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<p>Let's discuss about $SU(3)$. I understand that the most important representations (relevant to physics) are the defining and the adjoint. In the defining representation of $SU(3)$; namely $\mathbf{3}$, the Gell-Mann matrices are used to represent the generators
$$
\left[T^{A}\right]_{ij} = \dfrac{1}{2}\lambda^{A},
$$
where $T^A$ are the generators and $\lambda^A$ the Gell-Mann matrices. In adjoint representation, on the other hand, an $\mathbf{8}$, the generators are represented by matrices according to
$$
\left[ T_{i} \right]_{jk} = -if_{ijk},
$$
where $f_{ijk}$ are the structure constants.</p>
<p>My question is this, how can one represent the generators in the $\mathbf{10}$ of $SU(3)$, which corresponds to a symmetric tensor with 3 upper or lower indices (or for that matter how to represent the $\mathbf{6}$ with two symmetric indices). What is the general procedure to represent the generators in an arbitrary representation?</p> | 1,039 | [
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<p>So in the context of a set of notes I am reading about acoustics I get to equation (23) in this <a href="http://www3.nd.edu/~atassi/Teaching/ame%2060639/Notes/fundamentals_w.pdf" rel="nofollow">paper</a>. Basically it comes down to showing that (<strong>note the dots above the a's meaning time derivative!)</strong></p>
<p>$$
f(t) = -a_0c_0\int_{-\infty}^t \dot{a}(t'+\frac{a_0}{c_0})e^{-\frac{c_0}{a_0}(t-t')} dt'\ = -a_0^2\dot{a}(t),.
$$</p>
<p>under the assumption that $\frac{c_oT}{a_0} >>1$. $T$ can be thought of as a characteristic time scale for this problem. For those interested in more than just the math, please see the paper. Now I show this in the following way:</p>
<p>Integrating by parts:
$$
\begin{align}
f(t) = & \\
=& -a_0^2\dot{a}(t+\frac{a_0}{c_0})+a_0c_0\int_{-\infty}^t \ddot{a}(t'+\frac{a_0}{c_0})\frac{a_0}{c_0}e^{-\frac{c_0}{a_0}(t-t')} dt'\\\
&
\end{align}
$$</p>
<p>By expanding $\dot{a}$ as a taylor series we get:</p>
<p>$$
\dot{a}(t+\frac{a_0}{c_0}) = \dot{a}(t) +\frac{a_0}{c_0}\ddot{a}(t)+O\left(\frac{a_0^2}{c_0^2}\right)
$$</p>
<p>Now if we make the order of magnitude estimate:</p>
<p>$$
\frac{a_0}{c_0}\ddot{a}(t) \simeq \frac{a_0}{c_0T}\dot{a}(t)
$$</p>
<p>then I hope it is clear by applying this reasoning to the equation for $f(t)$ that we have shown $f(t) = -a_0^2\dot{a}(t)$ for $\frac{c_oT}{a_0} >>1$. </p>
<p><strong>Now here comes my question:</strong> How can I justify the time derivative operator behaving as $1/T$, where $T$ was just given as the length scale of the problem (which seems so arbitrary)? Is my reasoning above correct? I was hoping someone could put my mind at ease about the "hand wavy" nature of these order of magnitude approximations. </p> | 1,040 | [
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<p>I was wondering with a question for a quite long time, thought to ask here.</p>
<p>I need to know is there any material or element which can block magnetic field? I mean I am searching for such material or element that cannot allow magnetic field though itself?</p>
<p>The practical scenario is, there are two permanent magnets and those are positioned within each other's magnetic field. I want to put something so that both the magnets become free of interference withing themselves.</p>
<p>Hope I could clarify my question.</p>
<p>Can anyone help me of give me some suggestion on this aspect please?</p> | 0 | [
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] |
<p>Why is the gravitation force always attractive? Is there a way to explain this other than the curvature of space time? </p>
<p>PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve with concavity?</p> | 81 | [
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<p>I am wondering what will be the physics to explain how two neutral, chemically nonreactive objects stick. I know that using van der Waals formalism, we can treat neutral body electrodynamic forces and arrive with attractive forces that pull the objects together. </p>
<p>Now, once the objects touch (say a mechanical cantiliver in a MEMS sensor like the one used in an iPhone), what happens to the forces? A quantitative answer or some estimate on how strong the attractive force is for simple cases will be very appreciated.</p>
<p>in response to anna's comment :
Let us consider what happens in vacuum for ultra smooth surfaces, with no residual electrical charge and fully chemically stablized surfaces (example, silicon crystals with stabilised surface bonds). </p> | 1,041 | [
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<p>I have a question about the relation:
$\exp(-i \vec{\sigma} \cdot \hat{n}\phi/2) = \cos(\phi/2) - i \vec{\sigma} \cdot \hat{n} \sin(\phi/2)$.</p>
<p>In my texts, I see $\phi\hat{n}$ always as c-numbers. My question is whether or not this relation can be generalized for $\hat{n}$ being an operator?</p>
<p>If so how exactly would the expression be different?</p>
<p>Thanks.</p> | 1,042 | [
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<p>The second postulate of <a href="http://en.wikipedia.org/wiki/Special_relativity" rel="nofollow">special relativity</a> deals with constancy of light in inertial reference frames. But, how did Einstein came to this conclusion? Did he knew about the <a href="http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment" rel="nofollow">Michelson-Morley experiment</a>? </p> | 1,043 | [
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<p>Does one need to invoke quantum mechanics to explain casimir force or vander waals force. I see that textbooks show derivation of vander waal force with no QM but casimir force is typically described with QM. </p>
<p>Is there a difference between vanderwaal and casimir forces ? Are there distinct examples of these two forces in real life. Is there a way to prove a given force is vanderwaal and not casimir or vice versa. </p> | 1,044 | [
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<p>Einstein has suggested that light can behave as a waves as well as like a particle i.e, it has dual character. In 1924, Louis de Broglie suggested that just as light exhibits wave and particle properties, all microscopic material particles such as electrons, protons, atoms, molecules etc. have also dual character. They behave as a particle as well as wave. This means that an electron which has been regarded as a particle also behaves like a wave. Thus, according to de Broglie, </p>
<blockquote>
<p><em><a href="http://en.wikipedia.org/wiki/Matter_wave" rel="nofollow">all the material particles in motion possess wave characteristics</a>.</em> </p>
</blockquote>
<p>Although the dual nature of matter is applicable to all material objects but it is significant for microscopic bodies only. For large bodies, the wavelengths of associated waves are very small and cannot be measured by any of the avalable methods. Therefore, practically these bodies are said to have no wavelengths. Thus, any material body in motion can have wavelength but it is measurable or significant only for microscopic bodies such as electron, proton, atom or molecule. And it is to be noted that electrons won't orbit the nucleus in the sense planet orbits around the sun, but instead exist as <em>STANDING WAVES</em>. </p>
<p>I am finding difficulty in understanding electron or other material particles to behave as wave. In order to understand electron in the form of wave, I considered the following example of high racing car. See the picture below.<br>
<img src="http://i.stack.imgur.com/zcvaI.jpg" alt="enter image description here"></p>
<p>You can see that car is under very less speed compared to electron, it shows some what blur appearance, I considered this to be wave like in nature. And car has comparatively very very less wavelength, so that it can be considered as negligible, even though I considered it to understand. I assumed electron to be under similar position as the car, to understand what actually is called wave nature. Is it that as like car appears to be blur or some what wavy, electron also under immense speed show wave like appearance. But in reality car stays as material particle, even if it appears to be wavy. Is it the same thing what actually meant by wave nature of electron? i.e is it that electron seems to be wavy under high speed, but it will remain as particle during its motion? </p>
<p>This is what I have understood about wave nature of electron. I don't know whether I have misunderstood or not. If I have misunderstood the concept about wave nature, please explain what actually is meant by wave nature of material particles. </p> | 1 | [
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<p>When studying the de Broglie relations, I have stumbled across the following problem:</p>
<blockquote>
<p>Consider an electron with known velocity $v$ and assume: $v \ll c$. Calculate the corresponding wavelength of the electron.</p>
</blockquote>
<p>Using the de Broglie relations from Wikipedia:
$$\begin{align}
\lambda &= h/ p \\
f &= E/h
\end{align}
$$</p>
<p>I would use the first equation and arrive at wavelength $h / (m v)$</p>
<p>However (and this is where my problem is situated) when using the second equation and assuming the only kinetic energy ($ E = m v^2/2$) is present, I arrive at the solution wavelength $2 h / (m v)$, therefore twice as much as in the first case.</p>
<p>My question is: Why do the results differ? I believe it is due to a wrong assumption (kinetic energy not being the total energy) but I can't really prove it.</p>
<p>Thanks in advance!</p> | 2 | [
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<p>Are there any known uses of modeling with elastic fractals in current physical applications? (Especially uses concerning with self-similarity)</p> | 1,045 | [
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<p>When something falls down, potential energy is transformed into kinetic one. Furthermore, you can attach a pulley and extract some energy (like in an aljibe), for example with a dynamo... If the object fall in a deep well, we can extract more energy, till the object reach downhole..</p>
<p>But what if we could have a "bottomless pit"... Endless energy?.. Well we would need an infinite earth... Or like a "black hole aljibe"... hmmm that doesn't seem an option..</p>
<p>Oh, wait a minute, we already have one! A satellite is like always "falling down"...
