title: Why is color conserved in QCD?

text: According to Noether's theorem, global invariance under $SU(N)$ leads to $N^2-1$ conserved charges. But in QCD gluons are not conserved; color is. There are N colors, not $N^2-1$ colors. Am I misunderstanding Noether's theorem? ~~My only guess (which is not made clear anywhere I can find) is that there are $N_R^2-1$ conserved charges, where $N_R$ is the dimension of the representation of SU(N) that the matter field transforms under.~~ **EDIT:** I think I can answer my own question by saying that eight _color combinations_ are conserved which _do_ correspond to the colors carried by gluons. Gluon number is obviously not conserved, but the color currents of each gluon type are conserved. An arbitrary number of gluons can be created from the vacuum without violating color conservation because color pair production {$r,\bar{r}$}, {$g,\bar{g}$}, {$b,\bar{b}$} does not affect the overal color flow. Lubos or anyone please correct me if this is wrong, or if you want to clean it up and incorporate it into your answer Lubos I will accept your answer.


title: Does an object's color change its rate of cooling?

text: The motivation for this question comes directly from this thread. The proposition is that the color of something changes how fast it cools (note: specifically the rate of cooling, not taking into account the change in heat absorption). For example: **would adding black food coloring to an otherwise light-colored fluid, make it cool faster?** The proposed explanation for such an effect is that the fluid becomes a 'better' black body. Is this true? Why or why not? My current thinking:   The 'color', per se, has no effect on the rate of cooling. For two bodies at the same temperature, any difference in color is due to it's reflection properties, not its emission properties - which will be identical, because they have the same temperature.


title: Photon detection rate for pure / mixed states coming from single mode point source

text: Let the pure states be in superposition of horizontal and vertically polarized basis states. They are arriving at the point detector one at a time. So, a pure state is $|\Psi\rangle = \alpha|V\rangle + \beta|H\rangle$. The formula for counting rate is as follows, $$G({\bf r},t) \propto \langle \Psi_F^{(0)}|\hat{E}^{(-)}({\bf r}, t) \hat{E}^{(+)}({\bf r}, t) | \Psi_F^{(0)} \rangle $$. Let us assume it is a single mode field. So, $$\hat{E}^{(+)} = iE_V \hat{a}e^{-i\omega t}$$ and $$\hat{E}^{(-)} = -iE_V \hat{a}^\dagger e^{i\omega t}$$. So,$$G({\bf r},t) \propto \langle \Psi_F^{(0)}|(-iE_V \hat{a}^\dagger e^{i\omega t}) (iE_V \hat{a}e^{-i\omega t}) | \Psi_F^{(0)} \rangle $$ $$ G({\bf r},t) \propto E_V^2 \langle \Psi_F^{(0)}| \hat{a}^\dagger \hat{a} | \Psi_F^{(0)} \rangle $$ $$ G({\bf r},t) \propto E_V^2 \langle \Psi_F^{(0)}| \hat{a}^\dagger \hat{a} | \Psi_F^{(0)} \rangle $$ $$ G({\bf r},t) \propto E_V^2 \langle \Psi_F^{(0)}| \hat{n} | \Psi_F^{(0)} \rangle $$ $$ G({\bf r},t) \propto E_V^2 \langle \Psi_F^{(0)}| \begin{pmatrix} 1 & 0 \\\ 0 & 0 \\\ \end{pmatrix} | \Psi_F^{(0)} \rangle $$. Let's consider the other case now. If the incoming states are mixed states, let's assume the density matrix to be $$\rho = \frac{1}{2} \left( \begin{pmatrix} 1 \\\ 0 \\\ \end{pmatrix} . \begin{pmatrix} 1 & 0\\\ \end{pmatrix} + \begin{pmatrix} 0 \\\ 1 \\\ \end{pmatrix} . \begin{pmatrix} 0 & 1\\\ \end{pmatrix} \right) = \begin{pmatrix} \frac{1}{2} & 0 \\\ 0 & \frac{1}{2} \\\ \end{pmatrix}$$. Here, $|H\rangle = \begin{pmatrix} 1 \\\ 0 \\\ \end{pmatrix}$ and $|V\rangle = \begin{pmatrix} 0 \\\ 1 \\\ \end{pmatrix}$. So, the formula for photon detection rate is as follows. $$G({\bf r},t) \propto Tr[\rho \hat{E}^{(-)} \hat{E}^{(+)}]$$ $$G({\bf r},t) \propto E_V^2 Tr[\rho \begin{pmatrix} 1 & 0 \\\ 0 & 0 \\\ \end{pmatrix}]$$ $$G({\bf r},t) \propto E_V^2 Tr[\begin{pmatrix} \frac{1}{2} & 0 \\\ 0 & \frac{1}{2} \\\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\\ 0 & 0 \\\ \end{pmatrix}]$$ $$G({\bf r},t) \propto E_V^2 Tr[\begin{pmatrix} \frac{1}{2} & 0 \\\ 0 & 0 \\\ \end{pmatrix} ]$$ $$G({\bf r},t) \propto \frac{1}{2} E_V^2 $$. The observable for both pure and mixed state look same. Does it mean that the physical design of the detection system will be same for both pure and mixed state?


title: Time evolution of states - Is total energy constant or not?

text: Suppose the state of the particle is given as follows: $$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$ Where the wavefunctions are: $\phi_0 = \left( \frac{1}{a^2\pi} \right)^{\frac{1}{4}} e^{-\frac{x^2}{2a^2}} $ and $\phi_1 = \left( \frac{4}{a^6\pi} \right)^{\frac{1}{4}} x \space e^{-\frac{x^2}{2a^2}} $. I have found the expected position and momentum with time: $$\langle \psi_{(t)}|x|\psi_{(t)}\rangle = \frac{a}{\sqrt 2} cos (\omega t)$$ $$\langle \psi_{(t)}|\hat p |\psi_{(t)}\rangle = \frac{i\hbar}{a \sqrt 2} cos (\omega t) $$ Then I try to compute the rate of change of energy: $$ \frac{d}{dt} \left[ \langle \psi_{(t)}|\hat p |\psi_{(t)}\rangle + m\omega^2 \langle \psi_{(t)}|x|\psi_{(t)}\rangle \right] = - \left( \frac{i\hbar}{a\sqrt 2} + \frac{a}{\sqrt 2} m\omega^3 \right) sin (\omega t) $$ This means that the total energy fluctuates with time, which is strange as shouldn't the total energy be conserved?


title: Is this how a Universe could be created from nothing?

text: So this is a rather simplistic understanding but I want to know if it's fundamentally correct: We have nothing which we'll represent as $0$. Then we need a Universe which we can represent as any number of parts that sum to $0$ $$ 0 = -1 + 1 $$ In this way we can take $0$ and make a "Universe" that is now equal parts $1$ and $-1$. Nothing in this Universe could be created or destroyed because then it wouldn't equal 0 and yet we've now taken $0$ and made "something" of it by expressing it as different parts. Is this a valid, albeit simplistic understanding? Or am I missing something here? This would even mean that an infinite number of Universes could exist, $0$ would just be expressed in different ways, or repeatedly in the same way.   1. $0 = -1 + 1$   2. $ 0 = -2 + 2$


title: Quantum gate: Phase shift

text: I dont undestand how to apply a phase shift gate to a qubit. By example how to map $|\psi_0\rangle = \cos (30^\circ) |0\rangle + \sin (30^\circ) |1\rangle$ to $|\psi_1\rangle = \cos(-15^\circ) |0\rangle + \sin(-15^\circ) |1\rangle$


title: What is the path integral exactly?

text: I asked a question here about path integrals and QFT. I just want to confirm something. Is the path integral in quantum field theory a mathematical tool only? I thought the path integral meant that particles literally do take all paths as stated by Victor Stenger and Stephen Hawking in their interpretation of what happens in reality? If particles are field excitations, as QFT says, does that mean Stenger and Hawking's interpretation is right and in reality particles do take all paths (sum over histories)?


title: Irreversible expansion and time reversal symmetry

text: Suppose there are N non-interacting classical particles in a box, so their state can be described by the $\\{\mathbf{x}_i(t), \mathbf{p}_i(t) \\}$. If the particles are initially at the left of the box, they can eventually occupy the whole box according to the Newtons law. In this case, we call the gas expand and this process is irreversible. Nevertheless, Newtons laws also have time-reversal symmetry, so we should be able to construct an initial condition such that particles occupy the whole box (i.e. not all particles in the left chamber) will all move to the left chamber. Here are the simple questions:   1. How to select the initial condition $\\{\mathbf{x}_i(0), \mathbf{p}_i(0) \\}$ if you can solve the set of equation of motion.   2. Why the gas expansion is irreversible even though you can select the initial condition above.


title: Does heat kernel factorize on product spaces?

text: I have a doubt regarding whether the trace of the vector heat kernel on a product space factorizes into the corresponding heat kernel traces on each manifold in the product space. I know this holds for the spin-0 as well as spin-1/2 cases, but I am not sure about the spin-1 case. Can somebody please help me out in this regard?


title: Why do helium filled balloons move away from the Earth?

text: From my understanding objects do not fall but are pulled to the earth from gravity. With this in mind, I can't understand why if helium filled balloons are not pulled by gravity then shouldn't they float in place (or drift like objects in space with no gravity) rather than actively move away from the earth? Is gravity pushing it rather than pulling it? Why is this?


title: Speed of Universe Expansion

text: How much is the universe expanding every second (Earth second)? If you could be a bit simple in the answer that would be good.


title: Canonical form representation of a Linear Gaussian CPD

text: I'd like to know how a linear Gaussian conditional probability distribution can be represented to a canonical form. For example, let X and Y be two sets of continuous variables, with |X| = n and |Y| = m. Let p(Y | X) = _N_ (Y | _a_ \+ BX; C) where _a_ is a vector of dimension m, B is an m by n matrix, and C is an m by m matrix.


title: Educational applications of a small Michelson interferometer?

