title: Does the entropy of an adiabatically isolated system stays the same after we decrease/increase the pressure?

text: > Even if the system is isolated and there is no heat exchange with > surroundings, shouldn't the decrease/increase of pressure result in > increase/decrease of entropy? > > Does this property of an isolated system means that increase of pressure > means equivalent decrease in volume, so after the process the $pdV$ i.e. the > work is still the same? > > Can someone explain this in more detail? Quoted text above was original question. Please read on further, I clarified the question. To be more specific, situation is this. We have an adiabaticaly isolated system and (for example) a piston inside it. If there is piston, that means that volume work or $PdV$ can be done. **So if we do work on the system, by some force from surroundings which pushes piston(compresses the gas) we would get increase of the pressure inside the system... right?** **If we increase the pressure in the system, the entropy will decrease as well, altough we haven’t bring in the heat .. right?** So if we increased the pressure and mutually decreased the volume, the molecules of the gas are packed more tightly and they have lesser space to move in...This is followed by system having more internal energy now, then it had before process...Since the activity of molecules inside is higher, they are moving at higher speeds(cause of smaller volume)... **That means that system should have more internal energy after the process, right**? Taking that into consideration, and if we follow the rule that increase in pressure is also increase in temperature, that is molecules have higher activity on higher pressure and that kind of activity is measured with temperature... **that means that system also has more amount of heat?** _Correct me if I am wrong._ **And even if it is adiabatically isolated (dQ=0) shouldn't the internal energy increase and thus amount of heat in the system increase as well when we do the work on the system (move the piston to compress the gas )?** I am talking that higher activity leads to more heat inside the system even if it is prevented (isolated) to exchange heat with its surroundings. I hope you understand me now.


title: Why is a book on a table not an example of Newton's third law?

text: My textbook explains Newton's Third Law like this: > If an object A exterts a force on object B, then object B exerts an equal > but opposite force on object A It then says: > Newton's 3rd law applies in all situations and to all types of force. But > the pair of forces are always the same type, eg both gravitational or both > electrical. And: If you have a book on a table the book is exerted a force on the table (weight due to gravity), and the table reacts with an equal and opposite force. But the force acting on the table is due to gravity (is this the same as a gravitational force?), and the forcing acting from the table to the book is a reaction force. So one is a gravitational, and the other is not. Therefore this is not Newton's Third Law as the forces must be of the same type.


title: Why does a ball bounce?

text: If an object is acted on by equal and opposite forces then it will be in equilibrium, and it's acceleration or velocity (and so direction as well) will not be changed. So when a ball bounces, it exerts a force on the floor, which matches the magnitude of the force in the opposite direction (the ball is bouncing perfectly vertical), up. So how is it's velocity/direction changed? If the forces are equal and opposite to each other. In order for it bounce, surely the force acting from the floor to the ball must be greater than the force acting from the ball to the floor?


title: About binary stars and calculating velocity, period and radius of their orbit

text: I saw somewhere about being able to measure the velocity, period and radius of a binary star orbit by looking at red shift and blue shift. I understand it but can someone give me an example of calculations etc done to calculate the velocity, period and radius of a binary stars in orbit?


title: Is the meander ratio of a river $= \pi$?

text: To get from point $A$ to $B$, a river will take a path that is $\pi$ times longer than as the crow flies (I think this result is from Einstein). What is the proof of this, and how well does it hold water against experiment? What assumptions are made during the proof? **Edit:** I was also sceptical when I first heard this, asking this question in part to potentially mythbust. To be clear, I assumed that the statement was: > For a river that is formed in the 'usual way' on a flat plain under 'usual > conditions' (whatever those are, assume the conditions of Princeton as > Einstein was possibly involved), as the arc length of the river from a > starting point tends to infinity, $\frac{\text{arc length}}{\text{crow's > length}}\rightarrow \pi$. I am aware this is equivalent to the hypothesis that the meandering of the river can be well approximated as semi-circular as the length increases above a certain value.


title: Particle Spacing in a Vacuum

text: Four questions: (To start off, I know very little about physics it isn't even funny (I probably use a ton of wrong terms here and leave out vital information, if so I will try to edit it in as you point it out!), Anyways, I have been always curious about this scenario though so I finally figured I'd ask some people who actually know what they are talking about) First question: Theoretically speaking, if I have a vacuum that is a closed system, it's made of walls that are 100% nonreactive and have no electromagnetic force, strong force, weak force, gravitation force, or anything, but they can still restrain things placed inside the box (without any interaction on the particles, but instead just by the laws of this system) If I place 1 particle in the box will it stay exactly where I placed it, or will it somehow move to the center of the box thus distancing itself as far as possible away from each wall? (I've drawn a 2D demonstration) ![enter image description here](http://i.stack.imgur.com/DX2s1.png) Now, if I place a second particle in, will they both stay where I placed them, will they divide the box in half in some way (I just gave 1 example, there are many) and then both move to the center of those halves, will they space themselves as far apart as possible (pushing themselves against the nearest corners of the box), or is there some ring of energy around particles that I don't know about that has a statistical radius and they would just distance themselves as much as the "reach" of this ring so that the rings aren't touching? ![enter image description here](http://i.stack.imgur.com/KdhRb.png) Or will something else happen? * * * Second question: I'm assuming different things will happen if the particles are different things, for example if Particle A was a proton and Particle B was an electron they would just pull together? Or the electron would begin orbiting the proton (Or would they even both start orbiting each other?) (If either of these are the case what determines if the orbiting is clockwise or counter-clock-wise (Or whatever the 3D equivalent of these would be, does spin determine?)? What are some particles that would react different ways, for example 2 photons, a proton and a proton, a proton and an electron, etc. Or would it not matter? * * * Third and Fourth questions: What would happen in the same scenario except the walls are reactive, they are made of steel, or carbon fiber, or something strong enough to support a vacuum?


title: Salisbury Screens and Energy Conservation

text: From Wikipedia on how a Salisbury Screen works: > _1\. When the radar wave strikes the front surface of the dielectric, it is > split into two waves._ > > _2\. One wave is reflected from the glossy surface screen. The second wave > passes into the dielectric layer, is reflected from the metal surface, and > passes back out of the dielectric into the air._ > > _3\. The extra distance the second wave travels causes it to be 180° out of > phase with the first wave by the time it emerges from the dielectric > surface._ > > _4\. When the second wave reaches the surface, the two waves combine and > cancel each other out due to the phenomenon of interference. Therefore there > is no wave energy reflected back to the radar receiver._ The Wikipedia article focuses on use to absorb radar, but the same concept can be applied to other wavelengths of light. The energy of a photon is found with the equation $E=hc/\lambda$ My question is: When the waves cancel each other out, is the energy converted to heat, or is it expressed some other way?


title: Calculate energy from an reaction

text: I'm trying to calculate the _released energy from a reaction_. The radioactive substance polonium decays according to this formula: $$^{210}_{84}\mbox{Po} \rightarrow \mbox{X}+^4_{2}\mbox{He} $$ At first I solved X to be: $^{206}_{82} \mbox{Pb}$ Now when I have the whole reaction, what formula should I use?


title: Causal structure of the inflationary multiverse

text: In the multiverse as it is described by eternal inflation, it is not clear to me what is its causal structure and in particular if the bubble-universes are causally connected. We start from a de-Sitter space-time and a scalar field whose almost constant potential plays the role of a positive cosmological constant, but after decay of the scalar field in certain regions it is not clear to me what becomes of the causal structure. If they are considered universes on their own, it should mean that the bubble are causally disconnected from everything else; however I already heard of bubble colliding and leaving hypothetical imprints in the CMB. There also the different probability measure defined in inflationary cosmology, are there probabilities seen by an observer in one of the bubbles, or by an outside (super)observer seeing the whole space-time foliation and every bubble, no matter the causal structure of the space-time?


title: Can you calculate unix timestamps (universal clock) from any planet?

text: Given the definition of unix timestamp as the number of seconds elapsed since January 1st, 1970 as GMT+0, without leap seconds, is it possible to create a universal clock that will generate the correct timestamp? Is the current definition of a second > the duration of 9192631770 periods of the radiation corresponding to the > transition between the two hyperfine levels of the ground state of the > caesium 133 atom.[1] In 1997 CIPM added that the periods would be defined > for a caesium atom at rest, and approaching the theoretical temperature of > absolute zero (0 K), and in 1999, it included corrections from ambient > radiation.[1] Absolute zero implies no movement, and therefore zero external > radiation effects (i.e., zero local electric and magnetic fields). sufficient for this?


title: Finding the electric field around an electric eel

text: I'm having problems solving for the electric field and the current that an electric eel generates. Would I use Gauss's law and treat it as a long charged wire? How would I find the charge of the electric eel? I'm really confused about what I can do and where to start. In the second part of the project I need to find the current and electric field that affects the prey of an electric field also. Would this calculation be like adding a (human) resistor in a circuit?


title: Could the collision of two pairs of quantum entangled protons cause a temporary "wormhole"?

text: I recently read this article from MIT News. I then started thinking about how a particle accelerator creates a temporary microscopic black hole. My question is: > _If quantum entangled pair $A$, consisting of $A_1$ and $A_2$, and a second > pair $B$, consisting of $B_1$ and $B_2$, were collided, $A_1$ collides with > $B_1$, and $A_2$ collides with $B_2$, would this create a temporary > wormhole?_


title: What is an "Interaction Hamiltonian"

text: I'm an undergraduate reading up on some quantum physics so that I can help out more in the lab that I'm working in this summer. In the book I'm reading (Shankar's "Principles of Quantum Mechanics") I just came across the term interaction Hamiltonian in describing how orbiting electrons interact with a magnetic field. I have an idea of what it might mean, but I can't find a good explanation anywhere. What is an "interaction Hamiltonian", and how does it differ from a standard Hamiltonian?


title: Stupid yet tricky question: Why do we actually see the sun?

text: I haven't yet gotten a good answer to this: If you have two rays of light of the same wavelength and polarization (just to make it simple for now, but it easily generalizes to any range and all polarizations) meet at a point such that they're 180 degrees out of phase (due to path length difference, or whatever), we all know they interfere destructively, and a detector at exactly that point wouldn't read anything. So my question is, since such an insanely huge number of photons are coming out of the sun constantly, why isn't any photon hitting a detector matched up with another photon that happens to be exactly out of phase with it? If you have an enormous number of randomly produced photons traveling random distances (with respect to their wavelength, anyway), that seems like it would happen, similar to the way that the sum of a huge number of randomly selected 1's and -1's would never stray far from 0. Mathematically, it would be: $$\int_0 ^{2\pi} e^{i \phi} d\phi = 0$$ Of course, the same would happen for a given polarization, and any given wavelength. I'm pretty sure I see the sun though, so I suspect something with my assumption that there are effectively an infinite number of photons hitting a given spot is flawed... are they locally in phase or something?


