title: Is it possible to bind an entangled electron to the outer shell of an atom or just a proton?

text: Say we start with an entangled electron--positron pair and we separate them. I want to take the entangled electron and bind it to a proton or the outer shell of an atom. Is it possible to do this while having the entangled electron still remain entangled with the positron? Would the electromagnetic interaction with the proton "make a measurement" of the entangled electron's spin because of its magnetic field? Assuming my above example is possible, say we then measure the spin of the positron that's far away. Whatever it may be, the entangled electron bound to the atom will now assume the opposite spin state. The total angular momentum of the atom will now be at a different value than what it was before, and therefore something must happen to conserve this, emit a photon? Again, if my above example has a measurable effect whether it emits a photon or changes in some way, then this suggests to me that if we have an ensemble of these special atoms, we will be able to send information via quantum entanglement. By having these special atoms in one station and the entangled positrons in another station, we can measure or not measure, 1 or 0, the spin of the entangled positron and thus produce a measurable effect at the other station via the special atoms.


title: Oscillation of a rolling sphere in a bowl

text: This is a homework task. I already came to a result but I am very unsure. The task: In a bowl with the shape of a semi-circle ($R$ = 0.5m) a sphere (there is no specification for the size of the sphere) is rolling without friction. Calculate the period of the oscillation. Use approximation for a small amplitude.   Original problem description (German):               In einer Wanne mit halbkreisförmigem Querschnitt (Radius $R = 0.5m$) rollt ein kleines     Kügelchen reibungsfrei hin- und her.      ![Sphere oscillating by rolling in bowl](http://i.stack.imgur.com/PSXoS.jpg)               Berechnen Sie die Schwingungsfrequenz dieser Bewegung für kleine Auslenkungen.      My solution: Using Newton 2 I make up the equality of torque: The first term is the inertia of the sphere according to the linear movement, the second term is the torque caused by gravity and the third term is the inertia according to the rotation of the sphere, $r$ is the radius of the sphere. $$mR^2*\frac{d^2\phi}{dt^2} + mgR*\phi + \frac{d\phi}{dt^2}*\frac{R}{r}*\frac25mr^2 = 0$$ $$\frac{d^2\phi}{dt^2}*(mR^2+\frac25mr^2\frac{R}{r}) = - mgR*\phi$$ $$\frac{d^2\phi}{dt^2} = - \frac{mgR}{(mR^2+\frac25mr^2\frac{R}{r})}*\phi$$ $$\frac{d^2\phi}{dt^2} = - \frac{g}{R+\frac25r}*\phi$$ This is a standard differential equation for oscillation and solving it yields: $$T = 2\pi\sqrt{\frac{R+\frac25r}{g}}$$ for the period $T$. It makes sense that $T$ is higher the bigger the radius of the sphere is but I can't calculate it with $r$ being unspecified. Is there an error in my solution or is the task unsolvable?


title: Non-Equilibrium Statistical Mechanics

text: Can anybody please explain what is the difference between equilibrium state and steady state, as quoted by book by Degroot and Mazur. Also, does violation of Principle of Detailed Balance means the system will not reach equilibrium?


title: Generators and electric motors

text: In a diesel generator, instead of using diesel engine can we use an electric engine and connect the motor and generator to the battery? will this make it an infinitely moving system or will the electricity get reduced due to resistence? can we connect the generator to another generator and thus increase the energy generation. Will this make it an infinitely running system.[my knowladge of physics is very less, please dont laugh!!] ![enter image description here](http://i.stack.imgur.com/0CS20.jpg)


title: Merger happening tangentially, but dark matter at both sides?

text: According to this news, > The expectation of "unaccounted energy" comes from the fact the merger of > galaxy clusters is occurring tangentially to the observers' line-of-sight. > This means they are potentially missing a good fraction of the kinetic > energy of the merger because their spectroscopic measurements only track the > radial speeds of the galaxies. Read more at: http://phys.org/news/2014-04-hubble-team-monster-el- gordo.html#jCp What strikes as a surprising is that the picture shows the dark matter distribution (inferred from weak lensing) in blue hue, and it shows a similar pattern that Bullet Cluster, even though in the Bullet Cluster case, the collision happen perpendicular to the line-of-sight Any idea why so much discrepancy between normal matter and dark matter distribution along that axis?


title: How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system?

text: I have a material specified by a permittivity tensor written in Cartesian coordiantes: $$\begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz}\\\ \epsilon_{yx} &\epsilon_{yy} & \epsilon_{yz}\\\ \epsilon_{zx} & \epsilon_{zy} &\epsilon_{zz} \end{pmatrix}$$ I want to write all my equations in cylindrical coordinates and so I have to write the permittivity tensor in cylindrical coordinates too. How can I do that?


title: How to isolate and verify the source of optical absorption/effects?

text: This is similar to my other question, but not the same -- that one was about the energy ranges of various absorption mechanisms, and this one is more about experimental techniques to find them. Let's assume you've taken transmission and reflection spectra from the mid-IR to the UV of a sample. It has various dips and peaks, to keep it very general. Now of course you can use the information my previous question was about, for clues -- if an absorption peak occurs around 0.1eV, you might assume it's a phonon mode. But for most real samples, all these different effects are usually very broad, overlapping, of different relative magnitudes, and not at the exact places they are in this image: ![enter image description here](http://i.stack.imgur.com/DqoEW.png) So, how do you confirm that it's what you suspect, in the case of, for example, phonon absorption? What experimental techniques exist for isolating and confirming the other major absorption mechanisms too?


title: Why is there an emission of gamma rays?

text: When a spontaneous radioactive reaction happens, there is an emission of gamma rays (in most cases) What causes this emission?


title: What is the most effective dielectric material known to man?

text: What is the most effective dielectric medium/material (highest breakdown voltage) known to man? Is it the vacuum? or are there any materials which could be more effective than the vacuum?


title: Kinematics - ball bouncing off vertical walls

text: So here's my best attempt to state the question: A ball is thrown from a 100 meter tall tower, which has a vertical wall in front of it. The initial velocity of the ball is 10m/s and the distance between the tower and the wall is 10m. Find the distance from the foot of the tower that the ball falls. (The ball obviously comes down bouncing, alternating between tower and wall. g = 10m/s^2) What I did: sqr() means square root Range of the ball I found to be 20sqr(5) Thus the ball does 4 bounces, and falls at a distance 20sqr(5) - 40 meters from the base of the tower. Any problem in this? Also, this is a very.... ummm..... temporary sort of way to find it (if it is correct). Any other, more beautiful and elaborate way to find it? Thanks :D


title: Covariant derivative-Differential

text: I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be $\dfrac{\partial}{\partial t^\prime}=\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial t^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial t^\prime}$ $\dfrac{\partial}{\partial x_1^\prime}=\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial x_1^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial x_1^\prime}$ I was trying to understand but I'm afraid I can't.. if oyou for instance add the terms in the second part of the equations you get $\dfrac{\partial}{\partial t}\dfrac{\partial t}{\partial t^\prime} + \dfrac{\partial}{\partial x_1} \dfrac{\partial x_1}{\partial t^\prime}=\dfrac{\partial}{\partial t^\prime} + \dfrac{\partial}{\partial t^\prime} =2 \dfrac{\partial}{\partial t^\prime}\neq \dfrac{\partial}{\partial t^\prime}$ How is that possible?


title: Applications of a certain wave equation in Physics?

text: I am doing research in the field of number theory and as part of this looking for correspondencies to other discilines and particularly physics. I am searching for examples in physics where the equation below finds its application: $$\frac{\partial^2\psi}{\partial x^2}=\psi(x)+ x \frac{\partial\psi}{\partial x} \qquad (*)$$ under the constraint $$\Psi(x)=\sum_{k=-j}^j \psi(x,k,j)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$$ Particularly inetresting also whether this equation $(*)$ comes in other notations such as in classical quantum mechanics. Please see this as a research type of question rather solving a mathematical problem. I would really appreciat your help. Thanks a lot PS: although this question relates to Number theoretical function applied in physics? but it is not the same, here it is really about the differential equation.


title: Guitar strings and temperature

text: I am investigating Mersenne's law with a guitar by varying tension (hanging weights) and string length. Will temperature change (room temperature to ~4°C) effect the frequency noticeably? If so, is the string oscillating differently or is the change due to a variation of the speed of sound? The strings have a free end so the contraction of the string will not increase tension. Any help would be appreciated.


title: Classification of small extra-solar system bodies

text: A Small Solar System Body (SSSB) is an object in the Solar System that is neither a planet, nor a dwarf planet, nor a satellite. This encompasses all comets and all minor planets, as well as all asteroids (that are classed as minor planets). I want to know if there is a term for classifying Small Extra Solar System Bodies. What about comets and minor planets in systems other than our own. Do they have a classification? Is there also a recognised categorisation for Small Bodies regardless of whether they are in the solar system or not? In the same way the 'solar system' pertains to our system but a 'planetary system' means any system including but not limited to our own.


title: superhorizon mode and cmb

text: During the period of inflation, perturbations went out of the Hubble horizon (Hubble radius, R = 1/aH) because the Hubble radius was decreasing during that time. After inflation, the Hubble radius increased in size and perturbations re-entered the horizon. Well, other perturbations are just re-entering right now while the Hubble radius continues to increase. Perturbations with short wavelength entered earlier and therefore became subjected to microphysical processes. Perturbations with large wavelength entered late and if they entered late enough during the matter-dominated era, they effectively become constant. Superhorizon perturbations are those with wavelength far larger than the Hubble radius and a given perturbation maybe superhorizon long time ago but became sub horizon at some point in time. What I'm confused about after reading some papers, is that superhorizon perturbations with wavelengths far larger than the current Hubble radius can leave an imprint on the current CMB radiation that we are receiving. How is it possible that those CURRENTLY superhorizon-mode perturbations have an effect on the CMB that we are observing now? This idea came out of various suggestions on the possibility of explaining the dipole asymmetry by using superhorizon modes that are far larger than the current Hubble radius.


title: Stellar remnants in a state of matter denser than neutron-degenerate

text: When discussing the stellar life cycle, it's often stated that if the collapsing core of a star is bigger than the mass limit for a stable neutron star, it must collapse to a black hole. However, could there be additional states of matter, denser than neutron- degenerate matter, that could form a stable stellar remnant without collapsing all the way to a black hole? For example, I've seen a bit about hypothetical "quark matter" on Wikipedia, but there's not much info about it. Could there also be some type of "GUT matter" based on the unknown fundamental physics at the GUT scale? Or does current knowledge make this unlikely?