text: The community college where I teach has some nice old Michelson interferometers. There appear to be a bunch of versions of these that used to be sold (may still be sold?) which were all probably knock-offs of some manufacturer's original model. Ours are Cenco 71990-549, and are identical to the one in the photo here (but without the stepping motor). Here and here are manuals for a couple of other similar ones. (We no longer have the manuals for ours.) The arms are about 20 cm, the movable mirror can travel about 25 mm, and it has a micrometer that is accurate to about 2 um. What is a good educational application for these sweet old instruments? Here are the things I've tried, both by myself and using students as guinea pigs:   1. Crank the micrometer while counting fringes with a Na discharge tube. Determine the wavelength.   2. Like 1, but with laser light, projecting the fringes rather than squinting into the instrument.   3. Equalize the arms by finding the condition where fringes are visible with white light. Insert a sample of glass with an unknown index of refraction, and reacquire the condition of equal optical path length. Determine the unknown index of refraction.   4. The Na line is actually a close doublet, so as you move the micrometer you go back and forth from a condition of high-contrast fringes (the bull's eye patterns for the two wavelengths agree) to a condition of little or no contrast (the patterns disagree and counteract each other). If $\Delta d$ is the distance traveled between two successive positions of maximum contrast, then the splitting of the doublet is $\Delta \lambda\approx \lambda^2/2\Delta d$. Comments on how these went:   1. Incredibly tedious, and nearly impossible to do with high precision because you pass through the conditions of low contrast described in #4. The same result could be obtained with much higher precision in five minutes with a diffraction grating.   2. Not as bad as #1, but still doesn't use the instrument for an application to which it's uniquely well suited.   3. This was very cool. It was amazing how accurately the equal-length condition could be detected using this technique (to within 1 um). However, it doesn't seem to be a high-precision way to determine an unknown index of refraction, because you have to use a thin sample like a microscope slide, and your precision ends up limited by the determination of the thickness of the slide using Vernier calipers (e.g., 1.8 mm, so only 2 sig figs).   4. This is sublimely tricky, but I doubt that most students understand it. Measuring the spacing of a close doublet, while impressive to a spectroscopist, is of little intrinsic interest to a student in a freshman survey course. Other than the Michelson-Morley experiment (which obviously can't be done in a 3-hour student lab with a tabletop instrument), the classic historical application seems to have been the determination of the meter in terms of atomic standards. (See Hardy and Perrin, p. 584.) This doesn't seem practical for us. I've seen references to gas cells for use with this setup, presumably using the technique of #3. A back-of-the-envelope calculation seems to suggest that you would get a low-precision measurement, so I don't really see the point. Is it possible to use a technique sort of like #3 to determine the ratio of two wavelengths from the same source with high precision? This would have some intrinsic interest for hydrogen, whose wavelengths are in the ratios of small integers.


title: Gyrocompas equation

text: Ok, need a little help for part (b) of the following problem. I set my axis similar to figure 113 and have expressed both $\omega$ and $\Omega$ in terms of the new coordinates. I have been told to re express (a) in terms of this basis set. Then what? Also, what about the $r_p$ vector? I am a little shaky on rigid bodies so any help would be appreciated.


title: Math / Physics Help - Barrel Pressure and Velocity

text: Back in 1993 I derived the following equations to calculate projectile velocity and barrel pressure. Recently, I have noticed that I need to **double** the calculated results in order to obtain real word results. I need some help to find out why I have to **double** the derived results to make the calculation work. The doubled results appear at the bottom of the image. A Microsoft Excel Spreadsheet Download using the **doubled** (2x) results can be obtained here: question.xls (removed) Thanks in Advance. ![question](http://i.stack.imgur.com/h3dK0.gif)


title: How far could the LHC "fire" a particle

text: Very simple question, and frankly quite a silly one, but I'm currently writing a lecture for secondary school kids and I'd love to tell them how far the Large Hadron Collider could fire a proton. The talk is actually about space, so what I'd really like to know is how would the velocity of the proton would compare to, for example, the necessary escape velocity for the proton to escape the Milky Way? As the talk is mostly about energetics and kinematics I'd be keen to ignore the effect of the protons charge and just look at the kinetic energy being converted to gravitational potential.


title: Relationship between material and fringes behind an edge

text: The double-slit experiment shows fringes on a screen. Closing one of the slits there is still an interference pattern on the screen behind the slit. Making the slit wider we still see fringes between the shadow and the exposed area. Even with single photons or electrons. On the other hand, using a polarisation filter it is possible to let through about 50% of the light. In our case the electric field of the photons may be equally distributed. Using a second filter, 90° rotated to the first, no light (suitable wavelength) goes through. The amazing fact is that using a third filter between the other two - best under 45° - some light goes through. That means that there is influence of the material of the edges. In the experiment with electrons an electrostatical potential changes the fringes dimensions. Does in an experiment with light the use of different materials of the slit plate or the use of a material with different temperatures change the fringes dimensions too?


title: Ising model observables

text: Is there a formula or equation relating $\langle E\rangle$ and $\langle M\rangle$ (average spin per site) and $\langle E^2\rangle$ to temperature $T$ for the square lattice Ising model at zero magnetic field? I have done some simulations and know what the graphs are supposed to look like approximately, but is there some neat expression for them? And is there known what the exact functions are for finite grid sizes?


title: Maximum theoretical efficiency of internal combustion engine

text: This Wikipedia article states that: > Most steel engines have a thermodynamic limit of 37%. Is it correct? If so then it is not clear where this number comes from. Is it derived somehow from Carnot's theorem? I tried to google it but found nothing usefull. If this limit of 37% is wrong then what is correct limit?


title: How to understand the matrix behind a Hamiltonian?

text: thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which _resembles_ a second quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: $$H_i=\sum_{q=1}^\infty \gamma_{q,i} \; \lambda_q$$ while $q$ and $i$ positive integers. What we find is that $\gamma_{q,i}$ is an infinite matrix, and we have all elements of this infinite matrix from observations. $\gamma_{q,i}$ contains only positive integers (and zero), it is a lower triangular matrix with all diagonal elements equal zero except of $\gamma_{11}=\gamma_{22}=1$. The matrix is not random. I am not a physicist, and wonder if someone in this community may help me to understand where and how such matrices are dealt with in physics? Is there a specific name/notation for them? Is there a certain intuition behind of such Hamiltonian and such tzpe of matrix? I want to analyse $\gamma_{q,i}$ further from a physicist's point of view. Thanks


title: About the standard derivation of the gravitational redshift

text: The objective is to derive the gravitational redshift ONLY from the Einstein's equivalence principle (E.E.P.), without using the whole theory of Relativity. **This is the standard "informal" derivation of the gravitational redshift** (For example Carroll in his book follows this way): Consider an emitter, $E$, e.g. a vibrating atom, at rest at a point near the Earth's surface, say, of gravitational potential $\phi$. Let it send light, or any other electromagnetic, signals to a receiver $R$ at rest directly above $E$ and distance $h$ from it; the gravitational potential at $R$ is $\phi+\Delta\phi$, where $\Delta\phi = gh$, $g$ being the acceleration due to gravity. Let $\nu_E$ be the frequency of the signal as measured at $E$, and $\nu_R$ the frequency of the signal received, and measured, at $R$. Then it is used the relativistic Doppler effect, _in the case where the receiver is moving with constant relative velocity $V$ respect to the emitter_ , to show that: $$ \frac{\nu_R-\nu_E}{\nu_E}=-\frac{\Delta\phi}{c^2}=-\frac{gh}{c^2} $$ By the E.E.P. will follow easly the gravitational redshift. **And now my trouble:** In the derivation of the basic formula for the classical Doppler shift (which, it may be recalled, is the first approximation in $\frac{V}{c}$ of the corresponding special relativistic formula), on which the standard arguent is so decisively based, the emitter and the receiver move with constant velocities relative to an inertial frame and $V$ is the constant velocity of the receiver relative to the emitter and away from it. That is, the velocity of the emitter is the same at the instant of the emission and, likewise, the velocity of the receiver is the same at the instant of the reception. This is not the case when $E$ and $R$ are accelerating relative to an inertial frame. So, should I conclude that the above argument is wrong?


title: Bravais lattice with sublattices : why multiple bands?

text: I have a very naive question : given a tight-binding model (with nearest- neighbor hoping) on a lattice defined by a Bravais lattice with a number of sublattices (for instance the honeycomb lattice is a triangular lattice with two sublattices), why is there a band associated with each sublattice ? For instance when a lattice is defined by a Bravais lattice with a 2 sublattice basis, the tight-binding model will have 2 bands.. But why is that ? There must be something very simple that I am not getting here... is it just because that to define a band structure, you need some translation invariance ?


title: What coordinate system is used to describe planets positions in the universe?

text: How are planets positions described in the space and in respect to what? For example is Sun the origo and right now at this moment Earth has [coord_X, coord_Y, coord_Z]? or maybe [lng, lat]? Edit1: I found out that the universe has no center point. Any chose point in the universe can be a center point.


title: How and why can random matrices answer physical problems?

text: _Random matrix theory_ pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive example or a basic introduction to the topic? I would also appreciate hints on relevant literature.


title: Detail of deriving Berry Curvature From Berry Connection

text: The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines: $B^n(\vec{R}) \equiv \nabla \times A^n(\vec{R}) = -Im \left[ \left< \nabla n(\vec{R}) | \nabla n (\vec{R}) \right> \right]$ where $A^n(\vec{R})\equiv -Im \left[ \left< n(\vec{R})| \partial_{\vec{R}} n(\vec{R}) \right> \right]$ is the Berry connection. I don't know exactly how perform this step in the calculation in detail: $\nabla \times A^n(\vec{R}) = -Im \left[ \left< \nabla n(\vec{R}) | \nabla n (\vec{R}) \right> \right]$ Help would be greatly appreciated. Thank you very much.


title: Spoon released out of a satelite moving with orbital velocity

text: ![enter image description here](http://i.stack.imgur.com/VwmZU.png)Suppose we are in a satelite revolving around the earth with a velocity equal to orbital velocity. Now if we release a spoon from the satelite then will it reach the earth or will it just revolve around the earth like the satelite with a velocity equal to orbital velocity?


title: Understanding voltage

text: I understand that water flows because of height difference and current flows because of potential dfference. Now, in this analogy, I can understand height but I fail to realise what potential is. Is it the number of electrons? Say, A has 10 electrons, B has 20 and C has 100 electrons. So, can I say AC has more potential difference than AB and can I also say C has the highest potential?


title: Reunion condition in the Twin Paradox

text: In the Twin Paradox two twins initially at rest in the same reference frame are separated and take different journeys through spacetime. Eventually they are reunited. What is the condition for the reunion? In Newtonian physics, they are reunited when they have the same spacetime coordinates. From a Special Relativity viewpoint they have different spacetime coordinates. In other words, how can the twins decide on a rendezvous point? It is OK to assume that one twin remains stationary.


title: Is it possible to sky dive without a parachute and land safely?

text: Let's assume an averaged sized man (1.8 meters height 80 kg) who's sky-diving from a 5000 m height. Let's also assume he's using tight clothes and no parachute. The idea is: Is it possible for him to survive the fall if he tries to get the maximum velocity he can get by free falling and later, by positioning his body horizontally in a way that resembles a wing shape, get enough upwards impulsion so he can land safely? EDIT: I was not looking for surviving stories that happened by chance. I was looking for the possibility of a technique that would allow someone (trained) to land safely multiple times, not saying without a scratch but at least without broken bones, and without a special impact area.


