Horizontal projectile motion, finding the height of the object at a given time.

A canon is fired Horizontal to the ground 80 meters above ground The canon ball is fired a $T=0$ and hits the ground at $T_g$. Calculate the height at the time $T_g/2$ So far I have calculated time. which gives me the equation $$y=y_i+v_{iy}(4.05) - (0.5)(9.82)(4.05^2)$$ Now my problem is that when I try to calculate $v_{iy}$ I am getting $-37\text{ m/s}$ which I know is not the case; it wouldn't make any sense. $Y$ initial is equal to $80\ \ V_{iy}$ I don't know $4.05$ is the time it takes the projectile to hit the ground. $9.82$ is gravity which is my acceleration.   So my question is two parts. First, is this the best equation to find what I'm trying to find? Second, how do I find velocity in the $y$ direction when $\theta$ is $180$ degrees? This is an edited question I do apologize for being unclear. Thank you for responding and requesting clarification. I was using the incorrect equation a simpler way to find the answer would be $y=.75\times H$. Because change in $y = -.5\times g\times t^2$ sense initial velocity in the $y$ direction equals $0$. Thank you for the help in structuring my question better.


Statistical Mechanics treatment of the reaction process?

I'm searching for an at least semi-rigorous Statistical Mechanics description/treatment of a (spatially resolved) chemical reaction process of a macroscopic portion of at least two different species along the lines of $$A_1+A_2\longrightarrow A_3,$$ (with $A_3=A_1A_2$, necessarily) or $$A_3\longrightarrow A_1+A_2.$$ Might be a description here or a reference. I'm aware of the reaction kinetics description of combustion processes, where the idea is to modify diffusion equations with a reaction rate term $$''\frac{\partial}{\partial t}[A_i]=D\ \Delta [A_i]+\Sigma_j\ R_{ij}(T,[A_k]),''$$ where the last term in the end is coming from a Boltzmann'esque theory, but I'm interested in "more modern" approaches or a self contained treatment from first principles, which might, for example, model the whole thing in therms of ensembles, obviously as linkage between thermodynamics and non-equilibrium physics.


Why has Earth's core not become solid?

The Earth is billions of years old, yet its core has not yet cooled down and become solid. Will this happen in the foreseeable future?


Angular momentum and spin

I am having problems with this excercise. We look at a system where the total angular momentum is given by an electron with $l=1$ and $s=\frac{1}{2}$. Now I am supposed to calculate the eigenfunctions $|\frac{3}{2},m_j\rangle, |\frac{1}{2},m_j\rangle$ with respect to $J^2 $ and $J_z$, where $J=L^{(1)}+S^{(2)}.$ In part a) I derived the equation $J^2=L^{(1)2}+S^{(2)2}+2L_3^{(1)2}S_3^{(2)2}+L_+^{(1)2}S_-^{(2)2}+L_-^{(1)2}S_+^{(2)2}$. Maybe this one is useful here. Also a hint was given: I should check that $|1,1\rangle \otimes |\frac{1}{2},\frac{1}{2}\rangle $and $ \sqrt{2/3}|1,1\rangle \otimes |1/2,-1/2\rangle-\sqrt{1/3}|1,0\rangle \times |1/2,1/2\rangle $ are eigenfunctions with respect to $J_z$. This was clear to me how to do this and I checked it, but I don't know what this has to do with the actual excercise. Is there anybody who could help me with this excercise?


Difference between mirror reflected light rays and rays of a screen

What is the difference between the light rays reflected from a mirror and rays directly emmited from a screen if the screen shows the same image as the mirror would.


The reference frame of $c$

I don't have a lot of knowledge of special relativity and associated topics; some of the few things I know are that "all motion is relative" (that is, there is no 'stationary reference frame'), and the speed of light ($c \simeq 3 \cdot 10^8~\mathrm{m~s^{-1}}$) is the absolute asymptotic speed limit ( _asymptotic_ meaning that you can never equal it, only get arbitrarily close). What escapes me is how those concepts work together - to my naive understanding, an object would never move in its own reference frame (and so would never reach $c$). What reference frame is $c$ measured against? ( _Is_ it measured against a reference frame?) Or am I looking at this the wrong way?


Why does my infrared thermometer say the sky is at -2 °C?

I just got myself an infrared thermometer. I wouldn't have been able to predict what temperature it would give me when pointing at the sky, but it turned out to be -2 °C the first time I measured, and slightly going up and down with the temperature on the ground (say from -6 °C to 2 °C in the course of a day in which the ground temperature goes from 20 °C at night to 30 °C during daytime). Some details: this was for pointing straight up (as long as its not pointing too closely to the sun), a clear sky and a humidity of 63% according to internet. The range in which it measures radiation is from 8-14 μm and the specified range of temperatures is -50 to 550 °C. What is it that causes this reading?


Computing fine structure of terms in LS coupling,

In order to compute the fine structure of the terms in LS coupling (Russell- Saunders coupling), we must treat the hamiltonian $$H_2 = \sum_{\mbox{open subshells}} \xi(r_i) \vec{l_i}\cdot\vec{s_i}$$ as a perturbation to the perturbation given by the hamiltonian $$H_1 = \sum_{\mbox{open subshells}}\frac{1}{r_{ij}}$$ which splits in differents terms labeled by $^{2S+1}L$ the energies obtained in the central field approximation. $H_2$ can be written as $$ H_2 = \xi(\vec{L}\cdot\vec{S}) \vec{L}\cdot\vec{S}$$ and then the correction given from $H_2$ is then readily obtained. **My question is, how can this form of $H_2$ be derived from Wigner-Eckart's theorem?**


Force exerted on a cone

This is probably a pretty elemental question, but I can't figure out what the concept is called or how to calculate it. Say you have a solid cone shaped piece of ice, you hold it in your hand and squeeze the cone and it slips up as you continue the same force until it slips all the way up and out of your grip. What is this called and how do you calculate the speed of the cones movement with a given amount of force?


Gribov ambiguities for splitting strings in the BRST conformal gauge

The standard textbook treatment of the conformal gauge for a free string with BRST ghosts is fine, but when a string splits into two, we have Gribov ambiguities for the conformal gauge. We can impose the conformal gauge for the initial string before the split, and also the final two strings after the split, but not for the interpolating region. How do we deal with this?


What role does gravity have on flame composition?

In 2000, Nasa conducted an experiment and found that gravity plays an indirect role in flame formation and composition (see the wikipedia article). What role does gravity play? Why does a flame extend in the opposite direction of gravity? Assuming the experiment could be re-created, are there any other forces that would cause a flame to extend beyond a spherical shape?


Why is the Taub-NUT instanton singular at $\theta=\pi$?

Consider the following metric $$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$ where $$V=1+\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been told that it is singular at $\theta=\pi$ but I don't really see anything wrong with it. So, why is it singular at $\theta=\pi$? EDIT:: I have just found in this paper that the metric is singular "since the $(1-\cos\theta)$ term in the metric means that a small loop about this axis does not shrink to zero at lenght at $\theta=\pi$" but this is still too obscure for me, any clarification would be much appreciated.


Vapor-compression refrigeration

In a vapor-compression refrigeration system I would like to understand the following :   1. Why is there a need for the compressor ? Can't the saturated vapor from the evaporator be cooled directly in the condenser ? Why does it need to be a superheated vapor in order to be condensed ?   2. How an expansion valve works exactly (physically speaking) ? I understand it uses flash evaporation to transform part of the liquid to vapor without heating it. But I don't understand how is this mixture temperature lowered below the temperature of the environment to be cooled. There is some kind of auto-refrigeration that is mentioned in the books I have read that I don't grasp. Can anyone explain to me how this happens ?


How to calculate the supporting force for a cantilever beam?

How do I find the supporting force $R_A$ and the moment $M_A$? Should I use the sum of moments or?


