Electric scooter acceleration

I tested two electric scooters with almost same power (1500 W), but very different performances: A "Zem Star 45" (1500 W, 60 Nm) can reach 50 km/h in "a few seconds" (not measured actually), An "Ingaeta Ingo" (1500 W, ??? Nm) took more than 10 seconds to reach 50 km/h. How much does torque affect acceleration performance? Given two identical scooters (same weight, same engine power, same wheels size), rgardless of wheels and air friction, how can I calculate acceleration given the torque on a flat street?


Is eternal inflation Lorentz invariant?

Start without general relativity. Consider a metastable vacuum over good ol'-fashioned Minkowski space. It decays. A bubble forms and the domain wall expands. The domain wall is timelike, and accelerates, driven by the difference in pressure between both phases. Its speed approaches the speed of light asymptotically. Far away, it looks like an expanding light cone, but it stops looking so around the moment of decay. Roughly speaking, this breaks Lorentz covariance, and this picks out an approximate frame of reference. It might be argued that the metastable phase prior to decay is Lorentz invariant. If so, there is only one Lorentz invariant measure over the space of future- oriented unit 4-velocity vectors up to a multiplicative factor. This space has infinite measure, and the typical decay velocity of a decay would have to be infinite. Nonsense. The point to note is we have to specify the initial conditions of the metastable vacuum. That can't be infinitely in the past, as the lifetime, while exponentially long, is still finite. A Lorentz invariant initial boundary can only be an expanding light cone, or a hyperboloid. However, both have infinite volumes, and to regulate, we need to cut off these surfaces at some finite radius. This radius has to be less than the decay lifetime. This cutoff roughly picks out a frame of reference. At any rate, affine parameters on light cones are tricky. Boost them enough, and you can get an exponential change in the affine parameter, and in this boosted frame, the likelihood of crossing a bubble nucleation approaches one in probability. Turn on general relativity. Does eternal inflation break Lorentz invariance?


If time isn't continuous, what is the best-known upper bound on the length of time intervals?

There have been several questions about whether time is continuous or not and it seems like the answer isn't currently known. I know quantum mechanics treats time as continuous and any mathematics that involves integrating over some time interval treats time as continuous too. Surely though there are experiments that are quite sensitive to discrete time with large intervals. It seems the shortest laser pulse so far is only 67 attoseconds ($67 \times 10^{-18}\: \mathrm{s}$) but wouldn't this experiment actually constrain time intervals to much less than that? Are short laser pulses even a good experiment to determine if time is discrete or not? **So, assuming time isn't continuous, what is the best-known upper-bound on time intervals? Also, which experiments have done the best to constrain how non-continuous time could be?**


Ricci tensor of direct product of manifolds

Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\\ 0&g_{mn}\\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal without any mixing of the two manifolds?


Is Force equal to components in different dimensions of Force or distance of those components

I'm trying to understand to find my homework problem. Its a simple concept of finding force. I wonder if you have two formulas then is the total Force = force from x + force from y. Or $\sqrt{forceX^2+forceY^2}$. And why. It would make sense to me for it to be the sum of two forces if its not so why not?? I have the actual example: 3.00-kg object is moving in a plane, with its x and y coordinates given by $x=5t^2 - 1$ and $y = 3t^3+2$, where x and y are in meters and t is in seconds. Find the magnitude of the net force acting on this object at $t = 2.00 s$ so I either get 66 by addition of the forces. Or 46.861 by distance formula.. Which one is correct and why. (feel free to use the example i provided or just explain the concept without it. I just wonder which way of finding force is correct and why)


What physical phenomenon occured in this experiment?

I noticed a strange behavior of my experiment. I have a glass container with a fully saturated solution of CuSO4, which I use to grow CuSO4 crystals, by waiting for the water to evaporate. Recently I noticed that crystals were build above water surface and outside of the container. After some time I noticed that half of my solution ended up on the table and floor creating a big blue pool. It looked almost like the behavior of a super-fluid, which can creep out of its container. I removed the crystals using a paper towel, but after some time the same thing happened. Once the crystals go over the edge and below the water line on the other side, the fluid goes outside quite quickly. Just like if you put an inverted U-pipe into a container. Once you force the liquid to start going, it will continue to flow by itself. ![CuSo4 creeping out of a container](http://i.stack.imgur.com/juqYQ.jpg) Can you explain what physical phenomenon(s) is/are happening here? Is it some sort of capillary action?


Deriving the Hamiltonian density for a free scalar field

I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the subsequent derivation of the components of the energy-momentum four-vector in terms of the new field operators. Could you point me in the right direction? $${\mathcal{H}~=~\frac{1}{2}\left[\left(\partial_t\phi\right)^{2}+\left(\nabla\phi\right)^{2}+m^{2}\phi^{2}\right]\textrm{.}}$$ References:   1. M Dasgupta, _AN INTRODUCTION TO QUANTUM FIELD THEORY,_ Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009.


Equation 7.22 in Peskin & Schroeder: writing the Fourier transform of a two-point function as a series of 1PI diagrams

In Peskin and Schroeder's QFT book, on page 219, there is the following equation:![enter image description here](http://i.stack.imgur.com/nC46a.png) The heading to the equation is: "The Fourier transform of the two-point function can now be written as". Could someone help me formally prove this formula? I don't really get why the perturbation expansion decomposes into 1PI's.


two distant galaxies seen from earth

From the Earth If we observe two galaxies that are diametrically opposed and each 1000ly far from the earth. the separation distance between the galaxies will be 20000ly? really question is: if the galaxies were separated from Earth 10Gly. then the separation distance between galaxies would 20Gyr?


Finite difference formulation of the heat equation with thermal conductivity in 1D

This may seem trivial, but I'm having some trouble deriving the finite difference form of the heat equation with a thermal conductivity function $a(x)$ depending on $x$: $$\frac{\partial u(x, t)}{\partial t} = \frac{\partial }{\partial x}[a(x)\frac{\partial u(x, t)}{\partial x}]$$ I find some trouble deriving the right hand side. Let $h$ be the space step, and $x_{i−1}=x_i−h$ and $x_{i+1}=x_i+h$. I want to evaluate the right hand side on node $i$. If I take centered differences I get: $$\begin{align*}\left\\{\frac{\partial}{\partial x}\left[ a(x)\frac{\partial \, u(x, t)}{\partial x}\right]\right\\}_i &= \frac{\left[a(x)\frac{\partial\, u(x, t)}{\partial x}\right]_{i+1/2} - \left[a(x)\frac{\partial \, u(x, t)}{\partial x}\right]_{i-1/2}}{h} \\\ &= \frac{a_{i+1/2}\left[\frac{u_{i+1}-u_{i}}{h}\right] - a_{i-1/2}\left[\frac{u_{i}-u_{i-1}}{h}\right]}{h} \end{align*}$$ Is this ok? The conductivities should be in between the nodes I'm calculating? (intuitively it seems this is reasonable). Is there a way, for example by using forward or backward differences, to use conductivity nodes that overlap with the nodes I'm calculating (I calculate only the interger indexed nodes)? Can I mix a centered difference approach for one derivative and a backward or forward approach for the other one? I would really appreciate any hint. Thanks in advance, Federico


Modelling of nuclear motions (Classification) after invoking the BO approximation

I know that after invoking the Born-Oppenheimer approximation, the nuclei will move on the adiabatic potential provided by the electronic energy (also called potential energy surface (PES)). Such nuclear motions, can be described following quantum-mechanical, classical or hybrids methods on a fitted potential energy surface (PES) or when we want to avoid any analytical representation or interpolation of the PES, we can resort in “on the fly” calculation of the potential. My question is, what are those quantum, classical or hybrid theories/methods that are in connection with an statistical description? and how can i make a general classification o each of them?, at least the most important methods.


Blackbody and standing waves

I'm reading articles about black body radiation and why classical mechanics fails to explain it. My question is: Why do EM waves have to be standing wave in a cavity?


A Derivation of Ehrenfest's Theorem in a particular case

What are the missing lines in the integration? $$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ $$= \frac{\text d}{\text d t} \int\limits_{-\infty}^{\infty} \Psi^* \left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi ~\text d x$$ $$= \frac{\hbar}{i}\int\frac{\partial }{\partial t} \left(\Psi^*\frac{\partial\Psi}{\partial x}\right)~\text d x $$ $$= \frac{\hbar}{i}\int \frac{\partial \Psi^*}{\partial t}\frac{\partial \Psi}{\partial x} +\Psi^*\frac{\partial }{\partial x}\frac{\partial \Psi }{ \partial t} ~\text {d} x$$ $$= \frac{\hbar}{i}\int\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}+\frac{i}{\hbar} V\Psi^*\right)\frac{\partial \Psi}{\partial x}+\Psi^* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x$$ $$=\int\left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}\right) \frac{\partial \Psi}{\partial x}+\Psi^*\frac{\partial}{\partial x}\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}-V\Psi\right)\text d x$$ $$=\left. \left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{ \partial x^2} \right) \Psi \right|_{-\infty}^{\infty}-\int\left(\frac{\partial}{\partial x} (V\Psi^*)-\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}\right)\Psi \text d x+ \left.\Psi^*\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} - V\Psi\right)\right|_{-\infty}^\infty-\int\frac{\partial\Psi^*}{\partial x} \left( \frac{\hbar^2}{2m}\frac{\partial ^2 \Psi}{\partial x^2} - V \Psi \right)\text d x$$ $$=0 + \int\left(\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}-\frac{\partial}{\partial x} (V\Psi^*)\right) \Psi \text d x+0+\int\frac{\partial\Psi^*}{\partial x}\left(V\Psi - \frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}\right) \text d x$$ $$=\int\frac{\hbar^2}{2m} \left(\frac{\partial^3\Psi^*}{\partial x^3}\Psi -\frac{\partial\Psi^*}{\partial x} \frac{\partial^2\Psi}{\partial x^2}\right)+\frac{\partial\Psi^*}{\partial x}(V\Psi )-\frac{\partial}{\partial x}(V\Psi^*)\Psi\text d x $$ $$\vdots$$ $$=\int \frac{\hbar^2}{2m}\left( \Psi^* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)+\left( V \Psi^* \frac{\partial \Psi}{\partial x}-\Psi^* \frac{\partial}{\partial x} (V \Psi) \right)~\text d x$$ $$\vdots $$ $$= \int \left( V \Psi^* \frac{\partial \Psi}{\partial x}- \Psi^* \frac{\partial V}{\partial x} \Psi-\Psi^*V \frac{\partial \Psi}{\partial x} \right)~\text d x$$ $$= \int\limits_{-\infty}^{\infty} -\Psi^* \frac{\partial V}{\partial x} \Psi ~\text {d} x$$ $$=\left\langle - \frac{ \partial V }{\partial x} \right\rangle $$


2 Time Dimensions

Our universe has 4 dimensions, 3 spacial and 1 temporal. We can add another spacial dimension and use math to explore it. I cannot think of any reason why we cannot add add another temporal dimension in the same way. So, what would a universe with 2 temporal dimensions be like? These are some thought I had about physics in 2 temporal dimensions. With one spacial dimension, forwards is the only movement possible. With two spacial dimensions, movement in many different ways is possible. It would follow then that with two temporal dimensions time travel in many different 'directions' would be possible. The formula for velocity is displacement/time. With two times, how would velocity be effected? What would happen to light, which must travel at a constant speed? Frankly, I have no idea a universe with 2 times is like, but I very much would like to know.


