Why isn't the wave equation $\nabla^2 \psi - 1/c^2 \partial_{tt} \psi = (\frac{mc}{\hbar})^2\psi$

Special relativity was well established by the time the schrodinger equation came out. Using the correspondence of classical energy with frequency and momentum with wave number, this is the equation that comes out, and looks sensible because this is of the form of a wave equation, like the one for sound etc, except with an inhomogeneous term $$\nabla^2 \psi - 1/c^2 \partial_{tt} \psi = (\frac{mc}{\hbar})^2\psi$$ Instead, we have schrodinger's equation which reminds of the heat equation as it is first order in time. **EDIT** Some thoughts after seeing the first posted answer What is wrong with negative energy solutions? When we do freshman physics and solve some equation quadratic in, say, time... we reject the negative time solution as unphysical for our case. We do have a reason why we get them, like, for the same initial conditions and acceleration, this equation tells usthat given these iniial velocity and acceleration, the particle would have been at the place before we started our clock, since we're interested only in what happens _after_ the negative times don't concern us. Another example is if we have areflected wave on a rope, we get these solutions of plane progressive waves travelling at opposite velocities unbounded by our wall and rope extent. We say there is a wall, an infinite energy barrier and whatever progressive wave is travelling beyond that is unphysical, not to be considered, etc. Now the same thing could be said about $E=\pm\sqrt{p^2c^2+(mc^2)^2}$ that negative solution can't be true because a particle can't have an energy lower than its rest mass energy $(mc^2)$ and reject that. If we have a divergent series, and because answers must be finite, we say.. ahh! These are not real numbers in the ordinary sense, they are _p-adics_! And in this interpretation we get rid of divergence. IIRC Casimirt effect was the relevant phenomenon here. **My question boils down to this**. I guess the general perception is that mathematics is only as convenient as long as it gives us the answers for physical phenomenon. I feel this is sensible because nature can possibly be more absolute than any formal framework we can construct to analyze it. How and when is it OK to sidestep maths and not miss out a crucial mystery in physics.


What parity has an electron?

I couldn't find anything about the parity of an electron. Neither in the german, nor in the spanish and nor in the english version of Wikipedia. I only found one sentence in the parity article of Wiki: > One way to fix a standard parity operator is to assign the parities of three > particles with linearly independent charges B, L and Q. In general one > assigns the parity of the most common massive particles, the proton, the > neutron and the electron, to be +1. - Wiki But I cannot find other sources to confirm it. I do not need it for a special exercise, it's just to understand the whole thing...


A simple way of calculating Euler Angles from Rotation Matrix --- help!

This is a follow up of this question : I have the rotation matrix $$ \left( \begin{matrix} a_{11} & a_{12} & a_{13}\\\ a_{21} & a_{22} & a_{23}\\\ a_{31} & a_{32} & a_{33}\\\ \end{matrix}\right) $$ I'm using pre-multiplying rotation matrix (that operates on column vectors) for intrinsic rotations (i.e. I make rotations about the axes of the plane that rotates). And since the fixed frame is my reference frame --- $$ \left( \begin{matrix} 1 & 0 & 0\\\ 0 & 1 & 0\\\ 0 & 0 & 1\\\ \end{matrix}\right) $$ My rotation matrix is nothing but the column unit-vectors of the axes of the rotated frame, i.e. $$ \left( \begin{matrix} x_{1} & x_{2} & x_{3}\\\ y_{1} & y_{2} & y_{3}\\\ z_{1} & z_{2} & z_{3}\\\ \end{matrix}\right) $$ So therefore I have the values of a11,a12,a13,a21,a22,a23,a31,a32,a33 as x1, x2, x3, y1, y2, y3, z1, z2, z3. Now if I consider a particular set of rotation (say X first, then Y , then Z), with the corresponding Tait-Bryan angles \--- a,b and c. My rotation matrix will be the following. Rx(a)*Ry(b)*Rz(c) \--- $$ \left(\small{ \begin{matrix} \cos(b)\cos(c) & -\cos(b)\sin(c) & \sin(b)\\\ \cos(a)\sin(c) + \cos(c)\sin(a)\sin(b) & \cos(a)\cos(c) - \sin(a)\sin(b)\sin(c) & -\cos(b)\sin(a)\\\ \sin(a)\sin(c) - \cos(a)\cos(c)\sin(b) & \cos(c)\sin(a) + \cos(a)\sin(b)\sin(c) & \cos(a)\cos(b)\\\ \end{matrix}} \right) $$ Now if I have to solve for the above angles a, b & c (pitch, yaw and roll), I basically have nine equations but three unknowns. Following are the equations ---               a11 =  cos(b)∗cos(c)     a12 = −cos(b)∗sin(c)     a13 =  sin(b)     a21 = cos(a)∗sin(c)+cos(c)∗sin(a)∗sin(b)     a22 = cos(a)∗cos(c)−sin(a)∗sin(b)∗sin(c)     a23 = −cos(b)∗sin(a)     a31 = sin(a)∗sin(c)−cos(a)∗cos(c)∗sin(b)     a32 = cos(c)∗sin(a)+cos(a)∗sin(b)∗sin(c)     a33 = cos(a)∗cos(b)       This is where I learnt about it. Now I am using a non-linear least squares curve fitting method to solve the above set of over-determined equations. There are two major problems that I am encountering   1. The final values of a,b, c change as I change the initial values in the iterative algorithm. I get different results if I start from [50 50 50] and different results with [0 0 0].   2. Secondly I don't think that the values obtained are correct; since the angles of pitch, yaw & roll seem pretty much different in the video. I'm using the lsqcurvefit(click for the question I asked on stackoverflow) command in Matlab. (click for documentation) I have been on this problem of how to calculate pitch, yaw & roll for quite some time now. I think this post will give you all the details of what I have tried. I need your help to know if what I am doing is the best approach to tackle my problem. If so, please point out what is wrong in my method. If you think there are other simpler methods, please let me know about them. I'm sure there has to be a better method, since this seems like a pretty simple thing to do. Should I change my Matlab algorithm? Anyone know any special Matlab/Mathematica toolbox that calculates the yaw, pitch, roll? Thanks!


Kallen–Lehmann spectral representation for an arbitrary spin

Let's have Kallen–Lehmann spectral representation for the scalar theory: $$ \tag 1 D(p) = \int \limits_{0}^{\infty} d(\mu^{2})\frac{\rho (\mu^{2})}{p^{2} - \mu^{2} + i\varepsilon}. $$ We can represent $(1)$ in a form $$ D (p) = D_{free}(p, m)Z + \int \limits_{m^{2}}^{\infty}d(\mu^{2})\rho (\mu^{2})D(p, \mu), $$ where $\mu$ refers to the mass of the multiparticle states and $m$ refers to the mass of one-particle free state. From here there is interesting result: it can be showed that $0 \leqslant Z \leqslant 1$. How to get analogous result for the arbitrary spin field (or at least for fermions with spin $\frac{1}{2}$ and for photons)? I know the general structure of the propagator of the free theory (for simplicity I assume massive case): for field $\Psi $ with spin $s$ $$ \hat {\Psi}_{l} (x)= \sum_{\sigma = -s}^{s}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2E_{\mathbf p}}}\left( u^{\sigma}_{l}(\mathbf p )e^{-ipx}\hat {a}_{\sigma}(\mathbf p ) + v^{\sigma}_{l}(\mathbf p )e^{ipx}\hat {b}^{\dagger}_{\sigma}(\mathbf p ) \right) $$ it will be $$ D_{lm}(p) = \frac{F_{lm}(p)}{p^{2} - m^{2} + i\varepsilon}, \quad F_{lm} = \sum_{\sigma}u^{\sigma}_{l}(\mathbf p )\left(u^{\sigma}_{m}(\mathbf p) \right)^{\dagger}. $$ For the derivation of spectral representation for the scalar field see the reference above. Here also you can see the problem with this derivation which appears for the case of field with nonzero spin: it is impossible to introduce the scalar density function $\rho (\mu^{2})$. Also there is bigger problem: there was using relation $\langle |[\hat {\Psi}(\mathbf x , t), \frac{d}{dt}\hat {\Psi}^{\dagger}(\mathbf y, t)]|\rangle = i\delta (\mathbf x - \mathbf y)$ for getting relation $0 \leqslant Z \leqslant 1$ (look to the Weinberg's "QFT" (chapter "Kallen–Lehmann spectral representation" of the section "Nonperturbative methods") or to Greiner's "Field Quantization" (pp. 278-282)). But this relation is correct only if one field is canonical conjugated to another (one field is canonical coordinate while the another one is canonical momentum). But for the arbitrary spins the canonical momentums must be chosen "individually".


Does entropy decrease through measurement?

For an electron in its rest frame, we have an entropy $$ S = \log 2, $$ which comes from the 2 possible spin directions along z-axis. If the measurement $S_z$ changes its state to $\left| + \right>$, the entropy goes to zero. Does this violate the second law of thermodynamics?


How many years would it take the Pioneer space probe to travel to Proxima Centauri with its current speed?

Pioneer space probe moving at a speed of $30km/s$. Assuming its heading for Proxima Centauri, which is situated at $4.2ly$ from earth, calculate how long it would take to get there in years, to the nearest year? Could you please help.


What's the Noether charge associated with Kaehler invariance of SuGra?

What is the Noether charge associated with Kahler invariance of supergravity (SUGRA)? As the question is rather tangential to what I need to do, I have not tried explicitly calculating it myself, but I'm sure that I'm not the first one to wonder.


How can a conductor be grounded yet there are induced charges on it?

A classic example for the method of images is the following, quoted from Griffiths's Introduction to Electrodynamics, page 121: "Suppose a point charge $q$ is held a distance $d$ above an infinite grounded conducting plane. Question: What is the potential in the region above the plane?" Griffiths continued on solving the example using the method of images setting V=0 on the plane as one of the boundary conditions saying "since the conducting plane is grounded". Now, of course there will be an induced surface charge density. My question is, how can this be since the plane is grounded? Does the word grounded have different meanings? sometimes it means not charged and the others it means the potential there is 0?


Does reactive force require the two force involved have to have two different medium for reactive force to occur?

Does reactive force require the two force involved have to have two medium for reactive force to occur? I know the fuel-thruster is working on vacuum space, but we human could not use arm to swim in space? For example, two baseball hitting together have two medium


Scattering problem: Converting the two-body lab frame problem into a one-body center-of-mass frame problem

I'm reading the section on scattering in Goldstein's Classical Mechanics, and I have a rather basic question about this. It says that scattering in the laboratory is a two-body problem because of the recoil of the scatterer. Therefore, we convert the obtained data into center of mass coordinates in order to find the true scattering angle, the angle that we would get if the scatterer didn't recoil. However, I'm having a hard time understanding what exactly this center of mass system is. The figure (3.25) below shows the path of two particles moving towards each other, but it looks like it is simply the laboratory problem shown in the frame of reference where the center of mass is not moving. My question: It still looks like a two-body problem to me, especially since the scatterer is deviating from its path. How does this simply the situation? Figure 3.25 still doesn't show a situation where the scatterer's position is fixed. Neither does figure 3.24. ![enter image description here](http://i.stack.imgur.com/9OSm9.jpg) ![enter image description here](http://i.stack.imgur.com/Fl79e.jpg)


Gauge symmetry is not a symmetry?

