title: Ice skating, how does it really work?

text: Okay, some textbooks I came across, and a homework assignment I had to do several years ago, suggested that the reason we can skate on ice is the peculiar $p(T)$-curve of the ice-water boundary. The reasoning is that due to the high pressure the skates put on the ice, it will melt at temperaturs below $273 K$ and thus provide a thin film of liquid on which we can skate. It was then mentioned as fun fact that you could ice-skate on a planet with lakes of frozen dioxide because that gas has the $p(T)$-curve the other way round. My calculations at that time told me that this was, pardon my french, bollocks. The pressure wasn't nearly high enough to lower the melting point to even something like $-0.5$ degrees Celsius. I suppose it is some other mechanism, probably related to the crystal structure of ice, but I'd really appreciate if someone more knowledgeable could tell something about it.


title: Being in a solid state, why is ice slippery?

text: Saying that ice is slippery is like saying that water is wet -- it's something we've known for as long as we can be said to have known anything. Presumably, humans as a species knew ice was slippery before we knew fire was hot, or that it existed. But ask anyone why, and they won't be able to give you any better explanation than one of those cave people would have.


title: What exactly is the composition of radiation in interplanetary space?

text: The articles I found on radiation in the solar system mostly dealt with solar wind, I wonder about other types. Is there a breakdown that tells me, withhin an order of magnitude, at least what intensity I can expect for a the different spectra - hard gamma through to radio frequencies, and chargede particles - and how this varies depending on my position?


title: Does the Nuclear reaction happen inside the fuel rods?

text: I've been reading up on nuclear reactors, and understand explanations of how it works, how water is heated to steam, which turns the turbines, etc.. I understand all of the safety features, and how control rods are used, and what they do with spent fuel. However, what I can't figure out (after lookin at many websites and videos) is where the reaction actually starts. The fuel rods contain Uranium pellets, and then they're put in the reactor, where the reaction starts and neutrons start hitting each other to create heat? So is this happening all over the reactor, or just in each fuel rod assembly? Also, you can hold a Uranium pellet in your hand (ideally wearing a glove) and it's not dangerous, so what _starts_ a reaction? A lot of people say the reaction can start by itself, so why do uranium pellets not suddenly heat up and start spreading radiation by themselves? Is it because they haven't been _enriched_? What if you dropped a Uranium pellet on the floor?


title: Can you magnetize iron with a hammer?

text: We know that a piece of ferromagnet, such as iron, can be magnetized by putting in a strong magnetic field to get domains parallel to the field grow. I also remember from pop. culture and MacGyver old tv series that you can magnetize a piece of iron by hitting it hard, with a hammer say, along the same direction. 1-Is this way of magnetizing iron scientific? or is it pseudoscience? and if it is scientific then: 2-what is the physical principle that will allow iron to get magnetized by hitting? and 3-how about nonferromagnetic materials?


title: Time relative to pressure?

text: We all know the effects of speed on time, and we have a formula for that, but I have a hunch, that time also is relative to pressure, and that by increasing the pressure on matter, in fact time will speed up for that matter. I am not a physics expert ( I have a Masters Degree in Computer Engineering, and have hence taken several physics, math and chemistry classes ) so don't slaughter me, I would appreciate a talkative tone. So I will say some "ideas" in my head that might be stupid, or assumptions from my understanding of how things are that are in fact wrong from an expert view, but please try to understand my **intention** or follow my idea of what I am thinking instead of focusing on details. So, my thinking is that once something starts travelling near the speed of light, or the speed of light, it will start to dissolve, before turning into energy and becoming the lightest of all while dissolving = light, and hence is why it can travel at the speed of light. So if something is to travel at the speed of light, it actually have to be dissolved into energy. The idea I have with pressure, is that if matter, in a pressure chamber, where pressure is turned up, the matter will eventually turn into pure energy as well = light. I also have this thought, that once pressure is applied to matter, the atoms starts to vibrate, move faster, and a reaction that would normally take hundreds of years, can be achieved with quicker results in a pressure chamber because time for all that materia increases, not because of the pressure, but because time is happening faster, which supports my idea of time actually speeding up for those particles. What happens with used reactor fuel in a pressure chamber today? How much can we speed up the halftime? It appears to me that if this was true, it will be the contradiction to speed. A twin X travelling near the speed of light, will come back to visit an older twin Y on earth. However, with pressure, it might just be the opposite, that twin X in the pressure chamber will in fact be older than twin Y outside. Surely turning up pressure on a twin might not be so healthy, but neither is acceleration from 0 to the speed of light in 1s. So pressure is probably not a healthy way to speed up time for matter, it will probably destroy certain things in it. Am I on something here? No?


title: Goodwins "proof" and space time

text: I read about Goodwin's "proof" that $\pi = 3.20$, it's BS and I know that. What I am wondering is whether his technique may have stumbled on something ( a warped space) years before Einstein. So I guess my question is this: Is it possible that somewhere in the universe (say the vicinity of a neutron star) where (due to the warped space) the ratio of the circumference to the radius is measured as something other than 3.14...? Following from that, is it possible that we only measure $\pi$ as 3.14... because of the way _our_ local space time is curved by the earth?


title: Capacitors in parallel and capacitors in series

text: How can I recognize clearly when I deal with capacitors in parallel or capacitors in series? Can follow a "rule" or a intuitive method? I have to do difficult exercises regarding system of capacitors so I must understand when I have capacitors in parallel or in series. Can you explain me in a simple way?


title: How to explain (pedagogically) why there is 4 spacetime dimensions while we see only the 3 spatial dimesions?

text: I have been asked this question by a student, but I was able and in the same time incapable to give a good answer for this without equations, so do you have ideas how one can explain this in a simple way? (Answers like we can take time as an imaginary, or our space is actually pseudo-Euclidean will be hard to grasp for new students.) Note that the problem is _not_ in visualizing the 4th dimension, that an easy thing to explain. The problem is more related to why we are in 3D that moving along 1D time dimension? In Differential geometry this interpreted by fiber bundles, but how to explain it to usual student.


title: How much space to simulate a small Hilbert space?

text: I'm thinking about trying to do a numerical simulation of some very simple QM problems. How much space do I need? To simulate the Hilbert space? I'd like to eventually simulate the absorption or emission of a photon by a hydrogen atom. So at least three particles (two fermions, one boson). Let's generalize that to three particles with arbitrary spin, so I can look at three photons or three electrons if I want to. In order to do a numerical simulation I need to replace the continuous spacetime with a rectangular grid or lattice. I'd like to eventually get more precision, but let's start with just ten cells per dimension to begin with. So including time, I need ten thousand cells in a four dimensional lattice. ` How many cells do I need to simulate the Hilbert space? and what goes in each cell? If I put restrictions on the shape of the wavefunction, does that help?


title: Why is a hole in a ball?

text: I wanted to know why there is a hole in the ball (basketball, volleyball, handball) to fill the ball with air. Why can't the ball come with filled in air and fully sealed so that there is no loss of air?


title: Textbook about the handiwork of a HEP analysis?

text: I'm wondering if there is a textbook that describes the handiwork of a particle physics analysis. There are a bunch of books about theory, about the experimental aspects like detectors, and about statistical methods, but I haven't seen one that focused on the actual work of an analyzer. I mean stuff like:   * How to conduct a search vs. a measurement   * What backgrounds to consider for a final state, and how to model them   * What distributions to look at -- pT of various objects or MET are obvious, but there are things like transverse masses, $\cos \theta^*$, aplanarity, the Florida variable (seriously), or ptBalance, that you first have to know about   * How to choose signal regions, sidebands   * How to determine systematic errors concretely   * How to perform a cutflow   * How to do tag-and-probe   * How to perform a multivariate analysis, and how to choose btw. techniques   * How to set limits, and how to perform combinations Most of this stuff is covered in various books, but mostly from a different perspective. For example, its nice to know how to solve an integral with Monte Carlo methods, and what the factorization theorem does. But for someone working on an analysis its more useful to know what the main differences between MC generators are, and how to deal with negative event weights in various situations. Similarly, there are a couple of textbooks about statistics for high energy physics, but those I've found tend to focus on derivations, instead of practical issues of the analysis. Does anyone know of a book that fits my description? _(Note I don't believe this fits very well under the currentbook policy. Resource recommendation questions tend to be fairly broad, and thus the answers have to be very descriptive. In this case, the question is already descriptive, so a brief answer would also be OK, even a negative answer (a la "I've been a expert for 20 years and can say for certain that such a book doesn't exist").)_


title: Does the wavelength always decrease in a medium?

text: I was studying a GRE Physics Test problem where optical light with a wavelength of 500 nm travels through a gas with refractive index $n$. If we look at the equations for wave motion and index of refraction $$c=\lambda_0\nu\quad\text{(in vacuum)}$$ $$v = \lambda\nu\quad\text{(in medium)}$$ $$n = c/v$$ we see that, if the frequency is constant, the **_wavelength decreases in the medium compared to vacuum_**. Is this a consistent property at all frequencies and for all mediums with refractive index real and greater than 1? Are there dielectrics which change the frequency (still for `n` > 1), and is there an example of that?


title: Why higher frequencies in Fourier series are more suppressed than lower frequencies?

text: One can expand any periodic function in sines and cosines. When calculating the coefficients $a_0$, $a_n$, and $b_n$ one find that $a_1>a_2>...>a_n>...$, similarly for $b_n$. Is there an intuitive reason to understand this, I mean why would one expect this to happen? It looks miraculous and mysterious to me that the coefficients came out "ordered" in a way that made the convergence manifest.


title: What is the potential difference between point X and point Y?

text: Here is the problem: ![Here is the problem](http://i.stack.imgur.com/iojG5.jpg) In the above figure I want help on finding the potential difference between X and Y. It is getting quite confusing due to the battery in the middle. I found the current in both the loops using Kirchhoff's Voltage law but then I'm confused on the proper method to find the potential drop between the two loops.


title: Do all light rays from a point passing through a thin lens converge at the image?

text: I have often seen diagrams, like this one on Wikipedia for a thin convex lens that show three lines from a point on the object converging at the image. Do all the other lines from that point on the object that pass through the lens converge at the same point on the image? * _Updated question: *_ to say, "from that point on the object"


title: Sparks from van de graff generator

text: I was at a lecture yesterday and there was a demo of a van de graff generator. He held the smaller metal globe which is attached by a wire, about 4cm from the main globe. This created a spark between the two globes. At the same time the spark happened the speakers in the hall would click loudly, the projector (on the other side of the room) would turn yellow and the text moved down, then flashed black and back to normal. And the plasma globe that was next to the van de graff generator which was switched off (but on at the wall) would briefly turn on. Is it possible for a spark to do that? Or is there something else happening at the same time?


