title: Is this constraint non-holonomic?

text: I am working on a variational problem involving elastic stability of a beam. The deformation of the beam is given by six functions of the material coordinate along the beams longitudinal axis. The governing functional looks like $$ \int_{s_1}^{s_2} F \left( u(s),v(s),w(s),\phi(s),\chi(s),\psi(s) \right) ds. $$ (It represents the beam's strain energy.) Now I want to impose the condition that transverse shear deformation and elongation are zero. The consequence is that the 6 degrees of freedom of the beam are reduced to essentially 3. It can be shown that the three relations $$ \frac{du}{ds} = \sin(\chi), \quad \frac{dv}{ds} = -\sin(\phi)\cos(\chi), \quad \frac{dw}{ds} = \cos(\phi)\cos(\chi) -1 \qquad (*) $$ between the coordinates are valid in case transverse shear deformation and elongation are neglected. My understanding is that there are now two ways to incorporate this constraint condition:   1. eliminate variables in the original functional by substitution    2. include equations (*) by means of Lagrange multipliers I would like to use the latter method. The book _The variational principles of mechanics_ by Lanczos explains this for both holonomic and non-holonomic constraints (and isoperimetric ones as well). Though I can see that equations (*) are clearly not some algebraic relation $$ f_i(u,v,w,\phi,\chi,\psi)=0, \quad i=1\dots3 $$ between the coordinates, does this means the constraint qualifies as non-holonomic? I am confused because it seems integrable. Or does it then become an isoperimetric constraint? ## Edit I am getting more confused; can we only call constraints non-holonomic if derivatives with respect to _time_ are involved? (Meaning that all equations (*) above just represent holonomic constraints, since it is all static?) In _The enigma of nonholonomic constraints_ by Flannery (2005), it is explained that for the typical action $$ S = \int_{t_1}^{t_2} L({q},\dot{q},t) dt $$ the Lagrange multiplier rule can only be used for holonomic and semi- holonomic (exact linear) conditions. Here, the independent variable clearly is time, but in my case it is some spatial coordinate. If we assume that the independent variables space and time are interchangeable, my (*) equations appear non-holonomic (so, not to be incorporated with Lagrange multipliers), as they cannot be expressed as exact differential.. But, is this correct reasoning?


title: What accelerates the electrons in the thin part of a current-carrying wire?

text: Consider a thick wire that narrows for some part of its length, connected to the terminals of a battery, hence carrying some current. Presumably the electrons have to speed up as the wire gets thinner - if the current is constant everywhere, then if the wire cross-section is less, they have to go through faster in order for the same number of electrons to pass any point in a given time. What accelerates them as the wire narrows? And what decelerates them again as the wire widens out again after the narrow bit?


title: Kepler's third law doesn't give earth's orbital period! Why?

text: I tried to calculate earth's orbital period using Kepler's third law, but I found 365.2075 days for the orbital period instead of 365.256363004 which is the correct value. I checked everything, and I couldn't find what's the problem. I used these values for my calculation:   * Semi-major axis, a: 149,598,261 km   * Gravitational constant, G: 6.67*10-11 N·(m/kg)2   * Solar mass, M: 1.9891*1030 kg


title: semiclassical approximation for the exponential sum (partition function)

text: for any real and positive 's' and in the sense of semiclassical approximation is this valid ?? $ \sum_{n}exp(-sE_{n})\sim \iint_{C}dxdpe^{-s(p^{2}+V(x))}$ valid for every 's' here simply both the sum and integral are convergent, so in this case the approximation can be seen as the replacement of a sum by an integral.


title: A person weighing 100 N stands on some bathroom scales in a lift. If the scales show a reading of 110 N, which way is he going

text: Can someone answer and explain it please. Here are the options:   * Moving downwards and decelerating.   * Moving downwards with a constant velocity.   * Moving upwards and decelerating.   * Moving upwards with a constant velocity.


title: Again about all-win lottery

text: I suggest the following thought experiment that describes a machine which makes everybody happy. Suppose a lottery is conducted. The winner is awarded a billion dollars plus the title of eternal Earth dictator. The winner is determined in the following way.   * There is a thorough isolated black box inside which a pretender is placed.   * Inside the box a quantum experiment is conducted with equal probability of the two outcomes: 0 or 1 (it may be based on an atom decay)   * The pretender measures the result with an apparatus.    * The black box is connected by a quantum link (in form of an optical fiber) to an external quantum computer   * After the measurement by the pretender is done the result is automatically transmitted by the optical link to the quantum computer   * The transmitted qubit is used by quantum computer as an initial state of the quantum register instead of usual Hadamard transform before performing a Shor factorization algorithm.   * After the algorithm is performed, the result is checked whether it is correct.   * If it is correct the pretender is declared the looser. If the factorization returned a wrong result, the pretender is declared the winner. What happens. Any average person placed in the black box will measure the result of the atom decay and similarly to a Schroedinger cat will enter the superposition of two states: the one which saw 0 and the one which saw 1. Similarly the measuring apparatus will enter the superposition. So the qubit sent via the optical fiber will be a superposition of two states: $$\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$$ As this value is exactly the same what Hadamard gate returns after being applied to an empty qubit, this will allow the quantum computer to perform the factorization as usual and return the correct result, dispite that the initial step of applying the Hadamard gate has been replaced by loading the qubit obtained from the black box. Now suppose the observer himself is placed in the black box. After the atom decay he will perform measurement which will collapse the wave function. The observer will be in either state where he saw 0 or where he saw 1. So either basis state 0 or basis state 1 is sent via the cable to the quantum computer. Since this is different from the usual value which should be loaded in the register before performing the factorization, the factorization will return a wrong result. So the pretender will be declared the winner. * * * It follows that any person, when he or she is observing the lottery from outside will see all other pretenders to loose. At the same time any participant that tries the lottery himself will find himself a winner. This is a fascinating lottery which will make all the participation pretenders the winners in their own eyes. Each person who goes through this machine will see other people to congratulate them, to proclaim them the eternal emperor and so on. I would like to see whether there any mistake in the reasoning.


title: How to calculate the energy freed in the reaction: $^{10}_5Be +\space ^2_1H \rightarrow \space^{11}_5B + \space ^1_1H$?

text: I have the following reaction: $^{10}_5\mathrm{Be} +\space ^2_1\mathrm{H} \rightarrow \space^{11}_5\mathrm{B} + \space ^1_1\mathrm{H}$ And I know that I have to use the formula: $E = \Delta m\cdot c^2 = \Delta m \cdot \frac{931,5MeV}{u}$. So I just need $\Delta m$ which is equal to: $\Delta m = m_b - m_a$ where $m_b$ represents the mass "before the reaction" and $m_a$ the mass "after the reaction" so we have:   $m_b = m(^{10}_5\mathrm{Be}) + m(^2_1\mathrm{H})$   $m_a = m(^{11}_5\mathrm{B}) + m(^1_1\mathrm{H})$ The book which contains this problem contains the following table: http://i.imgur.com/esoGDVf.png but from this table, I only know $ m(^1_1\mathrm{H})$ and $m(^2_1\mathrm{H})$ i.e.   $m_b = m(^{10}_5\mathrm{Be}) + 2.01410u$   $m_a = m(^{11}_5\mathrm{B}) + 1.00783u$ How do I calculate $m(^{10}_5\mathrm{Be})$ and $m(^{11}_5\mathrm{B})$ ? P.S. I don't know if the tag is correct. The chapter in the book where I found this exercise is called "Basics of nuclear physics".


title: A question about Feynman diagram and symmetry factor

text: Consider a $\varphi^3$ theory: $$ Z_1(J) \propto \exp\left[\frac{i}{6} Z_g g\int \mathrm{d}^4 x \left(\frac{1}{i}\frac{\delta}{\delta J}\right)^3\right] Z_0(J), $$ where $$ Z_0(J) = \exp\left[\frac{i}{2} \int \mathrm{d}^4 x \mathrm{d}^4 x' J(x)\Delta(x-x')J(x')\right]. $$ That is $$ Z_1(J) \propto \sum_{V=0}^\infty \frac{1}{V!}\left[\frac{i}{6} Z_g g \int \mathrm{d}^4 x \, \left(\frac{1}{i}\frac{\delta}{\delta J}\right)^3\right]^V \times \sum_{P=0}^\infty \frac{1}{P!}\left[\frac{i}{2} \int \mathrm{d}^4 y \, \,\mathrm{d}^4 z\, J(y)\Delta(y-z)J(z)\right]^P. $$ In particular, we can consider the term when $V=2, P=3$. Calculation shows that \begin{equation} \- i \frac{1}{2!}\frac{1}{3!} \frac{(Z_g g)^2}{6^2*2^3} \left[\int \mathrm{d}^4\, x_1 \, \mathrm{d}^4 x_2 \, \left(\frac{\delta}{\delta J(x_1)}\right)^3 \left(\frac{\delta}{\delta J(x_2)}\right)^3\right] \left[\int \mathrm{d}^4 y \, \mathrm{d}^4 z \, J(y)\Delta(y-z)J(z)\right]^3\\\= \- i \frac{1}{2!}\frac{1}{3!} \frac{(Z_g g)^2}{6^2*2^3} \int \mathrm{d}^4 x_1 \, \mathrm{d}^4 x_2 \, \left[3^3*2^4 \Delta(x_1-x_1)\Delta(x_1-x_2)\Delta(x_2-x_2) + \\\ 3^2\times 2^5\Delta(x_1-x_2)\Delta(x_1-x_2)\Delta(x_1-x_2)\right]\\\= -i(Z_g g)^2 \int \mathrm{d}^4 x_1 \, \mathrm{d}^4 x_2\, \times \left[\frac{1}{2^3} \Delta(x_1-x_1)\Delta(x_1-x_2)\Delta(x_2-x_2) + \\\ \frac{1}{2\times 3!}\Delta(x_1-x_2)\Delta(x_1-x_2)\Delta(x_1-x_2)\right], \end{equation} where $\Delta(x_1-x_1)\Delta(x_1-x_2)\Delta(x_2-x_2)$ and $\Delta(x_1-x_2)\Delta(x_1-x_2)\Delta(x_1-x_2)$ correspond to their Feynman diagram. Then the question is why $\frac{1}{2^3}$ and $\frac{1}{2\times 3!}$ are just the reciprocal of symmetry factors of the corresponding Feynman diagram respectively? In the general case of $V, P$, why the coefficients of the terms in the result of calculation are just the reciprocal of symmetry factors of the corresponding Feynman diagram respectively?


title: Description of the heat equation with an additional term

text: I have the following equation: $$\frac{\partial U}{\partial t}=k\frac{\partial^2 U}{\partial x^2}-v_{0}\frac{\partial U}{\partial x}, x>0$$ with initial conditions: $$U(0,t)=0$$ $$U(x,0)=f(x)$$ In the problem is requested to give an interpretation of each of the terms in the above equation, and noting that such systems can model, besides solving by Fourier Transform. The Fourier Transform solution is quite simple to do; however, I can not give a physical interpretation of the terms of the equation not to mention a system that can model it. So I wanted to ask your help to answer this question. Thank you very much for your help and attention.


