Parseval's Theorem on a Random Signal

NB - I'm re-posting this question in physics because I haven't had any luck getting a response from the maths StackExchange site - it's a rather applied problem so is probably better suited here anyway. ** I'm struggling with Parseval's Theorem. I'm trying to relate variation in the time domain to the average value in the frequency domain. To do this, I'm performing the Fourier Transform on an arbitary random signal that I've generated with 2048 points (although the graph below only shows 100 of them) and a standard distribution (in this particular case) of 0.58: ![Time domain trace of random signal - NB only showing 100 of 2048 points!](http://i.stack.imgur.com/O0zlF.png) The frequency domain, after an FFT, then looks like this: ![Frequency domain trace of random signal](http://i.stack.imgur.com/coWig.png) with an average value of 0.022 - although this depends on the number of samples used. Now when I try to apply Parseval's Theorem, where: $\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{n=0}^{N-1} |X[k]|^2 $ I run into a problem. When I do the summations, I get 678 for the time domain and 0.662 for the frequency domain. When I apply the factor of $1/N$ I end up with 0.0003 for the frequency side - clearly a long way off! Obviously I'm going wrong somewhere, but can't see where. This many orders of magnitudes off isn't very encouraging... Thanks for your help!


(Exercise review) Find the moment of the electric dipole

I have solved an exercise, but the result I have obtined is wrong and I can't understand why. If you can help me, I'll be so grateful. Let's consider a charged cylinder, its radius is equal to R and its height=4R. Its center of is in the origin of the cartesian axes and its axis is parallel to z axis; we know the volumic density of charge: $\rho= 2az \epsilon$ where $a, \epsilon$ constant $>0$. There is an electric field inner to the cylinder, described by: $E_{ox}=0, E_{0y}=0, E_{0z}=az^2$ We want to find the electric potential to a great distance from the origin of the axes. I have thought as follow. Using Maxwell's first equation, I have found the volumic density of charge: $\rho=2\epsilon_0 a z$. At a very big distance to the origin, I can consider the cylinder like an electric dipole, with the positive charge concetrated in the center uf the upper part of the cylinder and the negative one in the center of the lower part: So $Q_+$ will be in (0,0,R) and $Q_-$ in (0,0,-R). Now, I need the moment of dipole, ${\bf p}=q{\bf a}$ where ${\bf a}$ is the vector from the negative charge to the positive charge. I have integrate $\rho$ and I have obtained $q=\int_0^{2R}\int_0^R\int_0^{2\pi}2az\epsilon r d\theta dr dz$ and I have obtained $q=4\pi a \epsilon_0 R^5$ Then I have posed a=2R, so I have obtained $p=8\pi a \epsilon_0 R^4$.The correct result is $p=32/3 \pi a \epsilon_0 R^5$ I can't understand where I was wrong and I wouldn't like to have made an error of proceeding.


Black bodies, energy per unit volume

While reading about black bodies, I found that the average energy per mode per unit volume is $KT$ classically, but I don't understand why this is per unit volume. The linear oscillators are considering to be in the walls of a cavity modeling a blackbody, so shouldn't it be modes per area?


Newtonian gravity vs. general relativity: exactly how wrong is Newton?

Is there a simple function I can use to describe the difference between simple Newtonian dynamics and the actual observed motion? Or maybe some ratios for common examples of, say, the motion of stars and planets? I know for example that Newtonian physics is sufficient to fire a rocket at the moon, so the error there must be miniscule. What exactly is that ratio, and could a layperson calculate it generally for other bodies? The context is this: in a debate regarding how drastically wrong science can be, I want to make the case that even though Newton was wrong regarding gravity in the bigger picture, he was only ___ % wrong regarding what he'd observed. I know science can often be wrong, but I want to emphasize the fact that our system of observation isn't fundamentally dysfunctional.


Max. speed that a space probe can communite with the Earth?

**Question** : _Which is the average or maximum speed of a telematically sent message that a space probe can send_? **Is as simple as radio waves travel at the speed of light in a vacuum?** **Context** : I have been asked to make a first estimation about how much time is needed to send a space probe and then receive the signal. I am supposing that the space probe is sent at $15 \ km/s$ but I have not idea of the telematic signal speed. Thank you in advance!


Quadratic versus other types of degrees of freedom

In my text book I have the following question and after thinking about this one for hours, I can't figure out what they're getting at: In Section 2.5* I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of such a system is proportional to $U^{Nf/2}$, where $Nf$ is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula $\Omega$ cannot be valid when the total energy is very small? * * * (*) = In section 2.5 of the text book, the formula for multiplicity $\Omega$ is given to be $$\Omega\left(U,V,N\right)=f\left(N\right)V^NU^{3N/2}.$$ with $f\left(N\right)$ some function of $N$. (This formula is for three degrees of freedom.) * * * What I got so far is: The initial expression (for $f$ degrees of freedom) is $$\Omega_N\approx\frac{1}{N!}\frac{V^N}{h^{3N}}\frac{\pi^{Nf/2}}{\frac{Nf}{2}!}\left(\sqrt{2mU}\right)^{Nf}.$$ Using $S=k\ln\Omega$ and solving $T=\left(\partial S/\partial U\right)^{-1}$ for $U$. the final answer for $U$ becomes $$U=\frac{1}{2}NfkT.$$ (Which is really just the equipartition theorem all over again???? Or did I make a mistake?) Now, I don't get why this formula can somehow tell me that my initial formula for multiplicity is not sufficient when the energy is small:   * Both $N$ and $k$ are constants.   * Although $f$ sort of "jumps" per the rule of Dulong and Petit, assuming that more degrees of freedom freeze out would seem baseless to me. Do I have to treat this as a constant as well?   * From this I conclude that $U$ is proportional to $T$, which really doesn't seem so problematic to me. How do I have to interpret the role of the quadratic vs. the non-quadratic degree of freedom here? What is the consequence?


Changes in planetary orbit?

Forgive a novice and her naive curiosity, but I was wondering about planetary orbit and the aging of planets. As a **planet ages** , does its transit around its sun/star **decrease** or **increase** in **speed**? ## Does the speed and proximity of a planet's orbit around the star, have more to do with the age and type of the star, and less to do with the age of the planet and its distance from the star? Also, is it possible for a planet to be **pushed** or **pulled** closer to its star, into a closer orbit, for example by increased gravitational pull of the star, or the planet being knocked off orbit by a comet or asteroid, or even a shockwave from an implosion -- one far enough away so that no debris comes near, but near enough that the shockwave is powerful enough to knock the planet. I realise such an occurance would probably cause catastrophic upset on the planet, but if it remained relatively intact, could it find itself on a new path around its star??   Or could the comet/asteroid/shockwave hit the planet from the opposite direction of its transit, perhaps glancing it at such an angle as to cause it to slow in pace around the star, or to turn faster or slower on its own axis? And would its poles be reversed? Living in a world with sci-fi like **Doctor Who** , **Eureka** , **Stargate** , **Star Trek** , **X-files** , **Fringe**.. it makes you wonder what is **theoretically possible**. Often the explanations they give sound **far- fetched, yet plausible**. Just as every myth evolves from a truth, you tend to wonder how much truth exists within these stories. Hope one of you can shine a light on my wonderings and find a kernel of truth among them.


Are "phase-specific" masks ever used in the context of double exposure lithography? Can these sorts of things exist?

I was recently reading an 08' paper on double exposure lithography: and I was wondering if there existed _some_ material that could be used to create a mask for e.g. a 193 nm light source that was _only opaque to light with a specific phase_ $\theta$. The idea, in the context of double exposure lithography, would be to overlay two masks, one opaque for light with some phase $\theta_a$ and another opaque for light with some phase $\theta_b$, where (in radians) $$\theta_a = \theta_b + \pi\cdot n$$ where $n$ is any non-zero integer value. Thus, one could just phase-shift the source of coherent light used for lithography by $\pi$ radians to pattern with one mask and then the other. This would save an alignment step for the substrate being patterned. Is there any precedent for this sort of thing? Or is this question nonsense?


Doing a Gaussian Integral

When you integrate over p you get: ![enter image description here](http://i.stack.imgur.com/yEUbh.png) by using ![enter image description here](http://i.stack.imgur.com/9gXXP.png) What are the steps to this? Do you integrate by parts?


Why aren't units with powers, like cm³, surrounded by parentheses?

Since $\renewcommand{\unit}[1]{\,\mathrm{#1}} 1\unit{dm} = 10^{-1}\unit{m}$, it follows that $1\unit{dm^3} = 10^{-1} \times 10^{-1} \times 10^{-1} \unit{m^3} = 10^{-3} \unit{m^3}$. However, in regular mathematics the following equation holds true: $$a\,b^{3} = a\,b\,b\,b$$ By the above, the cube unit should expand as follows: $$\mathrm{dm^3} = \mathrm{dmmm}$$ While in actual usage (as seen in the second equation) the expansion is $\mathrm{dddmmm}$, which would arise from using $\mathrm{(dm)^3}$ instead. $$\mathrm{(dm)^3} = \mathrm{dddmmm}$$ So shortly: why aren't parentheses (commonly?) used in units?


Is the 74.2km/sec per megaparsec accidental?

The rate of expansion of the universe is 74.2km/sec per megaparsec The age of the universe is 13.798 bln years That places the furthest theoretically observable edge of universe at 13.798 bln light years away - this is how long light would take to reach us from there, and there were no light sources before that date. Now, how fast does the observable part of the universe expand? A bit of unit conversion and multiplication: 13798000000 light years in megaparsecs = 4 230.39567 megaParsecs [1] 74.2km/sec in light years/year = 0.000247504559 light years / year [2] Multiply the two, and I get the expansion rate: 1.04704221 light years per year per distance to the observable edge of the universe. Speed of 1 light year per year is the speed of light. The 0.047 seems awfully small. Probably within error margin of the universe expansion speed and age of the universe estimates. Is this a coincidence, or is this somehow bound - and if so, what are the consequences?


Integrating the divergence of a velocity field (Fluid Mechanics)

I'm looking more for verification of my answer as I'm not 100% sure how valid my integration is here: If I have a 2-D incompressible fluid velocity field given by $\vec{U}(x,y,t)=u(x,t)\vec{x}+v(y,t)\vec{y}$, I know that $\nabla \cdot \vec{U} = 0$, call this eqn (1), by the definition of incompressibility. In order to find out more about the velocity components $u(x,t)$ and $v(y,t)$, i have been told to integrate eqn (1). Is the following integration w.r.t. space valid?: \begin{align} &\int \nabla \cdot \vec{U} \mathbb{d} r = \int 0 \\\ &\int \left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) \mathbb{d}r = C = \text{const.} \\\ &\int \frac{\partial u}{\partial x}dx+\int \frac{\partial v}{\partial y}\mathbb{d}y = C \\\ &u(x,t) + v(y,t) = C(t) \end{align} In other words, the magnitude of the sum of the fluid velocity in the $x$ and $y$ directions are constant everywhere in the fluid at any instant in time? Is there a better way of obtaining more information?