title: Probability and probability amplitude

text: What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the statistical mathematics a similar equation where we calculate probability like $P=\left|x\right|^2$ where $x$ is whatever?


title: Probability and probability amplitude

text: The equation: $$P = |A|^2$$ appears in many books and lectures, where $P$ is a _"probability"_ and $A$ is an _"amplitude"_ or _"probability amplitude"_. What led physicists to believe that the square of modulus connects them in this way? How can we show that this relation really holds? In a wonderful QM lecture by J. J. Binney at Oxford University (minute 12:00) he states that this equation holds but doesn't explain why or where it comes from.


title: A particular supersymmetry transformation

text: While reading a paper I ran into this particular way of writing a $\cal{N}=3$ fields (in a theory with $N_f$ hypermultiplets) that I couldn't relate to anything I had seen before in the text-books (typically Weinberg's) The claim is that $q^{Aa}$ and $\psi_{\alpha}^{Aa}$ are scalar and fermions such that $a,b$ are $SU(R)_R$ indices, $A,B$ are denoting them to transform in the fundamental representation of $USp(2N_f)$ (why?) and $\alpha, \beta$ are $SO(2,1)$ spinor indices (theory is in $2+1$) Apparently these fields satisfy some reality conditions given by, the following equations, $(q^\dagger)_{Aa} = \omega_{AB}\epsilon_{ab}q^{Bb}$ and $\bar{\psi} ^\alpha _{Aa} = \omega_{AB} \epsilon_{ab} \epsilon ^{\alpha \beta} \psi _{\beta} ^{Bb}$ (where $\omega$ is some symplectic form, which I have no clue about) I find the above equations very unfamiliar. For one thing why do the scalar and the fermion have $R$ charge indices? (I am only familiar with the supercharges having a R-symmetry when the central charges are absent) Also I don't understand the $USp(2N_f)$ index. Apparently this can be related to usual $\cal{N}=2$ fields $(Q,\tilde{Q})$ as, $q^{A1} = (Q,\tilde{Q})$ and $q^{A2} = (-\bar{\tilde{Q}}, \bar{Q})$ I am completely unfamiliar with such a relationship or this thinking of $\cal{N}=2$ fields as being thought of as a pair of $Qs$. Once I understand the meaning of the above I will post another about its curious way of writing the supersymmetry transformation and the lagrangian with this field content.


title: How can I find the potential difference of a plastic sphere in a capacitor?

text: A thin spherical shell made of plastic carries a uniformly distributed negative charge of -Q coulombs. Two large thin disks made of glass carry uniformly distributed positive and negative charges S coulombs and -S coulombs. The radius of the plastic spherical shell is R1, and the radius of the glass disks is R2. The distance from the center of the spherical shell to the positive disk is d. I want to know how to find two things:   * The potential difference V1-V2. Point 1 is at the center of the plastic sphere, and point 2 is just outside the sphere.   * The potential difference V2-V3. Point 2 is just below the sphere, and point 3 is right beside the positive glass disk.


title: Can one measure the Higgs field?

text: Can one measure the Higgs field? Or is the higgs field not measurable? I know that the higgs boson is a indication for the presence of the higgs field. But it is the only way of the presence of the higgs field? When the higgs field is not measurable, what is the reason?


title: What is meant by polarised protons?

text: Really short question, but I cannot find anything on the internet. What is meant _proton polarisation_? Is it to do with the spin of the proton? I guess the spin of the proton is obtained from the vector addition of the 3 quarks' individual spins, so it can't be 0... Can we have _unpolarised_ protons then? Thanks!


title: Doppler effect and light

text: Approaching the speed of sound in an aircraft is relatively difficult, because the closer you get to Mach 1, the denser the pressure is around you (sound accumulates causing vibrations). Is there a similar effect as you approach the speed of light? Does the Doppler Effect apply for light as it does with sound? Will light accumulate around your aircraft causing heat? Speaking strictly hypothetical.


title: Infinitely many planets on a line, with Newtonian gravity

text: (I apologize if this question is too theoretical for this site.) This is related to the answer here, although I came up with it independently of that. $\:$ Suppose we   have a unit mass planet at each integer point in 1-d space. $\:$ As described in that answer, the sum   of the forces acting on any particular planet is absolutely convergent. $\;\;$ Suppose we move planet_0   to point $\epsilon$, where $\: 0< \epsilon< \frac12 \:$. $\;\;$ For similar reasons, those sums will still be absolutely convergent.   Now we let Newtonian gravity apply. $\:$ What will happen?       If it's unclear what an answer might look like, you could consider the following more specific questions:         planet_0 will start out moving right, and all of the other planets will start out moving to the left.   Will there be a positive amount of time before any of them turn around?   (As opposed to, for example, each planet_n for $\: n\neq 0 \:$ turning around at time 1/|n|.) Will there be a positive amount of time before any collisions occur? "Obviously" (at least, I hope I'm right), planet_0 will collide with planet_1. $\:$ Will that be the first collision? How long will it be before there are any collisions? $\:\:$ (perhaps just an approximation for small $\:\epsilon\:$)


title: Refraction through prism

text: We know that the light gets refracted when it enter a medium low/high refractive index. But why light is not refracted when it comes out from the vertex of a prism.


title: mixture of maximally mixed and maximally entangled state

text: Consider the quantum system $\mathcal{B}(\mathbb{C}^d\otimes\mathbb{C}^d)$ and $|\psi\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i,i\rangle$ be the (standard) maximally entangled state. Consider the state $\rho_\lambda=\lambda \frac{\mathbb{I}_{d^2}}{d^2}+(1-\lambda)|\psi\rangle\langle\psi|.$ Now for some values of $\lambda$ this state is entangled (example $\lambda=0$ it is $|\psi\rangle\langle\psi|$) and hence its entanglement can be detected by partial transpose operation. Can $\rho(\lambda)$ be an entangled state which is positive under partial transpose (known in literature as PPT entangled state) for some values $\lambda$? My intuition tells me that this is the case. However, I was told (without reference) that this is not the case and for the values of $\lambda$ we get only separable states or not PPT entangled states. I could not find the corresponding paper. May be I am not giving the proper string for searching. Advanced thanks for any suggestion, reference or comment.


title: Quick question on degeneracy of harmonic oscillator states

text: I'm currently learning about symmetry between particles. For a simple case of two non-interacting particles at $x_1$ and $x_2$, we know that the wavefunction can be written as $\psi_{n_1, n_2} = \phi_{n_1}(x_1) \phi_{n_2}(x_2)$ Let's consider two cases in ground state and first excited state. (a) Two particles are not identical, and have spin zero (b) Two identical particles, each with spin zero We know that the energy is given by $E_n = (n_1 + n_2 + 1)\hbar \omega$. **For case (a)** Since they are not identical, the wavefunction at ground state would be $\phi_1(x_1) \phi_1 (x_2)$ right? Then the degeneracy would be 0 right? Since $n_1 = n_2 = 1$. For the first excited state, the wavefunction would be $\phi_2(x_1)\phi_1(x_2)$, assuming $E_2(x_1) < E_2(x_2)$, meaning we just take whatever is lower. The degeneracy would be 0, since they are distinguishable? **For case (b)** Now they are identical, ground state wavefunction would still be $\phi_1(x_1)\phi_1(x_2)$ right? Degeneracy of ground state would be 0, since $n_1=n_2=1$? But for the first excited state, the degeneracy would be 2, since either $n_1=2$ and $n_2=1$ or $n_1=1$ and $n_2=2$? Spin 0 particles are bosons, so the wavefunction must be symmetric right? So I'm thinking $\psi = \frac{1}{2} \left[ \phi_2(x_1)\phi_1(x_2) + \phi_1(x_1)\phi_2(x_2)\right]$ The concept of symmetry and exchange symmetry is still new to me, I'd appreciate any response!


title: Torque for rotation

text: The force for revolution of earth is provided by the gravitational force of attraction between earth and the sun. What provides the force(torque) for the rotation of earth?


title: Developments originating from $E=mc^2$

text: What kind of significant impacts have originated from $E=mc^2$. Generally, it is regarded as the most famous equation of all time. Except for nuclear energy (fission and fusion) I do not know any other way in which this equation has made an impact on the world. Can somebody list some developments and impacts based on this equation?


title: Why photon-electron energy transfer can't occur in steps or does it?

text: The process of exchange of energy between a photon and an electron only occur after a specific energy called work-function of the material. Thus, the energy transferred is quantised due to the fact that the electron can reside only in quantised orbits. But, why doesn't the process happen in steps? I mean the electron getting excited to a higher orbit and before the electron gets de-excited another photon gives it energy to escape from the metal. Is it theoretically possible?


title: Why is electron presented in books, pictures as a sphere?

text: Why is electron presented in books, pictures as a sphere, when in fact it's not?


title: Travel direction of lightning

text: Is it possible to know whether a lightning bolt travel from the ground to the sky or from the sky to the ground? Alternatively, it could be both sides approaching


title: What does it mean to find acceleration in terms of g?

text: I'm having trouble understanding what a problem I have is seeking. To simplify the problem: > A particle reaches a speed of 1.6 m/s in a 5.0 micrometer launch. The speed > is reduced to zero in 1.0 mm by the air. Assume constant acceleration and > find the acceleration in terms of g during a) the launch and b) the speed > reduction. The basic strategy to find acceleration I am using is to calculate two velocity equations: one between (0 m/s, 0 m) and (1.6 m/s, 5.0 micrometers); the second between (1.6 m/s, 5.0 micrometers) and (0 m/s, 1.0 mm). Then I will derive the acceleration value for each. Because acceleration is constant I can expect a linear velocity equation. What is confusing me is that we are to assume constant acceleration. Thus the acceleration equation will merely be some real number. So, what exactly is expected if it is to be in terms of g? Is my strategy to find acceleration incorrect?


title: Nonlinear absorption coefficient and the band gap

text: How does the nonlinear absorption coefficient depend on the band gap? How can that coefficient be calculated theoretically? (Preferably with an example)


title: Latent heat of vaporization at non-boiling temperature

text: The latent heat of vaporization of water is what we normally use to calculate the heat transfer that occurs at 373 K when liquid water transits to vapor phase. But there are curves for latent heat of vaporization at different temperatures. What is meant by latent heat of vaporization at a temperature different from its boiling point?


title: Half-integer Spin and "natural conformal dimension"

text: If we consider a classical field theory for a massless particle of integer spin $s$, in a curved space-time, one finds that it is "naturally" conformal in a space-time of dimension $2+2s$ For instance, the free massless scalar field theory is conformal in $2$ dimensions, the electromagnetic field theory is conformal in $4$ dimensions, etc... But does it work for half-integer spin, for instance is a spin $\frac{1}{2}$ field theory conformal in a $3$ dimensional space-time , and is a spin $\frac{3}{2}$ field theory conformal in a $5$ dimensional space-time ?