Then is exchanging "potential energy"(who knows from what reference system) into kinetic energy..</p>
<p>What about tied it to a pulley (a long rope I see) to a dynamo, or an earth winding with a magnet satellite, or a satellite winding with the earth magnet, or whatever..
The idea is to USE the "bottomless pit" energy..</p>
<p>Where is the mistake? (just in case there be any =P)</p> | 1,046 | [
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<p>Say I have a linear motor [aka rail-gun] and use a x amount of electrical power. I fire the gun and the object exits at velocity v. I then reuse the same object as my projectile and fire the rail-gun a second time this time with 2x the electrical power. My lessons on momentum suggest it will go 2v. The Work-Energy Theorem says 1.414v. Which is right and why?</p> | 1,047 | [
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<p>Assume a spherical metallic shell over which a charge $Q$ is distributed uniformly. </p>
<p>Applying Gauss's law $\displaystyle\oint\textbf{E}\cdot d\textbf{a}=\frac{Q_{\text{enc.}}}{\epsilon_0}$ by considering a gaussian spherical surface concentric with the shell but with smaller radius, one can easily show that the electric field inside the shell is 0 since there is no enclosed charge within the gaussian surface.</p>
<p>What would change in the above analysis and result had the electrostatic interaction been different from the inverse square behavior (i.e., $1/r^{2+\epsilon}$ instead of $1/r^2$)? </p> | 1,048 | [
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<p>Broadly speaking how do ideas of leading singularities and Grassmanian help in curing infrared divergences when calculating N=4 scattering amplitudes? My understanding is that one gets infra red divergences because the external gluon momenta becomes collinear with the the loop momenta. I am confused as to what Nima's and Freddie's collaboration are doing to avoid this? If people can clarify my confusion and direct me to the appropriate literature I would be grateful.Also, is there a clear way to understand this for theories like planar QCD?</p> | 1,049 | [
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<p>In multipole expansion, we use monopole, dipole, quadrupole or octupole to describe an electromagnetic field. But I saw someone use <a href="http://www.google.com/search?as_q=sextupole" rel="nofollow">sextupole</a> to describe transition states. </p>
<p>If we expand an electromagnetic field by spherical harmonics, $\ell=0,1,2,3$ represent monopole, dipole, quadrupole and octupole. Do we sextupole in electric field? For different $m$ in the expansion, are they also different modes?</p> | 1,050 | [
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<p>I am writing this question here because I have a problem in understanding the Wigner Threshold law in Photodetachment and Photoionization.</p>
<p>The Wigner Threshold Law is given by:</p>
<p>$\sigma$=$E^{L+1/2}$.</p>
<p>where $\sigma$ is the photodetachment cross section, E is the kinetic energy of the detached electron from the anion, and L is the detached electron angular momentum. </p>
<p>I have the following questions:</p>
<p>What is the threshold in this law? For this threshold law to be valid, should the energy be above or below the threshold? Where the literature says "near threshold", does this mean above or below the threshold - and how far?</p>
<p>I found the threshold law for photodetachment but I couldn't find a threshold law for photoionization. Is there any threshold law for photoionization? It is known that the ejected electron and the neutral core (in the case of photodetachment) have an effective potential which is a sum of the interaction potential and a centrifugal potential:</p>
<p>$Veff$=$V(r) + h/2mr^2 [l(l+1)]$.</p>
<p>Is this centrifugal potential (second term) still valid in the case of photoionization? </p>
<p>PS: If anyone can suggest a textbook or any other reference, having the derivation and details about the Wigner law on a fundamental level (a graduate level so that I can understand the basics very well), that would be good.</p> | 1,051 | [
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<p>I have an intrinsic silicon layer sandwiched in between two aluminum contacts. I'm trying to figure out the band diagram of the entire device when a positive bias (much larger than the work function of the aluminium) is applied to one of the contacts. Intuitively, I would expect the i-layer part to be tilted (like in a p-i-n device), but I can't figure out what the connections between the i-layer and the metal contacts looks like. </p> | 1,052 | [
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<p>I had an argument about the most cost-effective way to keep the energy bill low in the winter (here, temperature usually have an average of -20°C (-4°F)).</p>
<p>He thinks that it's more effective to keep a temperature at a constant temperature at say 24°C (75.2°F), because It would cost as much or more to re-heat the house if you let the temperature drop.</p>
<p>I simply think that the lower the temperature the better, because heat from the house will dissipate less and you need less energy to maintant the temperature.</p>
<p>Am-I right? if so, which scientific theory could I look at to my make point?</p> | 1,053 | [
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<p>I know that a hydraulic jump is formed when a zone of supercritical flow discharges into a zone of subcritical flow, but why exactly is a hydraulic jump formed?</p>
<p>From a mechanical point of view, are the only forces acting in the diagram weight and friction?
<img src="http://i.stack.imgur.com/3H1eU.gif" alt="enter image description here"></p>
<p>If someone could actually label the diagram with forces and explain them to me, I'd really appreciate it.</p> | 1,054 | [
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<p>I want to know whether all solids are mono-atomic or not, and if there was diatomic solids or not,and if there was compounds of solids. </p> | 1,055 | [
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<p>I purchased a 90x refractor telescope with below configuration. </p>
<ul>
<li>Objective: 50mm diameter with 360mm focal length</li>
<li>2 Eye pieces: 6mm, 20mm (10-1/ 8" scope)</li>
<li>1.5x erecting eye piece</li>
</ul>
<p>It measures 13- 5/8” long x 2-1/2" diameter.</p>
<p>I was trying to see a building half Km far from my balcony. But I can't see anything with any eyepiece other than white lighting circle. Moreover when I tried an eyepiece of toy binocular, I could see something, not much larger but at least visible.</p>
<p>Is it something minimum distance of view for a telescope? Or do I need to focus it? (I don't know how)</p> | 1,056 | [
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<p>I asked in this thread <a href="http://physics.stackexchange.com/questions/119923/time-dependent-schr%C3%B6dinger-equation-with-v-vx-t">Time-dependet Schrödinger equation</a> how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was the Fourier transform, which is why I want to quote his example: </p>
<hr>
<p><strong>Example</strong></p>
<p>As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes,</p>
<p>$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$</p>
<p>We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space:</p>
<p>$$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$</p>
<p>which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution.</p>
<hr>
<p>Now, my question would be: What are meaningful initial conditions for this ODE? I mean, what you probably want to look at is how a wavefunction $\Psi(t=0,x)$ propagates in time? So how do you set up meaningful initial conditions for this Fourier-transformed Schrödinger equation? You don't need to refer to this particular ODE(with this potential). My question is rather: When you solve this ODE, what are appropriate initial/boundary conditions for this Fourier transformed ODE, cause this is were my imagination fails.</p>
<p>If anything is unclear, please let me know.</p> | 1,057 | [
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<p>I have two weights, weighing one ten times the other. I should figure out a way to make a pulley system that can lift the heavier mass with the smaller one. I would like to use the least number of pulleys possible because of the friction, but I think that anyway the rope would suffer from the same problem.
I found this:
<a href="http://science.howstuffworks.com/transport/engines-equipment/pulley.htm" rel="nofollow">http://science.howstuffworks.com/transport/engines-equipment/pulley.htm</a></p>
<p>But I cannot figure out how to continue. I would like to start from a system of 5\6 pulleys and make up a reasonable system. I believe that stress can be negligible.</p>
<p>Can anyone show how to go through such problems starting from adding pulley and decreasing the needed force, pulley by pulley, and then reduce the number of them adding sheaves or more complicated shapes?
