title: Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

text: I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my questions are based off (30:09). At that point, he has circled in black the step up and step down operators and how they act on the wavefunction $\psi$. He does not explain how he obtains the normalisation constants $\sqrt n$ and $\sqrt{n+1}$. Here's my attempt starting with the step down operator (C is my normalisation constant) : $$\int{(a^-\psi)^*a^-\psi}.dx=1$$ $$\int{ C^*\psi^*_{n-1}a^-\psi}.dx=1$$ $$\int{C^*\psi^*_{n-1}C\psi_{n-1}}.dx=1$$ $$\mid C\mid^2\int{\psi^*_{n-1}\psi_{n-1}}.dx=1$$ Where do I go from here ? EDIT : I would also like to know whether the equation for the energy of the harmonic oscillator is derived from pure observation of pattern or is there a general method to derive it ?


title: Entropy with infinite baths

text: I'm struggling with the following problem: (Stephen J. Blundell, Concepts in Thermal Physics S. 154 Problem 14.5): > A block of lead of heat capacity 1kJ/K is cooled from 200K to 100K in two > ways: > > a) It is plunged into a large liquid bath at 100K. > > b) It is first cooled to 150K in one liquid bath and then to 100K in > another. > > Calculate the entropy changes in the system comprising block plus baths in > cooling from 200K to 100K in these two cases. Prove that in the limit of an > infinite number of intermediate baths the total entropy change is zero. This is my solution so far: Bath: $dS = \frac{dQ}{T_{bath}} \Rightarrow \Delta S = \frac{\Delta Q}{T_{bath}} = \frac{C* \Delta T}{T_{bath}}$ Block: $dS = \frac{dQ}{T}; C = \frac{dQ}{dT}$ $\Delta S = \int_{T1}^{T2}\frac{C}{T}* dT = C * ln(\frac{T2}{T1})= C* ln(\frac{T2 * T_{m}}{T_{m}*T1}) = C*(ln(\frac{T2}{T_{m}})+ ln(\frac{T_{m}}{T1}))$ in both cases: $\Delta S_{Block} = 693,1 \frac{J}{K}$ a) $ \Delta S_{bath} = C * \frac{100K}{100K} = 1000 \frac{J}{K}$ b) $\Delta S_{bath} = C* (\frac{50K}{150K} + \frac{50K}{100K} ) = \frac{5}{6}*1000 \frac{J}{K}$ Is this so far correct? I'm struggling with the infinite case: $ \frac{\Delta S}{C} = \sum_{i = 1}^N \frac{100K}{n*(200K- i * \frac{100K}{n})}= \sum_{i= 1}^N \frac{1}{2n-i}$ I could not find any solution for $N \rightarrow \infty$. Where did I go wrong?


I'm taking solid state physics, subject mostly based on Ashcroft-Mermin's "solid state physics". Yesterday I sat for an exam and there was a question I couldn' t even answer: "Which interaction is the responsible for long-range magnetism?" Some said it was spin-spin interaction, but that doesn't make any sense for me, I don't even know what that means. Some others said it was wrong to say dipole-dipole... not a clue any idea? thanks in advance


title: dynamical operator and $SU(n+1)$

text: I want to know precisely by example   1. what is dynamical operator?   2. what is the relationship between dynamical operators and the $SU(n+1)$   3. How to show all the eigen states of a dynamical operator form a complete set of basis vectors of a Hilbert Space?(Any quantum state of a quantum system can be regarded as a vector in the Hilbert space)


title: A step in the derivation of the magnetic momentum of the electron in Zee's QFT book

text: In chapter III.6 of his _Quantum Field Theory in a Nutshell_ , A. Zee sets out to derive the magnetic moment of an electron in quantum electrodynamics. He starts by replacing in the Dirac equation the derivative $\partial_\mu$ by the covariant derivative $D_\mu = \partial_\mu - i e A_\mu$, where $A_\mu$ is a (classical) external electromagnetic field. We have $$ (i \gamma^\mu D_\mu - m) \psi ~=~ 0. $$ From that he derives $$ \left(D_\mu D^\mu - \frac{e}{2} \sigma^{\mu \nu} F_{\mu\nu} + m^2\right) \psi~=~ 0, $$ where, as usual, $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$ and $\sigma^{\mu \nu}$ are the commutators of the Dirac $\gamma$ matrices: $$ \sigma^{\mu \nu}=\frac{i}{2}[\gamma^\mu, \gamma^\nu]. $$ My problem deals with a apparently simple step that Zee uses in the derivation. He claims that $$ (i/2) \sigma^{\mu \nu}[D_\mu, D_\nu] = (e/2) \sigma^{\mu \nu} F_{\mu\nu}. $$ However, I get $$ [D_\mu, D_\nu] = -ie\partial_\mu A_\nu + ie\partial_\nu A_\mu -ieA_\mu\partial_\nu + ie A_\nu \partial_\mu = -ieF_{\mu\nu} -ie A_\mu\partial_\nu + ie A_\nu \partial_\mu, $$ but I do not see right now why the last two terms vanish when multiplied by $\sigma^{\mu \nu}$. I even tried to use the explicit expressions for $\sigma^{\mu \nu}$ and got a nonzero value. I have the feeling that I am missing something really simple here. Does somebody see what I did wrong?


title: boundary limit conditions in 3D water surface simulation

text: As is discused on this post, taking some assumptions, the water surface can be simulated by a discrete aproximation of a grid of heights using this formula ![enter image description here](http://i.stack.imgur.com/2h2Vg.png) Where: **HT** is the new height grid **HT-1** is the grid mapped in the previous step of the simulation **N** is a damping factor The 2D simulation works pretty well. I want now to move to 3D, but I'm concerned about the bundary limit conditions. Unless I'm missing something, the document doesn't explain how to calculate the height on the limits, where there is no **x+1** or **x-1**. I could mirror the point on the other site as if the boundaries were mirrors, but I'm not sure. The following image depicts what's described above ![enter image description here](http://i.stack.imgur.com/2UDSc.png)


title: Would it be possible to create a hadron collider in space?

text: Would it be theoretically possible to create a hadron collider in space by orbiting particles around a planet?


title: Starcraft II: Estimate how much air a viper would have to move to abduct an ultralisk

text: This is a physics question about an in-game world scenario. In Starcraft II, there is a flying unit called "viper". It has the ability to abduct huge, massive enemy units - while remaining stationary in the air. The question is, if one were to apply very basic physics, how much air would that viper need to move to oppose the force of pulling a massive object? Original problem description found here: http://www.reddit.com/r/starcraft/comments/277xz5/burrowed_ultralisks_can_be_abducted/chys5wx The game Starcraft 2 Zerg unit: viper Zerg unit: ultralisk Demo: a viper pulling a colossus (while itself remaining stationary in the air) I can do the basic calculations, but I found the more I thought about it the more intricacies I encountered, until the basic problem looked harder than it is. Now I'm looking for people with a clearer head. Or actually, I really think I know how to solve this (can't get more basic as far as the fundamental calculations are concerned) but would still hear your opinion/strategy, mostly for the "edges" of the problem.


title: Does anisotropic expansion of the universe imply quintaessence?

text: a Recent paper establishes under solid grounds anisotropy in the expansion acceleration rate in the universe. My question is very simple: **can this anomaly be explained entirely in terms of a cosmological constant?** I suspect that by adjusting the $\alpha$ scale factor to depend not only on time but also on a unit direction it seems that one could explain the above anisotropy without appealing to any exotic field (other than a cosmological constant) but then you _only_ do not know why the asymmetry in the boundary condition (early universe) exists in the first place


title: Since the universe is expanding, does anything ever occupy the same point in space?

text: Let's say we can observe expansion in a supercluster.   We define origin of our frame of reference at the center of the supercluster.   We observe an object/atom at point A at time T. The object is motionless relative to the origin.   We wait for expansion until T+ΔT and again observe the object.   Is the object at A or somewhere else?


title: Physics-based derivation of the formula for entropy

text: > **Possible Duplicate:**   >  Proof of $S=-\sum p\ln p$? I am looking for a derivation of the formula $$S~=~-\Sigma_ip_i \log (p_i).$$ for entropy, from first principles. I only wish to assume the laws of physics, and without involving concepts in information theory. (After all, the concept of entropy and Boltzmann's formula for it is far older than information theory.) What is a good definition of entropy? What assumptions are needed to arrive at this? What is the justification to maximizing entropy of a system to arrive at thermodynamics?


title: Equation of Motion in a Non-Inertial (Rotating) Frame

text: Let me preface my question by informing you this is for an assignment, so I would rather not have explicit answers but rather be given guidance in arriving at the correct solution. The question is thus; I have a mass on a frictionless rotating turntable. I need to adopt a co-rotating frame to find the equation of motion for this mass. I am expected to solve this using only Newtonian mechanics. As we are restricted to a plane, there is no component of motion in the $\vec{k}$ direction. The only forces acting on our mass (in the rotating frame) are the Centrifugal force, and the Coriolis force. My attempt at a solution; $\vec{F} = m\vec{a}$ $\vec{F_{cent}} = -m\vec{\Omega}$ x $\vec{\Omega}$ x $\vec{r}$ $\vec{F_{cor}} = -2m\vec{\Omega}$ x $\vec{v}$ $m\vec{a} = -m\vec{\Omega}$ x $\vec{\Omega}$ x $\vec{r} - 2m\vec{\Omega}$ x $\vec{v}$ As I'm working in the rotating frame, I don't see any reason to not use cartaesian coordinates (please note this is my first time using latex, so in this case $\vec{i},\vec{j},\vec{k}$ are unit vectors, and x,y,z are magnitudes in the respective directions); $\vec{\Omega} = \Omega\vec{k}$ $\vec{r} = x\vec{i} + y\vec{j}$ $\vec{v} = \vec{\dot{r}} = \dot{x}\vec{i} + \dot{y}\vec{j}$ Substituting this in, evaluating the cross-products and simplifying yields; $\ddot{x}\vec{i} + \ddot{y}\vec{j} = \Omega^2x\vec{i} + \Omega^2y\vec{j} - 2\Omega\dot{x}\vec{j} + 2\Omega\dot{y}\vec{i}$ And so we have two coupled second order differential equations; $\ddot{x} = \Omega^2x + 2\Omega\dot{y}$ $\ddot{y} = \Omega^2y - 2\Omega\dot{x}$ A method we had used previously in class to solve coupled equations was to set $\Omega = 0$ and solve, then substitute this solution back in for $\dot{x} and \dot{y}$. I attempted this, however it yielded two cubic equations. The solution I am told this system has, for the initial conditions $(x(0),y(0)) = (x_0,0)$, is a spiral when mapped parametrically, namely; $x(t) = (x_0 + v_{x0}t)\cos\Omega t + (v_{y0} + \Omega x_0)t\cos\Omega t$ $y(t) = -(x_0 + v_{x0}t)\sin\Omega t + (v_{y0} + \Omega x_0)t\sin\Omega t$ To me these appeared to be solutions gained from solving the non-homogenous linear second-order differential equation, however this did not work either. Is my derivation of the original vector equation of motion correct? If not, where did I go wrong? If so, what method should I use to solve these equations to find the appropriate solutions?