title: Compute impulse response of a cavity for sound waves

text: Given a (closed or not) surface and a point emitting a spherical sound wave, how can I calculate the wave amplitude in any point of space, considering reflections on this surface ? The idea is to determine the response of the cavity to a dirac impulse, for use in convolution reverb afterwards (and also mainly because the question interests me). The main idea is answering the question "If I am at this point of the cavity and say 'Hello World', what echo will I hear ?". Remembering some lessons on waves some years ago, I tried to go with Huygens- Fresnel principle, by computing the value of wave after exaclty 1, 2, 3 etc.. reflexions and summing up everything, but I couldn't find out how to properly formulate it in this case. Going deeper into Wikipedia, the Kirchhoff integral theorem, but my attempts to use it where quite unsuccessful as well. Is there some formula of this kind that would stand for my problem (and maybe that could be exactly resolved in cases with a lot of symmetries), or should I just get down to the wave equation and compute numerically the value (and thus, how should it be done with a dirac impulse as initial conditions) ?


title: Are there critical constants for liquid-solid transition?

text: I'm aware of the critical constants for gas-liquid transition but is there such constants for liquid-solid transition? Like the maximum temperature above which a liquid cannot be solidified.


title: How are accelerated reference frames non-symmetrical?

text: As in, if I'm accelerating away from the Earth, then does the Earth _also_ appear to be accelerating away from me at the same rate? Or is there something to "break" this type of symmetry? My question is inspired by the below discussions: http://www.quora.com/Special-Relativity/When-I-am-traveling-at-high-speed- near-speed-of-light-am-I-time-traveling-or-slowing-my-aging-slowing-my- particles-speed-interactions/answer/Anthony-Yeh http://en.wikipedia.org/wiki/Twin_paradox#Resolution_of_the_paradox_in_special_relativity


title: Dimensional regularization and the finite part

text: Let be a dimensional regularized integral $$ \int d^{4-\epsilon}kF(k,m,s)= \frac{2}{\epsilon}+\frac{m^{2}}{3}(\gamma +log(4\pi)-\frac{1}{\epsilon}))$$ then formally if we elmiinate the divergent quantities we may have $$ \int d^{4-\epsilon}kF(k,m,s)_{reg}= \frac{m^{2}}{3}(\gamma +log(4\pi)+log\mu) $$ here $m$ and $s$ are parameters and $ \mu $ and energy scale but is it that enoguh is renormalization simply 'deleting' teh divergent quantities proportional to $ \frac{1}{\epsilon ^{k}} $ ?


title: Meaning of the Poisson bracket as a coordinate transformation

text: Well, the Possion bracket: $ \\{ A(q,p),B(q,p) \\} \equiv \sum_{s} \left( \dfrac{\partial A}{\partial q_{s}} \dfrac{\partial B}{\partial p_{s}} - \dfrac{\partial A}{\partial p_{s}} \dfrac{\partial B}{\partial q_{s}} \right) $ is a coordinate transformation according to the words of this Wikipedia page: (link: http://en.wikipedia.org/wiki/Poisson_bracket) _...".It places mechanics and dynamics in the context of coordinate- transformations: specifically in coordinate planes such as canonical position/momentum, or canonical-position/canonical transformation."..._ The Jacobian of, for example, $ A(q,p) $ and $ B(q,p) $ has the same form as the terms in the parenthesis. From what to what coordinates does the bracket transform? Does it make any sense to make comparison (Poisson bracket and the Jacobian)?


title: Closed formula for product of gamma matrices

text: I was asking myself if there is a closed formula for the following product of gamma matrices: $$\gamma_\mu\gamma_\nu \gamma_5.$$ I would like to express this matrix in terms of the basis $${\mathbb{1}, \gamma_{5}, \gamma_{\mu}, \gamma_{\mu}\gamma_{5},[\gamma_{\nu},\gamma_{\mu}]} .$$ Thank you very much for any help.


title: Anomalous target space diffeomorphisms for one-loop world-line integrals

text: The Schwinger effect can be calculated in the world-line formalism by coupling the particle to the target space potential $A$. My question relates to how this calculation might extend to computing particle creation in an accelerating frame of reference, i.e. the Unruh effect. Consider the one-loop world-line path integral: $$Z_{S_1} ~=~ \int^\infty_0 \frac{dt}{t} \int d[X(\tau)] e^{-\int_0^t d\tau g^{\mu\nu}\partial_\tau X_\mu \partial_\tau X_\nu},$$ where $g_{\mu\nu}$ is the target space metric in a (temporarily?) accelerating reference frame in flat space and the path integral is over periodic fields on $[0,t]$, $t$ being the modulus of the circular world-line. If the vacuum is unstable to particle creation, then the imaginary part of this should correspond to particle creation. Since diffeomorphism invariance is a symmetry of the classical 1-dimensional action here, but not of $Z_{S^1}$, since it depends on the reference frame, can I think of the Unruh effect as an anomaly in the one-dimensional theory, i.e. a symmetry that gets broken in the path integral measure when I quantize? This question would also apply to string theory: is target space diffeomorphism invariance anomalous on the worldsheet?


title: Dark energy before radiation or after?

text: In the cosmic history of the universe, does the dark energy comes before the radiation epoch, or only now, in the 'matter' universe (matter dominated era)? Because, we now know that like 75% of our visible universe is made up of dark energy, about 20% is dark matter and the rest is the 'ordinary matter', but does that mean that this dark energy came after the radiation and matter domination of the universe or was it there in the beginning, at the BB, and then was just spread out via inflation, while radiation era and matter domination era followed?


title: Optical Waveguide's "Base Bandwidth"

text: * * * Consider a dielectric slab waveguide (lossless, isotropic) illuminated transversally from the vacuum (with coherent, monochromatic light). We define the _base bandwidth_ of a waveguide (or optical fiber), $AB$, to be the inverse of the time retardation, $\Delta t$, at 1 km of the waveguide between the energy of a guided mode (transmitted following the _zig-zag_ model) with a $\theta_{c}$ critical angle and the energy transmitted without total internal reflections. Let $n_{f}$ and $n_{s}$ be the refractive indexes of the film and the substrate respectively. Prove that if $\Delta n=n_{f}-n_{s}<<1$, then: $$ {\rm{AB}} = {\left( {\Delta t} \right)^{ - 1}} \simeq \frac{{2c{n_s}}}{{{{\left( {{\rm{AN}}} \right)}^2}}} = \frac{c}{{{n_f} - {n_s}}} $$ where $AN$ is the numerical aperture of the guide. * * * This problem has been on my mind for 2 days now and it seems I can't find a method to calculate that time difference... Any ideas? _**My thinking so far:_** We have to look at the Ray Optics picture of Dielectric Waveguide Theory (see e.g. Tamir et al: _Integrated Optics_ (chapter 2)). A guided mode is propagated through the waveguide following a series of total internal reflections at an angle $\theta_{c}$ with the normal to the film-substrate or film-cover surfaces, and therefore its energy is "trapped" in the film. A mode which is _not_ a guided mode will travel through the waveguide suffering reflections and refractions and therefore radiating some energy to the cover and substrate. ![Dielectric slab waveguide](http://i.stack.imgur.com/9LTzS.png) The rays travel through the film at the same speed $c/n_{f}$ but follow different paths, so it will take different times for them to advance 1 km in the waveguide. We could then try to find the components of these velocities in the direction of propagation ($z$ axis), $v_{i}$, and use the simple relation $t_{i}=d/v_{i}$. The trouble with this, is that the problem doesn't specify the angle at which the non-guided mode incides in the surfaces of the film. Am I missing something here? ![Total internal reflection in the slab waveguide](http://i.stack.imgur.com/24Kix.png) On the other hand, the numerical aperture of the waveguide is: $$ n\sin \left( {{\theta _{\max }}} \right) = {n_f}\sin \left( {90 - {\theta _{\rm{c}}}} \right) = {n_f}\cos \left( {{\theta _{\rm{c}}}} \right) $$ where n=1, and $\theta_{\max}$ is the angle of incidence of the illumination beam with the normal to the $x-y$ plane so that doing some work we find: $$ {\rm{AN}} = \sin \left( {{\theta _{\max }}} \right) = \sqrt {{n_f}^2 - {n_s}^2} $$ * * * **UPDATE:** Some calculations: ![Rays](http://i.stack.imgur.com/7mSHt.jpg) The effective refractive indexes of the 3 rays (ray 1: guided mode, ray 2: radiation mode, ray 3: no reflection at all) are: $$ {N_1} = \frac{{{\beta _1}}}{{{k_1}}} = \frac{c}{{{V_1}}} = {n_f}\sin \left( {{\theta _{\rm{c}}}} \right) $$ $$ {N_2} = \frac{{{\beta _2}}}{{{k_2}}} = \frac{c}{{{V_2}}} = {n_f}\sin \left( \theta \right) $$ $$ {N_3} = {n_f} = \frac{c}{{{v_3}}} $$ So that the retardations would be: ($d=1km$) Between rays 1 and 2: $$ \Delta {t_{1 - 2}} = {t_2} - {t_1} = d\left( {\frac{1}{{{V_2}}} - \frac{1}{{{V_1}}}} \right) = \left[ {...} \right] = \left( {d\frac{{{n_f}}}{c}} \right)\left( {\sin {\theta _c} - \sin \theta } \right) $$ Between rays 1 and 3: $$ \Delta {t_{1 - 3}} = {t_3} - {t_1} = d\left( {\frac{1}{{{v_3}}} - \frac{1}{{{V_1}}}} \right) = \left[ {...} \right] = \left( {d\frac{{{n_f}}}{c}} \right)\left( {\sin {\theta _c} - 1} \right) $$ And the respective base bandwidths: $$ {\rm{A}}{{\rm{B}}_{1 - 2}} = \frac{{\frac{c}{{d{n_f}}}}}{{{\rm{AN}} - \sin \theta }} $$ $$ {\rm{A}}{{\rm{B}}_{1 - 3}} = \frac{{\frac{c}{{d{n_f}}}}}{{{\rm{AN}} - 1}} $$ Are any of these results equal to the equation given at the beginning for $AB$? How would one use the approximation $n_{f}-n_{s}<<1$?