Thanks.</p>
<p>EDIT: Maybe adding some trusses among pulleys, can it be an advantage?</p> | 1,058 | [
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<p>Imagine you are a common man and want to prove that the Earth is Round, how would you prove it without any mathematical derivation or without the theory of the ships.</p> | 1,059 | [
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<p>Why is the earth shaped like a sphere and not any other shape: cube, prism?</p> | 3 | [
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<p>We know that in an LC circuit the voltage across the capacitor is the same as the voltage across the inductor (the emf induced in it). This is a direct conclusion of KVL (Kirchoff's Loop Rule) applied to the circuit. But can we use KVL in such circuits, where we have things like capacitors and inductors which affect the voltages and current?</p>
<p>For example, in a circuit with a battery and a capacitor, KVL cannot be applied. In fact, the voltages and current asymptotically approach the values predicted by KVL as the capacitor charges. </p>
<p>Similarly, while deriving the time-dependence of current in an ac circuit with a capacitive load, the argument commonly presented is as follows:</p>
<p>Voltage of source = Voltage across capacitor </p>
<p>$$V_{cap}(t) = V_0\sin(\omega t)$$</p>
<p>$$Q_{cap} = CV_{cap}(t)$$</p>
<p>$$i = \frac{d(CV_{cap}(t))}{dt} = C\frac{d(V_0\sin(\omega t))}{dt} = CV_0\omega\cos(\omega t)$$</p>
<p>thus concluding that the current and emf are phase shifted by $\pi/2$. But isn't there a flaw if we use KVL in such circuits?</p> | 1,060 | [
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<p>Is it true that in Lattice Boltzmann method particles have only one degree of freedom even in 3D case? Can someone explain that fact or provide a link?</p>
<p>Thanks!</p> | 1,061 | [
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<p>In Green's superstring theory, Chapter 3, from equation 3.3.13 to 3.3.18, the author says that the singularity behavior of the ghost field $c^+$ must be no faster than $z^2$ as $z$ goes to infinity. And $b_{++}$ is required to approach $0$ as $z$ goes to infinity.</p>
<p>May I ask what is the reason for such constraints?</p> | 1,062 | [
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<p>Using a simple lattice model of conduction, where electrons are accelerated by an electric field, and are slowed down by bumping into the lattice, you get the following equation for current density:</p>
<p>$\vec{J}_n=nqμ_n\vec{E}$</p>
<p>Let's imagine an ideal DC voltage source connected with perfectly conducting wires to a resistor. Just from intuition, as the electrons reach the lattice of the resistor, you'd think that there'd be a pileup of electrons, since they don't have as much mobility in the resistor (almost like a traffic jam at tight roads). Do electrons or other charge carries collect at the end of resistors? If they do, is this what creates a voltage drop across resistors (or equivalently, an electric field across a resistor)? This idea of a collection of charge seems to imply a capacitance to the resistor. Do real resistors display any in-built capacitance?</p> | 1,063 | [
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<p>Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?</p> | 1,064 | [
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<p>I understand that the strings in string theory are posited to be many, many orders of size smaller than say, a quark, electron or any other particle. But if this is so, how does the string "expand" to produce the gargantuan, by comparison, particle? Is this expansion caused by its vibrational pattern or is there another way by which the extremely small string manifests itself into a much larger object? I am, of course, under the impression that each "elementary" particle is generated by the vibrational pattern of only one individual string, i.e., one string produces one electron, etc.</p> | 1,065 | [
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<p>Given an atomic transition with associated E-field $E(t) = E_{0}\cos(\omega_{0}t)e^{-t/\tau}$ where $\omega_{0}$ is the natural line frequency and $\tau$ is the decay constant of the simple harmonic oscillator. Find an expression for the line flux, that is $I(\omega)/I(\omega_{0})$.</p>
<p>I'm trying to do the following Fourier transform:
$$
f(\omega)=\int^{\infty}_{-\infty}E_{0}\cos(\omega_{0}t)e^{-t/\tau}e^{-i\omega t}dt
$$</p>
<p>I'm not really sure how to do this integral. I tried calculating it from $-\tau/2$ to $\tau/2$ but I get an answer with the sine of a complex number in the numerator that will not go away when I square it to find Intensity. Any help would be appreciated. Thanks</p> | 1,066 | [
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<p>Basically, are the atoms that make up my body right now something that has existed since the big bang? </p> | 1,067 | [
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<p>I was hoping that someone could give me the more fundamental reason that we take as the temporal part of a quantum wavefunction the function $e^{-i\omega t}$ and not $e^{+i\omega t}$? Clearly $e^{-i\omega t}$ solves the time dependent Schrödinger equation whereas $e^{+i\omega t}$ does not. </p>
<p>However, the Schrödinger equation, when it was first developed, was merely a hypothesis. It was new physics and, as such, could not be derived from previous work. Hence, why did Schrödinger and his contemporaries choose $e^{-i\omega t}$ and, thus, why does an antiparticle with wavefunction temporal dependence $e^{+i\omega t}$ correspond to backwards time travel or negative energy?</p> | 1,068 | [
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<p>Even though the thermal velocity of electron in a conductor is comparatively high, the thermal velocity is not responsible for current flow? Why is this the case?</p> | 1,069 | [
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<p>In time-dependent perturbation theory where $H=H_0+V$ and $V$ is considered small and has <em>no explicit time dependence</em>, the standard text-book treatment of the leading order probability amplitude for the system to make a transition from $|i\rangle$ to $|f\rangle$ is</p>
<p>$$
P_{f\leftarrow i}(\Delta t)=\big|\langle f|V|i \rangle\big|^2\frac{4\sin^2(\omega_{fi}\Delta t/2)}{\hbar^2\omega^2_{fi}}
$$</p>
<p>If I consider transitions between two states of the same energy, I take the $\omega_{fi}\rightarrow0$ limit, giving me</p>
<p>$$
P_{f\leftarrow i}(\Delta t)=\big|\langle f|V|i \rangle\big|^2(\Delta t)^2.,
$$</p>
<p>which grows without bound in time, as $\Delta t^2$. I would take this to mean that perturbation theory fails at long times. Why then am I allowed to take the large time limit to derive Fermi's Golden rule, without risking failure of perturbation theory?</p> | 1,070 | [
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<p>This question is relevant to Euler's angles and Euler's equations for a rigid body.