Laser frequency tripling

I've heard that common way to get 354nm laser is to triple 1064nm Nd:YAG one. But how to make non-linear crystal to do tripling instead of doubling? What are the best crystals for tripling and what's the efficiency?


Defining a Riemannian manifold - made easy?

In the context of GTR spacetime, I'm trying to get the basic idea of a Riemannian manifold clear in my mind. Apologies for the longwindedness. Question 1. Is this a reasonable, simplified summary of the steps needed to define such a manifold?   1. Start with an amorphous collection of points.   2. Find the minimum number of parameters needed to locate each point in space. That number will be the dimension of the manifold.   3. Construct a coordinate system to locate each point in space. I may need more than one coordinate system to to cover the whole manifold. I need a “good” coordinate system so that each point is uniquely described.   4. Decide on a metric that gives the infinitesimal distance between two adjacent point. The metric need not be the same all over the manifold (as $g_{\mu\upsilon}$ isn't in spacetime). In that case I would have a metric tensor field defined over the manifold.   5. The manifold has to be continuous and differentiable (not quite sure what this means, but I assume it's the obvious - I have to be able to differentiate it everywhere).   6. I can now start constructing scalar fields, covariant and contravariant vectors, tensors etc. Question 2. Assuming I do all the above with a flat plane (a table top for example), using Cartesian coordinates. Can I impose any kind of metric to my new “table top” manifold? Could I just make a weird one up: $$\left(\begin{array}{cc} x^{2} & \sin xy\\\ x+y & 13y \end{array}\right)$$ for example, instead of the Euclidean $$\left(\begin{array}{cc} 1 & 0\\\ 0 & 1 \end{array}\right)$$ Thank you.


Easy proof of Noether's theorem?

Where could I find an easy proof of Noether's theorem? I mean I know that the variation must be $ 0=\delta S = (EULER-LAGRANGE)+ (CONSERVED\, \, \, CURRENT) $ for the case of a particle $q(t)$. I know how to obtain it by I have doubts for the case of fields $ \phi (x) $, any hint? I have checked several books but I cannot find any easy proof of Noether's theorem anywhere; they use too complicated methods.


Higgs field in space around us?

Is the higgs field in space around us? I understand it as that the higgs field has a constant value on every space time point, is that right? And this value is the vacuum expectation value. This value is responsible for giving mass to the elementary particles, like the fermions, right? In the Lagrangian for e fermion there is a term $g \phi_0 \bar \psi \psi$. $g$ is the Yukawa coupling constant and $\phi_0$ is the vacuum expectation value, right?


Does brightness of the light remains constant with the variation of the distance with respect to the source in vaccum?

I have this question going in my mind from many days, i.e why brightness of light emitted from any light source around us decreases with distance? The brightness of light from tube light, streetlight, etc or any light from any source around us, decreases with the distance. As my teacher has taught me that light is a form of radiation, I thought this decrease in brightness may be due to absorption of radiation in the medium. I don't know whether it is correct or wrong. If it is wrong, please explain _why there will be decrease in brightness of light with increase in distance from the light source?_   If we assume that decrease in brightness of light is due to absorption of radiation in the medium, then _in vaccum (where we can assume no energy dissipation), would there be no decrease in brightness of light with respect to distance from the light source?_   ![enter image description here](http://i.stack.imgur.com/x6nhs.jpg)


Higher order covariant Lagrangian

I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-Gordon-Lagrangian) or non-covariant (e.g. KdV) ones. Also some pointers to the literature about general properties of such systems are welcome. Thanks


What is a simple physics application of the reciprocal of a square root?

I learned yesterday that the inverse of square roots is used to calculate the vectors of surface normals in 3d graphics. It seems like such a mind- bogglingly simple idea, and it leads me to wonder if it hasn't found use elsewhere. Is there anywhere in physics where one might need to calculate the reciprocal of a number's square root?


What is the Lorentz tensor with a superscript and subscript index?

I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar with $\Lambda$, the Lorentz matrix. If the indices are in superscript then it is the inverse of $\Lambda_{\mu \nu}$. However, what is $\Lambda^{\mu}_{\nu}$ in terms of the Lorentz matrix? I have chosen mathematical physics as the tag, as I do not think any discipline of pure mathematics is appropriate based on the context. Please correct me if I am wrong.


Cosmological triangle with PLANCK results

Is there an updated version of the cosmological triangle with recent PLANCK results included? ![](http://i.stack.imgur.com/Q31Pq.png)


Maximum potential of a van de graaff generator

I know that we can calculate the maximum potential of a van de graaff generator by ( radius* electric field in which corona discharge begin to form in the surrounding gas( according to wikipedia 30 kv/cm)). What if i covered the sphere with Mica with a dielectric strength ( according to wikipedia) 1.18 Mv/cm , will that increase the maximum potential of the sphere? I know it will be hard to discharge the sphere in this case , but is it possible?


Calculating partition function of ultra-relativistic 1D gas

This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) consisting in $3N$ particles moving in one dimension. I know that the partition function is given by $$Q_{3N}=\frac{1}{(3N)!h^{3N}} \int e^{-\beta H(q,p)}d\omega,$$ where $d\omega$ denotes a volume element of the phase space. In this case $$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$ and $$H(q,p)=\sum_{i=1}^{3N}cp_i.$$ Then, making the substitution I find that $$Q_{3N}=\frac{L^{3N}}{(3N)!h^{3N}} \left[\int_{-\infty}^{+\infty} e^{-\beta c p_j}dp_j\right]^{3N}.$$ $L$ being the "length" of the space available. But I'm pretty sure that this integral does not converge. Where am I wrong?


How much power would a space craft's magnetic shield require?

I've read over the decades that a magnetic shield might protect a spacecraft from cosmic radiation. Its a fascinating idea that might only be theory or science fiction at the moment. In regards to that here is an article just in case someone isn't sure what I mean: http://physicsworld.com/cws/article/news/2008/nov/06/magnetic-shield-could- protect-spacecraft The article even mentions simulations about midway into the article. Does anyone know if research has continued? My big question though, is how much electrical power would be required to protect a space vessel in this way? How large would the field be in order to be useful? Better yet, please use some practical objects that anyone can wrap their head around: What if you were using similar technology to protect the Apollo Command Module? Or the Discovery One from 2001 (140 meters long). _In laymans terms, how much electrical power would it take to create a magnetic shield?_


how do compression and expansion of air transfer energy(sound) and why it is adiabatic not isothermal?

Suppose a vibrating fork exerts force to the air particles to compress which leads to the increase of internal energy (heat).This heat or energy given by the fork is given to the next layer of air and in this way, this energy makes sound. But if it was so, it will be isothermal. And Laplace told it was adiabatic. How can it be? I am confused. The heat energy infact is the energy given by the fork. If it is adiabatic, how can the air pass this energy ?


propagation of sound energy

In my previous questions like 'how do compressions and relaxation carry energy...' and 'how are they not isothermal...' and 'where does the heat come from in the compressed gas...',i've made enquiries reagarding the phenomenon of sound energy propagation and got answers from experts.So i am now writting my concept:when a vibrating fork compresses the air,it does work on it increasing the K.E. of the layer.When the fork goes opp. side,it works on the compressed layer and relaxes it increasing theP.E.Meanwhile,during expansion,the layer works on the next layer and compresses it using the excess kinetic energy.And the next layer does the same to its next layer by compressing using that K.E. given to it by the fork via the first layer.Ultimately,the last layer works on our ear drum to produce sound using the kinetic energy.The question is am i right in my assumption?Plz help...


Equation of motion for the reduced density matrix

The equation of motion for the density matrix of a many body isolated quantum system is the von Neumann's equation: $\dot{\rho }(t)=i[\rho (t),H]$. How about the equation of motion for the reduced density matrix of one particle, is there a general procedure to obtain it?


Is this possible? LCD right in front of the eye

I came across these swimming goggles with a LCD display integrated right on the goggle itself. http://www.psfk.com/2011/09/smart-goggles-enable-swimmers-to-track-and-record- swim-pace.html/my_pace_goggles2 Does this look possible as shown in the design? Would I need a lens over the LCD to make it easier to focus on? How thick and/or curved would that lens need to be? Any advice appreciated!


How do I find equations for the distance of the planets relative to one another?

I know I should be able to piece together some basic Newtonian equations for this, but I'm not sure where to start. I want to be able to choose one planet as the center and calculate its distance from another planet over time as they both rotate the sun. First I need to even find equations for the motion of the planets, then... magic? Thanks for any guidance!


Are neutrino stars theoretically possible?

Since neutrinos   1. have a small mass and   2. are affected by gravity, wouldn't it be theoretically possible to have such a large quantity of them so close to each other, that they would form a _kind_ of a stellar object, i.e. one that would keep itself from dissolving due to the large gravity. If such objects were possible, how would they interact with rest of the world? Would they be invisible (dark matter?) because of neutrinos' lack of electromagnetic charge? Would this lack also make ordinary matter pass through them, or would the Pauli exclusion principle prevent this passing through due to the high density of neutrinos?


Physics Equivalent of Physical Chemistry (P-Chem)

Before I take P-Chem I would like to understand how physicists view the same material. What Physics course(s) should I take and/or books should I read to learn the same material from a physicist's perspective? Would thermodynamics be equivalent? Background: I have done the math: Calculus 1-3 and differential equations and have already taken general physics 1+2 and 2 semesters of modern physics (relativity, QM...).


