title: In spontaneous symmetry breaking, global symmetry broken by gauged subgroup?

text: My question is simple. Given a group $G$ broken to a subgroup $H$, gauging a possibly different subgroup Hg breaks explicitly the global symmetry $G$, generating what is known as pseudo-Goldstone bosons. Why is this? The usual answer I get is that the gauging determines a specific direction in field space, but I really don't understand this statement, how having a subgroup $Hg$ gauged can break the global $G$ explicitly?. What I have in mind is that inside the gauge transformations $Hg$ there are included global ones as well, so this is what is confusing me.


title: What is the alternative of PMG (Permanent Magnet Generator) in a Horizontal Axis Wind Turbine?

text: I need to make a report about a HAWT (Horizontal Axis Wind Turbine) and list the parts with their costs, but as a Permanent Magnet Generator is a costly thing so what can I use in alternative of this to make the turbine less expensive? P.S. Due to the short time, I didn't do any kind of research on this PMG part's alternative.


title: Projectile motion question

text: A ball is struck such that its max height is equal to its range. determine the angle its struck at. Air resistance is ignored and the ground is horizontal. my reasoning is that: using vectors Xsin(angle) = Xcos(angle) so tan(angle) = x/x so inverse tan of 1 is 45degrees Would this be the right approach to the question Id ask my lecturer but im at home Cheers


title: Tunneling Rate Constant

text: I am trying to "decode"/derive an expression for the macroscopic rate constant for the tunneling of protons through a potential energy barrier that I read in a journal article: $$ k_{\rm tun}(T)=(2\pi\hbar)^{-1}\int_0^{V_{\rm max}} Q(V,T) P_{\rm tun}(V)\ dV. $$ So basically: the authors say work out the probability of tunneling at each point up the potential energy barrier (from 0 to $V_{\rm max}$), multiply this by the Boltzmann factor ($Q(V,T)$), integrating over all energies (i.e. we are taking a Boltzmann average )and then convert from energy in J to rate by multiplying by $(2\pi\hbar)^{-1}$. The authors use $(2\pi\hbar)^{-1}$ to convert from an energy in J to a rate in per second. This implies that the relationship between energy and frequency being used is: $$ E=\hbar\omega $$ I thought this just applied to electromagnetic radiation and free particles not in a potential. Is it OK to use this relation for protons tunneling in a potential? Or should you use some info about the shape of the potential? Many thanks N26


title: Physical interpretation of equation for relativistic aberration

text: I'm working in a book on relativity. The author states that if $u$ and $u'$ are a velocity referred to two inertial frames with relative velocity $v$ confined to the $x$ axis, then the quantities $l$, $m$, $n$ defined by $$ (l, m, n) = \frac{1}{|u|}(u_x, u_y, u_z) $$ and $$ (l', m', n') = \frac{1}{|u'|}(u'_x, u'_y, u'_z) $$ are related by $$ (l', m', n') = \frac{1}{D}(l - \frac{v}{u}, m\gamma ^{-1}, n\gamma ^{-1}) $$ and that this can be considered a relativistic aberration formula. The author gives the following definition for $D$, copied verbatim. $$ D = \frac{u'}{u}\left( 1 - \frac{u_xv}{c^2}\right) = \left[1 - 2l\frac{v}{u} + \frac{v^2}{u^2} - (1 - l^2)\frac{v^2}{c^2} \right]^{\frac{1}{2}} $$ My question is why the author finessess the expression into the third/final form. I was able to get it but it seems like a pain. Why is that better than the second expression? It seems more difficult to calculate and not at all clear or meaningful. * * * Also, in case it's not clear, $\gamma =1/ \sqrt{1 - \frac{v^2}{c^2}}$ and $|u| = |(u_x, u_y, u_z)| = \sqrt{u_x^2 + u_y^2 + u_z^2}$


title: Ohm's Law and Space-Charge Limited Current

text: I am trying to understand Space-Charge-Limited-Current (SCLC) and the resulting Mott-Gurney Law. My main problem is conceptual; I don't get why SCLC doesn't produce Ohm's law (j $\propto$ V).


title: Have the correlation functions of the XY spin chain model been calculated using a functional partition function with source terms?

text: Have the correlation functions of the XY spin chain model, \begin{equation} H=-\sum_l (J_x \sigma_l^x \sigma_{l+1}^x+J_x \sigma_l^y \sigma_{l+1}^y)-B\sum_l \sigma_l^z \end{equation} been calculated using a functional partition function with source terms? By functional partition function I mean the partition function plus source terms (or generating functional) which allow correlation functions such as $\langle \sigma_l^x \sigma_{l+1}^x \rangle$ to be calculated analytically: \begin{equation} Z[\bar{\omega},\omega] = Tr \left[Exp \left[-\int_0^\beta d\tau H(a^{\dagger},a) - \int_0^\beta d\tau \sum_l (\bar{\omega} a + a^{\dagger} \omega) \right] \right] \end{equation} where $H(a^{\dagger},a)$ is the XY Hamiltonian expressed in terms of spinless fermions (i.e. after a Jordan-Wigner transformation). **Edit:** Anyone? An Ising chain would be fine too!


title: Why does this screwdriver roll in a curved trajectory?

text: I am trying my level best to give this screwdriver a horizontal push only, but it is going in a circular path. I thought of uneven friction due to non- uniform normal reaction...what do you think? I have read up torque a bit, but I dont have a clear understanding of it..does it have anything to do here? here is the video: https://docs.google.com/file/d/0BzGx6LKK8aWkU2pYaGJMdFZZZkk/edit?usp=sharing if the above doesn't work: http://www.youtube.com/watch?v=B21BBV0PUXo&feature=youtu.be


title: Relevant operators in two dimensional O(n) models

text: The most general hamiltonian of a two dimensional $O(n)$ and $Z_2$ invariant statistical model can be written: $$ H=\int d^2 x \left[\frac{\nabla \mathbf{\phi}^2}{2} + \frac{m_0^2}{2}\mathbf{\phi}^2 +\sum_{j,k,l}g_{k,l}(\mathbf{\phi}^2)^k\phi_j\Delta^l \phi_j\right] $$ In two dimensions the vector field $\phi$ is dimensionless at the gaussian fixed point, therefore I found difficult to guess which are the relevant operators and the corresponding constants $g_{k,l}$, I would like to ask if there is a simple way to estimate which operators are important to consider near the Wilson-Fischer fixed point without solving the problem.


title: Violation of Newton's 3rd law

text: I'm just expressing my guess. Let two particles A and B experiences forces $F_1$ and -$F_2$ between them and let guess also there are two observer, one is stationary and other is moving with constant speed. Now my measurement time for the collision will not same for the moving observer. So If I am stationary, the moving observer may say that my measurement is wrong that means Newtons third law is violated? Is it right?


title: Physical interpretation of changing variables $S(T,V)$ to $S(T,p)$

text: I'm confused in this situation. In an example of this book (Example 16.5. Thermal Physics, by Blundell) they ask to prove that $$C_P - C_V = VT\frac{\alpha^2}{\beta_T}\tag{1}$$ but using the fact that $S=S(T,V)$. Then they use the fact that $$C_V = T\biggl(\frac{\partial S}{\partial T}\biggr)_V\tag{2}$$ and that $$C_P = T\biggl(\frac{\partial S}{\partial T}\biggr)_P\tag{3}$$ My **question** is, the relation (3) is assuming that $S=S(T,p)$, so what happened with the initial assumption that $S=S(T,V)$? Or is there a way to obtain (3) without using the fact that $S=S(T,p)$? Finally, if the only way of obtain (3) is assuming that $S=S(T,p)$, then what is the physical meaning of the change $S=S(T,V)$ to $S=S(T,p)$? I mean, how can I justify that change in the variables?


title: Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

text: By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots a_pb_1\cdots b_{n-p}}$ Here, I assume the manifold is equipped with a metric $g_{ab}$ and the volume element is compatible with the metric, ie., $\epsilon_{a_1\cdots a_n}=\sqrt{|g|}\tilde\epsilon_{a_1\cdots a_n}$ with $\tilde\epsilon_{a_1\cdots a_n}=+1$ when $a_1,\cdots,a_n$ is an even permutation of $1,2,\cdots,n$; $=-1$ when old permutation; $=0$ when there are repeated indices. From this formula, I guess we can calculate the Hodge dual by first take the tensor product of $\omega_{a_1\cdots a_p}$ with $\epsilon_{a_1\cdots a_n}$ and then contract the first $p$ indices. Am I correct?


title: Naive Uncertainty principle for string theory

text: Is it possible, in some sense, that a naive uncertainly principle for string theory could be expressed as : $$ \Delta x_i \Delta p_j \Delta \sigma ~=~ \delta_{ij} \hbar \ell_s$$ where $\ell_s$ is the string length scale, and $\sigma$ a proper distance?


title: No cloning theorem and exclusive-or (XOR) operator

text: According to IBM's website, > _[...]where we would [classically] have done an assignment (x=y), we instead > initialize the target (x=0) and use exclusive or (x^=y)._ This sounds like x is a copy (clone) of y, however cloning is impossible in quantum mechanics. What's going on?


title: Long/short-range interaction

text: A potential of the form $r^{-n}$ is often considered long-range, while one that decays exponentially is considered short-range. Is this characterization simply relative/conventional, or is there a more fundamental reason?


title: Breaking of E6 to SO(10) in heterotic string theory

text: Some of the heterotic string models have an $E_6\otimes E_8$ symmetry. Examples include some orbifold models, some free fermionic models and Gepner models. We can break the gauge symmetry by including Wilson lines. Does anyone know if it is possible to break $E_6$ to $SO(10)$ while maintaining spacetime SUSY? I have seen examples in Gepner models where $E_6$ is broken to a smaller group than $SO(10)$ while keeping SUSY but none where $E_6$ is broken to $SO(10)$. Is this pure coincidence or is there a reason behind it? Any insights are appreciated!


title: Mass of particle near light speed in a medium

text: I am trying to get a common understanding from these two previous questions:   1. Why does the mass of an object increase when its speed approaches that of light?   2. What happens if light/particles exceeded the speed of light for a particular medium (sic)  Does the increase of mass occur only if the particle approaches _c_ (speed of light in a _vacuum_ ) or if it simply approaches the speed of light in its current medium? For example, does the mass of charged particles increase during Cherenkov radiation?


title: Zero Point Fluctuations

text: The total energy of a mode in a quantum mechanical resonator is given by $E_n ~=~ (n+ 1/2)hf$ where $n$ is the number of modes. So when there are no modes or vibrations, i.e. $n=0$, the energy is called the zero-point energy. What I don't understand is, if there are no modes, then what is this energy associated with? So what exactly is a mode? Also, is there a way of measuring this zero-point fluctuations? Why is the 1/2 photon introduced?


title: Hermitian Operators in time and Measurements

text: Consider an observable that can be described by a hermitian operator $A$ . No explicit relationship with time is given. What would happen to the probability if the quantity is measured a few days later?


