title: Why is the potential energy of a particle $mgh$ regardless of force?

text: Consider a particle raised to height $h$ by applying a constant force of $F$ vertically opposite to the direction of force of gravity on it. The potential energy of the particle is $$W = mgh$$ But, according to the definition of work, the work done on the particle is $$W' = Fh$$ According to conservation of energy, $W = W'$. But what if $F>mg$?


title: Different implementation of Feynman's clock

text: In equation 7 of the paper, Feynman's clock, a new variational principle, and parallel-in-time quantum dynamics, the clock register is different from Feynman's original proposal. According to this paper, for a four clock steps quantum circuit, the sequence of the state of the clock register will be $|00\rangle \to |01\rangle \to |10\rangle \to |11\rangle$. It infers that we will need a two qubit register. But according to Feynman's original proposal, it should be $|000\rangle \to |100\rangle \to |010\rangle \to |001\rangle$ i.e. a three qubit clock register is needed. I failed to understand if there is any specific reason behind this modification. Any help?


title: Dead time in data acquisition

text: I am creating a data acquisition software, based on Sparrow's Kmax. There I would like to add a feature that will show the system's dead time. I have already a text field that shows the real time and live. My question has to do, with the dead time. Is it right to say that the dead time is $\text{dead time}=\dfrac{\text{total time of recording events}}{\text{total time of acquisition}}=\dfrac{\text{live time}}{\text{real time}}$


title: Can Flow of time become still?

text: 1. According to theory of time dilation, flow of time slows down significantly at the speed of light.Is there any conditions practically or theoretically when flow of time is reduced to zero means it comes to still?   2. Whether time & space are infinite?


title: Simpler quantum "paradox" implying supraluminal connection

text: Executive summary: "Collapse of the wave function" is inherently supraluminal. I suggest an easier thought experiment to demonstrate the apparently supraluminal (or FTL) aspect of a quantum mechanical measurement, without spin and polarizers. A photon is sent through a beam splitter. The photon traverses both arms of the splitter, and is "everywhere" (or "nowhere") until it is detected at one end or the other. According to quantum mechanics, there are no hidden variables. The two arms of the splitter extend a significant distance (hundreds of meters) in opposite directions. A single photon is injected at a specified time. A short time later the photon is detected at one or the other of the two detectors, labeled A and B, each a random half of the trials. Clearly the two detectors do not have time to confer on the outcome. Suppose the photon was detected at A. How did detector B instantly "know" that the photon must not appear there?


title: Quantum Anomalies for Bosons

text: We know that there is Adler and Bell-Jackiw(ABJ) type anomalies for fermions. In some case, the ABJ anomaly affecs particle physics pheonomelogy, such as pion decays or kaon decays(in the case of pion, we still have a calculation on left/right chiral fermions running on the 1-loop triangle diagram). In some other case, there is 1+1D QED Schiwinger or axial anomaly for chiral fermions. The commutation of fermionic anomaly is usually done by, either a 1-loop Feynman diagram, or a Fujikawa path integral method. The above may be some examples of anomalies for fermions. **Is there any example of quantum anomalies for bosons (pure bosonic systems)?**


title: A small contradiction between periodic boundary condition and first Brilliouin zone

text: In condensed matter, one usually considers Bloch states inside the first Brilliouin zone, which, for 1d system with lattice constant $a$, is $-\pi/a<k<\pi/a$. But the basis of this, Bloch theorem, is based on periodic boundary condition, which implies $k=\frac{2\pi n}{Na}$ with $n=0,1,\cdots N-1$ and $N$ is the number of unit cells. This interval of $k$ is not the same as the interval given by first Brilliouin zone, which is fine if one just wants to consider the dispersion because dispersion is periodic. However, there may be some problems. For example, if we do a Fourier transformation numerically, we need to use $k=\frac{2\pi n}{Na}$ with $n=0,1,\cdots N-1$ instead of $-\pi/a<k<\pi/a$, otherwise we are not really using periodic boundary condition. Is this right? And is there any more subtle point on which one may easily make mistake?


title: How Would an Alcubierre Drive "ride" a Wave of Spacetime?

text: The colloquial explanation is that the spacetime in front of a ship contracts and the spacetime behind expands. I see how one could _think_ that this would bring you closer, but I don't see that it actually does. Let's look at the front of the bubble. So spacetime "contracts". If you look at a differential volume being run over, I would think it contracts to a minimum, _but then must expand again_ to pass under the ship. If the bubble is moving, but the ship inside is not accelerating (let's assume it's also motionless in its space), then spacetime must be "piling up" in a (compressed) region in front, because you can't throw away the extra spacetime. To paraphrase _A Wrinkle in Time_ 's analogy: think of some fabric and a bug that wants to get from where it is to somewhere else. You can fold the fabric and jump across the gap, but what I'm thinking of the Alcubierre drive doing is "scrunching up" the fabric: the bug still needs to walk over the same length of material, it's just that it _looks like_ less distance. Is all the above bogus? I don't really know what I'm doing with the physics here, as I have more of a mathematical background. I'm not seeing a way around this without introducing some discontinuity in space, which (I don't think) was the point. I also read somewhere that the expansion/contraction analogy could be misleading; is this why?


title: Moment of inertia of a system in different cases

text: ![enter image description here](http://i.stack.imgur.com/7NVNH.png) > A rod of mass $m$ and length $l$ is pivoted at one end to ceiling and free > to rotate in the vertical plane. A disc of radius $R$, which is less than > $l$, can be fixed at its other end in 2 ways : > > $1$ Disc is free to rotate about its centre > > $2$ Disc cannot rotate about its centre > > Compare moment of inertia of the 2 systems about the pivoted end. I have been thinking for several hours and can't think why they would be different. There is no torque on disc(about centre) and hence there should be no difference between 2 cases I found a similar question here but that uses Lagrangian which I have not studied (I'm in XII standard). Please explain why the moment of inertia will be different in 2 cases.


title: Classical disintegration of particles, Landau-Lifshitz series on Physics

text: i read Landau's book recently. In this book p.43 Landau says from (16.1) (16.2) can be write down $T_10$= $p_0^2$/2$m_1$=($M-m_1$)($E_i-E_1i-E_i'$)/$M$ For me, it is hard to understand the factor ($M-m_1$)/$M$. Can someone give me a hand to let me understanding master's idea? If you can give me a easy example, i will appreciate you. thx for your help.


title: Finding the constant of gravity in a game

text: Minecraft is a game of mining where one digs and finds ore (to put it simply). the world is composed of "blocks" 3 dimensional cubes that have different "pictures" on them, that make them appear like grass or rock or Steele or diamonds...ect. In minecraft there is also TNT which is a block just like all others except it explodes. That does not relate to the question however which is, How do i find the gravitational constant? I mostly need the formula, which my 10th grade physics class has not taught me yet so i can find the neccesary factors to solve it. Aside from just giving me the formula what ways do you think I could find G? Also, just for more info, most block dont fall but sand and gravel do so they could be key to this problem perhaps.


title: Electric Field Change Freezing Temperature of Water?

text: I assume that the reason water freezes is because as you decrease the temperature, the kinetic energy of the water molecules decreases and the dipole bonding potential eventually over comes the escape velocity of the molecules and they form a crystal structure. What would happen if you attempted to freeze water in the presence of a very powerful electric field? Will the water molecules align to the field and make it harder to form a solid or easier to form a solid?


title: Zinc is more electronegative than copper,than also why it happens?

text: Despite the fact that zinc is more electronegative than copper,why it will lose electrons more readily than copper in the voltaic cell due to sulphate ions.


title: Could quarks and leptons mix if they carried flavor charges?

text: If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?


title: Absolute value sign when normalizing a wave function

text: I have solved the following problem from Griffiths "Introduction to Quantum Mechanics". Consider the wavefunction: $\Psi (x,t) = A e^{-\lambda |x|} e^{-i\omega t} $ Normalize $\Psi$. Now, we want $ \int_{-\infty}^\infty |\Psi (x,t)|^2 dx = 1$ It is fairly straightforward, where the modulus is $|\Psi (x,t)|= r = A e^{-\lambda |x|}$. Therefore I square $r$ and integrate. I deal with the absolute value sign by multiplying by $2$ and integrating from 0 to $\infty$, while dropping the absolute value sign, to get: $ 2\int_0^\infty (A e^{-\lambda x})^2 dx$ This should give me a factor of $A^2$ which I can take outside the integral sign. However, instead of a simple $A^2$, the solution gives an $|A|^2$. I don't understand where the absolute value sign came from. After all, taking the above expression $r$ as being equal to $|\Psi(x,t)$|, the modulus has already been dealt with.


title: Calculating Average Velocity

text: I understand that the concept of an average of a data list means finding a certain value 'x', which ensures that the sum of the deviations of the numbers on the left of 'x' and on the right of 'x' yields the value 0. Assume we are given the values 2,6,9 and 12. I label these values from left to right in the sequence I have provided as: a,b,c and d respectively. The definition of finding these values' average (let's call this average value 'x') in mathematical form is:      $${(x - a) + (x - b) = -(x - c) - (x - d)}$$      So this equation can be reduced to:      $$\Large x = \frac{(a + b + c + d)}{4}$$      This will provide you with the value for the average of any number of values. Similarly there is a formula in physics that calculates the average velocity from a graph by dividing the sum of the initial and final velocities by 2.      $$\Large\frac{v_f + v_i}{2} = \bar{v}$$      There is also another formula which calculates the average velocity from a graph by calculating the slope of the line through two points on the graph. This is done by dividing the change in position by the change in time:      $$\Large \frac{x_f - x_i}{t_f - t_i} = \bar{v}$$      I can understand the third equation from the top of the page, as I can understand this calculation of average velocity in terms of what I explained in the first two equations from the top of the page. I am unable to reconcile the calculation of the average of velocities as seen in the last equation with the method as listed above in the first two equations. Can someone help me solve this dilemma?


