatterer to scatter the incident X-rays. I have been instructed that Acrylic Glass slab are the best for this purpose. I am not sure if it is because the acrylic glass slab does not have any free electrons present or any other reason. Please explain.


title: liquid polymers of high molecular weight

text: We often see a limit of melting point as a function of polymer chain length. Polyethlyene tops at about 190C, and polyethlyene glycol saturates at only 67C no matter how long the chain is: http://www.sigmaaldrich.com/materials-science/material-science- products.html?TablePage=20204110 Although it takes more energy per molecule to slide with respect to it's neighbors and be able to move, it appears that the number of degrees of freedom (and available thermal energy per molecule) increases proportionately. How does viscosity scale with molecular weight? My reasoning: Viscosity ~ size because larger molecules would have more relative velocity with respect to each-other. Size would be ~ length^(1/3) if they tended to coil randomly, or ~length if extensive shearing leads to chain straightening. Also, polymers tend to be non-newtonian; would the non-newtonian-ness of the fluid show a similar scaling behavior?


title: Why does a bicycle come to rest after travelling a distance?

text: According to Newton's law of motion, an object that is in motion will not change its velocity unless an external force acts upon it. In the case of a bicycle, if it is in motion with a certain velocity, it should not come to rest unless a force acts on it. In practice, we see the bicycle come to rest after a while. Does a force bring the bicycle to rest?


title: If I roll an elastic plate into a cylinder, does it shrink?

text: Suppose I start with a rectangular elastic (to keep things simple, zero Poisson's ratio) sheet of length $2\pi R$, thickness $h$, and (immaterial) width $W$. I roll it up into a cylinder of radius $R$, and allow it to relax. By symmetry, if it deforms at all, it will do so by uniformly scaling of the cross section to a cylinder of radius $R+\Delta R$. I'm trying to study the behavior of the cylinder using Kirchoff-Love plate theory, but have gotten myself confused. If I use Cauchy strain, I get that the strain is proportional to $$\frac{z+\Delta R}{R}$$ where $z\in (-h/2, h/2)$ is the normal coordinate. I then get that the energy density of the cylinder depends on $(\Delta R)^2$ but not $\Delta R$, and therefore setting $\Delta R=0$ minimizes energy -- the cylinder neither grows nor shrinks. However, if I instead use Green strain, the deformation energy has a nonzero linear term in $\Delta R$, and I get that the cylinder _shrinks_ ever so slightly. I find this very counter-intuitive. Why is using Cauchy strain inappropriate for approaching this problem? If I perform this experiment with a real sheet, will I really observe (very slight) shrinking? Is there intuition for why this occurs?


title: Stress in horizontal bars

text: Imagine we have an horizontal bar. My teacher expresses the tensions along the longitudinal axis by this way $\sigma_{xx}=A(x)y+B(x)$ He doesn't give any motivation behind this. So, is this general? and how is it justified?


title: Does the Delayed Choice Quantum Eraser experiment prove consciousness causes collapse?

text: https://www.youtube.com/watch?v=H6HLjpj4Nt4 This question comes from the video above. If you don't want to watch the video, I'll cut to the chase. Essentially, it claims the Delayed Choice Quantum Eraser experiment proves that time is an illusion, consciousness causes collapse of the wave function, and that the past is dependent on a future observer. I guess meaning that future observations can effect or change the past. Is this true? If not, what does the DCQE prove?


title: Why is there no way to make electric field at point P zero by changing quantity and sign of charge of $q_1$?

text: ![enter image description here](http://i.stack.imgur.com/dqDAf.png) q1 = -2.4μC and q2 = 8.9 μC, q3 = 2.3 μC, d is 6.2cm. Suppose that we want to make electric field at point P zero. We leave everything about q2 and q3 as they were before, but we are allowed to change the sign and magnitude of charge of q1. (distance and d cannot be changed for every charge) But it is told that there is no way to make electric field at point P zero just by changing q1. Why is it like that? All my calculations seem to show that it is indeed possible to make electric field at point P zero... using Coulomb's law..


title: How can one learn to visualize 4 dimensional spacetime?

text: I was watching Professor Leonard Susskind's lecture on Quantum Entanglements and he said that there are physicists and people who study physics, who 'rewire' themselves so they can visualize 4 dimensional spacetime in their heads. How can I learn to do this?


title: Monochromatic wave

text: If we have an EM wave like this one: $$E=\begin{pmatrix}1\\\i\\\0\end{pmatrix}e^{-i(\omega t-kz)}$$ The wave has clearly only one frequency $\omega$, but is it monochromatic? My doubt is that it's circularly polarised so the amplitud is changing during time, so the $E$ field is the composition of two periodic oscillations. I don't know if this implies it's not monochromatic, even when there's only one frequency involved, or if it's still monochromatic.


title: Quantum Mechanics Notation for BRA KET

text: I've been given this homework problem, but I do not understand its notation. > Please perform the following where the wavefunctions are the normalized > eigenfunctions of the harmonic oscillator Hamiltonian. > > i) Compute $\langle x\rangle, \langle p \rangle, \langle x^2\rangle$ , and > $\langle p^2\rangle $ for $| 0 \rangle, |1 \rangle$ , and $|12\rangle$. > > ii) Compute $\langle T \rangle$ for $n = 1$ and also compute $\langle V(x) > \rangle$ where, the potential energy $V(x) = \frac{m w^2 x^2}{2} $. Do these > expectation values add up to what you'd expect? I'm not understanding the BRA-KET notation. For n = 1: $\int \Psi_1^* (x) x^3 \Psi_1(x) dx$ - How does one "evaluate" this, without a function for $\Psi_1$? for $|1\rangle$, I am doing: $\langle1|x|1\rangle, \langle 1|x^2|1\rangle , \langle 1|p|1\rangle, \langle 1|p^2|1\rangle$ In which case I do not know how to proceed from that form..? And where does the wavefunction being the normalized eigenfunctions of the harmonic oscillator Hamiltonian come into play? for ii) Is it correct to asume that the "expectation" is that $\langle T\rangle + \langle V \rangle = \langle H \rangle$?


title: Find $x$ operator given $p$

text: This is problem $1.2$ of _Molecular Quantum Mechanics_ by Atkins, 4th edition. I'm given the momentum operator $$p=\sqrt{\frac{\hbar}{2m}}(A+B)$$ with $$[A,B]=1$$ and I need to find $x$ in this particular representation. A suitable solution, inspired by the formalism of the quantum mechanical harmonic oscillator is: $$x=-i\sqrt{\frac{m\hbar}{2}}(A-B) $$ Checking it: $$ [x,p]=-i\sqrt{\frac{m\hbar}{2}}\sqrt{\frac{\hbar}{2m}}\left( [A+B,A-B]\right)=-\frac{i\hbar}{2}\left( [A,A]-[A,B]+[B,A]-[B,B]\right)=\frac{i\hbar}{2}2[A,B]=i\hbar$$ **How to derive (perhaps algebraically) another particular solution for $x$ without having the harmonic oscillator hint?**


title: Electric potential of a spheroidal gaussian

text: I'm looking for results that compute the electrostatic potential due to a spheroidal gaussian distribution. Specifically, I'm looking for solutions of equations of the form $$ \nabla^2\Phi=N\exp\left({-\rho^2/2\sigma_r^2}\right)\exp\left({-z^2/2\sigma_\ell^2}\right), $$ where $\rho^2=x^2+y^2$, possibly with some polynomial factor of the form $\rho^{|m|}\cos(m\phi)z^n$ in front. This has an easy solution when the two variances $\sigma_r$ and $\sigma_\ell$ are equal, in which case there is spherical symmetry applies and Gauss's law readily yield $\Phi$ in terms of error functions. However, I'm having trouble finding results for the general case. Using spheroidal coordinates doesn't seem to help, as the constant-$\mu$ surfaces are confocal spheroids that taper out to spheres as they get larger, instead of maintaining their ellipticity as the charge density does. This looks standard enough that it ought to have been done before (right?), but it is also messy enough that I'm unsure it made it to any textbook. Has anyone seen the likes of this before?