title: Calculating the (on-shell) action of a free particle

text: I am having difficulty with the first problem from Feynman and Hibbs' book. > _For a free particle $L = (m/2)\dot{x}^2$. Show that the (on-shell) action > $S_{cl}$ corresponding to the classical motion of a free particle is > $$S_{cl} ~=~ \frac{m}{2}\frac{(x_b - x_a)^2}{t_b - t_a} $$ where we have > that $x(t_a) = x_a$ and $x(t_b) = x_b$._ I understand that the action is $$S ~=~ \int_{t_a}^{t_b} \frac{m}{2}\dot{x}^2 \,dt.$$ But I do not know how to solve the integral $\int \dot{x}^2\,dt $. Any help is appreciated. Following Noldig's comment, integrating by parts we have that: \begin{eqnarray} \int_{t_a}^{t_b} \dot{x} \dot{x} \,dt & = & \left. \dot{x} x\right|_{t_a}^{t_b} - \int x \ddot{x} \,dt \\\ & = & \dot{x}x(t_b)-\dot{x}x(t_a) \end{eqnarray} As the velocity is constant it is given by $\dot{x} = (x_b-x_a)/(t_b-t_a)$ and the result follows.


title: Superconducting Wavefunction Phase (Feynman Lectures)

text: In Volume 3, Section 21-5 of the Feynman lectures (superconductivity), Feynman makes a step that I can't quite follow. To start, he writes the wavefunction of the ground state in the following form (21.17): $\psi(r)=\rho(r)e^{i\theta(r)}$ If the density $\rho^2$ is approximated to be constant throughout a superconducting block, then Feynman says (21.18) that the (probability) current density can be written $J=\frac{\hbar}{m}\left( \nabla\theta-\frac{q}{\hbar}A \right)\rho$ By insisting that the divergence of the current be zero, Feynman shows that the Laplacian of the phase is zero. (Assuming $A$ is chosen to have zero divergence). $\nabla^2 \theta =0$ **I follow everything up to here.** Then he states that, for a single lump of superconducting material (by which I assume he means finite and simply-connected) this implies $\theta=0$. **I don't understand that step**...I recognize that the Laplace equation has $\theta=0$ as its unique solution if the boundary conditions are $\theta=0$. But the implied boundary condition for the superconducting chunk I would assume is only $J=0$ normal to the boundary (so no current flows in/out), which is not equivalent to $\theta=0$. **Now, for concreteness, let me choose a B-field** $B=B_0\hat{z}$. Then one choice of $A$ is $A=B_0x\hat{y}$. This choice makes $\nabla\cdot A=0$. In fact, if we use this B-field, then we can set $\nabla\theta=\frac{qB_0x}{\hbar}\hat{y}$, so that $J=0$ everywhere. The divergence of $\nabla \theta$ is zero, so Laplace's equation is satisfied, and we can integrate this up to get a wavefunction $\psi(r)=\rho e^{i qB_0xy/\hbar}$ So what have I done wrong? Why does Feynman say $\theta=0$? This seems important as the next step results in the a London equation. Thanks!


title: How can people do music with Tesla coils?

text: I saw a lot of videos of Tesla coils doing music on YouTube. And I wonder how can they do that sort of things. How they can calculate what tone it is going to do? And what are the factors to consider?


title: What chemistry course to take for a physics undergraduate

text: This question is mainly aimed at people who know about the science market in the UK, but any suggestions would be much appreciated. I am a third year physics undergraduate, studding for an MSci (4-year master course). For my A-levels I took: Maths, Further Maths and Physics. This was fine up until now, when I realised that a lot of cool physics applications revolve around chemistry, biology and medicine - which is an area I am considering going into. My chemistry knowledge is very patch, probably barely GCSE level and I do not feel confident using it at all. I would really like to fix this so am considering taking a course in chemistry in my own time. So my question is:   * A) Do people think taking a course is necessary to go into medical physics or a similar field?   * B) What sort of course should I take? Should I study a A-level or does anyone know a course specifically tailed for people in my situation. Again thanks in advance for any suggestions.


title: A question on the doped Kitaev-Heisenberg model?

text: Recently, some groups have studied the effects of doping the Kitaev model on honeycomb lattice(e.g.,http://arxiv.org/abs/1109.6681 and http://arxiv.org/abs/1109.4155) and their calculations show the existence of a topological p-wave superconducting phase. My question is as follows: As we know,one of the amazing properties of the Kitaev model is that the low energy physics(ground states properties and excitations above it) of the model does not depend on the signs of the coupling constants in the Hamiltonian. So for the (hole)doped case, is the existence of topological p-wave superconducting ground states still insensitive to the signs of the couplings we choose? Thanks in advance.


title: The critical point of Bose-Hubbard model

text: The Hamiltonian of Bose-Hubbard model reads as $$H=-t\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)-\mu\sum\limits_in_i$$. In the limit $t\ll U$, the ground states is Mott insulator (MI) states; in the limit $t\gg U$, the ground state is superfluid (SF) state. How to calculate the critical value of $\frac{t}{U}$ of the phase transition between SF and MI?


title: When is classical mechanics valid for describing motion of atoms?

text: In Molecular Dynamics simulations, the Newton's equation of motion is used to calculate the time evolution of system. Once, I read in an introductory text that when the thermal de Broglie wavelength $$\Lambda=\frac{h}{\sqrt{2\pi mkT}}$$ is much smaller than the interparticle distance, using classical mechanics is justified and it can be used instead of quantum mechanics. Why? I mean I'd like to start from the Schrodinger equation or a theorem which is based on it (e.g. Ehrenfest's theorem) and using the above criterion obtain the Newton's equation of motion. May you help me?


title: Can a moon have another large body as a satellite, and are there any examples of such?

text: In my mind, I'm comparing it to the Sun-Earth-Moon system. After all, the Earth is primarily a satellite of the Sun, but the Moon is still gravitationally bound to the Earth. Could something like this occur on a smaller scale? And are there any examples?


title: What equipment is needed for equatorial drift scanning/TDI with a Dobsonian?

text: It should be possible to do a "parked scan" along the celestial equator ("stripe 82" and "stripe 10") with a Dobsonian telescope, as no tracking is required. I can foresee two issues, though: first, the camera software must support drift scanning; second, it must be possible to make fine adjustments to the orientation of the camera, so the columns are precisely aligned with the celestial equator. Are these practical with "off-the-shelf" amateur equipment, such as an SBIG camera and a Celestron or Orion Dobsonian? If there is stuff I need to build by hand (a rotator, for example), how much skill and effort should I expect it to require? Are there any other major issues have missed?


title: Metal Dilemma: Only very few civilizations per galaxy

text: This is a question in the area of extraterrestrial life. While life may be possible without it, space travel is probably impossible without metal. Metal is created in stars and heavy metals (above iron) are created in super novae. In a German posting, someone mentioned that there are only very few stars outside of the galactic core which contain a lot of metal. There is little hope for life in the core itself but if metal is so rare outside the core, that surely limits the options for aliens to build civilizations. My question: We can easily measure the amount of metal in a star by looking at its spectrum. Does that give a good indication of the metal distribution of dark bodies? Is there a theory about the relation between elements in the star and its planets? I also read somewhere that there seem to be a lot of "free" planets (which don't orbit a sun). Does this also apply to them?


title: Does every hermitian operator represent a measurable quantity?

text: In Quantum mechanics, observables are represented by hermitian operator. But does every hermitian operator represent a observable? If not , how do we know that whether a hermitian operator represent observable or not? What is the precise definition of the term "observable"?


title: What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?

text: Ehrenfest 1931 gives an argument to the effect that the application of the spin-statistics theorem to composite systems is valid, but only as an approximation and under certain conditions. Unfortunately this article is behind a paywall. The abstract reads as follows: > From Pauli's exclusion principle we derive the rule for the symmetry of the > wave functions in the coordinates of the center of gravity of two similar > stable clusters of electrons and protons, and justify the assumption that > the clusters satisfy the Einstein-Bose or Fermi-Dirac statistics according > to whether the number of particles in each cluster is even or odd. The rule > is shown to become invalid only when the interaction between the clusters is > large enough to disturb their internal motion. The quaintness of the abstract is enhanced by the mysterious failure to mention neutrons -- this being explained by the fact that the neutron was not discovered experimentally until the following year. Pretty amusing that the American Physical Society still owns the copyright on a paper this old, one of whose authors was probably paid through my grandparents' and great- grandparents' income taxes. Googling quickly turns up various loose verbal characterizations of this result, without proof. Can anyone supply an outline of how the argument works and possibly a more precise statement of what the result really is? Some sources seem to refer to this as the Wigner-Ehrenfest-Oppenheimer (WEO) result or Ehrenfest-Oppenheimer-Bethe (EOB) rule. Historically, Feynman seems to have used this result to show that Cooper pairs were bosonic. Although I've tagged this question as quantum field theory, and the spin- statistics theorem is relativistic, the 1931 date seems to indicate that this would have been a result in nonrelativistic quantum mechanics. This answer by akhmeteli sketches what seems to be a similar relativistic result from a book by Lipkin. In the limit of low-energy experiments, the EO rule says the usual spin- statistics connection applies, and this seems extremely plausible. If not, then it would be too good to be true: we would be able to probe the composite structure of a system to arbitrarily high energies without having to build particle accelerators. Bethe 1997 has the following argument: > Consider now tightly bound composite objects like nuclei. It then makes > sense to ask for the symmetry of the wavefunction of a system containing > many identical objects of the same type, e.g., many He4 nuclei. This > symmetry can be deduced by imagining that the interchange of two composites > is carried out particle by particle. Each interchange of fermions changes > the sign of the wavefunction. Hence the composite will be a fermion if and > only if it contains an odd number of fermions[...] Note the qualifier "tightly bound," which implies a restriction to low-energy experiments. The abstract of Ehrenfest 1931 seems to say that this is a necessary assumption, but it is never used in Bethe and Jackiw's argument, and this suggests that their argument is an oversimplification. Fujita 2009 recaps the Bethe-Jackiw argument, but then says: > We shall see later that these arguments are incomplete. We note that Feynman > used these arguments to deduce that Cooper pairs [5] (pairons) are bosonic > [6]. The symmetry of the many-particle wavefunction and the quantum > statistics for elementary particles are one-to-one [1]. [...] But no one-to- > one correspondence exists for composites since composites, by construction, > have extra degrees of freedom. Wavefunctions and second-quantized operators > are important auxiliary quantum variables, but they are not observables in > Dirac's sense [1]. We must examine the observable occupation numbers for the > study of the quantum statistics of composites. In the present chapter we > shall show that EOB's rule applies to the Center-of-Mass (CM) motion of > composites. Unfortunately I only have access to Fujita through a keyhole, so I can't see what the references are or any of the details that they promise in this beginning-of-chapter preview. Related: Huge confusion with Fermions and Bosons and how they relate to total spin of atom Bethe and Jackiw, Intermediate Quantum Mechanics P. Ehrenfest and J. R. Oppenheimer, "Note on the Statistics of Nuclei," Phys. Rev. 37 (1931) 333, http://link.aps.org/doi/10.1103/PhysRev.37.333 , DOI: 10.1103/PhysRev.37.333 Fujita, Ito, and Godoy, Quantum Theory of Conducting Matter: Superconductivity, 2009


title: Stress due to the mismatch of thermal expansion coefficients of two different attached materials

text: I'm simulating the thermo-electro-mechanical behavior of a copper wire which is surrounded by silicon dioxide. In other words, the wire segments is under mechanical and thermal loads and at the same time an electrical current is flowing in it. In order to perform the simulation accurately, I think that I should consider an initial stress due to the mismatch of thermal expansion coefficients between copper and oxide. I am using COMSOL to do my simulation. The options that COMSOL provides me with are "body load" and "boundary load". Can anyone advise me which one I should choose? Which one is a more realistic assumption. Thanks.