Meaning of the Reduced Density Operator

I am confused about what it is exactly that a reduced density operator describes. To illustrate, I came across the following seemingly paradoxical argument. Consider a biparte system $AB$, described by the pure state: $$| \psi \rangle = \sum_{a,b} \psi(a,b) |ab \rangle$$ Its density matrix is defined as: $$\rho \equiv | \psi \rangle \langle \psi |$$ With reduced density operators: $$\rho_A = tr_B(\rho) \qquad \rho_B = tr_A(\rho)$$ We can construct a new operator: $$\rho_{AB} = \rho_A \otimes \rho_B$$ Which is only equal to $\rho$ if $|\psi\rangle$ is a seperable state (no correlations between $A$ and $B$). Another way to say this is: $$0 = S(\rho) \leq S(\rho_A) + S(\rho_B) = S(\rho_A \otimes \rho_B) = S(\rho_{AB})$$ Because if $A$ and $B$ are entangled, then $\rho$ contains information about these correlations, but $\rho_{AB}$ does not (is this wrong?). So say $A$ and $B$ are entangled, then $\rho_{AB}$ does not contain that information, but I was under the impression that $\rho_A$ ánd $\rho_B$ do contain (some, all?) of this correlation information; otherwise both $A$ and $B$ should be described by pure states. I realise that $\rho_{AB}$ has no physical meaning in the case of entanglement, but even as a purely mathematical construct I don't see how that information can just vanish. I guess my question is this: what information exactly does the reduced density operator encompass? And, if any of it is related to the correlations, how can this be reconciled with the argument in the paragraph above?


How can I simulate a rigid bounced from a wall?

How can I simulate a rigid sword bounced from a wall and hit the ground(like in physical world)? I want to simulate a simple animation. The sword is controlled by a center/mass point.(Actually finally it't in a game engine). I can detect the figure and the size of the sword(maybe needed in doing bounce). And the position is $\\{x,y,z\\}$ rotation can be controled by quaternions/matrix/euler angles.[In mathematica, by default is matrix] # Step1 * * *               point1 = {0, 0, 500};point2 = {500, 0, 500};      Fly 500 miles. (Here I can add an acceleration)               Graphics3D[{Red, Cuboid[{600, -100, 400}, {510.1, 200, 600}],   GeometricTransformation[swordX[[1]],    TranslationTransform /@ {point1, point2}]}, ViewPoint -> Front]      ![enter image description here](http://i.stack.imgur.com/tYdjg.png) As you see in the picture, my sword inserted into the `Red` wall. # step2 * * * sword bounced from the wall, It should turn the head and do rotations and fly to the ground. Something like a projectile How can I simulate this physical process? Maybe what I need is an equation and some parameters? Any references? I've played with this _Mathematica_ Demonstration model, and I've made a video before. And I need make it more physical, especially how the sword hit and bounced from the wall and fly like a hammer in this model... video[03:07] # step3 * * * hit the ground, (maybe can bounce from the ground and hit again...) Some model maybe the same as that in step2. * * * I have no physical engines in Mathematica, if you know, welcome to tell me something related.


Hamilton formalism for Dirac spinors

Let's have the Dirac free lagrangian: $$ L = \bar {\Psi} (i\gamma^{\mu}\partial_{\mu} - m) \Psi . $$ I can rewrite it as $$ L = i\Psi^{\dagger}\partial_{0}\Psi - H_{d}, \quad H_{d} = \Psi^{\dagger}(-i(\hat {\alpha} \cdot \nabla ) + \beta m)\Psi , \quad \hat {\alpha} = \gamma_{0}\gamma . $$ Here $H_{d}$ is the Hamiltonian density. Is it possible to build Hamilton equations and Poisson brackets for spinors (for example, $\Psi$ may be the canonical coordinate and $\Psi^{\dagger}$ be the canonical impulse)?


If light is considered as wave, then different colored light can be explained as waves of different wavelength/frequency. How colored light can be explained if light is considered as emission of photon?


Circuit Simulation

I am making a circuit simulation program, and I am not sure how to simulate the behavior of Diodes. Take this circuit:                                                          \------100 Resistor -----LED------   Positive 5V-----470 Resistor-----| |---------Negative 5V                                                          \-----------200 Resistor----------   How would you calculate the voltage and current through all of these components? Also, is it true that with a simple circuit like this: Positive 5V------470 Resistor------LED--------Negative 5V That you would subtract the forward voltage of the LED from the Battery's voltage, then use Ohm's law to calculate the current and resistance through the circuit? voltage = 5;   newVoltage = voltage - forwardVoltage;   current = newVoltage / 470;   resistance = voltage / current;   Thanks for the help.


Do Bell inequality violations appear instantly when the source is turned on, or do they increase over time?

This experimental Question is a result of reading a particular article on Bell violations. I addressed the e-mail below to the corresponding authors —because who knows, they might reply— but it is not specific to them, it's just that their article crystallized the Question, and perhaps someone on Physics SE can tell me what the results of the experiment would be without doing it (although I think these time-dependent properties are not part of the usual data sheets for off the shelf components). There is a subsidiary question: have such experiments been done, and I haven't seen them in the literature? > Dear Johannes and Anton, > >   I've read your PNAS article "Violation of local realism with freedom of > choice" as a result of a blog posting by Sabine Hossenfelder at > http://backreaction.blogspot.com/2011/06/nonlocal-correlations-between- > canary.html _[Physics SE readers can find PNAS and arXiv links there]_. I > was impressed by your three key remarks on the second page, which seem to me > nicely balanced. > >   Reading your article I wondered as I have many times whether Bell > inequality violations appear _instantly_ when the source is turned on, or to > what extent they increase over time. I am partly led to the questions below > because it seems that Bell violating experiments run continuously, whereas > it seems that technological applications may well be intermittent. > >   I imagine, specifically, an experiment _based on_ an experiment in which > we observe Bell violations routinely (presumably in CHSH form). The crucial > physical modification is to block both paths from the source to the two > detectors physically, at a point near the source, on a time scale of, say, > about a second, so that light travels from the source only for half a second > at a time. A steady state condition of the source, the quantized > electromagnetic field, and the detectors will presumably not be instantly > established, but, I suppose, would be established in half a second, so that > if we considered data only from the last quarter second of the on-phase we > would see the usual violation of Bell inequalities. > >   The crucial modification of the analysis is to consider the way Bell > violation changes over time, microsecond by microsecond (or more or less > finely time-sliced as experience indicates), after the moment that the > physical block is removed. Clearly this is a statistical analysis, since we > would expect to see approximately one photon pair every four microseconds > (at a local production rate comparable to the 250,000 photon pairs per > second that you report in your PNAS article). > >   I suppose that a number of questions emerge, some of which may be > characteristics of the source and detectors; others might be characteristic > of the experimental apparatus more generally considered. Firstly, most > elementarily, how does the rate of single photon detections change over time > in the microseconds after the physical block is removed? We would expect > that there would be a rapid approach to the steady state, but not > instantaneous. > >   Secondly, how does the rate of photon pair detections change over time in > the microseconds after the physical block is removed? Does the approach to > the steady state have the same relaxation rate as the relaxation rate for > the single photon approach to the steady state? > >   Thirdly, how does the violation of Bell inequalities vary over time in the > microseconds after the physical block is removed? Again, does this have the > same relaxation rate for the approach to the steady state? > >   It seems a matter of technological interest as well as of foundational > interest whether relaxation of Bell inequality violations towards the steady > state immediately after a light source is exposed is the same as the > relaxation rates of single photon detections and/or double photon > detections, or whether we have to wait longer for usable Bell violations to > emerge. > >   Because I suppose these questions to be of general interest, I have also > posted them at http://physics.stackexchange.com. I don't imagine the results > of such an experiment will go against quantum theoretical expectations, but > I'm curious whether characterizing these time-dependent properties of the > sources and detectors might give some surprises.