What is intrinsic gravitational entropy? Does it have to do with dark matter or coarse graining in the universe? Is it unique to general relativity, or there are predictions from quantum mechanics as well? Please explain in detail.


title: D-brane Lagrangian?

text: As I understand it from the threads I read, D-branes are viewed as somewhat secondary to strings: If I know what all the open strings do, then I know what the D-branes do as well. But if the D-brane is a "big" and higherdimensional object, then there should also be a part of its surface, which is not covered by strings ending there. Why or how is the whole thing still characterized by the strings? Do the moving string ends drag the D-branes with them? Is there something like a Lagrangian and dynamical rules for the D-branes alone? Do I only need the D-brane after quantization and/or is there a classical concept of it? In the former case, does a classical limit of a D-brane and an effective action for it make sense? What is the precise reason one doesn't write down Lagrangians for non-point particle/non-string/non-field quantities?


title: What´s the electric field in the entire space generated by a uniform (but not constant) magnetic field?

text: Let me explain in details. Consider a region in space with no free charges and no free currents, so that the charge density is $\rho=0$ and the volumetric current is $\vec{J}=\vec{0} \text{ }$ in the entire space. Let $\vec{B}=\vec{B}(\vec{x},t) \text{ }$ be the magnetic field in the space. Suppose that $\vec{B} \text{ }$ is uniform in space, that is, suppose that $\vec{B} \text{ }$ is only a function of time: $\vec{B}(\vec{x},t)=\vec{B}(t) \text{ }$. Moreover, suppose that $\vec{B}(t)=at\hat{e}_z$ for some constant $a$. Notice that although $\vec{B} \text{ }$ is uniform, $\vec{B} \text{ }$ varies with time. Under these circumstances, what is the electric field generated by $\vec{B} \text{ }$ in the entire space?


title: Tritium decay is spontaneous even if the binding energy of tritium is higher than the binding energy of 3He. Why?

text: Given this nuclear reaction: $^3_1\mathrm H\to {}^3_2\mathrm{He}+e^-+\bar{\nu}$ and knowing the binding energies: $BE(^3_1\mathrm H)=8.48 \,\mathrm{MeV}$ $BE(^3_2\mathrm{He})=7.72 \,\mathrm{MeV}$ If I calculate the mass defect (obviously considering the binding energies in the mass calculation) I obtain a positive value: $M(^3_1\mathrm H)c^2=2809.08 \,\mathrm{MeV} > M( ^3_2\mathrm{He})c^2+M(e^-)c^2=2808.991 \,\mathrm{MeV}$ as expected for a spontaneous decay. Considering the binding energies I have written above I expect the $^3\mathrm{H}$ to be more stable than $^3_2\mathrm{He}$. My question is: why does this decay occur?


title: Why is the movement of electrons random in electric field?

text: Suppose an uncharged conductor is present isolated and there is no net electric field applied. Then by assumption we have the motion of electrons random,i.e. isotropic. But now suppose there is a closed conductor present. We apply a net electric field and the electrons starts moving. Suppose steady state current is achieved. We notice that there is has been created a velocity of electrons on the opposite direction of the applied field. I know that there is random movement of the electrons and they perform Brownian motion. My question is that is it biased in one direction because of the field? > Is the motion of elections in a particular direction more probable or they > have more velocity in that direction?


title: Energy in orbit of satellites around the earth lost?

text: If the total mechanical energy in a satellite's orbit (assuming circular) is greater when it is closer to the earth, and hence smaller when it is farther from the earth, then we can say that as the moon drifts from the earth, the moon loses energy in translational speed and gravitational potential energy. If only those two are taken into consideration, then there is a net energy loss from the moon. I had first thought that the energy a satellite has increases as it goes on a larger orbit, but I ran some numbers and it didn't appear so. If I went wrong somewhere, please someone, correct me. Here are my numbers: For a geostationary satellite (r = 42 164 km, v = 11 068 km/s, m = 1 kg), its total energy is PE + KE. PE = mgh, but g = 0.22416 m/s^2. The result is PE = 9 451.650 kJ, KE = 4 726.582 kJ For a satellite at r = 45 000 km , m = 1kg, then v = sqrt(GM/r) = 2 976.06 km/s. g at that height is g = 0.19680 The result is PE = 8 856.094 kJ, KE = 4 428.047 kJ At the larger orbit, both PE and KE are lower than if it was at a lower orbit. Is this right? Now, the earth slows down its rotation, which allows the moon to go into a larger orbit by conservation of angular momentum. Since the moon goes into a larger orbit, it loses energy. But, since the spin of the earth has slowed down, it also loses energy. Moreover, the moon is still tidally locked with the earth, so its rotational speed isn't increasing. All in all, there seems to be an energy loss that's going on. How is this being compensated? Is it in the translational speed of the moon (so that the moon is actually moving faster than it should be to maintain a stable orbit)? That seems reasonable - there could be an increase in translational and rotational speed to compensate for the energy loss, maintaining the moon to be tidally locked. But that's just me. What really happens? How does the energy transfer occur, and are there mathematical equations describing this exchange?


title: Detailed procedures of numerical calculations in lattice models

text: I am looking for a reference to show me a detailed procedure of numerical calculation of a physical quantity of a lattice. I have knowledge of the physics part and I am able to find it. My problem is reproducing the correct calculations to be sure that I am on the right track.


title: What's the meaning of a pressure coefficient in fluid mechanics?

text: I have a jet of air leaving a contraction from a pressurized tank at a velocity of 28.8 m/s and it's being directed towards a small object equipped with static pressure holes on its surface that give a reading of -310 Pa. I then had to calculate the maximum velocity on the body. I found it to be 36.5 m/s but I'm not really sure what the physical interpretation of this maximum velocity. I know the air must be traveling around the object and the stagnation point is at the center of the object where the air velocity if 0. Does this maximum velocity occur on the outside of the object as the air is curving around it? Also, what is the term "pressure coefficient"? I have to find the pressure coefficient at the stagnation point but I'm not really sure what that term is...


title: Is there any anti-gravity material?

text: I want to know if there is any anti-gravity material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround?


title: Correct formula to express the potential generated by a single layer charge distribution

text: Assume that the closed surface $S$ encircles a volume $V$, and that a surface charge with density $\sigma$ ("single layer") is distributed over $S$. My question regards the electrostatic potential $\phi$ generated inside the volume $V$ by this charge distribution: on the one hand, by using the superposition principle and Coulomb's law, I get $$\tag{1} \phi(x)=\frac{1}{4\pi \epsilon_0} \int_S \frac{\sigma(x')\, dS'}{\lvert x-x'\rvert}. $$ On the other hand, I know that $\phi$ solves Laplace's equation on $V$ with Neumann boundary conditions on $S$: $$\begin{cases} -\nabla^2 \phi =0 & \text{on }V\\\ \frac{\partial \phi}{\partial \nu} \propto \sigma &\text{on }S \end{cases} $$ so that it may be expressed in an integral form by means of a suitable Green function (see Jackson, 3rd ed., equation (1.46) pag.39): $$\tag{2} \phi(x)=\langle \phi\rangle_S + C\int_S \sigma(x') \frac{\partial G}{\partial \nu'}(x, x')\, dS(x').$$ > **Question**. Formulas (1) and (2) do not agree in general, otherwise all > volumes would share the same Green function, and that's not true. So one of > the two must be wrong. Which one is wrong, and for what physical reason?