title: A discrete approach to the catenary

text: ![Look at this.](http://i.stack.imgur.com/CvXe0.png) I'm trying to work out a model for the system above, that is, $N$ particles of unitary mass subject to the constraints: $$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf r_i-\mathbf r_{i-1}|^2,\qquad 1\leq i \leq n+1,$$ and limited to the $xy$ plane ($\mathbf r_0$ and $\mathbf r_{n+1}$ are two fixed point on the $x$ axis). I'd like to write the Lagrangian of the above system (the particles are subject to gravity). I thought that one good choice for the coordinates could be to take, for $1\leq i\leq n+1$, the angle $\theta _i$ that $\overrightarrow{P_{i-1}P_i}$ makes with the $x$-axis. Note that there are $2n-(n+1)=n-1$ degrees of freedom and so $n-1$ angles should suffice. With this choice we can write the positions as $$\mathbf r_{i}=\mathbf r_{i-1}+(\cos \theta _i, \sin \theta _i)=\mathbf r_0+\sum _{k=1} ^i(\cos \theta _k ,\sin \theta _k).$$ In this line of reasoning, the two constraints are given by: $$\mathbf r_{n+1} -\mathbf r_0=\sum _{k=1} ^{n+1} (\cos \theta _k,\sin \theta _k),$$ that are equivalent to: $$\cos \theta _n+\cos \theta _{n+1}=\ell - \sum _{k=1} ^{n-1} \cos\theta _k,$$ $$\sin \theta _n+\sin \theta _{n+1}=-\sum _{k=1}^{n-1}\sin \theta _k,$$ assuming that the system is solvable for those two (by the implicit function theorem and the second equation it must be solvable locally for a couple of $\theta$s). Given this, I don't know how to go further. The velocities $\mathbf v_i$ and generalized velocities $\dot \theta _i$ satisfy similar relations to those of the positions, but I don't see any way to use these facts to write down the kinetic energy. So my question is: is it possible to write the lagrangian of this system without some sort of approximations (i.e. small angles etc.)? Is it possible to do it following my line of reasoning, completing my analysis? Or maybe a totally different approach would work? Thank you in advance.


title: Internal kinetic energy and center of mass kinetic energy

text: For a given system, how can you tell which one is kinetic energy for center of mass and which one is internal kientic energy? K = Kcm + K int For example, "A 150 g trick baseball is thrown at 63 km/h. It explodes in flight into two pieces, with a 39 g piece continuing straight ahead at 85 km/h. How much energy do the pieces gain in the explosion?" I found the answer using conservation of momentum. I solved for the velocty and then taking the difference in kinetic energies before and after, the change was .984 J. However, relative to the equation, how do I know which one is internal kinetic energy and which is center of mass kinetic energy. Does K in the equation mean to total energy? How would you find Kcm and Kint? Can someone please clarify?


title: Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$

text: While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting:   * $\int \phi^\dagger \hat A \psi \:\mathrm dx$ as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$ when one tries to observe the observable corresponding to $\hat A$"   * $\int \phi^\dagger \psi \:\mathrm dx$ as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$". All integrals are over all space here, and all $\phi$s and $\psi$s are normalized. Now, I realize that interpretation of the first integral doesn't really make sense when put side by side with the second one. For example, when $|\phi\rangle$ is an eigenstate of $\hat A$, I get two different expressions for "the square root of probability that one will get the corresponding eigenvalue of $\psi$ when one tries to observe the observable corresponding to $\hat A$". As far as I can tell, $\int \phi^\dagger \hat A \psi \:\mathrm dx \neq\int \phi^\dagger \psi \:\mathrm dx $, even if $|\phi\rangle$ is an eigenstate of $\hat A$. I am inclined to believe that the second integral is the correct answer here (it comes naturally when you split $|\psi\rangle$ into a linear combination of basis vectors). But I am at a loss as to the interpretation of the first integral. So, my question is, **what are the physical interpretations of $\int \phi^\dagger \hat A \psi \:\mathrm dx$ and $\int \phi^\dagger \psi \:\mathrm dx$?** (While I am familiar with the bra-ket notation, I have not used it in this question as I don't want to confuse myself further. Feel free to use it in your answer, though)


title: Basic Interpretation of Compostion of Observables and their Measurement

text: Given two (or more) observables $A, B$ which commute one can construct a third observable $C= A \circ B$. If $\psi$ is a common eigenvector of $A, B$ with eigenvalues $\lambda_1, \lambda_2$ then it is clear that the measurement of $C$ of the state $\psi$ gives the measurement result $\lambda =\lambda_1 \lambda_2$, i.e. the result of the measurement of the observable $C$ is the result from the measurement of $A$ times the result from the measurement $B$. But what if $\psi$ is an eigenvector of $C$, but not of $A$ and $B$? Is there any connection between the measurement results of $A$, $B$ and $C$? Example: Let there be three observers which measure a spin state with the corresponding observables $A = \sigma_x \otimes \mathbb{I} \otimes \mathbb{I}$, $B=\mathbb{I} \otimes \sigma_y \otimes \mathbb{I}$ and $C=\mathbb{I} \otimes \mathbb{I} \otimes \sigma_y$. They commute and we can construct $D= A \circ B \circ C = \sigma_x \otimes \sigma_y \otimes \sigma_y$. Now the GHZ-state $\psi = \frac{1}{\sqrt{2}} ( | +z, +z, +z \rangle - | -z, -z, -z\rangle)$ is an eigenvector of $D$ with eigenvalue $\lambda =-1$ but it is not an eigenstate of $A, B$ or $C$. Each of the observers will get a result $\pm 1$. Is there any connection between this individual results and the eigenvalue of $\psi$ (respectively the expectation value $\langle \psi | D | \psi \rangle = -1$)? Intuitively I would say that the product of the results should give the eigenvalue of $\psi$ but I can't see how this should follow from any quantum mechanical postulate or mathematical reasoning like in the case of the common eigenvector.


title: How could they be hit twice by debris?

text: In the movie _Gravity_ the two astronauts (Bullock and Clooney) are hit _twice_ by the fast moving debris that bring havoc to their locations. I think this is a bad scientific mistake in the movie plot. Indeed, different objects cannot travel the _same_ orbit at different speeds. The shape of the orbit is defined by the total energy which depends essentially on velocity. So, objects that happen to be at the same point with different velocities will follow different orbits. It is true that orbits are closed (in classical mechanics, not counting disturbing effects) so that an object will go over and over again through the same points, and the speedier objects will be again at the point where it met the slower one, but in the meantime the slower has also moved and the two won't pass through the meeting point at the same time again. Any further meeting would occur only after a certain number of orbits, which makes way more than the mere 90 minutes claimed in the movie. Is there a scientifically sound explanation for this?


title: Photon particle/wave question

text: Imagine a source of photons at the center of a spherical shell of detectors at radius $R$. Assume the photons are emitted one at a time. Now if photons are particles that are highly likely to travel on straight paths at velocity $c$ then one would expect the following behavior: At time $t=0$ as the photon is emitted in a particular direction the source recoils in the opposite direction. Later at time $t=R/c$ the photon is absorbed by one of the detectors which recoils as it absorbs the photon. But quantum mechanics says that the photon is actually emitted at time $t=0$ as a spherical wave that expands out to the detectors at the velocity $c$. While the spherical wave is in transit from the source to the detectors the source cannot recoil in any particular direction as no direction has been picked out yet by the photon detection. So does the source only recoil when the photon is absorbed at time $t=R/c$ or is its recoil somehow backdated to $t=0$ to be consistent with the particle picture?


title: Counting of brownian particles: Point Process

text: Imagine a point process defined by the passage time of purely brownian particles through a given point (in 1D), line (2D) or plane (3D). I'm interested in the variance of the counts (number of particles passing the points) as a function of sampling time. Unlike brownian motion, I expect my signal to show non-vanishing correlation, because a particle that just crossed the point, has a high probability to cross it again. Thus the correlation should look like a dirac delta function (classic for discrete point process) plus a decreasing time function. However, I'm a bit confuse with the way to calculate it analytically. I search in areas of photon counting, geiger counter, and gaz dynamics, but I didn't find any explicit calculation of the variance. Two idea I had: 1/ Studying the Spatio-Temporal correlation function of a field of Brownian particles (see Gardiner, Stochastic Methods, eq. 13.3.22 for instance): $G(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-x^2/(4Dt)\right)$ in 1d. We should have: $Var(T)=\lim_{x\rightarrow 0} \int_T\int_T G(x,t) dt dt'$. However, the last expression is hard to compute. 2/ Using the probability of return time to origin of Brownian motion (the correlation arise from successive passage of the same particle) Any idea or suggestions would be welcomed !


title: What distinguishes the particles we chose as matter from their antimatter equivalent?

text: Back before we knew about antimatter we just called everything matter. Ignoring CP-violation for a moment, there is nothing special about matter versus antimatter. Once we knew about antimatter it was easily to label all of the common particles as matter. How do we do it for some of the more exotic stuff though? For the Muon and Tau, do we call the matter version matter simply because they have negative charges like the electron? Do we use conservation of lepton number? For things like the Top quark, how can we distinguish between the two and determine which is the Top and the other the Anti-Top? Did we decide that the matter one was the one with the positive charge because the Up quark has a positive charge? How about for the W+ versus W- bosons? Does it matter which is the matter W and which is the antimatter W? Have we designated which a matter W? For Neutrinos it seems extra tricky. Did we go based only on lepton number preservation? For Mesons which have a quark and a different anti-quark, is the choice of the matter versus antimatter meson completely arbitrary?


title: Is there a difference between "two photon absorption" and "double quantum transitions"?

text: Wikipedia has articles on two photon absorption. And a lot of NMR literature refers to double quantum transitions. But is there a difference? As far as I can tell, a double quantum transition is has an intermediate step. $m = 1 \rightarrow m=2 \rightarrow m=3$. But two photon absorption just absorbs two photons and skips the $m = 2$ state.


title: Dropping an object from a plane

text: I've done this before, but it's been a long time. Using trigonometric functions, I've been asked to solve the following problem. However, I'm at a complete loss as to how to do it. I have 5 questions similar that I know I can figure out if given an example. A plane moving at $180$ mi/hr comes down to a $300$ ft elevation, flying straight and level. It releases a package to be caught in a net on the ground. Neglecting air friction, what should the horizontal distance between the plane and the net be at the time of release?


title: Quantum Locking Superconductors on Earth's Magnetic Field

text: Is the earth's magnetic field strong enough to hold a quantum locked superconductor, or the superconductor wouldn't be able hold its own weight? How strong should be the earth's magnetic field to hold quantum locked superconductors?


