title: Estimating quantum efficiency of gated avalanche photodiode

text: I have a photon counting system that uses a gated avalanche diode to detect single photons. The repetition frequency of the gates is $f_1$ and the temporal gate width is $\tau_1$ (so the duty cycle is $\tau_1 f_1$) I want to get an estimate of the quantum efficiency $\eta_{\lambda}$, i.e., the probability of detecting single photons, for this diode at some wavelength $\lambda$. Note $\lambda$ is not anywhere close to the peak wavelength $\lambda_p$ of the diode, so $\eta_{\lambda}$ is expected to be small. I have a pulsed laser system with repetition frequency $f_2 >> f_1$, pulse width $\tau_2 << \tau_1$, and center wavelength $\lambda$. I connect the output of this laser to the avalanche diode through a variable optical attenuator (VOA) already characterized at this wavelength before. I control the VOA setting so that the power impinging on the diode is $P$ and observe a probability of detection per gate $p_d$. The main thing to note is that in this experiment the **laser pulses are not synchronized to the detection gates** because the laser and photon counting system do not have/cannot share the same clock. In this case: Q1. Can I assume the diode essentially sees the laser light as a quasi-CW source? As in, is the power P more or less uniformly distributed throughout the train of gates? Q2. If the answer to the above is yes, the mean photon number seen inside a gate should be $\mu_p = P\lambda\tau_1/\hbar c$. Can I then invert the equation $p_d = 1 - e^{-\mu_p \eta_{\lambda}}$ to obtain an estimate for $\eta_{\lambda}$ (ignoring dark counts here)? Q3. If the above is correct, can it be said that the actual value -- found using a more precise method -- would be _surely_ larger/smaller than the estimated one? As in, does the estimate provide lower/upper bound in this experiment?


title: Yang-Mills CP violation

text: Why does a term proportional to $\left(F,\,\tilde{F}\right)\propto Tr\left[ F_{\mu\nu}\tilde{F}^{\mu\nu}\right]$ in the Lagrangian of the pure Yang-Mills theory violate CP?


title: Gaussian Integral over Grassman variables

text: I need to evaluate two Grassmann integrals, one over "real" Grassmann variable another one over complex variables. Lets start with the real one first : The prototype we have for $n$ real Grassmann variables : $$ \tag{1}\int d^n\psi \exp{[\frac{1}{2}\psi^T M\psi]} ~=~ (\det M)^{1/2} $$ Now we can shift the integration variable and use the shift invariance of the integration : If we replace $\psi\rightarrow \psi - M^{-1}\eta$ Then argument inside exponential becomes : $$ [\frac{1}{2}\psi^T M\psi] ~\longrightarrow~ \frac{1}{2}(\psi - M^{-1}\eta)^T M(\psi - M^{-1}\eta) $$ $$ =\frac{1}{2}[\psi^T M\psi + \eta^T (M^{-1})^T MM^{-1}\eta \space\underbrace{-\eta^T (M^{-1})^T M\psi - \psi^TMM^{-1}\eta}] $$ $$ =\frac{1}{2}\psi^T M\psi+\frac{1}{2}\underbrace{\eta^T (M^{-1})^T MM^{-1}\eta}_{\text extra}+\eta^T\psi. \tag{2} $$ This "extra" term should vanish, but how? I will run into same problem is do the same Gaussian integral over complex Grassmann variables.


title: Seiberg Duality Applications

text: I am interested in why people are interested in Seiberg duality? Can it be used for something? Does it (or may it) correspond to any 'real world' physics? I don't have enough background to understand the details and I would like to know if it is worth the effort.


title: Can Planck's temperature ever be reached?

text: Planck's temperature is theorized to be the highest possible temperature in the universe. But, has it been reached? Or, will any object or phenomenon ever reach this absolute temperature?


title: Naive questions on the classical equations of motion from the Chern-Simons Lagrangian

text: Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-\partial _ia_0$ and $b=\partial _1a_2-\partial _2a_1$. From the Lagrangian, we obtain the classical equations of motion for $a_\mu$ fields ($\mu=0,1,2$): $(\partial _0^2-\partial _1^2-\partial _2^2)a_\mu+\partial _\mu(\partial _1a_1+\partial _2a_2-\partial _0a_0)\pm g\epsilon^{\mu \nu \lambda}\partial _\nu a_\lambda=0$; or the 'Maxwell' equations for the 'electromagnetic' fields: $\partial _0b-\partial _1e_2+\partial _2e_1=0, \partial _1e_1+\partial _2e_2+gb=0, \partial _0e_1+\partial _2b+ge_2=0, \text{and } \partial _0e_2-\partial _1b-ge_1=0.$ It's easy to solve the four 'Maxwell' equations with a gapped dispersion relation $\omega ^2=\mathbf{k}^2+g^2$. My questions are: (1) Is it possible to solve the equations of motion for $a_\mu$ fields (maybe under some gauge fixing) where we get the same dispersion relation $\omega ^2=\mathbf{k}^2+g^2$? As we know, when $g=0$, it's easy to solve the equations (under Lorenz gauge) to show that each $a_\mu$ field and each $e_i,b$ field fulfil the same wave-equation $(\partial _0^2-\partial _1^2-\partial _2^2)\phi=0$ with the same gapless dispersion relation $\omega ^2=\mathbf{k}^2$. (2)If we further modify the Lagrangian to $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda+m(a_0^2-a_1^2-a_2^2)$, $m\geqslant 0$, then the resulting equations of motion become $(\partial _0^2-\partial _1^2-\partial _2^2)a_\mu+\partial _\mu(\partial _1a_1+\partial _2a_2-\partial _0a_0)\pm g\epsilon^{\mu \nu \lambda}\partial _\nu a_\lambda\pm ma_\mu=0$, and it seems impossible to write down some 'Maxwell' equations which are ONLY in terms of the 'electromagnetic' fields. Thus, for this case, how to solve the equations for $a_\mu$ fields and get the dispersion relation? (3)Following Q(2), do ONLY the gauge-invariant Lagrangians (up to a total derivative term) can produce the classical equations of motion containing ONLY the 'electromagnetic' fields? Thank you very much.


title: Gibbs equation and third law of thermodynamics

text: I have seen a discussion of what would happen for an ideal gas expands irreversibly and adiabatically until absolute zero degree K. The entropy change is like that: Δs=Cv ln(T2/T1)+R ln(V2/V1) It is impossible for T2 to be zero K in the equation and so it becomes one justification of the third law. I wonder whether it is valid and whether an irreversible process can be represented by the above equation. For irreversible process the heat exchange is not equal to Tds so can the Gibbs equation be applied? The video is in https://www.youtube.com/watch?v=r4fGG_7NQr8 , 46:12


title: Problem understanding basic sail mechanics

text: I am trying to understand the physics behind a single sailed boat. I have read the following questions Sailing against the wind. Is this a fair model? What exactly is the 'lift' of a sailboat as explained by Bernoulli principle I've also been following this explanation which seemed to be helping but I am still confused. Full disclosure: my physics is pretty weak and I've never sailed, but my maths should be sound. I am (now) aware of the two components that provide the force that moves the boat: The _"Newtonian aspect"_ , where the wind collides with the sail and this produces a force and the _"Bernoulli aspect"_ resulting from pressure differences resulting from the air passing around the sail. (If people think this would be better as two questions, happy to break it down, but I suspect one or both will be silly) **Update:** It has been said by @CarlWittoff that the Bernoulli aspect is pretty trivial, so I would like to focus on how the Newtonian part of the force can achieve a force that goes into the wind. **Newtonian Aspect** I understand this in the most simplest sense, a sail perpendicular to the wind will produce a force on the boat proportional to the relative wind speed. As the sail is angled in to the wind, you get more "lift" and less "drag" (in plane terms). In the sailing explanation, it implies that the Newtonian aspect can be used to explain going into the wind. This is also hinted at in the first question. However, I don't understand how this can be. Surely the force applied to the sail, and hence the ship, is always in the direction of the wind. Hence it can never provide a force into the wind. The article seems to suggest you can get away without the Bernoulli effect, but I don't see how. ![Diagram of the air flow](http://i.stack.imgur.com/16B4F.png) This is my attempt at how this translates into sail. The air is pushed around the sail, which exerts a force (in purple) on the boat. However, I can see no way for this force to be into the wind. **Bernoulli Aspect** This is the area I am even more lost with. My understanding is that somehow, you can angle the sail in such a way that the air that goes past on one side knows to go faster which reduces the pressure, giving a force. However, I cannot seem to connect that intuition to any kind of applicable formula. Specifically, what determines the direction of the force and what is it proportional to? I'm trying to simulate a boat for a game and any help on which formulae I actually need to be plugging in would be hugely appreciated :)


title: Why does a flat universe imply an infinite universe?

text: This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.


title: Color-charge conservation in proton decay

text: In some extensions of the Standard Model of particle physics (Supersymmetry with R-parity violation being a prominent example), the proton is allowed to decay, e.g. via $p\to e^+\pi^0$: ![Feynman diagram of proton decay in a GUT](http://i.stack.imgur.com/piYKh.png) While this decay is obviously violating the conservation of baryon and lepton number, I take it that color charge should still be conserved. My problem is that I don't see how this would work: We have color-neutral objects in initial (the proton) and final state (the pion), but when I try to keep track of the color flow this fails, basically because I have to get rid of the third color charge somehow. Is the simple picture of assigning each (anti-)quark an (anti-)color too much of a simplification to be applied here? What would be the color assignment of the intermediate particle $X$ in the proton decay?


title: Spontaneity / Free Energy of Non-Isothermal Process

text: I'm trying to determine a lower bound for the work input necessary to make an entropy-reducing process "spontaneous" in the sense that the 2nd law is not violated. For a constant temperature and volume process, I'd calculate the change in Helmholtz free energy and say ΔA = -W. When considering a non-isothermal process (suppose we have no thermal reservoir), I can still calculate dA = dU - TdS - SdT, but can I say anything about what that dA means for the spontaneity of such a (infinitesimal) process? EDIT: I think the following is equivalent, fundamentally -- How do we determine how much work we can get out of a system undergoing an entropy- increasing process, when that process is non-isothermal?


title: Proof involving tensor product

text: I am trying to prove when the following holds: $$|a\rangle |b\rangle \langle c|\langle d| = |a\rangle \langle c| \otimes |b\rangle \langle d|$$ where $\otimes$ stands for tensor product and the $a,b,c,d$ are $N$ dimensional state vectors. I can see that for $N=2$ it holds true for $a,b,c,d\ \in\ \lbrace |0\rangle , |1\rangle \rbrace$. Are there any other values?


