title: Wave functions for three identical fermions

text: I would like to express the wave functions for three identical particles, each with orbital angular momentum $L=1$ and spin angular momentum $S=1/2$, in terms of single-particle wave functions. In other words, I would like to obtain the Clebsch-Gordan coefficients for this problem. The problem is discussed in Sakurai's Quantum Mechanics textbook around p. 375 and in Greiner and Muller's "Quantum Mechanics: Symmetries, 2nd Edition" on p. 300. I know I have to find the spin wave functions and the orbital angular momentum wave functions separately, and then combine them to get fully antisymmetric wave functions. I have the spin wave functions (four symmetric, 2 mixed symmetric under exchange of particles 1 and 2, and 2 mixed antisymmetric under exchange of 1 and 2), but I haven't been able to get a small enough number of angular momentum wave functions to get just 20 fully- antisymmetric total wave functions. In Sakurai's book, p 376, eq'n (6.5.20), we see that the 20 states can be decomposed into 2 states with total angular momentum $j=1/2$, 4*3=12 with $j=3/2$, and 6 with $j=5/2$. Could anyone fill in _how_ Sakurai got 6 for $j=5/2$, 12 for $j=3/2$, and 2 for $j=1/2$? Most importantly, referring to my comments on Y Macdisi's answer below, could anyone answer the following question: Is the orbital angular momentum state with single particle wave functions $\alpha=1$, $\beta=0$, and $\gamma=0$ related in some way to the wave function with $\alpha=-1$, $\beta=0$, and $\gamma=0$, or $\alpha=1$, $\beta=-1$, and $\gamma=-1$, and so on? I would love it if I could just keep the first set of values for $\alpha$, $\beta$, and $\gamma$, but I see no good reason to do that. If anyone happens to know where this problem is discussed more fully, I would appreciate a reference. Or, if anyone knows how to do this, I would appreciate help in knowing what the $j=3/2$ and $j=5/2$ states could be in terms of single-particle orbital ang. momentum states and spin states. I think I actually have the normalization factors already, I just need to know what the single-particle states are.


title: Where inside a large uniformly dense, symmetrical sphere would its gravity toward the center be the strongest?

text: Imagine a sphere of uniform density with similar volume and average density of our Earth. There is a bore leading to the center of the sphere from the surface with a scale at regular intervals. At what part of your journey towards the center would the scale read your greatest weight? There are some caveats to consider. The sphere is in a very isolated region of space and has no angular motion. * * * I wrote this question down last night while I was having trouble sleeping. My intuition tells me that the surface of the sphere will have the greatest force of gravity. Any distance beyond the surface and the mass of the sphere above you will be reducing your weight. Is this correct? * * * If however the core is much more massive similar to Earth, one would feel a greater strain to stand upright as they descend.


title: Higgs-Boson/Graviton

text: The Higgs boson gives particles mass. And the graviton is the theoretical force-carrier of gravity. Gravity depends on mass. So if the Higgs Boson gives things mass, it therefore gives them gravity. Is the Higgs Boson the same thing as a Graviton? Or is there a difference? The only thing I know is that the Higgs Field is something very different from the Gravitational Field. Yet, I'm not satisfied with that fact. I want to know why the Higgs Boson is not the Graviton.


title: Is Thirring model a particular case of Gross model?

text: Look at this: http://en.wikipedia.org/wiki/Gross-Neveu_model Wikipedia sais "When N=1 it reduces to the integrable Thirring model". but the aditional term in thirring model is $\frac{g}{2}\left(\overline{\psi}\gamma^\mu\psi\right) \left(\overline{\psi}\gamma_\mu \psi\right)$ I think this is different from $\frac{1}{2}g \left(\overline{\psi} \psi\right)^2$ the aditional term of Gross.Neveu model. So I think wikipedia is wrong. Am I right? if you do not mind I would like that you respond to this question also Could this model have soliton solutions? Thanks in advance.


title: Does the cavity magnetron in a microwave oven produce x-rays?

text: It seems like it should due to bremsstrahlung, since we're talking about electrons with 5-7KeV of energy slamming into the walls of the device, but I've found no information about this online, so I'm assuming something else must be happening. CRT-style TVs and computer monitors both have this problem, so my guess is that most of the energy is being bled out of the electrons to form microwaves. Am I right? Also, what happens to the production of microwaves as the magnets on the magnetron slowly lose strength? Thanks


title: Stationary Perturbation Theory : Estimating higher order corrections for anharmonic oscillator

text: Note $\hbar = 1$. $$H = H_0 + \lambda V =\frac{p^2}{2m} + m\omega^2x^2 + \lambda m^2\omega^3 x^4$$ Supposedly the perturbation expansion diverges. We are supposed to estimate for what order we have a correction to the ground state energy which is comparable in magnitude to the unperturbed ground state energy itself. That is find $\eta$ such that $\lambda^\eta E^{\left(\eta\right)}_0 \approx \frac{\omega}{2}$. I only know how to find corrections up to second order. The first order correction to energy is the expectation of the perturbation in the unperturbed state $$ \lambda E_n^{\left( 1\right)} = \langle n\mid \lambda V \mid n \rangle$$ The second order correction to energy is ( where $E^{\left(0\right)}_n = \omega\left(n + 1/2 \right)$) $$\lambda ^2 E^{\left(2\right)}_n = \lambda ^2 \sum_{m\neq n} \frac{\left|\langle m \mid V \mid n\rangle\right|^2}{E^{\left(0\right)}_n- E^{\left(0\right)}_m}$$ expressing V in terms of ladder operators $$V = m^2\omega^3x^4 = \frac{m^2\omega^3}{4m^2\omega^2}\left(a^\dagger + a\right)^4 = \frac{\omega}{4}\left(a^\dagger + a\right)^4 $$ then expand and manipulate to get $$\left(a^\dagger\right)^4 + a^4 + 6N^2+6N + 3 + \left(4N + 6\right)a^2+\left(4N - 2\right)\left(a^\dagger\right)^2 $$ then obtain the 1st and 2nd order corrections to energy $$\lambda E_n^{\left( 1\right)} = \lambda \omega\frac{3}{2}\left(N^2 + N + \frac{1}{2}\right)$$ $$ \lambda ^2 E^{\left(2\right)}_n = -\frac{\lambda ^2\omega}{8} (2 n+1) (17 n^2+17 n+21) $$ I'm not sure how to make any estimation of the higher order corrections however. I get the feeling they may accept something extremely crude.


title: Nomenclature clarification concerning solitons

text: My experience with solitons is restricted to the classical setting, namely solutions to the quartic interaction $\phi^4$, the Sine-Gordon equation, and Korteweg–de Vries equations. I was explicit to show the boundary on my solitonic knowledge so that I could ask for an interpretation of the recent paper http://prl.aps.org/abstract/PRL/v112/i2/e025301 In particular I do not understand the difference between solitons and vortex rings. My confusion stems from the introduction to the video Baths and Quarks: http://www.youtube.com/watch?v=Ederft9dkag In the intro David Tong is blowing toroidal shaped bubbles, which I thought were examples of solitons. A phenomena which dolphins exhibit for play. However, in the watered down discussion of the aforementioned paper http://www.washington.edu/news/2014/02/03/solving-a-physics-mystery-those- solitons-are-really-vortex-rings/ the nomenclature vortex ring is used. Moreover, the so called "heavy soliton" posited in the Fermi gas is shown to be a quantized vortex ring. My guess is that the difference in word usage stems from the language of superfluid density functional theory. However my lack of expertise in the subject allows me to not go further than guessing at this stage. Could someone explain the difference, if there exists any, between the usage of the words soliton and vortex ring?


title: What does $\Psi^*$ mean in Schrodinger's formulation of Quantum Mechanics?

text: I am not a physics student. In one of my courses, some fundamental concepts of Quantum mechanics were needed, so I was going through them when I stumbled upon this. It says $$\text{probability} = \int_a^b\Psi^*\Psi\mathrm{d}x\quad\biggl(= \int_a^b\Psi^2\mathrm{d}x\text{ if }\Psi\text{ is a real function}\biggr)$$ Is the $\Psi^*$ in this expression the wave function's conjugate, or something else?


title: Finite corrections to the Higgs mass in SUSY

text: I am worried here about specific supersymmetric scenarios in which an energy scale above the TeV is introduces. An example would be the Seesaw extension of the Minimal Supersymmetric Standard Model. My questions is: In such a scenario, and even though SUSY guarantees a cancellation of the quadratic divergences afflicting the Higgs mass, isn't it true that large _finite_ corrections to the Higgs mass appear nonetheless - for instance proportional to the Majorana mass in the Seesaw example? Can SUSY cancel those as well?


title: Why does charge distribute evenly between two halves of a conducting sphere?

text: Always while studying I was told that if two identical spheres, one charged and the other uncharged, are brought together the charge in each one of them is halved. However, why does it happen? Is it because the electrons repel each other in the the charged sphere and shift to the other sphere, which is uncharged? For example lets consider two spheres of radius $12\text{ cm}$ and $6\text{ cm}$, each having a charge of $3\times10^8\text{ C}$. Then why would the current flow from the sphere of $6 \text{ cm}$ to the sphere with $12\text{ cm}$ of radius? Till what time will a current flow? i.e till the charge density is equal, the electric field is equal or the the force of repulsion is equal.


title: Light and Prisms

text: If you matched the yellow light of one prism to that of the green light of another prism, say through a thin glass pane, what would be the resulting color?


title: Can the proportion of dark energy change?

text: > This post generalizes the posts:   >  \- Proportion of dark matter/energy to other matters/energy at the > beginning of the universe?   >  \- How do people calculate proportions of dark matter, dark energy and > baryonic matter of the universe? Can the proportion of dark matter/energy (to other matters/energy) change with time? If so, is there a measurement of this evolution up to now, and a prediction for future?   How measure such a change?


title: rotating an object without changing its orientation

text: I have a link hinged at one end and attached to a rectangular object via another hinge in the other end as shown. Now, I was wondering how would I maintain the orientation of the rectangular object regardless of the rotation of the link. I know this sounds like a vague question but I am just out of ideas on achieving this effect. ![enter image description here](http://i.stack.imgur.com/muMGv.png)


title: Oscillation of a Bose Einstein condensate in a harmonical trap

text: We were asked to try to make a theoretical description of the following phenomenon: _Imagine a 2D Bose Einstein condensate in equilibrium in an harmonical trap with frequency $\omega$. Suddenly the trap is shifted over a distance a along the x-axis. The condensate is no longer in the center of the trap and will start oscillating in the trap._ First I thought about using a 2D trial wavefunction in the Gross-Pitaevski equation or the hydrodynamical equations for condensates, but then we were told that we should actually look at how the energy of the condensate depends on certain parameters (position, width,...) and do something with the fact that, for small deviations of such a parameter, a second order expansion can be made, which will introduce a restoring force. This makes sense for classical motion, but in this case it confused me, cause I don't know if the energy that is meant here is the original potential energy of the harmonic trap or the Gross-Pitaevskii energy that is calculated with the GP energy functional. This last one, that was calculated in an earlier exercise for a variational Gaussian wave function, turned out to be $E = \hbar \omega \sqrt{1+Na_s}$ (with $a_s$ the scattering length for the interaction energy) and so it doesn't even depend on the position. Does anyone has got any idea how I should start or approach this theoretical description?