Why aren't $\omega_1$, $\omega_2$ and $\omega_3 = 0$ in the body frame? How can we measure $\vec\omega$?</p> | 1,071 | [
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<p>I have a hard time understanding whether or not a wing placed in a potential flow, assuming there is no viscosity and no friction with the wing, will produce a lift. I've seen several contradictory (to me) versions of what happens in this case, in particular:</p>
<ul>
<li>In Landau - Lifshitz, Vol 6, §11 it is shown that the force vector on any object in a potential flow is zero (i.e. there is neither lift nor drag). The mathematics behind this derivation are above my understanding, thus I can't really follow how this is shown and what assumptions are made along the way.</li>
<li>The Wikipedia article on the <a href="http://en.wikipedia.org/wiki/D%27Alembert%27s_paradox">d'Alambert's paradox</a> states that a body in a potential flow does not experience drag. On the other hand the article does not say anything about lift forces and the German version explicitly states that it does not apply for lift forces.</li>
<li>A common explanation for the lift produced by wings seems to be that its asymmetric shape causes a higher velocity of the flow above the wing and a lower below. Due to Bernoulli's equation that would result in a pressure-difference and thus a force. Honestly I don't quite get this explanation as I don't see where the different velocities are supposed to come from (without friction and thus without the circulation resulting from turbulences behind the wing).</li>
</ul>
<p>So now I'm wondering, do or do not wings have lift in irrotational, inviscid, incompressible fluids.</p> | 1,072 | [
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<p>How much lift does the average helium filled party balloon produce? (not including any extras like ribbon string)</p> | 1,073 | [
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<p>I am stuck on how to connect the ideas that two spin half particles must form a anti-symmetric wavefunction. Is there a proof on how to show that two spin one half particles must form a anti-symmetric wavefunction? </p>
<p>\begin{align}
\Psi(x_1,x_2)
=
\Psi_1(x_2)
\Psi_2(x_1)
-
\Psi_2(x_1)
\Psi_1(x_2)
\end{align}</p>
<p>This would indeed prove that spin half particles must obey the Pauli Exclusion Principle, because if two fermions were in the same state, then the wavefunction would equal zero.</p> | 1,074 | [
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<p>The <a href="http://www.straightdope.com/columns/read/1886/does-black-clothing-keep-you-cooler">Straight Dope</a> ran an explanation of why nomads often wear black clothing - it absorbs heat better from the body. On the other hand, white clothing reflects sunlight better. Is it possible to get the best of both worlds and wear clothing that is black on the inside and white on the outside?</p> | 1,075 | [
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<p>One topic which was covered in university, but which I never understood, is how a spinning top "magically" resists the force of gravity. The conservation of energy explanations make sense, but I don't believe that they provide as much insight as a mechanical explanation would.</p>
<p>The <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/top.html">hyperphysics link</a> Cedric provided looks similar to a diagram that I saw in my physics textbook. This diagram illustrates precession nicely, but doesn't explain why the top doesn't fall. Since the angular acceleration is always tangential, I would expect that the top should spiral outwards until it falls to the ground. However, the diagram seems to indicate that the top should be precessing in a circle, not a spiral. Another reason I am not satisfied with this explanation is that the calculation is apparently limited to situations where: "the spin angular velocity $\omega$ is much greater than the precession angular velocity $\omega_P$". The calculation gives no explanation of why this is not the case.</p> | 920 | [
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<p>I think that relativity and quantum mechanics would provide some good examples.</p> | 1,076 | [
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<p>Does the scientific community consider the <a href="http://en.wikipedia.org/wiki/Loschmidt%27s_paradox" rel="nofollow">Loschmidt paradox</a> resolved? If so what is the resolution?</p>
<p>I have never seen dissipation explained, although what I have seen a lot is descriptions of dissipation (i.e. more detailed pathways/mechanisms for specific systems). Typically one introduces axioms of dissipation for example:</p>
<p>entropy $S(t_1) \geq S(t_0) \Leftrightarrow t_1 \geq t_0$ (most often in words)</p>
<p>These axioms (based on overwhelming evidence/observations) are sadly often considered proofs. I have no problem with usefull axioms (and I <em>most certainly believe they are true</em>), but I wonder if it can be proven in terms of other (deeper and already present) axioms. I.e. is the axiom really independent? or is it a corrollary from deeper axioms from say logic (but not necessarily that deep).</p>
<p>(my opinion is that a proof would need as axioms some suitable <em>definition</em> of time (based on connection between microscopic and macroscopic degrees of freedom))</p> | 344 | [
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<p>I just took a rare earth magnet out of an old hard drive. Lacking an appropriate screwdriver, force was used, and the magnet broke into two pieces; one about a quarter of the original size and one about 3/4 the original size.</p>
<p>Let's say this is the magnet:</p>
<pre><code>>>>>>>>>
</code></pre>
<p>and the arrows are pointing north, in the original magnet.</p>
<p>It broke into two pieces:</p>
<pre><code>>>>>>} }>
</code></pre>
<p>(where } represents the rough edge of the break).</p>
<p>I'd expect that the two magnets would keep their relative directions, so that the magnets would fit back together at the break, like this:</p>
<pre><code>>>>>>}}>
</code></pre>
<p>But instead, the broken edges repel, and the magnet wants to come together so that formerly "north-facing" sides are now connected, like so:</p>
<pre><code>>>>>>}<{
</code></pre>
<p>The obvious deduction here is that one of the pieces flipped polarities when the magnet was broken. I don't understand why this would be the case.</p>
<p>So why did one of the pieces flip polarities?</p>
<p>EDIT: To clarify, the actual magnet is a 120-degree arc of a circle. Thus, it's obvious what the original orientation of the pieces were. This isn't simply a case of me getting the pieces flipped over relative to each other.</p> | 715 | [
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<p>What would theoretically happen to an observed object's length (let it be a 5 meter line) moving at some speed greater than that of light in a straight path?</p> | 1,077 | [
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<p>While writing these answers: <a href="http://physics.stackexchange.com/questions/16354/hypercharge-for-u1-in-su2xu1-model/16685#16685">Hypercharge for U(1) in SU(2)xU(1) model</a> and <a href="http://physics.stackexchange.com/questions/16340/is-there-a-concise-but-thorough-statement-of-the-standard-model/16684#16684">Is there a concise-but-thorough statement of the Standard Model?</a> , it occured to me that the unification prediction for Georgi-Glashow is relying on high-scale renormalizability, if you have a nonrenormalizable SU(5) coupling between the SU(5) Higgsing field and the gauge field, you can make the couplings of the three subgroups different.</p>
<p>In general, after breaking, you can make different couplings for SU(2),SU(3) and U(1), depending on the Higgsing details, by allowing nonrenormalizable terms of sufficiently high order. If you Higgs in a natural way, using an SU(5) hermitian tensor field (spacetime scalar) with expectation (a,a,b,b,b), a term of the form $tr(\phi GG)$ where G is the SU(5) gauge field considered as an antihermitian SU(5) matrix, gives an SU(3) coupling correction which goes as (b-a)/M, where M is the higher string or planck scale. Considering that the string scale can be as low as $10^{17} GeV$, while the GUT scale can be as high as $10^{16} GeV$, depending on the details of low energy matter and high energy compactification, is it obvious that the failure of couplings can't be 10%?</p>
<p>Question: Considering that the GUT scale is close enough to the Planck scale that renormalizability is not exact, what are the precise natural errors in coupling unification?</p>
<h3>EDIT: Clarification</h3>
<p>Lubos Motl gave an answer which misses the point, so I will give a more detailed description of the question. The question is: how bad can coupling constant unification fail <em>at the GUT scale</em> just from lack of renormalizability, ignoring running.</p>
<p>When you have a big gauge group break into a smaller gauge group, the field strength tensor for the big gauge group is multiplied by $1\over g^2$, and decomposes into the field strength tensors for the smaller gauge groups, which are then also multiplied by 1/g^2 so that the couplings are the same (at the unification scale--- forget about running--- this is a classical question).</p>
<p>But you have a scalar VEV doing the breaking. Normally scalars can't interact with gauge fields to alter the coupling--- the coupling is nondynamical. But the reason for this is renormalizability--- you can't have a renormalizable scalar-scalar gauge-gauge derivative interaction.</p>
<p>But ignore renormalizability, and suppose that there is an additional term in the Lagrangian of the form (the obvious space-time indices on F are suppressed) </p>
<p>$$ A \mathrm{tr}(H F F) $$</p>
<p>Where F is the field strength tensor $F^{\mu\nu}$ for SU(5), A is a coupling constant of dimensions inverse mass, and H is an SU(5) adjoint scalar, which acts to Higgs SU(5), whose VEV is diag(a,a,b,b,b) in some gauge. The field H is invariant under SU(2) rotations of the top two components (because the H VEV is $\delta^a_b$ on the upper two-by-two block, and this is an invariant tensor of SU(2)), SU(3) rotations of the bottom 3 components (because the H VEV is $\delta$ on the bottom 3 by 3 block), and any diagonal phase rotations, including U(1) rotations that make up the Georgi-Glashow hypercharge. So this VEV breaks SU(5) to the standard model gauge group.</p>