Homework: A parachutist jumps from a plane

I need help with the following problem. I think I almost have it . . . A parachutist bails out and freely falls a distance of $y_1$. Then the parachute opens and thereafter, the parachutist decelerates at a rate of $a_2$. She reaches the ground with a speed of $v_2$. Find her average speed for the fall. (The answer should be $18.5\ \text{m/s}$.) Note: In the diagram, replace "x" with "y". :) ![http://i.imgur.com/OPTZfWI.png](http://i.stack.imgur.com/5lnSe.png) $ \text{Let upwards be the positive y direction.}\\\ \text{Givens:}\\\ y_1 = -59.7\ \text{m}\\\ v_0 = 0\ \text{m/s}\\\ v_2 = -3.41\ \text{m/s}\\\ a_1 = -9.8\ \text{m/s$^2$}\\\ a_2 = 1.60\ \text{m/s$^2$}\\\ \text{Solve for $t_1$:}\\\ y_1 - y_0 = \frac{1}{2}a_1t^2+v_0t\qquad\text{for}\ \ y_1 = 0; v_0 = 0; t=t_1.\\\ t_1 = \pm\sqrt{\frac{2y_1}{a_1}}\\\ t_1 \approx \pm 3.4905\ \text{s}.\\\ \text{Plug $t_1$ into the velocity equation for $v_1$ to find $v_1$:}\\\ v_1 = a_1t_1 + v_0\\\ v_1 \approx -34.207.\\\ \text{Plug $v_1$ into the velocity equation for $v_2$ to find $t_2$:}\\\ v_2 = a_2t_2 + v_1\\\ t_2 = \frac{v_2 - v_1}{a_2}\\\ t_2 \approx 19.248\ \text{s}.\\\ \text{Plug $t_2$ into the position equation for $y_2$ to find $y_2$.}\\\ y_2 = \frac{1}{2}a_2t_2^2+v_1t_2 + x_1\\\ y_2 \approx -422\ \text{m}.\\\ \text{Now solve for $\lvert\overline{v}\rvert$:}\\\ \lvert\overline{v}\rvert = \lvert\frac{x_2 - x_0}{t_2 - t_0}\rvert\\\ \lvert\overline{v}\rvert \approx 21.9\ \text{m/s} \neq 18.5\ \text{m/s}. $ My solution is incorrect. What am I doing wrong?


Independent systems and Lagrangians

**Definition 1:** The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is equal to the probability of finding the system ($S_1$) in the configuration ($C_1$) times the probability of finding the system ($S_2$) in the configuration ($C_2$). **Definition 2:** However, in we consider fields systems, the practical tool is Lagrangians. So I should say that 2 systems are independent if : $$ Lagrangian (S_1S_2) = Lagrangian (S_1) + Lagrangian (S_2)$$ **The question:** Now, what is the relationship between these 2 definitions? They could be only compatible, or they could be equivalent in the field domain. Is there a way to "demonstrate" the latter from the former ?


Where do high-energy neutrinos come from?

Last week the IceCube South Pole Neutrino Observatory published a press release reporting the possible discovery of two neutrinos with energies of over 1 PeV. Would anyone here be willing to help me understand the significance of this announcement? Although it was reported in the press that these neutrinos originated from outside our galaxy, the original press release seemed to say that this might possibly be the first sighting of 'astrophysical' neutrinos. Are there other observations of extragalactic neutrinos? Are there any conjectures as to processes which might generate? How would a neutrino even be accelerated if it is unaffected by magnetic fields?


Mathematical explanation of quantum teleportation

I am now studying quantum teleportation. I get what the process is like but I'm wondering why it happens this way. You've got two entangled particles A and B whose wavefunctions are entangled. You also have a third particle C, which is the one you want to teleport. You get C entangled with either of the two and final result is that the wavefuction of one of the two A and B (depending on which you entangled C with) becomes the same as the C's wavefunction is.   1. Why does it occur this way (mathematically explained, if possible)? Why does the wavefunction of A or B becomes that of C?   2. What is the equation explaining this process (addition of wavefunctions, maybe)?


Electric Field in Dieletric

I have a simple doubt. The textbook I'm working with says that if $E_0$ is the electric field in some region without diletric, then if we introduce a dieletric of constant $k$ the new field will simply be $E=k^{-1}E_0$. Is it always true? I mean, suppose we have concentric spheres of radii $a$ and $b$ being $a$ the radius of the inner sphere and $b$ the radius of the outer sphere and charge $Q_a$ in the inner and $Q_b$ in the outer. If there's no dieletric between them the field in the space between at a distance $r$ from the center will be given by Gauss' Law as: $$E_0=\frac{1}{4\pi\epsilon_0} \frac{Q_a}{r^2}$$ Now if we insert a dieletric on the space between is it true that the field on the dieletric will _simply_ be: $$E=\frac{1}{4\pi k \epsilon_0}\frac{Q_a}{r^2}$$ Thanks very much in advance.


How can I model buckyballs sticking together? (Tiny spherical magnets)

Buckyballs are a desktoy that consist of 216 tiny spherical magnets. You can makes all kinds of interesting shapes and structures out of them (do a Google search). I want to model them in a physics simulation, but I'm not sure how to do it. Right now, I'm thinking calculate the sum of the forces between each of the buckyball's poles and each of the other buckyballs' poles, and update its linear momentum. Then calculate the torque that each of those forces causes about the magnet's center, and use that to update angular momentum. The problem I'm having is how to model the magnets touching. It's easy in real life to make a long chain or a hexagon. How should I model the magnets actually running into each other to mimic this behavior?


Alcubierre Drive - Clarification on relativistic effects

On the Wikipedia article on the Alcubierre drive, it says: > Since the ship is not moving within this bubble, but carried along as the > region itself moves, conventional relativistic effects such as time dilation > do not apply in the way they would in the case of a ship moving at high > velocity through flat spacetime relative to other objects. And... > Also, this method of travel does not actually involve moving faster than > light in a local sense, since a light beam within the bubble would still > always move faster than the ship; it is only "faster than light" in the > sense that, thanks to the contraction of the space in front of it, the ship > could reach its destination faster than a light beam restricted to > travelling outside the warp bubble. I'm confused about the statement _"conventional relativistic effects such as time dilation do not apply"_. Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?


Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory?

I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre Calabi-Yau manifolds? I have Yau's semi-popular book but haven't read it yet, nor, obviously, String Theory Demystified :)


Is dark matter really present around the sun?

Recently I read an article that there is dark matter around the sun but if it is so, than why can we see it clearly. If it is called matter than it shall show some hindrance in radiation we receive and the solar flares must bounce back against them though they are coming straight to us. To stay close to sun it has to be rotating at a very very high speed, so does it? Does it pose some kind of gravitational effect on the rest of the system? Is it constant or does it increase or decrease in size?


How is Momenergy frame independent?

momenergy = mass * spacetime displacement/proper time for that displacement What I dont understand is how can the momenergy be frame independent? The unit-4 vector always points in the direction of the worldline? Agreed, the magnitude of momenergy will be invariant wrt to any frame. But wont the worldline of a particle be different in different frames? So wont the direction of the spacetime interval be frame dependent?


Does electromagnetic radiation make sense in one dimension?

I'm trying to do a simple simulation of a 1D charged quantum particle, which gets irradiated by an electromagnetic wave — in context of non-relativistic QM. The Schrödinger equation for such a particle would be $$i\hbar\frac{\partial \Psi(x,t)}{\partial t}=\frac1{2m}\left(-i\hbar\frac{\partial}{\partial x}-qA(x,t)\right)^2\Psi(x,t)+q\left(\phi_0(x)+\phi(x,t)\right)\Psi(x,t),$$ where $A(x,t)$ is (what I believe to be) an $x$ component of vector potential $\vec A$ of radiation, $\phi_0(x)$ is static scalar potential of non- irradiated system, and $\phi(x,t)$ is scalar potential of radiation. What I can't seem to understand is how electromagnetic radiation should be incorporated in this equation. I suppose it shouldn't depend on coordinate, i.e. in non-relativistic context speed of light should be considered infinite, thus wavelength should be infinite. From basic knowledge of electromagnetism I suppose that such a wave would be represented by: $$E=E_0 \cos(\omega t)$$ $$B=B_0 \cos(\omega t)$$ But simulation is to be 1D, while electromagnetic wave has three dimensions: one for direction of propagation (which I guess is irrelevant because of infinite wavelength), second for $\vec E$ and third one for $\vec B$. To add the complexity, I need to determine how vector potential $\vec A$ and scalar potential $\phi$ would look in such case, which I might understand once I realize how to orient the radiation in terms of $\vec B$ and $\vec E$. So, my question is: does it even make sense to talk about external electromagnetic radiation in 1D case? If yes, how should the field be oriented in this case to make a sensible quantum simulation? What would its vector potential $\vec A$ and scalar potential $\phi$ look like?


Considerations and questions for a DIY heat pipe; how to choose a working fluid?

I plan to build a DIY heat pipe! For more info on what a heat pipe is, see Wikipedia. I would like to use it to cool a desktop graphics card I have. I am having trouble deciding how to choose the working fluid for the pipe. I plan to build it based on this design. Using gravity to return the fluid to the hot end. Working fluid considerations My graphics card consumes 250 watts at 100% load and 15-20 watts at idle, the maximum temperature is 90º Celsius and the temperature at idle is 30º Celsius.   * How does this translate to a choice of working fluid? I would assume the fluid has to have a phase transition temperature just lower than that of the lowest operating temperature of the card, which can be changed by based on the PV=nRT law? So just whatever substance will boil between 30–90ºC at the given pressure in the pipe?   * How can I decide how much fluid to use? As long as the pipe is filled with only the vapor of the working fluid at the proper pressure, then would the only amount of fluid you need be just enough to cover the surface area of the boiling point on the hot end? Would it be a good idea to add a little more?


Raising a toolbox with a rope

Suppose you use a rope to hoist a box of tools vertically at constant speed v. The rope exerts a constant upward force of magnitude F_up on the box, and gravity exerts a constant downward force (the weight of the box). No other forces act on the box. To raise the box twice as fast, the force of the rope on the box would have to have:- A) the same magnitude $F_{up}$ B) a magnitude of $2F_{up}$ The answer seems to be (A), and I don't understand why! It says, if it's constant velocity, both upward and downward force = 0. Since the downward force is always the same, the upward force must be the same, too, regardless of the box's speed. My take on that: But how are you gonna change the speed of the box without exerting force? If the force is the same, while there is one in the other direction, there would be no change of speed. So, can you explain to me the correct answer? EDIT: "Consider ... (1) starting the motion, (2) maintaining the motion, and (3) slowing the toolbox to a stop ..." (1) Starting. The object needs force greater than weight to go upward. (2) Maintaining. The object needs force equal to the weight to maintain motion upward. (3) Slowing. The object needs force less than weight for it to stop. Ummm.. I don't know what to conclude with that? I understand one important thing: It can't be $F_{up} = W$, or in other words $F_{up} = F_{down}$, because there would be no acceleration in any direction. If there is no acceleration at any direction, I can't get the velocity to be twice as fast, or any amount faster. EDIT: Thank you all. @Manishearth I could imagine this, but the book doesn't mention the "jerk", nothing about it in the question or in the explanation for the answer. (This is an example in the book). That's why I got confused. Anyway, again, thank you all for the clarification.


supressing certain decay paths and enhancing others with interference

In a scattering reaction, there are many possible final states for the products, each with different production rates. > **Question:** Is there a way in which we could in general supress certain > rates of products of the reaction and enhance others? can we use coherence > and interference of the initial states to modulate such rates? As an application of such possibility, i'm wondering if we could enhance the production rate of antimatter (for the purpose of the question, let's consider antihydrogen) by a few orders of magnitude by such manipulations?


How do Anamorphic images work?

