Bra-ket notation, Bits, & Superposition

I am a quantum computing enthusiast, and recently I stumbled upon this the following two propositions: $$ \alpha|1\rangle + \beta|0\rangle$$ What does this mean? My understanding of this is that: the two bits, 1 and 0 are represented in a state of superposition, hence the bra-ket notation (which is commonly used for quantum mechanics), i.e., this is a qubit. Or is there a more concise explanation of this? Also: $$(\alpha|1\rangle + \beta|0\rangle)^N$$ What does it mean to raise this quantity (of superimposed bits, i.e., qubit) to the $N$th degree? If we take $2^N$ where $N$ is the number of qubits then this tells us the number of bits in the desired number of qubits. Is what I have stated in this post, generally correct?


Which way do spiral galaxies rotate?

Is it known whether spiral galaxies typically (or exclusively?) rotate with the arms trailing or facing? Intuitively it feels weird to think of the arms as facing the direction of rotation, but that's silly, of course -- it seems to have its origin in an instinctive assumption that some sort of friction with the intergalactic vacuum would make it hard for the galaxy to rotate against the barbs, which is nonsense. I assume that for galaxies that we don't view head-on, we can measure rotation curves spectroscopically. But that's not enough; it must also be possible to determine which end of the minor axis is the far one. Dust in front of the bulge can help with that -- but in the images I have seen where it is clear which way the galaxy is tilted, it's not easy to see which way the spiral curls. Still, I suppose there must be some galaxies where both the spiral and the rotation can be determined. What's the verdict for them?


Determining cloud cover from observer to near the horizon

Does there exist a Clear Sky Chart with the following enhancements?: 1 - Actual Cloud Cover (Offered Visually and not just Colors with a Legend, Over Time/Past & Predictive) 2 - Simulate/Predict Cloud Cover taking into account the direction from Observer to Observed Object and Angle of view \- Close to the horizon (May be helpful to know when you can reasonably start/end tracking something you want to catch that night that is close to the horizon) The reason I'm curious is: A) I wonder if it's just not feasible for any/many reasons. B) It would be great help to know this information. In general, does anyone know of other Earth Weather, Clear Sky Clocks and Charts or anything else that gives more information?...anything related will be helpful. EDIT: I would love to find this lecture "You can do better than Clear Sky Chart" mentioned: http://stjornuskodun.blog.is/blog/stjornuskodun/entry/966941/


Why is radiation for an ultrarelativistic charge zero on axis?

I attribute it to the fact that for an ultrarelativistic charge the field is contracted and essentially there are only fields in the transverse direction and nothing longitudinally (wrt the charges velocity). Any thoughts?


What is meant when it is said that the universe is homogeneous and isotropic?

It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?


If electrons are magnets do they attract each other?

It is said every electron is negatively charged and so they repel each other but if electrons are tiny magnets(which are responsible for atomic attraction and how solid magnet works) does it mean opposite electron poles attract each other?


Geodesic curvature and Weyl transformations

The geodesic curvature is given by $$k=\pm t^a n_b\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I don't understand why under the Weyl transformation $$\gamma_{ab}\rightarrow e^{2\omega}\gamma_{ab},$$ $t^a$ and $n_b$ transform as $$t^a\rightarrow e^{-\omega}t^a,~~~~~n_b\rightarrow e^{\omega}n_a.$$ Is this really as trivial as a normalisation? Also, what do "time-like boundary"$(+)$ and "space-like boundary"$(-)$ mean? I appreciate any discussion related to this. The geodesic curvature was somehow never mentioned in my GR class.


Electric Field and Magnetic field of Guassian beam

The center of the gaussian beam in glass has a beam power of 1 mW and a beam radius of 20 μm (micro meter). The wavelength of the light $\lambda_0$ is 1550 nm. How can I find the electric field E and magnetic field H?


Time dilation as an observer in special relativity

I've been having a discussion regarding time dilation relating to special relativity and how it should be observed from the FoR (Frame of Reference) of "the person moving" : **I assert** \- If we have a time is slower in the moving FoR than when I observe the stationary frame of reference (earth) then I should perceive events as occurring faster, i.e., if I observe 15 years passing and 90 years pass in the stationary frame then I should observe events on earth happening at a rate of 6:1. So if I observe a 1 min event on earth I watch that entire event occur in 10 seconds. My thought process is similar to the idea that if I was near a massive object and looked out I would observe events further away as moving faster. My teacher seems to believe this line of reasoning is incorrect but will not (or cannot) give me the correct thought process. Hence my question is, "is my logic correct, and if not why?"


What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment?

The Harvard-Smithsonian Centre for Astrophysics held a press conference today to announce a major discovery relating to gravitational waves. What was their announcement, and what are the implications? Would this discovery be confirmation of gravitational waves as predicted by general relativity (even though Sean Carroll links to the Nobel website implying that G-waves was detected decades ago, while my book on GR (B. Schutz) says they're still looking...I'm confused) Also, regarding inflation theory, would this discovery confirm inflation or refute it? Or something else, a la string theory?


Can a neutron be created from pure energy

Is it possible to create a neutron out of pure energy, i.e. not by bringing a bunch of already-existing quarks together? (A quick calculation using E = mc2 shows the energy required would be about 1.5 e-10 J.) Would an equivalent antiparticle always need to be created?


Does magnetic geometry determine the scaling of a Polywell fusor? Forgive imprecise terminology here - by "magnetic geometry" I mean the configuration of the magnets, the configuration that creates the "Christmas star" geometry of the electron field in the fusor. On to my question... I've read that the rate of fusion scales at $r^5$, with $r$ being the radius of the reaction chamber. Is this because of the aforementioned magnetic geometry? Or is it because of something more essential, like Bremsstrahlung, or something else?


What are the consequences in high-energy of the non-interaction of the Higgs Field?

At high-energies when the Higgs field won't affect (interact with) particles, when the symmetry breaking won't occur, what would be $\rm W\pm$ or $\rm Z^{0}$ bosons speed if they would then have a $0$ rest mass? What about Fermions?


How can I explain why the weak nuclear interaction between individual nucleons is 'weak'?

> By considering the energy-time uncertainty principle, estimate the range of > the weak nuclear interaction at low energies. Compare this range to the size > of a typical nucleon (for example, a proton) and therefore explain why the > weak nuclear interaction between individual nucleons is 'weak'. I'm a bit stuck with this question: to estimate the range of weak nuclear interactions do I need to put the energy levels of a quark into the energy- time principle? Beyond doing this I am a bit lost! Any comments would be much appreciated; this is an assignment so the work has to be original.


What is the curvature scalar $\Psi_{4}$? Is it related to the scalar curvature $R$? What does its real and imaginary parts represent?


Magnetic field when moving into the influence of a ferrous object

Suppose I have a magnetometer. I am on the Earth's surface, away from the influence of other magnetic sources. Assuming the magnetometer is aligned with the reference frame, it should read only the Earth's magnetic field vector at that point. Suppose now I walk towards a large ferrous object, for example, a large parked truck. As I approach the truck, what will happen to the magnetic field that I measure? Is the distortion to the magnetic field able to be modelled as additive as a function of position? If not, how could one mathematically describe the distortion? Are there any canonical references on the subject? * * * EDIT: In response to the answer(s) below, here are some additional details:   * Assume that I know what the magnetic field in the local geodetic frame _should_ be to within some desired tolerance, e.g. by using EMM2010 or similar.   * Ultimately, I do want to calculate attitude, but I have other sensors at my disposal (IMU and GPS) where heading is observable under motion. My desire to to "learn" the magnetic deviation on-the-fly to help stabilise the heading during near-stationary periods. My intuition says that the deviation should be observable when yaw is observable. But I need to start with some sort of model as to what the deviation looks like :)


What is the gist of the equations on Alejandro Guijarro's "2/14" "Momentum" blackboard?

The artist Alejandro Guijarro has an exhibit of photographs of a variety of blackboards with physics equations, drawings and text on them, as covered at Quantum Chaos on Display in Top Physicists’ Chalk Scrawls | Raw File | WIRED What is the one in the first closeup photograph (numbered "2/14") about? The circled text "non SUSY" suggests that at least one of the topics is Supersymmetry (Wikipedia), but what is the gist of it? For clarity, this question is about the blackboard in the first closeup image on the Momentum exhibit web site, labeled "2/14". ![enter image description here](http://i.stack.imgur.com/YhrRI.jpg) Caption: $\uparrow$ Click image to enlarge it. For example, the equation in the upper left starts with > $T_{ds} \sim 1/R$ For extra credit - where is the blackboard / who wrote on it / etc? (Note that I am also interested in the other blackboards, and suggest separate questions here for each of those. There is some commentary on some of the others at The Beautiful Blackboards at Quantum Physics Labs | MetaFilter)


Electric field for two adjacent infinite layers of width h charged + and − uniformly

I'm trying to figure how to calculate the electric field created by two adjacent infinite layers of width h charged + and − uniformly. I don't understand why isn't the field outside of the layers isn't simply zero. Why aren't the 2 cancelling each other? And how can I calculate the field anywhere between the two using gauss law? Thanks


How do I derive the Lorentz contraction from the invariant interval?

Reviewing some basic special relativity, and I stumbled upon this problem: From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using $x=vt$: $$c^2d\tau^2=c^2dt^2-v^2dt^2=c^2dt^2\left(1-\frac{v^2}{c^2}\right)\rightarrow d\tau = dt\sqrt{1-\frac{v^2}{c^2}}=t/\gamma$$ Now, I would very much like to be able to derive the length contraction formula in a similar fashion, and feel strongly that this should be possible. The definition of the invariant interval is: $$ds^2=dx^2-c^2dt^2$$ using $t=\frac{x}{v}$ I tried: $$ds^2=dx^2-\frac{c^2}{v^2}dx^2=dx^2\left(1-\frac{c^2}{v^2}\right)\rightarrow ds=dx\sqrt{1-\frac{c^2}{v^2}}$$ This is where I'm stuck: I don't see how this can be converted to a Lorentz factor... Any help that will allow me to arrive at the desired result $ds=\gamma dx$ would be very much appreciated.


Confusion With Neutrino Oscillation

Neutrinos produced in the weak interaction are eigenstates of the weak interaction hamiltonian or flavor eigenstates. Right? If this is so, then $\nu_e,\nu_\mu,\nu_\tau$ should be the stationary states of weak hamiltonian and weak hamiltonian is diagonal in this basis. Then, why do they oscillate during the propagation from one flavor to another? **_Is the hamiltonian different during propagation (i.e., no longer weak hamiltonian?) and $\nu_e,\nu_\mu,\nu_\tau$ are no longer eigenstates of the new hamiltonian_**?