Contracting Indices in General relativity

I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ \mathrm{g}^{\mu\sigma}{\Gamma}^{\nu}_{\rho\sigma} -\frac{1}{2}( {\Gamma}^{\sigma}_{\sigma}+{{\Gamma}^{\sigma}_{\sigma\rho}} ) \mathrm{g}^{\mu\nu} &=0, \tag1 \\\ \mathrm{g}^{[\mu\nu]}_{,\nu} +\frac{1}{2}( {\Gamma}^{\rho}_{\rho\nu}-{\Gamma}^{\rho}_{\nu\rho} ) \mathrm{g}^{(\mu\nu)} &=0, \tag2 \end{align} $$ Where the connection is not symmetric with respect to the lower Indices. and it says that by contracting equation (1) once with respect to ($\mu,\rho$), then with respect to ($\nu,\rho$), then subtracting the resulting equations we can get equation (2), however I can't see how that's possible.


When has the speed of light been measured, recently?

Yes, it is weird, absurd, but I can't stop thinking that the would-be superluminal neutrino speed has been computed by an arithmetic operation (space/time) and not by direct comparison with a simultaneous light ray running in "parallel". So the "unspeakable", outrageous question: is the speed of light increasing in the last months? When it has been measured the last time? Also: is the speed of light measured in a one-way or two-ways (forward and back) method? It is a "politically incorrect" question, but its logic is rock-solid, I think.


George Green's definition of Green's function

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x−x′)$ function as their source on the RHS. But the Dirac-delta was defined about a century after Green's work. I just want to know, how George Green have defined his function (in case he defined it)? In case he didn't define his Green's function, how did he give a hint about this method (without Dirac-delta function) ?


Is the Energy Sharply or Fuzzily Defined in Quantum Mechanics?

According to quantum mechanics, energy of a state is uncertain within a small range in hydrogen atom. But we also know that energy of a state is quantized which is contradictory to the first. Which one is correct?


Confusion with curl of Lorentz magnetic force

Since the magnetic force is a no work force, $dW=\vec F\cdot d\vec r=0$ for $\vec F(\vec r)=q(\vec v(\vec r) \times \vec B(\vec r))$, therefore $\oint \vec F \cdot d\vec r=0$ by Stoke's theorem. Therefore, $\vec\nabla \times \vec F=0$. If this conclusion is true then expanding $\vec\nabla\times[\vec v(\vec r) \times \vec B(\vec r)]$ explicitly using the relevant vector identity one must also obtain zero. But it seems to be non-zero in general. So my question is whether $\vec\nabla\times\vec F=0$ or not?


Liouville's theorem and conservation of phase space volume

It can be proved that the size of an initial volume element in phase space remain constant in time **even for time-dependent Hamiltonians**. So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?


High Current at first and then steady from a battery

I just came up with a question, When a battery is connected to a circuit, the current first suddenly peaks up and slowly goes down to a steady value. Why is it so? I have not actually observed this, but this question has been asked in exams, so it should be that way I hope.


Are valence electrons located solely in the s and p subshells?

Or are they in all subshells??


Interesting relationship between diffraction and Heisenberg's Uncertainty Principle?

I recently came across an interesting explanation of diffraction through an aperture which does not use Huygens' Construction but instead relies on Heisenberg's Uncertainty Principle: > The Uncertainty Principle states that trying to pin a particle down to a > definite position will create uncertainty in its momentum, and vice-versa. > Therefore when you are confining a particle to go through a narrow aperture, > one is very certain about its position (the aperture is very narrow and > hence the particle must have been somewhere in that extremely small gap) So, > by Heisenberg's Principle, the particle will now have a crazy momentum... so > it can go in any direction. However, it seems we cannot use the same principle to explain the case where diffraction occurs when light passes around an obstacle. We are not locking the photons in a small position in that case, are we? Keeping the Uncertainty Principle in mind, diffraction through an aperture makes sense but in the case of a solid obstacle, my intuition says the wave should just be deflected. Is there a way to relate the Uncertainty Principle to diffraction around an obstacle? Besides, given that the Uncertainty Principle is defined in terms of particles, can it be used to explain diffraction in mechanical waves such as sound which are not composed of photons? _Do note that my question has nothing to do with diffraction patterns or the process of interference, and is rather about the relationship between Heisenberg's Principle and diffraction._


Neel order and O(3) model

The coarse grained fluctuations of the Neel order parameter in the half integer spin anti-ferromagnetic Heisenberg model is described by the O(3) non- linear sigma model with a strange berry phase term. See for example : Eq-2 in http://arxiv.org/pdf/cond-mat/0311326.pdf I was wondering if someone could explain to me where the Berry phase term comes from. Apart from the origin of the term, it is also not clear to me how such a Berry phase term is allowed in a Lagrangian without any periodicity in the time direction. Thanks in advance for any inputs !


determinism, and quantum computing

It might seem logical to presume that the exact state of the world around us could have been predicted given a big enough computer to analyse all the particle's interactions and what not. Determinism seems to be logical. Then comes quantum computing. Since a collection of entangled qubits are only coherent when isolated from their environment, is this new information being fed into our known universe system? Whats going on here? Since you could hypothetically have hardware that starts manipulating particles as a result of these isolated quantum wave collapses, does this put the nail in the coffin of the deterministic universe? Or are we just awaiting a mechanism by which these wave function's chose the value to collapse to (quantum foam geometry or something)? As far as I understand we don't yet know how the wave function choses its collapsed spin state except that statistically it follows a predictable probability spread.


Knowing when wavefunction collapses

So I learned that after a measurement of, lets say the position of the wavefunction of a particle is made, if another measurement of the position of the particle is made right away, you should get the same result because the wavefunction has collapsed. How can you tell the difference between the observer measuring the wavefunction which causes the collapse vs. something else that measured the wavefunction and collapsed it already and you are measuring it the second time around?


Two paths having the same phase in the path integral approach

In the path integral approach to Quantum Mechanics, can two distinctly different paths of the possible infinite paths have the same phase, i.e can there be a bimodal distribution of the phases associated with each path? And Why?


What is the math/physics behind maximizing tinker's damage?

Tinker is a hero in the popular game Dota 2. He has a spell called 'March of the Machines' that creates a stampede of little robots in a rectangular area. The robots do damage on impact of an enemy unit. Basically this can be considered a vector field and enemy units are surface area. We want the maximum flux in order to optimize damage. If you're really interested in what this looks like: https://www.youtube.com/watch?v=AmXRvkaYhIE Anyways, you often end up in static situation where enemy units aren't moving in this common pattern. It makes sense that a diagonal march will provide the most damage: ![enter image description here](http://i.imgur.com/8hqUH4D.png) A vertical march would have one enemy 'shielding' damage from the one behind it (depending on if it were downwards or upwards). A horizontal march does negligable damage to the bottom due to way less surface area being hit. What kind of confuses me is once things start moving. The common advice is(artwork not mine): ![http://cloud-2.steampowered.com/ugc/596998708777054720/3ED611DEA2B3245C834C28C12DE9D0188A72C1C0/](http://i.stack.imgur.com/0RpEH.jpg) Things tend to run roughly single file so you can assume the stuff running through the march is a wall of surface area. My intuition always made me think that when things are running through march, going fully perpendicular would maximize damage. The sadder part is I've taken vector calculus before, so I've encountered a great deal of this stuff. But this is certainly the area where my brain had a melt down! I'd love to see a mathy justification for this angled approach being optimal.