title: Database of scattering amplitudes

text: I want to check whether my result for the invariant amplitude of the electron- electron scattering (to lowest order in $\alpha$; t+u channels) is correct or not. I can't find any reference that has the result explicitly. Can someone point out some kind of database of scattering amplitudes? **Edit:** for completeness I post my result (which might have an error) $$|\mathcal{M}|^2=\frac{2e^4}{u^2t^2}\left((s^2-8m^4)[(t+u)^2+u^4+t^4]+8m^2ut(4m^2-3s)\right)$$ **Update:** I managed to do the calculation using CompHEP+Mathematica. On CompHEP I selected the QED model and calculated the diagrams for $e^-e^- \rightarrow e^-e^-$ from which I get the two contributions to the process (t+u channels). Then I exported the symbolic computation of the squared diagrams to Mathematica code which gives $$ \frac{2e^4}{t^2 \left(-4 m^2+s+t\right)^2} (64 m^8+16 m^6 (t-6 s)+4 m^4 \left(13 s^2+3 s t+3 t^2\right)-4 m^2 \left(3 s^3+3 s^2 t+3 s t^2+2 t^3\right)+\left(s^2+s t+t^2\right)^2)$$ where the denominator is clearly $t^2u^2$ by using $s+t+u=4m^2$ but the rest is not so trivial to put in the same form as my equation for $|\mathcal{M}|^2$. So if I substitute the value of $u$ on my first equation I get something that doesn't look anything like the second equation. Therefore my first equation is wrong. Note: I compared the second equation with the expression for the differential cross section of the Møller scattering from a book and it is consistent.


title: Time evolution of Gaussian wave packet

text: I'm slightly confused as to answer this question, someone please help: Consider a free particle in one dimension, described by the initial wave function $$\psi(x,0) = e^{ip_{0}x/\hbar}e^{-x^{2}/2\Delta^{2}}(\pi\Delta^2)^{-1/4}.$$ Find the time-evolved wavefunctions $\psi(x,t)$. Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\frac{\sqrt{2mE}}{\hbar}$, and the time evolution of the Schrödinger equation gives $$\psi(x,t)=e^{-\frac{i}{\hbar}Ht}\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\psi(x,t)$. In short - how do I find the initial state in terms of the energy eigenfunctions $\psi_E(x,t)$ so that I can find the time evolved state wavefunction.


title: Does the Lorentz transformation not apply to light?

text: Since you would know that light always travels at the constant velocity with respect to all frame of reference ....according to relativity whenever we are traveling at speed of light our time with respect to relative rest observer would become stopped ..if. it means light travels with respect to all frame of reference at light speed, so it implies that light from the sun would never reach us ..but sadly it would reach us within 8 minutes ..How is that possible?


Here is a question about the canonical momentum that I had asked some days ago, but I still have one point that I am not understand. Considering a particle moves in a magnetic field with charge $q$ and mass $m$, its hamiltonian is $$H=\frac{\vec{P}^2}{2m}=\frac{(\vec{p}+q\vec{A})^2}{2m}$$ where $\vec{p}$ is the momentum of the particle, $\vec{A}$ is the vector potential of the magnetic field and $\vec{P}$ is the canonical momentum of the particle. I think, because of the expression of the hamiltonian, the canonical momentum $\vec{P}$ is a conserved quantity. But by the answer in the previous link, it seems that the canonical momentum is not conserved even in a simple example that a particle moves in a homogeneous magnetic field. I am confused about this question. Is the canonical momentum conserved when a particle moves in magnetic field?


title: Manufacturing Fresnel Lenses with Sound

text: I understand fresnel lenses are manufactured using CNC machines. I was wondering, if it would be possible to use sound to vibrate liquid silicon and then fast cool it into the standard form of a fresnel lens? This occurred to me when I noticed that ripples on a pond resemble the lenses in question, and that certain frequencies can cause liquids to vibrate. So, hypothetically, could it be done, and if so, would it be faster than machining?


title: Does string theory and preons exclude each other?

text: Does string theory contradict the theory of preons, especially the Harari- Shupe one?


title: Combining uncertainties - multiple measurements

text: I am trying to understand how to combine uncertainties when they are dependent and independent from each other. Using this formula : $$\delta z = \sqrt {\Biggl(\dfrac{\partial f}{\partial x} \delta x\Biggr)^2+\Biggl(\dfrac{\partial f}{\partial y} \delta y\Biggr)^2+2\Biggl(\dfrac{\partial f}{\partial x}\cdot \dfrac{\partial f}{\partial y}\Biggr)\text{cov}(x,y)}$$ Intuitively, if the covariance between the two is zero, the last term will disappear and the equation just becomes the square root of sums for combining uncertainty. My question is does $\delta z$ then need to be divided by 2 to get the final uncertainty value.(i.e. divide by $N$)


title: Estimating parameters to Falkenhagen equation for viscosity of solutes

text: In this answer to a question about viscosity of water in the presence of solutes, the Falkenhagen relation is given: > $$\frac{\eta_s}{\eta_0}=1+A\sqrt{c}$$ where $\eta_s$ is the solution > viscosity, $\eta_0$ the solvent viscosity, $A$ a constant that depends on > the electrostatic forces on the ions, and $c$ the concentration of the > solute. Now, in what order of magnitude would I assume A? Can one make general statements which group of solutes will be around which order of magnitude, like "expect A to be around $10^{-8}$ for single atom salts and $10^{-5}$ for long chained organics" (numbers plucked out of thin air) or similiar? p.s. The relation is quoted from a paywalled paper that I can't access.


title: From where (in space-time) does Hawking radiation originate?

text: According to my understanding of black hole thermodynamics, if I observe a black hole from a safe distance I should observe black body radiation emanating from it, with a temperature determined by its mass. The energy from this radiation comes from the black hole's mass itself. But where (in space-time) does the process of generating the Hawking radiation happen? It seems like it should be at the event horizon itself. However, here is a Penrose diagram of a black hole that forms from a collapsing star and then evaporates, which I've cribbed from this blog post by Luboš Motl. ![enter image description here](http://i.stack.imgur.com/Qtjrx.png) On the diagram I've drawn the world-lines of the star's surface (orange) and an observer who remains at a safe distance and eventually escapes to infinity (green). From the diagram I can see how the observer can see photons from the star itself and any other infalling matter (orange light rays). These will become red-shifted to undetectably low frequencies. But it seems as if any photons emitted from the horizon itself will only be observed at a single instant in time (blue light ray), which looks like it should be observed as the collapse of the black hole. So it seems that if I observe photons from a black hole at any time before its eventual evaporation, they must have originated from a time before the horizon actually formed. Is this correct? It seems very much at odds with the way the subject of Hawking radiation is usually summarised. How is it possible for the photons to be emitted _before_ the formation of the horizon? Does the energy- time uncertainty relation play a role here? One reason I'm interested in this is because I'd like to know whether Hawking radiation interacts with the matter that falls in to the black hole. There seem to be three possibilities:   1. Hawking radiation is generated in the space-time in between the black hole and the observer, and so doesn't interact (much, or at all) with the infalling matter;   2. Hawking radiation is generated near to the centre of the black hole, at a time before the horizon forms, and consequently it does interact with the matter.   3. The Hawking radiation is actually _emitted_ by the infalling matter, which for some reason is heated to a very high temperature as it approaches the event horizon.   4. (With thanks to pjcamp) you can't think of them as coming from a particular point, because they are quantum particles and never have a well-defined location. All these possibilities have quite different implications for how one should think of the information content of the radiation that eventually reaches the observer, so I'd like to know which (if any) is correct. The fourth possibility _does_ sound like the most reasonable, but if it's the case I'd like some more details, because what I'm really trying to understand is whether the Hawking photons can interact with the infalling matter or not. Ordinarily, if I observe a photon I expect it to have been emitted by something. If I observe one coming from a black hole, it doesn't seem unreasonable to try and trace its trajectory back in time and work out when and where it came from, and if I do that it will still _appear_ to have come from a time before the horizon formed, and in fact will _appear_ to be originating from surface of the original collapsing star, just before it passed the horizon. I understand the argument that the infalling matter will not experience any Hawking radiation, but I would like to understand whether, _from the perspective of the outside observer_ , the Hawking radiation appears to interact with the matter falling into the black hole. Clearly it _does_ interact with objects that are sufficiently far from the black hole, even if they're free-falling towards it, so if it doesn't interact with the surface of the collapsing star then where is the cutoff point, and why? In an answer below, Ron Maimon mentions "a microscopic point right where the black hole first formed," but in this diagram it looks like no radiation from that point will be observed until the hole's collapse. Everything I've read about black holes suggests that Hawking radiation is observed to emanate from the black hole continuously, and not just at the moment of collapse, so I'm still very confused about this. If the radiation _is_ all emitted from this point in space-time, it seems like it should interact very strongly with the in-falling matter: ![enter image description here](http://i.stack.imgur.com/OaYEI.png) In this case, crossing the event horizon would not be an uneventful non- experience after all, since it would involve colliding with a large proportion of the Hawking photons all at once. (Is this related to the idea of a "firewall" that I've heard about?) Finally, I realise it's possible that I'm just thinking about it in the wrong way. I know that the existence of photons is not observer-independent, so I guess it could just be that the question of where the photons originate is not a meaningful one. But even in this case I'd really like to have a clearer physical picture of the situation. If there is a good reason why "where and when do the photons originate?" is not the right question, I'd really appreciate an answer that explains it. (pjcamp's answer to the original version of the question goes some way towards this, but it doesn't address the time-related aspect of the current version, and it also doesn't give any insight as to whether the Hawking radiation interacts with the infalling matter, from the observer's perspective.) **Editorial note:** this question has been changed quite a bit since the version that pjcamp and Ron Maimon answered. The old version was based on a time-symmetry argument, which is correct for a Schwartzchild black hole, but not for a transient one that forms from a collapsing star and then evaporates. I think the exposition in terms of Penrose diagrams is much clearer.


title: How big (compared to our moon) would Saturn look from Titan's surface?

text: How large in the night sky would Saturn look from Titan's surface? I believe they are tidally locked.


title: Why is 1 AU the distance between the Sun and the Earth?

