Vertical and horizontal components of forces and vectors

I'm getting a bit confused when finding components of vectors and forces. In problems for vectors, I've always known that if you want to get the components of a vector, you would use the following: ![enter image description here](http://i.stack.imgur.com/UWNQM.png) * * * Recently we've started working with Forces in homework problems. This diagram is from one of my homework problems, which they want me to draw the free body diagram: ![enter image description here](http://i.stack.imgur.com/4zGQN.gif) Here is the free body diagram given by the homework solution: ![enter image description here](http://i.stack.imgur.com/KGMsO.gif) * * * What I don't understand is why would I be using $m_{2}gsin(\theta)$ instead of $m_{2}gcos(\theta)$ when finding the X-component of the gravitational force of $m_{2}$. Force _is_ a vector, so why wouldn't you use $m_{2}gcos(\theta)$? Are the rules different when dealing with forces?


Creating small polarised scientific lenses?

How would I go about creating small polarised scientific lenses? Thinking 5mm by 5mm, with a light-source behind it for the purposes of reducing (polarised) surface reflection (assisted with a cross-polarised viewer). Is the idea that I just need to apply magnesium fluoride; and if so, at what consistency and against what type of lens?


Don't understand the integral over the square of the Dirac delta function

In Griffiths' _Intro to QM_ [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on p. 101). He then says that these eigenfunctions are not square integrable because $$\int_{-\infty}^{\infty}g_{\lambda}\left(x\right)^{*}g_{\lambda}\left(x\right)dx ~=~\left|B_{\lambda}\right|^{2}\int_{-\infty}^{\infty}\delta\left(x-\lambda\right)\delta\left(x-\lambda\right)dx ~=~\left|B_{\lambda}\right|^{2}\delta\left(\lambda-\lambda\right) ~\rightarrow~\infty$$ (cf. second formula on p. 102). My question is, how does he arrive at the final term, more specifically, where does the $\delta\left(\lambda-\lambda\right)$ bit come from? My total knowledge of the Dirac delta function was gleaned earlier on in Griffiths and extends to just about understanding $$\tag{2.95}\int_{-\infty}^{\infty}f\left(x\right)\delta\left(x-a\right)dx~=~f\left(a\right)$$ (cf. second formula on p. 53). References:   1. D.J. Griffiths, _Introduction to Quantum Mechanics,_ (1995) p. 101-102.


How accurate are constants in cgs units?

I am just curious that if constants in cgs units change the answer of an equation. For example, Coulomb's constant, in SI units it equals to $8.98...\times 10^9 \,\mathrm{N\,m^2\,C^{-2}}$. However in cgs units it equals to 1. I think the difference between an answer calculated with Coulomb's constant in SI units and Coulomb's constant in cgs units would be a lot. I don't know if my logic is right.


How does electron gun accelerates electrons?

I know that in electron guns we see in TV's and lots of other places, we have electron emitter (cold/hot W needle in the simplest case) and electrons are accelerated using lattice with high-voltage potential. But the question is why doesn't this lattice slow down electrons once they passes through it? PS. Please correct if lattice is not the correct word here :-D


Quantum shot-noise and the fluctuation dissipation theorem

Classically, shot noise observed in the signal generated by a laser incident on a photodiode is explained as being due to the quantization of light into photons, giving rise to a Poisson process. In quantum optics, on the other hand, the shot noise is said to arise from interference with the vacuum field, which leaks in at points of optical loss (absorption). Meanwhile, mechanical oscillators are subject to the fluctuation-dissipation theorem, which says, roughly, that the thermal excitement of the various mechanical modes of the oscillator are proportional to the dissipation in that mode. In the context of electronics, Johnson noise is an example of this effect. I recently read a thought-provoking statement (in this paper) that these effects are really "the same"; just as thermal motion arises at points of mechanical dissipation, shot noise arises at points of optical dissipation. **Is this analogy physically/theoretically significant?**


Best way to keep food cool indoors without using a fridge/freezer

I've just moved into my first place, and for reasons I do not condone or recommend I don't have any method of refrigeration right now. Luckily I now live very close to shops, so it isn't a big deal, but it made me think: if I really had to, what would be the most efficient way to keep food cool without using any electrical (or any other means which would be impractical) method of refrigeration?


How to prove that we are living in a 3D world?

Is there any scientific experiment that can lead us to conclude we live in 3 dimensions without the premise of the conception of limited dimensions? Thank you all who helped in the improvement of this question (which was not clear at first).


Can classical orders coexist with quantum orders?

For example, the ground state of the antiferromagnetic(AFM) Heisenberg model $H=J\sum_{<ij>}\mathbf{S}_i \cdot \mathbf{S}_j(J>0)$ on a 2D square lattice is a Neel state, which is a classical order described by conventional order parameters $\left \langle \mathbf{S}_i \right \rangle$. On the other hand, if we use the Schwinger-fermion mean-field theory to study $H$, we will get a mean-field Hamiltonian $H_{MF}=\sum (f_i^\dagger \chi_{ij} f_j+f_i^\dagger \eta_{ij} f_j^\dagger+H.c.)$ which is studied in Wen's papers of PSG, and now we can study the PSG of the mean-field ansatz $(\chi_{ij},\eta_{ij})$ and associated quantum order. So according to the above example, can we say that the AFM SDW phase(classical order) _possesses_ the quantum order(PSG)? Thank you very much.


Tycho's stellar parallax measurements

How, exactly, did Tycho Brahe do his measurements of (no) stellar parallax? All the descriptions of parallax that I have been able to find seem to talk about the change in position of a nearby star compared to background stars farther away, but as far as I know the Ptolemaic system assumed that the fixed stars were all at the same distance? I guess I am looking for a description of how parallax measurements where actually historically done at Tycho's and Copernicus's time.


Breaking of conformal symmetry

I am wondering something about the breaking of conformal symmetry: I know that it can be broken at the quantum level, anomalously, but I never encountered or heard about a model where it is broken "à la Higgs" with a potential whose true minimum would spoil this invariance (e.g. making appear a particular energy scale). I guess we would then get some Goldstone bosons, would there be something special about them?


Yearly onshore wind turbine energy production

I am trying to work out the yearly energy production of an onshore Siemens SWT-3.0-101, hub height 94m, total height 144,5. Here is the power curve at each wind speed level worked out through a software: ![power curve](http://i.stack.imgur.com/s9yri.jpg) I am trying to work out the average yearly production at wind speed= 8 m/s. My calculations are the following: 1138(=Kw at 8 m/s wind speed)*3600(=sec in 1hr)=4096800 KW/h; 4096800Kw/h*6765,81(hours in one year)=27718170408Kw/h in one year=27718,17GW/h in one year. This is my result, but everywhere is said that an yearly yeld of an onshore wind turbine stays around 5.5-6 GW/h, which is very far from my result. Where am I wrong? Thank you very much in advance for your help


the search for calabicles

Pop-sci ST books say, "At every point of space, there is a Calabi-Yau space..." or similar. But uncertainty p. and noncommgeom say, no points in space, only planckons (not my coinage). Is 'point of space' (meaning real, not abstract/mathematical space) inexact terminology, or a category error, like "an atom of mayonnaise'- only worse? So I speculate that C-Y space can not be a continuum either, but must consist of discrete calabicles (my coinage).


What are the current experimental restrictions of the possible speeds of gravitation?

Somewhere I read that the Hulse-taylor binary pulsar can not differentiate between competing theories assuming different speeds of gravity. Is it mathematically true in general, that the orbital decay of two orbiting bodies cannot possibly favor certain speeds of gravity over others? If so, why is it the case? Any references would be much appreciated! Edit: I found the page where I read it: > The speed of gravitational radiation ($C_{gw}$) depends upon the specific > model of Gravitation that you use. There are quite a few competing models > (all consistent with all experiments to date[~1992]) including of course > Einstein's but also Brans-Dicke and several families of others. All metric > models can support gravity waves. But not all predict radiation travelling > at $C_{gw} = C_{em}$. ($C_{em}$ is the speed of electromagnetic waves.) So, **is it possible to construct a metric theory of gravitation with arbitrarily chosen $C_{gw}$** , so that it will be consistent with orbital decay experiments to date? If not, what is the possible range. I believe it must have been discussed in scientific literature, but I can't find anything. I know that it can be a very hard question to answer with mathematic rigor, but a non-technical overview should be sufficient.


A force opposing Gravity

_Every Action Has An Equal and Opposite Reaction_ (Newton's Third Law.) If this is the case, does gravity have an equal-opposing force? From asking around I still haven't got a very clear answer; those who I've talked to seem to believe there isn't one - that gravity is actually a singularity [a one way force] which somehow "just works", others think it differently - believing there **is** an opposing force of which prevents gravity from compressing masses more than it already does. So which one is the right answer? (if either!)


Why are there no true oneway mirrors?

"One way mirrors" are used in interrogation rooms etc. The wikipedia article states that: > A true one way mirror does not, and cannot, exist. Light always passes > exactly equally in both directions. However, when one side is brightly lit > and the other kept dark, the darker side becomes difficult to see from the > brightly lit side because it is masked by the much brighter reflection of > the lit side. Which I think I understand. What I find strange is that this surely can't be any deep principle of physics, as it's easily "violated"; you could just place a camera on one side and a TV on the other side, and it would be perfectly "see-through" from one side but not the other. Couldn't you imagine a sort of microscopic version of the above "violation"?


Trapping EM radiation

Is there a material which can allow light (or any other EM radiation) to pass through from one side as if it is transparent but its other side reflects light like a mirror? ![material](http://i.stack.imgur.com/SSOC7.png)


Physics-oriented books on fractals

I'm looking for some good books on fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis on the physics, even at the expense of mathematical rigor. Hope that was clear and specific enough.


What is the physical significance of the fermi-field asymmetric stress-energy tensor?

Using the ideas from a previous question here it can be shown that if one takes the boson spin 1 stress-energy tensor of the form \begin{align} T^{\mu\nu}_{\text{spin one}} = \begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) && S_x/c && S_y/c && S_z/c \\\ S_x/c && -\sigma_{xx} && -\sigma_{xy} && -\sigma_{xz} \\\ S_y/c && -\sigma_{yx} && -\sigma_{yy} && -\sigma_{yz} \\\ S_z/c && -\sigma_{zx} && -\sigma_{zy} && -\sigma_{zz} \end{bmatrix} \end{align} created from a boson spin 1 electromagnetic field $F=E+iB$ and changes the spin of $F$ from spin one to spin half, $\\{F,A\\} \rightarrow \\{ \phi_+ , \phi_- \\} $ using here, then $T^{\mu\nu}_{\text{spin one}}$ changes to \begin{align} T^{\mu\nu}_{\text{spin half}} = mc^2 \begin{bmatrix} I && 0 && 0 && i\gamma^{0123} \\\ \gamma^{01} && 0 && 0 && i\gamma^{23} \\\ \gamma^{02} && 0 && 0 && i\gamma^{31} \\\ \gamma^{03} && 0 && 0 && i\gamma^{12} \end{bmatrix} , \end{align} which I can only understand $T^{\mu\nu}_{\text{spin half}} $ as a fermion spin half stres-energy tensor. What is the physical significance of $T^{\mu\nu}_{\text{spin half}} $ compared to $T^{\mu\nu}_{\text{spin one}}$? I am struggling to understand how to interpret $T^{\mu\nu}_{\text{spin half}} $. This fermi-field stress-energy tensor is asymmetric. The divergence of $T^{\mu\nu}_{\text{spin half}} $ is zero.