Negative sign in the Dirac term from the SUSY Kahler potential

I want to calculate the Dirac term from the canonical Kahler potential, \begin{equation} K = \Phi ^\ast \Phi \tag{1} \end{equation} but I'm coming across a pesky negative sign in the final result. I am finding (see derivation below), \begin{equation} \- i \bar{\psi} \bar{\sigma} ^\mu \partial _\mu \psi \tag{2} \end{equation} This agrees with lecture notes by Matteo Bertolini (pg. 87) but is off by sign from lecture notes by Fernando Quevedo (pg. 50), while they both supposedly use the same conventions. I would be okay with deciding Quevedo's notes have an error but my result also seems to contradict the regular Dirac Lagrangian, which in four-vector notation is \begin{equation} {\cal L} _D = + i\bar{\Psi} \gamma ^\mu \partial _\mu \Psi \tag{3} \end{equation} Any ideas where this problem is coming from? Here are my steps: The chiral and antichiral superfields take the form (I am using the (+---) metric), \begin{align} & \Phi = \phi + \sqrt{2} \theta \psi + \theta ^2 F + i \theta \sigma ^\mu \bar{\theta} \partial _\mu \phi - \frac{ i }{ \sqrt{2} } \theta ^2 \partial _\mu \psi \sigma ^\mu \bar{\theta} - \frac{1}{4} \theta ^2 \bar{\theta} ^2 \Box \phi \tag{4}\\\ & \Phi ^\ast = \phi ^\ast + \sqrt{2} \bar{\psi} \bar{\theta} + \bar{\theta} ^2 F ^\ast - i \theta \sigma ^\mu \bar{\theta} \partial _\mu \phi ^\ast + \frac{ i }{ \sqrt{2} } \bar{\theta} ^2 \theta \sigma ^\mu \partial _\mu \bar{\psi} - \frac{1}{4} \theta ^2 \bar{\theta}^2 \Box \phi ^\ast \tag{5} \end{align} Calculating the Dirac term involves the product of the second term of $ \Phi ^\ast $ and the fifth term of $ \Phi $ and vice versa. I find: \begin{align} \- i \bar{\psi} \bar{\theta} \theta ^2 \partial _mu\psi \sigma ^\mu \bar{\theta} & = - i \partial _\mu \psi ^\alpha \sigma ^\mu _{ \alpha \dot{\alpha} } \bar{\psi} _{\dot{\beta}} \bar{\theta} ^{\dot{\beta}} \bar{\theta} ^{\dot{\alpha}} \theta ^2 \tag{6}\\\ & = \frac{ i }{ 2} \partial _\mu \psi ^\alpha \sigma ^\mu _{ \alpha \dot{\alpha} } \bar{\psi} ^{\dot{\alpha}} \bar{\theta} ^2 \theta ^2 \tag{7}\\\ & = - \frac{ i }{ 2} \bar{\psi} \bar{\sigma} ^\mu \partial _\mu \psi \bar{\theta} ^2 \theta ^2 \tag{8} \end{align} where in the last step I used the spinor identity, $ \psi \sigma ^\mu \bar{\chi} = - \bar{\chi} \bar{\sigma} ^\mu \psi $. Repeating the calculation for the product of the fifth term of $ \Phi ^\ast $ and the second term of $ \Phi $ and summing the two results gives: \begin{equation} \- i \bar{\psi} \bar{\sigma} ^\mu \partial _\mu \psi \bar{\theta} ^2 \theta ^2 \tag{9} \end{equation} which after stripping off the $\theta,\bar{\theta}$ is the result I quote above.


Violation of conservation of energy and potential energy between objects

I would like to clarify my question. I have numbered them to be independent questions   1. For any conservative fields, $\vec{F} = -\nabla U$. Which means the restoring force is opposite to the increasing values of the potential energy. If it so happens that $F = \nabla U$, does that mean the restoring force (not even called that anymore I guess) would be in the same direction as the increasing values of energy? Is this a violation to the first law in thermodynamics.   2. For attracting bodies, the closer the objects come together, the potential energy drops as $r \to 0^-$ and $U \to -\infty$, but for repelling bodies we get $r \to 0^-$ as $U \to \infty$ which means the potential energy actually increase. Am I right?


Wave functions for three identical fermions

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.


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

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.


Higgs-Boson/Graviton

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.


Nomenclature clarification concerning solitons

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?


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

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 ?


Defining simultaneity with a central light vs with clocks

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?


Flux through a Mobius strip

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".


Is there a limit to loudness?

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?


Trying to turn a nonlinear differential equation into a linear one

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.


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

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.


How to choose Gaussian surfaces while solving problems?

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?


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

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.


Two photons transition

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.


Relative wind velocity explanation - understanding Irodov problem 1.6

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


How is the sun's equatorial circumference measured?

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


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

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.


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

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.


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

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?


Relative linear velocity of a particle to a rotating object

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?


Classical and Semi-classical treatments of the ideal gas

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.


Stationary and Static

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.


Is a non-Newtonian gas possible?

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.


What is the phase space for outgoing photons?

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?


A question about speed

> 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: $u^2=-2as=-2(f/m)s=-2(-0.4N/m)s=-2(0.4mg/m)s=2*0.4*10*200=1600$ $u=40ms^{-1}$


Higgs mass and the hierarchy problem

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?


Partition Functions in (A)dS/CFT

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?


How many points are required to make a black box

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.


Optimal way to heat liquid from the sun?

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.


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

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.


Derivation of polytropic process equation

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?


Casimir effect plate experiment

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?


Is weak lensing the statistical effect of microlensing?

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?


When can i separate spin from the wavefunction?

I am currently working on a Tight-Binding model and for the derivation of nearest neighbor spin interactions I have terms like $$ \left<x\uparrow\right|\eta\;\mathbf{l}\cdot\mathbf{s}\left|z\downarrow\right> $$ where $\left|x\uparrow\right>$ is an atomic orbital.For the simplification it is very important that i can assume $$ \left|x\uparrow\right> = \left|x\right>\left|\uparrow\right> $$ My question is:   * is this always possible in a solid body or do i neglect some effects with that? (like spin polarization or something?)


Adiabatic filling of a container

Suppose a thermally insulated container is filled with atmospheric air until the pressure reaches 5000 psi. This could represent the filling of a diving cylinder, before thermal dissipation becomes significant. Initially, then, some mass of air has atmospheric temperature $T_1$ and $p_1$. When forced in to the tank adiabatically, we expect its final temperature to be \begin{align} T_2 &= T_1\left(\frac{p_2}{p_1}\right)^{1-1/\gamma} \\\ &= (300\,\mathrm{K})\left(\frac{5000\,\mathrm{psi}}{14.5\,\mathrm{psi}}\right)^{1-1/1.4} \\\ &\approx 1590\,\mathrm{K} \end{align} This final temperature seems far too hot. A diving cylinder would likely melt at this temperature. More likely, though, something is wrong with my reasoning. Is this not the correct use of the temperature-pressure relations, assuming an ideal gas? If not, what other information is needed to predict the final temperature of the container?


Bigger anti-matter particles

I have learned about the existence of positrons as a decay product from uranium fission - if I'm not mistaken. Is there any evidence for higher 'mass' anti-matter, or is that mere speculation or rationalisation? i.e. Is the argument that there anti-matter based planets in the universe, or just a soup of positrons?


What would we observe from thompson's lamp thought experiment?

The thought experiment goes like this: > Say there is some circuit which turns a lamp on/off with just a flick of a > switch. Say its off; you flick it, it turns on; flick it again it turns off, > and so on. > > So say you are conducting the experiment for two minutes. When the remaining > time halves, you flick the switch; and when that remaining time halves, you > flick again...This process goes on infinitely, until two minutes is reached. Mathematics says that the sum of all those remaining half-times will eventually tend to two minutes. And the outcome i.e. the final state is always unpredictable. We can't really know if its on/off. Lets not concern about the final state. What would we actually observe towards the end, few micro seconds towards the two minute mark? Wouldn't the lamp appear turned on? What would happen if the time between two flicks tend to zero?


Special Relativity: What differential equation describes an accelerated object from a non-inertial reference frame?