title: Calculating outside temperature based on indoor measurements

text: If I take three consecutive measurements $T_0$, $T_1$, $T_2$ at equal intervals, I can end up with two (Newton’s law of cooling) equations where the $e^{-kt}$ term is the same. $$T_{n+1} = T_{outside} + (T_n - T_{outside}) × e^{-kt}.$$ So, without knowing the temperature constant of the building, can I calculate the outside temperature? $$T_{outside} = \dfrac{T_0 × T_2 - T_1^2}{T_2 + T_0 - 2T_1}.$$ Have I missed something? Or is this really possible? (Provided no other sources of cooling and heating)


title: Lower bound for the spin in particle decay process

text: Is my logic right? Suppose there is a particle $p$ that can either decay into $ \\{$a spin-1 and a spin-0 particle$\\}$ or two spin-0 particles, then the lowest possible spin of $p$ is 2. This is because we need the spin to be even and large enough to accommodate the spin-1 product. * * * ADDED: $p$ is such that $p\to \pi^-+\rho^+$ and $p\to \xi+\xi$ where $\xi$ is a particle of spin 0. I want to know the lowest possible spin for $p$. I am assuming that conservation of total angular momentum and parity holds. By the way, would the intrinsic parity of $p$ be negative, since I think each of $\rho^+$ and $\pi^-$ have negative intrinsic parity?


title: Problem evaluating moles in a an isochor transformation

text: I have a problem with an isochor transformation. Me and my group of study made an experiment that want to check Gay-Lussac’s law. We registered the equilibrium states and fitted the $P = nRT / V$, in the $P(T)$ graph. The moles found are $n_f=(5.63 \times 10^{-3} \pm 1.24 \times 10^{-4})$ mol. The problem is that, initial moles are $n_i=(3.19 \times 10^{-3} \pm 6.97 \times 10^{-5})$ mol. How can the initial moles be smaller? _Details of the experiment_ : We started stated with a certain number of moles of air inside a metal cylinder closed with a rubber top and connected, through it, to a temperature and pressure sensor. The starting gas conditions are $V, P_0, T_0$ assuming the volume to be constant (the container is, in a good approximation, hermetic). We heated an amount of water in a calorimeter with a resistance and a temperature $T' \gg T_0$, then we put the cylinder in contact with the hot water plotting the trend of $\Delta P-T$ on a graph. From a certain point the trend gets linear, and with a fit we calculated the slope $m = nR/V$ and from the latter we calculated the number of moles $n$ assuming the air as an ideal gas because of the higher temperature. We checked the consistency between this value and the one calculated from the initial conditions $n' = P_0V/RT_0$ and surprisingly this value is smaller than the one calculated from the fit. Now what is really happening? the difference $n'-n < 0$ suggests that the number of moles in the cylinder is growing? Or maybe it might be that one of our assumptions is incorrect.


title: Decay of a particle

text: Would someone please explain the following found on P. 125 of these notes? > On the other hand, two $π^0$’s cannot be in an $l = 1$ state. The reason for > this is that pions are bosons and so the wavefunction for two identical > pions must be symmetric under interchange, whereas the wavefunction for an > $l = 1$ state is antisymmetric if we interchange the two pions. This means > that the decay mode $$\rho^0\to \pi^0+\pi^0$$ is forbidden. I don't understand why the wavefunction of $l=1$ must be antisymmetric. Perhaps I have forgotten something? Thanks.


title: A dielectric sphere in an initially uniform electric field and representation theory of SO(3)

text: I learned recently that the highest order spherical harmonic required to represent the spatial distribution of decay products of a particle can be used to determine its spin, by using arguments involving the representation theory of SO(3) / addition of angular momentum. I seek a similar argument for the well-studied problem of a dielectric sphere in an otherwise uniform electric field. This is, for example, solved in Griffiths' Electrodynamics, section 4.7 on the Third Edition (I'm using a Low Price Edition from India, where it is on Page 205). He concludes at the end of that problem that "the field inside is (surprisingly) uniform". I have a feeling that this result is less surprising when one applies ideas from SO(3) representation theory, but I'm not sure how to formulate the argument precisely. Here is my loose train of thought -- the field is a vector field with a specific direction, so it has one unit of angular momentum (or is given by an $l = 1$ spherical harmonic). Therefore, the induced surface charge on the dielectric sphere must be given by $P_1(\cos \theta) = \cos(\theta)$. The field inside is a vector field originating from these charges, so it must again be uniform. I'm also trying to use this to determine the highest order (i.e. highest $l$) component of $Y_{lm}$ in the total electric field after accounting for the polarization of the sphere.


title: Charge Distribution on a Parallel Plate Capacitor

text: If a parallel plate capacitor is formed by placing two infinite grounded conducting sheets, one at potential $V_1$ and another at $V_2$, a distance $d$ away from each other, then the charge on either plate will lie entirely on its inner surface. I'm having a little trouble showing why this is true. In the space between the two plates the field $E = ( V_1 - V_2 ) / d$ satisfies Laplace's equation and the boundary conditions, from which I can derive the surface charge density is $\pm E / 4 \pi$. But how about the space above and below the capacitor? Certainly I can't just use superposition of the inner surface charge distributions to say that the field outside the capacitor is zero, (and thus the surface area charge density is zero), for this assumes there is no charge on the outer surfaces to begin with. Any help clearing up this mental block would be greatly appreciated, thanks.


title: What changes occur while an atom approches the speed of light?

text: My knowledge of atomic dynamics is a little superficial. But to my understanding an electron travels an orbital path around a nucleus of an atom. "correct?" So let's say that if a hydrogen atom were to travel "extremely close to" the speed of light (relative to some laboratory system). With the electron circling the proton approximately already traveling "extremely close to" the speed of light, how is it that the electron can circle the proton without exceeding light speed? (or does it continue to orbit at all?)


title: Entanglement and relativity

text: Two observers A and B measure a quantum entangled state and obtain correlated results, even if their separation is space-like (each is out of the light cone of the other). A possible interpretation is that the observer who makes the first measurement (say A) collapses the quantum state, thus fixing the result of the other observer's (B) measurement. But there are frames of references in which B's measurement comes first. This seems a paradox. What is the solution?


title: Why do we feel the passing of time?

text: Why do we feel the passing of time? Why do we feel the time is changing with increasing speed $d\tau=\gamma^{-1}dt$? In other word why Lorentz factor (or scientifically relativistic velocity function $\gamma_v$,) affect over our feel of time?


title: Elastic Spring And Energy Issue

text: I`m trying to write an expression ( Work & Energy ) of spring that is not loose and it has a box on it with a weight of 1Kg.   how its should be? this is what I tried to do:   $$\frac{kx^2}{2}-mg$$ this is an illustration of it.   ![enter image description here](http://i.stack.imgur.com/IjSfg.png) the $m$ box is $1kg$ and $k=200$ and the state is that the spring is not loose. I would like to get some advice how to do that.   thanks!    **EDIT**   Here is my full problem. I want to find the maximum contraction of the spring, The first mass falls on the mass already in the spring. so I chose to solve it with Work and Energy.   $$mg(3+x_{0})+\frac{k{x_{0}}^2}{2}-mg=\frac{k{x_{0}}^2}{2}-3g$$ this is the right expression to write?   E$_{end}$=E$_{start}$?    ![the full problem](http://i.stack.imgur.com/ygh77.png)


title: Why are *high redshift* measurements of supernovae required to measure dark energy?

text: Why are _high redshift_ measurements of supernovae required... in order to measure the equation of state parameter of dark energy? The luminosity distance can be written as \begin{equation} d_{L}(z) = \frac{c}{H_{0}}\left(z-\frac{1}{2}(1+q_{0})z^{2}\right), \end{equation} where \begin{equation} q_{0} = \frac{1}{2}\left(1+3w_{\small{DE}}\Omega_{\small{DE}}\right). \end{equation} Here, $c$ is the speed of light, $H_{0}$ is the Hubble constant, $z$ is the redshift of the galaxy hosting the observed supernova, $w_{\small{DE}}$ is the equation of state parameter of dark energy, and $\Omega_{\small{DE}}$ is its density parameter. I was taught that because $q_{0}$ doesn't enter until quadratic order, we need to measure _high_ -$z$ supernovae. However, there are a lot of practical issues in measuring _high_ -$z$ supernovae. Now consider this. We can also write \begin{equation} d_{L}(z) = c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H_{0}\left[ \Omega_{\small{M}}(1+z')^{3}+\Omega_{\small{DE}}(1+z')^{3(1+w_{DE})}\right]}. \end{equation} Here, I cannot see any reason to prefer high-$z$ supernova of low-$z$ supernovae. It would seem, then, that we can avoid the practical difficulties by simply using a different formula. Where am I going wrong?


title: Question related to work done and energy

text: Suppose you lift a box from the ground over your head. You will be using energy as there is work done. But when you keep the box lifted over your head, aren't you using any extra energy?


title: Different approaches to calculating the Christoffel symbols

text: I would be very grateful to whoever can debug the following calculations... We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$ I want to find the associated Christoffel symbols. I know of 2 ways: The **_first_** is to use the formula $$\Gamma^a_{bc}={1\over 2}g^{ad}(g_{dc,b}+g_{db,c}-g_{bc,d})$$ Using this formula, I get $$\Gamma^0_{0i}={\partial U\over \partial x^i}$$ for $i=1,2,3$. However, by using a **_second_** way -- by reading off the Euler-Lagrange equation ,I got an extra factor of $2$ and I can't spot where I have made a mistake. Here is what I did: The Lagrangian $$L=\exp(2U(\vec x))\dot t^2-h_{ij}(\vec x) \dot x^i \dot x^j$$ So by the Euler-Lagrange equation for the $t$ component, $${d\over d\tau}(\exp(2U(\vec x))\dot t)=0$$ which leads to $$\ddot t +2{\partial U\over \partial x^i} \dot x^i \dot t=0$$ Therefore giving $$\Gamma^0_{0i}=2{\partial U\over \partial x^i}$$ What went wrong, where? Are my approaches and formulae correct? Thanks.


title: Is the concept of a field necessary to electrodynamics?

text: I've read (in Griffith's text) that it is "possible, though cumbersome" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory. What exactly is meant here? Since fields physically exist, why is this permissible? Edit: In light of the answers below, I'm curious about the apparent contradiction here - if fields physically exist, then how can one dispense with the field concept?


title: What states are satisfying an entropic area law and why do they satisfy it? More specificly why do matrix product states satisfy it?