Tuned Mass Damper

I was learning about the different applications of resonance and one of them is the tuned mass damper used in buildings. One thing I am confused about is as to why the mass attached to the building oscillates $180^\circ $ out of phase when in resonance. Is there any mathematical reason behind this or an intuitive way of understanding the concept?


Determine the acceleration in a double Atwood machine

Trying to solve the following double Atwood machine: > _Suppose there is a mass of 12kg hanging on an ideal rope that wraps around > an ideal pulley, and that the other end of the rope is attached to another > ideal pulley's axis. This second pulley has a rope around it which on one > side holds a mass of 4kg and on the other a mass of 8kg. What is the > acceleration of the first mass?_ So here's what I've done so far: First I've determined that the tension in the first rope is twice the tension in the rope for the "subsystem", and that the acceleration of the first mass is the average of the accelerations of the two other masses but in the opposite direction. That is to say $$T' = T/2$$ $$a_{1}=-\frac{a_{2}+a_{3}}{2}$$ From free-body diagrams we can determine that $$-12a = T-12g$$ $$4a_{2} = T/2 - 4g$$ $$-8a_{3} = T/2 - 8g$$ However, when I solve this system I obtain $T=32a$ which seems impossible that the tension would be greater than the greatest force of gravity of any mass. Part of me wonders if this has anything to do with it: How do I know which direction the third mass is traveling? The first mass is causing the subsystem to accelerate upwards but the third mass is accelerating downwards relative to the subsystem. So how do I know the direction of acceleration relative to the ground?


Electromagnetic radiation and quanta

since electromagnetic radiation possess the property of both wave and particle(photon). and both theory are applicable but how we have to find out that which theory is suitable or applicable in particular explanation. for example in laser traping of atom we use photon concept rather than wave why?


CFT Entanglement Entropy - relation between translations and the stress-energy tensor

In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric: $$ ds^2_{H_q^{d-1}} = d\tau^2 + du^2 + \sinh^2 u \; d\Omega_{d-2}^2 $$ Here $d\tau$ is probably a Wick-rotated time, $u$ is a radial variable and $d\Omega$ is the spherical area measure. Then they define a partition function $ Z_q = \mathrm{tr}(e^{-2\pi q H_\tau}) $ where $H_\tau$ "generates translations along the $S^1$". What does that mean? Could it mean this? $$ H_\tau = \frac{d}{d\tau}$$ This is the generator for translations along the $\tau$-axis. However, they also say this is related to the stress-energy tensor: $H_\tau = \int_{\mathbb{H}^{d-1}} dx^{d-1} \sqrt{g} T_{\tau\tau}$ This seems like a very complicated way of describing translations. Could there be another meaning for the phrase "generates translations along $S^1$? * * *   1. Jeongseog Lee, Aitor Lewkowycz, Eric Perlmutter, Benjamin R. Safdi   Renyi entropy, stationarity, and entanglement of the conformal scalar


Can the Kramers–Kronig relation be used to correct transfer function measurements?

In experimental physics, we often make measurements of linear transfer functions; these are complex-valued functions of frequency. If the underlying system is causal, then the transfer function must be analytic, satisfying the Kramers-Kronig relations. Our measurements, however, are corrupted by random (and perhaps systematic) errors. **Is it possible to improve a measurement of a linear transfer function of a causal system in the presence of noise by applying some kind of constraints derived from the Kramers-Kronig relations?**


Which of these two textbook equations of geodesic deviation is correct?

My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner _et al_ 's _Gravitation_ (p34) gives the geodesic deviation equation as$$\frac{D^{2}\xi^{\alpha}}{D\tau^{2}}+R_{\phantom{\mu}\beta\gamma\delta}^{\alpha}\frac{dx^{\beta}}{d\tau}\xi^{\gamma}\frac{dx^{\delta}}{d\tau}=0,$$ with the right-hand side $\xi$ index $\gamma$ equal to the _second_ lower index on the Riemann tensor. Lambourne's _Relativity, Gravitation and Cosmology_ (p185), on the other hand, gives $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\alpha\beta\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ with the right-hand side $\xi$ index $\alpha$ equal to the _first_ lower index on the Riemann tensor. My question is, which of these two equations is correct? I tried to answer this question myself by using the two equations to calculate the geodesic deviation on the surface of a unit 2-sphere. With Misner's equation (substituting $\lambda$ for $\tau$) I got $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}$$ and $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=\xi^{\theta}\left(u^{\theta}u^{\phi}\right)-\xi^{\phi}\left(u^{\theta}u^{\theta}\right).$$ You can see my calculation on my previous question Geodesic deviation on a 2-sphere - is this the right track? With Lambourne's equation I got $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=0$$ and $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=0.$$ This didn't seem right to me so I concluded that Lambourne's equation is incorrect.


Polarization vectors of massive and massless particles

I read from Mandl & Shaw that when quantizing massless vector particles such as photons in Lorentz gauge, there are 4 linearly independent polarization vectors (2 of them being able to "gauged away"), while when quantizing massive vector particles such as W bosons, there are only 3 linearly independent space-like polarization vectors. It also says that when W and Z gauge bosons "eat" the Goldstone bosons from Higgs field, they become massive and acquire longitudinal polarization. So apparently there is some rule that says massive vector particles have longitudinal polarization, while massless particles don't. Why is that? From what principle is it derived? Their Euler-Lagrangian equations are essentially the same (both look like Klein-Gordon equation), just one of them has $m=0$. Is it the case that when $m=0$, it has four linearly-independent solutions while only three when $m\neq0$ ?


Initial pressure of explosion

I would like to know, if given a TNT equivalent of explosive (or its Energy) is it possible to determine what would be the initial pressure that will be the result of that explosion ? To be more precise let's say I have a vessel and given the vessel volume can I determine pressure inside that vessel (assuming vessel won't break).


CPT symmetries for a free Klein-Gordon equation and in minimal coupling

I'm studying for an exam on relativistic quantum mechanics and one of the issues to prepare is about symmetries of Klein-Gordon equation concerning $C$, $P$, $T$ transformations for a free field, and with an electromagnetic interaction. So could you help me out a bit? As I understand the general free solution can be written as: $$ \Psi(\mathbf{x},t) = \int\limits_{ \mathbb{R}^3 }\frac{d^3p}{(2 \pi \hbar)^3} \frac{1}{2 E_p} \left( A_\mathbf{p}^{(+)}e^{-\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \+ A_\mathbf{p}^{(-)}e^{\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \right) $$ where: $E_p := \sqrt{\mathbf{p}^2c^2-m^2c^4}$ For a special solution (in a box): $$ \psi^{\pm}_{\mathbf{p}}(\mathbf{x},t) = \frac{1}{\sqrt{V}} e^{\mp\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar} \mathbf{p} \cdot \mathbf{x}} $$   * $P$ transformation gives: $P\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{-\mathbf{p}}^{\pm}(\mathbf{x},t)$,   * $T$ transformations gives: $T\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{\mathbf{p}}^{\mp}(\mathbf{x},t)$ and   * $C$ transformations gives: $C\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{-\mathbf{p}}^{\mp}(\mathbf{x},t)$ //Corrected a mistake thanks to a comment by TwoBs So what conclusion (if any) I can draw from this about the general solution? And do the conclusions stay valid for a KG equation with minimal coupling ($\partial_{\mu} \rightarrow D_{\mu} := \partial_{\mu} + \frac{i e}{\hbar c}A_{\mu}$)?


Low frequency electromagnetic waves

Some frequencies of electromagnetic waves are used for transmission of information, like radiowaves, microwaves, light, but some not.. What about Low Frequencies?: Perhaps low frequencies aren't used for transmision simply because the waves haven't got enough energy to reach destiny, but is well known that if you go near a high voltage transmission tower, you can turn on a fluorescent lamp http://www.electric-fields.bris.ac.uk/ Perhaps the intensity of the high voltage gives more photons to interact there, than in a home installation, but the fact is waves are traveling anyway at 50/60 Hz without need any modulation. There is a limit on how low a frequency can be to put photons into travel?


Is the charge of an ion evenly distributed?

This question relates to: Gauss' law and ions? Is the charge distribution in an ion spherically symmetric due to quantum mechanical effects or do we assume it when using Gauss's law, as in the linked question, to make the calculation easier? I think it is the former but I am not sure.


Can we multiply an electromagnetic field?

![photomultiplier](http://i.stack.imgur.com/vNvVw.jpg)   * can we multiply light by a sequence of reflections in a closed vessel?   * also, is light a part of EM wave?   * can we multiply the radio wave or EM wave from a transformer?


For semi-insulating GaAs is the bandgap bigger or smaller than the pure GaAs?

The carrier concentrations are lower in semi insulating semiconductors, therefore, can we say that their bang gap is wider in terms of energy (eV)? If they have wider band gap how much bigger is it approximately and what parameters affect it? i.e. is temperature one of them?


Reason for strange "canyons" of dust on the cooling platform

I took apart my cooling platform Something like this :![Cooling Platform](http://i.stack.imgur.com/kIPFr.jpg) And I found something like this inside : ![Canyon of dust 1 ](http://i.stack.imgur.com/IyZes.jpg) ![Canyon of dust 2 ](http://i.stack.imgur.com/4yX7K.jpg) ![Canyon of dust 3 ](http://i.stack.imgur.com/udFJ5.jpg) I wonder why around the hollows are the "canyon like" structures. PS: I also drop this platform on the floor, maybe this could be a reason ? EDIT : Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in plastic itself. It is dust on the ON the plastic. Proof : ![Dust canyon](http://i.stack.imgur.com/SCmXW.jpg) Completely wiped out with my finder : ![Dust canyon wiped](http://i.stack.imgur.com/my18U.jpg)


Representations of gamma matrices

I have to do this exercise for homework. Find a representation of the gamma matrices unitarily connected to the standard representation for wich the spinors $u(p)$ that satisfy the equation $(p_\mu \gamma^\mu - m)u(p)=0$ of motion for a generic momentum $p^2=m^2$ have only the first two components different from zero. I tried to find the transformation $V$ such that $Vu(p)$ meet the demand but was not able to complete my idea. Can anyone help me?