title: Nuclear Weapon Effects: Cloud Rise

text: I am interested in modeling the rise of a nuclear cloud to stabilization height post detonation. I am interested in any ideas you may have on the approach. I have collected some research on the topic but I am curious as to how someone might think through the problem without being exposed to one possible solution.


title: cooling glass of milk glass and a jar of water

text: When im given a steel glass of boiling milk which I cannot even touch, I used to put that glass in a jar surrounded with room-temp water and give it 10 mins. At the end, I used to find the steel glass is very mildly warm but the water itself is very hot. I never understood is water continuously absorbing heat from the milk glass even when the water itself gone warmer? I thought they should ideally be at the same temp at the end.


title: Do bifacial solar panels operate at lower temperature than normal solar panels?

text: Bifacial solar panels have a glass backplane instead of the usual aluminum and are designed to absorb and convert to electricity light incident on either face. Many manufacturers claim that their bifacial panels operate at lower temperature than normal (monofacial) panels because bifacial panels transmit the 20% of the solar spectrum that is too low-energy to be absorbed by silicon. See for example this manufacturer's datasheet. Meanwhile, the only reputable study I could find (apologies for the paywall) was of a lab-scale cell and not a commercial module. That study argues that convection is the dominant heat transfer mechanism and so the differences in optical absorption and emission aren't relevant. However, temperature differences of a few degrees centigrade can have a noticeable impact on solar cell efficiency and useful life. Anybody have a compelling argument or know of other relevant experimental work on this topic?


title: Kinetic energy when $F(t)$ graph is given

text: ![enter image description here](http://i.stack.imgur.com/jBPTL.png) The very basic problem here is that the point where this graph ends is ambiguous. I mean, what is the corresponding $y$-value for the point whose $x$-valu is $4.5s$? Then I think we should find the area, then that will be the momentum, so use $\frac{p^2}{2m}$, as mass is given. So if theres any error in this, pls let me know...


title: Eigenvector Grover Operator

text: I have a question about the eigenvectors for the evolution operator of Grover's algorithm. Let $U=R_DR_f$, where $$\begin{align*} R_D &= 2|D\rangle\langle D| -I_N , \\\ R_f &= I-2|x_0\rangle\langle x_0| \end{align*}$$ where $x_0$ is the (unique) marked element to find, and $|D\rangle$ is the uniform superposition of standard basis states. I have three eigenvalues: $e^{\pm i(1-2/N)}$, and $-1$ with multiplicity $N-2$. My lectures say: Assuming that $x_0 = 0$, one eigenvector associated to $-1$ is $$|\alpha\rangle = \dfrac{|1\rangle-|2\rangle}{\sqrt{2}}.$$ I can verify this: $$\begin{align*} \Bigl(2|D\rangle\langle D| -I_N\Bigr)&\Bigl(I-2|0\rangle\langle 0|\Bigr)\left(\dfrac{|1\rangle-|2\rangle}{\sqrt{2}}\right) \\\&= \Bigl(2|D\rangle\langle D| -I_N\Bigr)\left(\dfrac{|1\rangle-|2\rangle}{\sqrt{2}}-\dfrac{2|0\rangle \langle 0| 1\rangle}{\sqrt{2}}+\dfrac{2|0\rangle \langle 0| 2\rangle}{\sqrt{2}}\right) \\\&= \Bigl(2|D\rangle\langle D| -I_N\Bigr)\left(\dfrac{|1\rangle-|2\rangle}{\sqrt{2}}\right) =-\dfrac{|1\rangle-|2\rangle}{\sqrt{2}}. \end{align*}$$ My question is: in the denominator of $|\alpha\rangle$, can we change the $\sqrt{2}$ to $\sqrt{y}$ for any $y\in \mathbb{R}$? I ask this because the particular scalar factor does not seem to affect the operation.


title: What will the universe look like for anyone falling into a black hole?

text: I've heard that, from the perspective of an external observer, something falling into a black hole will eventually look "frozen": light waves will move to the infrared and further into lower frequencies. But what about that unfortunate individual falling into such a black hole? What will he/she see when looking back at our universe? For instance, I wonder, will it be the ultimate end of the Universe? Can someone who just crossed events horizon of the black hole still receive messages from our side?


title: How much up will an object of mass $m$ will be vertically lifted if a constant acceleration $a$ is given so that resulting force is more than weight?

text: Yesterday My friend asked me a question which put me into a confused state. If a body of mass m is given a vertical thrust of $F$ such that $F > mg$, and the $F$ is allowed to remain applied on the body with constant acceleration, how high will the mass be lifted off ground? He confused me asking that, as the body gains height, its potential energy should increase. So the body should stop going upward at some stage. Though I said he is wrong, as energy is scalar and not vector, a doubt haunted me. If I am able to give the force $F$ as said above constantly, will the body keep on going to outer space, as the weight $mg$ will never increase, but only decrease due to decrease in $g$ with altitude.


title: Standards for scientific metadata in common-use image formats?

text: I have a bunch of JPEG, JPEG2000 GIF and PNG files that in theory could be used for some types of science analysis if there were temporal and spatial information in them. Years ago, I had looked into trying to put some of the information as EXIF into the files, and I could enter timing information (DateTimeOriginal, SubsecTimeOriginal, ExposureTime), and even location of the main object of interest (SubjectArea), but it had some issues with distance to the object (SubjectDistance), as it stored it as a rational number (LONG/LONG), in meters, and so the farthest distance I could specify was about 0.014 AU. Is anyone else aware of people embedding scientific metadata in common-use graphical formats? And if so, can you point me towards the specification, project, or implementation? (and other than GeoTIFF or the like ... we just don't use TIFF except for sending to museums; I'm interested in trying to put the critical metadata into the browse images that we're generating for the scientists)


title: Sky projections in sky view software

text: I'm playing with an idea of making a little program to draw the sky. Nothing fancy, just for fun. And I'm reading about projections. Now I know that I must use different projections depending on my field of view. My idea is to start with fairly large field of view  (FOV), say 45 deg (and therefore with a large distortion) and be able to reduce the FOV ("zoom in"). What should I use? Another thing is with transitions between two projections. They should look almost the same for at least one FOV, to make it smooth.


title: Closed cage with an helium balloon?

text: Does a closed cage with an helium balloon weigh the same as an empty closed cage? What if the balloon isn't touching the ceiling ? (It's a tall cage and we're weighing it in the process of the balloon elevating)


title: Atom state vectors kets

text: An atom with two energy levels has 2 states (excited and ground), represented by kets $|e\rangle$ and $|g\rangle$ respectively. The atom has energy $\frac{1}{2}E_\theta$ when excited and $-\frac{1}{2}E_\theta$ when in ground state. Suppose it is prepared in the state $$|\Psi\rangle = \frac{1}{5}((2-i)|e\rangle + (4+2i)|g\rangle) $$ How do I show that the physical state of the system can be equally well described by the state vector written in the form $$|\Psi\rangle = \cos(\frac{1}{2}\theta)|e\rangle + \sin(\frac{1}{2}\theta)e^{i\phi}|g\rangle $$ i.e determine the values of $\theta$ and $\phi$ please help me if you can as this was in my textbook which has no solutions


title: Reflection vs indices of refraction

text: A magician during a show makes a glass lens with refractive index 1.47 disappear in a trough of liquid. What is the refractive index of the liquid? Could anyone please demonstrate with a ray diagram?


title: state occupation rate $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+{[1/-1/0]}}$ & density matrix $\rho _{m}=\frac{e^{-\frac{E_{m}}{kT}}}{Z(T)}$

text: Three kinds of distributions. The **states occupation rates** : ## **F.D.** $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+1}$ **B.E.** $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}-1}$ **Boltzmann** $n_{i}=e^{-\beta (\varepsilon _{i}-\mu )}$ $(i$ labels different energies for a single particle, for the same $i$ there are still $G_{i}$ degeneracy.$)$ While, the **density matrix** for **all three kinds of distribution** is $\rho _{m}=\frac{e^{-\frac{E_{m}}{kT}}}{Z(T)}$ $(m$ labels different states of the $N$ particle system.$)$ **It seems that Boltzmann distribution is more close to the density matrix expression. why is that?** $1$. If there is only one particle in the system, then $\varepsilon _{i}=E_{i}$ $n_{i}=e^{-\beta (\varepsilon _{i}-\mu )}=\rho _{i}=\frac{e^{-\frac{E_{i}}{kT}}}{Z(T)}$ $Z(T)=e^{-\mu }$ $2$. If there are $N$ particles in the system, then $\sum_{all\, particles} \varepsilon _{i}=E_{n}$ $\rho _{n}=\frac{e^{-\frac{E_{n}}{kT}}}{Z}= \frac{e^{-\frac{\sum \varepsilon _{i}n_{i}G_{i}}{kT}}}{Z}=\frac{\prod \left (e^{-\beta \varepsilon_{i} } \right )^{n_{i}G_{i}}}{Z}=\prod \left ( e^{-\beta (\varepsilon _{i}-\mu )} \right )^{n_{i}G_{i}}=\prod_{all\, particles}n_{i}$ $Z(T)=e^{-\mu N }$ $(i$ labels both different particle and its own energy number.$)$ Boltzmann distribution has more obvious analogy with density matrix distribution. **My question is: Is the analogy right? Can we still get the result** $Z(T)=e^{-\mu N }$ $\sum_{all\, particles} \varepsilon _{i}=E_{n}$ $\rho _{n}=\prod_{all\, particles}n_{i} $ **for F.D. and B.E. distributions?**


title: Lagrangian Field Entropy?

text: Has anyone investigated the inclusion of a field's entropy in its Lagrangian? I read from information theory that the entropy $H$ of a scalar field $\phi$ distrusted across space is defined as \begin{align} H = \int -\lambda^2|\phi|^2 \ln(|\phi|^2) dx_0 , \end{align} where \begin{align} \int |\phi|^2 dx_0 = 1 \end{align} and $\lambda$ is an unknown real scalar. I know that $|\phi|^2$ is phase invariance, thus it can be included in its Lagrangian. I can define a Lagrangian of a complex scalar massive field as \begin{align} \mathcal{L} = (\partial_n \phi)^*(\partial^n \phi) - m^2 |\phi|^2 + \lambda^2 |\phi|^2 \ln(|\phi|^2) , \end{align} which produces the following field equations \begin{align} \partial_n\partial^n \phi \+ (m^2- \lambda^2)\phi \- \lambda^2\phi \ln(|\phi|^2) =0 . \end{align} I choose the sign of the entropy $H$ in the Lagrangian to be negative so that if the Lagrangian action is minimized when the kinetic term $(\partial_n \phi)^*(\partial^n \phi)$ is minimized, then the Lagrangian action also has the option to minimize the action by maximizing the field entropy $H$. I am not sure if the sign chosen for the entropy is correct.