text: Why 1 AU is defined as the distance between the Sun and the Earth? (approximately if you like to be precise) > An astronomical unit (abbreviated as AU, au, a.u., or ua) is a unit of > length equal to about 149,597,870.7 kilometres (92,955,807.3 mi) or > approximately the mean Earth–Sun distance. Shouldn't astronomical units be defined within metric units (that is, $10^x$), so we can understand massive distances a little easier?


title: Is There Reddening on Objects Other than the Moon Upon Entry into Earth's Umbra?

text: With the recent lunar eclipse, for some reason this question came to me: The reason the Moon turns red is that the only appreciable sunlight hitting it when it's in Earth's umbra is refracted through Earth's atmosphere and shorter wavelengths are scattered out. I couldn't think of any reason why this shouldn't happen with, say, the space shuttle, ISS, or another satellite, but I've never seen any photograph nor discussion of it. It _should_ happen, right? And if so, have there been any photos of it?


title: Do gravitational waves cause time dilatation?

text: The effect of gravitational waves in transverse traceless gauge on matter is represented by the expansion and contraction of a ring of test particles in the direction of polarization of the wave. This result is obtained by choosing a gauge which in GR is a choice of a coordinates. Does that mean that choosing another coordinate system(moving observers) would experiment time dilatation plus the effects of streching and contracting? EDIT: When the weak field approximation is used and the metric is split into $g_{ab}=\eta_{ab}+h_{ab}$ the gauge transformations are no longer coordinate transformations and instead define equivalence classes of symmetric tensors in the flat Minkowski background. Then two Lorentzian frames are related by $h_{ab}=\Lambda_{a}^{\mu}\Lambda_{b}^{\nu}h_{\mu\nu}$, so uniform moving observers will measure time dilatations different from that of Minkoswki when the wave is passing. Is this correct? What about accelerated observers? Are there any extra effects in the full non- linear case?


title: Heuristic equation for Friction force between materials

text: I'm programming a game where different types of objects will be sliding over different types of terrains (Top-down in two dimensions). At my current level of physics education we are given the coefficient of friction between any pairing of surfaces, but that's not practical for the amount of objects and terrains this game (will) have. So my questions are,   * How do I find the force of friction between two objects, when I have their surface area and any other constants I need to add?    * How can I somewhat accurately simplify that to something on the order of $z=x*y+a*b$ ?    * I'd like to account for static vs kinetic friction, is there an easy way to do that? Edit: failing any established approximations, if I make up some abitrary "grippiness" constant for all of the surfaces, what's the best way to combine them into a coefficient of friction?


title: Are photons absorbed by expanding space?

text: Can a photon/EM-field-excitation redshifted by spatial expansion be completely dissipated? Does the energy reach a minimum value (Planck's constant) and continue on as normal? Does expansion also cause energy to be lost from travelling massive particles, as they too have a "wavy" aspect? From an informational point of view, where does the info go?


title: EMF & Magnetic force at the same time?

text: A conductor is placed in a magnetic field, current starts to flow, and the wire experiences the Lorentz force of $x$ Newtons. However, what confuses me is that $x$ is a **value** that is calculated from $F = IL \times B$. If $-EMF$ is induced due to the change in exterior magnetic field(due to the current now flowing in the conductor), shouldn't $I$ be reduced thus total $F$ (which is $x$) should be less than initially calculated? Also,if two wires have current flowing within them, as they initially "start" when current "beings" to flow, shouldn't there be $EMF$ induced? And what occurs first? The magnetic force or $EMF$ or do they occur simultaneously? I myself am confused about this notion, so this might sound a bit absurd please do state if it is unclear. I will review this and try to make it as clear as possible.


title: Omni-directional motion, resolving three or more vectors?

text: In robotics, there exists drive-trains that can move omni-directionally (that is in any direction). These come in many shapes and sizes, but most come in a three wheel or four wheel configuration, to keep things simple lets assume three wheels. ![Three wheel "kiwi" drive](http://i.stack.imgur.com/Pt7ZH.png) These drives are made possible by what is know as an omni-wheel, that is a wheel with casters along it which allows power to be applied in the direction the wheel spins, but not in the direction perpendicular to the wheel. ![enter image description here](http://i.stack.imgur.com/0MnOX.png) You can think of the "power" direction as having a lot of friction, and of course the part that rolls with the wheel, while the "slip" of the casters provides a very low friction, non-powered direction. I come from a programming background, not physics but I want to better understand the physics at work here. So let's look at a simplified picture: ![enter image description here](http://i.stack.imgur.com/K8Z0e.png) I know that friction and the center of mass are important factors. That's about it. So my question is, given a direction (green arrow), can a set of forces (red arrows) be calculated such that they equal the overall resulting force. IE, given blue and green can you calculate the various red forces?


title: Are O-planes not dynamical?

text: In Polchinski's book "String theory", he is saying "Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent fluctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identifications (8.8.3) become boundary conditions at the fixed plane, such that the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling." I want know how to check "For O-plane,the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling." Is there any papers or books or Web sites calculating it?


title: Operators explaination and momentum operator in QM

text: I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this. $$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, \mathrm d x $$ **Q1:** Is it ok if i interpret the equation above like quoted: > If i want to get expected value for position $\langle x \rangle$ i have to > multiply _probability_ $|\Psi|^2$ with a _position operator_ $\hat x$ which > in this case is equal to x. **Q2:** How can i derive a _momentum operator_ $\hat p$ and an expectated value $\langle p\rangle$ using the equation above?


How does a Fresnel rhomb work (half and quarter wave plate)? I am aware of birefringence, which creates a phase shift of $\Delta\phi=\dfrac{2\pi\Delta nL}{\lambda_0}$. But this doesn't explain how a plane polarised light shifts it's polarization angle after a half-waveplate or how a linearly polarised light becomes elliptical after a quarter-waveplate. For the half one let's assume that wikipedia has an answer http://en.wikipedia.org/wiki/Waveplate but why this electric field $E(\hat{f}+i\hat{s})e^{i(kz-\omega t})$ describes an elliptical polarised light? What I need is a mathematical description of how the retarders work...


title: Slit width for minimum spot size in electron slit diffraction if involving uncertainity principle

text: I don't believe the following is an accurate description of the physical but a homework problem to help understanding. > A beam of electron of energy 0.025 eV moving along x-direction, passes > through a slit of variable width w placed along y-axis. Estimate the value > of the width of the slit for which the spot size on a screen kept at a > distance of 0.5 m from slit would be minimum. I have the following idea: $$\theta \sim \frac{\lambda}{d}$$, where $\lambda$ is the wavelength, d is slit width, and $\theta$ represents diffraction spot size as an angle measure. For any $L$, the distance to the screen, this quantity goes minimum for $d$ tending to $\infty$ I have an intuition that the uncertainty principle shall be used to get a upper bound on $d$ as effects of Heisenberg's uncertainty exceed effects of diffraction. That way, I would have $$v_y = \Delta v_y = \frac{\hbar}{m_e \Delta y} = \frac{\hbar}{m_e d}$$ Taking $$v_x = \sqrt{\frac{2 K}{m_e}}$$, we have the spot size in angles as $$\frac{v_y}{v_x} = \frac{\hbar}{d \sqrt{2 K m_e}} = \frac{\lambda}{d}$$ That leaves me nowhere only when it starts seeming I have solved it. Please help.


title: What do the colors in false color images represent?

text: Every kid who first looks into a telescope is shocked to see that everything's black and white. The pretty colors, like those in this picture of the Sleeping Beauty Galaxy (M64), are missing: ![M64](http://i.stack.imgur.com/jCLh5.jpg) The person running the telescope will explain to them that the color they see in pictures like those isn't real. They're called "false color images", and the colors usually represent light outside the visual portion of the electromagnetic spectrum. Often you see images where a red color is used for infrared light and purple for ultraviolet. Is this also correct for false color astronomy images? What colors are used for other parts of the spectrum? Is there a standard, or does it vary by the telescope the image was taken from or some other factor?


title: How do you clean a dusty lens?

text: The lenses of my telescope and binoculars are dusty. What is the best way to clean them without damaging the optic coating?


title: Flat space limit of the Schwarzschild metric and Hawking temperature

text: The Schwarzschild metric reduces to the Minkowski metric in the limit of vanishing $M$, but the Hawking temperature which is proportional to $1/M$ diverges in the same limit. This would imply that flat spacetime has infinite rather than zero temperature. What am I missing? EDIT: This question is back up on the front page because I've posted an answer of my own \- let me know if I'm thinking about this right.


title: Interpreting the results

text: I have preformed the muon lifetime experiment at my uni's lab, and got the data. It's text file with 8190 numbers. My TDC unit was set so that the time gates were at 10 $\mu s$, and it has 8192 channels (first and last contained some kind of noise so they were not included in data analysis). Now, I made a program in python that will sum every n data, and make a new list that I can then plot in bar plots and from that calculate the mean muon lifetime, because it decays by exponential law. I also found that I can estimate the muon lifetime by using this formula: $$<t>=\frac{\sum\limits_{i=n_l}^{n_u}N_i t_i}{\sum\limits_{i=n_l}^{n_u}N_i},$$ where $N_i$ is the number of counts in the bin, $t_i$ is the 'time bin', and the bins go from $n_l$ (lower) to the $n_u$ (upper) bin. Using that as a guide, I found that for my data I get the best estimate if I set the bin count at 7 (7 bins), and if I put 10, I get higher result. Now if I set the bin number to 20, I get even higher number. But that sounds wrong. I mean, there must be some kind of catch. If I divide the time I get for 20 bins by two I get the correct result (~2.5 $\mu s$). The pictures are like this: ![enter image description here](http://i.stack.imgur.com/hZ1Z4.png) ![enter image description here](http://i.stack.imgur.com/QgNQJ.png) ![enter image description here](http://i.stack.imgur.com/eS8a0.png) So what am I doing wrong with interpreting this?


title: Ducted or open fluid flow, which is best for aerodynamics and lift

text: I'm designing a copter and trying to decide if the propellers should be ducted or open axial flow. I've read some theory on ducted and open air flow but I can't find any where that compares the two. I would prefer to use ducted over open for safety (ie put a safety guard round the propellers so they can't be run into with fingers etc). So are there any major advantages or considerations I should take into account, with ducted vs open axial flow before ploughing ignorantly ahead with one or the other type of flow?


title: Coefficients of friction for steel, aluminum & polymers

text: I am trying to analyse the problem of sticking an aluminum piece to a stainless steel piece only with pressure. The holding capacity of the system is determined by the static friction between the two pieces. To improve improve the holding capacity I would like to introduce a polymer layer between the two pieces. **My problem is that I am having trouble finding reliable values for the coefficient of static friction for:**   * aluminum-steel   * aluminum-rubber-steel   * aluminum-silicone-steel   * ... and other polymer metal combinations An example of the problems with the reliability of the numerical values are the steel-on-steel coefficient of static friction. The value is reported as follows:   * English language Wikipedia gives 0.8   * German language Wikipedia gives 0.15   * French language Wikipedia gives 0.2(0.15) Similarly the English language Wikipedia gives steel on aluminum as 0.61 which seems equally outlandish.


title: Static friction force on a block in a tunnel

text: Linked to this question: Comparing Static Frictions Suppose there is a cuboidal vertical tunnel, and a cubical block in it such that all surface of the block are in contact with the four walls of the cuboidal tunnel respectively. All walls of the tunnel are rough with different $\mu_s$'s. And the weight of the block is $m$ and it is a point mass such that it has only 3 translational degree of freedoms. The block is in mechanical-equilibrium. Now the block is being pulled vertically downwards by gravity, whose force is $mg$ and upwards by the four walls of the tunnel which apply static friction. How to measure static friction provided by four walls? Can we even measure?