Normal force of ball sliding on concave surface

Imagine a ball is sliding along a surface shaped like $y=x^2$. Like ![y=sin\(x\)](http://i.stack.imgur.com/7u0dr.png), but please ignore the fact that the center of the ball is on the surface instead of the edge. When the ball is stationary, I can calculate the normal force fairly easily. $F_\text{net}$ in the direction normal to the surface is 0. The normal force is normal to the surface, so it is the force that is exactly strong enough to cancel the normal component of the force of gravity. The normal force comes out to $<\frac{-a}{(1 + a^2)}, \frac{1}{(1 + a^2)}>$, where $a=f_{\text{surface}}'(x)$ (or in this specific case: $a=2x$) I initially expected this to be true even after the ball begins sliding down the surface, but I found that it was not. Intuitively, I understand that a velocity with a non-zero component in the normal direction affects the strength of the normal force. This seems like the same mechanism by which the normal force is able to stop a falling object as it hits the ground, for example. In the falling ball example, the relationship between velocity and normal force seems a bit sticky mathematically. Assuming neither the ground nor the object compress, the normal force would have to be infinite initially, right? It would only be infinite for an infinitesimal amount of time (so the integral would still be finite). However, in that case, the $v(t)$ curve is discontinuous, while it seems like $v(t)$ would be continuous for a ball sliding on $y=x^2$, so hopefully the relationship is less obnoxious in the continuous case. **Is there a mathematical formulation of this relation between normal force and velocity?** Can I add a velocity parameter to my equation for $F_\text{net}(x)$ (making it $F_\text{net}(x, v)$) to take this relationship into account? **Note** : I also see a conservation of energy solution that sidesteps the normal force issue altogether, but I'd like to understand the forces aspect.


Cellular automata rules for quantum mechanics

My limited understanding of quantum theory is that a **quantum system is completely described by its wave function** , which **deterministically evolves according to Schrödinger's equation** until wave function collapse. Schrödinger's equation is essentially just a differential equation, which means (as far as I can see) that the **state at every point in space changes depending only on its surroundings** (i.e local rules). Correct me if I'm wrong but it's at the point of wave function collapse that the non-locality comes in, which is what cellular autonoma (CA) models can't deal with very well. I'm not hypothesizing or suggesting that the universe **IS** a cellular autonoma (as Wolfram has suggested), but **what would a CA _model_ of Schrödinger's equation be like**? (Ignoring wave function collapse for the minute) Providing my understanding described above isn't wrong, I want to know, for the CA model:   1. What values describe the state at a given point? (i.e. what information would be in each CA cell?)   2. What rules are followed every discrete time tick to progress these values? I ask because I've gained a good intuition (an "understanding" that I'm happy with at least!) of heat flow, wave propagation and the rules of fluid dynamics through **interpreting their differential equations as local cellular automata rules** , and writing little toy simulation programs to see how they behave and look. I'm a computer science/mathematics undergraduate and I've always said that I don't understand something until I can write a program that simulates it, so being a little too ambitious for my own good I want to write a simulation for a QM system! My background is I've read a few Penrose/Feynman books, and I've got the complete Feynman lectures on physics (not read them yet...), and physics has always been a side hobby, but I'm not too hot yet on the mathematical language that is used for QM.


Is the spin-singlet state also a Resonating-Valence-Bond(RVB) state?

The _spin-singlet state_ of a lattice spin-1/2 system is defined as $S_x\Psi=S_y\Psi=S_z\Psi=0$, where $S_\alpha=\sum S_i^\alpha(\alpha=x,y,z)$ are the total spin operators, in other words, a _spin-singlet state_ is a spin state with _gobal $SU(2)$ spin-rotation symmetry_. On the other hand, a RVB state is the superposition of various configurations of singlet-product states (no need for equal weight superposition here), thus, a RVB state is of course a _spin-singlet state_ according to the above definition. For the reversed statement, is the spin-singlet state also a RVB state? How to prove it? For the extreme 2-spin system, it's direct to show the equivalence between the spin-singlet state and the RVB state.


What is the nucleon axial charge?

Can someone point me to a short definition of what the nucleon axial charge is?


Physical Quantities

What are the physical quantities which most relate to their definition? Could you give some examples? The definition I found is: " it is any property which can be measured". In the case of mass the given example is "when we go buy apples we purchase two kilograms of apples. Please give another example.


Quantifier problems of equations in physics

Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. I am wondering shouldn't it be written rigorously as something like: For all $E$ in a given range of values and all $m$ in a given range of values, we have $E =mc^2$? If not, then logically the equation $E = mc^2$ is sometimes valid and sometimes invalid as $E, m$ run through their given ranges, so that one cannot use "$E = mc^2$" to express what it is intended to mean. Then how does one incorporate this phys-mathematics writing style, as illustrated in the beginning, with the authentic mathematical writing style, as illustrated in the second paragraph, in the sense of logical rigor?


Are Both figures correct?

Actually I am in high school I just want to confirm that whether two figures of this question possible: Question is : A car is moving along a straight line OP.It moves from O to P in 18 seconds and returns from P to Q in 6 seconds , where OP = 360 metres and OQ = 240 metres.What are the average velocity and average speed of the car is going (a) From O to P and back to Q? My physics teacher made this figure from the question: ![by My teacher](http://i.stack.imgur.com/SCZjb.jpg) and I this figure is also right: ![by me](http://i.stack.imgur.com/URVXD.jpg) But When I solve the above Question by first figure I got Average speed of 20 m/s and 40 m/s from 2nd figure. Which one is correct? (When I ask my Question to my physics teacher , He said "First one is correct,Second one doesn't make any sense")


Where does the energy flow differ between a forward and reverse process?

![PV diagram of cycle](http://i.stack.imgur.com/Yn1d3.png) Mathematically ($W=\int PdV$) and by the First Law, I understand that $1 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 1$ and $1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1$ are not symmetric; the first requires work, whilst the latter gives it to the environment. What I am having trouble with is bellyfeeling _why_. A physical example: imagine a machine that works thusly: > **(1-2)** A piston loaded with 3 bricks in a hot reservoir isothermally > expands when 1 brick is removed. > > **(2-3)** Then, removed from contact from any reservoir, it adiabatically > expands when the 2nd brick is removed, cooling to the temperature of the > cold reservoir, where > > **(3-4)** the 2nd brick is replaced, isothermal compression takes place, > then > > **(4-1)** away from reservoir the 3rd is added, the temperature rising to > the intial hot temperature. **Given that the two possible cycles are asymmetric, _at what point (1,2,3,4) does the internal energy of the gas or the potential energy of the bricks differ for the two different cycles?_**


Angular momentum of a black hole

I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects it? I googled it but I couldn't understand much, all I found was about Orbital Velocity of a planet...


what forces are present when slicing an object.

I found here:What happens when we cut objects? that it is due to fracture wood splits open when using a wedge. Does it only apply when the wedge is already in the wood? How about the moment before a crack is made, do you need to overcome the compressive and shear stress of the wood to start a crack? I want to compute for the forces needed to cut the wood, along the grain, using a wedge. but Im confused whether shear and compressive strength or fracture initiation is involved to start the cut. and when the wedge in the the wood should I compute for fracture propagation or the cosine of the normal force on the wedge that is pushing the grains apart. I hope I'm making sense.


How do you calculate the position of the antinodes on a wave?

if for example i have a waveform with the formula $y=\sin(1.2x)+\sin(1.8x)$. The first 6 antinodes on my graph come up at around: $$ x=\begin{cases} 1.01\\\ 2.97\\\ 4.65\\\ 5.80\\\ 7.48\\\ 9.46\\\ \end{cases} $$ but I can't find a way to find the exact positions.


Gaussian surface question

There is an infinite slab of charge, and a (Gaussian surface) cylinder whose ends are both outside of the slab. $\phi_A$ is the flux through this cylinder, by symmetry the component of the flux through the top end is $\frac{ \phi_A}{2}$. Now, lift bottom end of the cylinder until it is _inside_ the slab. The net flux is now $ \phi_B$. Proposition 1: The component of the flux through the top end is unaffected by the position of the bottom end (even though the total flux _is_ ), as the electric field and area at and of the top end are both unaffected by the movement of the bottom end.Thus the flux through the top end is _still_ $\frac{ \phi_A}{2}$. Proposition 2: Therefore the flux through the bottom part is all the flux that is remaining (that is, $\phi_B-\frac{ \phi_A}{2}$). This must be balderdash, as with by extension of my logic, the flux through the bottom in the second case is$\phi_B-(\phi_B-\frac{ \phi_A}{2})=\frac{ \phi_A}{2}$: that is, the net flux is unaffected as the top and bottom flux add up to $ \phi_A $, in contradiction with $\phi=\frac{Q}{\epsilon_0}$. What is incorrect with propostion 1, both physically and mathematically?


Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given by Ricci Scalar) and in 3d it's 6 (i.e. Riemann Tensor given by Ricci tensor). But why does that constrain the Riemann tensor to only be a function of the metric? Why not a tensorial combination of derivatives of the metric? What I mean is why is the Riemann tensor in 2D of the form \begin{align} R_{abcd} = \frac{R}{2}(g_{ac}g_{bd} - g_{a d}g_{b c}) \end{align} and in 3D, \begin{align} R_{abcd} = f(R_{ac})g_{bd} - f(R_{ad})g_{bc} + f(R_{bd})g_{a c} - f(R_{bc})g_{ad} \end{align} where $f(R_{ab}) = R_{ab} - \frac{1}{4}R g_{ab}$? Wikipedia says something about the Bianchi identities but I can't work it out. A hint I got (for the 2d case at least) was to consider the RHS (the terms in parenthesis) and show that it satisfies all the required properties of the Riemann tensor (sraightforward) and proceed from there - but, I have not been able to come up with any argument as to why there must be a unique tensor satisfying those properties. Of course I could brute force it by computing $R_{abcd}$ from the Christoffel symbols etc., but surely there must be a more elegant method to prove the statements above. Help, anyone? I haven't been able to find any proofs online - maybe my Googling skills suck.


How can I calculate the pressure against the sides of a container?

While answering this question on DIY.SE, I wanted to calculate the amount of pressure (in psi) that would be exerted on the walls of the container. However, I don't know much about how to calculate such pressures. # Given: **The container:**   ![Hot Tub Frame with Banding and Bracing](http://i.stack.imgur.com/9Coxs.jpg) _Assume the interior of the container is solid (e.g. covered in plywood and a rubber liner)_   **Container outside dimensions:** 8' x 8' x 3'   **Container inside dimensions:** 89" x 89" x 36"   **Container volume:** 165.020917824 cubic feet (`(89x89x36) x 0.000578704`)   **Amount of water held by container :** 1234.521486241344 gallons (`165.020917824 x 7.481`)   **Weight of water held by container :** 10,297.3052722176 pounds (`165.020917824 x 62.4`) How would I calculate the amount of pressure on the walls in pounds per square inch?