title: Focusing light into an optical fiber cable

text: I'm trying to focus light from a flashing light lamp throught two lens into a optical fiber cable which is around 0.8cm in diameter.   * flashing light: a bulb and then covered by a glass of 7.5cm x 7cm   * Fresnel Lens 310 x 310mm, Thickness: 2mm, Focal length: 220mm I tried to use a piece of white paper to focus the image, however, my teacher said that the focal point might have more intense light. But focal point is not somewhere I can see. Is his theory right about focal point have more intense light than the image point? I wonder if I could use more convex lens to focus the spot of light into a smaller diameter?


title: Is there any operator behind probability, in quantum mechanics?

text: In Quantum mechanics, the probability of finding a particle at position $x$ is given by $|\psi(x)|^2$, where $\psi$ is the wave function. Wonder what is the operator which gives this probability? Is probability the result of any operator acting on $\psi$?


title: possible values of quantity in matrix mechanics

text: Suppose that there is a quantity described by the matrix as the following: $M = \begin{bmatrix} 3 & 0 & -i \\\ 0 & 1 & 0 \\\ i & 0 &3\end{bmatrix}$ How we determine possible values of this quantity? (According to the book, it says that some possible values are 1,2 and 4, but it does not say how it determined this.)


title: How did QED diverge from quantum mechanics mathematically?

text: We have either Heisenberg or Schrodinger picture of quantum mechanics world. So, how did quantum electrodynamics come from mathematical formulations of quantum mechanics? Also, QED seems to have Lagrangian formulation, while quantum mechanics does not. Why is it like this?


title: Task with three metal plates

text: There are three parallel metal plates. Two boundary plates are fixed and connected with a galvanic battery, maintained a constant potential difference $\Delta \varphi$. Middle plate was initially in contact with one of the end plates. Then, using an insulating handle, it began to move toward the other plate. Ignoring boundary effects, find the field strengths E1, E2 in the gaps between the plates, depending on the varying distance x. Answer: $$ E_{1} = \frac{\Delta \varphi x}{d^{2}}, \quad E_{2} = \frac{\Delta \varphi (x + d)}{d^{2}}, $$ where d is the distance between boundary plates. I don't know how the moving plate affect to the measure of field strengths. Can you help me?


title: Length Contraction And Simultaneity

text: I wanted to derive length contraction from the Lorentz transformation, but I keep getting stuck on problems with simultaneity in my derivation. At some point, I came across Wikipedia's article on Length Contraction where, in the section labeled "Derivation", it states > In an inertial reference frame $S^\prime$, $x_1^{\prime}$ and $x_2^{\prime}$ > shall denote the endpoints for an object of length $L_0^\prime$ at rest in > this system. The coordinates in $S^\prime$ are connected to those in $S$ by > the Lorentz transformations as follows: > > $ x_1^{\prime} = \frac{x_1-vt_1}{\sqrt{1-\frac{v^2}{c^2}}} $ and $ > x_2^{\prime} = \frac{x_2-vt_2}{\sqrt{1-\frac{v^2}{c^2}}} $ > > As this object is moving in $S$, its length $L$ has to be measured according > to the above convention by determining the simultaneous positions of its > endpoints, so we have to put $t_1=t_2$. Because $L=x_2-x_1$ and > $L_0^\prime=x_2^\prime-x_1^\prime$, we obtain > > $ L_0^{\prime} = \frac{L}{\sqrt{1-\frac{v^2}{c^2}}}. $ I understand everything except for the line which reads $L_0^\prime=x_2^\prime-x_1^\prime.$ After setting $t_1=t_2$, it is possible to write $L=x_2-x_1,$ since events $1$ and $2$ are simultaneous. However, in the primed inertial reference frame, which is moving with a nonzero velocity with respect to the laboratory frame, the events are not simultaneous. Therefore, how can it be that $ L_0^\prime=x_2^\prime-x_1^\prime, $ since $ t_1^\prime \neq t_2^\prime? $


title: Can non-conductive materials can be grounded by a wire?

text: Suppose that I comb my hair. The plastic comb would become negatively charged and would attract small paper pieces. But if I touch the comb in bare hand and in bare foot standing on the ground, or if I connect a copper wire with the comb into my floor, would the charges in the comb be neutralized? If not, how can be the comb be neutralized again? It seems the charges will remain in the comb forever. Please explain, I am confused about this matter. Another question: How can the charged comb attract small pieces of paper if they are neutral?


title: Thermometer reading

text: I quickly plunged a room-temperature thermometer into very hot water, the mercury level went down briefly before going up to a final reading. Why?


title: Angular momentum of light

text: Can someone explain the classical angular momentum in electromagnetic theory of light? If I shine elliptically polarised em wave on a black disc it rotates. I would like to know how to calculate torque in classical picture.


title: Why are the antimatter compositions of neutrons and protons different? Why by about 1%? References?

text: In his blog at http://motls.blogspot.com/2010/09/can-antimatters-gravity-be- repulsive.html Lubos Motl writes "...neutrons contain a slightly different mass contribution from the antiquarks (antimatter!) than the protons ..... where the difference is something of order 1%, ..." I would like some reliable references to this derivation, the simpler the better, assuming the usual standard model, of course. I assume it is somewhere in the computation of the masses of the baryons, but where, and why?


title: Filtering and extracting turbulent data from instantaneous dataset

text: I have collected data on flow velocity at 1 Hz sampling rate for 25 hours using a high resolution acoustic Doppler current profiler (HR-ADCP). how I can decide up to which period is the mean velocity? How I can exclude fluctuations with the above period using wavelet filters?


title: Closed form for shape/tension of an elastic cable slung between two points

text: Given the 2D coordinates of two points, $a$ and $b$, between which an elastic cable of known length, $l$, mass per unit length, $m$, and the spring constant, $e$, is slung, I need to compute the shape of the cable, and also the horizontal tension, $t$, in the cable. So far I have the equations for the x and y coordinates of the cable, parameterized by $p$, which is the distance along the unstretched cable: $$ f_x(p) = \frac{t}{mg} \sinh^{-1}\left(\frac{mgp}{t}\right) + \frac{tp}{e} + c_x.\\\ f_y(p) = \sqrt{\left(\frac{t}{mg}\right)^2 + p^2} + \frac{mgp^2}{2e} + c_y.\\\ $$ where $c_x$ and $c_y$ are constants, which leads me to the following triplet of simultaneous equations: $$ f_x(q) - f_x(r) = a_x - b_x.\\\ f_y(q) - f_y(r) = a_y - b_y.\\\ |q-r| = l. $$ in three unknowns, $t$, $q$ and $r$ (given that the constants cancel), where $q$ and $r$ are the values of the parameter $p$ at the points $a$ and $b$ respectively. How would you compute those unknowns, and can it be done in closed form?


title: Virtual Particles and Quantum Brownian Motion

text: Does the frequency of virtual particles (creation/annihilation) match with nucleus random paths - known as Quantum Brownian Motion?


title: Why did my liquid soda freeze once I pulled it out of the fridge and opened it?

text: This isn't a duplicate to "Why did my liquid soda freeze once I pulled it out of the fridge?". My question is why soda froze after it was opened. Opening a can or bottle seems to have a larger effect than just jostling it. Is it because of the disturbance noted in the previous question? Is it related to the pressure decrease? Is it because of the release of some CO2 when it was opened?


title: Spin degeneracy in perturbation theory

text: In pag. 270 of Griffith's "Introduction to Quantum Mechanics" a perturbative method for finding relativist correction to the energy levels of the Hydrogen athom is exposed. It is asserted, if I understand well, that the operator consisting in $\hat{L}^2\hat{L}_z$ commutes with the Hamiltonian (both the perturbed one and the unperturbed) and has different eigenvalues for each of the $n^2$ eigenstates that have the same $E_n$. This justifies the use of the nondegenerate theory. However, it seems to me that the degeneracy is really $2n^2$, because of the spin. This problem can indeed be solved if we consider the operator $\hat{L}^2\hat{L}_z\hat{S}_z$, but Griffith makes no mention of this. Am I missing something?


title: Exact diagonalization by Bogoliubov transformation

text: I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, $$ H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 & 0\\\ \- U_2 & 0 & \xi_{\mathbf{k}+(\pi/2,0)} & 0\\\ \- U_2 & 0 & 0 & \xi_{\mathbf{k}+(0,\pi/2)} \end{pmatrix} $$ And my Nambu operator is given by, $$ ψ_\mathbf{k} = \begin{pmatrix} c_{\mathbf{k},\sigma} \\\ c_{\mathbf{k}+(\pi,\pi),\sigma} \\\ c_{\mathbf{k}+(\pi/2,0),\sigma} \\\ c_{\mathbf{k}+(0,\pi/2),\sigma} \end{pmatrix} $$ I tried to diagonalized by making three Bogoliubov transformations, the first to diagonalize the upper right submatrix of H, and then the two others (a sort of nested transformations). But I get a lengthy result, what I would like to know if there is a smart transformation which allows me to write $$ H = A_1^\dagger A_2^\dagger A_3^\dagger D A_3 A_2 A_1 $$ or simply $$ H = U^\dagger D U $$ Or the only way is to use just brute force? My question may be rephrase as, is there any elegant representation or expression of the diagonalization as for example the one obtained in BCS Superconductivity? Thanks


title: Earth's Temperature

text: I have a question regarding earth's temperature... How come the average temperature in the middle east (Israel, Saudi Arabia, Sudan or lower) is always so much significantly higher than in Europe (say Germany, England etc.)? I know that the sun rays pass a greater distance to Europe than the middle east, but is that the only factor influencing? And also, the distance isn't that much greater so how come the sun is weakened so significantly in Europe in comparison to the middle east?