title: Is angular momentum conserved if you move off at a Tangent?

text: Lets imagine a binary system of two astronauts in space connected to one another via light rope. The rope is taut and they're spinning round and round with their axis of rotation being the the axis perpendicular to the their centre of mass. Now, my question is this. Lets say they each let go of the rope; they move off at tangents. Is angular momentum conserved ? And what is their subsequent motion ? My hypothesis : They will move off at constant velocity at a tangent to their circular motion. I believe that because they now under the effects of no external force by the rotational analogue of newton's second law (dL/dt = T) their angular momentum will be of equal magnitude but opposite sign. Does this make sense ?


title: Electric dipole moment, which charge is the $q$ for?

text: Electric dipole moment says $p = qd$. Which charge does the $q$ equal to?


title: How to find outflow pressure/force of air through a pipe

text: I have been trying to find the correct method for determining the outflow force resulting from a certain amount of air being forced into a pipe of a certain dimension. Essentially i'm trying to determine muzzle velocity/force/pressure. I'm sure this is incredibly simple, but I have been looking around and cannot find the proper formula. I essentially already know the length of the pipe, the volume of the air, and the diameter of the pipe. I don't think i'll need this to be precise enough for temperature to play a huge part, though I'll assume room temperature if needed. I'm not a physics person, so what I'm sure is a simple question like this is still outside of my abilities.


title: Physics (measurement)

text: In october,1707 four British warships ran aground because of error in position,sparking the effort to produce a marine clock that would be accurate to locate a position with in 30 miles after sailing from England to the West Indies & back again (a)What daily accuracy for such a clock would be required? (b)What dalilt accuracy in aclock is required to fix position to with in 0.5 miles after 1 year at sea?


title: Terminal velocity of all falling objects is same?

text: Ideally 2 objects of different mass and weight will hit the floor at same time because they face same gravitational pull and accelerate. Will the terminal velocity of both objects be same also? and with that theory does terminal velocity of every object is same ideally?


title: How do you come up with a POVM?

text: This is a made-up example, just to understand a concept. If changing the probability values aids your explanation, that's fine by me. Say you have a physical quantity $E$ that can take values 1, 2, 3 with probabilities 0.4, 0.25, 0.35 respectively (working in a quantum framework). You have a positive operator valued measure, $E_1, E_2, E_3$, with $E_i$ corresponding to your measurement resulting in value $i$. If $\rho$ is the density operator representing the current state, then you have: Tr($\rho E_1$) = 0.4, Tr($\rho E_2$) = 0.25, Tr($\rho E_3$) = 0.35 Given just these probability values, is it possible to construct $E_1, E_2, E_3$ "backwards"?


title: Defining simultaneity with a central light vs with clocks

text: So there's the classic example of the relativity of simultaneity involving two people on a train, with a light source exactly between them. Moments after the lights turn on, observers on the train will say the light struck the passengers simultaneously, while folks on the ground looking into the train will see the light hit one of them first. Now let's say the passengers start out next to each other, synchronize their clocks, and then slowly proceed to their respective ends of a 2 lightsecond long table. At 2:59:59 PM according to their clocks, the light between them turns on. Now there are 4 events: 1) The light hits Passenger A (the one closer to the front of the train). 2) The light hits Passenger B. 3) Passenger A's clock ticks 3 PM. 4) Passenger B's clock ticks 3 PM. Passengers on the train should all agree the four events are simultaneous, but what will people outside the train see? Will the clocks stay simultaneous with each other, or will 2 and 4 happen simultaneously, followed by 1 and 3? And regardless of the answer, can you justify it in terms of the fixed speed of light?


title: Flux through a Mobius strip

text: I was sent here from mathoverflow, hoping for a complete answer to this: === A friend of mine asked me what is the flux of the electric field (or any vector field like $$ \vec r=(x,y,z)\mapsto \frac{\vec r}{|r|^3} $$ where $|r|=(x^2+y^2+z^2)^{1/2}$) through a Mobius strip. It seems to me there are no way to compute it in the "standard" way because the strip is not orientable, but if I think about the fact that such a strip _can_ indeed be built (for example using a thin metal layer), I also think that an answer _must_ be mathematically expressible. Searching on wikipedia I found that http://en.wikipedia.org/wiki/Mobius_resistor A Möbius resistor is an electrical component made up of two conductive surfaces separated by a dielectric material, twisted 180° and connected to form a Möbius strip. _It provides a resistor which has no residual self- inductance, meaning that it can resist the flow of electricity without causing magnetic interference at the same time._ How can I relate the highlighted phrase to some known differential geometry (physics, analysis?) theorem? Thanks a lot! === Now, I'm convinced that there are no way to apply Gauss law since there are no ways to bound a portion of space with a Mobius strip. Nonetheless I would like to "see" some equations showing me that "[the mobius resistor] can resist the flow of electricity without causing magnetic interference at the same time".


title: What does "along the handle of the cart" mean? And how does one calculate weight?

text: > A horizontal force is used to pull a $5.0\text{ kg}$ cart at a constant > speed of $5.0\text{ m/s}$ across the floor. The force of friction between > the cart and the floor is $10\text{ N}$. > >   1. What is the magnitude of the horizontal force along the handle of the > cart? >   2. Calculate the weight of the cart? >   3. Calculate the coefficient of kinetic friction between the cart and the > floor. > I'm not sure what they mean by "along the handle of the cart". Is that still the same as $F_{net}$? Also, I have no idea how I would calculate the weight.Is the weight just the normal Force? Because in order to do anything with $F_N$, I need $F_f$ and $\mu$ which I do not have.


title: Is there a limit to loudness?

text: Is there any reason to believe that any measure of loudness (e.g. sound pressure) might have an upper boundary, similar to upper limit (c) of the speed of mass?


title: How to derive the Mott gap mathematically

text: From the one-band Hubbard model, $H=-t\sum\limits_{<ij>, \sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum\limits_{i}n_{i\uparrow}n_{i\downarrow}$, we know if $U\gg t$, the energy cost of two electrons staying on the same site is too high, so there the band will be split into two bands with gap U. How to prove this mathematically from the Hubbard Hamiltonian?


title: High Frequency radio wave propagation time

text: What is the maximum time for an HF radio wave to reach its destination regardless of propagation mode? Since electromagnetic waves' speed is close to the light speed in the air, can we say it is always lower than e.g. 1 second? Or, atmospheric conditions, reflection etc. may cause it to be as long as 10 seconds or more?


title: Deriving tensor in Euler's equations for rigid body rotation

text: The answer to physics.stackexchange.com/questions/104513 gives the following derivation of tensor $I$: $\begin{align} \frac{\text{d}}{\text{d}t} I &= \frac{\text{d}}{\text{d}t} (E\,I_{\text{body}}\,E^\top) \\\ &= \frac{\text{d}E}{\text{d}t}\,I_{\text{body}}\,E^\top + E\,I_{\text{body}}\,\frac{\text{d}E}{\text{d}t}^\top \\\ &= ([\omega]_\times\,E)\,I_{\text{body}}\,E^\top + E\,I_{\text{body}}\,([\omega]_\times\,E)^\top \\\ \frac{\text{d}}{\text{d}t} I&= [\omega]_\times\,I_{\text{body}} + I_{\text{body}}\,[\omega]_\times \end{align}$ But, I want to see the derivation happens element-by-element according to the definition of a tensor as I attempted below. $\begin{align} \frac{\text{d}}{\text{d}t}\left(\begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\\ I_{2,1} & I_{2,2} & I_{2,3} \\\ I_{3,1} & I_{3,2} & I_{3,3} \\\ \end{bmatrix}\right) &= \begin{bmatrix} \frac{\text{d}\,I_{1,1}}{\text{d}t} & \frac{\text{d}\,I_{1,2}}{\text{d}t} & \frac{\text{d}\,I_{1,3}}{\text{d}t} \\\ \frac{\text{d}\,I_{2,1}}{\text{d}t} & \frac{\text{d}\,I_{2,2}}{\text{d}t} & \frac{\text{d}\,I_{2,3}}{\text{d}t} \\\ \frac{\text{d}\,I_{3,1}}{\text{d}t} & \frac{\text{d}\,I_{3,2}}{\text{d}t} & \frac{\text{d}\,I_{3,3}}{\text{d}t} \\\ \end{bmatrix} \end{align}$ Then, I calculated the above one as follows: $\begin{align} \frac{\text{d}I_{i,j}}{\text{d}t} &= \frac{\text{d}}{\text{d}t}\left(\sum_p m_p\,\left(\delta_{i,j} \sum_k r_k^2 - r_i\,r_j\right)\right) \\\ &= \sum_p m_p \frac{\text{d}}{\text{d}t}\,\left(\delta_{i,j} \sum_k r_{p,k}^2 \right) - \frac{\text{d}}{\text{d}t}\,(r_{p,i}\,r_{p,j}) \\\ &= \sum_p m_p \delta_{i,j} \frac{\text{d}}{\text{d}t}\, \left(\sum_k r_{p,k}^2\right) - \frac{\text{d}}{\text{d}t}\,(r_{p,i}\,r_{p,j}) \\\ &= \sum_p m_p \delta_{i,j} \sum_k \frac{\text{d}}{\text{d}t}\,(r_{p,k}^2) - \frac{\text{d}}{\text{d}t}\,(r_{p,i}\,r_{p,j}) \\\ &= \sum_p m_p \delta_{i,j} \sum_k \frac{\text{d}\,r_{p,k}^2}{\text{d}r_{p,k}}\frac{\text{d}\,r_{p,k}}{\text{d}t} - \frac{\text{d}}{\text{d}t}\,(r_{p,i}\,r_{p,j}) \\\ &= \sum_p m_p \delta_{i,j} \sum_k 2\,r_{p,k}\,\omega_k - (\omega_i\,r_{p,j} + r_{p,i}\,\omega_j) \\\ \frac{\text{d}\text{I}_{i,j}}{\text{d}t} &= \sum_p m_p \begin{bmatrix} 2\,r_{p,2}\,\omega_2 + 2\,r_{p,3}\,\omega_3 & -(r_{p,1}\,\omega_2 + r_{p,2}\,\omega_1) & -(r_{p,1}\,\omega_3 + r_{p,3}\,\omega_1) \\\ -(r_{p,2}\,\omega_1 + r_{p,1}\,\omega_2) & 2\,r_{p,1}\,\omega_1 + 2\,r_{p,3}\,\omega_3 & -(r_{p,2}\,\omega_3 + r_{p,3}\,\omega_2) \\\ -(r_{p,3}\,\omega_1 + r_{p,1}\,\omega_3) & -(r_{p,3}\,\omega_2 + r_{p,2}\,\omega_3) & 2\,r_{p,1}\omega_1 + 2\,r_{p,2}\,\omega_2 \end{bmatrix} \end{align} $ But, after trying some factorizations, I can't seem to get to the following: $[\omega]_\times\,I_{\text{body}} + I_{\text{body}}\,[\omega]_\times$ $= \sum_p m_p \left( \begin{bmatrix} 0 & -\omega_3 & \omega_2 \\\ \omega_3 & 0 & -\omega_1 \\\ -\omega_2 & \omega_1 & 0 \end{bmatrix}\begin{bmatrix} r_{p,2}^2 + r_{p,3}^2 & -r_{p,1} r_{p,2} & -r_{p,1} r_{p,3} \\\ -r_{p,2} r_{p,1} & r_{p,1}^2 + r_{p,3}^2 & -r_{p,2} r_{p,3} \\\ -r_{p,3} r_{p,1} & -r_{p,3} r_{p,2} & r_{p,1}^2 + r_{p,2}^2 \end{bmatrix} + \begin{bmatrix} r_{p,2}^2 + r_{p,3}^2 & -r_{p,1} r_{p,2} & -r_{p,1} r_{p,3} \\\ -r_{p,2} r_{p,1} & r_{p,1}^2 + r_{p,3}^2 & -r_{p,2} r_{p,3} \\\ -r_{p,3} r_{p,1} & -r_{p,3} r_{p,2} & r_{p,1}^2 + r_{p,2}^2 \end{bmatrix}\begin{bmatrix} 0 & -\omega_3 & \omega_2 \\\ \omega_3 & 0 & -\omega_1 \\\ -\omega_2 & \omega_1 & 0 \end{bmatrix} \right)$ $= \sum_p m_p \left( \begin{bmatrix} r_{p,1}\,r_{p,2}\,\omega_3 - r_{p,1}\,r_{p,3}\,\omega_2 & \ldots \\\ \vdots & \ddots \\\ \end{bmatrix} + \begin{bmatrix} -r_{p,1}\,r_{p,2}\,\omega_3 + r_{p,1}\,r_{p,3}\,\omega_2 & \ldots \\\ \vdots & \ddots \\\ \end{bmatrix} \right)$ $= \sum_p m_p \begin{bmatrix} 0 & \ldots \\\ \vdots & \ddots \\\ \end{bmatrix} \ne \sum_p m_p \begin{bmatrix} 2\,r_{p,2}\,\omega_2 + 2\,r_{p,3}\,\omega_3 & \ldots \\\ \vdots & \ddots \\\ \end{bmatrix}$ So, I should have made a mistake in the differentiation. But, where? And, how to proceed? Or, is it because the definition of tensor cannot be used in performing the derivation? Why?