Coordinate system

**Quoting from 'Nuclear Physics - Theory and Experiment' by RR Roy, BP Nigam 2005 edition** Link to text * * * > How did the author arrive at equations (23a, 23b,23c)? * * * _Chapter 8 Nuclear model II, 8.7 **Quadrupole Deformation** , page 286_ In the body-fixed reference frame in which the coordinate axes coincide with the principal axes, we denote the deformation parameters $\alpha_{2\mu}$ by $a_{2\mu}$. The relationship between deformation in the two coordinate systems is $$\begin{align} \sum_\mu a^*_{2\mu}Y_{2\mu}(\theta,\phi)&=\sum_\nu a^*_{2\nu}Y_{2\nu}(\theta,\phi)\\\ &=\sum_\nu a^*_{2\nu}\sum_\mu D^2_{\mu\nu}(\theta,\phi,\psi)Y_{2\mu}(\theta',\phi') &(22c) \end{align}$$ so that $$\begin{align} \alpha_{2\mu}&=\sum_\nu a_{2\nu}D^{2*}_{\mu\nu}(\theta,\phi,\psi) \qquad\qquad\qquad\qquad\qquad\qquad&(22d) \end{align}$$ Since, in terms of the principal axes, the product of inertia is zero, we define the following $$\begin{align} a_{20}&=\beta\cos\gamma \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad&(23a)\\\ a_{21}&=a_{2-1}=0&(23b)\\\ a_{22}&=a_{2,-2}=\tfrac1{\sqrt 2}\beta\sin\gamma&(23c) \end{align}$$ where $\beta$ and $\gamma$ are new parameters whereby the $a$'s are defined. The deformations $\delta R_j$ along the principal axes $j=1,2,3$ (nuclear body-fixed axes) are obtained from $$\delta R(\theta,\phi)=R_0 \sum^2_{\mu=-2}a^*_{2\mu}Y_{2\mu}(\theta,\phi)$$ * * * > and how can we write the equations $\delta R_1$? * * * _page 287_ and are given by $$\begin{align} \delta R_1\left(\frac\pi2,0\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma-\frac{2\pi}3\right)\\\% i am so depressed \delta R_2\left(\frac\pi2,\frac\pi2\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma-\frac{4\pi}3\right)\\\ \delta R_3\left(0,\phi\right)&=\sqrt{\frac5{4\pi}}\beta R_0\cos\left(\gamma\right)\\\ \end{align}$$ * * *


Nillson Potential

We define a Hamiltonian to derive Nilsson potential, $$\hat{H}=-\frac{\mathbf{\hat{p}}^2}{2m}+ \frac{1}{2}m\left[\omega_\bot^2(\hat{x}^2+\hat{y}^2)+\omega_z^2\hat{z}^2\right]$$ Nilsson model is a famous model, but I didn't find the potential pertaining to it, so I wrote the Hamiltonian and Asked that , is there any way to get the Nilsson potential from the Hamiltonian. **EDIT: according to Thomas** So are you trying to say that, when symmetric potential is broken, that is the Nilsson potential? if someone ask me to write down the potential, then I can not write that down, because it is very subtle amount? **What are the failures of collective model that Nilsson potential introduced? I mean what are the difference between these model?**


The boundary between polycrystalline and crystalline

My current understanding of solid crystalline-like materials (please correct me if I'm wrong!) is that it is a continuum in terms of crystallinity, from amorphous (basically no periodicity) to single crystals (complete periodicity), with "polycrystalline" pretty much covering everything in the middle, like the wiki article shows. So my question is, what should you expect from a polycrystal in terms of properties you know about a single crystal of the same material, such as dispersion relation, band structure, etc? Obviously, it's a continuum, so you should expect a polycrystal that's _almost_ a single crystal (maybe like, a single crystal split in two, and put back together the tiniest bit off) to behave pretty damn similar to a single crystal. It seems like each single crystallite in a polycrystal should act as a tiny crystal with the band structure of a perfect crystal. But what do you do as the grains get smaller? Is there some way to predict how close to a single crystal you should expect the polycrystal to behave, given its average and variance of grain sizes? As the grains get smaller, it seems like you should expect the thin layers of amorphous solid holding the crystallites together to be more and more important. In fact, is there any sort of formalism for working with amorphous solids? Thanks!


Does the equivalence principle mean that the Earth is expanding?

Popular books suggest that for an observer in the Einstein lift the following situations are equivalent: 1, the lift hangs motionless (relative to the Earth) on a cable in the gravitational field of the Earth 2, there is no gravity but the lift is accelerated up The natural question then is: "If I stand on the Earth and feel its gravity does it mean that there is no gravity in fact and the Earth is expanding with acceleration?" I know the short answer ( **no** ) but as a devil's advocate I would appreciate a longer version.


Are there any electro-optic crystals that are also pyroelectric but not birefringent?

As the title says, a crystal that is electr-optic and pyroelectric can it be non-birefrigent?


Sign of Feynman rules with derivative couplings

Feynman rules for derivative couplings always make me confused. For example, the derivative in $gV^\mu\phi^+\partial_\mu\phi^-$ will give you $\pm ip_{-\mu}$, where $\pm$ depends on whether the particle is outgoing or incoming. I found "$-ip_{-\mu}$ for incoming" in Peskin & Schroeder's QFT book, BUT I also found "$-ip_{-\mu}$ for outgoing" in some other literature. So I'm wondering if the latter one is wrong. Or are they just in different conventions so that we can choose either of them?


Effect of small variable transverse force on satellite orbital precession

Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar = G.M/d^2 $$ Next imagine that a small, time-varying force acts upon the satellite which produces acceleration of the satellite in the direction transverse to the instantaneous radial direction. The general pattern of variable transverse acceleration (At) which I am interested in is given by the following equation:- $$At = (K/c^2)*Vr*Vt*Ar $$ I have modelled such effects in a reiterative computer model and would like to validate those measurements. The principal effect is the introduction of a rotation of the aspides axis of the orbit ("precession of the line of aspides"). QUESTION Is there an analytical technique I can use to produce a formula for the rate of orbital rotation? TERMS & ASSUMPTIONS   * K is a constant (e.g. +/- 3.0).    * Ar is the radial acceleration of the satellite towards the Sun due to Newtonian gravitational influence of the Sun.   * At is the time-variable acceleration of the satellite in the direction transverse to the Sun:satellite radial direction (positive when directed forward along the future orbital path of the satellite, negative when directed backwards)   * c is speed of light.    * Vr is satellite radial velocity relative to the Sun (positive when satellite is moving further away from the Sun)    * Vt is satellite transverse velocity relative to the Sun (always forward)    * G is Newton's Universal Gravitational Constant.    * M is mass of the Sun.    * d is distance of satellite from centre of the Sun (assume at least 10 million km).   * Assume satellite mass and diameter are extremely small relative to the Sun (e.g. 20kg and 2m respectively).   * Satellite is spherical with uniform density.    * No other bodies are in the system.   * All motions and accelerations are confined to a two-dimensional plane.    * Transverse acceleration is produced by a very small nuclear-powered mass-reaction rocket attached at the centre of the satellite.    * Consider the mass ejected by the rocket to be insignificant. Thus kinetic energy is added to and taken away from the system at different times, but there is no significant loss in mass.    * Assume a purely Newtonian System (no General Relativity please). I have seen various treatments of small _radial_ accelerations which use Lagrangian and Hamiltonian principles but I do not think these are extendable to _transverse_ accelerations (e.g. http://www.mathpages.com/home/kmath527/kmath527.htm). UPDATE I have accepted Walter's excellent answer at http://astronomy.stackexchange.com/questions/632/determining-effect-of-small- variable-force-on-planetary-perihelion-precession. (He used peturbation analysis to produce a precise formula for the rotation) But second opinions are welcome.


The difference between heat and temperature

So as I understand it, heat energy of an object is the SUM of all the kinetic energies of the molecules of the object (upto constant factor). The temperature on the other hand is the AVERAGE of the kinetic energies of all the molecules of the object. Now when ice is melting at 0 degrees Celsius, the temperature as measured on a thermometer does not go up. The common explanation is that any heat being absorbed by the ice is being used to break the somewhat strong solid solid bonds between the molecules of the ice. Here is my question. if heat of an object is what I defined above, then since all the molecules are increasing in kinetic energy, the average of the kinetic energy should also increase, meaning the temperature should increase. But at 0 degrees celsius for ice that does not seem to be the case. Where am I going wrong in my understanding?


The Physics Behind Chemical Bonding

Ok so here's the problem: say I'm a sodium atom. It is 'charged' at +1e. A partner Chlorine atom is flying about, also 'charged' at -1e. According to chemistry (or rather the measurement of the reaction) energy is released if the two bond together. The problem is, no electrons changed energy states. Both electrons in the shared band have the same state as before. I've heard a number of conflicting answers. One is that since the extra space is given to the shared electrons, there is a relaxing of the shared electrons which gives energy. That makes sense, but I have a few more questions: Why is a valence band such a limiting factor? Not all atoms bond together to share electrons, mainly just the ones closest to a full shell, yet why does the atom care? Shouldn't the extra space for the electrons always be a good thing? Why do covalent reactions occur solely for these atoms with near- perfect valence shells? Also, sometimes atoms just swap electon partners, in redox reactions specifically. Why do opposite spins attract heavily enough to do work against the coulomb potential? Also, even if electrons with opposite spin attract, since when do we forget the coulomb attraction? That is, electrons repel each other on the average. Too many questions? That's fine. Let's also have too many answers. I want a clear and well-explained response, and would not encourage a rushed reply. Thank you.


Does the dimensionality of phase space go up as the universe expands?

Ever since Hubble, it is well known that the universe is expanding from a Big Bang. The size of the universe had gone up by many many orders of magnitude as space expanded. If the dimensionality of the quantum phase space is finite because of spatial cutoffs at the Planck scale, does it go up as space expanded? If yes, how can this be squared with unitarity? If no, would this lead to what Tegmark called the Big Snap where something has got to give sometime. What is that something which gives?


Can you give me some examples of recreational physics? Thx


net displacement and path dependence

reading the paper about spacetime swimming by Wisdom   (something related to this has been previously asked here) can't help but think that there is more to this than what is on the paper. Basically the paper just proves that inertially, an object can get a net displacement by using the fact that phases are path-dependent due to non- commutativity in space paths, and he uses the rotational argument to properly ambient his point. The punchline here is that this same argument seems to be applicable when the masses have also for instance, a magnetic dipole, and there is a geometric phase in the magnetic field due to curvature Do you think that the same argument are applicable to obtain a similar effect due electric charges or magnetic dipoles in a curved manifold? i'm aware the question is vague but i hope this can entice (or stir) at least some deep discussion on the subject As a side (but very important) question; is not clear from the paper if the displacement is constrained to go only down in the gravitational potential well, but most discussions and papers referencing it says it should be so (check the conclusions section and comments)


Question about an integration by parts in Feynman's Quantum Mechanics

I have begun reading Feynman & Hibbs _Quantum Mechanics and Path Integrals_. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I saw it done once methodically it would clarify a lot. I have no problem with integration by parts in general. On p. 27 he says that upon integration by parts the variation in $S$ becomes $$ \delta S = \left[\delta x \frac{\partial L}{\partial \dot{x}} \right]_{t_a}^{t_b} - \int_{t_a}^{t_b}\delta x\left[\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}\right] dt. \tag{2-6} $$ Now $$ S = \int_{t_a}^{t_b}L(\dot{x},x,t) dt \tag{2-1}$$ in which $L$ is the Lagrangian $$ L = \frac{m}{2}\dot{x}^2 - V(x,t)\tag{2-2}$$ and he says that to a first order $$\delta S = S[\bar{x}+ \delta x] - S[x] = 0. \tag{2-4}$$ He shows $S[x+\delta x]$ explicitly in (2-5) and from this derives (2-6).