I have read before in one of Seiberg's articles something like, that gauge symmetry is not a symmetry but a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations. Regarding this I have a few questions:   1. Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc?   2. Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)?   3. Are there analogs or other examples to this idea, of introducing fake degrees of freedom to facilitate the calculations or to build interactions, in classical physics? Is it like introducing the fictitious force if one insists on using Newton's 2nd law in a noninertial frame of reference?


How does the quantum path integral relate to the quantization of energy?

So, the quantum path integral is a generalization of the classical principle of least action- but here we know that all paths contribute something finite to the probability density. What confuses me is that this doesn't seem to involve the quantization of energy (action) at all. How does this come into play? Is it a separate assumption, only coming into play when we worry about the non-commutativity of certain operators?


Polarized sunglasses: should the axes in both lenses be parallel?

See the pictures below. A pair of sunglasses I recently purchased has the polarization axis in one lens offset about 20 degrees (by eyeball estimation) from the other. I don't have much experience with other polarized sunglasses, but this seems very obviously wrong to me. And it's very noticeable while wearing the glasses. When facing into glare from certain directions, one eye filters our considerably more glare than the other, which is very distracting and uncomfortable. I can see arguments for making the polarization axis angle just about anything (though horizontal or vertical seem most logical), but what I cannot see is any argument for having it different in each lens on the same pair of glasses. Thoughts? ![enter image description here](http://i.stack.imgur.com/TIu4z.jpg) ![enter image description here](http://i.stack.imgur.com/TW0Fd.jpg)


Recommendation request for a book explaining string theory to a common idiot (me)

I find the idea of string theory fascinating, but I'm not a student of physics, mathematics or science in general. Are there any books which effectively break down the concepts in a way that laymen like me can understand without being overly simple but also overly technical?


phase difference between incident plane wave incident on a dipole and radiation fields from dipole

i have an incident plane wave and a dipole, consider that plane wave incident on dipole. at this moment what happen for dipole ? we know that after incident of plane wave on dipole, the radiation have occurred, and i want derive the **phase difference between radiation field and incident plane wave field** ? please introduce me a reference


What is the Laughlin argument?

The fundamental question is > Why is Hall conductance quantized? Let's start with the Hall bar, a 2D metal bar subject to a strong perpendicular magnetic field $B_0$. Let current $I$ flow in the x-direction, then the y-direction develops a voltage $V_H$. The Hall conductance is $\sigma_H = I/V_H$ ![Hall bar \(http://members.home.nl/skoric/quantum/setup.gif\)](http://i.stack.imgur.com/0bhnE.gif) > To make Laughlin's charge pump, how should we wrap the Hall bar? Identify > the left and right, or top and bottom sides? Based on my understanding, we should paste top and bottom side together. (correct? Figure 1. left of the Paper maybe a little confusing.) Laughlin assumes the Fermi level is in the middle of the gap, so that the ring is an insulator. But the changing flux will induce an current by taking "adiabatic derivative" of total energy w/r flux $$I = c\frac{\partial U}{\partial \Phi}$$ which flows in y-direction and where $c$ is speed of light. Following Laughlin's calculations, as one threads one flux quantum, there will be $p$ (number of filled Landau levels) electrons transported. Then $$U=peV$$ where $V$ is the potential difference of two edges. From the current formula, we find the quantized Hall conductance. The heart of the problem is > What is an adiabatic derivative? Why is ${\bf j} = \partial {\cal > H}/\partial {\bf A}$ valid?


Fermi's Golden Rule and Density of States

I know Fermi's Golden Rule in the form $$\Gamma_{fi} ~=~ \sum_{f}\frac{2\pi}{\hbar}\delta (E_f - E_i)|M_{fi}|^2$$ where $\Gamma_{fi}$ is the probability transition rate, $M_{fi}$ are the transition matrix elements. I'm struggling to do a derivation based on the density of states. I know that under certain circumstances it's a good approximation to replace $\sum_f$ with $\int_F \rho(E_f) \textrm{d}E_f$ to calculate the transition probability, for some energy range $F$. Doing this calculation I obtain $$\Gamma_{fi} ~=~ \int \rho(E_f) \frac{2\pi}{\hbar}\delta (E_f - E_i) |M_{fi}|^2\textrm{d}E_f.$$ Now assuming that the $M_{fi}$ are constant in the energy range under the integral we get $$\Gamma_{fi} ~=~ \rho(E_i) \frac{2\pi}{\hbar} |M_{fi}|^2.$$ Now this is **absolutely not** what is written anywhere else. Other sources pull the $\rho(E_f)$ out of the integral to obtain Fermi's Golden Rule of the form $$\Gamma_{fi} ~=~ \rho(E_f) \frac{2\pi}{\hbar} |M_{fi}|^2$$ for any $f$ with $E_f$ in $F$ which makes much more physical sense. But why is what I've done wrong? If anything it should be more precise, because I have actually done the integral! Where have I missed something?


Can sound be used to purify/filter water?

I remember reading about all chicken at a poultry farm being violently sick/dying for apparently no reason. It turned out the culprit was machinery at a nearby factory that emitted sound at a frequency equivalent to the resonant frequency for brain cavity of the chicken.   * Could the story be true?   * Could sound be used to purify water of bacteria/viruses (preferably filter, else cull)?


Is this a photograph of an electron-positron annihilation?

With degrees in Mechanical and Electrical engineering but no advanced education in physics, I submit a query based on ellipsometric macro photography of TEMS supplied by FDA/NIH. In one TEM a triangle of RNA/DNA indicates ionic activity.   PHOTO 1. ![100nm RNA/DNA Traingle](http://i.stack.imgur.com/mMdGD.jpg) This piqued my curiosity, so I photographed the 100nm size triangle with a macro lens in order to enlarge the 35-42,000x TEM further and obtained unexpected images upon which the question is based. I angled the macro lens camera to the light source hoping to enhance the images within the photographic gel. The black-gray shades in TEMS are not ink but silver atoms crystallized in the original film negative by electron beams then projected into a silver-halide photographic gel. Electrons convert silver halide over 10,000 times more efficiently than photons. The biological samples, dark stained with nano-gold, provide a plethora of particle targets and possibly emit surface plasmon resonances that generate latent photon induced images hidden due to the shortened photographic development time required to form primary images with electron precipitated silver atoms. Collisions between electrons and particles must have occurred as can be seen in the spiraling burst emanating from a point above the bottom angle of the triangle that impressed me as having ionic interactions. The triangle legs are about 2nm wide with 100nm between vertices. Microvilli sized ion channels appear to be in the picometer range and may be waves generated from electron collisions. PHOTO 2. ![Electron-particle collision erupts above lower angle](http://i.stack.imgur.com/djUBK.jpg) An invisible clear cloud seemed to lie between the left two vertices and I took hundreds of photos in an attempt to illuminate it. ![A spiraling cloud illuminated](http://i.stack.imgur.com/RFgEb.jpg) PHOTO 3 is the result of those efforts. I assume the cloud to be one of those hypothetical latent images of energy associated with electron-particle collision. Unfortunately the color forces in QED do not relate to colors. PHOTO 4 was taken in the same area as PHOTO 3 with a different lens-light configuration. ![Annihilation?](http://i.stack.imgur.com/2407g.jpg) Might the simultaneous Lycurgus Cup-like red/green~emission/absorption seen in PHOTO 4 correspond to an electron travelling back in time (positron) colliding with an electron beamed or displaced in the TEM vacuum and depict an annihilation?


Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy to the corresponding 1-dimensional case. This is because the Laplace operator in spherical coordinates can be written in the form $$L = \frac{1}{r}\,\frac{d^2}{dr^2}r.$$ Hence my making the substitution $\psi\to r\psi$, the 3D radial wave function $r\psi(r)$ satisfies exactly the same Schroedinger equation as the wave function $\psi(x)$ in the 1D case. So far so good. However, at this stage some authors point out that this substitution is merely feasible, as long as the condition $r\psi\to0$ in the limit $r \to0$ is met. If not, then the wave function diverges at the origin, and this is unacceptable on physical grounds. [One may counter this argument, by pointing out that in the calculation of expectation values the square of the absolute value of the wave function is always multiplied by the spherical shell $4\pi r^2\,dr$. The factor $r^2$ neutralizes the above mentioned divergence.] Indeed there appears to be a slight difference between the 3D and 1D case, when one observes the elementary case of a particle in a box. In both cases, outside of the box the wave function is exponentially decreasing. Inside, where the particle is "free", the solution is oscillatory. In the 1D case, the wave function can be written as $$\psi(x) = A\sin(kx) + B\cos(kx).$$ In the 3D case, the wave function is given by the zeroth order Bessel function $$\psi(r) = C\sin(kr)/r.$$ [Apparently the term $D\cos(kr)/r$ is omitted since it is considered unphysical, see previous paragraph.] Now if the wave function inside the box has only one free parameter $C$, all one can really do is determine its value by demanding continuity of the wave function on the boundary of the box. On the other hand, QM textbooks claim that one should always seek a solution in which not only the wave function itself, but also its first derivative is continuous on a boundary! All in all I find the omission of the $D\cos(kr)/r$ term on physical grounds confusing. First of all it creates a difference between the 3D and 1D case. This seems odd to me since the corresponding wave functions satisfy essentially the same Schroedinger equation. Secondly, by omitting this term one loses an adjustable parameter $D$, which plays an important role in meeting the full set of boundary conditions.


Is this picture correct from the point of view of physics?

![enter image description here](http://i.stack.imgur.com/Hgytn.jpg) This picture violates some laws of physics. Am I right?


how does different colour wavelength intefere?

Say a red colour with 400nm mixes with blue colour with 700nm, but out of phase by a little. What would be the resultant colour? I would know how to represent the resulting wave mathematically, but what colour is it? Is the colour only determined by the resulting wavelength?


Higgs boson in LHC

Recently,the higgs bosons are discovered in LHC. My question is How did they come to know that the particle that are created are actually Higgs boson? On the basis of which properties,they confirmed them as Higgs boson?


Could we enable ourselves to send messages to and receive messages from the future?