title: How do we know that heat is a differential form?

text: In thermodynamics, the first law can be written in differential form as $$dU = \delta Q - \delta W$$ Here, $dU$ is the differential $1$-form of the internal energy but $\delta Q$ and $\delta W$ are inexact differentials, which is emphasized with the replacement of $d$ with $\delta $. My question is why we regard heat (or work) as differential forms? Suppose our system can be described with state variables $(x_1,\ \cdots ,\ x_n)$. To my understanding, a general $1$-form is written as $$df = f_1\ dx_1 + f_2\ dx_2 + \cdots + f_n\ dx_n$$ In particular, differential forms are _linear_ functionals of our state variables. Is there any good reason to presuppose that $\delta Q$ is linear in the state variables? In other words, if the infinitesimal heat transferred when moving from state $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_1$ is $\delta Q_1$ and from $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_2$ is $\delta Q_2$, is there any physical reason why the heat transferred from $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_1 + d\mathbf{x}_2$ should be $\delta Q_1 + \delta Q_2$? I apologize if the question is a bit unclear, I am still trying to formulate some of these ideas in my head. P.S. I know that for quasi-static processes, we have $\delta Q = T\ dS$ (and $\delta W = p\ dV$) so have shown $\delta Q$ is a differential form in this case. I guess my question is about non-quasi-static processes in general.


title: Doesn't the Schwarzschild metric combined with Hawking radiation imply that nothing ever gets past the event horizon of a black hole?

text: According to the General Theory of Relativity, the coordinate time distance per spacetime distance traveled by a particle freely falling into a black hole gets closer and closer to $0$ as the particle approaches the event horizon $$\mathrm{d}s^2 ~=~ -\left(1-\frac{2M}{r}\right)\mathrm{d}t^2 + \left( 1-\frac{2M}{r} \right)^{-1}\mathrm{d}r^2 + \dots.$$ Equivalently, an observer looking from the outside will never see the particle cross the event horizon, even if he/she looked for an arbitrarily long time. However, Hawking radiation implies that black holes don't last forever, but instead shrink and fade away? What happens to everything that is "currently" falling into the black hole during that time?


title: Limits of integration for the radial wave function of the Hydrogen atom in the WKB approximation

text: I am working a problem where we have to find the energy eigenvalues for the radial wave function of the hydrogen atom for $\ell=0$ using the WKB approximation. I am sure that I set up the integral properly, I am just a little confused as to how to find the limits. My integral is $$ \int \sqrt{2m \left( E + \frac{e^2}{r}\right)}dr = \left( n + \frac 1 2\right) \hbar \pi. $$ would it be $0\ to\ r_0? $


title: Is a Betelgeuse supernova able to neutralise earth's nuclear arsenal?

text: According to an article on newscientist.com, a neutrino beam could neutralise nuclear bombs by inducing a slow meltdown of the nuclear fuel. The neutrino generator > would need to be more than a hundred times more powerful than any existing > particle accelerator Seems not much to me, compared to the "$10^{46}$ joules, approximately 10% of the star's rest mass," which, wikipedia says, "is converted into a ten-second burst of neutrinos, which is the main output of a supernova" So, is a supernova able to neutralise earth's nuclear arsenal?


title: Smoothed Particle Hydrodynamics, Chemical Reactions and Stiffness

text: Dear people from stackexchange physics. I have been using SPH for quite a while to simulate free boundary flow, and just recently we tried to include some kind of (simple) chemical reaction in our code. We are still doing some tests, but one curious thing that we found was that we needed much smaller time steps than we usually used to be necessary. I have never dealed with chemical reactions, but I don know that some of them fall on the category of 'Siff Equations', and as far as the wikipedia description goes, it seems to be my case, but I would like to ask if anyone tried anything similar and also found that their equations were stiff. I never studied stiff equations, I just knew that they existed and recently read a bit more about them, but that's it. Last but not least, do anyone know how to check if a set of SPH equations is stiff or not? The wikipedia recomendation seemed at least complicated to calculate for SPH.


title: Interpretation in Fourier-Laplace domain

text: The Green's function describing the distribution of particles sent from $\def\v#1{\boldsymbol{#1}}\def\u#1{\hat{\v#1}}\v r=0$ at $t=0$ uniformly in every directions is, in two dimensions $$ G(r,t)=\frac{\delta(r-ct)}{2\pi r}.$$ Taking the Hankel transform, the equivalent of Fourier transform for function depending only on $r$ in two dimensions, and the Laplace transform for time, we find $$\widehat{\widetilde G}(k,s)=\frac{1}{\sqrt{s^2+c^2k^2}}.$$ Consider now a function $f$ that has a Laplace transform $\widehat f$. Then the function $$ \frac{f\left(\sqrt{t^2-r^2/c^2}\right)}{2\pi c\sqrt{t^2-r^2/c^2}}$$ has for Hankel-Laplace transform the function $$ c\frac{\widehat f\left(\sqrt{s^2+c^2k^2}\right)}{\sqrt{s^2+c^2k^2}} =\widehat{\widetilde G}(k,s)c\widehat f\left(\frac1{\widehat{\widetilde G}(k,s)}\right).$$ In other words, in a two dimensional space, any function of $\sqrt{t^2-r^2/c^2}$ (which looks like a relativistic proper time) in the space-time domain is a function of the isotropic propagator in the conjugated space. Can someone see a physical interpretation for this ?


title: Trapped Surfaces. Any good articles?

text: I'm currently writing a dissertation on trapped surfaces as minimal surfaces. I have exhausted all of the resources I have, and the internet is pretty limited (in that it is fairly repetitive on just defining what a trapped surface is). I wondered if anyone could recommend any articles/papers which delve into the more mathematical side or even more detail on these surfaces (in particular as the apparent horizon).


title: Absolute permittivity and permeability of free space

text: I'm reading about the propagation of EM waves in vacuum. We define the wave equation using two constants: $\epsilon_0$ and $\mu_0$. Of course they are absolute permittivity and absolute permeability of free space. I've read they have capacitive and inductive properties, I want to know how do we describe it **physically**? Or, if I could put it this way: I want to understand the purpose of the inductive and capacitive properties of these constants in the propagation of the waves.


title: Converting between brilliance, intensity, and flux

text: This one should be a bit of a softball, but I can't find it explicitly stated anywhere on the internet, and my basic unit analysis doesn't seem to work. Suppose you have a beam of synchrotron radiation with a brilliance $B$ at a given energy. (For instance, the brilliance of the Advanced Light Source at Berkeley is given in a graph form here). If I know the distance $d$ from the source and the area $A$ of the (for simplicity, unfocused) spot on my sample, how do I calculate the flux (units photons/sec) to my sample? Another way to phrase the same question is given the distance from the source and the brilliance, how do I calculate the intensity (units photons/sec/area) of the photon source at my sample? Am I missing some parameter?


title: What are the main differences between these three quantum theories: QM, QFT & QG?

text: What are the main differences between these three quantum theories?   1. Quantum Mechanics (QM),    2. Quantum Field theory (QFT),   3. Quantum Gravity (QG).


title: Special designs to reduce the electrical resistance of a wire

text: The numerical simulation of this nerdy question shows that the resistance decreases with the number of nodes along longest side, and converges to a finite value when the # of nodes approaches infinite. Here comes my question: **Can such grid or mesh design be utilized to reduce the resistance in wires/cables for long distance transport?** Let's say one unit-length segment of copper wire can be treated as one unit resistor, so a copper wire mesh is basically equivalent to a grid of resistors. Of course there exist many different ways to reduce the resistance, such as using a thicker wire (increase the cross section area), lowering the temperature, or even changing the material. But here I specifically wonder what simple circuit or topological design can do the trick, and whether it is practical or not.


title: Does decoherence need non-determinism?

text: Consider standard quantum mechanics, but forget about the collapse of the wavefunction. Instead, use decoherence through interaction with the environment to bring the evolving quantum state into an eigenstate (rspt arbitrarily close by). Question: Can this theory be fundamentally deterministic? If one takes into account that the variables of the environment are not known, then the evolution is of course 'undetermined' in a probabilistic sense, but that isn't the question. Question is if fundamentally quantum mechanics with environmentally induced decoherence can be deterministic. Note that I'm not saying it has to be. I might be mistaken, but it seems to me decoherence could be followed by an actual non-deterministic process still, so the decoherence alone doesn't settle the question of determinism or non-determinism. Question is if one still _needs_ a non-deterministic ingredient? Update: Please note that I asked whether the evolution can _fundamentally_ be deterministic, or whether it has to be non-deterministic. It is clear to me that for all practical purposes it will appear non-deterministic. Note also that my question does _not_ refer to the prepared state after tracing out the environmental degrees of freedom, but to the full evolution of system and environment. Does one need a non-deterministic ingredient to reproduce quantum mechanics, or can it with the help of decoherence be only apparently non- deterministic yet fundamentally deterministic?


title: Feynman's derivation of Bernoulli (part of it)

text: ![enter image description here](http://i.stack.imgur.com/w3e54.png) ![enter image description here](http://i.stack.imgur.com/YwZK2.png) OK, so what's up with the $\Delta M$ here? I don't get it. In fact, my textbook doesn't even put it in. I don't think it is a type-o because he continues on with the explanation. If anything I got it from this link (it's from the Feynman lectures) http://www.feynmanlectures.caltech.edu/II_40.html#Ch40-S3. Also, why do people assume these pressures are the pressures of the fluid itself? These pressures act upon the system of fluid, and are therefore outside of it! If this is true then the traditional explanation of "it has a lower pressure because it's speed is faster" is completely invalid; the correct version should be "because there was a pressure gradient the fluid accelerated". But if we stick to the latter explanation, how can one then explain Venturi's effect? What caused the pressure gradient in this case? I think in this case one just needs to assume conservation of energy per unit volume, and do consider the pressures of the fluid. Some people insist on the former explanation, but if that were true, then when deriving Toricelli's law, why do they consider the outside pressures and not the pressures in the fluid? I'm very, very confused.


title: Kinematics, why must the final velocity be 50km/hr?

text: but I am reading a paper on optimization which involves physics. http://www.optimization-online.org/DB_FILE/2011/04/2997.pdf In page 17 (pg 16 bottom where it starts), just before the end of the first paragraph in the top it says > Similarly to the previous subsection, the final velocity Vf is fixed but in > this case it has to be equal to 50 km/h. Why is that? Acceleration is not discussed in this paper. Is it because the final position is 100m that forces it to be 50km/hr?