title: Why is “dark matter” a more parsimonious explanation than “more normal matter than we thought”?

text: > **Possible Duplicate:**   >  Why isn't dark matter just matter ? The evidence usually presented in support of dark matter is larger-than- expected masses in galaxy clusters: the mass required for the gravitational lensing is far larger than the mass detected in the galaxies themselves. So why, then, is the assumption that this undetected extra mass is **a previously-unknown kind of matter**? Why is that new concept required, instead of proposing that the estimates of total normal matter are wrong? Another way of asking the same question could be: Why are we **more sure of our estimate of the sum of normal matter** than of the evidence that there is more mass in galaxy clusters than estimated?


title: Wick theorem applying to partly ordered operator

text: I symbolize $T$ as the time-ordered operator and $::$ as normal order symbol. I know that in quantum field theory generally we have: $$T\phi_1(x_1)\dots\phi_n(x_n)=:\phi_1(x_1)\dots\phi_n(x_n):+A$$ where $A$ roughly means all possible contractions. I have read a convincing proof. However when I encounter terms like: $$T:\phi_1(x_1)\phi_2(x_2)::\phi_3(x_3)\phi_4(x_4):\phi_5(x_5)$$ I don't know how to apply Wick theorem. I guess that $A$ now contains only the contractions of pairs not included in the same normal order symbol $::$ (for example the contraction of $\phi_3\phi_4$ is eliminated) What is the general rules? How can I understand it intuitively? EDIT: Actually I have found a referrence that states the rules: http://www.fysik.su.se/~kardell/QFT/Tutorial-8.pdf


title: Modeling a potential well

text: I attempted to simulate the interaction of a moving particle and a potential well in Mathematica. The particle should experience a force of -$1/r^2$, if the equation for the potential well is -$1/r$. The inputs for the NDSolve function are:                 x''[t] == -x[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2),        y''[t] == -y[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2),       x[0] == -2,       y[0] == -2,       x'[0] == 2 ,       y'[0] == 2,      A kind of spiraling motion or trapping should happen when the particle approaching to the well in a certain angle. But it does not happen in this simple simulation. The particle must experiences some centrifugal force when it is close to the well so that its direction is changed. What other terms should be added to represent this force in the NDSolve function? Any idea or any help would be much appreciated. What I got is: ![enter image description here](http://i.stack.imgur.com/I1tUR.gif) What expected to see is: ![enter image description here](http://i.stack.imgur.com/hyU9S.gif) By adopting the suggestions given below, the terms for the NDSolve were now modified as follows:               soln[a_, b_, alp_, sp_] := Return[First@NDSolve[{           x''[t] == x[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2) - alp*x'[t],            y''[t] == y[t]/((x[t] - a)^2 + (y[t] - b)^2)^(3/2) - alp*y'[t],           x[0] == -2,           y[0] == -2,           x'[0] == sp,           y'[0] == sp},          {x, y}, {t, 0, 3}]];      and sliders were added for adjusting. Still, no conic trajectory was observed. The particle moves as if in a straight line from the top view. ![enter image description here](http://i.stack.imgur.com/4vr85.gif)


title: BTZ Black Hole Central Charge and Conformal Weight

text: I have been trying to reproduce a calculation (equation 4.12) in this paper http://arxiv.org/pdf/1107.2678v1.pdf by Carlip reviewing the derivation of the effective central charge of the BTZ Black Hole so that one can use the microcanonical form of the Cardy formula for the entropy. He is evaluating the boundary variation of the Hamiltonian, which for this case is given by: $\delta H = -\frac{1}{16\pi G} \int d\phi(rn^r\sigma^{\phi \phi}[N\xi^t(D_{\phi}\delta q_{r\phi} - D_r\delta q_{\phi \phi}) + (D_rN\xi^t) \delta q_{\phi\phi} + 2\xi^{\phi} \delta \pi^r_{\phi})$ Here $q_{ij}$ is the spatial metric, $D_i$ is the spatial covariant derivative, $n^r$ is the normal to the boundary ($n^r = N$ where $N$ is the lapse for this case) and $\sigma^{\phi \phi} = r^{-2}$ is the inverse of the induced metric on the boundary. The boundary is at spatial infinity and the boundary conditions are $g_{tt} = -\frac{r^2}{l^2}$ + order one terms $g_{rr} = \frac{l^2}{r^2}$ + order $r^{-4}$ $g_{\phi \phi} = r^2$ + order one $g_{t \phi} =$ order one The rest of the terms of the metric are required to be on order $r^{-3}$. Carlip finds that the above expression for the variation gives: $\delta H = -\frac{1}{16\pi G} \int d\phi(\xi^t\delta(\frac{r_+^2 + r_-^2}{l^2}) + \xi^{\phi} \delta (\frac{2r_-r_+}{l}))$ I can easily see where the second term comes from because $\pi^r_{\phi} = \frac{r_-r_+}{l}$ which is also proportional to the angular momentum $J$ but I cannot see how he gets $\frac{r_+^2 + r_-^2}{l^2}$, which is the mass of the black hole. Also, in the non-asymptotic limit the lapse is: $N = (-8GM + \frac{r^2}{l^2} + \frac{16G^2J^2}{r^2})^{1/2}$ Now I understand why the mass should be in the above variation of the Hamiltonian but I still can't seem to derive it. Any help on seeing where that term comes from would be very helpful!


title: Control system with equation C = A*x + B*dx/dt

text: This question came up in a computer science / robotics exam and I still don't know the solution for it. I figured out that it's classical mechanics related, so I thought this might be the best place to ask it. > Suppose a control system is described by the equation >   >   >     C = A*x + B*dx/dt >   > > where B is proportional to the mass of the robot. The behaviour of the > system can be characterised by the steady state (e.g. the asymptotic > velocity of the robot) and the half-life time of the decrease of the > distance to the steady state. **Explain how the steady state and the half- > life change if the mass of the robot is doubled.** I've figured out that the steady state doesn't change, as when dx/dt is 0 B is not affecting the solution. And this is how far I understand it. Can you explain to me, what kind of movement is this, what is the real-life meaning of the steady state and half- life for this movement and that how to calculate the change in the half-life?


title: Schematic design of the apparatus (photoelectric effect)

text: I think I understand the basic idea of photoelectric effect but there are two things in the schematic diagram of the apparatus for the investigation of the photoelectric effect which I do not understand. Firstly, the emitting electrode is connected to a positive terminal of a battery. This implies there is an electron deficiency there, so where are the 'loose' electrons are coming from? I understand there might be some to start with but with the intensive light shone on the emitter shouldn't the emitter run out of those loosely bound electrons quite soon? Second problem is what happens at the collecting electrode. It is connected to the negative terminal of the battery to create the retarding potential difference and I understand as the photoelectrons travel across the gap they experience a repulsive force. Now the book says that those with NOT enough kinetic energy will be stopped and no current will be registered. I am however interested in details as to what happens to those who do make it through. I suppose they would not make it all the way up to a galvanometer, so how exactly is the current produced? Do the photoelectrons 'push' the free electrons in the metal by repulsive force, just like it happens in a regular wire which is connected between the terminals of the battery? What eventually happens to those free electrons which are located near the negative terminal - do they get pushed 'inside' the battery - but this looks as if the battery is connected with the wrong polarity? Also I do not understand why the photoelectrons could push the free electrons in the metal - there are many more free electrons and their combined repulsive force should be much greater, shouldn't it? This I do not understand. ![Apparatus for the investigation of the photoelectric effect](http://i.stack.imgur.com/KpaeS.png)


title: Greatest volumetric heat capacity

text: Is there any substance with bigger volumetric heat capacity than water? According to this table water has the biggest known VHC. But I can't believe that in the 21. century we have no special material with larger VHC.


title: If the observable universe keeps expanding would we eventualy see light 24/7.... clouds permiting?

text: If we can see more observable universe could there be a time when day and night looked the same?


title: Is space infinitely divisible?

text: As a child I remember hearing the popular paradox presented by Zeno proposing that Achilles could never catch a tortoise in a race since he would have to traverse the infinite space between himself and the tortoise. After some pondering I concluded that the obvious answer to the dilemma is that the space is not infinitely divisible. Although in later years I learned that this paradox can be explained by a finite infinite series, I have always found it difficult to accept that just because a concept can be supported mathematically, it must therefor exist in reality. In the case of Zeno’s paradox, although finite infinite series do exist, I believe the simple explanation I constructed as a child to be a better answer since it is both a viable solution to the problem, while also being comprehensible at a practical and intuitive level that is often lost with mathematical constructs. A few years ago I learned that the current scientific position is that space is in fact infinitely divisible and I have been disturbed ever since. Does Infinity exist? In the literature the overwhelming answer to this question by the scientific community is yes... on paper. It is interesting to consider that in spite of a prolonged search for infinity in nature, it remains elusive. Although the importance of the concept of infinity in the field of mathematics is indisputable, ironically, it represents perhaps the most compelling example of a mathematical constructs that is “true” yet non existent in reality. Although many will argue that infinity exists in the singularity at the heart of a black hole, or perhaps at in some fields, when these phenomena are examined they become logically incomprehensible, mathematically error ridden, and physically non existent. If one were to weigh the evidence with respect to the physical existence of infinity, the overwhelming weight points to its non existence. If infinity doesn’t exist than space must be finitely divisible. If so, of what is space divisible into, and what are the consequences of its finite divisibility? In searching for a possible constituent of the fabric of space one could consider whether the oscillations of electro magnetic states observed by the motion of matter/energy through space are a reflection of the matter/energy “train” moving through space, or perhaps an illumination of the “tracks” on which it travels. As a photon of light, buckyball, or large celestial body travels through space, could it be passing through (occupying), a mesh like network of electro magnetic “cells”? These cells could consist of an electro core surrounded by a magnetic space that causes oscillations in state as the matter moves from one cell to another. Perhaps this process is similar to the duel particle process by which electrons jump between orbitals... i.e.) The energy is particulate in nature while in the middle of each cell but it never occupies the space between each cell as it moves from cell to cell... instead it is exhibits wave like properties between cells. Dr Richard Feynman once stated that the double slit experiment contains the only mystery of quantum mechanics. If one considers the double slit experiment from within the framework of an electro magnetic mesh through which matter and energy travel the experiment may be able to be accounted for in a more conventional manner than current postulations. Imagine the borders of the electro magnetic cells constantly moving in relation to the material containing the double slits as the earth, laboratory, apparatus, and particles being fired at the screen constantly move through the mesh like network. Depending on the position of the cell abutting the screen relative to the slot, the particle could be in a state of solid, defined, particulate matter, or in a state of diffuse energy (electro or magnetic). If the particle hits the slot in the screen in a state of diffuse energy it will diffusely enter both slits and interfere with itself. If it hits the screen during the solid particle phase it will pass through one of the two slits or hit the barrier and travel through none at all. Although I am not able to support these ideas mathematically due to a lack of training, the model I am proposing has some interesting intuitive possibilities. These include; In such a model could movement from cell to cell produce the electromagnetic radiation seen as matter changes velocity/direction when acted on by a force, or by the oscillation of atoms? Perhaps there is an electro magnetic “lag” of the vector a particle was on before force was applied to change its direction. Could the motion of matter through a network of electromagnetic cells produce gravitational forces? Interesting... the more matter massed together the more cells being “activated” the more gravity being produced. If this was true one would also expect an increase in velocity to result in an increase in gravitational forces since more cells would be activated per unit time. Could time itself be a function of motion through electro magnetic cell with time being relative to motion through cells similar to the example of gravity mentioned above? Perhaps some of these questions have been addressed with research done on electron orbitals and energy states?