title: Resolution in a Fourier transform spectroscopy setup

text: I am a bachelor physics student and as an assignment we had to perform measurements on an FT spectroscopy setup. **Context.** Our setup consisted of a Michelson interferometer through which the light of two sources namely a HeNe-laser and a sodium lamp, passed at the same time. The light of the interference pattern was split into two. One of the beams was passed through a polarizer to filter out the laser light (partially). Both beams were sent to their respective detectors which measured the intensity in a single spot of the interference pattern. One of the mirrors of the interferometer was attached to a moving stage which was moving at a constant velocity during the measurement. Here is a drawing of the setup: ![Fourier transform spectroscopy setup](http://i.stack.imgur.com/fbS8c.png) The detectors measured the intensity during the time the stage was moving. We therefore received intensity vs. time as data. Detector 1 measured the intensity of the sodium light + the weakened laser light. Detector 0 measured the intensity of both the laser light and the sodium light (the latter was negligible since the laser light was so much stronger than the sodium light). The data from detector 0 was used for calibration. We assumed the laser light consists of a single wavelength of light. We measured however that there were varying frequencies in the intensity. These variations must therefore be caused by flaws in the setup (like faulty calibration or inconsistency in the speed of the stage). The variations were corrected by moving the data points along the time axis such that a nice sinusoid remained. The frequency of that sinusoid was chosen to be the average frequency of the original data (I am not sure if this is true. We were not told the exact details of the data correction procedure.) To prepare the data for Fast Fourier Transform the data points were moved along the sinusoid such that they were spaced equally horizontally. Here is an illustration of the procedure: ![Data correction procedure](http://i.stack.imgur.com/zMMug.png) Because the variations caused by flaws in the setup were the same in the signals from both detectors (since they were measuring at the same time) the corrections of the data from detector 0 were applied to the data from detector 1 as well. The data from detector 1 was Fourier transformed (using the FFT algorithm). The frequencies found from this were scaled down by a factor of $\frac{2v}{c}$, where $v$ was the speed of the stage. (I am leaving out the derivation.) We scaled the frequencies back up such that the frequency of the laser peak matched its optical frequency and turned those into wavelengths. This gave the following spectrum: ![Sodium spectrum](http://i.stack.imgur.com/XYq15.png) Now onto the main problem. How do I determine the spectral resolution of this setup? Spectral resolution is the smallest difference in wavelength the spectrometer can detect at a certain wavelength. The time for measurements has long passed so it's no longer possible to make new measurements. **Question.** We were told that because the laser peak is sharpest in our spectrum its width gives us the resolution. But I do not understand how these are related. Can someone explain that? If the width of the laser peak does not give the resolution, what would and why? What I am also interested in are the factors that influence the resolution. **Attempt.** I was thinking about a hypothetical case where the detectors could measure infinitely fast producing a continuous signal. The setup would also have a special laser with a variable wavelength instead of a sodium lamp. By collecting the data from detector 0 which is measuring the intensity of both lasers and applying a continuous Fourier transform (we are skipping the data correction procedure) a spectrum is obtained. Because of the variations in the frequencies each laser will produce a normal distribution (a Gaussian) in the spectrum. The spectrum then consists of a sum of two Gaussians. One method of finding out how many peaks there are in a spectrum is counting the maxima. If the Gaussians are far apart we can see two maxima. But when two Gaussians get closer and closer together (the distance between their maxima decreases), at some point their sum will only have a single maximum before the tops of the two Gaussians have overlapped. At this point the two peaks can no longer be distinguished and our method will tell us there is only one peak. For the following plot we took the sum of two identical Gaussians $e^{-(x-a)^2}+e^{-(x+a)^2}$. The two Gaussians are a distance $2a$ apart. By taking the derivative with respect to $x$ and setting it equal to $0$, the extrema can be found. In the plot the $x$-coordinate of the extremas are plotted against $a$. WolframAlpha plot The minimum distance that the Gaussians have to be apart such that two maxima can be seen could be called the resolution. This distance is a function of the amplitudes and standard deviations of the two Gaussians which are themselves determined by the quality of the setup. To find the resolution of this hypothetical setup one could decrease the difference in wavelength between the two lasers till only one maximum is visible and repeat this for all wavelengths. However, this is just a hypothetical setup and not the real one. In the real one, the detectors measure a finite amount of points per second, there is data correction, FFT and a finite amount of point per nanometer in the spectrum and it's not clear how the resolution is determined.


title: Ascent rate and size of balloon

text: I am part of a school project, Project Stratos to send a balloon to the edge of space (the closer side :P) and was wondering how you would work out the accent rate of a large balloon (roughly 1m^3 of helium with 100g of mass) and the size of it as it increases its Altitude. I am creating a live map (that will be based on predictions rather than its actual location) and want to know the speed it will float up into the atmosphere. Currently we are assuming the ascent rate will be about 5m/s but I doubt that is very accurate and would this speed increase as it gets higher? Edit: I would also quite like to know the burst height of the balloon.


title: Does a simple double pendulum have transients?

text: Suppose, we have the most simple double pendulum:   * Both masses are equal.   * Both limbs are equal.   * No friction.   * No driver.   * Arbitrary initial conditions (no restriction to low energies) Does this pendulum have transients for any initial conditions or is it immediately on its respective attractor? I have seen several time series that suggest that there are no transients, however, I could not find any general statement on this. * * * I here say that a system has no transients, if its trajectory comes arbitrarily close in phase space to its initial state again, i.e., for a given initial state $x(0)$ and for all $ε>0$, there is a $T>0$ such that $\left | x(T) - x(0) \right | < ε$.


title: Artificial Gravity - Spinning Station Questions

text: I've done a little research but I can't seem to find the answers to these questions in regards to a spinning space station simulating gravity. 1) If we imagine the space station as a hollow doughnut and I'm floating in the center of the tube while the station is stopped what happens when the station starts? How are forces translated to my body that push me towards the floor or would I just remain floating there while the station flies by? 2) Does the spinning station rely on friction to simulate gravity? If the inner surface of the station were frictionless would I still move along with the station or would the station simply fly by me? Thanks!


title: Understanding the cause of the big bang

text: Ok, as I understand the expansion of the initial singularity was caused by quantum fluctuations like the ones predicted by the Heisenberg Uncertainty Principle. But how can these fluctuations occur within a singularity? And how can they cause the expansion of that singularity? Or do I have a terrible misunderstanding of the theory?


title: Why is it concluded that the cosmos is expanding when in fact the observations are for events further back in time?

text: Why is it concluded that the cosmos is expanding at an ever increasing rate when looking further out and therefore further back in time? Surely, if the time were reversed to play forwards, the conclusion would be that the expansion of the universe is actually deaccelerating, and the need to postulate 'Dark Energy' would be avoided.


title: Does a Lorentz-contracted object float or sink?

text: Consider the following thought experiment: Imagine an object of a certain mass density which allows it to float in water. Now if this object is viewed from a moving frame with high speed, it will look Lorentz contracted hence its density will look greater than the rest density. So from the rest frame it floats but from the moving frame it sinks! What am I missing?


title: Why is spacetime near a quantum black hole approximately AdS?

text: In this link, one of the answers contains the statement > If you examine the space-time near a finite area quantum black hole, you > will see an approximate AdS space. Presumably "approximate" means this is only true to some order in the distance from the horizon ? Could someone outline where the result comes from, or provide a reference.


title: Theoretical Upper Bound on Processor Speed?

text: Barring aside considerations such as heat dissipation, capacitance, etc... (aka any sort of technological issue) what is the fastest speed of a processor? I am told that at distances of 1 planck length $L =1.616$ x $ 10^{-35}$ m Heisenberg's uncertainty principle hovers around an error factor of 1. Does that suggest then CPU speeds (in this case measured via period) theoretically cannot exceed something along the same order of magnitude of: $L/c = 5.387$ x $10^{42}$ calculations per second? (pretending a single cycle of a lightwave bouncing across a single planck length of distance amounts to an entire calculation)


title: Where is radiation density in the Planck 2013 results?

text: I've been looking at the Planck 2013 cosmological parameters paper, trying to update my toy cosmology simulator with the most recent data. Most of the interesting values such as $H_0$, $\Omega_m$, and $\Omega_\Lambda$ can be found in Table 2 on page 12, but the one thing I didn't find was an estimate of the energy density of radiation. Can this be derived from some other parameters in these data?


title: Why does tossing a coin in a train and on a train differ?

text: Suppose that I am inside of a moving train. I have a coin in my hand and I am standing still. If I toss this coin straight up, it will fall back into my hand. Now, suppose that I am on that moving train's roof. Now if I toss the coin straight up, the coin will not land in my hand, rather it will fall toward the place where the train was when I tossed it. Why is there a difference?


title: Relation involving the Lorentz transformation and the inverse of its transpose

text: The relation I was referring to in the title is $${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$ where ${\Lambda_a}^b$ is the inverse transpose of $L$, the Lorentz transformation. I was wondering about the significance of such an equation? What does it imply in general relativity?


title: Units of the PDF in the Lattice Boltzmann Method

text: In the Lattice Boltzmann method we require (based on mass conservation) that the sum of the distribution functions for a node is equal to the density, i.e. $$ \sum_i f_i = \rho $$ But what units do $f_i$ have that make their sum equal to the density, which has units of $kg/m^3$? I guess it means that $f_i$ must have same units as the density, but how can I interpret $f_i$ in that case?


title: Why objects of different masses accelerate at the same speed under gravity

text: If you push massive object it accelerates slowly compared to object which is of same size but less density/mass. However, all objects do Accelerate at the same rate. `F=ma`....according to this if gravity is a force than spherical ball made of cotton shouldn't also accelerate toward earth with acceleration of 9.8 m/s2 just as the ball made of steel would also do at the same acceleration. But they both do. All i can think of is that force experienced by object increases or decreases linearly according to the mass an object has, is that true. Force object experiences changes according to its mass. I guess the whole confusion could be that if gravity wasn't called a force, in my mind force is something that has magnitude which is same regardless of the mass of object it's touching. Word forcefield would make more sense even though i won't have any idea what does that really mean.


title: Can capillary forces be used to make hydro electricity?

text: Could a device with very thin columns of glass or something that attracts water more be used to pull water up and then release it to drive generator and perhaps add vacuum. I have been wondering about this on and off for like 8 years.


title: Components of forces?

text: Let's say we have an object lying on a slope, the weight of the object can be divided into two components one horizontal and one vertical. My question is does the weight act only in these two directions or does it spread evenly between the the both components?