text: I am currently reading some papers concerning the question why the density matrix renormalization group (DMRG) method is working well for simulating one dimensional systems and bad for higher dimensional systems. Usually the literature claims that certain states created by DMRG, the matrix product states, satisfying an "entropic area law" are simulatable in one dimension and not in higher dimensions. As far as I understand obeying an "entropic area law" means that the entanglement entropy of a reduced subsystem is proportional to the border of the subsystem and not the volume of the subsystem. It is unclear to me, why this is a reasonable assumption. I know of one big review article concerning the subject (http://arxiv.org/abs/0808.3773). Unfortunately especially for the matrix product states it is said that the entropic area law, follows trivially from the definition, but i can not follow that. Regarding the other examples given in the article, i am somewhat overwhelmed and find it hard to see what they have in common and where the differences are. What states feature an entropic area law and more importantly why do they exhibit it? More specifically why does the ground state of a matrix product state exhibit the property, while other states don't (I read that matrix product states do that, but I don't see why)? P.s.: I am also thankful for counter examples, that could help illustrate the qualitative difference between states satisfying the are law and states not satisfying it.


title: What force creates ions out of neutral atoms?

text: Consider the reaction between Na and Cl to form NaCl. Na loses an electron and "gives" it to Cl because this makes both atoms more stable. But what forces "pulls" the electron from Na to Cl? Both atoms are electrically neutral at the beginning, so the electron that is going to be transferred isn't acted on by electrostatic forces. Since its not electrostatic forces that pull the electron from Na to Cl (since both atoms are neutral) what force pulls the electron from Na to Cl to make both atoms more stable?


title: Neutron stars and black holes

text: The official limits for a neutron star is $1.4 - 3.2\;M_\odot$. But I read that the limit depends on the particular structure of a star to estimate which mass it must have. I also read that neutron stars with less than $1.4\;M_\odot$ were observed. Given this information, I wonder if we can be sure that our Sun has definitely not enough mass to become a neutron star. Are there absolut limits (without the need of further information) for a star to become a neutron star or a black hole ?


title: How to smooth the spectrum of a light source?

text: Could somebody please tell me if there's a reasonably cheap substance or device can I use to smooth the spectrum of a light source? For example, if the spectrum has spikes as in the blue graph below, is it possible to smear in terms of wavelength (not spatially, as in ordinary diffusers) so that it resembles the red graph more? ![smooth_spectrum.png](http://i.stack.imgur.com/LFaeM.png) The actual shape of the smoothing (convolution kernel) is not really important, but it is preferable that most of the energy is transmitted. I've searched for "spectrum flattening" filters, but they instead attenuate certain wavelengths rather than spread them out. I suppose it's possible to rapidly move the light source forward and back so that the Doppler effect would cause a similar shift, but the speeds required would be tremendous!


title: Magnetic monopole

text: I've been watching a lot of videos claiming to make monopoles either by dropping the temperature to 5k (-268° C) and splitting the magnet or by pouring molten metal through an induction system and was just wondering, how true these systems are?


title: What physical conditions would allow for this kind of perpetual seesaw

text: I'm working with a simulator(Box2D) and need to create these conditions. I have a perpetual seesaw with two objects on either side. I'd like for the following conditions to be met. > The first object will be on the seesaw and the second object will be dropped > from a height. This will launch the first object and vice versa. > >   1. When their masses are the same, and the first object is dropped from a > height, the second object should be launched to the same height. >   2. If a heavier object is dropped from a height, the lighter object has to > be launched higher >   3. If a lighter object is dropped from a height, the heavier object has to > be launched lower. > Under what conditions would this work?


title: Tensor decomposition under $\mathrm{SU(3)}$

text: In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning that it is also traceless, $v^j_j =0$), equal $$ \dfrac{1}{2}\left(u^{i}v^{j}_{k}+u^{j}v^{i}_{k} - \dfrac{1}{4}\delta^{i}_{k}u^{l}v^{j}_{l} - \dfrac{1}{4}\delta^{j}_{k}u^{l}v^{i}_{l} \right) + \dfrac{1}{4}\epsilon^{ijl}\left(\epsilon_{lmn}u^m v^{n}_{k} + \epsilon_{kmn}u^m v^{n}_{l}\right) +\dfrac{1}{8}\left(3\delta^{i}_{k}u^l v^{j}_{l} - \delta^{j}_{k}u^{l}v^{i}_{l} \right). $$ I understand that initially (the first term in the above equation) he symmetrizes with respect to the two upper indices while at the same time he subtracts the traces. In the second term above, I can see that he lowers two indices with the invariant $\epsilon_{lmn}$ but I don't understand why he then symmetrizes! ($\epsilon_{lmn}u^m v^{n}_k \propto A_{lk}$ and he symmetrizes $A_{lk}$). While at the last term he adds the traces he subtraced with a spurious 3 and signs I don't understand how he got. Can someone elaborate on how exactly one can generally decompose tensor products into irreducible parts? _Is there a connection with the Young Tableaux? With the Young tableaux one can easily conclude that $3\otimes 8 = 15 \oplus \bar{6} \oplus 3$, **however how does one translate the Young tableaux into tensor components to get the above expression?**_ Thanks in advance.


title: How to prove the relations between the variables in bispherical coordinates?

text: When I was reading papers about the forces between two charged spherical conductors in a uniform electric field, I was confused with bispherical coordinates the author used to describe the spherical conductors. The schematic diagram is as follows: ![enter image description here](http://i.stack.imgur.com/DW2JI.png) where $β= \ln(\text{PL}_1 / \text{PL}_2) , \text{a = OL}_1 = \text{OL}_2$; Here is a link about the bispherical coordinates. I want to know why all the points on the sphere (for example sphere 2) satisfy $β=β2$. What's more, the relations between $\text{xyz}$ and $\text{aβα}$ are not entirely clear to me.Thank you!


title: A paradox to Lenz's law

text: I have read that in simple words, Lenz's law states that: > _The direction of current induced in a conductor is in such a fashion, that > it opposes its cause._ This validates law of conservation of mass-energy. I arranged the following thought experiment: Let there be a pendulum with its bob being a small bar magnet. The pendulum is oscillating in a direction parallel to the horizontal axis of the bar magnet on which the North and South poles lie. Also, the pendulum is in complete vacuum. (But gravity is there to make the pendulum oscillate.) At one of the extreme positions of the pendulum, we keep a solenoid, ends of which are connected to a load resistance. As the North pole of the bar magnet approaches the solenoid, current is induced in the solenoid in such a fashion that a North pole is formed at the end near to the bar magnet's North pole, and the bar magnet gets repelled towards the other side. The bar magnet then goes to the other end and then comes back (as a pendulum does) and again the same process is repeated. This _should_ go on forever, and current _should_ keep appearing across the load resistance. How does the law of conservation of energy hold here?


title: velocity function in slip effect

text: ![enter image description here](http://i.stack.imgur.com/piW2Y.jpg) $$\frac{dp_l}{dx}-\mu_l\frac{\partial^2 u}{\partial y^2}=0$$ where $\mu_l$ and $p_l$ is the liquid phase viscosity and pressure, respectively; and $u$ is the flow velocity. The boundary conditions of Eq. above can be readily determined by : $$y=0 \rightarrow u=u_s$$ and $$y=\delta \rightarrow \frac{\partial u}{\partial y}=0$$ and $$y=0 \rightarrow u_s=\beta \frac{\partial u}{\partial y} $$ how can i find $u$ as i function of $y$ ,$\delta$, $\beta$ and $\frac{dp_l}{dx}$?


title: What are the arguments for gravity not being a force? (in quantum gravity)

text: In quantum gravity the standard assumtion is that gravity _is_ a force, although there is a small but persistent group of theorethical physicists who think otherwise. What gives us the motivation to dismiss the _graviton_ and go for the unortodox theories such as loop quantum gravity, causal dynamical triangulation or any emergent gravitation? What is wrong with mass being a charge analoguosly to electromagnetism? To me it does not seem undesirable just yet. Is the non-renormalizability regarded as a technical issue or a fundamental one? Most importantly: Is it reasonably proven(or assumed) that one cannot quantize gravity preserving _Lorentz-invariance_ and the _equivalence principle_ without introducing further non-quantum ugliness(such as compactified extra dimensions)? _reason for asking is:I am trying to learn the motivations for alternative theories of gravitation(those that do not contain a graviton at all)_


title: Confusion about Dirac mass term

text: In chiral basis, $\psi=\begin{pmatrix} \psi_L\\\ \psi_R \end{pmatrix}$ and therefore, $\overline\psi=\psi^\dagger\gamma^0=\begin{pmatrix} \psi^\dagger_L & \psi^\dagger_R \end{pmatrix}\gamma^0=\begin{pmatrix} \psi^\dagger_R & \psi^\dagger_L \end{pmatrix}$. Hence, by matrix multiplication, $\overline \psi \psi=\psi^\dagger_R \psi_L+\psi^\dagger_L \psi_R$. Again using chiral projection operators it can be shown that, $\overline\psi\psi=\bar\psi_R \psi_L+\bar\psi_L \psi_R$. Therefore, these two relations suggest that, we can also write, $\overline\psi$ as $\overline\psi=\begin{pmatrix} \bar\psi_R & \bar\psi_L \end{pmatrix}$.   1. Am I right? If yes, how can I show $\overline\psi=\begin{pmatrix} \bar\psi_R & \bar\psi_L \end{pmatrix}$starting from the $\overline\psi=\psi^\dagger\gamma^0=\begin{pmatrix} \psi^\dagger_R & \psi^\dagger_L \end{pmatrix}$? Moreover, this implies: $$\psi^\dagger_R \psi_L+\psi^\dagger_L \psi_R=\bar\psi_R \psi_L+\bar\psi_L \psi_R$$   2. Am I right? If yes, how can I prove the last relation starting from either side of it?


title: Dielectric Tensor vs. Conductivity Tensor in (Cold) Plasmas

text: I'm studying Waves in Cold plasmas right now, but I guess my question is generalizable. It's about the 4th Maxwell Equation in polarizable / conductive media: $\nabla \times H = \frac{1}{c} \frac{\partial D}{\partial t} + \frac{4\pi}{c}J $ Now from our electrodynamics lectures we know that $D = \epsilon E$ with $\epsilon$ being the dielectric tensor, but also the crucial modellization of Ohm's law for conductive media $J = \sigma E$ is well within our memory. So my first reflex would now be to just plug in everything and obtain an equation for $E$ in Fourier-space and see where I get to. However this seems never to happen in the literature, and ppl as Stix (Waves in Plasmas, Chap1) or Jackson establish the relationship $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$. This is now what confuses me severly:   $D$ arises from the argumentation that if we have **bound** charges in a medium, they will linearly answer with a polarisation $P$ so that $D = E + 4\pi P$. The conductivity involves a (to be specified) steady-state modell of **free** charges in response to a electric field. Although fundamentally different views, the bound and free modell seem to be connected by $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$, but I'd be glad for some physical motivation. In the end (for cold plasmas at least), we always use $J = n m v$ and $v$ from the equation of motion to obtain a relationship between $J$ and $E$, thus obtaining $\sigma$. But there are also sources which do stuff with the dielectric tensor which I dont really understand (like Padmanabhan, Theoretical Astrophysics Vol I., Chap 9.5). So concluding, my questions would now be the following:   * Naively plugging in $D = \epsilon E$ and $J = \sigma E$ into Maxwell 4 seems to be wrong, also due to the physical reasonings behind those modells. We can't have both in the equation. True/False?   * Why can such a relation as $\epsilon = 1 - \frac{4 \pi \sigma}{i \omega}$ exist between a static and a dynamic modell? Or am I mislead here and the $i\omega$ hints at a dynamic origin for $\epsilon$ too? Thanks in advance, for anyone taking the time!