Based on John Isaacks' question, "If you view the Earth from far enough away can you observe its past?" and the responses, it appears that we could use mirrors to see into the past. Using Vintage's example, a single mirror on the moon would only allow an observer to see 0.1μs into the past. Would it therefore be theoretically possible to align a massive series of mirrors that sent a single beam of light back and forth over a larger span of time (e.g., months, years, decades)? If so, would it be possible to send messages to ourselves in the past via such an arrangement? Or would we always be looking into the past and only able to send messages to the future?


Physics of a Railgun

I have a couple of questions about how railguns actually work, and the mathematics behind them. I understand that the projectile is driven forward by the Lorentz force caused by high currents traveling through the projectile. Because of these high currents, when the projectile initially comes in contact with the rails it can be welded into place instead of pushed forward, and thus needs an initial velocity. So...   1. How do you calculate what the initial velocity needs to be in order to counteract the high current that wants to hold the projectile in place?   2. Would there be any difference in the Lorentz force in a pure-translational velocity versus a rotational and translational velocity?   3. How high do the currents need to be compared to the mass of the projectile?    4. Where would I find these kinds of formulas for these calculations?  Thanks!


Why does time dilation cause you to age slower? And is time considered relative to the observer?

I understand that the higher your velocity the slower light will move. But how does time itself slow down while you are moving faster?


Why is mass renormalization insufficient to explain electron mass?

In the Standard Model, I understand that the mass of the electron is assume to arise from two effects:   1. A bare mass given by Yukawa interaction with the Higgs field, and   2. A mass correction from mass renormalization effects In this framework, why do we need to assume 1? Could mass renormalization explain the mass of fermions in general?


Does specific heat change with pressure? If so, why?

As pressure is increased, do we require more energy to increase the temperature from given temperature for the same mass. For example, if we heat water at 1 atm to raise its temperature by 20 degree Celsius, will the heat required be the same when pressure is 3 atm?


How does smoke move in the air and how can I direct it so it will go to a place I want it to go?

Let's assume a close room with 1-2 people who only one of them smoking cigarette. What is the equation describe the smoking spreading? is it diffusion? what are the parameters is so? Is there a way/ device I can build or put that make the smoke go in some specific way, like a vacuum that will suck all the smoke even if it is far away?


Can we determine the force an object exerts by its mass and acceleration?

I understand that the objects acceleration is determined by the force exerted _on it_ , and that the force exerted _on it_ is determined by its acceleration. But, does an object's (named A) acceleration (and mass) tell us anything about how much force the object will exert on another object (named B)?


Will the Universe eventually stop expanding

Sorry if this is a naive question, not being even a part qualified physicist in any way shape or form. I've read that the universe is expanding and the rate of expansion is increasing. The assumption being that it will continue expanding indefinitely. However isn't there another possibility. Let me illustrate my question prior to posing it. If I throw a ball in the air.. it accelerates away from my hand and keeps accelerating, until momentum is lost and it starts to slow down, eventually falling back to my hand. Is it possible that the expanding universe is also in the throws (excuse the pun) of an initial acceleration phase, prior to that acceleration slowing and eventually the universe compressing? Again sorry if its a stupid observation!


Applications of quarks or string theory?

We wouldn't have computers if we didn't know about quantum physics. I understand understanding of general relativity is needed to make GPS work well. Has knowledge of quarks or string theory resulted in useful things? **EDIT:** I suspect quarks and string theory could be used by people who study traces of the Big Bang and similar topics. That aside have they resulted in technological advances that could be useful to people (besides improving our understanding of nature)? Perhaps advances in medical imaging have been possible. There might also have been improvements in nuclear fission, nuclear fusion. Other than that I will be surprised to hear of applications. I suspect many applications are possible, but I expect it will take years to get there.


Clarify formula in quantum perturbation theory

I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\psi_n$, that is $\psi_n^{(1)}$, I shall use the formula: \begin{equation} \psi_n^{(1)}=\sum_{m,m\ne n} \frac{V_{mn}}{E_n^{(0)} - E_m^{(0)}}\psi_m^{(0)} \end{equation} where $E_n^{(0)}$ is the zeroth-order correction (i.e., the unperturbed), and $V_{mn}$ are given by: \begin{equation} V_{mn} = \left(\psi_m^{(0)}, V\psi_n^{(0)}\right) \end{equation} and V is the "weak" potential that represents the physical disturbance to our system. $(x,y)$ represents inner product. **My question is** : What values does $m$ take in the above summation ? Say I'd like to calculate the first-order corrected $\psi_1$, that is to calculate $\psi_1^{(1)}$. What values would $m$ assume ? Infinite values except for $1$ ? Values $0,\ldots,n-1$ ?


So gravity turns things round

It makes sense, since gravity tends to push the surface of a body towards it's center. Unless I'm mistaken, everything with mass has it's own gravity, every atom and for instance, our own bodies should also have their own gravity. The question is: how strong is our own gravitational pull? I know it must be extremely weak, but is there actually anything at all that gets attracted to us, like maybe, bacteria or molecules? And finally (this will sound ridiculous, but I'd really want to get an answer or at least a way of calculating it myself): What size would a human body have to reach in order for it to collapse into a sphere?


Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book _Quantum Field Theory in a Nutshell_ , a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of a non-linear $\sigma$ model (up to some order) $L=\frac{1}{2}(\partial\vec{\pi})^2+\frac{1}{2f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2+...$, where $U(x)=e^{\frac{i}{f}\vec{\pi}\cdot\vec{\tau}}$ is a matrix-valued field belonging to $SU(2)$, $\vec{\pi}$ is a three components vector, $\vec{\tau}$ are Pauli matrices. Maybe it's not hard but I meet some problems to derive it. I suppose the first step is the Taylor expansion of $U$, $U=1+\frac{i}{f}\vec{\pi}\cdot\vec{\tau}-\frac{1}{2f^2}(\vec{\pi}\cdot\vec{\tau})^2+...$, and then $\partial^{\mu}U=\frac{i}{f}\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})-\frac{1}{f^2}(\vec{\pi}\cdot\vec{\tau})\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})$, then $(\partial_{\mu}U^{\dagger})(\partial^{\mu}U)=\frac{1}{f^2}[\partial(\vec{\pi}\cdot\vec{\tau})]^2+\frac{1}{f^4}.[(\vec{\pi}\cdot\vec{\tau})\partial(\vec{\pi}\cdot\vec{\tau})]^2$. Now there are my questions, (1) Can I write $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$? Then by $\vec{\tau}^2=1$, I get $L=\frac{1}{4}(\partial\vec{\pi})^2+\frac{1}{4f^2}(\vec{\pi}\cdot\partial\vec{\pi})^2$, which is almost correct but differ to the wished answer by a pre-factor $\frac{1}{2}$. (2) Suppose $\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ is correct, however, if I do $\partial^{\mu}U=\frac{i}{f}U\partial^{\mu}(\vec{\pi}\cdot\vec{\tau})=\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}$ first, it seems $\partial^{\mu}U^{\dagger}\cdot\partial^{\mu}U=|\frac{i}{f}U\partial^{\mu}\vec{\pi}\cdot\vec{\tau}|^2=\frac{1}{f^2}(\partial\vec{\pi})^2$, say, only the first term of the wished answer. I probably made something wrong somewhere, can anyone hit me?


What does the term liquid mean in condensed matter physics?

In condensed matter physics, people always say quantum liquid or spin liquid. What does liquid mean?


$E=mc^2$ why is it $c^2$ and not just $c$?

Why is constant for the conversion of mass to energy square of the ligths speed? is it bedside it's the fastest real matter?


what this Lagrangian stands for?

i saw this Lagrangian in notes i have printed: $$ L(x,dx/dt) = (m^2(dx/dt)^4)/12 + m(dx/dt)^2*V(x) -V^2(x) $$ what is it? is it physical? it seems like it doesn't have the right units of energy, thanks


Application of plasma actuators for flow control of moving objects

I've recently been to a converence on plasma physics were, to my surprise, a lot of presentations were concerned with plasma actuator. Could someone, preferably in the field, tell me how long people have been reseaching this now and, more importantly, give an honest opinion on how far (in years) this technique is from regular industrial use?


How is it possible for astronomers to see something 13B light years away?

In a NPR News story from a few years back: > "A gamma-ray burst from about 13 billion light years away has become the > most distant object in the known universe." I'm a layman when it comes to physics, so cut me some slack if this is an ignorant question, but assuming the universe is around 14B years old, and has been expanding since the Big Bang, how is it that we can see events so far back in time? I understand how it would work if you had a static universe and the GRB happened 13B years ago at 13B Light years away and the light just arrived. However, at the time of the burst wouldn't we (or at least the matter that we are made from) have been much closer to the source of that burst, and wouldn't the light have blown by us eons ago? How is it we are seeing it now? If we were expanding away from it at close to light speeds it would seem to make sense for why it took so long for it to get here, except for that whole notion that light moves at the same speed relative the to the observer, which I think would blow that idea out of the water. Perhaps gamma rays travel at sub-light speeds? But, I'd still think the math would require that they travel MUCH slower than light for this scenario to play out. Another, possibility is that the light has wrapped around a finite universe a few times before reaching us. Of course if that were the leading theory, there wouldn't be any remaining controversy about the finite vs. infinite universe models. What am I missing here?


"Hard wall"/ "soft wall"

I have encountered those terms in various places. As I understand it, "soft wall" can correspond to a smooth cutoff of some spacetime, while "hard wall" can be a sharp one, which can be described in terms of D-branes. Could somebody please explain the terminology, and in which context it can occur?


On Bolte's semiclassical law

i have seen on internet the following, for $ E >> 1 $ the Eigenvalue Staircase can be approximated by $ N(E)= \frac{1}{\pi}argZ(1/2+i \sqrt E ) $ http://books.google.es/books?id=hLEJkA34GMQC&pg=PA35&lpg=PA35&dq=quantum+chaos+bolte%27s+formula&source=bl&ots=jqMMu5KK-L&sig=L9ZlQ_Wq7XQLDvEHyf1nvIc1TdE&hl=es&sa=X&ei=oqHMT4zdEoO_8APlk4UM&ved=0CGUQ6AEwBzgK#v=onepage&q=quantum%20chaos%20bolte%27s%20formula&f=false PAGE 34 section 1.28 (of course you may use $ \lambda =1$ ) of course the situation is very familiar if we put $ Z(s)= \zeta (s) $ Riemann zeta, but how could i prove Bolte's Law ?? also from the reconstruction of the potential could we deduce that $ V^{-1} (x)=A \frac{d^{1/2}}{dx^{1/2}}argZ(1/2+i \sqrt E) $ where could i find some more info and a proof of Bolte's Law ?? thanks.


Is the electric field at the edge of a uniformly charged disk infinite?