title: Convolving Parton Distribution Functions

text: I have the cross-sections as a function of $\sqrt{s}$ for a process with a $u$-quark and $u$-antiquark in the initial state (eg.: $u \bar{u} \to e^- e^+$). I have a standard parton distribution function table (say, CTEQ). With these, how do I find the corresponding cross-sections for a process with protons in the initial state (e.g. $p p \to e^- e^+$ as at the LHC)? Specifically, should I use some sort of convolution? How?


title: The universe is expanding but why the galaxies have no motion?

text: It is said that the universe is expanding and the galaxies are moving apart. I understand that the space between every two galaxies is increasing. Doesn't this seem to imply that the galaxies will have relative motion and so have a moving velocity. According to relativity and the modern belief in physics, it is said that everything depends on relative motion and nothing else (I am talking about the concepts such as Time Dilation etc.). So in brief I would like to ask that why is it said that galaxies do not move even though the distance between them is increasing.


title: classical quantum particles in grand canonical ensemble

text: To derive Bose-Einstein and Fermi-Dirac distribution, we need to apply grand canonical ensemble:$Z(z,V,T)=\displaystyle\sum_{N=0}^{\infty}[z^N\sideset{}{'}\sum\limits_{\\{n_j\\}}e^{-\beta\sum\limits_{j}n_j\epsilon_j}]$. There is a constraint $\sideset{}{'}\sum\limits_{\\{n_j\\}}$ for quantum particles(bosons and fermions) in grand canonical ensemble:$\sum\limits_{j}n_j=N$, but why is there no such a constraint for classical particels?


title: The Voltage and Current of a Discharged Inductor at the Instant the Switch is Closed

text: The figure shows an LR circuit with a switch and a 240-volt battery. At the instant the switch is closed the current in the circuit and the potential difference between points a and b, Vab, are ![enter image description here](http://i.stack.imgur.com/Xs9ak.png) Choices :               a.     0 A,    0 V     b.     0 A, -240 V     c.     0 A, +240 V     d. 0.024 A,    0 V     e. 0.024 A, +240 V      Answer : c. 0 A, +240 V * * * **My Questions**   * When the battery discharging, the voltage will drop continuously, the voltage across a and b at the start instant should be equal to the original voltage of the battery 240V. However, how can we determine whether Vab is +240V or -240V?   * Moreover, if the potential different at the start instant is 240V, why should the current equals 0A? * * * Thank you for your help.


title: Time-like Killing vector in FRW metric?

text: The spatially flat FRW metric in cartesian co-ordinates is given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ As I understand it there are Killing vectors in the $x$, $y$, $z$ directions implying that momentum is conserved but there is no Killing vector in the $t$-direction which implies that energy is not conserved. I don't know much about Killing vectors. Is this correct? If I transform the time co-ordinate $t$ to conformal time $\tau$ using the relationship: $$d\tau = \frac{dt}{a(t)}$$ the metric now becomes: $$ds^2 = a^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ Does this metric now have a Killing vector in the $\tau$-direction as well as in the $x$, $y$ and $z$ directions?


title: Noether currents for the BRST tranformation of Yang-Mills fields

text: The Lagrangian of the Yang-Mills fields is given by $$ \mathcal{L}=-\frac{1}{4}(F^a_{\mu\nu})^2+\bar{\psi}(i\gamma^{\mu} D_{\mu}-m)\psi-\frac{1}{2\xi}(\partial\cdot A^a)^2+ \bar{c}^a(\partial\cdot D^{ab})c^b $$ where the metric is $(-,+,+,+)$, and the conventions are the following: $$ [D_{\mu},D_{\nu}]=-igF_{\mu\nu},\quad D_{\mu}=\partial_{\mu}-igA^a_{\mu}t^a, \quad D^{ab}_{\mu}=\delta^{ab}\partial_{\mu}-gf^{abc}A^c_{\mu} $$ Let $\epsilon$ be an infinitesimal anticummuting parameter, and consider the BRST transformation: $$ \delta\psi=ig\epsilon c^at^a\psi,\quad \delta A^a_{\mu}=\epsilon D^{ab}_{\mu}c^b,\quad \delta c^a=-\frac{1}{2}g\epsilon f^{abc}c^bc^c,\quad \delta\bar{c}^a=\frac{1}{\xi}\epsilon\partial^{\mu}A^a_{\mu} $$ I have calculated the corresponding Noether current as $$ j_{BRST}^{\mu}=-g\bar{\psi}\gamma^{\mu}c^at^a\psi-F^{a\mu\nu}D^{ab}_{\nu}c^b- \frac{1}{\xi}(\partial\cdot A^a)D^{ab\mu}c^b+ \frac{1}{2}gf^{abc}(\partial^{\mu}\bar{c}^a)c^bc^c $$ I am not sure whether the result is correct or not, so I would like to check that $\partial_{\mu}j^{\mu}_{BRST}=0$. Even though I have used the equation of motion $$ \partial_{\mu}F^{a\mu\nu}=-g\bar{\psi}\gamma^{\nu}t^a\psi- gf^{abc}A^b_{\mu}F^{c\mu\nu}-\frac{1}{\xi}\partial^{\nu} (\partial\cdot A^a)-gf^{abc}(\partial^{\nu}\bar{c}^b)c^c $$ $$ (i\gamma^{\mu}D_{\mu}-m)\psi=0,\quad \partial^{\mu}D^{ab}_{\mu}c^b=0 $$ and spent about four hours, I still cannot get it right. Could someone help me check this? Thanks a lot.


title: Finite or ∞ set of masses & ∃ gravity center?

text: Any finite & non empty set of masses has a computable center of gravity: $\vec{OG} = \frac{\sum_i m_i \vec{OM}_i}{\sum_i m_i}$ . Does the contrapositive permits to conclude that a mass system with physical evidence that it doesn't have a gravity center is an infinite set of mass (i.e. of cardinal larger than $\aleph_0$) ? On the other hand, an infinite set of masses may have a computable center of gravity. Ex. : within a 2D infinite plan, an infinite set of equal masses linearly distributed along the x and y axis will have a gravity center at its origin O. Unfortunately, this example doesn't have a center of gravity since the integral of masses in this particular topology doesn't converge.


title: Helicopter wings

text: Why do the propellers on helicopter appear to be so slow when the propellers is going at full speed? Can you please explain this particular optical illusion to me?


title: Help me to understand this conversion (4-vectors)

text: $u^{\mu}$ - 4-velocity $b^{\mu}$ - 4-vector of magnetic field $ u_{\mu}u^{\mu}=-1, \qquad u_{\mu}b^{\mu}=0 $ $$ u_{\beta}u^{\alpha}\nabla_{\alpha}b^{\beta}-u_{\beta}b^{\alpha}\nabla_{\alpha}u^{\beta}+\nabla_{\alpha}b^{\alpha}=0 $$ I don't understand why this equation gives this $$ u^{\alpha}u^{\beta}\nabla_{\alpha}b^{\beta}+\nabla_{\alpha}b^{\alpha}=0 $$ Help me please!


title: Help getting my head around frames of reference in special relativity

text: OK, so my textbook says that in time dilation and length contraction, the proper time and the proper length is "That which is in the frame of reference of the observer at rest relative to the event". This means that the proper time/length is that which is observed by the observer moving close to the speed of light, and the dilated time and contracted length is that which is observed by an observer moving relative to the event, i.e. is watching something move close to the speed of light. Am I correct so far? So, for example, if we have a beam of protons moving with a velocity of 0.99c and a stationary electron watching them, the proper time/length is that which the protons observe whereas the dilated time/contracted length is that which the electrons would see. Assuming I am correct so far (which I very well may not be, please tell me if I'm wrong), I have just been confused by an exam question. Here it is: > In an experiment, a beam of protons moving along a straight line at a > constant speed of $1.8$x$10^8$$ms^{-1}$ took 95ns to travel between two > detectors at a fixed distance $d_0$ apart. Calculate the distance $d_0$ > between the two detectors in the frame of reference of the detectors. Well, according to my understanding above, this would be the contracted distance, since the observer (detectors) are in motion relative to the event. However, they calculate the distance using $d=vt$, which would mean it's the proper distance. Someone please explain.


title: How to prove that a motion is Simple Harmonic Motion (SHM)?

text: I would like to know how one could show and prove that a given motion is simple harmonic motion. Once given an answer, I'll apply that technique to an example I am trying to figure out. Thank you in advance! I believe a motion can be proved simple harmonic, if the relation between its is as such: $$ a_x = - \omega^2\cdot x $$ And as such the period time is: $$ T =\frac{2\pi}{\omega} $$ Question-so-far: How do you prove such for a given force $F = \frac{G\cdot m_e \cdot M}{R_E} \cdot r$ ? Or any force that has non-trivial constants?


title: Simple question about a gas in a box with a moving wall

text: David Albert is a philosopher of Science at Columbia. His book "Time and Chance" includes this example (p 36). A gas is confined on one side of a box with a removable wall. "Draw the wall out, _slowly_ , and _perpendicular_ to its surface, like a piston. Now it happens to be the case, it happens to be a consequence of the Newtonian laws of motion, that a billiard ball which bounces off a receding wall will move more slowly _after_ the collision than _before_ it. (Footnote. Proof: consider the fame of reference in which the wall is at rest.) And so gas particles that bounce off the wall as it it's being drawn out will have their kinetic energy somewhat _depleted_." Why is that? We are assuming that an externally applied force is pulling the wall out. It's not the gas that is pushing the wall out. So why is energy lost. Why doesn't that violate conservation of energy. Another way to look at it is to assume a series of walls very close to each other. Remove them one by one parallel to the surface. There will be no change in the total kinetic energy of the gas. Why is this different?


title: Hydrostatic equilibrium of a star derivation

text: I've been told to revise the derivation that proves $\frac{\mathrm{d}P}{\mathrm{d}r} =\frac{GM(r)p(r)}{r^2}$ where brackets indicate a function of, P is pressure and p is density. Rather helpfully he hasn't given us it to revise, so if anyone knows it I'd be really grateful. Thanks.


title: Do Galactic Black Holes and Extra-Solar Planets change the mass balance of the universe?

text: With all the recent discoveries of supermassive black holes being at the center of nearly every galaxy, and the proposed "at least one planet" around (probably) every star. Does this affect the estimated total mass of our universe? Is Dark Matter necessary when planets and black holes are included in the calculation?


title: How much energy does lowering an object into a black hole generate?