<p>Now, if you decompose the field strength F into the SU(2) field strength $W$ and the SU(3) field strength $G$, the nonrenormalizable term above decomposes into</p>
<p>$$ A a \mathrm{tr}(W^2) + A b \mathrm{tr}(G^2) $$</p>
<p>So that the VEV of the scalar alters the <em>coupling constant</em> of the two gauge groups in the decomposition by the Higgsing, and this happens at the GUT scale, ignoring any running.</p>
<p>This scalar-gauge-gauge nonrenormalizable term is in addition to the ordinary kinetic term, so the full W and G action is</p>
<p>$$ ({1\over 4g^2} + Aa) \mathrm{tr}(W^2) + ({1\over 4g^2} + A b) \mathrm{tr}(G^2) $$</p>
<p>The coupling constants do not match just from the breaking. The difference in their reciprocal-square coupling at the GUT scale is</p>
<p>$$ 4A(a-b) $$</p>
<p>Now A is order ${1\over M_\mathrm{PL}}$, and $(a-b)$ is order $M_\mathrm{GUT}$, so you get an order $M_\mathrm{GUT}\over M_\mathrm{PL}$ difference between the coupling constants already at the GUT scale, before any running.</p>
<p>I know that a term like this is ignored in running calculations, since these are RG flows in renormalizable low energy theories only. It is not included in higher loop corrections, or in any threshhold corrections, and to calculate it requires a full string background (and numerical techniques on the string background). The question is whether this term is typically small, or whether it is typically large enough to make the arguments against Georgi-Glashow from failure of unification moot.</p>
<h3>EDIT: more clarification</h3>
<p>Unification relies on the coupling constant of a subgroups being equal to the coupling constant of the big group. The reason is that the $FF$ kinetic term breaks up to the subgroup kinetic terms.</p>
<p>But this depends on renormalizability. If you have <em>nonrenormalizable</em> scalar FF interactions, you break the equality of subgroup couplings. These terms can be neglected at ordinary energies, but not at 10^16 GeV. This is for Lubos, who downvoted the question, I am not sure why.</p>
<p>If you downvote this, you might also wish to downvote this:
<a href="http://physics.stackexchange.com/questions/15005/does-the-ruling-out-of-tev-scale-susy-breaking-disfavor-grand-unification/29408#29408">Does the ruling out of TeV scale SUSY breaking disfavor grand unification?</a></p>
<p>But I urge you to think about it instead, as I am 100% sure that they are both saying correct things.</p> | 1,078 | [
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<p>Sorry to go on about this scenario again but I think something is going on here.</p>
<p>Imagine a stationary charge $q$, with mass $m$, at the center of a stationary hollow spherical dielectric shell with radius $R$, mass $M$ and total charge $-Q$.</p>
<p>I apply a force $\mathbf{F}$ to charge $q$ so that it accelerates:</p>
<p>$$\mathbf{F} = m \mathbf{a}$$</p>
<p>The accelerating charge $q$ produces a retarded (forwards in time) radiation electric field at the sphere. When integrated over the sphere this field leads to a total force $\mathbf{f}$ on the sphere given by:</p>
<p>$$\mathbf{f} = \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}\mathbf{a}.$$</p>
<p>So I apply an external force $\mathbf{F}$ to the system (charge + sphere) but a total force $\mathbf{F}+\mathbf{f}$ operates on the system.</p>
<p>Isn't there an inconsistency here?</p>
<p>As the acceleration of charge $q$ is constant there is no radiation reaction force reacting back on it from its electromagnetic field - so that's not the answer.</p>
<p>Instead maybe there is a reaction force back from the charged shell, $-\mathbf{f}$, to the charge $q$ so that the equation of motion for the charge is given by:</p>
<p>$$\mathbf{F} - \mathbf{f} = m\mathbf{a}\ \ \ \ \ \ \ \ \ \ \ (1)$$</p>
<p>This reaction force might be mediated by an advanced electromagnetic interaction going backwards in time from the shell to the charge so that it acts at the moment the charge is accelerated.</p>
<p>Now the total force acting on the system is the same as the force supplied:</p>
<p>$$ \mathbf{F} - \mathbf{f} + \mathbf{f} = \mathbf{F}.$$</p>
<p>If one rearranges Equation (1) one gets:</p>
<p>$$\mathbf{F} = (m + \frac{2}{3} \frac{qQ}{4\pi\epsilon_0c^2R}) \mathbf{a}$$</p>
<p>Thus the effective mass of the charge $q$ has increased.</p> | 1,079 | [
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<p>I am looking for examples of physical indeterminacy impacting the macroscopic world. By physical indeterminacy, I mean physical sources of randomness such as quantum indeterminacy or brownian motion.</p>
<p>One example is of particular interest here: whether such randomness influences which sperm cell fertilizes a particular ovum, or whether such biological systems are too large to be affected by random perturbations, either at the atomic level (quantum) or molecular level (brownian motion).</p> | 1,080 | [
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<p>Is the curvature of space caused by the local density of the energy in that area?Could gravity be a separate phenomenon only arising from the curvature of space? For instance if the density of energy in a particular area cause that area of space to ”curve” but the effect that we understand as gravity, (causing anything with mass to be attracted to each other) is only arising as a consequence of that space being curved. I guess it seems to me that things other than mass can cause the curvature of space (electromagnetic fields, an enormously high density of photons in a small area or at least I think so, but I'm not sure about the photons, and if a black hole rotating causes frame dragging (which I'm assuming means the surrounding physical metric of space is probably some mechanism, or thought experiment where you could ball up space tight enough to become a black hole even without any matter in it. I guess it's another question I Could ask. </p> | 1,081 | [
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] |
<p>Title says it all really.. Why is the XX spin chain a free fermion (non-interacting) model, and the XXZ chain not? </p>
<p>Is it right that $\sum_l a_l^\dagger a_{l+1}$ isn't an interaction between fermions because it's creating a fermion on one site and destroying it on another? But why is $\sum_l a_l^\dagger a_l a_{l+1}^\dagger a_{l+1}$ an interaction term?</p>
<p>Is something like</p>
<p>\begin{equation}
H_1 = -\sum_l (J+(-1)^lK) ( \sigma_l^x \sigma_{l+1}^x +\sigma_l^y \sigma_{l+1}^y)
\end{equation}</p>
<p>a free fermion model? If not, why not?</p>
<p><strong>Edit</strong> I don't have enough reputation to set a bounty, but if anyone could answer this question, I'd be very grateful! </p>
<p><strong>Edit 2</strong> Anyone?</p> | 1,082 | [
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<p>Proof of gravitational potential energy. </p>
<p>Work done by gravity in bringing mass from infinity to a distance of $r$ between masses. </p>
<p>When we use the integration formula and arrive at the answer we get $-GMm/r$ taking lower limit as infinity and upper as $r$.</p>
<p>But this work should be positive as force and displacement are in same direction.</p>
<p>Please explain.</p>
<p>If my proof was wrong, then tell any other satisfying proof for GPE.</p> | 1,083 | [
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<p>I have a question about this motto used by Sean Carroll in his <a href="http://www.preposterousuniverse.com/blog/" rel="nofollow">blog</a>: </p>
<blockquote>
<p><em>In truth, only atoms and the void.</em> </p>
</blockquote>
<p>Can you explain what this sentence means? My interpretation is that the sentence does not makes sense because in physics, "atom" has multiple meanings, the same goes for "void". These words have so many meanings that, used without qualification, they mean nothing. The word truth is also a notoriously loaded and undefined word. Does he mean the absolute truth (which has no place in physics) or is he using it colloquially, we don't know.</p>
<p>Let's assume that he is using "atom" to mean "absolutely indivisible unit" and "void" to mean "absolutely empty space that contains nothing but atoms". This view is absurd because defines "void" to be "empty" and "not-empty" at the same time. But besides that, if in truth, there is nothing but atoms and the void, does Sean Carroll deny the existence of fields? </p>
<p>Here are a few meanings that physicists give to "void" (a search of titles in arxiv containing "void"): rigid void, relativistic void, magnetic fields in voids, empty voids, nano void, dynamics of void and so on. So, in physics void can mean anything but "void".</p>
<p>Then, what does "In truth, only atoms and the void" mean? Does it really have such a deep meaning to included as a motto of a blog?</p> | 1,084 | [
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<p>I'm trying to get my head around a problem (I should have checked whether I had the answer in class, the exams are coming up now and I don't know if I'll get a lecturer response over the holidays)</p>
<p>I can't get figure out the relationship between $\theta$ and $2\theta$ in the diagrams supposedly making clear how a crystal lattice diffracts.</p>
<p>From what I've read just now my understanding is that Bragg diffraction is actually transmission part-way into the crystal, then reflection off of an atom inside, hence the angle is twice the incident angle. </p>
<p>The textbook we've been referred to for further reading makes no note of a $2\theta$ (it can be read <a href="http://books.google.co.uk/books?id=qbHLkxbXY4YC&lpg=PA350&ots=c3NJjhJf4s&dq=%22denote%20the%20dislodging%20of%20an%20electron%22&pg=PA350#v=onepage&q&f=false" rel="nofollow">here</a>).</p>
<p>The example used in the lecture is below</p>
<p><img src="http://i.stack.imgur.com/VmKym.png" alt="enter image description here"></p>
<p>I obtained an answer by using a right-angled triangle made by the X-ray detector at the middle arrow, as this is the only perpendicular angle I can see. I really don't get how this would give the angle at the crystal as $2\theta$ (and I'm not at all confident that I should halve this angle to use in calculation of $\theta$</p>
<p>Can someone explain why I need to do so (as despite my misgivings, this is clearly indicated as the correct procedure). I've obtained an answer and am confident with the theory, maths etc., I just don't see the derivation of the $2\theta$ as opposed to $\theta$.</p> | 1,085 | [