Could dark energy be heat? Could dark matter be stuff that is not seen because of its refraction limits?

Heat has to go somewhere. Would the light and heat of the stars in the universe amount up to being dark matter ...and dark energy ... Dark matter would just be small pieces of stuff. Dark energy would just be heat that has gone into space. If dark energy was just heat then a warp-drive could be as simple as a sphere with one very hot side.


Position Representation in Quantum Mechanics

How does the 3d position operator look like in position representation? I know that in 1d the position operator $\hat{x}$ is just multiplication by $x$.


Refraction of light and frequency dependence

Why do higher frequency waves refract more, both ocean waves and light waves? Also why is energy stored in the frequency as opposed to the wavelength.


Pulley system with rope

> Two masses $m_1$ and $m_2$ are connected through a weighty, uniform and an > unstretchable rope of length $L$. The surface is frictionless. Masses of the > objects are $m_1=m$, $m_2=\frac{2}{3}m$, mass of the rope is > $m_r=\frac{1}{3}m$. At which length $x_1$ of the vertical part of the rope, > the tension forces exerted by the rope on the bodies would be equal? What > acceleration does the system have? What is the type of the acceleration > (increasing/decreasing). ![http://oi42.tinypic.com/e9956e.jpg](http://i.stack.imgur.com/nMkXx.png) I had no particular problems with this question, however, the book wrote the first second law applied to the rope as: $m_r a = N_1-N_2+\frac{x_1}{l}m_rg$ What I got is a bit different: $m_r a = N_2 - N_1 + \frac{x_1}{l}m_rg$ In this particular question, the signs of the tension forces aren't important, as we have to calculate the acceleration when both of them are of the same magnitude, and thus cancel each other out ($N_1=N_2=N$), but I want to know, how it should really look like. Also, I have a conceptual question - if we have a similar question, where the rope has some mass, and there's no information regarding the uniformity of the rope, can I assume that the rope is actually uniform (assuming all the other details are the same as in this question)? And what if the rope is non- uniform? What happens to the acceleration if the rope isn't uniform? Is it possible that the acceleration of such a system wouldn't be increasing nor decreasing, but rather be some arbitrary function (e.g. the acceleration may increase and decrease randomly)? And what if I have only a rope (no masses, no pulley), lying on such a frictionless surface, where part of it is hanging (vertical part) - will the hanging part always pull the horizontal one, and the whole rope will slide down? Will it just be the same equation as above, just without $N_1$ and $N_2$?


Inner products in relativity

In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined between two vectors (“contravariant vectors”). What is the mathematical justification for redefining the dot product? I understand why we do it and I know that it works, but I'd like to understand how to justify it from the point of view of mathematics. I'm asking because I've been told that it is not mathematically _a priori_ correct to do it that way.


Interesting identity on $SU(3)$

In arXiv:hep-ph/1307.5414 Grabovsky use an interesting identity which is not derived in the paper: $$\tag{2.30}\varepsilon^{ikm}\varepsilon^{jln}\left((U_{2}U_{4}^{\dagger}U_{1})_{ij}+(U_{1}U_{4}^{\dagger}U_{2})_{ij}\right)(U_{4})_{kl}(U_{3})_{mn}=\\\ -B_{123}+\frac{1}{2}(B_{144}B_{324}+B_{244}B_{314}-B_{344}B_{214}).$$ Where: $$\tag{2.6}B_{123}:=\varepsilon^{ikm}\varepsilon^{jln}(U_{1})_{ij}(U_{2})_{kl}(U_{3})_{mn},$$ which is known as "Baryon". I'm looking forward for some help to derive this identity and generalize it.


Impulse, relative velocties

![enter image description here](http://i.stack.imgur.com/TbAeS.jpg) Why is (m)(v)=Impulse or as they put it here, Vs = I/Ms? Shouldn't (m)(v) be equal to momentum, not I? I don't understand why that is the solution. I was trying to solve for relative velocities, V2f - V1f, using momentum conservation but I failed hopelessly. I was thinking perhaps I should set momentum equal to zero so I have a center of mass frame, but I do not know how I should approach that. How and Why is this solution true?


Effective medium approximation for calculating relative permeability of cylindrical shells of alternating magnetic/nonmagnetic layers

I'm trying to apply the effective medium approximation to my own research. http://ieeexplore.ieee.org/ielx5/20/30863/01430899.pdf?tp=&arnumber=1430899&isnumber=30863 As far as I understand, these approximations are for elliptical magnetic materials in a non-magnetic medium. I would like to have an approximation for a cylinder consisting of alternating layers of ferromagnetic and non-magnetic material. I have yet to find one. Can someone help me? Many thanks in advance. -Raphael


Dirac sea entanglement

The theory of the Dirac sea is in disrepute, but I still wonder about it sometimes. In this theory, the particles are all electrons, and obviously an electron with positive energy can be entangled with an electron of negative energy. But how, on this theory, can an electron with positive energy be entangled with a hole?


Gravitational Stark Effect

Could gravity induce line splittings in the optical spectrum of a molecule similar to the Stark or Zeeman Effects? Naively, a gravitational potential would be a simple addition to the Hamiltonian that should behave similarly to a constant electric field (i.e. the simplest condition for the Stark Effect). Is there a reason that things should not be so simple? If such splittings do occur, what would the typical energy difference be between the states? Would they be observable at tenable energies?


In a universe where the speed of light is infinite, are relativistic models and Newtonian models equivalent?

Take our universe. Observations are consistent with relativity, but not consistent with Newtonian mechanics. Assume that our current (relativistic) model of gravitation is correct. Now increase $c$ to infinity. Experimental observations in this altered universe are still consistent with relativity (by the assumption that relativity is correct). Are they now consistent with Newtonian mechanics as well?


Does hidden variable quantum mechanics imply the collapse of atoms(electrons falling into the nucleus)?

why do atoms not collapse on themselves. Doesnt this problem rule hidden variables as invalid as the heisenburg uncertainty is the solution to the problem because it says electrons exist in a probability cloud and is in many places at once, not at a single location. If hidden variables are right the electron has a well defined position orbiting the nucleus and where does the energy come from? Doesnt this mean if HV are right it would collapse in again. whats the answer?


Poles, wavefunctions, transmission

Why is it said that $\operatorname{sech}x$ (a transmission amplitude) has a simple pole on the imaginary axis?


Calculating the deflection of an arbitrary beam of light from a thin lens

Suppose we have a lens with known focal length and are using the thin lens approximation. Is it possible to determine how an arbitrary beam of light will be deflected by it? Or, is it the case that there are multiple kinds of thin lenses, and so more information is required. I'm asking this question just so I can solve first year university optics problems.


How do we know Dark Matter isn't simply Neutrinos?

What evidence is there that dark matter isn't one of the known types of neutrinos? If it were, how would this be measurable?


Angular Velocity and Translation velocity after ball bounce

A ball with radius $r$, angular velocity $\omega$ and linear velocity $v$ hits a surface with coefficient of friction $\mu$ at an angle $\theta$ with the surface. I want to calculate the final angular velocity and translation velocity after the ball leaves the surface. Let $dt$ be the impact time.


Why $e$ in the formula for air density?

I am reading a book that says that the density of air is approximately $D = 1.25 e^{(-0.0001h)}$, where h is the height in meters. Why is Euler's number $e$ used here? Was a differential equation used in deriving this formula?


-Thought Experiment- At superluminal speeds, to what degree would vision be impaired?

I was thinking about superluminal travel when thinking about advanced civilisations in the future. I was also thinking about the Doppler effect; so I decided to think about how light may be affected in a similar way (besides redshift). I was thinking about thing appearing to go back in time (when looking from behind a superluminal object), but then I found a fundamental flaw with that thought - any light that had previously hit the object before it travelled at such superluminal speeds would not somehow radiate out of the object again, therefore rendering everything behind invisible (shown in this diagram): ![v=c](http://i61.tinypic.com/2957lp4.jpg) (The circles represent light waves that would radiate from the source if it were luminous.) But then I thought: "If I went past light speed, and onto higher speeds altogether, would I have further limitations of vision?" Here is another diagram of that: ![v=xc](http://i58.tinypic.com/334l0f7.jpg) **My question:** If superluminal travel were possible, then would the "funnel of vision" be possible, and if so, how would we see it? Would we see it as a small, bluish (due to the Doppler effect) circle with darkness surrounding it or would light seem to bend, blueshift and contort itself?


Pseudo-Superluminal Motion and the Synchronization of Clocks

Let's consider two points A an B separated by a finite distance in curved space time. A light ray flashes across an infinitesimally small spatial interval at B. Metric: $$ds^2=g_{00}dt^2-g_{11}dx^2-g_{22}dy^2-g_{33}dz^2$$ ----------- (1) We may write, $$ds^2=dT^2-dL^2$$ Where, $dT= \sqrt{g_{00}}dt$ and $dL=\sqrt{ g_{11}dx^2+g_{22}dy^2+g_{33}dz^2}$ dL is the physical infinitesimal(spatial) at B. Observer at A records the same value of dL But for dT the observers at A and B record different values: $$dT_{A}=\sqrt{g_{A(00)}}dt$$ $$dT_{B}=\sqrt{g_{B(00)}}dt$$ Local Speed of light[observation from B]$$=c=\frac{dL}{dT_B}=\frac{dL}{\sqrt{g_{B(00)}dt}}$$ Non-Local Speed of Light:[speed measured from A] $$C_{non-local}=\frac{dL}{dT_A}=\frac{dL}{\sqrt{g_{A(00)}}dt}$$ Therefore : $$c_{non-local}=c \times \sqrt{\frac{g_{B(00)}}{g_{A(00)}}}$$ --------- (2) For a particle moving across an infinitesimally small spatial interval at B wehave in a similar fashion: $$V_{non-local}=V_{local} \times \sqrt{\frac{g_{B(00)}}{g_{A(00)}}}$$ -----(3) Since $V_{local}<c$ $$V_{non-local}<C_{non-local}$$ ----------- (4) From (2) we find that the non local speed of light may exceed the local barrier. The non_local speed of a particle may also exceed the local speed barrier. _But the light ray is always ahead of the particle [in purely local or non- ocal observations]according to relation (4)_. There is no violation of relativity as such. _Average speed of light for a specified path_ : Let's consider a light ray is passing from P to Q along some path.Observation is being made from the point P.Time interval observed from P is given by: $$dT=\frac{dL}{C_{non-local}}$$ ------(5) $$=\frac{dL}{c\times \sqrt{\frac{g_{00}}{g_{P(00)}}}}$$ $g_{00}$ is the value of the metric coefficient at some arbitrary point of the path and $g_{P(00)}$ is the value of the metric coeff. at P. The average speed of light for the path between P and Q obtained by integrating relation (5),to obtain the time of travelof the light ray, may be different from the constant standard value of “c” _Query_ : Is it important to take care of such differences[differences in the local and non-local speed of light] in the synchronization of clocks in sensitive experiments?