title: Quick question on derivation of mass of star

text: How do I show that for a binary star system, if one star has mass $M_s$, speed $V_s$, period $P$, the mass of the other star is given by: $$M_P^3 \approx \frac{V_s^3}{2\pi G} PM_s^2$$ I showed it using Kepler's second law and using centripetal force motion. But I made the **huge** assumption of $(M_p+M_s) \approx M_s$. Starting with centripetal force equation: $$\frac{GM_pM_s}{(a_p+a_s)^2} = \frac{M_s v_s^2}{a_s} $$ Substituting in $PV_s = 2\pi a_s$: $$ m_p = \frac{2\pi(a_p+a_s)^2V_s}{PG} $$: Using kepler's second law: $P^2 = \frac{(a_p+a_s)^3(2\pi)^2}{G(M_p+M_s)} $ : $$M_p^3 = \frac{V_s^3}{2\pi G} P (M_p+M_s)^2 $$ I feel like I'm missing something here..


title: Extract the temperature dependence of the integral

text: Actually, I have a question regarding this Physics StackExchange question: Ashcroft Mermin Solid State Eq. 22.15 I cannot understand the change of variables. Can anyone give a further explanation on this?


title: Work done in non-uniform circular motion

text: In non-uniform circular motion, the work done by force should not be zero because tangential force acts on body. Is this correct? If I am wrong, please give a proper explanation.


title: Can energy be harnessed and collected safely from a magnetic/electric field within a tube with the following scenario?

text: Scenario: Is it possible to safely collect energy from the following: Friction of a metallic object moving around a circular tube that is filled with water Inside the tube, magnetic forces are at play to move the metallic (magnetic) object around and around. There is just enough air in the tubing to allow for supercavitation. When the metallic object picks up enough speed to create supercavitation, there would be small channels cut out of the tubing to collect the energy from a conductor of electricity that can be used immediately or stored. The only time the energy is released through the channels to the conductor is when the air bubble moving along with the metallic object moves past this channel, so that energy does not build up within the tube. The water would be the containment of the electrical field so it does not continue to grow rapidly. I would imagine that the water will eventually evaporate due to heat buildup, but you could ensure that water is added to the tube when needed and controlled by a level sensor. Is this possible to safely collect energy?


title: Thomas-Fermin approximation of cold atoms in a harmonic trap

text: In Bose-Hubbard model of cold atoms, one can use Thomas-Fermi approximation to get a rough number of total particle number. In Thomas-Fermi approximation, the site density is $n(R_i)=（\mu-\kappa R_i^2）/U$,where $\mu$ is the chemical potential and $\kappa$ is the trapping potential. The atomic cloud extends to the Thomas-Fermi radius $r_{TF}=\sqrt{\mu/\kappa}$. Why in Thomas-Fermi approximation, one can neglect the kinetic energy. Does it mean it only work for strong interaction regime? When I use Gutzwiller mean field method to calculate the total particle number in a trap, I find the total particle number varies a lot as the hopping amplitude varies.


title: buckling of tube - shell thickness vs. momentum of inertia optimum

text: is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly elastic modulus and length would also affect it ). I mean - certainly there is a compromise between:   * **maximizing area moment of inertia** in euler formula for long rod.   * **keeping walls thick enought** to prevent local buckling of wall shell (such as Yoshimura / Donnell's buckling ) In case you want to keep weight (crossectional area) as low as possible ( let's say constant ) and achive maximal buckling-limited-strength of the tube under axial compression - to which extend it would be be better to increase diameter (moment of inertia) and thus decrease wall thickness until the problems with local buckling start to prevail. I would expect that in practical structural engeneering this should be very common problem, and there should be some well developed cook-book solution. I found a lot of deep theoretical papers discussing various forms of local buckling from first principles and emperically, but I did not found simple Ready-to-wear prescription what parameters ideal thin-shell tubular pilar should have.


title: Can electromagnetic fields be used to shield electromagnetic radiation?

text: Can electromagnetic fields be used as shielding for electromagnetic radiation?


title: What are good resources for someone looking to become an amateur astronomer?

text: What resources would you consider to be essential for an amateur astronomer?


title: Finding angular momentum about the center of mass?

text: If we have a couple of particles of an equal, unknown mass: $$r_{+} = (c + e^{-Bt} \cos({\theta}))\textbf{x} + (d + e^{-Bt} \sin({\theta}))\textbf{y}$$ $$r_{-} = (c - e^{-Bt} \cos({\theta}))\textbf{x} + (d - e^{-Bt} \sin({\theta}))\textbf{y}$$ where $c$, $d$, $B$ are positive constants and $\theta$ is a function of $t$. How do we find the angular momentum about the center of mass? I get $(0, 0)$ and if it is constant, how do we find $\theta$ and the system's kinetic energy?


title: Magnetic field of uniformly charged rotating hollow sphere

text: I want to compute the magnetic field due to a homogeneously charged, rotating sphere with radius $R$, angular velocity $\vec{\omega} = \omega\hat{z}$ and total charge $Q$. I want to use Biot-Savart law, so I first computed the current density $\vec{j}(r,\theta,\phi) = \frac{\omega Q}{4 \pi R} \delta(R-r) \cdot (-\sin\theta \sin\phi,\sin\theta\cos\phi,0)^T$. After taking out all the constants with respect to integration and removing the integral over the radial component with the $\delta$ function, what stays in the Biot-Savart integral is $$\vec{B}(\vec{r}) \propto \iint_{\theta,\phi}^{\pi,2\pi}\frac{\vec{j}(\vec{y}) \times (\vec{r} - \vec{y})}{|\vec{r}-\vec{y}|^3}\sin{\theta}\;\text{d}\phi\;\text{d}\theta$$ where $\vec{r} = |\vec{r}|\cdot(\sin\theta_r\cos\phi_r, \sin\theta_r\sin\phi_r,\cos\theta_r)^T$ is the point where the $B$-field is evaluated, $\vec{y} = R\cdot(\sin\theta\cos\phi, \sin\theta\sin\phi,\cos\theta)$ is a point on the sphere. and the $\vec{j}$ is now missing the $\delta$ part, and some constants. The problem I have is, that the vector inside the integral is $$\begin{pmatrix} \sin\theta\cos\phi(|\vec{r}|\cos{\theta_r} - R\cos\theta )\\\\\sin\theta\sin\phi(|\vec{r}|\cos\theta_r-R\cos\theta)\\\\-|\vec{r}|\sin\theta\sin\phi\sin\theta_r\sin\phi_r-|\vec{r}|\sin\theta\cos\phi\sin\theta_r\cos\phi_r+R\sin^2\theta\end{pmatrix}.$$ Since all terms involving $\sin\phi$ or $\cos\phi$ will integrate to zero, this implies that I will get a $B$-field that is along the $z$ axis, no matter where it's evaluated. This can't be, because in the next part I'll show that far away the field looks like that of a dipole. I have been sitting over this for hours, can someone help me clear this up? Where am I going wrong?


title: Why is the Hodge dual so essential?

text: It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric field as a 1-form, then why do they show up in Ampere's and Gauss' as laws as their duals, i.e. $$ \int_{\partial M} \star \mathcal B^2 = \int \int_M \left( 4\pi j^2 + \frac{\partial \star \mathcal E^1}{\partial t} \right)$$ $$ \int \int_{\partial U} \star \mathcal E^1 = 4 \pi Q_{\mathrm{enc}} $$ Similarly, angular momentum is considered nearly everywhere as a pseudo-vector instead of as a 2-form. Do these laws have formulations that do not use Hodge duals? Is this just for the sake of simplicity, since tensors are less familiar to the physics community than vectors?


title: How to convert a FITS file to .xls Excel file?

text: We are trying to determine the isophots in elliptical galaxies in order to check De-Vaucouleurs law. To do so, we want to convert the data from a FITS file to Excel and analyze it using Excel math capabilities. Does someone know how to make such a conversion?


title: What will happen to water at $0^\circ$ Celsius kept in large evacuated chamber

text: Suppose some water is kept at $0^\circ$, in a glass (ideal container, if necessary). Now it is placed in a large evacuated chamber. **What will happen?** I think some water will vaporize, some will freeze and rest will remain liquid. But then I think how can water exist in liquid phase in such a high volume. Then I think I am confused, so I must ask it here.


title: Sensor Temperature with time constant

text: I am having trouble with this question on my study guide. I can't seem to find anything on my previous lectures to help me out with it. It is part of my study guide and I think it will be on my exam. Here is the question: ![enter image description here](http://i.stack.imgur.com/mayOg.png) If someone could walk me through this problem I would greatly appreciate it!


title: Why there is no pressure regulator on helium/hydrogen tanks?

text: So I've had people filling party balloons for me with helium tanks which had no regulators. Lately I've also seen local meteorologists filling large weather balloons from hydrogen cylinders (180 atm) which also didn't have a pressure regulator. Are they doing a really bad thing, or is no regulator not a danger for such cases where you aren't doing other things at the same time and can control the flow rate as needed? And why?


title: Streamlines for steady flow problem

text: I need to calculate the streamlines for the steady flow given by the velocity field $$\mathbf{u}=(u,v,w) = \left( 0,\frac{y}{y^2 +z^2} , \frac{z}{y^2 +z^2}\right)$$ Now I've been able to notice that $\dfrac{ \partial y}{ \partial z} = \dfrac{y}{z}$ therefore $y = ze^{c}$ but I'm not sure how I should continue. I've tried to solve $\dfrac{ \partial y}{ \partial s} = \dfrac{y}{y^2+z^2}$ but I get a $y^{z^2}$ term in my answer which makes me doubt whether I'm doing it correctly.


title: Nonequilibrium thermodynamics in a Boltzmann picture

text: The Boltzman approach to statistical mechanics explains the fact that systems equilibriate by the idea that the equillibrium macrostate is associated with an overwhelming number of microstates, so that, given sufficiently ergotic dynamics, the system is overwhelmingly likely to move into a microstate associated with equilibrium. To what extent is it possible to extend the story to nonequilibrium dynamics? Can I make concrete predictions about the approach to equilibrium as passing from less likely macrostates to more likely macrostates? (Wouldn't this require us to say something about the geometric positioning of macrostate regions in phase space, rather then just measuring their area? Otherwise you'd think the system would immediately equilibriate rather than passing through intermediate states.) Can the fluctuation-dissipation theorem be explained in this way? **Edit:** Upon a bit more poking around, it looks like the fluctuation- dissipation theorem _cannot_ be explained in this way. The reason is that this theorem discusses the _time-independent distribution_ of fluctuations in some macroscopic parameter (e.g. energy of a subsystem) but, as far as I understand, it does not describe the _time dependence_ of such a parameter. In particular, I'd really like to understand is if it's possible to explain things like Fourier's Law of thermal conduction (that the rate of heat transfer through a material is proportional to the negative temperature gradient and to the cross-sectional area) with a Boltzman story. According to these slides, it's surprisingly hard.


title: Non-Hermitian operator with real eigenvalues?

text: So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\in \mathbb{R}$) because they correspond to observables. What about a non-Hermitian operator which, among the others, also has real ($\mathbb{R}$) eigenvalues? Would they correspond to observables? If no, why not?


title: Magnetism due to relativity?