Can someone please explain the "infrared catastrophe"?

In my readings I've run into this idea of an "infrared catastrophe" associated with 1/f noise. As far as I can tell it is because when you graph the periodogram of the 1/f signal you see the PSD goes to infinity as frequency goes to 0. Not sure what that means practically though. If we are talking about a sound wave, does that mean the sound becomes infinitely loud at low frequencies? What is the "catastrophe"?


Does the Night Mode of the screen display (LCD) save more energy?

In some cases, we can enable the Night Mode (reversing the bright and dark color of the display; such as White Text, Black Background) for the screen display. LCD(Liquid-crystal display) seems to be a common monitor display devise nowadays, so let us consider the LCD at this moment. > (1) Will the Night Mode save more energy? > > (2) How can we evaluate the energy efficiency ratio between, for example > **the Day Mode and the Night Mode**? > > (3) A bonus question: whether the energy efficiency (of Day Mode and Night > Mode) varies for CRT(Cathode ray tube), LCD, Plasma, and OLED(organic light- > emitting diode)? ![enter image description here](http://i.stack.imgur.com/SuwPE.jpg)


What does this feynman diagram represent?

This feynman diagram represents the elastic proton scattering of electrons. ![enter image description here](http://i.stack.imgur.com/gKizX.png) The $e^-$ has an initial momentum $k$ and a final $k^\prime$. The circle represents that the target(the proton that is) has a structure and $q$ is the momentum transfer between the electron and the proton, which is mediated by a photon. What does the $p$ and $p^\prime$ lines mean?


Reliability of time dependent measurements

How is a "baseline" established. When an experiment is performed to verify the reliability of something like carbon dating. How do we know the results we get over the period of the test is still valid for an item older than one billion years?


Changing Momentum

![](http://i.stack.imgur.com/ohx7t.jpg)![](http://i.stack.imgur.com/yLA6b.jpg) Okay, so I am having a hard time wrapping my head around this. Before looking at the answer, I worked the example and got the answer only because I used dimensional analysis to get me to the right units for force. However, to be honest, I don't think I even understand the concept here. I know mass is changing and velocity is constant, but what is throwing me off is where it says "The water strikes the window at $32\, m/s$ so each kilogram of water loses $32\, kgm/s$ of momentum. Water strikes the window at the rate of $45\,kg/s$, so the rate at which it loses momentum to the window is $1400\, kgm/s^2$" How is it losing $32\, kgm/s$ of momentum and then at the same time also losing it at a rate of $1400\,kg m/s^2$? Why does it even mention "The water strikes the window at $32\,m/s$ so each kilogram of water loses $32\,kgm/s$ of momentum" what is the purpose of that sentence. That is really confusing me. **EDIT:** I added this simple diagram to show how I see this system. Where is $32\,kgm/s$ of momentum come in? (I am aware that the window is now showing)


What temperature will create amorphous TiSi?

At what temperature will create amorphous TiSi? I think it's around 500 Celsius but I'm not sure.


Problem in applying Takens correlation dimension

I am trying to implement Takens' correlation dimension for noisy time series formula given in Eq (8) in (Estimation of the dimension of a noisy attractor. Schouten JC, Takens F, van den Bleek CM Physical Review E, vol 50, number 3, 1994) . The actual derivation and explanation is given in Takens' maximum likelihood estimation (F. Takens, On the numerical determination of the dimension of an attractor, in: D. Rand, L.S. Yong (Eds.), Dynamical Systems and Turbulence, Warwick, 1980, Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 1981.) and summarized below according to my understanding. My time series are noisy with additive white Gaussian noise, `Z = x+noise` Step1 = Perform phase space embedding of the 1D time series. This becomes U. Step2: Find euclidean distances or maximum norm distances of the higher dimensional space. Find a scaling distance, $l_0$ (or $r_0$) called as the maximum scaling distance. Step3: Then find those distances, $l_z$, which are $\le l_0$. This step converts distances into $l_z$ that now belongs to range $[0,1]$. Sometimes, the maximum scaling length $l_0$ is taken as the $l_0$ = average absolute deviation of the time series by the formula $l_0 = 1/{\rm total\ elements}[ {\sum (u_i - {\tt mean}(u)}]$. Step 4: In Takens' paper, it is mentioned under Section2, that the distances can be made independent of $l_0$ by dividing all distances $l_z$ by $l_0$. Step5: Take the logarithm of each element so found by Step 4. ${\rm Takens\ dimension} = -(\frac{2}{n(n-1)} \sum [\log (||X_i- X_j||/r_o)]^{-1}$ (Formula for Takens) $n$ = number of samples. **Questions:** (A) It is not clear to me what should be $l_0$ - the upper bound of the scaling distances. Due to the term "distance", I initially thought them to be the average standard deviation of the euclidean distances calculated. But, then this doe snot follow what is mentioned under Eq (12). When I took $l_0$ as the average absolute deviation of the time series and divided the distances by it, the result of correlation dimension is incorrect. So, if I follow what is mentioned in the paper1, then the value of the dimension is coming out to be incorrect. What should be $l_0$ ? (B) How do I formulate the dimension for noisy case , I mean will the final formula change and do I have to include the noise effects as done by Ellner http://rullf2.xs4all.nl/msct/node28.html I really do not understand what should be $l_0$, the upper bound of scaling distances. Should be the maximum distance, or the maximum standard deviation of the distance vector or the average standard deviation of the time series ? Please help, I am not sure if I understood correctly.


Consequences for causality if superluminal neutrinos were explained by extra dimensions

One suggestion for explaining superluminal neutrinos (assuming for the sake of argument that the OPERA results should hold up) is that the neutrinos have taken a route through extra dimensions, with an appropriate metric of the form $\mathrm{d}s^2=-\mathrm{d}t^2 + f(r)\mathrm{d}x^2 + \mathrm{d}r^2$. This would be consistent with ordinary physics, but we would still have faster-than-light travel in our 4 dimensions, so what would be the consequences for causality?


Boundary conditions on wave equation

![wave equation question](http://i.stack.imgur.com/cvSGd.png) I am having trouble understanding the boundary conditions. From the solutions, the first is that $D_1(0, t) = D_2(0, t)$ because the rope can't break at the junction. The second is that $\dfrac{\partial D_1}{\partial x} D_1(0, t) = \dfrac{\partial D_2}{\partial x}(0, t)$. How can I interpret this physically? I'm not quite sure how to think about $\partial D /\partial x$.


Energy conserved... or not? Confused!

I am confused. Could someone kindly explain what's going on in this question? > A particle of mass $m$ and charge $e$ moves in the $x,y-$ plane. There is a > constant magnetic field $B$ that points in the $z$ direction. Initially the > particle has energy $\gamma m$ in the rest frame of the observer. After a > time elapse of $\Delta t$, the observed energy is $\gamma' m$ (again in the > rest frame of the observer). How are $\gamma,\gamma',t$ related? Some thoughts: Since we have an accelerated charge, energy is radiated. Since the question does not specify that the motion is non-relativistic, I shall use the relativistic equations. Larmor's formula: $$P=\frac{\mu_0 q^2}{6\pi m^2}\frac{dp^a}{d\tau}\frac{dp_a}{d\tau}$$ To get $\frac{dp^a}{d\tau}$ I need the equation of motion: $$\frac{dp^a}{d\tau}=eF^a{}_b\frac{dx^b}{d\tau}=\frac{e}{m}F^a{}_bp^b$$ where $$F^a{}_b=\begin{pmatrix}0&0&0&0\\\ 0 &0& B&0\\\0 &-B& 0&0\\\0&0&0&0\end{pmatrix}$$ This gives 4 coupled ODEs. Actually the first 2 are trivial such that $$p_0=p_{0, init}\\\p_4=p_{4,init} $$ and $$\frac{dp_1}{d\tau}=\frac{qB}{m}p_2\\\ \frac{dp_2}{d\tau}=-\frac{qB}{m}p_1$$ But then doesn't $$p_0=p_{0, init}=\gamma m$$ say that the energy is conserved? It is also not physically surprising if it were since the magnetic field induces a force perpendicular to the velocity of the particle, therefore does no work on it. What's gone wrong? * * * **[Update 1]** With Mike's and Michael's help, I have proceeded to figured out how to do it for the non-relativistic case (but I'm not very sure how to extend it to the relativistic case :-/ ) -- if I'm not mistaken, the NR case should be something of the following: $$F=e \vec v\times \vec B$$ Assuming circular motion and the magnetic field being perpendicular to the plane of motion, $$r=\frac{mv_{init}}{eB}$$ Therefore acceleration is $$a=\frac{v^2}{r}$$ Then I can substitute it into Larmor's formula and so on. How do I generalize it to the relativistic case? **[Update 2]** Do I simply add a $\gamma$ factor in front of the $v$'s?


Question about Classical Transport Theory

With a distribution function of the form $f=f_{0} + \vec{v} \cdot \vec{g}$, one can obtain the current density. My question is about $\vec{g}$; we assume a general solution to $\vec{g}$ of the form $\vec{g}= \alpha \vec{E}+\beta \vec{B} + \gamma(\vec{B} \times \vec{E}) $. By substitution into $$\frac{\mu}{c}(\vec{v} \times \vec{B})\cdot \vec{g} + \vec{v} \cdot \vec{g} = e \tau \frac{\partial f_{0}}{\partial \epsilon} \vec{E} \cdot \vec{v}$$ and by equating both sides and equating the coefficients of $\vec{v} \cdot \vec{E}$, $\vec{v} \cdot \vec{B}$, and $\vec{v} \cdot (\vec{B} \times \vec{E})$ we are supposed to get $$\vec{g} = \frac{\partial f_{0}}{\partial \epsilon} \frac{e \tau}{1+ (\omega_{c}\tau)^{2}}[\vec{E}+(\omega_{c}\tau)^{2}(\hat{z}\cdot\vec{E})\hat{z}+\omega_{c}\tau(\vec{E} \times \hat{z})]$$ My question is how? I'm afraid this isn't apparent to me. I don't know where to start to try and prove this!


Focusing Light with Flat Mirrors

What is the best way to focus (sun)light using flat mirrors? My goal is to start a fire. Cutting the mirrors is easy.


The time component is $\gamma m c$, so shouldn't $E=mc$?