I found the following anamorphic image on this site ![enter image description here](http://i.stack.imgur.com/c2f3G.jpg) Can anyone explain the physics behind the working of such images? Can we create our own anamorphic images? If so how? Thanks in advance.


Why is Mosely screening in heavy atom K-shell 1 unit?

The Wikipedia article on Moseley's law seems to show that the screening of heavy atoms is by 1 electron charge _exactly_ (in the limit of large Z, experimental precision, within nonrelativistic limits, and). But why is this exactly one unit? The other K-shell electron is not screening exactly one unit, and this seems to be a conspiracy of other electrons. I suspect it is because of an unappreciated hole-picture of deep holes in heavy atoms (electrons missing in deep shells), and I will describe this theory briefly. If you remove an electron from close to the nucleus, the electron-hole behaves as an object with _positive charge_ and _negative mass_ (this is why it orbits the nucleus that it is repelled by). The state is not a vacuum quite, because of the presence of other electrons, but the rigidity of the Fermi liquid near the nucleus of a heavy atom means that the hole behaves as a single particle. This single-particle behavior is in the potential background of the nucleus and the other electrons, and it is possible that the result can give an exact 1 unit screening. I developed the formalism a little bit to see what the form should be, but I did not see any reason for 1 unit screening. Perhaps there is none, but it looks to be more than a coincidence.


Predicting the direction of sound incorrectly?

When i was sitting outside my house yesterday, I heard a strange sharp sound. I could not point out it's exact direction but apparently looked like coming from left but later my friend told me that an accident had occurred (due to which the strange sound came) and it totally came from different direction? How did this happen? Why can not our ears point out the exact direction of sound sometimes?


Covariant derivatives

I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \\{X_i, Y_a\\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \\{X_i\\}$ Lie algebra of $G$ has the following structure: $$[X, X] \sim X, \quad [X, Y] \sim Y, \quad [Y, Y] \sim X + Y.$$ The coset space $G/H$ is parametrized by coordinates $\xi^a$ corresponding to the generators $Y_a$. In the exponential parametrization an element of this coset has the form $$G/H: \quad \Omega = exp\\{i\xi^a Y_a\\}$$ The ordinary derivatives $\frac{\partial}{\partial\xi^a}$ of $\psi(\xi)$ ($\psi(\xi)$ - field defined on $G/H$) do not transform covariantly. I was told once that it is quite easy to define covariant derivatives using the Cartan form $\Omega^{-1}d\Omega$. I looked through the bunch of the mathematical books about Lie groups and Lie algebras but I can't find a clear description of the procedure. Can anyone advise me a proper book (paper) or, if it's not take much time, tell it here? Thank you. Sorry for my English)


Is the symmetrisation postulate unnecessary according to Landau Lifshitz?

The symmetrisation postulate is known for stating that, in nature, particles have either completely symmetric or completely antisymmetric wave functions. According to these postulate, these states are thought to be sufficient do describe all possible systems of identical particles. However, in Landau Lifshitz Quantum Mechanics, in the first page of Chapter IX - Identity of Particles, he comes to the same conclusion without needing to state any ad-hoc postulate. It goes like this: Let $\psi(\xi_1,\xi_2)$ be the wave function of the system, $\xi_1$ and $\xi_2$ denoting the three coordinates and spin projection for each particle. As a result of interchanging the two particles, the wave function can change only by an unimportant phase factor: $$ \psi(\xi_1,\xi_2)=e^{i\alpha}\psi(\xi_2,\xi_1) $$ By repeating the interchange, we return to the original state, while the function $\psi$ is multiplied by $e^{2i\alpha}$. Hence it follows that $e^{2i\alpha}=1$ or $e^{i\alpha}=\pm1$. Thus $$ \psi(\xi_1,\xi_2)=\pm\psi(\xi_2,\xi_1) $$ Thus there are only two possibilities: the wave function is either symmetrical or antisymmetrical. It goes on by explaining how to generalize this concept to systems with any number of identical particles, etc. Im summary, no symmetrisation postulate was ever stated in this rationale. Is "shifting by an unimportant phase factor" a too strong requirement for ensuring identity of particles?


Centrifugal Force

When a body is in circular motion, realistically, it experiences only the centripetal force, created by gravitational pull, tension etc. , which gives it acceleration towards the centre. Now, assuming we were to fill a bucket with water, and whirl it in the vertical plane. There will be a certain velocity when the water doesn't fall out even when the bucket is upside down. At that instant, if we were to write down the forces that act on the water, they would be the Normal reaction by the bucket, and it's weight. This means that the net force is downwards. Why doesn't the water fall down, since there's no force in the upward direction? Centrifugal force is a psuedo force, so it shouldn't be applicable here, as the water won't wall down even if I watch from a non inertial frame. How can one explain this phenomenon? Also, say we were to fit a ring in a rod, and rotate the rod, the ring will eventually slip and fall off. Why does this happen if there is no outward force?


Crystal magnetic response only skin deep?

The Hamiltonian for a single electron in a magnetic field reads $$H=\left(\frac{{\bf p}^{2}}{2m_{e}}+q_{e}\phi\right)+\mu_{B}\left({\bf \hat{L}}+g{\bf \hat{S}}\right)\cdot{\bf B}+\frac{e^{2}}{8m_{e}}\left({\bf B}\times{\bf r}\right)^{2}$$ where symmetric gauge ${\bf A}=\frac{1}{2}{\bf B \times r}$ is used and where the last two term are responsible for paramagnetism and diamagnetism respectively. The many-body Hamiltonian, weak electron-electron interaction considered, reads $${\cal H} = \sum_{\sigma}\int d{\bf r}\,\left[-\left(\psi_{\sigma}^{\dagger}({\bf r})\frac{\hbar^{2}}{2m^{*}}{\nabla}^{2}\psi_{\sigma}({\bf r})\right)+\boldsymbol{\mu}({\bf r})\cdot({\nabla}\times{\bf A})+\hat{n}_{e}({\bf r})\frac{e^{2}}{2m^*}{\bf A}({\bf r})^{2}\right]$$ where $\boldsymbol\mu({\bf r})=\sum_\sigma \mu_B \psi^\dagger({\bf L}+g{\bf S})\psi $ is local magnetic moment density of electron of total electron angular momentum and define $\mathbf M_\text{para} \equiv \langle \boldsymbol \mu\rangle$ Now that we vary $\bf A$, we should obtain the response current $${\bf j}={\nabla}\times{\mathbf M_\text{para}}+\frac{n_{e}e^{2}}{m^*}{\bf A}$$ It is rather strange to me that (i) the last term is essentially the same as the one in London's equation, which gives a penetration depth $\lambda$, and that (ii) the first term is non-zero only on the crystal surface if $M$ is uniform deep inside, which I think is just the magnetization current we saw in textbooks. If (i) is true, one may conclude that magnetic response for all crystals (neglect interactions) happens in a thin layer near to the surface (a few nanometer) since magnetic field can only penetrate that far. The length scale that electron density drops to zero is a few angstroms, so we are able to neglect the first term in an appropriate region.


Why does earth cool?

This question is boggling me for some time. We know that heat can be transferred from matter to matter and heat is nothing more than tiny atoms vibration intensity (correct me if I'm wrong). But space is a vacuum, and so arises the question: Can you heat vacuum? :)) How can it be that earth loses it's heat to space itself? Thanks


Did I see two Milky Ways next to each other?

I am from the northern hemisphere and as far as I remember, the Milky Way was "single lane", just one stripe. At least, this is what I recall the Milky Way to look like when seeing it from near Irkutsk, Siberia, in late August 1997. Yesterday I went to Blackbutt, Australia (somewhere near 27° South, 152° East) and the Milky Way seemed to have a large black stripe in the middle. I was wondering, if these were two spiral arms of the galaxy that we are part of.


How "big" objects can WISE and NEOWISE detect?

I mean WISE is monitoring near-Earth objects, but cannot see the latest Russian meteor and others. Why can it not detect small objects? What is the limit of it's infrared detectors?


Question on the hydrostatic paradox

The hydrostatic paradox, states that the > height of the water in any container is independent of the shape of the > container. This does make sense since it is only the height of the fluid relative to ground which determines the pressure, and it is the pressure equilibrium that decides the shape of the fluid. But then consider the (B) container in the figure below.   ![enter image description here](http://i.stack.imgur.com/DoLKb.png) The pressure at a particular depth say, $h$ is same as in the case of the cylindrical container. But in the cylindrical case (A), any element of fluid is in equilibrium because the difference in pressure of the fluid element above it and that below it exactly balances the weight of the element. Extending this argument to (B), for any element at the edge, the pressure of the fluid above is the same as in the case of the cylindrical container, but below it is the surface of the container. This container, is in equilibrium with the atmospheric pressure outside, and therefore, would be exerting pressure equal to the atmospheric pressure onto the fluid within, which would not be enough to maintain the equilibrium of the fluid element. **How then can the equilibrium of the fluid element at the edge be explained?**


What is the logic behind the Fahrenheit scale?

The Fahrenheit scale is defined by fixed points on the scale. What interests me is the apparent arbitrary chosen numbers in these fix-points. First wikipedia wites from 32 to 212. -and later in the _History_ part we have from 0 to 32 to 96(based on a quadrupling of 7.5 to 22.5 to 60). I am stille confused. Does any of the numbers make sense in an other numeral system? Did Rømer or Fahrenheit use their lucky numbers or what is going on? **Edit:** I am not interested in what solutions or objects, that is used for the fix-points. I am asking what the logic is behind the chosen numbers?


Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power

Wien's fifth power law says that emissive power is proportional to the temperature raised to the fifth power. On the other hand, the Stefan–Boltzmann law says emissive power is proportional to the temperature raised to the fourth power. How can both of these be true?


What is the frequency of occurrence of stellar classifications off the HR main-sequence?

An alternative version of this question would be: "if was to pick a star from the $10^{11}$ or so in our galaxy at random, what are the probabilities of it being various kinds of star?" (and I do mean "in our galaxy" and not "visible in the night sky"). There is a nice table I've seen in various places (e.g wikipedia) which goes a long way towards answering this. It tells me that 0.00003% are 'O' types and 0.13% 'B' and 0.6% 'A' through to the 76% rump of 'M types. Unfortunately there is some small print associated with that table which is that it's only for stars **on the main sequence** (hence its 'M' types are all low-luminosity dwarfs - red giants aren't included - and its 'A's and 'B's are all young giants - white dwarfs aren't included). However, I've no idea what proportion of the stellar population are **off** the main sequence. One in a hundred? One in a million ? What I'd really like to find (or gain enough information to compile) is a 2D version of that table with the same axes as a HR diagram where I can look up the frequency of white dwarfs by looking in the cells in the A-B spectral and the luminosity 0.001 - 0.1 ranges, or the frequency of stars like Betelgeuse by looking in the M, $10^5$ cell. (Of course another issue with the cited table is that it claims to be frequencies in "solar neighbourhood". For example it'd be nice to have different versions specific to stellar populations of, say, globular clusters, the galactic disk and central "bulge". But I'll settle for some pan-galactic numbers initially).