Just what the title states. An artificial satellite will orbit Earth with some defined angular momentum. An aircraft,on the other hand, relies upon it's engines to provide continuous forward velocity & lift. I understand the mass of the satellite, shape of the orbit may be a factor here. As also the factors mentioned in dmckee's comment below. To clarify, the intent here was indeed to determine atmospheric drag. Mike & Peter, I proffer my apologies. How closely may an artificial satellite orbit Earth without decaying appreciably each orbit? Is there a definite formula/formulae to determine this by direct substitution of values, or is each such calculation unique?


title: Where does the equation $E_q = \frac{kq}{r^2}$ come from in this problem?

text: > A point charge is placed at each corner of a square with side length $a$. > The charges all have the same magnitude $q$. Two of the charges are positive > and two are negative, as shown in the following figure. > > What is the magnitude of the net electric field at the center of the square > due to the four charges in terms of $q$ and $a$? I'm trying to understand this question by using the solution my instructor gave me, but I'm confused. I found $r=\frac{a}{\sqrt{2}}$, which makes sense. However then it says: > The magnitude of the electric field due to each charge is the same and equal > to > > $$E_q = \frac{kq}{r^2} = \frac{2kq}{a^2}$$ How did they come up with this? I only know the equations $F = \frac{k\lvert q_1q_2\rvert}{r^2}$ and $E = \frac{F_0}{q_0}$ and I can't seem to work out how they got to this point.


title: Do eigenvalues in a cylindrical symmetric problem tell us anything about the Fourier spectrum?

text: During a lecture we were solving the Helmholtz equation for particular boundary conditions, corresponding to different shapes of an oscillating drum, as in the famous Mark Kac's problem http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum. For Cartesian boundary conditions we used the Fourier eigenfunctions (sines and cosines), as for the Cylindrical b.c. we used the Bessel functions (http://en.wikipedia.org/wiki/Bessel_functions). We then compared the spectra in both of those bases, interpreting the different result as a difference in the pitch and timbre, that a particularly shaped drum would have. My question is: what do we really "hear" in a sound? I thought, that we only "hear the Fourier eigenvalues". So is there easy way of calculating Fourier eigenvalues, knowing the eigenvalues in the other basis? Or am I missing something obvious? PS. I know that technically I'm referring to a orthonormal set of functions not a basis, but I hope that the physicists' forum won't mind.


title: What is the advantage of supersymmetry?

text: what are the advantages of SUSY with respect to other theories? for example DIMENSIONAL REGULARIZATION can also deal with the UV and IR divergences too :D so what other advantages has supersymmetry?


title: What technological advance would allow LEP3 to surpass LEP2?

text: I learned that for electron accelerators synchrotron radiation and acceleration are the limiting factors. This article, that I found in one answer to this question mentions that one would **not** use the superconducting acceleration elements that were developed in the meanwhile, but rely on regular normal-conduction acceleration. Then I wonder, at the same radius, what technological advance would allow LEP3 to surpass LEP2 in a significant ammount?


title: Equilibrium - uniform circular motion

text: Maybe this is a bit of a silly question, but let us pretend we have a pendulum in a ideal universe with no friction, drag, or anomalous forces there to affect it. Additionally, our pendulum is subjected to an ideal gravitation force, the force vector of which is unaffected by the position of the ball (hence the responsible gravitating body is so massive that its own spherical geometry does not affect the forces on the pendulum-ball. Now we have our pendulum spin around so that it experiences uniform circular motion and some angle theta is formed between the resultant orientation of the spinning pendulum string and its previous stationary form. My question is, is this system in equilibrium? I know that no work is being done even though there is a force. And that from a relativistic (non-Newtonian) point of view, there is no centrifugal force to cancel the centripetal. So please, could you explain the "answer" from a Newtonian (high-school physics) POV and a relativistic POV.


title: Why do tropical cyclones not tear themselves apart?

text: A tropical cyclone is the generic term for a hurricane, typhoon, or tropical storm. Tropical cyclones derive their energy from evaporation of water at the ocean surface which ultimately recondenses when it cools to the saturation point. The rotation of a tropical cyclone is caused by the Coriolis effect. What is unclear to me is: **Why do these massive bodies of warm, moist air clump together over thousands of kilometers?** What physical process causes them to be drawn in towards the center of the storm? ![enter image description here](http://i.stack.imgur.com/cGc1i.jpg) *image taken from the Wikipedia article on tropical cyclones


title: Why is using Newton's First Law failing me?

text: A factory worker pushes a 29.7kg crate a distance of 5.0m along a level floor at constant velocity by pushing downward at an angle of 32$^\circ$ below the horizontal. The coefficient of kinetic friction between the crate and floor is 0.24. What magnitude of force must the worker apply to move the crate at constant velocity? So because constant velocity \begin{align}\sum F&=0\\\ F_x\times \cos{(360^\circ-32^\circ)} - f_k &= 0\\\ F_x\times \cos{(360^\circ-32^\circ)} - (mg)\times 0.24 &= 0\\\ F_x\times \cos{(328^\circ)} - 69.8544\,\mathrm N &= 0\\\ F_x &= 69.8544\,\mathrm N/\cos{(328^\circ)}\\\ F_x&=82.3708\,\mathrm N \end{align} That turns out to be wrong, the expected answer is $f_k$ the force of kinetic friction. Why? **Shouldn't I be able to use $\sum F=0$ in this problem to find the answer?** I also tried to find $\sqrt{F_x^2+F_y^2} -f_k$ but that didn't help either.


title: Approaching of Milky way to M87

text: The Andromeda Galaxy is approaching the Milky Way at about 110 kilometres per second (68 mi/s). What is the velocity that Milky way is approaching M87 ?


title: Why was theory of cosmic aether discarded?

text: Can cosmic aether be considered analogous to higgs field? Higgs field remains (almost) consistent throughout.


title: How does Higgs field relate to Aether theories?

text: I am an amateur learning about the Higgs because I was interested in what the LHC's purpose is. I read that as a particle passes through space, it is actually passing through a Higgs field and there are little Higgs particles that accumulate on the moving particle, which is where mass and momentum come from. But that's where my main question comes from: This all sounds similar to a "medium"; and mediums usually impart resistance on particles moving through them so the particles would slow down even in space. As far as I know, this is the reason why aether/ether theories don't work. Please let me know.


title: How viable is reuse of nuclear waste at this point, and what are future prospects?

text: I keep seeing documentaries describing the pile-up of nuclear waste from nuclear power stations in the US and in other countries. Countries such as Finland are working to deposit waste in old mines in geologically stable regions, and I was under the impression that the US was heading down that track as well, but this has now apparently been put on hold. A case of NIMBY I presume (not in my back yard). The media story is that the waste is piling up, and is being stored in pools at nuclear power stations all over the US - which sounds dangerous considering the prevalence of natural disasters. I wonder what the real truth is? I am not quite sure what the situation is in France and England, but I got the impression they are experimenting with methods of re-use of the waste products as well as storing some material under ground. What is the current state-of-the-art in terms of nuclear waste reuse? Are new reactors capable of burning discarded fuel from older reactors? Are there realistic expectations to be able to eliminate deposition of nuclear waste altogether in the foreseeable future?


title: Good introductory papers and books on laser physics and pulsed lasers

text: I am looking for good introductory papers and/or books on the principles of lasers. In particular, I am interested in pulsed laser technology. I understand that > Gould, R. Gordon (1959). "The LASER, Light Amplification by Stimulated > Emission of Radiation" was one of the principal papers published by the disputed inventor himself. What are other good sources?


title: Basis transformation between eigenstates of harmonic oscillators with different frequency

text: Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$. Is there a compact expression for the basis transformation that relates the $\Psi_n$ to the $\Psi'_n$? For the ground state, I think I have an idea of how to do that: The matrix element $\langle \Psi'_0 | \Psi_0\rangle$ is easy to calculate as it's just a Gaussian integral, and for the matrix elements of the type $\langle \Psi'_n | \Psi_0 \rangle$ I can use $$\langle \Psi'_n | a | \Psi_0 \rangle = 0$$ and then use the fact that the operator $a$ must be related to operator $a'$ for the other oscillator via a unitary transformation of the type $a = u a' + v a'^\dagger$. That gives me a recursive relation for the coefficients of a relatively simple form that I can solve. But I'm not sure if that's the most elegant way, and how I'd generalized that to get _all_ the matrix elements.


title: Does charge distribute itself uniformly on a conductor?

text: An excerpt from a beginning E&M book > [...] In other words, **_the surface of a conductor is an equipotential > surface under static conditions._** [...] Summarizing the _boundary > conditions_ at the conductor surface, we have $$\Tiny > \color{maroon}{\boxed{\begin{array}{c} \color{black}{\text{Boundary > Conditions}} \\\ \color{black}{\text{at a Conductor/Free Space Interface}} > \\\\[5pt] \hline \color{black}{E_t=0} \\\ \displaystyle \color{black}{E_n = > \frac{\rho_s}{\epsilon_0}} \\\ \end{array}}}$$ This challenges my intuition because this would mean a charged gourd at equilibrium would be in stress. (Presumably the skinny side would be pushed away from the base.) Is this another approximation for the sake of computational simplicity? ![a charged gourd](http://i.stack.imgur.com/ZhkCp.jpg)


title: Two people pushing off each other

text: If there are two people on ice skates named $A$ and $B$, and $A$ pushes on $B$ with a force $F_A$, then both of them will experience a force of magnitude $F_A$ in opposite directions. I'm confused as to what will happen if $B$ is also pushing at the same time with a force $F_B$. If $A$ pushes on $B$, he experiences an opposite force because of his 'crime' (the act of pushing on $B$), but now $B$ is also pushing at the same time with a force $F_B$, so maybe the forces add up on each of them? I'm not sure. I would _really_ appreciate any help!


title: How can one calculate the gamma-ray energy spectrum for proton-proton collisions?

text: How does one calculate the the energy spectrum of gamma-rays produced by a proton-proton collision? I'm at a complete loss.


title: Mathematical definitions in string theory

text: Does anyone know of a book that has mathematical definitions of a string, a $p$-brane, a $D$-brane and other related topics. All the books I have looked at don't have a precise definition and this is really bugging me.


title: How do eigenstates of harmonic oscillators with different frequencies compare?

text: Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ground-state of one of them be expressed in terms of eigenstates of the other one? An application of my question would be a harmonic oscillator whose frequency can be controlled. Suppose then I start out in the ground state and then suddenly change the frequency. I'd expect that I'm then not in a ground state of the (new) oscillator any more, and I'd be interested in the time evolution of my state. For that, I need to do a basis transform of my groundstate. The problem seems basic enough to me that there should be previous work done on it. A brute force solution would probably be to perform integrals over the eigenstates in real space, but I have hope that an algebraic solution in terms of creation and destruction operators exists.