title: Finding dimensional formula

text: $$y(x,t)=2A\sin(Kx)\cos(\omega t)$$ $A$ and $x$ are in metre, $\omega$ is angular frequency. Then find dimensions of $A$ and $K$. In this equation how can I find the dimension of $K$?


title: Cosmic Microwave Background (CMB) and its relation to Inertial Frames

text: We know that the CMB is isotropic when viewed outside of the spinning and revolving earth. Is it homogeneous? Can we relate the CMB to an inertial frame in the Newtonian sense (in which space and time are homogeneous and isotropic)? Or can it just provide an idea to build upon a new theory in which global (privileged) inertial frames exist? Einstein's General Theory of Relativity could be useful in an answer, but some outside-the-box thinking would be appreciated; feel free to question GR.


title: Elastic collision and spring

text: > Bodies $A$ and $B$ are moving in the same direction in a straight line with > a constant velocities on a frictionless surface. The mass and the velocity > of $A$ are $2 \text{kg}$ and $10 \text{m/s}$. The mass and the velocity of > $B$ are $6 \text{kg}$ and $4 \text{m/s}$. A spring is connected to the back > of the body $B$ and its rate is $800 \text{N/m}$. > >   * What is the velocity of $A$ relatively to $B$ before the collision and > after the collision? > > (Answer: $6 \text{m/s}$ and $-6 \text{m/s}$) > >   * What is the relative velocity between the objects when the spring > achieves its maximum contraction? > > (Answer: $0$) > > ![Illustration](http://i.stack.imgur.com/lB3mI.png) I had no problem with the first question. We know that in elastic collision the coefficient of restitution is $1$, therefore the relative velocities before and after the collision are equal but are opposite in sign. The relatve velocity before the collision is $10-4=6\text{m/s}$ . My trouble is with the second question. I though a lot about it and came to a contradiction in my thoughts (of course it is because I'm missing something): On the one hand, it seems to me that when the contraction of the spring is maximal, all the kinetic energy of $A$ was transformed to a potential elastic energy and to some portion of kinetic energy of $B$ (it will move faster). So it means that $A$ has no velocity at that moment, but $B$ has, so it's impossible that their relative velocity is zero. Intuitively, it means that $A$ has some kinetic energy. So it means that $A$ will be slowed down by the spring, until its velocity is zero, and the spring can finally return to its normal state. So it seems that the spring will be contracted maximally for some relatively short period of time until the kinetic energy of $A$ becomes zero. But something tells me that I'm wrong. Anyway, I've tried to play with the law of conservation of energy and to wrap it around my thoughts, but it didn't quite work well. So what am I missing? I will appreciate any help. * * * Proposed solution: **Relative speed during the maximal contraction** The moment $A$ reaches the spring, the spring will push both of the bodies as a reaction to the $A$'s push. Therefore, the $A$ will start to decelerate while $B$ will start to gain more speed. Although $A$ is losing its kinetic energy, its velocity won't be zero at the moment of the maximal contraction. This is because $A$ will continue to contract the spring as long as its speed is higher than the speed of $B$ (and therefore the speed of the spring itself). The moment their velocities are the same, $A$ will no longer be "overtaking" the spring, therefore its contraction will be maximal at that point. Therefore, the relative speed is zero. **Finding the maximal contraction:** The net momentum is conserved at any point in the time, so: $m_A v_A + m_B v_B = (m_A + m_B) V$ In this particular question, $V$ would be: $5.5 \text{m/s}$ We also know that the total mechanical energy is conserved (the collision is perfectly elastic), therefore, the mechanical energy before the collision (which only consists of kinetic energies) will be equal to the total mechanical energy during the max contract. Therefore: $\frac{1}{2} m_A v_A^2 + \frac{1}{2} m_B v_B^2 = \frac{1}{2} (m_A + m_B) V^2 + \frac{1}{2} k (\Delta x_{max})^2$ The energy when the contraction is maximal consists of kinetic energies of $A$ and $B$ (which are moving with the same velocity) and of elastic potential energy due to the fact that the spring is contracted. In this particular question $\Delta x_{max}$ would be: $\Delta x_{max} = \sqrt{\frac{27}{400}} \approx 26 \text{cm}$


title: Notational techniques for dealing with creation operators on Fock space

text: This question is trying to see if anyone has some simple notation (or tricks) for dealing with operators acting on coherent states in a Fock space. I use bosons for concreteness; what I'm interested in might not be applicable to fermions. If I have a multiparticle state defined by $$|\phi\rangle=|n_1 n_2 ...n_k\rangle=\frac{(a_1^\dagger)^{n_1}}{\sqrt{n_1 !}}\frac{(a_2^\dagger)^{n_2}}{\sqrt{n_2 !}}...\frac{(a_k^\dagger)^{n_k}}{\sqrt{n_k !}}|0\rangle$$ I can write this compactly as $$|\phi\rangle=\prod_{i=1}^{k}\frac{(a_i^\dagger)^{n_i}}{\sqrt{n_i !}}|0\rangle.$$ Now, I would like to act on this state with some operator consisting of creation and annihilation operators; this could be quite complicated, like $$a_ja_ka^\dagger_l.$$ Now, IF the above product was a sum, I could find the answer very easily using commutation relations and delta functions: \begin{align} a_ja_ka^\dagger_l|\phi\rangle&=a^\dagger_la_j\sum_i\frac{(\delta_{ki}+a^\dagger_{i}a_k)}{\sqrt{n_i!}}|0\rangle\\\ &=a^\dagger_la_j\left(\frac{1}{\sqrt{n_k!}}|0\rangle \right) \end{align} ...continue until you get it to normal form, and you are done. Of course, this is a different problem but I am just illustrating how neat this notation is. You can't just throw around delta functions in the product above, since they would make the entire product vanish. What I want is a clean way to denote "I have passed $a_k$ through all $i\neq k$, used the commutation relations to get $1+a^\dagger_ia_k$, and am now ready to hit this with another creation opertaor $a_l$." I think it's fairly clear what I am looking for; anyone have any notation or tricks to make calculations using arbitrary operators like the one I have above easier?


title: In the expansion of the scattered wave function, why do these two functions have the same index?

text: See Griffiths Quantum Mechanics, eq. 11.21. Evidently, $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m}^{\infty}C_{l,m}h_{l}(kr)Y_{l}^{m}(\theta,\phi).$$ But I don't see why the $l$th Hankel function necessarily needs to multiply the $l$th spherical harmonic. Seeing as we arrived at this equation through separation of variables, my feeling is that the equation should look something like: $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m,n}^{\infty}C_{l,m,n}h_{n}(kr)Y_{l}^{m}(\theta,\phi).$$


title: About unitarity and R-charge in 2+1 superconformal field theory

text: * How does unitarity require that every scalar operator in a $2+1$ SCFT will have to have a scaling dimension $\geq \frac{1}{2}$ ?    * Why is an operator with scaling dimension exactly equal to $\frac{1}{2}$ said to be "free (i.e decoupled from the rest of the theory)" ?    * Let us say that we know in some such theory the R-charge is monotonically decreasing with increasing coupling constant (say $\lambda$). Then define $\lambda _n ^f$ to be that value of the coupling constant at which the R-charge of the operator $Tr[\phi ^n]$ becomes $= \frac{1}{2n}$. Let $\lambda _n ^m$ be that that value of the coupling at which the R-charge of the same operator becomes $= \frac{2}{n}$ (..marginal ?..) Then clearly $\lambda _n ^m < \lambda_n ^f$.  How does this imply that there has to exist a $\lambda _c \leq \lambda_2^f$ where the theory might undergo a phase transition?


title: Cosmology and TOE

text: I have read an answer on this site regarding the change of laws over time . However a physisct told me that the laws did evolve at planck era and then stopped evolving after it , is that true even in case of a TOE ( namingly as the best candidate string theory )


title: Time dilation in a gravitational field and the equivalence principle

text: A clock near the surface of the earth will run slower than one on the top of the mountain. If the equivalence principal tells us that being at rest in a gravitational field is equivalent to being in an accelerated frame of reference in free space, shouldn't the clock near the earth run _increasingly_ slower than the other clock over time? If those two clocks are considered to be in two different frames of acceleration, the one near the earth will have a greater acceleration than the one on the mountain, and thus over time, their relative velocity will increase over time which will increase the time dilation. Or is this not a proper use of the equivalence principle?


title: Is length/distance a vector?

text: I have heard that area is a vector quantity in 3 dimensions, e.g. this Phys.SE post, what about the length/distance? Since area is the product of two lengths, does this mean that length is also a vector quantity, and why?


title: Graphene as optical and UV mirrors

text: One usually hears about graphene as a good thermal conductor, and good light absorber due to its tunable bandgap properties. But i haven't heard about its aplicability as an optical mirror. In fact, mostly the opposite is true: the optical transmissivity of monolayer graphene is very high ($\approx$ 98%) Since there are material engineering tricks to tune the separation between two or more graphene layers, i would expect that multi-layer destructive interference can be achieved with optical wavelengths, enhancing reflectivity I know that probably graphene is not the best material to do this, but allow me to insist. Why? Well, graphene mantains it's properties well over 3000K, and a couple of layers of it can be quite strong and very light. All these properties make it the ideal material for a laser-pushed sail. By the way, i'm asking about optical and UV frequencies, i know that monolayer graphene is reflective on the microwave region, and i'm not asking about that. Thanks


title: Composition of squeeze operators?

text: I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of the annihilation $a$ and creation $a^{\dagger}$ operators of some bosonic modes. I define the squeezing operator as $$ S\left(\zeta\right) = e^{\Sigma} \;\; ; \;\; \Sigma = \frac{\zeta^{\ast} aa -\zeta a^{\dagger}a^{\dagger}}{2}$$ for any complex parameter $\zeta$. **I would like to know the rule for $S\left(\zeta_{1}\right)\cdot S\left(\zeta_{2}\right)$ ?** For instance, we know that $$D\left(\alpha\right)\cdot D\left(\beta\right)=D\left(\alpha + \beta\right) e^{\left(\alpha \beta^{\ast}-\beta\alpha^{\ast} \right)/2}$$ for the coherent / displacement operator $D\left(\alpha\right)=e^{\Delta}$ with $\Delta = \alpha^{\ast} a - \alpha a^{\dagger}$. A reference for $S\left(\zeta_{1}\right)\cdot S\left(\zeta_{2}\right)$ would be sufficient.


title: Why does an electric motor burn up when you physically stop it?

text: As an electric motor spins, the energy from the electricity is 'conducted' to the rotor by the magnetic fields. However, when the motor is stopped, the energy becomes heat and burns up to motor. What causes this heat to be formed? Is it purely generated by the current flowing through the wire or is the magnetic field form the permanent magnets involved somehow? Conversely, why is there less heat when the motor is spinning? I'm hoping this is not just because the motor contains a fan.


title: What is the resolution to this apparent contradiction?