Basically, the book is Brian Cox's _Why Does $E=mc^2$?: (And Why Should We Care?)_. I just finished Chapter 5, where we derived the spacetime momentum vector (energy-momentum four vector, as he establishes the physics jargon). Let $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ So, as we found out, the vector's spatial component is $\gamma m v$, leaving the time component as $\gamma m c$. He went on, under the guise of making the outcome more intellgible, to saying we could happily multiply the time component by $c$ without changing it's meaning. Cool, I thought, no problem. Next, he pointed out $\gamma\approx1+\frac{1}{2}\frac{v^2}{c^2}$, so $\gamma m c^2 \approx mc^2 + \frac{1}{2}mv^2$. Et voilà, $mc^2$. Granted, he's obviously trying to simplify things so I can reach an intuitive understanding, but from that point onwards, he uses $mc^2$ as the conversion value. I'm confused. Could someone explain why we use $mc^2$ and not the version scaled down by a factor of $c$?


Can we solve the particle in an infinite well in QM using creation and annihilation operators?

The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can be solved elegantly algebraically using the creation and annihilation operators to find its spectrum. Is it possible to do the particle in a box problem using creation and annihilation operator and how?


What is Schrödinger's wave equation?

What is Schrödinger's wave equation? We recently studied about the Schrödinger wave equation in school. I understand that because of Heisenberg's Uncertainty principle, and De Broglie's Hypothesis, it was not possible to consider an electron as entirely a particle, and also know about it's velocity and position accurately. I also know the bare basics of the Schrödinger equation, and that it tells us what the probability of finding an electron within an atom. I really need to know how the quantum numbers come into the picture, and what the mathematical aspect to the equation is. I'm not looking for derivations, and just want to understand the background of the equation. Lastly, i know that Psi is the amplitude of the electron wave, which is essentially a length. Length should ideally be a positive rational number, however small. Why then, is Psi, a complex number? Also, it would be much appreciated if the explanation was a bit simple. Thanks a lot. I really appreciate any help you can provide to me.


Kepler problem in time: how do two gravitationally attracting particles move?

Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that $$ {d^2r_1\over dt^2} = - { m_2(r_1-r_2)\over |r_1-r_2|^3} $$ $$ {d^2r_2\over dt^2} = - { m_1(r_2-r_1)\over |r_1-r_2|^3} $$ (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time? After reading a lot on Wikipedia, I've come to the defintion of center of mass and relative coordinates: $$R(t) ~=~ \frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$ $$\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$ Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation?


Nature of orbit due to central force!

> **Possible Duplicate:**   >  Kepler problem in time: how do two gravitationally attracting particles > move? How do we get the shape of orbit under the condition that   1. Force is centrally directed   2. Force follows inverse square law   3. Angular momentum is conserved I read somewhere under these three constraint the orbit becomes elliptical. I tried to put together these conditions but couldn't find the solution! Any help in derivation of elliptical orbit would be appreciated. Thanks


How would a black hole power plant work?

A black hole power plant (BHPP) is something I'll define here as a machine that uses a black hole to convert mass into energy for useful work. As such, it constitutes the 3rd kind of matter-energy power (formerly "nuclear power") humans have entertained, the first two being fission and fusion. Putting aside the fact that the level of technological advancement needed for this is far beyond modern day humans, it seems to be pretty well established that this is possible (and maybe someday inevitable) among physicists. Personally, I feel like I don't understand the proposal because no one has really created a coherent picture of how it would work. To keep this focused, here are my objective questions: Related to the conversion itself:   * **What mechanism of energy release are we talking about?** Possibilities include radiation from accretion of material, Hawking radiation, magnetic conversion of rotational energy, and maybe others (EDIT: Manishearth's answer revealed that I don't know the possibilities myself). Can we rule some of these out pragmatically? How well are these mechanisms understood? Do we really have the physics/observations to back them up, or is it still on the forefront of physics?   * **How would the energy be harvested?** Some possibilities include a thermal cycle (like a Rankine cycle), radiation to electricity conversion (like photovoltaics), magnetically induced currents, and charge movements. Could the particles of emission be _too_ high energy to harness with known materials as a part of the machine structure? That would certainly apply to Hawking radiation from small BHs, do other options have the problem of overly-energetic emissions? Related to application:   * **Is there _anything_ we could use to change the Hawking radiation power output other than the BH mass?** Would we be dancing on a knife's edge by using Hawking radiation as a power source? As you should know, a BH evaporation would be a fairly major cosmic event, and to get more power out, one would have to use a smaller BH, which presents a greater risk of catastrophe. This problem could be mitigated by using several BHs so a lower energy throughput would be needed for each, but then that risks BH collision which is also a catastrophic event. Does the spin, charge, or local gravitational potential affect the output? Is there any way for Hawking radiation to be a reasonably safe method of matter to energy conversion (even for an advanced civilization)?   * **What are the parameters for the total mass and energy output?** Specifically, what would the total energy output divided by mass be, and what would the mass-to-energy conversion efficiency of new in-falling material be? For accretion, I've heard numbers like 30% of the in-falling material can be converted to energy and ejected back into space at the poles... but could we change and control that? How would a BHPP's parameters compare to our sun and current Earth-based power plants? Sorry I couldn't wrap this up into a single question. I tried. To address the "on topic" concern, I hear people mention that tokomak _fusion_ power research still has many unknowns that are in the domain of physics, as opposed to engineering, which will come later in its development. I tried to edit my question above to be actually physics in nature. I think I speak for most people when I say that I don't understand the vast majority of the _physics_ of BHs. My question here is generally how those physics would play out in a machine that does useful matter to energy conversion.


If a measured value is oscillating so fast between two numbers, what does the detector read?

There is a picture usually depicted in introductory quantum mechanics about how the probability pattern appears in the classical limit, that the oscillation is too fast and that it cannot be resolved by the experimental apparatus, hence the detector will detect only the average value. ![Image of oscillation probability](http://i.stack.imgur.com/SXnFl.gif) I do not see how the detector "decides" to give the average value if the observed value oscillate too fast between 2 numbers!


Circular motion question

> _(Q) Find the maximum speed at which a truck can safely travel without > toppling over, on a curve of radius 250m. The height of the centre of > gravity of the truck above the ground is 1.5m and the distance between the > wheels is 1.5m, the truck being horizontal._ I actually can't figure out which force acts as the centripetal force in the question. Also since it is a truck so it can't lean, so I can't apply the method we generally use in leaning of cycle. Also no friction is described here between the road and wheels. ![My progress till now](http://i.stack.imgur.com/A19c4.jpg)


Could a real-life X-Wing fly in Earth's atmosphere?

From an aerodynamic point of view, could a full-size aircraft of X-Wing design fly in Earth's atmosphere? Assuming you were free to add control surfaces here and there, could the wings in open configuration create lift ? ![X-Wing](http://i.stack.imgur.com/GS4Mt.jpg)


What is the color of Water?

What is the color of Water? I am not talking about large water bodies like Ocean or lake but a glass of water. If it has no color how can we see it?


Quantum systems with real structure

A lot has been said as to why quantum mechanics needs complex numbers. However, all measurements produce real values. Expectation values are real, the observables form a real Lie-algebra (use $-\mathrm i[*,*]$ as Lie bracket instead of the raw commutator) and any state $\mathbf z$ is uniquely determined by the real-valued function $P_{\mathbf z}(*) = |\langle \mathbf z,*\rangle|^2$ on the space of states. The space of states itself, however, is a complex projective space, and even though phase factors are irrelevant when taking expectation values, they do matter when combining states - linear superposition $α\mathbf z + β\mathbf z'$ depends on the ration $|α|/|β|$ as well as the phase difference $\arg(α) - \arg(β)$. But there is at least one important quantum system which can be modeled as a real system: the qubit, ie the space of spin states of a single electron. In addition to the standard representation as $\mathbf z=(z_1,z_2)\in\mathbb C^2,\langle \mathbf z,\mathbf z\rangle =1$, states can also be represented as $\vec z\in\mathbb R^3,\vec z^2=1$ via $$ \vec z = \begin{pmatrix} \langle\mathbf z,\pmb σ_1\mathbf z\rangle\\\ \langle\mathbf z,\pmb σ_2\mathbf z\rangle\\\ \langle\mathbf z,\pmb σ_3\mathbf z\rangle \end{pmatrix} = \begin{pmatrix} 2\Re(\bar z_1z_2)\\\ 2\Im(\bar z_1z_2)\\\ |z_1|^2 - |z_2|^2 \end{pmatrix} $$ where $\pmb σ_1, \pmb σ_2, \pmb σ_3$ are the Pauli-matrices. This is the well-known Hopf fibration. What makes this useful is the fact that $P_z$ takes a particular simple form: $$ P_{\mathbf z}(\mathbf z') = \frac 12 (1 + \vec z*\vec z') = \frac 12 (1 + \cos\measuredangle(\vec z,\vec z')) $$ This also means that states $\mathbf z,\mathbf z'$ are orthogonal in $\mathbb C^2$ exactly when $\measuredangle(\vec z,\vec z')=180°$ in $\mathbb R^3$, ie $\vec z'=-\vec z$ The expection value of any observable $\mathbf A$ can be expanded in terms of $P_{\mathbf a_i}$ after a choice of orthonormal eigenvectors $\\{\mathbf a_i\\}$, ie $$ \langle\mathbf z,\mathbf{Az}\rangle = \sum_i α_iP_{\mathbf a_i}(\mathbf z) $$ where $\\{α_i\\}$ denote the eigenvalues. Substituting our definition of $P_{\mathbf a_i}$ and using the fact that $\mathbf a_1=-\mathbf a_2$, this is simply $$ \langle\mathbf z,\mathbf{Az}\rangle = \frac 12(α_1 + α_2) + \frac 12(α_1-α_2)\,\vec a_1*\vec z $$ Restricting ourselves to $\mathfrak{su}(2)$, ie the traceless matrices with $α_2=-α_1$, yields $$ \langle\mathbf z,\mathbf{Az}\rangle = \vec A*\vec z $$ where $\vec A=α_1\,\vec a_1$ is the representation of our observable. We arrived at a quantum system which can be described with nothing more than school math. After this lengthy exposition, finally my question: Are there any other quantum system with a similar real structure? If not, is there a particular reason why? ### Update To get a better understanding of the problem, I read up on complex manifolds, and it seems that the qubit is indeed special. In particular, the projective spaces $P_n\mathbb C$ _are_ complex manifolds, but spheres $S^k$ do not even permit almost-complex structures for $k\not= 2,6$ (Borel, Serre 1951) and are out as state spaces in the general case. The finite-dimensional complex projective spaces can be realized as different homogeneous spaces. In particular, we have the obvious $$P_n\mathbb C \cong \mathbb C^{n+1}\setminus\\{0\\}\:\big/\:\mathbb C^*$$ the not-so-obvious $$P_n\mathbb C \cong U(n+1)\:\big/\:U(n)\times U(1)$$ and the one which is probably the most clear-cut $$P_n\mathbb C \cong S^{2n+1}\:\big/\:S^1$$ but does not convey that we are dealing with complex spaces. This quotient also only ever results in a sphere for $n=1$, ie our qubit case. The Wikipedia page on Hopf fibration links to this paper, where the fibration $$S^7\xrightarrow{S^3}S^4$$ is used to model the state space $P_3\mathbb C$ of two qubits. Even though there _might_ be some insight into the structure of the state space in this fibration, it is less natural (and less useful, for that matter) than the single qubit Hopf fibration $$S^3\xrightarrow{S^1}S^2$$ where actually $S^2 \cong P_1\mathbb C$, whereas obviously $S^4\not\cong P_3\mathbb C$ as they do not even have the same dimensions when considered as real manifolds.