title: Time relativity - 'philosophical' question

text: I'm aware that time is relative to the observer. But, does this also extend to the perception of time? For example, let's consider a fly (there's research suggesting that flies perceive time passing more slowly). Thus if a fly had knowledge of physics - and access to a laboratory -, it could perform an experiment to estimate the gravitational acceleration and the mass of objects. If it used 'human' units, it would obviously get the same 'numbers' we get. However, wouldn't the fly reality be different than ours? As in, wouldn't the fly reality be similar to the fly traveling in time? (If it used 'fly' seconds for the experiments - the time that a fly perceives just as a human perceives a second - the fly would get lower gravitational acceleration and higher masses for the objects)


title: Does the transmission axis matter for sending polarized light through polarized glass?

text: If I have polarized light and I send through only one polarized glass plane, does the transmission axis matter, or will the intensity be halved no matter what.


title: Is String Theory proven to be finite?

text: I read Lee Smolin's book "The trouble with physics" and the book says that the finiteness of string theory ( or string pertubative theory) is by no means a proven mathematical fact, despite that the string community widely believes it to be so. However, some string theorists do pronounce in a very strong term that the string theory is indeed proven to be finite, such as this website: > The names associated with the available proofs of the finiteness include > Martinec; Mandelstam; Berkovits; Atick, Moore, Sen; d'Hoker, Phong, and > others. Some of these papers are more complete - or quite complete - or more > constructive than others and there are various causal relations between the > papers. Many of these results are secretly equivalent to each other because > of the equivalences between the approaches that are demonstrated in other > papers. Many of these papers were preceded by less successful papers or > papers with flaws - flaws that were eventually fixed and settled. > > Also, I assure Jacques that he has met people who consider Mandelstam's > proof to be a proof, and besides your humble correspondent, this set > includes Nathan Berkovits who confirms Mandelstam's proof on page 4 of his > own proof in hep-th/0406055, reference 31, even though Nathan's proof is of > course better. ;-) > > At any rate, the question of perturbative finiteness has been settled for > decades. Many people have tried to find some problems with the existing > proofs but all of these attempts have failed so far. Nikita will certainly > forgive me that I use him as an example that these episodes carry human > names: Nikita Nekrasov had some pretty reasonably sounding doubts whether > the pure spinor correlators in Berkovits' proof were well-defined until he > published a sophisticated paper with Berkovits that answers in the > affirmative. So? Who is right on this? Are there rigorous proofs that show that string theory is always finite, as opposed to proofs that **_only_** show the second, or third term of the series is finite? Edit: This website says that, in Remark 1: > The full perturbation series is the sum of all these (finite) contributions > over the genera of Riemann surfaces (the “loop orders”). **This sum > diverges** , even if all loop orders are finite. So I guess this says-- in a very strong term--that String Theory is proven to be infinite... am I right? Edit 2: > According to here, **it is a good thing that the string theory is > infinite**, because if the sum is finite, this would indicate negative > coupling constants which are not physical. But I still don't get it. The reason why we use an infinite series to represent a physical quantity is because we believe that after summing up the series, we will come to get a finite number. If not, we would say that the theory breaks down and the physical quantity is not computable from the theory. So in order to avoid negative coupling constant which is unphysical, then we allow the sum to be infinite? Then what does this tell us of string theory's predictive power? If a theory can't predict physical values, then it is quite as useless as any meta-reasoning.


title: What is maximum width of a plank which is penetrated by bullet?

text: How to find max width of a plank which is penetrated by bullet taking into account the force of opposite reaction: $$m = 0.09 kg$$ << should be 0.009 $$v_{bullet}=800\;m/s$$ $$F_{reaction}=-200*v$$ * * * My attempt: $$F_{reaction}=m*a$$ $$-200*\frac{dS}{dt}=m*\frac{d\frac{dS}{dt}}{dt}$$ $$\frac{dS}{dt}=\frac{-200}{m}\int\frac{d\frac{dS}{dt}}{dt}$$ $$\frac{dS}{dt}=\frac{-200}{2*m}*(\frac{dS}{dt})^2$$ $$\frac{dt}{dS}=\frac{-100}{m}$$ $$v^{-1}=-100/m$$ going further to get `S`: $$S=\frac{m}{-100}\int1*dt$$ $$S=\frac{m*t}{-100}$$ What's next? * * * Trying to solve using formula: $$m\, \frac{{\rm d} v}{{\rm d} x} = - \beta$$ $$dx=\frac{m*dv}{-\beta}$$ $$x=\frac{m}{-\beta}\int dv=\frac{m*v}{-\beta}=\frac{0.009*800}{-200}=-3.6E-2$$ * * * The correct answer is $$3.6E-2$$


title: Variational Derivation of Schrodinger Equation

text: In reading Weinstock's _Calculus of Variations_ , on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't think page 262 is showing so I'll explain the gist of it: "In his initial paper" he considers the reduced Hamilton-Jacobi equation $$\frac{1}{2m}\left[\left(\frac{\partial S}{\partial x}\right)^2 \ + \ \left(\frac{\partial S}{\partial y}\right)^2 \ + \ \left(\frac{\partial S}{\partial z}\right)^2\right] \ + \ V(x,y,z) \ - \ E \ = \ 0$$ for a single particle of mass $m$ in an arbitrary force field described by a potential $V = V(x,y,z)$. With a change of variables $S \ = \ K\log(\Psi)$, (where $K$ will turn out to be $\hbar=h/2\pi$) it reduces to $$\frac{K^2}{2m}\left[\left(\frac{\partial \Psi}{\partial x}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial y}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial z}\right)^2\right] \ + \ (V \ - \ E)\Psi^2 \ = 0.$$ Now instead of solving this he, randomly from my point of view, chose to integrate over space $$I = \iiint_\mathcal{V}\left(\frac{K^2}{2m}\left[\left(\frac{\partial \Psi}{\partial x}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial y}\right)^2 \ + \ \left(\frac{\partial \Psi}{\partial z}\right)^2\right] + (V - E)\Psi^2\right)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ then Weinstock extremizes this integral which gives us the Schrodinger equation. Apparently as the book then claims on page 264 it is only after this derivation that he sought to connect his idea's to deBroglie's wave-particle duality. Thus I have three questions, 1) What is the justification for Feynman's famous quote: > _Where did we get that [Schrödinger's equation] from? It's not possible to > derive it from anything you know. It came out of the mind of Schrödinger._ > The Feynman Lectures on Physics in light of the above derivation. I note that all the derivations I've seen of the Schrodinger equation doing something like using operators such as $i\hbar\partial/\partial t = E$ to derive it always mention it's merely heuristic, yet what Schrodinger apparently originally did seems like a roundabout way of solving the Hamilton-Jacobi equation with no heuristic-ness in sight. What subtleties am I missing here? Why would I be a fool to arrogantly 'correct' someone who says Schrodinger is not derivable from anything you know? Note I've read a ton of the threads on this topic on this forum & none even go near the calculus of variations, in fact the above contradicts this explanation here that even refers to Schrodinger's Nobel prize lecture, so hopefully it's not a duplicate. 2) Is the mathematical trick Schrodinger has used something you can use to solve problems? 3) Why can't you use this exact derivation in the relativistic case? **Edit:** On the topic of complex numbers: On page 276, 14 pages after he explains what I posted, & after he goes through how Schrodinger linked his work with DeBroglie's work, it is only then that Weinstock says: > _In a more complete study of quantum mechanics than the present one the > admissibility of complex eigenfunctions $\Psi$ is generally shown to be > necessary. If $\Psi$ is complex, the quantity $|\Psi|^2$ is employed as the > position probability-density function inasmuch as $\Psi^2$ is not restricted > to real nonnegative values._ Apparently Schrodinger was able to do what I have posted using real-valued functions & have K as I have defined it, without i. If you're following what Weinstock is saying he shows how the hydrogen atom energy levels are explainable without complex numbers, i.e. he is able to derive a physical interpretation of the eigenvalues (discrete energy levels) of the Schrodinger equation that were in accord with experiment (see Section 11.3 Page 279 on). As far as I understand it it is in trying to find a physical interpretation of the eigenfunctions that one is forced into complex numbers, though apparently, according to the book, it can be shown to be necessary. That is something to take into account!