title: PSF Measurements In Fluorescence Imaging

text: Quite a technical question! I have measured the Point Spread Function of 100nm fluorescent breads with my Olympus scanning head. I'm two-photon exciting the beads with a wavelength of 800nm and focused in the sample with a 100x with N.A.: 1.4 The theory suggests me the resolution of such a system to be: $$d=\frac{0.7\cdot\lambda}{N.A.}\approx 400nm$$ Now that value should be equal (with at most a 18% correction) to the FWHM of the 2D gaussian I obtain on the image. But from my analysis of the images I obtain a FWHM close to 2um. Now surely the formula is for an _ideal_ optical setup but a factor five seems to me quite strange! Is that possible to obtain such a result, in the case of **very not ideal** optical elements, or should I look for some sort of problem with the acquisition sw that tells me the pixel dimension of the images?


title: How to compute speed without knowing mass or charge values?

text: > A gold nucleus is 460 fm (1 fm = 10^(-15) m) from a proton, which initially > is at rest. When the proton is released, it speeds away because of the > repulsion that it experiences due to the charge on the gold nucleus. What is > the proton's speed a large distance (assume to be infinity) from the gold > nucleus? (Assume the gold nucleus remains stationary.) My solution: suppose the gold nucleus' charge is $Q$, and the proton's charge is $q$, then $\int_{R}^{\infty} {KQq\over (r^2)}=\frac{1}{2}mv^2$, where $R=460~\text{fm}$. That's my solution. But the question doesn't tell me the mass of the proton and the quantity of charge for the gold nucleus. How can I solve it? I am confused!


title: Is it possible/correct to describe electromagnetism using curved space(-time)?

text: Comparing the simples form of the forces of both phenomena: the law of Newton for gravitation $V\propto \frac{1}{r}$, and the Coulomb law for electrostatics $V\propto \frac{1}{r}$, one might think that if one can be extended relativistically to a curvature in space-time, that the other one would lend itself to a similar description. Is this so? Thanks for any useful thoughts and/or suggestions!


title: Is there any relationship between Gravity and Electromagnetism?

text: We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force . Now, is there any relationship between Electromagnetism and gravity?


title: Theories that Relate Gravity, Electricity, and Magnetism

text: There are some people who (without having a stated theory that I know of) insist that Gravity, Electricity, and Magnetism are related. Some point to symmetry in Maxwell's Equations as a potential indication of this (intuitive) connection. Is it possible that there are "Maxwell-like" equations that relate Electricity and Magnetism directly to Gravity? All forms of electromagnetic energy do interact with gravity. Are there any physicists who are working on this sort of theory?


title: Invariant interval and gravity

text: Gravity is inversely proportional to the _distance_ between objects. Do we use Euclidean distance or the invariant interval for that distance? Using the invariant interval makes everything a bit more complex: An object that is one lightyear away, is also one year "younger" when we observe it. Does that temporal distance contribute, so that when we calculate trajectories and physical phenomena on a distance - they are different from what's observed locally? Does this affect observed things such as the trajectory of two equally distant (from us) objects and rotational velocity of distant objects? An example: When we look towards the center of the milky way, we're looking about 27 000 years back in time; i.e. the temporal distance is 27000 years, and spatial distance is 27 kly When we look at the center of the andromeda galaxy, it's about 25 Mly away. When we look at another object in the andromeda galaxy, which is also 25 Mly away - the temporal distance between those objects, _relative to us_ is 0. So; _is the relative temporal distance between objects and the observer incorporated in Einsteins field equations?_


title: Magnetic deviation of a uniform external magnetic field in the presence of a well-defined iron object

text: Suppose initially I have a uniform magnetic field. Now, in this formerly-uniform magnetic field, I place an un-magnetised cylindrical body of iron into this field of known dimension, with the axis of rotation aligned with the (formerly uniform) magnetic field. How would one go about mathematically describing the magnetic field? How does this solution change as the cylinder is rotated about a different axis? * * * **Note:** This isn't a homework problem, but I'm guessing that there is a classic solution in ferromagnetism or magnetostatics to this problem. I'm not quite sure where to start looking - it's a bit beyond my freshman physics textbook that was gathering dust on my shelf. My vector calculus is a bit rusty, but I can probably work it out :) The actual shape isn't too important, so if an object other than a cylinder helps illustrate the solution, then I am happy to use it.


title: How to calculate the dispersion relation for a wave equation with non-constant speed of wave propagation?

text: Specifically, it is a one-dimensional wave equation for waves on a string with a non-constant cross-section, i. e. $$S(x)=S_1+S_2 \cos{2x}; \qquad c(x)=\sqrt{F/\rho\, S(x)}.$$ Separating the variables yields the Mathieu differential equation $$\frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0,$$ $a$ and $q$ are parameters of the string. One form of the solution is the Floquet solution $$F(a,q,x) = \exp(i \mu \,x) \, P(a,q,x),$$ where $P(a,q,x)$ is a periodic function and $\mu$ is a complex exponent (another form of the solution are the Mathieu sine and cosine, which are odd and even functions respectively, if that is of any use). The temporal part of the solution is sinusoidal. My question is how to get the dispersion relation $\omega(k)$ from the equation. With the normal wave equation one would plug the straight wave solution $e^{i(kx-\omega t)}$ into the equation, which would yield the dispersion relation $\omega=c\,k$.


title: Additive Trichromatic Color: Seeking Answer for a Fun Application

text: I'm not the brightest person and I was wondering if it was possible, in the same way that it's possible to make ink that's visible under UV light, if it's possible to make ink that requires a distinct combination of colours of light to work. If there are different coloured UV lights would it be possible to make an ink (or buy one) that reacts only when all three are present and overlapping?


title: Sound velocity in ideal Bose gas

text: Speed of sound is related to the derivative of pressure with respect to density: $v_s=\sqrt{(\frac{\partial P}{\partial \rho})_S}$ where S tells us the derivative must be taken while keeping entropy constant. For an ideal Bose gas, $P$ and $\rho$ can be expressed in terms of Bose- Einstein functions: $g_v(z) = \frac{1}{\Gamma(v)}\int \frac{x^{v-1}}{z^{-1}e^x -1}$ where in the given context $z=e^{\frac{\mu}{T}}$ and $\lambda = \lambda (T)$ is the thermodynamic wavelength Those expressions are: $ P = \frac{T}{\lambda^3} g_{\frac{5}{2}}(z)$ $ \rho = \frac{m}{\lambda^3} g_{\frac{3}{2}}(z)$ Having all this in my hands, I still fail to perform the differentiation. Could you please hint what should my first step be?


title: Is it always possible to express an operator in terms of creation/annihilation operators?

text: I'm referring to _"Path integral approach to birth-death processes on a lattice"_ , L. Peliti, J. Physique 46, 1469-1483 (1985), available at: http://people.na.infn.it/~peliti/path.pdf The article is about a reformulation of the master equation for a Markov process in terms of the path-integral formalism. However my question is mainly about Quantum Mechanics. The author defines a Hilbert space $\mathcal{H}$, an orthogonal basis of which is given by $\mid n \rangle$, $n \in \mathbb{N}$, with: $$ \langle n \mid m \rangle = n! \delta_{n,m} $$ The creation/annihilation operators are defined on $\mathcal{H}$ as follows: $$ a \mid n \rangle = n \mid n - 1 \rangle $$ $$ \pi \mid n \rangle = \mid n + 1 \rangle $$ and they are easily to be seen each other's hermitean conjugates, according to the scalar product just defined. The conventions are a little bit different from Quantum Mechanics, but this is not really relevant for my question. The author implies that it is possible to rewrite every operator $O: \mathcal{H} \rightarrow \mathcal{H}$ only in terms (sums of products) of creation/annihilation operators. I cannot demonstrate this assertion. I have tried taking the matrix elements of a generic operator $O$, and demonstrating that everything can be rewritten in terms of $a$ and $\pi$ but actually this is not working. Thank you in advance for any replies.


title: What are some fields where physics is leading mathematics?

text: I know developments in physics and mathematics often nourish each other, and at different points in history, physics and math have taken turns leading each other. An example of a physicist "inventing" the required mathematics is Newton and his calculus. At other times, the mathematics have been discovered as a curiosity, and the physicists come along to show where it is used in nature. My question is: In the current state of affairs, in which fields does the physics lead mathematics?


title: Solving a Statically Indeterminate Truss with Energy Equations (Tricky)

text: Okay, first off, I know that there is a very similar problem about statically indeterminate trusses here, but my problem was a bit different and I didn't know how to adapt the answer for that problem to mine. ![enter image description here](http://i.stack.imgur.com/Ucm9F.png) So my problem is this: I'm trying to solve for the displacement at node 7 using the energy method, and then I need to solve it using the Finite Element Method. Since the Energy Method seemed the easiest, I went for that first. Through the sum of moments at node 7 and at node 1, I found out that the vertical reaction force at 1 and 3 are the same, and their horizontal reaction forces are equal but in opposite directions, thus cancelling out. 2 doesn't have a horizontal reaction force. Thus I found out that this was statically indeterminate, since now I had $P = 2\cdot(R_{1y}) + (R_{2y})$. So my next step was to use the energy method from the question I linked to earlier, in order to solve the forces in each member. However, being the idiot that I am, I don't know how to solve the forces in the members to begin with. I don't know which forces point where. I know that if the forces at the ends of a member point away, it's tension. Otherwise, it's compression. But since I know that the reaction forces at 1, 2, and 3 point up, that would mean that the force in the members at 1, 2 and 3 must point down, which means that it is in compression....at least in my head it does. In short, could someone help me get this so that I can go on with my Energy method equation?


title: Centripetal force and circular motion

text: I have a doubt, say when a body moves in a circular loop, there are basically two types of acceleration acting on the object. One is the linear acceleration which is basically tangent to the circle, and there is an acceleration that acts inward which is basically centripetal acceleration. These two acceleration vectors are perpendicular to each other and they have a net acceleration vector pointing inward. My question is since there is a net acceleration due to these two acceleration vectors then the net force must also be acting in that direction. Then why does the object not move in the direction of that net force rather than moving along the circular path. Please understand that i am not talking about the centripetal force, rather i am taking about the net resultant force(centripetal + tangential force). I feel that along the tangential direction of my circular path, the force would be basically the force of my engine which is driving my car. This force kind of propels me forward in the linear direction, and during the circular turn, the friction force basically would provide me the centripetal force and prevent me from flying off my turn. So, what would be the meaning of the resultant of these two forces would imply in my real world because the resultant of these two force would be parallel to the X-axis and would act inward. I am basically moving the circular arc due to the combined effects of these two forces. Please I am standard 10 student, so my physics concepts are not really very advanced. Can someone explain this to me in a little conceptual manner.


title: Why is potential energy negative when orbiting in a gravitational field?