1) The bent rod $ABCD$ rotates about the line $AD$ whit a constant angular velocity of $90 rad / s$. Determine the velocity and acceleration of the vertex $B$ when the rod is in the position shown in Figure. 2) Determine the velocity and acceleration of the vertex $B$ assuming now that the angular velocity is of $95rad / s$ decreasing the rate of $380rad / s^2$. ![enter image description here](http://i.stack.imgur.com/bTPHf.png) My attempt was to apply the formula $\vec{v}=\vec{r}\times \vec{\omega}$ but I think it is not valid if the rotation axis is not aligned with any of the axes in the $xyz$ space. Maybe I'm talking nonsense. I was thinking about making a change coordinatas to the rotation axis coincide with the $z$ axis. But it seems very counterproductive and does not seem to be the purpose of the exercise.


title: Effective mass in Spring-with-mass/mass system

text: Suppose you have a particle of mass $m$ fixed to a spring of mass $m_0$ that, in turn, is fixed to some wall. I'm trying to calculate the effective mass $m'$ that appears in the law of motion of the particle (suppose the system is isolated):$$m'\ddot x=-k(x-x_0).$$ I've read somewhere that this should be $m'=m+m_0/3$, but I'm getting a different result. My reasoning is as follows. Suppose the particle is at position $x$. The lenght of the spring is $x$ and we can suppose that its center of mass is at $x/2$. So the spring/particle center of mass is at: $$X= \dfrac {\frac{m_0}{2} + m}{m+m_0}x$$ Differentiating two times we get $$\ddot X = \dfrac {\frac{m_0}{2} + m}{m+m_0}\ddot x$$ Now, the only external force causing acceleration to the center of mass is the ceiling reaction to elastic force, that is exactly $-k(x-x_0)$. Thus:$$-k(x-x_0)=(m+m_0)\ddot X=(\frac{m_0}{2} + m)\ddot x $$and so I'm getting:$$m'=\frac{m_0}{2} + m.$$ Could you please point out where am I wrong (if I am) and possibly how is the result demonstrated?


title: Silicon-based life

text: My question may not be suitable here, because it's more of astrobiology. Life as we know it is carbon based.   1. Is life based on silicon possible?   2. What would the conditions for habitability for silicon based life be?   3. Can both carbon- and silicon-based life co-exist?   4. Is any silicon-based life found on Earth? Can any be grown in the laboratory?


title: Variance of daylight over short periods of time?

text: I've searched various publication archives but I'm a signal processing engineer and not an applied physicist, so I might not be using the right terms. While I can find various average values for the flux of the sky during the day (no clouds) (in both psychometric and radiometric units), what I'm really interested in is the spatial variance. With sufficiently-high dynamic range imaging equipment, what short-time-scale variance would I see for a small patch of sky? How homogeneous is the scattering? What is the spectra? I imagine many of these experiments may have been done long ago. Any pointers would be greatly appreciated.


title: Is building your own reflector telescope worth it?

text: I have always fancied building my own reflector telescope. I am wondering - For a certain budget, can you get better results with a home made rig? Or is this a folly, and really it would be better to buy a ready made unit?


title: What is the fundamental reason for noise?

text: I have read that noise is a result of there being "no such thing as a perfect one-way valve". That energy transfer is never perfectly one-way; there will always (at some level) a finite flow of energy from output to input. I understand this much, but what is the fundamental reason for this? Does it have its roots in thermodynamics? Edit: The idea of a "one-way valve" is merely a construction in the mind, it doesn't refer to an actual valve, like in a car engine or something. It's just a general way of describing 'what connects two systems'. Additionally, although I am looking for the _fundamental_ reason for noise, it may be easier to let people know that I am referring specifically to _measurement noise_.


title: Newton's cradle

text: Why, when one releases 2 balls in Newton's cradle, two balls on the opposite side bounce out at approximately the same speed as the 1st pair, rather than one ball at higher speed, or 3 balls at lower speed?


title: A simple question

text: This is a simple question but it seems to me that both the explanations are acceptable. Let say the least distance of least distinguished vision is 25 cm. An object is placed 12.5 cm in front of a plain mirror. Where should my eyes be placed such that I can see the image clearly? Actually, there are only two options:- A) My eye should be placed 12.5cm from the mirror. \----[Because object distance = 12.5 cm and image distance = 12.5cm .] B) My eye should be placed 25cm from the mirror. \----[Because the image is “printed” on the mirror. It is just like looking at a picture on the wall.] So, which is correct?


title: Information of things inside a black hole

text: **Can we get the information of things that are gone in black hole?**


title: Is the expectation value always an eigenvalue?

text: Does the expectation value of an observable must be equal to an eigenvalue of the corresponding operator? I already know that 0 is not an eigenvalue, but is there any other examples?


title: Boundary conditions in AdS/CFT

text: This question is in reference to this very famous paper of Witten.   * In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to take into account all the other fields and the complicated Lagrangian in the bulk (a Type IIB superstrings?)    * To construct the example in equation 4.1 (middle of page 6) why did the author choose half-BPS operators and is there a simple way to see that an example of ${\cal O}$ written down is a half-BPS operator? (..what are other such?..is there a classification?..)    * How generic is the argument in equation 4.8 (top of page 8) to get the RG flow equation? Or is this a special case which works here for some special reason?  With a change in the mass/renormalization scale/cut-off one usually asks for the connected n-point functions or the effective potential to be invariant - but here the author seems to want to have the scalar field's boundary asymptotics to be invariant - I found this renormalization condition very new and mysterious.   * I guess the most exciting analysis in this paper is the argument in the first paragraph on the top of page 9. Can someone help understand that?      * To start off how one know that the operators ${\cal O}_1$ and ${\cal O}_2'$ related to the boundary values of the two scalar fields are actually (super?)conformal primaries of the boundary (S?)CFT?      * I did not understand how one sees that the deformation as stated in equation 4.12 (and the line before it) preserves quantum conformal invariance.      * and the main point about the structure of equation 4.12 and the conformal invariance of the boundary being maintainable for $f \neq 0$..


title: Bicycle Wheel Drag in Slipstream

text: I was recently driving behind a car that had a bicycle mounted on a carrier over the rear bumper. The bicycle wheels were not bound so they were rotating in the slip-stream of the car. I wonder, the fact that the wheels are turning; does this theoretically increase or decrease the drag on the car?


title: Motivation behind studying the asymptotic structures

text: I am trying to explain to myself the motivation behind studying the asymptotic structures at null, time-like and space-like infinities (For the purposes of this post, I will stick to four dimensional Minkowski spacetime). I think I have the motivation behind the first two of them down, but not the third.   1. Null Infinity is interesting since radiation and fluxes of massless particles can reach it. One must understand these in order to answer questions such as energy loss due to gravitational radiation, or charge fluxes. Further, information can travel from the bulk to null infinity only, and is therefore an important thing to study in that context as well. Geroch's paper notes that > At null infinity, one's description involves "what the system is doing > through time", i.e. the dynamics of the system. I agree with this.   1. Time-like infinity corresponds to the starting and ending points of massive particles travelling along geodesics. I would presume that these would be of importance when describing "In" and "Out" states of a quantum theory.   2. Space-like infinity is a little harder for me to motivate. In Geroch's paper, he notes > At spatial infinity, "what the system does" is never even recorded. Rather, > one sees only the state of the system", once and for all. Here are my questions: >   1. Are the motivations I provided above for null and timelike infinity > correct? Is there anything more to add? > >   2. What does Geroch's comment mean? Surely, in general, a system will > evolve in time. What then describes the "state of a system"?? Is it the > initial conditions? What, precisely is the information about the system that > is stored at space-like infinity? > > I have often seen null and spacelike infinities being discussed thoroughly without much literature on time-like infinity. While the extensive study of null infinities does not surprise me, the study of spacelike infinity does. I therefore feel like I am missing something quite important here. EDIT: I just had a thought. It seems to me that both null and spatial infinity can be reached in a finite value of coordinate time (appropriately defined). On the other hand, time-like infinity can only be reached asymptotically. This implies that study of time-like infinity requires taking two limits $r \to \infty$ and $t \to \pm \infty$, whereas, study of null and spacelike infinity involve single limits like $r \to \infty,~v$ fixed and $r \to \infty,~t$ fixed respectively. Maybe this has something to do with my answer, but I'm not quite sure how.