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<p>Imagine that a pebble is placed on a uniformly rotating, frictionless disk. What will happen to this pebble? Will the disk slide under it and the pebble stay as is? Or will there be a centrifugal force on the pebble and it'll be thrown off the disk?</p> | 680 | [
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<p>I'm making a scratched hologram and I'm interested in we have to make curved lines to get a 3D image.</p> | 1,086 | [
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<p>If we have an array of telescopes attached one after another, would the resultant magnification be multiplied?Also would such a contraption be feasible to make telescopes with amazing magnification?</p> | 1,087 | [
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<p>I'm trying to understand Newton's Shell Theorem (Third)</p>
<p><a href="http://en.wikipedia.org/wiki/Shell_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Shell_theorem</a></p>
<p>However this applies to a sphere of <em>constant density</em>. How is this formulated for sphere of <em>varying density</em>, e.g., a ball of gas bound together by gravity? </p>
<p>Actually, this requires another question: how does the density of ball of gas bound by gravity diminish with radius? </p>
<p><strong>EDIT</strong></p>
<p>as DMCKEE pointed out "You can't answer the problem ignoring thermodynamics", so I've removed that proviso.</p> | 1,088 | [
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<p>As always the caveat is that I am a mathematician with very little knowledge of physics. I've started my quest for knowledge in this field, but am very very far from having a good grasp.</p>
<p>General relativity assumes that spacetime is a $4$-dimensional manifold. Ultimately, I know that physicists deal with Calabi-Yau manifolds (in string theory, I imagine). These are holomorphic complex manifolds.</p>
<p>How, and when, is this change of hypotheses justified? Is one way to view the general relativistic space time that spacetime is a variety of dimension $2$ over $\mathbb{C}$? Or is this change done later? Why is it ever done?</p> | 1,089 | [
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<p>As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.</p>
<p>This falls into the category of things I'm curious about. Have people considered whether spacetime is <a href="http://mathworld.wolfram.com/SimplyConnected.html" rel="nofollow">simply connected</a>? Similarly, one can ask if it contractible, what its Betti numbers are, its Euler characteristic and so forth. What would be the physical significance of it being non-simply-connected?</p> | 503 | [
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<p>Almost every book on physics that I read have some weird and non-clear explanations regarding the potential energy. Ok, I do understand that if we integrate a force over some path, we'll get a difference in some origin-function values ($\int_{A}^{B} Fdx = U(B) - U(A)$). This function is the potential energy. Of course, whether we can define this term or not depends on the force.</p>
<p>Now, here's an example of explanation (to be more precise - lack of explanation) regarding the GPE from one of the books:</p>
<blockquote>
<p>"...When a body moves from some point A to point B, gravity is doing work: $U_A-U_B=W_{A \to B}$. The magnitude can be calculated using an integral:
$W_{A \to B}=\int_{r_A}^{r_B} F(r) dr = \int_{r_A}^{r_B} \left(-\frac{GMm}{r^2} \right)dr=(-\frac{GMm}{r_A})-(-\frac{GMm}{r_B})=U_A-U_B$</p>
<p>...</p>
<p>Thus, when $r_A>r_B$, the magnitude is positive and therefore $U_A>U_B$. In other words: when the distance between the bodies is being increased - the gravitational potential energy of the system is also being increased.</p>
</blockquote>
<p>Giving absolutely no explanation on why all of a sudden they put a minus sign into the integral.</p>
<p>From another book:</p>
<blockquote>
<p>Work done by Coulomb's force: $W_{el}=\int_{r_1}^{r_2}\frac{q_1q_2dr}{4 \pi \epsilon_0 r^2}=\frac{q_1q_2}{4 \pi \epsilon_0 r_1}-\frac{q_1q_2}{4 \pi \epsilon_0 r_2}$</p>
<p>... Calculating the work gravity is doing is no different from the calculation of the work done by an electric field, with two exceptions - instead of $q_1q_2/4 \pi \epsilon_0$ we should plug $G M m$, and we also should change the sign, because the gravitational force is <strong>always a force of attraction</strong>.</p>
</blockquote>
<p>Now, this is not satisfying at all. So what if it is an attraction force? How this should influence our calculations, if the work is defined as $|F| |\Delta x| \cos \theta$, so the sign only depends on the angle between the path vector and the force vector? Why they put a minus sign? Is is just a convention or a must thing to do?</p>
<p>Some say the sign is important, others say the opposite. Some explain this as a consequence of that we bring the body from infinity to some point, while others say it is a consequence of an attractive nature of the gravitational force. All of that is really confusing me.</p>
<p>Also, in some of the questions like "what work is required to bring something from point $A$ to point $B$ in the field of gravitational/electric force", the books sometimes confuse $U_A-U_B$ and $U_B-U_A$ - as I understand it - the work that <strong>I</strong> must do is always $U_B-U_A$. However the work that <strong>the force</strong> that is being created by the field do is always $U_A-U_B$, am I correct?</p> | 1,090 | [
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/2175/is-it-possible-for-information-to-be-transmitted-faster-than-light">Is it possible for information to be transmitted faster than light?</a> </p>
</blockquote>
<p>Consider the following thought experiment. You have a long perfectly rigid beam (for the sake of simplicity, suppose it is one light-second long) which is placed on a fulcrum in the middle, so it is like an extremely long see-saw. There are two participants on either side of the beam.</p>
<p>Suppose participant A flips a coin and, based on the flip, chooses to place either a light or heavy boulder on his end of the beam. Participant B, one light-second away, sits down on his end of the beam.</p>
<p>What happens to participant B for the first second of his sit? If participant A randomly decided to place a light boulder on his end, then participant B would lower the beam with his weight; conversely, if there's a heavy boulder on the other end, he'll stay up in the air. <em>Either way, he will know in that first second what the outcome of participant A's coin flip was, and therefore gain one bit of information faster than the speed of light!</em> This is, of course, patently impossible.</p>
<p>My guess, as you can probably tell from the title of the question, is that this entire hypothetical situation cannot happen because there is not really such a thing as a "perfectly rigid body". The reason that one end of a lever moves at all in relation to the other is because the electromagnetic forces between the atoms in the beam push each other up. But what does this "look like" for absurd lengths like one light-second? Does the upward motion travel like a wave through the beam? Does the speed of this wave depend on the material, the magnitude of the forces involved, or something else entirely? Is there a name for it? And what, if anything, does participant B feel in that first second, and what would the beam "look like" to an external observer who can see the whole beam at once? I can't intuitively visualize any of this at all.</p>
<p>Standard disclaimer: I only know high-school level physics, if that helps when aiming answers.</p> | 4 | [
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<p>What is relativistic canonical transformation? I need every piece of information about it.
Does anyone know a reference or an article about relativistic canonical transformation?
For example, in classical mechanics, under one and only one condition, you can say that a transformation is canonical and that is:</p>
<p>$J\cdot M\cdot J^T= M$<br>
where $J$ and $M$ are two matrices which are represented in <em>Goldstein (3rd Edition) - Page 342</em></p> | 1,091 | [
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<p>I have read that it's not possible.</p> | 5 | [
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<p>In classical physics if one reverses time then energy does not change sign. For example in the formula for kinetic energy one has:</p>
<p>$$E = \frac{1}{2}m v^2$$</p>
<p>If you reverse time the velocity $v$ changes to $-v$ but $E$ stays the same.</p>
<p>What happens in quantum mechanics?</p>
<p>In QM one has the formula:</p>
<p>$$E = h \nu = \frac{h}{\Delta t} $$</p>
<p>If $\Delta t$ changes to $-\Delta t$ then in QM the energy should change sign if the time is reversed.</p>
<p>Why is there a difference between classical and QM?</p> | 1,092 | [
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<p>Up to which precision has the coulomb law proven to be true?
I.e. if you have two electrons in a vacuum chamber, 5 meters appart, have the third order terms been ruled out? Are there any theoretical limits to measure the precision ( Planck's constant?). Obviously there are practical limitations ( imperfect vacuum, cosmic rays, vacuum fluctuation). Still, does anyone know what was the smallest amount ever correctly predicted by that law?</p>
<hr>
<p>Edit : Summary</p>
<p>On the high end of the energy spectrum a precision of 10^-16 has been shown ( 42 years ago )</p>
<p>For electron point charges at large distances the law might brake down due to practical reasons.</p>
<p>For moving particles QED gives a correction to the law: <a href="http://arxiv.org/abs/1111.2303" rel="nofollow">http://arxiv.org/abs/1111.2303</a></p> | 1,093 | [
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<p>Does this photograph depict surface plasmon resonance? </p>
<p><img src="http://i.stack.imgur.com/RRN4e.jpg" alt="PHOTO 1 - Ellipsometric style photograph produces blue-green and purple resonance waves encircle nanogold-like tubule.">
PHOTO 1 - Ellipsometric style photograph produces blue-green and purple resonance waves from nanogold-like tubule.</p>
<p>PHOTO 1 was cropped from PHOTO 2...