Factors that make beam divergence worse after refocusing

A beam of light of width $W$ and wavelength $\lambda$ with divergence that is diffraction-limited is refocused with an optical element placed at a distance $D$ from the beam source. Will the refocused beam have worse divergence than the one imposed by diffraction limits? how much worse? Is there an optical way to improve the beam divergence of the refocused beam? If a 2nd refocusing element, very similar to the 1st one, is placed on front of it, will the divergence get progressively worse? by how much? Finally, does it matter if the beam is a Gaussian or Fresnel beam?


How can we get this formula for Gravitational potential?

I just read the wikipedia page http://en.wikipedia.org/wiki/Gravitational_potential But I don't understand how to get this formula: $$\rho(\mathbf{x}) = \frac{1}{4\pi G}\Delta V(\mathbf{x})$$ Can anyone tell me why?


Show that the energy levels of a particle in a specific potential are $E_n=(n+\frac{1}{2})h\omega-\frac{1}{2}\frac{F^2}{m\omega^2}$

A particle of mass m moves on the x-axis under the inﬂuence of the potential $$V(x)=\frac{1}{2}m\omega^2x^2+Fx$$ Can anyone help me, using Schrödinger's equation in one dimension that the energy levels are: $$E_n=(n+\frac{1}{2})h\omega-\frac{1}{2}\frac{F^2}{m\omega^2}$$ Where n is a non negative integer?


Statistical interpretation of Entropy

I'm preparing my statistical physics course, and while writing the lecture notes it says that a system with non distinguishable particles has much less microstates asociated with a particular macrostate. Hence, a system with distinguishable particles has much more microstates asociated with a particular microstate. Entropy $S$ is related with the number of microstates $\Omega$ via: $$S=k \ln (\Omega) $$ where $k$ is Boltzmann's constant. Then my question is **Given the same macrostate for both systems, the entropy of the first is much lower than the entropy of the second?** Thanks for your time.


Electric field and electric potential of a point charge in 2D and 1D

in 3D, electric field of a piont charge is inversely proportional to the square of distance while the potential is inversely proportional to distance. We can derive it from Coulomb's law. however, I don't known how to derive the formula in 2D and 1D. I read in a book that electric potential of a point charge in 2D is proportional to the logarithm of distance. How to prove it?


Is there an atomic level explanation for diffraction of light around opaque aperture?

It has always been a hard time understanding the phenomenon of diffraction of light around opaque objects. Hoe does it happen? Why does transparent objects do not diffract light? What quantum mechanics operate that bends light? I used to have an intuitive understanding that surface atoms of the aperture edges reflect the light that interferes and forms the fringes.


Drake's equation

Some years ago i read about this equation that helps us estimate the number of active, communicative extraterrestrial civilizations in the Milky Way galaxy. ![enter image description here](http://i.stack.imgur.com/yqMHa.png) Now, i am wondering if there are results from Drake's equation (given that some parameters are extremely hard to be calculated) and if do these reflect the real world?


Optical devices for micro meter photons?

Are there all the common optical devices like beam splitters, mirrors, polarizers, fibers, etc for photons with a wavelenght in the range of micro meters? If so, how broad can the bandwidth be for the optical devices to work properly? Consider for example a photon with a mean frequency of $10^{12}\,\text{Hz}$ (i.e. a wavelength of $300\,\mu\text{m}$). Will problems occur if the bandwidth of the photon is also in the range of $10^{11}\,\text{Hz}$ or $10^{12}\,\text{Hz}$?


Neil deGrasse Tyson says that electrons "teleport" between energy levels?

This page: https://blog.afach.de/?p=62 Discusses the error Neil deGrasse Tyson made when talking about electronic transitions (video included there). Tyson clearly said in his Cosmos series that electrons disappear from one level and appear in the other one. The guy in the page discusses that this is wrong, because there is no proof of that, and because it breaks simplest rules of relativity, and because it contradicts the simplest quantum model. Is his argument correct? And why would Tyson do such a horrible mistake? For more information please visit the link.


Does the possibility of large scale entanglement mean 2-Body Problems are also unsolvable?

Experiments are showing that larger and larger objects can be entangled whereby proving that this quantum feature has no upper limit. Assuming this is true, does entangled celestial bodies mean even the 2-Body and not just N-body problems are also unsolvable?


How do you tell what forces do no work?

> The total mass of the children and the toboggan is 66 kg. The force the > parent exerts is 58 N (18 degrees above the horizontal). What 3 forces/ > components do no work on the toboggan? I said the vertical component of the force being applied, since the displacement and the force will be perpendicular. I also said gravity and the $x$ component of the force being applied since it will also be perpendicular. I don't know any other things that will make work $0$.


How can the Earth keep spinning with a liquid core?

In regards to the 'conservation of angular momentum' being the explanation of why celestial objects spin... If you fill a ball or any other container with a liquid and try to spin it, you will not see any more than 5 or 6 revolutions because of the frictional losses of the liquid inside the container. I first discovered this during lunch in grade school when I tried to spin my milk carton by throwing it up into the air while giving it a 'spin'. At best, I could get only 3 rotations out of it. This same principle can be seen when you spin a raw egg and a hard boiled egg on a table top. The cooked 'solid' egg will spin the raw one won't. Because of my early experiences with milk cartons and eggs, somewhere in the intervening years I found it hard to believe some of the accepted 'facts' about our planet: So here is my question: **If the age of the earth 4.5 BILLION years, how can it be spinning freely in space with a liquid or semi liquid core for that length of time?** Combine the effects of the liquid core with the effects of liquid oceans and a gaseous atmosphere, all of which are creating resistance to rotation, these frictional losses would have stopped any rotation long ago. If the earth had a solid core, I could understand it... If the earth was less that 4.5 billion years old I might understand it... But given the accepted age of 4.5 billion years with a liquid core and a fluid outer shell I say there is a fly in the ointment somewhere!


Local fermionic symmetry and GS action

I have a trouble understanding an argument which I think has a simple answer but I am not getting it. The question is that if you don't impose local fermionic symmetry the GS action has only one term and one of the equation of motion looks as $Γ\cdot p\partial_0\psi=0$, where $\psi$ is a Majorana-Weyl spinor in 10 dimensions and $Γ$ are just generalized gamma matrices. Now $(Γ\cdot p)^2 = p^2= 0$ for the on-shell condition, but now they say that since the square is zero hence half of the eigenvalues of $Γ\cdot p$ is zero and this is where I am not getting it. If the square of a matrix is zero then how can we say that half of it's eigenvalues are zero. Since the eigenvalues of a squared matrix is square of the eigenvalues of the matrix whose square is taken, so how can we say that half of the eigenvalues are zero? May be it can happen when spinor contain complex component but here the spinor is Majorana too ( thinking in some different way that how it will impose constraint on half of the component of $\psi$ but this is not working either). (For reference, see chapter 5 of GSW Vol.1.)


Derivation of Universal Variables

Does anyone know of link to the derivation of the Universal Variables? This is in regards to Kepler's equations. The Wikipedia link didn't have enough information for me, and the Colorado page doesn't explain how they came up with $S$ and $\chi$. A universal variable is the universal anomaly in the universal Kepler equation. It is traditionally marked as chi. There is a derivation in Battin books `An Introduction to the Mathematics and Methods of Astrodynamics`.


Could a hard drive actually have been erased as described in Cryptonomicon?

In chapter 80, _The Primary_ , Neal Stephenson has one of his character describe how the drives of a computer carried through a door (by the police, who where raiding the facility) will have been erased.   Basically, there's an enormous electromagnet around the door frame: > Cantrell is now drawing an elaborate diagram, and has even slowed down, > almost to a stop, the better to draw it. It begins with a tall rectangle. > Set within that is a parallelogram, the same size, but skewed a little bit > downwards, and with a little circle drawn in the middle of one edge. Randy > realizes he’s looking at a perspective view of a door-frame with its door > hanging slightly ajar, the little circle being its knob. _STEEL FRAME_ , > Cantrell writes, _hollow metal channels_. Quick meandering scribbles suggest > the matrix of wall surrounding it, and the floor underneath. Where the > uprights of the doorframe are planted in the floor, Cantrell draws small, > carefully foreshortened circles. _Holes in the floor_. Then he encircles the > doorframe in a continuous hoop, beginning at one of those circles and > climbing up one side of the doorframe, across the top, down the other side, > through the other hole in the floor, and then horizontally beneath the door, > then up through the first hole again, completing the loop. He draws one or > two careful iterations of this and then numerous sloppy ones until the whole > thing is surrounded in a vague, elongated tornado. _Many turns of fine > wire_. Finally he draws two leads away from this huge door-sized coil and > connects them to a sandwich of alternating long and short horizontal lines, > which Randy recognizes as the symbol for a battery. The diagram is completed > with a huge arrow drawn vigorously through the center of the doorway, like > an airborne battering ram, labeled B which means a magnetic field. _Ordo > computer room door_. > > "Wow," Randy says. Cantrell has drawn a classic elementary-school > electromagnet, the kind of thing young Randy made by winding a wire around a > nail and hooking it up to a lantern battery. Except that this one is wound > around the outside of a doorframe and, Randy guesses, hidden inside the > walls and beneath the floor so that no one would know it was there unless > they tore the building apart. Magnetic fields are the styli of the modern > world, they are what writes bits onto disks, or wipes them away. The > read/write heads of Tombstone’s hard drive are exactly the same thing, but a > lot smaller. If they are fine-pointed draftsman’s pens, then what Cantrell’s > drawn here is a firehose spraying India ink. It probably would have no > effect on a disk drive that was a few meters away from it, but anything that > was actually carried through that doorway would be wiped clean. Between the > pulse-gun fired into the building from outside (destroying every chip within > range) and this doorframe hack (losing every bit on every disk) the Ordo > raid must have been purely a scrap-hauling run for whoever organized it - > Andrew Loeb or (according to the Secret Admirers) Attorney General > Comstock’s sinister Fed forces who were using Andy as a cat’s paw. The only > thing that would have made it through that doorway intact would have been > information stored on CD-ROM or other nonmagnetic media, and Tombstone had > none of that. Would this actually work?