title: How can I make a simple climate simulator?

text: Given an Earth-like planet with arbitrary terrain, how can I create a simple climate simulator that can output parameters such as humidity and temperature, so that I can assign biomes to areas of land? My knowledge of climate modelling approaches zero, so to take a step back: is such a simple model even possible (i.e. one accessible to hobbyists), if so are there examples, if not where can I get started? My motivation is that, it seems to me that a lot of fictional worlds were created without applying climate models. Their creators often apply a very simple model (e.g. things are wet near the seas and get drier inland), most don't even try to determine things like rain shadows. I thought that we could improve a lot on this situation by applying some simple climate science principles. With this motivation in mind, I'm aiming more for verisimilitude than rigor. This recent story about some climate scientists modelling the climate of Middle Earth is in line with what I'm aiming for, although I read that they used HadCM3L which doesn't seem like something accessible to laymen. * * * To show what I've tried so far, I've done a bit of reading and come up with the following. It's been difficult for me because I'm a novice, and also a lot of available material seems to be focused on the climate change issue. Perhaps Paleoclimatology is more relevant? I have no idea.   * Prevailing winds, caused by Hadley cells, and affect cloud movement and creates surface currents and gyres   * Deep ocean currents, caused by salt and temperature differences near the poles and circulates globally, which combines with the gyres to form warm current flows   * Air masses, whose temperature and humidity are determined by whether they're over land or sea, create weather fronts and lots of precipitation between them   * Mountain ranges that, in the presence of humidity-carrying winds, cause orthographic lift and rain shadows afterwards I have no idea whether this is on the right track, and even if so, how far the rabbit hole goes (e.g. do I have to worry about the influence of moons/tides? What about effects of vegetation like forests?)


title: Constructing a Toffoli gate with 2-and 1-qubit gates?

text: I'm looking through Nielson's book on quantum computation and information and in part of it he says that any $C^2(U)$ gate can be constructed from two qubit and one qubit gates. I can't figure out how to do this, or how to verify it (fig 4.8 in his book) I've attached a photo of the diagram: ![](http://i.minus.com/i1JWvF4bKP1N1.png) Also: Is there an easier way to do this than multiplying 8x8 matrices? Right now I represent the first gate as $ I_1 \otimes\begin{pmatrix} I & 0 \\\ 0 & V \end{pmatrix}_{23}$ where $I$ is the identity matrix in for one qubit, and $V$ satisfies $V^2 = U$. $U$ is the unitary matrix being applied.


How is possible for current to flow so fast when charge flows so slow? We know electrons travel very slowly while charge travels at ~the speed of light.


title: Loop integral using Feynman's trick

text: I am trying to show for the one-loop integral with three propagators with different internal masses $m_1$, $m_2$, $m_3$, and all off-shell external momenta $p_1$, $p_2$, $p_3$ the following formula appearing in 't Hooft(1979), Bardin (1999), Denner (2007): (unfortunate metric $-,+,+,+$) $$\int d^d q\frac{1}{(q^2+m_1^2)((q+p1)^2+m_2^2)((q+p_1+p_2)^2+m_3^2)} $$ $$=i\pi^2\int_0^1dx\int_0^xdy\frac{1}{ax^2+by^2+cxy+dx+ey+f}$$ where $a$, $b$, $c$, ... are coefficients depending on the momenta in the following way: $a=-p_2^2$, $b=-p_1^2$, $c=-2p_1.p_2$, $d=m_2^2-m_3^2+p_2^2$, $e=m_1^2-m_2^2+p_1^2+2(p_1.p_2)$, $f=m_3^2-i\epsilon$. I don't really care about factors in fromt like $i\pi^2$. My **simple problem** is: I am totally unable to reproduce coefficients $d$, $e$ and $f$. The problem is, when I integrate over the third Feynman parameter, $m_3$ appears in all three coefficients $d$, $e$ and $f$. How do I squeeze the denominators to reproduce this formula?


title: Simple harmonic motion

text: A uniform straight rod of length $L$ is hinged at one end. It is free to oscillate in vertical plane. Time period of oscillation with small angular amplitude when a point mass of mass equal to that of rod is fixed with lower end is : $1$. ![enter image description here](http://i.stack.imgur.com/eqjPb.jpg) $2$. ![enter image description here](http://i.stack.imgur.com/Qjk1K.jpg) $3$. ![enter image description here](http://i.stack.imgur.com/w2WSl.jpg) $4$. ![enter image description here](http://i.stack.imgur.com/Evpiy.jpg) Hey,please could you help me solve this question. Actually,this seems to be a torsional pendulum and the time period in such a case is $2\pi\sqrt{I\over C}$ where C is the restoring torque.Now,my doubt is what should the restoring torque be?Should it be 2mg(mass of the rod as well as the one suspended)times the length..or something else because that is not giving me my answer. High school student,so please keep it simple.


title: Twistor theory?

text: There is a great deal of basic information available about String Theory, but very little about alternative theories such as Twistor theory. Can anyone give me, in layman's terms, an idea of the basic premise of Twistor theory and how it works?


title: metastable energy levels

text: can someone give me a good definition of "metastable energy levels" in a helium plasma (glow discharge). Am I right when I say : "by adding some neon in my helium plasma, I will obtain a molecular plasma of NeHe" ? I guess the metastable energy levels are the reason of forming NeHe, but I don't know why, can someone explain to me ? If both my guesses are wrong, please give me some explanation, if you can. thx


Jacobian, Lorentz and Fourier Transformation. I am confused with the physical interpretation/meaning of all these transformations.   1. As far as I understood, Jacobian transforms from one coordinate space to another (there are examples for Jacobian from cartesian to spherical so and so...). But Lorentz transform also do similar transformation from one coordinate space to another right? So, Jacobian is also a kind of Lorentz transform! Or else what is the physical difference?   2. Finally, I got confused with the Fourier transform as well! It also transform from one space to another!.  How can we explain the physical difference and physical usefulness of all these transforms? How we decide what transformation can be done in some physical situation? Can anyone please explain with physical example and comparing these three transformations?


title: Collision of two asteroids in the main belt

text: > If there are two asteroids $A$ and $B$ of equal mass (and density of > 3000kg/m^3), with the same semi major axis of $a=$2.2 AU with asteroid $A$ > having a circular orbit and no inclination, and asteroid $B$ has an > elliptical orbit with an eccentricity of $e=0.05$ and an inclination of > $i=5$ degrees..   1. Whats the best equation to use to calculate their collision speed?    2. How can this be related to the gravitational binding energy of an asteroid?


title: Center of mass of a cylinder from which liquid is flowing out

text: A cylindrical container is of mass M, radius R and height H. The cylinder is filled with a liquid of density D. The base of the cylinder has an outlet tap of radius r through which the liquid can flow out of the cylinder. Initially the center of mass of the cylinder and the liquid in the cylinder system is at a height of H/2 along the axis of the cylinder. At time t = 0, the outlet tap at the base of the cylinder is opened and the liquid starts flowing out. We want to study the locus of the center of mass of the system. As liquid flows out, the center of mass will start coming down till it reaches a minimum point ans then start rising up again until it returns back to the height of H/2 along the axis when the cylinder is completely empty. **Assumptions** :   1. The upper surface of the cylinder is closed.   2. The experiment is done at an altitude A above sea level. **Questions** :   1. Where will the center of mass of the system be at time t = T?   2. Find T at which the center of mass of the system will be lowest.   3. How much liquid will be present in the cylinder when the center of mass of the system is lowest?


title: Resonance and a tuning fork

text: I carried out this experiment in class: I struck a tuning fork with a hammer. The sound lasted for some time. However, when I connected the tuning fork onto a wooden sounding box, the sound lasted for a few seconds (much shorter than the previous case) But why is that? I know the reason is because of resonance but can someone please explain the reasoning behind these observations with detail.


title: Calculating Tr(log($\Delta_F$))

text: I am stuck with this problem for quite sometime. I have a propagator in the momentum representation (from this question), which looks like $$ \widetilde\Delta_F(p) = \frac{1}{(p^0)^2-\Big(\Big(\frac{n\pi}{L}\Big)^2+m^2\Big)+i\epsilon} $$ I wish to know how do go about calculating $Tr(\log(\Delta_F))$ for in general these kind of propagators. The propagator in the position representation would look like, $$ \Delta_F(x-x') = \sum_{n=1}^\infty\int\frac{dp_0}{(2\pi)^2}\frac{e^{ip_0(x^0-x'^0)}e^{i\frac{n\pi}{L}(z-z')}}{(p^0)^2-\big(\big(\frac{n\pi}{L}\big)^2+m^2\big)} $$ where I have replaced the integral over $p_z$ with a sum over $n$. EDIT 1 : With the given propagator I can write the Trace to be, $$ \text{Tr}\log{\Delta} = - \sum_n \int dp_0 \log{\bigg(p_0^2 - \bigg(\frac{n\pi}{L}\bigg)^2 + m^2\bigg)} $$ but this is divergent in both the limits of $p_0$ I suppose. I have not introduced any cut-off too. How do I renormalise this given the context of this problem. PS: Sorry, I am a beginner with QFT and path integral calculations. It would be helpful if I could get quite an explicit answer. **More precisely, I wish to know what is the meaning of $\text{Tr}(\log(\Delta_F))$.**


title: Name of unknown effect where liquid moves when placed on a jagged surface

text: I recently saw a video in which a water droplet, when dropped on a jagged surface (see photo), and whilst under the Leidenfrost Effect, moved. Does anyone know the name of this effect? ![Water droplet on a jagged surface](http://i.stack.imgur.com/T17yL.jpg)


title: Quantum State Function $\psi$

text: If I write in QM at an instant, that the quantum state that describes the particle completely at an instant $\psi(x)=\cos(6\pi x)$. Does that mean $|\psi(x)|^2dx$ after normalisation gives me the probability that particle will be positioned between $x$ and $dx$ ? What if instead of position it was a function for state of momentum/energy, could I apply born's rule for getting wave function for position ?


title: Inertial Frames in Special Relativity

text: According to the postulates of special relativity, all inertial frames are equal in all respects. Then how does it follow from this, that the space is isotropic and homogenous for an inertial frame and time is homogenous. Or does the former follow from the latter? Or are they two independent postulates about inertial frames? Can isotropy and homogeneity of space and homogeneity of time for an inertial frame follow from a perhaps even more fundamental postulate in physics?