title: Working with atomic (?) units in solid state physics

text: I'm having some troubles understanding the units used in solid state physics paper. In the paper I read $\Lambda a \sim 1$ where $\Lambda$ is a momentum cutoff and $a$ is the lattice spacing of a crystal. Questions: 1) What kind of units are customarily used in solid state physics scientific articles? Can I be confident that they are atomic units? 2) The aforementioned equation is not (strictly speaking) dimensionally correct. If this was particle physics, I would say that one could set the speed of light and the Planck constant to 1 and make the equation correct. However in a low energy treatment in solid state physics, I don't see the point in using the speed of light or the Planck constant, and making the physics of the system dependent upon these quantities.


title: Multiple meanings of polarizability?

text: There's a concept in linear response theory called the polarizability $\chi$ $$\tag{1} \rho_{ind}=\int \chi(\vec{r},\vec{r}',t-t') V_{ext} \: d\vec{r}' dt' $$ where $\rho_{ind}$ is the induced charge and $V_{ext}$ is an external potential. The polarizability is sometimes called the density response function (because according to Kubo, it is a density-density correlation function). This Wikipedia article features a concept $\alpha$ called "polarizability" given by $$\tag{2} \vec{p}=\alpha \vec{E} $$ where $\vec{p}$ is an atomic dipole moment and $\vec{E}$ is an external electric field. I know $\chi$ and $\alpha$ are related, but are they the same thing? They don't look the same to me, but I find it confusing that we would name two different concepts (both related to dielectric screening) "polarizability". Furthermore, adding to the confusion, we denote the electric susceptibility $\chi_e$ given by $$\tag{1} \vec{P}=\int \chi_e(\vec{r},\vec{r}',t-t') \vec{E} \: d\vec{r}' dt' $$ Despite sharing the same symbol, the polarizability $\chi$ and the electric susceptibility $\chi_e$ are also not the same thing, right? If I remember correctly, in Fourier space (or in real space with appropriate integral symbols included), the dielectric constant is $\epsilon = 1+\chi_e$ and $1/\epsilon = 1 + V_{Coulomb} \chi$. Also, $\epsilon$ and $\alpha$ are related by Clausius–Mossotti. So, to summarize my question: Are either $\alpha$ or $\chi_e$ the same as $\chi$, or are they three distinct (but related) entities? When I say "distinct", I mean that they aren't related by trivial factors (such as multiplication by $4\pi$ or addition of constants). I suspect that they are all different things.


title: Are the forces I'm using correct? (ball in air)

text: I have been working on a program which should simulate all the forces on a spherical object. Right now those are balls, but they can easily be changed to something else. These are the formulas I use for the forces so far. My question is, if all of them are correct for this situation and if I get the correct resultant force. $$\matrix{ F_{zw} &=& g m \\\ F_d &=& \frac12 \rho v_y^2 C_d A \\\ F_b &=& \rho V g \\\ Fres &=& F_{zw} + F_d - F_b \\\ } $$ Would this work when the ball is falling down? Will I have to change the sign of one of these forces when the ball is going up? (It's a bouncing ball) **EDIT:** $$\matrix{ m &=& 0.0425\,\text{kg} \\\ \text{radius} &=& 0.125\,\text{m} \\\ } $$ To get the speed I get the acceleration ($F_{res} / m$) and then multiply that by the difference in time. $F_{zw}$ is gravity, $F_b$ is buoyancy, and $F_d$ is drag.


title: At near light speeds, are small objects in your path a problem?

text: A mental exercise: A space ship is accelerating toward me from far away. As it approaches me, I can observe its acceleration. This high speed ship keeps accelerating faster toward the speed of light. At some point, its apparent acceleration changes to a deceleration and slows more and more. Say, for the sake of argument, that the ships top speed is such that from my perspective, it only seems to be moving around, say, 500 mph (or something relatively slow). As it nears me and flies by, I move along side of it and match its speed flying side by side. As we fly along there are two small pebbles (1/4 inch in diameter) in our path ahead. We collide with them. The small pebble I collide with bounces off my ship.   1. Does the small pebble the other ship collides with destroy the ship (assuming it doesn't have any way to deflect it)?   2. If I flung a baseball at the ship from my ship and it hit the other ship, would it destroy the ship as well with a huge impact?  EDIT: In essence what I'm asking is what's the difference between me hitting the pebble and the other ship hitting the pebble (which is at near light speed). Clearly there would have to be a different outcome. The other ship is hitting it at near the speed of light which I would think would spell disaster for that ship, but not mine, even though we're flying along side each other.


title: Force between two charged particles

text: Consider two positively charged particles, one of charge $q_0$ (particle 0) fixed at the origin, and another of charge $q_1$ (particle 1) fixed on the y-axis at $(0,d_1,0)$. What is the net force $\vec{F}$ on particle 0 due to particle 1? Express your answer (a vector) using any or all of $k$, $q_0$, $q_1$, $d_1$, $\hat{i}$, $\hat{j}$, and $\hat{k}$. Using Couloumb's law, I computed $\frac{k q_0 q_1}{d_1^2} \hat{j}$. I'm not sure if this is correct. I would like some to correct it for me if it is.


title: Low-temperature hobby rocketry

text: I work with a lot of Boy Scouts who want to complete the Space Exploration merit badge. It's a great program, and one of the coolest requirements is building, launching, and recovering a model rocket. Unfortunately, my Scouts are in the city and finding a location to launch rockets is ... well ... tricky. In the past, we'd drive out to the countryside and use either a park or a friendly farm to launch the rockets off. It never used to be a big deal ... but someone got it into their head recently that these rockets pose a significant fire hazard (despite over a decade of fun without a single problem). The biggest argument I've heard against the rockets is the nature of the engine. Basically, model rocket engines are black powder-based. A small electric igniter lights the engine and it produces a lot of heat and smoke to propel the rocket. The parks and farms that have since banned us from launching rockets finally conceded that, if I could find a way to launch rockets _without_ burning anything, they'd let me bring the Scouts back. Every reference I can find to "cold rocket engines" seems to refer to older 1970s freon-based engines. I don't want to go that route, but I'm looking for some ideas for cold or low-temperature rocket engines. Even if it's something I'd have to build myself ... Any ideas or suggestions?


title: Do magnetic fields interact with electromagnetic fields?

text: I'm not sure if this is even a possible scenario, but what would happen with a permanent magnets magnetic field, when current flows through the magnet? Do the magnetic/electromagnetic force-lines add up?