<img src="http://i.stack.imgur.com/0hV22.jpg" alt="enter image description here"></p>
<p>...an accumulation of cellular debris contained in a 1976 TEM, gratuitously provided by FDA/NIH. </p>
<p>The original TEM images were produced on silver-halide photographic gel paper and the darkness of the biological sample can be attributed to staining the biological with nanogold. (NOTE - The dark images in the TEMS are silver atoms precipitated by electron impact with the target biological into film and then and duplicated by the development process in the photographic gel. The TEM images are not ink...this technology preceded digital printers.)</p>
<p>This photographing technique is not novel and a variation of the medium is used for nuclear emulsions according to 'Anna.v' who answered my first question as to whether or not PHOTO 3 depicted a particle annihilation.</p>
<p>PHOTO 3 <img src="http://i.stack.imgur.com/qQeHk.jpg" alt="enter image description here"></p>
<p>Answer from Anna.v:</p>
<blockquote>
<p>There exist pictures of positron annihilations and creation of electron positron pairs. Here is one:
(Sorry, unable to duplicate image as the question about particle annihilation.. was closed, fortunately after this insightful answer.)</p>
<p>A positron in flight annihilates with an electron into two gammas, which are invisible. One of them materializes at a certain distance from the track stop, resulting in a new electron-positron pair (marked with green)</p>
<p>These are taken in bubble chambers with a magnetic field perpendicular to the plane. What you are photographing is much more like what is seen in nuclear emulsions.</p>
<p>Chemical and biological energies are of the order of eV. To create a positron with a mass of ~500.keV is not possible. From what I see the TEMS has at most keV energies in the electron beam You may be seeing muons from the continuous muon background at sea level. The flux if I remember correctly is 1 per centimeter square per second (all energies). These could kick off electrons and even have enough energy to create electron positron pairs but it would not be a repeatable phenomenon since the flux is random. Also to see how elementary particle tracks would look in your material you would need to calibrate it at some accelerator lab.</p>
</blockquote>
<p>The photographic gel emulsion that produced the original images has yet to be calibrated, however, the initial impression that PHOTO 1 depicted a blue-green purple resonance Googled me into awareness of SPRs which have been studied extensively and detected elaborately but lack photographic depictions, at least from an online search. I suspect the photographic gel may have been doped with gold vapors, but coming from a government lab, the paper lacks commercial markings.</p>
<p>The predominant issue in my analysis arises from information that blue-green resonance changes the color of 50 nm gold to red. The color of colloidal gold depends on the size and shape of the nanoparticle, so possibly this shape generates an anomaly with blue-green purple resonance and gold coloration. </p>
<p>The underlying paradigm may depend on whether or not the questioned SPRs were generated by the original electron beam from the TEM or arise from interaction with the possible presence of gold vapor in the gel when photographed. </p>
<p>Any insight would be greatly appreciated.</p>
<p>Thank you,</p>
<p>Walter Kyle</p> | 1,094 | [
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<p>I'm working on this problem; however, I cannot seem to get anywhere.</p>
<p>Given information:</p>
<p>The rectangular loop in the figure has 2.1x10^-2 ohm resistance.
What is the induced current in the loop at this instant?</p>
<p><img src="http://i.stack.imgur.com/IAVF7.jpg" alt="[Picture of Problem](http://session.masteringphysics.com/problemAsset/1075420/4/34.P38.jpg)"></p>
<p>It asks for the answer of the current of the loop in terms of I of the rod.</p>
<p>I attempted the problem:</p>
<blockquote>
<p>I_loop = emf/R = vLBA / R</p>
<p>emf = d(magnetic flux)/dt = d(B * A)/dt</p>
<p>B = (mu_0 I)/(2 pi d)</p>
</blockquote>
<p>From here I'm not exactly sure what to do.
Any ideas?</p> | 1,095 | [
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<p>Suppose you have a theory of maps
$\phi: {\cal T} \to M$
with $M$ some Riemannian manifold,
Lagrangian
$$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i D_t\psi^j-D_t\overline{\psi}^i\psi^j) -\frac{1}{4}R_{ijkl}\psi^i\psi^j\overline{\psi}^k\overline{\psi}^l,$$</p>
<p>where $g_{ij}=g_{ij}(\phi)$ is the metric, $R$ its riemann tensor,
and covariant derivative</p>
<p>$$D_t\psi^i~=~ \partial_t \psi^i +\Gamma^i_{jk}\dot{\phi}^j\psi^k.$$</p>
<p>(Details taken from the book <em>Mirror Symmetry,</em> written by Vafa et al., paragraph 10.4.1.)</p>
<p>Taken for granted that the above Lagrangian is classically supersymmetric,
with susy transformations given by</p>
<p>$$ \begin{aligned}\delta\phi^i &= \epsilon \overline\psi^i-\overline\epsilon \psi^i \\
\delta\psi^i &= i\epsilon (\dot\phi^i-\Gamma^i_{jk}\overline\psi^j \psi^k)\\
\delta\overline\psi^i &= -i\epsilon (\dot\phi^i-\Gamma^i_{jk}\overline\psi^j \psi^k). \end{aligned}$$</p>
<p>How can I quantise the classical supercharges</p>
<p>$$Q=i\overline\psi_i\dot\phi^i, \qquad \overline Q=-ig_{ij}\psi^i\dot\phi^j$$</p>
<p>in a way such that</p>
<p>$$ \delta F=[\epsilon Q+\overline\epsilon\overline{Q},F]_{\pm}$$</p>
<p>where $F$ is either a fermionic or bosonic field and $\pm$ is used appropriately?</p>
<p>The natural answer would be something like calculate conjugate momenta</p>
<p>$$ p_i=\frac{\partial L}{\partial\dot\phi^i}, \qquad \pi_{i\psi}=\frac{\partial L}{\partial\dot\psi^i},$$</p>
<p>and impose canonical commutation relations</p>
<p>$$ [\phi^i,p_j]=i\delta^i_j, \qquad \{\psi^i,\pi_{\psi,j}\}=\delta^i_j.$$</p>
<p>Since in doing this I face non-trivial ordering issues, which the book doesn't seem to be worried about, and moreover its quantized version of conjugate momentum to $\phi$ seems wrong to me, as well as its quantized $Q$ doesn't seem to reproduce the correct transformations for the fields, I ask if someone could clarify this.</p>
<p>Moreover, looking in the paper <em>Constraints on supersymmetry breaking</em>
by Witten, in the neighbourhood of eqs. (90), (91), he seems to claim that the correct definition of conjugate momentum is derivative with respect to covariant derivative instead of time derivative, and this is another thing which leaves me with some doubts.</p> | 1,096 | [
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<p>I came across many recommendations for both of these books, but I'm not sure which one should I use to study calculus...</p>
<p>I know most of the methods used in calculus and I use them frequently, but I'm shooting for mathematical physics and I was wondering which of these books would be better for someone aiming for something which is both mathematically rigorous and written in a good style for a physicist.</p>
<p>Also, if there is some other book that fits the description, I would appreciate the recommendation! :)</p> | 1,097 | [
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<p>if it takes a 4 door sedan x amount of feet to go from 65 mph to 45 mph (and i don't know how many feet that is) on a dry straight road,
how many feet would it take for a large Mack truck transporting a full load of fluid, which would be most likely twice as heavy, to slow down from 65 to 45?