Interpretation of theoretical Schrodinger's cat experiment

I do appreciate how to interpret the famous theoretical experiment of Schrodinger's cat experiment in the following situation: > _A Lab has a closed box with a dead or live cat and and a person who have no > idea what has inside that box open it for the first time and nobody has > opened it before. How its conscious cood make a choice - dead or live cat - > if he even don't know that there is a cat inside it?_


Proton scattering off electrons

I'm looking at a question which starts off as: A proton is travelling through a material and scattering the electrons in the material. Express the scattering angle in terms of the impact parameter, $b$, ther reduced mass $\mu$, the relative speed $v$, and the scattering angle in the ZMF. Hence show that the momentum transfer is $q=\frac{2\mu{v}}{\sqrt{1+z^2}}$ where $z=b\mu{v^2}/\alpha$. My attempt: Somethings are confusing me straight off, I'm not sure whether this is inelastic scattering or elastic scattering and I'm not sure whether $v$ is the relative speed in the LAB or ZMF frame. Also, assuming that $\alpha$ is meant to be the fine structure constant, this means that $z$ has units which seems a bit of an issue to me... I've also assumed that this is all none relativistic. By looking at Rutherford scattering, we have that (in the electron rest/LAB frame) the scattering angle $\chi$ is given by $cot(\frac{\chi}{2})=\frac{mv^2b}{\alpha\hbar{c}}$ I have assume that $v$ is the relative speed in the LAB frame (not sure how correct that is). From here I'm really not sure where I am meant to be going, I am pretty sure that this is pretty basic mechanics but I can't see how I should be going about this so any tips/ advice would be much appreciated!


Speed and transparency of light

I have been puzzled with a fact that as an object moves faster, it ceases its property of opacity. I mean to say that as an object moves faster we can see right through it (more clearly than in a situation when his speed is low). Now with the fact that the speed of light has been controlled upto 61kmph in Bose- Einstein condensate, does it imply that light also becomes more and more opaque as its speed slower down?


Magnetic field strength of a hydrogen electron

Does the rotation of the electron around the hydrogen nucleus cause a resultant magnetic field in the atom? If so, the current density must be very high and the resulting field should affect the stability of the atom. How is this field counteracted so that the atom remains stable? Thanks


Elastic collision of photon

Consider an elastic collision of a photon with 100 eV energy hitting a mirror. How much momentum is exchanged in the collision? Also, can one model the reflection of an elastically colliding photon with a single electron in the mirror? It would seem that the photon would be absorbed by the electron, so that the collision is necessarily inelastic.


How do horseshoe orbits work?

An asteroid was recently discovered that is in a horseshoe orbit with respect to the earth. Is there an intuitive explanation for these orbits? It seems that the earth acts as a repulsive force where one would expect gravity to be attractive. From the above linked article: Accepted in MNRAS, Apostolos A. Christou, David J. Asher, 1104.0036, _A long- lived horseshoe companion to the Earth_ ![enter image description here](http://i.stack.imgur.com/lesXN.png) A better description of the horseshoe orbits, along with the tadpole orbits and Lagrange points:   ![enter image description here](http://i.stack.imgur.com/FrWhl.jpg)


What happens to light in a perfect reflective sphere?

Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape. If you could observe the inside of the sphere, what would you observe? A glow? And would temperature affect the outcome? Seems silly, it's just something I've always thought about but never spent enough time (until now) to actually find an answer.


How can there be a Current and an Electric field in an idealized wire with no voltage drop?

![enter image description here](http://i.stack.imgur.com/GEnoN.gif) In an ideal circuit, How can there be a current b/w points a & b, when there is no potential difference and thus no electric field between a & b? If there is no current, then where does current across the resistor come from, because that means no charge is coming from the battery (?).


How are the distances to the most distant $\gamma$-ray bursts measured?

This question came to my mind when I read on NASA's website that an explosion (gamma ray burst, GRB 080913) took place 12.8 billion light years away from us. How do they measure such large distances?


If there were fundamental forces weaker than gravity, would we know about it?

We know that gravity is a very weak force compared to electromagnetic forces and the nuclear forces. We know about the other forces because they're necessary to explain atoms, and we can detect gravity easily, because unlike the other forces it is always attractive, so the weak gravitational force from every particle in a planet can add up to a measurable effect. However, is it possible that there are other fundamental forces that are much weaker than gravity, or even of a similar magnitude but with "charges" that attract their opposite and therefore cancel out over large scales? It seems that we wouldn't necessarily have detected such a force if it did exist, and it seems different from the kind of thing you can probe using a particle accelerator. I realise this question is kind of naïve. I know a little quantum mechanics but never studied quantum field theory or particle physics. I'm curious about whether those formalisms provide a way to rule out such additional weak forces.


Why does this model fall apart when angular velocity is small?

I'm doing a physics problem in which a marble spins around a spinning bowl and both have angular velocity $\omega$. It rotates with radius $r$ around the central axis and the hemispherical bowl has radius $R$. I've solved for the radius $r$ in terms of $\omega$: $$r=\sqrt{R^2-\frac{g^2}{\omega^4}}$$ But I can't figure out what this means when $\omega$ is really small ($<\sqrt{\frac{g}{R}}$ ). Some hypotheses:   * Something to do with friction   * It falls through the center of the bowl (if it were hollow)   * We're missing something in this model Are any of these right? Is this more complex than it seems?


Radiation from a pair of charged objects orbiting each other

This question on binary black hole solutions, led to me think about the similar question from the perspective of what we know about the Hydrogen atom. Prior to quantum mechanics, it was not understood what led to the stability of the Hydrogen atom against collapse of the electron orbits due to Bremsstrahlung, i.e. the emission of radiation due to the fact that it is in a non-inertial (accelerated) frame of reference. Bohr and Sommerfeld came up with a somewhat _ad-hoc_ procedure \- the first ever instance of quantization - according to which in the quantum theory only those classical orbits are allowed the value of whose action is quantized in units of $\hbar$. $$ \int_i p dq = 2 \pi n_i $$ where the integral is over the $i^{th}$ (closed) classical orbit. Now what I'm thinking of next has probably been thought of before but I haven't done a literature review to find out. Classically, we expect the accelerating electron to radiate resulting in the catastrophic collapse of its orbit. However, in a _complete_ description we must also take the proton into consideration. It is also a charged object and as is well known from the two-body, inverse square law, central force problem (see eg. Goldstein) both the proton and electron orbit each other. Therefore the proton, being a charged object, must also radiate if we don't ignore its (accelerated) motion around the electron. An observer sitting at a distance $d \gg r$, where $r$ is the mean size of the two-body system, will measure radiation which is a superposition of that coming from both the electron _and_ the proton. The question is this: **a).** What is the phase difference between the two contributions ($\mathbf{E}_e$ and $\mathbf{E}_p$) to the net electric field $\mathbf{E}$ as seen by this observer?, **b).** What is the value of the total energy $E$ emitted by the electron-proton system, given by the integral of the Poynting vector across a closed surface enclosing the system, as seen by this observer? [ My motivation is to see if we can learn more about the Bohr-Sommerfeld quantization condition by considering the classical electrodynamics of the **full** electron-proton system. The quantity $E$ will depend on the size and shape of the classical orbits of the two charged objects or more simply of their mean separation $E:=E(r)$. As we vary $r$ from $0$ to some value $r_{max} \ll d$ we would expect $E(r)$ to oscillate and have local minima for some classical orbits. If these classical minima occur for orbits which satisfy the Bohr-Sommerfeld condition then we would have established a connection between the full classical problem and its quantization. ]


Does $\langle\Omega|\mathrm{e}^{\mathrm{i}Px}=\langle\Omega|\mathrm{e}^{\mathrm{i}0x}$? $(\langle\Omega| =$ ground state of the interacting theory)

Let $\langle\Omega|$ be the ground state of an interacting theory, just as Peskin & Schroeder(PS) describes on page 82 and page 213. On page 213 PS do the following $$\tag{1}\langle\Omega|\phi(x)|\lambda_{\vec{p}}\rangle = \langle\Omega|\exp(+\mathrm{i}Px)\phi(0)\exp(-\mathrm{i}Px)|\lambda_{\vec{p}}\rangle \\\ =\langle\Omega|\phi(0)|\lambda_{0}\rangle\exp(-\mathrm{i}px)\bigg|_{p^0=E_{\vec{p}}} $$ In the above $|\lambda_0\rangle$ is an eigenstate of the full interacting hamiltonian $H$. The ket $|\lambda_{\vec{p}}\rangle$ is the boost of $|\lambda_0\rangle$ with momentum $\vec{p}.$ Furthermore $E_{\vec{p}} = \sqrt{p^2+m_\lambda^2}$ and $\vec{P}|\lambda_0\rangle = 0.$ I assume PS used $$\tag{2}\langle\Omega|\mathrm{e}^{\mathrm{i}Px}=\langle\Omega|\mathrm{e}^{\mathrm{i}0x}$$ in $(1)$. Is $(2)$ true? I thought $\langle0|\mathrm{e}^{\mathrm{i}Px}=\langle0|\mathrm{e}^{\mathrm{i}0x}$ where $\langle0|$ is the ground state of the free theory. I hope I have provided enough info for the question to make sense.


Do we need inertial frames in Lagrangian mechanics?

Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose because I was thinking how could you describe the motion of pendulum with an oscillating suspension point, something like this: ![enter image description here](http://i.stack.imgur.com/g5LT3m.gif) If you take your coordinate frame fixed in the suspension point, you would get the motion of a simple pendulum, which is false (because it's a non-inertial frame and you should have virtual forces). But if it were moving at constant speed, a moving coordinate frame should give the same equations of motion. **So where does the restriction to non-inertial frames come from in the Lagrangian formalism?** I've read that in non-inertial frames the lagrangian would be $L(q,\dot q, \ddot q, \dddot q..., t)$, but I didn't see a detailed explanation. And I'm not sure if the situation would be similar to Kepler's problem in polar coordinates.


Is there a limit on how hot you can superheat water at 1 atm?

Is there a limit on how hot you can superheat water at 1 atm in a perfect container? What about in a microwave?


Deriving Rabi rotation matrix

I want to understand where the matrix: $$ \left|\psi(t)\right> = \binom{a(t)}{b(t)} = \begin{bmatrix} cos(\Omega t/2)&-ie^{i\phi_L t}sin(\Omega t/2) \\\ -ie^{-i\phi_L t}sin(\Omega t/2) & cos(\Omega t/2) \end{bmatrix}\binom{a(0)}{b(0)}$$ comes from. I've already derived $$\dot{a} = \frac{\Omega}{2}e^{i\phi_L t} b$$ and its counterpart $$\dot{b} = \frac{\Omega}{2}e^{-i\phi_L t} a$$ where $\Omega = \frac{1}{\hbar} \left<1\right|-d\cdot\varepsilon\left|0\right>$ and, I think, $\phi_L$ is the frequency of the incident radiation. But how can I get from the above two relations to the matrix?