Angular momentum operator in terms of ladder operators

I wanted to show that the angular momentum of the particle state with zero momentum $| \vec{0} \rangle$ is $0$, that is to say the intrinsic spin of a scalar field is $0$ using a mode expansion. There is a clever argument which is essentially that $e^{i \vec{\theta} \cdot \hat{\vec{J}}} \, | \vec{p} \rangle = | R(\vec{\theta}) \vec{p} \rangle$ and thus we have that for any $\vec{\theta}$ $e^{i \vec{\theta} \cdot \hat{\vec{J}}} \, | \vec{0} \rangle = |\vec{0} \rangle + i \vec{\theta} \cdot \hat{\vec{J}} |\vec{0} \, \rangle + \cdots = |\vec{0} \rangle$ and hence it is clear that $\hat{\vec{J}} |\vec{0} \, \rangle = 0$ which shows the desired result. I would like to show the same thing by using $\hat{\vec{J}} = -\int d^3x \, \vec{x} \times ( \hat{\pi} \, \nabla \hat{\phi})$ and use the mode expansions for $\hat{\phi}$ and $\hat{\pi}$: $\hat{\phi} = \int \frac{d^3p}{(2 \pi)^3} \frac{1}{\sqrt{2 E_{\vec{p}}}} (\hat{a}_{\vec{p}} + \hat{a}^{\dagger}_{-\vec{p}})e^{i \hat{p} \cdot \hat{x}}$ $\hat{\pi} = \int \frac{d^3p}{(2 \pi)^3} \sqrt{\frac{E_{\vec{p}}}{2}} (\hat{a}_{\vec{p}} - \hat{a}^{\dagger}_{-\vec{p}})e^{i \hat{p} \cdot \hat{x}}$ Doing this (and normal ordering) I get the following $\hat{\vec{J}} = - i \int \frac{d^3p}{(2 \pi)^3} \hat{a}^{\dagger}_{\vec{p}} ( \vec{p} \times \nabla_{\vec{p}}) \hat{a}_{\vec{p}}$ I am unsure of how to proceed with derivatives of the ladder operators and how to apply them to states in the Fock space. Could someone argue as to how one can apply this mode expansion to the state $| \vec{0} \rangle$ and show that it annihilates it, as it must.


3D coherence of light speckles- difference between Airy and gaussian speckles

In this article, http://www.dfm.uninsubria.it/laboferri/web_page/articoli_pdf/PRA_2009-Magatti- Three_dimensional_coherence_of_light_speckles-experiment.pdf, the 3D-coherence properties of light speckles are studied at different distances from the source. What I don't understand is the distinction drawn between what the authors call Airy speckles and gaussian speckles. In particular, it's not clear under which experimental conditions Airy or gaussian speckles are produced. How should I choose my setup so that I can know a priori that the speckles will be Airy or gaussian?


Dynamics - Beginner question

I am trying to understand this question, but I keep coming to a dead end: Two identical rectangular masses rest on top of one another, the top is tied to a wall on the left, while the one on the bottom has a roped attached to it on the right side. What is the max force that can be applied to the rope on the bottom before the block will begin to slide out from underneath the top one. (the coefficient of friction is equal between the block and the ground, and the bottom block with the top block). Ok, so I am pretty sure this question is just asking me to find the net force on the bottom block, as applying a force larger then this will result in an acceleration of the lower block: $$ F_{net}=ma$$ $$ 2F_f=ma$$ $$2F_n\mu=ma$$ $$2mg\times2\mu=ma$$ Ok, I am totally lost as to how solve this question. My intention is to find the net force acting on the lower block. I am pretty sure the the acceleration will be zero, resulting in the right side of the equation equaling zero. But I am not sure how to handle the coefficient of friction.


Continuum limit for solid mechanics

Is there a rigorous derivation of the limits for continuum properties in solid mechanics? For instance, the stress-strain relationship may be linear for large samples (the slope being the Young's Modulus) but at what limit does that break down? Even if a linear relationship exists for the stress-strain curve between two atoms, or two crystals, the slope likely isn't the same as for the large sample. What spatial scale separates the two regimes and is it similar to fluid dynamics where there is the Boltzmann equation at the non-continuum regime, the Navier-Stokes equations in the continuum regime, and a bridge between the two regimes (Chapman-Enskog equations)?


Why meet at the center of mass?

If two objects of different masses are held at a distance $d$ and then I let them go, they will meet at the center of mass of the particle system due to mutual gravitational attraction My question (which may be rather silly) is that why do they meet a the center of mass, why not anywhere else? Is there some sort of mathematical proof behind this?


Friction acting on mass on string

> _Assume we have some stationary point mass that would slide, if not for some > coefficient of friction, along a light, inextensible string, attached to two > points at different heights. Considering forces on the mass whilst it is > stationary, which direction does the frictional force act?_ My first thought is that it will act along the steeper of the two strings, but the concept of a reaction force (which surely friction must still be acting perpendicular to) seems less clear for a string? ![enter image description here](http://i.stack.imgur.com/293gN.gif) This diagram gives an idea of the kind of set up i'm thinking about (from a random webpage with a sufficiently useful diagram, not specific to this problem), although the string can be of any length and thus don't assume that either string is horizontal.


Quantum circuit decomposition

I need to construct a universal quantum circuit decomposition for a three- qubit operation where one qubit is the control bit, and a two-qubit unitary operator acts on the other two depending on the control bit. I can do the decomposition when there's a one-qubit unitary operation with as many control bits as needed; this two-qubit unitary with one control bit I am not clear about. Anyone has any clue what the recipe for this construction is?


Can I feel (the results of) electromagnetic induction from overhead rail lines in the human body?

A few weeks back, I was standing at a Hornsey rail station (in the UK) which uses overhead lines, and particularly has a number of parallel rail lines, all close to the platform. While I was standing there, I went to hold hands with my girlfriend, and experienced a peculiar sensation in my fingertips, almost as if there were very large ridges on her hands. I've felt this before, touching a metal bodied laptop computer that was seemingly not earthed - dragging my finger across the surface my finger felt as if it was jumping along the surface, despite it being a smooth aluminium surface. Is this a result of the overhead lines inducing a reciprocating current through my arm and/or fingertips? Is there some other feature of the railway that could cause this effect? And were we in any danger of any sort?


Why are smaller animals stronger than larger ones, when considered relative to their body weight?

I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat. It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects/lifeforms to support several times their own weight, but this is harder as objects/animals become larger?


Correlation/relationship between human magnetic field and body temperature

Is there any relationship between human magnetic field and body temperature? I am looking for studies on this topic.


Collision of 2 neutrons

If two neutrons collide in 3D space and we want to determine the final velocities of both nuetrons (3 components for each neutrons), we can use the conservation of momentum equations and the conservation of energy equation. Those equations are 4 while the number of unknowns we have is 6. There are 2 free variables. For classical perfectly spherical objects, the solution is strait forward as explained here. However, neutrons because of their quantum nature can't be treated as classical perfectly spherical objects. We can't define a precise point of collision, so the trick we use for classical objects won't work for neutrons or any quantum level particle, or will it? If it doesn't work, does that mean we can't avoid having free two degrees of freedom in this system?? NOTE: What is meant by a collision is a momentum-exchange event


Why trimming the mainsail gives a larger torque to head up

I am learning sailing on a 5m catamaran (Nacra 5). I am familiar with basic aerodynamics and the physics of the sail and keel. We learned that when sailing closed hauled, too tight a mainsail tends to bring the boat up to the wind. And that the opposite is true for the jib. For example, one may steer up to the wind to come about, using the mainsail alone, by trimming it tight. My question is why the trimmed mainsail in the above setup gives a larger torque. This also seems opposed to the rule: "The more the mainsail is sheeted out the more the boat tends to come up.", as explained in http://www.sailtheory.com/mandf.html#sailsteering Edit: To my understanding, there are several possible competing effects involved: (1) Effects that tend to INCREASE the mainsail torque to head up: (1A) The direction of the sail force becomes more perpendicular to the boat. This increases the heeling torque. Since heeling motion happens faster than turning, the boat will heel more. This moves the sail force out, which increases the lever arm and the torque to head up. see diagram in: http://www.sailtheory.com/mandf.html#hellingstuur (1B) The sail force moves backwards since the sail is stretched backwards. This increases the lever arm and the torque. (2) Effects that tend to DECREASE the mainsail torque to head up: (2A) The direction of the sail force becomes more perpendicular to the boat. Assuming that the force is perpendicular to the sail, and that the center of rotation is between the mast and the center of force, one sees that pulling the sail in, reduces the torque to head up, as can be seen in the following diagram: ![diagram of the sail force torque](http://i.stack.imgur.com/9I2TX.jpg) As Theta gets smaller, the torque Tau is reduced. This is opposed to what is stated in some of the answers below. (2B) The sail force is reduced since the sail was pulled beyond its optimal angle of attack, thus losing lift and reducing torque. (2C) Due to (2B) heeling torque is also decreased. With a similar reasoning to (1A) this decreases the lever arm and torque. (2D) The sail force moves forward since the aft part of the sail is too flat and stalls. This shortens the lever arm and reduces torque. This is described in http://www.sailtheory.com/mandf.html#sailsteering We know from reality that 1A+1B > 2A+2B+2C+2D. Considering heeling alone, we also know that 1A > 2C. It remains to be explained why in general this is the case.