text: Momentum is defined as $$p = \gamma m_0 v$$ And here is another law $$E^2=(m_0c^2)^2+(pc)^2$$ And this website says the energy of a red photon is $1.9074 eV$. Also, light has a rest mass of $0$. The problem is that by the first equation implies momentum is $0$, and then the second equation would imply energy is $0$, and $0 \ne 1.9074 eV$. What is wrong?


title: Types of circular acceleration?

text: To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:   1. Angular acceleration : this is the rate of change of angular velocity.   2. Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration.   3. Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle. I also think there are two types of velocity:   1. Linear velocity: this is its velocity in the tangential direction and is constantly changing   2. Angular velocity or angular frequency: this is the rate of change of angle. Is the above correct? And have I missed anything?


title: Matter Waves Interference

text: When an EM wave diffracts, I can imagine that its EM field interacts with the charges in a certain obstacle thus inducing a wave behaviour on the charges of the matter that will interact with the EM of the photon. However, I am having difficulty to make the analogy with matter waves. Since a matter wave is a result of a momentum, which implies kinetic energy, how does a matter wave is created in order to interfere with an electron with, for instance, 200 keV? Since a single electron can be diffracted, this means that its wave must interfere with other induced matter waves, right? But wouldn't that imply the creation of high kinetic energy particles?


title: Noether Charge Operators In Electroweak Theory

text: 1. In electroweak theory, for the first generation of leptons, in the doublet $$\psi_L=\begin{pmatrix} \nu_{eL}\\\ e_L \end{pmatrix}$$ we assign a non-abelian charge $I=\frac{1}{2}$. Is this a Noether charge?    2. A $SU(2)$ transformation on the Lagrangian: $$\mathcal{L}^{ferm}=i\bar\psi_R\gamma^\mu D_\mu\psi_R+i\bar\psi_L\gamma^\mu D_\mu\psi_L$$ leads to this conservation of this charge. Right?    3. When that corresponding charge operator acts on the one-particle (or multi-particle states) it gives us the corresponding charge. Right?    4. The weak hyper charge $Y$ appears in the $U(1)$ transformation: $e^{\beta(x) Y/2}$ and $I$ appears in the $SU(2)$ transformation $e^{i\vec\alpha(x)\cdot\vec I}$? One can define, Noether charge (operator) for both these transformations, for $U(1)$ case it will be related to $Y$ and for $SU(2)$ case it will be related to $I$. But do we have one-particle and many-particle states here?    5. On what states will these Noether charges act on?


title: Differentiation and delta function

text: Need help doing this simple differentiation. Consider 4 d Euclidean(or Minkowskian) spacetime. \begin{equation} \partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ? \end{equation} where $a_\mu$ is a constant vector and the indices are summed over since one really doesn't need to bother about upper and lower indices in flat space. Also $(a-x)^2=(a-x)_\mu(a-x)_\mu$. A simple minded calculation gives me the result $0$. But I think the answer may contain Dirac Delta function like the relation below in 3 dimension \begin{equation} {\bf{\nabla}} . \frac{\hat{r}}{r^2}=4\pi \delta ^3(r) \end{equation} A corollary question. What will be $\partial_{\mu}\frac{1}{(a-x)^2}$ ? Is it $\frac{2(a-x)_\mu}{(a-x)^4}$ or does it also involve the Delta function?


title: Bottom magnetic levitation

text: There are many decorative devices, that keep a small metal sphere levitating in air by attracting it to a top electromagnetic coil while monitoring their height by optical sensors. There is also the 'levitron' where a passive permanent magnet spinning top is hoovering above some permanent magnets, it's orientation is controlled by gyro forces. And there are bottom based electromagnetic devices, which keep a disk hoovering in air above them by rotating magnetic fields. Like this one: http://usables24.de/shop/article_GL02.001/Gravity-Lifter-Mini.html On the first look, it seems simple to exchange the rotation of the top by the rotation of the field. But: A rotating field must employ the orientation and position control by some electrodynamic means I guess, as the disk is spinning quite slow in these devices, and thus is not stabilized by gyro forces. What mechanisms are behind this? Maybe the 'levitron' is stabilzed by magnetic forces too, not by the gyro force alone?


title: Infinite potential well with barrier in the middle- symmetric

text: So I'm having problems with the double infinite potential well given by $$V(x)= \left\\{\begin{array}{ll} \infty & -\infty < x < -a-b \\\ 0 & -a-b< x < -a \\\ V_0 & -a < x < a \\\ 0 & a < x < a + b \\\ \infty & a+b < x < \infty \\\ \end{array}\right.$$ I have to use the fact that the potential well is symmetric about x=0. I have solved the Schrödinger equation in all the regions and end up with $$ψ1=A\sin(k_1x)+B\cos(k_1x)$$ $$ψ2=Ce^{k_2x}+De^{−k_2x}$$ $$ψ3=E\sin(k_1x)+F\cos(k_1x)$$ where $k_1=\sqrt{\frac{2mE}{\hbar^2}}$ and $k_2=\sqrt{\frac{2m(V_0-E)}{\hbar^2}}$. I can use the symmetry argument and get C=D which means cosh for even and sinh for odd wave functions. I know that I'm meant to get $A\sin(k_1(a+b−x))$ between the $-a-b$ and $-a$ and between $a+b$ and $a$ but I'm not sure how. If I know how to do that it may help me normalise the wave function which is key at the moment. Many thanks in advance!


title: Given that matter cannot escape a black hole, how did the big bang produce the universe we see today?

text: Extrapolation of the expansion of the Universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. If the matter contained within our galaxy were concentrated within a small radius wouldn't that lead to the entire universe being a black hole? If so how is it that all the matter of our observable universe could have originated from a region of infinite density?


title: wave-particle duality and entanglement

text: By fundamental definition of a entangled system we can say that if we know the quantum state of one subsystem then we can describe the state of another subsystem. A particle possess wave-particle duality. If one experiment verify the wave nature of particle then we can not see its particle behaviour in same exp and vice-versa. Can we say that wave and particle behaviour are in some sort of entanglement state.


title: Separable states of maximum non-classical correlations

text: Although there is no standard measure of entanglement, the GHZ states $|GHZ\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$ are often deemed as maximally entangled states of $n$ qubits. We know by now that even separable states can exhibit correlations that are impossible in the classical world. This makes me wonder: is there a similar characterization of _separable_ states that exhibit maximum non-classical correlations? For example, I imagine that states of the form $\rho=(1/4)(|0\rangle\langle 0|\otimes|+\rangle\langle +|)+(|1\rangle\langle 1|\otimes|-\rangle\langle -|)+(|+\rangle\langle +|\otimes|1\rangle\langle 1|)+(|-\rangle\langle -|\otimes|0\rangle\langle 0|)$ are two-qubit separable states with maximum correlations. Is this true? Is there a general notion for $n$ qubits?


title: Heat of vaporization of water - dependence on relative humidity?

text: Does the heat of vaporization of water depend strongly on the relative humidity of the gas into which it evaporates? Some context: If we want to calculate the dew point of water, we find the temperature at which the _partial pressure_ of the water lies on the liquid/vapor boundary of the water phase diagram. This is why water can evaporate from our bodies even though we do not heat it to anywhere near its boiling point. The heat of vaporization should be pressure dependent (in addition to temperature dependent). Yet, when specifying the heat of vaporization, most references only specify the _total_ ambient temperature, usually 1 atmosphere. Why is the total pressure used in this case instead of the partial pressure? And if the partial pressure is what matters after all, then wouldn't relative humidity be important when calculating heats of vaporization? Of course, relative humidity governs the rate and the total amount of evaporation, which is why we can't cool ourselves by sweating in humid weather, but that's not what my question is about.


title: Inductor's energy

text: I have some knowledge about LC circuits, like I know Maxwell's equations, the differential equations of circuit oscillator and so on. I am armed with equations to solve physical problems involving inductors and their behavior. But I always need to understand things as they really are. And I can't find anything on the Internet about how the inductor really works. So the alternating current in the coil gives rise to a magnetic field in the coil and as I read the energy is stored in it in a magnetic field (but magnetic field cannot do work so I guess it is stored only for split second and it just changes to electric field instantly in a way opposing the current that created this effect and treating the magnetism as the relativistic effect of moving charge which in fact is nothing more how there could be energy stored in it?). But how exactly does it happen? So we have LC circuit. We put the electrons on one plate of a inductor. The electrons push the electrons on the other side away creating plus charges and the potential difference which drives the current what in fact makes the capacitor plate be more and more charged, when it is fully charged the current stops. Now we have also the coil in our circuit. And when the capacitor is fully charged for some reason starts to make current in the other direction (i get that it can be reversed when there is emf and we change its direction but i dont get how it happens only with capacitor and coil). As it does so it moves the electrons through the coil creating magnetic field, which creates electric field in which energy is stored which can be then once more changed into charges moving to the capacitor. And then it repeats. But how an electron on a microscopic level creates this magnetic field inside the coil, how is energy stored in such inductor, how it gets there, is it stolen from kinetic energy of electron it slows down and gives energy to the coil? How the magnetic field then holds the energy or the electric field that happanes just after? In a static electric field it happens because of plus and minus charges which create the field around but how it happens in that case? I really lack of good explanation of that process and I can't really think of any electron behavior that would create such effects. Please explain it to me or point me to some readings so I can finally get how it works.


title: Hydrogen ground state energy calculation?

text: We want to find the energy of a hydrogen atom ($Z=1$) in the ground state $$ \psi_{100} = \frac{1}{\sqrt{\pi}}e^{-r}\ \ \ \ \ \ (\mbox{atomic units}) $$ with Hamiltonian $$ H = -\frac{1}{2}\nabla^2-\frac{1}{r} $$ Then $$ \begin{align*} \langle \psi_{100}|H|\psi_{100}\rangle &=\int_0^\infty \int_0^{2\pi} \int_0^\pi r^2\sin\theta\frac{1}{\sqrt{\pi}}e^{-r}\left(-\frac{1}{2}\frac{d^2}{dr^2}-\frac{1}{r}\right)\frac{1}{\sqrt{\pi}}e^{-r} d\theta d\phi dr \\\ &=\int_0^\pi \sin\theta d\theta \int_0^{2\pi}d\phi \int_0^\infty \frac{r^2e^{-r}}{\sqrt{\pi}}\left(-\frac{1}{2\sqrt{\pi}}e^{-r}-\frac{1}{r\sqrt{\pi}}e^{-r}\right)dr \\\ &= 4\pi \cdot \frac{1}{\pi}\int_0^\infty \left(-\frac{r^2}{2}e^{-2r}-re^{-2r}\right)dr \\\ &= 4\left(-\frac{3}{8}\right) \\\ &= -\frac{3}{2} \end{align*} $$ However, I've read everywhere that $E = -\frac{Z^2}{2n^2}$, and so for a hydrogen atom in the ground state we should have $E=-\frac{1}{2}$. So why am I getting $-\frac{3}{2}$? I've double-checked with Mathematica.