title: Voltage Drop and Charge Flow

text: I am hoping someone can help me with understanding voltage drop and charge flow. Here is what I think I know. The voltage drop through a circuit must equal the voltage of the battery source, and there is also an electric field that is generated throughout the circuit which provides a force to move the electrons. If a voltage drop across a resistor causes the net voltage to be zero, then how does a current still exist? I first thought to answer this due to the electric field N/C providing a force on the electrons, but the force times a distance would equal Joules. Since J/C represents the units for voltage, and there is a voltage drop equal to the voltage of a battery, then how can charge still be flowing towards the end of the circuit?? I apologize if this question has already been asked on this site.. If so please point me that threads direction :)


title: Does limit $\hbar \rightarrow 0$ in Quantum Mechanics mean anything?

text: Assuming that I learn Quantum Mechanics first, and then I approach Classical Mechanics as a special case of Quantum Mechanics, I will definitely find the relationship between Quantum Mechanics and Classical Mechanics very confusing. I don't know how to make sense of what happens when $\hbar \rightarrow 0$. For one, you can't recover classical mechanics from quantum theory by setting $\hbar \rightarrow 0$. However, it is possible to recover classical mechanics from Schrodinger equations. So, does limit $\hbar \rightarrow 0$ in Quantum Mechanics mean _anything_? How should we interpret it? Or does the above contradiction reveal yet another flaw in the fundamentals of Quantum Mechanics?


title: Applications non-Newtonian fluid

text: Are non-Newtonian fluid difused in the "real world"? Which are most common or fantastic applications?


title: Metallic and glass sphere of same size released at a height

text: This question was in my exam today : A metallic and glass sphere of same size were dropped at same height. Which sphere would hit the ground first and why? I have thought about several things and cannot come to conclusion. An emf will be developed as electrons will move due to earth's magnetic field. So we have a charged sphere? What now? Magnetic force will be perpendicular to velocity hence no work done by it. If only someone could just point me in the right direction.


title: How do we determine what distant planets, stars etc are made of?

text: I remember this being covered somewhat back in school and I have casually read about it. I know it involves inferring from spectral analysis what physical properties an object may have right? Though I was stumped when trying to give a quick explanation to some friends who ridiculed the idea astronomers can know for certain what composition a distant object has. I'd love to personally understand what different test are involved but also a quick rebuttal to my friends. It's depressing when people belittle science just because they don't understand something. :/


title: Refraction problem - limiting radius of observation

text: I solved this problem but I don't know if the solution is correct. Please, correct me if I'm wrong. There are two mediums, 1 and 2, with absolute refraction indexes $n_1$ < $n_2$, and an observer in medium 1 with height H (measured from surface to his eyes) . What is the limiting radius R that the observer can look at the surface and see an object located in medium 2? (as a function of $n_1$, $n_2$ and H) Let $\theta_c$ be the critical angle of refraction. Then light that comes from medium 2 leaves the surface only for incident angles $\theta_2 < \theta_c$. Call $\theta_2 = \theta_c - \epsilon$, with $\epsilon > 0$. The refrated angle will be, by Snell's Law: $$\theta_1 = \sin^{-1}(\frac{n_2}{n_1}\sin(\theta_2))$$ Such that: $$R = \frac{H}{\tan(\frac{\pi}{2} - \theta_1)} = H\tan{\theta_1} = H\tan{(\sin^{-1}(\frac{n_2}{n_1}\sin(\theta_2)))}$$ But in the limit $\epsilon \rightarrow 0$: $$\theta_2 \rightarrow \theta_c = \sin^{-1}(\frac{n_1}{n_2}\sin(\frac{\pi}{2})) \Rightarrow \theta_2 \rightarrow \sin^{-1}(\frac{n_1}{n_2})$$ And therefore: $$R \rightarrow H\tan(\sin^{-1}(1)) = H\tan(\frac{\pi}{2}) \Rightarrow R \rightarrow \infty$$ But this conclusion seems absurd to me since you can only see the mirror of an lake at long distances.


title: What exactly is three phase power?

text: What is three phase power? How does it differ from normal AC current?


title: Capacitance of an arbitrary electrode configuration

text: I have to find the capacitance of a circuit element in three dimensions. I have the potential and charge in this cross sectional slice calculated using numerical methods. Does this value of charge have any useful physical meaning in 3D so that I can calculate the capacitance of a length of electrode? I will ignore edge effects in the third dimension for ease of calculation.


title: Which clock is the fastest inside an accelerating body?

text: The picture shows an accelerating spaceship with two clocks inside it. It is so far away from all other bodys that gravity is of no importance. Will the bottommost clock be slower than the topmost one? Or will both clocks have the same speed? ![Spaceship with two clocks inside it.](http://i.stack.imgur.com/yMaWm.png?editPost)


title: Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?

text: Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation? Does the latter not describe a spin-$\frac 32$ field? Why is the gravitino given by the Rarita-Schwinger-type representation rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation? This is related to a recent question I asked on gauge invariance of the Rarita-Schwinger field. Thanks!


title: Is renormalization associated with a volume scale or with an energy-momentum and length scale?

text: Given that real-space renormalization blocks together small volume elements to construct larger volume elements, is it more appropriate/helpful to consider the renormalization scale to be a volume scale instead of an energy scale?


title: Numerical Analysis of Elliptic PDEs

text: I am looking for an elementary reference regarding issues of stability in numerical analysis of non-linear elliptic PDEs, particularly using the finite difference method (but something more comprehensive, covering more sophisticated methods, is even better). Ideally, something aimed at a physicist or applied math audience would be easiest for me to read, but that is not a strict requirement. To be clear, implementation of the method (using your favourite software or programming language) is not the issue, I am more interested in the mathematics involved in numerical stability.


title: Relevance of SIC-POVMs to quantum information

text: What is the real relevance of SIC-POVMs (symmetric informationally complete POVMs) to concrete tasks in quantum information theory? A lot of work has been put into giving explicit constructions, and yet it seems that in many concrete applications that at first glance require SIC-POVM-like objects, one can do without them using e.g. suitably chosen set of random projectors. So, my question is - if the problem of constructing SIC-POVMs in arbitrary dimensions was solved one day, would that enable solving problems which can't be solved or approximated using different techniques now? SIC-POVMs seem to play a role in some approaches to quantum foundations (see e.g. Chris Fuchs, Bayesian approaches to QM that rely on the existence of SIC- POVMs), but it's unclear to me how "mainstream" such applications are.


title: What is the relationship between the second law of thermodynamics and evolution?

text: On one hand evolution seems to drive against the second law in that it creates a state of (locally) higher order.   On the other hand the second law seems to drives evolution - in the sense that it provides a constant fitness measure - essentially evolutionary fitness of life is a measure of its ability to maintain an ordered state in a system that naturally drives towards disorder. No entropy -> no evolution.   Wikipedia states: "and the second law follows because random chance alone practically guarantees that the system will evolve towards such thermodynamic equilibrium".   Which actually uses the word "evolve". But Evolution actually gives us a system that locally moves away from thermodynamic equilibrium giving us system of very high complexity (us!). What is the relationship between the 2nd law and evolution? Update:   Applogies the question is vague. I understand that taken as a closed system the 2nd law holds - hence my careful use of "locally" - I never meant to suggest that evolution => 2nd law doesn't hold.   The essence of what bugs me I think is that it is odd that a law which essentially state that disorder increases seems to be dirrectly linked to (i.e be a driving force behind) evolution - which has given rise to some highly complex systems (albeit within a closed system that overall obeys the 2nd law). Without entropy and a natural tendance for things to decay there would be no selection. The 2nd law seems to provides a fitness function that is constantly applied in realtime to provide selection necessary to drive evolution.   It seems ironic that a law that implies an overall move to a state of disorder seems responsible for peaks of extreem complexity - indeed for some of the most complex systems known to science. So maybe my question becomes:   Does entropy and the 2nd law drive evolution? Its it true that a law that states that a closed system moves to a state of increasing disorder can also be responsable for extreem peaks of order within that system (that overall obeys the law)?


title: Replacing an operator with its expectation value

text: While dealing with a _circling particle in an spherical symetric potential_ our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value - he denoted it just $L_z$ - of the angular momentum if $L_z$ is constant. Why is that? So we first had this equation: \begin{align} \underbrace{\psi (r,\varphi,\vartheta)}_{\llap{ \text{wave function in spherical coordinates}}} &= \exp\left[\hat{L}_z \frac{i}{\hbar}\, \varphi\right] \underbrace{\psi (r,0,\vartheta)}_{\rlap{\text{wave function in spherical coordinates at $\varphi=0$}}} \end{align} and we got this one (notice that there is no operator over an $L_z$): \begin{align} \psi (r,\varphi,\vartheta) &= \exp\left[L_z \frac{i}{\hbar}\, \varphi\right] \psi (r,0,\vartheta) \end{align} Anyway here is the spherical coordinate system we ve been using all the time (the blue spherical aure is supposed to be a spherical potential...): ![enter image description here](http://i.stack.imgur.com/G8aUE.png)


title: The role of state space composition in quantum computation

text: In a paper by Richard Josza and Noah Linden they argue that the way state spaces of composite systems are formed is a key aspect in the benefits of quantum computers. In (classical) phase space, two systems are composed by the Cartesian product, whereas in (quantum) Hilbert space, they're composed by the tensor product. This means that the amount of physical resource (for example, 2 level systems) required to get a state space of a given dimension can be exponentially smaller for quantum over classical states. So my question is whether _all_ benefits of quantum mechanics in computing rely on this feature. One way of asking this is, if you restrict yourself to a paradigm where your entire state can only consist of a single qudit (ignoring the unfeasibly large number of levels this would require), would all quantum complexity classes become equal to their classical equivalents?


title: Forms of the first law of thermodynamics

text: The first law of thermodynamics states that $$\frac{D}{Dt}(K+U)=W+H,$$ where K is the kinetic energy, U is the internal energy, W is the power of the external forces and H is the heat flux. I have seen (in my textbook) how this was reduced to the local law $$\rho\frac{De}{Dt}=\sigma:D-\nabla\cdot q+\rho\mathbb E,$$ where $\rho$ is the density, $e$ is the energy per unit mass, $\sigma$ is the stress tensor, $D$ is the rate of deformation tensor, $q$ is the heat flux vector, $\mathbb E$ is the internal heat generation per unit mass. And ":" is the double dot product. I want to show that the above is equivalent to the following: $$\rho\frac{D}{Dt}(e+\frac{v^2}{2})=\nabla\cdot(\sigma\cdot v)+\rho f\cdot v+\rho\mathbb E-\nabla\cdot q$$ Here, $v$ is the velocity and $f$ is the net body force per unit mass. (The book is _An Introduction to Continuum Mechanics_ by J. Reddy, a Cambrige edition.)


title: Where does the potential energy go when mass is lifted out of Earth's gravitational field?