title: Orthogonality of summed wave functions

text: **Problem.** I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to $\Psi_4=\Psi_1-\Psi_2$. **My naive solution.** From the premises, we know that $$\int_{-\infty}^\infty \Psi_1^*\Psi_1 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_2 dx=1$$ and $$\int_{-\infty}^\infty \Psi_1^*\Psi_2 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_1 dx=0$$ We also have $(z_1+z_2)^*=z_1^*+z_2^*$ $$\int_{-\infty}^\infty \Psi_3^*\Psi_4 dx = \int_{-\infty}^\infty (\Psi_1+\Psi_2)^*(\Psi_1-\Psi_2)dx \\\ =\int_{-\infty}^\infty(\Psi_1^*+\Psi_2^*)(\Psi_1-\Psi_2)dx\\\ =\int_{-\infty}^\infty(\Psi_1^*\Psi_1-\Psi_1^*\Psi_2+\Psi_2^*\Psi_1-\Psi_2^*\Psi_2)dx\\\ =1-0+0-1=0\,,$$ which is equivalent with what we wanted to prove. Is this a legitimate proof? Is there any simpler way to do this? I am afraid I still haven't grasped how wave functions behave mathematically, so I may have missed somethings very obvious here. _Edit_ : The solution manual somehow uses normalization factors for $\Psi_3$ and $\Psi_4$. How are these factors when you don't actually know the exact functions? And how does this relate to the concept of orthogonality?


title: Why do we use functional integration in QFT?

text: Recently I learned functional integral's formalism in QFT. I have realized that I don't understand why exactly do we introduce it. We have the expression for $S$-matrix, then we may rewrite it in through functional integral. But why do we use functional integration? I read that it helps to "conserve" gauge symmetries for massless fields and that it avoids infinite results after integration through integnal lines. But these are formal arguments. I don't understand it clearly. Can you add some information which can help me?


title: Understanding boundary conditions on slices of AdS5

text: This is a thing Iïve seen on many papers dealing with Warped Extra Dimensions, specifically on slices of AdS5. But the one where it appears more clearly is a lecture by Tony Gherghetta: http://arxiv.org/abs/hep-ph/0601213 Essentially what is done is that one builds a 5-dimensional theory in a slice of a Anti-De-Sitter space, the slice meaning that the fifth dimension has a small size and ends with a 4D brane at each extreme. The principle of least action applied to the 5D action leads us to two terms (eq. 7 in the paper above), one on the bulk (the vanishing of this one leads us to the bulk equation of motion) and one on the branes. The vanishing of this second term can be accomplished in two ways: 1) the variation of the field on the branes is zero 2) the term multiplying the variation is zero on the branes (let?s call it B) Now, all authors affirm that, in absence of extra terms on the brane, this two conditions lead to Neumann or Dirichlet conditions for the field on the branes (in the paper above this is said just after equation 10). My question is: why is that? For scalar fields it can be shown that B equals the derivative of the field, so I can see the Neumann condition there (am I wrong?). But the 1st condition says the variation of the field is zero not the field itself. Clearly I am misunderstanding something here...


title: Two masses connected to a spring separated are charged to q Coulombs

text: I am working through Yale's Physics 201, and on the first problem set there is a question that is puzzling me. From my reasoning, I think the answer they provided is incorrect hence why I am here. The question is as follows: > Two sphere of mass m and negligible size are connected to two identical > springs of force constant k as shown in Figure 1. The separation is a. When > charged to q Coulombs each, the separation doubles. (i) What is k in terms > of q,a,and ε0? (ii) Find k if the separation goes to a/2 when the charges > are ±q. (iii) In case (i) suppose the charge on the right is held fixed > while that on the left is displaced by a tiny amount x and released. Find > the resorting force F = −kex and the (angular) frequency ω of small > oscillations. (I call the effective force constant for oscillations as ke to > distinguish it from the k for the springs.) For part (ii) they provided the answer: $$ k = \frac{4q^2}{\pi\epsilon_0a^3} $$ Their reasoning is that because the separation was reduced to $$\frac{a}{2}$$ the length of the spring was in turn reduced by $$\frac{a}{4}$$ and that's how they arrived at the answer. However, I thought that of a system where the length of both springs is 1 and the separation is also 1, so the system is 3 units long. If I were to reduce the separation by half (like the question), each spring would increase its size by a fourth, so that it would be $$\frac{5a}{4}$$ equalling to 1.25 for one spring, 1.25 for the other, and .5 for the separation, giving the total 3 units. So my question is: is my reasoning incorrect? If so, how do I make sense of their answer? Edit: My answer is $$k = \frac{4q^2}{5\pi\epsilon_0 a^3}$$ Edit2: ![Figure 1](http://i.stack.imgur.com/uQe0w.jpg)


title: Why is charge not taken as a fundamental unit?

text: According to the definition of electric current, it appears to be a derived quantity. Charge on the other hand seems more fundamental than electric current. Then why is current taken as fundamental quantity instead of charge? Is it arbitrary choice? Is it because we can measure current more efficiently than charge or some other reason?


title: Is electron volt an alternate unit for electric potential?

text: My question is: Can an electron volt be considered an alternate unit for electric potential?


title: Function of air conditioner

text: What i always thought that air conditioners blow the cool air without knowing that they actually take the warm air from indoor and from outdoor. But whats the point of taking air from indoor and outdoor, Shouldn't they just be blowing the cool air inside and out air outside? How does it really make the difference? Also, Does all Air conditioners function the same way as in shown in this figure, even split A.Cs? ![enter link description here](http://i.stack.imgur.com/iXrvX.png) I've run out of **tags** for questions :\


title: Fluid Mechanics from a variational principle

text: It is posible to define a good variational principle to describe Fluid Mechanics? if so, wath is the correct tratement of the issue. I guess something like: $I=\int d^4x (\frac{1}{2}\rho v^2-P-\rho g x)$


title: Reference area of a parachute

text: I am trying to do an aerodynamic drag equation on a descending parachute (the round variety) and have no idea what the reference area on one would be. I know for a sphere, you can use radius*radius*PI to get the reference area. Is that the same for a parachute?


title: How and why will the Milky way collide with the Andromeda?

text: Hubble's law says that the universe is expanding.How come the milky way and the andromeda are on a collision course?How will they end up colliding with each other?


title: Problems with putting mass on Yang-Mills theory by hand

text: When Yang-Mills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. Later people invented spontaneous symmetry breaking and Higgs mechanism to give the gauge field mass. The Higgs particle is almost confirmed at LHC. My question is, since there is symmetry non-conservation (P/CP) in nature, why not simply put a mass on Yang-Mill's theory directly, got a non-abelian Proca action, say gauge symmetry breaking (although gauge may not be a symmetry actually Gauge symmetry is not a symmetry?; actually this point is more subtle, one can also take a Stueckelberg action then fixing the gauge, it leads to the same Lagrangian)? Is there any theoretical reason for not adding mass by hand on Yang-Mills theory? Or just because Higgs particle was found, it works, that's it. My friend has a guess, that gauge invariance implies BRST symmetry, which restricts the possible form of Lagrangian. If one did a renormalization flow transformation to lower energy scale without BRST symmetry, there will be other coupling in the effective Lagrangian at lower energy scale. I am not sure about this reasoning, because BRST symmetry can restrict the possible counter terms, can it also restrict the possible terms in the effective Lagrangian?


title: Is Pluto made out of ice?

text: Is Pluto made out of ice--as in frozen water? If so, wouldn't that provide evidence for alien life in the cosmos?


title: How does the Milkovic Two-Stage Mechanical Oscillator Pendulum-Lever System work?

text: See http://peswiki.com/index.php/Directory:Milkovic_Two- Stage_Mechanical_Oscillator The Two-Stage Mechanical Oscillator Pendulum-Lever System is very simple, yet very puzzling because it appears that more energy is going out (since it is able to pump water) than is put in (only lift one lift of the pendulum and an occasional push). Where is the apparent excess energy coming from? The reason I am asking this question is because all of the explanations I have seen claim that somehow Newton's Laws are violated, which I cannot believe.


title: Can Gases conduct Electricity?

text: Liquid electrolytes ionize and hence a current can pass through them. So if a gas can ionize, can it conduct electricity too? If so, what are a few such gases?


title: Microphones, Loudspeaker and their analogies to spring mass system

text: I have just started studying Microphones and Loudspeakers. I need a good text to refer which can explain their mechanical analogies with simplicity and basics too.


title: Why is a stack of N D-branes equivalent to an extremal black brane?

text: A stack of N D-branes can have open strings ending on them. There is a U(N) brane gauge field, and r adjoint Higgs fields, with r equal to the number of transverse spatial dimensions. The eigenvalues of the Higgs fields gives the (noncommutative) brane displacements. Surprisingly, this is equivalent to an extremal black brane with a p-form flux of N, with only closed strings. No gauge symmetry is manifest. Why are they equivalent? What is the exact nature of this correspondence? It's true we can have "closed" strings which stretch all the way to the horizon an infinite distance away. Their total energy is actually finite though, because of gravitational time dilation. However, no trace of any Chan- Paton factors appear anywhere.


title: Why do we think of light as a wave?

text: I've read that light travels in a straight line and has a wavelength of 400nm to 700nm. But I don't understand why does it have a wavelength and what creates its wavelength? I agree with the concept of sound which also has wavelength, thus called sound waves which are created by the vibrational movement in air. I'm not aware of calling light a wave. Does the light vibrate too? If so, then how?


title: Why is a white cloud formed in the hot water from the pot when salt is added?

text: Why is a white cloud is formed in the hot water from the pot when salt (NaCl) is added?


title: Black hole - white hole (collision)

text: A non-spinning, equally massive black hole and white hole experience a direct collision. What shall happen? What shall be the result of such a collision?


title: Farady's law and div B = 0

text: I'm reading a book on electromagnetism and I am a bit confused about some things in Maxwells equations. This is what I don't like about many physics books: they are very wordy, but at the end you don't know what is an experimental fact, what is a "theorem", what is an assumption and so on,... Anyway the questions are:   1. Is Faraday's law a consequence of Coulomb's and Biot-Savart's (or Ampere's) law?   2. Is divB = 0 consequence of Biot-Savart's law i.e. if we found the magnetic monopole, would that mean that Biot-Savart's law is not true? (I think so, but I am not 100% sure)


title: Would we be able to hear the sun if space were full of air?

text: I was wondering if the sun could be audible from earth in an air-filled space scenario. We can ignore all the other disastrous consequences! Thanks!