title: High-School Physics with inclines

text: > _The figure below shows two blocks with a rope that runs over a pulley. The > mass of $m_2$ is 5.0 kg and the incline is 35 degrees. The coefficient of > static friction between $m_1$ and the inclined plane is 0.25. The > coefficient of kinetic friction between $m_1$ and the inclined plane is > 0.17. Determine the largest mass $m_1$ such that both blocks remain at > rest._ Picture can be found Here . The answer I obtained was 6.42 kg although I'm not sure whether that is correct or not. Any help would be appreciated.


title: Chemical Potential with Pressure at Constant Temperature

text: ![Gibbs free energy of a solid, liquid and gas vary with pressure](http://i.stack.imgur.com/Tq6iP.jpg) The above picture shows how the Gibbs free energy of a solid, liquid and gas vary with pressure. I have found no similar graphs on the internet that match the above. Instead I often find these graphs : ![Chemical Potential Variation with Pressure](http://i.stack.imgur.com/Qbjug.gif) My questions are : 1.) How can both the two graphs correct ? Since their gradients are different. 2.) In the first picture; why is the gas curve a curve in the first place ? Why isn't it a straight line ?


title: Coffee cooling in a cup

text: Will coffee in two differant size styrofoam cups in the same room conditions cool at the same rate? One cup is 8 oz. and the other would be 12 oz.


title: Proof of S-duality between Type IIB, IIB and Type HO, I string theories

text: About every source on string theory I've read which do mention S-duality state that: $$\begin{array}{l} \operatorname S:\operatorname{IIB} \leftrightarrow \operatorname{IIB}\\\ \operatorname S:\operatorname{HO} \leftrightarrow \operatorname{I} \end{array}$$ However, how does one prove that the Type IIB string theory is self-S-dual and even more bizzarely, that the Type HO string theory is S-dual to the Type I string theory?


title: Frank Hertz experiment and different jumps

text: Why is it assumed that in this experiment, the jump will be between the second and the first states. Couldn't it be that when the electrons have enough energy, an atom absorbs enough to get to the third state and then jumping only to the second, emiting less energetic light? Isn't this right? Or is it just that the number of electrons that do this is very small compared to the ones that excite the atom to only the second level? I am saying this because I've seen the experiment, and a blue-violet glow is clearly visible when the potential is very high. Unless I have super powers, I cannot explain this, because the light emited from the second to the first level is 4.9 eV is equivalent to light of $\lambda \approx$ 250 nm, clearly out of the visible spectrum.


title: 1D Kinematics - Relative Motion involving non-inertial frames of reference

text: I recently came across a question involving non-inertial frames of reference. I didn't quite understand the way it had been solved due to some conceptual confusion regarding certain deductions made to solve the problem. The question is this:      > _QUESTION: An elevator is ascending with an acceleration $0.2 \ m s^{-2}$. > At the instant the elevator's velocity is $1 \ m s^{-1}$, a loose bolt on > the elevator's ceiling drops down. The height between the ceiling and floor > of the elevator is $5 \ m$. Find out the time taken for the bolt to reach > the floor and the distance it covered._   >   > MY ANSWER: Since with respect to the ground, the initial velocities of both the elevator and the bolt are $1 \ m s^{-1}$ at that particular instant, the bolt's velocity relative to the elevator is $0 \ m s^{-1}$. The relative displacement of the bolt with respect to the elevator floor will be $5 \ m$ after the time taken for the bolt to reach the floor. Since the elevator is an accelerating frame of reference, there will be a fictitious (pseudo) force acting in the opposite direction to the acceleration of the elevator but equal in magnitude. So not only does the bolt undergo free fall due to acceleration due to gravity (which I have taken as $9.8 \ m s^{-2}$), but also experiences an additional acceleration due to this pseudo-force. If I have considered the direction the elevator is moving in to be in the positive direction, then the total downward acceleration of the bolt is $= -9.8 + (-0.2) = -10 \ m s^{-2}$. But I need to consider the bolt's relative acceleration with respect to the floor of the elevator $= (-10 + 0.2) = -9.8 \ m s^{-2}$. This is as far as I can come. When I try substituting this into an equation, mathematical trouble brews and I am unable to solve the problem.      Please enlighten as to whether my ideas are correct and if not how would one go about solving this particular question using the ideas of relative motion?


title: Einstein Field Equations in other space-time dimensions than 3+1?

text: This question is apparently quite simple but I can't seem to find an answer to it, so I was hopping anyone could clarify me. Are the Einstein field equations (EFE) only valid for a 3+1 dimensional space- time? I've read somewhere, which I can't remember or find, that there were problems with the EFE in a 2+1 dimensions...Why would that be? What about 1+1?


title: Block and inclined plane (INPhO Problem)

text: The figure shows two blocks on an inclined plane of mass 1kg each.The coefficient of static as well as kinetic friction is $0.6$ and angle of inclination is $30^\circ$ . Find the acceleration of the system. (string and pulley are ideal). Take $g=10m/s^2$. ![](http://i.stack.imgur.com/u86mg.jpg) [This is not the real image given in the question and they have not specified the direction of acceleration] I attended the question like any other block-inclined plane problem assuming block $m_2$ to be moving down and got the answer to be $-5.1m/s^2$. But, the answer says if you take the other block to be moving down then you get an acceleration of $-0.1 m/s^2$. Since there are two negative values for acceleration we can conclude that the acceleration is zero. It doesn't make any sense to me. Please explain this. The answer is given in such a way that the static friction has no role. Why is it so?


title: Why are lab Magnets painted red?

text: Wy are lab magnets painted red in color? I tried searching everywhere but couldn't get a satisfactory answer.


title: Shouldn't the addition of angular momentum be commutative?

text: I have angular momenta $S=\frac{1}{2}$ for spin, and $I=\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \begin{align} \lvert 1,1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2}\tfrac{1}{2}\rangle,\tag{4.21a} \\\ \lvert 1,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle + \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21b} \\\ \lvert 1,-1\rangle &= \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2},-\tfrac{1}{2}\rangle,\tag{4.21c} \\\ \lvert 0,0\rangle &= \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle - \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21d} \end{align} $$ where $F=I+S$, so this is the basis $\lvert F m_F \rangle = \sum_m \lvert\bigl(I S\bigr),m_I m_S\rangle $. Now since adding angular momenta is commutative, the exchange between $I$ and $S$ shouldn't mathematically introduce any kind of difference. In other words, in the basis described in those equations, it shouldn't matter whether I write it as $\lvert\bigl(I S\bigr),m_I m_S\rangle$ or $\lvert\bigl(S I\bigr),m_S m_I\rangle$, right? Now the problem is the following: I have created the Hamiltonian matrix $H=-\vec{\mu}\cdot \vec{B} = -2 \mu B_z S_z/\hbar$ in the $\lvert F m_F \rangle$ representation, and actually the result depends on how you call those angular momenta, so the result could be $$H = \begin{pmatrix} \mu_B B & 0 & 0 & 0 \\\ 0 & \- \mu_B B & 0 & 0 \\\ 0 & 0 & 0 &\mu_B B \\\ 0 & 0 & \mu_B B & 0 \end{pmatrix}$$ Or could be $$H = \begin{pmatrix} \mu_B B & 0 & 0 & 0 \\\ 0 & \- \mu_B B & 0 & 0 \\\ 0 & 0 & 0 &-\mu_B B \\\ 0 & 0 & -\mu_B B & 0 \end{pmatrix}$$ Depending on how you "label" them, $I$ or $S$... which is very confusing! This happens because the off-diagonal terms $$\left\langle 1 0 \right| S_z \left| 0 0 \right\rangle = \frac{1}{2} \left( \left\langle (\frac{1}{2} \frac{1}{2}) \frac{1}{2} -\frac{1}{2} \right| + \left\langle (\frac{1}{2} \frac{1}{2}) -\frac{1}{2} \frac{1}{2} \right| \right) S_z \left( \left| (\frac{1}{2} \frac{1}{2}) \frac{1}{2} -\frac{1}{2} \right\rangle - \left| (\frac{1}{2} \frac{1}{2}) -\frac{1}{2} \frac{1}{2} \right\rangle \right)$$ will be either $\hbar/2$ or $-\hbar/2$ depending on your convention whether it's $\lvert\bigl(I S\bigr),m_I m_S\rangle$ or $\lvert\bigl(S I\bigr),m_S m_I\rangle$. How can I understand this physically and mathematically? Shouldn't the addition be commutative and the process be blind to which labels I use?