title: Dielectric constant of water

text: I need the dielectric constant of water from $10^{-2}$ Hz to $10^4$ Hz. As stupid as it may seem, I cannot find much info. I've googled for days. All I can find is close to GHz. And the only info close to Hz, ($100$ Hz) shows a great variation. A relative dielectric constant at $100$ Hz of about $4000$. So, I cannot interpolate back in frequency (I put a link to the paper at the end). Does anyone have any info about where I could find this data? I know that for constant current and about $20$ C the constant is $80.1$. What about at $50$ Hz? I need the complex dielectric constant to test a program. Any lead would be really appreciated. http://arxiv.org/abs/1010.4089


title: Symmetries of relativistic Lagrangian and Hamiltonian systems

text: In non-relativistic mechanics, the conserved quantities found using Noethers theorem in Lagrangian mechanics are the same as those quantities which are conserved under canonical commutation with the Hamiltonian in Hamiltonian mechanics. Does this carry over straight-forwardly to relativistic mechanics?


title: Explanation of the classical coupling of the Higgs Field to Electromagnetism

text: I'm interested in learning about the classical coupling of the Higgs Field to Electromagnetism. There are numerous sources explaining the Higgs mechanism quantum mechanically, i.e. How does the Higgs mechanism work? In addition to the question about classical coupling I'm also interested in how the Mexican hat describes charged particles?


title: How do charge carriers move thermal energy? (Peltier effect)

text: I am having hard time understanding how the charge carriers (electrons and holes) are able to move thermal energy. I am on a high school physics level, so I will probably have a hard time understanding it if there are advanced quantum physics involved.


title: Why does my body create interference with my clock radio while in bed?

text: I typically listen to NPR in the morning from a clock radio next to my bed. There are days when just moving my body, even my arm or leg, is enough to cause interference with the radio to dramatically decrease the volume, or it appears as if I lost the radio station altogether. I know that my body acts like an antenna because, from experience with an oscilloscope probe, I can grab one end of the probe and amplify the AC signal coming from the overhead fluorescent lights in the physics lab room. So I am assuming that this is the same effect as my clock radio.   1. How is this interference created? That is, how is my body acting like an antenna, causing interference to occur with my clock radio?   2. I assume that this interference of my body is (i) size- and (ii) distance-dependent as well as (iii) frequency-dependent. Can someone explain the connection between these things?


title: What sets AdS radius of the Vasiliev dual to the O(N) vector model?

text: In $\mathrm{AdS}_5$/$\mathrm{CFT}_4$ the AdS radius $R$ is determined in terms of the string length by the gauge theory t'Hooft parameter as follows \begin{equation} \frac{R}{l_{\rm s}} \sim \lambda^{1/4} \end{equation} Consequently large t'Hooft parameter corresponds to dropping derivative corrections to the spacetime effective action. In the free O(N) model there is no such parameter. What sets the AdS radius in this case?


title: Neutrinos arrived before the photons (supernova)

text: A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment.


title: How do electrons get the energy to jump from one orbital to the next when in stationary orbits the electron does not radiate energy

text: My Question is how do electrons get energy/lose energy to jump up or down an orbital as in a stationary orbit they do not gain energy and their energy is finite?


title: How does the Higgs Particle Decay?

text: **Background** : It is my understanding that the Higgs Boson is a fluctuation of the Higgs field. It also is not very stable and decays into a plethora of other particles. If all other particles interact with the Higgs field "to give them mass", then it suggests that all other particles are also fluctuations in the Higgs field as well. To me this sounds like all particles interact with themselves (i.e. particles are fluctuations in Higgs field and interact with Higgs field??) **Questions** : 1) I am assuming this is not true or flawed. If so how does a fluctuation in the Higgs field (Higgs Boson) decay into a some other particle. 2) Is it the frequency of the fluctuation that determines what particle it is? Or is any fluctuation of Higgs field a Higgs Boson? Edit: @dgh has informed me (in the comments) about particles and their respective fields. And that if a field is coupled to another one then there is always a probability of decaying into the other one, unless breaking conservation laws. I think that answers both questions. Thanks @dgh! But any more insight is welcomed!


title: Point charge moving towards a conducting plane

text: A point charge $q$ of mass $m$ is released from rest at a distance $d$ from an infinite grounded conducting plane. Show that the charge hits the plane after an amount of time given by: $ \Delta t= \frac{\pi d} {q} \sqrt{2\pi\epsilon_0md} $ I can't seem to start this one. I'm assuming I need to start by finding the force by the charge as a function of $z$, and setting it equal to $\frac{md^2z}{dt} $ giving a differential equation. The diff eq. seems hard to solve, but a hint I recieved was to multiply both sides by $\frac{dz}{dt}$, but I don't see how this simplifies the problem.


title: What's the proper interpretation of canceling infinitesimals?

text: In most textbooks of physics I've found this demonstration of work-kinetic energy theorem: $$\begin{align} W &= \int_{x_{1}}^{x_{2}} F(x)\ dx \tag{1}\\\ &= \int_{x_{1}}^{x_{2}} m\cdot a\ dx \tag{2}\\\ &= m\int_{x_{1}}^{x_{2}} \frac{dv}{dt}\ dx \tag{3}\\\ &= m\int_{x_{1}}^{x_{2}} \frac{dv}{dx} \cdot \frac{dx}{dt}\ dx \tag{4}\\\ &= m\int_{x_{1}}^{x_{2}} \frac{dv}{dx} \cdot v \ dx \tag{5}\\\ &= m\int_{v_{1}}^{v_{2}} v \ dv \tag{6}\\\ &= \frac{1}{2}mv_{2}^{2} - \frac{1}{2}mv_{1}^{2} \tag{7} \end{align}$$ I don't fully understand how they go from (5) to (6). It seems that they cancel the dx as if they were algebraic elements. I know from calculus that $\int f(g(x))\cdot g'(x)\ dx = F(g(x))$, where $f(x)=F'(x)$ because we are applying the chain rule reversed. And in order to do so, Leibniz's notation can help us. If we define $u=g(x)$ and the "differential" $du=g'(x)dx$ the former integral turns in $\int f(u)\ du = F(u)$. But du and dx doesn't exist, they are just a notation we use to realize more easily that we can find the antiderivatives applying the chain rule reversed. So, what are the true mathematical operation they are doing between (5) and (6)?


title: Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?

text: I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did next is correct: I took the normalized eigenvectors, placed them in matrix form, and did matrix multiplication with the basis set of solutions. Let me try to be more precise since I am not sure I am using the right language when mentioning the basis solutions. In the problem we are using the set of solutions of the particle in a box model as our basis. I can increase the number of basis elements in the calculation of the matrix of the Hamiltonian (which amounts to doing $<\psi_n|H \psi_k>$ over a specified range of $n$ and $k$) in order for some of my smallest eigenvalues to begin to converge. Once I have this $H$ matrix built, and that I see that my eigenvalues are converging to some degree, I take the eigenvectors of the $H$ matrix, format them to be in matrix form, and multiply them by the set of basis solutions. I hope that makes things clearer.


title: Experimental evidence for the relic neutrinos

text: What are the experimental (indirect) evidence for the cosmic neutrino background? Where can I read more about this? The discussion on the wikipedia page about the C$\nu$B seems to me to be more about the evidence of the number of generation of neutrinos, than about the cosmic neutrino background...


title: Projection operators and their subspaces (of Hilbert space)

text: I've been watching Susskind's lectures on Quantum Entanglement, and something he said regarding (non-)commuting projection operators confused me. Consider two subspaces {$|a>$} and {$|b>$} of Hilbert space, with operators $K$ and $L$ for which:   * $K |a> = \lambda |a> (1)$   * $L |b> = \mu |b> (2)$ Now considers operators $P_K $ and $P_L$ that project any vector in Hilbert space onto their respective subspaces, that is:   * $K (P_K |\psi>) = \lambda (P_K |\psi>) $   * $L (P_L |\psi>) = \mu (P_L |\psi>) $ We want to find simultaneous eigenstates of both $K$ and $L$. If $P_K$ and $P_L$ commute: $P_K (P_L |\psi>) = P_L (P_K |\psi>)$. Now the left-hand satiesfies $(1)$, and the right-hand side satisfies $(2)$, so these are the required states. In fact, if $P_K$ and $P_L$ operators commute, they share a complete set of eigenstates. The eigenstates of projection operators are those that span the subspace they project onto, so apparently $P_K$ and $P_L$ project onto the same subspace, which means they're the same operator? Then, is the statement: "projection operators commute $\rightarrow$ they're the same" correct, or do they somehow project states onto the same subspace in a different way? Furthermore, we can imagine the subspaces geometrically as 'planes', and where these planes intersect we can find states that satisfy both $(1)$ and $(2)$. Now, according to Susskind, if $P_K$ and $P_L$ do not commute, finding such states is impossible. If the previous paragraph holds (does it?), then them commuting implies the intersection of their subspaces is the entire subspace. I don't know what non-commuting means geometrically, but shouldn't there be a case where the intersection of their subspaces isn't the entire subspace (for example, imagine two 2D perpendicular planes intersecting eachother on a 1D line)? Susskind's comment seems to contradict that, and can't see exactly where I'm going wrongly.


title: Interpretation of Heisenberg's uncertainty principle

text: Heisenberg's uncertainty principle is one of the most fundamental principles on which quantum mechanics is based on. But it is also one of the most confusing laws we encounter. My doubt is whether the uncertainty is due to observation error or due to dual nature of matter ? Dual nature of matter states that a particle exhibits properties of both particle and wave. My question is whether at any given time a particle exists both as a particle and a wave combined. I mean quasi-state. Statistically we can say a given ensemble for a given time interval is 80% matter and rest wave. But my question is in terms of extremely precise measurements i.e., particle is observed for time interval which is given by an impulse function. If time interval is large we wont get result in absolute(discrete) terms but as a statistical graph. In more technical terms does a particle follow a Gaussian distribution as mass and wave for a given time interval. The significance of gaussian distribution is that it is applicable only for mutually exclusive events. Is the existence of particle nature and wave nature mutually exclusive events for time period tending to zero?


title: Accuracy of the Boltzmann equation

text: I have had this question for some time now. Hopefully someone can answer it. I know that the Boltzmann equation is widely regarded as a cornerstone of statistical mechanics and many applications have been explored with a linearized version. I also know that it's extremely hard to obtain exact solutions, which has started a considerable amount of investigation looking for an equally good (or acceptable) formalism to analyse systems that otherwise would be impossible or would take a great deal of computational resources to obtain a solution using the Boltzmann equation. In spite of this, I never heard a precise description about the degree of accuracy (in comparison with experiments) that can be drawn from the Boltzmann equation. Obviously, I expect that accuracy depends on the system at hand, however, it would be great to hear about some specific examples. Recommended readings would also be appreciated. Thanks in advance.


title: How to obtain band dispersion from a band structure diagram?

text: Reading about bands dispersion, I came across the following (Computational Chemsitry of Solid State Materials):               The amount of covalent interaction looks small, which may be deduced from the,     likewise small, dispersion of the oxygen 2p bands which is less than 3 eV.      How is the dispersion obtained (in this case 3 eV) just by inspecting the band structure diagram?