title: Rocket engines: air & vacuum

text: Could you please help me understand what is the difference between rocket engines designed to work in air (first stage) and vacuum (later stages)?


title: Where can I get the most accurate measurements of parton distribution functions?

text: Where would I look to get the most accurate experimental values of parton distribution functions for the proton? I know these functions aren't measured _directly_ , but I'd basically like to find a fit that involves recent experimental data. I was reading a paper on PDFs which briefly discussed how the theoretical functions are fit to data and included some sample figures, but it dates from 1996 and I'm looking for something more recent.


title: Open shells in Quantum mechanics of multielectron atoms

text: This question: How do electron configuration microstates map to term symbols?   And the discussion of multielectron effects here: Quantum Computing and Animal Navigation   Inspired me to try to understand quantum mechanics of multi-electron atoms better. As it is aparrently difficult to solve anything in closed form, I found a numeric multi-electron quantum mechanics calculation package that is free for students! (Called GAMESS) My goal is to understand the basics at least at the level of the 'molecular orbital' picture, and be able to calculate some energies. From the discussion in the other question and reading wikipedia I think I understand the basic idea behind Hartree Fock and the wavefunction made from a Slater determinant. And I was able to use the program to calculate the orbital energies and ground state energy for a ${}^1S_{0}$ term symbol atom like Neon. These make sense to me as each orbital filled with two electrons: $$ \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow} \ \underline{\uparrow \downarrow} \ \underline{\uparrow \downarrow} \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array}$$ My main question is regarding atoms with open shells:   **How is the wavefunction represented in the orbital picture for open shell atoms?** For example Lithium with term symbol ${}^2S_{1/2}$ $$ \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \ } & \underline{\quad \ } \ \underline{\quad \ } \ \underline{\quad \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array}$$ versus Boron with term symbol ${}^2P_{1/2}$ $$ \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\uparrow \ } \ \underline{\quad \ } \ \underline{\quad \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array}$$ It seems like these should be described exactly the same way in molecular orbitals: one open shell below which all the remaining shells are closed. However this apparently isn't the case. The Lithium atom is a "doublet" and can be described by ROHF with GAMESS, while the Boron case is apparently not just a doublet since somehow the degeneracy needs to be taken into account (does this 'kind of doublet' have a name besides the term symbol?) although this GAMESS document says they are both 'technically ROHF' states ( http://haucke-jena.de/gamess/gam_grund_1/gvb_ref.htm ). Question "part 2":   **Is this trying to say we need more than one slater determinant for these degenerate states?** That is, something like: $$ \frac{1}{\sqrt{2}} \left[ \left( \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\uparrow \ } \ \underline{\quad \ } \ \underline{\quad \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array} \right) \+ \left( \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\quad \ } \ \underline{\quad \ } \ \underline{\uparrow \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array} \right) \right] $$ So this would be called a "multi configuration" state I think (MCSCF?), but the quantum calculation documentation (previous link) says MCSCF isn't necessary for either of these states. So are those two "pieces" of the wavefunction somehow not different slater determinants? So I'm very confused and would appreciate any help. Please clearly explain the terminology you use in an answer, as the difference between RHF, ROHF, GVB, MCSCF, etc. seem to be a large part of my confusion here.    UPDATE:   Some example calculation results for an Oxygen atom.   energy = -74.8002167288 Hartrees --- triplet (ROHF with MULT=3)   energy = -74.6780126893 Hartrees --- singlet (ROHF with MULT=1)   energy = -74.6780126893 Hartrees --- singlet (RHF), should be same as ROHF MULT=1   energy = -74.7986255288 Hartrees --- ??? (GVB with setting specified by GAMESS doc for ${}^3P_2$ term symbol) The GVB output also says:                    MULLIKEN ATOMIC POPULATION IN EACH MOLECULAR ORBITAL                                1          2          3          4          5                            2.000000   2.000000   1.333333   1.333333   1.333333      Which seems to me like a uniform mixing of $$ \left( \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\uparrow \downarrow } \ \underline{\uparrow \ } \ \underline{\uparrow \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array} \right), \left( \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\uparrow \ } \ \underline{\uparrow \downarrow } \ \underline{\uparrow \ } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array} \right), \left( \begin{array}{ccc} \underline{\uparrow \downarrow} & \underline{\uparrow \downarrow } & \underline{\uparrow \ } \ \underline{\uparrow \ } \ \underline{\uparrow \downarrow } \\\ \underset{1s}{\ } & \underset{2s}{\ } & \underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{2p} \end{array} \right) $$ But how can that center one be a ${}^3P_2$ state? Shouldn't it be a ${}^3S_1$ state? This all seems very conflicting. Especially that trying to include the effects of the degeneracy actually raised the energy instead of lowering it. Here's the GAMESS input I used if someone else wants to play with it:                $CONTRL UNITS=ANGS COORD=unique RUNTYP=energy $END     ! p**4 3-P state      $CONTRL SCFTYP=GVB MULT=3 $END      $SCF NCO=2 NSETO=1 NO=3 COUPLE=.TRUE.         F(1)= 1.0 0.66666666666667         ALPHA(1)= 2.0 1.33333333333333 0.83333333333333         BETA(1)= -1.0 -0.66666666666667 -0.50000000000000 $END      $BASIS GBASIS=N311 NGAUSS=6 NPFUNC=1 NDFUNC=1 $END      $DATA      Oxygen atom      C1      O           8.0   0.0000000000   0.0000000000   0.0000000000      $END         EDIT: My latex is kludgy. I had trouble making things line up. If someone could edit this so I can see the proper way to do things, that would be great.


title: Is there a limit to the resolving power of a mirror telescope?

text: Like, if you hammered out the asteroid 16 Psyche into a 1 mm thick iron foil disc telescope mirror with 2.4x the radius of the Sun, could you resolve details on the surface of an exoplanet? At what resolution? Could you make the mirror arbitrarily bigger and continue to get better resolution? What are the formulas for calculating this and what are the limits, if any?


title: What happens to a rotating rod that breaks in two?

text: I know that the approximation for the moment of inertia of an infinitely thin rod of mass $m$ and length $L$ spinning around an axis perpendicular to its own axis at its center is $\frac{mL^2}{3}$: ![enter image description here](http://i.stack.imgur.com/LVgxB.png) What happens when this rod is instantaneously broken in the middle with no loss or gain of energy? Rather than some external event triggering the breakage, imagine some sort of latch that held it together before, and now it is simply released. Easier to pretend with than an infinitely thin rod with mass, at any rate. I'd like to know how I should determine the motion (absent gravity) of the two halves. Suppose I view the scene from directly above (viewing in the same the direction as the axis of rotation), the rod is rotating clockwise from that perspective (viewing from below, in the picture above), and the event occurs when the rod is vertically aligned from that view. Based on this I assume a coordinate system: The rod was rotating within the x-y plane with its center at the origin. The axis of rotation is the positive z axis. My camera is pointing in the positive z axis from some position $(0,0,-z)$. The top half of the rod travels to the right. Its center of mass is at its middle point, which is at position $(0,\frac{L}{4},0)$ at $t=0$. The bottom half is going to travel to the left, in a symmetrical fashion. What are their angular velocities? Do they spin faster or slower, or at the same speed as the original rod? The length has halved, which means the moment of inertia has been quartered. But conservation of angular momentum states each rod carries half of the rotational energy (is this correct application of conservation of angular momentum? Is there such a thing as conservation of rotational kinetic energy?) Is part of the original angular kinetic energy now split between resultant angular kinetic energy and linear kinetic energy? It seems so, because the original rod had zero linear kinetic energy (and momentum) but after breaking, the parts now both have linear kinetic energy since they are flying apart. At this point I'm almost positive I can stumble my way to the solution for the split happening exactly at the middle. How might things change if the split is for example 1/10th of the way down the rod? Surely the little piece would go off much faster but would it spin at the same rate also?


title: Is it necessary to embed a 4D surface in 5D space?

text: Lets consider the line element: $$ds^2=dr^2+r^2[d\theta^2+\sin^2\theta d\phi^2]$$ There are three variables r,theta and phi. If we use a surface constraint like r=constant the number of independent variables is reduced by one--now we have two independent variables.These surfaces[corresponding to r=const] may be embedded in a three dimensional space. Now lets consider Schwarzschild's metric: $$ds^2=(1-2m/r)dt^2 - (1-2m/r)^{-1} dr^2 - r^2[d \theta^2+\sin^2 \theta d\phi^2]$$ If we use a surface constraint[for example: t=constant] we have three independent variables.The resulting time slices are three dimensional surfaces which are naturally embedded in a 4D space. The General Relativity metric has four variables: one relating to time and three relating to the spatial coordinates.Any surface constraint would reduce the number of variables to three. In fact any arbitrary spacetime curve[world-line] may be made to lie on a 3D- Surface obtained by applying some suitable constraint on 4D space.The constraint may not be a simple one like t= constant or r= constant. It may be of a complicated nature. For the purpose of embedding in GR a 4D space seems to be sufficient. Queries:   1. Is it essential the we should consider a 4D surface embedded in 5D space to understand or interpret GR?   2. It appears that the curve[4D path] is more important than the surface on which it is lying since the same curve may be made to lie on several distinct surfaces at the same time[you may extend this to the case of 4D surf embedded in 5D space].Is this interpretation correct?