Can bad guys really hide "dirty bomb" material via bananas?

This one probably has some cross-over with Skeptics SE, but I thought it fits here a little better. The concept comes from the TV show NCIS:LA second season episode "Empty Quiver" Here, the villains steal some nuclear material from a hospital's radiology department (in itself dubious). Then in order to transport it across the country without detection, they surround it in a truck full of bananas, as if it's nothing more than a produce delivery to a grocery store. Ok, so let's accept that bananas contain potassium, that there exists an isotope of potassium that is radioactive and found normally in nature (including bananas) and other things as indicated here. Is this collectively, i.e. an entire truck load, enough to fool a trained inspector?


How is a Lichtenberg figure created?

Related: Breakdown voltage of a dielectric When I think of the dielectric breakdown of air, an electric field of $3×10^6$ V/m is enough to accelerate initially ionized electrons created primarily from muons via cosmic radiation. These ionized electrons then collide with neutral air molecules, which in turn generate an avalanche of charge that produces lightning. Using Wikipedia’s language, lightning is a three-dimensional Lichtenberg figure. Reading Wikipedia’s section on Modern 3D Lichtenberg figures, a piece of acrylic (or glass) is impregnated with electrons from a linear accelerator. When this acrylic is made to discharge, a “beam tree” is formed that is absolutely beautiful. ![enter image description here](http://i.stack.imgur.com/sy0sY.jpg) What is the role of the pointed conductive object (or nail) that initiates the dielectric breakdown of the acrylic? The electrons trapped inside the acrylic create a charged dielectric and create some potential. I assume that the nail grounds out this acrylic so that electrons burn a path towards the nail to form the beam tree. However, **I want to know what is happening at the atomic level**?


Mass is a stuff or not?

Einstein said, there can be no clear definition can be give for moving body mass – instead of giving its name as “rest mass”, it is better to mention, “momentum and energy”. Matter is a stuff, particles and fields are stuffs – but “mass” is not a stuff?


How to restore the potential energy?

How to restore the potential energy, of a particle with variable mass and variable coordinate, by сartesian coordinates and mass of the particle? ? Dimension - one dimensional. Data that is to compute, only the mass and Cartesian coordinate of the particle, like this:                x        m      100.0  2710988       99.8  1000395      100.42 1669902       99.74  927829       99.74  927829      101.3  2153945      etc.      x and m are time-dependent ! No other data to calculate potential energy.


Speed of birth of a star

A star forms when enough material gets close enough. But how fast does the star start to shine? Is it gradually heating up (similar to an oven) slowly shining more and more? Or is it more like an explosion (similar to an atom bomb) that happens the second the critical mass is reached?


Speed of light, lasers and mass

Hopefully this isn't a bad question. Light travels at the speed it does and nothing else can travel that fast because things have mass, correct? Or at least correct on an elementary level. And if lasers, which are light as I understand it, have the ability to destroy other things. Doesn't that mean light has energy? But if light has energy shouldn't it have mass? I thought $E = mc^2$ and $E$ is energy, $m$ is mass. 0 mass times speed of light squared would equal zero energy, right? It's probably obvious that I'm no physicist but hopefully its also obvious that I enjoy science and physics.


There is absolute silence?

Silence is a relative or total lack of sound. But it is possible for real? Is there any way to achieve this?


Renormalizing Chaos: Transition in a Logistic Map

I am currently trying to understand the analysis of a logistic-like map $$f_\mu (x) = 1-\mu x^2$$ after section 2.2 in "Renormalization Methods" by A. Lesne. As I understand it, the physical situation is that $f_\mu$ has exactly $2^n$ attractors $x_{i\dots 2^n}$ in a certain region $\mu\in [0,\mu_c)$. Since one can order them such that $f_\mu(x_i) = x_{i+1}$ and the number of attractors grows in some way always by a factor of two, it is called "double-periodic scenario". However, at $\mu_c\approx1.401$, this behavior cannot be found any more and one might interprete this as critical point since the system appears to be **chaotic** from there, or, in some sense, the number of iterations it takes until a point $x$ is reached again becomes infinity. Here is a scetch of the situation: ![Logistic-Like- Map](http://i.stack.imgur.com/zNq95.png) At certain points $\mu_i$, $i=[i,\infty]$, the number of attractors doubles until $\mu_c$ is reached where this periodicity cannot be found. $N$ has been chosen to be sufficiently large. To understand the transition from the periodic point of view, Lesne conjectures that some delta exists such that $$\lim_{i\rightarrow \infty}\delta^i (\mu_c - \mu_i)=A\neq0\ .$$ Then, it is stated that $\delta \approx 4.67$ is somehow universal and can be derived using a **renormalization** approach of the form $R\left[ f\right] \propto f^k$ with $R$ beeing an operator that contains in the end all the informations about the system. Two things are unclear to me: > ### How can one use the renormalization operator $R$ to analyze the critical > behaviour, thus $\delta$? and > ### Is there a systematic way to find an appropriate $R$ better than just > looking at self-similarities of some graphs? Thank you in advance


Unitary Lorentz transformation on quantized Dirac spinor

I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be completely wrong). A unitary operator $U(\Lambda)$ acts on states as follows: \begin{equation} |p,s\rangle \rightarrow U(\Lambda)|p,s\rangle \end{equation} and therefore any operator, such as a Dirac field, transforms as: \begin{equation} \psi'(x) = U(\Lambda)\psi(x)U^{-1}(\Lambda) \end{equation} Now, from equation (3.109): \begin{equation} U(\Lambda) a_p^s U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda p}}{E_p}}a^s_{\Lambda p} \end{equation} we can find the transformation of the positive frequency solution of $\psi$: \begin{equation} U(\Lambda) \psi U^{-1}(\Lambda) = U(\Lambda) \int \frac{d^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_p}} \sum_s a_p^s u^s(p) e^{-ip\cdot x} U^{-1}(\Lambda) \end{equation} \begin{equation} \Rightarrow U(\Lambda) \psi U^{-1}(\Lambda) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_p}} \sum_s U(\Lambda) a_p^s U^{-1} (\Lambda) U(\Lambda) u^s(p)U^{-1}(\Lambda) e^{-ip\cdot x} \end{equation} and using equation (3.109) this becomes: \begin{equation} \Rightarrow U(\Lambda) \psi U^{-1}(\Lambda) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{2 E_p}\sqrt{2 E_{\Lambda p}} \sum_s a^s_{\Lambda p} U(\Lambda) u^s(p)U^{-1}(\Lambda) e^{-ip\cdot x} \end{equation} and from this point I have no idea how to get to equation (3.110). If anybody could push me in the right direction, then this will be greatly appreciated. (I am aware that the integration measure is Lorentz invariant.) Another question: does anybody have any other references/notes/books where they discuss how the quantized Dirac _operator_ field transforms? I find P&S explanation thoroughly confusing (as may have become clear from the questions I have been asking recently on this forum :) ), but I cannot find any other book that treats this stuff.


Is negative mass for a bound system of two particles forbidden?

Is there any theorem that forbids the bound system of two massive particles to have negative mass?


What are the key properties of and differences between classical and quantum statistical mechanics?

I'm studying different ensembles and different statistics (M-B, B-E, F-D), and I have some ambiguities about which of these models are applicable to quantum systems and which are usable for classical systems or both. (For example, the applicability of grand canonical ensemble to classical systems, M-B statistics to quantum mechanical systems, etc.) So I arrived at the question of what are the key properties of classical and quantum statistical mechanical systems, and what are their differences?


What are these rays that appear in photograph of sun?

In many images of light emitting objects we see such rays. Why do they appear ? What is the math behind their number and direction? ![enter image description here](http://i.stack.imgur.com/eXMj0.jpg)


Thoughts on the ice cube from orbit problem

Let's say we have a really exquisite cocktail party somewhere in New Mexico, and we just ran out of ice cubes. To the rescue comes this new service provided by Orbital Glacier Inc. They provide ice cubes around the world within only 5 minutes (!). How do they do it? How big must an ice cube be when you drop it from the orbit, so that it will have the size of a typical ice cube when it hits your glass of scotch that you're holding in your hand. Assuming the ice cube in the orbit needs to be of an enourmous size, would Orbit Glacier Inc. inevitably impact the climate conditions on planet earth when celebrating several such cocktail parties? Disclaimer: We are in no way related to our competitor Moon Ice Now, Inc.


Anomalous Total Transmittance

I am measuring with a Spectrometer UV-Vis-NIR Lambda 900 the total transmittance of a sample made by a sheet of Ethylene vinyl acetate (EVA) between two glasses of 3 mm thickness. I am obtaining an apparent total transmittance greater than 100% (up to 120%) for wavelengths 500-1000 nm. Can this measurement be ascribed to a physical phenomenon? Or there is something wrong in my apparatus?


Can the Large Hadron Collider break the speed of light?

Can Large Hadron Collider experiments break the speed of light at some point in the future? ![enter image description here](http://i.stack.imgur.com/i438z.jpg)


Problem based on Rotational Motion

> A spool of mass $\mathsf m$ and inner radius $\mathsf r$ and outer radius > $\mathsf{2r}$, having moment of inertia $\Large\mathsf{\frac{mr^2}{2}}$ is > made to roll without sliding on a rough horizontal surface by the help of an > applied force $\small\mathsf{(F=mg)}$, on ideal string wrapped around the > inner cylinder (Shown in the figure). > > ![](http://i.stack.imgur.com/qOKD2.png) > > Find the minimum Co-efficient of Friction $\small\mathsf{(\mu_{k})}$ > required for Pure Rolling. My Work For Pure Rolling, $$\mathsf{ω.r=v \tag 1}$$ $$\mathsf{α.r=a\tag 2}$$ Two torques are acting on the Spool about it's center (Circle's center), one due to the applied force F and other due to the friction generated by the rolling of the spool on the rough horizontal surface. $$\mathsf{∑τ=2r.f_k−F.r=Iα \tag 3}$$ Translation motion's equation, $$\mathsf{\sum F_{ext} = F−f_k=Ma\tag 4}$$ Solving equations $(2), (3)$ and $(4)$ and using $\mathsf{μ_k=\frac{f_k}{N}}$. I am getting $\mathsf{μ=\frac{3}{5}}$. While the options are $\mathsf{\frac{2}{9},\frac{4}{9},\frac{5}{9}}$and $\mathsf{\text{'none of these'}}$.


What makes up a resonator of radio?