title: Trying to turn a nonlinear differential equation into a linear one

text: The problem I have today is to determine the differential equation for a square fin protruding from a wall that experiences surface to ambient radiation and has an internal heat generation of $\dot q$ per length. The fin does not experience convection. I performed an energy balance on a differential element of the fin and reduced the differential equation to this: $$\frac{d}{dx}(kA_{c}\frac{dT}{dx})+\dot q=\frac{P \sigma}{kA_{c}}(T(x)^{4}-T_{sur}^{4})$$ Because of constant corss-sectional area and constant specific heat of the fin, the equation reduces to: $$\frac{d^{2}T}{dx^{2}}-\frac{P \sigma}{kA_{c}}T(x)^{4}+\frac{P \sigma}{kA_{c}}T_{sur}^{4}+\frac{\dot q}{kA_{c}}=0$$ When a fin is normally subjected to convection, we can say that $\theta(x)=T(x)-T_{\infty}$ and $\frac{d \theta}{dx}=\frac{dT}{dx}$ and $\frac{d^{2}\theta}{dx^{2}}=\frac{d^{2}T}{dx^{2}}$ . In that scenario, the $h(T(x)-T_{\infty})$ term reduces to $\theta(x)$ and the differential equation becomes second-order and linear. In this case with radiation, the differential equation is second order and highly nonlinear. I saw that I could "linearize" the radiation heat loss term but it still looks like there is a $T^{2}$ dependence which suggests it is still nonlinear. I am confused on the next steps I should take. If the differential equation that I created is fine then that is also great. Any Help is appreciated.


title: What is the logic of not regarding perturbative renormalizability as a fundamental requirement?

text: I read a statement in Becker and Becker's _String Theory and M-Theory_ page 2. After pointing out the non-renormalizablity of GR by the dimension of gravitational constant, it is said: > Some physicists believe that perturbative renormalizability is not a > fundamental requirement and try to quantize pure general relativity despite > its nonrenormalizability. Loop quantum gravity is an example of this > approach. Whatever one thinks of the logic, it is fair to say that despite a > considerable amount of effort such attempts have not yet been very fruitful. I am curious, what is the logic behind "perturbative renormalizability is not a fundamental requirement"?


title: Parity of annihilation and creation operator - Real Klein-Gordon field

text: In calculating the total-momentum operator of the real Klein-Gordon field, I end up with an equation like $$ \vec P = \frac{1}{(2\pi)^3}\int d^3p \mspace{9mu} \vec p\Big(a_pa_{-p} + a_pa^\dagger_{p}-a_{-p}^\dagger a_{-p} - a_{-p}^\dagger a_p^\dagger\Big) $$ and it seems the first and the last integrals vanish because they are antisymmetric w.r.t $ \vec p \rightarrow -\vec p $. I am unable to see this anti-symmetry. Also I am not able to see how to relate $ a_{-p} $ to $ a_p $ or $ a_p^\dagger $ other than some implicit integral relations like(which also am not very sure how to prove), $$ \int d^3p a_p^\dagger e^{-i\vec p.\vec x} = \int d^3p a_p e^{i\vec p.\vec x}$$ Thanks for any inputs.


title: Crash course on algebraic geometry with view to applications in physics

text: Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)? What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.


title: Variance and Intermittency in turbulence

text: In the turbulent transport of a scalar field, $\Phi$, decomposed into mean and fluctuating components, $\Phi=\left<\Phi\right>+\phi^\prime$, the scalar variance is defined as $\left<{\phi^\prime}^2\right>$. Today I got a bit reamed out in an academic meeting for using the variance to synonymously describe the intermittence of the flow. I realize the difference between variance and intermittence, but in this context are the two connected? If so, can somebody show this mathematically? If not, can somebody explain how they are clearly separated?


title: How to choose Gaussian surfaces while solving problems?

text: I have a doubt regarding this problem: > _Two large identical flat metal plates are placed parallel to one another, > seperated by a small distance compared to their linear size. One plate is > given a charge per unit area $-5.50 \times 10^{-5} Cm^{-2}$ and the other a > charge per unit area $+5.50 \times 10^{-5} Cm^{-2}$. Using Gauss's law find > the electric field at an arbritrary point between the two plates._ I did it with the plug and chug formula without much trouble. However, when I am trying to use Gauss' Law, I don't know what surface to choose. I drew a cylinder but I'm not sure of its orientation (should I set the circular faces perpendicular or parallel to the field?). As well, I'm not sure on which surface to use when confronted with new problems. Is there a foolproof way of doing it or any set forms recommended?


title: The most price-efficient experimental setup involving SPDC, single-photon counting etc

text: I would like a suggestion on a price-efficient way of experimenting with spontaneous down-conversion and single-photon counting. The simple dual-slit experiment could be one part of an application although there are many, but I'm mainly asking about the key components which are similar - the laser that pumps a SPDC-crystal, the crystal itself, and photon-counters (the other standard optical bench items I'm more familiar with). For example, is a 405 nm (blue) diode-laser module the way to go, with a BBO- crystal, and what to use for detectors - are avalance photo-diodes good enough? What kind of quality does the laser need to have (wavelength stability etc)? I know there is a Journal of american physics teachers or something which have articles covering these fundamental experimental setups now and then, but I thought I'd ask here in case anyone who works at a lab (with limited funding :) with a good setup could share some ideas. If this gets many answers, it should be turned into a community wiki entry I guess.


title: Can electrons move through vacuum?

text: I studied that electronics is the branch of physics which deals with the practical applications of moving electrons through vacuum, semi-conductors and gasses. Can electrons move through vacuum?


title: Two photons transition

text: if an atom in its ground state is coupled to an electromagnetic field it can absorb a photon if the EM field contains one with the right frequency. These transitions depends on $⟨f|H_i|i⟩$ (from Fermi's golden rule) with $|i⟩$ the initial state, $|f⟩$ the final one and $H_i=d.E$ the interaction between the field and the atom (with d the dipolar momentum and E the electric field). If we use $_F$ for the field and $_A$ for the atom we have : $⟨f|Hi|i⟩=_F⟨f|E|i⟩_F⋅_A⟨f|d|i⟩_A$ If $|⟨f|H_i|i⟩|^2=0$ the transition is not possible, for example when $_A⟨f|d|i⟩_A=0$. But is it also possible that $_F⟨f|E|i⟩_F=0$ ? If we look at transition with one photon (with the right impulsion p) we have $_F⟨0|E|1,p⟩_F\neq0$. Is $_F⟨0|E|2,p'⟩_F$ (with p'=p/2) also different from zero? I have tried to make these calculations for a Klein-Gordon field : I find $_F⟨0|\phi(x)|1,p⟩_F=e^{ipx}\neq0$, but I'm not sure whether $_F⟨0|\phi(x)|2,p⟩_F=0$ or if I've made a mistake. I think it's strange since I only heard that two photon linear absorption was forbidden by consideration on the atom states, but I know that non-linear two photons absorption is possible and it seems to make sense with $⟨0|\phi(x)|2,p⟩=0$ and $⟨0|\phi^n(x)|2,p⟩\neq0$ So here is my question, am I right? What prevents linear two photons transition? EDIT : Thanks to Emilio Pisanty for making everything more clear, I'm not very good at notations.


title: How can photons exert gravity if they are wave-like?

text: As a reference, see this question: Does a photon exert a gravitational pull? It turns out the answer is "Yes" -- but this does not seem consistent with light being wave-like. I am imagining a extremely high-energy photon -- about a Solar Mass's worth in eV -- flying through space at light speed. How would it exert a gravitational force on objects it passes without exposing it's exact path/position? On a smaller scale, imagine the dual slit experiment. If you had a sensitive enough gravitometers placed all over the room, could they not detect which path the photon took?


title: Getting the AdS metric from maximally symmetric spaces

text: I am familiar with the way we derive the form of the FRW metric by just using the fact that we have a maximally symmetric space i.e the universe is homogeneous and isotropic in spatial coordinates. Similarly, how do I get the Poincare patch of $AdS_{p+2}$ i.e \begin{equation} ds^2 = R^{2}\left(\frac{du^2}{u^2}+u^2(-dt^2+d\mathbf{x}^2)\right) \end{equation} by using the property of maximal symmetry _only_.


title: Electrical conductivity of an intrinsic semiconductor

text: On which factor does the electrical conductivity of an intrinsic semiconductor depend? It doesn't have an excess of charge carriers in fact, does it?


title: Why is gravity such a unique force?