text: There is a finite amount of energy needed to be spent on a rocket or a rock that is taken out of the Earth's gravitational field. Lets suppose the rock is taken far enough to have no effect of Earth's gravity. What happens to this energy? Who gets this? Where does this energy go? Assume that the rock went just outside the gravitational field with zero velocity. Now. Lets say at the gravitational boundary ( place were the gravitational effect is very small) you decide to convert the mass of the rock to pure energy. What will be the energy released ? $$ E = m c ^ 2 $$ or $$ E = m c ^ 2 + mgd $$ where d is the distance of the mass from Earth.


title: Why can x-rays pass and not radio waves?

text: If we stand in an elevator or other metal containers we lose radio signals etc the very same reason for which we cannot see across a block of metal that light cannot penetrate it. However x-rays and gamma rays can. Now if you see the working of Faraday's cages then from weak to very strong no electric field passes through them which is why they are safe! Now I would draw the simple conclusion that atleast one part of EM radiation must not be penetrating any metal surface and since EM radiations cannot survive solely as E or M radiation any sort of radiation penetrating metal surface must not be possible. But in fact x-rays do pass metal surfaces. How can this happen when clearly _atleast_ electric field part of radiation **must** not penetrate the block?


title: How can you distinguish between projections of quantum states?

text: Consider this problem in quantum cryptography: We have two pure states $\phi_1,\phi_2$ as input and constants $0 \leq \alpha <\beta \leq 1 $, where "Yes instances" are those for which $$\left|\left<\phi_1,\phi_2\right>\right| \leq \alpha$$ and "No instances" are those for which $$\left|\left<\phi_1,\phi_2\right>\right|\geq \beta$$ I need to devise a quantum circuit that accepts "Yes instances" with probability at least $p$, and "No instances" with at most probability $q$ for some $q<p$. How can I do this? Is there a reference in which this problem is solved? Thanks in advance.


title: Calculating a Deflection of Light Integral from Einstein's 1907 paper on Gravity and Light

text: $\frac{1}{c^2}\int_{\theta=-\pi/2}^{\theta=\pi/2} \frac{kM}{r^2}cos\theta ds=2\frac{kM}{c^2\Delta}$ An observation: $\cos\theta=\frac{\Delta}{r}$ ![enter image description here](http://i.stack.imgur.com/amILk.jpg)


title: Does a tidally locked planet have seasons?

text: Could a tidally locked planet have seasons? According to my understanding, a tidally locked planet rotates around itself exactly once per rotation around its sun. However, if the axis of rotation of the planet is significantly off from the axis of rotation around the sun, wouldn't it still have seasons? Or does tidal locking require that the axis of rotation of the planet about itself be perfectly orthogonal to the axis of rotation of the planet about its sun? If not, what would happen to a "tidally locked" planet with an axis of rotation "pointing towards" its sun (sort of like Uranus)?


title: Is the spin and charge of an atom a quantum or classical concept?

text: I have no idea whether these properties of an atom fall under quantum or classical physics, or perhaps both. Some clarification would be helpful.


title: weird terms in W boson self energy

text: I calculated the correction to the self energy of the W boson due to a fermion doublet (below I have n(e) for neutrino (electron), but it could be up and down quarks or just any 4th generation contribution). My result checks with the results from section 21.3 in Peskin, so I am fairly confident in the form below. I think of this as a fairly straight-forward calculation, but I am getting inconsistent results. The interaction comes from the term $$\mathcal{L}= \frac{g}{\sqrt{2}}W^+_\mu \bar{n}P_R \gamma^\mu P_L e + \frac{g}{\sqrt{2}}W^-_\mu \bar{e}P_R \gamma^\mu P_L n $$ so we get a single diagram with a electron (neutrino) running in the top(bottom) leg of the loop. For the self energy correction I get (with $\bar{MS}$ scheme): $$\Pi^{\mu \nu}(q^2) \sim \eta^{\mu \nu} \int^1_0 dx \left( x(1-x)q^2 - \frac{1}{2} ( x m_e^2 +(1-x)m_n^2)\right) \log \left[ \frac{(1-x)m_n^2 + x m_e^2 +(x-1)x q^2}{\mu^2} \right] $$ where I have dropped the $q^\mu q^\nu $ term.This seems innocuous enough however if I perform the x integration and then expand in the external momentum $q^2$ I get weird terms like $$\Pi(q^2) \approx(m_n^3-m_e^2)^3 \log \frac{m_e}{m_n}\frac{1}{q^2}$$ which seems incorrect since this should correspond to a term in the effective action, but there shouldn't be $\frac{1}{q^2}$ terms in the action. Furthermore the original integrand is not singular in $q^2$ so I could have just as well expanded in $q^2$ first and then performed the Feynman parameter integral and no such $\frac{1}{q^2}$ terms would be present. Any idea why the Feynman parameter integral and the momentum expansion don't seem to commute, or why I get these weird terms if I integrate first? Thanks a lot.


title: Electric Dipoles and Spherical Coordinates

text: I've become confused over the use of spherical coordinates when working with dipole moments. It would probably be best o use an example to show where I'm confused. If we have a pure dipole, with a dipole moment pointing angle $ \theta $ from the vertical axis (which we'll call $x$), it's dipole moment is given by the vector $ \vec{p} = pcos( \theta) \hat{x} + psin( \theta) \hat{y} + 0 \hat{z}$ in Cartesian coordinates, but my book claims that it is given by (in spherical coordinates) $ \vec{p} = pcos( \theta) \hat{r} + psin( \theta) \hat{\theta}$ . How is that possible? If you want to change from Cartesian to spherical, you use the conversation equations, which equal $\vec{p} = p \hat{r} + \theta \hat{ \theta} + 0 \hat{ \phi}$. The rest of the problem asks to find the torque on this dipole moment due to a conductor a distance away (method of images). I get the process, but the coordinates are really causing problems for me. I'm not asking for help solving the problem, I'm mostly confused with the coordinates, I think my book may have made a mistake.


title: Describing the surface of a sphere in Huygens' Principle

text: $U(P) = \frac {iU(r_0)}{\lambda} \int_{S} \frac {e^{iks}}{s} K(\chi)\,dS$ Where $S$ describes the surface of the sphere, how is the surface of the sphere described? Sorry if you feel this is quite a simple question, I couldn't find the information anywhere on the internet. (I'm not entirely sure about the tag either) so any edits would be much appreciated


title: How do High Electron Mobility Transistors (HEMT) work?

text: I am studying High Electron Mobility Transistors (HEMT), but I simply cannot understand how they work in the way described by the references I've read on the Internet. This is what I understand so far: The idea is to decrease scattering and therefore increase mobility by separating the conducting electrons by the ionized donor atoms that are necessary to provide these electrons. Therefore, these transistors are made up of a metal gate, a highly doped n++ semi- conductor (used to supply the mobile carriers), an undoped n spacer region, and a lightly p-doped region. When put together, the undoped n spacer region and lightly doped p region form a heterojunction as both Fermi levels need to be the same AND the ionization energies need to remain constant too. This heterojunction is in the shape of a triangular well on the side of the p-doped semiconductor. So, electrons are able to tunnel in from the n side into the triangular well on the p-side where they stay trapped and separated from ions and can therefore reach high velocities. This triangular region of high mobility electrons is called a 2DEG (2D electron gas) I hope so far all I've said is correct. (Please correct me if I'm wrong). This is where I get puzzled:   1. This heterojunction always exists no matter what voltage is applied on the gate. Consequently, there will always be a 2DEG region at the interface between the p and n semiconductors, so a conducting channel between source and drain will always exist (the transistor will always be on). HOW does the gate control this channel ? What bias should be applied on the gate and what are its effects on the 2DEG (and on the conductivity of the device)?   2. The source and drain regions extend vertically across the different regions (n++, spacer, p). In order to have high mobility, we want the channel between drain and source to be in the 2DEG. So we want the electrons to flow there because that is where they are able to reach the highest speeds because there is hardly no scattering there. However, what stops the electrons from flowing through the n++ region between gate and spacer? It seems to me there are 2 paths for the electrons to follow.


title: Are multipole fields, multipole expansion, and multipole radiation the same thing?

text: Interaction between electromagnetic radiation and nuclei can be written in terms of multipole radiation. Are multipole fields, multipole expansion and multipole radiation the same thing? I have found this topics in Nuclear physics book written by R.R. Roy and B.P. Nigam.


title: "Find the net force the southern hemisphere of a uniformly charged sphere exerts on the northern hemisphere"

text: This is Griffiths, _Introduction to Electrodynamics,_ 2.43, if you have the book. > The problem states _Find the net force that the southern hemisphere of a > uniformly charged sphere exerts on the northern hemisphere. Express your > answer in terms of the radius $R$ and the total charge $Q$._ Note: I will > say its uniform charge is $\rho$. My attempt at a solution: My idea is to find the field generated by the southern hemisphere in the northern hemisphere, and use the field to calculate the force, since the field is force per unit charge. To do this I start by introducing a Gaussian shell with radius $r < R$ centered at the same spot as our sphere. Then in this sphere, $$\int E\cdot\mathrm{d}a = \frac{1}{\epsilon_0}Q_{enc}$$ Now what is $Q_{enc}$? I feel like $Q_{enc} = \frac{2}{3}\pi r^3\rho$ , since we're just counting the charge from the lower half of the sphere (the part thats in the southern hemisphere of our original sphere). (Perhaps here is my error, should I count the charge from the entire sphere?, if so why?) Using this we get $$\left|E\right|4\pi r^2 = \frac{2\pi r^3\rho}{3},$$ so $$E = \frac{r\rho}{6\epsilon_0}.$$ Using these I calculate the force per unit volume as $\rho E$ or $$\frac{\rho^2 r}{6\epsilon_0}$$ Then by symmetry, we know that any net force exerted on the top shell by the bottom must be in the $\hat{z}$ direction, so we get $$ F = \frac{\rho^2}{6\epsilon_0} \int^{2\pi}_0\int^{\frac{\pi}{2}}_0\int^R_0 r^3\sin(\theta)\cos(\theta) \mathrm{d}r\mathrm{d}\theta\mathrm{d}\phi$$ integrating we get $F = \frac{1}{4}\frac{R^4\rho^2\pi}{6\epsilon_0}$. Now Griffiths requests us to put this in terms of the total charge, and to do so we write $\rho^2 = \frac{9Q^2}{16\pi^2R^6}$ Plugging this back into $F$ we get $$F = (\frac{1}{8\pi\epsilon_0})(\frac{3Q^2}{16R^2})$$ Now the problem is that this is off by a factor of $2$ ... I tried looking back through and the only place I see where I could somehow gain a factor of $2$ is the spot I mentioned in the solution, where I could include the entire charge, however, I can't see why I should include the entire charge, so if that's the reason I would be very grateful if someone could explain to me why I need to include the entire charge. If that is not the reason, and perhaps this attempt at a solution is just complete hogwash, I would appreciate if you could tell me how I should go about solving this problem instead. (but you don't need to completely solve it out for me.)