Calculating engine starter’s energy use

During a discussion on start-and-stop vehicle technology some bloke began pushing the point that re-starting the car uses stored energy from the battery, which needs to be replenished by increased fuel usage once the engine is moving. Well, this is obvious, the question is that of significance of such additional usage, especially in comparison with savings from reduced idle. Common sense tells me that this is splitting hairs: if ICE efficiency outlays put all accessory usage at 2 to 3% of overall energy consumption, which includes AC, and all electricals, then it can’t be that much. More over, this dissertation explores idle-reduction technologies for long- haul commercial trucks, and one of the systems exampled is battery-powered (p. 14) that stores enough energy to power AC or heater overnight, and takes about six hours of charge while driving. Author acknowledges increased fuel consumption due to necessity of higher-amperage alternator, although there are no specific numbers provided. However, the mere existence of such commercial application available on the market leads me to believe that such added load is still better than idling. But, in the interest of science, I need some hard numbers (besides, the fellow just wouldn’t go away). I have some ideas on what to consider, but I’m not well versed in electrical and mechanical engineering, so I do not think I can account for most major factors. Energy use of the starter motor can be calculated by using the amount of current used per start (which would be 2 to 3 seconds) by the motor itself and the solenoid that engages the starter gear onto the flywheel. Both current and power demand can be found in the starter’s specifications, but I am not sure as to how reliable those numbers would be in the real-world application. Then there is fuel consumption of the start itself which is estimated at 10 to 15 seconds worth of idling (and Florida chapter of ASME even calculated six seconds for 6-cylinder engine (in a simple, non-rigorous field experiment), but the original link is broken). Now, how to calculate the increased consumption due to the charging of depleted battery from the starter motor itself, and, additionally, to account for all the accessories that were running while the engine was stopped? Is it a simple matter of using the same number calculated for energy usage of starter motor, and arriving at it by calculating the amount of fuel needed to produce that much extra energy given the losses in the engine itself and in the charging circuit? And, ultimately, how significant are those considerations in the bigger picture?


Is the potential energy in a compressed spring a Lorentz invariant?

The total energy of an object comes from the time part of the four-momentum, and so isn't a Lorentz invariant. On the other hand, is the potential energy of a compressed spring a Lorentz invariant?


What is the position and orientation of the Earth as a function of time

I would like to know the position and orientation of the Earth, relative to the sun, as a function of time. I.e. select any coordinate system with the Sun at the origin so that the earth lies in the x-y coordinate plane. Provide a function which takes as input the time and produces   1. The x-y position of the center of the Earth   2. The unit vector from the center of the Earth to the North Pole.   3. The orientation of the Earth around this vector This function is chaotic in general. I would be happy with a decent approximation. This problem is solved as part of the "what is the position of the sun in the sky" problem. Solutions to this problem tend not to be well separable (at least I have trouble separating my question out of the complete position-of- sun-in-sky solution.)


Relationship between the cosmic microwave background (CMB) radiation and the visible universe

I know that the cosmic microwave background radiation (CMBR) is the leftover radiation from the Big Bang. Have we been able to make a concrete map of the radiation and the objects in the visible universe? (Something like "this point in the CMBR map evolved into this particular region of the visible universe".)


Pressure at a bend in a pipe

This is a question from a study guide for a military officer exam: I have fluid flowing through an elbow in a pipe (the elbow is in the shape of an upside down and backwards "J") and the flow goes from south to north to east. If holes are drilled at the upper left outside part of the bend and the lower right inside part of the bend, at which point will pressure be lower? Please see attached image for a picture of the question. When I have thought about this problem, I keep coming back to pressure being transmitted equally to all parts of a fluid, so I don't understand how the pressure can be different at one of the holes. The book gives the answer as the inside bend which I can force myself to accept since I can picture the fluid attempting to go straight to the outside bend hole and thus hitting that part harder, but I don't really understand it.


Fermions in the same state

I need some clarification of what is meant when someone says "fermions cannot occupy the same quantum state". Consider two bosons: $$\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = \frac{1}{\sqrt{2}} \left( \phi_A(\vec{r_1}, s_1)\phi_B(\vec{r_2}, s_2) + \phi_A(\vec{r_2}, s_2)\phi_B(\vec{r_1}, s_1) \right)$$ This is one wavefunction of two particles. A wavefunction directly corresponds to a state, and since this is only one wavefunction, it seems there is only one state -- that two bosons occupy. But now consider two fermions: $$\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = \frac{1}{\sqrt{2}} \left( \phi_A(\vec{r_1}, s_1)\phi_B(\vec{r_2}, s_2) - \phi_A(\vec{r_2}, s_2)\phi_B(\vec{r_1}, s_1) \right)$$ Again, one wavefunction (=> one state) and two particles that occupy it. Yeah, $\psi(\vec{r_1}, s_1, \vec{r_2}, s_2) = -\psi(\vec{r_2}, s_2, \vec{r_1}, s_1)$, but it's still just one state -- occupied by two fermions. Could someone clarify?


Boundary between supercritical fluid and gaseous phase

As far as I understand, going around the critical point, it is possible to transform a liquid into a gas without seeing a phase transition. Does this also mean that there is no clear separation between a gaseous phase and a supercritical fluid ? This is more or less what can be understood for the dotted line between the two that usually appears in a phase diagram such as the following: ![Phase diagram](http://i.stack.imgur.com/lbHET.png) So my question is: Is there a standard physical measure for deciding whether a given fluid is in gaseous phase or in supercritical state? Alternatively, is the boundary between the two clear-cut or somewhat fuzzy?


Deriving the Sommerfeld expansion by contour integration (Le Bellac p. 277)

In Le Bellac's statistical physics book he derives the Sommerfeld expansion by a contour integral. The idea is to expand integrals of the type $I(\beta)\equiv \int_{0}^{\infty}d\epsilon\, \frac{\varphi'(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$. Integrate by parts and expand $\varphi(\epsilon)$ $$\varphi(\epsilon)=\sum_{m=0}^{\infty}\frac{(\epsilon-\mu)^{m}}{m!}\left(\frac{d ^{m}\varphi}{d \epsilon}\Bigg |_{\epsilon=\mu}\right)$$ Make a change of variables to write $I(\beta)=\sum_{m=0}^{\infty}\frac{\beta^{-m}\varphi^{(m)(\mu)}}{m!}I_{m}$ where $I_{m}=\int_{-\infty}^{\infty}dx\, \frac{x^{m}e^{x}}{(e^{x}+1)^{2}}$. The idea then to evaluate the $I_m$ is to consider the following integral $$J(p)=\int_{-\infty}^{\infty}dx\, \frac{e^{ipx}}{(e^{x}+1)(e^{-x}+1)}=\sum_{m=0}^{\infty}\frac{(ip)^{m}}{m!}\int_{-\infty}^{\infty}\, dx \frac{x^{m}}{(e^{x}+1)(e^{-x}+1)}= \sum_{m=0}^{\infty}\frac{(ip)^{m}}{m!}I_{m}$$ Integrating $J(p)$ gives $J(p)=\frac{2\pi pe^{-\pi p}}{1-e^{-2\pi p}}=1-\frac{1}{6}(\pi p)^2+\ldots$ hence, $$I(\beta)= \varphi(\mu) + \frac{\pi^{2}}{6}(k_{B}T)^{2}\varphi^{(2)}(\mu)+O(k_{B}T^{4})$$ So far so good, my problem comes when he writes the expansion of the Fermi- Dirac distribution itself as $$\frac{1}{e^{\beta(\epsilon-\mu)}+1}\simeq \Theta_{H}(\mu-\epsilon)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\delta'(\epsilon-\mu)+O(k_{B}T^{4})$$ which is meant to be valid under an integral. It appears to mimic the expansion of the generating function, at least where the signs are concerned. If we consider the definition of the Sommerfeld expansion from Wikipedia, we have a formula in the form $$\int_{-\infty}^\infty \frac{H(\varepsilon)}{e^{\beta(\varepsilon - \mu)} + 1}\,\mathrm{d}\varepsilon = \int_{-\infty}^\mu H(\varepsilon)\,\mathrm{d}\varepsilon + \frac{\pi^2}{6}\left(\frac{1}{\beta}\right)^2H^\prime(\mu) + \mathrm{O} \left(\frac{1}{\beta\mu}\right)^4$$ If we give Le Bellac's form a try, $$I(\beta)\simeq\int_{0}^{\infty}d\epsilon\, \varphi'(\epsilon)\left[\Theta_{H}(\mu-\epsilon)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\delta'(\epsilon-\mu)+O(k_{B}T^{4}) \right] \\\ =\varphi(\mu)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\underbrace{\int_{0}^{\infty}d\epsilon\, \varphi'(\epsilon)\delta'(\epsilon-\mu)}_{K}$$ but integrating $K$ by parts gives $$K=\left[\delta(\epsilon-\mu)\varphi'(\epsilon) \right]_{0}^{\infty}- \int_{0}^{\infty}d\epsilon\, \varphi''(\epsilon)\delta(\epsilon-\mu)=-\varphi''(\mu)$$ So the signs work out, and the negative sign is probably not a typo. On the other hand, the next term in the expansion would be $+\frac{7\pi^4}{360}(k_{B}T)^{4}\delta^{(3)}(\epsilon-\mu)$ which would take a negative sign when integrated. The alternating behavior of the signs, as far as I can tell, is due to the imaginary unit that was included in the exponential for the contour integral. A very similar approach is done on p. 255 of the following notes except there is no imaginary unit in the exponential. Also, the expansion is meant to operate only on $\varphi$ instead of actually correcting the Fermi-Dirac distribution for small $T$. My questions are as follows:   1. Are the corrections to the Fermi-Dirac distribution given by Le Bellac correct to all orders? They seem to work for the 0th and 1st order terms.   2. Assuming (1) is correct, how do we "know", or what's the reasoning, that the series expansion of the generating function would give the corrections to the Fermi-Dirac distribution? Because it seemed to me that the generating function was only to determine the $I_m$ coefficients.


What does (001) Silicon mean?

If someone gives me a thin film of Si, and they tell me it's (001) Si, does that mean that the (001) planes of Si are the ones making up the surface of the film?


Why do some sunsets have a green flash?

Some sunsets exhibit a phenomenon called "green flash," where there is a quick flash of green light right as the sun is setting. I have seen this myself several times, and sort of understand it. I am a bit frustrated at my inability to explain it better to my family. I understand how Rayleigh scattering would make the dominant color shift towards the blue just after the sun passes below the horizon. How does mirage cause the green light to refract more than the blue? Is there an easy way to use the two effects to estimate the wavelength of the flash?