title: Modelling the fluidity of a fluid (grease) based on temperature

text: I am trying to create a statistical model of a lubrication system. A central grease pump takes grease from a tank and injects it into some cavities (via grease lines) until a pressure set-point is reached. I noticed that the system takes longer to pump grease through all the lines when the temperature is colder. I assume this is because the grease's fluidity reduces in cold temperature, therefore it travels slower through the lines and is pumped less efficiently. Back to my statistical model: I am trying to do a regression to correlate temperature records with observations of the time required to lubricate all parts. I am wondering which regression model I should use:   * a linear function `time = a + b x temperature` doesn't work well.    * a second-order polynomial fits the data quite well   * a log `time = a+ b x log(temperature)` fits the data quite well too Can you give me an insight in the underlying science behind my observations? I assume there are a lot of other factors, such as pressure, velocity of the fluid inside the lines, type of lines, diameter, etc... But for a start, which law determines the fluidity of a lubricant like grease, based on temperature ? For instance, is it a polynomial, an exponential, 1/x, or something else ? **EDIT** I just stumbled upon this: Wikipedia Temperature dependence of liquid viscosity. I guess Grease is not exactly a fluid, but all these laws seem to be exponential or log.


title: Are derivative indices summed in indicial notation?

text: A paper I'm reading uses indicial notation and the convention that $u_{j,k}$ means the derivative with respect to $x_k$. Which one of these interpretations are correct? $$A_{ijk}u_{j,k} = \sum_{j}A_{ijk} \frac{\delta u_j}{\delta x_k}$$ $$A_{ijk}u_{j,k} = \sum_{j,k}A_{ijk} \frac{\delta u_j}{\delta x_k}$$


title: When Venus is the brightest when viewed from Earth?

text: Nowadays Venus is very bright. I can spot it during broad daylight without problem. It's because it's near Earth and appear as a crescent. This made me think: as it's reaching the inferior conjunction the crescent becomes slimmer, so I expect that it will fade a bit. So there should be an optimal phase angle for optimal brightness. So my question is at which phase angle Venus is the brightest when viewed from the Earth?


title: Why is the Bohr's idea of defined circular orbits overruled?

text: * If we consider a thought experiment for determining position of an electron by using photons of light. According to principles of optics, if we use light of wavelength $\lambda$, then the position of electron cannot be located more accurately than + or - $\lambda$. The shorter the wavelength, the greater is the accuracy. Therefore, to observe the position of the electron accurately, light of approximately small wavelength should be used. But the photons of radiations of smaller wavelength will have higher momentum. When even a single photon of this light strikes against it, a large amount of momentum will be transferred to the electron at the time of collision. This will result into greater uncertainty in velocity or momentum.    * On the other hand, in order to minimize the change in momentum we have to use light having photons with small values of momentum. This will require radiations of larger wavelengths (low momentum), the velocity or momentum will not change appreciably but we will not be able to measure the position accurately with larger wavelength. Therefore, uncertainty in position will increase. Thus, we cannot simultaneously measure the position and momentum of a small moving object like electron accurately. However, in case of macroscopic objects, the position and velocity of the objects can be determined accurately because in these cases, during the interaction between the object and the measuring device, the changes in position and velocity are negligible.  What I explained above is actually the Heisenberg's Uncertainty principle which states that > _it is not possible to measure simultaneously both the position and momentum > (or velocity) of a microscopic particle, with absolute accuracy._ According to Bohr, the electrons revolve around the nucleus in certain well defined circular orbits. But the idea of uncertainty in position and velocity is said to overrule the Bohr's idea of uncertainty picture of fixed circular orbits. We may not be able to design an experiment (until now) to measure simultaneously both the position and momentum for the electron. But we cannot overrule Bohr's idea of fixed orbits because of this reason. Because, we may not know whether electrons are revolving in fixed orbits, if are able to locate electrons without using photons. Thus, this could not be the exact reason for overruling the idea of fixed orbit. So, is there any reason for overruling the idea of fixed orbit? or is there any thing wrong in my opinion about the concept, if so please explain, so that I would not proceed with that wrong thinking? _EXTERNAL LINKS_   * Is uncertainty principle a technical difficulty in measurement? (This is a very good question by gotaquestion, which discusses inability of knowing electron location without disturbing it, this link strengthens my last claim on the opposition of rejecting fixed orbit)


title: Gravity and objects at 0K

text: Let's say I put 2 objects at rest relative to each other with 0 Kelvin. Will they move towards each other because of gravity? Where they will get energy to move? I don't have education in this field, just wondering.


title: Quantum explanation of Newton's Third Law of Motion

text: Newton's law states that for every action there is equal and opposite reaction. This law explains how rockets fly in space and accounts for the the majority of the lift action generated by a airplane's wings. > Is there a fundamental quantum explanation of the third law of motion?


title: A simple derivation

text: During the study of a paper I see that its author defines $$\frac{dh^{ab}}{d\tau}:=h^{am}h^{bn}\frac{dh_{mn}}{d\tau}$$ and from this concludes that $$\frac{d}{d\tau}(h^{ab})=\frac{d}{d\tau}(h^{am}h^{bn}h_{mn})=-\frac{dh^{ab}}{d\tau}$$ where $h_{ab}$ is a Riemannian metric that depends on the parameter $\tau$. I Can not derive last equation. Can someone point me in the right direction? Sorry, if my question is very obvious!


title: Strength of strong nuclear force vs distance?

text: Is there at least an approximation of the decrease in strong nuclear attraction vs distance from the center of the nucleus?


title: How long would it take a projectile accelerating twice as fast as another to catch up?

text: Given a projectile with initial velocity $u$, and constant acceleration $a$, travelling in the one direction. How long would it take a second projectile the same as the first with the same initial velocity $u$ and twice the acceleration $2a$ but launched after the first by $t$ amount of time? I thought I could do this by calculating the displacement of the first projectile travels after time $t$: $s$. Then take the difference in acceleration $2a - a = a$, and find how long it would take for the second projectile to travel $s$ and that would be the answer? OK so I know to calculate the displacement at time t I need the final velocity at time $t$: $v$. $v$ can be found by: $$v = u + at$$ then $s$: $$s = ut + \frac{1}{2}{at^2}$$ Now I have the displacement of the first projectile after time $t$ I simply need to find the time the second projectile will take to travel $s$, but how can I find that? (I do not know what the second projectiles final velocity will be when it travels $s$...


title: Why doesnt this violate 2nd law of thermodynamics?

text: Consider an ideal gas in a cylindrical container in a gravitational field, with a piston on top pushing down by gravity. The piston has some locking mechanism that locks it in place if it is displaced upwards enough that the temperature of the gas is some $1>>\epsilon>0$. If we wait long enough, enough particles will line up due to their random motion and push the piston upwards. We can make the piston arbitrarily thin and arbitrarily heavy. Eventually it will lock in some final top position and some arbitrarily small portion of kinetic energy is left in the gas. Since we can make the volume change arbitrarily small, while momentum phase space goes to zero, the entropy must have decreased in this process, but that cant happen, so what is wrong with this reasoning?


title: how does the mirror equation works and what lead to using of sign convention?

text: EVERYTHING HERE IS FOR CONCAVE MIRROR Everywhere I see the derivation of the mirror equation is given by placing an object before the focus and then proving similarity of the triangles to get to the form $\frac 1u + \frac 1v = \frac 1f$ .......(1) ( $v$ is toward the reflecting side of mirror) It absolutely true and i completely agree with it , it should be valid for all the cases where you keep an object before the focus , my question is that how is it valid for the cases where object is kept between focus and pole ?? The triangles whose similarity we had used to get to the equations are not even formed here .....how then are we using the equation here ?? Moreover the '$v$' that we used here is actually the side length of a triangle , it cannot be negative this clearly shows that the equation is invalid for object kept between the focus and centre ... Because we get a negative '$v$' After thinking for some time and calculations i have come to an answer for the requirement Of a sign convention If we keep an object between the focus and the pole and extend the rays backward to form a virtual image , we can prove (again by similarity of different triangles ) that for object placed between the focus and pole $\frac 1u - \frac 1v = \frac 1f$ ........(2) > ( $u$ and $v$ are the object and image DISTANCE respectively from pole) ($v$ > is towards the opposite of mirror) Now we can see that for an object kept before the focus we can use eq(1) and for object between focus and pole we can use eq(2) to get image distance respectively Where is the sign convention now ?? What if we want to generalise the formula i.e use a single formula for both cases .. Can we do that ?! Lets see First case For cases of objects before focus we can clearly use equation 1 Second case For objects between focus and pole we can use equation 1 to get v What if we use equation 1 for the second case It us logically absurd but lets see what happens .. Acc. To equation 1 $v=\frac {uf}{(u-f)}$ ( v is towards reflecting side) And acc to equation 2 $v=\frac{uf}{(f-u)} = -\frac{uv}{(u-f)}$ ($v$ is opposite to reflecting side) So we see even though it is absurd to use equation 1 for the second case logically but due to some type of similarity between the two equations the result we get from using equation (1) for the second case is negative of the result from equation (2) (which is correct equation for the second case) for the second case So if we ise equation (1) in second case we get a negative answer (I)Negative answer means that we are using wrong equation and we have to use eq(2) (II)By using eq(2) we get the same answer of $|v|$ but this time opposite side So we can generalise equation (1) by using a method that negative answer of '$v$' means image is at distance $|v|$ opposite to the mirror This method is sign convention This is the reason for involvement if a sign convention system in mirrors .... Clearly it is a method to generalise all equations to a single mirror equation .. I just want reviews and want to know that isnt it right ?


title: Intuition contradicting answer for calculating pressure

text: I have a cylindrical container say of some base area $A$. The atmospheric pressure is $P_0$. The cylinder is moved at an acceleration $a$ horizontally right. I need to find which base would have to face more internal pressure. The cylinder is filled with some ideal gas. My intuition, even though I doubt it, says that in the frame of the cylinder the fictitious forces act towards left, so it should be the left face. But on some calculations using the laws of motion, I get equations as $$(P_0-P_1)A=ma$$ $$(P_2-P_0)A=ma$$   ($m$ is the mass of each base of cylinder) On solving I have $$P_1=P_0-{ma\over A},P_2=P_0+{ma\over A}$$ which means the right face faces more internal pressure. What is going wrong? P.S. just give me a hint. ![enter image description here](http://i.stack.imgur.com/3euyA.png)


title: Difference between primitive unit cell and the associated basis?

text: As I understand it, the basis is the group of atoms whilst the primitive unit cell is the unit space that fits the total space without any gaps, and only containing one lattice point? How do the two relate to each other? Thanks.


title: What is $T_{\mu\nu}T_{\mu\nu}$ for the electromagnetic stress-energy tensor?

text: Given the electromagnetic stress-energy tensor components \begin{align} T_{\mu\nu} = \begin{pmatrix} u_{00} && s_{0 \nu} \\\ s_{\mu 0} && \sigma_{\mu\nu} \end{pmatrix}_{\mu\nu} \end{align} where $u_{00}$ is the energy density, $s_{0 \nu}$ is the poynting vector, and $\sigma_{\mu\nu}$ is the Maxwell stress energy tensor: What is \begin{align} \sum_{\mu =0}^{3}\sum_{\nu =0}^{3} T_{\mu\nu}^2 = ? \end{align}


title: Can a photon survive a collision? If so, is it at rest during the process?