title: Understanding Black hole information paradox?

text: I am not a physicist, I am a enthusiast trying to understand thinking behind "Holographic Principle" by Leonard Susskind. Recently I saw program on DS Through the wormhole - The riddle of the black hole, there was one analogy about Bob and Alice near black hole. 1) First they were in the space ship near black hole and Alice jumps in. Bob sees her getting closer to black hole but slowing down and frozen near event horizon. But from Alice's point of view she falls in the black hole passes event horizon with no traces at all. 2) Second time they each have an aeroplane with String theory propeller. Alice flies through the black hole and meets the same horrible faith. She could only see central hub of her propeller. But when Bob approaches black hole he could see increasing number of propeller. I have few questions about above analogy 1) In the first analogy: Bob sees Alice frozen at the event horizon, If this is true why don't we see stars revolving around black hole slowing down when they approach black hole at the center of our galaxy? (I am talking about observations that indicate Black Hole at the center of our galaxy with stars revolving around it, which show stars speed up while they approach black hole) 2) In the second analogy: Does the propeller indicate vibrations of atom? If so why Alice does not see more propellers as she approaches event horizon but the Bob sees it ? Also can the plane without air in the space or it was just to imagine atom to propeller ? Links: Here is the video on the you tube http://www.youtube.com/watch?v=1dwM_0kxYp8


title: Where does the transformation of $\gamma$ come from?

text: I was reading Tipler, and in his book he says that in order to transform energy, we must write gamma in terms of quantities in the non-primed frame. However his derivation makes no sense to me, it's on page 73 in the fifth edition if anyone can find it. It says that $$\frac{1}{\sqrt{1-u'^2/c^2}} = \gamma \frac{1-vu_x/c^2}{\sqrt{1-u^2/c^2}}$$ I see that it has the velocity transform with some other things mixed in, but I don't know where it all comes from.


title: Understanding Poynting's Theorem intuitively

text: I understand that, broadly speaking, Poynting's theorem is a statement of conservation of energy. Energy density of a volume of current and charge decreases proportionally to work done on charges within the volume and by energy flux through the surface bounding the volume. The energy flux term is the area integral of the Poynting vector. My question is this: is this energy flux that is described by the Poynting vector a form of electromagnetic radiation? If not, how is energy radiating out of the charge and current distribution if not by work done on charges?


title: Von Neumann Entropy: varying definitions

text: I have seen different authors define von Neumann entropy in different ways. In particular, some use the natural logarithm and others log to base 2. What is the reasoning for this? Does it make any difference? What is the accepted definition? Thank you in anticipation of your help.


title: How do sunflowers rotate?

text: Is there a physics mechanism to explain how sunflowers rotate to always face the sun? I tried to find more information or references using google search, but no luck.


title: Origin of motion and relative speed of bodies in the universe

text: Charged particles can hit the earth at relativistic speeds. But it seems that all large bodies have fairly low relative speed. Of course, speed can increase considerably when a body orbits close to a massive object, but then it will not travel very far, and it can be discounted by averaging the speed over a long enough time (maybe a year). The Earth cruises around the Sun at 30 km/s, the sun cruises at 200 km/s and the Milky Way at 600 km/s. Not that much. I have two somewhat opposite questions:   * What do we know of the relative speeds of massive bodies in the universe, small or large (not particles)? I am not sure whether it was the proper way to state the question. Also, do relative speeds get very high if we correct out the part due to universe expansion? Are there braking phenomena?   * On the other hand, what initiated that motion? If the initial soup had been homogenous, the coalescence of randomly moving particles should have produced structures essentially at rest (ahem: where is the energy going?) with respect to each other. There were small variations in temperature, but how should that create speed differences? Or did it cause large scale streams that coalesced into moving bodies? Even if some speed is due to contraction of large scale rotating structures, it does require initial momentum to exist. To put it together, do we have measured speed statistics in conformance with universe evolution models? What does it say about speed? Sorry if the questions are not well stated; that is my best. A reference to a paper for non-specialists would do too. * * * **Added after 3 weeks, considering the answer by @BenCrowell and the comment of @ChrisWhite** (thank you both). Please, forget the part about cosmological speed comparisons. **This is my own answer** to my question; I leave it as the question, since it raises other issues. The first point is: does my answer make sense? It is only guesswork on my part. I wrote this answer because, though really helpful, Ben Crowell's answer did not really answer the heart of my initial question. Though I suggested it in my initial question, I realize even more now that the problem is only the origin of momentum, probably only angular momentum. This was confirmed by the answer of Ben Crowell regarding the fact that structures were essentially at rest to start with. Probably my main error was to think that there might be another "source of speed and momentum", and my awkward attempt to discount observed very high speeds as going nowhere because in tight orbit. I do not see how large scale momentum could come from some form of accretion of spontaneously emerging angular momentum quanta. I would think they would balance on average without any visible macroscopic effect, outside very special and anisotropic places like a black hole horizon (and even then). I feel the same regarding linear momentum quanta (if such a thing exists). My guess is that high scale momentum arises from momentum exchange between large collapsing structures. It is well known that celestial structure, such as planets and satellites, can exchange angular momentum. Though I no longer try to follow the mathematical analysis, I also read that much of the exchange can be mediated by tidal effects. But tidal effect should be even stronger when it is between structures that are not yet collapsed and are hence very deformable. Angular Momentum exchange is not necessarily one structure slowing while another is speeding up. It is vectorial and may be two structures both speeding up in opposite directions, provided extra energy gets in from somewhere, such as potential energy from collapse. So it would be the case that, as they are collapsing, structures are being deformed under the gravitational influence of other structures, so that instead of converging to some global center of gravity, some subparts collapse separately around their own centers of gravity, and rotate around the main center of gravity. Is it actually in that way that the originally small variations of temperature (density?) created the various structures of the universe? Observed momentum arising from such momentum exchanges, there is no reason it should have the same orientation in all subparts of a larger structure, and this is indeed observed in the solar system. (see Can the axis of rotation of a celestial body point in any arbitrary direction?) What about other structures in the universe? The variations in speed (earth in solar system, sun in galaxy, etc.) are obviously just the consequence of variations in mass and size of the structures, very massive structure permitting higher speed as observed. The speed formula being $v = \sqrt{\frac{G\cdot M} r}$, and the mass most likely increasing like the cube of the (initial) structure size. Hence the "average speed" in a structure should grow more or less like the initial linear size of the structure (assuming an almost uniform initial density). As reported in the initial question above, this is not quite what is observed. Where do I err? Could it be that very large structures collapse less than smaller ones because of the angular momentum of the subpart? This could reduce the influence of the total mass on individual subparts, and increase the orbital radius, thus reducing the speed. Is there any way to measure the degree of collapse as a function of initial size of a structure, and some notion of reference speed of substructures? Licence CC BY-SA 3.0 from the author.


What is the age of the Universe from the big-bang light perspective? We all agree that the universe is around 14 billion years, from the perspective of a hypothetical observer moving in a galaxy, e.g. ourselves. But what would be the age of the universe from the perspective of the first light that emerged from the big bang, given that it has been traveling at the absolute $c$ speed of light?


title: consequences of universe in which second law of thermodynamics does not hold

text: What would happen if the second law of thermodynamics was not there for governing in every day processes. Can a universe exist and evolve if the second law of thermodynamics does not exist in first place?


title: Is there a mechanism for time symmetry breaking?

text: Excluding Thermodynamic's arrow of time, all mathematical descriptions of time are symmetric. We know the arrow of time is real and we know the equations describing physics are real so is there any insights into what breaks the symmetry between real time and the mathematical description?


title: Will photon's energy be exactly same after million years?

text: * If photon will travel for million years without collisions, what subtle effects can be accumulated ?    * Gravity fields affect trajectory, but is energy completely intact after fly by ?   * Photon has its own equivalent mass of energy, does it affect itself over long long period of time and integrate into frequency shift ?    * Is uncertainty principles applicable for speed of free flying photon ?   * What if uncertainty of time/location of photon violates speed of light (exceeding it with very small amount of violation) or this violation only occures in lag just a little bit below hard speed limit ?   * What if there is photon-to-photon interaction when they travel in packs or cross each other individually with some tiny losses scattered ?   * As photons and mass affect each other, can it be that photons in packs are affecting each other as well ? Can it cause some lens, scatter, frequency change effects ?   * If some of this effects exist, can it be accumulated over enourmous age and be percepted falsely as doppler red-shift of very distant objects ?


title: How to explain the Moon halo phenomenon?

text: Today, here in Brazil, I have observed (and is still observing) an interesting phenomenon. The Moon is near to a big star in the sky, but this is normal. The interesting part is what's around them. A huge circle in which the moon and the star is inside. Why does this circle appear? What is it? (This isn't a cloud. It looks like more something like a light reflection of a lamp.) Unfortunately, I hadn't a good camera to catch this. I would appreciate having an image posted for this.


title: Motion of a rod struck at one end

text: Imagine a strong metal rod of uniform density and thickness floating in a weightless environment. Imagine it lies on an X-Y plane, with one end (A) lying at 0,0, and the other end (B) at 0,1. Then it is struck at A in the direction of increasing X. Please describe the trajectory of the bar. In particular, will the point B move in the opposite direction (i.e. decreasing X) momentarily? Is there a website or article that goes through the physics of this problem? EDIT: some specific questions: What would be the trajectory of the centre of gravity? Would it just travel horizontally along the line y = 0.5? What would be the ratio of the velocity of the centre of gravity to the rotational velocity of the ends of the bar? ![enter image description here](http://i.stack.imgur.com/8Tbq6.png)


title: Temperature of WIMPs

text: As a dark matter candidate, what should be the temperature and kinetic energy (or also the speed) of the WIMPs (weakly interacting massive particles) to agree with the observed distribution of dark matter and not to accumulate so much in the inner part (40 AU radius) of the Solar System that the change in the orbits of the planets would be observable? Is it in equilibrium with the microwave background radiation or not yet or never will? EDIT: I have replaced some part into another question, _Evidence of dark matter in our galaxy_ , and I have deleted some other part.


title: Convert ECI coordinates to latitude/longitude?

text: I have been given output in (what I believe to be) ECI format (from OrbitTools):               TSINCE            X                Y                Z             0.00     4733.08693007    5370.64476064       0.09227655      360.00    -3928.94737512   -4784.93071195   -3602.13095174      720.00     2078.23905091    2928.47218732    6181.06427083     1080.00      343.60889986    -292.76288815   -7150.74620799     1440.00    -2682.96778230   -2415.73481378    6170.18900818      and need to convert the x,y pairs to latitude/longitude (to plot on a map where the object is above). I can't reason through the conversion. Is there a document that a laymen could get through or formula that would take a given time/point and yield a terrestrial latitude and longitude?