text: I had to do a problem, and part of it was to find the mechanical energy of satellite orbiting around mars, and I had all of the information I needed. I thought the total mechanical energy would be the kinetic energy+ the potential energy, or $KE+PE$. However, I had the answer sheet and it said that I had to do $KE-PE$, because when you integrate $Gm_1m_2/r^2$ you get a negative sign. I can see why it works out mathematically, but I don't understand why you are actually losing energy when orbiting in a gravitational field.


title: The physicality of the photon propagator

text: The equation for the photon propagator is straightforward $$ D_{ij} = \langle 0 |T \\{ A_{i}(x')A_{j}(x) \\}|0 \rangle $$ However, $A_{i}(x)$ is gauge- dependent and therefore unphysical (in the arguable sense). Then, since the propagator is dependent on the vector potential, the propagator is unphysical. Sadly, my whole understanding of what amplitudes mean may be skewed, but I would assume the probability amplitude for a photon to propagate between $x$ and $x'$ is something we would want to be gauge- _independent_. Edit: I guess I wasn't clear enough. By computing the probability amplitude for a process, we obtain a complex number that when multiplied by it's complex conjugate we obtain a probability for such a process to occur (when normalized). Here, the physical process is propagation, and the probability is $|\langle 0 |T \\{ A_{i}(x')A_{j}(x) \\}|0 \rangle|^2$. However, this probability is gauge dependent, and hence, the usual physical interpretation of $|\langle 0 |T \\{ A_{i}(x')A_{j}(x) \\}|0 \rangle|^2$ is questionable to me. Where has my interpretation gone astray?


title: Cosmology questions from a novice

text: These ideas/questions probably represent a lack of understanding on my part, but here they are: 1) Cosmologists talk about the increasing speed of expansion of the universe and talk of dark energy as the cause. But, I keep thinking that the farther out we observe, is farther back in time and the light we observe is from a time billions of years ago. We don't know what they look like or how fast they are moving now. Who's to say they haven't slowed down? How can we know now? So, how can we say that the universe is increasing in it's rate of expansion? All we can say for sure is that it was expanding at that rate. Another thought- on black holes: 2) If Black holes where not composed of highly dense matter, why would there be different "sizes" of Black holes. if all black holes were collapsed to a "singularity", there should be no difference in "size" (the diameter of the light free area). Therefore, I have trouble with the singularity concept and think they are just another form of dense matter that just happens to have enough gravity to hold back even photons.


title: Integral in Peskin and Schroeder

text: I'm having a bit of a slow day, and can't see how to do the following integral in Peskin and Schroeder (page 107 for anyone with the book). We've derived in the centre of mass frame the integral over 2 body phase space $$\int d\Pi_2 =\int \frac{dp_1p_1^2d\Omega}{(2\pi)^3)2E_12E_2}(2\pi)\delta(E_{cm}-E_1-E_2)$$ where $E_1^2=p_1^2+m_1^2,\ E_2=p_1^2+m_2^2$ and $E_{cm}$ is the total initial energy. Apparently one can then "integrate over the final delta function" to get $$\int d\Pi_2=\int d\Omega\frac{p_1^2}{16\pi^2E_1E_2}(\frac{p_1}{E_1}+\frac{p_1}{E_2})^{-1}$$ How is this done? I suppose it's some kind of clever change of variables or trick but I can't see it! Apologies if I'm missing something easy! To be honest I'm just puzzled as to why we're left with any factors of $p_1$ after doing the integral, because it's just an integration variable, isn't it? A clear argument would be very much appreciated, as would some intuition as to why I'm misunderstanding the notation! Many thanks in advace.


title: How connected thermodynamical stability and dynamical stability for black holes?

text: Criteria for thermodynamical stability is the convex of entropy. But for black hole entropy is non-additive.


title: Determine date of birth from "astrological" signs?

text: Suppose someone tells me what constellation the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn were in when they were born, and the information is astronomically accurate (ie, not astrological signs). Could I use this information to estimate their date of birth (including year) to a reasonable approximation? My thoughts:   * 12 constellations, 7 planets (incl Sun/Moon) yields 35,831,808 possibilities, and we know the moon changes constellations daily. If the planets were in purely random constellations every day, it would take 98104 years for all possible combinations. Using this, and additional information that the person was born in the last 200 years, it would seem easy to find their birth date.   * Of course, planets aren't in random constellations. Additionally, the birthday paradox suggests there will be many repeats despite the 35,831,808 combinations.  However, it still seems like you could do a fairly good job of approximating someone's birthday with this information?


title: Would other intelligent life in the Universe be likely to construct the same periodic table of the elements?

text: Is the structure of the periodic table fundamental enough that other forms of intelligent life would likely structure their own listing of the elements in the same way (i.e. are there other ways of arranging the periodic table (in 2D) which would be equally useful, or is the current structure the most efficient way to convey the properties of the elements)?


title: Is the Doppler effect for sound symmetrical for observer or source movement?

text: It makes intuitive sense to me for the apparent frequency of a sound as modified by the Doppler effect to be based entirely on the speed at which the observer and the source get closer or farther apart, regardless of whether the source or the observer is moving, based on the principle of relativity. However, the general equation I receive from every source $$f'=f((v±v_o)/(v∓v_s))$$ does not have this property. For example, if the speed of sound in the medium is 2 units and the observer and source are coming closer at a rate of 1 unit, then if we take the observer to be moving then we get $$f'=f(3/2)$$ but if we take the source to be moving then we get $$f'=f(2/1)$$ What is the difference between these two cases?


title: Why does a human body only emit infra red radiation and not other types of electromagnetic radiation?

text: What causes humans to emit infra red radiation and why don't we also emit other types of electromagnetic radiation such as ultra violet or microwave? Sunlight contains UV and our body takes it in, but when we give off radiation, do we not radiate it back? I know that the constant motion of molecules embodies heat, and that when an electron is excited by colliding with another, it jumps to a higher energy level and falls back again, releasing energy in the form of photons. My question is, then, does this energy 'in the form of photons' carry heat? Or is heat another form of energy release that is separate from EM radiation?


title: How much weight can a person carry to avoid being blown over by wind?

text: I wonder if there is a certain amount of weight that I could carry in a rolling bag to help prevent me from being knocked over by the wind. I have balance problems and some lack of muscle strength in my legs and sometimes it feels like strong wind gusts here in Boston, MA may knock me over, especially when I walk in a wind tunnel between buildings. I have a rolling bag that I pull along that weighs 6 lbs. when it has a pair of shoes in it. I also fill it with kids' books and toys for work but they don't usually weigh very much. The handle extends 20" above the bag and I can keep a good grip on it. Would the bag have to weigh more than I do in order to be of any help? Thanks for any ideas. Donna


title: Meaning of kinetic part in the Lagrangian density?

text: What is the **physical meaning** of the kinetic term in the classical scalar field Lagrangian $$\mathcal{L}_{kin}~=~\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)~?$$ It gives how does the field change from one space-time point to the other. Right? What is the significance of the relative minus sign between the square of the time gradient and square of the space gradient?


title: What are the current size limitations of NIR spectroscopes?

text: So few days ago I came across this new tool for smart phones http://www.consumerphysics.com/myscio/. It says it uses NIR spectroscopy, which I don't know lot about. My question is: **Is it possible for such a small device (73x25x16.5 mm) to be accurate? What are the size limitations of this technology?** They claim that scio will be able to recognize molecules, which concentration is above 1%. Is it possible? How deep under surface can NIR spectroscopy go? What are the material limitations of this technology? I know I have lot of questions, but I am really interested in this topic. If anyone could suggest me some literature about this as well, I will be glad.


title: Functional derivative in Lagrangian field theory

text: The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{align} where $'$ is $d/dt$. Question: What is \begin{align} \frac{\delta q(t)}{\delta \dot{q}(t')}? \end{align} I'm asking this because in QFT, the lecturer defined the canonical momentum field to $\phi$ by \begin{align} \pi(x,t) ~:=~ \frac{\delta L(t)}{\delta \dot{\phi}(x,t)}, \end{align} where $L$ is the Lagrangian, a functional of the field: $L[\phi,\partial_\mu \phi] = \int d^d x \mathcal{L}(\phi,\partial_\mu \phi)$. I know I should get \begin{align} \pi(x,t) ~=~ \frac{\partial \mathcal{L(x,t)}}{\partial \dot{\phi}(x,t)}. \end{align} (Note it's now a partial derivative with respect to the Lagrangian density.) But doing it I get: \begin{align} \delta L = \int d^dx \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial \partial_\mu \phi} \delta\partial_\mu \phi. \end{align} So somehow we ignore the first term $\int d^dx \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi$! Why is that? It can't be that we are treating $\delta \phi$ and $\delta \dot{\phi}$ as independent because if I were to take the functional derivative w.r.t. $\phi(x')$, I would have to move the dot from $\delta \dot{\phi}$ over to $\frac{\partial \mathcal{L}}{\partial \dot{\phi}}$ which will give me $$\int d^dx (\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu\frac{\partial \mathcal{L}}{\partial \partial_\mu \phi}) \delta \phi$$ i.e. the functional derivative gives the Euler-Lagrange equations. So how to I take the functional derivative of a functional w.r.t to the derivative of a function?


title: Are the second and third generations of matter required or optional?

text: If the second and third matter families (so-called generations: muon, muon neutrino, tau, tau neutrino, ...) didn't exist, would that affect how the universe runs? Are they optional or required?


title: Lorentz Transformation via Geometry

text: Today, I was tutoring and explained the space-time. I explained how one can convert North-South into West-East by rotating, and how you can convert time into space with velocity. Below the Energy-Momentum stuff the book had some problems. One was the following: > Given an event at (1 Lightsecond, 2 s), give it's coordinates in a frame of > reference that is moving with 0.6 of speed of light to it. This the diagram I came up with, in blue there is the event. (Not really to scale.) ![](http://wstaw.org/m/2011/11/11/m8.png) Then I projected the point onto the orange axes, and I get smaller values on each. This makes sense to me, as a moving observer would see thing smaller and slower, resulting in less values in either measurement. But how do I get the orange lines? I cannot see how I get this angle. If I got them, I could tell you the speed of the reference frame, but the speed is given here … And did I do the projection of the speed right?


title: How is quantum superposition different from mixed state?

text: According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Now consider state $\left|\Psi\right>=(\left|\psi_1\right>+\left|\psi_2\right>)/\sqrt2$, which is a superposition. Let $\left|\psi_i\right>$ be eigenstates of Hamiltonian. Then measurements of energy will give $50\%$ chance of it being $E_1$ and $50\%$ of being $E_2$. But this then corresponds to the definition above of mixed state! At the same time superposition is defined to be a pure state. So, what is the mistake here? What is real difference from mixed state and superposition of pure states?