text: An object of mass m is slowly lowered into a black hole of mass 1000 m. Is the amount of braking energy larger than $0.6 mc^2$? Now what if, after lowering the mass close to the event horizon, we wait the black hole to shrink to mass 100 m, by emitting Hawking radiation, and then we lower the mass some more? Is the braking energy of the second lowering process larger than $0.5 mc^2$? I wonder can we extract more than $E=mc^2$ energy from mass m? (In this thought experiment we use a wire and a winch, or anything that works)


title: Why does sand stick to my shoes?

text: Well, that's easy: the sand is wet, and my shoes are wet, and hydrogen bonding adheres the wet sand to my wet feet and to my shoes. But then I walk home, and my shoes dry, and the sand on them dries, and some of the sand falls off. But some does not. It's really stuck: even several days later I can turn the shoe upside-down and it won't fall off. What holds it on? Sand sticks to my feet after my feed dry and the sand dries. Is this the same?


title: Infinite-dimensional Hilbert spaces in physical systems

text: Can anyone give an example of when infinite-dimensional Hilbert spaces are required to describe a physical system? The standard answer to this question is yes, and I'm sure some of you will be quick to point out several clear examples: Ex. 1: Photon number states $\left\\{ |0\rangle,|1\rangle,...,|n\rangle \right\\}$ Ex. 2: Harmonic oscillator number states (same as above). Ex. 3: Continuous-variable basis of a single free particle $\left\\{ |x\rangle \right\\} \forall x \in \mathbb{R}$ From the infinite number of basis vectors in these examples, it is generally concluded that the dimension of the Hilbert space is also infinite. However, it is clear that a photon number state such as $|\psi\rangle=|100\rangle$ is unphysical (to be clear, this is the Fock number state with $n=100$, not a tripartite state with one photon in one channel). If you dispute this, I would suggest you try preparing such a state in a single- mode fiber in an optics lab. For this reason, we can apply a (somewhat arbitrary) cutoff value of $n$ and recover a finite-dimensional Hilbert space. Similarly, the cardinality of the basis set for a free particle is actually $\mathbb{N}$, not $\mathbb{R}$ (due to the properties of the function space), and although this set of orthogonal functions is infinite, we can also apply a similar cutoff for any reasonably behaved wavefunction. Can anyone give an example of a system described by an infinite-dimensional vector space for which such a cutoff cannot be applied?


title: Does electron go through a forbidden state when annihilate with positron?

text: Let's consider an electron-positron pair with total spin equal to zero. When it annihilates it can not emit only one photon because it would have zero momentum and nonzero energy. The pair emits two photons with opposite momenta but on the momentum-energy plain it looks like the particle goes through a forbidden state (red path on the picture below). The first question is: How is it possible? I suppose this is because of the energy-time uncertainty. The annihilation process is instant (at least looks like on Feynman diagram) and the energy of the intermediate state is not determined. Is it correct? If we can go through any forbidden state, why doesn't the annihilation go the blue path? This is the second question. And the third question: Why do electron-hole pairs in semiconductors always emit photons with energy equal to the band gap? Is it just because the interaction with one photon has higher probability or there is a fundamental difference? ![enter image description here](http://i.stack.imgur.com/XtNu2.png)


title: Calculating pressure in accelerated fluids in closed and open vessels?

text: ![enter image description here](http://i.stack.imgur.com/v7mfP.png) The question asked was "what should be the acceleration such that the pressure at both the points marked by thick dots be equal? the vessel is **open** and cubic with side 5m?" Initially i considered the diagram to be resembling the figure B where $$\tan(\theta)=a/g$$ but then i realized since the both points are the midpoints of the respective sides, the height of the liquid above the lower point must be half of the height above the point on the side. I took the diagram then to resemble D. Then i got the answer as $a=2g$. Is this CORRECT?   **If this had been a closed vessel what would the answer have been?** Figure C That is my question.


title: How can one determine at which distance the Lennard-Jones potential reaches a given value?

text: My question is fairly simple, but I do need clarification on how to get the inverse of the Lennard-Jones potential V(x). I am working with the following expression: $$ V(x) = e\times[(R/x)^{12} -2\times(R/x)^6] $$ So given a value $V$, how can I find $x(V)$ ?


title: The unitary time-evolution in the interation picture

text: I'm currently consuming a course on QFT where we need to define the unitary time-evolution to get the time evolution of the wave function in the interaction picture: $\hat{U}(t_1,t_0) = \exp\left(\frac{i}{\hbar}\hat{H}_0t_1\right)\exp\left(-\frac{i}{\hbar}\hat{H}(t_1-t_0)\right)\exp\left(-\frac{i}{\hbar}\hat{H}_0t_0\right)$ . Now one can show that this operator follows a Schrödinger equation by simply taking the derivative to time: $i\hbar\frac{d}{dt}\hat{U}(t,t_0) = \hat{H}_1^I(t)\hat{U}(t,t_0)$ . Where $\hat{H}_1^I$ is the perturbation to our free-field Hamiltonian $\hat{H}_0$. Now I started wondering whether $\hat{U}(t_1,t_0)$ shouldn't also follow a Heisenberg equation since it's an operator. I believe it shouldn't since $\hat{U}(t_1,t_0)$ gives a unitairy time- evolution which is a transformation, while the Heisenberg equation applies to observables. I was woundering if someone could confirm my reasoning or disprove it?


title: Victorian cosmology after the second law of thermodynamics but before relativity?

text: In the 19th century, most astronomers adopted an island universe model, in which our galaxy was the only object in an infinite space. They didn't know that the "spiral nebulae" were other galaxies. This model had the advantage of being dynamically stable (unlike Newton's infinite and uniform cosmology), and of avoiding the fact that Poisson's equation doesn't have unique solutions for an infinite and uniform universe. Ca. 1830, the geologist Lyell advocated a theory of uniformitarianism, in which the universe would have had uniform conditions going back infinitely far in time. There was some debate between Lyell and physicists such as Kelvin, who objected that the sun would have run out of energy, the earth would have lost its internal heat, and so on, but this was inconclusive because they didn't know enough thermodynamics and didn't know about the atomic nucleus. But by 1850, the second law of thermodynamics had been formulated, and this would seem to have been solid proof that the universe could not possibly have existed for an infinite time in the past. (This level of understanding of thermodynamics would also, I'd imagine, allow them to show that stars would gradually evaporate out of the galaxy, since the velocity distribution would have a high-velocity tail.) Kelvin did attack Lyell with the second law, but it doesn't sound like people at the time appreciated that this was a decisive argument. With the benefit of hindsight, we can say that it was decisive, and from 1850 to 1905, people should have been considering only cosmological models that used Newtonian mechanics and that stretched back a finite time into the past. Are such models possible? For example, I suppose you could make a Newtonian big bang, in which, as many students today imagine, there was an explosion at a specific point in otherwise empty space. Extrapolating the motion of all particles back in time using the laws of physics, one would find their trajectories all converging on a point, then diverging from it again on the other side, i.e., it would look like a big crunch/big bang "singularity," -- it would be a singularity of the matter density, temperature, etc. Thermodynamically, it would look like an extreme thermodynamic fluctuation, which is no worse than the thermodynamic implausibility of the low-entropy big bang in modern GR-based theories. There would be an upper bound on ages of objects in today's universe, because everything was destroyed in the singularity. Another possibility I can imagine is that you could start off with a universe containing an infinite amount of matter and an infinite amount of energy thermodynamically available to do work. After an infinite time, you would have expended an infinite amount of energy to do work, but you would still have an infinite amount left. The modern counterpart of this kind of thinking would be calculations of whether you can do an infinite calculation in various cosmological models (Dyson 1979, Krauss 1999). Related: Why didn't Newton have a cosmological constant _References_ Dyson, Time without end: Physics and biology in an open universe, Reviews of Modern Physics 51 (1979), pp. 447–460, doi:10.1103/RevModPhys.51.447; described in http://math.ucr.edu/home/baez/end.html Krauss and Starkman, 1999, Life, The Universe, and Nothing: Life and Death in an Ever-Expanding Universe, http://arxiv.org/abs/astro-ph/9902189


title: Why should SUSY be expected naturally?

text: In the last 40 years (approximately) people have been "discovering", "rediscovering" and "studying" SUSY as a powerful tool and "symmetry principle". Question: What if SUSY is not realized in Nature at the end? Is SUSY the only path to "relate" fermions and bosons or what else? Remark: SUSY has not been discovered yet, so keep you totally conservative. What if there is no SUSY? Bonus: What are the merits of SUSY? What are its main issues? I do know some answers to this, but I think it could very enlightening if we "listed" pros and contras of current supersymmetric theories to see where we are NOW.


title: Why doesn't intensity of light affect the emission of electrons?

text: So electrons of specific atoms have a minimum amount of energy needed to escape the atom, called the work function, W. Now let's say that you emit a certain frequency of light, and $hf<W$. However, from my understanding, this work function is just a form of kinetic energy, and if an electron has enough kinetic energy, its inertia will be greater than the centripetal force and the electron will escape. So even though $hf<W$, the object being illuminated will still heat up from the light. This heating up means that the material is gaining internal energy, which means that the electrons are gaining kinetic energy. If the material gets hot enough and the electrons keep gaining speed, couldn't the they escape anyway even though $hf<W$? Thanks!


title: Irreversible process

text: I have this problem. I have an ideal gas that goes through an irreversible adiabatic decompression. I have the initial state (P,T,V), and the final pressure, and I have to calculate the entropy difference of the proccess. So, what I know is that I can make up any reversible proces bewteen the initial and final state because entropy is a state function, and integrate the heat over that process, but I can't solve it. I'm aplying $P_1V_1^\lambda=P_2V_2^\lambda$ to get the final state, but as I am creating the final point of the adiabatic, I get entropy difference of $0$ for an invented reversible path. That way of getting the final state is not correct, right? I have $PV=nRT$ but I only have the final $P$ so I need another equation. EDIT: I have tried also to use $\Delta U=W\Rightarrow C_v\Delta T=P_{ext}\Delta V$, get $T_{final}$, and go on, but I don't get the correct result either. I thought this was general: $dU=C_vdT$, when can I apply that equation?


title: Does EM radiation (any, i.e. RF), or sound, radiate everywhere at once?

text: I am having trouble understanding electromagnetic radiation (or waves in general, be it EM or sound). If I have a 1 Watt speaker, is it infinitely divided and spread out so that everyone in every direction around the speaker can hear it? I do not believe they have "height", to reach more than one person at once, but if they did they would probably collide at one point. How do sound waves travel "backwards" (i.e. you are behind a speaker), are they scattered by air particles or itself so that people behind could hear it (at reduced amplitude)? I am just unsure how to wrap my head around it.


title: What does net mechanical efficiency mean?

text: I often see the term "net mechanical efficiency" used in literature, but I am not quite sure what it means, and what the difference is between it and "normal" efficiency. Take this sentence for example: `... increased the effectiveness, while reducing net mechanical efficiency.` What does exactly does this mean?