title: What material can a lens be made from?

text: I need to make a camera lens that can easily withstand temperatures of about 1000 Fahrenheit. What should I make the lens out of?


title: Why do we consider the application of force as well as bending moment in finding the moments of a beam?

text: ![enter image description here](http://i.stack.imgur.com/rwrAc.png) In the standard techniques of beam related problems, solving for moment at any cross section includes counting the torque due to shear force as well as some extra moment, which according to me should be due to the same shear force, so why do we take the moment into account twice, or what is the difference between the moment and torque due to shear force? **In essence:- isn't M produced due to V itself?**


title: Why does acceleration seem not to be the gradient of gravitational potential?

text: Consider a spherically symmetric distribution of density $\rho(r)$. We can define the mass enclosed within each radius $r$ using $\frac{dM(r)}{dr} = 4\pi r^2 \rho(r)$, with the condition that $M(r=0) = 0$. The gravitational acceleration and potential can then be given as, $$\begin{align} a_g &= -\frac{GM}{r^2}\\\ \Phi_g &= -\frac{GM}{r}. \end{align}$$ But then the acceleration wouldn't seem to be the gradient of the potential! $$a_g = - \frac{GM}{r^2} \neq -\nabla \Phi_g = \frac{d}{dr}\left(\frac{GM(r)}{r}\right).$$ What am I missing? * * * This comes up, for example when writing the equations of stellar structure, or the Lane-Emden equations. I've seen these written as both, $\frac{1}{\rho}\frac{dP}{dr} = -\frac{GM}{r^2}$, and as   $\frac{1}{\rho}\frac{dP}{dr} = -\nabla \Phi_g$ for example, the Wikipedia article on Lane-Emden includes both of these expressions. This seems to suggest that $\Phi_g \neq \frac{GM(r)}{r}$...


title: Spin-Position Coupling/Interaction

text: Firstly, is there a general procedure/algorithm for attacking Hamiltonians having spin-position coupling, e.g., $$\frac{\mathbf{\sigma_1}\cdot\mathbf{\sigma_2}}{|\mathbf{r_1}-\mathbf{r_2}|}$$ I do not know where to begin. Secondly, I would like you to recommend a textbook which discuss such a kind of systems. I looked out and found none.


title: Questions on the elementary excitations in the resonating-valence-bond(RVB) states?

text: It is known that the RVB states can support spin-charge separations and its elementary excitations are spinons and holons. But it seems that there are some different possibilities for the nature of the statistics of the spinons and holons. Specifically, (1) spinons are _fermions_ while holons are _bosons_ (Kivelson et al);(2)spinons are _bosons_ while holons are _fermions_ (Read et al);(3)spinons and holons are both _semions_ obeying fractional statistics(Laughlin et al). Do we now have a commonly accepted view for one of the above three possibilities? Or do the quasiparticle statistics depend on the details(e.g. which kind) of the RVB wave functions? Furthermore, assume RVB states as the ground-states of the antiferromagnetic Heisenberg model, and when we use $t-J$ model to study the **lightly doped** Mott insulator, for the possibility (1), it's reasonable to adopt **slave- boson** formalism, while for the possibility (2), it's reasonable to adopt **slave-fermion** formalism, right? But for the possibility (3), what kind of **slave-particle** formalism we should adopt? This part may be related to my previous question. Thanks in advance.


title: $D$ and $H$ in macroscopic Maxwell's equation: auxiliary or constitutive?

text: I'm not a physicist. I want to understand the macroscopic Maxwell's equations. But after reading Wikipedia and other Googled stuffs, I got very confused. In particular, $D$ and $H$ have two different equations, respectively. One group is called auxiliary fields: $$B = \mu_0(H+M),$$ $$D = \epsilon_0E+P.$$ The other group is called constitutive relations: $$B = \mu H,$$ $$D = \epsilon E.$$ Which group of equations are relevant to the macroscopic Maxwell's equation if we consider the case with polarization ($P$ and $M$ present). My aim is to learn computational electromagnetic.


title: Where to find detailed measured emission spectra of all chemical elements?

text: I'd like to have something like this, but for single atoms and with more extended range of wavelengths. All I could find e.g. for hydrogen was lots of talks about Rydberg formula etc. and plots of spectrum in visible region. But I need _experimental_ data to look at, not something plotted from formulas like Rydberg's one. Similarly for other elements, I mostly found only plots of spectral lines in visible range. So, where can one find detailed measured spectra for all (or most) chemical elements, preferably with UV and IR parts? It'd be most useful to have it in numeric form (spreadsheet etc.) instead of pictures.


title: Does the strength of a smell depend on the volume or surface area of an agglomerate of material?

text: I just switched from Timberland to Adidas shoes, and noticed a horrible trait of the Adidas tread: instead of being composed of externally-protruding bumps like the Timberland, it is composed of channels striated _internally_. This has the horrible trait of trapping and making difficult to remove small pebbles, mud, and other foreign objects that one might inadvertently step on. Assuming that one were to step on an unusually aromatic organic viscoplastic, **would the 'strength' of the smell depend on the surface area of the exposed agglomerate, or of its mass**? My instinct leads me to think that the surface area would be the major factor, but empirically I see that larger massed agglomerates, even if wedged deep within an Adidas tread with little exposed surface area, seem to be more easily detected by human olfactory organs. However I contend that my sensing apparatus and testng methodology are not scientifically sound, nor do I have access to facilities which would allow me to make more objective observations. Of course, the principle of "a month in the lab could save an hour in the library" applies, so I searched online but found surprisingly little (read: none) literature on the subject.


title: Why do the plugs in Europe and US have different voltage?

text: Is there a reason why plugs tend to be 110 V in America but 220 V in Europe? Is one of them better than the other?


title: Escape velocity problem

text: I was given a problem at school: > How much Energy do we need to make a rocket of mass $m$ faster than the > escape velocity so that it can travel in outer space? Here's how I worked:   I know that the escape velocity from a object of mass $M$ that in this case is the Earth is $$v_f=\sqrt{\frac{2GM}{r}}$$ Then I calculated the Work that I need using the formula for Kinetic Energy: $$ K=\frac{1}{2}mv^2 \rightarrow K=\frac{1}{2}mv_f^2=\frac{1}{2}m\sqrt{\frac{2GM}{r}}^2=\frac{1}{2}m\frac{2GM}{r}= \frac{mMG}{r} $$ Is this right? Now I know that $W=F\cdot S$ and if I want to know the force I need to change my equation to $F=\frac{mMG}{rS}$ right? Thanks! p.s. I noticed that $\frac{mMG}{r}$ is similar to the equation for the gravitational force: $F_g=G \frac{mM}{r^2}$ but is equal only that is equal to $\frac{F_g}{r} $


title: Lifetime of Positron and its existence

text: Since we know in condensed matter the lifetime of positron is about 10^-10 seconds, how do we collide positrons with electrons? How many positrons were there during the LEP experiment? I know positrons can be produced in beta decays or when two photons 'collide', but do positrons exist naturally such as electrons and protons? And also the positrons used in LEP were produced via beta decays? If not, how?P


title: when water falls to the ground and forms a puddle, where does most of the energy go?

text: When water falls from some height to the ground and forms a puddle, where does most of the initial potential energy go? Sound, heat, surface tension, or motion of the earth?


title: Why Change Of Magnetic Flux Induces Current In circuit?

text: Why change of magnetic flux with the time through a coil induces current in it? Please tell me that what happen to charge in the coil if the magnet or electromagnet wave around this? Also, why a constant magnetic flux does not induces current?


title: What Would Negative Mass Do To Spacetime?

text: It's known that positive mass bends space-time to create a curvature. But if something had negative mass what would it do? Make it flat or like a crest?


title: Helicity operator in Non relativistic limit

text: Helicity operator in Dirac equation is given by $$H=\frac{\vec{S}\times \vec{P}}{P^{2}}$$ This operator commutes with dirac hamiltonian.We can also define a helicity(with same form) operator in case of Schroedinger equation.Which will commute with Schroedinger hamiltonian and have eigenvalue $\frac{1}{2}$ or $-\frac{1}{2}$. We know spin also have eigenvalues same as helicity operator.Now spin is the projection of spin angular momentum on $z$ axis but $H$ is the projection of spin angular momentum on momentum axis.My question is how do I relate these two operators in non relativistic case? It looks like helicity operator does all the things spin operator normally do in Non relativistic case also.So isn't more sensible to use helicity operator in non-relativistic limit also instead of spin operator?


title: Static electrical attraction

text: Coulomb's law is used to calculate the electrical attraction between 2 charged particles, what formula do I use to calculate an electrical attraction magnitude between 2 plates? Let's assume the first plate holds a charge of +3x10^-4 coulombs and the other one holds the same negative amount of charge (-3x10^-4), both plates area is 5 square centimetres and they are fully exposed and facing to each other. Also let me assume that the distance between them is 2 meters.


title: Based on what does Hawking give 1000 years of time for humans to escape the planet earth?

text: I read it news that: > Hawking Gives Humans 1,000 Years to Escape Earth How did he arrive at this figure? Whats gonna make human race extinct in 1,000 years?


title: Henry's Law in metals

text: For fixed temperature the concentration of a gas dissolved in a solution is directly proportional to the gas pressure $p$, i.e. $c_s (T,p) = f(T)p$. Since $H_2$ only dissolves in metals as single atoms this law needs to get modified to: $c_s(T,p) = f(T) \sqrt{p}$, which is what I want to show. This is what I've got so far: at first there needs to be a chemical reaction $H_2 \rightarrow 2 H$, which with the chemical potential gives: $- \mu_{H_2} + 2 \mu_H = 0$. Since $\mu_H$ is dissolved in the metal there is an extra mixterm i.e. $\mu_H = \mu_H^° + RTlog(c_{H})$ where $\mu_H^°$ is the chemical potential of atomic hydrogen outside of a solution. If we derive the above expression w.r.t. $p$ at constant $T$ we get: $-(\frac{\partial \mu_{H_2}}{\partial p})|_T + 2 (\frac{\partial \mu_H^°}{\partial p})|_T = 0$ because the mixterm doesn't depend on $p$. Now from the Gibbs-Duhem relation ($SdT - Vdp + Nd\mu = 0$) I know that $N(\frac{\partial \mu}{\partial p}) = V$ which transforms the above expression into: $-\frac{V_{H_2}}{N_{H_2}} + 2 \frac{V_{H} }{N_H} = 0$ This is where I'm stuck. My assistant told me I should arrive at a differential equation for the concentration in the end, but I don't know where I went wrong / how to go on. Thanks in advance!