Particle/antiparticle annihilation and entanglement

This is a basic question. Suppose that A and B are completely entangled particles and so are C and D. If B and C are antiparticles that annihilate each other will A and D be entangled as a result. It seems they should be in order to satisfy conservation laws. Also if the initial pairs were the results of pair creation events then the Feynman diagram of the entire can also be interpreted as a single particle starting as A emitting and absorbing photons and emerging as D. More generally if the state of the 4 particle system just before annihilation was represented as $$\alpha_{ij}\, |A_i\rangle\, |B_j\rangle\; \beta_{kl}\, |C_k\rangle\, |D_l\rangle$$ would the state of A and D just the after the B-C annihilation be $$\sum_j\alpha_{ij}\beta_{jk}\,|A_i\rangle\, |D_k\rangle$$ in a product state with the state of the emitted photons. Is this correct?


Can a planet form before the parent star ignites?

I'm unable to find an answer to my questions via my searches. This questions pertains to the timeline of a star system creation. My question: During the creation of a star system, can satellite planets form before the star has ignited and begun to burn? In more detail, I understand the basic creation of a star, with the gravitational collapse of a large cloud, leading eventually to fusion. Is this transition from protostar to maturity (or to where "ignition" occurs) long enough that planets have already started to form, even in a primitive state (just a big rock)? Or has the star long since been burning when planets finally start to form?


Equivalent Dose absorption from Na22 source

No idea what exactly the rules are or how I should ask this question, but here's the verbatim quesiton: "A Na$^{22}$ source has an activity of 100 $ \mu $C. If you handle the source, how big is the equivalent dose on your hands, if all the radiation were absorbed? Assume the volume and density to be 10 x 5 x 1 $cm^3$ and ~1 $g/cm^3$, respectively." I'm using the SI-system mostly since I live in Norway, but seem to be using $\mu$C for medical radiation. I presume the volume and density apply to the tissue of the hands, and Sodium 22 has mostly positron decay with (I think) energies of 546keV, which makes it have a quality factor of 1. And as far as I've gathered the equation I'm supposed to use is $$D_q = qD_E=q\frac{1}{\rho}\frac{W}{V}$$ where $D_q$=equivalent dose, q=quality factor, $\rho$=density, $W$=absorbed energy, and $V$=volume, but I honestly don't know how I'm supposed to find $W$. If it is as simple as to multiply the activity with the energy value of the positrons, or if I need to find something else entirely. But if it's the first, then when everything is inserted gives me a dosage of: $2.32*10^{-7} mSv / hour$, seems a bit too low dosage, I feel like it should be higher. I'm also wondering if this says that I have to consider both the positron decay, the gamma decay AND the two annihilation photons when I calculate my total energy? With the different weightfactors on the decay with the % listed there? Namely: positron decay=0.546MeV (89,9%), gamma=1.275MeV (100%), annihilation photon=0.511MeV (180%) Haven't had any classes about this so far so I'm sorta stuck trying to find things on the internet. So any help would be much appreciated!


Can we represent Simple Harmonic function as triangular waves?

Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : ![enter image description here](http://i.stack.imgur.com/iGqpV.gif) and the equations governing the motion would be ![enter image description here](http://penguinphysic.files.wordpress.com/2009/11/image0151.png) So I've been wondering why can't simple harmonic motion be represented in form of triangular waves.Although the equations above involve angular momentum so I may be contradicting myself but fundamentally the velocity time is sine function : $$-\sin(x)$$ and the gradient represents the acceleration is non-uniformly increasing and decreasing. What if instead of that you use $$-(\arcsin(\sin(x))$$ Which would represent a triangular wave whose gradient would depict that the acceleration is uniformly accelerating and decelerating.So would this represent harmonic motion or is it fundamentally incorrect.


Work to Create a Vacuum

I'm trying to determine the amount of work required (in Joules) to create a vacuum of a certain pressure in a tube. A pneumatic cylinder will be used to remove the air from the tube to create the vacuum. I understand the standard formula for the work to create a vacuum is: $\mathrm{pressure \; difference} \times \mathrm{volume}$. However, I've also read that when considering the use of a cylinder and piston there is actually some work that can be reclaimed since the piston and cylinder provides additional volume for the exhausted gas to expand into. Therefore, the correct formula to calculate work under this scenario is: $$W = V \left( P_{\mathrm{ambient}} - P_{\mathrm{new}} \right) + V P_{\mathrm{new}} \ln(P_{\mathrm{new}}/P_{\mathrm{ambient}})$$ Could someone validate which is the correct work formula to use for this scenario?


Is momentum conservation for the classical Schrödinger equation due to non-relativistic or due to some more exotic invariance?

I had no problem appliying the Neothers theorem for translations to the non- relativistic Schrödinger equation $\mathrm i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \;=\; \left(- \frac{\hbar^2}{2m}\Delta + V(\mathbf{r},t)\right)\psi(\mathbf{r},t)$ $\Longrightarrow\ \mathcal{L}\left(\psi, \mathbf{\nabla}\psi, \dot{\psi}\right) = \mathrm i\hbar\, \frac{1}{2} (\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{*} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{*}\psi$ $\Longrightarrow\ \pi=\frac{\partial \mathcal{L}}{\partial \dot{\psi}} \propto \psi^{*}$ $T[\psi]\propto \mathbf{\nabla} \psi$ $\Longrightarrow\ I_{\ \psi,{\ T_\text{(translation)}}}=\int\text d^3x\ \pi\ T[\psi]\propto \int\text d^3x\ \psi^{*} \mathbf{\nabla} \psi = \langle P \rangle_\psi$ But I actually wonder why that works out, given that the Schrödigner equation is not invariant under Galileian transformations. It might well be that the Schrödinger group, which I'm not familiar with, is close enough to the Galileian group, that the fourth line $T[\psi]\propto \mathbf{\nabla} \psi$ is just the same and that's the reason. I'd like to know if the evaluation of the infinitesimal transformation is the only point at which one has to know the transformations one is actually dealing with. Is my guess right? * * * Also, regrding the "trick" to establish Galilei-invariance after the conventional transformation via multiplication of the Schrödinger field by a phase (a phase which, among other things, is mass dependend): Some authors change $\psi(r,t)$ to $\psi(r',t')=\psi(r-vt,t)$, like here in the paper referenced on wikipedia (there is also a two year old version of it online (google)), but other authors, like the writers of the page in the first link, also transform $p$ to $p+mv$ in $\phi$ (which doesn't change the fact that they still have to add a phase). This is all before the phase multiplication. So what is the "right way" here? If I do this transformations involving a multiplication of the phase, do I only transform the actual arguments of the scalar field $\psi(r,t)$ or do I also transform the objects like $p$, which classically transform too, but are really just parameters (and the Eigenvalues) or the field - and not arguments?


relativity - equivalent frames do not have equivalent clocks and rods

In problems about relativity we have a stationary frame S and a moving frame M and two observers initially at the same reference point. Then the two frames pass each other and we measure the distance and time with a second reference point. The two reference points are at rest in the S frame. The conclusion is that the observer in S reference point measured a longer time and distance than the observer. But as S and M are equivalent (Galilean relativity) these two observers must when they compare each other measurements have the same results. How can you explain this difference in measurements and the equivalence of both frames at the same time.


How do you calculate the change in temperature of an adiabatic system?

I really don't even know how to start doing this problem. I understand PV=nRT and I understand that if either T, V, or P is kept constant, then the other 2 change at the same rate, and I understand that if either T, V, or P is kept constant while the other two change, then the system experiences a change in heat. I also understand that the internal energy is kept constant and that internal energy = heat - work. However I do not know how to relate the change in temperature to a change in heat. Can somebody please walk me through this? Here is a diagram to better visualize the problem. ![enter image description here](http://i.stack.imgur.com/Q7Qu2.jpg) My homework question asks to find the temperature at point c I have already calculated that n=0.41 and that the pressure at point b is 2.01966 atm You do not need to solve for c, just please tell me how. Okay so I couldn't handle it anymore and I just looked at the answer which happened to be 810J. This make me even more confused. Where did the extra 110J of energy just appear from?


Why were the fathers of quantum mechanics so sure radioactive decay was indeterministic?

The classic example of an indeterministic system is a radioactive isotope, e.g. the one that kills Schrödinger's cat. I get there are arguments against hidden variables in quantum mechanics, but how could they be so sure, back in the twenties, that the strong nuclear forces involved in radioactivity were not governed by hidden variables rather than true randomness? Einstein was very unhappy about the indeterminism of quantum mechanics regarding even well understood effects like Young's slit experiments, but it seems kind of ideological and brash on behalf of Heisenberg & Co to extend the indeterminism over to phenomena they hadn't even begun to understand, like alpha decay. Is there a reason for this early self-assuredness in postulating indeterminsm?


Could the contents of a bottle affect the surface temperature of that bottle?

e.g. If I had two bottles one with Vodka in and one with Water in (both the same glass bottle spec/size) If I then put these two bottles in a freezer would the surface temperature of the bottle be the same or different? Thanks


Sources for new experimental limits on susy models?

I know the LHC people are publishing new limits every now and then, but as a non-expert in reading experimental papers (yet), I was wondering if there's a friendly website that collects and presents the limits in a readable and accessible way for non-experts?


Magnetic field of solenoid in cone shape

I have done some drawings of the magnetic fields of cone shaped solenoid electromagnets. I based this on the work of Tesla and some explanations of Bashar, known as the Bashar Cones. I found very strange patterns. ![enter image description here](http://i.stack.imgur.com/jMAun.png) ![enter image description here](http://i.stack.imgur.com/dRcF7.png) ![enter image description here](http://i.stack.imgur.com/XIO5p.png) ![enter image description here](http://i.stack.imgur.com/l62x5.png) ![enter image description here](http://i.stack.imgur.com/DPQLO.png) ![enter image description here](http://i.stack.imgur.com/yxexk.png) ![enter image description here](http://i.stack.imgur.com/UA9o6.png) Can someone study and analyze them. I think some of those fields can be used to generate rotation as others do have strong fields to generate electricity. http://s1316.photobucket.com/user/amtriorix/library/Bashar%20Cones I do see circular magnetic fields, as I do see on some strong fields to do more experiments on it. http://www.youtube.com/watch?v=7Ii_Bb6YYQU Walter Russell Cones ?


Coulomb gauge and two degrees of freedom of EM field

The EM field has two possible polarizations, which is caused by spin-one nature of field (leads to the Lorenz gauge) and massless of the field. Really, the Klein-Gordon equations for the EM field $$ \square A^{\mu} = 0 $$ does not leave a $A^{\mu}$ uniquely determined: $$ A^{\mu} \to A^{\mu}{'} = A^{\mu} + \partial^{\mu}\varphi , \quad \square \varphi = 0 \Rightarrow \varphi^{0}{'} = 0, \quad \partial_{\mu}A^{\mu} = (\nabla \mathbf A ) = 0, $$ which is called Coulomb gauge. So EM field has two components. But after using Lorentz- transformations the new $A^{0}$ component isn't equal to zero. So I'm not imagine, what does it mean. It looks like that EM field begin to have three independently components in the new frame. Of course, this is not true, but I want to know, can the Coulomb gauge be an argument of existence only two independently components. Can you help me?