title: Thermodynamics and cross entropy

text: I am facing with the concept of cross entropy. I would like to know the thermodynamic and statistical meaning of cross entropy (if exists)?


title: Trains travelling at different speeds towards a station C

text: Here's the question: > Consider two stations A and B located 100 kilometer apart. There is a > station C, located between A and B. Now trains from station A and B start > moving towards station C at different speeds. The train starting from A > moves at 40 km/h while the train starting from B moves at 60 km/h. These > trains have to stop at station C. A bird also starts from station B and > flies at a speed of 80 km/h towards station A. When it reaches the train > coming from station A, it switches direction – flying back towards station B > and when it reaches the train coming from station B, it turns around again, > heading towards the direction of station A. This goes on for some time till > both the trains reach station C. What is the total distance covered by the > bird in the time it takes for the two trains to reach station C? This is what I thought: The distance travelled by the bird should depend on location of station C. Reason: If Ta and Tb is the time taken by the trains to reach station C respectively, and the bird will have to keep on flying until _both_ trains reach station C, the time for which the bird will have to fly is **max(T a, Tb)**. So the answer should be **max(T a, Tb) x 80**. The fact that the answer will depend on location of _C_ can also be verified by taking 2 cases. In the first case, let _C_ be **1km** from station A, and in the other one, let _C_ be **1km** from station B. You will get different answers. However, apparently my answer is wrong, as the answer was asked in an exam and had a unique answer (independent of any variable). Can someone please explain where I went wrong?


title: Problem in a truss

text: ![enter image description here](http://i.stack.imgur.com/9jFoH.png) For the truss above known are the following: l= 1 + 0.02*n [m], h = 2 + 0.01*n [m], F_1 = 200 + 2*n [N], F_2= 500 + 6*n [N], F3 = 400 + 5*n [N], Young's modulus E = 2*10^11 [N/m2], allowable stress σ_a= 8*10^8 [N/m2] In the amounts above, n a natural number. Find:   1. Reaction forces at A and E .   2. Tension forces in all members of the truss. Draw free body diagrams for all joints and members; indicate extension or compression for the member forces.   3. Diameters of all the members considering they all have circular cross-sections. The diameters have to be selected from the following sequence of diameters (in mm): 2, 5, 8, 10, 12, 16, 20, 24, 30, 36, 40, 50, 60, 70.   4. The axial deformation (Δl) of bars BC and FG.


title: Why is there a 90˚ phase angle between particle velocity and sound pressure in spherical waves?

text: My text says that in a plane sound wave (or in the far field), particle velocity and pressure is in phase. As we move closer to the sound source (to near field and more spherical waves), the phase angle between these two quantities will gradually shift towards 90˚. Why does this phase shift occur? I haven't been able to find an intuitive explanation for why there is a phase shift in spherical waves, but not in plane waves, and vice versa.


title: on Brownian motors

text: From this review on Brownian motors, there is such a statement without detailed explanation: (I think this statement is general enough so that one does not need to read the article) "Apart from transients, directed transport in a spatially periodic system in contact with a single dissipation- and noise-generating thermal heat bath is ruled out by the second law of thermodynamics." I don't have any idea how to relate the second law thermodynamics to the absence of directed transport. The first obstacle, I think, is which description of the 2nd law should we use as the starting point? And then, how to relate that particular description to the absence of directed transport? How does the nature of the heat bath relate to the problem?


title: Why does a bottle shrink when you pour hot water in it?

text: I think the title says pretty much. Last night I poured hot water in a half full bottle of water and closed the cap. The bottle immediately shrunk. Why? My girlfriend suggested that the hot water has less oxygen in it and thus consumes some of the oxygen in the bottle.


title: Focal length, power and magnification

text: In refracting (lens not mirror based) telescopes, to have a large magnification the objective lens needs to have a large focal length. if $\text{Power} = \frac{1}{\text{focal length}}$ then that means telescopes that have a large magnication have a low powered objective lens? Also if a large focal length provides a greater magnification then a lens which refracts (bends) the light the least is better, so surely that's no lens at all (or just a pane of glass)? How does a longer focal length (and so low power lens) provide a greater magnitude? It seems to my common sense (obviously wrong) that a lens that bends the light more would provide a greater magnification? (As the rays would be spread over a greater area for a shorter length)


title: How do you interpret the product of two components?

text: I hope this isn't borderline philosophy. It is really difficult for me to articulate the question and so Google was of no use to me. The question spawned when I was learning laws of conservation of energy, because although I can calculate the answers I don't understand why things like mgh (mass times gravity times height) are able to give us numbers that we can use to plug into other formulas for combining energies into totals. Let's say we have 2 components, A and B. A/B is the number of A per B, such as m/s. Therefore if he have A= 10m and B = 5 seconds, we can determine that A/B is 2 meters per second. I used this try and understand what products yield. A times B is A of B's or B of A's. Such as A=5, B=6, A times B is 30, or five sixes or six fives. Now if we go back and multiply 10m by 5s, we get 50 ms. How do you interpret 50 meter seconds or 50 second meters. What is that? Any references for me to read would be cool too. Thanks.


title: Why are there two quasi Fermi levels and only one Equilibrium Fermi level?

text: I am reading a book and I'm trying to understand the concept of quasi Fermi levels. _**For example,_** A steady state of Electron Hole pairs are created at the rate of $10^{13}\ \mathrm{cm}^{-3}$ per $\mu$s in a sample of silicon. The equilibrium concentration of electrons in the sample is $n_0 = 10^{14}\ \mathrm{cm}^{-3}$. Also, it gives $\tau_n = \tau_p = 2\ \mu\mathrm{s}$. I am not sure what this is but I think this is the average recombination time. The result is that the new levels of carrier concentrations (under the described steady state) are $n = 2.0 \times 10^{14}$ ($n_0 = 1.0 \times 10^{14}$) $p = 2.0 \times 10^{14}$ ($p_0 = 2.25 \times 10^6$) I follow until here but I get a bit confused after this. The book goes onto say that this results in two different _virtual_ Fermi levels which are at: $F_n-E_i = 0.233\ \mathrm{eV}$ $E_i-F_p = 0.186\ \mathrm{eV}$ The equilibrium fermi level ($E_F$) being at $E_F-E_i=0.228\ \mathrm{eV}$ _**My question:_**   1. Why are there two different quasi fermi levels now created?   2. Why do we not consider two different ones at equilibrium conditions?   3. Why is it that due to a steady state input of electron hole pairs that we now consider two quasi Fermi levels?   4. What is the relevance of these new quasi fermi levels?


title: Is all angular momentum quantized?

text: Angular momentum is definitely quantized in elementary particles and electrons in atoms.   Molecules also have characteristic rotation spectra.   Is it true that all angular momentum is quantized, including big things like automobile tires, flywheels and planets?   If so what is the largest object for which this quantized rotation has been verified/observed/measured?