Consider a disk with a radius $R$ (I'll use $R=1$ at various points here) that has a constant surface charge density $\sigma$. Unlike the similar problem of the field in the vicinity of a infinitely thin ring, the field directly above the disk is very well behaved. Like the case of an infinite sheet of constant surface charge, the field limits to a constant as you get closer to the surface, and a point on the surface has a defined electrical potential. This recent paper gives equations for the specific case of points on the plane of the disk. Since we have rotational symmetry, there is only one variable, $\rho$, the distance from the axis of rotational symmetry. Graph from the paper: ![Potential](http://i.stack.imgur.com/PvFoJ.jpg) As you can imagine, the point of the highest slope is the edge of the disk. Is the slope (field) at this point infinite? From what I can make out of their equations, signs seem to point to "yes". The Elliptic functions are tricky, however, and I don't trust my conclusions. Up to the point $\rho=R$, the form of the above function is: $$ V(\rho) = V(0) \frac{2}{\pi} E(\rho)$$ Here, $E(\rho)$ is an elliptic function, and is different from the paper, because it uses the "Matlab" convention while I use the "Wikipedia" convention. The slope of E() doesn't _look_ convincingly upright, but the derivative seems to indicate it should be. I'm also troubled by the fact that I can't seem to numerically observe a divergence at $\rho=0.9999999999$. The physics at play are even more perplexing to me. I expected that a 2D surface, any 2D surface, would have a finite field on its face. Not so for a line, I understand that. But the ring that defines the edge of the disk only has a differential amount of charge on it. Consider another thought experiment: start at $\rho=1$ and $z=0$, I'll denote it $(1,0)$. Now move upward, but keep $\rho$ the same. If you're at $(1,0.001)$ you should have a finite field. But what happens as $z\rightarrow 0$? Does it just jump from a defined finite value to infinity? What on Earth is going on with the electric field in the vicinity of this edge, or ANY edge of a 2D surface for that matter?


dose physics affectthe biological like this

According to neural sciences the brain process information and reacts in 1 teen of a seconds the numbers add up to be 10.596674 loest is 30 to 20 years the course of time in reality recorded by our time keeping devices. but whant to know if perhaps i estimated wrong or something or if it really is what it sound like and what i mean by this that like stars from far out galaxies we to are seeing what happen years. Back in the past


Deflection of light by the Sun

Can you give details of a recent experiment of deflection of light by the Sun? What is the distance from the surface of the Sun and what is the exact value of the angle of deflection?


Does the sun "drag the solar system through space"?

In this video: https://www.youtube.com/watch?v=0jHsq36_NTU#t=55 at 0:55, it is claimed that "the sun is [...] dragging the planets in its wake". Is this true? My understanding is that the sun and the planets are all moving together through the solar system at the same velocity, and there is no friction involved, so there's no dragging or indeed motor force required. What's really going on?


Different units in the equations of electrostatics in macroscopic media

I have the usual equations of electrostatics in macroscopic media expressed in Gaussian units: $$\nabla\cdot\vec{D}=4\pi\rho$$ $$\nabla\times\vec{E}=0$$ $$\vec{D}=\vec{E}+4\pi\vec{P} \tag1$$ My teacher wants me to convert these Equations to the SI system using the following transformation rules: > $$\vec{E}\rightarrow\frac{\vec{E}}{k_1}$$ > $$\vec{D}\rightarrow\frac{\vec{D}}{k_1k_4}$$ > > Where $k_1=\frac{1}{4\pi\epsilon_0}$ and $k_4=\epsilon_0$, which gives: $$\vec{E}\rightarrow 4\pi\epsilon_0\vec{E} \tag2$$ $$\vec{D}\rightarrow 4\pi\vec{D}. \tag3$$ Applying this to my electrostatics equations I get: $$\nabla\cdot\vec{D}=\rho$$ $$\nabla\times\vec{E}=0$$ $$\vec{D}=\epsilon_0\vec{E}+\vec{P},$$ which is consistent with what I find in books. If now I suppose that $\vec{P}=\chi\vec{E}$, doing the appropriate manipulations to the equation (1) I obtain: $$\vec{D}=\epsilon\vec{E} \quad , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+4\pi\chi$$ If now I apply the transformation rules (2) and (3) to this in order to express it in SI units I get: $$\vec{D}=\epsilon\epsilon_0\vec{E} \,\,\ , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+4\pi\chi$$ But this is not consistent with what I find in books, which is: $$\vec{D}=\epsilon\epsilon_0\vec{E} \,\,\ , \,\,\, \mbox{Where I define:} \,\,\, \epsilon=1+\chi$$ The only way to obtain the solution that appears in the books is to assume that $\vec{P}=\epsilon_0\chi\vec{E}$ (This is the accepted equation in SI units) and do the same I did before (Replace this in (1) and use the transformation rules).   1. My question is: is there any way of obtaining the equations in SI units without using the relation $\vec{P}=\epsilon_0\chi\vec{E}$? (using only the transformation rules)?   2. And if this is not possible, how do I convert between $\vec{P}=\epsilon_0\chi\vec{E}$ (SI units) and this $\vec{P}=\chi\vec{E}$ (Gaussian units)?


Why can't we know the speed, $\vec{v}(t)$, and position, $\vec{r}(t)$, of an electron (the two) at the same time $t$?

I've read something about this and I conclude that it happens because of the uncertainty principle. But I don't understand very well the meaning of that. I mean, it's very abstract that the speed, $\vec{v}(t)$, and position, $\vec{r}(t)$, of a particle can't be known at the same time. I don't understand that statement.


How much is 1 electron-volt (eV)?

I am interested in knowing how much is one eV of energy. Everywhere I found are the technical definitions. Can anybody please tell me how much is this much energy. I need something which I can feel. I mean how much work I can do with 1 eV? Can I drive a 1000cc car for 1hour? Any of example in context of real life usage would be interesting.


Wi-Fi in the presence of very strong magnets?

A friend of mine claims to have been able to surf the Internet without fuss on a Wi-Fi connection while performing NMR on samples he was analyzing. I would have thought the strong magnets needed for this would have washed out Wi-Fi signals due to the radio waves emitted. How plausible is his claim? Or, to be more general, what kind of (electro?)magnet would it take to interfere with mobile devices?


How can we detect a black hole?

If black holes are phenomena of very high density (gravitational singularities) which don't emit radiation **how can we detect them** so far away from us where so much **other radiation can hide the black hole**? Aren't there many objects and things in front of black holes that would obstruct our view?


$\pi$ and the Curvature of Space

If one draws a circle on a sphere and measures the ratio of the diameter to the circumference, that value varies depending on the diameter of the circle compared to the diameter of the sphere it is drawn on (for a circle much smaller than the sphere it's drawn on, the ratio will converge to $\pi$, whereas for a circle the same size as the sphere, the ratio will be 2). Does the same or similar hold for curved 4-dimensional space? As I approach a massive body, will the ratio of the circumference of a circle to its diameter change? I found a number of sometimes conflicting statements online, including http://www.physicsforums.com/archive/index.php/t-9869.html http://mathforum.org/library/drmath/view/55198.html http://www.last-word.com/content_handling/show_tree/tree_id/2339.html


Difference between single mode and multi mode optical fibres?

What is the difference between single mode and multi mode optical fibres? First off, I guess that by _modes_ we mean the spatial modes of the electric (or magnetic?) field right? Now: what makes a fibre able to support more than a single mode? I mean, what aspect of its structure corresponds to which mode(s) can be transmitted?


Boiling when I plunge my french press?

Occasionally when I make coffee in my french press I experience something odd. It happens pretty infrequently but certainly enough to be curious about. I have the grounds ready in the carafe. The water just heated in an electric kettle. I pour the water over the grounds and place the plunger on top. Then, maybe 1/4 inch into plunging, the water "explodes" out the top like a busted sprinkler head. I'm curious what's going on here. Is the slight pressure increase (it can't be much) - or simply the plunging action - adding enough energy to the system to induce a flash boil? I've ruled out that the additional volume that the plunger itself adds is not enough to make the coffee overflow (this is a normal setup in every other respect where plunging would simply deliver a refreshing hot beverage).


Energy levels in disordered organic semiconductors?

Now in disordered organics, the band picture is thrown out the window, from what I can tell (due to lack of symmetry). But don't HOMO/LUMO levels basically take the place of conduction/valence bands in molecules? In a organic system (a lot of molecules with no order), then, I am correct to believe that the HOMO and LUMO levels broaden into a Guassian density of states? However, the LUMO 'band' does not act like a conduction band in that the states are still localized. From my reading, it appears that the Fermi Level is in the LUMO 'band'. My question is, how can the LUMO 'band' have filled states since a single molecule of course has no electrons in the LUMO. What am I missing?


How to do calculation in relativity of simultaneity

I have great trouble in understanding simultaneity in special relativity. Let me illustrate it with a concrete example. Assuming there is a train, its two end points are $A$ and $B$, the length of the train is $x$. The train moves at speed $v$. Assuming the train is moving in the direction from $A$ to $B$. For a ground observer observing the train movement, he notices that two lightnings strike simultaneously at $A$ and $B$ when the middle part of the train $O'$ passes through right in front of him. In other words, the ground observer is located at the middle part of the train($O'$) when the lightnings strike simultaneously at $A$ and $B$. Now there is another moving observer sitting inside the train and he sits right in the middle of the train ($O$, equidistant from $A$ and $B$). Does this moving observer think that the lighting happens at the same time? If no, how much time has passed before he notices lightning at $B$, after he had observed lightning at $A$?


Light absorbs and Color

I'm curious about how the material absorb the light and reflect the light back as colors in a sense of Quantum Mechanics (Quantum Electro Dynamics)   Does Hadron related to the absorbs of photon ? or are there any other factors ?


Why is oxygen the third most abundant element?

I was reading the article _Oxygen finally spotted in space_ today in which it stated > Oxygen is the third most abundant element in the cosmos, after hydrogen and > helium. Why would oxygen take the third spot when it is so heavy (relative to the five elements ignored for it)? It would seem logical to me that the third most abundant element would be lithium or beryllium as hydrogen and helium smash into each other.


Can an inhomogeneous magnetic field be built to slow down and catch a neutron?

Neutrons have a measureable magnetic dipole momentum from their intrinsic spin. Is it possible to slow down and catch the neutron by imposing a force by an inhomogeneous magnetic field. I think the force the neutron experiences in a magnetic field is \begin{align} \bar{F} = \bar{\nabla}(\bar{m}\cdot\bar{B}). \end{align}


What is entropy of the universe?

What is entropy of the universe? What is the change in entropy of our universe? Is this change increases or decreases? If the entropy of universe keeps on increasing then what are the consequences? Is there any limiting factor to this increase? One practical application of entropy where it is applied in day to day life?


Faster than light galaxies/clusters?