title: Why is the inertia ellipsoid of a higher symmetry than the rigid body?

text: I was always puzzled by this fact. A uniform cube has a sphere-shaped inertia ellipsoid. The sphere has a higher symmetry then the cube. Is there any deep reason or implication behind it?


title: Other units of resistance

text: Resistance is $\frac{V}{I}$, and we get the unit $\Omega$. Another unit is $\frac{W}{A^2}$. How do you derive that unit?


title: Vibration transmission using spring and damper model

text: I want to simulate the vibration transmission between two colliding objects. I use a Finite Element Model to simulate the vibration on each object, but I haven't found any good mathematical model to transfer the vibration between two FEM models that collided and keep in contact. ![enter image description here](http://i.stack.imgur.com/LcbS8.jpg) In the figure, a sphere is dropped into a plate. the change in color represent the displacement on the FEM models. Currently, I am using a spring/damper model to transmit the vibration from one object to the other. But, I haven't found any good model that give any theoretical foundation to my simple spring/damper model.


title: How to calculate differential pressure

text: A ventilator fan mounted on the wall of an air tight room(except for an air intake with a given cross sectional area) is blowing air out of the room. If air displacement rate of the fan is given(CFM or m3/hr), is it possible to calculate the pressure differential that it can create between inside the room and outside the room ?


title: Rod slipping against block due to gravity?

text: A uniform rod of mass $m$ and length $l$ is pivoted at point O. The rod is initially in vertical position and touching a block of mass M which is at rest on a horizontal surface. The rod is given a slight jerk and it starts rotating about point O. This causes the block to move forward. The rod loses contact with the block at $\theta = 30^\circ$. All surfaces are smooth.   1. The value of $M/m$?   2. The velocity of block when the rod loses contact with the block?   3. The acceleration of center of mass of rod, when it loses contact with the block?    4. The hinge reaction at O on the rod when it loses contact with the block? ![enter image description here](http://i.stack.imgur.com/uJWRv.jpg) Here is my attempt at the problem The component of angular speed of the tip of the rod in horizontal direction is same as that of the block. Let V be the speed of the block when rod loses contact with block. So,Horizontal component of angular speed of the tip of the rod = $\omega l\cos(60)=V$ So, $\omega l/2=V$ Thus, $\omega = 2V/l$ Applying Conservation of Energy, Loss in Potential Energy of Rod = Gain in Rotational Kinetic Energy of Rod + Gain in KE of Block $$\frac{Mgl}{2}(1-\sin60^\circ) = \frac{1}{2}\frac{ml^2}{3}(ω ^2) +\frac{1}{2}MV^2 $$ $$\frac{Mgl}{2}(1-\sin60^\circ) = \frac{1}{2}\frac{ml^2}{3}\frac{4V ^2}{l^2} +\frac{1}{2}MV^2 $$ $$\frac{Mgl}{4} = \frac{2mV^2}{3} +\frac{1}{2}MV^2 $$ The forces on the rod are Hinge force in x direction = $ H_{x} $ Hinge force in y direction = $ H_{y} $ Force due to gravity = mg Normal contact force from the block = $N$ The force on the block are Normal contact force from the rod =$ N$ Now, I am not able to proceed further.


title: Eigenstates of a density matrix of continuous variables

text: Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = \int_{-\infty}^{+\infty} dx_1 \int_{-\infty}^{+\infty} dx_2 \int_{-\infty}^{+\infty} dx_2' \,\psi_0(x_1, x_2)\psi_0^*(x_1, x_2') |x_2\rangle\langle x_2'|$$ So the entries are: $$\rho_2(x_2,x_2') = \int_{-\infty}^{\infty} dx_1 \psi_0(x_1,x_2) \psi_0^*(x_1,x_2')$$ $x_1$ and $x_2$ are continuous, so I interpreted $\rho_2$ as a matrix of (uncountably) infinite dimensionality: it has an entry for every $(x_2, x_2')$. Now we want to find the eigenvalues. A paper I'm reading defines these as follows: $$\int_{-\infty}^{+\infty} dx' \rho_2(x, x') f_n(x') = p_n f_n(x)$$ A system of $n$ equations, where $n$ is an integer running from zero to infinity. Now my question is three-fold: 1) Where do these equations come from? Are they simply the generalisation to continuous variables of the discrete case $$\sum_j O_{ij} \psi_j = \lambda \psi_i$$ for each $n$? 2) For a matrix to be diagonalisable, it needs to have a number of eigenvectors equal to its dimensionality. $n$ takes only countably infinite values, while I thought $\rho_2$ had uncountably infinite rows and columns. I suppose my interpretation of $\rho_2$ is incorrect, but I don't see why, since $x$ and $x'$ can clearly take non-integer values. Now the author 'guesses' the following solution: $$p_n = (1 - \epsilon)\epsilon^n \\\ f_n = H_n(\alpha^{1/2}x) e^{-\alpha x^2 / 2}$$ Where $H_n$ is the (physicist's) Hermite polynomial, $\alpha = (\gamma^2 - \beta^2)^{1/2}$, $\epsilon = \beta / (\gamma + \alpha)$, and $n$ runs from zero to infinity. In order to define these new quantities, I'll backtrack a bit. The system has Hamiltonian: $$H = \frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2]$$ Uncoupling the Hamiltonian, where we use that $x_± = \frac{1}{\sqrt{2}}(x_1 ± x_2)$, $\omega_+ = k_0^{1/2}$ and $\omega_- = (k_0 + 2k_1)^{1/2}$, we obtain the normalised ground state: $$\psi_0(x_1,x_2) = \pi^{-1/2} (\omega_+ \omega_-)^{1/4} e^{-(\omega_+ x_+^2 + \omega_- x_-^2) / 2}$$ Now we define $\beta = \frac{1}{4} (\omega_+ - \omega_-)^2 / (\omega_+ + \omega_-)$ and $\gamma - \beta = 2 \omega_+ \omega_- / (\omega_+ + \omega_-)$, and write the reduced density matrix entries as: $$\rho_2 (x_2, x_2') = \pi^{-1/2} (\gamma - \beta)^{1/2} e^{-\gamma (x_2^2 + x_2'^2)/2 + \beta x_2 x_2'}$$ 3) I want to check the author's solutions. Using Mathematica I've evaluated the integral to a mess of Gamma and Krummer's fuctions: rewriting that in terms of the eigenstates did not seem doable. Besides, I'd rather do a more explicit check. A second idea was to use the generating fuction: $$e^{2xt - t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}$$ And use orthogonality of Hermite polynomials to get rid of the sum. This seemed doable, since the left-hand side of the eigenvalue equation contains an exponential of this form. I've run into some problems using this method though, the biggest of which is that I've failed to make the arguments of the generated polynomial and the polynomial in the eigenstate equal, and am thus unable to use orthogonality. I've been stuck on this for days, so any help would be greatly appreciated!


title: Calculate divergence of vector in curvilinear coordinates using the metric

text: In a curved $(3+1)$ dimensional spacetime with metric components $g_{\mu \nu}$, the covariant derivative of a $4$ vector $\mathbf V = (V^0, \vec V)$ is given by $$\nabla_\mu~ V^\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}~V^\mu).$$ I expect that this relation can also be used to derive the expression for divergence of the $3$ vector $\vec V$ in a flat spatial hypersurface in a curvilinear coordinate system, eg. the cylindrical polar coordinates $(r,\phi,z)$. We will then need to replace the $\sqrt{-g}~$ by $\sqrt g~$ as the metric of the spatial hypersurface has a positive determinant. This will then give $$\vec \nabla \cdot \vec V = \nabla_i V^i = \frac{1}{r}\partial_r(r~V^r) + \partial_\phi V^\phi + \partial_zV^z.$$ However, the actual expression for the divergence of a $3$ vector in cylindrical polar coordinates is $$\vec \nabla \cdot \vec V = \frac{1}{r}\partial_r(r~V^r) + \frac{1}{r}\partial_\phi V^\phi + \partial_zV^z.$$ Can you please point out and explain where I am going wrong?


title: A question about multiverse

text: I have a question that I would like to ask here since I'm not an expert or any near to that. Actually I just came up with this question when I was watching The fabric of cosmos. I couldn't find where I can ask this kind of question. So hopefully this is the right place.   * What is the "background" of multiverse? Let me elaborate a little more what that means clearly. As far as I understand, multiverse is a cluster of many universe and it seems like (well from what I saw in documentary) that bubbles in a cup of soda. What I am curious is that then what is the mechanism of the background or surrounding, I should say, of the multiverse? Since if multiverse "exists" somewhere, then that means the surrounding has to have some kind of characteristics, such as space-time. I hope that I explained comprehensibly, and somebody can give some reasonable explanation. Even better, any book recommendation would be appreciated! Even though I studied physics for 1 year and half, I forgot many of them except curiosity. ;)


title: What is the stress energy tensor?

text: I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page: Is the "stress-energy" something that can be derived from the pseudo- Riemannian metric (like the Ricci tensor, scalar curvature, and obviously the metric coefficients that appear in the equation) or is it some empirical physics thing like the "constants of nature" that appear in the equation? Or do you need some extra mathematical gadget to specify it? Thanks and apologies in advance if this is utterly nonsensical. Also, as a non-physicists I'm not sure how to tag this either so sorry for that as well.


title: How is/do they plan on having fusion energy collected at the National Ignition Facility?

text: I was just watching a program on television showing how the method they use at the National Ignition Facility to attempt to achieve fusion is to fire extremely powerful lasers at a small pellet of fuel sitting in a vacuum at the center of a large 10 m diameter sphere. To people familiar with the design of the setup they have there, I ask: what is the mechanism by which they intend to collect the energy that would result when that pellet releases its energy?


title: Why the original BB84 paper "Quantum cryptography: Public key distribution and coin tossing" has 'withdrawn' status?

text: The original paper proposing quantum key distribution protocol (now known as BB84):   * Charles H. Bennett, Gilles Brassard, Quantum cryptography: Public key distribution and coin tossing seems to have `WITHDRAWN` status with a note: > This article has been withdrawn at the request of the author(s) and/or > editor. The Publisher apologizes for any inconvenience this may cause. The > full Elsevier Policy on Article Withdrawal can be found at > http://www.elsevier.com/locate/withdrawalpolicy. Is is known why? Is it known if it is the same paper (the date is 28 September 2011 rather than anything in 1984)?