I am looking for a set of differential equations (to be solved numerically for an educational program) that would describe the position and apparent time of an accelerated clock relative to a non-inertial reference frame. For now, all movement is along a single spatial dimension. I am _not_ interested in considering the time light takes to travel between observers; assume that they are clever enough to account for this delay. As inputs to this system of equations, I would have:   * An arbitrary, time-dependent acceleration a(t) as experienced by an observer in the non-inertial frame. In practice, this would be a simple list such as a(t=0) = 0; a(t=1) = 0.2; ...   * The initial position x(t=0), clock time t'(t=0), and velocity v(t=0) of the accelerated clock as seen in the non-inertial reference frame   * Another time-dependent acceleration a'(t') which describes the acceleration experienced by an observer traveling with the clock. To make this (hopefully) more clear, I'll give an example: an observer A, initially at rest, sees a relativistic rocket pass by carrying observer B. When A's watch reads t=0, she notes that B's position is x(t=0), B's speed is v(t=0) and B's (prominently displayed) clock reads t'(t=0). A has a list of instructions a(t) which tell her to set her rocket engines to achieve a specific acceleration at specific times. B has a similar set of instructions a'(t'). Both A and B consult their _own_ watches when determining when to change their acceleration. I have seen this question: What is the displacement of an accelerated and relativistic object?, which tells me enough to model the situation for an _inertial_ observer, but I have not figured out how to reapply this for a non- inertial observer. Similarly, I can compute the worldlines of non-accelerated objects from the perspective of a non-inertial observer, but it seems something sneaky happens when both the observer and observed are accelerating. I have also tried computing both worldlines from an inertial reference frame and then using the Lorentz transformation [x' = g (x - vt); y' = g (t - vx/c^2)] to ask what B's worldline would look like from A's perspective, but this did not work (I can describe more about this, but for now it should suffice to say that the worldlines were not remotely correct).


Total scattering rate of light for an atomic transition given a laser field with an intensity distribution?

I have a laser with an intensity distribution $I(\lambda)$ over a broad wavelength range (arroun 40 nm), I want to calculate the total scattering rate due to the full spectrum of the laser; the the scattering rate is given by $\gamma_p(\lambda) = \gamma \rho_{ee}(\lambda) = \frac{\gamma}{2}\cdot\frac{A(I(\lambda))}{1+A(I(\lambda))+B(\lambda)}$ where $\gamma$ is the linewidth of the atomic transition and $\rho_{ee}$ is the population in the excited state. I was directly integrating $\int\gamma_p(\lambda)d\lambda$, but the result makes no sense. Any idea?


How to compute the change in entropy of s system when the temperature is decreasing? Such a question may be too general,then we may ask a special case:How to compute the change in entropy of Annealing system when such a process may be regarded as Markov Chain ?


Consider quantum Hamiltonian of free massive scalar particle: $$\hat{H} = \int d^3x \left[\frac{1}{2} \hat{\pi}^2 (t, \vec{x}) + \frac{1}{2} \partial_i \hat{\phi}(t, \vec{x}) \partial_i \hat{\phi}(t, \vec{x}) + \frac{m_0^2}{2} \hat{\phi}^2(t, \vec{x}) \right].$$ I'd like to show that Heisenberg evolution equation for $\hat{\phi}$ holds, i.e.: $$\partial_t \hat{\phi} (t, \vec{x}) = i [\hat{H}, \hat{\phi}(t, \vec{x})].$$ My problem is to compute the following term: $$\int d^3 y \left[\partial_i \hat{\phi}(t, \vec{y}) \partial_i \hat{\phi}(t, \vec{y}), \hat{\phi}(t, \vec{x})\right].$$ I would appreciate some help on how should I rearrange this commutator to see it vanishes.


Conservation of Linear Momentum at the point of collision

This is a pretty basic conceptual question about the conservation of linear momentum. Consider an isolated system of 2 fixed-mass particles of masses $m_1$ and $m_2$ moving toward each other with velocities $v_1(t)$ and $v_2(t)$ respectively. Now conservation of momentum says that at any point during the particles' motion the quantity $$m_1v_1(t) + m_2v_2(t) =constant$$ With non-zero velocities and non-zero masses this constant will be **non- zero**. Let us say the particles collide at time $t_0$. At the point of collision, both particles have velocity **zero**. which would mean that the constant above will be zero. Contradiction. I realize I might be going wrong in my reasoning at the point of collision. In fact, I feel defining velocity at that point would not even make sense, since if one considers the displacement functions $x_i(t)$ $i=1,2$ of the particles, then $t_0$ would represent a point of **non-differentiability** of $x_i(t)$ for $i=1,2$. So **assuming** there are no collisions, by following the text-book derivation I can see why $$m_1v_1(t) + m_2v_2(t) = C_1$$ before the collision and $$m_1v_1(t) + m_2v_2(t) = C_2$$ after the collision would hold true, but not why $C_1=C_2$ Can someone help me in clearing this up?


Lorentz invariance of the action for free relativistic particle

I tried to check the Lorentz invariance of the standard special relativity action for free particle **directly** : ($c=1$) $$ S=\int L dt=-m\int\sqrt{1-v^{2}}dt $$ Lorentz boost: $$ dt=\frac{dt^{'}+udx^{'}}{\sqrt{1-u^{2}}}$$, $$dx=\frac{udt^{'}+dx^{'}}{\sqrt{1-u^{2}}}$$, $$v=\frac{dx}{dt}=\frac{u+v^{'}}{1+uv^{'}}$$ Substitute these expressions into the action: $$ S^{'}=-m\int \sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}} dt=-\int \sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}} \left(\frac{dt^{'}+udx^{'}}{\sqrt{1-u^{2}}}\right) $$ $$ S^{'}=-m\int \left(\frac{\sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}}}{\sqrt{1-u^{2}}} \right)dt^{'}-m\int \left(\frac{u\sqrt{1-\left(\frac{u+v^{'}}{1+uv^{'}} \right)^{2}}}{\sqrt{1-u^{2}}}\right)dx^{'} $$ But I know that action of this type is invariant under lorentz transformations, so we want to get that $$ S^{'}=-m\int \sqrt{(1-v'^{2})} dt^{'} $$ But it doesnt seem to me that these expressions for action are equal. So I'm confused. Maybe there is a mistake in my calculations?


Hydrogen atom in partial differential equations, show it is is independent of time.

For the hydrogen atom, if $\int|u|^2\ \mbox{d}x=1$ show that the same is true at all later times. Hint: Differentiate the integral with respect to t, taking care about the solution being complex valued. assume that $u$ and $\nabla u\rightarrow \infty$ fast enough as $|x|\rightarrow\infty$.


Angle needed for object A to intercept with object B

Object B is 15 degrees East of North at a distance of 20km/h. Object B is moving at an average speed of 30km/h in the direction 40 degrees East of North. If object A is capable of moving at 100km/h, at what angle does it need to move to intercept object B? ![obligatory sketch](http://i.stack.imgur.com/O2qUC.jpg) I've tried calculating it as vectors, but got stuck. In the end I just tried using good old geomatry, where $$ \frac{\sin A}{a} = \frac{\sin B}{b} $$ So I called the angle I'm searching for A, and the angle from the line between A and B, and the direction of object B, as angle B. 'a' and 'b' are the average speeds of object a and b. $$ \frac{\sin A}{30 \times t} = \frac{\sin(155°)}{100\times t} $$ $$ A = arcsin\left(\frac{30\times t\times \sin(155°)}{100\times t} \right) = 7.284° $$ So the final angle should be $15°+7.284°=\mathbf{22.284°}$ Is this correct? Isn't there a better way of doing this, like using vector products? PS. I've found a very similar question here, but didn't quite understand how i.e. $\alpha + \beta = 40° $, because my calculations show $ =35° $. And trying to read through comments and answers, confuses me more due to updates being made inconsistent to comments about the updates done. And due to me not having enough reputation, I'm not able to ask counter-questions. BONUS Question: I have been searching high and low on how to make a simple graph of my problem. Like this here question, with a graph of the problem hosted at stack.imgur.com which I guess was generated by LaTeX on stackexchange.com, right? EDIT: Added obligatory sketch!