text: So I have been reading in some books that magnetism does not have to be assumed a priori, but can be obtained from the electric field + special relativity. And I have seen how this leads to the common formula for the magnetic field of a current carrying wire. Fine. What about materials that are inherently magnetic? Such as iron, or magnetite? Surely their magnetic field is not a consequence of relativity? (if yes, who's moving and with respect to whom?)


title: Confusion with LSZ reduction formula

text: LSZ reduction formula relates the matrix element of the scattering operator to the n-point Green's function $$\langle 0|\phi(x_1)\phi(x_2)...\phi(x_n)|0\rangle$$ My question is:   1. Is the vacuum on the left same as that of the right of this expression? Or these are different in the sense that one is the vacuum of "in" Fock space and the other is the vacuum of the "out" Fock space? These are the vacuum of the free fields. Right?   2. How is the above expression related to  $$\langle \Omega|\phi(x_1)\phi(x_2)...\phi(x_n)|\Omega\rangle$$ where $|\Omega\rangle$ is the vacuum of the interacting theory. Are these to quantities same? Which of the above expression really called n-point Green's function. In Peskin and Schroeder, they used $$\langle \Omega|\phi(x_1)\phi(x_2)|\Omega\rangle$$ as the 2-point Green's function and not $$\langle 0|\phi(x_1)\phi(x_2)|0\rangle.$$ I'm little confused.


title: Superconductivity in graphene with spin orbital coupling, is it proper to let the order parameter on two sub-lattice equal?

text: I am reading this article: Edge superconducting correlation in the attractive-U Kane-Mele-Hubbard model. Considering just the first part of the article, where a **negative-U Hubbard model with the intrinsic Spin-Orbital** term is considered on the **infinite** single layer graphene. In deriving the **gap equation** and the so-called **number equation**. The author assumed that the superconductivity order parameter in sub-lattice A and sub-lattice B is equal. which is $$\Delta_A=\frac{U}{N}\sum_k\langle a_{-k\downarrow}a_{k\uparrow}\rangle=\Delta_B=\frac{U}{N}\sum_k\langle b_{-k\downarrow}b_{k\uparrow}\rangle=\Delta$$ However, according to my calculation, using the mean-field Hamiltonian with $\Delta_A=\Delta_B=\Delta$ in the article, I found that $\Delta_A \neq \Delta_B$. The gap-equation in the paper can only obtained when I let $\Delta=\frac{\Delta_A+\Delta_B}{2}$; similar to number equation, I have to average $n_a$ and $n_b$. Because this is a rather lengthy calculation, I am not 100 percent sure that my calculation is right. I confirm myself that the Spin-Orbital term is not invariant under exchange of A and B, so there is no A,B sub-lattice symmetry. So my question is: am I right? If so, is it proper in the article to let the order parameter on the two sub-lattice equal?


title: What may be effect of air friction to the velocity of satellite?

text: What is the effect of air friction to the velocity of satellite? I have heard satellite's speed increases with air friction. But I'm in confusion how is it possible?


title: Same number of independent parameters for $SO(n)$ and $O(n)$

text: Why is the number of independent parameters for $SO(n)$ and $O(n)$ same, in spite of an additional constraint of unit determinant for $SO(n)$?


title: Trouble with derivation in an equation for Newton's Law of Angular Motion

text: I'm an autodidact and can't follow the part after "it is easily seen that"... which is the 31st equation: ![mass x displacement x second derivative of displacement over time = stuff](http://i.stack.imgur.com/34df3.png) Shouldn't it be: $m_i\,{\bf r}_i\times \frac{d^2{\bf r}_i }{dt^2}= \frac{d}{dt}(m_i r_i \times \frac{dr_i}{dt}) - \frac{d(m_i r_i)}{dt} \times \frac{d(r_i)}{dt}$ According to the rule of derivation: $(u v)' = u'v + v'u$ where $u(t)=m_i r_i$ and $v(t)=(r_i)'$ Furthermore, I would be most grateful if you could point me to resources that could help me with this sort of problems.


title: What would we see in the sky if it weren't blue

text: What would we see during the day when we look into the sky (other than clouds), if it weren't for Rayleigh scattering making the sky blue? Would the sky be dark, like at night?


title: Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction

text: The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$ Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim \limits_{h \to 0} [\frac{\psi(t+h)-\psi(t)}{h}]$, and because $\psi(t)$ is a Hilbert space vector, the limit is defined using convergence via the norm. (In other words, for any $\epsilon>0$ there exists an $h_\epsilon>0$, such that $\big|\big|\psi'(t)-(\frac{\psi(t+h)-\psi(t)}{h})\big|\big| < \epsilon $ for all $|h|<h_\epsilon$). So the convergence depends on the vector _as a whole_. But in the wavefunction realization (for a single particle), the Schrodinger equation is expressed as $$\frac{\partial}{\partial t} \psi(x;t) = \frac{-i}{\hbar}\left [ - \frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V \right ] \psi(x;t). $$ Here, however, $\frac{\partial}{\partial t} \psi(x;t)$ is a _pointwise_ partial derivative with respect to $t$, and so the convergence depends only on each individual point $x$ of $\psi(x)$ separately. If we take the abstract Hilbert space expression as definitive (axiomatically), then how can it be shown that the wavefunction realization actually expresses the same thing, given the different meanings of $\frac{\partial}{\partial t}$?


title: Rotating and moving reference frame

text: I've looked through your forums and can't find exactly what I need. I have a two objects whos dynamics can be described in discrete time as follows: $x_A(k+1) = x_A(k) + sin(\theta_A(k))v_A(k)\Delta{}t\\\ y_A(k+1) = y_A(k) + cos(\theta_A(k)))v_A(k)\Delta{}t$ $x_B(k+1) = x_B(k) + sin(\theta_B(k))v_B(k)\Delta{}t\\\ y_B(k+1) = y_B(k) + cos(\theta_B(k))v_B(k)\Delta{}t$ and my goal is to describe the movement of object B in the reference frame of A. Now in the reference frame of A we say that $x_A(k+1), y_A(k+1)$ and $\theta_A(k+1)$ are all zero, so we can describe that the movement of object B in reference frame of A as $x_{AB}(k+1) = x_B(k) + sin(\theta_B(k))v_B(k)\Delta{}t \\\ y_{AB}(k+1) = y_B(k) + cos(\theta_B(k))v_B(k)\Delta{}t-v_A(k)\Delta{}t$ This is pretty straightforward, but now I want to add the rotational part of A, so essentially $\theta_A(k+1) \neq 0$ and this is where it gets tricky. I tried to solve this in the following way: Let $r_{AB}(k+1) = [x_{AB};y_{AB}]$, then for a yaw rate of $\dot{\theta_A}(k)$ we multiply this vector with the rotational matrix (see here) with the angle $\dot{\theta_A}(k)\Delta{}t$ but I'm not sure if this is the correct way. Any ideas ?


title: Specific Heat equations - paradox!

text: We know that enthalpy is $h = u + pv$. Where $u, p, v, h$ are internal energy, pressure, volume, and enthalpy respectively. Now specific heat at constant volume is calculated as $c_v = \frac{\partial u}{\partial T}$, and specific heat at constant pressure as $c_p = \frac{\partial h}{\partial T}$. I am pretty confused at this point, in volume case it is internal energy, and in pressure case it is enthalpy. Because if $pv$ is a form of energy independent of internal energy, then it should figure in $c_v$ as well, because at constant volume when I am putting energy in the system, the pressure can not stay constant. What is wrong in the above argument? Appreciate your thoughts!


title: Simulate / plot electrostatic field

text: I have something like this: ![](http://i.stack.imgur.com/zv3xh.jpg) For given voltage (9 V red wire, 0 V blue wire) I want to simulate electrostatic field. I already know, for example that in the point (10.5;0) there are 10 V and etc. I want to simulate and plot electrostatic field for future comparison and for my report. Is there any software for doing that? I like to work with Maple, so if there is any toolboxes, I would be glad to hear about them. For slightly different position plot looks awful now: ![enter image description here](http://i.stack.imgur.com/xe32n.png)


title: Has energy in 'transit' been incorporated into missing matter calcs.?

text: Has energy in 'transit' been incorporated into missing matter calcs. ?It would seem that, although very small in mass, the sheer number of particles shhoting about from one end of the universe to the other would have to add up to a fair bit of mass. sorry for method of question im an imagineer not a physisisisist :)


title: Does a light wave that has been cancelled by another light wave continue traveling forward?

text: I imagine that if a light wave is cancelled out by another light wave, it would still continue to exist as a photon that is traveling at the speed of light--only without a wavelength. Would it behave only as a particle after it has been cancelled or does it exhibit some other kind of behavior?


title: Is it physically possible to convert matter into the electromagnetic spectrum (specifically x-rays)?

text: I was considering the problem with teleporting organic matter, and was curious to know if I can use this to avoid killing anything I teleport?


title: Simplifying a biquadratic filter  equation

text: I am trying to calculate input impedance of a multiple feedback low pass filter. What I need is the symbolic expression as simple as possible so that later I fill in the values and get the impedance itself: ![enter image description here](http://i.stack.imgur.com/QBGOb.png) I assume that I am on right track in calculation, the thing is I can not go any further and simplify the equation more. Can someone please help with putting calculated V2 (down in the writings) into equation 2 and then put Vin into equation 1 and simplify it so that **i1** will be removed? P.S: I posted this question in electronics and math of stack exchange but they suggested to be posted in physics! ![enter image description here](http://i.stack.imgur.com/wfCgz.jpg)


title: Electromagnetic wave reflection vs. light reflection

text: Related: x-ray interaction with atmosphere I know that electromagnetic waves of particular frequencies reflect from the ionosphere. And the light (which from one perspective is an electromagnetic wave) also reflects from the.. let's say snow. These types of reflection have the same nature or different mechanisms are involved here?


title: Finding the contact force in a mechanical problem

text: what is the proper FBD for this system. I cant decide since the weight is given, but there is tension and contact forces. Things get mixed up. Determine the contact force between the smooth 200-lb ball B and the horizontal bar, and the magnitude of the pin reaction at A. Neglect the weights of the bar and the pulley. ![](http://i.stack.imgur.com/ysifR.png)


title: Do $x$ and $Q^2$ associate with particular directions in the infinite momentum frame?

text: In deep inelastic scattering, you describe a collision using the variables $Q^2 = -q^2$ (probe virtuality) and $x = Q^2/2p\cdot q$ (Bjorken x, parton momentum fraction). Now, I seem to remember reading somewhere that in the infinite momentum frame, one of these variables primarily depends on momentum in the transverse direction and the other primarily depends on momentum in the longitudinal direction, but I can't remember which, and checking my most likely sources hasn't turned up anything. Is this actually the case, or is my memory playing tricks on me? And if it is true, which is which? Apologies for asking a vague question, but I'm really drawing a blank on what it is I might have read, and asking here seemed like the quickest way to figure it out ;)


title: Alma correlator for visible light and for space based telescopes?