title: Why do humans have bilateral symmetry?

text: About the eyes I know that it requires for gauging distance as in Modern 3D cameras have two sensors. And two ears for sound source localization using differences in levels and timing (But not yet two microphones in mobile phones/other devices). What about other organs? Even not all organs are in pair. If it is about about redundancy then why not more than two? Thanks for your answer.


title: Formula for potential for 2 coaxial tubes

text: Could someone remind me of the formula for potential function for a system consisting of 2 coaxial tubes maintained at a fixed potential difference and the in between medium has uniform conductivity $c$? (I have forgotten how to calculate these things, have been a long time since high school!)


title: Is there a proof from the first principle that the Lagrangian L = T - V?

text: Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it? On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the answer, and if possible, the primary source (who published the answer first in the literature). * * * Notes added on Sept 22:   \- Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of.   \- We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider.   \- But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion.   \- Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.


title: How do you respond to questions like "Have you ever observed a UFO?"

text: This is a question that astronomers get asked by the public regularly, and I'm curious to see how others respond.


title: What is the current status of Pluto?

text: Pluto has been designated a planet in our solar system for years (ever since it was discovered in the last century), but a couple of years ago it was demoted. What caused this decision? And is there a chance that it could be reversed?


title: Questions about compressing a liquid in a piston

text: We were discussing super-critical fluids with a colleague and as were thinking about different thought-experiments, it occurred to me that you cannot really have only liquid in a piston in a real situation. I want to make sure I'm not missing something. Are there liquids which have zero vapor pressure under certain conditions? If not, am I right to think even if you would perfectly enclose a volume of liquid in an hermetic container, you would always have a tiny fraction of vapor due to evaporation and vapor-pressure equilibrium? If that is the case, ato turn this liquid into a supercritical fluid, I will only need to reduce the volume of the container by a very small amount and this can be explained in two ways:   * from the liquid point of view, since it is almost incompressible, I need a very large force to change that volume even by a small amount. Thus the pressure in my piston is large and likely over the critical pressure of the liquid, turning it into a super-critical fluid   * from the gas point of view, since the volume occupied by the gas is very small and since it will "feel" the compression first, it is easy to see the gas pressure increasing very rapidly for small changes in volume, once again leading to a supercritical fluid. Thanks in advance for your answers !


title: What limits range of gravitational attraction concerning space expansion?

text: Wikipedia says: > Metric expansion is a key feature of Big Bang cosmology and is modeled > mathematically with the FLRW metric. This model is valid in the present era > only on large scales (roughly the scale of galaxy clusters and above). At > smaller scales matter has become bound together under the influence of > gravitational attraction and such bound objects clumps do not expand at the > metric expansion rate as the universe ages, though they continue to recede > from one another. ...why? All gravity equations I know are continuous and smooth - gravity never drops to zero. Reading the above though sounds as if there was a cut-off point somewhere; bodies far enough cease to be bound entirely. It's no longer several forces overlapping and struggling, like electromagnetic and strong, struggling between binding the atom nucleus and tearing it apart. The statement makes it sound as if there was a clear-cut border between where gravity works and where metric expansion works, clusters of galaxies with their gravitational field reaching a finite distance, like free-falling snowflakes able to clump together when they meet. Is there a border limiting range of gravity? How is the separation between gravitationally bound and independent systems explained?


title: What is the physical meaning of $|f''(\xi )|\leq\frac{4(f(b)-f(a))}{(b-a)^{2}}$ where $f(t)$ is position as a function of time?

text: Here is the full problem: ![enter image description here](http://i.stack.imgur.com/DD5Dy.png) I was able to prove the given expression; but the interpretation part is giving me a bit of trouble. Here is my interpretation: The average acceleration in $(a,b)$, is given by: $\frac{f(b)-f(a)}{(b-a)^{2}}$ Thus, by the definition of the average, there has to be some $\xi$ in $(a,b)$ S.T. $$|f''(\xi )|\leq\frac{f(b)-f(a)}{(b-a)^{2}} \leq\frac{4(f(b)-f(a))}{(b-a)^{2}}$$ assuming $f(b) \geq f(a)$ However, I am just arbitrarily throwing that 4 in the numerator, and can't attach it to any physical interpretation. Maybe I need to make use of the fact that $f'(a)=f'(b)=0$? Any insight would be appreciated!


title: Decomposition of SU(N) adjoint representation under SU(2)

text: I am having trouble in demonstrating that under SU(2) transformations the adjoint representation of SU(N) transforms as one spin 1, 2(N-2) spin $\frac12$ and singlets. I am trying to demonstrate it from $N \otimes \bar{N} = 1 + A$ where $A$ is the adjoint representation; and the fact that an N vector of SU(N) decomposes as one $j=\frac12$ plus two $j=0$. So basically: $(1\oplus^{N-2}0)\otimes(1\oplus^{N-2}0)= ?$. But I end up with one spin 1, $2N-3$ spin $\frac12$, and $(N-2)^2+1$ spin 0. Embarrassing. Is my equation with the question mark wrong?


title: Gravitation in a space that is topologically toroidal

text: In my scant spare time I'm building an Asteroids game. You know - a little ship equipped with a pea shooter and a bunch of asteroids floating around everywhere waiting to be to blown up. But, I wanted to add a little twist. Wouldn't it be cool if Newtonian gravity was in effect and you could do thinks like enter into an orbit around an asteroid and fire at it, or shoot gravity assisted bankshots around a massive asteroid so that you can shoot one behind it. But the problem is that in asteroids, space is topologically toroidal. If you fly off of the top of the screen, you reappear at the same x coordinate at the bottom of the screen (and similarly for the right of the screen). So how does one calculate the distance between two bodies in this space? Really, I realize that this question doesn't make sense because body A would pull upon body B from a variety of directions each with their corresponding distances. But anyway, the main questions: How would Newtonian gravity work in toroidal space? AND Is there any applicability to the answer to this question outside of my game?


title: I think there is a mistake in solutions to this problem. plus and minus sign

text: ![](http://img526.imageshack.us/img526/2593/49699289.jpg)! I pretty much had what the solutions had, but we disagreed with one thing, a minus sensitive sign. $\sum W = \Delta K$ $\int_0^{-s} - ks^2 ds = \frac{ks^3}{3} = \frac{1}{2}m(v^2-4^2)$ Rearranging, I get $v^2 = 4^2 - \frac{2k(-0.2)^3}{3m} = 19.2$ So v = 4.38m/s But the key did $v^2 = 4^2 - \frac{2k(0.2)^3}{3m} = 12.8$ where they let $\int_0^{s} - ks^2 ds = \frac{ks^3}{3}$ **EDIT** Apparently I am stupid and I made a question that didn't even need answering... I had this on paper and I typed up the wrong thing, that's why I was confused. $\sum W = -\int_{-s}^{0} ks^2 ds = \frac{1}{2}m(v^2-4^2)$ Sorry everyone


The lubricant is needed only on the hair surface? How can the blade cut the slippery surface? Why does soap make shaving with razor less painful?


title: Is aluminium magnetic?

text: From high school, I remember that Aluminium has 13 electrons and thus has an unpaired electron in the 3p shell. This should make Aluminium magnetic. However, the wiki page of Aluminium says its non-magnetic at one place(with a citation needed tag though) and at another place says it's paramagnetic. Doing a google search shows up some contradictory results. So what is the truth? Note:The context of the question is this answer on scifi.SE about magneto.


title: how to determine if a vortex is laminar or turbulent

text: In a cylindrical chamber with a high diameter-to-height ratio; a fluid is tangentially injected. there is an axial exit to the cylinder. how do I determine if the vortex so formed is laminar or turbulent? what parameter will indicate this? (this is a simulation; so I can measure whichever velocity/turbulent KE that is needed.)


title: Feynman diagrams in three dimensions

text: Is there such a thing as an extension of Feynman diagrams into three dimensions? Canonical Feynman diagrams use one space and one time dimension to visualize processes involving elementary particles in space-time. This is very convenient for presenting them on a two-dimensional surface, such as in an academic paper. It seems to me that with computer displays one could now employ two space and one time dimension in three-dimensional Feynman diagrams, which could be rotated on a computer screen, etc. I'm sure that this is technically feasible. I am less sure that there are relevant processes whose presentation would sufficiently gain in substance from showing two spacial dimensions instead of one, so three-dimensional Feynman diagrams (or whatever one would call those) may amount to a solution without a problem. In terms of answers to the question I would appreciate pointers to academic papers or visualization tools, if there are any.


title: If 100% of the energy from the sun is reflected back into space

text: 100% of the energy from the sun is reflected back into space, it's just shifted from a low-entropy state to a high-entropy state, and from a high frequency (ultraviolet) to a low frequency (infrared). But does the total number of photons change as a result? Are there more low frequency photons, then, in order to make the same amount of energy?


title: How to get the angle needed for a projectile to pass through a given point for trajectory plotting

text: I am trying to find the angle needed for a projectile to pass-through a given point. Here is what I do know:   * Starting Point $(x_0,y_0)$   * Velocity   * Pass-through point $(x_1, y_1)$ I also need to incorporate gravity into the equation. Anyone have any ideas? I haven't had much luck so far, so any ideas/suggestions would be great.


title: Excitons in metals-do they exist?

text: Recently I red an article "Surface Enhanced Fluorescence". It is a topical review by Emmanuel Fort and Samuel Gresillon. Here it is: http://turroserver.chem.columbia.edu/surfaceplasmons/pdf/53_JPD_41_1_2008_SEFSRev.pdf On the 8th page of pdf there's such thing stated: > The energy is transferred to electron–hole pairs of the metal (excitons) > which act as energy acceptors. How is it possible? I was searching for knowledge about excitons and it seems that they can't exist in metals. Even princeton says so: https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Exciton.html Not a single word about metals. So... can I treat this article seriously? Or maybe there's something I'm missing? Can an exciton exist in metal?


title: Einstein +Maxwell 's tensor

text: Why is it true that we can deduce that Einstein's GR equations coupled with Maxwell's EM equations may be written in the form $$R_{ij}=C(F_{ik}F_j^{\,\,k}-{1\over 4}g_{ij}F_{mn}F^{mn})$$ without knowing EM? I wonder if it's a degree of freedom thing. Please enlighten! Thanks.