A few years ago in an astronomy course, we calculated some (transverse?) velocity of a moving object and got super luminal results. The answer was apparent and not physical velocity of the object. Hence no problem. But at the moment, I don't recall the solution to this apparent issue. Anyone?


Is this configuration correct for this spherical capacitor?

This capacitance contain 4 dielectric as shown in the figure dielectric 1 in half sphere and 2,3 in for 1/4 of the sphere and the fourth one in the last 1/4 of the sphere as shown and I want to find total capacitance. i think that 2nd dielectric parallel with potion of 1st and portion of 4th then this group series with (3rd dielectric and other portion of 1st and 4th one which are also parallel) is the correct answer as the voltage between a,b is the same at any dielectric as the electric field is the same from boundary condition ET1'=ET2=ET4' AND ET1''=ET3=ET4'' ![enter image description here](http://i.stack.imgur.com/4tJ6n.png) is it correct


The Helium mass fraction from the Big Bang Nucleosynthesis

In Perkin's book Particle Astrophysics (page 144): I do not understand how one comes to the following expression (the second equality with $r$) for the Helium mass fraction due to the Big Bang Nucleosynthesis: $$Y= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}= \frac{2r}{1+r} $$ where $r=N_\text{n}/N_\text{p}$. The first equality follows from the fact that He is (approximately) 4 times heavier than H: $$Y = \frac{m_\text{He}}{m_\text{He}+m_\text{H}}= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}.$$ However I can't derive the second equality relating $Y$ to $r$: $N_\text{He}= 2N_\text{p} + 2N_\text{n}$ and $N_\text{H}= N_\text{p} + N_\text{n}$ $$Y= \frac{4N_\text{He}}{4N_\text{He}+N_\text{H}}= \frac{8(N_\text{n}+N_\text{p})}{9(N_\text{n}+N_\text{p})}\quad???$$


Fourier transform between $x$ and $p$

On this page right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space: $$ \begin{split} f(x) &= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} A(k) e^{ikx} dk\\\ A(k) &= \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^{\infty} f(x) e^{ikx} dx \end{split} $$ while the second set is connection between $x$ (position) and $p$ (momentum): $$ \begin{split} \psi(x) &= \frac{1}{\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty} \phi(p) e^{i\frac{p}{\hbar}x} dp\\\ \phi(p) &= \frac{1}{\sqrt{2\pi \hbar}} \int\limits_{-\infty}^{\infty} \psi(x) e^{-i\frac{p}{\hbar}x} dx\\\ \end{split} $$ * * * **Q1:** How do i derive the second set out of first one? I know De Broglie relation $p = k \hbar$. Hence from $\exp[\pm ikx]$ in the first set we get $\exp \left[\pm i \frac{p}{\hbar} x\right]$ in the second set of equations. This is clear to me. What i dont know is how do we get from $1/\sqrt{2\pi}$ in the first set to $1/\sqrt{2 \pi \hbar}$ in the second set. Where does a $\hbar$ come from?


Bernoulli's theorem: $\frac{p}{\rho}+\frac{1}{2}u^2+\phi$ is constant along a streamline

I am trying to understand **the Bernoulli's theorem:**   * **$\frac{p}{\rho}+\frac{1}{2}u^2+\phi$ is a constant along a streamline** I got that: $\frac{\partial u}{\partial t}$ + ($\nabla \times u)\times u$ = $-\nabla(\frac{p}{\rho}+\frac{1}{2}u^2+\phi)$ For a steady flow: $\frac{\partial}{\partial t}$ = 0 and then: ($\nabla \times u)\times u$ = -$\nabla H$ with the scalar: $H$ = $\frac{p}{\rho}+\frac{1}{2}u^2+\phi$ **now, I didn't understand the next steps:** Taking the “dot product” of ($\nabla \times u)\times u$ = -$\nabla H$ the left hand side vanishes, as ($\nabla \times u)\times u$ is perpendicular to $u$ and we get: $(u \cdot \nabla)H = 0$ This implies that $H$ is constant along a streamline **Can someone explain me this thing in other words please?**


Why can we analyse force balance on a dislocation?

Dislocation (like screw or edge dislocation) is not a 'real' thing, while Newton's laws only apply to a real object (no matter macroscopic, like stars, or microscopic, like atoms). In the derivation of Peach-Koehler force (stress acting on a dislocation), I understand that the force is actually acting on those atoms around the dislocation, which is equivalent to acting on the dislocation in the mathematical sense. However, based on the above point (treating dislocation as a 'real' thing), some books directly use force balance and other mechanical analysis on dislocation. Can we treat dislocation as a 'real' thing in all mechanical cases just as the void in a solid when analysing electrical properties? Is there some way to think about it easily other than a complicated mathematical argument?


Why should space be empty?

> **Possible Duplicate:**   >  How vacuous is intergalactic space? The _emptiness_ of space is explained in many articles... But, space does contain some matter due to these possible reasons:   * During the explosion of supernovae, some elements are created. Not all supernovae nucleosynthesis are successful. So, there should be some scattering of matter into outer space.   * Cosmic rays are coming from interstellar space (and even from sun) in all directions. Would all of the particles have enough efficiency to reach their destination (I mean, take earth or any other interstellar object)?. Some protons or neutrons could be scattered into space.   * Even the gaseous molecules in atmosphere of celestial bodies have possibility to reach escape velocity and go into space. Are my assumptions correct? If so, then space must contain elements up to some extent. Isn't it? Or is it due to the infiniteness of space, that these scattered particles are ignored?


Thin lens formula workout method (Query)

![enter image description here](http://i.stack.imgur.com/0nuvB.jpg) In order to workout the method for establishing the formula of thin lens, my teacher says that the optical path is: $PA + AQ = PS_1 + nS_1S_2 + S_2Q$ ($n$ is the refractive index of the lens)   Why is she multiplying $S_1S_2$ by $n$? Also, she says that for spherical surface $S_1$, with centre of curvature $C_1$, we can write from geometry,   $h^2 = 2(S_1C_1 - OS_1) OS_1 $ How is she deriving this?


Where does the dust on the moon come from?

From the Apollo missions we know that the moon is covered with dust. Where does it come from? Is it from the erosion of the moon rock? By what? Or by accretion of dust from space? Which comes from where?


When combining three spin $\frac{1}{2}$ particles what are the corresponding states?

I want to combine three spin half particles and this is what I have so far. I used the lowering operator $J_{-}$ on the top states and found the following states fine: $$|\frac{3}{2},\frac{3}{2}\rangle , |\frac{3}{2},\frac{1}{2}\rangle , |\frac{3}{2},0\rangle , |\frac{3}{2},-\frac{1}{2}\rangle , |\frac{3}{2},-\frac{3}{2}\rangle $$ So that is the combination of three spin $\frac{1}{2}$ particles is equivalent to a spin $\frac{3}{2}$ particle, right? In the case where I did it for combining two spin $\frac{1}{2}$ particles I found this was equivalent to a spin 1 particle and an additional spin zero particle. So my question is, is there a singlet or anymore that accompany what I found for the case where I combined three spin $\frac{1}{2}$?


Streamlines tangent to velocity vector

As from the title, I'm not too sure how they are related. Definition is that streamlines are instantaneously tangential to the velocity vector of the field. Why would a streamline that shows direction be a tangent to the velocity?


LC Oscillator and relativity

There are two identical LC oscillators with electronic counters attached indicating how many times they have oscillated (from the time they are turned on). They are turned on simultaneously and one is kept on earth and other one is hurled to outer space at very large speeds on a rocket and brought back to earth after traveling several million miles. Does relativity say they will show two different counts ? (One traveled will show less than the one on earth)


Does the geomagnetic field rotate?

The Earth rotates about it's own axis. Do the geomagnetic field lines rotate due to this rotation or not?


How was Newton's third law discovered?

How did Newton's third law came into being? Was it his original finding like the second law? Or was it more of a restatement of someone else like the 1st one coming from Galileo? In either case what initiated the thought of what is now known as Newton's 3rd law?


Thermal equilibrium and kinetic energies

Is temperature solely a function of a kinetic energy? If a solid and a gas are at thermal equilibrium at a temperature of 20 degrees Celsius, the solid has much less kinetic energy than the gas. How can the temperatures of both be the same? What is keeping the solid at the same temperature as the gas?


Angular momentum equations

I do not understand this because angular momentum is $L=I\omega$ ($I$ is moment of inertia;$\omega$ is angular velocity) but it I have also seen equations where $L= rmv\sin(x)$. I do not understand how these are related, could someone please explain the connection?


Conveyor scales modeling

> **Possible Duplicate:**   >  Conveyor scales modeling Assume we have a conveyor scales. Which consists of scales, and motor with conveyor belt placed above, so that the boxes can be measured (weight) while moving above. What I want is to create the model of the oscillations of that scale. The equation of the oscillations of the spring is well known for everybody. How to include the influence of the motor, conveyor and other important effects to the system? Is there any papers (articles) about this thema? I think that the influence of the motor in simle case can be modelled as the force: $F(t) = A\sin(\Omega t)$ Where $A$ is a constant and $\Omega$ is the parameter which depends on the motor (frequency), which is also constant (depending on the speed of the motor). How to add the ocsillations created by conveyor belt and may be some other oscillations such as box moving above can create additional vibrations?


Finding two dimensional critical point

I'm reading an article about bi layered membranes which state that for the free energy function   $f(\theta) = \theta \ln \theta + (1-\theta)\ln(1-\theta) + \chi \theta (1-\theta)$ Where $\phi_i$ is the mole fraction of certain types of lipids in layer $i$ and $\chi$ is the energy of interaction between two lipids molecules composing the layer. The article states that a critical point exist at $\chi=2$ and $\theta=0.5$. Reading in wikipedia, I concluded that removing the layers interaction part, I simply need to take the second and third derivative of the free energy of the membrane. How do I do this with the interactions? Where can I read about this?


The gravitational potential of ellipsoid

In the literature (Kirchhoff G. - Mechanic (1897), Lecture 18 or Lamb, H. - Hydrodynamics (1879)) one can find the following analytical closed form expression for the gravitational potential of homogeneous ellipsoid of unit density, whose surface is given by \begin{equation} \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \;. \end{equation} Gravitational point for internal points is \begin{equation} \Omega=\pi abc\int_0^\infty\left(1-\frac{x^2}{a^2+\lambda}-\frac{y^2}{b^2+\lambda}-\frac{z^2}{c^2+\lambda}\right)\frac{d\lambda}{\Delta} \end{equation} and for external points \begin{equation} \Omega=\pi abc\int_u^\infty\left(1-\frac{x^2}{a^2+\lambda}-\frac{y^2}{b^2+\lambda}-\frac{z^2}{c^2+\lambda}\right)\frac{d\lambda}{\Delta} \;, \end{equation} where \begin{equation} \Delta=\sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)} \end{equation} and $u$ is the positive root of equation \begin{equation} \frac{x^2}{a^2+u}+\frac{y^2}{b^2+u}+\frac{z^2}{c^2+u}=1 \;. \end{equation} The expressions in these formulas appear similar to confocal ellipsoidal coordinates. How can these formulas be derived? (perhaps something more readable than the original papers) Can they be derived in terms of ellipsoidal harmonics?