Silly question ... An antenna is necessary to receive a transmitted signal from atmosphere, or otherwise. The transmitted signal weakens as it travels through the atmosphere. When the signal encounters an antenna (subject to antenna gain) a corresponding signal is induced in the antenna. Is the inducing signal weakened to the extent it induces a signal in the receiving antenna?


Mirror Transportation

Why do people transport mirrors in inclined position? I frequently find people carrying mirrors in slant position on mini trucks.I guess it has some advantage of carrying them without breaking,and I am now so curious to know the reason behind it ![Either sides of the wooden piece are aligned by mirrors](http://i.stack.imgur.com/J1S63.jpg)


Which experiment gave scientists reason to believe nuclear fission/fusion produced energy?

Every piece of knowledge in science has a beginning lying in someone's experiment. I would like to know which experiment gave scientists the reason to believe nuclear fission/fusion existed and was instrumental in the development of the field of nuclear energy. I would also accept a thought experiment as an acceptable answer, as long as it answers the question.


What is the most distant object from the Earth that a spacecraft has visited to date?

What is the most distant object from the earth that a spacecraft visited has visited so far? What was the mission and when did it happen?


For observing planets what kind of telescope offers the best view?

I've heard that using a refractor is better than a reflector when it comes to planets to best reproduce their colors. But perhaps other factors can weight in too? For example, do you want a slow or fast f-stop to get the best view of a planet? What about eye-pieces, what style of eyepiece construction will give the best views?


Radioactive decay - What mechanism decides when an unstable nucleus decays?

My first question on Stackexchange (if it is formatted wrong or something please tell me so I know in future) - here it is: **Given an unstable nucleus (exactly which nucleus is not particularly pertinent) - what decides _precisely_ when it will decay?** I am somewhat familiar with the concept- a nucleus becomes 'unstable' when the mutual repulsion between the protons exceeds the binding force of the strong nuclear force. That being said, if we were to theoretically isolate a single unstable atom, there are a few possibilities (correct me if I am wrong):   1. atom immediately decays   2. atom decays x seconds later   3. atom doesn't decay There is an unpredictable nature (which I assume arises from something to do with quantum uncertainty). But getting back to my question, what is it that suddenly makes an 'unstable nucleus' decay? Any further reading would be appreciated, although nothing too complicated (university level is my limit I think). UPDATE Thanks all, my first time using this forum and I was not let down :) With regards to the question itself, I'm a bit disappointed that there is no precise mechanism, but I guess Einstein was wrong when he quoted: "God does not play dice"


When to pull out a negative sign from a variable

I get confused about when there should be a negative sign in certain equations or not. I will give three short examples (that I will make long with explanation) that show my confusion. Example 1: Rockets. Consider a rocket of mass $m$ traveling in the positive x direction with velocity $v$. The ejected fuel has a velocity relative to the rocket $v_{ex}$. At time $t$, the rocket's momentum is $P(t) = mv$. In a short time later, $t + dt$, the rocket's mass is $m + dm$ (where $dm$) is negative and momentum $P(t + dt) = (m + dm)(v + dv)$. The fuel ejected in a time $dt$ has mass (-$dm$) and velocity $v - v_{ex}$ relative to ground. The total momentum at time $t + dt$ is $$P(t+dt) = (m+dm)(v+dv) - dm(v-v_{ex})$$ $$ = mv + mdv + dmv_{ex}$$ 1a) Why is the negative contained in the $dm$ of the rocket mass while the negative is pulled out of the $dm$ for the fuel? To reconcile this, I thought of something like, if ejecting mass is a positive, then use positive $dm$ as they did in the text from which I'm reading out of. So ejecting means positive, while the fuel gains mass in a way. So I can define gaining mass with a $-dm$ as they did. However, if I was doing this problem, I would have made ejecting mass negative and gaining mass positive. But if I change the sign on all of the $dm$'s I don't get the same result in the text. Example 2: I'm not reading this one out of a text but I can explain the question. Often times you'll see something like $$\frac{d(Q)}{dx} = ax$$ where $Q$ is some function. The book or whatever will then say, but as $x$ increases $Q$ decreases so the expression should really be $$\frac{d(Q)}{dx} = - ax$$ 2a) Why can't the negative sign be contained in the constant of proportionality? Does it change the result? All it does by putting a minus sign there is force $a$ to be positive. Example 3: Projectile motion with quadratic drag. The equation of motion for this is given by $$m\ddot{\vec{r}} = m\vec{g} - cv^2\hat{v} = m\vec{g} - cv\vec{v}$$ where $v$ is the magnitude of the velocity of the projectile. If we only look in the vertical direction, with y measured vertically upwards, my text gives $$m\ddot{v_y} = -mg - c\sqrt{v^2_x + v^2_y}v_y$$ 3a) So $\vec{g} = -g\hat y$ and they pulled negative out of the $g$. So it makes sense that its $-mg$ since y is pointed upwards. Why isn't it $+ c\sqrt{...}$? Is the reason because $v_y$ is negative, so a negative times a negative is a positive? If so, why did the negative come out of the $g$ while the negative was kept inside the $v_y$?


Electron and photon interaction

Assuming I shoot a beam of light to an electron, the electron will take the energy and reemit EMR at all direction randomly? If it will, is it happening at the same time?


Superluminal travel without time travel

Can superluminal travel or communication be possible without leading to the possibility of closed timelike curves (CTCs) and causality violations? I have seen conflicting opinions regarding this issue, and whether all possible methods of superluminal travel (e.g. wormholes, Kraskinov tubes, and warp drives like the Alcubierre drive) necessarily open the door to causality violations. I understand that wormholes may lead to CTCs in certain conditions (i.e. through time dilation or Roman rings, as well as Kraskinov tubes). My question is basically this: Are there possible (hypothetical) methods of superluminal travel that do not open Pandora's box of causality violation?


Can an electromagnet repel a strong magnet?

When the direction of current is reversed, the polarity of an electromagnet should be reversed. I am trying to understand why this did not happen in an experiment I conducted today. An electromagnet was connected to a 12v battery and I verified that it was working. I then touched the electromagnet to a strong regular magnet (rated to hold over 50 kg). There was a strong force between the magnets. I then disconnected the electromagnet from power, reversed the wires, and turned it back on. I expected to feel the electromagnet repel the other magnet, but instead they two magnets got stuck together again. It seems like reversing the wires had no affect on the polarity of the electromagnet. Here is a picture of the electromagnet I used for reference: ![enter image description here](http://i.stack.imgur.com/9mea3.jpg) And here is the strong regular magnet.


Is the stability of the nucleus due to pions or gluons?

> **Possible Duplicate:**   >  Is it pions or gluons that mediate the strong force between nucleons? (I'm currently studying particle physics in my physics classes at high school so simple language would be appreciated.) We have learnt that the proton is the only stable hadron and the neutron has a mean lifetime of approximately 20 minutes. This leads to the question of how nuclei are stable. As I currently understand it, pions are exchanged between the neutrons and protons which turn the proton into a neutron and the neutron into a proton. My question is how gluons are involved in this process and what simplifications have been made in my understanding? My initial guess is something to do with quark color.


Optics Brewster's Angle Reflection Intensity

An incident unpolarised light beam of intensity $I_{0}$ strikes glass plate B at Brewster's Angle. The reflected light travels vertically and strikes a second glass plate A, again at Brewster's Angle. (We ignore the light transmitted by the glass plates.) Plate A is then rotated about the z-axis as shown. Briefly explain how the intensity of the light reflected by the apparatus varies with the angle of plate A. Illustrate with a qualitative sketch of intensity vs. angle of plate A. ![Diagram](http://i.imgur.com/2AZstU1.png?1) We have an unpolarised beam incident on plate B at Brewster's Angle which upon reflection should become s-polarised or polarised perpendicular to the incident plane. My understanding of Brewster's Angle then says that upon the second reflection at plate A if the plate is oriented such that the incident plane is the same as for plate B there will be no reflection. Otherwise as it rotates there will be light reflected, is this correct? Also does the intensity of the light reflected off of A increase gradually to a maximum when the beam is parallel to the incident plane at A? Any help would be much appreciated.


Domain walls intersection

I was reading this article( _On domain shapes and processes in supersymmetric theories_ ). In the paragraph about domain walls intersection (paragraph $4$, page $7$) the authors say: > In a one-field theory it is known that an intersection of domain walls is > unstable They explain it as follows: > Indeed the configuration shown in Fig. 2a has translational zero modes, > whose shape is given by the gradient of the field. Since there are > directions in the plane, where the field approaches the same values at both > innities, the component of the gradient in such direction necessarily has > zero. Thus the zero mode cannot be the lowest in the spectrum and a negative > mode exists, leading to a separation of the walls. So, i have the theory of one scalar field in $2+1$ with mexican hat potential: $V=\frac{\lambda}{4}\left( \phi^{2}-v^{2}\right)^{2} $. I consider configuration (domain walls intersection) with boundary conditions: $$ \phi_{d}(+\infty, +\infty)=-v $$ $$ \phi_{d}(+\infty, -\infty)=v $$ $$ \phi_{d}(-\infty, +\infty)=v $$ $$ \phi_{d}(-\infty, -\infty)=-v $$ Here $\phi_{d}$ satisfies the stationary equation of motion (recall that $\dot{\phi}=0$): $$ \Delta \phi_{d}-\frac{\partial V}{\partial \phi}(\phi_{d})=0. $$ I consider the small excitation over $\phi_d$: $\tilde{\phi}=\phi_{d}+\phi$. After linearizing, I obtain the following equation ($\phi=e^{i\omega t}f_{\omega}(x_1,x_2)$): $$\left[-\Delta + \frac{\partial^{2}V}{\partial \phi^{2}}(\phi_{d})\right]f=\omega^{2}f \tag{1}$$. I have found **zero modes** : its directional derivative : $f_{0}(x_{1},x_{2})=(\nabla\phi, \mathbf{n})$. It satisfies the equation $(1)$ with $\omega=0$. Now I can consider the directions along "diagonals", at their ends the field takes the same value $v$ or $-v$ respectively (see figure 2 at page 10 in the file). So at this diagonals there is a point where $f_{0}=0$, as authors said. But I cant understand the last sentence of their explanation: **How the existance of such point (where $f_{0}=0$) implies that we also have negative modes?** Can you explain me this? Thanks in advance. _PS. I will accept another approaches to show the existance of negative modes (and therefore instability of such configuration) too._


$E=mc^2$ resembles kinetic energy formula?

The simplest equation expressing mass–energy equivalence is the famous $E=mc^2$ where $c$ represents the speed of light. Compare this with $E_K = \frac{1}{2}mv^2$. Since $E=mc^2$ can be applied to rest mass ($m_0$) and rest energy ($E_0$) to show their proportionality as $E_0=m_0c^2$, I ask whether this resemblance is just a coincidence created by the need of any equation to be homogeneous for units, or are the two equations fundamentally related? I know about Mass-Energy Equivalency but I could not find the answer I'm looking for there.