title: Viscosity of water in the presence of solutes

text: Some physical properties of water change in the presence of solutes: vapor pressure, boiling point, freezing point and osmotic pressure. In particular, these four properties are called _colligative properties_ because they depend only on the ratio of concentrations of solute and solvent, and not on the nature of the solute. I am interested in the viscosity of water. I have two questions:   1. Does the viscosity of water change in the presence of solutes?    2. Does the change depend on the nature of the solute? On the concentration of the solute? How? Of course, if the answer to the first question is in the negative, the second question is nullified.


title: What is the relation between (physicists) functional derivatives and Fréchet derivatives

text: I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books: $$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\epsilon \delta(x-y)]-F[f(x)]}{\epsilon}$$ from the definitions of Functional Derivatives used by mathematicians (I´ve seen many claims that it is, in effect, the Fréchet derivative, but no proofs). The Wikipedia article says it´s just a matter of using the delta function as “test function” but then goes on to say that it is nonsense. Where does this $\delta(x-y)$ comes from?


title: Double Slit Experiment: How do scientists ensure that there's only one photon?

text: Many documentaries regarding the double slit experiment state that they only send a single photon through the slit. How is that achieved and can it really be ensured that it is a single photon?


title: Lever Counterweight of uneven weight distances

text: So, I have a camera crane that has a pivot leaving 2 distances, 0.5mts and 1.5mts respectively. Plus a small L-shaped piece of metal and the end of d2 (1.5) which holds a camera, which I suppose I have to add it to the force applied on that side. One additional variable is that there's a piece of metal attached to the fulcrum and then connected to the second distance (d2, 1.5) at the end, which leaves me the doubt of where to add that weight (which is 580grs, let's say d2*) to the force, or add it as another variable in the equation. The current equation that i'm using (which was without the second piece of weight on distance 2) is: $$ w = \dfrac{f * d2}{d1} $$ ![Lever example](http://i.stack.imgur.com/1P35v.png) So i'm dubious if this is the right equation taking in count the variables I have explained above. Thanks!


title: Standard Heat of Combustion

text: When considering tabulated standard values of heat of combustion (∆H° combustion) or any other similar value, the results are reported for species in their standard states (1 bar pressure, 25°C, etc). When something combusts, however, macro state variables such as pressure and temperature change. For the tabulated values, are the conditions kept at constant temperature and pressure for the duration of the reaction? Or is the reaction started at standard conditions and the temp/pressure/etc allowed to fluctuate as the system reaches equilibrium?


title: Noether's Theorem in Field Theory

text: This question is regarding Noether's Theorem in general, but also in the application to an example. The example is: > _Find the conserved current for the Lagrangian_ > $$L=\bar{\psi}(\frac{i}{2}\gamma^{\mu}\partial_{\mu}-m)\psi.$$ Is my first step to find a transformation that leaves the action invariant? But isn't there more than one symmetry meaning this question has several answers? From Peskin and Schroeder, it says we need to find $$\partial_{\mu}j^{\mu}(x)=0 \; \text{ for } \; j^{\mu}(x)=\frac{\partial L}{\partial (\partial_{\mu}\psi)}\Delta\psi-J^{\mu}.$$ So I think I'm right in saying this $J^{\mu}$ is dependent on the transformation we're making? e.g. if it's just $\psi \rightarrow \psi +a$ then $J^{\mu}=0$. The Lagrangian changes under the transformation of the field as $$L \rightarrow L+\alpha \partial_{\mu}J^{\mu}.$$ But this doesn't help me because plugging $\psi \rightarrow \psi +\alpha \Delta \psi$ (an arbitrary transformation of the field $\psi$) into the Lagrangian at the top gives me (after no more than one line) $$L \rightarrow L+\alpha \bar{\psi}(\frac{i}{2}\gamma^{\mu}\partial_{\mu}-m)\Delta \psi$$ and how do we use this to find $J^{\mu}$? (Something I've assumed throughout all of this is that $\psi$ and $\bar{\psi}$ are treated completely separate and here we're only considering $\psi$. I hope I am right in doing this.) I haven't really pin pointed a question here as my understanding breaks down at many points and when I think I finally understand it, I'm given a new Lagrangian and become stuck again. There must be some general procedure?


title: Simple units question (School)

text: A tank has 2000 litres of capacity (Ct). It fills at 5 l/s. It is filled with 800 l at time 0. a) How many seconds will it need to fill half the tank. So I did this: X = time.               5 l   *  X     2000L      ----       = -------   -  800L       s           2               X =   200L        ---------          5  L            ---             s               X =   200          -------             5         ---------             s      Eventually, I get to `X = 40 / s`, but I guess it should be `X = 40s`. However, I fail to see what I should do to get the seconds multiplying the 40. PS: Shouldnt I give the X a unit at the beggining, or is it ok to be just X?


title: Numerical Ising Model - Wolff algorithm and correlations

text: I'm doing some numerical Monte Carlo analysis on the 2 dimensional Ising model at the critical point. I was using the Metropolis 'single flip' evolution at first with success, though it suffers from critical slow down and makes studying large lattices unlikely possible. I'm now looking at cluster flip algorithms, specifically the Wolff algorithm. I managed to implement it, and it looks to be working as it should (flipping a unique spin at $T = +\infty$, the whole lattice at $T=0$, matches the right energy density in the thermodynamic limit...) but I don't get the right behaviour for the two point $<\sigma_i\sigma_j>$ correlation function. According to CFT it should behave like: $$<\sigma_i\sigma_j> \;\propto \;\frac{1}{|i-j|^{\frac{1}{4}}}$$ I'm more and more convinced that it has to do with boundary conditions, I use non periodic free boundaries. The literature on the subject doesn't say much on this point. Am I missing a subtlety (or an evidence) in this procedure, or in the use of this algorithm?


title: Error propagation of statistical error

text: I have a pulse profile (binned photon counts versus phase) of a star, and for each count rate I have its statistical error. I want to calculate the so-called pulsed-fraction $P_{frac}=\frac{F_{max}-F_{min}}{F_{max}+F{min}}$, where $F_{max}$ and $F_{min}$ are the maximum and minimum count rates, respectively. Is it possible to do propagation error to find the error for $P_{frac}$? If yes, how?


title: Computation theory and the simulation argument

text: 1. Can physical states be treated as information (strings over some alphabet)?   2. If (1) is true, isn't this a trivial conclusion that the universe can be simulated by a Turing machine or a cellular automaton or any other computational model, given that differential equations that determine the evolution of physical states (the bits in digital physics) are simulated by the transition function of the Turing machine?


title: Question about teeter totters or see saws

text: I have a basic enough question. Assume that one has one of those ideal see saws, i.e. the teeter-totter pivots on point source, the balance is of uniform mass etc. Now assume that one places an object of mass m on one end. This will force the balance to tilt on side. Now suppose that one adds another mass of m on the other side. What happens? It seems that everyone agrees that the the balance would move to the stable equilibrium, that is the middle. What I don't understand is, what is the Force that moves the teeter-totter to the middle position? Once the second mass has been placed on the other side, the two net forces acting on the pivot is the same. And since the distance is the same, the torque is the same. So what force is it that moves the balance to the middle? And if no such force exists then every position of the balance is valid if the two weights are equal, then how do those mass balances work?


title: Understand equations of a conducting sphere

text: Can somebody explain to me, when the following two equations (equations 2.48 and 2.50 in this document) are applicable and what $\Phi_s$ and $\Phi$ actually are? The thing is, I want to find general equations that determine the field produced by conducting spherical sphere in an external field and was wondering whether these are the equations I am looking for. $$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{a}{r} \right )^{l+1}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r>a$$ $$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{r}{a} \right )^{l}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r<a$$ Or is it rather this equation (equation (17.3) in this document), probably they are one and the same: $$\Phi (\mathbf{x})=-\frac{1}{4\pi}\int_S \Phi(\mathbf{x}')(\hat n.\nabla')G_D(\mathbf{x},\mathbf{x}')dS'$$


title: Collision of two deformable bodies - limitation of work-energy theorem

text: I am attempting to model the impact of a deformable body (clean, dry sand characterized as a continuous fluid) with flexible diaphragm (in this case a sheet of rubber). Assuming that this is a closed system (no sand escapes) do you think work-energy theorem is adequate for approximating the total force imparted on the fluid by the diaphragm? Disregarding heat loss, can I take the KE of the flow (assuming uniform flow), look at the total deflection of the barrier and get an approximation of the force? I guess this comes down to my lack of knowledge about work. I know the force on the flow from the barrier increases linearly with deflection, but does that matter if I am only concerned about the total force required to take the flow from it's initial velocity to rest? Sorry for the long winded description. Any guidance would be appreciated.


title: Large-$N$ Yang Mills

text: I've bumped into the study of the $SU(N)$ theory in the large-$N$ limit. I'm wondering in which way the study of this Yang-Mills theory, can give contribution to QCD with gauge group $SU(3)$, i.e. $N=3$ Any suggestion?


title: Homemade salad dressing separates into layers after it sits for a while. Why doesn't this violate the 2nd law of thermodynamics?

text: The oil, vinegar and other liquids in homemade salad dressing separate into layers after sitting for a while, making the mixture become more organized as time evolves. Why doesn't this violate the 2nd law of thermodynamics? I assume that the answer is that since the separation is due to gravity, the effect is due to an external force and so the system isn't closed which is necessary for the 2nd law, but I'm not sure. If that's the answer, then what happens if I consider the whole system include the salad dressing, gravitational field and whatever mass is generating the gravitational field? The entropy of the salad dressing seems to decrease and so the entropy of some other component of this system must be increasing, but it's hard to see what this other component would be.


title: What does it mean to have infinite negative conformal time?