A basic question: what is accelerating voltage?

Or would it be acceleration voltage? Acceleration sounds like it makes more sense, but my paper says accelerating. What are possible ways you could go about calculating it?


Composition of solar spectrum

I read some where that there are three types of UV and infrared rays namely UV-A, UV-B, UV-C and near infrared, mid infrared and far infrared. Which is the most abundant among the the three in Ultraviolet and infrared radiation from sun? I mean there is a total of 1000 W per unit area illumination by sun among which more than 500 watts is infrared, around 450 watts in visible and the rest part is ultraviolet so in the infrared rays which are the most abundant means which contribute to most of the 500 watts? Also, what are the sources of infrared at night and what is the power density of infrared rays at night!.


Three polarizers, 45° apart

If light is passed through two polarizing filters before arriving at a target, and both of the filters are oriented at 90° to each other, then no light will be received at the target. If a third filter is added between the first two, oriented at a 45° angle (as shown below), light will reach the target. Why is this the case? As I understand it, a polarized filter does nothing except filter out light--it does not alter the light passing through in any way. If two filters exist that will eliminate all of the light, why does the presence of a third, which should serve only to filter out additional light, actually act to allow light through? ![Image of three polarizers, target is at the right](http://i.stack.imgur.com/OogSm.png)


Strings and QFT: particles moving backward in time?

New question: In string theory and QFT, do particles travel back in time? Not related to antimatter: Do they travel back and forth in time in reality or are these just interpretations of mathematical formulas used to make sense of calculations? This is a different question (from this one) as it concerns strings and not antimatter...


Why is the turbulent energy cascade described as function of a wavenumber?

In all the literature I've seen the turbulent energy spectrum described as $E(k)$ instead of $E(L)$, i.e. as a function of a wave number not eddy size. The connection via $k=2\pi/\lambda$ is clear, but exactly what wave process is meant here. Is the idea that turbulent flow can be viewed as a superposition of waves? Waves of what? Or is this just a common notation used for energy spectra?


Better explanation of the common general relativity illustration (stretched sheet of fabric)

I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to _process and understand_ basic ideas behind most of these. However for general relativity there is this one illustration, which is being used over and over (image from Wikipedia): ![Spacetime curvature \(from Wikipedia\)](http://upload.wikimedia.org/wikipedia/commons/2/22/Spacetime_curvature.png) I always thought that general relativity gives another way how you can describe gravity. However for this illustration to work, there needs to be another force, pulling the object down (referring to a direction in the attached image). If I put two non-moving objects in the image, what force will pull them together? So where is my understanding incorrect? Or is general relativity not about explaining gravity and just describes how heavy objects bends spacetime (in that case the analogy is being used not correctly in my opinion)? * * * **UPDATE** Thank you for the answers and comments. Namely the XKCD comics is a spot on. I understand that the analogy with bent sheet of fabric pretty bad, but it seems that it can be fixed if you don't bent the fabric, but just distort the drawn grid. Would you be so kind and answer the second part of the question as well - whether general relativity is explaining gravitational force. To me it seems that it is not (bending of spacetime simply can not affect two non-moving objects). However most of the time it is being presented that it does.


potential energy of an object due to other two objects

consider object A with mass $m_{A}$ and positional vector $\overrightarrow{r_{A}}$ object B with mass $m_{B}$ and positional vector $\overrightarrow{r_{B}}$ object C with mass $m_{C}$ and positional vector $\overrightarrow{r_{C}}$ since reference frame is inertial (assumed) so $m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F}{}_{AB}+\overrightarrow{F}{}_{AC}$ $\Rightarrow$$m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})$ ....[1] for potential energy equation is to be integrated with a positional vector and i am trying hard to figure out what that vector could be, but still no success. please help * * * Potential energy of a object due to another object $m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{A}}-\overrightarrow{r_{B}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})$ assuming $\overrightarrow{r_{0}}=\overrightarrow{r_{B}}-\overrightarrow{r_{A}}$ $\Rightarrow m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}=-(1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})$ $\Rightarrow\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}\right).d\overrightarrow{r_{0}}=-\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left((1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})\right).d\overrightarrow{r_{0}}$ $\Rightarrow\left.\frac{1}{2}m_{A}v_{0}^{2}\right|_{v_{0}(\overrightarrow{r_{0i}})}^{v_{0}(\overrightarrow{r_{0f}})}=\left.G\frac{m_{A}(m_{A}+m_{B})}{r_{0}}\right|_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}$ So What positional vector should be integrated with equ [1] ? How it is decided ? (i mean what kind of property(s) that vector should possess ?) EDIT 1 $\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}\right).d\overrightarrow{r}=\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})\right).d\overrightarrow{r}$ .....[2] questions a) what is $\overrightarrow{r}$ in terms of $\overrightarrow{r_{A}}$,$\overrightarrow{r_{B}}$or $\overrightarrow{r_{C}}$ ? b) solve equ [2].


How does gravitation propagate along curved spacetime?

In this wikipedia article it is described how a beam of light, with its locally constant speed, can travel "faster than light". That is to say it travels a distance, which, from a special relativistic point of view, is surprisingly big. I wonder if a gravitational wave on such a curved spacetime (of which the wave is actually part of) behaves equally. > Does a gravitational wave also ride on expanding spacetime, just as light > does? Do the nonlinearities of gravitation-gravitation interaction influence > the propagation of a wave (like e.g. a plasma) such that light and gravity > are effectively not equally fast? > > If I want to send a fast signal in this expanding universe scenario, in what > fashion do I decide to I send it?


What is light localisation?

Reading about plasmonic nanoparticles I faced the term "localised light". How can one localise light? What are applications of it?


Do black holes have infinite areas and volumes?

How to calculate the area / volume of a black hole? Is there a corresponding mathematical function such as rotating $1/x$ around the $x$-axis or likewise to find the volume?


Does the kinetic theory of gases means gases mix almost instantaneously?

This theory has bugged me ever since my first physics class on the subject. If this (http://en.wikipedia.org/wiki/Kinetic_theory) is true, it leads me to a few weird conclusions. Opening the rear window in a pickup truck at 10 m/s doesn't empty the cabin of all its air. This means that gas molecules can catch up with us from behind so we can conclude that gases move much faster than 10 m/s at normal temperatures. Then, shouldn't gases mix almost instantaneously? For example, shouldn't a fart's smell reach us faster than a car on an highway and then dissapear almost immediately into such a large area that the concentration is too low for our noses to detect?


How to get distance when acceleration is not constant?

I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant acceleration is: $$d=ut +\frac{1}{2}at^2$$ which can also be expressed $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}$$ (where x(t) is the position of the object at time t) That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed? The acceleration is obviously not constant, but what about the change in acceleration? Is it constant? I suspect not. And then what about the change in the change of acceleration, etc. etc.? In other words, how does one know how many additional terms to add in the series? $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}+\frac{\mathrm{d^3}x}{\mathrm{d}t^3}\frac{t^3}{3}+\frac{\mathrm{d^4}x}{\mathrm{d}t^4}\frac{t^4}{4}\cdot etc. \cdot ?$$


Getting pairs of angle and velocity for a projectile to a given destination

I'm trying to calculate the initial velocity $v_0$ and angle $\theta$ for a given destination $(x, y)$ with a launch height of $y_0$. Obviously there will be a set of pairs of velocity and angle that will pass through the destination point. This set is given by $$ \left\\{(v_0,\theta) \middle|y = - \frac{g}{2 v_0^2 \cdot \cos^2(\theta)} \cdot x^2 + \tan(\theta) \cdot x +y_0 \right\\} \ .$$ I have already looked at this questions: Solving for initial velocity required to launch a projectile to a given destination at a different height How to get the angle needed for a projectile to pass through a given point for trajectory plotting So for example I could rewrite the set to $$\left\\{ (v_0,\theta) \middle| v_0 = \frac{1}{\cos(\theta)}\sqrt{\frac{\frac{1}{2} g x^2}{x \tan(\theta)+y_0}} \right\\}\ .$$ Is there an easy way to compute this set for one given destination without iterating over all possible velocities or angles?


TISE for a triangular potential

I want to solve the TISE of a particle of charge $q$ and mass $m$ in a one dimensional triangular potential, with an infinitely high potential wall at $x = 0$, i.e. $$\hat{H}=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x),$$ with $$V\left(x\right)=\begin{cases}qEx&{\rm if}\,x>0 \\\ \infty &{\rm if}\,x\leq0\end{cases}$$ For $x>0$, this gives the TISE $$-\frac{\hbar^2}{2m}\frac{\partial^2\Psi_n}{\partial x^2}+qEx\cdot\Psi_n = E_n\Psi_n.$$ I know that the solutions of $$\frac{d^2y}{dx^2} - xy = 0$$ are given by the Airy functions, however I don't know how I can transform the TISE to this form. I know it's possible, because I have the solution to the problem, but I don't know how I can myself come up with this, I don't see the aim which has to be reached in order for a substitution to be successful. Also, I don't know how the differential will transform under a substitution including $x$. Any help? The substitution used in the solution is $$u=\left(\frac{2mqE}{\hbar^2}\right)^\frac{1}{3}\left(x-\frac{E_n}{qE}\right).$$


What would ACTUALLY happen to a person jettisoned into space?

_[insert obligatory statement of my lack of knowledge in physics]_ Alright, so we have all seen the movies where someone gets blasted out of the airlock on their starship, or their suit decompresses while on a space walk. The poor schmoe usually either decompresses so violently that blood is oozing out of every orifice in their body, or they freeze instantly. From this I have two questions:   1. **Would the decompression really be that violent?**     1. Clearly the drastic difference in pressure from a normal "earth" like environment to space would be bad, but would it be _that_ devastating.     2. I vaguely remember that standard atmospheric pressure was something like 15 psi, which doesn't seem like enough to mess you up that bad.   2. **Would you _actually_ freeze instantly in space?**     1. Heat, or lack thereof is a measure of internal energy, but in a vacuum there wouldn't be anything to have internal energy, so does space even have a temperature?     2. Wouldn't some form of matter have to be present in order to cool off? If there were no matter besides yourself and a few stray particles here and there, it seems like it would take a very long time to cool off.


Traditional Kirchoff voltage law in AC circuit?