text: ## Background Irving Kaplan, in Article 6.7: The Compton Effect of _Nuclear Physics_ (2nd Ed.) explains the Compton effect as follows: > Compton (1923) was able to show that when a beam of monochromatic x-rays was > scattered by a light element such as carbon, the scattered radiation > consisted of two components, one of **the same wavelength as that of the > incident beam** , the second of slightly longer wavelength. The explanation for the observation of unshifted line is that > If the collision is such that the electron is not detached from the atom, > but remains bound, the rest mass $m_0$ of the electron must be replaced by > the mass of the atom, which is several thousand times greater. The > calculated value of $\Delta \lambda$ then becomes much too small to be > detected. A photon which collides with a bound electron, therefore, does not > have its wavelength changed, and this accounts for the presence of the > unshifted spectral line. Now consider a carbon atom (vibrating in a bulk) moving towards X-ray photon (having frequency $\nu \neq(\delta E_n)/h$, where $\delta E_n$ refers to any possible excitation of electron) and photon is moving towards the carbon atom. photon strikes the electron of carbon atom. Their velocities are such that same energy is given back to photon which is given to carbon atom when photon collides and is thrown back at 180 degrees. The photon has same energy as before hence same frequency. (Everything is taken in a particular laboratory frame of reference.) The energy of atom and photon are not changed only their direction of motion are changed. * * * $%Note the visible separation between the background and the questions$ With this in mind, then   1. Is the photon ejected after collision the same photon which was thrown in? If not, what happened to the previous photon? What is going on with the energy carried by the previous photon? Since energy cannot be absorbed, where does the energy go?   2. When this photon collides (perpendicularly), is it at rest at the very moment of collision?


title: How can Superconductivity materials levitate permanent magnet?

text: I have thought that by eddy current. But how eddy current in superconductivity materials can be generated by using permanent magnet?


title: Volume quotient in Carnot-cycle

text: Problem: > One kilomole of an ideal, monatomic gas undergoes a reversible Carnot- > processes between temperatures 300 °C and 20 °C. The work done during one > cycle is 1500 kJ. > > a) Find the entropy-change in every process and show that total sum of > entropy-changes is zero > > b) What is the ratio between the largest and smallest volume that the gas > takes on during the whole process? I have already done a), it is b) I am struggling with. I realized that the greatest and smallest volume is assumed in the adiabatic processes of the Carnot-cycle, so I applied $T_1V_1^{\gamma -1} = T_1V_1^{\gamma -1}$ but to no avail. How can one do it?


title: What limits laser pulse duration to single cycle?

text: 1. Why pulses shorter than single cycle cannot propagate in free space ? Is there fundamental reason or it is engineering ?   2. Is it possible for a laser pulse to have bandwidth that can support half cycle pulse and how it will propagate in free space ?


Is the universe a perpetual motion machine?


title: What is the effect of humidity in temperature?

text: Suppose we have a building in an area with weather conditions $W_t$ at time $t$, and suppose we could change the relative humidity $\phi_t \in W_t$. Would the change in relative humidity $\phi_t$ have effect on some room/office within that building? I'm making assumptions which can't be made in real situations, and am aware of that. But just solely to get some insights; this question. Isolation? I don't make any assumptions. (notice that the sun is at the same position during the humidity change, along with the "non-moving" clouds etc.)


title: Centripetal Force Acceleration

text: In uniform circular motion, acceleration is $\frac{v^2}{r}$ and time which it acts $\rightarrow 0$. So $\Delta v = 0$, but then why/how does direction change, when the acceleration should be producing no effect at all .


title: Ground states in the shell model for odd-even nuclei

text: > I understand that even-even nuclei (Z and N number) have zero spin because > of pairing. > > Even-odd nuclei have the spin of the odd nucleon, and parity is given by > $(-1)^L$ - so my question is, how do we work out the state which this odd > nucleon is in? As an example: $^3_2 \text{He}$ the odd nucleon, is a neutron which is the state $1\text{s}_\frac{1}{2}$ ($l=0,$ $s=1/2$) so the answer for the ground state spin-parity is $\frac{1}{2}^+$... Or with $^9_4 \text{Be}$ the most energetic neutron is the $1\text{p}_{\frac{3}{2}}$ giving $\frac{3}{2}^-$ but how is this worked out? I assume it is something to do with the nucleons occupying all the lowest energy states and finding the highest energy spare nucleon - How will I know which quantum numbers are the lowest energy? Edit: for example, do I need to refer to this table? http://en.wikipedia.org/wiki/File:Shells.png


title: Why don't metals bond when touched together?

text: It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.). If two pieces of the same metal are touched together, why don't they bond? It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)?


title: What is baryon loading in the context of gamma ray bursts (GRBs)?

text: I've read that with short hard gamma ray bursts (shGRBs) associated with the coalescence of NS-NS and NS-BH binaries are expected to be beamed along the axis of the orbital angular momentum (i.e. perpendicular to the orbital plane) because the " _baryon loading_ is lowest there". Here, NS = neutron star, BH = (stellar mass) black hole.   1. What is meant by baryon loading?   2. Why is it lowest along the direction of the orbital angular momentum?


title: Faddeev Popov Gauge Fixing in Electromagnetism

text: Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. To "fix" this, we recognize that _anyway_ we would not have liked to get contributions from pure-gauge field configurations, because field- configurations in the same gauge-orbit correspond to identical physical field configurations. Ultimately, we would like to make a functional integral which ranges over only distinct gauge-orbits, taking each time only one representative from each gauge orbit. The way to do this technically is to insert a functional-delta function into the functional integral, where this delta function is always zero unless the field configuration obeys a particular gauge-condition, which is non-zero only _once_ in each gauge-orbit. So far, so good. However, then Peskin chooses as the gauge condition the Lorenz gauge condition. I'm wondering: why is that valid? _The Lorenz gauge condition does not completely fix the gauge_ : one can still make further gauge transformations by harmonic functions. What gives?


title: What is a symmetry of a physical system?

text: If I understand correctly, in many context in physics (quantum mechanics?), a physical system is specified by giving its Hamiltonian. I also hear that symmetries are rather essential. As far as the most natural symmetries are concerned, it seems fairly easy to recognise them. They should apparently be linear self-maps of the underlying Hilbert space, which seems natural enough, and they should surely commute with the Hamiltonian (if you transform the system by symmetry and then use the Hamiltonian to figure out how it evolves, the result should be the same as when you first look at time evolution, and then apply symmetry). For a while I thought this is just what a symmetry is supposed to be doing, but then I realised that any map that keeps the eigenspaces of the Hamiltonian invariant would do, and I suspect this is not quite right. Also, there seem also to be some other operators around, often used when defining the Hamiltonian --- I'm not sure if a symmetry is supposed to commute with these as well... I would be extremely grateful if someone could explain to me what symmetries of a physical system (viewed from the point of view of the Hamiltonian) actually are, and why is it that they are defined like this and not differently. An explanation in layman's/mathematician's terms would be even more appreciated. Disclaimer: I am a mathematician, not a physicist.


title: Physics of the point of contact for a spinning top

text: I understand how spinning tops don't tip over, cf. e.g. this and this Phys.SE questions. What I'm more interested is in identifying the factors that determine the direction the spinning top moves to?


title: is boiling of water an isobaric process?

text: Is the conversion of water into steam at 100°C an isobaric process? It is an isothermal process, but does pressure rain constant during this process?Please explain as well..


title: How to calculate the supporting force exerted by a stationary object (cupboard) and support (brackets) for it?

text: I have a wall mounted Stuva Ikea cupboard, which measures 60x64x30cm. The base is 64x30cm. It is already supported by 2 holes at the back base, (64x60cm). ![enter image description here](http://i.stack.imgur.com/SGPBQ.png) But I need to add in L shape brackets underneath for extra support. Only, the brackets can support a max load of 15kg only. The cupboard is 15kg, I need to load another 15kg of books on, making the total 30 kg. Can anyone help to calculate the distance that I have to put the brackets as additional support? 15kg max load brackets (28cm x 2cm) base of cupboard (64x30cm) total weight to support (30kg) - alr 2 holes wall mounted on base-to-wall (60x64cm) Appreciate any help. How do I calculate F=ma, as the cupboard and brackets are stationary? @all who answered, more info here, until i figure out how to add images on here lol: Cupboard frame 8.6kg http://www.ikea.com/sg/en/catalog/products/70165163/ Cupboard doors 5kg http://www.ikea.com/sg/en/catalog/products/90169103/ The cupboard backing board comes with 2 holes predrilled, and 2 plastic backing boards.the backing board is 2mm thick, doubt the screws i screwed in will actially hold it up. Ikea is no help at all! I hv two of this cupboards side by side (but not joined)


title: Canonical Commutation Relations

text: Is it logically sound to accept the canonical commutation relation (CCR) $$[x,p]~=~i\hbar$$ as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the position basis? I understand QM formalism works, it's just that I sometimes end up thinking in circles when I try to see where the postulates are. Could someone give me a clear and logical account of what should be taken as a postulate in this regard, and an explanation as to why their viewpoint is the most right, in some sense!


title: Why is an electric motor more efficient at higher loads?

text: My question is driven by the plot below. We see that acceptable operating range of a motor is between 50-100% of the rated load. Below 40% or so the efficiency of the motor drops off dramatically. What is the cause of this phenomenon? ![Motor Efficiency Plot](http://i.stack.imgur.com/uGERX.png)


title: Interpretations of quantum mechanics

text: It is well known that there are many interpretations of quantum mechanics. I'm wondering if there is a specific reason why the Copenhagen interpretation is the most popular. Why is it that the wavefunction interpreted as a probability distribution is such a useful description. This question came to me when I was trying to write a paper and got a strange wavefunction (it wasn't normalizable, had singularities etc.). I'm a mathematical physics grad student so I was usually more concerned with mathematical elegance than physical interpretations, but this question has been bothering me.


title: Mass of a superstring between two branes?

text: A open bosonic string between two parallel branes seems to obey formulae such as $M^2 = \big((n + {\theta_i - \theta_j \over 2 \pi}) {R' \over \alpha'}\big)^2 + {N-1 \over \alpha'} $ So that the difference $\theta_i - \theta_j$ is the distance between branes. Now I wonder, which is the formula for the superstring stretched between two parallel branes? Is it the same?