text: In the context of Inflationary Cosmology, it is postulated that there was a period of shrinking Hubble Sphere radius $(aH)^{-1}$. $$ \frac{d}{dt} (aH)^{-1} < 0 $$ Then the regions of the universe which appear to us to not be in causal contact can indeed have been in causal contact at some (conformal? physical? I'm not sure. My intuition would be physical) time in the past. This solves the Horizon Problem, namely that we observe a homogenous isotropic universe to one part in 10,000, yet it doesn't seem that all regions of the universe were in casual contact to achieve such a uniformity. $\\\$ Anyway, when this shrinking Hubble sphere radius $(aH)^{-1}$ is invoked, it has the result of pushing the Hot Big Bang singularity back to negative infinity conformal time. It is this statement that I am having trouble grasping. $\\\$ Explicitly $$ \chi_{PH}(\tau) = \tau - \tau_i = \int_{t_i}^{t} \frac{1}{a(t)} dt = \int_{a_i}^a (a H)^{-1} d\log(a) $$ where by Friedmann $$ (aH)^{-1} \propto a^{\frac{1+3w}{2}} $$ giving $$ \chi_{PH}(a(\tau)) = \tau - \tau_i \propto \frac{2}{1+3w} \left[ a(\tau)^{\frac{2}{1+3w}} - a_i^{\frac{2}{1+3w} } \right] $$ So for a strong energy violating fluid such that $$ 1 + 3w < 0 $$ this becomes $$ \tau_i \propto \frac{2}{1+3w} a_i^{\frac{2}{1+3w}} \rightarrow -\infty $$ $\\\$ I understand the maths behind it, and can't argue against that the initial conformal time goes to minus infitity, but I don't know how to think about that. In a standard Hot Big Bang the singularity occurs at $\tau_i = 0$ conformal time. This seems fine to me. I am aware that conformal time is unphysical, but unless the scale factor goes negative, then negative conformal time suggests to me negative physical time. And a negative scale factor would bring a whole other conceptual problem! Namely, they are directly related by $$ d\tau = \frac{1}{a(t)} dt $$ $\\\$ So my questions are; What does it mean to have negative conformal time? And what does it mean to have negative infinity conformal time? $\\\$ Thanks


title: Area moment of inertia of regular $n$-gons over polygon center $O$

text: Is it possible to consider the regular polygons ($n$-gons) as deformed circles and use a pseudo-polar coordinate system to calculate their moment of inertia over its center $O$. Inasmuch as I know (I am a solid mechanics beginner), we can find the area moment of inertia $J_O$ of the circle $C(O,R)$ considering a circular disk of thickness $dr$. Can I apply the same to regular polygons, namely if their area is equal to: $$A = \frac{nr^2}{2 \sin{(2 \pi / n})}$$ then the differential area element $dA$ will be equal to the area of the outer polygon with radius $r+dr$ minus the area of the inside one of radius $r$: $$dA = A(r+dr) - A(r) = \frac{n}{2 \sin{(2\pi/n})}dr (2r+dr)$$ so the moment of inertia at the center O of the solid polygon can be calculated as: $$J_O = \int^R_0 r^2 dA = \int^R_0 r^2 \frac{n}{2 \sin{(2\pi/n})}dr (2r+dr)$$ where $R$ is the radius of the polygon (the distance between its center and its vertices). Considering the above $J_O$ tends to be equal to the one of the perfect circle: $$\lim_{n \to +\infty} J_O = \pi r^4 / 4$$ Are my claims right?


title: Can low end hepa vaccums still clean near all of the air?

text: In the US HEPA filters will remove 99.97% of particles sized 0.3 microns. In Europe HEPA filters do not have to meet this exact figure and there are different classes of HEPA filters e.g. h10(removes 85% at .3 microns, h11 removes 95% at .3 microns etc, h14 removes 99.97% etc.). Suppose you have a vacuum cleaner that is h10(removes 85% of particles at .3 microns), when vacuuming 15% of .3 microns will pass through the filter. If we continue to vacuum the area these will go into the vacuum again and be filtered again. Therefore even though the filter performs less, is it correct to say eventually(with continued vacuuming) the filter will remove the remaining 15% so its not a big deal if you get one with 85% or 99.9% filtration so long as you are bothered to continue vacuuming? Thanks


title: The "replica trick" initial formula?

text: In Spin-glass theory for pedestrians by Castellani and Cavagna, the initial formula used to introduce the replica trick is written as: $$\overline{\log Z}=\lim_{n\rightarrow0}\frac{1}{n}\log\overline{Z^{n}}\qquad(1)$$ where the overbar denotes average over quenched disorder. I don't know how to prove this formula. In other treatments I have seen of the replica method (wiki, for example), one starts from: $$\log Z = \lim_{n\rightarrow 0} \frac{Z^n-1}{n}\qquad(2)$$ which I understand. How are (2) and (1) connected? What's the proof of (1)?


title: Where does the potential energy associated with the field go if it is removed?

text: I have an electric field and a certain charged particle in it that has a certain potential energy associated with it. Where does the energy go if I remove the field?


title: Why is the orbital resonance of the Galilean moons stable?

text: It is well known that the orbits of Ganymede, Europa and Io are in a 4:2:1 resonance. Most online sources (including but not limited to Wikipedia) say that such an orbital resonance, along with the 3:2 resonance, is "stable and self-correcting", but fail to explain why this is so. The textbook _Fundamental Astronomy_ says that this phenomenon is due to "tidal forces" but does not elaborate further. I presume it refers to the tidal deceleration which causes the orbits of the moons to evolve outwards, which, other sources say, caused the moons to eventually enter into resonance, but this also does not explain why the resonance is stable. I am aware of a similar question here but I'm more interested in stability rather than instability in this case. In short, why is the orbital resonance of the Galilean moons stable, and how is it different from other cases of orbital resonance that are unstable? I don't mind (and would prefer) if the answer is mathematical in nature.


title: Direct observations of a black hole?

text: I'm not very knowledgeable about physics generally, but know that nothing can escape a black hole's gravitational pull, not even light (making them nearly invisible?). My question is: **What has been obtained from direct observation of a black hole to prove their existence?** Please note that I'm not questioning the existence of black holes, but am just curious as to how we have come to realize it.


title: Is it possible to use the parabolic shape of a rotating fluid to measure the angular frequency of the rotation of the Earth?

text: A fluid in a rotating bucket will take on a parabolic shape (for example of some simple derivations of this result see http://en.wikipedia.org/wiki/Bucket_argument). The assumptions that play into the derivations that I've seen do not take into consideration a Coriolis effect. Suppose one did not know that the earth was spinning. Is it possible to measure the angular frequency of the earth using a bucket of fluid by considering how the Coriolis effect modified the curvature of the paraboloid in question? Perhaps it's possible to optically measure the curvature of the paraboloid setup by the spinning process and from that determine the angular frequency of the planets rotation. I am concerned though that the effect of the rotation of the earth is so minuscule on the parabolic shape that it would be impossible to physically measure it.


title: How does blood/saliva boil in outer space?

text: Why would the blood boil? Is it because of the temperature or pressure? Because I really can't figure it out. I thought space didn't have a temperature above freezing unless close to a star or the sun. But if it is the pressure how would that work?


title: Will quantum computers ever work?

text: > **Possible Duplicate:**   >  Why do some physicists believe that scalable quantum computing is possible? The idea of a quantum computer is that a quantum system can be in a Quantum Superposition of many states. then the same calculation can be done on each of the states of the superposition, so the system can do many calculations in parallel. This idea leads to the possibility of much more powerful computers than would be possible with ordinary processors.


title: Solving for motion of rotating rod using only Newton's laws?

text: I have a question that's been bothering me for years. Given a rod of uniform mass distribution with total mass $M$ and length $L$ that lies on a horizontal table (with one end fixed to the table around which the rod is free to rotate in the horizontal plane, and a force F applied perpendicular to the rod at the other end), how do you solve for the motion of the rod (and the internal forces) using only Newton's laws and the assumption that the rod is a rotating rigid body? By that I mean only using the most basic conception of Newton's laws and the system's constraints, without the ideas of torque and moment of inertia, energy and momentum, and even without the idea that the net force on the rod gives the acceleration of the center of mass--so only using Newton's laws for point particles, or in this case infinitesimal $dm$ sections of the rod. I've tried to solve this by breaking the rod down into these small $dm$ components and using an idea I've seen (at least I think I've seen) where you set $F(x+dx)-F(x)=dm(a)$ and then are able to find $F'(x)$ and integrate and then use the boundary conditions on the force. I did this for both tangential and radial components, with radial acceleration equal to $x(\omega(t))^2$ and tangential acceleration equal to $\omega\ '(t)x$, but was unable to obtain the right answer. I used the force at one end of the rod as a boundary condition (is this correct?), but was unable to even solve for the force at the pivot, let alone the angular velocity as a function of time, and have no idea if this technique is even valid. I feel like at a certain point it might be that my force equation switches sign--as the net force that accelerates the infinitesimal mass starts coming from the inward instead of outward side. I'd also be interested to know more generally how to solve for internal forces and motion of a rigid body using only these most basic assumptions, such as for a free uniform square on a horizontal table with a force applied perpendicularly to one side into one corner.


title: The Hermiticity of the Laplacian (and other operators)

text: Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator? Alternatively: is the matrix representation of the Laplacian Hermitian? i.e. $$\langle \nabla^{2} x | y \rangle = \langle x | \nabla^{2} y \rangle$$ I believe that $\nabla^{2}$ is Hermitian (if it was not, then the Hamiltonian in the time-independent Schroedinger equation would not be Hermitian), but I do not know how one would demonstrate that this is the case. More broadly, how would one determine whether a general operator is Hermitian? One could calculate every element in a matrix representation of the operator to see whether the matrix is equal to it's conjugate transpose, but this would neither efficient or general. It is my understanding that Hermiticity is a property that does not depend on the matrix representation of the operator. I feel that there should be a general way to test the Hermiticity of an operator without evaluating matrix elements in a particular matrix representation. Apologies if this question is poorly posed. I am not sure if I need to be more specific with the definitions of "Hermitian" and "Laplacian". Feel free to request clarification.


title: Magnitude of the Fourier Transform of White Noise

text: Say you have two white noise signals with different variation amplitudes A1 and A2 as shown in this beautiful Excel graph: ![White noise signals](http://i.stack.imgur.com/6ASjL.png) Ignoring the DC offset as it's been represented here, how do you relate the amplitudes A1 and A2 to the magnitude of the Fourier coefficients after a Fourier transform (as shown in the diagram below)? ![Fourier transforms of white noise signals](http://i.stack.imgur.com/VeXgm.png) In other words, is it possible to relate A1 to Mag1 and A2 to Mag2? Can this even be done analytically or will it require a bit of simulation? Any help is much appreciated!


title: Time period of torsion oscillation

text: For the oscillation of a torsion pendulum (a mechanical motion), the time period is given by   $T=2\pi\sqrt{\frac{I}{C}}$ which is a result of the angular acceleration $\alpha=\frac{d^2\theta}{dt^2}=-(\frac{C}{I})\theta$ where $C$ is the restoring couple of the string. Do we relate $T=\frac{2\pi}{\omega}$ and $\alpha$ for finding the time period of torsion pendulum? I can't really understand how they arrived at $2\pi\sqrt{\frac{I}{C}}$ ?


title: Wrong positioned Ampere-meter and Voltmeter

text: I'm dealing with a problem here and even that I'm trying to solve it i can't It says: > _In what figures the voltmeter and ampere-meter are wrong positioned?_ ![enter image description here](http://i.stack.imgur.com/fKKPJ.jpg) I think that all the the others are correct except the second one. Can anyone help me?


title: What is lambda R in Richardson's Law?

text: I've got to calculate the thermionic emission through a diode, so I need to use Richardson's Law. However, one thing's got me confused - according to the Wikipedia page: $$J = A_GT^2e^\frac{-W}{kt}$$ I could live with that, but for $A_G$. Apparently, I'm not the only one; Wikipedia's a bit cryptic about what $A_G$ is, mentioning that physicists have struggled with it for decades, "but there is agreement that $A_G$ must be written in the form:" $$A_G = \lambda_RA_0$$ "Where $\lambda_R$ is a material-specific correction factor that is typically of order 0.5, and $A_0$ is a universal constant." That's the last real mention of $\lambda_R$, and the only reference for it is in French. So, what is $\lambda_R$, really? How can I figure out its numerical value so that I can actually use Richardson's Law?


title: Coupling constant is turned off adiabatically?