title: Discreteness of Spacetime and Violation of Lorentz symmetry

text: It is usually said that existence of discrete spacetime violates Lorentz symmetry. What quantity is used to quantify such violation? I mean could someone points a reference for a derivation that shows such analysis. My other question is that: if, and this is a big fat if, OPERA result was true (and I believe it is not) would that prove that spacetime is discrete? (this is like the inverse to the 1st part of this post, if space-time is discrete then Lorentz symmetry is violated, what if Lorentz symmetry is violated, does that imply space-time is discrete? or not necessarily?)


title: 2nd order pertubation theory for harmonic oscillator

text: I'm having some trouble calculating the 2nd order energy shift in a problem. I am given the pertubation: $\hat{H}'=\alpha \hat{p}$, where $\alpha$ is a constant, and $\hat{p}$ is given by: $p=i\sqrt{\frac{\hbar m\omega }{2}}\left( {{a}_{+}}-{{a}_{-}} \right)$, where ${a}_{+}$ and ${a}_{-}$ are the usual ladder operators. Now, according to my book, the 2nd order energy shift is given by: $E_{n}^{2}=\sum\limits_{m\ne n}{\frac{{{\left| \left\langle \psi _{m}^{0} \right|H'\left| \psi _{n}^{0} \right\rangle \right|}^{2}}}{E_{n}^{0}-E_{m}^{0}}}$ Now, what I have tried to do is to calculate the term inside the power of 2. And so far I have done this: $\begin{align} & E_{n}^{1}=\alpha i\sqrt{\frac{\hbar m\omega }{2}}\int{\psi _{m}^{*}\left( {{{\hat{a}}}_{+}}-{{{\hat{a}}}_{-}} \right)}\,{{\psi }_{n}}\,dx=\alpha i\sqrt{\frac{\hbar m\omega }{2}}\left( \int{\psi _{m}^{*}\,{{{\hat{a}}}_{+}}{{\psi }_{n}}\,dx-\int{\psi _{m}^{*}\,{{{\hat{a}}}_{-}}{{\psi }_{n}}\,dx}} \right) \\\ & =\alpha i\sqrt{\frac{\hbar m\omega }{2}}\left( \sqrt{n+1}\int{\psi _{m}^{*}\,{{\psi }_{n+1}}\,dx-\sqrt{n}\int{\psi _{m}^{*}\,{{\psi }_{n-1}}\,dx}} \right) \end{align} $ As you can see, I end up with the two integrals. But I don't know what to do next ? 'Cause if $m > n$, and only by 1, then the first integral will be 1, and the other will be zero. And if $n > m$, only by 1, then the second integral will be 1, and the first will be zero. Otherwise both will be zero. And it seems wrong to have to make two expressions for the energy shift for $n > m$ and $m > n$. So am I on the right track, or doing it totally wrong ? Thanks in advance. Regards


title: Postulate of a-priori probabilities

text: In Statistical Mechanics, we often postulate that for an isolated system, the phase-space density of all accessible microstates (i.e all microstates consistent with the energy) is the same. This is equivalent to assuming that the system is ergodic. This postulate leads us to the assertion that at any given time, the system is most likely to be found in that macrostate which has the maximum number of consistent microstates, and from then on, we calculate the entropy for this macrostate and get the fundamental relation for entropy, and hence, other thermodynamic quantities. My question now is: Is this assumption of equal a-priori probabilities too strong to obtain the second assertion? In other words, can we not say that the macrostate with the maximum number of microstates _is_ the observed thermodynamic state, while being non- committal as to whether indeed they are all equally probable or not? Or am I losing some information by not considering their probability distribution (maybe, say fluctuations)?


title: Does $c = 0$ implies that the theory is "empty"?

text: I'm wondering if there is more than the empty theory (no local fields, identically vanishing stress energy tensor) that can have central charge $c$ equals to $0$? My intuition tells me no, the stress energy tensor would transform as a primary field and so would only contain primary fields but there are none except for the identity operator. So there could only be (topological) degrees of freedom on the boundary. Is this reasoning wrong? ps: I'm only thinking at unitary theories, otherwise, sure it could be possible to combine in a balanced way fields carrying positive and negative central charge.


title: Why can't single LEDs produce white light directly?

text: Why does production of white light using a LED require combining a short wavelength LED such as blue or UV, and a yellow phosphor coating? Why can't a single LED produce pure white light?


title: Pressure difference in u tube

text: Suppose you have a U tube filled with a liquid. The height of the liquid column in both the arms of the U tube is not the same. Will the pressure be the same at two points in the two arms of the U tube which are at the same height from the base of the U tube or will it vary depending on height of liquid column above these two points?


title: Neutrino mass as counted in Dark Matter

text: If I try to add up neutrino masses (let's assume 1 eV rest mass equivalent each) to count as DM, do I use the rest mass or relativistic mass?


title: The end of theoretical physics?

text: About the meta questions 1193 and 2609, I've heard parallelly, that the complete branch theoretical physics is already done and that there isn't any thing else to do in this field, how true is it?


title: Complex coordinates in CFT

text: **The Setup** : Let's say we want to study a Euclidean $\mathrm{CFT}_2$ on $\mathbb R^2$ with coordinates $\sigma^1$ and $\sigma^2$ and metric $ds^2 = (d\sigma^1)^2+(d\sigma^2)^2$. It seems to me that in the usual discussion (e.g. di Francesco, Ginsparg, Polchinski), one proceeds to consider an analytic continuation of the CFT to $\mathbb C^2$ with coordinates $z^1, z^2$ and complex metric $ds^2 = (dz^1)^2+(dz^2)^2$ and then, one performs the coordinate transformation $z = z^1+iz^2$ and $\bar z = z^1-iz^2$. In this way the coordinates $z$ and $\bar z$ can be considered "independent" because they are coordinates on a complex two-dimensional manifold. Also, in these coordinates the metric becomes $ds^2 = dz\,d\bar z$ and it becomes clear that conformal mappings consist of mappings: $(z, \bar z)\to (f(z), g(\bar z))$. **My confusion is this** : Since our original theory was on $\mathbb R^2$, books say that when we do calculations, we should consider the physical theory as living on the copy of $\mathbb R^2$ embedded in $\mathbb C^2$ given by the condition $\bar z = z^*$. But consider the mapping $(z, \bar z)\to (z^2, \bar z)$. This is a conformal mapping on $\mathbb C^2$, but it does not map the surface $\bar z = z^*$ to itself; for example the point $(z, \bar z)=(2,2)$ gets mapped to the point $(z^2, \bar z) =(4,2)$ and $2$ is clearly not equal to $4^*$. In particular, it seems to me that analytic continuation to a CFT on $\mathbb C^2$ enlarges the set of mappings one can have, so what relevance does it really have to the original CFT on $\mathbb R^2$? (my apologies for the long post)


title: How to use the WKB approximation to find wave functions?

text: I'm trying to learn how to apply WKB. I asked a similar question already, but that question was related to finding the energies. Here, I would like to understand how to find the wave functions using WKB. > _An electron, say, in the nuclear potential_ $$U(r)=\begin{cases} & -U_{0} > \;\;\;\;\;\;\text{ if } r < r_{0} \\\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > > r_{0} \end{cases}$$ _What is the wave function inside the barrier region > ($r_{0} < r < k/E$)?_ Shouldn't the wave function have the following form? $$\psi(r)=\frac{A}{\sqrt{2m(E-U(r))}}e^{\phi(r)}+\frac{B}{\sqrt{2m(E-U(r))}}e^{-\phi(r)}$$ where $$\phi(r)=\frac{1}{\hbar}\int_{0}^{r} \sqrt{2m(E-U(r))} dr'$$


title: Impurity scattering

text: > **Possible Duplicate:**   >  Impurity scattering temperature dependence Is there any temperature dependence of relaxation time in impurity scattering of conducting electrons? It seems to me that there is none. But, some people claim that there is. So if you could explain, how temperature dependence comes into play?


title: Finding Rotational Kinetic Energy Of A Clock

text: The problem I am working on is: > _Big Ben, the Parliament tower clock in London, has an hour hand 2.70 m long > with a mass of 300 kg, and a minute hand 4.20 m long with a mass of 100 kg > (see figure below). Calculate the total rotational kinetic energy of the two > hands about the axis of rotation. (You may model the hands as long, thin > rods rotated about one end. Assume the hour and minute hands are rotating at > a constant rate of one revolution per 12 hours and 60 minutes, > respectively.)_ (Converted) Angular Speed of Clock Hands: Hour Hand $1.45\cdot10^{-4}~rad/s$ Minute Hand $1.75\cdot10^{-3}~rad/s$ Rotational Moment of Inertia: Hour Hand $I=1/3(300~kg)(2.70~m)^2=729~kg\cdot m^2$ Minute Hand $I=1/3(100~kg)(4.20~m)^2=243~kg\cdot m^2$ Rotational Kinetic Energy: $K_{rot}=1/2(729~kg\cdot m^2)(1.45\cdot10^{-4}~rad/s)^2+1/2(243~kg\cdot m^2)(1.75\cdot10^{-3}~rad/s)^2$ When I calculate this, it comes out incorrect, what has happened?


title: The Physical Meaning behind a Commutator

text: I've just been introduced to the idea of commutators and I'm aware that it's not a trivial thing if two operators $A$ and $B$ commute, i.e. if two Hermitian operators commute then the eigenvalues of the two operators can be measured with certainty simultaneously. But what is the physical significance when two operators do not commute such as to give a certain value? For example the position and momentum operator do not commute and give a value of $i\hbar$. What is the significance of the $i\hbar$?


title: What is Electromagnetic Induced Transparency?

text: Please can anyone explain the concepts of electromagnetic induced transparency? I am having problem with the technicality of the explanation on wikipedia. Please I am an engineer with a physics background though and would love if this can be explained with little to no jargons . I am working on optical buffers and studying recent technologies and would love and explanation on this.


title: Momentum of Light Question

text: Since light beams carry momentum, why would a person holding a flashlight not feel a recoil similar to that of a rifle being fired?


title: Laws of physics and general relativity

text: I'm reading that general relativity let's us describe physics from the point of view of both inertial and accelerated observers. What does that actually mean in terms of doing actual physics? For example, say a physicist performed Millikan's oil drop experiment or Young's double slit experiment or decomposed white light with a prism: (1) in deep space, far from any gravitational field, (2) on Earth, (3) near a black hole. I'm assuming she would get different results, and I'm assuming that in some way this is near the heart of what's important regarding GTR, but how exactly? In other words, what are the implications of GTR for understanding these experiments? Thank you


title: pros and cons of the anthropic principle

text: The anthropic principle has become a very popular explanation among theoretical physicists lately. Life is unlikely, but only life can observe, so we find ourselves in an unlikely position, so to speak. What I would like to ask is just how unlikely is life? Is the evolution of life generic for most laws of physics, or extremely unlikely with extreme fine-tuning? What is so controversial about the anthropic principle? No one is surprised to find themselves on an Earth-like planet orbiting a long lived star at just about the right distance from the star with plenty of liquid water, a thick atmosphere and plenty of carbon amidst the sterile antiseptic reaches of space. What are the major pros and cons? String theory conveniently provides a multiverse and a landscape.