The traditional (not taking into account phasor addition or complex addition) application of Kirchoff Voltage law, i.e. $\Sigma\Delta V=0$ along a loop, does not work for AC circuits. We can sum the voltage drops to zero if we take into account their phase differences. But at some particular time, no matter what the phase differences in the general equation for the voltages across various components, there will be only one numeric value of $\Delta V$. > **Why doesn't this sum to zero over the loop in case of AC circuits with > non-zeroreactance?** Is this because there is some contribution to the $\oint E.dl$ due to changing magnetic flux? If it is, even in absence of any inductance, the voltage drops still do not sum to zero. why?


Exploiting the Heisenberg Uncertainty Principle as a means to communicate

It seems as though I've come across a rather unusual conclusion that could either simply be a misinterpretation or a contradictory discovery. I seem to have found a way to utilize the Heisenberg Uncertainty Principle (HUP) to our benefit to communicate faster than the speed of light (FTL). I am aware of many proposals that try to utilize entangled spin particles and try to communicate, but this schemes fail because the measurement outcome can NOT be controlled. Therefore, even if two people share an entangled pair of particles, and Alice measures spin up, although she knows Bob has a spin down particle, Bob didn't measure yet and has no way of knowing that Alice has measured her particle. Hence, FTL is impossible based on the reason that the measurement outcome can not be controlled. This leads to my proposal of using position and momentum entangled pairs. Consider **Alice and Bob hold an ensemble of entangled particles in position and momenta** , where each particle is trapped in a separate harmonic potential. At some specified time agreed upon in the future, Alice measures all of her particles' position to high precision. **Alice and Bob have synchronized clocks** and Bob measures all of his entangled particle's momentum to whatever accuracy he chooses. Alice can calculate the average value and also the standard deviation of position. The standard deviation of position would be extremely narrow, i.e, the spread of her measurements would be very small. From the entanglement relation, **x1 = x2 and p1 = -p2** , we know that when Bob measures his particle's momentum, the spread of momentum will be very large. This must be true because position and momentum cannot be measured to arbitrary accuracy at the same time. This is very similar to Einstein's proposal to violate the HUP, however I am exploiting HUP. What this all means is that the if Bob measures a very large momentum spread, it must mean that Alice has made her measurements. If Bob measures a relatively moderate momentum spread, then he knows Alice did not measure her particles. **Since the position measurement can be made to arbitrary accuracy, we are "controlling the spread or the standard deviation as the means to communicate."** (MAIN RATIONALE) Say Alice and Bob have multiple ensembles of entangled particles. Alice can relay a message by simultaneously measuring her first ensemble, meaning its a "1" and not touch her second ensemble meaning its a "0", and perhaps she chose to measure the third ensemble, "1", etc. Hence generating the series 101..., where **each ensemble of entangled particles represents one bit of information.** What is flawed in this proposal? Entanglement in position and momenta is well established. We can also choose to measure the position of a particle to arbitrary precision. The HUP must hold for Bob and everyone involved.


Would the inside of a black hole be like a giant mirror?

As any light reflected or emitted from objects inside a black hole (if it is possible to be there) does not leave the event horizon and comes back inside, would it be like seeing yourself? What I mean is that would the light we might emit/reflect return to our eyes and make us sort of look at ourselves?


force on a moving charge in magnetic field

Need help in understanding the direction of magnetic force in the magnetic field!Totally confused by directions. Why is it that magnetic force is perpendicular to the direction of magnetic field and velocity of charged particle. Why is it(force) not in the same direction as the magnetuc field


Electron distribution around atom when moving

I do not have much experience on this but if an atom has some electrons around nucleus and the atom itself it is moving at some speed does that affect the distribution of electrons around? I am presuming that the interaction between the nucleus and electrons has a constant speed $c$. Anything I found so far is a calculation for interactions that presume an infinite speed. As an argument I am thinking of relativistic Doppler effect that does not change proportional with $v/c$. So I am thinking that maybe the speed does affect the distribution and so that is why the difference in the emission energy.


Shor's Algorithm: Why throw away the f(x)?

I'm having a little trouble understanding Shor's algorithm - namely, why do we throw away the result f(x) that we get after applying the F gate? Isn't that the answer we need? My notation: $\newcommand{\ket}[1]{\left|#1\right>}$ $F(\ket x \otimes \ket0) = \ket x \otimes \ket{f(x)}$ , $f(x) = a^x \mod r$


S-operator lorentz invariance

How to show that $\hat {S}$-operator must be lorentz-invariant operator? $$ |\Psi (t)\rangle = \hat {S} | \Psi (0) \rangle , \quad \hat {S} = \hat {T}e^{-i\int \hat {H}_{I}d^{4}x}. $$ I have read that this result follows from the unitarity of the Poincare group operator $U_{0}(\Lambda , a)$ and the covariance of S-matrix $S_{out, in} = \langle out| \hat {S}|in \rangle$, but I don't understand how do we conclude that from this follows that $U_{0}(\Lambda , a)\hat {S}U^{-1}_{0}(\Lambda , a) = \hat {S}$. I'm not interesting in the derivation of lorentz-invariance of S-operator from causality principle at this moment.


How do the different types of energy apply to voltage in a circuit

I was just wondering what happens in a circuit in terms of different types of energy transformations. If you apply a voltage to a circuit then electrons start moving (very slowly). Since the electrons want to flow to the positive terminal they will have electrical potential. But the electrons are also now moving once the voltage is applied as there is a current so would they have kinetic energy as well? Some information I have found online says that electrons collide with atoms in a bulb and this is why the filament heats up. But this would imply that kinetic energy is being transformed into heat and this can't be correct because surely any change in kinetic energy would alter the current flowing? Also, in a series circuit, if you measure the potential difference between any 2 points after all of the loads then you always get zero. I was just wondering why it is zero because don't the electrons keep moving even after passing through all the load in order to get back to the power source so surely they cannot do this without some form of energy?


Active and passive transformations and the change in potential energy

Under active transformation, the particle moves. On the other hand, for a passive one, the coordinate is just relabel. I've read that the passive one will not affect the potential energy and the active one will change it. This is the case for two massively different size objects, like earth and a ball. But basically there will be no difference between passive and active transformation if the two objects making the system are comparable in size. My question is how to prove that and which mathematical framework one should use?


Superconducting nanowire in a parallel magnetic field

Consider a nanoscopic wire (with radius $R$) of superconducting material. The wire lies along the $z$-axis and a magnetic field $\mathbf{H}_a = H\mathbf{e}_z$ is applied. The magnetic field is too weak to destroy superconductivity or to induce vortices and we assume that the superconducting parameter $\psi$ can be put equal to the constant value $\psi_{\infty}$ in the material (and of course, zero outside). Now, I want to use the second Ginzburg-Landau equation to find the vector potential $\mathbf{A}$ (and the magnetic induction field $\mathbf{B} = \nabla\times\mathbf{A}$ from this potential). With the assumptions mentioned above the second Ginzburg-Landau equation simplifies to   $\nabla\times\left(\nabla\times\mathbf{A}\right) = \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2\mathbf{A}$   where   $\psi(\mathbf{r}) = \left\\{\begin{array}{lr} \psi_{\infty} & \mathrm{in\,nanowire} \\\ 0 & \mathrm{outside} \end{array} \right.$   This expression is simplified even further by using the Coulomb gauge $\nabla\cdot\mathbf{A} = 0$, leading to the following vector laplace equation in the case outside the wire:   $\nabla^2\mathbf{A} = 0$   Since the system has cylindrical symmetry, I'm working in cylindrical coordinates. The scalar equations associated with the general vector equation are then:   $\left\\{\begin{array}{rcl} \dfrac{\partial^2A_{\rho}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\rho}}{\partial\phi^2} + \dfrac{\partial^2A_{\rho}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\rho}}{\partial\rho} - \dfrac{2}{\rho^2}\dfrac{\partial A_{\phi}}{\partial\phi} - \dfrac{A_{\rho}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\rho} \\\ \dfrac{\partial^2A_{\phi}}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_{\phi}}{\partial\phi^2} + \dfrac{\partial^2A_{\phi}}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_{\phi}}{\partial\rho} + \dfrac{2}{\rho^2}\dfrac{\partial A_{\rho}}{\partial\phi} - \dfrac{A_{\phi}}{\rho^2} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_{\phi} \\\ \dfrac{\partial^2A_z}{\partial\rho^2} + \dfrac{1}{\rho^2}\dfrac{\partial^2A_z}{\partial\phi^2} + \dfrac{\partial^2A_z}{\partial z^2} + \dfrac{1}{\rho}\dfrac{\partial A_z}{\partial\rho} & = & \dfrac{\mu_0Q^2}{M}\left|\psi\right|^2 A_z \end{array} \right.$   while for the case outside the wire the right hand sides are zero. These equations are coupled and I'm not sure how to efficiently solve them. So my question is: could anyone point me in the right direction as to how I should try to solve this vector laplace equation in cylindrical coordinates? Or are there perhaps any possible further assumptions I failed to make? I'm not looking for a complete answer by the way, just a push down the right path, a method. Note: I've asked for a method to solve the vector laplace equation in cylindrical coordinates with less context on math.se yesterday (Solving the vector Laplace equation in cylindrical coordinates) but I thought it might be better to put this here, so I can describe the full physical problem. Perhaps the question on math.se can/should be deleted then?


Calculating the height objects fall from

* Imagine an object of a given mass.   * The object falls from a certain height.   * On contact with the ground, the object has a certain force. If the force of the object and the mass of the object are know, what formula would be used to find the height fallen from?


Relation of fraction of binary stars with spectral class (mass)

What is relation of fraction of binary stars with spectral class (mass)? For example, how many binary stars are among O,B,A,F,G,K,M stars separately?


What Cat States of light have been experimentally produced?

I'm specifically looking for Schrödinger's Cat states involving superpositions of two, or if it's been done more, coherent states, i.e. monomodal states of the form $$|\psi\rangle=a|\alpha\rangle+b|\beta\rangle.$$ What states of this form have been produced in experiment? How even can the weights be? What regions of the $\hat{a}$-eigenvalue $\alpha$ and $\beta$ are accessible? If more than two coherent states can be superposed, how many? What phase-space geometries are possible so far? I'm also interested in what techniques are currently used to generate these states.


Confused about indices of the Ricci tensor

In an intro to GR book the Ricci tensor is given as: $$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu \lambda}^{\lambda}+\Gamma_{\nu \sigma}^{\lambda}\Gamma_{\mu \lambda}^{\sigma}]$$ I have gotten to the point where I can work out a given Christoffel symbol, but I am still having trouble working out the above tensor as a whole (just algebraically speaking). If I'm not mistaken, $R_{\mu\nu}$ should end up a $\mu$x$\nu$ (i.e. 4x4) matrix just like the energy-momentum tensor on the other side of the field equations. In the above rendering $\sigma$ is clearly a dummy index to be summed over, and I can see how $\lambda$ is also a dummy index in the first term. But the $\lambda$s in the other terms seem to be free indices, which would then introduce incompatible dimensions in the matrix operations. I appreciate it if someone can point out the error of my ways.