title: Why is the Dirac operator so important - in both physics and mathematics?

text: Why is the Dirac operator considered so important - in both physics and (pure) mathematics?


title: Homogeneity and isotropy of stress energy tensor

text: Given the energy momentum tensor in E&M: $T_{\mu\nu} = -F_{\mu\alpha} g^{\alpha \beta} F_{\beta \nu} +\frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma}$ I want to show it is homogenous and isotropic in Robertson-Walker spacetime. How would you go about performing a rotation and a translation to show that $T_{\mu \nu}$ is left invariant?


title: Why does charge accumulate at points?

text: This is a very trivial question, but I can't seem to reason it out, again, as to why charges gather at points and edges.


title: Work done by an expanding gas

text: Today in an engineering thermodynamics lecture, the professor gave an example of a gas doing work. We had a cylinder full of helium at a pressure of something like 200 kPa absolute and the valve was opened so that the gas would flow out against the atmospheric pressure until the pressures were equal. Also the cylinder was assumed to be in thermal equilibrium with its surroundings so the temperature of the gas was equal to the temperature of the ambient air. However, the way he calculated the work perturbed me. He said that this was an isobaric process because the gas was expanding against a constant atmospheric pressure. I was under the assumption that an isobaric process means that the working fluid stays at constant pressure throughout the process which is not the case in this expansion. And in this case, the gas pressure is dropping as it leaves the cylinder. The professor then proceeded to calculate the work as $W = P_\text{atm}\Delta V$. But I didn't think that was right and that simple. I should also note that the system was modeled as a closed system, so the gas inside the cylinder and the escaping gas were the system. If this is not an isobaric process why is the work calculated with a constant pressure? Am I correct, or is the professor?


title: Physicality of claims regarding AC ice energy storage units

text: Many of us may remember the problems from thermodynamics where we had to determine if a pretend air conditioning unit supplier's claims violated the 2nd law. I want to pose a similar question regarding a new "green" Air Conditioning (AC) technology. An ice energy storage unit for AC works by making ice during the night and using the latent heat (mostly) to cool the building during the day. The major motive for doing this is to move the energy demand from the peak energy use times to the low-demand times at night when it is cheaper. The cost of electricity is not relevant to this question. The claims of a vendor of this technology, **Ice Bear Energy Storage System** , include the following: > Bottom line: The Ice Bear system reduces total net energy consumption for > most buildings under virtually all operating conditions and installations. **Question:** Is this claim physically plausible, given reasonable assumptions about climate? There are some example cities below, I imagine that considering an average location or just the extremes would be sufficient. Here are a few things to note, but these are certainly not comprehensive. Creativity may be important for identifying all the factors at play.   * The temperature of ice is $0^{\circ} C$ ($32^{\circ} F$)   * Room temperature is $22.2^{\circ} C$ ($72^{\circ} F$)   * The outside ambient temperature is lower at night than during the day. Exactly how much depends on the location and humidity particularly. Locations mentioned on their website include Glendale, CA; Howell, NJ; Las Vegas, NV; Fort Collins, CO; Orange County, CA; and Redding, CA.   * The ice storage duration is 6 hours and since these are multiple MW products, I'll say it's fair to assume almost perfect insulation unless someone wants to challenge this.   * You _can_ obtain useful work while exchanging the heat from the office building to the ice. I believe this would be important considering the wording of "total net energy consumption". # Temperatures I have looked up the temperatures for most of the mentioned cities. For some I couldn't get data for I used a nearby city instead.   * **(City/State) (night-day difference) (seasonal difference) (hottest month high)**   * Redding CA, 27.4, 44, 99   * Las Vegas NV, 26.5, 48, 104   * Denver CO, 27.8, 45, 88   * Lakehurts NJ, 27.8, 45, 88   * Santa Anna CA, 20.5, 18, 83 These seem to be fairly consistent for the night-day difference in particular. I hope that simplifies the problem. It seems clear that running at night "buys" you about 27.5 degrees F. Now, the absolute temperatures as well as the seasonal variation will affect the efficiency, but to a much smaller degree. My conflict: Can a $27.5$ degree difference really make up for a $72-32=40$ degree penalty on the thermal cycle? _Really_? It's not a trivial problem, but I am skeptical. I will certainly send them an email letting them know they violated the 2nd law depending on the outcome here.


title: Cyclic rotation of a Matrix

text: What is meant by a cyclic rotation of a matrix, specifically in proving that j*k-k*j=i*l where j,k,l are cyclic rotations of the Pauli spin matrices sigma x, sigma y, and sigma z


title: Cascade in relativistic turbulence

text: The Kolmogorov theory of turbulence indicates an energy cascade in turbulence. Is there a corresponding version of relativistic fluid?


title: Stern-Gerlach-Experiment with j=1 atoms?

text: Suppose you do a Stern-Gerlach experiment with atoms in a $j=1$ state. There would be three separate beams ($m_j = -1, 0, 1$) coming out of the apperatus. But what would be the relative distribution of atoms in these beams? On first thought I would say that the distribution should be $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{3}$. But considering the "classical" distribution of the projections of the angular momentum onto one axis I could imagine that there could be more atoms in the $m_j = 0$ state. This would still be compatible with symmetry considerations. To state the question differently: how does the density matrix of an unpolarized beam of $j=1$ atoms look like?


title: Objective measure of anisotropy

text: If we know a function $f(\phi, \theta)$ in $\mathbb R^3$ only over a convex surface (which for simplicity let's assume a sphere of radius $r$), is there any measure for the degree of anisotropy over this surface? That is, given two functions, $f_1, f_2$ is there any objective measure $A(f)$ where we can make qualitative comparisons like $A(f_1) > A(f_2)$? Thoughts:   * For an electrostatic field produced by a centered monopole we should get $A(f_{\text{monopole}})=0$.   * For sufficiently large $r$, the field produced by a possibly non-centered monopole versus that of a possibly non-centered dipole should be $A(f_{\text{dipole}}) > A(f_{\text{monopole}})$.   * An expansion in spherical multipole moments seems like a good start, but how would you compare the two expansions?   * Would the comparison generalize to a non-spherical but concave surface provided you normalized for the surface area?   * This may be related to the calculation of the degree of anisotropy of the CMB, but I'm not familiar with how they define anisotropy.


title: Why is a coherent state an eigenfunction to the annihilation operator?

text: In class when we talked about the harmonic oscillator in QM we noticed that the eigenfunctions to the annihilation operator are coherent states in the sense that they have minimum uncertainty in momentum and position. My question is: What is the physical meaning behind this? Why are coherent states eigenfunctions to the annihilation operator and: Was this only a lucky punch and a special property of the harmonic oscillator or is this true for more physical problems?


title: A block in motion explodes

text: > A 9.5 kg block (A) is traveling in the positive x-direction with a speed of > 3.0 m/s. At some point, it explodes and breaks into two pieces. After the > explosion, block C, which is 6.0 kg, moves off in the negative y-direction, > and block B, which is 3.5 kg, moves off at an angle 35° with respect to the > x axis. ![sorry for the crappy recreation](http://i.stack.imgur.com/KZk8n.jpg) The two parts of this question that I am stuck on are: > **a)** find the momentum of the 2-block combination after the explosion and > write it in vector notation   * Here I know that momentum is conserved, so Pa = Pbc = 28.5 kg*m/s   * I am confused about how to separate the x and y components of the momentum after the explosion: can I use trigonometry with the mass of block B as the hypotenuse? > **b)** Find the speeds of blocks B and C after the explosion   * Since momentum is conserved, and Mass(A) = Mass(B)+ Mass(C), I know that V(A)=V(B)+V(C), but I don't understand how to split them (I know it's not in half...)


title: Maxwell equations and symmetry

text: Do the full inhomogeneous Maxwell equations obey parity (P) and time reversal (T) symmetry separately or only the full CPT symmetry? I believe the homogeneous Maxwell equations obey parity and time reversal symmetry separately - is that right? The homogeneous Maxwell equations reduce to a wave equation in which space and time appear as second order derivatives. Here's my effort at answering my first question. In the first inhomogeneous equations one has: $$\mathbf{\nabla . E} = \frac{\rho}{\epsilon_0}$$ If I change the sign of x,y,z, in $\mathbf{\nabla . E}$ then I must change the sign of $\rho$ to balance it. In the third equation one has: $$ \mathbf{\nabla \times E} = - \frac{\partial \mathbf{B}}{\partial t}$$ As the sign of x,y,z has changed then the sign of t must change to balance it. Thus we must have CPT symmetry. This can be checked in the last equation: $$\mathbf{\nabla \times B} = \mu_0 \mathbf{j} + \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$ The sign of x,y,z in $\mathbf{\nabla \times B}$ has changed. This is balanced by a sign change of t in $\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$. The sign of $\mathbf{j}$ has changed because the charge has changed sign. Thus the inhomogeneous Maxwell equations obey CPT symmetry rather than C,P or T alone. Is this reasoning right?


title: A key relation in Di Francesco's book on Conformal Field Theory

text: Recently I am reading Di Francesco's book Volume I on Conformal Field Theory. In order to reduce the number of fields in a correlator, the calculation of Operator Algebra is extremely important. To do this, equation (6.175), which is derived from (6.174), is necessary. The author writes in the book below equation 6.174: "Applying this relation (here the author refers to eq.6.174) on the r.h.s of Eq.(6.169), we find:" ![enter image description here](http://i.stack.imgur.com/HDFai.png) But I still can not understand how 6.174 is applied to deduce this equation presented above. Nor can I understand why the holomorphic dimension h appears on the right hand side of this equation. Since this is very important for the discussion on the calculation of the Operator Algebra, I can not go on without an understanding of this.


title: An electron falling into a black hole

text: If an electron falls into a black hole. How can the Heisenberg uncertainty principle hold? The electron has fallen into the singularity now so it has a well defined position which means that it doesn't have a well defined momentum? Furthermore, the electron can't have a well defined position in space because space eigenkets are unphysical. It's momentum must certainly cause it to move. Another question, Can one calculate the amount of new mass(the relativistic mass) that the black hole acquire after quantum particles fall in the singularity? Doesn't this mean that the electron can't be described by a wave packet at the singularity of the black hole? If we want quantum mechanics to be applicable inside a black hole,the wavefunction should leak outside right ?