title: Constant velocity in siphon?

text: I was recently reading the explanation for the behaviour of a siphon on Wikipedia, which uses Bernoulli's equation in its proof. The argument is generally pretty easy to follow, except for one key point: how can we assume that the velocity of the water is constant at all points throughout its flow through the tube? This tacit assumption is made at various point, not least in the terming of "the siphon velocity" (implying there's only one). I would be grateful if someone could clarify this. Note: I've done a hefty amount of classical mechanics in my time, but barely any fluid mechanics I'm afraid. First principles explanations would be appreciated.


title: Do amorphous metals undergo conchoidal fracture?

text: Amorphous metals are often referred to as metallic glasses due to their quenched atomic disorder. Do they fracture in the same fashion as silicate glasses? If not, what failure mode(s) do they have?


title: What's the difference between dim and bright light?

text: When comparing two light sources, for example, a light bulb at 20W and a light bulb at 100W, what is it about the incoming light that makes the latter look brighter than the former? Are there different reasons why different light sources looks different in brightness (High five for cramming three instances of "different" in the same sentence)? For example, in this thread, it is stated that the human eye is most sensitive around 555nm, something that I guess translates to meaning that given a light of the same intensity (whatever that means, hence my question), it is going to be perceived as most bright when hitting 555nm. Does this question have different answers depending on if you're seeing light as a particle vs a wave?


title: Are the neutrons in a neutron star on orbitals as in atomic nuclei?

text: Does the internal structure of a neutron star resemble anything like that of an atomic nucleus? I.e., are the neutrons arranged in a shell like structure with different energy levels, and without a distinct location for each neutron? I know that the force holding a neutron star is gravity, compared to the strong interaction inside atomic nuclei, but does this matter?


title: If a car appears in horison and within 2 seconds passes you by, whats the speed it's doing?

text: While watching the first 4 seconds of driving at 745 km/h is ludicrous from any angle wondered 1)If we knew the curvature of the earth in a "flat" desert, what would be the speed of the car? 2)Assuming we don't know the curvature of the earth, how long would it take for the car to go around the earth ( assuming a "flat" desert all the way travelling on a great arc)? Using my ninja pause skills it seemed it took 2 seconds from the time the car appeared ( as a dot ) in horizon to the time it passed by the camera, although 2 second window of observation seems to lead to a great error in overall estimates, using the 745 as the value for 1 and 2 the error of human observation could be calculated (?)


title: How come an anti-reflective coating makes glass *more* transparent?

text: The book I'm reading about optics says that an anti-reflective film applied on glass* makes the glass _more_ transparent, because the air→film and film→glass reflected waves (originated from a paraxial incoming wave) interfere destructively with each other, resulting on virtually no reflected light; therefore the "extra" light that would normally get reflected, gets transmitted instead (to honor the principle of conservation of energy, I suppose?). However, this answer states that _"Superposition is the principle that the amplitudes due to two waves incident on the same point in space at the same time can be naively added together, but the waves do not affect each other."_ So, how does this fit into this picture? If the reflected waves actually continue happily travelling back, where does the extra transmitted light come from? * the film is described as (1) having an intermediate index of refraction between those of air and glass, so that both the air-film and film-glass reflections are "hard", i.e., produce a 180º inversion in the phase of the incoming wave, and (2) having a depth of 1/4 of the wavelength of the wave in the film, so that the film-glass reflection travels half its wavelength back and meets the air-film reflection in the opposite phase, thus cancelling it.


title: Total power consumption of electric device

text: For some electric devices, like a fan or air conditioner, I read about their power consumption in watts on their specification guide. Does it tell about the power at normal or full speed? or Does the speed even affect the power consumption? Can I find the consumption at different speeds?


title: Why is electricity not transmitted wirelessly?

text: Why is electricity not transmitted wirelessly such that we don't need to span cables on the earth's surface? As in: electricity is transmitted wirelessly from the power plant to the household.


title: Why don't waves with different wavelengths interfere with each other in white-light?

text: The book I'm reading about optics says at some point that "each color (wavelength) contained in the white light interferes only with itself". But why is this so? Edit: I moved the rest of the question elsewhere.


title: Review of Gas Chromatographs?

text: Could anyone suggest a recent review article or book on gas chromatography instrumentation? My interest is in the devices themselves, rather than using them for a particular application. This article, "Gas Chromatography" in Review of Scientific Instruments is the sort of thing I'm going for; I'm just checking to see if there's a more recent or superior introduction.


title: How do I determine $\phi$ in a hollow circular cross section using torsion equivalence?

text: I know that the maximum shear stress $$\tau = \frac{T}{J}\rho$$ where $\rho$ is the radial distance from the center of the cross section. I have also determined the torsional constant $J$, which is equal to $\frac{\pi}{2}(R_o^4-R_i^4)$ for this particular cross section. The following boundary condition applies: $$\nabla^2\phi = F$$ I know that $\phi$ would be zero for a solid cross section, but I do not understand how I should calculate it with torsion equivalence. Any help is appreciated.


title: Can computers accurately model all of the details (to the subatomic level) of macro objects in collisions?

text: Frequently when trying to solve cosmology questions physicists turn to computer simulations of the universe (albeit massively simplified) in order to verify or disprove their hypotheses. This got me thinking. My question is about the theoretical maximum possible complexity of these systems. Let me give an example, if we imagine a tennis ball bouncing on a flat surface if we want to accurately simulate and measure the results of every single facet of the collision right down to the atomic and quantum effects you could actually find a tennis ball and drop it over your surface. In this case the universe is "simulating" the collision for you. Would it be possible to simulate this same event just as accurately using a computer? Is there a theoretical reason why the computer would need to have more mass than the two colliding objects? (in this case a tennis ball and the planet!) Now I have always assumed that the answer to this question is "yes you need a more massive computer to simulate any object with total physical accuracy" because if that were not the case there would be no reason why a computer less massive than the universe could not simulate the entire universe with total accuracy, which seems to me to be counterintuitive. I hope I was clear with my question! I would like to find a hard and fast answer to this question with a reason preferably, been mulling it over for a little while now. Thanks for any help.


title: Boundary Condition for Generating Evanescent Waves

text: I'm reading through Novotny and Hecht's book on Nano-Optics (Principles of Nano-Optics), and I've come across a subtlety in the boundary conditions for evanescent wave generation (via total internal reflection) that I don't understand. In the 2nd edition of the book the equation I'm struggling with is labelled 2.104, but exactly the same content is available online here where it's labelled 2.91. My question is this: where does the negative sign come from in the x component of the transmitted field $\textbf{E}_2$? I understand that the parallel components of the electric field must be conserved across the boundary, but where does the factor of $-1$ arise? Many thanks!


title: Photon propagator in terms of creation/annihilation?

text: As far as I understand it the photon propagator, $P(A\rightarrow B)$, described in Feynman's QED book, gives the amplitude that a photon moves from spacetime point A to spacetime point B. I was wondering if in quantum field theory terms $P(A\rightarrow B)$ is made up of the product of the following two amplitudes:   1. the amplitude that a photon is created at $B$ given that there is a photon at $A$.   2. the amplitude that a photon is annihilated at $A$ given that there is a photon at $B$. Is this the correct approach to describing a photon moving from $A$ to $B$ using creation and annihilation operators?


title: What are the strings in string theory made of?

text: This is a follow-up to an intriguing question last year about tension in string theory. What are the strings in string theory composed of? I am serious. Strings made of matter are complex objects that require a highly specific form of long-chain inter-atomic bonding (mostly carbon based) that would be difficult to implement if the physics parameters of our universe were tweaked even a tiny bit. That bonding gets even more complicated when you add in elasticity. The vibration modes of a real string are the non-obvious emergent outcome of a complex interplay of mass, angular momentum, various conservation laws, and convenient linearities inherent in of our form of spacetime. In short, a matter-based vibrating real string is the _outcome_ of the interplay of most of the more important physics rules of our universe. Its composition -- what is is made of -- is particularly complex. Real strings are composed out of a statistically unlikely form of long-chain bonding, which in turn depend on the rather unlikely properties that emerge from highly complex multiparticle entities called atoms. So how does string theory handle all of this? What are the strings in string theory made of, and what is it about this substance that makes string-theories simple in comparison to the emergent and non-obvious complexities required to produce string-like vibrations in real, matter-based strings? * * * **Addendum 2012-12-28 (all new as of 2012-12-29):** OK, I'm trying to go back to my original question after some apt complaints that my addendum yesterday had morphed it into an entirely new question. But I don't want to trash the great responses that addendum produced, so I'm trying to walk the razor's edge by creating an entirely new addendum that I hope expands on the intent of my question without changing it in any fundamental way. Here goes: The simplest answer to my question is that strings are pure mathematical abstractions, and so need no further explanation. All of the initial answers were variants of that answer. I truly did not expect that to happen! While such answers are sincere and certainly well-intended, I suspect that most people reading my original question will find them a bit disappointing and almost certainly not terribly insightful. They will be hoping for more, and here's why. While most of modern mathematical physics arguably is derived from materials analogies, early wave analogies tended towards placing waves within homogeneous and isotropic "water like" or "air like" media, e.g. the aether of the late 1800s. Over time and with no small amount of insight, these early analogies were transformed into sets of equations that increasingly removed the need for physical media analogies. The history of Maxwell's equations and then SR is a gorgeous example. That one nicely demonstrates the remarkable progress of the associated physics theories _away_ from using physical media, and _towards_ more universal mathematical constructs. In those cases I understand immediately why the outcomes are considered "fundamental." After all, they started out with clunky material-science analogies, and then managed over time to strip away the encumbering analogies, leaving for us shiny little nuggets of pure math that to this day are gorgeous to behold. Now in the more recent case of string theory, here's where I think the rub is for most of us who are not immersed in it on a daily basis: The very word "string" invokes the image of a vibrating entity that is a good deal more complicated and specific than some isotropic wave medium. For one thing the word string invokes (perhaps incorrectly) an image of an object localized in space. That is, the vibrations are taking place not within some isotropic field located throughout space, but within some _entity_ located in some very specific region of space. Strings in string theory also seem to possess a rather complicated and certainly non-trivial suite materials-like properties such as length, rigidity, tension, and I'm sure others (e.g. some analog of angular momentum?). So, again trying to keep to my original question: Can someone explain what a string in string theory is made of in a way that provides some insight into why such an unusually object-like "medium of vibration" was selected as the basis for building all of the surrounding mathematics of string theory? From one excellent comment (you know who you are!), I can even give an example of the kind of answer I was hoping for. Paraphrasing, the comment was this: > "Strings vibrate in ways that are immediately reminiscent of the harmonic > oscillators that have proven so useful analytically in wave and quantum > theory." Now I like that style of answer a lot! For one thing, anyone who has read Feynman's section on such oscillators in his lectures will immediately get the idea. Based on that, my own understanding of the origins of strings has now shifted to something far more specific and "connectable" to historical physics, which is this: > Making tuning forks smaller and smaller has been been shown repeatedly in > the history of physics to provide an exceptionally powerful analytical > method for analyzing how various types of vibrations propagate and interact. > So, why not take this idea to the logical limit and make space itself into > what amounts to a huge field of very small, tuning-fork-like harmonic > oscillators? Now _that_ I can at least understand as an argument for why strings "resonated" well with a lot of physicists as an interesting approach to unifying physics.