title: Why is $\rho_m$ proportional to the deviation from critical temperature in critical phenomena?

text: In Peskin and Schroeder's chapter 12 about the renormalization group, it is stated that the parameter $\rho_m=m^2/M^2$, where $m$ is the mass and $M$ is the renormalization scale, is proportional to $T-T_c$. But this assertion is not derived in any detail, except for a handwaving one-line explanation "$\rho_m$ is just the parameter that one adjust finely to bring the system to the critical temperature". Could someone elaborate this? In particular, why doesn't the parameter depend on the square or some other power of $T-T_c$?


title: Gamma-ray bursts: A supernova connection

text: Can the gamma ray bursts connected to a new supernova prevent the black hole from forming? > Quoting the abstract of the linked reference "...has become clear that > probably all long GRBs are associated with type-Ic supernovae resulting from > massive stars **that suffered substantial mass loss**...". If enough mass is converted into radiated energy then the condition for BH formation can vanish? Edit Add:   Due to the lack of the expected neutrinos, the IceCube Experiment puts to trash the connection of GRB and SNe.   quoting Cosmic Varience: > More recently, a consensus had grown up that GRB’s (as they are called) are > associated with intense beams of particles created by newly-born supernovae. > That’s a model that seems to fit most of the data, anyway, and it also makes > a pretty good prediction for the production of associated neutrinos. But a > new paper by the marvelous IceCube experiment has thrown a spanner into the > works: they should have been able to see those neutrinos, and they don’t.


title: A question about Gell-Mann Low theorem?

text: Let $|\Psi_0\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\epsilon=H_0 + e^{-\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\epsilon \rightarrow 0^+ $ and $|t|\rightarrow\infty$. Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\epsilon\rightarrow 0^+$ of $|\Psi^{(\pm)}_\epsilon \rangle = \frac{ U_{\epsilon I} (0,\pm\infty) |\Psi_0 \rangle}{\langle \Psi_0 | U_{\epsilon I}(0,\pm\infty)|\Psi_0\rangle}$ exists, then $|\Psi^{(\pm)}_\epsilon \rangle$ are eigenstates of $H$. Here is my question: In interacting field theory, whether the infinite past state $|p_1p_2>_{in} = a^{\dagger}_{in}(p_1)a^{\dagger}_{in}(p_2)|0>$ is some kind of $|\Psi_0\rangle$ I think it may be wrong. Where is my fault? Thanks!


title: Nilsson Hamiltonian - looking for problem and answers

text: I have been looking, unsuccessfully, online for academic problems involving the Nilsson nuclear hamiltonian. I understand the nature of the terms in the equation, but have never seen actual calculations from given data. Does anyone here know of any sites I could find these?


title: How are the calories in food calculated?

text: This is intended to be a fun question. Calorimetry used for calculating the heat generated from chemical changes has been around for centuries, however, I suspect the process for calculating food calories is much more complicated. Can anybody provide a clear succinct, knowledgable answer? Although I will accept answers with links, an independent answer would be preferred.


title: energy field of a newly created particle

text: The question is: How to calculate the energy of electrostatic field of a newly created particle. The usual formula involves integration _'all over space'_. According to GR I believe that _'all over space'_ means the interior volume with radius c*t_now, where t_now is the lifetime of the particle at the time of calculation and c is the light speed. Thus the energy stored in the field is a quantity dependent on time evolution. this is a clarification of the question of PS: conservation of energy and shrinking matter


title: Why is this entangled?

text: I am studying a book of quantum computing and the author gives an example of a four qubits separable! He writes: > Let $\left|ψ\right> = \frac 1 2(\left|00\right> \+ \left|11\right> \+ > \left|22\right> \+ \left|33\right>) = 1/2 \, (\left|0000\right> \+ > \left|0101\right> \+ \left|1010\right> \+ \left|1111\right>)$. > > There are other decompositions with respect to which this state is > unentangled. For example, $\left|ψ\right>$ can be expressed as the product > of two two-qubit states: > > $\left|ψ\right> = \frac 1 2(\left|0_1\right> > \left|0_2\right>\left|0_3\right>\left|0_4\right> > +\left|0_1\right>\left|1_2\right>\left|0_3\right>\left|1_4\right> > +\left|1_1\right>\left|0_2\right>\left|1_3\right>\left|0_4\right> \+ > \left|1_1\right>\left|1_2\right>\left|1_3\right>\left|1_4\right> = \frac 1 > {\sqrt 2}(\left|0_1\right>\left|0_3\right> > +\left|1_1\right>\left|1_3\right>)⊗\frac 1 {\sqrt 2} > (\left|0_2\right>\left|0_4\right> +\left|1_2\right>\left|1_4\right>)$, > > where the subscripts indicate which qubit we are talking about. So > $\left|ψ\right>$ is not entangled with respect to the system decomposition > consisting of a subsystem of the first and third qubit and a subsystem > consisting of the second and fourth qubit. On the other hand, the reader can > check that $\left|ψ\right>$ is entangled with respect to the decomposition > into the two two-qubit systems consisting of the first and second qubits and > the third and fourth qubits. I do not understand why $$ \left|\psi\right> = \frac{1}{\sqrt 2}\left(\left|0_1\right>\left|0_3\right> \+ \left|1_1\right>\left|1_3\right>\right)\otimes \frac{1}{\sqrt 2} (\left|0_2\right>\left|0_4\right> +\left|1_2\right>\left|1_4\right>)$$ And why it is true that "$\left|ψ\right>$ is entangled with respect to the decomposition into the two two-qubit systems consisting of the first and second qubits and the third and fourth qubits."?


title: Needle Floating on a Water Surface

text: So, I'm trying to find the largest radius of a needle before it ends up sinking, and I know that it's a balance of needle weight vs surface tension. However, I'm not really sure how the surface tension force is defined. In a lot of resources, I see the surface tension force define as gamma, which is the force / length. Also, I am seeing surface tension defined in terms of curvature give by the Young-Laplace equation. Which one am I supposed to use? If I can use the gamma one, it would be straightforward, but I'm not clear on what length to use. Obviously, it's the portion of the circumference in contact with the needle, but this length can vary based upon a lot of factors. Aside from that, I'm thinking that I would have to set the contact angle equal to zero (for the maximum surface tension force), but does that mean that I use gamma*2*pi*r? Wouldn't that imply that the needle is completely submerged? Any help would be appreciated!


title: Ionization by heating

text: I would like to ask what happens if an atom exposed to a very high temperature - say millions of degrees (Kelvin). Can we use heating to separate electrons from their nucleus? And what happens to the nucleus if its electrons are stripped off? Thank you,


title: Approaches to Quantum gravity

text: I'm going to start my graduate studies in theoretical physics. My supervisor wants me to work on quantum gravity. He gave me the liberty to chose a particular approach to Quantum gravity (Excluding string theory, he does not like string theory). So I'm thinking which approach to Quantum gravity should I chose? I have a fascination about twistor model http://en.wikipedia.org/wiki/Twistor. But I think it's not that developed fully due to lack of attention from physicists.


title: Is the gravitational potential of a planet in orbit always equal to minus the squared velocity?

text: Say a planet (mass $m$) is orbiting a star (mass $M$) in a perfect circle, so it is in circular motion. $F=ma$ and the gravitational force between two masses $F=\frac{GMm}{r^2}$ so $\frac{GMm}{r^2}=ma$ $\frac{GM}{r^2}=a$ And in circular motion $a=\frac{v^2}{r}$ so $\frac{GM}{r^2}=\frac{v^2}{r}$ $\frac{GM}{r}=v^2$ And gravitational potential $V=-\frac{GM}{r}$ So $v^2=-V$ Is there a (qualitative/less math/wordy) reason why this is the case? (or have I got this wrong?) and is this limited to the specific case of perfect circular motion?


title: Circular motion when F=ma'

text: I apologize in advance if this question is deemed too general or too similar to this and this question. How would mechanics be different if $F=mx'''$ instead of $F=ma$? I feel like I have intuition about what kind of place an $F=mv$ world would be like, since, after all, this is part of our everyday experience dominated by friction. Let me make my question more specific. If I pick a ball attached to the end of a string in a $F=mv$ world, I cannot make it move in uniform circular motion by swinging it around. I can separate the ball and the end of string by pulling the two apart, but then pulling on the string simply pulls the ball towards me. $F=mv$ is like living in a very viscous liquid. I've heard that at a very small scale (like that of bacteria), water is like this and this has consequences for the propulsion of microorganisms. In reality, it is possible to give an initial velocity to the ball and then to tug on the string to rotate the velocity vector so that it is moving in uniform circular motion. Please correct me if any of the above is wrong. But what happens if $F=mx'''$? I have no intuition about what would happen with the string and ball example in this case. You could give the ball a constant velocity and zero initial acceleration. But then how does it move constrained by a string that is applying a force which changes acceleration. Would it spiral? What kind of setup of forces could force a particle into circular motion? Is there any way to gain intuition about the consequences of $F=mx'''$?


title: Do I need mass to calculate the rms speed of diatomic Nitrogen?

text: Calculate the rms speed of diatomic Nitrogen ($N_2$) at an altitude of $600$ km, where the temperature is $1500$ K. I'm confused because only an altitude and a temperature is given, and I feel like I need the mass if the Nitrogen to complete the problem. $v_{rms} = \sqrt{\frac{3K_bT}{m}}$


title: While holding an object, no work done but costs energy (in response to a similar question)

text: I read the answer to Why does holding something up cost energy while no work is being done? and wanting to know more, I asked my teacher about it without telling him what I read here. Instead of referring to muscle cells and biophysics, he answered my question in terms of entropy. He told me that while my arm muscles are stretched when I hold the object, they are more ordered. When my arm is at rest and muscles are not contracted, the muscles are less ordered (more entropy). So his conclusion was that the energy is required to keep the system (my arm muscles) from going to a state of higher entropy. However, the answer in terms of muscle cells doing work on each other (i.e the answer to the hyper-linked question) made more sense to me. Could someone please give me some intuitive sense to my teacher's answer or explain the link between the two answers if there are any...


title: Can we have a static point in the universe?

text: Big bang theory states that the universe began from a single point (which we may consider as a central frame of reference for the objects in the Universe). In 1929, Edwin Hubble confirmed that the Universe is expanding (consider that with respect to that reference point). Does that imply that we cannot have a single static four dimensional point (i.e., space and time changes ever) in the Universe?


title: Is the $N$ factorial in the Partition function for $N$ indistinguishable particle an approximation?