text: My knowledge on this particular field of physics is very sketchy, but I frequently hear of a theoretical "graviton", the quantum of the gravitational field. So I guess most physicists' assumption is that gravity can be described by a QFT? But I find this weird, because gravity seems so incredibly different from the other forces (yes, I know "weirdness" isn't any sort of scientific deduction principle). For relative strengths:   * Strong force: $10^{38}$   * Electromagnetic force: $10^{36}$   * Weak force: $10^{25}$   * Gravity: $1$ Not only does gravity have a vastly weaker magnitude, it also has a very strange interaction with everything else. Consider the Standard Model interactions: ![Standard model interactions](http://i.stack.imgur.com/asCAF.png) No particle (or field) interacts directly with all other fields. Heck, gluons only barely interact with the rest of them. So why is it then that anything that has energy (e.g. _everything that exists_ ) also has a gravitational interaction? Gravity seems unique in that all particles interact through it. Then there's the whole issue of affecting spacetime. As far as I'm aware, properties such as charge, spin, color, etc. don't affect spacetime (only the energy related to these properties).


title: How to choose a $\alpha$, $\beta$, $\gamma$ measurement detector?

text: There are many different detectors for different radiation,such as NaI,HpGe,CsI for $\gamma$ detection,and ionization chamber,proportional counter,Geiger counter for $\alpha$, $\beta$ detection,but how to select a suitable detector for different radiation?I mean compare between them,what is the advantages and defects of them?


title: Why can $\beta$ not be linearly proportional to $T$, that is $\beta = constant \times T$?

text: $\beta$ in statistical mechanics is equal to $\frac{1}{k_BT}$ in in thermodynamics, but I do not understand why $\beta\propto T^{-1}$ instead of, say, $\beta\propto T$?


title: Is the Lagrangian density in field theory real?

text: As the Lagrangian in classical mechanics corresponds to energy, it must be real. But is that the case in quantum field theory? I mean, it should still correspond to some sort of energy, but what about all the "$i$"s here and there, such as in the Dirac Lagrangian $i\bar{\psi}\gamma^\mu\partial_\mu \psi$ and the current density $J_\mu = ie[\dots]$ (see Griffiths for example)? Another question is, how can it be hermitian, $\mathcal{L} = \mathcal{L}^\dagger$, when we have those "$i$"s? Wouldn't I get a minus sign if I complex-conjugated the interaction term and the Dirac field term? I'm really confused and hope someone can help


title: How does a warp field interferometer work?

text: Assume I have a solid grasp of undergraduate physics. From what I've read the warp field interferometer is supposed to be a sort of Michelson interferometer, except instead of adjusting the displacement of one mirror, you have a toroidal capacitor that... somehow expands space? Edit: I am talking about the work of Harold White at NASA. He's been apparently working on a way to implement an Alcubierre drive by generating a negative energy density with a torus of positive energy (but I don't know how this would work).


title: Potential Energy tends to infinity on the N-Body Problem

text: I need help to solve this problem related with the N-Body problem, i dont understand quite well what I need to define or to express in order to solve it. _We assume a particular solution to the N-Body Problem, for all $t >0$, and $h>0$, where $h$ is the total energy of the N-Bodies, show that $U\rightarrow \infty $ as $t\rightarrow \infty $. This mean that the distance between a pair of particles goes to infinity? (No.)_ In the N-Body problem $U$ is given by $U=\sum_{1\leq i< j\leq N}\frac{Gm_{i}m_j}{\left \| q_i-q_j \right \|}$, where $G$ is the gravitational constant. The Kinetic energy is $T=\sum_{i=1}^{N}\frac{\left \| p_i \right \|^2}{2m_i}=\frac{1}{2}\sum_{i=1}^{N}m_i{\left \| \dot{q}_i \right \|^2}$ The vector $q_{i}$ define the position vector of the $i$ particle. So Basically $U$ is like the sum of all the potential energies between all the $N$ particles. Also by the Lagrange Jacobi Formula, we have that $I$ is the moment of inertia, $T$ the kinetic energy so we can express: $$\ddot{I}=2T-U=T+h\quad,$$ where $h$ is a conserved quantity. I think that if $U\rightarrow \infty $, then $T\rightarrow \infty$ (because $h$ is constant), the problem is that the only way that i see to $U\rightarrow \infty $ is when the distance between all the particles $\left \| q_i-q_j \right \| \rightarrow 0$, but it means that it will be a collision, so if we have a collision then $t\rightarrow t_1$ and not to $\infty$, because a collision takes a finite amount of time **(Sundmanns theorem of total collapse)** , as I said i dont know what i have to define to show that $U\rightarrow \infty $ as $t\rightarrow \infty $, or maybe i need to define a $q_i(t)$ that in some way that $\left \| q_i-q_j \right \| $ goes very near to zero, but never zero, so $t$ can $t\rightarrow \infty$? Also, what about the question of a pair of particles going to infinity? It is clear that they should not go to $\infty$ because then $U\rightarrow 0$, and we are trying to prove the other case.


title: Why can't there be a trap door under nuclear reactors in case of meltdown?

text: This may be a naive question, but after the Fukushima Daiichi partial meltdown and studying the aftermath of Chernobyl it seems they could be helped by this idea. In Chernobyl, the liquidators that cleaned up the disaster tunneled concrete under the reactor core and covered the whole complex in a big containment unit. Why could they not have a 30 meter pit below a reactor filled with water with a trap door holding the reactor up? This pit could be very thick reinforced concrete similar to a missile silo If there were a meltdown or imminent meltdown and there were no other options the reactor core could be dropped into this pit by triggering the trap door and then perhaps have lead shot dumped on it or sealed in concrete. This seems very simple but there must be some reason why this is not practical. I know that the core would still be hot and it would still be a problem but would this not be better than it being exposed to air?


title: Relative wind velocity explanation - understanding Irodov problem 1.6

text: I am having trouble understanding the reasoning behind the solution in this Irodov General Physics problem. The problem is 1.6: > 1.6. A ship moves along the equator to the east with velocity vo = 30 > km/hour. The southeastern wind blows at an angle Θ = 60° to the equator with > velocity v = 15 km/hour. Find the wind velocity v' relative to the ship and > the angle Θ' between the equator and the wind direction in the reference > frame fixed to the ship. My calculations: ![My diagram](http://i.stack.imgur.com/u4AlL.jpg) After drawing the initial diagram I consider that in order for the wind direction to be relative to the boat, I need to subtract the boat vector from the wind vector. In my diagram I have repositioned the boat vector to be subtracted from the wind vector and using trigonometry I solve as follows: First the cosine law to determine the magnitude The magnitude = $\sqrt{30^2 + 15^2 - 2\cdot 15\cdot 30\cdot \cos{60}}$ = $\sqrt{675} \approx 25.98$ Then for the angle I can use sine law $\frac{sin(\Theta)}{30 km/h} = \frac{sin(60)}{\sqrt{675}}$ $\Theta= \arcsin{30km/h \cdot\frac{sin(60)}{\sqrt{675}}}$ $\Theta = 90°$ The answer I get is therefore that the wind, in reference to the boat (as felt by the people standing on the boat), is approx 26 km/h at an angle of 60° South of West (in relation to the equator). However the answer given in the solutions is always that the approximate speed of the wind is 40km/h and the angle is approx 19°. I have looked at the solutions and I think the best explanation is given in the following diagrams obtained from link #3: ![Diagram](http://i.stack.imgur.com/9rxFQ.jpg) ![Formula](http://i.stack.imgur.com/VFCDY.jpg) I do not understand, however, why he is using cos(180+$\Theta$). I want to understand why my solution is incorrect and yields a different answer. What is my mistake and what is the proper intuition for understanding this problem if I have done so incorrectly? Solutions I have viewed:   1. This Youtube Video   2. This online blog   3. This online blog


title: Optimal temperature for Mpemba Effect

text: Is there an pptimal temperature for the Mpemba Effect? Is there an optimal initial water temperature for speedy freezing? If so, what is it? If a specific example is needed: There is a 100 milliliter sphere of water floating in air at -10 °C. At what initial temperature greater than ~10 °C1 will the temperature of said water drop to 0 °C the fastest? 1Or whatever the local maximum nearest to 0 °C in a temperature/time to freeze graph is.


title: T duality under a small fluctuation of the compact dimension

text: How do small perturbations around the compact dimension affect T duality. What happens if I chose a compactification of the nature $r+\delta r$. And what keeps the compact dimension stable, i.e from expanding into a large dimension or imploding into itself.


title: What is the meaning of pressure of a gas

text: I understand the meaning of pressure on a surface to be force acting per unit area. But when it comes to understanding pressure in the context of fluids in motion or at rest I think I am having some trouble grasping the concept. When we say the pressure of a gas in a container (which may be of any arbitrary blobby shape) is say 2 Pascals, what does it really mean? 1) Does it mean the pressure of the gas on **ANY** small area considered on the **surface** of the container == 2 pascals? 2) Or Does it mean the pressure on a thin small wafer (possible imaginary?) placed anywhere inside the gas == 2 pascals? I am really confused! To add to my problem of understand pressure, is a sentence from Chorin and Marsden's a Mathemtical introduction to fluid dynamics when they define an ideal fluid. *"Lets us define an ideal fluid with the following property: For any motion of the fluid there is a function `p(x,t)` called the pressure such that if S is a surface in the fluid with a chosen unit normal n, the force of stress exerted across the surface S per unit area at x in S at time t is p(x,t) n i.e.  force across S per unit area=p( **x** ,T) **n** "* Here is my problem: Suppose we have an ideal fluid and a time and position varying pressure function as above. For a thin small wafer placed at x at time t, there are two equal and oppsite forces acting across its surface i.e. p( **x** ,t) **n** and -p( **x** ,t) **n** since the surface has 2 forces in the direction of the two normals to it. This reasoning means the force on the thin small wafer is zero. Which means that the force at any point the the fluid at any time is zero. Which is totally counterintuitive. Why would they define an ideal fluid like that?


title: Contact force betweeen 2 objects

text: Suppose that we have 2 objects next to each other.The object at the left is A and the object at right is B.now we have these situations: 1) we push B to the left. 2) we pull A to the left.(in this case A and B are stuck to each other.) Do the forces A exert on B have the same magnitude and direction in both situations?


title: From differentials to differential equations

text: Suppose I have a function of time $t$ and position $(x,y)$ such that \begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + p_y \,dy\end{equation} where the subscript denotes a differentiation. In this case, I am able to derive a (partial) differential equation from this form. I'd love to have your help to address the case in which, for example, $dy$ appears also with higher orders. Something like: \begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + (dy)^2 (p-(1-y)p_y). \end{equation} or simpler (the key point is the presence of $(dy)^2$). I expect that in this case the pde will be second order... Any idea? P.S. I posted yesterday a similar question on the math.stackexchange but maybe it is more a physics-like question :)


title: Is Poincare recurrence relevant to our universe?

text: If the theory of everything indicates a singularity-free and finite universe, will Poincare recurrence be relevant to the universe? If so, is there any interesting physical consequence, e.g. in superstring or quantum gravity?


title: Amount of energy to separate Gases - relationship to concentration

text: I want to understand the efficiencies of separating mixed gases, and for that I want to understand the thermodynamic limit case. Looking at the wikipedia page for entropy of mixing, I find the following formula: $$\Delta S_{mix}=-nR \sum_{i}^rx_i\ln x_i $$ with $r$ being the number of phases, $x_i$ With gases, the Gibbs free energy is entropy x temperature, no enthalpy. Now, the change of entropy with mixing/separating is highest for a two phase mixture with both components at equal concentrations. Now I'm confused: I know that removing small traces from a mixture requires more effort/energy per mass/mole than a bulk. Thermodynamics tell me that this is not necessarily so. While the specific energy is higher to remove trace amounts, the Gibbs energy still poses an upper bound to the energy necessary to separate gases assuming a 100% efficient process. Now I assume that the relatively larger effort for trace amounts stems from engineering difficulties (PSA needs higher pressure, centrifuges spin faster, more passes through membranes...) but there is no law of thermodynamics stating: Thou shalt not easily remove trace amounts from mixtures of gases. Or did I miss something?