I was reading this article about resonators. **Quote** : > The sine wave that matches that particular frequency will get amplified by > the **resonator** , and all of the other frequencies will be ignored. As far as I know, in picture below, the capacitor and inductor below makes up an **oscillator circuit** which oscillates at some frequency so that it can be matched and amplified by resonator. So what is this resonator that the article is talking about? ![enter image description here](http://i.stack.imgur.com/O0xxy.gif)


Relationship between inductance and number of turns in a coil in loudspeaker

I am in the process of designing a loudspeaker and have a question regarding the number of turns in the multilayered solenoidal coil and the speaker impedance of $8\:\Omega$. I understand that the Impedance is equal to the real part $R$ (resistance) added to the inductance and that the inductance is partly a function of the number of turn of the copper wire. I am having difficulty with understanding which equation describes this relationship between inductance and number of turns. Thanks.


What are some interesting coupled harmonic oscillators problems?

That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks.


Quantum Computing, Qubit Creation/Entanglement

I am currently a high school student researching quantum computing. I was referred to this site by Google and a friend. Currently I am researching the qubit part of quantum computing. My question is exactly how are qubits created in the lab, and how are they entangled? I don't expect the answers to be incredibly specific but a general overview would be of a great deal of help.


parallel/anti-parallel vs. triplet/singlet description of two spins

If we consider two spins, we can think of the spins as being either parallel (up|up or down|down)or anti-parallel (up|down or down|up). Or we can think of them as being in the triplet or singlet configuration. Is one description more correct than the other? Or is it just a matter of choice between two basis sets? It would seem to me that using T/S is correct because it accurately reflects the symmetry needed in the wavefunction.


Does the magnitude of pressure on the sides (not the top and bottom) of a volume depend on depth?

I understand that at a point in a fluid, such as water, the fluid will be in equilibrium as the sum of the pressure from all directions will equal to zero. For a volume of water, the reasoning should be the same otherwise unequal pressure would mean the volume would move. However, are the magnitudes of different points along the side of the volume the same for all depth or are they different? A picture would help illustrate my problem ![box](http://i.imgur.com/ZOUm7.jpg) The box can be a solid object or just an imaginary boundary enclosing water (it shouldn't change the question I believe). Since the volume is at rest in the picture, I believe that Pressure - Side.1 and Side.2 will have the same magnitude and cancel out, and the same goes for Side.3 and Side.4. However, is there a difference in magnitude for Side.1 in comparison with Side.3 or Side.4? If so, is there an equation? It's been a long time since I read about it and I tried skimming an introductory physics book but didn't find it.


What is enstrophy?

In turbulence, the enstrophy of a flow in a domain $\mathcal{D} \subset \mathbb{R}^{D}$ $$ \mathcal{E} = \int_{\mathcal{D}} |\vec{\nabla} \times \, \vec{v}|^2 d^{D}x $$ appears sometimes, it's cool and has cute properties in 2d (see this answer about turbulent flows), etc. Now, what does it represent physically? Okay, it's the integral of the vorticity squared, but has it a physically understandable interpretation? Why is it interesting to study _that_?


First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high machinery from algebraic geometry such as étale cohomology and motives appears in such books, however I could not guess where these structures arise in physical situations. How algebraic geometry and motives appears in physics? Why do physicists needs to use a projective scheme? When this scheme (or other structure) needs a noncommutative analog?


Color of a Metal's Threshold Wavelength?

How do I find the color of the threshold wavelength if the metal has a threshold wavelength of $\mathrm{6.5\times 10^{-7}m}$? I know that converts down to $\mathrm{650\ nm}$, but can I still use the visible color spectrum's wavelengths to find the color or is it different for metal? I'm sorry I'm new to this and am just making sure it still applies.


Scaling an electron microscope to fit within 1 cubic cm when running multiple parallel scans

Yes, I'm serious.   * What components correspond to an individual scan vrs what components can be cascaded across all objects? Ex: The vacuum chamber can be cascaded ( colocate the sample chambers within a single vacuum enclosure). Whereas there will need to be individual detectors for each sample   * Can there be a singular electron source? Using multiple magnetic focal lenses to direct those electrons to the individual samples?   * Can this idea be extrapolated to significantly downsize the components of an electron microscope?


How to find the wavefunction that solves an infinite square well with a delta function well in the middle?

Solutions for the wavefunction in an infinite square well with a delta function _barrier_ in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an infinite square well with a delta function _well_ in the middle. The setup is the bottom of the infinite square well is defined to be zero energy. I realize that there will be two situations, one where the particle's energy is less than zero and will therefore be bound to the delta function well and one where the particle's energy is greater than zero and is bound to the infinite square well. What are the wavefunctions for these two situations?


Why is boiling water the second time more quiet than boiling it the first time?

First of all: This is a different question than Why is boiling water loud, then quiet?, although the answers might be similar. When I wake up, I boil some water for a cup of tea. It happens quite often that I do something else for 10-15 minutes and forget the water, so I boil it again. I noticed that boiling the water the first time is much louder than boiling it the second time. Why is this the case? Did the air dissolved in the water disappear and therefore it became more quiet? How long would it take until the water has again air (I'm sorry, I know this is linguistically wrong, but I don't know how to write it correct). Is there any simple way I could get air again into the water to check if it's only the air?


Special Relativity

Could someone explain to me how special relativity works? I know there are thousands of sources and databases of knowledge out there, but I find it difficult to understand, even after reading up on those sources. (Note: if you're an admin to close my question down, would you please be so nice as to point out something to help me with this question?)


$E^2 = (mc^2)^2 + (pc)^2$: What units are used to measure $E$, $m$, $c$ and $p$?

\begin{equation} E^2 = (mc^2)^2 + (pc)^2 \end{equation} If I am using this equation to figure out the energy of something, what units would I use? Would it be the metric system? I.e. kilograms for $m$, meters per second for $p$, kilometers per second for $c$? And what units of measurement are used for $E$?


Why in professional spectroscopes diffraction gratings are used instead of prisms? I guess transparency is not an issue, as we need optics anyway.


Gauss Law for Electric Fields

What is the integral form for the **Gauss Law for Electric Fields**? ![a](http://i.stack.imgur.com/to4Wa.png) or ![b](http://i.stack.imgur.com/i0fa9.png) ?


Why is a general Green's function invariant under translations?

I am struggling to understand Green's functions, as used in Quantum Field Theory. One of my main problems is that the source I have been reading has a definition which is certainly correct, but involves enough integrals to obscure any intuition I initially had. Could someone explain what a Green function $G^{(n)}(x_1,x_2,\dots,x_n)$ intuitively represents (or provide a reference to a readable explanation)? In particular, I am being told that the Green function enjoys the invariance $$G^{(n)}(x_1+y,x_2+y,\dots,x_n+y) = G^{(n)}(x_1,x_2,\dots,x_n)$$ Supposedly, this is very simple and intuitive, but I can't see it (other than going a few levels down in the definitions, which in not very enlightning). Could someone please explain this property? _(my usual disclaimer: I am not a physicist, but a mathematician who occasionally comes into contact with physics. Please forgive my lack of physical insight.)_


Microsoft Excel not graphing $x = y^{1/2}$

The experiment was relating the period of one "bounce" when you hang a weight on a spring and let it bounce. I have this data here, one being mass and one being time. The time is an average of 5 trials, each one being and average of 20 bounces, to minimize human error.               t          0.3049s     0.3982s     0.4838s     0.5572s     0.6219s     0.6804s     0.7362s     0.7811s     0.8328s     0.869s      The mass is the mass that was used in each trial (they aren't going up in exact differences because each weight has a slight difference, nothing is perfect in the real world)               m          50.59g     100.43g     150.25g     200.19g     250.89g     301.16g     351.28g     400.79g     450.43g     499.71g      My problem is that I need to find the relationship between them, I know $m = \frac{k}{4\pi^2}\times T^2$ so I can work out $k$ like that but we need to graph it. In this equation, $k$ is the spring constant, $m$ is the mass hanging on the spring and $T$ is the period. I can assume that the relationship is a `sqrt` relation, not sure on that one. But it appears to be the reverse of a square. Should it be $\frac{1}{x^2}$ then? Either way my problem is still present, I have tried $\frac{1}{x}$, $\frac{1}{x^2}$, $\sqrt{x}$, $x^2$, none of them produce a straight line. The problem for SU is that when I go to graph the data on Excel I set the y axis data (which is the weights) and then when I go to set the x axis (which is the time) it just replaces the y axis with what I want to be the x axis, this is only happening when I have the `sqrt` of $m$ as the y axis and I try to set the x axis as the time. The problem of math/physics is that, am I even using the right thing? To get a straight line it would need to be $x = y^{1/2}$ right? I thought I was doing the right thing, it is what we were told to do. I'm just not getting anything that looks right.


't Hooft limit of coupling fundamental fermions to Chern-Simons theory

This question is in reference to this paper: arXiv:1110.4386 [hep-th].   * I would like to know what is the derivation or a reference to the proof of their crucial equation 2.3 (page 12).    * In their analysis of fundamental fermions coupled to Chern-Simons theory, why have they been able to ignore the ghosts altogether? Does it have something to do with working in the light-cone gauge in which the self-interaction of the gauge field probably disappears and hence that removes the necessity to have ghosts?    * Their Lagrangian is 2.1 (page 11) is massless and they say that they can always tune the bare mass to be $0$ and they can always ignore the mass of the fermion. Can someone elaborate on this? Why was this possible? Isn't this tantamount to assumption of conformality which they want to prove in the 't Hooft limit? (Then isn't the argument becoming circular?)  Similarly if this were a scalar field theory then in the same strain one might want to say that the quadratic and the quartic scalar interactions can always be held at zero - but again wouldn't that be an assumption of conformality in the 't Hooft limit? What would be a genuine proof of conformality for this theory or its scalar version?   * Is the scalar version of this theory somehow uninteresting or known?    * In this paper whatever is claimed as the higher-spin currents seem to have their conservation laws broken in in $1/k$ or $1/N$, then isn't the theory interesting only when $k$ and $N$ are both infinite and then isn't that a trivial theory?  What sense does it make to say the Fermion $2$-point function depends on the 't Hooft coupling (as in equation 2.23 page 16)? Isn't that a non-physical quantity to talk of?


Could someone jump from the international space station and live?

Felix Baumgartner just completed his breathtaking free-fall skydiving jump from $120,000\,\text{feet} = 39\,\text{km}$ above the Earth, breaking the speed of sound during the process. I was wondering if the next step could be jumping from the international space station. The person would have to overcome the orbital velocity of the station, re-enter the Earth's atmosphere, and land on the Earth with their feet, assisted by a parachute. Would such a stunt be survivable by a human? Could the person wear a space suit like the one depicted in the movie "Sunshine"? Is such a space suit possible? ![enter image description here](http://i.stack.imgur.com/W4fSO.jpg) **UPDATE** : I came across an interesting news article about a company that is working on a skydiving suite that would survive freefall from outer space. Read More Here ![enter image description here](http://i.stack.imgur.com/tbqPi.jpg)


1. How does Baumgartner slow his velocity so that the parachute can open up?   2. What is Baumgartner's speed just before his parachute opens?   3. What causes his speed to slow down just before the chute opens?   4. Is it air-resistance or is it altering his position from pencil to delta? The reason for this question is the supposition his body can endure only a maximum of "anti-force`" when the chute opens up.