Hamiltonian form of Noether's Theorem

I understand that Noether's Theorem has a Hamiltonian form, whereby {X, H} = 0 iff {H, X} = 0. The proof of this is trivial, as it follows from the antisymmetry of the Poisson Brackets. First question: It remains a question on my mind whether every symmetry, which transforms q(t) into q(t) + k(q(t)) truly has a representation as X(q,p) on the phase space. In the Butterfield paper, they rely on recitfication to make q a cyclic coordinate, in which case the problem is trivial. Is there an easier way to see it in the general case? Second question: What about general non-symmetry transformations? Do they always have a representation X(q,p) on phase space? In other words, suppose I have a know L(q, q'), and p(q, q'). Suppose I define a transformation that changes q(t) into q(t) + k(q(t)), and therefore changes q'(t) into q'(t) + q'(t)dk/dq. Does there exist a corresponding X(q, p) on phase space such that dX/dq = -k(q) and dX/dp = q'(t) dk/dq (after changing variables from q' to p)


Different solutions for floating a DIY torii gate deeper than it's original floating level

I'm facing a dead end with my physics experience on a new build I am working on. Any help would be appreciated. Basic problem: I am trying to build a floating wooden Torii gate to be placed on water in a way that the structure's original floating level is pulled/pushed downwards in some way - Maybe 15-30 cm below surface. Reasons for a build like this: 1) It can't be there all year due to weather conditions 2) I want to go through it with my kayak 3) It's flipping cool! Questions: A) How can I achieve a STABLEish structure floating under water surface? I guess the best way would be to somehow pull it downwards with weights? How would YOU do it? B) How can this be calculated? C) Tips, tricks? Ask questions too. ![Mockup image](http://i.stack.imgur.com/svOEo.png)


Parallel capacitors without battery. Does charge flowing after a dielectric input?

If I charge two capacitors which are connected parallel $[$the minus (-) of the one opposite to the minus (-) of the other and the plus(+) of the one opposite to the plus (+) of the other.$]$, will I get current (charge flowing) after adding a dielectric material to one of them ?


Impedance formula for an edge coupled differential stripline in an asymmetrical stackup?

I've been searching and have not been able to find a formula for calculating the impedance of differential lines on inner layers where the dielectric above and below it is not symmetrical. I've seen plenty of examples of symmetrical stackups, but that would not help in my case. Does anyone know what the formula would be for this? Thanks! Edited for more information: For those that don't know, a stackup consists of all the different layers of a PCB (Printed Circuit BOard). You can have, for example, a 4 layber PCB which has individual copper material & glass epoxy stacked on top of each other. This allows the board to have more layers for routing the nets of the circuits.


How much energy would the Human Torch need?

I wasn't sure whether to put this in the SciFi section or here, but since I'm interested in a physical quantity,so I decided to ask it here. I was wondering about approximately how much energy would the Human Torch need to reach X temperature. I was wondering if $Q=mC\Delta T$ would be useful, but I'm not sure if there's such thing as a $C$ for human skin?


What are the theoretical prerequisites for experimental work with SEM and TEM?

What level of knowledge and in what areas would be required to perform experimental work with an SEM (Scanning Electron Microscope) and a TEM (Transmission Electron Microscope) respectively? E.g. would a strong background in quantum mechanics be necessary for EELS (Electron Energy Loss Spectroscopy) using a TEM? Assume that both the SEM and TEM machines are modern (made within the last 3 years).   Samples to be observed in the microscope would be metal alloys for materials science research.   Imaging methods of particular interest are EBSD and Surface Imaging for the SEM, and BSE, EELS, and Electron Diffraction for the TEM.


Why is Helium-3 better than Deuterium

I see that many websites and magazines with physics thematic are pretty excited about mining Helium 3 isotope on the Moon. But this seems to be a very hard-to-get resource. For more than one reason:   1. There is not too much of it on the moon. It actually seems to be 0.007% of the moon soil.   2. Transport to earth would be also very VERY complicated in my opinion.   3. People would be required to take care of the harvesters. Deuterium can be found in the seas as "heavy water". No spaceships needed. Now even when we get the Helium-3, I have a strong suspicion that it contains less energy than Deuterium but it will require more energy to start the fusion. ## Why I think it needs more initial energy to start the fusion?   * Deuterium contains one proton and one neutron. That makes a +1e core.   * Helium 3 has 2 protons and one neutron - That's +2 core. Now the worst problem with fusion reaction is, that once you pull the atoms so close, that the electron shells no longer shield the polarity of the protons they start repelling with electric force. Now if you're trying to pull +1 and +1 together it seems much easier than pulling +2 and +2 together. Just think about the standard equation: $F_e = k*\frac{|Q_1*Q_2|}{r^2}$ where $k$ is the permittivity constant and $Q_{1,2}$ are the proton charges. Repelling force must be **4 times as large**. ![enter image description here](http://i.stack.imgur.com/gORNB.png) ## Why I think it contains less energy? This is even simpler to explain. In the fusion process we merge smaller cores to create larger. We normally start with hydrogen and create helium. Now when we already have Helium fused, it means that we **missed one step** in the fusion. Why would we do that?   Can't actually Helium 3 core be one of cores that are created in Hydrogen fusion?


How can an an ant lift 50 times its own weight and pull 30 times its weight?

According to many sites like this one, an ant can apparently lift 50 times its own weight and pull 30 times its weight. Is it true? Can it be proved using physics? Though most sites agree that an ant can lift many times its own weight, not all agree with exactly _how many_ times its weight. The explanations provided if any are usually vague and do not use specific numbers. Can a specific numerical value be calculated? Secondly, how is it possible for the ant to pull 30 times its weight? I find it unbelievable. Can anyone explain this?


What makes an Antenna special?

The most important property of an antenna, beyond its conductivity(?) seems to be it's length (or more generally, its shape). But the antenna device is connected to the receiver by an arbitrarily-long connector. What is the fundamental difference between an antenna and the connector? Why doesn't the connector's length, straying from some 1/2, 1/4 length of the targeted band, interfere with the reception?


Boundary conditions for static electric field

Consider a surface that carries surface charge density. In electrostatics, boundary conditions are studied by showing that there is a discontinuity in the normal component of the electric field across the charged surface and that there is no discontinuity in the horizontal component. My question is aren't the electric fields at the surface always perpendicular to the surface? I mean where does the horizontal component come from if wherever we look locally, close enough to the surface, it looks like a flat infinite patch and there is no horizontal components!


The nature of time, according to quantum field theory

I will try my best to ask the question that best fits something I have been pondering on for a few days. > **Are virtual particles really constantly popping in and out of existence? > Or are they merely a mathematical bookkeeping device forquantum field > theory?** > > Gordon Kane, director of the Michigan Center for Theoretical Physics at the > University of Michigan at Ann Arbor, provides this answer. > > Virtual particles are indeed real particles. Quantum theory predicts that > every particle spends some time as a combination of other particles in all > possible ways. These predictions are very well understood and tested. > http://www.scientificamerican.com/article.cfm?id=are-virtual-particles-rea We define a second by the " _duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom_ " (from wikipedia). Or in other words, we define a second by movement of a certain amount of waves/particles. Since virtual particles pop in and out of existence, could you still claim that these things have a movement on the quantum level? If not, then how do you explain time/spacetime in the eyes of quantum field theory? And how does quantum entanglement fit in to this picture? I am not a scientist, so please forgive me if my understandings of these concepts are flawed.


Why do I still get static shocks when hands are wet?