title: Maxwell-Faraday Equation and Electric Fields

text: I have a question regarding, as the title says, this equation: $\nabla \times \textbf{E}=-\frac{\partial \textbf{B}}{\partial{t}}$ So, the above equation says that the curl of an electric field is proportional to the rate of change of a magnetic field. However, since electric fields have a potential associated with them (i.e. voltage), they should be conservative and thus have 0 curl. Then this would mean that all magnetic fields are constant in time. This seems like a very strong result, though, and one that I never hear mentioned at that. So is my assumption above wrong? Or am I just misunderstanding the equation?


title: Nuclear fission difference in energy calculation?

text: If you have a uranium atom of radius $10^{-14}m$ that undergoes fission into two fragments each with 46 protons and radius $8\times10^{-15}m$. There are two ways you can calculate the electrostatic energy realised (according to my textbook):   1. Find the electric potential energy of the two fragments as if they where point charges at a distance of $2\times8\times10^{-15}m$ apart (i.e. find the potential energy one of these charges would have in the field of the other). This gives an answer of around 180 MeV.   2. Find the electrostatic energy of the original nucleus by imagining the nucleus was built layer by layer each of thickness $dr$ and then integrating over it to find the energy of the whole nucleus. If this is then repeated for the two fragments and the sum of the electrostatic energies of the fragments is taken from that of the original nucleus that is the electrostatic energy released. But this gives an energy of around 270MeV. Here is my question, why is there a difference between these two values (the 180MeV and the 270MeV)?


title: Example of Hamilton's Principle to Systems with Constraints (Goldstein)

text: I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints). I'd like to understand the example he gives. Here it comes: * * * Consider a smooth solid hemisphere of radius $a$ placed with its flat side down and fastened to the Earth whose gravitational acceleration is $g$. Place a small mass $M$ at the top of the hemisphere with an infinitesimal displacement off center so the mass slides down without friction. Choose coordinate $x, y, z$ centered on the base of the hemisphere with $z$ vertical and the $x$-$z$-plane containing the initial motion of the mass. Let $\theta$ be the angle from the top of the sphere to the mass. The Lagrangian is $L = \frac{1}{2}\cdot M \cdot (\dot x^2 + \dot y^2 + \dot z^2) - m\cdot g\cdot z$. The initial conditions allow us to ignore the $y$ coordinate, so the constraint equation is $a - \sqrt{x^2 + y^2} = 0$. Expressing the problem in terms of $r^2 = x^2+z^2$ and $x/z = \cos(\theta)$, Lagrange's equations are $$ Ma\dot\theta^2 - M g \cos(\theta) + \lambda = 0$$ and $$ Ma^2\ddot \theta + M g\, a \sin (\theta) = 0$$ Solve the second equation and then the first to obtain $$ \dot\theta^2 = -\frac{2g}{a}\cos(\theta) + \frac{2g}{a}$$ and $$ \lambda = M g (3\cos(\theta)-2)$$ So $\lambda$ is the magnitude of the force keeping the particle on the sphere and since $\lambda = 0$ when $\theta = \cos^{-1}(\tfrac{2}{3})$, the mass leaves the sphere at that angle. * * * I have the following questions:   1. Shouldn't it be $x/z = \tan \theta$?   2. Could it be that he's mixing up $r$ and $a$? My guess is that from "Lagrange's equations are" it should say $r$ instead of $a$. I get confused whether $a$ is a system parameter or a Lagrangian multiplier.   3. Could you give me a) an explanation or b) a good read on why setting $L' = L + \lambda\cdot f$ gives us an analogue of Hamilton's principle on constraint systems? I don't understand Goldstein's derivation. ($L$ is the original Lagrangian, $f$ is the constraint and $\lambda$ is the Lagrangian multiplier.)   4. Why can $\lambda$ be thought of as the constraint force? When I understand 3., I understand the example -- I reverse engineered the supposedly Lagrangian equations to see that $L'$ needs to be of form $$\frac{1}{2}M r^2 \dot\theta^2 - Mrg\cos(\theta) + \lambda \cdot f$$ with generalized coordinates $\theta$ and $r$. Then everything works out just fine.


title: Will overlapping two different beams of coherent light cause interference?

text: I have two laser beams with same polarization running parallel to each other. Will they interfere? If yes, then what are the conditions (perpendicular distance etc) and how can I observe the interference pattern?


title: $\exp(i\alpha\hat {\bf n}\cdot{\bf \sigma} )=\cos\alpha I+i(\hat {\bf n}\cdot{\bf \sigma})\sin\alpha$

text: ![I don't understand this proof](http://i.stack.imgur.com/zH5ft.png) Could anyone tell me $\hat {\bf n}\cdot{\bf \sigma}$ is defined in such way? In the book they have not defined what is $n_z,n_x,n_y$. It is from _Quantum Computing: From Linear Algebra to Physical Realization_ by Nakahara, First Chapter, page 23.


title: Regarding derivation of Probability Current

text: The question for the full derivation of Probability Conservation -> Probability Current was already asked here: Probability current. I apologize for not retyping it out, but it's already beautifully done in latex which I have yet to become proficient at. I can follow most of the derivation. However, I'm still confused as to why one can take the complex conjugate of the actual Schrodinger Equation in the second step. What allows one to do this? It's not as if it's a wave-function, which can have a complex conjugate because it describes a state. I guess my question refers to how one can take the conjugate of a (momentum) operator, because that appears to be what's happening. And if one is allowed to take complex conjugates of operators, why isn't this prevalent anytime we do use Schrodinger or other Equations?? Shouldn't they also be part of the full solution? Thanks in Advance!


title: Basics of osmosis. What about excluded volume?

text: I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given volume of solute in one of the compartments (say right). At first, the level of the right compartment will increase, to accommodate the extra volume of solute. Because of this, the concentration of water in the right compartment has decreased, and is no more at equilibrium with the left compartment. Thus a net flux of water from left to right occurs until equilibrium is reached. That's what I understand from what I read so far, but I have a problem with this. In this particular example, it is true to say that the water concentration on the right side decreased when the solute is introduced, but because of the excluded volume (from the solute), the water pressure should remain the same, and thus I would not expect a net flux. Unless there is other effect I don't consider? Thank you for your explanation.


title: Where can I learn the meaning of variables in a formula?

text: I am in a low level physics class that is taught in high school. We were given a couple formulas as seen on the board, but I don't know what the variables stand for (their names), and what units correspond to each variable. I have searched a few formulas on Google word for word (4), but nothing positive came up. Can any one help me layout what each Variable stands for and what units they correspond to? Or maybe a website? ![enter image description here](http://i.stack.imgur.com/zAnxN.jpg)


title: Does the speed of medium affect the path of light?

text: Let's say I shine a laser from a stationary medium into a moving medium (suppose the water is moving very quickly) perpendicular to the interface and back to a stationary medium like this: ![Scenarios](http://i.stack.imgur.com/SiL6P.png) (Note: left and right sides of the image are stationary mediums, center is a medium moving in the direction indicated by the arrow) Which of the above scenarios (A, B, C, or "I'm way off") correctly reflects the path the light will take (even if the translation is incredibly small)? **Edit:** To answer some good questions (and things I left out of the original question):   * The center (moving) medium is water   * The left and right medium (stationary) is air   * The first angle of incidence (on the left) between the air and the water is perpendicular


title: Why do some capacitors leak and other capacitors hardly leak at all?

text: I just found out that electrolytic capacitors hardly leak whereas non- electrolytic capacitors leak and at times, leak a lot. Questions: (a) what causes capacitors to leak? (b) why do electrolytic capacitors hardly leak when wired correctly whereas if they are reversed, they leak a fair amount? (c) why do non-electrolytic capacitors leak more than electrolytic capacitors?


title: When one monkey accelerates up a rope, what happens to the other?

text: The question is: Two equal weight monkeys each hang from the ends of a rope passing over a weightless, frictionless pulley. If one accelerates up the rope, what happens to the other? I'm going to assume here that if one monkey climbs, the other monkey goes up (I tried to imagine how it would look in real life). But in physical terms, why would this be? Because of the additional force exerted by the climbing monkey?


title: Optimization of a functional defined implicitly

text: I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's equations, but in this case $T$ is defined implicitly by an equation of type: $$f(t,\gamma,\dot\gamma,\ddot\gamma,T)=0$$ So, how can I do to find a path $\gamma$ minimizing $L$? * * * A possible way could maybe be the following (I will skip all formality _à la_ physicist): Assuming $f$ to be well behaved (i.e. a small variation in the path $\gamma$ leads to a small variation of the solution $T$), take a variation $\delta\gamma$ of the path. It will lead to a variation of $T$ in the following way: $$0=f(t,\gamma+\delta\gamma,\dot\gamma+\delta\dot\gamma,\ddot\gamma+\delta\ddot\gamma,T+\delta T)$$ And this (in first order in the variation) gives: $$0=f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma+f_T\delta T$$ where $f_x=\frac{\partial f}{\partial x}$ and $f=f(t,\gamma,\dot\gamma,\ddot\gamma,T)$. Assuming $f_T$ to be invertible (in particular this gives us unicity of the solution, by the implicit function theorem), then we find: $$\delta T = -f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)$$ And thus we obtain: $$\begin{array}{ll}\delta L=L[\gamma+\delta\gamma]-L[\gamma]&=\int F(T+\delta T)-F(T)dt\\\&=\int F_T(T)\delta Tdt\\\&=\int -F_T(T)f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)\end{array}$$ Integrating by parts (and omitting boundary terms) the term in the integral becomes: $$\left(-F_T(T)f_T^{-1}f+\frac{d}{dt}(F_T(T)f_T^{-1}f_\gamma)-\frac{d^2}{dt^2}(F_T(T)f_T^{-1}f_{\dot\gamma})+\frac{d^3}{dt^3}(F_T(T)f_T^{-1}f_{\ddot\gamma})\right)\delta\gamma$$ By variational principle, $\delta L=0$ if, and only if the term in the big brackets is $0$. Is this correct? And does this give us an effective way to find $\gamma$ (at least numerically)?