Supersymmetric cancellation of loop contributions in a SUSY gauge theory

It is known that in SUSY models, loop contributions are automatically zero leading to a **_technically natural_** solution of the Higgs mass hierarchy problem. In many SUSY books/notes, it is often shown that, with a Lagrangian consisting chiral superfields, $$ {\cal L} = \int \left\\{\Phi_i^{\dagger}\Phi_i+\left[\frac{1}{2} m_{ij}\Phi_i\Phi_j+\frac{1}{3}\lambda_{ijk}\Phi_i\Phi_j\Phi_k\right]\delta^2(\bar{\theta}) +h.c.\right\\}d^2\theta d^2\bar{\theta} $$ All the loop diagrams of the superfields are identically zero. Thus in terms of component fields, the loop contributions are canceled out. Now, the same kind of vanishing loop contributions should be true in SUSY gauge theory as well. Without any direct/explicit calculation of the loop supergraphs, is there a simple argument/way to see how they are zero? Is that the same reason as the chiral model above, each supergraph contains a factor of $\delta^4(\theta)$ or higher order, which is $0$ ? Even if you provide references, your own interpretation/explanation is appreciated.


Calibrating the relative intensity of a spectrometer with a blackbody source?

I am trying to compare the relative brightness of H-Beta and H-Alpha lines using a CCD spectrometer. In order to correct for the different grating efficiency at the two different wavelengths. I took the spectrum of a 100W bulb to use as a blackbody source. I also know the pixel location that correlates to the H-alpha and H-beta line. I am wondering how to translate the blackbody spectrum to a relative intensity calibration. I thought to try to approximate the peak and use Wien's law to get the temperate of the filament and use that temperature in the Planck's law equation. However, I was unsure if there was a standard method of obtaining a relative intensity calibration for a spectrometer.


Is $E=mc^2$ false?

Recently I have found a Youtube video saying that $$E=mc^2$$ is false. It is said that the real one is $$ E^2=m^2 c^4 + p^2 c^2 $$ where $E$=energy, $m$=mass, $c$=speed of light, $p$=momentum. Is there anything true in it? It says that if it was $E=mc^2$ then as light had only momentum it would not have any energy. Youtube link Youtube Search result


Calculating sum of heat transfered

I have a formula of thermal conductance heat transfer rate. Here it is: $$ Q = \lambda{S (T_1 - T_2) \over L} \Delta t $$ For my calculations I have got some constant values available $$ Q = 0.58{1 (T_1 - T_2) \over 0.01} \Delta t $$ I can express difference of $T_1$(90°C) and $T_2$(0°C) as (note: $c_1$ and $c_2$, $m_1$ and $m_2$ are equal for my values, so I join them) $$ T_1 - T_2 = {Q_1\over c_1m_1} - {Q_2\over c_2m_2} = {Q_1 - Q_2 \over c_1m_1} $$ $$ 90 - 0 = {172431270 - 0\over2090\times 916,7} $$ for beginning values. Now the problem: $T_1 - T_2$ is not constant, because $Q$ is just a rate of energy in one moment. In next moment, the difference of $T_1$ and $T_2$ will be smaller by $Q$ $$ T_1 - T_2 = {(Q_1 - Q) - (Q_2 + Q)\over c_1m_1} = {Q_1 - Q_2 - 2Q\over c_1m_1} = {172431270 - 2Q\over 2090\times 916,7} $$ This happens every following moment. So, completing everything into one formula, $$ Q = {58{172431270 - 2Q_{moved} \over 2090\times 916,7} \Delta t} $$ My question: Is there any way to do sum of all moments in 10 seconds to get heat transmitted ($Q$) between two solids? The only problem I have there is that $T_1$ - $T_2$ is not constant. I can't evaluate it without using $Q$. Please help.


Why does the blade height goes on decreasing in a multi-stage axial flow compressor?

The axial compressors used in jet engines have their blade heights in decreasing order in the direction of the flow.


Why doesn't air freeze?

I am in no way experienced in the Physics field so this question may seem a bit silly but i'd appreciate an answer :) Why doesn't air freeze?


Fermi's Golden Rule

It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial system in an eigenstate of the unperturbed Hamiltonian $H_0$ and considering the time-dependent perturbation $V(t)=\begin{cases} V(\underline{r}),& \mbox{if } t_i<t<t_f \\\ 0, & \mbox{otherwise} \end{cases}$ where $t_f-t_i$ is the interval of interaction, and then considering the limit for $t_i \rightarrow -\infty$, $t_f \rightarrow + \infty$. Now, what is the range of validity of this approximation?


Parity Transformation Behavior of (Weyl-)spinors

Suppose $u^\pm (p)$ are Weyl-spinors with helicity $h = \pm$, i.e. solutions of the Dirac equation $p_\mu \gamma^\mu u(p) = 0$. How do these spinors behave under parity transformation? In Peskin&Schroeder, chapter 3.6, "Parity", the authors write $$u(p) = \gamma_0 u(\tilde{p})$$ where the tilde is understood to implement the parity on the coordinates: $\tilde{p} = (p_0,-p_1,-p_2,-p_3)$. The helicities are not explicity mentioned, but I guess this means that they are preserved. If I google for "parity spinors", however, the first result[1] suggest they do (scroll down the page). I also found references suggesting that the phase may change. My Question: Is the formula above correct? Is there an additional phase factor? Does it depend on the representation of the Gamma matrices? [1] http://www.ippp.dur.ac.uk/~krauss/Lectures/QuarksLeptons/Dirac/RealDirac_6.html


What does the Fermi Energy really signify in a Semiconductor?

In understanding the behavior of semiconductors, I'm coming across a description of the Fermi Energy here and at Wikipedia's page (Fermi Energy, Fermi Level). If I understand correctly, the _Fermi Level_ refers to the energy state at which there's a 50% chance of finding an electron. This varies with temperature. The _Fermi Energy_ is the highest occupied energy state of fermions at absolute zero. I'm a little confused as to the relation of the two terms. Additionally, in a semiconductor, the _Fermi Energy_ falls between the valence band and the conduction band. However, my understanding is that electrons cannot exist between the two bands -- so why isn't the _Fermi Energy_ the top of the valence band?


Stresses put through joints on a steel climbing wall

Being a fan of stackoverflow, I thought I could try this question here, hope its not too off topic. I'm wanting to build a free standing climbing wall in my garden and dont know where to start working out its strength and so construction. It will be made from interlocking (using bolts) 50cmx50cmx50cm mild steel cubes stacked 3m tall with a base of 1mx1m. The 4 base cubes will then be bolted to a concrete foundation. If I then bolt other cubes to the side of the main stack to create an overhang how can I work out the stress put onto the joints between the overhang and the main stack and between the stack and the foundations? The bolts between the cubes will be 3cm long and 2cm diameter. I havent decided what fixings to use between the stack and concrete as I believe this is where the problem will lie. Thanks for any help its much appreciated!


Is the cross section of a relativistic water hose or string always a perfect circle?

Given is a very long tube, such as a water hose or a tubular string with finite thickness, that has a constant circular cross section of radius $r$ along the length and that is at rest in an inertial frame. The hose is either   1. straight or    2. curved with radius $R$, or    3. irregularly shaped. Is the shape of the cross section always a circle for a relativistically fast observer of velocity $v$? What is the observed cross-section radius $r(v)$ of the hose? How does $r(v)$ depend on the geometry of the situation, i.e., on the angles between v and the hose? Is it constant along the length of the hose? The overall shape of the hose will vary for each observer. But will its radius $r(v)$ vary or not? We suspected that the cross section remains circular in all cases, and that $r(v)=r$ in all cases, for all observers, along the full length of the hose. The answer is a bit unexpected, though. P.S. This is not homework. It is a question we discussed at lunch yesterday. P.P.S. If the "hose" would be a string with some radius, the result might also apply to string theory?


Determining Average Tidal Effects

Maximum tidal heights vary widely across the globe, from 16 m in the Bay of Fundy to mere centimeters elsewhere. These variations are due to coastline and shoreline differences. This makes it difficult to determine a global average ocean tidal maximum height. Let's assume the Earth is a landless planet covered throughout by an average one km of ocean. How can we calculate the maximum ocean tidal height from a new moon or full moon syzygy, assuming semi- major Moon & Sun distances?


Clarifying constraint forces in Lagrangian dynamics

In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form $f_k(q,t)=0$, we have $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial {q}}+\sum{\lambda_k \frac{\partial f_k}{\partial {q}}}=0.$$ So, the questions:   1. Are the $\lambda_k$ the constraint forces (and therefore equal for each $q_i$), or is $\lambda_k \frac{\partial f_k}{\partial {q_i}}$ the $i$th component of the $k$th constraint force?   2. Should the signs of the constraint forces be interpreted as positive along the $\hat{q}$ direction and negative opposing? I have worked out the Lagrangian for a car going around a banked curve at constant speed, radius, etc., to see if I could derive the constraint forces. $$L = \frac{m}{2}(\dot{z}^2+R^2\dot{\theta}^2+\dot{R}^2)-mgz$$ with constraint $f = z-R\tan(\alpha)=0$. I get, from the $z$ equation (after applying $\ddot{z}=0$), $$\lambda = mg$$ and, from the R equation (after applying $\ddot{R}=0$), $$\lambda=-\frac{mR\dot{\theta}^2}{\tan(\alpha)}.$$ From the FBD, it's easily seen that $tan(\alpha) = \frac{m\omega^2R}{mg}$, which is consistent with the idea that what I really had above were $\lambda_z$ and $\lambda_R$, leading to $F_z=\lambda \frac{\partial f}{\partial {z}}$ and $F_R=\lambda \frac{\partial f}{\partial {R}}$.


Taylor series: Epsilon not differentiated?

Why isn't epsilon differentiated with respect to time? (see my question on the right) ![enter image description here](http://i.stack.imgur.com/AlAgL.png)


Can rain evaporate?

Are there any atmospheric condition under which a visible raindrop can fail to hit the ground by evaporating first? I imagine this would require a large vertical temperature difference, and possibly the rain forming very high up. Has anything like this been observed experimentally, or if not - is it possible to perform a calculation to show whether this is plausible?