Some questions about the logics of the principles of independence of motion and composition of motion

In high-school level textbooks* one encounters often the principles of independence of motion and that of composition (or superpositions) of motions. In this context this is used as "independence of _velocities_ " and superposition of _velocities_ (not of forces). This is often illustrated by the example of the motion of a projectile, where the vertical and horizontal motions are said to be independent and that the velocities add like vectors. Now, if $x\colon \mathbb{R} \to \mathbb{R^3}$ describes the motion of the of the considered object, it is clear that one can decompose the velocity $v = \dot x$ arbitrarily by $v = v_1 + v_2$ where $v_1$ is arbitrary and $v_2 := v - v_1$. This leads me to my first question: Am I correct that this is purely trivial math and contains no physics at all? If so, it would not deserve to be called "principle of composition of motions" or something like that and said to be fundamental. However it seems that one could interpret the decomposition above such that $v_1$ is the velocity of the object with reference to a frame of reference moving with $v_2$. If so, how can one see, that this goes wrong in the relativistic case? Now suppose you have two forces $F_1$ and $F_2$ which you can switch on and off, suppose that $F_i$ alone would result in a motion $x_i$ ($i=1,2$). Newtonian Mechanics tells us the principle of superposition of forces, i.e. if you turn both forces $F_1$ and $F_2$ on, the resulting motion $x$ is the solution of the differential equation $\ddot x = \frac1m F(x, \dot x, t)$ (where m is the Mass of our object) with $F = F_1 + F_2$. One might interpret the principle of composition of motions such that always $\dot x = \dot x_1 + \dot x_2$ holds true. This is clearly the case if $F_i$ depends linearily on $(x,\dot x)$. However I think that it doesn't need to be true for nonlinear forces. This leads me the my third question: Is there any simple mechanical experiment where such nonlinear forces occur, which shows that in this case the "principle of composition of motions" doesn't hold? *I have found this in some (older) german textbooks, for example: Kuhn Physik IIA Mechanik, p. 107, Grimsehl Physik II p.16,17 compare also   http://sirius.ucsc.edu/demoweb/cgi-bin/?mechan-no_rot-2nd_law and Arons


Which magnetic force is stronger? Electron or proton?

Assume the particles generates the magnetic field with their own spin. No motions except spin.(they have their intrinsic angular momentum.) Then which one generates stronger magnetic force? I've seen the formula which represents the magnetic moment of the 1/2 spin particles with charge $q$, mass $m$, $\vec{\mu}=\frac{g_sq}{2m}\vec{S}$ where $g_s$ called g-factor. But I can't understand the formula. Let me ask a question, which magnetic force is stronger between electron and proton?


Finding coefficient of proportionality

Recently in my AP Physics class I did a lab in which I measured k for a spring by setting up an oscillating system with it, and timing the period, repeating for different masses. Since $T=2\pi\sqrt{\frac mk}$, we were supposed to relate $T^2=\frac{4\pi^2}km=Cm$ for some $C$. This is a proportionality equation in $T^2$ and $m$, which we were to plot on a graph and find the slope $C=\frac{4\pi^2}k$, from which we can find $k=\frac C{4\pi^2}$. Now my lab partner and I found the $k$ in two different ways; while I simply averaged $k_i={C_i\over4\pi^2}$ for each trial, he used statistics software to find the equation $\hat{T}^2\approx C_am+\beta$, where $\beta\approx0$ and $\bar{k}={C_a\over4\pi^2}$. He claimed that the $\beta$ term helped to eliminate error, leading to a better $k$. We recalculated $k$ by direct use of Hooke's Law and found that the percent difference was lower in my partner's case, leading me to believe that he may be correct in his method. So, my question is, physics.stackexchange: How does using the linreg analysis on the data produce better results? Or, if this is a fluke of data and my results should have been better, why shouldn't it? Thank you in advance.


First class and second class constraints

Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second class constraints. Suppose, from the Lagrangian, I found two primary constraints $\Phi_a$ and $\Phi_b$. Let $\dot\Phi_a$ and $\dot\Phi_b$ both leads to the secondary constraints $\Sigma_a$ and $\Sigma_b$ respectively. From the consistency condition $\dot\Sigma_a$ leads to a tertiary constraints $\Theta_a$ but $\dot\Sigma_b$ become zero. Now How can I check which one of them are first-class and which one of them are second-class?


What's the role of field equation in QFT?

For free field theory, it seems the solutions of a field equation are used to give a representation of Poincare group, and the field equation is still satisfied after quantization. However for an interacting theory, such as QED, the field equation: $$\partial_\nu F^{\nu \mu} ~=~ e \bar{\psi} \gamma^\mu \psi.$$ I don't see any possibility of satisfying this equation using operators since LHS will only act nontrivially on bosonic sector of the Hilbert space and RHS only on fermionic sector, and I don't think perturbation technique can fix this since it's qualitatively not satisfied. Yet nobody seems to be bothered by this, so I'm wondering what's the role of field equation in QFT, especially for interacting ones.


How does one prove:$\nabla(\vec{\mu_m}\cdot\vec{B})\cdot\vec{dr}=0$?

Work done by a magnetic force(even over an infinitesimally short displacement)=0 _Net Force_ on a current loop in an external magnetic field is given by: $$\vec{F}=\nabla(\vec{\mu_m } \cdot \vec{B})$$ How does one prove: $$dW=\nabla(\vec{\mu_m} \cdot \vec{B})\cdot\vec{dr}=0$$ $\vec{\mu_m}$: Magnetic Moment of the current loop.


How do electrons jump orbitals?

My question isn't how they receive the energy to jump, but why. When someone views an element's emission spectrum, we see a line spectrum which proves that they don't exist outside of their orbitals (else we would see a continuous spectrum). Electrons can be released in the form of beta decay, thus proving that they are capable of traveling outside of orbitals contrary to the statement my teacher said that they stay within orbitals. Then, to add to the confusion, the older model of rings floating around a nucleus has, from what I can tell, been outdated, which would support this model. My teacher's explanation was that the electrons made a quantum jump of some kind. **_How_** do electrons move between orbitals or do we know how they jump, excluding the reason that energy causes them to jump, and why are positrons formed sometimes instead of electrons in Beta decay? When I'm asking "how do electrons jump" I would like to know how an electron can jump between each orbital such as how it moves and how it knows where to jump since it appears to be a jump where the electron doesn't slow into a orbital position. Specifically how they jump what is this Atomic electron transition, I understand that they jump and that they do this through absorbing and releasing energy but what is this Atomic electron transition other than what is already on the wikipedia article http://en.wikipedia.org/wiki/Atomic_electron_transition.


Finding directions of the internal forces within a freely jointed rigid object?

Consider the the following problem: ![enter image description here](http://i.stack.imgur.com/utK3P.png) Here is a force diagram showing the situation: ![enter image description here](http://i.stack.imgur.com/eHwGx.png) For part (i) Taking moments about $B$ for $BC$ gives $84.5Lcos\beta=2LT$ So $T=39$N For part (ii) Resolving vertically upwards on $BC$ gives $39\cos\beta -84.5=-r$ which results in $r=48.5$N in the direction shown on diagram (downwards). I'm aware that the question asks for the forces acting _on_ $BC$ so I deduced that by newtons third law $Y=48.5$N (upwards). Everything so far is correct but here's the problem: To find the horizontal component I resolved leftward on $BC$ to obtain $p=39\sin\beta$ giving $p=15$N direction as shown in diagram (leftward). Again realizing that the question asks for forces _on_ $BC$, by newtons third law $X=15$N ( **rightward** ): Here's the official answer: ![enter image description here](http://i.stack.imgur.com/fvK3u.png) The marks-scheme is always correct. So could someone please kindly explain what I did wrong?


Can we explicitly solve the Hamilton–Jacobi equation for a particle in a uniform magnetic field?

HJE for nonrelativistic charged particle in an electromagnetic field is $$\frac{1}{2m}\left(\nabla S - q\mathbf{A}\right)^2 + q\phi + \frac{\partial S}{\partial t} = 0.$$ For a uniform magnetic field $\mathbf{B} = B_0 \hat{\mathbf{z}}$ and a particular choice of gauge this becomes something like: $$\left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y} - qB_0 x\right)^2 + \left(\frac{\partial S}{\partial z}\right)^2 = -2m \frac{\partial S}{\partial t}.$$ Can we solve this explicitly? It seems to me we can start by separating the variables $t, z$ by letting $S = f(x, y) + p_z z - Et$, giving $$\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y} - qB_0 x\right)^2 = 2mE - p_z^2,$$ but I'm really bad at solving differential equations, so I don't know how to proceed. I also have no intuition for what the solution $S$ is supposed to look like, even though I already know how a particle moves in a uniform magnetic field, so I don't have any idea how to guess a form for $S$.


How does $\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$ relate to the orbital angular momentum operator?

When I attempted to calculate the following by hand $$\bar{r}\times(\bar{\nabla}\times\bar{F}) - \bar{\nabla}\times(\bar{r}\times\bar{F}),$$ I noticed some of the terms I extracted looked similar to the terms that appear in the oribtal anglar momentum operator $$\bar{L} = -i\hbar(\bar{r}\times\bar{\nabla}).$$ Is there a condensed expression that uses $\bar{L}$? How does $$\bar{r}\times(\bar{\nabla}\times) - \bar{\nabla}\times(\bar{r}\times)$$ relate to the orbital angular momentum operator?


Physical Consequence of Global Charge Conservation versus Local Charge Conservation?

Gauge invariance of the electromagnetic field implies local charge conservation. If the gauge is no longer invariant, then the electromagnetic field must no longer conserve charge locally. Is it possible either physically or mathematically, to break local charge conservation, but still enforce global charge conservation? Charge can be created and destroyed at will, but the total amount of charge created at any given time is zero?


If charged particles always attach to black hole event horizons, how can ordinary matter fall in?