title: Can a "bowl" shaped liquid half-bubble be free-floating in the air?

text: **Intent:** Not looking to make-up something, I seek explanations which are possible to duplicate **Description of Phenomenon:** The halved side of the bubble would be horizontal to Earth's surface, and the bowl shape would be closer to the ground. **Image of Shape (not the bubble itself):** Here's a steel version of the bubbles shape * * * **Related Physics:** Here's the best explanation I've found related to the physics know to relate to the issue: > We consider half of the soap bubble. The forces on the hemisphere will be > the surface tensions on the two circles and the net force from the excess > pressure between the inside and the outside of the bubble. This net force is > the sum of all of the forces perpendicular to the surface of the hemisphere, > but must be parallel to the surface tension. Therefore we can find it by > finding the force on the circle that is the base of the hemisphere. The > total force must be zero, so we have > > 2(2¹r)g = (¹r2) ÆP, which gives ÆP = 4g/r. SOURCE and RELATED IMAGE * * * **CURRENT-STATUS:** I'm out of possible reasons why this would happen, except for one, that being, that any half-bubbles observed might in fact be a whole free-floating bubble with three walls, two of which are refracting near zero light, while the third is highly refractive. The three walls form a single sphere divided in half by single wall; meaning that there's a top-hemisphere, and lower-hemisphere, and the dividing wall. Would this be possible? **BACKGROUND:** This phenomenon is based on a question I saw years ago posted on another physics community forum. I'm unable to locate that page, but I've searched for it. On that page, more than three(3) people had independently observed the phenomenon; which to me gives some validity to it existence in some form; that, or someone was just having fun, and gaming the forum.


title: stress tensor in Kazama-Suzuki construction

text: This is a technical question about equation (2.42) of the original paper [KS] of the Kazama-Suzuki construction. I think the authors did a simple substitution $\tilde{J}^a=\hat{J}^a-\frac{i}{k}f_{a\bar{b}\bar{c}}j^{\bar{b}} j^{\bar{c}}$ in the equation (2.40) to get the equation (2.42). If this is the case, I think I will get plus signs for the last two terms of the equation (2.42). But in the paper these two terms have minus signs in front of them. I guess I have missed some important steps in the derivation, so any instruction/comment will be very helpful to me. Thank you very much.


title: Sign of $dr$ in Schwarzschild geodesics

text: There is an equation that relates energy $E$, angular momentum $L$ and other constants and variables to find $\left(\frac{dr}{d\tau}\right)^2$ in a plane. $$\left(\frac{dr}{d\tau}\right)^2=\frac{E^2}{m^2c^2}-\left(1-\frac{2GM}{c^2r}\right)\left(c^2+\frac{L^2}{m^2r^2}\right)$$ So, $\frac{dr}{d\tau}$ is: $$\frac{dr}{d\tau}=\pm\sqrt{\frac{E^2}{m^2c^2}-\left(1-\frac{2GM}{c^2r}\right)\left(c^2+\frac{L^2}{m^2r^2}\right)}$$ So, which sign should I use? Is there a method to find which sign is to be used? Perhaps, if $|d\phi|$ is increasing, so $r$ is decreasing and then $dr$ is negative, and vice versa. But I don't know how much this is true, especially near the event horizon.


title: Do all the 4 forces of nature act at the same speed?

text: It is believed that gravity, the weakest of the four forces propagates at the speed of light, cf. e.g. this Phys.SE post. One would expect (perhaps erroneously) that the other, stronger, forces acted at an even quicker speed. Of course, it is not like that. At least, it's not for electromagnetism. What do we know about the speed of the other two?


title: Wall visible after light source removed

text: Let's say you have a lamp pointed at a wall, in a dark room with no other light source, if you turned off that lamp #  Would the wall still be visible for any amount of time?


title: Finding the direction of the electric field of several charges?

text: A clock has negative charges -q, -2q, -3q,...., -12q fixed at a the position of the corresponding numerals on the dial. The clock hands do not disturb the electric net electric field due to pont charges. At what times does the hour hand point in the direction of the electric field at the centre of the dial?


title: Interference at the focus of a convex lens or a concave mirror

text: Now consider parallel rays travelling towards the convex lens. After emerging they get converged at the focus. Won't this lead to interference between light rays? Practically it should but I would like to know why this does not happen. ![http://www.passmyexams.co.uk/GCSE/physics/images/convex_lens.jpg](http://i.stack.imgur.com/bwiBG.jpg) Look at the focus in the above image. The rays get converged and shouldn't that lead to interference. Similar phenomenon should occur with a concave mirror at its focus.


title: Why are infinitesimal rotations commutative, whereas finite rotations are not?

text: Infinitesimal rotations commute and every finite rotation is the composition of infinitesimal rotations which should logically mean they also commute; but they don't. Why?


title: The necessity of the B field

text: It is fairly easy using basic special relativity to arrive at the conclusion that the magnetic force effect on nearby charges of wires carrying currents on nearby charges is only due to the length contraction in certain inertial frames from the point of view of charges, which gives rise to a perceived change in charge density. However, we structure the (classical field) laws of electromagnetism in Maxwell's laws using a B field. My question is, is it possible to formulate Maxwell's equations only in terms of Lorentz transformed E fields? If not, why not?


title: How temperature gradient is a vector?

text: Everyone knows Temperature gradient is a vector quantity having direction from cold to hot.My confusion: why is temperature gradient vector if its direction is always fixed (as in the case of pressure) (Don't say that it is because it follows vector law of addition,I am searching for more concrete answer...)


title: Atoms in our body

text: Our body of course exchanges some atoms with environment every day however are there some of atoms which stay with us a life time?


title: Perturbation theory in quantum mechanics

text: In perturbation theory perturbed eigenstates expanded by unperturbed eigenstates, but we know when the system perturbed its Hilbert space altered and hence its basis changed, then we can't state this basis spans the new Hilbert space and we can't expand perturbed eigenstates by them. How this problem resolved? Please don't answer this question to say that it is approximately true since in textbooks this is done in exact manner.


title: How long does a supernova last?

text: Just what the title indicates. Is a supernova over instantaneously? Or, does the (for want of a better word) explosion continue for a while? What is/are the order of timescales involved? EDIT: Following the posted replies/comments I figure what I mean to ask is the duration for which the supernova continues to release copious amounts of energy.


title: Energy stored in fields

text: How to intuitively think of energy stored in a magnetic/electric field? Kindly answer in a bit simple terms without referring to mathematics.


title: Can the laws of quantum mechanics be derived from a more fundamental theory?

text: String theory takes quantum mechanics and tries to make it compatible with gravity. If it turns out to be a theory of everything then would it explain why our world is described by the laws of quantum mechanics? I want to understand if the laws of quantum mechanics are the only logical possibility for a low-energy theory potentially derivable from a better understanding of string theory. The other possibility is that they are just strange laws that are discovered through experiments that can't be reached by pure theoretical reasoning. Which of the two possibilities is more reasonable from a physics point of view and why?


title: Vacuum and repulsive gravity

text: How can one show from General Relativity that gravity is attractive force, and under which conditions it becomes repulsive, also why positive energy vacuum drives repulsive gravity?


title: Significance for LQG of Sen's result on entropy of black holes?

text: Sen 2013 says, > ...we apply Euclidean gravity to compute logarithmic corrections to the > entropy of various non-extremal black holes in different dimensions [...] > For Schwarzschild black holes in four space-time dimensions the macroscopic > result seems to disagree with the existing result in loop quantum gravity. How serious a problem is this for LQG? Does this mean that LQG doesn't have GR as its semiclassical limit? Does that mean it's a dead theory, or maybe just that it needs to be modified? Is the technique using Euclidean gravity reliable? Since I'm not a specialist, I'd be interested in a hand-wavy explanation of what the Euclidean gravity technique is about. Sen, http://arxiv.org/abs/arXiv:1205.0971


title: In which direction do gravitons travel? Do they originate from mass or are they collected by masses?

text: This is a kind of silly question, that mostly goes unanswered: are gravitons detatched from the objects they act on, like photons in free space. Maybe gravity or gravitons are collected by mass (including black holes), as they are collected by the curve in spacetime caused by the mass. The origin of gravitons in such an arrangement seems phenominal in many respects, but could it be that space itself is just composed of evenly distributed graviton sources? these spill gravitons when a mass distorts it.


title: Cyclic integral?

text: Can anyone explain me what is cyclic integral and give me some instructions how to show that there exist cyclic integral for Lagrangian   $$L~=~\frac{1}{2}ma^2{\dot {\theta}}^2+\frac{1}{2}ma^2{\dot {\phi}}^2\sin^2\theta+mga\cos\theta?$$


title: Concentrating Sunlight to initiate fusion reaction

text: The idea is to collect sunlight over a large area, and concentrate it down to the nano scale.   For the sake of discussion lets say you concentrate all the light you collect down to 1000 nanometers (1µm). Questions is... how much surface area sunlight would you need to collect and focus to 1µm, in order to deliver enough energy density to initiate a fusion reaction.


title: Understanding Where I Went Wrong in a Linear Conservation Problem

text: I have the following physics problem. In the figure below, a stationary block explodes into two pieces L and R that slide across a frictionless floor and then into regions with friction, where they stop. Piece L, with a mass of 2.0 kg, encounters a coefficient of kinetic friction $μ_L$ = 0.40 and slides to a stop in distance $d_L$ = 0.15 m. Piece R encounters a coefficient of kinetic friction $μ_R$ = 0.50 and slides to a stop in distance $d_R$ = 0.20 m. What was the mass of the original block? ![enter image description here](http://i.stack.imgur.com/SHr6f.png) I have a solutions manual and can understand the solution to this problem, but that doesn’t seem to help me understand exactly what I did wrong to begin with, and for this reason I don’t think I’m really learning, I’m just learning how they solved it correctly. I was hoping that I could explain how I approached the problem (it was invalid), and somebody out there could explain to me why what I was doing was wrong intuitevley so that I will not make the same mistake. It’s just hard to learn if you don’t know what you did wrong, so I’ll explain my approach below. My Approach: So I know that there are no external forces acting on the system, so the linear momentum is conserved. So $P_i = P_f$. Now after the explosion there are two pieces that travel some distance in opposite directions, so I thought that our linear momentum could be modeled something like this: $mv = m_L v_L + m_R v_R $ which will equal zero because $mv = 0$ (ie the block is initially at rest) Next I noticed that the coefficient of kinetic friction and distance traveled is given so I thought that I could calculate the positive work in one direction and the negative direction in the other direction and this must be equal because the change in time for each block (ie left and right) will be the same. So I thought about it like this $\Delta t \cdot W_R = \Delta t \cdot W_L$. I then proceeded to solve for the mass of the right block given this equation (ie first cancel $\Delta t$ and then solve). Can someone please explain why this is an invalid approach?


title: How to comoving volumes depend on the evolution of the Universe?

text: I'm reading a paper which states that > Neutron star binary merger rate at redshift $z$ per unit observer time > interval per unit volume is $\dot{n}_{m} = \dot{n}_{0} (1+z)^{2} > (1+z)^{\beta}$, where $\dot{n}_{0}$ is the local neutron star binary merger > rate per unit volume, $(1+z)^2$ accounts for the shrinking of volumes with > redshift (assuming constant comoving volume density of the merger rate) and > time dilation, $\beta$ describes evolutionary effects. I don't understand where they get the $\beta$ part from, could anyone shead some light on this? Edit: The above is from this paper, and the paragraph is the one at the top of page 3. The author uses $\beta=0$ and $\beta =1$ in the plots.