title: How to determine if an emergent gauge theory is deconfined or not?

text: 2+1D lattice gauge theory can emerge in a spin system through fractionalization. Usually if the gauge structure is broken down to $\mathbb{Z}_N$, it is believed that the fractionalized spinons are deconfined. However in general, $\mathbb{Z}_N$ gauge theory also have a confined phase. The question is how to determine if the discrete emergent gauge theory is really deconfined or not? For example, I am considering a $\mathbb{Z}_3$ gauge-Higgs model defined on the Kagome lattice with the Hamiltonian $H=J\sum_{\langle i j\rangle}\cos(\theta_i-\theta_j-A_{ij})$, where $\theta_i=0,\pm2\pi/3$ is the matter field and $A_{ij}=0,\pm2\pi/3$ is the gauge field. If the matter field is in a ferromagnetic phase, then I can understand that the gauge field will be Higgs out. But the matter field here is a Kagome antiferromagnet, which is strongly frustrated and may not order at low temperature. So in this case, I would suspect that the effective $\mathbb{Z}_3$ gauge theory will be driven into a confined phase. Is my conjecture right? How to prove or disprove that? Thanks in advance.


title: EM Field having simultaneously minimum divergence and minimal transversal extension

text: I read that the field that have simultaneously minimum divergence and minimal transversal extension is the fundamental Gaussian mode. What are the good references in this subject, to prove the assertion above?


title: Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$

text: I know how to derive below equations found on wikipedia and have done it myselt too: \begin{align} \hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\\ \hat{H} &= \hbar \omega \left(\hat{a}\hat{a}^\dagger - \frac{1}{2}\right)\\\ \end{align} where $\hat{a}=\tfrac{1}{\sqrt{2}} \left(\hat{P} - i \hat{X}\right)$ is a annihilation operator and $\hat{a}^\dagger=\tfrac{1}{\sqrt{2}} \left(\hat{P} + i \hat{X}\right)$ a creation operator. Let me write also that: \begin{align} \hat{P}&= \frac{1}{p_0}\hat{p} = -\frac{i\hbar}{\sqrt{\hbar m \omega}} \frac{d}{dx}\\\ \hat{X}&=\frac{1}{x_0} \hat{x}=\sqrt{\frac{m\omega}{\hbar}}x \end{align} In order to continue i need a proof that operators $\hat{a}$ and $\hat{a}^\dagger$ give a following commutator with hamiltonian $\hat{H}$: \begin{align} \left[\hat{H},\hat{a} \right] &= -\hbar\omega \, \hat{a}\\\ \left[\hat{H},\hat{a}^\dagger \right] &= +\hbar\omega \, \hat{a}^\dagger \end{align} These statements can be found on wikipedia as well as here, but nowhere it is proven that the above relations for commutator really hold. I tried to derive $\left[\hat{H},\hat{a} \right]$ and my result was: $$ \left[\hat{H},\hat{a} \right] \psi = -i \sqrt{\frac{\omega \hbar^3}{4m}}\psi $$ You should know that this this is 3rd commutator that i have ever calculated so it probably is wrong, but here is a photo of my attempt on paper. I would appreciate if anyone has any link to a proof of the commutator relations (one will do) or could post a proof here.


title: Meaning of invariable

text: Invariable means which is not variable i.e. can't be changed. Recently I have seen a sentence when reading a chapter based on measurement: > _The accepted standards must be accessible to those who need to calibrate > their secondary standards, and they must be **invariable** to change with > the passage of time or with changes in the environment (temperature, > humidity, etc.)._ What does 'invariable to change' mean here?


title: Constant-dependent potential in radial Schrodinger equation

text: Studying quantum mechanics, I've found an exercise I don't know how to solve it. Given the radial Schrödinger equation, $$\left [ \frac{d^2}{dr^2}+k^2-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) \right ]\psi_\lambda {\left ( r \right )}= 0$$ and doing whatever integrals are required, I have to show that: $$\int_{0}^{R}\psi_\lambda {\left ( r \right )}U\psi_\lambda {\left ( r \right )}dr=\left [ \psi_\lambda \frac{\partial^2 \psi_\lambda}{\partial r\partial\lambda}-\frac{\partial \psi_\lambda}{\partial r}\frac{\partial \psi_\lambda}{\partial \lambda} \right ]_{0}^{R}.$$ I tried to see if integration by parts, or Hellmann-Feynman theorem may help, but no luck. Any ideas? Updated: Following the indications given by @joshphysics (thanks!), I get, for the first step: $$ \frac{\partial }{\partial \lambda} \frac{d^2\psi_\lambda}{dr^2}+k^2\frac{\partial \psi_\lambda}{\partial \lambda}-\frac{2m}{\hbar^2}\lambda U\left ( r \right ) \frac{\partial \psi_\lambda}{\partial \lambda}-\frac{2m}{\hbar^2}U\left ( r \right ) \psi_\lambda =0$$ And using the original equation, to eliminate the lambda- dependent term, I get: $$\frac{\partial }{\partial \lambda} \frac{d^2\psi_\lambda}{dr^2}+k^2\frac{\partial \psi_\lambda}{\partial \lambda}-\frac{d^2\psi_\lambda {\left ( r \right )}}{dr^2}-k^2\psi_\lambda {\left ( r \right )}-\frac{2m}{\hbar^2}U\left ( r \right ) \psi_\lambda= 0$$. How do I eliminate the k dependent terms?


title: How is antenna gain correlated to beam width?

text: Let's say you have two dipole type antennas. Antenna A has a gain of 2.15 dBi, a horizontal beam width of 360 deg and a vertical beam width of 45 deg. Antenna B is similar to antenna A, but has a horizontal beam width of 360 deg and a vertical beam width of 42 deg. Can you use the ratio of the vertical beam widths to predict the gain of antenna B? Note: In the application I'm asking about I'm not sure what method they used to calculate the beam widths. Maybe someone else knows which methods are most commonly used for dipole antennas.


title: What is the information content of a human being?

text: How much memory would we need to represent a human? How would each atom be stored as? Bytes? Something more complex?


title: Feynman Lectures on Physics

text: I want to develop a good intuition of concepts in Physics, and I've heard that the Feynman Lectures on Physics are very good for this. However, I've also heard that the Mathematics is very complicated. I was just wondering, just how much Math do I need to know? I love Physics, and I'm willing to put in the work. How much time do you guys reckon this will take? Thank you :)


title: Colder surface radiates to warmer surface

text: When radiation from a colder source arrives at a warmer surface there is some debate about what happens next. To make the question more concrete lets say that the colder source is at temperature 288K. The warmer surface is at 888K and has emissivity of 1. 3 possibilities   1. We ignore such radiation because it cannot happen.   2. The radiation is subtracted from the much larger radiation of every wavelength leaving the hotter surface.   3. The radiation is fully absorbed and its effect is to be re radiated at characteristic temperature of 888K (plus infinitesimally small T increase due to radiation absorption). I would have thought that 2. and 3 are more plausible than 1. Both 2 and 3 satisfy the Stephan Boltzmann equation. 3 however seems to imply that the radiation from colder object is transformed into much higher quality radiation and a possible second law of thermodynamics infringement.


title: How does the grid on the microwave oven window prevent microwave radiation from coming out?

text: If I look through the microwave window I can see through, which means visible radiation can get out. We know also that there is a mesh on the microwave window which prevents microwave from coming out. My question is how does this work? how come making stripes or mesh of metals can attenuate microwave radiation yet allow visible radiation? Looks like an electrodynamics problem to me with periodic boundary conditions (because of the partitions on the microwave oven window). Is it discussed in any textbook?


title: Why must the eigenvalue of the number operator be an integer?

text: My apologies if this question has already been answered. Using the notation of Sakurai, we have an energy eigenket N and an eigenvalue n. $$N | n \rangle = n | n \rangle$$ For the number operator $N = a^\dagger a $. My question is precisely why does n have to be an integer. I do understand that it must be positive definite due to the norm condition on $ a | n \rangle $: $$\langle n | a^\dagger a | n \rangle \geq 0$$ I can see that we can show: $$ a | n \rangle = \sqrt{n-1} | n-1 \rangle $$ and similarly for $a^\dagger | n \rangle$ Sakurai argues that the sequential operation of $a$ operators leads to the form which he does not write, but I will write here for clarity: $$ a^k | n \rangle = \left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right) | n - k + 1 \rangle $$ Since $$ a|0 \rangle = 0 $$ the sequence must terminate, but can only terminate if n is an integer. How can we make that jump? Is there a formal mathematical proof that the sequence $\left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right)$ will terminate if and only if n is an integer?


title: Why is the free energy minimized by the Boltzmann distribution?

text: Can someone show me, without glossing over anything, why $F = E - TS$ is minimized when $p_i = e^{-U_i/k_bT}/\sum_ie^{-U_i/k_bT}$? I understand it conceptually, but am having difficulty showing it formally.


title: Does gravitational force attract bodies with mass or with energy?

text: On my textbook is written that gravitational force is the force that attracts bodies with mass. But I've seen on a book that It actually attracts bodies with energy. I'm having a class tomorrow and I would like to know some argumments to use with - against my professor.


title: What is the definition of how to count degrees of freedom?

text: This question resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here:   * Are necessary1 derivatives such as velocities counted as individual DOFs or together with the respective coordinate2?   * Should complex valued DOFs be counted twice as in "two real-valued DOFs" or once as in "one complex-valued DOF"? (I mean, when one does not want to specify this explicitly) Please answer with some reference, unless it turns out this is actually a matter of taste rather than a strictly defined thing. * * * 1) I mean those a value of which is _required_ as an initial condition   2) I count fields in QFT as coordinates as well, while space-time-coordinates are parameters to me, if that matters. I know a field actually has $\infty$ (or rather, $2^\aleph$) DOFs itself, but let's say e.g. "one $\mathbb R^3$ continous DOF" in that case


title: Strict general mathematical definition of drag

text: Is there a formal definition of drag, say, as some surface integral of normal and shear forces? There seem to be a lot of formulas for specific cases, but is there a general one? I need to accurately calculate the drag of three cylinders placed between two plates.