title: Relativity of simultaneity and cause-effect

text: Relativity says that two events simultaneous in one reference frame might not be simultaneous in another reference frame. Can we extend this idea a little and say that the order of two events might be be reversed in going from one reference frame to another? And if so, what if they have a cause-effect relation ship in the first frame. Isn't reversing the two in the second frame a contradiction?


title: Is it fair to judge this speedskating race by only 3 thousands of a second?

text: I'm reading this article: > Zbigniew Brodka of Poland won the Olympic men's 1,500 meters speed skating > title by just 0.003 seconds at the Adler Arena on Saturday. > > Brodka clocked one minute, 45.006 seconds with Koen Verweij of Netherlands > crossing the finish line in the final heat just outside the Pole's time. > Denny Morrison of Canada took bronze. I'm a noob with physics, please forgive me. The first thing I think is: Can you ever measure this so precisely. I'm not sure how exactly this is measured, but normally in physics, I think that you must say that a measurement is 1.45.006 $\pm$ 0.002 or something. I think that someone would have to know how this is measured to judge this, and this is maybe not so relevant on this website. But if someone knows more about this, I would like to hear it. Now comes the physic question. Here is the start photo of Koen Verweij (in the orange suit, on the right): ![](http://snag.gy/pAU5o.jpg) The start shot is given by a man on the left, which you can't see on this photo. Here is the start photo of the winner (Zbigniew Brodka, in the red suit on the left): ![](http://snag.gy/pbZHk.jpg) As you can see, Zbigniew Brodka is much closer to the source of the start sound, than Koen Verweij is. I'm wondering how much the time differences would be between when Koen Verweij hears the start shot and when Zbigniew Brodka hears the startshot. Could you say that Koen Verweij was faster, if you would start the time at the moment the sounds has reached the skater ?


title: Chronological and normal ordering

text: I've realized I'm little bit confused when I want to treat elements like this $$\left<\phi_0|T\\{a_p(t)a_p^+(t')V(t_1)V(t_2)\\}|\phi_0\right>$$ with $$V(t)=\dfrac12 \dfrac{1}{(2\pi \hbar)^9}\int d^3p_1 d^3p_2 d^3q v(q):a_{p_1}^+(t)a_{p_2}^+(t)a_{p_2-q}(t)a_{p_1+q}(t):$$ There is chronological ordering ($T\\{\\}$) and if I would apply Wick's theorem, there will pop up normal ordering and Wick contractions. Without times it would be easy, but together its mess. How is this done? Also could you recommend me some book that would clarify how to treat this. Edit: This [NOTES ON WICK’S THEOREM IN MANY-BODY THEORY, Luca Guido Molinari] (part V) could help here. I guess I'll have to look up on commutation relations for operators with different times (depending only on $H_0$, $H=H_0+V$).


title: Tadpole symmetry factor

text: Can someone help me with symmetry factor of one-loop tadpole diagram (one loop correction to one point Green function in phi-3 theory)?


title: Is there a closed form expression for Landau-level eigenstates?

text: Is there a closed form expression for the Landau-level eigenstates (preferably in the symmetric gauge)? _This is the 2-dimensional quantum mechanical problem of a charged particle moving in a homogenous magnetic field transverse to the plane._ Because of the similarity to the quantum simple harmonic oscillator, which have closed-form eigenstate expressions in terms of Hermite polynomials, I would have expected something similar to happen for this problem.


title: Is energy lost in an elastic tunneling process?

text: Suppose we had two metal plates separated by an Angstrom. Now, apply a voltage $V$ between the two metal plates. There will be a tunneling current $I$ between the metal plates. Since $P=IV$, does that imply power is being dissipated by the system, even if the tunneling is elastic? Some additional thoughts: Perhaps this is related to why there is still resistance in ballistic transport.


title: How do people calculate proportions of dark matter, dark energy and baryonic matter of the universe?

text: The Wikipedia page on dark matter mentions that the Planck mission had revealed that in our universe ordinary baryonic matter, dark matter and dark energy are present in the ratio: 4.9%, 26.8% and 68.3% respectively. I do not understand exactly how this result had been got. Did the Planck mission scan the entire universe as it is today to get these figures or just a part of the universe? (Is it really possible to scan the entire universe if the universe is infinitely large?) Can anybody please explain the principle behind the calculation? I also notice that the above figures are given in terms of percentages. Can I get the absolute values of the individual quantities like, total dark energy = .... Joule in the universe? In case only a part had been scanned then is it possible that in future these figures may change if a larger portion of the universe is scanned by another mission?


title: Component of angular velocity along an axis inclined at $\theta$

text: If an arbitrary rigid body rotates with angular velocity $\omega_0$ about some axis, can it be said that the body will rotate with an angular velocity $\omega_0 \cos(\theta)$ about an axis which is at an angle of $\theta$ with this axis. If yes, what is the physical significance of such an equation?


title: What would a sufficiently dense sphere of uranium do first: blow up, or form a black hole?

text: I think it is safe to assume that humans are a long way off from having either the technology to compress a normal-sized sphere of uranium below its Schwarzschild radius, or the technology to create a sphere of uranium with a mass comparable to that of a star, so this is deep into idle-curiosity-land, but supposing that we _"called into existence"_ a sphere of uranium whose mass and density was sufficient for both the effects of nuclear fission and gravitational collapse to be relevant, > does the sphere of uranium **explode** due to nuclear fission, or **collapse > into a black hole** , first? Of course, perhaps there is a combination of effects - maybe the sphere starts to undergo fission, but forms a black hole before any explosion can take place.


title: The Sun Can Make Stuff Hotter Than Itself

text: We intuitively understand the direction that energy travels -- from the thing with energy to the thing with less energy. That's why the second law of thermodynamics is among the first things you learn in science class that makes you say, "Well, I could have told you that." If you're too hot, you move away from the campfire, not toward it. You don't need science to tell you that heat energy travels from the hot thing to the less-hot thing. Well, everywhere in the universe except the sun. There's a discrepancy between what science says should happen and what the sun actually does, and it's known as the sun's coronal heating problem. Essentially, when heat leaves the sun, the laws of thermodynamics just totally break down for a few hundred miles, and nobody can quite figure out why. The facts are pretty straightforward; the sun's surface sits comfortably at a blazing temperature of roughly 5,500 degrees Celsius. No problem there. However, as the heat travels from the sun's surface to the layer a few hundred miles away from its surface (known as the sun's corona), it rises to a temperature of 1,000,000 degrees Celsius. Which is 995,000 degrees Celsius, or 1,791,000 degrees Fahrenheit, or 1 billion gigawatts per 1/4 gigabyte jiggawatt hour (metric) hotter than it has any right to be. He's a loose cannon! The heat source (the giant ball of nuclear explosions and plasma) should be the hottest thing, not the empty vacuum of space around it. This is the only instance in the known universe where the thing doing the heating is actually cooler than the thing it's heating. And it's been plaguing solar physicists worldwide since they discovered the little disagreement reality has with our universe in 1939. How is it possible that the area around the sun is about 200 times hotter than its surface?


title: Can you give example of some problems with solutions in each of Newtonian, Lagrangian and Hamiltonian method?

text: I am a student from information system and just want to know about classical mechanics. I know Newtonian mechanics from high school and I have read about Lagrangian and Hamiltonian mechanics in intuitive manner. Can you use an example to outline the basic characteristics (and key differences) of Newtonian, Lagrangian and Hamiltonian methods in their approach to classical mechanics problems?


title: Rope wrapped around a cylinder

text: If a rope is wrapped aound a cylinder what is the relationship between the amount of wrap and the ability to resist slipping off of the cylinder? An example would be if I had a 180 degree wrap and attached a 100 pound weight on one end of the rope, what force would be required to hold the other end? What if the rope wrapped 360 degrees around the cylinder and so on?


title: Am I talking about the correct units here?

text: Paolo made a move I knew from dozens of townie soccer games played under lights on the fenced-in macadam pitch next to the elementary school in Badalucco, packed tight between Via Cristoforo Colombo and the bed of the Argentina River. Paolo twisted up his left leg; rotated that leg in an entirely autonomous and anatomically-impossible split-second burst of acetylcholine; power-rammed, with his left foot, the front bumper of our white cheese-tour van. A van, it now became clear, that Paolo had left precariously in neutral. With the parking brake disengaged. Tilted up along the road’s seriously steep grade. Its rear wheels inches from the verge, a sharp drop to the curve below. His cousin’s van, I remembered, as it lurched backward and gathered speed. Here’s the thing, often under-appreciated, about translational kinetic energy:   ET=(1/2)mv2 I’m quite certain that Hugo failed to fully apprehend the ramifications of this formula, given our weighty cheese van and his own position vis-à-vis an initially lumbering, but definitely accelerating, object. Perhaps Hugo had forgotten his boyhood physics class with its repeated exhortation that v = v0 + at.   Gravity, Hugo, gravity. 9.8 m/s² Hugo. But, most important: mass, Hugo. Mass. The van was heavy, no doubt about it, all by itself. Then you add all that luggage. Plus one 100 pound wheel of Parmigiano Reggiano. Plus Jack’s, or Lauren’s, or whoever’s six boxes of Canopic Cookie Jars. Plus whatever illicit items the Arbogasts had inserted into their collection. Plus nearly two weeks of accumulated detritus dumped on the floor by people who’ve spent way too many van-hours for the niceties of proper trash disposal to hold much sway. Plus Lauren. Maybe it was Lauren of whom Hugo was thinking when he jumped not away from but toward the rapidly-accelerating van, its front passenger door still gaping wide; a van that had by then about reached its tipping-point on the steep-shouldered roadside. Maybe it was Lauren. Or maybe Hugo had played linebacker for Texas A & M, back in the day, and still figured he was on top of his game, good to go, stop anything coming at him. But this was no 250 pound running back, Hugo. This was not even a steroidally-enhanced 350 pound running back. No. This was a stout door winged out from a 3000 kilogram white Fiat Ducato made weightier by its 20 gallon auxiliary tank full of used grease. So, let’s say three seconds elapsed between the moment when Paolo’s kick overcame the van’s inertia (and started ‘er rolling) and the instant when Hugo attempted to stop her. For ease of calculation, let’s put the inclined plane on which the Ducato rested at 45 degrees, i.e., pretty steep, and thus the parallel vector of gravity pulling her downhill at about 4.9 meters per second per second. After 3 seconds and change (we’ll ignore the rolling friction of axels and rubber tires) the Ducato would be chugging along at close to 15 meters per second. I think. Over 30 mph. Not so shabby. Which brings us back to ET=(1/2)mv2. 3000 kg at 15 meters per second. At the instant of contact, a direct hit would work on a firmly-planted Hugo with something in the vicinity of 337,500 Joules. Fortunately for Hugo, he was not planted, immovable, like the high-security bollard he believed himself to be, but more along the lines of a wobbly bowling pin. The door dealt Hugo a split-second glancing blow. What we probably need to figure here, as in football, is impact force per unit area. Newtons/m2. Stress. The specifics of this inelastic collision. Consult the law of cosines, maybe? You do the math, Hugo. Hugo was not going to be doing any math for quite a while. More or less simultaneously: I saw the open door plow into a Hugo determined to impede its progress. I heard the loudest noise I’d ever experienced. I saw the windshield explode in frost. I saw Paolo drop from sight. I heard Lauren scream. Then: I saw Hugo go down. I remembered Phil and Massimo. Under the backseat. Gliding toward oblivion.   Which must be when I made my own ill-advised lunge toward the driver’s seat, into the path of the open driver’s side door. A split-second later, as Pippa howled “Sybilla, don’t!” must be when I stumbled over a loose piece of rubble and myself went down. I’m told the bottom of the door just grazed my face. But, as noted, at 15 meters per second and rising, even a graze by a three ton vehicle can hit you with lots of Ns.