Equation of motion for a falling rod (with one end touching a frictionless surface)

I have a quick question about the equation of motion for a falling rod (with one end touching a frictionless surface). The end touching the surface is not fixed. I am given the moment of inertia about the center of mass. Only gravity is acting on the rod. The rod begins at an angle $\theta$ above the ground. I know that the rod will rotate about the center of mass, and the point touching the frictionless surface will slide along the surface, but I am having trouble calculating the torque. For reference, the left end of the rod is touching the surface, and the right end is in air. I have calculated the torque, $\tau$, from the right moment arm as $\tau=\frac{mg\cos{\theta}}{4I_G}$ because the right half of the rod contains half of the mass and half of the length. I don't really understand how to calculate the total net torque, though. Any hints would be appreciated.


Huggins Displacement Theory and Retrocausality

I was looking at the Wikipedia entries on Time Travel and the Grandfather paradox and noticed a paragraph on the so-called Huggins Displacement Theory. I haven't been able to find the source although I assume it's originator was the Victorian astronomer Sir William Huggins. Does anyone know of the precise reference? Normally sending information into one's past can lead to contradictions. For example based on some condition now I could send a message into my past that prevents that very condition from occurring in my present. Thus sending information into the past, and therefore arbitrarily changing it, seems to be forbidden physically. However the Huggins Displacement Theory says that there is a loop-hole in this argument (or more accurately a "corner case"!) I could send information along my past lightcone so that if I influence something a year in my past then it must be a light-year away from me. Thus no change in the past has a enough time to travel back to me to cause a contradiction in my present. For example imagine that I see a stock price going up today. The Huggins Theory says that I can beam knowledge of this stock price rise back in time provided that the message is also displaced by a comparable distance. Thus if I send it back a year in time then it must also travel away from me by a light-year in distance. If my twin gets the message then he might try to profit from the information by sending back a message to me to buy the stock. Unfortunately the message will only get back to me at the instant that the price rises so that neither myself nor my twin will be able to profit from this situation. Maybe physics does allow information to be sent back in time along a past lightcone so that this effect could be measured. My hunch is that perhaps if one charged up an electrode then electrostatic influences from it might travel back along its lightcone to cause measureable repulsion or attraction on distant electrodes. I don't think electromagnetic radiation itself could go backwards in time as in principle it could be reflected back onto an observer's past worldline and therefore trigger contradictions in his/her present.


How would one compute the angle of deflection, in a relativistic collision - underspecified system?

Consider the simplistic case of two identical mass particles colliding elastically with the second particle initially stationary and the first particle travelling with energy $E$. By conservation of 4-momentum we have: $$p_{1}^{\mu}+p_{2}^{\mu}=p_{1}'^{\mu}+p_{2}'^{\mu}$$ Taking the inner product of this with itself: $$\left\langle p_{1}^{\mu} \middle| p_{1}^{\mu}\right\rangle + 2\left\langle p_{1}^{\mu} \middle| p_{2}^{\mu} \right\rangle+\left\langle p_{2}^{\mu} \middle| p_{2}^{\mu} \right\rangle = \left\langle p_{1}'^{\mu} \middle| p_{1}'^{\mu}\right\rangle + 2\left\langle p_{1}'^{\mu} \middle| p_{2}'^{\mu}\right\rangle +\left\langle p_{2}'^{\mu} \middle| p_{2}'^{\mu}\right\rangle $$ Using the fact that $\left\langle p_{i}^{\mu} \middle| p_{i}^{\mu}\right\rangle = m_{0}^{2}c^{2}$, we can simplify this: $$2m_{0}c^{2}+2m_{0}E= 2m_{0}c^{2}+2\left(\frac{E_{1}'E_{2}'}{c^{2}}-\vec{p}_{1}'\cdot\vec{p}_{2}'\right) \implies m_{0}E=\frac{E_{1}'E_{2}'}{c^{2}}-\vec{p}_{1}'\cdot\vec{p}_{2}'$$ We note that $\vec{p}_{1}'\cdot\vec{p}_{2}'=\|\vec{p}_{1}'\|\|\vec{p}_{2}'\|\cos(\theta)$, where $\theta$ is the inner angle between $\vec{p}_{1}'$ and $\vec{p}_{2}'$. However, this results in a system with $\|p_{1}'\|$,$\|p_{2}'\|$ and $E_{1,2}$ unspecified and I cannot see how we could thus extract $\theta$ from the initial conditions; what have I misunderstood or misapplied here?


Are gravitons bound by the event horizon?

I understand that photons, even when traveling at the speed of light, cannot escape the event horizon of a black hole. Are gravitons and other virtual particles traveling at the speed of light also confined by event horizons? If so, it seems that the gravitational field created by the black hole would result only from the mass of the black hole beyond the event horizon, where gravitons are capable of escaping. As a result, would there would be a disparity between the apparent mass of the black hole due to its gravitational field on other celestial bodies and the total amount of matter contained within the black hole? Also, I was reading this question: Nature of gravity: gravitons, curvature of space-time or both?, which suggests that gravitons and curved space may be indistinguishable. However, if gravitons are bound by the event horizon it seems that a black hole would act differently based on whether gravity results from gravitons or curved space-time. The existence of bound gravitons would negate the gravitational field of mass within the event horizon, resulting in a significantly lower gravitational field outside of the black hole. Would this occur, or am I neglecting some effect of relativity upon the gravitational field?


Will an anti-neutron annihilate a regular neutron even though they have no charge

Since neutrons have no electric charge will an anti-neutron annihilate when it comes into contact with a regular neutron as protons and anti-protons do? if so what causes it to annihilate?


Locality in the scattering amplitude

Early in this talk by Nima Arkani-Hamed, he describes what locality means for scattering amplitudes. "Locality tells you that the only poles in the scattering amplitude occur when the sum of a subset of the momenta of the particles goes on shell." The talk then goes on to describe his recent research, but right now I'm not worrying about understanding that. I'm just trying to understand the statement quoted above. I know what locality looks like in the Lagrangian density (the terms of the Lagrangian density contain field operators which are all functions of the same spacetime coordinate), but it's not at all obvious to me how that is equivalent to the above statement about scattering amplitudes. Can you explain it (preferably at about the level of a first graduate course in QFT)?


S-matrix and it's exponential form

By using Dyson series for the representation of the $S$-matrix, it's expression can be written in a form $$ \hat {S}(\infty , -\infty) = \sum_{n = 0}^{\infty}\frac{(-i)^{n}}{n!}\int \limits_{-\infty}^{\infty}d^{4}x_{1}...\int \limits_{-\infty}^{\infty}d^{4}x_{n}\hat {P}(\hat {\Gamma }_{i}(x_{1}))...\hat {\Gamma}_{i}(x_{n})), \qquad (.1) $$ where $$ \hat {N}(\hat {H}_{k}(t_{i}), \hat {H}_{k}(t_{j})) = \left\\{ \begin{matrix} \hat {H}_{k}(t_{i}) \hat {H}_{k}(t_{j}), t_{i} > t_{j} \\\ \hat {H}_{k}(t_{j}) \hat {H}_{k}(t_{i}), t_{j} > t_{i} \end{matrix}\right. . $$ How to rewrite $(.1)$ into exponential form?


How is voltage increased from a battery?

I have recently watched a video about a powerful Octa-copter that uses 30,000 Volt brushless motors to propell, the batteries however look quite small about a 19v laptop battery. How is the voltage increased to suffice the voltage requirements? where does the extra electrical energy come from or what is the tradeoff for stepping up the voltage? and if the batteries are at most about 19V how does it obey ohms law?


Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?

A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions to check if a quantum gate (2x2 or 4x4) is in any of these groups? Or maybe there are matlab routines that generate quantum gates outside of this groups? Thanks..


Speed of light that is traveling away from the observer

The second postulate of Special Relativity states: > The speed of light in a vacuum is the same for all observers, regardless of > their motion relative to the source. Now imagine the observer that is looking at the light which is traveling away from him in a direction he is looking. In this case, the light should look somewhat like a "dot" to such observer - not really moving at any direction for him. However according to postulate above the light should move at the speed of c (if we are in vacuum to be precise). What is wrong with this reasoning? Or is it just some kind of exception from the rule?


What would the electromagnetic field of a massless electron look like

The Standard Model gives non-zero mass to the electron via the coupling to the Higgs field. Issues of renormalizability aside, this is fundamentally unrelated to the fact that the electron couples to the EM field. However, if the Higgs mechanism did not operate - that is, if there were no vacuum symmetry breaking, the electron field would have no effective mass term. In QFT perturbation theory, this model offers no special difficulty. My question is, what is the classical limit of this theory, if it has one? Does the electron acquire a purely EM mass? If the low energy renormalized mass is set to zero (is there an obstacle to doing that?) what do the classical field configurations look like? My puzzlement is related to the classical description of the EM field of a relativistic charged particle - namely, that it becomes "flattened" in the direction of motion, just like a Lorentz-contracted sphere. That is, the field is weaker than for a motionless particle in the longitudinal direction, and stronger in the transverse directions. The limiting case is that the field becomes concentrated on a 2-d surface transverse to the particle location, where it has infinite strength - obviously an unphysical situation. So what actually happens?


I would like to explore some ideas with different geometries exchanging energy mainly by radiation. This means solving Laplace equation for the steady state. However the boundary conditions are always Neumann's because the only constraints are fluxes. After many questions and attempts, I realized that Mathematica cannot (yet ?) treat PDEs with Neumann boundary conditions. To my great surprise I have also found no literature describing the treatment of the heat equation with the derivative at boundary proportional to $T^4$ even if I thought that similar problems would play a paramount role in engineering. What are some good books/papers to treat the Heat equation with Neumann boundary conditions and more specifically when the flux at boundary is proportional to $T^4$ ?


Equations of motion for bob-on-a-string -- am I missing some terms?