text: I suspect that the $N$ factorial in the partition function for N indistinguishable particles $$ Z = \frac{ Z_0^N } {N!} $$ is an approximation. Please someone correct me if I am wrong and why or why not. Thanks. * * * **A simple case** : each particle has two states with energy $0$ and $E$. The partition function for a single particle is $$ Z_0 = 1 + e^{- \beta E} . $$ If there are only two particles, there is the total partition function $$ Z = \frac{ Z_0^2 } {2}. $$ But regarding the whole system consisting of these two particles, we can also write $$ Z = 1 + e^{- \beta E} + e^{-2 \beta E} . $$ And it is certain that $$ \frac{ Z_0^2 } {2} \neq 1 + e^{- \beta E} + e^{-2 \beta E} $$


title: Is travel to the distant future possible without massive energy expenditure?

text: Well-known physics establishes that, compared to some reference frame (I'll denote as "stationary" here), one can do several things to start at rest, end at rest, and experience _less_ time than objects in the stationary reference frame do, although one can not experience _more_ time given the start and end at rest condition. The former can be accomplished by the following:   1. Take a trip at relativistic speeds    2. Approach to an object in a deep gravity well, approaching an event horizon potential **Question:** Can either of these be done for an object with mass $m$ while expending a small amount of energy to do it, formally using an amount of energy $E\ll m c^2$? To be sufficiently clear, the requirement for traveling to the "distant future" is satisfied if $\Delta t_m \ll \Delta t$, meaning that the time experienced for the object in question is much less than that experienced by the stationary reference frame. Conventional thinking is that #1 is simply not possible given my requirement. Any physical ship that let you planet-hop while living a single human lifetime would require spaceships that probably expend much more energy in the propellent than what the rest mass of the spaceship is. This is the relativistic rocket problem. Thinking about #2 is what caused me to ask this question. Imagine a large black hole and a spaceship that starts off at a great distance from it. It then uses a small amount of energy to enter into a very highly elliptical orbit that comes close to the event horizon. I think (although I am not sure) it would seem to the stationary reference frame that the flyby takes a small amount of time. The spaceship would be going very fast (large time dilatation), but only for a short time and thus could not travel to the distant future and make it back out without using huge amounts of energy. You can introduce other possibilities, like a stationary planet-like surface where a traveler could sit to pass the time, but the escape potential renders this useless too. Is it a fundamental reality that we can not use time dilation to go to the distance future without using extraordinary energy to do so, or do I simply lack the creativity to think of how?


title: Electric field in wire's cross section?

text: ![enter image description here](http://i.stack.imgur.com/cgdLI.png)￼￼￼ $\Delta V_{ABCDA} = - \int_A^A \vec{E} \dot{}d\vec{l}$ > The requirement that the round-trip potential difference be zero means that > $E_1$ and $E_2$ have to be equal. Therefore the electric field must be > uniform both along the length of the wire and also across the cross- > sectional area of the wire. Since the drift speed is proportional to $E$, we > find that the current is indeed uniformly distributed across the cross > section. This result is true only for uniform cross section and uniform > material, in the steady state. The current is not uniformly distributed > across the cross section in the case of high-frequency (non-steady-state) > alternating currents, because time-varying currents can create non-Coulomb > forces, as we will see in a later chapter on Faraday's law. Say the field is non-uniform but completely longitudinal. That would result in the integral above being non-zero, which is impossible, so the field always has to be uniform if its longitudinal. If the wire is made of different types of material, the different parts would act as dielectrics, which would cause the material to polarize and dilute the electric field at different points, would cause the electric field to be non- uniform but still longitudinal, which contradicts the point made in the paragraph above. How can I resolve this apparent contradiction?


title: What is concept of temperture in ion trap?

text: As only several thousands of particles are holding in the ion trap, how can we understand the temperature?


title: Calculating the uncertainty of $r$, when $1/r^2$ is used

text: I have tried to search through the forums and the Internet, however I haven't been able to find a source I was able to understand fully. I have a distance $r$, with an uncertainty of $\mathrm{±\ 0.003\ m}$, however in order to linearise my graph, I need to divide $1$ with $r$ squared ($1/r^2$), what would the final uncertainty for $r$ be? - Furthermore, what would the unit be? Would that still be $m$? or $m^2$?


title: Rijke tube: Why Sound is maximum when heated 1/4th length from bottom of tube

text: Why is it that the maximum sound coming from a Rijke tube happens only when heated wire gauze is placed at 1/4th the length of the tube from the bottom. According to Rayleigh: > If heat be periodically communicated to, and abstracted from, a mass of air > vibrating in a cylinder bounded by a piston, the effect produced will depend > upon the phase of the vibration at which the transfer of heat takes place. > If heat be given to the air at the moment of greatest condensation, or be > taken from it at the moment of greatest rarefaction, the vibration is > encouraged. On the other hand, if heat be given at the moment of greatest > rarefaction, or abstracted at the moment of greatest condensation, the > vibration is discouraged. Wouldn't the sound be maximal if heated gauze is placed at the bottom, where there exists an antinode?


title: Range Of An Interaction

text: Why is the Compton wavelength $\lambda_c=\frac{\hbar}{mc}$ used as a _sensible measure_ for the range of an interaction, where m is the mass of the corresponding mediator?


title: The Synchrotron at CalTech

text: In the Feynman Lectures Vol. 1, it says that the Synchrotron at the California Institute of Technology is capable of producing electromagnetic Radiations with a frequency of $10^{24}$ cycles per second. Since I'm not familiar with _cycles_ , I'd like to know where this lies in the UV Spectrum.


What is the code distance in quantum information theory? Code distance seems to be a very important concept in fault tolerant quantum computation and topological quantum computation.


title: Why droplets of water under oil explode when heated

text: In experiment, cold skillet with droplets of water was filled with layer of oil and heated. The droplets started exploding. Why water can not leave skillet less violently, like steady chain of steam bubbles ?


title: how affine connection follows from Two derivative operator

text: IN wald's GR book in chapter 3 This is stated behind the definition of affine connection : First He showed that if we have two derivative operator $\nabla_a , \tilde\nabla_a$ (both of which consistent with the definition of derivative operator) then at point $p\in M$, for any dual vector field $\omega_b$ $$ (\nabla_a - \tilde\nabla_a)\omega_b $$ will depend only on the value of $\omega_b$ at $p$. That is if $\omega_b^\prime=\omega_b$ at p then $(\nabla_a - \tilde\nabla_a)\omega_b=(\nabla_a - \tilde\nabla_a)\omega_b^\prime$ Then he implied from above that at $p$, $(\nabla_a - \tilde\nabla_a)$ defines a $(1,2)$ tensor like $$(\nabla_a - \tilde\nabla_a)\omega_b=C_{ab}^c\omega_c$$ I'm unable to see how his later assertion is connected to the earlier.


title: Is it possible to observe interference from 2 independent optical lasers?

text: It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true? How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers? What is the coherence length of the best- available laser? More specifically, does there exist any (optical single- wavelength) laser with coherence length exceeding 500 meters?


title: Evolution in the interpretation of the Dirac equation

text: As I understand, Dirac equation was first interpreted as a wave equation following the ideas of non relativistic quantum mechanics, but this lead to different problems. The equation was then reinterpreted as a field equation and it is now a crucial part of quantum field theory. My question is: could you provide me a reference (paper, book) that explains this evolution, including the different historical steps, etc. ? I have a good knowledge of QM and I studying field theories, but I would like to have a clearer view on this historical evolution.


title: Intuition behind the formula for macroscopic entropy

text: Wikipedia says that the 'macroscopic' definition of entropy is: $$ \Delta S = \displaystyle \int \dfrac{dQ_{\rm rev}}{T}$$ Where $T$ is the uniform absolute temperature of a closed system and $dQ_{\rm rev}$ is an incremental reversible transfer of heat into that system. I've always had trouble with derivatives and integrals in physical equations, and this is no exception. Why does the integral appear in this formula? I know that entropy has just been defined that way, but why? What is the logic behind dividing $dQ_{\rm rev}$ by $T$ and taking the integral of that?


title: Is the efficiency of a stirling engine affected by dead volumes?

text: Consider the p-V diagrams of an ideal and real stirling cycle, as found here The article suggests that dead volumes reduce the efficiency of a stirling engine. On the other hand, a simple thought experiment that comparas an ideal stilring engine A with a bigger real stirling engine B, where B has dead volumes so the amount of actually working fluid in both is the same, suggests that the thermodynamic efficiency should be the same. Only the power-per-cycle should suffer from dead volume. To clarify, dead volume means volume (in the hot or cold chamber or regenerator) not being swept by the pistons.


title: Wyckoff positions and lattice coordinations

text: Many papers use Wyckoff positions and Space Groups to report atom coordinates, making use of the structure's symmetries to save space in the notation (e.g. diamond = Fd-3m, only).   1. How can I reconstruct the whole lattices given such symbols? More specifically I'm interested in the lattices of table 1 from this arxiv paper.   2. Where is a good and short source to learn about such reconstructions?


title: logarithmic wind speed profile

text: Under some atmospheric stability condition, over flat terrain, it has been observed for a while that the ratio between wind speed at height $h_1$ above the earth and the wind speed at height $h_0$ is $\log\frac{h_1}{h^*}/\log\frac{h_0}{h^*}$ where $h^*$ is related to the terrain (called roughness length). (see for example http://en.wikipedia.org/wiki/Log_wind_profile) What are the theories (with some details or references please) that explain this rule. Please put only your prefered theory (and hence one per post). Thanks in advance


title: Conservation of angular momentum in propeller planes and helicopters

text: Consider a propeller plane with only one propeller in the front. If the propeller rotates, I would expect by conservation of angular momentum, that the body of the plane would spin in the opposite direction of the propeller but with much lower angular velocity because of the larger moment of inertia. However this doesn't happen in real world. Is it correct, that this would happen in empty space and that the air is responsible to prevent the spinning of the body of the plane? If so, how does this mechanism exactly work? How can one estimate the order of magnitude of the effects? What's the difference between the propeller plane and a helicopter where one actually needs for example a tail rotor to stabilize the helicopter. Would be great if your answer would contain some formulas and approximations which estimate the orders of magnitudes in those problems.


title: Meaning of heat sink in engine analysis

text: I am confused with the meaning of sink. What part of vehicle engine acts as reservoir and what does as sink?


title: Question Based On Direct And Alternating Current

text: A Wikipedia article on Power engineering reads, > Most refrigerators, air conditioners, pumps and industrial machinery use > **AC** power whereas most computers and digital equipment use **DC** power. As far as I know, all digital stuffs require **DC** power. **AC** power isn't particularly useful for digital computers, right? What does **_"most computers and digital equipment use DC power"_** imply or mean? Does that mean there are computers and other digital electronic gadgets out there which runs on **AC** power?


title: A true vacuum bubble will expand slower than our universe expansion rate?

text: Why our universe is supposed to expand faster than light beyond the Hubble limit while a bubble nucleation of true vacuum is supposed tho expand asymptotically to the speed of light ?


title: Will a football (soccer) diffract?