text: I was shocked to see that one can simulate interferometry on a computer like the ALMA Correlator described by this video (posted on Wikipedia's article on ALMA) Is is possible to do the same in visible light? Is only the CPU power the limiting factor or is there some physical limit? Would it be possible to use similar technology to launch 2 space-based telescopes and put one on the far side of the Sun with respect to the other to get and effective mirror size of about 2 AU? Is only money the limiting factor to do something like that or is there some physical limit? Or maybe there simply is no point to push the diffraction limit so far with a 2 AU mirror?


title: Time dilation in special relativity

text: Suppose a star ship is moving with some velocity. Two light pulses one in direction similar to star ship another opposite to it is shot towards the space ship. Then how time inside space ship adjust to make velocity of light constant?


title: Spin with Stern gerlach Experiment

text: Within a week, I am trying to conduct a seminar on the topic " Spin with Stern gerlach Experiment" at my University. It would be great if you guys could provide me some inputs regarding the things I should say about the experiment during the seminar. I am still unsure about the outlines I should draw in my mind about what I should present serially.......Since it would be the first seminar of mine infront of more than 100 people, I am really confused as well as afraid.


title: Is it possible to prove that planets should be approximately spherical using the calculus of variations?

text: Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Lets assume that a planet is 'born' when lots of particles ($N\rightarrow\infty$) are piled together and constrained only by the gravity of the $N$ particles with no external gravity. My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical?


title: Does it really make sense to talk about field lines?

text: Field lines should only provide a visual representation of a field. There is a rule for their construction: take an object subject to a field, move it by d$\mathbf{r}$ and draw the direction of the force due to the field. Do it indefinitely, connect all the d$\mathbf{r}$ and get a line. A _field line_. Some examples of common 'properties' of field lines: The density of field lines is related to the strength of a field. In plasma physics, there is an effect called _flux freezing_ , where the number of field lines crossing a contour stays constant under topological changes of the contour. **BUT** field lines do not occupy a physical position in space so we can draw how many we want? Does it make sense, then, to talk about the **number** of field lines?


title: How does the focal length of the lens change the distance from the camera to an object?

text: I am reviewing some old image processing notes and have the following problem: > I have a camera with a projection plane of $2cm\ by\ 2cm$ and a $15mm$ lens. > If you want an $1.5m$ object to take up half of the resulting image, how far > away should it be? It seems like the standard $x = \frac{fX}{Z}$ equation should be useful here, but I am not sure if this information gives focal length at all. How should this be solved?


title: Are Carnot engine efficieny and Fourier heat trasmission law related?

text: It just occured to me that the efficiency of Carnot cycles is $\eta= \frac{T_1 - T_2}{T_1}$, that is, the efficiency decreases as the difference between reservoir temperatures decreases. On the other side, Fourier's law states that the dissipation of heat is proportional to the temperature gradient, that is, to the temperature difference. My question, then, is: are these two results related? Do they both have a common cause?


title: Hawking radiation from point of view of a falling observer

text: This paper tells that Hawking claimed that the falling to a black hole observer will not detect any radiation. But **only because** the frequency of the Hawking radiation will be of the order $1/R_s$ so that the falling observer will not have a suitable detector. This argument seems fallacious to me. First, the falling orserver can communicate to a distant observer who has a suitable detector. That observer will tell him that the BH indeed evaporates. So its radius will decrease for both falling and distant observer. Before the falling observer will touch the hoizon, the radius of the BH will become smaller than his own dimentions so he will be able to detect the radiation. The smaller the BH will become, the higher frequency of the Hawking radiation will be. At some point the apparent temperature of the BH for a falling observer will reach infinity so that the BH will explode. Am I right that this claim by Hawking is fallacious and in fact all ouside observers will detect Hawking radiation sooner or later?


title: This is going to sound really stupid. But how do I change my normal force?

text: Suppose I am going to the doctor's and I want to do either of the following   1. Make myself lighter on the scale   2. Make myself heavier on the scale For 2), isn't it just relaxing and let your mass drop? For 1) how could I do this? Should I maybe exert a force on a table close by? Will that force (from my hand) transfer to the scale?


title: Relativity - time dilation

text: I'm learning about relativity and I'm having some issues with it and the twin paradox. I found many questions and answers on this subject but they did not answer my specific problem. In my thought experiment I added a third twin - triplet. So here it goes: Let's have triplets $A$,$B$ and $C$. sitting still at same point in space. They want to test relativity so they do this experiment.. They all reset their clocks, and $B$ and $C$ start moving together away from $A$ at some ridiculous speed $v_{365}$ which coincidentaly has lorentz factor $365$. they move at uniform motion in only one direction (let's say acceleration time nears 0). State 1: $$\begin{align}t(A)=t(B)=t(C)&=0 \\\ v(A,B) = v(A,C) &= v_{365}; \\\ v(B,C) &= 0\end{align}$$ $t(A)$ being A's clock, $v(A,B)$ being velocity at which $A$ and $B$ are moving away from each other, etc.. Now after some time, let's say when $t(A) = 1\text{ year}$, brother $B$ starts to decellerate to full stop. It also shows up that brother $C$ was sleeping through whole first part of the experiment. But now he wakes up and sees that his brother $B$ is accelerating away from him in the opposite direction than was planned but that doesn't bother him. State 2: $$\begin{align}t(A)&=1\text{ year} \\\ t(B)=t(C) &\stackrel{?}{=} 1\text{ day} \\\ v(A,B) &= 0 \\\ v(A,C)=v(B,C) &= v_{365}\end{align}$$ Now after another while lets say when $t(A)=2\text{ years}$, brother $C$ sees that brother $B$ isnt stopping any time soon so he decides to match his speed.. (brother $C$ stops now from point of $A$) State 3: $$\begin{align}t(A)&=2\text{ years} \\\ t(B)&= 1\text{ year }1\text{ day} \\\ t(C)&= 2\text{ days} \\\ v(A,B)=v(A,C)=v(B,C)&=0\end{align}$$ That's it.. now they stand still some light year away from each other, but they can communicate and compare their clocks.. here are my questions: 1] Is my assesment of time dilation effects correct in this scenario? if not which state contains first error and why? 2] If my assesment is correct, then from point of brother $C$ it was brother $B$ who was accelerating away, yet it was also brother $B$ who aged when they got in same speed - therefore it seems to me its always the one who is matching speed who doesnt age - is this a general rule? 3] If answer to questions 1 and 2 is yes, then can this effect be used to gather infomation that would take much longer to extract in normal speeds? Regards


title: How the radio-freqency magnetic field helps to record fluorescence signal

text: Suppose a sample of atoms (say rubidium), which is exposed to constant magnetic field, is irradiated with circularly polarized light so that the electrons are excited from lower S level (F=3) to P level (F'=4). Then the fluorescence signal is to be recorded while the frequency of a linearly polarized radio-frequency magnetic field is swept across the resonances. My question is: how does the sweeping of frequency of a linearly polarized radio-frequency field across the resonances help for fluorescence detection?


title: Initial vs Constant Orbital Velocity

text: I am working on some basic physics simulation for a game and need to simulate gravity. I have a system working that is behaving more or less correctly so far, but I want to see if I can send a projectile into a stable orbit by giving it an **initial** velocity. I understand that the following equation expresses the velocity of a stable orbit around a mass, $$v = \sqrt{\frac{GM}{r}}$$ I am able to use this equation to create a stable orbit but not from setting a starting velocity and then letting the gravity do the work. It only works if I calculate the {x, y} components of v and set that vector as the projectiles velocity each time I render the scene. The direction of the vector is always calculated to be perpendicular in direction to the line representing the gravitational force towards the center of the mass. So my question is essentially whether this equation is even intended for finding an appropriate **initial** velocity, and if not, is there another way? I am not clear on whether there is any meaningful distinction between calculating v as a stable orbital velocity versus v as an initial velocity that resolves into a stable orbit. Or alternatively might it be more appropriate to say that I need to give the projectile and initial force or acceleration that _resolves_ into a stable orbit (like an escape velocity, sort of). Thanks. EDIT -- here is how I calculate the gravitational acceleration vector at a certain point. Let (x, y) be the point of interest, (x1, y1) be the center of mass, dx be the x-component acceleration, and dy be the y-component acceleration:                       1) seperation of the points                  sep_x = x1 - x;             sep_y = y1 - y;                  2) effect of gravity given distance between points                  gravity = (G * mass) / radialDistance ^ 2                  3) break down into components                  dx = (gravity * sep_x) / radialDistance ^ 3             dy = (gravity * sep_y) / radialDistance ^ 3


What is the highest speed time dilation has been tested? How close to the Special Relativity prediction did it get?


title: Why does my kettle only make a noise when it is turned on

text: Almost as soon as I turn my kettle on it starts to make the familiar kettle noise, yet very shortly after turning off the power the boiling noise stops and the kettle is totally silent. The temperature of the water is (almost) the same as when I turned it off. So why is there only noise when energy is being added to the water?


title: Kirchoff Loop and current

text: The pic of the circuit/problem: ![enter image description here](http://i.stack.imgur.com/kS5Pm.jpg) I got the problem right, but i want to make sure my reasoning is correct for the current left of the 2.1 V battery. Its the same current as $I_1$ because the charge carriers do not split in anyway, and the battery only does work on them moving the carriers from a low potential to a high potential. However, if $I= \frac{dQ}{dt}$, wouldn't supplying energy via moving the charges from low potential to high potential cause the charges to move more quickly, i.e. increase the current? A little fuzzy on why it is the same current, if someone could enlighten me I would very much so appreciate it.


title: Why do ships lean to the outside, but boats lean to the inside of a turn?

text: Small vessels generally lean into a turn, whereas big vessels lean out. > **Why do ships lean to the outside, but boats lean to the inside of a > turn?** For example, a boat leaning into a turn: ![](http://i111.photobucket.com/albums/n131/Golf_Bravo_Zulu/T3449_1_hi1.jpg) Image Source; imgur backup. And a ship leaning out: ![enter image description here](http://i.stack.imgur.com/lbnrV.jpg) Image source


title: Spin-ice materials with strong quantum fluctuations

text: Spin-ice materials are insulating materials where spins form a 3D pyrochlore lattice and have a frustrated magnetic interaction. The spin dynamics in most spin-ice materials is very classical and has little quantum fluctuations. So the question is that which spin-ice materials have strong quantum fluctuations? What are the model Hamiltonian for the spins in those materials? Any interesting experimental results on those materials?


title: Feynman's 'diamond jumping out of a box' parody, how would this work?

text: I have been told that Feynman deduced from a path integral formulation an equation that predicts the amount of time it would take for a diamond to 'jump' out of a box: $t > \dfrac{x \Delta{x} m}{ h} $ where $x$ is the size of the box, $\Delta x$ is the distance the diamond must travel to leave the box and $m$ the mass of the diamond. How would it be possible for a diamond to leave the box? What are the processes that happen over the stupendous time gap to allow the diamond to do this?