Stress energy tensor of a perfect fluid and four-velocity

In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : $\boxed{g_{\mu\nu} = \begin{pmatrix} -c^2 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}} \qquad\qquad \boxed{ g^{\mu\nu} = \begin{pmatrix} \frac{-1}{c^2} & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}} $ The tensor of a perfect fluid is: $ \boxed { \begin{aligned} T^{\mu\nu} &= \left(\rho c^2+P\right)u^{\mu}u^{\nu}+Pg^{\mu\nu}\\\ T_{\nu}^{\mu} &= \left(\rho c^2+P\right)u^{\mu}u_{\nu}+P\delta_{\nu}^{\mu}\\\ T_{\mu\nu} &= \left(\rho c^2+P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu} \end{aligned} }$ where $\rho$ is the mass density. We can easily check that $\rho c^2$ and $P$ have the same unit. If we consider only the first component $00$, there is no pressure $P$ involved (I consider a cosmological case where the fluid is at rest in comoving coordinates). The only solution to have that is that $u^0u^0=-g^{00}$, so that the contravariant perfect fluid tensor become : $T^{00} = \left(\rho c^2+P\right)u^{0}u^{0}+Pg^{00}=\left(\rho c^2+P\right)\times{-g^{00}}+Pg^{00}=-\rho c^2 g^{00} = -\rho c^2 \times -\frac{1}{c^2}=\rho$ which is the expected result. So, to have the expected result, one should have $\boxed{\left(u^0\right)^2=+\frac{1}{c^2}}$. Now come the weird part. The metric can be written: $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=-c^2dt^2+dx^2+dy^2+dz^2=dt^2\left(-c^2+\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2}\right)\approx-c^2dt^2$ which is equivalent to : $\frac{dt^2}{ds^2}\approx-\frac{1}{c^2}$ which means : $\boxed{\left(u^0\right)^2=-\frac{1}{c^2}}$ So we clearly have a problem of sign here between the two expressions of $u^0$. **QUESTION: where is the mistake ?**


A question on an exercise from Gravitation by Misner, Thorne and Wheeler

My question is on problem 4.1 of Gravitation. In a generic case of electric field and magnetic field(i.e not $E=0$ or $B=0$ or $E$ and $B$ perpendicular), define the direction $\hat{n}$ unit vector , $$\hat{n}\tanh (2\alpha)=\frac{2\vec{E}\times\vec{B}}{\vec{E}^{2}+\vec{B}^2}$$ and $\vec{\beta}=\tanh(\alpha)\hat{n}$ is the velocity vector. Show in the frame of the rocket with velocity $\vec{\beta}$, the Poynting vector vanishes. I tried the following but I am stuck at the cancellation. Let the $\bar{\vec{E}}$ and $\bar{\vec{B}}$ be the field in the rocket frame and the field without bars be the field of the rest frame. direction parallel along the velocity of rocket is denoted as subscript $\parallel$ and direction perpendicular to velocity of rocket direction is denoted by $\perp$ as the subscript. By lorentz transformation. \begin{align} \bar{\vec{E_{\parallel}}}&=\vec{E_{\parallel}}, \\\ \bar{\vec{E_{\perp}}}&=\frac{\vec{E_{\perp}}+\vec{\beta}\times\vec{B_\perp}}{\sqrt{1-\beta^{2}}},\\\ \bar{\vec{B_{\parallel}}}&=\vec{B_{\parallel}}, \\\ \bar{\vec{B_{\perp}}}&=\frac{\vec{B_{\perp}}-\vec{\beta}\times\vec{E_\perp}}{\sqrt{1-\beta^{2}}}. \end{align} Note $\beta\times X_{\perp}$=$\beta\times X$ In barred frame, \begin{align} \bar{\vec{E}}\times\bar{\vec{B}} &=(\bar{\vec{E_{\perp}}}+\bar{\vec{E_{\parallel}}})\times(\bar{\vec{B_{\perp}}}+\bar{\vec{B_{\parallel}}})\\\ &=\bar{\vec{E_{\perp}}}\times\bar{\vec{B_{\parallel}}}+\bar{\vec{E_{\parallel}}}\times\bar{\vec{B_{\perp}}}. \end{align} Plug in the lorentz transformation. One gets $$\frac{\vec{E}\times\vec{B}-\vec{E}_{\parallel}\times(\vec{\beta}\times\vec{E})-\vec{B}_{\parallel}\times(\vec{\beta}\times\vec{B})}{\sqrt{1-\beta^{2}}}.$$ Now the last two terms looks generically like $\vec{X_{\parallel}}\times(\vec{\beta}\times\vec{X}$) simplifies to $\vec{X_{\perp}}(\vec{X}\cdot\vec{\beta})$. However, I could not see the cancellation at this stage. Did I do something wrong?


Wave function not normalizable

Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$ $$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad \text{or}\qquad \int_{0}^{\infty}|\psi(x)|^2 dx <\infty$$ What would be the physical implications if one (or both) of those integrals diverges. From the viewpoint that the Copenhagen interpretation is one of the most popular and the wave function is interpretated as a probability distribution in this case; will the wavefunction diverging be valid for any other interpretation of quantum mechanics. Does anyone know any wavefunctions which are not normalizable? What if there was a singularity at 0 that made it diverge at all times. For instance $$\int_{0}^{\infty}|\psi(x)|^2 dx \to\infty\quad \text{but}\quad \int_{a}^{\infty}|\psi(x)|^2 dx <\infty $$ where $a>0.$ Would the second integral count as a valid pdf?


What if the Moon was 100x more massive: Binary planets?

I was reading a couple of Earth-Moon related questions (Mars just collided with Earth! A question of eccentricity, Could the earth have another moon?) and they got me thinking about planet-moon systems in general. Binary star systems are pretty common. The types of the two stars in the binary can vary pretty widely (main sequence, puslar/neutron star, black hole, white dwarf, giant phases, etc, etc), but some are formed of a pair of roughly equal mass (within a factor of <10). I can't say I've ever heard of a binary planet system, though. Of course a planet with a moon is sort of a binary, but I've never heard of an equal mass binary planet. I think the closest thing in the solar system would be the Pluto-Charon system, with a mass ratio of about 10:1. Is there any reason a binary planet would be unstable? Obviously this is a three-body system, which has some inherent instability, but Earth-Moon-Sun seems pretty stable. Would increasing the mass of the Moon to match that of Earth make the system unstable? How about gas-giants? I think a Jupiter-Jupiter binary close to a star would be short lived because of three-body interactions, but what about further out? Would, for instance, a double-Jupiter or double-Saturn be stable in our solar system? Or is there some tidal effect that would cause the orbit to decay and the binary planet to merge? As an aside, it seems that binary asteroids aren't terribly difficult to find... perhaps we just haven't seen any binary planets yet because they're only stable relatively far away from their star, making them difficult to detect outside our own solar system?


Mass of particle w.r.t. dimension

I heard in a lecture recently - just as a comment - that a particle which is massive in say $D=4$ can be seen as a massless particle in higher dimensions and vice versa. Our prof didn't give any equations to explain this and I don't really see how this comes about. What's the trick?


Determinant of ray transfer matrix

Quick question on the wikipedia entry for the _ray transfer matrix_ , which is defined : ![enter image description here](http://i.stack.imgur.com/Jt51r.png) where $x$ is the distance of a point from the optical axis and $\theta$ the angle that the beam profile at that point makes with the optical axis. Now, wikipedia says:![enter image description here](http://i.stack.imgur.com/2eMFv.png) but does NOT provide a reference. Does anyone have any idea on how to get that results? Do you know a website/paper where this is discussed?


Under what conditions is a pure fusion nuclear bomb possible?

Is it possible to build a pure fusion and powerful nuclear bomb? (because due to the harsh conditions of enriched uranium, I would guess most of today's world countries will want to build pure fusion bombs.) A pure fusion bomb is a nuclear bomb that does not need fission so does not need enriched uranium.


Is it theoretically possible for a person to pass through a solid wall/object?

I understand that matter cannot pass through other solid matter because of the electrons that orbit an atom prevents this but I was curious to know if it is theoretically possible to somehow get passed this problem caused by the electron shell and by doing so would this allow th effective passage of matter through matter. My thinking was that on the scale of the atom the amount of space taken up by actual particle mass is small compared to the space between the nucleus and the electron cloud so if the electron problem can be circumvented then all that space could possibly be used to pass through. It would probably take some fanciful piece of tech based on unknown physics but i though i would ask!! Thanks.


Is it possible see an invisible light with something similar to a normal lens of an eyeglass?

Let's say that I am a spy in a room with a lot of people and I want to communicate with my friend using Morse code via a light signal. Does there exist a type of light (laser or otherwise) that is invisible to the human eye, but that I can see using a modified transparent lens? In other words, can I make an infrared or ultraviolet light visible using something similar to a normal lens of an eyeglass? If yes, is it easy to buy or make it?


Why are wires in simple circuits approximated as equipotentials?

I just answered this question: Voltage in a circuit by asserting that any two points joined only by a wire, but no other circuit elements, are at the same value of the electric potential. It is, of course, true that in the context of electrostatics, the values of the electric potential at any two points inside of a perfect conductor are the same, but the electrostatics argument seems irrelevant for circuits. **Why are we justified in using this same fact for circuits in which we are clearly no longer faced with electrostatics?** **Edit.** As prompted by user twistor59 in the comments below, let me include a bit more detail to clarify the question and hopefully indicate why I don't think the answer is obvious. If we consider any segment of the wire as a small ohmic resistor, then as the resistance of any segment drops to zero, the voltage drop across that segment will be correspondingly small for a given current. In particular, zero resistance implies zero voltage drop. However, if I just have a piece of a perfect conductor sitting somewhere, then it is totally unclear to me why, or that it is even true, that the electric field will vanish in its interior if the situation is non-electrostatic. Perhaps, therefore, the magic lies in the microscopic physics that (in particular) lead to Ohm's Law?


How fictitious are fictitious forces?

How fictitious are fictitious forces? More specifically, in a rotating reference frame i.e. on the surface of the earth does an object that is 'stationary' and in contract with the ground feel centrifugal and Coriolis forces? Or are these forces purely fictional and used to account for differences in observed behaviour relative to an inertial frame? To give a practical example a turreted armoured vehicle is sitting stationary and horizontally somewhere in the UK. The turret is continually rotating in an anti-clockwise direction. Do the motors that drive the turret's rotation require more power as the turret rotates from east to west and less power as the turret rotates from west to east? i.e. are the turret motors cyclically assisted and hindered by the earths rotation?


What exactly is the difference between advection and convection?

After reading Wikipedia articles on advection and convection, I still cannot determine whether there is a consensus on a difference between these two terms. Sometimes, the term convection seems to include advection and diffusion, sometimes not. After all, equations exhibit two different terms for advection and diffusion, and we don't call the sum of them a convective term. When we talk about a convective term, it is indeed the advective term! It seems:   * transport of pollutants in a river by bulk water flow downstream relates to **advection**   * boiling water for pasta relates to **convection** Maybe, a conserved property such as energy can be advected, but we only use convection for transport of "ensembles of molecules". So, is there a difference? Could we figure it out with examples?


Was the 2013 meteor over Russia stronger than an atomic bomb?

Reports of the Russian meteor event (2013) say that it released more energy than 20 atomic bombs of the size dropped on Hiroshima, Japan: > Scientists estimated the meteor unleashed a force 20 times more powerful > than the Hiroshima bomb, although the space rock exploded at a much higher > altitude. Amy Mainzer, a scientist at NASA's Jet Propulsion Laboratory, said > the atmosphere acted as a shield.   >  The shock wave may have shattered windows, but "the atmosphere absorbed the > vast majority of that energy," she said. > > http://abclocal.go.com/kfsn/story?section=news/national_world&id=8994311 Really? Wouldn't that have done more damage than was seen? How does the damage depend on how quickly the energy is released?


Torque, lever and mass

The Force used in a catapult is exerted near its axis. If we double the length of the arm of the catapult, but still use the same Force at the same point as before near the same axis, does the distance in the Torque equation change?Edit (In an experiment we have found that the longer arm/lever makes the ball fly a longer distance, but how do we compute this into the equation?) Also, how can one combine the mass of the ball used by the catapult with the calculation of the torque?


A bomb can be made to explode or implode?

A bomb can be made to implode or explode. Can one be made to explode while creating a vortex at the same time?


Question: if a nuclear reactor of a submarine explodes will it contaminate the surrounding waters?

After watching a Naval War movie I began to ask myself this question and I would like to know.


Is there really time reversibility in physics?