title: How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work?

text: The world is full of nuclear warheads being stockpiled. Controlled fusion power seems a long way away. Could we put these warheads to better use by exploding them in a controlled way and capturing the energy they produce? By useful work I mean the power is then available for the national grid to boil kettles or have a shower! Extra points for looking at the practicalities of building a facility to do this (although I guess that would get the question closed :-(... )


title: What barriers exist to prevent us from turning a baryon into a anti-baryon?

text: At present the only way we can produce anti-matter is through high powered collisions. New matter is created from the energy produced in these collisions and some of them are anti-matter particles such as positrons, anti-protons etc. My question is, normal matter and anti-matter are so similar. They both appear to react to the fundamental forces in the same way. Is it conceivable that we could find a way of "switching" matter to become "anti-matter" through some low energy technique? E.g. find a way to flick a few "quantum switches" and voila your proton is now an anti-proton, ready for annihilation and limitless energy for humanity! Would we be breaking a great deal of the fundamental laws of physics if this was found to be possible (i.e. should I give up now)?


title: Does the wind gust over the ocean?

text: It is easy to observe that on a windy day, the wind does not blow for several hours at constant speed, then gradually subside. Instead, on a time scale of seconds or tens of seconds, there are stronger gusts of wind followed by lulls. Presumably this is an effect of turbulence. If so, is this turbulence due to the complicated geometry around me - buildings, trees, hills, etc.? If we removed these features and had wind blowing over a flat ocean surface or a flat plain for hundreds of miles, would we still observe the wind to blow in the same sorts of gusts, or should I expect a more steady flow?


title: Finding the Steady State Charges

text: > > > > Here the problem states to find the steady state charges on the > condensers.<<<< ![enter image description > here](http://i.stack.imgur.com/5MwH2.png) According to me the charges on second at steady must be zero else there would be some current in the resistance in it's parallel(which can't be true because then $C$ too will have current in it and it will not be steady state) . Is this argument correct or there would be something else in this case? ## EDIT:additional question How does the charge initially that started accumulating on the $2C$ capacitor goes away at steady state? Initially the capacitor does not pose resistance so, maximum charges pass through the capacitor instead of the resistance.But finally the charge on it is zero,please explain what happens in mean time.


title: What is a good reynolds number for this process?

text: I’m trying to convince my boss that the mixers we are using are too much. I’m trying to prove that we are over-mixing our product. Our product is ink…just your basic ink found in your printer at home. We mix in a 23 inch diameter 50 gallon vessel using a 2.75 inch diameter axial flow impeller going at 1050 rpm with a 3 inch rotor stator 5 inches below it. The inks have a viscosity of around 3. What Reynolds number should be sufficient for this process or can you give me any further input? Any help would be GREATLY appreciated. Thanks.


title: How can I write a Gaussian state as a squeezed, displaced thermal state

text: I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\dagger(\alpha) \hat{S}^\dagger(\zeta) . \end{gather} Here, \begin{gather} \rho_{\bar{n}} = \int_{\mathbb{C}} P_{\bar{n}}(\alpha) |\alpha\rangle\langle\alpha|d\alpha \text{ with } P_{\bar{n}}(\alpha) = \frac{1}{\pi \bar{n}} e^{- |\alpha|^2 / \bar{n}} \end{gather} is a thermal state with occupation $\bar{n}$, \begin{gather} \hat{S}(\zeta) = e^{(\zeta^* \hat{a}^2 + \zeta \hat{a}^{\dagger 2}) / 2} \end{gather} is the squeezing operator, and \begin{gather} \hat{D}(\alpha) = e^{\alpha^* \hat{a} - \alpha \hat{a}^\dagger} . \end{gather} is the displacement operator. I prefer to use the convention $\hat{x} = (\hat{a} + \hat{a}^\dagger) / \sqrt{2}$ and $\hat{p} = (\hat{a} - \hat{a}^\dagger) / \sqrt{2} i$. I assume that the way to accomplish this is to derive the mean and variance of our Gaussian state $\rho$ and thereby determine $\zeta$ and $\alpha$. However, I have been unsuccessful in doing so. That is, given the mean and variance of our Gaussian state $\rho$, what are $\zeta$ and $\alpha$? On a side note, I was also wondering if there is a standard result for the commutator of $\hat{S}(\zeta)$ and $\hat{D}(\alpha)$?


title: Extent of coordinate freedom to set metric components along a spacetime path

text: If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ g_{\mu\nu}(x^\alpha) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\\ 0 &-1 & 0 & 0 \\\ 0 & 0 &-1 & 0 \\\ 0 & 0 & 0 &-1 \end{array} \right) $$ But our freedom is much greater than this. In curved spacetime, the equivalence principle suggests we can choose coordinates such that the metric is of that form for every point along a chosen time-like geodesic. And in flat spacetime, we can see an explicit example that it doesn't even need to be a geodesic, for the Rindler metric has that form on every point of a particular worldline with constant proper acceleration. I have a feeling this is possible for any time-like worldine. So my question is: Given a coordinate system and metric for a Lorentzian manifold and a time-like worldline on this spacetime, is it always possible to find a coordinate transformation such that _for every point on the world line_ the components of the metric in this coordinate system are just (1,-1,-1,-1) on the diagonal? I realize that even for simplified cases (say a geodesic on the Schwarzschild background), such a coordinate system could be incredibly complicated. So if someone creates an incredibly complicated explicit construction, please also show that the solution to the equations you setup exist with some kind of existence proof. This originally started from trying to find such a local coordinate chart for a free falling observer towards a blackhole, but realized I didn't know a good mathematical way to represent such coordinate freedom to get me started. Eventually I ended up pondering this current question. So even if you can't give a full answer, but can suggest some mathematical tools or where to read up on them, any help would be appreciated.


title: What makes background gauge field quantization work?

text: _[Again I am unsure as to whether this is appropriate for this site since this is again from standard graduate text-books and not research level. Please do not answer the question if you think that this will eventually get closed. Since otherwise the software doesn't allow me to delete questions once it has answers!]_   * A priori the technique of quantization using the background gauge field seems to be a "mere" redefinition of the good old Lagrangian whereby one "artificially" seems to double the number of variables by writing all of the fileds like say the fermions $\psi$, the gauge field $A_\alpha^\mu$, the ghosts $\omega_\alpha$ and $\omega_\alpha ^*$ as a sum of two fields one with a prime (the quantum fluctuation) and one unprimed (the classical background) - like replace every $A_\alpha ^\mu$ by $A_\alpha ^\mu + A_\alpha^{\mu'}$. Till this step nothing seems to distinguish the primed from the unprimed fields.    * Then one defines the a new set of covariant derivatives where the the connection is defined only by the classical background gauge field (the unprimed fields) only.  Then one rewrites the Lgrangian in terms of these new covariant derivatives and these doubled the number of fields. But I would think that the Lagrangian is still exactly the same and nothing conceptually has changed. Now two statements come in which seem to be somewhat "magical" and thats where it seems that the crux lies,   1. Firstly one says that it is possible to set the background fields such that all the unprimed gauge fields are constant and every other unprimed field is 0.    2. Then one says that the 1-loop contribution to the running of the coupling constant (and also the effective action?) is completely determined by looking at only that part of the orginal action which is quadratic in the primed fields ("quantum fluctuations"?) I would like to understand why the above two steps are consistent and correct and how to understand them. Also if there is some larger philosophy from which this comes out and if there are generalizations of this. {This technique and idea seems quite powerful since after one says the above it is almost routine calculation to get the beta-function of a $SU(N_c)$ Yang- Mill's theory with say $N_f$ flavours - and hence asymptotic freedom of QCD! - things which I would think are corner stones of physics! }


title: Graduate School for Theoretical Physics

text: First off, let me just say that I am unsure if this question is appropriate for this site, and if the community deems it necessary, the question should be closed. So right now I am a fourth year mathematics and physics major whose research interests lie in quantum field theory / quantum gravity. In particular, I would like to go into theoretical physics. However, I have found that most physicists do not achieve a level of rigor I would strive for in my work, and so I am reluctant to apply for graduate school in physics. On the other hand, it seems that in most mathematics departments in the U.S., there are at most a couple of mathematicians working in what you might all theoretical physics, and I would not feel comfortable applying to a graduate school banking on the fact I could work with a specific one or two persons. What possible routes are there that I could take if my interests lie in theoretical physics, but I wish to practice theoretical physics as a mathematician would? Furthermore, are there any (not necessarily American) universities with a joint mathematics-physics department (like the DAMTP at Cambridge), or, the next best thing, a mathematics and physics department that collaborate frequently? What about theoretical physicists working in quantum field theory/quantum gravity who don't shy away from rigor? Any and all input is greatly appreciated!


title: Connections and applications of SLE in physics

text: In probability theory, the Schramm–Loewner evolution, also known as stochastic Loewner evolution or SLE, is a conformally invariant stochastic process. It is a family of random planar curves that are generated by solving Loewner's differential equation with Brownian motion as input. The motivation for SLE was as a candidate for the scaling limit of "loop-erased random walk" (LERW) and, later, as a scaling limit of various other planar processes. My question is about connections of SLE with theoretical physics, applications of SLE to theoretical physics and also applications of (other) theoretical physics to SLE. I will be happy to learn about various examples of such connections/applications preferably described as much as possible in a non- technical way.


title: Space expansion looking like time dilation

text: Space looks like time depending on the motion of the observer so I was going to ask if space expansion was the same as the unfolding of time, but this was asked on physics.stackexchange before and the answer was that in GR time does not flow - there is no more a flow of time than there is a flow of space. So instead I'll ask: is space expansion the same as time dilation ?


title: 3-point correlation function for a massive scalar field

text: I am a little bit perplexed as to how to compute the three-point correlation function for a massive scalar field, I know that it should be equal to zero. I need to show that: $\lim_{T\rightarrow \infty (1-i\epsilon)} \int D\phi \phi(x_1) \phi(x_2) \phi(x_3) = 0$ How to show this? Edit: I forgot to mention without using the generating founction, $Z[J]$.


title: Why strong and weak forces are short range?

text: Why are the strong and weak nuclear forces short range? Are quarks confined or welded together? Why are elementary particles confined at short range? Or is color confinement color welding? * * * the quarks are glued together by strong nuclear force, (but not at high energy) it seems to me that its a kind of elementary welding process.