text: Apparently all objects have wavelike properties, so, if we kick a football (soccer ball, if you must) through a pair of posts, does the ball in any sense diffract? If this is ridiculous then let me know and I will delete the question.


title: Finite square well

text: I know how and why we use this form of stationary Schrödinger equation for finding $\psi$ outside the finite square potential well: $$\frac{d^2 \psi}{dx^2}=\kappa^2 \psi$$ I Also know that the general solution to this equation is: $$\psi = Ae^{\kappa\, x} + Be^{-\kappa\, x}.$$ But why do we use only left part $\psi = Ae^{\kappa \, x}$ for the left outer side $x<0$, and the right part $\psi = B e^{-\kappa\, x}$ on the right outer side part $x>0$?


title: Lorentz force in homopolar motor

text: There are a lot of videos showing the simple homopolar motors in action. Let's look at this one: http://www.youtube.com/watch?v=6jyraFLqfqE It is supposed to be working due to the Lorentz force. Electrons go through the magnet, deflect under magnetic field due to the Lorentz force and colliding with atoms in the magnet, rotate it. But what if electrons would travel not through the magnet itself but through the thin wire put between the battery and the magnet, directly connecting + and - of the battery. In this way electrons would still travel in the same direction in same magnetic field, but in the thin wire, not in the magnet. Would then the magnet rotate? Now the other video: http://www.youtube.com/watch?NR=1&v=zOdboRYf1hM&feature=fvwp Here electrons travel through the wire but still close to the magnet so we still can assume that electrons are deflected due to the Lorentz force even though magnetic field should be weaker on the side of the magnet. But what would happen if everything would be glued together: battery, magnet, wire. Would then this system rotate? While the electrons would still travel under the magnetic field they should be deflected and rotate the whole system. But it sounds unphysical. Can somebody explain how it works?


title: Polarizability and the Clausius-Mossotti Relation

text: There seems to be a fairly large inconsistency in various textbooks (and some assorted papers that I went through) about how to define the Clausius-Mossotti relationship (also called the Lorentz-Lorenz relationship, when applied to optics). It basically relates the polarizability of a material to it's relative permittivity (and hence it's refractive index). Now the confusion arises because in Griffith's _Introduction to Electrodynamics_ he defines the induced dipole moment of an atom/molecule in the presence of an external electric field to be given by $$\vec{p} = \alpha \vec{E}$$ where $\alpha$ is the polarizability. From this definition, he derives the Clausius-Mossotti relation to be $$\alpha = \frac{3\epsilon_0}{N}\left(\frac{\epsilon_r -1}{\epsilon_r + 2}\right)$$ but in Panofsky and Phillips' _Classical Electricity and Magnetism_ they've defined the induced dipole moment to be $$\vec{p} = \alpha \epsilon_0 \vec{E_{eff}}$$ where $$\vec{E_{eff}} = \vec{E} + \frac{\vec{P}}{3\epsilon_0}$$ the total electric field acting on a molecule. Using this definition, they've arrived at the relationship $$\alpha = \frac{3}{N} \left(\frac{\epsilon_r -1}{\epsilon_r + 2}\right)$$ which is missing by a factor of $\epsilon_0$. I've seen various sources deriving relations that both equations are right, but I can't quite figure it out. Everyone seems to be working in SI units as far as I can tell. The wikipedia article on the Lorentz-Lorenz equation (which is the same thing) has an extra factor of $4\pi$. I tried working it out, but got lost because I don't really understand how the two differing definitions of $\vec{E}$ and $\vec{E_{eff}}$ are related. How can all these different versions of the equation be consistent with each other?


title: Frank-Hertz experiment setup

text: The usual schematic representing the setup of Frank-Hertz experiment is the following: ![enter image description here](http://i.stack.imgur.com/YRuQi.gif) However, sometimes, you can see a bit different schematic: ![enter image description here](http://i.stack.imgur.com/7v3mT.png) My question is: what function does $V_{G_1K}$ serve? I understand that this constant offset is irrelevant to the experiment (because what matters is the difference in accelerating potential, not its absolute value), but I'm curious and want to check my intuition. Thanks


title: Do magnets lose their magnetism/force when they attract/repel each other?

text: So yea, if two magnets attract/repel multiple times(A LOT) will they lose their magnetism? If they don't break during the whole process?


title: Gas mixture flow rate equation

text: 2 gas flows connect and mix in a pipe. The following is known: $\dot{m_1}$; $\dot{m_2}$ - mass flow rate of the inlet gases $(\frac{kg}{s})$ $A_1$; $A_2$; $A_{mix}$ - the cross section of the pipes $(m^2)$ $\rho_1$; $\rho_2$ - the densities of the 2 gases $T_1 = T_2 = T_{mix} $ **We are looking for the following:** $Q_{mix}$ - flow rate of the mixture $(\frac{m^3}{s})$ So far I have been able to get only this: > $\dot{m_1} + \dot{m_2} = \dot{m}_{mix}$ (1) > > $\dot{m}=\rho*A*V$ (2) > > $\dot{m} = \rho*Q $ (3) > > $\dot{m_1} + \dot{m_2} = \rho_{mix}*Q_{mix}$ (4) > > $Q_{mix} = \frac{\dot{m_1} + \dot{m_2}}{\rho_{mix}} $ (5)


title: What exactly is the connection between gauge transformations and symmetry groups?

text: For a given gauge transformation, say, the electromagnetic field, where observable quantities aren't affected by transformations of the form $$\mathbf{A}' = \mathbf{A} + \nabla \chi,$$ $$\phi' = \phi - \frac{\partial \chi}{\partial t},$$ $$\Psi' = \Psi \cdot \exp(\frac{iq\chi}{\hbar}).$$ What exactly does the $U(1)$ symmetry Lie group have to do with these gauge transformation?


title: Where does the relation $u=h_f-RT+(c_p-R)(T-T_0)$ come from?

text: I'm working on a college project for a subject called _Motorcycle Engineering_ , and in this project I'm supposed to do some reverse engineering on a MATLAB program to model the engine of the motorcycle. In this program I just came across the following expression: $$u=h_f-RT+(c_p-R)(T-T_0)$$ where $u$ is the specific internal energy, $h_f$ is the enthalpy of formation per unit mol, $c_p$ is the constant pressure specific heat, $T$ is the temperature, $T_0$ is the room temperature and $R$ is the gas constant. Does anybody know where does it come from?


title: What if the LHC doesn't see SUSY?

text: A question in four parts.   1. What are the main problems which supersymmetry purports to solve?    2. What would constitute lack of evidence for SUSY at the proposed LHC energy scales (e.g. certain predicted superpartners are not in fact observed)?    3. Are there alternative theoretical approaches which would address the SUSY problem set and which would still be credible in such an LHC no-SUSY-scenario?   4. Where would LHC-disconfirmation of SUSY leave String Theory? I would like to think that these four points could be taken together as one question.


title: Celestron NexStar 6 SE vs. Orion XT8

text: I'm interested in purchasing a new telescope and have narrowed it down to the NexStar 6 SE vs. the Orion XT8. I'd like to be able to get into astrophotography and read that having the Go- To (auto-tracking) is a must -- this would eliminate the Orion XT8. I really like the large(r) aperture of the XT8, but having the Go-To seems nice. Is the image quality sharper on one or the other?


title: A Energy Eigenbasis for the Potential Step

text: We should all fondly remember this basic undergraduate problem: A quantum particle is incident (from the left) upon a potential barrier of height V and width L. Compute the transmission and reflection coefficients. However, this isn't the full story. To completely solve the potential barrier, we need a complete set of orthogonal and normalized energy eigenstates. The wavefunctions of the leftward traveling particles are just some of the energy eigenstates. The wavefunctions of the rightward traveling particles are some more energy eigenstates. Are there more possible states? What is a complete energy eigenbasis for the Hilbert Space of a single particle in a step potential? (I'm thinking about a positive potential barrier +V, but one could ask this question for a finite potential well too.)


title: Why did the june 2011 lunar eclipse last so long?

text: It was kind of hard to miss the lunar eclipse this week, although I didn't see it in person (Sod's law means that on every relatively major astronomical event clouds cover where I am). From what I understand it lasted about 100 minutes. I work that out as being about 9 minutes short of 1 degree of the Moon's orbit? How did it last so long? Surely 100 minutes is more than enough time for the Moon to move out of Earth shadow, or for Earth's shadow to "overtake" the Moon?


title: Universe Expansion

text: I know the galaxies are moving away from us (faster and faster) but when we say the universe is expanding, is that really only pertaining to the galaxies expanding or is the space itself expanding? I know we don't yet know what the boundary is, but I just wanted clarification around the commonly used phrase "Universe Expansion".


title: What is going on in the system? How are the formulas `mg sin(x)` and `mg cos(x)` derived?

text: When a load is resting on an inclined plane, there is force $mg \sin(x)$ that's vertical to the inclined plane and force $mg \cos(x)$ horizontal to the plane acting on it like this: ![enter image description here](http://i.stack.imgur.com/mfCsM.png) My textbook says the weight $mg$ has two components: $mg \sin(x)$ and $mg \cos(x)$. But when adding up $mg \sin(x)$ to $mg cos(x)$ it does not equal to $mg$. If `mg` and N are the only forces in this system, which means the net force should be 0, how should I explain the existence of $mg \sin(x)$ and $mg \cos(x)$? Does $mg \sin(x)$ and $mg \cos(x)$ have their own reaction force to cancel them out and turn the net force to be 0 too or is it wrong the say the system's net force is 0? And how are the formulas $mg \sin(x)$ and $mg \cos(x)$ that are used to describe these two forces derived?


title: Do all celestial objects have stationary orbit?

text: Do all celestial bodies of sufficient mass such big asteroids, moon, planets etc. have orbit such that the period of revolution is same as the sidereal period of that object (such as geostationary orbit in earth) ?


title: Why do neutrons repel each other?

text: I can understand why 2 protons will repel each other, because they're both positive. But there isn't a neutral charge is there? So why do neutrons repel? (Do they, or have I been misinformed?) The reason why I'm asking this is because, I've just been learning about neutron stars and how the neutrons are forced (as in, they repel) together according to my teacher (he's a great teacher btw, though what I just said doesn't make it seem so). So I wondered, why they have to be forced by gravity and not just pushed?


title: Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories

text: (v2) As I read e.g. in this question, the nice holonomy group features of Calabi- Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, not only the fields?) for quantization. But moreover, their features also seem to turn up in questions regarding the defomration of one two-dimensional world sheet quantum field field theory to another. I see the introduction of operators in the Lagrangian etc. and it's reminiscent of renomralization (keyword: moduli space?). Then I read something about the beta function being zero, because it would be proportional to the Ricci curvature of the target manifold, which is zero for Calabi-Yau manifolds. And that's another argument, which speak for their consideration. (Remark: Maybe there is a broader spectrum of theories for which these questions don't make sense. I guess I'm talking about non-linear sigma models here.) > In string theory, do you use the renormalization techniques as > representations of the geometrical deformations? > > We get different theories for neightboring metrics right? How closely is the > number of possible vacuum states is related to the number of Calabi-Yau > metrics?