text: To me, adiabatic processes are idealisation. What do people mean with statements such as: "turning off the coupling constant (in QED say) adiabatically"?


title: Are the implications of an infinite universe necessarily so unsettling

text: I have often heard it said (by professional cosmologists) that if the universe is infinite, then there necessarily exist infinitely many copies of me repeated throughout. The reasoning seems to be that any finite volume of space can contain at most finitely many fundamental particles existing in finitely many configurations. From here it is implied that all possible (permissible) configurations must both exist and occur infinitely often. This does not sound like a sound argument to me, so I have to believe that I have misunderstood the argument. Is this unsettling implication true in an infinite universe, and if so how is it justified.


title: Conservation of energy in a different frame of reference

text: Consider a rollercoaster that goes down a slope: ![rollercoaster](http://i.stack.imgur.com/94VBI.png) At the higher level it has speed $v_0$, then it goes down a slope and at the end it has speed $v_0 + \Delta v$. The carriage is not powered and has negligible friction. So by conservation of energy we should, after eliminating mass $m$ that appears in all terms, have $$ \frac{1}{2}v_0^2 + gh = \frac{1}{2}(v_0 + \Delta v)^2 \\\ \frac{1}{2}v_0^2 + gh = \frac{1}{2}v_0^2 + v_0\Delta v + \frac{1}{2}\Delta v^2 \\\ gh = \frac{1}{2}\Delta v^2 + v_0\Delta v $$ (left side is energy at the beginning, right side energy at the end). But what if we consider the same situation in a reference frame moving left with speed $v_0$. Then we should get just $$ gh = \frac{1}{2}\Delta v^2 $$ These differ by a $v_0\Delta v$ term, so solving for $\Delta v$ gives different result. But obviously the speeds should be the same. So where is the error?


title: how to solve this scaling equation

text: I am confronted with the following scaling equation (in Laplace domain if that matters): $\tilde\psi(s) = a \tilde\psi(s/b)$ I know the answer is such that $\tilde\psi(s) \sim s^{\beta} K(s)$ where $\beta = \ln a/\ln b$ and I don't really care what K is. I'm not sure if the functions $\tilde\psi(s)$ are need to solve this but I do know them if they are necessary. How is this type of equation solved?


title: What is replica symmetry breaking, and what is a good resource for learning it?

text: M. Mezard, G. Parisi and coworkers have written about replica symmetry and its breaking in spin glasses, structural glasses, and hard computational problems. I am just getting acquainted with this literature. Where are the best places to start understanding replica symmetry? Can anyone here explain the concept?


title: Is uncertainty principle a technical difficulty in measurement?

text: I have searched for an answer to this question on physics SE but I have not seen a question in which it is addressed properly. Please let me know if there is an answer already. My question briefly is, is the uncertainty principle a technical difficulty in measurement OR is it an intrinsic concept in Quantum Mechanics irrelevant of any measurement?? Everyone knows the thought experiment of measuring the position of an electron. One can detect electron's position by hitting it with a photon, due to Compton scattering the collision of the photon with electron will change electron's momentum. This experiment is used to explain uncertainty principle to layman, but it is over simplified, isn't it? It also gives an impression that if there was a better suited experimental method the uncertainty principle becomes irrelevant. I personally think it is intrinsic as it arises from the non-zero commutator of position and momentum operators irrespective of the measurement process. Am I right? **EDIT:** My question is similar to certain extent to this question and this question. The answers there are nice but they focus on explaining basics of quantum mechanics more than they comment on the technical difficulty part. In the answers of question 2 there are statements like "So, it's not a knowledge limit" and "you're sort of correct when you say it's an observational limit" without further comments To summarize, assume hypothetically we managed to find a way in the future where we can have a look at an electron without disturbing it by measurement or causing its wave function to collapse, would the uncertainty principle still hold in such a case?? Why/Why not?


title: What does Mass bend?

text: Mass and Energy can warp space-time around them, but that doesn't answer what space-time is, what is space?


title: What is clock or phase adjustment in projectors? Why its proper setting eliminates strange behaviour of 1px lines/

text: I'm doing experiments that require using a big, thin, contrast grid, for example 1 black pixel and then 1 white pixel alternately. I've met with a few DLP projectors and in each of them I had to manually adjust **clock adjustment/phase adjustment.** If I didn't do so, image was flickering or halos were present around contours. I have read few manuals looking for description of this setting, but none of them explained why it happens. Computer and projector are connected through VGA, and of course, the resolution is set properly to meet the projector's. Do you think you have an explanation for this? It's exactly what happens here, behind the grid: http://www.lagom.nl/lcd- test/clock_phase.php#clockcalib But why?


title: Probability of position in linear shm?

text: The problem that got me thinking goes like this:- > _Find $dp/dx$ where $p$ is the probability of finding a body at a random > instant of time undergoing linear shm according to $x=a\sin(\omega t)$. Plot > the probability versus displacement graph. $x$=Displacement from mean._ My work: $$v=dx/dt=\omega \sqrt{a^2-x^2}$$ Probability of finding within $x$ and $x+dx$ is $dt/T$ where dt is the time it spends there and T$$ is the total period.   Therefore $$dp=dt/T=\frac{dx}{\pi \sqrt{a^2-x^2}}$$ because $t=2\pi /\omega$ and the factor 2 is to account for the fact that it spends time twice in one oscillation. The answer matches the answer and also the condition that integration $-a$ to $a$ of $dp =1$. But when i try to find p as a function of x to plot the graph I get $$p=\frac{1}{\pi}\arcsin(x/a)+C.$$ But then I get stuck as there is no way to find $C$ (except the fact that for $C=0$ the probability at the mean position is $0$ and hence $C$ cannot equal 0) which I know of. So how can I get a restraint on $C$ to find its value and hence to properly graph it with the condition that the probability from $-a$ to $a$ be 1?


title: Can general wavefunctions be expressed as kets?

text: I am confused on bra-ket notation in quantum mechanics. My professor says that a ket is an eigenfunction of some operator. However, for some time now I thought a ket could represent a general wavefunction. Obviously any wavefunction should be able to be expressed as a linear combination of eigenfunctions (and thus a linear combination of kets based on my professor's definition), but I've always thought that a ket could represent any wavefunction, not just an eigenfunction. Which of these two viewpoints is right, or, if they are both right, how are they both right?


title: First Order Time Depenent Perturbation theory of particle in magnetic field

text: So I am dealing with the following hamiltonian, and the following perturbation: $$H=-\mu B_0\sigma_z$$ $$V=\mu B_1(\cos(\omega t)\hat x-\sin(\omega t)\hat y)\cdot{\bf \sigma}$$ I am asked for the probability that the spin will go from ground state at time 0 to the excited state at time t using first order time dependent perturbation theory. I know that the ground state is $\begin{pmatrix}1\\\0 \end{pmatrix}$ with energy $-\mu B_0$. This means the excited state is $\begin{pmatrix}0\\\1 \end{pmatrix}$ with energy$+\mu B_0$. So the probability I want is: $$P=|(0 1)U^{(1)}(t,0)\begin{pmatrix}0\\\1\end{pmatrix}|^2$$ Where I have $$U^{(1)}=\frac{1}{i\hbar}\int^t_0 dt_1e^{-iH_f(t-t_1)/\hbar} V(t_1) e^{-iH_it_1/\hbar}$$ So I have 2 questions. The first is if this is the correct expression I want to use for finding the probability. And secondly, how do I treat the matrices? Seeing as they are a constant, can I just pull them out and then integrate the remaining pieces?


Why can't we accelerate to the speed of light? It's just a speed and nothing else. Universe also crossed the speed of light at the time of big bang. Is this is just a interpretation or there is any fact behind it? I am asking it as I don't know much about this theory.


title: What really allows airplanes to fly?

text: What aerodynamic effects actually contribute to producing the lift on an airplane? I know there's a common belief that lift comes from the Bernoulli effect, where air moving over the wings is at reduced pressure because it's forced to travel further than air flowing under the wings. But I also know that this is wrong, or at best a minor contribution to the actual lift. The thing is, none of the many sources I've seen that discredit the Bernoulli effect explain what's actually going on, so I'm left wondering. Why do airplanes actually fly? Is this something that can be explained or summarized at a level appropriate for someone who isn't trained in fluid dynamics? (Links to further reading for more detail would also be much appreciated)


title: Can the lift generated by a helicopter be justified using Bernoulli theorem?

text: When the shaft of the helicopter rotates, it creates a low pressure. Because of the low pressure, the helicopter lifts. Is my understanding that this is just an application of Bernoulli's theorem?


title: Equal Transit Time Fallacy

text: I learned, in grade school, that lift was generated via the particles on either side of the wing having to reach the other end at the same time. Looking back, that indeed has no physicality to it. So what is the real explanation of how lift is generated? I've found references to Navier Stokes, but I don't know what that is. What is the best way to learn, from the ground up, why some curved object would generate a definite change in pressure? I can see the results, and I know that there is a pressure difference, but **why** is that pressure difference, and the two different velocities generated?


title: how to explain the upright force for the plane?

text: I remember in the high school physics, my teacher told us that the design of the plane wing is because we want the air above the wing flowing faster than the air flowing below so the pressure above and below will be different so the net force is pointing upright. I am thinking if it is possible to explain this by bernoulli equation? But I soon stuck there because I don't know if bernoulli equation applicable to the case of the air flow in the open region? If so, what cross section we should choose?


title: Why air above airplane's wing moves faster?

text: One explanation I read: Because of the wing's geometry, the "upper" side of the wing is longer, so the air has to travel faster: ![enter image description here](http://i.stack.imgur.com/HDgVA.jpg) **My wondering** : Who said (and what was his/her explanation) that air must travel the distance above and under the wing in the _same_ time? I'll try to be clearer. Considering the below image, does **m1** and **m2** meets at the same point exactly at the same time on point **T**? ![enter image description here](http://i.stack.imgur.com/j7Icc.png)


title: How does this help an aeroplane to fly?

text: I read it somewhere on the internet that wings of an aeroplane are designed in such a way, that they increase the velocity of air above the wings and so pressure above the plane becomes less than the pressure below it and therefore the aeroplane flies well in air, but how is velocity increased above the wings of an aeroplane?


title: Why is more intrinsic carriers bad for high temperature semiconductors?

text: I'm taking a solid state course, and is currently on the subject of dielectrics. In one of the sections, concerning "Impurities in Dielectrics" the books says: _"Impurities can also be used to make insulators conductive, exactly like doped semiconductors. Donor or acceptor levels have to lie close to the conduction band and valence band, respectively, in order to give rise to a finite density of electrons and holes and thereby conductivity. The important advantage over a usual semiconductor material is that high-temperature applications are possible. In a narrow-gap semiconductor, high temperatures are a problem because of the exponentially increasing number of intrinsic carriers. In a doped insulator, this is not an issue for practically relevant temperatures"_ Now, I'm not getting why more carriers are a bad thing ? I see, that if the gap is too small (Semiconductor), and the temperature is high, the you should be able to excite electrons into the conduction band, instead of just the doped bands, right ? But why is that a bad thing ? I see that high-temperature applications can be bad, since you have to get high temperatures, which is not always easy and great, but if you already are at that temperature, what is the problem ?