The dynamics of a type of physical system I am currently working on are modeled in most of the literature by replacing the moving parts of that system with an equivalent set of pendulums. Parameters and initial conditions for this approximate pendulum model are chosen such that the total mass and the center of mass of these pendulums describe those of the real system as accurately as possible. For one particular system however, I need to take this one step further because that system is losing mass _rapidly_. (Also, it is spinning about an axis that lies outside the system, but let's leave that out of consideration here). Therefore, I ended up modeling this particular system with a series of "specialized" pendulums:   * the pendulum masses are free to move in any direction (i.e., not constrained to move in any single plane)    * the point masses are attached to the pendulum hinge points by means of massless strings of infinite strength but finite elasticity (spring constant $k$)   * The pendulum mass $m$, the height of the hinge point $h$, the restoring force $m\mathbf{g}$ and the nominal string length $L$ are all functions of time. ![enter image description here](http://i.stack.imgur.com/RJv7C.png) In this figure, $h(t) = \left|\mathbf{h}(t)\right|$ and $\left|\mathbf{s}(t)\right| = L(t)$ when the string is _just_ stretched. $\dot{\mathbf{h}}$ and $\ddot{\mathbf{h}}$ are constrained to only the pure vertical direction. Leaving the fictitious forces out of consideration, I have come up with the following equation of motion for this kind of pendulum: $$ m\ddot{\mathbf{r}} = -m\mathbf{g} + \begin{cases} 0 & \text{if } |\mathbf{s}| < L\\\ m\left(\frac{\mathbf{s}\times\mathbf{\dot{r}}}{|\mathbf{s}|^2}\times\mathbf{\dot{r}}\right) - \gamma_1\left(\frac{\mathbf{s}\times\mathbf{\dot{r}}}{|\mathbf{s}|^2}\times\mathbf{r}\right) & \\\ \ \ \ \ \ \ \ \ \ - k\left(|\mathbf{s}|-L\right)\frac{\mathbf{s}}{|\mathbf{s}|} - \gamma_2\left(\frac{\mathbf{s}}{|\mathbf{s}|}\cdot\mathbf{\dot{r}} + \dot{L}\right)\frac{\mathbf{s}}{|\mathbf{s}|}& \text{if } |\mathbf{s}| \geq L \end{cases} $$ where $\gamma_1$ is the torsional damping ratio, $k$ the string constant of the (flexible) string and $\gamma_2$ its linear damping ratio. The first two terms of the second case lie in the current plane of rotation of the mass $m$ about the hinge point. They describe the centripetal force towards the hinge point, and torsional damping, respectively. The last two terms are in the radial direction, and describe the stretching and damping of the stretched string, respectively. Numerical results of this equation are encouraging. However, this is all a bit messy (especially with the terms for the ficticious forces included), so I'd really appreciate if someone else took a look at it. Why I ask this is because I can't figure out why the $\dot{\mathrm{h}}$, $\ddot{\mathrm{h}}$, $\ddot{L}$ and $\dot{m}$ do not show up in this equation... Am I missing any terms anywhere?


Integral related to particle diffusion

In the context of particle diffusion, I am trying to understand the equations that describe Brownian motion as a macroscopic process. Assume $N(x,t)$ is a number concentration and $D$ is a diffusion constant, then we can write: $$\frac{\partial N(x,t)}{\partial t} = D \frac{\partial^2}{\partial x^2} N(x,t)$$ next, we can multiply by the mean square displacement $x^2$ and integrate over $x$ from $-\infty$ to $\infty$: $$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = \int x^2 D \frac{\partial^2}{\partial x^2} N(x,t) dx $$ Now the script that I'm reading writes that the left-hand side can be written as: $$\int x^2 \frac{\partial N(x,t)}{\partial t} dx = N_0 \frac{\partial \langle x^2 \rangle}{\partial t} $$ where I assume that $N_0$ is the initial number concentration of particles and $\langle \rangle$ is the mean square displacement. I do not understand the last equation. Could somebody explain why I can express the integral as the partial derivative wrt to time of the mean square displacement (multiplied by $N_0$). **EDIT: $\langle x^2 \rangle$ is the mean square displacement (contrarily to what I have written earlier)**


Static friction

Just a quick question, When a force on an object is equal to its static coefficient * normal force , would the object move or would it stay stationary? Or does the force always need to be greater than this value for the object to move? My guess is that it need to be greater but im not completely certain.


Stress-Energy Tensor

As of recent, I've been doing a bit of self education in GR, equipped with a working knowledge of the key elements of the differential geometry in GR, and in looking at the Einstein-Rosen bridge, ![Einstein-Rosen Bridge](http://i.stack.imgur.com/bMT6L.jpg) I see that geometrically it is a hyperboloid of one sheet. Now when using this manifold to calculate things like curvature and geodesics, we need a metric, which we can somewhat easily derive from the equation of the aforementioned surface. Now per $ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^{4}} T_{\mu\nu} $ ( setting $ \Lambda = 0 $ for the simple case ), one should be able to solve for the stress-energy tensor, albeit the grotesque mathematics. By this logic, based on the predetermined shape of the local spacetime, we should be able to calculate the mass and geometry of the object required to create such a metric. My questions then are: To what extent is this a valid statement if at all? Given the fact that there are going to be initial value problems embedded in the PDE's with conditions that require the input of some properties of the geometry of the aforementioned object: How is it that one goes about finding physically viable objects that satisfy the EFE under the constraints that we've applied if any exists. How does one test to see if there are even any solutions that work? For an attempt at the second questions, I was thinking that if we have to apply constraints to get rid of the unknown constants that resulted from the PDE, we just put constraints on what the solution can be, to ensure that it fits with what we want. Ie I want to make sure my sphere of matter is of this size, but can't exceed this mass. Am I at least on the right track with this?


Definition of a non-pertubative Quantum field theory

How do you define a non-perturbative Quantum field theory. What does it mean? I was just digging around some math about the meaning of $Z_1$, and such in terms of probabilities. It turns out these and associated counter terms are derived via perturbation theory. So the question is , if you associate meaning to things like probability of self interaction and such, how do you make these things make sense if you are working non-perturbatively. What is a self interaction outside of perturbation theory? I mean considering some probable sketch, for some arbitrary calculations , something looking like the lines below might appear, then some cleaver physics is said. $1 + \Sigma_a$ $\Gamma^{n} \rightarrow Z_{b} \bar{\Gamma^{m}}$ ok so what is a self interaction in non-pertubative models like. Is a non-pertubative model real? Can it make sense. . . . I don't know .. . . . .


Question regarding waves squeezed

in one MOOC I am taking, the professor had a slide stating "when squeezed into a narrow space wave is amplified". I am not sure I understand what he meant by that, and I am trying to think into terms of wave equations. Could anyone explain it to me? Thank you.


How was the formula for kinetic energy found, and who found it?

My questions mostly concern the history of physics. Who found the formula for kinetic energy $E_k =\frac{1}{2}mv^{2}$ and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way. My guess is that someone thought along the following lines: > Energy is conserved, in the sense that when you lift something up you've > done work, but when you let it go back down you're basically back where you > started. So it seems that my work and the work of gravity just traded off. > But how do I make the concept mathematically rigorous? I suppose I need > functions $U$ and $V$, so that the total energy is their sum $E=U+V$, and > the time derivative is always zero, $\frac{dE}{dt}=0$. But where do I go from here? How do I leap to either   * a) $U=\frac{1}{2}mv^{2}$   * b) $F=-\frac{dV}{dt}$?  It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.


Length of the missing part

I have got a very interesting question. A car is driving this way on the picture down. In the looping is a missing part. The radius of the looping is r = 40cm. The car starts at the height h = 3r. If the car jumps from the interruption of the looping it flys and lands an the other side of the interruption and goes ahead to drive. Now I have to find out, how long the missing part of the looping is. phi has no value. I don't have to calculate the friction. ![enter image description here](http://i.stack.imgur.com/mAVxR.png) The car has got a velocity of 17,5 km/h in B, right?


A beam of light, traveling around a planet

Some time ago, I happened to have found one problem: > The refractive index of the planet's atmosphere decreases with height above > the surface under the following law: > > $$n=n_0-\alpha h$$ > > with condition $\frac{\alpha h}{n_0}\ll1$ > > Find the altitude $H$ above the surface of the planet, where a ray, emitted > horizontally, will bypass the planet, remaining at this elevation. Radius of > the planet is $R$. Since optics is not my field I decided to ask here. What would be the height $H$ for the Earth? (approximately)


Application of Ampère's law in low $Re_m$ MHD

Ampère's law states that $$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} \tag{1}$$ Simultaneously, from Ohm's law, we know that $$\mathbf{J}=\sigma\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}\right)\tag{2}$$ When equating both currents $\mathbf{J}$, and applying Faraday's law, and knowing that the magnetic field $\mathbf{B}$ is solenoidal, one arrives at the transport equation of the magnetic field $$ \frac{\partial\mathbf{B}}{\partial t}=\nabla\times(\mathbf{u}\times\mathbf{B)}+\frac{1}{\sigma\mu_0}\nabla^2\mathbf{B}\tag{3}$$ In the low magnetic Reynolds numbers limit $Re_m=\mu_0\sigma u L \ll 1$, the externally imposed magnetic field does not change and equation $(3)$ is not explicitly solved, because the diffusional term becomes negligible. In this so-called one-way coupled magnetohydrodynamics (MHD) (valid for most liquid metals), the electric field $\mathbf{E}$ has a potential $\Psi$, which is calculated from the divergence free ($\nabla\cdot\mathbf{J}=0$) condition, using equation $(2)$, with a Lorentz force $\mathbf{J}\times\mathbf{B}$. My question is how should one look at Ampère's law in the context of one-way coupled, or low $Re_m$ MHD? Specifically considering the following two examples **Example 1** Suppose a situation with an insulating boundary (with $\mathbf{n}=\mathbf{\hat{y}}$). This means that $J_y=0$. Furthermore, the imposed magnetic field is constant in time, but varies in space. From $(1)$ $$J_y = \frac{dB_x}{dz}-\frac{dB_z}{dx},$$ which, for an arbitrary, non-uniform, steady magnetic field, would be non- zero. **Example 2** Suppose the externally imposed magnetic field $\mathbf{B}(x,y,z,t)=\mathbf{B}_0$. According to $(1)$, $\mathbf{J}=\mathbf{0}$, and thus there would be no Lorentz force? What is exactly happening at the interface, and am I applying Ampère's law correctly. What am I overlooking?


Subjectivity of decoherence

I read that quantum decoherence is subjective, in the sense that two observers may not have the same "environment" and after each one has traced over those degrees of freedom they will end up with a different density matrix for the couple system-apparatus. I kind of understand the idea, but has an example of this calculation already been done? With the same system looked at by two different observers, and both density matrices calculated/compared.


Maximum speed of a water skier

Water skiing is a sport where an individual is pulled behind a boat or a cable ski installation on a body of water, skimming the surface. Consider an idealized case where the boat is moving at a constant velocity $\overrightarrow v_0=\text{const}$ (relative to the water), independently of the skier. The boat and skier are connected by a massless, unstretchable rope. Surface of the water is assumed to be smooth. Then, what is the maximum possible (instantaneous)speed of the skier $ v_\text{max}$ (relative to the water)? My first guess, based on intuition, is that $v_\text{max}=2v_0$. But I'm not at all sure. **Edit:** Skier's velocity may vary. Consider vector projections of velocity vectors $\overrightarrow v_1 $ and $\overrightarrow v_2 $ of the boat and skier in the direction of the rope. Because the rope is unstretchable, these projections must be equal. That means, must hold the equality: $$v_1\cos{\alpha}=v_2\cos{\beta}$$ $\alpha$ is an angle between $\overrightarrow v_1 $ and the direction of the rope   $\beta$ is an angle between $\overrightarrow v_2 $ and the direction of the rope So for example, if $\alpha<\beta$ then $v_2>v_1$. I.e. the skier is moving faster than the boat relative to the water surface.