Special Relativity moving in space

Given that time is 'just another' dimension and people get hung up on the fact that we cannot go back and forth in time like the other dimensions. Is there any proof that the corner of 8th Avenue and 14th Street at 3pm is the same place as as 8th and 14th at 4pm? Consider that in a 3 dimensional space, coordinates (8, 14, 3) isn't the same spot as (8, 14, 4) why can we assume that in a 2D place with Time that (8, 14, 3pm) is the same place as (8, 14, 4pm) only that the time has changed? Could it be that we cannot move back and forth in the other dimensions either, i.e. if we walk forward 10 meters and back 10 meters, it isn't really the same place since the time component has changed.


distance of electron from proton

An electron is projected, with an initial speed of $1.10 \times 10^5 \text{m/s}$, directly towards a proton that is essentially at rest. If the electron is initially a great distance from the proton, at what distance from the proton is its speed instantaneously equal to twice its initial value? i know it can be solved by equating the total energy as K.E+P.E(electron)=K.E+P.E(Proton) kinetic energy is $ \frac{1}{2}mv^2$. how can i find out potential energy of each particle?


What was the first ionization radiation?

While trying to make somewhat of a timeline of the history of ionizatig radiation, i am wondering about the following questions: The first photoelectrical effect was found 1839 by Alexandre Becquerell - the photoelectrical effect and therefore UV-light, would be the first ionization radiation. Is that right? While i am quite sure about it for electromagnetical radiation, i am not sure about particle radiation. Was there any ionizating particle radiation before the discovery of uran-radiation by Antoine Becquerell? A source for proof would be quite a nice extra, thanks!


Metric to describe an expanding spacetime from coordinates reflecting the perspective of a local observer

The FLRW metric describes the metric expansion of spacetime from the perspective of comoving coordinates. Given the way this metric is usually formulated, comoving distances stay constant, and the size of the unit ball "shrinks" as you translate it along the time axis (but not the spatial axes). I'm interested in seeing how to reformulate this metric given a choice of coordinates that lines up with the perspective of an observer who is in the space. From the observer's perspective, everything is falling along timelike geodesics that are moving away from it, and the further objects are from the observer, the faster they seem to be falling, and the size of the unit ball around the observer remains constant with time. To keep it simple, I only care about the case where curvature is 0 and we set $c=1$. So the FLRW metric in those circumstances is $$\mathrm{d}s^2 = -\mathrm{d}t^2 + a(t)^2 \cdot (\mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2)$$ Where $a(t)$ is the time-dependent scale factor. I thought that to flip this on its head, you'd want a metric that looks like this: $$\mathrm{d}s^2 = -b(x,y,z)^2 \cdot \mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$$ Where we now have the space-dependent scale factor of $b(x,y,z)$, and it affects the time coordinate. I was thinking that for some such suitable function $b$, which is presumably monotonic, you'd end up with a metric which stays constant in time rather than in space, and timelike geodesics that are all constantly moving away from one another. (They're also moving away from the line $(t,0,0,0)$, representing the observer I was talking about.) Am I on the right track here? I'm looking for something which is basically exactly the same as FLRW, but just reflects the different choice of coordinates.


Why don't planets have Circular orbits?

This might be a completely wrong question, but this is bothering me since many days ago. Given the mass (Sun) curves the space around it, gravitation is the result of such curved space (Correct me if I am wrong, source: A Documentary film). Given any point on a circle with center same as center of the mass, curvature in the space should be equal (Intuition). Planets rotate around the Sun because of the curve in the space they should follow a circular path and the distance between planet and Sun should be at a distance. Given the fact that earth has a elliptical orbit around the sun, and the distance between Earth and Sun varies according to position of the earth. Why do we have a elliptical orbit not a circular orbit.


In spontaneous symmetry breaking, global symmetry broken by gauged subgroup?

My question is simple. Given a group $G$ broken to a subgroup $H$, gauging a possibly different subgroup Hg breaks explicitly the global symmetry $G$, generating what is known as pseudo-Goldstone bosons. Why is this? The usual answer I get is that the gauging determines a specific direction in field space, but I really don't understand this statement, how having a subgroup $Hg$ gauged can break the global $G$ explicitly?. What I have in mind is that inside the gauge transformations $Hg$ there are included global ones as well, so this is what is confusing me.


What is the alternative of PMG (Permanent Magnet Generator) in a Horizontal Axis Wind Turbine?

I need to make a report about a HAWT (Horizontal Axis Wind Turbine) and list the parts with their costs, but as a Permanent Magnet Generator is a costly thing so what can I use in alternative of this to make the turbine less expensive? P.S. Due to the short time, I didn't do any kind of research on this PMG part's alternative.


Tunneling Rate Constant

I am trying to "decode"/derive an expression for the macroscopic rate constant for the tunneling of protons through a potential energy barrier that I read in a journal article: $$ k_{\rm tun}(T)=(2\pi\hbar)^{-1}\int_0^{V_{\rm max}} Q(V,T) P_{\rm tun}(V)\ dV. $$ So basically: the authors say work out the probability of tunneling at each point up the potential energy barrier (from 0 to $V_{\rm max}$), multiply this by the Boltzmann factor ($Q(V,T)$), integrating over all energies (i.e. we are taking a Boltzmann average )and then convert from energy in J to rate by multiplying by $(2\pi\hbar)^{-1}$. The authors use $(2\pi\hbar)^{-1}$ to convert from an energy in J to a rate in per second. This implies that the relationship between energy and frequency being used is: $$ E=\hbar\omega $$ I thought this just applied to electromagnetic radiation and free particles not in a potential. Is it OK to use this relation for protons tunneling in a potential? Or should you use some info about the shape of the potential? Many thanks N26


Have the correlation functions of the XY spin chain model been calculated using a functional partition function with source terms?

Have the correlation functions of the XY spin chain model, \begin{equation} H=-\sum_l (J_x \sigma_l^x \sigma_{l+1}^x+J_x \sigma_l^y \sigma_{l+1}^y)-B\sum_l \sigma_l^z \end{equation} been calculated using a functional partition function with source terms? By functional partition function I mean the partition function plus source terms (or generating functional) which allow correlation functions such as $\langle \sigma_l^x \sigma_{l+1}^x \rangle$ to be calculated analytically: \begin{equation} Z[\bar{\omega},\omega] = Tr \left[Exp \left[-\int_0^\beta d\tau H(a^{\dagger},a) - \int_0^\beta d\tau \sum_l (\bar{\omega} a + a^{\dagger} \omega) \right] \right] \end{equation} where $H(a^{\dagger},a)$ is the XY Hamiltonian expressed in terms of spinless fermions (i.e. after a Jordan-Wigner transformation). **Edit:** Anyone? An Ising chain would be fine too!


Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots a_pb_1\cdots b_{n-p}}$ Here, I assume the manifold is equipped with a metric $g_{ab}$ and the volume element is compatible with the metric, ie., $\epsilon_{a_1\cdots a_n}=\sqrt{|g|}\tilde\epsilon_{a_1\cdots a_n}$ with $\tilde\epsilon_{a_1\cdots a_n}=+1$ when $a_1,\cdots,a_n$ is an even permutation of $1,2,\cdots,n$; $=-1$ when old permutation; $=0$ when there are repeated indices. From this formula, I guess we can calculate the Hodge dual by first take the tensor product of $\omega_{a_1\cdots a_p}$ with $\epsilon_{a_1\cdots a_n}$ and then contract the first $p$ indices. Am I correct?


Zero Point Fluctuations

The total energy of a mode in a quantum mechanical resonator is given by $E_n ~=~ (n+ 1/2)hf$ where $n$ is the number of modes. So when there are no modes or vibrations, i.e. $n=0$, the energy is called the zero-point energy. What I don't understand is, if there are no modes, then what is this energy associated with? So what exactly is a mode? Also, is there a way of measuring this zero-point fluctuations? Why is the 1/2 photon introduced?


Problem evaluating moles in a an isochor transformation

I have a problem with an isochor transformation. Me and my group of study made an experiment that want to check Gay-Lussac’s law. We registered the equilibrium states and fitted the $P = nRT / V$, in the $P(T)$ graph. The moles found are $n_f=(5.63 \times 10^{-3} \pm 1.24 \times 10^{-4})$ mol. The problem is that, initial moles are $n_i=(3.19 \times 10^{-3} \pm 6.97 \times 10^{-5})$ mol. How can the initial moles be smaller? _Details of the experiment_ : We started stated with a certain number of moles of air inside a metal cylinder closed with a rubber top and connected, through it, to a temperature and pressure sensor. The starting gas conditions are $V, P_0, T_0$ assuming the volume to be constant (the container is, in a good approximation, hermetic). We heated an amount of water in a calorimeter with a resistance and a temperature $T' \gg T_0$, then we put the cylinder in contact with the hot water plotting the trend of $\Delta P-T$ on a graph. From a certain point the trend gets linear, and with a fit we calculated the slope $m = nR/V$ and from the latter we calculated the number of moles $n$ assuming the air as an ideal gas because of the higher temperature. We checked the consistency between this value and the one calculated from the initial conditions $n' = P_0V/RT_0$ and surprisingly this value is smaller than the one calculated from the fit. Now what is really happening? the difference $n'-n < 0$ suggests that the number of moles in the cylinder is growing? Or maybe it might be that one of our assumptions is incorrect.


A dielectric sphere in an initially uniform electric field and representation theory of SO(3)

I learned recently that the highest order spherical harmonic required to represent the spatial distribution of decay products of a particle can be used to determine its spin, by using arguments involving the representation theory of SO(3) / addition of angular momentum. I seek a similar argument for the well-studied problem of a dielectric sphere in an otherwise uniform electric field. This is, for example, solved in Griffiths' Electrodynamics, section 4.7 on the Third Edition (I'm using a Low Price Edition from India, where it is on Page 205). He concludes at the end of that problem that "the field inside is (surprisingly) uniform". I have a feeling that this result is less surprising when one applies ideas from SO(3) representation theory, but I'm not sure how to formulate the argument precisely. Here is my loose train of thought -- the field is a vector field with a specific direction, so it has one unit of angular momentum (or is given by an $l = 1$ spherical harmonic). Therefore, the induced surface charge on the dielectric sphere must be given by $P_1(\cos \theta) = \cos(\theta)$. The field inside is a vector field originating from these charges, so it must again be uniform. I'm also trying to use this to determine the highest order (i.e. highest $l$) component of $Y_{lm}$ in the total electric field after accounting for the polarization of the sphere.


Why are *high redshift* measurements of supernovae required to measure dark energy?