title: How does the winter solstice relate to the "longest night"

text: The winter solstice just gone was on the 22nd at 11:12 UTC. Does that mean the longest night was the night before or the one after? Or more generally, given the time of the winter solstice of any given year, and my position on earth, how can I know which night is the longest one of the year?


title: How to make neutron activation analysis online or with free software?

text: i want to make a neutron activation analysis for different crystals. (MnWO4, TbMnO3, and others) how can i do this with free or open source software? Can someone recommend a online calculator for this task? I do not know the precise isotope ratio, so i have to make my estimation with natural ratio of isotopes in the compound. I want to estimate the activity after investigation in a neutron beam. Thank you and kind regards


title: Distribution of point charges on a line of finite length

text: How will $N$ freely moving charges confined to a line with length $L$ be distributed? What are their equilibrium positions?


title: Finding the EOM for a charged relativistic particle

text: For an exercise sheet of a course in general relativity I'm asked to derive the equations of motion for a charged particle in an EM-field given by a potential $A^\mu$. I am give the action: $$S = -m\int ds-e\int A_\mu dx^\mu$$ With parametrization in proper time I get: $$S = -\int d\tau\left(mg_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta+eA_\mu\dot{x}^\mu\right)$$ where $\dot{x}^\alpha =\frac{d x^\alpha}{d\tau}$. Euler-Lagrange gives: $$0=\frac{\delta L}{\delta x^\alpha}-\frac{d}{d\tau}\frac{\delta L}{\delta\dot{x}^\alpha}$$ The first term is zero, so: $$0 = \frac{d}{d\tau}\left(2mg_{\alpha\beta}\dot{x}^\beta+eA_\alpha\right)=2mg_{\alpha\beta}\ddot{x}^\beta+e\dot{A}_\alpha$$ where in the first equality we used the fact that $g$ is a _symmetric_ $(0,2)$-tensor. Now I'm asked to write the equations of motion in terms of the field tensor $F_{\mu\nu}=\frac{\partial A_\nu}{\partial x^\mu}-\frac{\partial A_\mu}{\partial x^\nu}$. I have tried to use the identity $$\dot{A}_\alpha = \frac{\partial A_\alpha}{\partial x^\nu}\dot{x}^\nu$$ but I couldn't get anywhere. Can someone help me out see how to do it (it's probably some dumb algebra trick) or say me where I'm wrong (if I'm wrong somewhere in the steps above)?


title: Rotation of parabola

text: I'm designing a parabolic solar concentrator and am doing ray-tracing on Matlab from scratch. I'm beginning to look into compound concentrators and would need to have an equation for a parabola rotated by x degrees centered at a specific (non origin) point. The equation I am using for my parabolas is $ a*(x-X)^2 - Y $. I am not sure how to rotate this equation besides using a rotation matrix, solving for y using quadratic equation and choosing one term. Even this method gives me another parabola in $y(x)$. Any suggestions on a simple way to get half of a parabola rotated by $ \theta $? Should I be looking into parameterization? Thanks


title: models for astrophysical relativistic jets from compact objects

text: what is the simplest way to understand the physics of relativistic jets? we know that they have axial symmetry with very tight angular spread, presumably aligned with the axis of rotation of the black hole. It seems to be a mechanism not entirely related with mass accretion because that tends to be constrained in the rotation plane


title: PV cell for invisible spectrum only

text: Is it possible to make a photovoltaic cell that would only absorb the invisible part of the electromagnetic spectrum, while letting visible light pass through or bounce off its surface? I guess that if the n-type Si and p-type are added special dopants so the energy gap needed for the electron to escape would be the same as the one provided by the photons of such radiations, it would work. But this is when it gets tricky as visible light stands in the middle of IR and UV light (what you would normally look for in the non-visible spectrum, ignoring the other waves), so if the cell would have a specific band gap that would be satisfied from UV light, it wouldn't be from IR as it has lower energy (provided visible light is let intact).


title: eigenvalue staircase and hamiltonians

text: Let two Hamiltonians $H_{1}$ and $H_{2}$ be defined in such a manner that their eigenvalue staircases satisfy $ N_{1} (E) = N_{2} (E)+A +O(E^{-1})$ What can we say about their potentials $ V_{1} (x) $ and $ V_{2} (x)$ ?? Are they the same at least for $ x \rightarrow \infty$ ?? If we consider that the energy function $ E^i_{n}=f^i(n) $ as the inverse of $ N(E) $, will the energies be the same $ E_{n}^{1}=E_{n}^{2}$ in the limit $ n \rightarrow \infty$ ??


title: If Earth was the size of an orange, what consistency would it be?

text: I understand that the Earth is composed of a thin crust "floating" over a semi-solid layer of mantle. I was wondering how it would feel at a human scale, say the size of an orange in your hand. Could you somewhat squeeze it or would it rather be totally solid?


title: Terms allowed in Lagrangian density

text: I never fully understood why the Lagrangian density (let's say for a scalar field) was restricted to have only first order derivatives of the field in time and space in QFT. One reason I can think of is to ensure locality. But that doesn't restrict one from having a finite number of higher-order derivatives of the field, does it? Thanks!


title: Why does a moving electric charge have nonuniform electric field lines?

text: The electric field of a stationary electron is uniform, radially symmetric with straight field lines. However, the electric field of a moving electron still has radial and straight field lines, but they are no longer uniformly distributed, and the effect becomes more pronounced at higher speeds (that is, the field lines become more and more perpendicular to the direction of motion). I found this effect in this applet. Can someone physically explain to me the cause of this nonuniformity in the field lines? Thanks in advance.


title: Relation between Higgs boson and force of gravitation

text: I don't know and I can't handle it. I am so much curious about the recent discoveries in science. European Space Agency and NASA are doing great work. Hats up to these institutions. My nights are sleepless just thinking about the consequences of Higg's Boson. I am so excited for the thing that all the science fictions turning into reality will be happening in my generation. I am able to see it. Currently I am 28. So I am very optimistic about it when I went through the NASA and ESA's vision and missions down the years till 2025. I would be very much happy and grateful to know if someone can explain me a little about the relation between Higgs Boson and Force of Gravitation(Gravitons if exists). By any means does Higgs Boson supports String Theory?


title: Can project LISA will bring us closer to time travel?

text: After looking at the current discovery of Higgs Boson and the next long term plans of NASA and ESA(European Space Agency), I can't stop myself asking about the possibilities in near future. According to my little knowledge, time travel is possible if we can bend the space-time fabric. The bending or curving of space-time fabric is possible only by black hole or worm hole(Although I don't know the specific difference between these two). Now to think about black hole which is a gravitational sink of the universe which has infinite mass where masses sink and on the other side the black hole there is white hole which is big bang. So this seems the game of gravity. But to really get any clue any information about gravity of cosmic objects, the so called gravitational waves are the medium. To get all the information about presence of black-hole NASA and ESA are planning for the project LISA which will detect the gravitational waves from distant cosmic objects. I would happy to know from theoretical physicists that how far the project LISA can help in this direction. If anybody can share their view about this and can throw some light on possibility of time travel, I would be grateful.


title: Are voltages discrete when we zoom in enough?

text: Voltages are often thought of as continuous physical quantities. I was wondering whether by zooming in a lot, they are discrete. I feel like the answer to the above question is yes as voltages in the real world are generated by actions of electrons. Can someone give me a more formal proof or a disproof? Whether voltages are discrete of continuous can have some impact the correctness of devices such as the analog to digital converter. For example, if voltages in the real world are continuous, then the Buridan principle[1] says that there cannot be a perfect analog to digital converter because such a device makes a discrete decision from continuous inputs. [1] : Lamport, L. (2012). Buridan’s Principle. Found Phys 42, 1056–1066. http://link.springer.com/article/10.1007/s10701-012-9647-7 (It would be great if someone could also answer a related question http://electronics.stackexchange.com/questions/126091/is-there-an-adc-with-a- finite-bound-on-measurement-time)


title: Two trains at the speed of light

text: Okay. Two trains travelling towards each other at the speed of light. So, from one train (let's call it train A), the other is moving towards it at the speed of light. The other train shines a torch. They are both moving at the same speed so they both reach train A at the same time. From train B (the other train), A is moving towards it at the speed of light. It shines the torch, and the torch is moving faster than the train itself (since the train is stationary from it's own frame of reference). So the light should hit train A before train B does. Obviously both of these scenarios cannot be true, so what happens from each frame of reference?


title: Vector Boson Fusion

text: I have been reading about the production mechanisms for the Higgs at the LHC. It is always mentioned that for Vector Boson Fusion, the initial quarks cause jets that are back to back and with a higher transverse momentum pT than jets in other processes. Is there a kinematic reason for why the jets have such a high pT? Is it because the initial quarks have high pT due to the large energies at the LHC?


title: How can you calculate the exact position of the Earth taking into account the gravitational attraction of Mars and the Sun?

text: I think there must be a formula to calculate the exact position of the Earth taking into account the gravitational attraction of Mars and the Sun. Is it possible with vectors?


title: What is wrong with this derivation of moment of inertia of a sphere?

text: I know the moment of inertia of a solid sphere is $\frac 2 5 MR^2$, but I keep getting $\frac 3 5 MR^2$ when deriving it: For a sphere of mass $M$ and radius $R$, $$\rho = \frac {M}{\frac 4 3\pi R^3}$$ $$I = \int_0 ^R (\rho 4 \pi r^2)(r^2) dr = (\rho 4\pi)(\frac {R^5} 5) = \frac 3 5 MR^2$$ What exactly is wrong with this?


title: Can the magnetic fields of EM radiation be harnessed or measured?

text: We use the electric component of EM radiation to create the EM radiation and to detect it (antennas and Etc.), but does anyone know of a situation where the magnetic component of EM radiation is used? I've read that it is so vanishingly small that it is almost non measurable, but this seems odd because they both hold the same amount of energy. Thanks!


title: Speed cannot be greater than speed of light. How about acceleration? Does it have limit?

text: We know that $c$ is the largest speed in the universe. How about the acceleration, does it has limit?


title: Where does the phase difference come from in a Josephson Junction?

text: When you separate two superconductors by a thin insulating film, a current $I(t)=I_0 \sin{\theta(t)}$ flows between the superconductors, where $\theta$ is the phase difference between the superconducting order parameters. Where does this phase difference come from? Does it come from initial conditions?


title: Diagonalization of this Hamiltonian: How do I transform the differential operators?

text: In order to find the eigenstates of this Hamilatonian $$ H = \sum_{j=1}^3 \left( - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x_j^2}\right) \+ \frac{1}{2} m \omega^2 \left( (x_1 - x_2)^2 + (x_1 - x_3)^2 + (x_2 - x_3)^2\right) $$ I want to do a coordinate transformation $y_1 = x_1 - x_2$, $y_2 = x_1 - x_3$, $y_3 = x_2 - x_3$, but I am not sure how to transform the differential operators. Are they just $\frac{\partial^2}{\partial y_j^2}$? Or do they transform to something else? How do I calculate this?