title: If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?

text: > Are the more recently discovered symmetries in string theory such that the > theories based on mirroring geometries are absolutely the same from an > observable point of view? I have mirror symmetry in mind, i.e. by symmetries I mean the dualities encountered in string theory. * * * (v2) Say you take e.g. take the Schwarzschild metric from general relativity and consider its quivalent in some theory of quantum gravity theory. There, if you don't look close enough, i.e. if you look at averages, the quantum- Schwarzschild space would look classical, namely like the general relativity Schwarzschild space. Now the target space in string theory is one manifold and it leads to a quantized theory. But if there are now actually two mirroring manifolds where this theory can come from, the I guess the classical version looks like neighter of these geometries. The following question comes from this observation, that there are totally different (mirroring) options with which Calabi-Yau you can start (considered in the Lagrangian) if you go in the quantization direction and the conclusion that neighter of will be the limit, or there are more limits. > If we assume that the world is described by such a theory, do we besically > live in two indestingushable geometries at once? To put it differently, I ask if string theory can be considered a quantum gravity in the sense that there is a classical limit down to classical differential geometry, especially regarding the extra dimensions. And if so, to what geometry in these higher dimensions does the limit lead? I do know that at least some of the geometry must come from the string excitation representing a quantizations of a metric/graviton. But I suspect different metrics in the non-linear sigma model Lagrangian will also affect the classical picture in some essential way, directly or not.


title: Can cannonballs go through water?

text: In the recent Spielberg/Jackson Tintin movie, there is a scene where Red Rackham and Captain Haddock's ships are fighting, and cannons are fired. The cannonball is shown at one point to go through a wave, and inflict serious damage on the other ship. I know that bullets stop in water; do cannonballs, with their greater weight, continue with enough force to inflict damage?


title: Domain of validity for semiconductor equations

text: If I understand correctly, the distinction between semiconductors and insulators is a matter of convention? A semiconductor is basically an insulator with an (enough) narrow band gap, usually around $3$ $eV$. I have a few equations used for semiconductors, will they still work for insulators? Or are they gradually starting to breakdown for larger band gaps? Which effects remains and which fades away? In this specific case I'm working on $TiO_2$ which is (if I'm not mistaken?) a wide-band semiconductor, which classifies between semiconductor and insulator. Example of semiconductor equations in use:   * Conductivity: $\sigma = ne\mu_e+pe\mu_h$   * Inner voltage: $V_{bi}=\frac{k_BT}{q}\ln{\frac{N_AN_D}{n_0p_0}}$   * Intrinsic carrier concentration: $n_i=p_i=2(\frac{k_BT}{2\pi h^2})^{3/2}(m_em_h)^{3/2}e^{-\frac{E_g}{2k_BT}}$   * Doping:  Baseline, is it okay to use semiconductor formula's on $TiO_2$? Any other wide-band semiconductor? Any insulator? What are the general rule?


title: Fine tuning: why is multiverse with anthropic principle considered a better explanation than anthropic principle alone?

text: I got the impression from several public lectures that, when it comes to the fine tuning problem, physicists tend to give multiverse together with anthropic principle as a plausible solution, but seldom anthropic principle alone. However it seems the core of this kind of solution is really anthropic principle, that is, once we decide to invoke anthropic principle, the concept of multiverse becomes irrelevant(if fine tuning problem is our only concern), adding multiverse to the explanation does not seem to strengthen the solution by much. Is there anything I missed or misunderstood? Of course it is possible that I overestimate the popularity of multiverse explanation in academia, after all I got this impression from public lectures and I never talked to a cosmologist in person, in this case I would be happy if anyone better informed can confirm this.


title: Refraction of light in water

text: Here's a question in a paper based on refraction of light. I can't seem to solve it for some reason. > A man looks down at a fish length of 20 cm. His eye is 2m above the surface > of the water ($\mu = 4/3$) an the fish is 2m below the surface shown in the > figure. The ratio of angular width $\Delta\theta_1$ of the fish as seen by > the man in presence of water to the $\Delta\theta_2$ in the absence of water > is:   >  _(A) 6/5 $\hspace{3cm}$ (B) 5/6 $\hspace{3cm}$ (C)7/8 $\hspace{3cm}$ > (D)8/7_ ![fish](http://i.stack.imgur.com/5zvpD.jpg) * * * _Here's what I've done:-_ When there is no water present: $$\frac{\Delta\theta_{air}}{2} = \sin^{-1}(10/400)$$ $$\therefore \Delta\theta_{air} = 2\sin^{-1}(1/40)$$ ...from the triangle obtained by the eye, and half the fish. When the water is present: ![diag](http://i.stack.imgur.com/qCTOk.jpg)   ($P.S:$ the equation written below the diagram is wrong, I've just realised. I've written the correct version below) Let the light rays at angle $\phi$ with the normal refract at the water surface and converge at the eye. Here the angle of refraction is $\theta$. So we have: $$\sin\phi . \mu_{water} = \sin\theta . \mu_{air}$$ $$\therefore\frac{\sin\theta}{\sin\phi} = \frac{\mu_{water}}{\mu_{air}}$$ $$\therefore\frac{\sin\theta}{\sin\phi}=\frac{4}{3}$$ After which I am stuck and have no clue how to go about this. Can anyone help me? * * * _**Addendum_** : Will taking $\sin x = x$ help? $x$ is very small anyway, and $\sin x \approx x$ is turning out to be true ($x = 0.025$, $\sin(0.025) = 0.024997...$). _**Addendum2_** : Another thing I figured out is that the water will increase the observed angular width, so the ratio is definitely $>1$. So we can safely rule out (B) and (C) which are $<1$.


title: Need clearing up on vector decomposition in motion physics

text: I've been pursuing physics on my own, and I need something cleared up. Say I have two arbitrary objects, I have their velocities, I know when the collide, I have their normal vectors, etc. I know where a force is applied, how much, which direction(lever arm, etc). I THINK that what I'm trying to do is vector decomposition. I know that torque is the force applied perpendicular to the normal(tangent?), and that linear force is the force applied directly through the center of mass. I believe I intuitively understand these interactions, but I'm having an issue... what is the mass with the splitting of the forces? How 'much' goes through the center of mass, and how much is perpendicular, or torque? I feel like trigonometry may be a simple answer, but I'm unsure of the implementation. My variables I have are the force vector(x, y coordinates scaled to magnitude), and vector(distance vector?) which represents it's offset position from the center of mass. I'm not asking how do I calculate the torque on the CM, but how do I simply determine the force that is perpendicular, not yet crossed by the distance vector(?), and how much of the force goes through the CM, and converts into linear velocity? P.S. This is in a 2D coordinate plane, in a computer simulation I'm making. Labeled as homework as I want to intuitively understand, not actually assigned by a school.


title: Is a falling, perfect sheet of fluid possible to create?

text: This is a bit of an abstract question so I'll try explain this as best I can from the bottom up. I would like to know if it is possible to observe a sheet of fluid, much like a sheet of glass, falling completely flat and level. Imagine a volume of water that is at rest on some sort of very, very fast trap door, which once opened, would allow the volume of water to retain its shape and fall through the trap door. Is this possible if it were done in a room containing a near-perfect vacuum to discount drag effects? would things like surface tension stop this happening? perhaps using a super-fluid? Thanks!


title: Cherenkov radiation and K-40

text: I am thinking if it's possible to detect Cherenkov radiation in water caused by dissolved natural potassium (KOH) in it without using photo-multipliers. The main question is does 1 beta particle generate 1 photon, or multiple? Specifically, how many photons of _visible light_ should I expect from 1.33 MeV electron in water? I know that blue light photon has energy of 2.76 eV, but I have no idea about efficiency of this process.


title: Why does the "Troll-Mobile" not work?

text: See image: ![](http://2.bp.blogspot.com/_vI6LGR352yE/TMapkx5gYsI/AAAAAAAAAFc/zza9ylA-9Ek/s320/1285770302993.jpg) If I'm not totally dumb the car would move when the right magnet would not be attached to the car, at same distance of the booth magnets. So why would it not move?


title: Please explain the flaw in this picture

text: So I saw this picture on my google+ feed and I immediately know why it wont work. But I'm having trouble explaining to myself and others exactly why. Without using anything overly complex, can anyone explain the flaw.   ![Magnet pulling truck](http://i.stack.imgur.com/lbFah.jpg)   P.S. Please don't say it won't work because of the law of conservation of energy. I know it can't work, I want to know why this specific scenario won't work.   Thanks for the help.


title: What happens to an astronaut who's floating in a spaceship (in space) when it begins to move?

text: I feel this is somehow a stupid question, but I don't know the true answer. What happens to an astronaut who's floating in a spaceship in space when it begins to move? Will the astronaut not move until he smashes onto a wall in the spaceship? Or will he move with the spaceship due to gravity? Or does it depend on the speed/acceleration of the spaceship; if it's too fast, the astronaut will smash onto a wall and if it's enough slow, he'll move with the spaceship? Thanks for helping me out!


title: Is the "consistent histories" interpretation of QM a "many worlds interpretation" in disguise?

text: The so called consistent histories interpretation is claimed to be a correction of the Copenhagen Interpretation. One of its aim, as much as I can see is to show that observers don't have any special role and a history is represented as a sequence of projection operators in the Heisenberg picture. It has been convincingly shown that due to environmental decoherence the density matrix will be almost diagonalized within a short time (expressed in terms of a certain basis of interest). It will smell just like a "collapse" of the wave function. However, if we want to see the whole picture, we see that superposition still exists in the entangled whole. My question is whether this interpretation is actually a "many worlds interpretation" in disguise? If not, then why not?


title: How do we resolve operator ordering ambiguities when quantizing generic nonlinear second-class constraints?

text: Dirac came up with a general theory of constraints, including second-class constraints. To quantize such systems, he first computed the Dirac bracket _classically_ , and only then "promoted" the classical Dirac bracket into a commutator. However, this leads to operator ordering ambiguities over and above what already exists for the Poisson bracket. Is there any more direct way of coming up with a quantum Dirac bracket and Hamiltonian operator which resolves such ambiguities? If the system is symmetrical, symmetry considerations can often single out the "correct" quantization, but what about asymmetrical systems? Ideally, the correct quantum prescription ought to match what we get from using path integrals. Similarly, if the second-class constraint is linear, the quantum commutator and Hamiltonian is also straightforward. However, even relatively simple models can have subtleties. Consider the toy model $$L=\frac{\dot{q}^2}{2} -f[q]\frac{F^2}{2}$$ where F is an auxiliary variable and $f$ is a function of $q$. The classical second-class constraints are given by $p_F \approx 0$ and $fF\approx 0$. Using path integrals as a guide, $$\int \mathcal{D}q\, \mathcal{D}F\, \exp\left( \frac{i\dot{q}^2}{2} - \frac{ifF^2}{2} \right) \propto \int f^{-1/2}\mathcal{D}q\, \exp\left( \frac{i\dot{q}^2}{2} \right),$$ which corresponds to the following operator ordering for the Hamiltonian: $$\widehat{H} = \frac{1}{2} \widehat{f}^{1/4}\widehat{p}\widehat{f}^{-1/2}\widehat{p}\widehat{f}^{1/4}.$$ But what about the generic nonlinear case? * * * On another note, is there a formalism where we can impose the Dirac bracket _after_ quantization, rather than before?