Why are _high redshift_ measurements of supernovae required... in order to measure the equation of state parameter of dark energy? The luminosity distance can be written as \begin{equation} d_{L}(z) = \frac{c}{H_{0}}\left(z-\frac{1}{2}(1+q_{0})z^{2}\right), \end{equation} where \begin{equation} q_{0} = \frac{1}{2}\left(1+3w_{\small{DE}}\Omega_{\small{DE}}\right). \end{equation} Here, $c$ is the speed of light, $H_{0}$ is the Hubble constant, $z$ is the redshift of the galaxy hosting the observed supernova, $w_{\small{DE}}$ is the equation of state parameter of dark energy, and $\Omega_{\small{DE}}$ is its density parameter. I was taught that because $q_{0}$ doesn't enter until quadratic order, we need to measure _high_ -$z$ supernovae. However, there are a lot of practical issues in measuring _high_ -$z$ supernovae. Now consider this. We can also write \begin{equation} d_{L}(z) = c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H_{0}\left[ \Omega_{\small{M}}(1+z')^{3}+\Omega_{\small{DE}}(1+z')^{3(1+w_{DE})}\right]}. \end{equation} Here, I cannot see any reason to prefer high-$z$ supernova of low-$z$ supernovae. It would seem, then, that we can avoid the practical difficulties by simply using a different formula. Where am I going wrong?


Is the concept of a field necessary to electrodynamics?

I've read (in Griffith's text) that it is "possible, though cumbersome" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory. What exactly is meant here? Since fields physically exist, why is this permissible? Edit: In light of the answers below, I'm curious about the apparent contradiction here - if fields physically exist, then how can one dispense with the field concept?


What force creates ions out of neutral atoms?

Consider the reaction between Na and Cl to form NaCl. Na loses an electron and "gives" it to Cl because this makes both atoms more stable. But what forces "pulls" the electron from Na to Cl? Both atoms are electrically neutral at the beginning, so the electron that is going to be transferred isn't acted on by electrostatic forces. Since its not electrostatic forces that pull the electron from Na to Cl (since both atoms are neutral) what force pulls the electron from Na to Cl to make both atoms more stable?


How to smooth the spectrum of a light source?

Could somebody please tell me if there's a reasonably cheap substance or device can I use to smooth the spectrum of a light source? For example, if the spectrum has spikes as in the blue graph below, is it possible to smear in terms of wavelength (not spatially, as in ordinary diffusers) so that it resembles the red graph more? ![smooth_spectrum.png](http://i.stack.imgur.com/LFaeM.png) The actual shape of the smoothing (convolution kernel) is not really important, but it is preferable that most of the energy is transmitted. I've searched for "spectrum flattening" filters, but they instead attenuate certain wavelengths rather than spread them out. I suppose it's possible to rapidly move the light source forward and back so that the Doppler effect would cause a similar shift, but the speeds required would be tremendous!


Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning that it is also traceless, $v^j_j =0$), equal $$ \dfrac{1}{2}\left(u^{i}v^{j}_{k}+u^{j}v^{i}_{k} - \dfrac{1}{4}\delta^{i}_{k}u^{l}v^{j}_{l} - \dfrac{1}{4}\delta^{j}_{k}u^{l}v^{i}_{l} \right) + \dfrac{1}{4}\epsilon^{ijl}\left(\epsilon_{lmn}u^m v^{n}_{k} + \epsilon_{kmn}u^m v^{n}_{l}\right) +\dfrac{1}{8}\left(3\delta^{i}_{k}u^l v^{j}_{l} - \delta^{j}_{k}u^{l}v^{i}_{l} \right). $$ I understand that initially (the first term in the above equation) he symmetrizes with respect to the two upper indices while at the same time he subtracts the traces. In the second term above, I can see that he lowers two indices with the invariant $\epsilon_{lmn}$ but I don't understand why he then symmetrizes! ($\epsilon_{lmn}u^m v^{n}_k \propto A_{lk}$ and he symmetrizes $A_{lk}$). While at the last term he adds the traces he subtraced with a spurious 3 and signs I don't understand how he got. Can someone elaborate on how exactly one can generally decompose tensor products into irreducible parts? _Is there a connection with the Young Tableaux? With the Young tableaux one can easily conclude that $3\otimes 8 = 15 \oplus \bar{6} \oplus 3$, **however how does one translate the Young tableaux into tensor components to get the above expression?**_ Thanks in advance.


How to prove the relations between the variables in bispherical coordinates?

When I was reading papers about the forces between two charged spherical conductors in a uniform electric field, I was confused with bispherical coordinates the author used to describe the spherical conductors. The schematic diagram is as follows: ![enter image description here](http://i.stack.imgur.com/DW2JI.png) where $β= \ln(\text{PL}_1 / \text{PL}_2) , \text{a = OL}_1 = \text{OL}_2$; Here is a link about the bispherical coordinates. I want to know why all the points on the sphere (for example sphere 2) satisfy $β=β2$. What's more, the relations between $\text{xyz}$ and $\text{aβα}$ are not entirely clear to me.Thank you!


A paradox to Lenz's law

I have read that in simple words, Lenz's law states that: > _The direction of current induced in a conductor is in such a fashion, that > it opposes its cause._ This validates law of conservation of mass-energy. I arranged the following thought experiment: Let there be a pendulum with its bob being a small bar magnet. The pendulum is oscillating in a direction parallel to the horizontal axis of the bar magnet on which the North and South poles lie. Also, the pendulum is in complete vacuum. (But gravity is there to make the pendulum oscillate.) At one of the extreme positions of the pendulum, we keep a solenoid, ends of which are connected to a load resistance. As the North pole of the bar magnet approaches the solenoid, current is induced in the solenoid in such a fashion that a North pole is formed at the end near to the bar magnet's North pole, and the bar magnet gets repelled towards the other side. The bar magnet then goes to the other end and then comes back (as a pendulum does) and again the same process is repeated. This _should_ go on forever, and current _should_ keep appearing across the load resistance. How does the law of conservation of energy hold here?


Loop integral using Feynman's trick

I am trying to show for the one-loop integral with three propagators with different internal masses $m_1$, $m_2$, $m_3$, and all off-shell external momenta $p_1$, $p_2$, $p_3$ the following formula appearing in 't Hooft(1979), Bardin (1999), Denner (2007): (unfortunate metric $-,+,+,+$) $$\int d^d q\frac{1}{(q^2+m_1^2)((q+p1)^2+m_2^2)((q+p_1+p_2)^2+m_3^2)} $$ $$=i\pi^2\int_0^1dx\int_0^xdy\frac{1}{ax^2+by^2+cxy+dx+ey+f}$$ where $a$, $b$, $c$, ... are coefficients depending on the momenta in the following way: $a=-p_2^2$, $b=-p_1^2$, $c=-2p_1.p_2$, $d=m_2^2-m_3^2+p_2^2$, $e=m_1^2-m_2^2+p_1^2+2(p_1.p_2)$, $f=m_3^2-i\epsilon$. I don't really care about factors in fromt like $i\pi^2$. My **simple problem** is: I am totally unable to reproduce coefficients $d$, $e$ and $f$. The problem is, when I integrate over the third Feynman parameter, $m_3$ appears in all three coefficients $d$, $e$ and $f$. How do I squeeze the denominators to reproduce this formula?


Simple harmonic motion

A uniform straight rod of length $L$ is hinged at one end. It is free to oscillate in vertical plane. Time period of oscillation with small angular amplitude when a point mass of mass equal to that of rod is fixed with lower end is : $1$. ![enter image description here](http://i.stack.imgur.com/eqjPb.jpg) $2$. ![enter image description here](http://i.stack.imgur.com/Qjk1K.jpg) $3$. ![enter image description here](http://i.stack.imgur.com/w2WSl.jpg) $4$. ![enter image description here](http://i.stack.imgur.com/Evpiy.jpg) Hey,please could you help me solve this question. Actually,this seems to be a torsional pendulum and the time period in such a case is $2\pi\sqrt{I\over C}$ where C is the restoring torque.Now,my doubt is what should the restoring torque be?Should it be 2mg(mass of the rod as well as the one suspended)times the length..or something else because that is not giving me my answer. High school student,so please keep it simple.


metastable energy levels

can someone give me a good definition of "metastable energy levels" in a helium plasma (glow discharge). Am I right when I say : "by adding some neon in my helium plasma, I will obtain a molecular plasma of NeHe" ? I guess the metastable energy levels are the reason of forming NeHe, but I don't know why, can someone explain to me ? If both my guesses are wrong, please give me some explanation, if you can. thx


Center of mass of a cylinder from which liquid is flowing out

A cylindrical container is of mass M, radius R and height H. The cylinder is filled with a liquid of density D. The base of the cylinder has an outlet tap of radius r through which the liquid can flow out of the cylinder. Initially the center of mass of the cylinder and the liquid in the cylinder system is at a height of H/2 along the axis of the cylinder. At time t = 0, the outlet tap at the base of the cylinder is opened and the liquid starts flowing out. We want to study the locus of the center of mass of the system. As liquid flows out, the center of mass will start coming down till it reaches a minimum point ans then start rising up again until it returns back to the height of H/2 along the axis when the cylinder is completely empty. **Assumptions** :   1. The upper surface of the cylinder is closed.   2. The experiment is done at an altitude A above sea level. **Questions** :   1. Where will the center of mass of the system be at time t = T?   2. Find T at which the center of mass of the system will be lowest.   3. How much liquid will be present in the cylinder when the center of mass of the system is lowest?


Resonance and a tuning fork

I carried out this experiment in class: I struck a tuning fork with a hammer. The sound lasted for some time. However, when I connected the tuning fork onto a wooden sounding box, the sound lasted for a few seconds (much shorter than the previous case) But why is that? I know the reason is because of resonance but can someone please explain the reasoning behind these observations with detail.


Does limit $\hbar \rightarrow 0$ in Quantum Mechanics mean anything?

Assuming that I learn Quantum Mechanics first, and then I approach Classical Mechanics as a special case of Quantum Mechanics, I will definitely find the relationship between Quantum Mechanics and Classical Mechanics very confusing. I don't know how to make sense of what happens when $\hbar \rightarrow 0$. For one, you can't recover classical mechanics from quantum theory by setting $\hbar \rightarrow 0$. However, it is possible to recover classical mechanics from Schrodinger equations. So, does limit $\hbar \rightarrow 0$ in Quantum Mechanics mean _anything_? How should we interpret it? Or does the above contradiction reveal yet another flaw in the fundamentals of Quantum Mechanics?