title: How does one calculate the quantum propagator for a massless photon

text: So I want to calculate the quantum massless photon propagator. To do this, I write $$ A_\mu(x) = \sum\limits_{i=1}^2 \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \left( \epsilon_\mu^i (p) a_{p,i} e^{-i p \cdot x} + { \epsilon_\mu^i} ^* (p) a_{p,i}^\dagger e^{i p \cdot x} \right) $$ where $\epsilon_\mu^i(p),~ i = 1,2$ are appropriate basis polarizations in the gauge we choose to work in. I then calculate the propagator, which is defined as $\left< 0 \right| T \left\\{ A_\mu(x) A_\nu(y) \right\\} \left| 0\right>$. Using the above formula, I calculate this. I get $$ \left< 0 \right| T \left\\{ A_\mu(x) A_\nu(y) \right\\} \left| 0\right> = \int \frac{d^4p}{(2\pi)^4} \frac{i}{p^2 + i\epsilon} \left( \sum\limits_{i=1}^2 \epsilon_\mu^i (p) {\epsilon_\nu^i}^*(p)\right) e^{-i p \cdot (x-y)} $$ To complete this calculation, I now have to show (in the general $\xi$ gauge), $$ \sum\limits_{i=1}^2 \epsilon_\mu^i (p) {\epsilon_\nu^i}^*(p) = -\eta_{\mu\nu} + \left( 1 - \xi \right) \frac{p_\mu p_\nu}{p^2} $$ This last step is where I am getting stuck. I know that this is the classical propagator for the photon in the general $\xi$ gauge. But how do I relate that to the polarizations exactly? Also, I know that for on-shell photons, (with $p^2 = 0$), the polarization sums give $$ \sum\limits_{i=1}^2 \epsilon_\mu^i (p) {\epsilon_\nu^i}^*(p) = -\eta_{\mu\nu}+ \frac{p_\mu {\tilde p}_\nu + {\tilde p}_\mu p_\nu}{p \cdot {\tilde p}} $$ where ${\tilde p}^\mu = (p^0, - \vec{p})$. I have no clue how to extend this for off-shell photons. Any help?


title: Is this a known entropy formula?

text: While playing around with a variant of the one-dimensional Ising model with periodic boundary conditions I came up with a formula, let's call it $F$, whose form looks suspiciously like an entropy formula, although as far as I can tell it shouldn't have anything to do with entropy. The form is:      $F = -\sum \frac{1}{b_i}ln(1 - \alpha^{b_i})$   where $\alpha$ is a fixed real number $0 < \alpha < 1$, and $b_i > 0$. Without getting into the gritty details, my question is a simple one: Has anyone encountered such a formula before, possibly related to entropy?


title: Why only vertical component of the stress tensor on vertically suspended bar?

text: EDIT: I am gonna rephrase the question entirely. Imagine we have a bar which we will analyze in the linear elastic regime. The shape of the cross section is irrelevant. The bar is suspended from a ceiling and it is under its own weight. I have been told that in a problem of this type, in every point of the bar, ONLY the longitudinal component of the stress tensor is non zero. Is this right? and if so how do you justify this?


title: Electrostatic field in a Dielectric equations misbehaving

text: Equation 1: $\int{\vec{E}.\vec{ds}} = \int\frac{\rho_{free} + \rho_{bound}}{\epsilon_0} dv$ (Gauss's Law) Equation 2: $\int{\vec{D}.\vec{ds}} = \int\rho_{free} dv$ (Gauss's Law) , but $\vec{D} = \epsilon_0 \epsilon_r \vec{E}$ , so equation 2 becomes $\:\:\:\epsilon_0 \ \epsilon_r\int{\vec{E}.\vec{ds}} = \int\rho_{free} dv$ or, $\int{\vec{E}.\vec{ds}} = \int\frac{\rho_{free}}{\epsilon_0 \epsilon_r} dv$ But if we compare this last equation with Equation 1 , we get $\rho_{free} = \epsilon_r(\rho_{free} + \rho_{bound}) $ which doesn't make sense because there are situations where there is only a "frozen in " volume bound charge and no free charges. Whereas the last relation I derived says there cannot be a bound charge without a free charge. What am I doing wrong?


title: quadrupole moment and higher for simple current loop

text: I'm working on a field configuration that needs to die off rapidly and I got to a $1/r^5$ dependence with canceling the dipole moment of the system cleverly, but to go get better arrangements I need to figure out the higher order moments of a simple loop as well. Griffiths (page 243, 3rd ed) refers to this integral $\oint (r')^{n}P_n(cos \theta ') d L'$, with $P_n$ as the Legendre polynomials but I am unclear on how to go about evaluating it for higher moments because for the dipole term Griffiths does a clever trick that I'm not super sure extrapolates. I feel like the current loop is such a simple, tidy system it's higher moments would be tabulated somewhere just as an example of how higher orders are calculated, but I'm unable to find them after a few a days of googling and librarying.


title: Why does $E=mc^2$ give results in Joules?

text: Why does $E=mc^2$ give results in Joules ? Why didn't Einstein need to create a separate unit and how come using two standard units (speed of light in km.s and mass in grams) give another result in Joules ?


title: Ohms Law Voltage/Potential Difference

text: Three 2.0 Ω resistors are connected in series to a 12 V battery. What is the potential difference across each resistor? � This question requires Ohms law because V is needed right? So I would use I = V/R and use Req = r1 + r2 + r3... then does I(Current) = 12? so 12 = V/6 Im confused.


title: How can a single quantum system be prepared in a particular mixed state?

text: Let's consider an electron in some sort of potential well. Suppose $|i\rangle$ is an $i$th eigenstate, and we want to prepare the electron in state with state vector (up to normalization) $|\psi_1\rangle=|1\rangle+|2\rangle$. We can prepare such a state arbitrarily precisely by e.g. making the barrier finite (but still large) in size, and shedding two quasimonochromatic wave packets from left and from right — one with frequency corresponding to state $|1\rangle$ and the other with frequency of state $|2\rangle$. After some time, because of resonance the waves will noticeably tunnel through barriers inside the well, and we have our pure state. Now, suppose we want to prepare a mixed state from same eigenstates: with density operator $\hat\sigma=|1\rangle\langle1|+|2\rangle\langle2|$. How could we go to prepare our electron in such a state? Do we have to prepare many such electrons, randomly selecting whether they'll be in $|1\rangle$ or in $|2\rangle$ and then measure randomly selected one, or can we just make that single electron in such a mixed state?


title: What are the parameters for Pauli's repulsion pseudo-force?

text: I have found the following formula for the repulsion potential due to the overlap of the electron clouds arising from Pauli's exclusion principle: $$V = A\exp(-r/\phi)$$ where r is the distance between ions. In terms of energy: $$E = A\exp(-r/\phi)/r$$ What would be the final force expression? I am calculating Pauli's pseudoforce (it's not a real force because the repulsion arises from electromagnetic interactions) for Na, Ca and K ions. I need the values for parameters $A$ and $\phi$, which are commonly fitted to energy curves. Do you know of any tables which have these parameters?. If so, could you please give me the references?


Why is mass the quadratic term in a Lagrangian?


title: Justification for new theories of Particle Physics and General Relativity

text: In reference to arxiv:1212.4893v3 and arxiv:1206.5078v2 papers of Ma and Wang, they have proposed new theories in particle physics, the weakton model where quarks and leptons are formed using these weaktons; modified Einstein's field equations taking care of Dark matter and Energy. Are these really major breakthroughs as they apparently seem to be, with proper experimental evidences of their accuracy ? Or are they yet to be verified ? What does the Physics community think of these ? I am curious.


title: Cloud chamber temperature

text: I'm planning to build a small cloud chamber. The design is the usual: alcohol evaporates at the top by gently heating it with a thin wire and small current. The vapour sinks to the bottom of the chamber, where the temperature is significantly lower than at the top, and causes the vapour to super-cool. Ion tracks caused by fast moving particles can be observed as mist tracks. My question is about the bottom plate temperature. The usual design uses a (near) room temperature solvent and very cold bottom plate to achieve a big temperature gradient. Could a cloud chamber be constructed that would use just a slightly cold bottom plate, and a more aggressively heated liquid? I'm thinking about ethanol or isopropyl alcohol evaporated at 60°C, and a bottom plate temperature at maybe only 0°C Could this work? Is it just the temperature gradient, or the temperature itself is important?


title: Projective Transformations in GR

text: A Thought Experiment: We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made to vary upto some final state. During this process of experimentation the physical separations change while the coordinate labels remain fixed to their own positions. [Coordinate separations remain unchanged]. The length and the orientation of a vector changes in this process both in the 3D and in the 4D sense. We are passing through different/distinct manifolds in our thought experiment and if the 4D arc length does not change we are simply having a transition between manifolds for which ds^2 is not changing but the metric coefficients are changing.We consider the option of $ds^2$ changing in this posting. Query: Our experiment indicates at projective transformations operating in the physical sense[considering changes in the metric and in the value $ds^2$].A time dependent field is being observed where the metric coefficients are not being preserved. Is it important to include projective transformations[concerned with the non-preservation of the metric] in the mathematical framework of GR?


**Question: Is the Avogadro's constant equal to one?** I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the `mole` and to `Avogadro's constant`. And, to be honest, it doesn't make any mathematical sense to me.               1 mole = 6.022 * 10^23     Avogadro's constant = 6.022 * 10^23 * mole^(-1)      What? This hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!


title: Pertinence of the wave function of the universe, or complete description of system with massive number of dof

text: I have heard couple of times about the concept of wave function of the universe, an object that would capture every degrees of freedom inside it (every particle, me, even you dear reader, etc...) and it always sounded fallacious or at least non pertinent, what would be the point of using that gigantic object to describe our universe. From my first classes of statistical mechanics, I learned that there is no point in trying to monitor $10^{23}$ and more degrees of freedom, that we need to look for emergent pertinent quantities (pressure, temperature, etc...). Even more, I am know reading an article where the author (there is no point in giving you the title/name) takes the example of a local QFT completely describing the Solar system, so roughly $10^{60}$ degrees of freedom, plus the possibility that an observer could be monitoring all those in real time, which is totally unrealistic. Does this make sense to you?


title: Can someone explain probability flux in the tunneling boundary condition of Vilenkin?

text: This is what's leading to the notion of a quantum universe tunneling from nothing into existence, right? The idea is that probability flux flows out of superspace (configuration space) at singularities and into superspace at non- singular geometries at past timelike infinity if I have it correct. If I were to picture it, it's a 2 dimensional Minkowski space (coordinates (a, phi)) where there could be, arbitrarily, a ring representing the boundary composed of singularities at which probability flux goes out towards past timelike infinity until it reaches the non-singularity areas in the center (a=0, phi=finite) where it then turns "ingoing" and gets injected into superspace... and eventually back to the singular boundary and then outgoing again. The outgoing wave should be present in the classically allowed region. So this means probability flux gets injected into superspace, universe tunnels into existence and as universe collapses back into singularity, probability flux goes back out of superspace?