title: Question about tide

text: Let's say a canal is built in the middle of Kansas in the shape of a ring that stretches 100 miles in diameter. The band of water is 100 feet wide and 10 feet deep throughout the entire canal, and the land where the canal is built is flat in every way aside from conforming to the curvature of the earth. Would a current develop from tidal forces? If so, in what direction would it flow? How rapid would it be?


title: Gas in movement

text: Suppose I have a gas contained in a solid box and I drop it from a certain height, is the temperature of the gas going to change because of the velocity that it acquires during the fall? If so, by how much?


title: Integration over the state space

text: I came across the concept of average fidelity $\int f(\psi)d\,\psi$ where the integration is with respect to the uniform Haar measure on pure states. I've only seen Haar measures in connection with topological groups, so I don't understand what measure is used in this context. Could anyone provide a reference where this is explained ?


title: How long will it take to fill a bath tub with oil compared to water?

text: As everyone knows, the oil is more viscous than water. But, I've been wondering how long it will take to fill a bath tub compared to water. It is quite obvious that oil will take longer. But, how much longer? If you wondering about what oil I'm talking about, let's say Cooking oil! You are free to make any assumptions about the tap or the pipes. As you know, the flow rate of water and oil would be different under these circumstances. This is not my homework; just my curiosity.


title: Do virtual photons/magnetic fields change speed in a medium?

text: Light can be slowed down to a walking pace and even stopped in a medium of hot rubidium vapour. If an electromagnet was placed in this medium and turned on would the popagation of the magnetic field be slowed down or stopped? Do virtual photons change speed in a medium?


title: Riemann curvature tensor symmetries confusion

text: In the context of spacetime, reading Schutz, I'm confused about the symmetries of the Riemann curvature tensor, which I understand are: $$R_{\alpha\beta\gamma\mu}=-R_{\beta\alpha\gamma\mu}=-R_{\alpha\beta\mu\gamma}=R_{\gamma\mu\alpha\beta}.$$ But using the metric to contract the Riemann tensor can't I also say $$R_{\gamma\mu}=g^{\alpha\beta}R_{\alpha\beta\gamma\mu}=g^{\alpha\beta}R_{\alpha\gamma\beta\mu}?$$ Which leads me to think that $$R_{\alpha\beta\gamma\mu}=R_{\alpha\gamma\beta\mu}.$$ But $R_{\alpha\gamma\beta\mu}$ isn't one of the above listed symmetries. Where am I going wrong?


title: Axial forces on a solenoid windings

text: I understand that the windings in a solenoid experience a Lorentz force $\mathbf{f} = \mathbf{J} \times \mathbf{B}$, which tend to cause an outward pressure where $\mathbf{B}$ is directed along the solenoid axis (near the middle), and axial compression, where $\mathbf{B}$ has a radial component (near the ends). In the case of a solenoid that is much longer than it's radius, the outward pressure is easily derived from energy considerations, but the same logic also seems to imply a force that tends to drive the solenoid apart axially. **Radial force** These forces can be derived from energy considerations. Consider an very long solenoid (or, at least, one where we neglect end effects) with radius $r$ and current $I$. Then $$\mathbf{B} = \frac{\mu_0 N I}{\ell}\hat{\mathbf{z}}$$ so the energy density inside is $$u_B = \frac{1}{2} \frac{B^2}{\mu_0} = \frac{1}{2} \frac{\mu_0 N^2 I^2}{\ell^2}$$ and energy is $$U_B = \frac{1}{2} \frac{\mu_0N^2 I^2}{\ell^2} \pi r^2 \ell $$ from which we derive a radial (generalized) force $$f_r = -\frac{\partial U_B}{\partial r} = \frac{\mu_0 N^2 I^2}{2\ell^2} 2\pi r \ell $$               Note: The above expression is missing a minus sign. See Art Brown's answer below.      or, equivalently, a pressure $$P = \frac{\mu_0N^2 I^2}{2\ell^2} = \frac{B^2}{2\mu_0} $$ **Axial force** It seems that, far away from the solenoid ends, there should also be an expansive force, which we can calculate. Consider a small segment of the solenoid with number of turns $\Delta N$ and length $\Delta \ell$: $$f_z = -\frac{\partial U_B}{\partial (\Delta\ell)} = \frac{\mu_0\Delta N^2 I^2}{2\Delta\ell^2} \pi r^2 $$ It seems counter-intuitive, both from the point of view of the axial compression due to the ends, and thinking of a solenoid as a linear arrangement of attractive current loops. Where am I going wrong?


title: Some small questions about quantum spin and rotations

text: I'm studying about quantum-spin (in a syllabus about non-relativistic quantum- mechanics though), but I have some trouble understanding everything. So I would like to ask some small questions, which may perhaps clear up the picture as a whole. So they are trying to determine the influence of a spatial rotation, to the state in the two dimensional spin-space. So they try to determine the operator $Û_R$ responsible for this rotation (which acts on the spin-space and thus is represented by a 2x2 matrix). To do this they first try to determine $Û_{\epsilon}$ which is the infinitesimal unitary operator for rotation about an infinitesimal angle $\epsilon$ and they state that such an operator can always be written as: $$Û_{\epsilon}= 1-i\epsilonÂ$$ with $Â$ hermitic. So $Â$ is also acting on the two dimensional spin-space and can be represented by a 2x2 matrix. So far so good. But now they state that $Â$ has to be constructed in such a way that $<\chi\vert\chi>$ and $<\chi'\vert\chi'>$ and $<\chi'\vert\chi>$ are rotation- invariant. Where $\vert\chi>$ and $\vert\chi'>$ are states of the two dimensional spin-space and $\vert\chi'> = Û_{\epsilon}\vert\chi>$. What I don't understand is: 1) What is meant by rotation invariance? Because if we rotate $\vert\chi>$ and $\vert\chi'>$ by taking the action of a unitary rotation-operator $Û_{\theta}$ , then these inproducts are necessarily conserved, whatever $Â$ may be. So I guess they mean something else. 2) Why must $Â$ be constructed so that this is true? Perhaps this will be clear if I understand 1) though. Furthermore they state that this can only be the case if: $$Â = \vec{1}_n\cdot\vec{Â} = \frac{1}{2}\vec{1}_n\cdot\vec{\sigma}$$ With $\vec{Â}$ and $\vec{\sigma}$ hermitical vector operators. So I wonder: 3) Why is this only the case if this is true?   4) Somewhat further they also mention that $Tr(\sigma_z) = 0$ but I don't really see here too why this has to be the case. I hope that someone can answer one (or more) of these questions, I think it will help a lot in understanding the other claims of the chapter about quantum-spin.


title: weak solution of Schroedinger equation .. are they useful?

text: We know that Schroedinger equation can be deduced from a variational principle (non relativistic Schroedinger equation). Assuming this I have 2 questions: a) Using variational methods, could we prove that there exist weak solutions (distributions) to the Schroedinger equation under certain conditions? b) Are 'weak solutions' of Schroedinger equation useful?


title: Convective Operators: Cartesian vs Spherical Coordinates

text: (This question may be more appropriate for Math Stack Exchange, but since physicists tend to be more well acquainted with vector calculus, I'm asking the question here). This question is about convective operators. Please open the Wolfram MathWorld page (accessed March 08, 2014). Lets assume that the vector field ${\bf{B}}(x)$ is constant in all space. Then according to equation (2) (which is the formula in Cartesian coordinates) the result of the operator is the zero vector field, since ${\bf{B}}$ appears only in derivatives. But in equation (4) (spherical coordinates) there are also terms where ${\bf{B}}$ is not in a derivative, and so the results could be different from zero. How is this possible? (By the way, the same formula appears in Wikipedia (accessed March 08, 2014))


title: Why doesn't the potential energy of any object equal 0

text: Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't the two cancel and no net work should be done on the particle? If they don't cancel, then where did the energy that came from the work done by the force of gravity go?