title: Ghosts on Torus worldsheet

text: Why after the expansion, only 0-mode of bc-ghost contributes to the 4-points ghost function on a torus worldsheet? $$<c(z_1)b(z_2)\tilde{c}(\bar{z}_3)\tilde{b}(\bar{z}_4)>_{T^2} ~\longrightarrow~ <c_0 b_0 \tilde{c}_0 \tilde{b}_0>$$


title: Good book on deriving approximate solutions from first principles?

text: I have always been excited by examples in which a few simple assumptions and first principles are used to characterize a system. For example, I did an exercise in which Crawford estimates a lake to be rectangular and then predicts the frequency of its waves through the harmonic motion of its center of mass. I'd love to come up with questions and simple answers about natural phenomena, but I am not very good at coming up with either the questions or the answers, and I'd like to see lots of examples. What are some of the books you like on the subject?


title: moment of inertia of a ring about an axis at 45° to the normal

text: I wanted to calculate the moment of inertia of a ring about an axis at 45° to its normal outside the plane of the ring . How do i calculate without using integration? I was thinking about using perpendicular axis theorom , but I need 2 axis in the plane of ring which I was unable to find . Please help


title: Unit determinant for relevant symmetry groups in QFT

text: When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. Moreover, it is invariant under Lorentz transformations which form the Lorentz group $O(1,3)$ with the proper transformations subgroup $SO(1,3)$ In this example $S$ stands for Special, meaning that these transformations are represented by matrices with determinant $1$. My questions are:   1. why is this requirement so important?    2. In particular, what would happen if we allow the determinant to be any real (or complex) number?


title: Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed

text: I am preparing myself for an exam and I got stuck with the following problem. If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I have to tackle this problem effectively? If I remove a point on the $2$-plane the standard solution is given by $A \propto \frac{1}{r^2}(-y,x,0)$, which is closed but not exact (due to the removed point at the origin). Can I lift this to $S^2$ and argue similarly?


title: What's the purpose of the speed of light in $E = mc^2$?

text: Is $E=mc^2$ not just $E=m$. What does the speed of light have to do with this other than to give it a really big number so it looks cool? What spectrum of light is used? How can we test the speed of light with out a stationary point to test it from?


title: woven appearance of water-streams by the side of the street

text: It's pretty rainy in Boston now, and there are streams of water by the sides of the street. Why do they have a 'woven' appearance? What's the formal name for this kind of fluid flow? With lots of diagonals criss-crossing each other. And is there a good way to simulate it?


title: Proving invariance of $ds^2$ from the invariance of the speed of light

text: I've started today the book of Landau and Lifshitz Vol.2: _The Classical Theory of Fields_ $\S 2$. They start from the invariance of the speed of light, express it as the fact that $$c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$$ is preserved when we change inertial frame, so they consider $$ds^2=c^2dt^2-dx^2-dy^2-dz^2,$$ and say > _We have observed that if $ds=0$ in one frame then $ds'=0$ in another frame. > But $ds$ and $ds'$ are infinitesimal of the same order. So it follows that > $ds^2$ and $ds'^2$ have to be proportional that is $ds^2=ads'^2$..._ and he goes on to prove that $a=1$. How to translate this argument in a rigorous one? I'm really interested in this, both to understand this deduction and also to be able in future to make similar ones.


title: Can the wavefunction of a system be reconstructed from density matrices of subsystems?

text: Suppose we have several interacting particles in pure state $\left|\psi\right>$. For each of particles we can extract density matrices via $$\rho_i(x_i,x_i^\prime)=\int \left<X_i,x_i\middle|\psi\right>\left<\psi\middle|X_i,x_i^\prime\right>dX_i,$$ where $x_i$ is coordinates of particle $i$ and $X_i$ is coordinates of the rest of system. Can the original wave function $\left<X\middle|\psi\right>$ of the whole system be restored from all the $\rho_i$?


title: Electric circuit. Slight problem with the sign on voltages

text: ![enter image description here](http://i.stack.imgur.com/UhwNG.jpg) Translation first!   The figure shows an electric circuit.   a) Use Kirchhoffs 2.law on the part-circuit abed and show that $I_3$ = 0,50A.   b) Explain that $I_2 + I_5 = 2,0A$. Use Kirchhoffs 2.law on the part-curcuit befc to find another equation with $I_2$ and $I_5$ as unknows. Solve the set of equations.   c) Find $I_1$ and $I_4$.   d) Find the resitance of R. a) Assume clockwise rotation of circuit. That makes 8v positive and 5,5v negative. 8-5,5=2,5v. This would be the V over the resitance at $I_3$. $\frac{2,5v}{5 ohm} = 0,5A$ I think this is correct. b) **My problem** is that I can't seem to find the equation properly. If I assume a clockwise current, that makes V at $I_3$ negative. And V at $I_2$ positive and V at $I_5$ negative.   $-2,5 = 5I_2 - 2,5I_5$   Solution in the book is $-2.5 = - 5I_2 + 2,5I_5 $ with $I_2 + I_5 =2$ It gives $I_2=1$ and $I_5 = 1$   My logic is wrong, but wrong how?


title: Will a precessing spinning wheel fall down if there is no friction at all?

text: If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down? ![Spinning wheel](http://i.stack.imgur.com/S3f2n.gif) I just asked another question about the same problem: Direction of torque precession of a spinning wheel Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions.


title: Direction of torque precession of a spinning wheel

text: Consider a spinning wheel, which is held up by one end of it's axis like this: ![Spinning wheel from hyperphysics](http://i.stack.imgur.com/bGYsu.gif) To explain why the change of angular momentum is directed as shown in the figure above, one usually says that there is an applied torque $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is in this case the radius vector from the point where the string is attached to the center of mass of the spinning wheel and $\vec{F}$ is just the gravitational force which acts on the center of mass of the wheel. So it seems to be clear that it must point into the direction as shown in the figure. However is it possible to see this without applying the Formula for the torque above. I am thinking of something like this: By applying the gravitational force one would move the free end of the wheel axis down a bit for a very small time, so we get the beginning of a rotation which angular velocity vector points into the same direction as the $\Delta L$ on the picture. **Thus** we get an additional angular momentum $\Delta \vec{L}$ into this direction, which changes the direction of the entire angular momentum vector $\vec{L}$. However I don't see why the last " **thus** " must be true, since in general the angular velocity doesn't have to point in the same direction as the angular momentum vector. This is true for the rotation about the wheel axis since this is a principal axis of the inertia tensor (which is needed to justify that the axis of rotation must turn, when the angular momentum vector turns), but the (beginning/infinitesimal) rotation above doesn't seem to be around a principal axis. So it would be great if someone could make the argumentation I tried more clear and more rigorous and clarify the problem described above.


title: Zero divergence of energy-momentum tensor and gravitational energy

text: Trying to teach myself general relativity and have just hit yet another confusion. I'm reading that in curved spacetime the energy-momentum tensor has zero divergence, ie $$\nabla_{\mu}T^{\mu\nu}=0.$$ But that this doesn't imply the total conservation of energy and momentum as there is an additional source of energy (the gravitational field) that isn't included in the EMT. If that's the case, and if the EMT doesn't describe the total energy of system, how is it valid to use the tensor to describe various systems. For example, $$T^{\mu\nu}=0$$ for the Schwarzschild solution, or assuming spacetime is a perfect fluid in cosmology? How are these assumptions valid if they don't include the energy contribution of the gravitational field? Seems a bit of an elephant in the room-type situation. Or is that energy so small it can be ignored? Thank you


title: Why is physical space equivalent to $\mathbb{R}^3$?

text: Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$? I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$ . The set of reals $\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. For reference see here http://en.wikipedia.org/wiki/Construction_of_the_real_numbers. Now my question is what is the reason behind our approximation of the physical space by this abstract set. Why is this approximation assumed to be most suitable or good approximation?


title: Can we have discontinuous wavefunctions in the Infinite Square well?

text: The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an infinite linear combination of those energy eigenstates. But since linear superpositions of solutions are solutions, does this mean that this square function is an allowable wavefunction that a particle can be in in this problem? The issue I'm having is the fact that the Schrödinger equation involves the derivative of the wavefunction, but this function has none, so how can it be a solution to the equation?


title: Negative probability and spin-0 scalar field in Klein-Gordon equation

text: Klein-Gordon equation in quantum field theory is known to suffer from the possibility of negative probability. So, the question is, despite this, Klein- Gordon describes spin-zero field. So, how can negative probability and scalar field co-exist?


title: Time of a ball going up and down with air resistance

text: A ball is shot directly upward, and then it comes back to the place where it was shot. Suppose we have air resistance. Suppose $t_1$ is the time period from the moment that the ball was shot to the moment that it reached its highest altitude, and $t_2$ is the time period from the moment it reached its highest altitude to the moment it reached its original position. Is $t_1=t_2$? Why or why not?


title: Why is there no oxidizer in space?

text: Just as a thought experiment. One factor in the economies of space exploration is that of fuel. This may be split as   * MSL - Earth Orbit    * Earth Orbit to Inner Solar System    * Beyond. In each of these cases, the rocket demands an oxidizer to be loaded at the point of launch as neither oxygen, nor oxidizers exist in free space. Why does space not have free oxygen/oxidizers? p.s. According to wikipedia Oxygen is the third most abundant element in the known Universe ...


title: State normalization in Dirac's formulation of quantum mechanics

text: > Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then > we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} > e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our states are normalized by > $\langle q' | q \rangle = δ(q' − q)$ with $δ$ the Dirac delta function. > ("Dirac's formulation, in the book "Quantum Field Theory" by A. Zee) 1) Why must $\langle q' | q \rangle = δ(q' − q)$ hold? 2) Is this related to $\int_{-\infty}^{\infty}\Psi^*\Psi dx = 1$? Also, > Now use the fact that $|q \rangle$ forms a complete set of states so that > $\int dq |q \rangle \langle q| = 1$. 3) Does this even make sense? 4) Shouldn't integral be $\int_{-\infty}^{\infty}$?


title: Schrödinger equation with complex potential

text: In 1 dimension what is the solution of the Schrödinger equation with potential $$ V(x) = V_r + i V_i $$ Potentials are constant.