title: Depression of Water Surface by a Needle

text: Given a needle of mass $m$ modeled by a cylinder of length $l$ and radius $r$ placed on an infinitely large water surface, what is   1. The maximum depression in the water surface; and   2. The equation of the shape of the water surface when depressed? I'm quite sure this is a simple question that already has been modeled theoretically somewhere, but I haven't been able to find a theoretical treatment of the problem with experimentally verifiable quantities. In addition, can the treatment of the problem be extended to thin films, for example a thin soap film?


title: How is the sun's equatorial circumference measured?

text: How is the sun's equatorial circumference measured by scientists?


title: Are QFT solitons expected to represent standard model particles? Or strings?

text: Is work on solitons in QFT's focused on finding solutions that could represent the fundamental particles of the Standard Model, or is the work focused on finding particles Beyond The Standard Model? If the work is primarily BTSM, is it still possible that solitons represent the SM particles? My related question is: is there any possible relationship between solitions in N-dimensional QFT's, and string theory? Is there a possible duality in which strings are just solitions in a QFT? Or is that just silly and impossible?


title: What is the constant $g_0$ used for calculating the specific impulse in seconds?

text: Wikipedia defines specific impulse as: $$ I_\mathrm{sp} = \frac{F_\mathrm{thrust}}{\dot{m} \cdot g_0} $$ The $g_0$ is said to be _the acceleration at the Earth's surface_. So is it actually the acceleration at Earth's surface, or is it a universal constant that just happens to be equal to that acceleration? In other words, is the specific impulse of the same rocket different on either of two planets which have different gravitational constants and no atmosphere? * * * Update: Let me restate the question in yet another way: If I was building a rocket on Mars, would I need to adjust my $I_{sp}$ values (eg. for delta-v calculations to make sense for combining engines)? Unless I am mistaken, I understand now that the answer is **_no_**. However, I don't think that any of the current answers state this clearly enough.


title: what is 't Hooft up to?

text: apart of the 't Hooft diagrams that you all love (and find all sort of dualities starting with them) one of the venues 't Hooft works nowadays is apparently some sort of "deterministic representation of quantum mechanics". I avoid saying "the world is classical" in a broad sense because it is my feeling that this is not what 't Hooft means by what he writes. So, what is 't Hooft up to? I tried today to think in terms of number theory about what he says but I ended up with the Universal Coefficient theorem popping into my head. What if some properties of a quantum space in the presence of gravity are best described using some "choice of coefficients" that looks more like classical than like "quantum" looks like at our "energy scale"? I would be very careful in dismissing completely poor 't Hooft just yet...


title: Photovoltaic IV data

text: I am looking for any available measured solar-cell datasets (Voltage/Current) especially for organic photovoltaic cells for the testing of a software unfortunately I am unable to get the measured values. Please mention if anyone knows about any on-line available data set ?


title: What happens if you try to freeze water in an water tight container

text: If I have a container that is full of water and I attempt to freeze the water by freezing the container, what would happen if the container is strong enough to prevent the water expansion? Could the water be in liquid state even though it is at or below freezing point? This is probably a daft question but its something that's been bugging me.


title: causal sketches

text: I don't have much of an idea of how to draw causal sketches. I know that you need to work out the gradient of the light cones, which can be done using a given metric and using null vectors. But how do you use null vectors to work out the gradient of metric? I've got a metric of $ds^2=-dt^2+(e^t)(dx^2)$ that I need to draw a causal sketch for. How do I find the gradient (using the null-vector), to draw light cones for a causal sketch?


title: Are leptons, baryons and energy the only products of radioactive decay?

text: I recently visited my child's elementary school to speak to a science classroom about rocks and minerals. While trying to explain what a crystal is, I got sloppy and mis-spoke that an atom was the smallest possible piece of matter (rather than an element!) I was quickly stopped and corrected by a 9-year-old that told me in fact atoms can be split into leptons and baryons. I told her she was right, and explained that if an atom of an element is divided it becomes a different element (overlooking isotopes!). My knowledge of particle physics is limited and later I began wonder what I might be leaving out, that should not be left out, when talking of the types of 'ordinary matter' in nature. What happens in high-energy physics experiments aside. I know that there are other elementary particles besides leptons and baryons, photons are obviously everywhere. But if we restrict the discussion to radioactive decay, fusion in stars, cosmic rays, is everything a lepton, baryon, or a photon?


title: Relating Schrödinger's Wave Equation and Heisenberg Uncertainty Principle

text: A homework question that I don't conceptually understand: A quantum particle of mass M is trapped inside an infinite, one-dimensional square well of width $L$. If we were to solve Schrodinger's wave equation for such a particle, we could show that the minimum energy such a particle can have is $$E_{min} = \frac{h^2}{8ML^2}$$ where $E$ is measured from the bottom of the well and $h$ is Planck's constant. Estimate the minimum energy of the same particle using the Heisenberg Uncertainty Principle and assess the consistency of your estimate with the exact result. What I don't understand is how can I equate the HUP to this since I don't know what the particle's momentum is in relation to it's uncertainty. Thanks for any clarification.


title: Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

text: Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: > ... With a generalization of the argument given there (section 7.4 for Ward > identity), on can show that the diagrammatic identity (7.68) holds for > digrams that include counterterm vertices in loops. Let's assume this is granted true, but I got lost in his following argument: > **Thus** , if the counterterms $\delta_1$ and $\delta_2$ are determined up > to order $\alpha^n$, the **unrenormalized vertex diagram** at $q^2=0$ equals > the derivative of the unrenormalized self-energy diagram on-shell in order > $\alpha^{n+1}$. To satisfy the renormalization conditions (10.40), **we must > then** set the counterterms $\delta_1$ and $\delta_2$ equal to order > $\alpha^{n+1}$. This recursive argument gives yet another proof that > $Z_1=Z_2$ to all orders in QED perturbation theory. What does he mean by **unrenormalized vertex diagram**?   Can somebody please explain the connections in his logic?   Thanks!


title: Pressure difference between 2 side of wings of aircraft

text: I understand why the pressure is reduced above the wings of aircraft, but when I read books and look for information, it says that the air is compressed below the wings, I do not understand why it is? Could you tell me what makes the air compressed below the wing, and how is it compressed?


title: Derivation of KLT relations

text: The KLT relations (Kawai, Lewellen, Tye) relate a closed string amplitude to a product of open ones. While I get the physics behind this I don't really understand the derivation in the original paper (see http://www.sciencedirect.com/science/article/pii/0550321386903627 ). Especially I don't understand the contour deformations starting around eq 3.4 and following. I'd be glad if someone could either explain that a bit more in detail or point to a reference where this is done in all gory detail :)


title: What is "species" in the context of an electrochemical cell?

text: From the Wikipedia entry on Electrochemical cell (http://en.wikipedia.org/wiki/Electrochemical_cell): > _An electrochemical cell consists of two half-cells. Each half-cell consists > of an electrode, and an electrolyte. The two half-cells may use the same > electrolyte, or they may use different electrolytes. The chemical reactions > in the cell may involve the electrolyte, the electrodes or an external > substance (as in fuel cells which may use hydrogen gas as a reactant). In a > full electrochemical cell, species from one half-cell lose electrons > (oxidation) to their electrode while species from the other half-cell gain > electrons (reduction) from their electrode._ My familiarity with the word species begins and ends with the classification of living things so I am utterly baffled as to what it means in the present context.


title: Large-N critical NLSM (equation 13.115 of Peskin and Schroeder)

text: Any opinions if the equation 13.115 of Peskin and Schroeder is true on arbitrary manifolds in arbitrary dimensions for the same Lagrangian? I a priori see no problem. The point I also want to ask is - if one evaluates that equation for spacetime >2 then one will miss the symmetry broken phase of the theory - right? - BUT isn't the saddle point obtained from it still the critical point of the corresponding theory on whatever is the (>2)-manifold? * * * Is this in general true that the large-N critical NLSM on a manifold $M$ is the same as the theory of $N$ non-interacting conformally coupled scalars on $M$? Or in otherwords - will the Lagrange multiplier field in NLSM have to have at the large-N saddle point an expectation value which is the same as the conformally coupled mass for scalars on that manifold?


title: Measurement and probability for quantum states

text: Suppose that the physical system is in generic state $|\psi\rangle$. Show that $\sum_{\lambda}p^2_{\lambda} = 1$ to an observable $O$, if and only if $\Delta O = 0$. ($\Delta O$ is a standard deviation). Using the probability measure definition $p_{\lambda} = \langle\psi|P_{\lambda}|\psi\rangle$. Using hypothesis, $$\sum_{\lambda}\langle\psi|P_{\lambda}|\psi\rangle^2 = 1,$$ the fact that $\sum_{\lambda}\langle\psi|P_{\lambda}|\psi\rangle = 1$ and the definition $\Delta O = \sqrt{\langle O^2\rangle - \langle O\rangle^2 }$. I tried $$\langle O^2\rangle - \langle O\rangle^2 = \langle\psi|O^2|\psi\rangle - (\sum_{\lambda}\lambda\langle\psi|P_{\lambda}|\psi\rangle)^2,$$ but dont know what else to do


title: Relative linear velocity of a particle to a rotating object

text: I am trying to calculate the "relative linear velocity" of a particle moving over a rotating object. According to this paper (section 2.2) I am reading the relative linear velocity is calculated by: $$ v'= v - \omega r $$ Where $v$ is the velocity of the particle, $\omega$ the angular velocity of the rotating object and $r$ the radius of the object. What I don't understand is **why subtract?** For high angular velocities with low particle velocities that will produce a negative velocity. The particle's surely not traveling backwards relatively, if the particle is traveling in the same direction as the rotation?


title: How should I REALLY have done this continuous moment formula derivation?

text: One of my more recent hobbies is making "swords" out of PVC conduit, foam, and duct tape. As all the weight is forward of the cross guards, I needed to put some amount of some substance inside the hilt to counter-balance it. Curious as to how much counter-weight I needed, I decided to see if I could develop a continuous version of the typical moment formula (moment = force * distance, or $M = F*D$) as there was no way the counter-weight would be close enough to a point mass to use the usual formula. Here's how I set about it... $m$ = mass   $l$ = length   $d$ = fraction of whole For example, if I have a rod where $m = 80 kg$ and $l = 16 ft$ and cut it into 8 pieces, $d = .125$ and the mass of each piece will be $m*d = 10kg$ and the length of each piece will be $l*d=2ft$. Thus, treating each piece as a point mass centered at the point closest to the pivot, adding up the moments gives: $\sum F*D$ $=\displaystyle \lim_{d \to 0} \; \sum_{k = 0}^{\frac{1}{d}} mld^2k$ $=\displaystyle \lim_{d \to 0} \;\; mld^2\sum_{k = 0}^{\frac{1}{d}}k$ $=\displaystyle \lim_{d \to 0} \;\; mld^2 \left( \frac{ \left( \frac{1}{d} \right) \left( \frac{1}{d}+1 \right) }{2} \right)$ $=\displaystyle \lim_{d \to 0} \; \frac{ml}{2} + \frac{mld}{2}$ $=\displaystyle \frac{ml}{2}$ The problem I've noticed with this formula is that it means that the further half of an object has the exact same moment as the nearer half (a rod twice as long has twice the moment, not four times as it should be). This is clearly wrong, but I don't know where I made an invalid assumption. Can someone else see it? By the way, I had a brain flash while writing this and realized the simplest (and correct) way to derive it is as follows: $\displaystyle \int_{a}^{b} (F*D)\;dD = \frac{F}{2}(b^2-a^2)$


title: How is the Breit-Rabi formula derived?