First, how i got to the question. I was randomly looking at this page: http://en.wikipedia.org/wiki/Langton%27s_ant describing some sort of cellular automaton. Here is the path of this automaton after more then 10000 steps: ![Langton's ant trajectory](http://upload.wikimedia.org/wikipedia/commons/d/d3/LangtonsAnt.png) You can clearly see that after some time ant gets stuck in the loop (actually its a highway) and will be going on it forever. Here comes the problem with many faces: If we turn back time and start ant from one of the loop(highway) points we will never get to the original messy part, so obviously this computation is not time reversible since there is no way out of the loop! :) (But there should be ! :D) Another thing is that information is also not time reversible - since if at some point in computation operation DELETE is used to erase all the previous data there is no way to get that data back. Ok. So far so good. Lets take this question one step further. How is that so? They always told us laws of physics are time reversible and so on. My intuitive answer is following. We can say that in the universe there are so called LAWS (physics and so on) and OPERATIONS. What is the difference? LAWS do not change the objects on which they are operating and hence probably are time reversible. OPERATIONS on the other hand can change objects, their number in the system and hence are not time reversible (if I delete some files there is no algorithm how to restore them again) I understand its very close to the idea and interpretation of information. So when they show us movie backwards and we think time got reversed its actually false! Yes you can reverse the motion of the people and objects, but you can not reverse speach for example since there is no logical connection between phrases being said. Here we can see how the content of the movie (life) is actually twofold - time revesible part (geometry in the poorest sense) and informational part (speech, computations and so on). From all above it seems that the universe overall is NOT time reversible, and there is no way to go back in time! :) Or is there?


Do photons in interferometer violate the law of conservation of mass?

I mean Mach-Zehnder Interferometer, where light split into two shortest paths. Is light after splitting wave or particle? Is it a particle? How could one photon change to two? If the wave, does the sum of the energies of both waves divided by the square of the speed of light equal to the weight of the original photon? ![Mach-Zender interferometer](https://upload.wikimedia.org/wikipedia/commons/d/d8/Mach_Zehnder_interferometer.svg) Thank you.


maximum positive electric charge of solid body

What are the limiting factors on the positive charge of a solid body? If I assume a 'perfect insulator' environment that would not exchange charge with my solid body, I would guess that I can remove electrons from the body until it's internal chemical bond structure made by them breaks down and the body disintegrates. Are there any other other limits that would be hit first? How to calculate the charge at which the body breaks away?


How do we measure $i[\hat\phi(x),\hat\phi(y)]$ in QFT?

What operational procedure is required to measure $i[\hat\phi(x),\hat\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test- functions, or give an answer in Fourier space, for $i\bigl[\widetilde{\hat\phi}(k),\widetilde{\hat\phi}(k')\bigr]$, to your taste; give electromagnetic field indices to $\hat\phi(x)$ and transpose the question to QED if the scalar field case seems unreal in an essential way]. In a non-interacting QFT, $i[\hat\phi(x),\hat\phi(y)]$ is a c-number, but in an interacting QFT it may presumably be an observable independent of $\hat\phi(x)$ and $\hat\phi(y)$. How do we measure it, however? If we cannot measure it, is it effectively chosen by convention? An answer that describes the (operational) difference between $A=\hat\phi(x_1)...\hat\phi(x_i)\hat\phi(x_{i+1})...\hat\phi(x_n)$ and $B=\hat\phi(x_1)...\hat\phi(x_{i+1})\hat\phi(x_i)...\hat\phi(x_n)$ might be equally good (though $A^\dagger\not=A$ and $B^\dagger\not=B$). As always, a reference would be good. EDIT: Following Luboš' answer, and more particularly his comments, for which again thanks, $i[\hat\phi(x),\hat\phi(y)]$ is zero if $x-y$ is space-like, both in non-interacting and interacting quantum fields. If a quantum field $\hat\phi(x)$ is the basis for an empirically adequate quantum theory, measurement operators at time-like separation from a given space-like hyperplane at time $t$ can in principal be expressed as a nontrivial function of the field and field momentum at time $t$, $\hat\phi(y)=\int[\hat\phi(t,\mathbf{x}')f_t(\mathbf{x}')+\hat\pi(t,\mathbf{x}')g_t(\mathbf{x}')]\mathrm{d}^3\mathbf{x}'$, for two functions $f_t$ and $g_t$, which for a given theory are determined by the Lagrangian. Given many implicit assumptions, measuring the commutator $i[\hat\phi(t,\mathbf{x}),\hat\phi(y)]=-g_t(\mathbf{x})$ (because $[\hat\phi(t,\mathbf{x}),\hat\pi(t,\mathbf{y})]=i\delta^3(\mathbf{x}-\mathbf{y})$) is equivalent to measuring $g_t(\mathbf{x})$. If we also measure $i[\hat\pi(t,\mathbf{x}),\hat\phi(y)]$, we can determine (or at least significantly constrain) the Lagrangian. _If we do not know the Lagrangian_ , therefore, it would seem to be desirable to have an operational way to measure $i[\hat\phi(x),\hat\phi(y)]$ in order to _measure_ the Lagrangian (instead of the usual approach of guessing the Lagrangian and seeing how close the consequent predictions are to the measurements; the haphazardness of this process is something of a crisis in HEP). I would welcome comments on this or an answer that takes this EDIT into account. Alternatively, an operational approach to _measuring_ the Lagrangian would implicitly be a measurement, in principle, of $i[\hat\phi(x),\hat\phi(y)]$; is there such a thing?


Does Kaluza-Klein theory successfully unify GR and EM? Why can't it be extended to the Standard Model gauge group?

As a quick disclaimer, I thought this might be a better place to ask than Physics.SE. I already searched there with "kaluza" and "klein" keywords to find an answer, but without luck. As background, I've been reading Walter Isaacson's biography of Einstein, and I reached the part where he briefly mentions the work of Kaluza and Klein. Also, I did my undergraduate degree in theoretical physics, but that was quite a few years ago now... The way I understand the original work of Kaluza and Klein is that you can construct a theory that looks like a 5-D version of GR that reproduces the Field Equations and the Maxwell Equations. An important bit is that the 4th spatial dimension is the circle group $U(1)$, which we now know as the gauge group of EM. I guess the first part of the question should be whether I've understood that correctly. Then, if you build a Kaluza-Klein type theory, but use the SM gauge group $U(1)\times SU(2)\times SU(3)$ instead of $U(1)$, what do you get? Is it gravity and the standard model together? If not, then what? The Wiki article on Kaluza-Klein theory says that this logic "flounders on a number of issues". The only issue is explicitly states is that fermions have to be included by hand. But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?


Does electric potential have a temperature?

When I took my first thermo class a tucked away chapter introduced Exergy in terms of electrical energy, meaning that the amount of electrical energy you could get from something is functionally its exergy. Wikipedia doesn't go so far, but this has bothered me for years. > Forms of energy such as macroscopic kinetic energy, **electrical energy** , > and chemical Gibbs free energy **are 100% recoverable** as work, and > therefore have an exergy equal to their energy. However, forms of energy > such as radiation and thermal energy can not be converted completely to > work, and have exergy content less than their energy content. Let's say that we have 1 Volt of potential. The smallest unit of energy we could isolate from this would be a single electron accelerated over the potential. Numerically and naively I could write: $$ 2/3 k T = V e$$ Where $e$ is the electron charge. Doing this I would get something like $17,000 K$ for the temperature. This is high enough to approximate to infinity for conversion processes where your sink is room temperature. I can imagine accelerating something with a greater charge than electron over that potential. This method would find a higher temperature, which presents a counter-argument. So what is it, **is there a theoretical limit to the energy you can get from electricity, giving it a functional temperature?** That would imply an unavoidable loss from DC-DC voltage converters. Another thing that bothers me is saying that radiation can't be converted completely to work. Thermal radiation, yes, but most nuclear radiation is at $keV$ or $MeV$ energies, which would give a dramatically higher temperature than electrical energy (if I've understood correctly).


Gravitational force from spherical shell

Say we have a point mass $\mu$ located at $(0,0,R)$ and a spherical shell ( _not_ considering it's volume!) of radius $R$ located in the origin. So we have a particle standing right on top of the sphere and we want to determine the total gravitational force exerted on the particle by the spherical shell. Since the force is directed along the z-axis, we only have to consider the z-component of the force. Moreover, suppose the shell has homogeneous density $\sigma$. We can then write $dM = \sigma dA = \sigma R^2 \sin \theta d\theta d\phi$. So we have $$ dF_z = \frac {(z-R)G\mu dM}{\sqrt{x^2 + y^2 + (z-R)^2}^{\ 3}} = \sigma \mu G R^2 \frac { (z-R)\sin \theta d\theta d\phi }{ \sqrt{x^2 + y^2 + (z-R)^2}^{\ 3} } $$ We then integrate over the entire sphere: $$ F_z = \iint \sigma \mu G R^2 \frac { (z-R)\sin \theta d\theta d\phi }{ \sqrt{x^2 + y^2 + (z-R)^2}^{\ 3} } $$ This turns out to equal $-2\pi G \sigma \mu$ and $\sigma = \frac M{4\pi R^2}$, where $M$ is the total mass of the spherical shell, which yields $-\frac {GM\mu}{2R^2}$. What I do not understand why there is a factor 2 in the denominator, should it really be there? Just for the protocol, we get the classical formula if the particle is outside the shell and zero if it is inside the shell.


Difference between weight of water and pressure of water

_(I didn't even have a basic formal education in physics. I'm learning through the internet out of my own interest, so if there are any silly mistakes, kindly bear with and guide me through.)_ Everywhere, everyone is saying that pressure will same be for a given height. How is it possible? When volume changes, doesn't pressure change in calculations? [Pressure will be 2.5 bars approximately (including atmospheric pressure) for a height of 15 metres from ground level. I checked on many sites that 2.5 bars equals 2.5 kg/cm².] Now let's take **base area as** **2000 cm² and height of 15 m as constant** in three scenarios: EACH SCENARIO IS INDEPENDENT AND TUBES ARE SEPARATE, I.E NOT CONNECTED WITH EACH OTHER In **scenario 1** a straight vertical tube from ground level will have capacity of 3000 litres with a weight of 2.5 kg per cm² [3000 litres / 2000 cm² = 1.5 kg/cm² + 1 kg/cm² atmospheric pressure]. In **scenario 2** the tube becomes narrow from the base area, resulting in a volume of 2000 litres. In **scenario 3** the tube becomes wide from the base area, resulting in a volume of 4000 litres Question: When they say pressure remains the same at a given height, does it mean that weight will be 2.5 kg/cm² in all 3 scenarios, where volume is 3000, 2000, or 4000 litres (base area 2000 cm², height 15 m constant)? How is it possible? Where am I wrong?