Quite a few times now I have washed my hands, and while my hands are still wet I will touch something metal and get a static shock. I thought static electricity doesn't really build up when there's a lot of moisture, so that's why you don't get static shocks in the summer when there's more humidity. Even though it's winter now, I figured I wouldn't feel any static shocks if my hands are wet, but I still do.


seemingly impossible query

While on an expedition to a lake, it was noticed that the wind speeds gusted violently in seemingly periodic wakes and the proposed explanation was Von Karman vortex shedding off of nearby mountains that the lake was in the lee of. Therefore I am attempting to calculate the amount of influence that a Karman vortex street would have on some wind speed readings obtained during that time in order to see if I can detect the influence in wind speed readings collected during that time that the observation of intense gusts was made. From some lengthy research into this topic I have found a cadre of equations which I think give me the desired results of the frequency and maximum wind speed within the Karman vortexes all based solely upon wind speed readings from the lake.I however have come across an issue in the calculations which reduces all parameters to either constants or to be exactly directly proportional to the presumed oncoming wind speed, a highly unlikely and illogical occurrence. The problem rests in the fact that the laboratory calculated kinematic viscosity of air doesn't hold up to large scale observations of vortex street formation, thus necessitation a new parameter as stated in this article. This new parameter is β and is defined as $$β=St/Re$$ with $Re$ being the Reynolds number and $St$ being the Strouhal number. Due to the fact that I don't have the shedding frequency but need to calculate it, I cannot calculate St except through the formula $$St = .212(1-\frac{21.2}{Re})$$ as found in this article. However, doing this and substituting into the parameter of $β$ yields a relationship between the constant $β$ and $Re$, which makes $Re$ constant and in turn $St$ constant as $St$ is defined in terms of $Re$ only. This causes all the parameters that I have to be either linear with respect to any oncoming flow velocity or constant, something obviously wrong. Although I have everything that I need after this point in order to obtain maximum wind speed and shedding frequency, this initial step has stopped me dead in my tracks due to the fact that every author whom I have seen on this subject had a picture to work with to calculate the distances between the vortices and use that and an assumed wind speed to calculate shedding frequency, plug that in to find Strouhal number and then use the $β$ parameter to calculate Reynolds number. All those working in lab tests similarly calculated shedding frequency and recorded it by video taping or counting the eddy shedding process. I, however, only have the wind speed to work with and so have problems by reducing out too many variables. My question therefore becomes how do I define a either a variable $β$ parameter or redefine either my $St$ or $Re$ number so that one can be calculated independently with the limited information at my disposal of only the wind speeds and the ability to assume other parameters such as landmass dimensions etc. Thank you for your time reading this.


Stable equilibrium given force

If a particle moves under the influence of a resistive force proportianal to velocity and a potential $U$, $$F(x,\dot x)=-b\dot x-\frac {\partial U}{\partial x}$$ Where b>0 and $U(x)=(x^2-a^2)^2$ My thoughts were to make $F=0$, which would result in: $$\dot x= \frac 1b(4x^3-4a^2x)$$ This means that the points of equilibrium are a function of time? EDIT:: So I set up as suggested, using $F=m\ddot x$ to get $$0=m\ddot x+b\dot x+4x(x^2-a^2)$$ I am trying to solve for $x(t)$. I know how to solve the equation that has only x, but I do not know how to solve for the x^3 term. I am curious how to combine these to get x(t)


Inductance of loop to a square wire? What is the induced current and the induced $\bar E$?

_I am practising for the elementary-physics -course Tfy-0.1064, this puzzle is a simplified version of the problem 3Ahere (sorry not in English). My basic idea here is to understand a scenario with one loop and then understand the solenoid case._ Suppose a circle around the square of 20-cm-side and $\frac{dB}{dt}=35\frac{mT}{s}$ in the circle, please, see the picture below. What is the induced current and the induced $\bar E$ to the square by the circle? ![enter image description here](http://i.stack.imgur.com/CoIpB.jpg)


vector-scalar gravity

Has anyone developed a vector-scalar theory of gravity using an acceleration field equation that is based on a ratio in which the numerator is the familiar Newtonian vector sum of all masses times the inverse square of their respective distances, and the denominator is the scalar (not vector) sum of the very same formula? As a scalar sum, the denominator would always be larger than the numerator because there is no cancellation from opposite directions, and the quotient could be interpreted as the proportion of "directionality" of the combined contributions. Or has anyone developed a relativistic theory based on the same ratio of vector to scalar sum of the same contributions?


An Equality in The Calculus of Many Instantons

I am reading the review on instantons. When I tried to derive formula (2.27) on page 17, I always get the different coefficient of $gF_{mn}$ term. My calculation is just directly expanding the first term $-\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn}$ in action to quadratic form, and leaving the second term $i\theta k$ invariant. Under the fluctuation $A_n \to A_n +\delta A_n $, the first term $-\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn}$ becomes $$ -\frac{1}{2} \int d^4x {\rm{tr_N}} F^2_{mn} \\\\-\frac{1}{2} \int d^4x {\rm{tr}_N }\left( 2(D_n\delta A_m)^2 -2D_n\delta A_mD_m\delta A_n +2g F_{mn} [\delta A_m,\delta A_n] \right) +...$$ Integrating by parts and disregarding the surface term, the quadratic term becomes $$ \int d^4x {\rm{tr}_N }\left( \delta A_m D^2 \delta_{mn}\delta A_n -\delta A_mD_nD_m\delta A_n+2g F_{mn} \delta A_n \delta A_m \right)$$ Using the equalities $D_n D_m=D_mD_n-gF_{mn}$, $D_n\delta A_n=0$, the quadratic form becomes $$ \int d^4x {\rm{tr}_N }\delta A_m\left( D^2 \delta_{mn} +g F_{mn}+2g F_{mn} \right)\delta A_n $$ So the coefficient of $gF_{mn}$ term is 3. What is wrong with the above calculation?


"Conceptualizing" neutrinos

Layman here. EE and BS physics. I am "content" in viewing photons/electromagnetic radiation as an "emergent" property of spacetime? due to the electrons ("particles..?") and all their activity jumping around in atoms. [or is this likely too simple?] ... in the "same" manner that sound and water waves are an "emergent" property of air and water particles. OK. So WHAT do neutrinos represent? Can I visualize them as an emergent property of neutrons and protons [and quarks?] and all their activity in the nucleus? Neutrinos are a real mystery to me, as if everything else weren't!


Centripetal acceleration for particle that moves in a circle at non-constant speed

How is $a_{R}=v_T^{2}/r$ derived for changing tangential speed? ($a_R$-centripetal component $r$-distance from axis $v_T$- tangential speed)


Right-angle lever paradox in special relativity

I remember to have read somewhere an interesting special relativity "paradox" considering two perpendicular rods $A$ and $B$ of equal proper length $L$ fixed at point $O$. ![rest frame](http://i.stack.imgur.com/ULxzd.png) In the "rest" frame equal (in magnitude) forces are applied on the edges of the rods, perpendicular to them, thus giving zero net angular force: $$-F_AL_A + F_BL_B = -FL + FL = 0$$ (assuming $z$-axis is towards us). In the frame moving parallel to rod $B$ with the velocity $v$ it's length is diminished due to Lorentz-contraction $$L_B' = L \sqrt{1-v^2/c^2}$$ and perpendicular component of the force is diminished also $$F_B' = F\sqrt{1-v^2/c^2}$$ while $L_A' = L$ and $F_A'=F$. So the net angular force in this frame is not zero: $$-F_A'L_A' + F_B'L_B' = -FL + FL (1-v^2/c^2) = -\frac{v^2}{c^2} FL$$ and the system should be rotating clockwise with angular acceleration. I don't remember the explaination of the paradox as I did not understand it at that time, but now it seems to be very simple: even in Newtonian Mechanics total angular force would not be zero in "moving" frame given the fact that the net force is not zero. If we take into account that rods are _fixed_ relative to $O$, we should assume that in the junction point $O$ there is an additional force which compensates the forces on the edges of rods. The same argument seems to be valid in the relativistic case too. However, I remember that the explaination was not that simple. Am I wrong with my explaination, or the statement of the paradox is wrong? Does anyone know similar "paradox"?