text: Does anyone know how the Breit-Rabi formula (energy dependence of the hyper- fine splitting in presence of any external magnetic field for alkali atoms when $F=I \pm 1/2$) is derived, or where to find a reference for it? It doesn't seem to be in the original paper from 1931, or in the textbooks I've looked at. EDIT: The low- and high-field solutions I understand, but in which step in the derivation is $F=I \pm 1/2$ used, and how does this lead to the exact solution? Thanks in advance!


title: Making symmetry between E and B fields manifest in Lagrangian

text: Maxwell's equations are nearly symmetric between $E$ and $B$. If we add magnetic monopoles, or of course if we restrict ourselves to the sourceless case, then this symmetry is exact. This is not just a discrete symmetry of exchange. A "duality rotation" of $(E, B) \rightarrow (E \cos \theta - B \sin \theta, B \cos \theta + E \sin \theta)$ preserves the form of the equations. (The charges and currents, if any, would have to similarly mix as well. The upshot is that if all particles have the same ratio between their electric and magnetic charges, it's indistinguishable from just having electric charges. See e.g. Jackson 6.12.) This shows up quite nicely in the Hamiltonian formulation as the invariance of the free Hamiltonian density $\mathcal{H} = E^2 + B^2$. The free Lagrangian density, however, is the Lorentz invariant scalar $\mathcal{L} = E^2 - B^2$, which does not appear to have the full continuous symmetry, though a quarter rotation leads to $\mathcal{L} \rightarrow - \mathcal{L}$, leaving the equations of motion unchanged. Is there an alternate free Lagrangian density that makes the full continuous symmetry manifest? EDIT: The normal Lagrangian expression is dependent on a gauge field A, whose existence depends on the non-existence of monopoles. As Peter Morgan points out, this obviously breaks the symmetry and explains why this Lagrangian needn't have this symmetry. However, that leaves the following questions:   1. Is there another Lagrangian giving the same equations of motion that does have this symmetry? If yes, can it be demonstrated? If not, is there a proof?   2. Even if the answer to 1 is no, is there a family of Lagrangians related by this symmetry that each give the same equations of motion? If yes, can it be demonstrated? If not, is there a proof?


title: Limit of Lorentzian is Dirac Delta

text: I have a quick question that just came up in my research and I could not find an answer anywhere so I thought I'd try here. So one of the definitions of the Dirac Delta is the limit of the Lorentzian function with $\epsilon$ going to zero. See here http://hitoshi.berkeley.edu/221a/delta.pdf for the expression on the first page. My question is, can I define the Dirac Delta just as well with this $$\delta(t) ~=~ \lim_{\epsilon\rightarrow 0} \frac{1}{\pi}\frac{\epsilon^2}{\epsilon^2+t^2},$$ where I have included an extra $\epsilon$ in the numerator. My hunch is that this is no problem since the limiting behavior looks the same to me.


title: Measurement using Vernier Callipers

text: > The diameter of a cylinder is measured using a Vernier Callipers with no > zero error. It is found that the zero of the Vernier scale lies between 5.10 > cm and 5.15 cm of the main scale. The Vernier scale has 50 divisions > equivalent to 2.45 cm. The 24th division of the Vernier scale exactly > coincides with one of the main scale divisions. The diameter of the cylinder > is ... **My attempt**   1 division of vernier scale is equivalent to 50/2.45 cm.   Least count of main scale = 0.05cm   Thus Vernier constant = 0.5-(50/2.45) cm   Surprisingly this is negative.?? I don't really know how to solve these kinds of problems. Please help me.


title: Heim Theory and FTL

text: Years ago there was some hype around a theory that would supposedly allow for FTL. But I have since heard nothing of this. Is anything happening with the theory? How would a magnetic field allow us alter spacial dimensions?


title: Alkali atom - photon interaction in zero magnetic field

text: An alkali atom has a single outer electron that interacts with incoming photons of the right wavelength (for alkalies it's in the visible & IR range). If there is an external magnetic field, the electron has a well defined quantization axis and the the incoming light can be separated into three components: linearly polarized and left/right-handed circular polarized parts. The interaction strength between the different atomic energy levels (levels described by different quantum numbers) and each of the polarization components is relatively straightforward to calculate (with the appropriate Clebsch-Gordan coefficients and 6-j symbols). What is then the situation if there is no external magnetic field to provide the quantization axis? Is "polarization" meaningful from the atoms point of view? How one sets out to calculate the interaction strength for the different polarizations in the laboratory frame?


title: Is it possible to create nothing?

text: I'm not sure whether I'm asking the right question here. I am simply a physics admirer! Alright. So my question is, is it possible to create nothing? Lets say you take a cube serving only as a a shell. Then expanded the cube. Or a balloon, the size of really really small. Then expanded the balloon using outside forces to pull on the exterior. Could it be possible to create a shell that harnesses... well.. nothing but space? No atoms inside other than the exterior? I mean nothing. Can you create 0? A vacuum isn't what I'm asking here. Of course trying to do this, in my eyes, would simply rip the exterior apart because there's nothing getting inside to fill in the gap. Any thoughts? (i've been watching too much Cosmos haha)


title: Classical and Semi-classical treatments of the ideal gas

text: In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the number of particles. It is semi-classical in the sense that we consider the indistinguishability of the particles, so we divide by $N!$. The resulting expression for the entropy of the system is $$S = Nk \left(\ln \left[\left(\frac{V}{N}\right) \left(\frac{2\pi mkT}{h^2}\right)^{3/2} \right] + \frac{5}{2}\right)$$ Now consider a fully classical analysis. There $Z(1) = \sum_{E} \exp (-\frac{E}{kT})$, where $E = p^2/2m$ (assuming no interaction potentials). The problem can be mapped to an integral over the phase space of the Hamiltonian to give $$Z(1) \rightarrow \int \exp \left(-\frac{1}{2mkT} (p_x^2 + p_y^2 + p_z^2) \right)\text{d}^3 \underline{p} \,\text{d}^3 \underline{x}$$ This can then be rewritten like $$ \int \exp \left(-\frac{p_x^2}{2mkT}\right) \text{d}p_x \int \exp \left(-\frac{p_y^2}{2mkT}\right) \text{d}p_y \int \exp \left(-\frac{p_z^2}{2mkT} \right) \text{d}p_z \cdot V$$ where $V$ is the volume of the container. Those are Gaussian integrals and so evaluation is immediate. The result is that $Z(1) = (2\pi mkT)^{3/2} V$ The corresponding entropy can be calculated and the result is that $$S = Nk \left(\frac{3}{2} + \ln\left(\frac{(2\pi mkT)^{3/2}}{V}\right)\right)$$. My question is: What is the significance of the factors $5/2$ in the semi- classical treatment and the factor $3/2$ in the classical treatment and why are they different? They look like the number of degrees of freedom a monatomic and diatomic molecule would have at room temperature, but I think this is a coincidence.


title: Solve Eigenvalues for Particles enclosed in a hard surface?

text: Is there a way to calculate the energy eigenvalues for a particle enclosed in an impenetrable enclosing surface? I tried evaluating the Fourier transformation of the Laplacian operator \begin{align} \int_V e^{-i\bar{k}\cdot \bar{x}} \nabla^2 \psi \ dV \end{align} to use the surface integrals to reveal the momentum eigenvalues dependent on the surface and I was able to obtain the following result \begin{align} \oint_S e^{-i\bar{k}\cdot \bar{x}} \bar{\nabla} \psi \cdot d\bar{S} = 0 \end{align} however I do not how to specify $\bar{\nabla} \psi$ at the surface. Any suggestions?


title: Collected Gravitational Field

text: I wasn't sure what to call this, I'm not a physicist. I basically have a collection of massive bodies. I to calculate the gravitational field of all those objects collected; how to do this? The reason I want to do this is because at the end I want to see how another object will interact with that collection objects, and it'll be a lot of work to calculate how it's influenced by every single body in the collection.


title: horizontal motion inside a cone (cylindrical polars)

text: I have a question from an example we done in lecture Suppose we have a particle moving inside the surface of a cone given by $r = wz$ where $w$ is a constant, and also suppose initially the particle is moving horizontally at height $b$ and speed $U$,the lecturer then stated these "facts" $\underline{\dot r} = U \underline{e_\theta}$ so $\dot r = 0$, $\dot z = 0$ I understand where he got $\underline{\dot r} = U \underline{e_\theta}$, however, I do not understand why $\dot r = 0$ and $ \dot z = 0$, all we know is that $z(0) = b$, and $\dot r(0) = U$, but that does not mean general $\dot z, \dot r$ are $0$, if someone could clear this up please edit: ![a](http://i.gyazo.com/9c5d292390687ecc4adf13bbcfc680c5.png) !


title: Stationary and Static

text: I have some confusion about the concept of stationary and static. A metric $g$ is called **stationary** if there is a time like Killing vector $K$. $g$ is called **static** is further HSO (hyper surface orthogonal), i.e. there exists a foliation of hyper surface orthogonal to the integral curve of $K$. The Schwarzschild solution requires $g$ to be static, and the Kerr solution only require $g$ to be stationary. Here comes my question. To my understanding, requiring $K$ to be HSO is to enabling us to slice the spacetime into pieces with fixed time. The Kerr solution is for the metric outside a rotating body. In the note I am reading, it says > Do not require $K$ to be HSO, after all we want a rotating body. Why a rotating body indicates that the metric cannot be HSO? I think I can also slice the spacetime into piece for every time.