title: Do all massive bodies emit Hawking radiation?

text: It is known that any accelerated observer is subject to a heat bath due to Unruh radiation. The principle of equivalence suggests that any stationary observer on the surface of a massive body should also experience heat bath. Indeed by substituting surface gravity g into the formula for Unruh radiation one can find the black body temperature for a hypothetical hyper-cool planet: $$T = \frac{\hbar g}{2\pi c k}$$ which is $3.9766×10^{-20}\ K$ for Earth one even can find the time which it will take for Earth to evaporate: $5.69×10^{50}$ years. Since the heat in the super-cold Earth cannot come out of nothing one should assume that it will come from decay of particles due to a certain mechanism. Sometimes I heared an argument that an event horizon is needed for Hawking radiation to exist. But this can be countered by assumption of possibility of decay due to quantum virtual black holes (which inevitably should appear due to uncertainty principle, and the more massive and dense body is the greather concentration of virtual black holes inside it will be, eventually becoming similar to the concentration of bobbles inside a body of boiling water). Or just suggest that any massive body due to uncertainty principle can quantum tunnel into a black hole state so to emit Hawking radiation. So what is the conclusion here?   * Can we say that all massive bodies are surrounded by the atmosphere of heated vacuum? This is a weaker preposition: thermal state of surrounding vacuum does not mean energy transfer if the system is in thermodynamic equilibrium.   * Any body gradually evaporates, i.e. transfers its energy to the surrounding vacuum until completely vanishes? This is a stronger preposition and suggests emission of radiation al loss of mass.


I have recently read that the dimensional regularization scheme is "special" because power law divergences are absent. It was argued that power law divergences were unphysical and that there was no fine-tuning problem. I was immediately suspicious. Let us take $\lambda\phi^4$ theory. For the renormalized mass $m$ (not a physical mass) with dimensional regularization, $$ m^2_\text{phys} = m^2(\mu) + m^2_\text{phys}\frac{\lambda}{16\pi^2}\ln({\mu^2}/{m^2_\text{phys}}) $$ This looks promising, but $m$ is a renormalized mass, not a true parameter of a Lagrangian that will be set by some new physics, like string theory or whatever it is. For the Lagrangian mass with a cut-off regulator, $$ m^2_\text{phys} = m_0^2(\Lambda) + \frac{\lambda}{16\pi^2} \Lambda^2 $$ which is basically what I understand to be the fine-tuning problem. We would need incredible cancellations for $m_0\ll \Lambda$ at the low scale. Here I understand $m_0$ to be a "real" parameter determining the theory, whereas $m$ in dimensional regularisation was just an intermediate parameter that scheme. I suspect that these two equations are related by the wave-function renormalization, $$ m_0^2=Z m^2 = (1+\text{const}\Lambda^2/m^2 + \ldots) m^2 $$ If I am correct, not much has improved with dimensional regularization. We've sort of hidden the fine-tuning in the wave-function renormalization. You don't see the fine-tuning in dimensional regularization because you are working with a renormalized mass. The bare Lagrangian mass $m_0$ is the one being set at the high-scale by some physics we don't know about. So it's $m_0$ that we need to worry about being fine-tuned. With dimensional regularization, we see that $m$ isn't fine-tuned, but that isn't a big deal. Have I misunderstood something? I feel like I am missing something. Can dimensional regularization solve the fine-tuning problem? Is dimensional regularization really special? EDIT I am not necessarily associating $\Lambda$ with a massive particle, just a massive scale at which $m_0$ is set to a finite value. It seems to me that the dimensional regularisation cannot help me understand how $m_0$ runs, or the tuning associated with setting it at the high scale, especially as it obliterates information about the divergences. I have no idea how quickly to take the $\epsilon\to0$ limit. I can do something like, $$ m_0^2 = Z m^2 = ( 1 +\lambda/\epsilon) m^2\\\ m_0^2(\epsilon_1) - m_0^2(\epsilon_2) = m^2 \lambda (1/\epsilon_1 - 1/\epsilon_2) $$ Now if I take $\epsilon_1$ somehow so that it corresponds to a low scale, and $\epsilon_2$ somehow corresponds to a high scale. $m_0^2(\epsilon_1) $ needs to be small for a light scalar. But then I need to fine tune the massive number on the right hand side with the bare mass at a high scale. The fine tuning is still there. Admittedly this is very informal because I have no idea how to really interpret $\epsilon$


title: Entropic force in polymers

text: According to my textbook, the elastic force in a rubber is caused to the tendency of the polymers to return to their initial disordered state of higher entropy. But isn't this looking at entropy on a larger level (I'm hesitant to say macroscopic) than it is usually looked at? Generally, we consider the entropy of individual atoms in an ordered structure (a crystal lattice, for instance). But here, we are looking at the entropy of entire polymer _chains_. Also, it just doesn't seem to make sense with the non-statistical view of entropy. The book explains it in the following way. The first law of thermodynamics, $$dE = dQ - dW$$ Making some substitutions, $$dE = TdS + Fdx$$ where $F$ is the elastic force, $dx$ is the displacement, $T$ is the temperature and $dS$ is the entropy change. Next, approximating $dE$ as zero, and solving for $F$, $$F = -T\frac{dS}{dx}$$ which is supposed to tell us the that the force is proportional to the rate $dS/dx$ at which the rubber band's entropy changes. But how is there any chage in entropy when no heat is being added? And, for that matter, if I take a string, lying all entangled, and try to straighten it, why doesn't _it_ exert an entropic force and try to go back to its higher entropic state?


title: Decompressing gas whilst driving a turbine - where does the energy come from?

text: Here's the situation: a friend is working on a civil engineering project to make some equipment to decompress liquefied natural gas before pumping to households. I don't know what the proper term for this bit of kit is calld but I will call it the **decompresser** Apparently what happens in these stations is that the gas is decompressed and because the gas cools down during expansion, it has to be heated back to something like room temperature before going on its way through the gas pipes to homes. They do this by burning kerosene. Now my friend says that they have designed a new decompresser that, in order to be more energy efficient, will generate electricity by decompressing the gas through a turbine. It is a long time since I did any related physics but the simpleton-scientist- thought-process that I have going on in my head is something like the following: -   * if the pressurised gas is having to do work turning the turbine, then this energy has to come from somewhere    * thus the temperature of the resulting de-pressurized gas will be lower.    * Because the temperature of the gas is lower, they would have to put more energy than they would have done to reheat the gas and they would not have any net energy gain. They would have electricity but they would be burning an equivalent amount of kerosene.  Now assuming that these civil engineers know a thing or two about thermodynamics, I readily admit that my thinking is wrong. I would very much appreciate it if someone could put me right on whether or not they can generate electricity as well as have the gas come out at the same temperature and pressure and could they not just have used the extra kerosene to generate the same amount of electricity (ignoring the various differing efficiencies of energy conversion that are going on). **PS.** Because I am clueless as to the actual answer, I will be relying heavily on the upvoting of + comments about the answers when it comes to marking an answer as the right 1!


title: What is the symmetry which is responsible for conservation of mass?

text: According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation. What kind of symmetry creates the conservation of mass?