(A friend at work kindly loaned me loaned me his copy of Kip S Thorne's "Black Holes & Time Warps". This may have been ill-advised... :) BH&TW 1994 paperback p.410 Figure 11.5: > ... all positive charge that enters the horizon from the outside Universe > becomes attached to the horizon, and moves around on it, until it exits back > into the outside Universe (in the form of of negative charge falling inward > to neutralize the positive charge. Whoa there buckaroos, QED this ain't. Protons are made of quarks, and quarks are _individually_ charged at fractions of 1e. So, if I'm reading the above very straightforward statement the way it seem to be saying, _every charge_ should be individually trapped by the horizon. So how does that nasty lumpy $-1e$ of a charge cancel out each of the fractional charges of the proton quarks, some of which (the _d_ quarks) are negative BTW? So while I'm mostly OK with protons and antiprotons doing this little dance and falling in, because in that case all you are really doing is shooting gammas into the interior of the horizon. Even for that case, wow, that one statement above would be skipping over a plethora of particle complexity for any real event. E.g. you really need to look at the _quarks_ cancelling to get a complete picture, not just two particles with actual spatial size. The Thorne book is nine years and intended to be a popular read, so am I safe to assume that someone has subsequently looked in gruesome detail at the troublesome issue how exactly you cancel the mismatched charges of ordinary matter in a way that allows out-of-balance all-matter (or all-antimatter) lumps and atoms to fall into the interior of an event horizon? I note that the conservation principles in play for this question are the same ones that keep hydrogen atoms from turning into gammas. So while the interior of a black whole is a thing of beauty and pure energy and simplicity, messy matter first has to _get_ there by getting, more or less, converted 100% into gamma radiation. Once you have converted matter into all gammas and shot those into the event horizon interior, you can spit back whatever matter-antimatter or radiation combo you want. But if horizon charge attachment is real and fully self-consistent, I fully confess that I flatly do not see how you can ever get _into_ that interior. Also, I assume the membrane model has not been dramatically modified or abandoned since that time? After all, the idea that horizons are conductive seems so relevant to how quasars are powered that I find it a bit hard to believe that the conductive- horizon idea could ever be abandoned, since in effect quasars qualify as decent experimental evidence for the idea. Also, for whatever it's worth, I found the arguments about misbehaving field lines leading to charge capture to be nicely self-consistent and persuasive. So, more crisply: > If charge cancellation is required for particles to fall past an event > horizon, how can ordinary matter with its odd mix of spatially separated > fractional charges ever reach the 100% self-cancellation of charge needed > for it to pass through the event horizon? If appropriate, a reference to some widely accepted paper that adequately explains the nature of the whole-fractional charge cancellation mechanism would be fine.


Muons Internal Clocks and Time Dilation?

I don't get how time dilation affects the half-life of muons? Time is just a tool that is used by humans how does that affect the internal clocks of elementary particles?


Massive ropes and tension

Consider the diagram bellow, a massive string hanging in equilibrium ![enter image description here](http://i.stack.imgur.com/Vdwh2.png) My question is this: Can we take any 2 random point on the rope, say A and B and then use $\sum F=0$ on the rope between them: i.e. say that $$T_0 \sin(\theta)+ T_0 \sin(\theta)=mg.$$ Where $\theta$ is the angle the two forces $T_0$ make with the horizontal and m is the total mass of the rope between these two points. If this is the case would is work with the points in green so that in resolving horizontally would get us say $$T_1 \cos(\theta_1) = T_2 \cos(\theta_2)$$ where $\theta_1$ and $\theta_2$ are the angles they make with the horizontal. If you need any more information please just ask, thanks.


Coordinate System vs. Angular Properties vs. Centroid

Please help me check my understanding related to the rotational motion of a 3D rigid body after reading some Physics textbooks and googling for some more materials (e.g., Wikipedia's Torque, Wikipedia's Moment of Inertia). The following points are to be checked against a situation where the rotational motion of a 3D rigid body happens about a single rotation axis in an inertial Cartesian coordinate system where the z-axis is taken to be the rotation axis but the origin $O$ of the coordinate system is **not** necessarily on the centroid of the 3D rigid body and can even be **outside** the 3D rigid body:   1. Each particle in the rigid body has $\vec{v} = \vec{\omega} \times \vec{r}$ where $\vec{r}$ is the position vector of the particle with respect to the origin $O$, and therefore, the speed of each particle depends on the choice of origin $O$.   2. With respect to $O$, the moment of force $\vec{\tau}$ and the angular momentum change $d\vec{L}$ about the z-axis have the same directions.   3. With respect to $O$, the angular acceleration $\vec{\alpha}$ and the angular velocity change $d\vec{\omega}$ about the z-axis have the same directions.   4. Moment of inertia $I$ is always calculated with respect to z-axis, not the centroid of the rigid body, such that the $s$ in $\int s^2 dm$ is always the distance between a point mass in the rigid body and z-axis.   5. Angular momentum of each particle in the rigid body $\vec{L} = \vec{r}\times\vec{p}$ where $\vec{r}$ is the position vector of $\vec{p}$ with respect to the origin $O$ (i.e., $||\vec{r}||$ is **not** the distance between $\vec{p}$ and z-axis), and therefore, the direction of $\vec{L}$ may be different from $\vec{\omega}$.   6. Total angular momentum of the rigid body is then $\sum \vec{L} = I \vec{\omega}$, and therefore, the direction of $\sum \vec{L}$ is the same as $\vec{\omega}$.   7. The torque experienced by each particle in the rigid body $\vec{\tau} = \vec{r}\times\vec{F}$ where $\vec{r}$ is the position vector of $\vec{F}$ with respect to the origin $O$ (i.e., $||\vec{r}||$ is **not** the distance between $\vec{F}$ and z-axis), and therefore, the direction of $\vec{\tau}$ may be different from $\vec{\omega}$.   8. The total torque experienced by the rigid body is then $\sum \vec{\tau} = I \vec{\alpha}$, and therefore, the direction of $\sum \vec{\tau}$ is the same as $\vec{\alpha}$.   9. The torque experienced by each particle in the rigid body is related to each particle's angular momentum rate of change $\vec{r}\times\vec{F} = \vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d(\vec{r} \times \vec{p})}{dt} = m\frac{d(\vec{r} \times \vec{v})}{dt} = m\frac{d(\vec{r} \times (\vec{\omega}\times\vec{r}))}{dt} = m\frac{d((\vec{r}\cdot\vec{r}) \vec{\omega} - (\vec{r}\cdot\vec{\omega})\vec{r})}{dt}$, and therefore, for each particle the directions of $\vec{\tau}$ and $d\vec{L}$ are the same but may be different from the direction of $\vec{\omega}$.   10. The total torque experienced by the rigid body is related to the total angular momentum's rate of change $I \vec{\alpha} = \sum \vec{\tau} = \frac{d\left(\sum\vec{L}\right)}{dt} = \frac{d\left(\sum I\;\vec{\omega}\right)}{dt} = \sum I \frac{d\vec{\omega}}{dt}$, and therefore, the directions of $\vec{\alpha}$, $\sum \vec{\tau}$, $\sum d\vec{L}$, and $d\vec{\omega}$ are the same. Any mistake? Because the rotation axis can be outside the 3D rigid body, I feel uneasy especially for Point 5 to 10.


Confused about elasticity and collisions

I was solving the following problem and the explanation to it confused me. > _There are two objects with mass $m$ and $M$, respectively. The object with > mass $m$ has a velocity of $\sqrt{2gl}$ and collides with the other object > that is initially at rest. If the collision is elastic, what is the velocity > of the object with mass $M$ right after the collision? Friction is > negligible._ My understanding is, for these kind of collision problems momentum is conserved so, $$mv_{m} +Mv_{M} =mv'_{m}+Mv'_{M}$$ or in other words, $$mv_m=mv'_{m}+Mv'_{M}$$ Since the initial velocity is 0 for the object with mass $M$. In addition, elastic collision means that kinetic energy is conserved and intuitively, I am imagining that the smaller object (say, $m$) will "bounce" to the opposite direction as its initial velocity while the bigger object starts moving in the direction that is as same as the initial velocity of $m$, just a little slower. However, the answer explanation tells me that right after the collision the momentum is preserved, so $$mv_m=(m+M)v'_{m+M}$$. I thought that this equation represents the situation where the collision is completely INELASTIC and the objects stick to each other. Does it have to do with the wording "right after the collision" ? Or, is the explanation not making sense ?


Uncertainty Principle for Information?

I'm not familiar (yet) on how Information theory can be emerged/used in QM/QFT but I was thinking about this question: While we have Heisenberg uncertainty principle on measuring coupled observables, can we express it using somehow a more fundamental/abstract concept like Information uncertainty (especially that we have Information conservation principle like for energy and momentum, and associated symmetry, i.e CPT)? in sense that because there is always some details that we can't measure/know, can we express that as some information uncertainty? And despite of the answer please explain why there is or there isn't connection/relation between the two concepts.


What are the biggest unanswered questions in physics today?

Are there fundamental big questions in physics that most scientists encounter no matter what they're working on, or are most questions details of smaller subtopics? What are the great unknowns of our day? Sorry in advance if my question is too general. I told my kid that there are some things science still doesn't have theories for, and she asked "like what" and I said I wasn't sure, so I'm asking here. So much science has been done since I was in school that I don't know what all has changed. I think what she's after is some real mysteries to wonder about, ones that don't have generally accepted theories yet.


Ultrafast photon detectors

Are there already photon detectors that can resolve the shape of photons in the time domaine if it looks like below where the two humps are separated by a time of the order of femtoseconds? What I know is that there are photon detectors that can resolve in the picosecond regime (see here). But this article is 6 years old... What is the best resolution that is possible today? ![enter image description here](http://i.stack.imgur.com/wYDlb.png)


Is it possible to tell whether the space ship is moving or not?

Consider a space ship which is not under any force. Being inside the space ship, I will make a robotic fly from the platform to crawl inside using my remote. For simplicity assume that space ship will be drifting at uniform speed only under forward direction with respect to me, or else it will be under rest. The following experiment is assumed to be done in vaccum. From the top, the first image is of space ship. Assume that I am at point A and the robotic fly to be beside me as shown. In the second image, I will make the insect just to lift up and I will stop its engine, as the robot is not under any force it will not fall down, it will stay where it was. As the fly will continue to be under same motion as the space ship, I will not be able to tell whether my ship is moving or not. In the third image and fourth image, I will apply force to the robotic fly to move little back and I will stop its engine.   * (Third image) If the ship was under forward motion, the fly, because of the applied force backward i.e against its acquired motion from the spaceship, the forward motion of fly will not be synchronzed with the forward motion of the ship, so it will crash the wall with in an interval of time. This ensures that space ship was under uniform velocity.    * (Fourth image) If the ship was at absolute rest, the fly, even though there was an force applied in the backward direction, the fly will not crash with the backside of the space ship. This ensures that space ship was at absolute rest.  ![enter image description here](http://i.imgur.com/S97AJfl.jpg) But, according to Feynman: > The principle of relativity was first stated by newton, in one of his > corollaries to the laws of motion: "the motions of bodies included in a > given space are the same among themselves, whether that space is at rest or > moves uniformly forward in a straight line." This means, for example, _that > if a space ship is drifting along at a uniform speed, all experiments > performed in the space ship and all the phenomena in the space ship will > appear the same as if the ship were not moving, provided, of course, that > one does not look outside_...........in a moving space ship the electrical > and optical phenomena should be different from those in a stationary ship. Hoping that my above experiment is neither optical nor electrical, my experiment is in contradiction with the Feyman's explanation. From my experiment, I am able to say whether the space ship is moving or not. How is it possible?