Why is my restaurant silverware magnetized? Specifically, if I lay all three flat on the table, and touch the bottoms of the handles together, I can pull the knife with the handle of the spoon. I have a video (if I can figure out how to upload it from my phone).


title: Speed of neutrons in a nuclear reactor

text: How fast (in $\frac{\mathrm{m}}{\mathrm{s}}$) are typical neutrons _before_ and _after_ slowing down in a nuclear reactor (with U-235 and $H_2O$ as moderator)? Do you have any reference for the values?


title: Containement system of a nuclear reactor: role of the water moderator

text: In a modern nuclear reactor for example a PWR there are multiple containment systems which prevent the release of radioactive material into the environment and shield the environment from the radiation. Here is a quote from the wikipedia article about this: > The reactor vessel is the first layer of shielding around the nuclear fuel > and usually is designed to trap most of the radiation released during a > nuclear reaction. The reactor vessel is also designed to withstand high > pressures. I think that the radiation in this part is shielded partially by the walls of the pressure vessel and partially by the water it contains. However how much percent of the radiation is shielded by the water alone and how much by the walls? I.e. how do the shielding effects of the walls and the water compare to each other (roughly) and why?


title: Why is the Dirac equation not used for calculations?

text: From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing actual ab initio quantum mechanical calculations uses the Schrödinger equation. If relativistic effects are important one adds a relativistic correction. If the Dirac equation is a more correct description of reality, shouldn't it give rise to easier calculations? If it doesn't, is it really a more correct description?


title: The definition of Lorentz transformation

text: I know that the Lorentz transformation, when two frames $\mathcal{S}$ and $\mathcal{S}'$ are in standard configuration (the axes are all parallel to their counterparts in the other inertial frame) is given by $$L~=~\left( \begin{array}{cccc} \gamma & \beta \gamma & 0 & 0 \\\ -\beta \gamma & \gamma & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{array} \right)$$ Is the general definition of the Lorentz transformation actually just $$[L^T][\eta] [L]~=~[\eta]?$$ Where $\eta=\text{diag}(-1,1,1,1)$? I am trying to find a fundamental definition of the Lorentz transformation so I can show that some transformations _are_ LTs, e.g. an ordinary rotation is obviously a Lorentz transformation but I'd like to show this explicitly and without any reference to the Lorentz group if possible. Is the defintion simply: a transformation in Minkowski space that preserves the length of 4-vectors under the Minkowski scalar product? If so then it would be easy.


title: Fading transition and rotation of and object in 2D

text: I'm looking for sources about I guess dynamics subject. The model I'd like to solve is reduced to a question of: How does a force applied on a certain point of an object results in both fading rotation around point's axis and transition of the body? My issue concerns objects on plane surface and I do prefer solution in that simpler matter but I'll be either grateful for pointing me a source or keyword, or any specific directions for the general case.


title: How to write QM operator if I know all of it's eigenfunctions?

text: Suppose I have selected enough orthogonal functions in representation of operator `A` and I want to derive operator `B` which has these functions as it's eigenfunctions. How to do that?


title: Does light escape from highly dense bodies

text: If it were possible to launch a projectile at the speed of light from a body with the mass equal to that of our sun but a density such that its escape velocity was the speed of light how far would it go into empty space before its velocity dropped to one metre per second and how long would it take? The question was only partly related to light escaping from dense bodies but before seeking the answer to light i will re-frame the question as follows; If a projectile were launched from a body with the mass equal to that of our sun but a density such that its escape velocity was only 1/100th of the speed of light how far would it go into empty space before its velocity dropped to one metre per second and how long would it take? This move the question away from relativity issues. I think the distance is actually 2*G*M but how long does it take?


title: Do black holes cause gravitational redshift?

text: Suppose a light ray is emitted by a light source very close to the black hole's singularity away from the black hole's gravitational center. Why won't the light escape? Shouldn't the massive gravity of the black hole just cause massive redshift in the light ray? How does it pull it back? I know there might be many questions similar to this but I just need a basic answer for the redshift part. I understand that if a light ray is passing nearby the black hole, then it bends and gets pulled inside. But what if the light is going perpendicularly away from the black hole? **The source is inside the event horizon of course. Otherwise this is just a normal case.**


title: The meaning of "heralded photon"

text: I am not a native English speaker, and I have just started to study physics in English. However, I came across the term **_heralded photon_** while I was reading a review article about optical quantum memory. I don't understand what it means. A dictionary explains herald as, "to be a sign that something is going to happen," but it does not make any sense. Below is the excerpt from the article: > If a single-photon detector is placed in one of the emission channels of > non-degenerate spontaneous parametric down-conversion, a detection event > indicates emission of a photon pair, and thus the presence of a single > photon in the other channel. Such a **_heralded photon_** is emitted at an > arbitrary time, however, making it unsuitable for many application. Would you help me understand this term?


title: Clebsch-Gordan in Fock Space?

text: When adding the angular momenta of two particles, you use Clebsch-Gordan coefficients, which allow you, in fancy language, to decompose the tensor product of two irreducible representations of the rotation group into a direct sum of irreducible representations. (I'm not exactly clear on what this means, so on a side note can someone suggest a good book on representation theory of Lie groups?) If we want to add the angular momenta of even more particles, then we have to use other coefficients, like the Wigner 3j, 6j, or in general 3nj symbol. But what if you have an unknown number of particles, like you do in quantum field theory? In this case quantum states live in a so-called Fock space made of a direct sum of infinitely many tensor products of Hilbert spaces (which is another thing I'm unclear on). So then how do you add angular momentum in Fock space? To put it another way, how do you decompose a tensor product of irreducible representations into a direct sum if you have a variable number of terms in the tensor product? Any help would be greatly appreciated. Thank You in Advance.


title: Expectation value for the time of a photon reflection

text: A photon is reflected by matter (by an electron in empty space). How long does the reflection take? (i.e. is there any infinitesimal time elapsing during the reflection process?), or more precisely, what is the time difference, the time retardation by the interaction compared to the light velocity in vacuum. A very approximate number would be sufficient for the answer.


title: Is a non-Newtonian gas possible?

text: I was wondering if it was possible to have a Non-Newtonian Gas. I was planning a science fair experiment to try to create one if its possible.


title: Entropy Maximization using undetermined multipliers

text: This is from Problems in Thermodynamics and Statistical Physics by P.T. Landsberg A system can be in any one of N states. Using the method of undetermined multipliers to show for the maximum entropy, $S = -k \sum_i p_i \ln p_i$ where $p_i = 1/N$, $$S = k \ln N$$ The solution is as follows, Using an undetermined multiplier $\alpha$, write $$f = -k \sum_i(p_i \ln p_i - \alpha p_i)$$ $$\frac{\partial f}{\partial p_j} = -k (\ln p_i + 1 -\alpha ) = 0$$ $\ln p_j = \alpha -1$for all $j$, therefore all $p_i$ are equal. The manxium entropy is $$S_{max} = -k \sum_i(\frac{1}{N}\ln \frac{1}{N}) = k \ln N$$ I don't understand the last part. How to go from the sum to the final expression given $\sum_i p_i = 1$


title: Why does wavelength affect diffraction?

text: I have seen many questions of this type but I could nowhere find the answer to "why". I know this is a phenomenon which has been seen and discovered and we know it happens and how it happens. But my question is why would wavelength affect the amount of diffraction? I am looking for a very simple logical explanation rather than a complex mathematical answer. Why will a blue ray bend lesser than a red ray through a slit of the size a little bigger than the wavelength of the blue ray? I need an answer that will answer "why" does diffraction depend on wavelength of light. ![http://www.olympusmicro.com/primer/java/diffraction/index.html](http://i.stack.imgur.com/2PmJk.jpg) ![http://www.olympusmicro.com/primer/java/diffraction/index.html](http://i.stack.imgur.com/HcZC2.jpg) Image sources: http://www.olympusmicro.com/primer/java/diffraction/index.html


title: What is the phase space for outgoing photons?

text: For a scattering process for which $n$ fermions are scattered, (by some conventions) the cross section acquires a phase space factor of: $$d\sigma \sim \prod_{i=1}^n\frac{d^3p_i}{(2\pi)^3 2E_i}$$   1. In this convention, what is the equivalent phase space factor for an outgoing photon?   2. How is it that different conventions differ by arbitrary factors of $2$ and $\pi$? isn't the cross section a measurable which has to "stay put" irrespective of the convention used?


title: Is mass an inherent property?

text: Suppose I have an electronic weighing machine placed in a uniform gravitational field. Now I put a mass above it and register the reading. Now I give the system (mass + machine) an impulse so that it travels at $0.99c$. Will the reading on the scales change and why?


title: Explanation of Superconductivity

text: I can't get a definitive explanation of why superconductivity happens and I am getting mixed explanations from my textbooks. I will tell you what I know and hopefully you can correct any misunderstandings I have:   * A metal consists of an ionic lattice because the electrons in the valence band have 'jumped' to the conduction band   * When a potential difference is applied across a conductor, electrons would move towards the positive potential but the lattice will become distorted as the ions are attracted to the electrons   * This creates a region that has dense positive charge. Electrons would be attracted towards this region. However, electrons repel each other and don't want to be close to each other, additionally, Wolfgang Pauli's exclusion principle states that no two fermions can occupy the same quantum state so every two electrons have to combine and form a Bose Einstein condensate as bosons can occupy the same quantum state.   * Cooper pairs arise because of the exchange of phonons. Phonons are a collection of excitations of atoms or molecules. The atoms or molecules have to be vibrating in some collective mode.   * The bond between two electrons in a boson is very weak, 10^-3 eV. Therefore, the temperature of the metal must be 10 Kelvin for the cooper pairs to exist (using E=kT where k is 10^-4 eV).   * Bosons do not interact with matter, hence, they aren't impeded by the lattice. There are three things I do not understand about this explanation:   1. Pauli's principle.   2. Why bosons can be in the same energy state and fermions can't.   3. Phonons.


title: A question about speed

text: > A object with mass 0.5 kg slides in straight line for 200 m, where the > friction act on it is 0.4 times the normal reaction acting on it. Find its > initial speed. I let $u$ be the speed. $u^2=2as=2*0.4mg*200=2*0.4*0.5*9.81*200,u=28.01ms^{-1}$. But the answer should be $40ms^{-1}$, can someone tell me what's wrong with my solution? Thank you. EDIT: Correct answer: $u^2=-2as=-2(f/m)s=-2(-0.4N/m)s=-2(0.4mg/m)s=2*0.4*10*200=1600$ $u=40ms^{-1}$


title: Line-shape asymmetry in undergraduate Hydrogen-Deuterium experiment

text: I'm working as an LA (undergraduate TA) for an undergraduate physics laboratory experiment where students test the Bohr model and use reduced mass to determine the approximate mass of the neutron. In the experiment, a discharge tube is filled with a mixture of H and D gas, and the data that come from the spectrometer (Ocean Optics HR-2000 with an optical resolution of about 0.04 nm) look fairly defined. However there's an asymmetry in the peak (the peaks are skewed to the left slightly), and this is bothering me a little bit. The results that students get from fitting gaussians to the peaks are pretty good in terms of agreeing with the literature but the models themselves don't get great reduced chi-square values. Where could the asymmetry come from? Doppler and natural lifetime broadening are symmetric and should yield a Voigt profile, which is symmetric. These contributions are also much smaller than the instrument resolution. Using the optical resolution as the standard deviation of a gaussian instrument error function, I fit these peaks with a model taking fine- structure into account (fine-structure is about half the size of the instrument error). This, however yields some very poor confidence intervals in the fit and still does not fix the high chi-squared value. The only things left that I can think of are instrument error (which I think could be asymmetric) or (grasping at straws here) something kind of complicated like a fano resonance. What could this be? Is there any reason to think that the peak line shape of H$_\alpha$ should be skewed?


title: How does one measure Earth's speed of revolution around the sun?