title: Homework - Mechanics

text: Just started mechanics and I'm not sure what I'm doing wrong on a question. > _Car accelerates uniformly from a speed of $50\: \mathrm{kmh}^{-1}$ to a > speed of $80\: \mathrm{kmh}^{-1}$ in $20$ seconds._ > > _A) calculate the acceleration of the car._ I have tried subbing into a SUVAT equation but might have made a mistake. $$U = 50\\\ V = 80\\\ T = 20\\\ A = ?$$ $$v = u + at\\\ 80 = 50 + 20t\\\ 30 = 20t\\\ t = 1.5$$ Which is wrong. Thanks for any help. Answer is in $\mathrm{m/s}^-2$


title: Where does this formula for prediction of a multiple wave come from?

text: On the slide I am reading, it says this: > Any surface related multiple can be construced of primaries: > > $\hspace{5cm}$Multiple = Primary$_1$ (green) + Primary $_2$ (red) ![enter image description here](http://i.stack.imgur.com/i2XtH.png) > $$T_2 = T_1 + \left( \frac{1}{v} \frac{2h}{\cos (\theta)} \right)$$ > > where > > $$\cos(\theta) = \frac{h}{\sqrt{\frac{x^2}{4} + h^2}}.$$ I think $v$ is supposed to represent the velocity in between the layers of the rocks. My question is where has that $\cos$ bit come from? Why does it equal that? Also, am I correct in saying that what this equation says is that the time it takes for our second primary to complete is the same as the time it takes the first primary to complete, plus that little bit on the end? This is all the information on the slide I am reading so sorry there isn't much information.


title: What is the difference between realism in locality, and counterfactual definiteness?

text: I understand the EPR-experiment and the Bell inequalities. I can see how dropping 'locality' solves the issue, and how dropping 'realism' solves the issue (e.g. there are really no hidden variables until you measure them) But the third option, counterfactual definiteness, I don't understand. What is the difference between counterfactual definiteness and realism exactly?


title: Why photoelectron imaging is a 'complete' measurement?

text: In many articles and books, it says that photoelectron imaging gives a 'complete' information. What is mean by 'complete' measurement or a 'complete' information? Through photoelectron imaging (recording the angular distribution of photoelectrons) one can obtain Photoelectron Kinetic energy, detachment cross section and the photoelectron angular distribution. I am not clear how these three forms a complete measurement!


title: Dark Energy and Conservation of Momentum

text: IF dark energy was not perfectly evenly distributed(as some theories suggest), in other words if some cubic centimeter had a very small difference from another cubic centimeter, would that imply that spatial translation is not invariant, and hence that momentum is not conserved? Or would the laws of physics be the same despite the tiny difference and thus momentum would still be conserved?


title: Are there general circuits that differentiate/integrate empirically?

text: Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?


Pretty much letting my mind free-wheel. Assume a fleet of air-supported hover-craft were to replace cars/etc on the streets. Assume also that the present traffic-signals/pedestrian rules remain unchanged. As I understand hover-craft come to a gradual stop; similar to a train. What would be the equivalent of disc-brakes on hover-craft? i.e. How would you improve braking capability on a hovercraft? p.s. Friction, Aerodynamics, Inertia hence the post here in the Physics forum but please feel free to vote as OT ...


title: How are possible microstates discerned in Gibb's entropy formula?

text: On the entry of Gibb's entropy formula on Wikipedia, the following definition is given: _"The macroscopic state of the system is defined by a distribution on the microstates that are accessible to a system in the course of its thermal fluctuations."_ (I will assume that this description is correct. If not, this question does not apply.) I'm having a hard time grasping exactly what this description tries to say. I initially thought that the distribution of microstates was given by which microstates that could instantiate the macrostate in question. However, both _"accessible"_ and _"in the course of its thermal fluctuations"_ makes me wonder about if this interpretation is correct. When reading the description above, it seems to me that you should start of with a certain microstate that instantiate the macrostate and then see how many states that are accessible, that is, that could be the case within the time course of some time unit defined by the thermal fluctuation...Ehh....I might have lost myself there. Could somebody deconstruct the description cited above?


title: How to measure resistance of a piece of wire?

text: My son is doing a science experiment on how varying temperature and diameter of wire impacts the resistance. We are assuming we can accomplish this by using different gauge wires, a home thermometer, and a basic digital multimeter (e.g. this one at amazon). Is this correct, or is the multimeter only used to measure resistance of batteries and circuits and not a plain piece of wire? If this setup is not sufficient, can you suggest an alternative? Regarding temperature, will a simple home thermometer (battery, not mercury) be sufficient?


title: Can one see radioactive substances with an X-ray detector?

text: I was wondering the other day an X-ray detector (like the ones used at airports) can detect gamma-rays lets say from a sample of uranium. I know its all electro-magnetic waves but I'm really unsure about how it works in practical terms. **Edit** Now out of curiousity, if you got an beta-radiator, might that go unnoticed?


title: Spin of an electron

text: I have a conceptual difficulty in understanding the electron spin. On the one hand, it is an experimental, observable feature of electrons. The problem is in understanding to what it belongs - to a bare electron or to an electron already coupled to the electromagnetic field. I think it is the latter. In other words, the electron spin characterizes a complicated object (a real system) rather than a "free" (non interacting) electron. It is like the total spin of a compound system. On the other hand, in theory we ascribe it to a "bare" electron solely. Do you think we have a conceptual problem here?


title: Do Neveu-Schwarz conditions make sense?

text: When putting fermions on the string, we have to choose boundary conditions for our spinor fields - Ramond or Neveu-Schwarz. NS conditions on the closed string have antiperiodic conditions such as $\psi_L(\sigma) = -\psi_L(\sigma+\pi)$ (where $\pi$ is the periodicity of the string). I'm confused by this - shouldn't $\psi$ be single-valued?


title: Mass of a compressed spring

text: I was wondering if:   1. The mass of a compressed spring is greater than that of an uncompressed spring    2. The mass of a body on the surface of the Moon is greater than that on Earth according to the equation $E= mc^2$


title: The things to consider when choosing a mesh for an electron detector (MCP detector)

text: I am designing a electron velocity map imaging setup. I need a copper mesh just infront of the detector to accelerate the electrons towards the MCP detector. I am doing the instrument design for the first time. Can anybody please tell me how to choose a mesh and what are the things should I keep in mind while choosing a mesh for the detector (regarding the number of wires per inch, its transmission etc.)


title: Is there a minimum mass is required for light to be noticeably bent?

text: The sun bends the trajectory of light slightly. And a black hole will bend the trajectory entirely. This is all dependent on the proximity to the source of gravity. For a given angle, is there some minimum mass required to bend light light at least that much?


title: Is it possible to rotate an angle on perfectly smooth ice?

text: You must start and end with the same pose. Prove it if you think you can't. ![enter image description here](http://i.stack.imgur.com/6CVJf.jpg)


title: Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

text: **Background:**   * Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$   * Starting from $F = ma$, integrating with respect to distance, and substituting $a\ ds = v\ dv$ (from calculus), one can derive $\int Fdx = KE_f - KE_i$ (Obviously I care about these results because, when combined with $F_{ab} = -F_{ba}$, they give conservation of momentum and a starting point for conservation of energy, but for this question I'm not going to consider this part of the derivation.) In 1-D, the two bulleted results make sense to me: the starting point is a 2nd-order ODE, and the two results form a coupled system of two 1st-order ODEs. This is precisely what math says should be possible - the number of mathematical constraints has been conserved. **Question:** A similar (slightly more involved) derivation is possible in 3 dimensions, but it's harder for me to classify the resulting mathematical constraints and reassure myself that the transformation doesn't add or remove constraints:   * Starting Point: $\vec{F}=m\ddot{\vec{s}}\ \rightarrow$ 3 coupled second-order ODES    * Result 1: $\int \vec{F}dt = m(\dot{\vec{s}_f} -\dot{\vec{s}_i})\ \rightarrow$ 3 coupled first-order ODEs (viewing initial state as a boundary condition)   * Result 2: $\int \vec{F}\cdot d\vec{s} = \frac{1}{2}m(\dot{\vec{s}_f} \cdot \dot{\vec{s}_f}) - KE_i\ \rightarrow $ Unsure how to classify this      * Rewriting this as $\int \vec{F}(x,y,z)\cdot d\vec{s} = \frac{1}{2}m(\dot{x}_f^2 +\dot{y}_f^2 +\dot{z}_f^2 ) - KE_i $ makes it clearer that this is a single non-linear ODE with the first derivatives of 3 different dependent variables in it.  Math says that a system of 3 coupled second-order ODEs can be rewritten as a system of 6 coupled first-order ODEs, but I obviously only have 4 equations. **What type of ODE is Result 2?** A single first-order ODE? A 'triple' first- order ODE with 3 dependent variables? I'm ultimately looking to reassure myself that commuting the problem 'solve $F=ma$' to the problem 'solve conservation of momentum and conservation of energy' doesn't add or remove mathematical constraints. If anyone can refer me to a textbook which addresses this idea, I'd appreciate that as well. **Edit:** On further consideration, I think I've made a mistake. $F=ma$ is a differential equation with time as the independent variable, so integrating with respect to time (as is necessary to derive conservation of momentum) changes the order, and it's not actually reasonable to expect two coupled first-order differential equations as a result any more. In fact, we could integrate Conservation of Momentum wrt $t$ again, and then we'd no longer have a differential equation at all. Since integrating is a valid technique for solving differential equations, it now kind of appears as though Conservation of Momentum could be viewed as a [/the] solution to $F=ma$, and could thus be expected to describes all the information which is present in the original ODE.   * That would make the $F=ma$-to-Conservation of Momentum derivation logical (from a Conservation of Constraints perspective) in both 1-D and 3-D, and then make the real question 'Where Conservation of Energy come from, from a constraints perspective?'   * On the other hand, the 1-D elastic collision problem can't be solved with Conservation of Momentum alone - Conservation of Energy is also required. Since Conservation of Energy can be derived from $F=ma$ and calculus alone, it must be representing a constraint which is in the original equation. Thus $F=ma$ holds more information than Conservation of Momentum. **Other things I've been thinking about:**   * This question may actually relate to integro-differential equations, which I know nothing about.    * There may be some subtleties related to $F$ which I'm missing, and writing it as $\vec{F}$ instead of $\vec{F}(\vec{s},\dot{\vec{s}},t)$ may be sweeping them under the carpet.