title: Lower bounds on spectral gaps of ferromagnetic spin-1/2 XXX Hamiltonians?

text: **Question.** _Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, ortree-graphs?_ I'm interested in frustration-free Hamiltonians such as the XXX ferromagnet on spin-1/2 particles, _i.e._ $$ H \;=\; -\sum_{j=1}^{n} \Bigl( \sigma^{(x)}_j \sigma^{(x)}_{j+1} \;+\; \sigma^{(y)}_j \sigma^{(y)}_{j+1} \;+\; \sigma^{(z)}_j \sigma^{(z)}_{j+1} \Bigr) $$ acting on $n$ spins (where we identify the spins labelled $1$ and $n+1$ to obtain periodic boundary conditions). The ground-state manifold here is the entire symmetric subspace on $n$ spin-1/2 particles, which has dimension $n+1$. In the usual manner of speaking, it has no "eigenvalue gap". What I am interested in, however, is the gap between the two lowest distinct eigenvalues $E_0$ and $E_1$ for such systems. **However** : I'm interested in such systems where the pair-wise interaction may occur over an arbitrary undirected graph -- such as a tree graph -- and not just the line. In my searches, I have only found articles addressing the problem for 1d spin chains or for rectangular lattices in 2,3 dimensions. The references I have found are:   * Koma, Nachtergaele (1997). _The spectral gap of the ferromagnetic XXZ chain_.   Lett. Math. Phys., **40** , pp. 1-16\. [arXiv:cond-mat/9512120]   * Starr (2001, PhD thesis).   _Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system_.   [arxiv.org:math-ph/0106024] where the first reference formally shows that for spin-chains of length $n$, the spectral gap is precisely $$(E_1 - E_0) \;=\; 1 - \cos(\pi/n) \;>\; \tfrac{\pi^2\\!\\!}{2} n^{-2} -\, \tfrac{\pi^4\\!\\!}{24} n^{-4}\,,$$ and the second reference discusses that result and others on rectangular lattices for spin-1/2 particles (for instance in its preliminary chapters), as well as for higher spins. These references represent the simplest cases of the problem that I would like to consider. To consider arbitrary graphs, it would be sufficient for me to know what results apply for trees, which are a reasonable generalization from spin chains topologically, and which I can use to obtain a lower bound for all graphs. However, I'm only interested in spin-1/2 systems.


title: What is a resonating valence bond (RVB) state?

text: There's something known as a "resonating valence bond" (RVB) state, which plays a role in at least some attempts to understand physics of high-$T_c$ superconductors. This, roughly, involves a state that's in a superposition (hence the "resonating" part of the name, if I understand correctly) of different ways to pair electrons into strongly-bonded spin singlets. My question is: what is a more precise definition of this sort of state? What's the underlying physics, when does it arise, and why is it interesting? Points an answer might address: is there a simple toy model for which this is the ground state, that sheds light on what sort of system it could arise in? Is there an interesting continuum limit, in which we can characterize this state in a more field-theoretic language? Are there particular kinds of instabilities such a state tends to be subject to? I think I know where I would start digging if I wanted to really understand this for myself, but mostly I'm asking it to probe the community and see what kind of expertise might be lurking here, since there haven't been so many condensed matter questions.


title: What is the experiment where subatomic particles appear to foresee the future?

text: I've seen a documentary, whose name I don't remember but I'm curious because it suggests that subatomic particles are able to "foresee the future". I'll try to describe it here: Some particles are shot through two slits in a plate, if the particles are not interfered by trying to measure its trajectory, they form an interference pattern on a wall where the beam is being projected, otherwise they just hit the wall in a single point according to the measured trajectory. The following image depicts what is described above: ![enter image description here](http://i.stack.imgur.com/E27zL.png) The dotted red lines are where waves are canceled, blue ones are where waves overlap. Then, they modify the experiment, so that they sort of turn on the detector after the particles have crossed the slit, where particles have allready being "forced" to behave as waves or single particles, to see if they change its behaviour after the measurement, just to find that the particles were measured as if they always knew that they would be measured, and they crossed the slit as particles and not waves. **Update:** This is the link to the documentary, it's titled "Microscopic Universe", is part of this TV series. They describe the experiment about 17 minutes in.


title: How do I define time-ordering for Wightman functions?

text: This is a follow-up question to What are Wightman fields/functions Ok, so based on my reading, the field operators of a theory are understood to be operator-valued distributions, that is, to be integrated over a smooth function: e.g. $\hat\phi(f)≡\int dx\, f(x) \hat\phi(x)$. But if the smooth function has support over such a large region in space-time that if I try to compute time-ordered products, I run into some ambiguity. How do I define the time-ordering symbol for Wightman functions?


title: Could a strong AC electric field adversely affect the capacity of a lithium battery?

text: I run some battery powered test equipment in close proximity (~10cm) to a conductor carrying 25kV (50Hz AC). I find the battery life is much shorter than expected, having checked the current consumption of the equipment. Could the electric field adversely affect battery operation? The battery is a primary lithium thionyl chloride 3.6V D-cell, rated at 17Ah.


title: A question to clarify the use of divergent series in calculating the casimir effect

text: Some time ago I posted a question here on this forum. I would like to ask some questions regarding the way the energy per unit area between metallic plates is calculated. The full calculation is on wikipedia. At some point in the calculation on the relevant wikipedia page (see the link above), we have the equation: $$\frac{ \langle E \rangle }{ A} = - \frac{ \hbar c \pi^2 }{6a^3}\cdot\zeta(-3) . $$ In the next step, it is written rather casually that $\zeta(-3) = - \frac{1}{120} \qquad (*) $. This is true when considering the analytic continuation of the riemann zeta function or the Ramanujan Summation method. Therefore, it is concluded, that $$\frac{ \langle E \rangle }{A} = - \frac{ \hbar c \pi^2}{720 a^3} . $$ I am wondering under which circumstances people decided to assume the $(*)$-marked equation is 'true'. I can think of a couple of scenarios:   1. The formula for $\frac{ \langle E \rangle }{A} $ was already derived by means of another method which did not require the use of (regularised) divergent sums. Therefore, physicists could infer that $\zeta(-3)$ _had_ to be equal to $ - \frac{1}{120} $, making the derivation of the formula by means of this method, which does use divergent series, correct.    2. The exact formula for $\frac{ \langle E \rangle }{A} $ was not already known. Physicist did have some data points that roughly showed them how the formula should look. Therefore, they tried some different constants for $\zeta(-3)$. At some point they guessed $\zeta(-3) = - \frac{1}{120} $, which yielded a formula that coincided with the known data points. They might have already known that $\zeta(-3) = - \frac{1}{120} $ by means of zeta function regularisation, making it easier to use this equation as a "guess" to find a suitable formula for $\frac{ \langle E \rangle }{A} $ .    3. Some other scenario.  Which scenario roughly describes how the formula for $\frac{ \langle E \rangle }{A} $ came into existence? If it was scenario 1, which other method did physicists formerly employ to derive the formula? If it was scenario 3, how _did_ this whole process unfold? Thanks a lot, Max


title: If energy is just a conserved quantity how can it have flux and density?

text: As I'm very well convinced for now, the better understanding of energy we have today in Physics is that of a conserved quantity, i.e, a numerical valued associated with a system that remains the same whatever process the system undergoes. This is all fine, but then we have areas in Physics where we talk of flux and density of energy in a way that seems to be like some really material thing, like a fluid for example. For instance, in electromagnetism one consider the electric energy density given by the $3$-form: $$dU = \dfrac{1}{2}\epsilon_0 |E|^2 dV$$ So how can a "simply mathematical object" have a density like material objects as mass and charge? I know how to derive this expression, but what is the underlying idea here? Also, we usually talk about energy flux, like when we deal with the energy- momentum tensor. In that case we are again in a certain sense imagining that energy in a certain sense flows like fluids and this seems a little strange if we consider energy simply a conserved quantity. How those ideas are to be interpreted?


title: How can I tell that circular motion is a solution for a particle confined to the surface of a cone?

text: I'm working on a problem where a particle of mass $m$ is confined to the surface of an inverted half cone (and is circling downwards due to gravity), with the cone's half angle $\alpha$. I chose to use cylindrical coordinates $(z,\phi,\rho)$ and I used the Lagrangian to solve this problem. After going through some math, I find the equation of motion for $z$, from which I can write that $$\ddot{z}\sec(\alpha) - \frac{p^2_{\phi}}{m^2z^3tan^2(\alpha)}+ g = 0$$ Here, $p_{\phi}$ is the angular momentum, which is conserved. Only $z$ depends on time, the other expressions are all constants. At this point, I've been told that it 'can be seen' that one solution to this is given by a circular motion at constant height $z_c$. I am then asked to impose a small perturbation $z = z_c + \eta$, and (keeping only first order terms in $\eta$) find the period with which $z$ will oscillate around $z_c$. Now I am pretty clueless how to do this. First of all, how can you see that there is a circular motion at constant height $z_c$? I mean, I can plug in $z = z_c$ and solve for it, but then I don't see how to find the period of something like that with the small perturbation. All the perturbation does is add some terms, but I don't see how they are time dependent and I certainly don't see how to extract a period from it. Could someone perhaps suggest a 'plan of attack'? If I do simply plug in $z = z_c$ I find that $$z_c=\left(\frac{p^2_{\phi}}{gm^2\tan^2(\alpha)}\right)^{\frac{1}{3}}$$ which at least has the right units. Moreover, plugging $z = z_c + \eta$ into the first equation and keeping only first order terms of $\eta$, I find that $$z = \frac{2z_c}{3} - \frac{p^2_{\phi}}{3z_c^2gm^2\tan^2(\alpha)}$$ But I don't see any period in that.


title: Difference between positron and electron scattering in Coulomb field

text: In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors) $$ S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(p_{f}, s_{f})\gamma_{0}u (p_{i}, s_{i}) \int \frac{d^{4}xe^{i(p_{f} - p_{i})x}}{|\mathbf x|}, $$ where $f, i$-indices mark correspondingly final and initial waves, $s_{i, f}$ marks the polarization. What is the difference for the case of positron scattering (the amplitude of scattering has different expression)?