title: The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$

text: Who can provide me some elegant solution for $$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$ other than brute force calculation? ================================================ O lala!!! Thanks for @Prathyush and @Qmechanic !!! I got the same result with Qmechanic... I think Prathyush's suggestion should be equivalent to the my suggestion of the correspondence up to a canonical transformation. Here is my calculation (I was not confident to post it...) $\begin{array}{c} \mbox{representation of }\left(\hat{a},\hat{a}^{\dagger}\right)\mbox{ on polynomial space }span\left\\{ \frac{x^{n}}{\sqrt{n!}}\right\\} _{n\ge0}\\\ \hat{a}\left[f\left(x\right)\right]=\frac{d}{dx}f\left(x\right)\;;\;\hat{a}^{\dagger}\left[f\left(x\right)\right]=xf\left(x\right)\;;\;\left[\hat{a},\hat{a}^{\dagger}\right]\left[f\left(x\right)\right]=id\left[f\left(x\right)\right]\\\ \left|0\right\rangle \sim 1\;;\;\left|n\right\rangle \sim x^{n}/\sqrt{n!} \end{array}$ $\begin{array}{c} \mbox{calculate the normal ordering }\left[\hat{a}^{M},\hat{a}^{\dagger}{}^{N}\right]\mbox{:}\\\ \sim\left[\frac{d^{M}}{dx^{M}},x^{N}\right]=\frac{d^{M}}{dx^{M}}\left(x^{N}\star\right)-x^{N}\frac{d^{M}}{dx^{M}}\left(\star\right)\\\ \sim\left\\{ \overset{min\left\\{ M,N\right\\} }{\underset{k=0}{\sum}}\frac{N!}{\left(N-k\right)!}C_{M}^{k}\left(\hat{a}^{\dagger}\right)^{N-k}\left(\hat{a}\right)^{M-k}\right\\} -\left(\hat{a}^{\dagger}\right)^{N}\left(\hat{a}\right)^{M}\\\ \end{array}$ ====================== One comment on 02-12-2012: The representation I was using is actually related to Bergmann representation with the inner product for Hilbert space (polynomials) being: $\left\langle f\left(x\right),g\left(x\right)\right\rangle :=\int dxe^{-x^{2}}\overline{f\left(x\right)}g\left(x\right)\,,x\in\mathbb{R}\,,\, f,g\in\mathbb{C}\left[x\right]$


title: How does the buoyant force on a cube at the bottom of a tank of water manifest itself?

text: Let's say a 10N cube (in air, on Earth) rests flat on a scale at the bottom of a tank of water, and the scale reads 8N, so there is 2N of buoyant force on the cube. How does the buoyant force manifest itself on the cube? I assume the force is normal to the surface of the scale, but if there isn't any water under the cube, how does the water push _up_ on the cube?


title: Hamiltonian in position basis

text: Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this: $\langle x \mid H \mid x' \rangle = \frac{-h^2}{2m}\frac{\partial^2}{\partial x^2} \delta(x -x')$. How do we get this? are we allowed to do $\langle x | \frac{\partial^2}{\partial x^2} \mid x' \rangle = \frac{\partial^2}{\partial x^2} \langle x \mid x' \rangle$? Why? It seems some thing similar is done above.


title: Is carbon dioxide a greenhouse gas?

text: > **Possible Duplicate:**   >  What experiments prove the greenhouse effect? I am seeking for a proof that CO2 is a greenhouse gas. I posted this on **Skeptic.SE** recently but found no help in seeking for proof: > I assisted to a physicist conference in my university a few years ago > against the case that carbon dioxide was a cause of global warming. The main > point was that CO2 is **not** a greenhouse gas. I did a research to find > evidence for either side and found absolutely nothing. > > So, is carbon dioxide a greenhouse gas? If yes, has it been demonstrated in > a scientific paper? > > Here are some discussion articles describing arguments against CO2 being a > greenhouse gas: Home experiments Ph.D. Tim Ball Evidence from temperature rise through history I understand that the sources do not prove that CO2 isn't a greenhouse gas...but I can't find evidence proving the contrary. Since a real physicist with many years of experience a demontrated to me that it is not possible by definition, I need to find proof to accept the other premise. I found this post on physics.SE, but the asker do not seek proofs and simply accept what the others are saying. Baically, to have a greenhouse effect, a gas needs to absorb only a portion of a wavelength and let pass another portion. Since CO2 absorbs some IR, a part of it is remitted to earth. The spectrum is absorbed by the earth (everything) and it is remitted on another wavelength which allows a different proportion to pass through the CO2 layer. The question is, why the heat would be remitted toward earth (it should go in the colder direction)? And also, if CO2 is all around the earth and blocking a certain wavelength, why would increasing it warm the earth? Am I missing something?


title: what is the difference between binding energy and adsorption energy

text: I know the equations, but I am having difficulties to get the difference between this two concepts, binding energy and adsorption energy.


title: Can a liquid insulator be electrically charged by touching a charged conductor?

text: > Can a liquid insulator be electrically charged by touching a charged > conductor? I understand that solid insulator will only be charge on the surface where it is touch, but the case is different from liquid which it circulates, so I imagine that slowly liquid insulator will be fully charge due to circulation.


title: What are the temperatures of objects in Low Earth Orbit (LEO)?

text: What is the temperature of objects in Low Earth Orbit? Consider LEO to be 600km to 800km.


title: Rotational invariance and operator-squares

text: My mind is drawing a blank right now. In systems with spin and orbital angular momentum, I know that rotational invariance implies that $[H, \mathbf{J}]=0$ where $\mathbf J=\mathbf L+\mathbf S$. But I don't see why rotationally invariant systems should necessarily commute with the _square_ -magnitude operators: $$[H,\mathbf{L}^2]=0\qquad\text{and}\qquad[H,\mathbf{S}^2]=0\,.$$ But the funny thing is, I am unable to come up with model Hamiltonians that are rotationally invariant, but do not commute with $\mathbf{L}^2$ and $\mathbf{S}^2$. So my question is: is there an easy algebraic reason for this commutation relation? or are there systems that do not conserve these operators? In the latter case, I need help coming up with examples.


title: Optical refocusing efficiency

text: What is the material for optical focusing that produces the less diffraction losses? Suppose one have a sequence of serial optical elements $R$ that keep refocusing a beam of collimated light, each a distance $D$. Now suppose that the wavelength $\lambda$, the optical element and the distance are such that the $\epsilon$ is the amount of radiation that falls outside the next optical element, which i'll take to be small. I'm concerned with other losses in this system, for instance, the optical material can have some intrinsic diffusion, or it absorbs a portion of the radiation What are the main losses that an optical element can have of this kind, and which are the best refocusing optical elements that minimise those losses?


title: What happens when the black hole at a galactic core eats the galaxy?

text: I'm making several assumptions, not sure if any are correct:   * there is a black hole at the center of a galaxy   * the black hole is eating the galaxy Eventually the galaxy will be gone, right? Has this been observed? Do we know what happens afterwards? Posting here since astronomy got merged into physics


title: How will the super massive black hole affect our galaxy?

text: I've recently learned that the general consensus is that several (if not, most) galaxies have super massive black holes in their center, in particular the Milky Way. This, at least to me, makes perfect sense seeing as we are in a spiral galaxy which means we need something to "spiral" around (a large body or a bunch of mass). But seeing as we're rotating around this massive black hole, won't we inevitably end up sucked in by it? Aren't we spinning towards the center of the galaxy, or are we staying steady where we are?


title: Can every particle be regarded as being a combination of Black holes and White holes?

text: Can the statement be regarded as true? That every particle, or element in the universe can be regarded as a combination of black hole and white hole in variable proportion.


title: Randomness, Chaos, Quantum mechanical probability functions

text: Can someone explain these 3 concepts into a unified framework.   1. Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision of the experiment) physics but, the complexity allows us to treat the system as random. (i.e we can always run an exact physical simulation to predict the coin's fall)   2. Chaotic randomness: randomness as seen in weather or fluid flow, where the system is highly sensitive to initial conditions and as a result no matter what the precision of measurement/simulation is, we cannot predict the out come.    3. Quantum randomness: The inherent probability distributions that we study in QM, where the electron's position probablity distribution is set by QM.  how do the concepts of randomness in coin toss, randomness in chaos and randomness in QM fit into a unified picture. Can someone please elaborate ?


title: Pauli matrix rotations

text: When doing physics with two-level systems and introducing rotations, a term that appears quite often is the rotation of a pauli matrix by another one: $e^{- i \sigma_i \theta/2} \sigma_k e^{i \sigma_i \theta/2}$ The way I know to evaluate this is by using $\exp(i \sigma_i \theta/2) = I \cos \frac{\theta}{2} + i \sigma_i \sin \frac{\theta}{2}$ and expanding the previous equation. If $i = j$ it's trivial, but in the other cases it get's quite tedious. Is there an easier way to do that (or some shortcut)?


title: Reachable area of cannonball given fixed initial speed

text: The trajectory of a cannonball fired from the origin with initial speed $v_0$ at an angle $\theta$ is given by $$y = x\tan\theta - \frac{g}{2v_0^2 \cos^2\theta}x^2.$$ For fixed $v_0$, at what angle $\theta$ should the canon ball be fired so that it will hit a target located at $(x_0,y_0)$? What is the area of the region in which the target can be located such that it is possible to hit the target?


title: Is it only the ground state of the quantum harmonic oscillator that has the minimum uncertainty product?

text: We know that the uncertainty product of general states is bounded by the inequality described by Heisenberg's uncertainty relation. And the ground state of the quantum harmonic oscillator falls under the minimum uncertainty product ($\frac{\hbar}{2}$). Do the excited states of QHO have also the minimum uncertainty product or higher?


title: Are black and white colours?

text: I am an artist, and I have heard that white is a 'tint' and black is a 'shade' in the artist world. I have also heard that they are colours in the normal world/scientific world. Which is correct and why?


title: Is black a color or absence of color?

text: Is black a color or absence of color? When there is no light, everything is black, so how can we say that black is a color?


title: Suppose we fill a membrane with ferro fluid, and alter the magnetic field around it in specific ways, what dynamics can come into play?

text: im trying to develop a flexible robotic arm filled with ferro fluid. Different flexible motions can be performed using this concept.


title: Motion in a straight line?

text: I am trying to solve the following problem that involves motion in a straight line. I am not quite sure how to go about it though. I know it will involve formulas and derivatives but which specific ones would I use? > An electron has a constant acceleration of 2.6 m/s^2. At a certain instant > its velocity is 12.1 m/s. > > What was its velocity 2.5 s earlier? What was its velocity 2.5 s later?


title: Why the name "optical phonon"?

text: Why do we have two different types of phonons, optical and acoustical? Why are they called optical phonons?