What do theoretical physicists need from computer scientists?

I recently co-authored a paper (not online yet unfortunately) with some chemists that essentially provided answers to the question, "What do chemists need from computer scientists?" This included the solution of theoretical problems, like combinatorial enumeration and the sampling of certain classes of graphs; and practical programming problems, like open-source implementations of algorithms that are currently only implemented in expensive software packages. This motivates me to ask: what about this field? Are there theoretical issues of combinatorics, algorithm analysis, that physics needs a theoretical computer scientist to solve? Or how about creation of practical tools that would allow a theoretical physicist to do a better job: "If only I had a program that would solve this type of problem for me!" Intended as community wiki.


Particle in a 1D box (momentum representation)

I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box. We know that the wave function in the position space is: $$Y_n(x) = A\sin{(n\pi x/L)}$$ Well, if I write the Schrödinger equation in the moment space I have: $$\frac{p^2}{2m}Y_n(p) = E_nY_n(p)$$ So, this equation doesn't give me any information about the wave function $Y_n(p)$ I know that I can solve this problem just using the Fourier tranformation, but I'm asking myself if there is another posibility to solve this problem. PD. If I use the Fourier transformation, do I have to integrate just from $O$ to $L$?


Balmer lines and their positions

I have two stars, both of which have Balmer lines at the same wavelength positions. The depths of the Balmer lines on each star are different, and the stars are different as well. Why does this pattern exist?


Compute an aerodynamic shape

I'm interested in building a Fairing for a recumbent bicycle. A fairing is just a rigid curved surface (usually plastic or fiberglass) that surrounds the bike in order to reduce wind drag. Apparently it can increase your speed dramatically (as much as 300% with zero wind relative to the ground). Questions of building the thing aside (which looks feasible actually), how can I compute the "most" aerodynamic surfaces to enclose a given space? In other words, if I have a given three dimensional "blob" (my body's range on motion + the bike itself) and a given direction of motion, what formulae and/or software can I use to give me some highly-aerodynamic shapes that are just slightly larger than the blob?


Deriving the transformation under Weyl rescaling in Polchinski eq. (1.2.31)

I have another question in Polchinski's string theory book volume 1, namely how to derive Eq. (1.2.32)? $$(-\gamma')^{1/2} R'=(-\gamma)^{1/2} (R-2 \nabla^2 \omega) \tag{1.2.32}$$ I have awared his Eq. (1.2.21) for the transformation of world-line metric $$\gamma_{ab}'(\tau,\sigma)=\exp(2 \omega(\tau,\sigma )) \gamma_{ab} (\tau,\sigma)$$ , the definitions of Ricci scalar, $R:=R_{\mu \nu} g^{\mu \nu}$, Ricci, and Riemann tensors. It is still far from obvious how to construct the transformation of $R'$ from metric->Christoffel symbols->Riemann->Ricci-> (1.2.32) My question is, how to derive Eq. (1.2.32)?


Conversion of QCD cross section formula

I'm writing a program to calculate NLO cross sections for semi-inclusive high-$p_T$ pion production in proton-proton-collisions for my bachelor thesis. I've got a paper describing the production cross section as function of the differential cross sections of the partons $\frac{d\sigma}{dv}$ and $\frac{d\sigma}{dv dw}$ where $v = 1+\frac t s$ and $w = - \frac{u}{s+t}$, with $s$,$t$,$u$ being the conventional mandelstam variables. And I found a paper tabulating all the partonic cross sections. But unfortunately they use the formula $$d\sigma = \frac{1}{32 \pi^2 s} \frac{d^{n-1}p_3}{p_3^0} \frac{d^{n-1}p_4}{p_4^0} \sum M_{fi} \delta \left(p_1+p_2-p_3-p_4\right)$$ How do I calculate the conversion from $\frac{d\sigma}{d^{n-1}p_3 d^{n-1}p_4}$ to $\frac{d\sigma}{dv}$? I know in principle how to deal with differentials, and that the conversion should somehow begin with $\frac{d\sigma}{dv} \frac{dv}{d^{n-1}p_3 d^{n-1}p_4}$ but I don't have a good ansatz how to deal with the posterior part...


Why wet is dark?

When something gets wet, it usually appears darker. This can be observed with soil, sand, cloth, paper, concrete, bricks ... What is the reason for this? How does water soaking into the material change its optical properties?


Time travel to the past in binary black hole system

I just saw this YouTube video (only 3 mins long) of Neil deGrasse Tyson explaining what happens when two black holes collide. He says that when the black holes are rotating around each other, there exists a path that you could follow that would cause you to travel back in time. Any chance that someone could explain this in an intuïtive way to me? I just finished my 1st bachelor physics, to give you an idea of my knowledge. I also understand the basics of special relativity.


Are temperature increases discrete?

Are temperature increases ever discrete in nature, or is it a continuous variable? If a discrete case exists, is there any material that exhibits particularly strange behavior?


Enthalpy Change in Reversible, Isothermal Expansion of Ideal Gas

For the reversible isothermal expansion of an ideal gas: $${∆H}={∆U}=0 \tag1$$ This is obvious for the case of internal energy because $${∆U} = \frac {3}{2} n R {∆T} = 0 \tag2$$ and $${∆U} = -C_P n {∆T} = 0 \tag3$$ For the case of enthalpy it is easy to see that $${∆H} = -C_v n {∆T} = 0 \tag4$$ I've also seen $${∆H} = ∆U + ∆(PV) = ∆U + nR{∆T} = 0 \tag5$$ Now for the part I don't understand. $$dH = dU + PdV \tag6$$ $$dH = dU + nRT \frac {dV}{V} \tag7$$ $${∆H} = {∆U} + nRT \ln\frac {V_2}{V_1} \tag8$$ $${∆H} = 0 + nRT \ln\frac {V_2}{V_1} = \ln\frac {V_2}{V_1} ≠ 0\tag9$$ Clearly, it is incorrect to make the substitution $ P = nRT/V$ in going from $(6)$ to $(7)$. Why is that? I thought equation $(6)$ was always valid, and integrating such a substitution should account for any change in the variables throughout the process. Why does this not yield the same answer as $(4)$ and $(5)$?


Can parallel universes constitute the missing mass aka dark matter?

I was pondering about this question for a while now. And with my limited knowledge of physics I can answer myself - if parallel universe theory is true and if indeed there is gravitation leakage from 1 universe to the other, there would be infinite number of universes and thus unimaginable amount leakage would occur making life or formation of any matter impossible. Does that sound like a plausible answer? :)


How can I see this equation describes advection?

My notes say that this is the equation for the "advection of a quantity **A** at speed **v** ": $$ \frac{\partial \mathbf{A}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{A}) .$$ Is this true? How can I see (mathematically) that this equation corresponds to advection?


Relation for Dirac-spinors of different helicities

Assume that we have massless spin-1/2 particles. The Dirac-spinor is the solution of the Dirac equation: $$ p^\mu \gamma_\mu u_\pm(p) = 0, \quad p^2 = 0$$ The subscripts $\pm$ denote two different solutions, belonging to two different helicities. Is it possible to find a representation for the $\gamma_\mu$ so that the following relation holds true? $$ u_+ = \left( u_- \right)^* $$ I know this possible for (polarization) vectors, i.e. one may choose $\epsilon^\mu_+ = \left( \epsilon^\mu_- \right)^* $, but I guess it is not for spinors (in the weyl representation it obviously is not). So here is my **Question:** Is it possible to find a representation of the Dirac-Gamma- matrices so that the spinors of different helicities are related by complex conjugation?