title: Deriving the group velocity of a wave produced by some basic cosine waves with unequal amplitudes

text: Consider some basic cosine waves of the form ${E_i} = {E_0}\cos ({\omega _i}t - {k_i}z)$ with different amplitudes, frequencies and phases. We know a combination of such waves could result in a wave which has an envelope that is traveling at a group velocity and also has a phase velocity. When there are some basic waves **with equal amplitudes** propagating in the same direction of propagation, deriving the expression for the resultant wave is easily possible using some simple trigonometric identities. say when there are two waves of the form: ${E_1} = {E_0}\cos ({\omega _1}t - {k_1}z)$ and ${E_2} = {E_0}\cos ({\omega _2}t - {k_2}z)$ we'll have: $$\begin{align} {E_1} + {E_2} &= {E_0}\bigl(\cos ({\omega _1}t - {k_1}z) + \cos ({\omega _2}t - {k_2}z)\bigr) \\\ &= 2{E_0}\cos\biggl(\frac{\omega_1 - \omega_2}{2}t - \frac{k_1 - k_2}{2}z\biggr)\cos\biggl(\frac{\omega_1 + \omega_2}{2}t - \frac{k_1 + k_2}{2}z\biggr) \\\ &= \biggl\\{ 2{E_0}\cos\biggl(\frac{\omega_1 - \omega_2}{2}t - \frac{k_1 - k_2}{2}z\biggr)\biggr\\} \cos(\frac{\omega_1 + \omega_2}{2}t - \frac{k_1 + k_2}{2}z\biggr) \end{align}$$ First part refers to envelope of the resultant wave and group velocity will be derived from this part and the other part deals with the phase velocity. Deriving such a expression gives a better physical viewpoint of what we are dealing with than when we represent it in the form of a summation of some basic cosine waves. Plus such a relation for the wave function could make it so easy to find out many charactristics of the wave in a brief look at the expression. Therefore it's so important and helpful to derive an expression for the wave which is caused by superposition of some basic propagating waves. But it was a simple situation.My question is that **what if basic waves amplitudes were not the same?** When we are aware of propagation of some basic waves with unequal amplitudes and frequencies, how can we derive the relation for the resultant wave (which actually represent the only changing field in the medium)? And of course I have the same question about the expression for the group velocity. Like what we did in relation to equal amplitude waves example, is it possible to derive the expressions only by use of simple trigonometric identities?


title: What is most infeasible about a sky hook (as in space travel not Batman)?

text: We need to get things into space but it costs too much. Lots of people talk about space elevators but no one talks about the skyhook idea. I'm referring to this: Skyhook (Wikipedia), not the thing in Batman. It is also illustrated in this low-budget video. Is this idea really (even) less feasible than a space elevator? What is the most difficult aspect of implementing it?


title: Anti particles: What exactly is inverted?

text: http://en.wikipedia.org/wiki/Antiparticle says "Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge." and What is anti-matter? goes along with that. But. Reading about the Standard Model I see several things sometimes called XXX- charge, like baryon-charge, lepton-charge - or baryon-number, lepton-number. And as far as I see in the examples I found (but until now never in any definition I found) the anti particles have a negative XXX-charge/-number too. So can you point to a list, what exactly is inverted in anti particles? (And why this list? Sure it has to do with the Standard Model, hasn't it? But Why? :)


title: Constructive and destructive interference in De Broglie matter waves

text: I'm a little confused about matter waves and especially interference. $\lambda = h/p$ The denominator is momentum and is thus reliant on mass. If we assume that the property holds for particles only, then I run into some problems with constituent "particles". What I mean by that are objects normally considered to be point masses, that are really made of several other particles. Examples being protons, neutrons, atoms, and occasionally molecules. Let's take a $ ^4He$ atom as our example. It consists of 2 protons and 2 neutrons. This gives it a mass of $4.002602amu$ or $6.6464 * 10^{-27}kg$. Let's give it a nice non-relativistic velocity of $15m/s$. The calculation of its De Broglie wavelength is trivial. $\lambda = \frac{h}{6.6464 * 10^{-27} * 15} = 6.646\mathrm{nanometers}$ Now, lets take one of the constituent protons in the helium atom and calculate its wavelength. $\lambda = \frac{h}{1.6726 * 10^{-27} * 15} = 26.426\mathrm{nanometers}$ Similar result for neutrons: $\lambda = \frac{h}{1.6749 * 10^{-27} * 15} = 26.390\mathrm{nanometers}$ Its apparent that the difference in these is substantial. The main question I have is how these differences can coexist? List of questions: Assuming that the difference is a result of incorrect application of the formula.   1. Why do these values differ, is it not appropriate to take the wavelength of something that is bonded to the larger whole?   2. If not then why not? Why does the binding energy prevent the nucleons from having their own waves?   3. Do we have a better model that computes for the wavelength of a particle being attracted or repelled in an arbitrary electromagnetic or strong/weak field?   4. Reasons other than binding energy as to why you can not apply the formula to an individual nucleon? Assuming that the difference is a result of constructive/destructive interference.   1. What happens to the binding energy, does its existence in the mass term of the atom make the calculation for the wavelength of the atom an approximation?   2. What happens to the little bit of the wave that might not constructively interfere but destructively interfere because the nucleons are not perfectly on top of each other? Does that make the atom wavelength only an approximation?   3. If the atoms wavelength is only an approximation then do we have a better model that takes into account all of the interference and produces a more accurate wavelength for a particle with constituents?   4. Has this been used to try to determine the binding energy of quarks, is this something that has been tried?


title: relativistic spaceship, CMB radiation and thermodynamics

text: **Scenario:** a spaceship is travelling at a high fraction of $c$. The interstellar gas and CMB radiation has blueshifted significantly and we are facing a possible melting of the front radiation shield! but the ship has good radiator area in the back so there might be a significant temperature difference between the back and the front shield. **Question:** can we use this thermal differential to impulse the ship without expending more fuel? or all we can do is slow down the radiation drag?


title: How to calculate Acceleration

text: Assume you are moving 10 meters in 10 seconds northward. What is the acceleration?. Using this equation: $a = (v_f - v_o)/\Delta(t)$ and filling it in with this data: $v_f=1$ which comes from $10$ $m/10s=1m/s$, $v_o=0$ because we are starting at a speed of $0$ and $t=10$ because that's how long it tooks. We find out that $a = (1-0)/(10)$ which equals $.1$ $m/s$ $N$. But when you use a different equation: $a=2d/t^2$ and filling in $d$ with $10m$, and $t$ with $10s$ you get $a=20/100^2$ which is $.2m/s/s$) Why is this?


title: How to "derive" the wave equation without refering to strings?

text: The wave equation in $3$ dimensions is simply: $$\nabla^2\psi = \dfrac{1}{v^2} \dfrac{\partial^2}{\partial t^2}\psi,$$ and the intuition behind this is that at each point of space with coordinates $(x,y,z)$ with have some quantity oscilating there. If it's a sound wave what is oscilating are molecules, if it's an electromagnetic wave what is oscilating are electromagnetic fields and so on. The important thing is: $\psi$ represents the association $t\mapsto \psi_t$ where $\psi_t$ represents a quantity on each point of space and this association from $t$ into $\psi_t$ is to be thought of "oscilatory". Basic texts of Physics take the oscilation of points on a string, derive from Newton's second law that the wave equation in $1$-dimension is obeyed and then say: "because of that we have good reasons already to call wave something that satisfies this equation". The problem is that I'm not yet convinced. Is there any other way to "derive" the wave equation without refering to the particular case of waves on strings? That is, starting from the fact that we want that association $t\to\psi_t$ as I've said, is there a way to conclude that the $\psi$ function should satisfy that equation? I've tried to reason with the harmonic oscilator. So at each point $(x_0,y_0,z_0)$ the function $t\mapsto \psi(t,x_0,y_0,z_0)$ should satisfy the harmonic oscilator equation, that is: $$\dfrac{\partial^2}{\partial t^2}\psi + \omega^2 \psi =0,$$ But I think that's not the way, since I can't see a way to put the laplacian in there. So how can we do it?


title: Chiral Landau's fermi liquid theory in 3+1D

text: In standard LFL theory, the effect mass of quasiparticle is different from the composite Fermion's mass. It seems that this is no longer true in Chiral case (e.g. a finite density system with interacting massless left handed fermions)? What is Landau's FL theory for chiral Fermions? Thanks!


title: Will Anderson's Poor Man's Scaling loose its effect when band width is small?

text: The s-d interaction Hamiltonian is as fellows $H_I=Js.S$, J is the coupling strength. We focus on the antiferromagnetic case, where $J>0$. According Anderson's poor man's scaling, the renormalized coupling strength $J$ increases as the band width $D$ decreases. Anderson originally did the scaling to the lowest order of $J$. But the situation is different as $D$ decreases and even tend to zero. In this case, higher order terms become more and more important, since $J$ becomes bigger or even divergent. These higher order terms typically are different from s-d type. To some extent, poor man's scaling loose its effect when $D$ is very small. Is it correct? Thanks very much.


title: Pair production in complete vacuum

text: I want to prove that pair production (electron-positron) cannot happen in complete vacuum. This is why I obeyed conservation of energy and got equation: $$h \nu = m_e c^2 \Bigl[ \gamma(v_1) + \gamma(v_{2})\Bigl]$$ I did the same for conservation of momentum and got an equation which is different: $$h \nu = m_e c \Bigl[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Bigl]$$ I noticed that parts $v_1 \cos \alpha$ and $v_2 \cos \beta$ will never equal $c$, so I cannot get same equation as above. **QUESTION:** Can I state now that pair production cannot happen? What here is the reason I can state this? I mean is it that energy of a photon should be the same in both cases?


title: Is it possible to write a Density Matrix in the following form?

text: Is it possible to write an arbitrary density matrix $\hat{\rho}$ in the following form ? $$\hat{\rho} ~=~ \frac{1}{N} \sum_{\ell=1}^N \left|x_{\ell}\right\rangle \left\langle x_{\ell}\right|,$$ where $\left\\{\left|x_{\ell}\right\rangle\right\\}_{\ell = 1}^{N}$ are normalized states (but not necessarily orthogonal). If yes, how can one prove this ?