title: Why use Fourier expansion in Quantum Field Theory?

text: I have just begun studying quantum field theory and am following the book by Peskin and Schroeder for that. So while quantising the Klein Gordon field, we Fourier expand the field and then work only in the momentum space. What is the need for this expansion?


title: Using gravity to send messages (at the speed of light)

text: I've been wondering what the current state of the art (both theoretically and practically) for sending messages using changes in a gravitational field is, and I have had a hard time finding information about it online. I've seen the Wikipedia page on gravitational waves, but it seems that gravitational waves _per se_ are not yet practical as a communication medium, since they have yet to be detected. However, a different (unoriginal, I know) idea is simply to cause a change in a gravitational field from some point, and measure the change at another. You could encode the magnitude and/or frequency of the change into bits and communicate wirelessly. Although this is still subject to the speed of light, it has other benefits over radio communication like easily penetrating all matter. A particularly slow and inefficient implementation of this might be to attach a large number of powerful rockets to the moon and alter its velocity enough that it changed the timing of ocean tides in a measurable way. How far can we extend this? Can I build a gravitational detector sensitive enough to measure a bowling ball oscillating near it? How about from the other side of the earth? How little mass can I use? Can I do it without any mass at all? We know that massless particles are still affected by gravity, and I believe we also know that massless particles have their own gravitational field (in accordance with mass/energy equivalence). Can we exploit the gravity of massless particles to generate high frequency changes in a gravitational field without having to worry about inertia? Can we then use the high frequency of these oscillations to isolate them from the high level of noise generally present in low frequency gravitational detectors?


title: What is temperature?

text: Recently I read an interesting article about negative temperature. I was puzzled because I thought before that temperature has definite meaning in thermodynamics: it tells about how fast atoms jiggle. Now if temperature can be negative, this means temperature has wider meaning... I wonder if there is any deep physical meaning of temperature, not mathematical one.


title: Does the axiom of choice appear to be "true" in the context of physics?

text: I have been wondering about the axiom of choice and how it relates to physics. In particular, I was wondering how many (if any) experimentally-verified physical theories require axiom of choice (or well-ordering) and if any theories actually require constructability. As a math student, I have always been told the axiom of choice is invoked because of the beautiful results that transpire from its assumption. Do any mainstream physical theories require AoC or constructability, and if so, how do they require AoC or constructability?


title: How can a Human voice or animal voice have unique frequency

text: Well this is pretty noobish question and I am not sure how to ask.   When We talk we don't talk in an uniform frequency. Then how can one measure frequency of ones sound/voice ?   I am asking this cause several times I've heard people to say this is his/her frequency of voice. Is this a vague term ? or it has any real meaning ?


title: Moment of inertia of a cylinder about its base

text: I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see here . But when I checked my results with different references ,I've found that it's incorrect!.I need a help to figure out where I did it wrong. since $$I=\int\limits x^2.dm$$ $$dm = \rho.dv$$ where $$dv= r.d\theta . dr .dh$$ & $$\rho=\frac{M}{\pi R^2 H}$$ $M$:cylinder total mass. $H$:total height of cylinder. $R$:the radius of the cylinder. The distance of each infinitesimal element from the axis of rotation would be : $$\sqrt{r^2+h^2}$$ Therfore, $$I= \frac{2M}{ R^2 H}\int\limits_{0}^{R} \int\limits_{0}^{H}\, r(r^2+h^2)dh\,dr$$ and this gives a result of : $$\frac{M~H^2}{3}+\frac{M~R^2}{2}$$ Which is obviously wrong since the second term should be multiplied by a factor of $1/2$.


title: Partition function of classical quadrupole in an electric field

text: The partition function of a dipole in an electric field is a well-known problem, analytical solvable (nice integral, can be calculated with pen and paper), for example in the Langevin treatment of paramagnetism. I'm right now looking at a classical quadrupole in an electric field. This looks very different, and I have not been able to solve it by myself. Before I try harder and reinvent the (broken) wheel, would anyone know if it's been addressed, and possibly solved, before? * * * **The way I see it:** I consider a quadrupole $\mathcal{Q}$ fixed in space but with free orientation. It interacts with an electric field $\mathbf{E}$, whose derivatives form the tensor $\mathcal{E}$ ($\mathcal{E}_{ij} = \frac{\partial E_i}{\partial x_j}$). The potential energy is thus: $$ U = \frac{1}{3} \mathcal{E}:\mathcal{Q} $$ and the partition function I want to calculate is: $$ Z = \int \exp\left(-U/kT\right) $$ where the integral is taken over all orientations of the quadrupole with respect to the field, i.e. a 3D rotation matrix determined by three angle $(\theta,\phi,\xi)$. I've started doing this integral, but neither my brain nor Mathematica can express it in a nice way. _I don't know if I'm missing a way to solve this in this fashion, or if it would be better attacked from another angle…_


title: Does spin-0 or spin-2 describe massive or massless particles?

text: spin-0 is massive or massless? How does we separate the massive and massless degrees of freedom for spin-2? What is the partially massive?


title: Maintaining local thermodynamic equilibrium (LTE) in radiating gas with a broad atomic transition line

text: ## Definitions / Background In LTE, Kirchoff's law for radiation holds: $$ \frac{j_{\nu}}{\alpha_{\nu}} = B_{\nu} (T) $$ where $j_{\nu}$ is the specific radiative emissivity, $\alpha_{\nu}$ is the monochromatic radiative absorption, and $B_{\nu} (T)$ is the Planck function evaluated at the temperature $T$. Consider a gas of two-level atoms with energies $E_u$ and $E_l$, with $E_u > E_l$, statistical weights $g_u$ and $g_l$, and number densities $n_u$ and $n_l$. The transition between these states has Einstein coefficients $A_{ul}$, $B_{ul}$ and $B_{lu}$ that we can use to write the emissivity and absoprtion of the transition: $$ j_{\nu} = \frac{h \nu}{4 \pi} n_u A_{ul} \psi({\nu})$$ $$ \alpha_{\nu} = \frac{h \nu}{4 \pi} [ n_l B_{lu} \phi({\nu}) - n_u B_{ul} \chi({\nu})]$$ where $\psi$, $\phi$, and $\chi$ are line profile functions accounting for line-broadening mechanisms such as thermal motion. Then, making use of the standard relations between the Einstein coefficients, we have $$ \frac{j_{\nu}}{\alpha_{\nu}} = \frac{2 h \nu^3}{c^2} \frac{\frac{\psi}{\phi}}{\frac{g_u n_l}{g_l n_u} - \frac{\chi}{\phi}}$$ ## Question I want to understand how the right-hand side of the last equation simplifies to the Planck function under conditions of LTE, over the entire width of the line, without assuming that the line is narrow. A standard discussion of this topic, as found in e.g. the astrophysics textbook by Rybicki and Lightman, does not seem to achieve this. In my reading, they proceed by making the following observations and/or assumptions: 1) There is complete redistribution of frequency between absorption and all types of emission, so that $\psi = \chi = \phi$. 2) In LTE at temperature T, the fraction $ g_u n_l / (g_l n_u)$ is equal to $\exp[h \nu_0/(kT)]$, where $\nu_0 = (E_u - E_l)/h$. If those are true, then we have $$ \frac{j_{\nu}}{\alpha_{\nu}} = \frac{2 h \nu^3}{c^2} \frac{1}{\exp[h \nu_0/(kT)] - 1}$$ But this does not have the correct exponential term in the denominator to match the Planck function, because $\nu_0$ is constant (frequency- independent). Or from another perspective, the problem is that $g_u n_l / (g_l n_u)$ is frequency-independent. So, what is going on? Do we have to break either of the assumptions/observations 1 or 2 above? If so, how? If not, then does Kirchoff's law simply not apply on a frequency-by-frequency basis over the width of a broad line, although it still might apply in a line-averaged sense? Is there some other possibility or detail I have overlooked?


title: How does phase modulation conspire to eliminate power variations?

text: A purely phase-modulated signal has no power modulation. This is obvious enough if you look at the time series, but I'd like to "see" it in the frequency domain. In physical terms, if we take a laser beam and apply phase modulation via an electro-optic modulator and then send this beam to a photodiode, we will see only a DC term on the photodiode. I would like to work out how the sidebands created by phase modulation manage to conspire to eliminate any beat frequencies. The operation of phase modulation at a frequency $\Omega$ consists of multiplication by $\exp\\{i\Gamma\sin\Omega t\\}$ which is easily seen to have unit amplitude at all times. Via the Jacobi-Anger identity, this can be written in terms of sidebands: $$\exp\\{i\Gamma\sin\Omega t\\} = \sum_{n=-\infty}^\infty J_n(\Gamma) \exp\\{i n \Omega t\\}$$ where $\exp\\{i n \Omega t\\}$ is interpreted as creating a sideband at a frequency of $n\Omega$ Hz away from the carrier whose amplitude is given by the nth Bessel function $J_n(\Gamma)$ where $\Gamma$ is the modulation depth. A photodiode will see the modulus squared of this signal. We can write down the result by grouping terms by like powers of $\exp\\{i\Omega t\\}$. It's easy to see that the energy is conserved at DC due to the relation $ 1 = \sum_{n=-\infty}^\infty |J_n(x)|^2 $. Likewise, it's easy to see that there is no signal at frequency $1\Omega$ on the photodiode, because the beat of the $n$th upper sideband with the $(n-1)$st upper sideband always cancels with the beat of the nth lower sideband with the $(n-1)$st lower side band, because of the property of the Bessel functions that $J_{-n}(x)=(-1)^n J_n(x)$. However, the same sort of nice cancellation does not appear for the $2\Omega$ signal, because the bessel functions always appear in products with two bessel functions of the same parity, i.e. $\cdots + J_{-3}J_{-1} + J_{-2}J_0 + J_{-1}J_1 + J_0 J_2 + J_1 J_3 + \cdots$. **How can we show that there is no beat signal at frequencies $2\Omega$ (and higher)?**