If microwave ovens and WiFi both operate on the same frequency, why doesn't WiFi cook things?

If we ignore 5GHz WiFi, then both microwaves and WiFi create photons at ~2.4GHz but one of them will boil water in a few seconds but the other doesn't have any effect. So what's the difference? Is it simply the number of photons created? Is that what the wattage of a microwave measures? If so, what would be the wattage of a wireless router? Does the enclosed space have anything to do with it? If it all has to do with power output could I put enough WiFi routers together in a room to cook a turkey (from microwaves and not waste heat)?


Reynolds number, turbulence regime, and drag force

I am trying to model a system in which cubes of about 2 cm in size are floating in a circular water thank of about 30 cm in diameter. The cubes move around under the influence of the fluid flow induced by four inlets that point toward the center of the tank, and are located at the positions $0$, $\pi/2$, $\pi$, and $3\pi/2$. The flow velocity ranges from 0 to 10 cm/s, with an average velocity around 6 cm/s. My questions are the following:   * What would be the Reynolds number of the system? In particular, should I take as characteristic length the size of the cubes, or that of the tank?   * For such a system, what is the limit Reynolds number for the turbulent regime?   * What would be the correct form of the drag force, and do you intuitively think that the orientation of the blocks is negligible from a drag coefficient point of view? Thanks for your help!


Similarity in two events from Dynamics point of view

What is similar from dynamic point of view between book lying stationary on horizontal table and rain drop falling down with constant speed? * * * I can find two similarities   1. Force acting on them is equal   2. Both have horizontal velocities zero Is there any other similarities possible? It is not mentioned how many similarities we have to give that means we should give all possible similarities. Answer should contain related basic physic like acceleration, velocity, displacement, force,etc. We can say stuff which can be understand by high school student.


Why does optical pumping of Rubidium require presence of magnetic field?

The optical pumping experiment of Rubidium requires the presence of magnetic field, but I don't understand why. The basic principle of pumping is that the selection rule forbids transition from $m_F=2$ of the ground state of ${}^{87} \mathrm{Rb}$ to excited states, but not the other way around ($\vec{F}$ is the total angular momentum of electron and nucleus). After several round of absorption and spontaneous emission, all atoms will reach the state of $m_F=2$, hence the optical pumping effect. But what does the Zeeman splitting have anything to do with optical pumping? Granted, the ground state, even after fine structure and hyperfine structure considered, is degenerate without Zeeman splitting, but the states with different $m_F$ still exists. In addition, how is the strength of optical pumping related to the intensity of magnetic field applied?


Differentiating inside an integral sign

I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation. Here is the part of the derivation that I didn't follow: ![enter image description here](http://i.stack.imgur.com/7B0wm.jpg) I don't get how he goes from 6.9 to 6.10 by partial-differentiating the term inside the integral. If this is allowed, I was probably missed my calculus class the day it was covered. Can someone tell me more about this? Which part of calculus is this from?


Newton third law. Is there a limit on how much force you can apply to an object?

Recently in class we went over Newton's Third Law. In the book they put an example of hitting a punching-bag with your fist and hitting a piece of paper, or an object with much less mass. It's clear that you cannot exert more force on the paper than what the paper can exert on you, otherwise it might stop your fist in the process, just like the punching-bag stops your fist. Is there a limit, however, on much force you can exert on a paper? This might sound silly but it got me thinking for the past couple of days. In order to make it a little more clear consider a Car with a paper in front of it. The car-system can continuously increase the acceleration, but this means that the paper will increase the acceleration as well. Ergo, by Newton 2nd Law, the force exerted on the paper has to increase. Right? That same force back to the car. Therefore, the car can continuously increase the force it exerts on the paper.


Expression of electrostatic field

An electrostatic field is characterized by the fact that it depends only by $r$, isn't it? If it is true, I don't understand why this expression, given in cylindrical coordinates, $${\bf E(r)}=\frac{\alpha}{z^2}{\bf u_r}-2 \frac{\alpha r}{z^3}{\bf u_z} $$ rapresents an electrostatic field.


Can we develop a new kind of principles?

We have a basic equation $F=ma$. Now can we change this equation to another variable and start a new era in physics? If so, then how? If not, then why?


Police laser speed detection guns

### Situation Police shooting a laser device into his rearview mirror, bouncing off the oncoming car license plate. I want to know if this second surface mirror, and the symmetric system--laser gun to mirror to license plate of moving car to mirror to laser gun---has a built in error because of the wavelength shift due to the glass of the mirror. Velocity and wavelength change for the raypath inside the mirror's glass, but would revert to the original wavelength for the "in air" portion of the raypath. ### Question Is this symmetry perfect and creates no error in measuring the car's speed?


How the Bloch sphere of a Hahn echo in NMR looks like? 90-t-90-t-echo

I have tried to find in the literature a proper nice and beautiful Bloch sphere to describe the trajectory of a nuclear spin, starting in z-axis, using a pulse sequence of an initial 90º pulse with zero phase (which means that the pulse is launched in the x-axis), wait for a time $\tau$, and then launch another 90º pulse with zero phase, and wait the same amount of time tau to measure an echo. Please notice that I am talking about Nuclear Magnetic resonance, and most of the books don't offer a Bloch sphere to explain this echo, which is supposed to be the first! echo, and only use a 90º and 180º pulse to explain that. I have read the original article of Hahn echo, but I saw that his diagram is not clear.


Why does everything spin?

The origin of spin is some what a puzzle to me, everything spin from galaxies to planets to weather to electrons. Where has all the angular momentum come from? Why is it so natural? I was also thinking do photons spin? we always think of the wave as a standard 2d sin wave but could this rotate in 3d? What implications would this have? And what about spacetime how does all the spinning effect? This has always been avoided in all lectures and classes I ever went to.


Why does each celestial object spin on its own axis?

AFAIK all the celestial objects have a spin motion around its axis. What is the reason for this? If it must rotate by some theory, what decides it's direction and speed of rotation? Is there any object that does not rotate about its axis?


Why planets are rotating only in one plane?

Since gravity is three dimensional why planets are rotating only in one plane around sun.


Why does earth spins?

I understand the governing force causing the earth's cycle around the sun is gravity. It can be described by Keplar's law. But what causes the earth's spin? what is its governing law?


Why does Venus rotate the opposite direction as other planets?

Given: Law of Conservation of Angular Momentum.   * Reverse spinning with dense atmosphere (92 times > Earth & CO2 dominant sulphur based).   * Surface same degree of aging all over.   * Hypothetical large impact is not a sufficient answer. Assuming any object large enough to alter a planets rotation or even orbit would likely destroy most of its shape, yet Venus has retained a spherical property with a seemingly flat, even terrain indicating no volcanoes,and few if any visible meteor impacts. It would be fragmented and dispersed for billions of years. Even the question of what meteor, comet, asteroid composition could survive traveling that close to the sun's temperature, radiation, electromagnetic energy, solar flares, or gravity to equal a mass reactionary change as to alter it's spin.


Why do we have a galactic and a solar plane

Why do all of the planets in our solar system move is (more or less) the same plane? It would seem that the planets should all have very different orbital planes, each at a different angel to each other. The fact that all of their orbits are so closely parallel suggests that a force is pulling and/or maintaining the orbits of the planets in a flat plane. What is that force? A similar force also seems to be working at a galactic level too.


Why are the orbits of the planets in our solar system along the same basic plane?

> **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc/plane/layer? After watching this video I realized that the orbits of the planets in our solar system basically reside along a similar 'flat' plane: ![enter image description here](http://i.stack.imgur.com/q4O5U.gif) Meaning that the angles of orbit are no greater than 20 degrees (if you still count Pluto as a planetary object). Why are there none of the planets at a 45 or even 90 degree orbit (in relation to the earth)? Is this due specifically to the gravitational pull from the Sun?


Single plane Ring system

> **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc/plane/layer? I've noticed this in many pictures, Planets are shown with a single ring around them (in some particular plane). Taking extreme case... As gravity should act in all the directions, such planets must be covered with asteroids all around them. Not just _single_ ring in some _single_ plane..! So, My question is: Why don't planets have many rings instead of just a single ring..?


Why the galaxies forms 2D plane (or spiral-like) instead of 3D ball (or spherical-like)?

> **Question** : As we know (1) the macroscopic spatial dimension of our > universe is 3 dimension, and (2) Gravity attracts massive objects together > and the gravitational force is isotropic without directional preferences. > **Why do we have the spiral 2D plane-like Galaxy(galaxies), (or more common > than), instead of spherical, 3D ball or elliptic like galaxies?** **Input** : Gravity is (at least, seems to be) **isotropic** from its equation of force law (Newtonian gravity). It should show **no directional preferences** from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically. If the gravity attracts massive objects together **isotropically** , and the macroscopic space dimension is **3 dimension** , it seems to be natural to have a spherical or ball like shape of massive objects gather together. Such as the Globular clusters, or **GC** , are roughly spherical groupings Star cluster, and its Wiki picture: ![Star cluster](http://i.stack.imgur.com/KKEUZ.jpg) However, my impression is that, even if we have observed some more-spherical or more-ball-like Elliptical galaxy, it is more common to find more-planer Spiral galaxy? ( **Is this statement correct? Let me know if I am wrong.** ) Such as our Milky Way Also, such as this NGC 4414 galaxy: ![Galaxy](http://i.stack.imgur.com/jrklh.jpg) **Is there some physics or math theory explains why theGalaxy turns out to be planer-like (or spiral-like) instead ball-like or spherical-like?**


Why does the earth rotate?

> **Possible Duplicate:**   >  Why does every thing spin? So why would the earth, or any planet for that matter, rotate along an axis? I know of no force which could come into play here, so i assume it just started off with an initial rotation. But where did this rotational energy come from? And are there any known planets which do not rotate? Assuming no force accounts for earth's rotation, would it be true that the angular velocity of the earth would decrease(by a very very small amount), when bombarded by meteors(Due to an increase in the moment of inertia by the increase in mass of the earth i.e. the additional mass of the meteors)?


On the origin of the rotation of celestial bodies

> **Possible Duplicate:**   >  Why does Venus rotate the opposite direction as other planets?   >  Why does every thing spin? As far as I can imagine, almost each celestial body, star, planet, solar- systems, galaxies do rotate on their center. Where this come from ? Is it the normal work of gravity? And why Venus has a retrograde rotation? (Or, why all the solar systems rotate in the same way - except Venus?)