What does it mean if a body has kinetic energy? Does it mean that the momentum vectors of each particle of that body has the same direction? What about angular momentum?


title: What is the highest useful magnification todays largest telescopes can offer?

text: I know that the maximum (useful) magnification is limited by the diffraction limit, but I was not able to find numbers for the highest useful magnification factors using modern large telescopes. How much magnification is possible, e.g. using the E-ELT?


title: Source of electrons in Crooks tube experiment (Inside the Discharge tube)

text: In the experiment, gases are put at low pressure inside the discharge tube. Now the issue that arises is a very simple one. http://highered.mcgraw- hill.com/olcweb/cgi/pluginpop.cgi?it=swf::100%::100%::/sites/dl/free/0072512644/117354/01_Cathode_Ray_Tube.swf::Cathode%20Ray%20Tube This is the animation for the experiment. Its pretty obvious that the electrons are originating from the Cathode. This basically means that the metal is 'releasing' its electrons so as to complete the circuit. However my chemistry teacher told me that the gases inside the tube are ionized, thus providing the necessary electrons to complete the circuit. However, shouldn't the metal/cathode/wire release its electrons as well to complete the circuit. I mean if it doesnt, then how is the kinetic energy of the electrons in the circuit transferred to the electrons released from the ionized gases, and why is cathode the source in that case. I have studied about thermionic emission by heating metals, that asserts a threshold for the metal. However if the battery drives the electrons then shouldn't they be released as well? My basic question to you is, where do the electrons come from and how their energy is transferred throughout the circuit. Thank you :)


title: Measuring flow rate of gases

text: Flow rates of liquids could be easily measured with a stopwatch and a known volume container. But gases, if entrapped in a container will create a back pressure, and may lead to decreased flow. I wonder how they measure the quantity of gases generated by a reaction or flow rate of fluids in turbines, etc., without creating back pressure. What/how back pressure corrections are providedif this cannot be prevented.? Are there flow meters for gases at all pressures similar to fluids.?


title: Chiral perturbation theory: what is the Quark Condensate? why expand in $U$ rather than Goldstone fields?

text: I'm studying Chiral Perturbation Theory ($\chi PT$) from Scherer's _Introduction to Chiral Perturbation Theory_. What I am currently having some trouble understanding are two things:   1. The quark condensate. What is this and why is it a sufficient condition for spontaneous breaking of chiral symmetry? What I do not really understand is where the operator $S_a=\bar{q} \lambda_a q$ comes from ($\lambda_a$ are the Gell-Mann matrices) and why the expectation value of this (which I gather is zero) gives us this thing called the quark condensate.   2. The formulation of the effective Lagrangian. There is some stuff in Scherer about the coset $G/H$ where in this case G is the full chiral group and $H$ is the vector subgroup which is left after spontaneous symmetry breaking, but I do not really follow how this discussion explains why the Lagrangian is given in terms of the SU(3) matrix $U=\exp{\frac{i}{F_0} \Phi} = \exp{\frac{i}{F_0} \phi_a\lambda_a}$ for (individual) Goldstone fields $\phi_a$? Why can't we write down the effective Lagrangian in terms of the actual degrees of freedom in the theory, i.e. the Goldstone fields? I've read something about them not transforming non-linearly (and the $U$ transforming linearly) but could not really follow so if someone could elaborate on this I would be very glad. A big thanks in advance for all help given! And another thing - if anyone has another tip for an introductory reference to $\chi PT$, I would be very grateful. Scherer works decently but it's always good to read about things from a different viewpoint.


title: Physics of a conductor gaining potential

text: Working on electrical engineering but thus far, the physics stack has proven to be a better place to read and ask questions in order to develop a better overall understanding. I am currently waist deep in _semi-conductor theory_. I am not confused about how a conductor gains the same potential as the lead it is connected to (in the case of a battery), what I am confused about however, is the underlying physics. Lets take a conductive copper wire and touch it to the negative lead on the battery and leave it at that. I have been taught that essentially, the free electrons have no path to take, so even though there are more of them, they don't disperse through the conductor unless there was potential on the other end of the conductor (excess holes). Sorry this is kind of long-winded, I am looking for what happens at the atomic level when a conductor gains a potential, but is not a completed circuit yet. For instance, will the other end of the wire instantly see excess holes (other end is connected to positive) or see excess electrons (other end is connected to negative)?


title: What exactly is the difference between a sychrotron light source and a free electron laser?

text: Cyclotron radiation is radiation emitted by a charged particle being bent by a magnetic field. The only difference between cyclotron and synchrotron radiation is that the latter occurs at relativistic speeds. Is this right? Now, AFAIK, synchrotron radiation is artificially produced by channeling the accelerated particle beam (I guess this would generally be electrons, but I suppose it could be protons like in the LHC) into an "insertion device." I know that wigglers and undulators are both examples, but I'm unclear on what exactly the difference between those two is. What I don't get is what is done in a free-electron laser that differentiates it from a regular old synchrotron radiation source. Apparently synchrotron radiation is coherent, so it clearly is not that. Moreover, I'm confused where the stimulated emission comes into play. According to wikipedia, a free- electron laser uses free electrons (thus the name) instead of, for example, chromium doped sapphire a.k.a. ruby, as the gain medium. Except, hold on. . .the electrons certainly aren't static, they're moving, quite rapidly, so how would the photons reflect back to stimulate emission if the electrons are still moving forward right past them? Am I correct in my assumptions, or am I totally wrong to begin with? Unfortunately, wikipedia is rather vague on the subject. I have tried to do some research on the subject, but I have found very little on the actual construction of a free-electron laser (at least, that wasn't behind a paywall). Thanks for your help!


title: What does it mean to "convert energy into time"?

text: In a recent article about creating electron-positron pairs by colliding photons in a laboratory, Andrei Seryi, director of the John Adams Institute at Oxford University, was quoted to said: > It's breathtaking to think that things we thought are not connected, can in > fact be converted to each other: matter and energy, particles and light. > Would we be able in the future to **convert energy into time** and vice > versa? Let's ignore for the moment that a physicist is amazed that $\gamma\gamma\rightarrow e^-e^+$ process is possible in a laboratory. But what does he mean by converting energy into time? Is this just a poetic sentence to emphasize his amazement, or is there some deeper meaning behind it? Only thing that comes up in my mind is that, because our currently best theory of gravity - general relativity - is describing the dynamics of spacetime, "creating time" might mean inducing some change in the metric tensor $g_{\mu\nu}$, so he might be talking about the prospect of converting ordinary Standard Model particles into gravitational waves and vice versa. However, he might be talking about something else. Am I missing something here?


title: Orientation in GaAs

text: I can't find the precise definition of what is the orientation of a GaAs lattice. Being the superposition of two fcc lattices (one of Ga, the other of As), I would think that it is the direction of the closest Ga atom to any As atom of the structure. Is it right?


title: Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group

text: So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can be expressed as one by denoting: $$J_{I}^{\pm} = \frac{1}{2} (J_{I} \pm ik_{I})$$ This gives two Lie algebra: $$[J_{I}^{\pm}, J_{j}^{\pm}] = i\epsilon_{ijk}J_{k}^{\pm}$$ But how is this related to our original product of SU(2)? I was under the impression if you have two Lie groups, $(g_{1},h_{1})$ and $(g_{2}, h_{2})$, then the product would be $(g_{1}g_{2},h_{1}h_{2})$. However, I tried that with the Pauli matrices and it doesn't work to be the same as $SO(4)$ Lie algebra. How should this be done?


title: What is the electric field part of an EM wave? Radiation field or the induction field?

text: Look at this image: ![Electromagnetic wave](http://i.stack.imgur.com/3i6dy.png) I wonder if the electric field is from the induction field from a vibrating electron or the radiation field? If it is from the radiation field, as I suppose, than can someone explain why they are perpendicular to me? As they have derived it from this link. It says that the magnetic field and the radiation field is in phase(in degrees! not in x-direction).


title: Some questions about car radio and cellphone antennas

text: 1-Why the antenna of the radio of cars is located outside the car and not inside? 2-If the answer to 1 is because that cars are like Faraday cages then how come my cell phone can receive signal inside the car? 3-To receive a strong tv/radio signal the antenna must be oriented in a specific way. Why the strength of the cellphone signal seems independent of the orientation of the cellphone? 4-It seems to me that the length of the antennas of cell phones or that of a tv or radio is of the same order as the corresponding wavelength used, why is that?


title: Wick rotation and spinors

text: I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; when we perform this Wick rotation, in a way we change the metric to (-,+,+,+) to (+,+,+,+), so the invariant group is no more SO(3,1) but SO(4) and (SO(4) being compact and the spinor representation non unitary) spinors don't carry finite dimensional representation of this group. So I feel like we shouldn't be talking anymore about this objects, but only about vectors of SO(4). Is my fear justified? or where am I wrong in my reasoning?


title: Intensity distribution of a LED light source

text: The light source: 190 high power LEDs in NIR. I have the datasheet of the LED. I want to make a graph showing the spatial distribution, and to measure the intensity of the beam on the whole field of view. Actually, I used to work with this light source without defining the exact distribution. According to the datasheet of the LED, I know that the radiant intensity is approx 500W/Sr. Someone can suggest me a setup or a protocol to get the informations I want? Furthermore if someone heard about a powermeter that could help me.