title: Dirac equation algebraic derivation, a gauge symmetry

text: Suppose i try to derive the most generic Dirac-like equation (that is, as factors of first-order expression in momenta and mass operator where we allow coefficients that are associative, don't necessarily commute to each other, but they commute with the space-time degrees of freedom) Usually the Dirac equation is obtained by taking this expression: $E^2 = P^2 + m^2$ with $E=i \hbar \partial_{t}$, $P=-i \hbar \partial_{x}$ and getting $E^2 = ( \beta m + \alpha_{i} P_{i} )^2$ and obtaining a satisfying algebra:   * $\beta \alpha_{i} + \alpha_{i} \beta = 0$   * $\alpha_{i} \alpha_{j} + \alpha_{j} \alpha_{i} = 0$   * $\beta^2 = \alpha_{i}^2 = 1$ However what happens if i don't assume a squared expression and just try to factorize the operator; I note that the most generic expression satisfying the above would be this: $$E^2 = ( \hat{\beta} m + \hat{\alpha_{i}} P_{i} ) ( \hat{\beta^{-1}} m + \hat{\alpha^{-1}_{i}} P_{i} )$$ where i use the hat to denote that these are more generic that the usual $\beta, \alpha_{i}$ that we deal with traditionally This system satisfies the original expression if we demand the following algebra;   * $\hat{\beta} \hat{ \alpha^{-1}_{i}} + \hat{\alpha_{i} } \hat{\beta^{-1}} = 0$   * $\hat{\alpha_{i}} \hat{ \alpha^{-1}_{j}} + \hat{\alpha_{j} } \hat{\alpha^{-1}_{i}} = 0$ The question here is: It seems to me that there is a gauge freedom in specifying these quantities; **what is the meaning of these gauge symmetries and what, if any, interesting invariants we should expected to get from them?** **Note 1:** if i assume the following representation of these hatted cliffords in terms of the well-known ones;   * $\hat{\beta} = \boldsymbol{L} \beta \boldsymbol{M} $   * $\hat{\alpha_{i}} = \boldsymbol{L} \alpha_{i} \boldsymbol{M}$ i notice that this satisfies the required algebra _(and makes the extra degrees of freedom explicit!)_ **Note 2;** it seems to me that the most generic solutions look like:   * $( \hat{\beta} m + \hat{\alpha_{i}} P_{i} ) \boldsymbol{u} = \lambda_{0} \boldsymbol{v} $   * $( \hat{\beta^{-1}} m + \hat{\alpha^{-1}_{i}} P_{i} ) \boldsymbol{v} = \lambda_{1} \boldsymbol{u} $ **is this just a rewrite of the left-hand and right-hand Weyl spinors, or is something different?** I have purposely not assumed any space-time or spinorial dimensionality in the above, but feel free to make your answers about concrete dimensions


title: Does the long axis of Earth's orbit precess? Does it nutate?

text: Get up off the plane of the ecliptic by a couple of million miles. Look back at the Sun and watch the Earth's orbit in time-lapse for a few centuries. The orbit is an ellipse tilted at 23.5 degrees from the ecliptic. Hold station with the Sun as it rotates about the galactic center. Question: Does that tilted ellipse -- Earth's orbit -- precess? Over millions of years, does the long axis rotate with the Sun's rotation in the galaxy? Or does it stay fixed in some larger frame of reference, such as the Local Group of galaxies? Or does it exhibit some more complex and subtle motion? Does the long axis nutate? Does the angle of the orbit to the ecliptic change in any periodic way? Why? What are the forces in play?


title: What Quark and Anti-quark are electrically neutral Pions made out of?

text: A positive pion is an up and an anti-down. A negative pion is a down and an anti-up. What's a pion with an electrical charge of 0?


title: Star Surface Temperature Vs. Mass

text: I am hoping that someone can clarify this for me. With these equations. (Boltzmann Law) and radius of Radius of star how does surface temp scale with mass > $R _x \approx R_\bigodot (\frac{M_x}{M_\bigodot})^.5$ > > $L _x \approx L_\bigodot (\frac{M_x}{M_\bigodot})^3.5$ > > $\frac{L}{\pi R^2} = \sigma T^4$ > > $T_\bigodot = 5800 K$ = Surface Temp of Sun What I have done is that I have substituted $R_x$ and $L_x$ in the 3rd equation and simplified it in terms of $T$ where my final expression is > $T = (\frac{L_\bigodot M_x^2.5}{\pi\sigma M_\bigodot^2.5 R_\bigodot})^0.25$ Am i done? What am I suppose to do next? Thanks


title: Does interference take place only in waves parallel to each other?

text: The title says it all but I will add some details. I believe that interference takes place only in waves which are parallel to each other. See the picture to understand what I mean by parallel. ![enter image description here](http://i.stack.imgur.com/omkAt.png) We see that in the image the interfering waves are parallel. If interference can happen in non-parallel waves then please give me an explanation but if it does not consider the double slit experiment. How are the waves parallel in the double slit experiment? ![enter image description here](http://i.stack.imgur.com/2GKvA.png) We see that in the double-slit experiment the waves are not parallel. Please answer accordingly as I have given both the situations.


title: QM - calculating expectation value for velocity of an electron

text: How do we calculate the expectation value for speed? I have heard that we must first calculate the expectaion value for kinetic energy. Someone please explain a bit what options do we have.


title: What is the geometrical interpretation of Ricci tensor?

text: In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So what's the geometrical meaning of Ricci tensor since it's been defined with the Riemann tensor as $$\mathrm{Ric}_{ij}=\sum_a R^a_{iaj}?$$


title: Why don't we see solar and lunar eclipses often?

text: Since we see the new moon at least once in a month when the Moon gets in between of the Sun and the Moon at the night and as far as I know if this happens during the day, you'll get to see a solar eclipse. Why don't we get to see this often or in the day? Does it mean that in some part of world there's a solar eclipse when we are seeing a new moon? I'm looking for a diagram or interactive way to understand this if possible as I'm not a native English speaker, but I'll try my best to do so.


title: Strange things about new moon

text: I have some strange and infantile questions about new moon. I want to know how is it possible that the Moon is not visible at night and also at day it is not Sun eclipse? I will explain the problem in a few images. ![enter image description here](http://i.stack.imgur.com/PURQc.png) ![enter image description here](http://i.stack.imgur.com/g0pBC.jpg) ![enter image description here](http://i.stack.imgur.com/ZijhQ.png) With these images, how it is possible that there is not solar eclipse at every new moon? Or is there somewhere on Earth?


title: Center of Mass of two planets

text: The formula for calculating the center of mass is $$ r_{center} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1+m_2} $$ Why can't I use it to calculate the barycentre of two planets? ![](http://i.stack.imgur.com/KVVK5.png) I understand how to use ratios to approach this problem. $$ \frac{r_1}{r_2} = \frac{m_1}{m_2} $$ $$ r_1 = \frac{m_2}{m_1+m_2}\cdot (d_{Sun->Earth}) $$ However, I do not understand why I can not use the $r_{center}$ formula.


title: Does relativistic mass exhibit gravitiational effects?

text: **Groundhog Day Update, 2014** The simple and dumb way to ask my main question is this: If something like a neutron start goes sailing by at very close to the speed of light, say fast enough to double its total mass-energy, do you "feel" the energy it is carrying as gravity as it passes by? I'm about 99% sure the answer is yes, but I'd sure appreciate some confirmation of that. I was surprised (and appreciative) at the complexity of the first two attempts at answers, but must confess I thought it was a simpler problem than that. For one thing, frame equivalence should not an issue, since in terms of gravitational effects it doesn't matter whether it was you or the star that doubled in mass. And since the mass used to speed up the star cannot simply disappear, its gravitational pull has to go _somewhere_ , so why not in the star itself? I now gather that such a simple argument gets very, very messy when expressed in tensor form. Ah... oops, but again, thanks. And I _think_ that's what John Rennie answered... * * * Contingent on relativistic mass actually having gravitational effects, I think I can now answer my second question myself. Imagine a self-contained cluster of masses within a compact region of space, all initially motionless relative to each other. At a large distances the gravitational behavior of the cluster will asymptotically approach that of a single large gravitational mass, one equal in magnitude to a simple sum of the individual masses in the cluster. Next, have the cluster go crazy, all by itself and without any external stimulus. Huge parts of it are annihilated into pure energy, which in turn drives other parts outward at relativistic velocities. (If that sounds a bit outrageous, look up black hole jets sometime.) Since there are no external influences, the outbound parts must have momenta that sum to zero, with the simplest case being two equal-mass objects moving in opposite directions at the same velocities. Now from a great enough distance, the cluster will continue to look asymptotically close to a single mass that is still equal to the sum of the original cluster masses. To that distant observer, the conversion of huge chunks of the cluster into pure momentum makes not a whit of difference: The cluster still has exactly the same mass-energy, with the only difference being that it is getting harder to approximate as a point. So _of course_ the few chunks of the original cluster that were never accelerated during the explosion will look a bit unique to the distant observer. In particular they will appear to have the lowest masses, since they did not acquire any of the converted mass energy during the explosion. Nothing profound, that... yet still interesting, especially as you broaden the argument to include larger and larger assemblages of mass-energy. The black holes at the center of most (all?) galaxies would be examples of mass-gravity minimum frames, and their dual jets examples of entities with excess gravitation. * * * Oh, and one other point: Will stars or mass (large jets?) moving at relativistic speed exhibit higher levels of gravitational lensing? I would assume so... My original more specific and inadvertently homework-like version of the above thought experiment is below. * * * Start with five large objects $\\{m_a,m_b,m_c,m_d,m_e\\}$ of equal mass $m$. Oddly, $m_b$ is made of antimatter. $m_a$ remains unchanged at the center of mass of the group. $m_b$ and $m_c$ are mutually annihilated. Their energy is used to launch $m_d$ and $m_e$ along ${\pm}x$ paths. The energy imparts to each a velocity as measured from $m_a$ of $(\sqrt{\frac{3}{4}})c$, which in turn gives $m_d$ and $m_e$ each a relativistic mass of $2m$, again as measured from $m_a$. The annihilated masses of $m_b$ and $m_c$ have in effect been "added" in the form of momentum to the rest masses of $m_d$ and $m_e$.   1. In terms of _gravity_ , where in the resulting three-body system do the masses of $m_b$ and $m_c$ reside?   2. If you answered "$m_d$ and $m_e$", what happened to frame invariance? * * * **Notes** A related (but definitely different) question is: Does the increase of (relativistic) mass, while flying near speed of light, has any impact on astronauts? I could not find any exact matches, but I also cheerfully acknowledge that my question search skills are not as good as some on this group.


title: Diagonalization of a hamiltonian for a quantum wire with proximity-induced superconductivity

text: I'm trying to diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor it's all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian in > Tudor D. Stanescu and Sumanta Tewari. “Majorana fermions in semiconductor > nanowires: fundamentals, modeling, and experiment.” Journal of Physics: > Condensed Matter **25** , no. 23 (2013): 233201. arXiv:1302.5433 [cond- > mat.supr-con]. given by $$ H = \eta_{k}\tau_{z} + B\sigma_{x} + \alpha k\sigma_{y}\tau_{z} + \Delta\tau_{x} $$ Here $\sigma$ and $\tau$ are the Pauli matrices for the spin and particle-hole space. Now the correct result is: $E^{2}_{k} = \Delta^{2} + \eta_{k}^{2} + B^{2} + \left(\alpha k\right)^{2} \pm 2\sqrt{B^{2}\Delta^{2} + \eta^{2}_{k}B^{2} + \eta^{2}_{k}\left(\alpha k\right)^{2}}$ Now, my problem is that I don't know how I can bring the Hamiltonian in the correct matrix form for the calculation of the eigenvalues. If I try it with the upper Hamiltonian I have completely wrong results for the energy spectrum. I believe my mistake is the interpretation of the Pauli matrices $\tau$ but I don't know how I can write the Hamiltonian in the form to get the correct eigenvalues.