title: Relativistic charge density in a closed loop

text: When charges of conductance are at rest, there is an average distance between them. The relativistic origin of magnetic field says that distances between electrons shrink when they are set into a motion (we get current). This means that the electron density is increased while total charge is conserved. I just try to relate the contracted distance between electrons with the conductor, which stands still and does not contract. If we had infinite-long wire, we could borrow any amount of charge from the edges of Hotel Infinity to increase the electron density without increasing it. But, real wires are of limited length. So, condensing charge `-*--*--*--*--*--*-` => `-----*-*-*-*-*---` cannot make one segment more dense without depleting density in the other parts. I cannot make the whole wire more charged (and keep it neutral in the lab frame, at that). The same applies to the loop.                * - * - * - *            |           |      * * * *      *           * ==>  *     *      |           |      * * * *      * - * - * - *             Here alternative physics draw a nicer picture to ask the same question ![enter image description here](http://i.stack.imgur.com/XTq96.gif) The image says that **the loop of electrons shrinks** as they start to flow. Yet, mechanically, electrons do not escape the solid wires, which stand still and do not contract! That is the question: how do inter-electron distances shrink without leading to logical absurd? This question stems from understanding why loop with current stays neutral, despite the electron density increase. I first guessed that the conductor stays neutral because positive charges flow symmetrically in opposite direction so that relativistic charge density increase in both directions compensate each other and keep the loop neutral. In this case the conducting loop would contract proportionally with the loop of electrons but it would not stay at rest in this case. Yet, the problem is that the loop of wire stays at rest and does not contract.


title: Relativistic origin of magnetic field

text: There is an explanation in the Wikipedia. Unfortunately the article is quite verbose and doesn't clearly explain why both positive and negative charges vary density even if only one is moving. It is also not possible to figure out with respect to which frame of reference they shrink/expand. Can this be made more clear or is there a source that already does so? Actually, I resorted to wikipedia after Feynmann because he said something like > model of a wire with a current of positive charges, separated by an average > distance, L. The wire has to be electrically neutral in the lab frame, so > there must be a bunch of negative charges, at rest, separated by the same > average distance. Therefore there's no electrostatic force on a test charge > Q outside the wire. I have got this quotation from the last answer. I do not understand why nobody, who discusses the relativistic density growth, cannot see the problem. How, after saying that   1. motion increases density and   2. positive charges are moving can they say that we have negative charges, standing still in the same frame, having the same density? I see that when positive charges are stopped, their density drops and we'll get the negative charge prevailing. No current = object is charged negatively! Do you see that? This means that all objects in the Universe must be negatively charged in order to be neutral when positive charge starts moving in them. But, we know that objects are normally neutral. So, when current appears in the loop, you must first explain where the extra charge is coming from and why the loop remains neutral. But, teachers do not do that. They have a conspiracy to avoid discussing this simplest case and jump immediately to the case of test charge moving at speed of current charges. So, is it right that all objects in your Universe are electrically charged when no current circulates in them?


title: Hydrogen wave function in momentum space

text: We can seperate the wave function of an hydrogen atom in a radial and an angle part: $$ \phi_{n,l,m} (\mathbf{r}) = R_{n,l,m}(r) Y_{l,m}(\vartheta,\varphi) \, , $$ where $Y_{l,m}$ are the spherical harmonics.   My question is: How does this look like in momentum space? Is the general form preserved? Do we get as well a radial and an angle dependent part?


title: The weight of a cavity of radiation

text: A sealed cavity full of light weighs more than an empty one. So consider a cavity consisting of parallel mirrors, with a pair of beams bouncing between the mirrors. The question is whether that weight will change if we arrange for the beams to completely cancel one another at every point along their length?


title: what compactifications of the Poincare group have been studied?

text: as we know the Poincare group is non-compact. Poincare invariance have been observed in velocities and energies up to $10^{20}$ eV in cosmic rays. The other day i was thinking in how $SU(2)$ homeomorphism in $SO(3)$ imposes a double cover, and i keep wondering if something like that could exist in the Poincare group, but of course the main problem is that the group is not compact. I wonder if it is possible at all to make a compactification of the Group that is consistent with low-energy physics and still preserves some form of isotropy of space-time. For instance, i considered indentifying the different connected components (either CP or PT inverted) of the group at some boundary consistent with energies of the order of $10^{28}$ eV, but with meaningful dimensional analysis, but have not succeeded analysising the symmetry properties of the resulting manifold and the algebraic properties of it (it is still a Lie group after such identification?) The physical interpretation of such identification is up to discussion, but i think that it would basically stand for a duality that maps _continuously_ (in the concrete example compactification i gave) particles with energies above $E_p$ (some abritrary boundary energy) with particles with energy below $E_p$ and $P$ or $CP$ reversed. This latter would make for instance, electric charge conservation an approximate symmetry. Has something like this been attempted? or are there good reasons known why this could not work?


title: Young's Double Slit experiment question

text: > _Q-A beam of light consisting of two wavelenghts 600 nm and 450 nm is used > to obtain interference in Young's Double Slit experiment (YDSE). Find the > least distance from the central maximum where the bright fringes due to both > wavelengths coincide. The distance between the two slits is 0.4 mm and > screen is placed at a distance of 1.0 m from the slits._ Now my problem is not with the question but the solution given in my book. It says > _Let at distance $x$ from central bright maxima, the bright fringes due to > both the wavelengths coincide for the first time. It is possible only if > within the distance $x$ there are $n$ fringes for light of 600nm and $(n+1)$ > fringes for light of wavelength 450 nm._ Okay so my problem is that I am unable to figure out how have they made this argument about $n$ and $n+1$ fringes? It's isn't mention anywhere in the question. Can't it be ($n$+any number)? If there's any better solution than this I would be really thankful for that


title: calculating focal length of meniscus lens

text: As I read about telescope, distance between objective lens and eyepiece must be equal to addition of their focal lengths.               D = F1 + F2      I used one of the eyepiece of 2cm focal length to calculate focal length of a magnifying glass . I seen clear distance view when I kept eyepiece 36cm far from magnifying glass. So it's focal length should be 34cm as per above formula. I purchased a meniscus lens of -3 power which is used in specs to correct distance view. I kept it approx 4cm far from magnifying lens for clear distance view and concluded that its focal length should be -30cm. I confirmed it with by using magnifying glasses of various focal lengths.               D = F1 + (-F2)      **Problem:** Now I tried the same experiment with a specs of -.25 power. I guess its focal length should be -2.5cm as I calculated for above mentioned meniscus lens. But I dint seen any difference in magnification or clear view when I used this specs with any lens. What's wrong I am doing? How can I confirm its focal length?


title: Electric field singularities

text: Is this the list of all possible types of singularities in electrostatic field $E$?   1. near a point charge: $\frac{1}{r^2}$   2. near a line of charge: $\frac{1}{r}$   3. near the edge (not surface) of a uniformly charged sheet: $-\log(r)$ (... and no singularities for finite charges smoothly spread over a finite volume.)


title: AC Circuit Theory Maximum Power Transfer

text: I am currently hitting a barrier with regards to the basics of this question. A 50HZ supply is connected to various given impedance's, calculate the maximum power transfer. Maximum power transfer I understand is basically the Thevenin/Norton equivalent. Though THERE IS NO EMF indicated in the question. Is there a method around this problem (Calculate the Norton/Thevenin with only Impedance values)? Any help would be superbly fantastic


title: Does a guitar sound different in zero (or micro) gravity?

text: Seeing a video of astronaut Chris Hadfield playing a guitar on the International Space Station made me wonder if a guitar or other stringed instrument played in zero-G would sound any different than on earth. It seems that on earth, the downward pull of gravity could cause an asymmetrical oscillation of the guitar string, with a larger amplitude as the string moves downward due to the pull of gravity. (of course, it wouldn't take zero-G to test this since a guitar could be held vertically) With the amount of tension that a guitar string is under, perhaps this effect is so minuscule so as to be unnoticeable?


title: Physics of a cold and hot top

text: Imagine two tops made up of exactly one thousand atoms. One is kept at 4 degrees Kelvin, the other at room temperature.   1\. Would they weigh the same given an arbitrarily precise scale in the Earth's gravitational field?   2\. If both were spun at exactly 100 rpm, would they have the same moment of inertia? Here is my own initial idea. Gravitational forces bind to energy density. As such, the hotter top would weigh more on a scale because heat contributes to the energy density of the top. The rest mass of a thousand atoms is the same no matter what its temperature. As such, the cold and hot top would have the same moment of inertia.


title: If there were two equal masses with one mass in the middle which way would gravity pull it?

text: If there were two masses with a third exactly in the middle of those two, which way would the middle mass be pulled?


title: Charging a metallic sphere positive by induction

text: > Bring a negatively charged rod close to a metallic sphere placed on a > insulated stand. As the rod is brought close to the sphere, the free > electrons in the sphere move away due to repulsion and start piling up at > the farther end. The near end becomes positively charged due to deficit of > electrons. This process of charge stops when the net force on the free > electrons inside the metal is zero. > > _Connect the sphere to the ground while the positive charges at the near end > will remain held there due to the attractive forces of the negative charges > on the rod._ Disconnect the sphere from the ground. The positive charge > continues to be held at the near end. Remove the electrified rod. The > positive charge will spread uniformly over the sphere. Is the above statements true? If true why is it true? I want a proof like "The potential of the sphere is greater than zero, hence there will be a flow of charge." If false what is the correct statement?


title: What does it mean that there is no mathematical proof for confinement?

text: I see this all the time* that there still doesn't exist a mathematical proof for confinement. What does this really mean and how would a sketch of a proof look like? What I mean by that second question is: what are the steps one needs to prove in order to "mathematically prove confinement". * * * *See e.g. Scherer's "Introduction to Chiral Perturbation Theory" middle of page 7


title: metric tensor of expanding universe

text: Why is the metric tensor of a expanding universe a function of time? Why is it not a function of distance between the galaxies? I heard this from a lecture. Can anyone help me understand?