Noncanonical commutation relation and noncanonical wave mechanics

Consider noncanonical operators $\hat{x}_1,\hat{x}_2,\hat{p}_1,\hat{p}_2$ satisying the following condition in the $q_1,q_2$ - basis ($\psi=\langle q_1,q_2|\Psi\rangle$)(similar to wave mechanics): $$\hat{x}_1\psi=q_1\psi-ia\hbar\frac{\partial}{\partial q_2}\psi$$ $$\hat{x}_2\psi=q_2\psi-ib\hbar\frac{\partial}{\partial q_1}\psi$$ $$\hat{p}_1\psi=-i\hbar\frac{\partial}{\partial q_1}\psi+cq_2\psi$$ $$\hat{p}_2\psi=-i\hbar\frac{\partial}{\partial q_2}\psi+dq_1\psi+i\hbar\gamma\frac{\partial}{\partial q_1}\psi$$ The commutation relations matrices are (similar to Hilbert-space quantum mechanics): $[x_i,x_j]=i\hbar\left(\begin{array}{cc} 0 & (b-a) \\\ (a-b) & 0 \end{array}\right)$   $[p_i,p_j]=i\hbar\left(\begin{array}{cc} 0 & (c-d) \\\ (d-c) & 0 \end{array}\right)$ $[x_i,p_j]=i\hbar\left(\begin{array}{cc} 1 -ac & -\gamma \\\ 0 & 1-bd \end{array}\right)$ Because $\hat{x}_1,\hat{x}_2$ are not commute so I have to introduce 2 other parameter $q_1,q_2$. My question is: is it true or is it not true that all wave mechanics theories with the same commutation relations will predict the same results?


Is frequency quantized?

I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum? The quantization of photon energies


Why does gravitational collapse occur suddenly in a supernova progenitor?

I was reading the Wikipedia article on Supernovae, and it says that one of the reasons why a supernova occurs is due to sudden gravitational collapse when the core of the star has little fusable material. But why is this collapse sudden? Shouldnt the collapse be gradual as the fusable material gradually decreases? Why is the sudden collapse like driving off a cliff? (if I've interpreted the article rightly.)


Variational wavefunctions and "spread" of potential in quantum mechanics

A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a "narrow and deep" potential is higher in energy than a variational solution for a "wider and shallower" potential, by appealing to the particle in a box as a special case. The former wavefunction is said to be "more confined" in the former than the latter, and the act of being given more space to spread out is said to lower the energy variationally. Question: are there _rigorous mathematical statements_ of this argument?


Is enthalpy defined under non-isobaric conditions?

My question is this, Does Enthalpy have a meaning under non-isobaric conditions? Is its existence as a property of a system independent of whether the system is under isobaric condition or not? Edit:- I wanted to know if enthalpy as a property of a system is valid only if from its creation to current state, it undergoes only isobaric processes.


What is the definition of a timelike and spacelike singularity? Trying to find, but haven't yet, what the definitions are.


How do we know that there isn't a classical solution to the measurement problem/Quantum Mechanical uncertainty?

It was mentioned to me that it can be shown that there is no classical explanation for the uncertainty in Quantum Mechanics -- i.e. that there are no hidden workings that we have just not yet seen, which could be explained classically and would explain the probabilistic nature of Quantum events in a 'deterministic' fashion. Can someone explain how this is known please?


Can we live forever by taking advantage of the time dilation effect?

I have been reading about Time dilation and twin paradox in Relativity. So,I thought that if It can be the difference of 10 yrs between the Girl in space and on Earth, then why can't anyone not age rather. Is it possible? [I'm 14 and new in the field of modern physics, so pls correct if I'm wrong. I regard All of You as teachers.]


Symmetry of Euler-Lagrange equations and conservation laws

1. Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law?    2. Also is symmetry of equations of motion also a symmetry of the action?  I assume that the equations of motion I am talking about follow from the action, i.e. the equations of motion are Euler-Lagrange equations. Related examples could help and references also.


Are there any materials that change the electromagnetic wavelength of visible light?

What i mean is for example exclusively red light comes in but gets reflected or refracted as blue or xray wavelength or whatever. Does this happen at all?


Are magnetic hysteresis losses relevant to alternating currents flowing in a wire?

Say we have an AC in a magnetically lossy material, like iron. Because of iron's relatively high permeability, skin effect will be more pronounced than it is in say, copper, so this iron wire isn't so great a conductor, practically speaking. However, skin effect arises due to eddy currents, which are themselves due to the time varying magnetic field due to the time varying current. So, this suggests there is a magnetic field at work, and there's the potential for magnetic hysteresis to be an additional source of loss in the material. Yet, I'm not sure if this is true or not, if somehow the geometry of the conductor and the fields around it make this a non-issue. I would think that if a time-varying magnetic field exists inside the iron, then there would be hysteresis losses. Is this true? Does such a field exist? To be clear: the issue is with iron _as an electric conductor_ , not as part of a magnetic circuit such as a transformer core as is the more common application.


Probability current density : Isn't there something wrong with this proof?

Is there something wrong with the following proof (see below)? To me, it seems like the third line should show $$\frac{dP_{ab}}{dt}=-\int_a^b \frac{\partial}{\partial t}J(x,t)dx$$ Am I missing something obvious? * * * ![enter image description here](http://i.stack.imgur.com/9L5fT.jpg)


Hermitian operator ?? is this possible

we know that the operator $ H= - \hbar ^{2} \frac{d^{2}}{dx^{2}}+ V(x) $ is hermitian isn't it ?? however what would happen if the potential were still real but it depends on the Wave function, for example huypothetically $ V(x) = |\Psi (x)|^{2} $ or $ V(x) = arg \Psi (x) $ since the functions $ |x| $ and $ arg (x+iy) $ are ALWAYS real, then the hamiltonian with potentials (1) and (2) should be Hermitian but they depend on the solution so i am not sure about the Hermiticity of a Hamiltonian in the form $ H =p^{2}+ |\Psi (x) |^{2} $ i believe that a real potential makes the Hamiltonian hermitian even in the case that we do not exactly know what potential is


Adjoint of Gamma Matrices - Dirac

I just started to learn how to quantise Dirac field. Meanwhile, as we can write the Dirac equation in terms of gamma matrices : $$ (i\hbar\gamma^\mu\partial_\mu - m)\psi = 0 $$ where $\gamma_\mu$ matrices obey Clifford algebra $$ \\{\gamma^\mu,\gamma^\nu\\} = \eta^{\mu\nu} $$ Now I just came across that the adjoint of the gamma matrices can be written as : $$ \gamma^{\mu\dagger} = \gamma^0\gamma^\mu\gamma^0 $$ I already checked this question, but it doesn't suggest a way to prove it in a representation independent manner. Also the Wikipedia page suggests that gamma matrices are chosen such that they satisfy the above relations, since they are arbitrary upto a similarity transformation. So is that the way this identity comes, or is there something else ?


Can entanglement with an inaccessible system be useful?

Quantum phenomena in bipartite pure state systems like teleportation are pretty well understood. What I'm interested in is the following situation: Alice, Bob and Charlie hold some general tripartite pure state $\rho_{ABC}$. As usual, Alice and Bob are trying to perform some task, and are allowed only LOCC. Charlie's subsystem, however, is completely "inaccessible", by which I mean that Charlie can't do anything at all- no local operations and no classical communication with either of the other two. His role is solely to hold onto his part of the system. So my question is, is there any situation where Alice and Bob, with full knowledge of the tripartite state, would be able to perform some task 'better' (e.g. with a greater probability of success) with this setup than they would if they just traced out Charlie's system and were left with $\rho_{AB}$? If the answer is yes, then I'd be interested to know of any general results in this area.


Does Saturation velocity in semiconductors have a relation with the wavelength in which the peak in the absorption spectrum occurs?

> Saturation velocity is the maximum velocity a charge carrier in a > semiconductor, generally an electron, attains in the presence of very high > electric fields. (source) I want to know if the absorption peak can be calculated with this parameter when a semiconductor like GaAs is under illumination.


How long a straw could Superman use?

To suck water through a straw, you create a partial vacuum in your lungs. Water rises through the straw until the pressure in the straw at the water level equals atmospheric pressure. This corresponds to drinking water through a straw about ten meters long at maximum. By taping several straws together, a friend and I drank through a $3.07m$ straw. I think we may have had some leaking preventing us going higher. Also, we were about to empty the red cup into the straw completely. ![alt text](http://i.stack.imgur.com/VCLN5.jpg) My question is about what would happen if Superman were to drink through a straw by creating a complete vacuum in the straw. The water would rise to ten meters in the steady state, but if he created the vacuum suddenly, would the water's inertia carry it higher? What would the motion of water up the straw be? What is the highest height he could drink from? Ignore thermodynamic effects like evaporation and assume the straw is stationary relative to the water and that there is no friction.


Temperature rate of change inside a cylinder

I was wondering about this today, mainly for baking a cylindrical cake. Say you have a solid cylinder which is at temperature `T0` inside. Say you then move the cylinder to a warm location (like an oven) which is at temperature `T1`. ![Solid Cylinder](http://i.stack.imgur.com/4hE0W.png) **Is it possible to achieve a rate equation for`dT/dt=f(t)` where `T` is the temperature exactly in the centre of the cylinder and `t` is time?** And therefore work out the time it would take this location to reach a temperature `T2` somewhere between `T0` and `T1`. To keep it as general as possible I would prefer to assume the cylinder is height `h` and radius `r`. I've been trying to use the thermal diffusivity equation: ![Diffusivity Eqn](http://i.stack.imgur.com/XZa4O.png) But my working gets messy quickly, I figured as this is quite a simple shape it should be less complicated than some similar examples I've found. * * * EDIT: So using my notes from a differential equations course a few years back I managed to do the following: ![Working](http://i.stack.imgur.com/0YeHf.png) But I'm stuck here..


Electrostatics:Basic question on electric current

I've got two questions for you.   1. Electric current is the flow of electrons across a conductor.Why has it always got to do with electrons and not with protons? (I know neutrons are not eligible for electricity as they carry no charge.)   2. Sometimes my computer starts properly after I disconnect all the cables and press and hold the power button for a few seconds and then re-connect all the cables (same thing with my modem). What's going on? I've heard that some amount of left over current get's drained. But where do they go? How did that excess current get accumulated in the first place?


Expectation of dipole is vector in x direction

So I am asked to find a state $|\Psi(t)\rangle$ in terms of the hydrogen wave functions, such that the expectation of the dipole operator -$e\hat r$ is a vector in the x direction. I am not completely sure how to approach this. I am having a hard time visualing what this is trying to say in a sense of wavefunctions. I know what a dipole is, but I would expect the dipole moment for hydrogen states to be a vector in two directions. Any ideas on where to start?


How exactly do protons and electrons interact with each other?

How do these particles exchange information about charge and position between themselves, even though there's mostly empty space between them? Also what happens if a free electron passes closer to a nucleus of a hydrogen atom than its own electron? My random guesses so far:   1. Proton won't react because it's somehow locked with its existing electron   2. Proton will catch it and release the previous one    3. Proton will catch it and keep both of them with half the force   4. Existing electron will simply repel incoming one   5. Something else?