text: I know that there are several formulae that one can plug numbers into to arrive an estimate of Earth's speed around the sun (Kepler's third law for instance), but I'm wondering how these things are measured. Since Earth is moving in an ellipse around the Sun, I thought doppler measurements of spectral lines would help tell where we were on the ellipse, since the velocity won't be constant everywhere. But other than that, I'm not sure how one measures Earth's velocity.


title: Convert state Vectors to Bloch Sphere angles

text: I think this question is a bit low brow for the forum. I want to take a state vector $ \alpha |0\rangle + \beta |1\rangle $ to the two bloch angles. What's the best way? I tried to just factor out the phase from $\alpha$, but then ended up with a divide by zero when trying to compute $\phi$ from $\beta$.


title: What is the significance of the branch cut in renormalization group logarithms?

text: What is the physical significance of the branch cut in renormalization group logarithms? (Is this just an avatar of the optical theorem, or is there something to be understood about these logarithms from the point of view of taking the scale as a complex parameter? Usually non-analyticity is a signal of something of physical importance, right?)


title: Pohlmeyer reduction of string theory for flat and AdS spaces

text: The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's  DDF and Pohlmeyer invariants of (super)string) is the following: $ Z^{\mu_1...\mu_N} (\mathcal{P}) = \frac{1}{N} \int\limits_0^{2\pi} d\sigma^1 \int\limits_{\sigma_1}^{\sigma_1+2\pi} d\sigma^2... \int\limits_{\sigma_{N-1}}^{\sigma_1} d\sigma^N \ \mathcal{P}^{\mu_1}(\sigma_1)\ \mathcal{P}^{\mu_2}(\sigma_2)...\mathcal{P}^{\mu_N}(\sigma_N) $ Another reference is eq-3.29 in Thomas Thiemann's  The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space. These formulations of the Pohlmeyer invariants is used to show its equivalence to DDF invariants of light-cone gauge fixed strings in flat-space (cf. eq-2.53 in Urs Schreiber's paper and chap-2.3.2 in Green, Schwarz & Witten vol-I) Now, what is the corresponding invariant for the Pohlmeyer reduction in AdS case (ref: Miramontes' "Pohlmeyer reduction revisited" arXiv:0808.3365)? I cant seem to find one that resembles the flat-space expression.


title: Higgs mass and the hierarchy problem

text: I was wondering what is the opinion about importance of the hierarchy problem in the hep community? I'm still a student and I don't really understand, why there is so much attention around this issue. 1 loop corrections to the Higgs mass are divergent - in the cut-off regularization proportional to $\Lambda^2$ and therefore require large fine tuning between the parameters to make those corrections small. But this kind of problem do not appear in the dimensional regularization. People like the value of $\Lambda$ to be very large, with an argument that it should correspond to same energy scale at which our theory breaks down. I don't think, that we should treat the scale $\Lambda$, as some kind of a physical scale of our model cut-off, as it is just a parameter to regularize the integral. Just like the $4+\epsilon$ dimension in the dimensional regularization is not a physical thing. Why do we apply a physical meaning to $\Lambda$? Not to mention the troubles with the Lorentz invariance. Maybe the hierarchy problem is an argument that the cut-off regularization scheme is just not right to use?


title: Partition Functions in (A)dS/CFT

text: I'm trying to understand some aspects of dS/CFT, and I'm running into a little trouble. Any help would be much appreciated. In arXix:1104.2621, Harlow and Stanford showed that the late-time Hartle- Hawking state in de Sitter space, \begin{align*} \Psi[h]=\int_{dS} Dg_{g|_{\partial dS}=h} \exp(iS(g)) \end{align*} is the analytic continuation of the infrared wavefunction from anti-de Sitter space near the boundary, \begin{align*} \Psi_{\text{IR}}[h]=\lim_{z\to \partial AdS}\int_{AdS}Dg_{g|_{z}=h}\exp(-S(g)). \end{align*} For simplicity I've taken the metric to be the only field in the theory. Maldacena's formulation of dS/CFT in arXiv:astro-ph/0210603 is that the partition function of the dual CFT is equal to the Hartle-Hawking state at late times, i.e. $Z[h]=\Psi[h]$. Now, if I take $h$ to be the metric on a sphere cross a circle, say, then $\Psi_{\text{IR}}$ computes the finite temperature partition function of the CFT dual to the AdS theory. This seems to imply that the partition function of the dual to the dS theory is just the analytic continuation of the partition function of the dual to the AdS theory. However, this seems to contradict the only known example of dS/CFT, which is the $SP(n)$ theory dual to Vasiliev theory on $dS_4$, conjectured in arXiv:1108.5735. In this case, the dS/CFT partition function is that of fermions at finite temperature, while the AdS/CFT partition function is that of bosons. These are definitely not analytic continuations of one another under $\Lambda\to -\Lambda$. Any ideas?


title: Convexity -- reference request

text: I've been reading a few papers on generalized probabilistic theories, and have been struggling through proofs of some results that involve use of convexity and group theory, e.g. this paper on bit symmetry and many others. Is there a set of introductory lecture notes on convexity (as used in quantum theory) online that anyone can refer me to?


title: How many points are required to make a black box

text: I have a black box with an arbitrary mass distribution inside it. I want to replace that object with n point masses without changing any mechanical properties of the box (center of mass, total mass, moment of inertia tensor). What is the fewest number of points I need and how can I find them? I count 10 constraints. 3 from center of mass, 1 from total mass, and 6 from the moment of inertia tensor. 3 point masses give me 12 degrees of freedom so at first glance it seems the answer should be 3. In practice however it does not seem possible to represent some objects (a sphere for instance) with only coplanar points. At least 4 are required.


title: Optimal way to heat liquid from the sun?

text: I understand that one way to tell how much a liquid will be heated up by a remote heat source is by looking at its color. I assume clear and colorless heats up least, then white, other colors and last black liquids will heat up the most. ( Is this right and can more detail be added?) However I don't understand the physics well enough to answer this simple sounding question. What other factors except for _visible_ color affect how much the liquid will be heated? If it is possible to give quantitive figures that would be great.


title: Larmor Precession - Determing frequency

text: Every time I go through some literature about Larmor Precession, i.e. the precession of orbit charged particle in the presence of a Magnetic Field. It doesn't give convincing arguements in calculating the Larmor frequency. Mostly in the books if the mathematics is taken care, it physically makes no sense to me or vice versa. Can somebody provide a nice insight into this that mathematically sufficient and also easy to see what is happening physically ?


title: Motivation of the Heisenberg model of ferromagnetism

text: In the Heisenberg model of ferromagnetism the atoms are assumed to be arranged in a lattice. The $i$-th atom has a spin operator $\vec S_i$ (here $i$ belongs to the lattice). The Hamiltonian is given by $H=-J\sum_{<ij>}\vec S_i\cdot\vec S_j$ where the sum runs over all pairs of atoms connected by an edge, $J>0$ is a constant. As far as I understand from the literature that this form of the Hamiltonian comes from the exchange interaction which, in turn, comes from the fact that each (neutral) atom is a boson (see e.g. vol. IX of Landau-Lifshitz, paragraph 72). **QUESTION. In addition to exchange interaction there exists a different spin- spin interaction which has a different form. Why it was neglected?** The spin-spin interaction between two atoms with magnetic moments $\vec\mu_1,\vec\mu_2$ and large separation $\vec R$ is equal to $$\frac{\vec\mu_1\cdot\vec\mu_2}{R^3}-3\frac{(\vec\mu_1\cdot\vec R)(\vec\mu_2\cdot\vec R)}{R^5}.$$ After quantization, $\vec \mu_i$ is replaced by $\vec S_i$ times a constant. As it is claimed in "Statistical mechanics" by R. Feynman, Section 7.1, the spin-spin interaction is very small in comparison to the exchange interaction. However this seems to be true only on short distances since the exchange interaction decays exponentially with the distance $R$ between the atoms, while the spin-spin decays as $1/R^3$. Hence when we are on 3d-lattice the sum over the lattice of terms of order $1/R^3$ diverges logarithmically. **This cannot be neglected from the experimental point of view too. When the ferromagnetic material is magnetized, it creates a magnetic field, which will interact with another magnet or moving charge. This interaction has already nothing to do with the exchange interaction, as far as I understand.**


title: Did the Feynman heuristic of "simple effects have simple causes" fail for spin statistics?

text: Someone here recently noted that _"The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated._ " "Spin statistics" is of course the shorthand way of referencing a most curious fact about the universe, which is this: If a particle of any type has a "spin" measured in whole units of quantized angular momentum, it will be a boson, a group that includes energy-like force particles such as photons. However, if its spin is (rather strangely) off-set by half a unit, it will instead be a fermion, which includes the particles that occupy space and that make up most ordinary matter. The rule is very simple. The explanation of it is arguably a bit more complicated. I find it fascinating that Nobel Laureate Richard Feynman worried over this simple theorem for decades, yet he never seemed to find an explanation for it that truly satisfied him. It was _not_ a lack of mathematical explanations, I should note. It was because Feynman deeply believed in a rather simple search heuristic: very simple relationships should in general also have simple, easily-conveyed explanations. Alas, Feynman's last attempt to explain spin statistics, in his _Dirac Lecture_ , always seemed to me one of his least clear bits of exposition ever. I am fairly confident Feynman would have assessed his Lecture that way himself, as he tended to be quite brutal in self-critiques on anything related to clarity of explanation. (I think there is an interesting family insight in that observation, incidentally: Richard Feynman's scientifically inclined father always hoped that his son, who had received the education he was never able to have, would someday explain all those little physics mysteries to him. The young Richard took that duty very seriously, and never really abandoned it, even towards the end of his own life.) So, my question and challenge: How is everyone doing on Feynman's spin- statistics challenge these days? Do you, fair reader, have in your hands some truly simple explanation for why whole-spin particles always seems to be bosons, and ones with half-spin offsets always seem to be fermions? I am not asking for twisty belts and wine glasses (please, _no!_ ), nor am I asking for something math free... though I do think anyone trying to answer this question should first look at how Feynman handled even complex numbers in his book _QED_. What I _am_ asking for is insight, the kind of explanation that makes the reader stop and think _wow, of course that's it, why didn't I see it what way before?_ So, anyone? I probably will not put an explicit bonus on this one, but if someone can provide an explanation that knocks everyone's +1 socks off, I guarantee I'll contribute at least a couple of hundred points to that overall consensus.


title: Derivation of polytropic process equation

text: I know that a polytropic process equation is often the result of an empirical fit of a P-V curve that allows one to derive some nice analytical equations for the process. Is there actually a derivation for the polytropic process equation?


title: Casimir effect plate experiment

text: i was a little confused about the casimir effect. my understanding is with the 2 plate experiment there was a force pushing the plates together because there were less virtual particles in between the plates than outside them. Why are there less particles in between the plates?


title: Is weak lensing the statistical effect of microlensing?

text: I am looking into the effects of gravitational lensing of gravitational waves. I know that gravitons travel along null geodesics, just as photons, and so they will suffer the same deflection angle by a massive close to their paths I think I know what _microlensing_ is, with the impact parameter, Einstein angle, and amplification factor. However, I have read that _weak_ lensing is likely to be a problem for advanced gravitational wave detectors. Is weak lensing just the statistical, overall effect of many different microlensing events, or is it something else?