Refraction problem - limiting radius of observation

I solved this problem but I don't know if the solution is correct. Please, correct me if I'm wrong. There are two mediums, 1 and 2, with absolute refraction indexes $n_1$ < $n_2$, and an observer in medium 1 with height H (measured from surface to his eyes) . What is the limiting radius R that the observer can look at the surface and see an object located in medium 2? (as a function of $n_1$, $n_2$ and H) Let $\theta_c$ be the critical angle of refraction. Then light that comes from medium 2 leaves the surface only for incident angles $\theta_2 < \theta_c$. Call $\theta_2 = \theta_c - \epsilon$, with $\epsilon > 0$. The refrated angle will be, by Snell's Law: $$\theta_1 = \sin^{-1}(\frac{n_2}{n_1}\sin(\theta_2))$$ Such that: $$R = \frac{H}{\tan(\frac{\pi}{2} - \theta_1)} = H\tan{\theta_1} = H\tan{(\sin^{-1}(\frac{n_2}{n_1}\sin(\theta_2)))}$$ But in the limit $\epsilon \rightarrow 0$: $$\theta_2 \rightarrow \theta_c = \sin^{-1}(\frac{n_1}{n_2}\sin(\frac{\pi}{2})) \Rightarrow \theta_2 \rightarrow \sin^{-1}(\frac{n_1}{n_2})$$ And therefore: $$R \rightarrow H\tan(\sin^{-1}(1)) = H\tan(\frac{\pi}{2}) \Rightarrow R \rightarrow \infty$$ But this conclusion seems absurd to me since you can only see the mirror of an lake at long distances.


Capacitance of an arbitrary electrode configuration

I have to find the capacitance of a circuit element in three dimensions. I have the potential and charge in this cross sectional slice calculated using numerical methods. Does this value of charge have any useful physical meaning in 3D so that I can calculate the capacitance of a length of electrode? I will ignore edge effects in the third dimension for ease of calculation.


Which clock is the fastest inside an accelerating body?

The picture shows an accelerating spaceship with two clocks inside it. It is so far away from all other bodys that gravity is of no importance. Will the bottommost clock be slower than the topmost one? Or will both clocks have the same speed? ![Spaceship with two clocks inside it.](http://i.stack.imgur.com/yMaWm.png?editPost)


Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?

Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation? Does the latter not describe a spin-$\frac 32$ field? Why is the gravitino given by the Rarita-Schwinger-type representation rather than the $(\frac 32,0)\oplus(0,\frac 32)$ representation? This is related to a recent question I asked on gauge invariance of the Rarita-Schwinger field. Thanks!


Relevance of SIC-POVMs to quantum information

What is the real relevance of SIC-POVMs (symmetric informationally complete POVMs) to concrete tasks in quantum information theory? A lot of work has been put into giving explicit constructions, and yet it seems that in many concrete applications that at first glance require SIC-POVM-like objects, one can do without them using e.g. suitably chosen set of random projectors. So, my question is - if the problem of constructing SIC-POVMs in arbitrary dimensions was solved one day, would that enable solving problems which can't be solved or approximated using different techniques now? SIC-POVMs seem to play a role in some approaches to quantum foundations (see e.g. Chris Fuchs, Bayesian approaches to QM that rely on the existence of SIC- POVMs), but it's unclear to me how "mainstream" such applications are.


Replacing an operator with its expectation value

While dealing with a _circling particle in an spherical symetric potential_ our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value - he denoted it just $L_z$ - of the angular momentum if $L_z$ is constant. Why is that? So we first had this equation: \begin{align} \underbrace{\psi (r,\varphi,\vartheta)}_{\llap{ \text{wave function in spherical coordinates}}} &= \exp\left[\hat{L}_z \frac{i}{\hbar}\, \varphi\right] \underbrace{\psi (r,0,\vartheta)}_{\rlap{\text{wave function in spherical coordinates at $\varphi=0$}}} \end{align} and we got this one (notice that there is no operator over an $L_z$): \begin{align} \psi (r,\varphi,\vartheta) &= \exp\left[L_z \frac{i}{\hbar}\, \varphi\right] \psi (r,0,\vartheta) \end{align} Anyway here is the spherical coordinate system we ve been using all the time (the blue spherical aure is supposed to be a spherical potential...): ![enter image description here](http://i.stack.imgur.com/G8aUE.png)


Does a tidally locked planet have seasons?

Could a tidally locked planet have seasons? According to my understanding, a tidally locked planet rotates around itself exactly once per rotation around its sun. However, if the axis of rotation of the planet is significantly off from the axis of rotation around the sun, wouldn't it still have seasons? Or does tidal locking require that the axis of rotation of the planet about itself be perfectly orthogonal to the axis of rotation of the planet about its sun? If not, what would happen to a "tidally locked" planet with an axis of rotation "pointing towards" its sun (sort of like Uranus)?


weird terms in W boson self energy

I calculated the correction to the self energy of the W boson due to a fermion doublet (below I have n(e) for neutrino (electron), but it could be up and down quarks or just any 4th generation contribution). My result checks with the results from section 21.3 in Peskin, so I am fairly confident in the form below. I think of this as a fairly straight-forward calculation, but I am getting inconsistent results. The interaction comes from the term $$\mathcal{L}= \frac{g}{\sqrt{2}}W^+_\mu \bar{n}P_R \gamma^\mu P_L e + \frac{g}{\sqrt{2}}W^-_\mu \bar{e}P_R \gamma^\mu P_L n $$ so we get a single diagram with a electron (neutrino) running in the top(bottom) leg of the loop. For the self energy correction I get (with $\bar{MS}$ scheme): $$\Pi^{\mu \nu}(q^2) \sim \eta^{\mu \nu} \int^1_0 dx \left( x(1-x)q^2 - \frac{1}{2} ( x m_e^2 +(1-x)m_n^2)\right) \log \left[ \frac{(1-x)m_n^2 + x m_e^2 +(x-1)x q^2}{\mu^2} \right] $$ where I have dropped the $q^\mu q^\nu $ term.This seems innocuous enough however if I perform the x integration and then expand in the external momentum $q^2$ I get weird terms like $$\Pi(q^2) \approx(m_n^3-m_e^2)^3 \log \frac{m_e}{m_n}\frac{1}{q^2}$$ which seems incorrect since this should correspond to a term in the effective action, but there shouldn't be $\frac{1}{q^2}$ terms in the action. Furthermore the original integrand is not singular in $q^2$ so I could have just as well expanded in $q^2$ first and then performed the Feynman parameter integral and no such $\frac{1}{q^2}$ terms would be present. Any idea why the Feynman parameter integral and the momentum expansion don't seem to commute, or why I get these weird terms if I integrate first? Thanks a lot.


Are multipole fields, multipole expansion, and multipole radiation the same thing?

Interaction between electromagnetic radiation and nuclei can be written in terms of multipole radiation. Are multipole fields, multipole expansion and multipole radiation the same thing? I have found this topics in Nuclear physics book written by R.R. Roy and B.P. Nigam.


Understanding Poynting's Theorem intuitively

I understand that, broadly speaking, Poynting's theorem is a statement of conservation of energy. Energy density of a volume of current and charge decreases proportionally to work done on charges within the volume and by energy flux through the surface bounding the volume. The energy flux term is the area integral of the Poynting vector. My question is this: is this energy flux that is described by the Poynting vector a form of electromagnetic radiation? If not, how is energy radiating out of the charge and current distribution if not by work done on charges?


Origin of motion and relative speed of bodies in the universe

Charged particles can hit the earth at relativistic speeds. But it seems that all large bodies have fairly low relative speed. Of course, speed can increase considerably when a body orbits close to a massive object, but then it will not travel very far, and it can be discounted by averaging the speed over a long enough time (maybe a year). The Earth cruises around the Sun at 30 km/s, the sun cruises at 200 km/s and the Milky Way at 600 km/s. Not that much. I have two somewhat opposite questions:   * What do we know of the relative speeds of massive bodies in the universe, small or large (not particles)? I am not sure whether it was the proper way to state the question. Also, do relative speeds get very high if we correct out the part due to universe expansion? Are there braking phenomena?   * On the other hand, what initiated that motion? If the initial soup had been homogenous, the coalescence of randomly moving particles should have produced structures essentially at rest (ahem: where is the energy going?) with respect to each other. There were small variations in temperature, but how should that create speed differences? Or did it cause large scale streams that coalesced into moving bodies? Even if some speed is due to contraction of large scale rotating structures, it does require initial momentum to exist. To put it together, do we have measured speed statistics in conformance with universe evolution models? What does it say about speed? Sorry if the questions are not well stated; that is my best. A reference to a paper for non-specialists would do too. * * * **Added after 3 weeks, considering the answer by @BenCrowell and the comment of @ChrisWhite** (thank you both). Please, forget the part about cosmological speed comparisons. **This is my own answer** to my question; I leave it as the question, since it raises other issues. The first point is: does my answer make sense? It is only guesswork on my part. I wrote this answer because, though really helpful, Ben Crowell's answer did not really answer the heart of my initial question. Though I suggested it in my initial question, I realize even more now that the problem is only the origin of momentum, probably only angular momentum. This was confirmed by the answer of Ben Crowell regarding the fact that structures were essentially at rest to start with. Probably my main error was to think that there might be another "source of speed and momentum", and my awkward attempt to discount observed very high speeds as going nowhere because in tight orbit. I do not see how large scale momentum could come from some form of accretion of spontaneously emerging angular momentum quanta. I would think they would balance on average without any visible macroscopic effect, outside very special and anisotropic places like a black hole horizon (and even then). I feel the same regarding linear momentum quanta (if such a thing exists). My guess is that high scale momentum arises from momentum exchange between large collapsing structures. It is well known that celestial structure, such as planets and satellites, can exchange angular momentum. Though I no longer try to follow the mathematical analysis, I also read that much of the exchange can be mediated by tidal effects. But tidal effect should be even stronger when it is between structures that are not yet collapsed and are hence very deformable. Angular Momentum exchange is not necessarily one structure slowing while another is speeding up. It is vectorial and may be two structures both speeding up in opposite directions, provided extra energy gets in from somewhere, such as potential energy from collapse. So it would be the case that, as they are collapsing, structures are being deformed under the gravitational influence of other structures, so that instead of converging to some global center of gravity, some subparts collapse separately around their own centers of gravity, and rotate around the main center of gravity. Is it actually in that way that the originally small variations of temperature (density?) created the various structures of the universe? Observed momentum arising from such momentum exchanges, there is no reason it should have the same orientation in all subparts of a larger structure, and this is indeed observed in the solar system. (see Can the axis of rotation of a celestial body point in any arbitrary direction?) What about other structures in the universe? The variations in speed (earth in solar system, sun in galaxy, etc.) are obviously just the consequence of variations in mass and size of the structures, very massive structure permitting higher speed as observed. The speed formula being $v = \sqrt{\frac{G\cdot M} r}$, and the mass most likely increasing like the cube of the (initial) structure size. Hence the "average speed" in a structure should grow more or less like the initial linear size of the structure (assuming an almost uniform initial density). As reported in the initial question above, this is not quite what is observed. Where do I err? Could it be that very large structures collapse less than smaller ones because of the angular momentum of the subpart? This could reduce the influence of the total mass on individual subparts, and increase the orbital radius, thus reducing the speed. Is there any way to measure the degree of collapse as a function of initial size of a structure, and some notion of reference speed of substructures? Licence CC BY-SA 3.0 from the author.