Also, as the truck slows, it is moving to the right. Because of the fluid in the tank pushing forward and to the right also, would this have an effect on how well the truck can travel straight or would it naturally pull to the right side?</p> | 1,098 | [
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<p>In the classic bicycle wheel gyroscopic precession, using the right hand coordinate system of xyz and right hand rule of rotation,</p>
<p>Such as
<img src="http://i.stack.imgur.com/0ERpO.jpg" alt="bicycle wheel"></p>
<p>X axis is in the longitudinal direction (spin axis)
z axis pointing in the direction of gravitation acceleration (downwards)</p>
<p>Using Euler's equation,</p>
<p><img src="http://i.stack.imgur.com/xdWeg.png" alt="Euler's equation"></p>
<p>Am I right to understand gyroscopic effect as:</p>
<p>The holding force of the string on the wheel produces a torque about the wheel which cause the wheel to have a natural tendency to swing downward. However, when there is a spin of the wheel in the x-axis, that force torque which will cause the wheel to drop actually produces a resultant torque that in the Y-axis, resulting in the wheel rotating about the Y-axis.</p>
<p>Due to this rotation in the Y-axis, coupled with the spin in the x-axis, it produces a secondary precession effect of torque which prevents the wheel from dropping down.</p>
<p>Is my understanding of precession correct?</p>
<p>The wheel produces this direction of precession because of the input spin (x-axis) of the wheel. If the spin is reversed, the precession direction will be reversed. Is this correct thinking too?</p>
<p>For a bicycle wheel, the moment of inertia in the spin (x-axis) is larger than the moment of inertia in the y and z direction. Now, my question is, based on the Euler's equation, if the moment of inertia is the other way round, what happens? For instance, instead of a bicycle wheel, what happens if we use a long rod where the moment of inertia in y and z direction is larger than the x direction? </p>
<p>Will the direction of precession be reversed too?</p> | 1,099 | [
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<p>I'm working my way through <a href="http://cua.mit.edu/8.422_S05/PHYSICS-henry-glotzer-a-squeezed-state-primer-am-j-phys-v56-p318-1988-AJP000318.pdf" rel="nofollow">A Squeezed State Primer</a>, filling in details along the way.</p>
<p>Let $a$ and $a^\dagger$ be the usual annihilation and creation operators with $[a,a^\dagger]=1$ and $|n\rangle=\frac{1}{\sqrt{n}}(a^\dagger)^n|0\rangle$.</p>
<p>With $\mu$ and $\nu$ complex numbers, define
\begin{eqnarray*}
b &=& \mu a+\nu a^\dagger \\
b^\dagger &=& \mu^\ast a^\dagger+\nu^\ast a
\end{eqnarray*}</p>
<p>Choose $\mu$ and $\nu$ so that $b$ and $b^\dagger$ satisfy $[b,b^\dagger]=1$, ie. $|\mu|^2-|\nu|^2=1$. So $b$ and $b^\dagger$ give a set of states 'isomorphic' to the usual number eigenstates.</p>
<p>Define generalised number states $|n'\rangle$ by
\begin{eqnarray*}
b|0'\rangle &=& 0\\
|n'\rangle &=& \frac{1}{\sqrt{n}}({b^\dagger})^n|0'\rangle
\end{eqnarray*}
(So the prime as attached to the state, not the $n$.)</p>
<p>With $N'={b^\dagger}b$ we have $\langle n'|N'|n'\rangle=n$.</p>
<p>Inverting the relationship between $a$ and $b$:
\begin{eqnarray*}
\mu^\ast b &=& |\mu|^2a+\mu^\ast\nu{a^\dagger}\\
\nu{b^\dagger} &=& \nu\mu^\ast{a^\dagger}+|\nu|^2a\\
\end{eqnarray*}
\begin{eqnarray*}
a &=&(|\mu|^2-|\nu|^2)a &=& \mu^\ast b-\nu{b^\dagger} \\
a^\dagger &=& (|\mu|^2-|\nu|^2){a^\dagger} &=& \mu{b^\dagger}-\nu^\ast b \\
\end{eqnarray*}</p>
<p>So the question is, what are the expected number of quanta in the $|n'\rangle$ states? I think I can compute this via:</p>
<p>\begin{eqnarray*}
\langle n'|N|n'\rangle &=& \langle n'|{a^\dagger} a|n'\rangle \\
&=& \langle n'|(\mu{b^\dagger}-\nu^\ast b)(\mu^\ast b-\nu{b^\dagger})|n'\rangle \\
&=& \langle n'||\mu|^2{b^\dagger} b+|\nu|^2b{b^\dagger}-\mu\nu({b^\dagger})^2-\mu^\ast\nu^\ast b^2|n'\rangle \\
&=& \langle n'||\mu|^2{b^\dagger} b+|\nu|^2b{b^\dagger}|n'\rangle \\
&=& \langle n'||\mu|^2N'+|\nu|^2(N'+1)|n'\rangle \\
&=& n(|\mu|^2+|\nu|^2)+|\nu|^2 \\
&=& n(2|\mu|^2-1)+|\mu|^2-1 \\
\end{eqnarray*}</p>
<p>Is that right?</p>
<p>For $n=1$ I get $3|\mu|^2-2$.</p>
<p>Page 323 of <a href="http://cua.mit.edu/8.422_S05/PHYSICS-henry-glotzer-a-squeezed-state-primer-am-j-phys-v56-p318-1988-AJP000318.pdf" rel="nofollow">the</a> paper appears to say it's $2\mu^2-1$ but I may be misunderstanding it. Where is my error?</p> | 1,100 | [
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<p>Is there a fundamental way to look at the universal constants ? can their orders of magnitude be explained from a general points of view like stability, causality, information theory, uncertainty? </p>
<p>for example, what sets the relative magnitudes of Planck's constant compared to say charge of electron or is it just a matter of choice of units. </p>
<p>Does the physics become more or less cumbersome, insightful if we set all fundamental constants to 1 in appropriate units.</p> | 1,101 | [
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<p>I'm trying to follow the steps in Eq. 2.60 of said book.</p>
<p>What I cant seem to figure out is how to change the integration variables from 'k' to 'E', as they state.</p>
<p>The equation is</p>
<p>$$\int \frac{d\textbf{k}}{4\pi^3} F(\epsilon(\textbf{k})) = \int_0^\infty \frac{k^2 dk}{\pi^2} F(\epsilon(k)) = \int_{-\infty}^\infty d\epsilon \, g(\epsilon) F(\epsilon)$$</p>
<p>I can follow the first transformation (why is $\textbf{k}$ suddendly $k$?),</p>
<p>$$\int\frac{1}{4\pi^3} k^2 F(\epsilon(k)) \, dk \int_0^\pi \sin \theta \, d\theta \int_0^{2\pi} d\phi = \int_0^\infty \frac{k^2 dk}{\pi^2} F(\epsilon(k))$$</p>
<p>But what's happening in the second step is unclear to me.</p>
<p>In the book it says, "one often exploits the fact that the integrand depends on $\textbf{k}$ only through the electronic energy $\epsilon = \hbar^2k^2/2m$,...", but I'm unsure how this is used.</p>
<p>Could anybody point this out to me?</p> | 1,102 | [
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<p>What defines the <a href="http://en.wikipedia.org/wiki/Adiabatic_flame_temperature" rel="nofollow">adiabatic flame temperature</a>?</p>
<p>In a case I have to solve, I need to describe the combustion of natural gas (Groningen natural gas, to be specific). However, I am having some problems understanding the adiabatic flame temperature, since on the internet I find some temperatures, but to me it seems that it would depend on the initial conditions, such as the start temperature and the fuel-oxidizer mixture.</p>
<p>The combustion should take place inside a ceramic foam, and I am assuming that this would happen isobaric at atmospheric pressure, and the inserted gas mixture before combustion would be at room temperature.
I need to describe a stationary situation, in which the flue gas will lose a certain amount of heat according to a given formula to foam (which will emit the same amount of energy through thermal radiation).</p>
<p>Would the adiabatic flame temperature be equal to the temperature of the exhaust gas before the heat loss to the ceramic foam?</p>
<p>$$
{\Delta}T=\frac{Q}{c_p},
$$
with ${\Delta}T$ the increase in temperature of the exhaust gas (relative to the temperature of the inserted fuel-oxidizer mixture) and $Q$ the net calorific value of the gas (taking in to account that the amount of moles change during the combustion). Or is the adiabatic flame temperature something completely different?</p> | 1,103 | [
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<p>Air at 20 degrees Celsius is compressed adiabatically from 1 bar to 10 bar, what will its temperature be?</p>
<p>With
$$P_1 = 1,$$ $$P_2 = 10$$
$$T_1 = 293K,$$ $$T_2 = unknown$$
using $$\dfrac{P_1}{P_2}=\dfrac{T_1}{T_2}$$
my solution was
$$\dfrac{1}{10}=\dfrac{293}{T_2}$$
giving
$$T_2 = 2930$$</p>
<p>My physics tutor said this is wrong and I should use $P_1V_1^\lambda=P_2V_2^\lambda$ and then use $\dfrac{V_1}{V_2}=\dfrac{T_1}{T_2}$ to find the temperature.</p>
<p>The only problem I now have is that $V_1 = V_2 = unknown$ which is making me think that he may have forgot to add that information to the question. </p>
<p>I would like to know if there is anyway to solve this (I do not want the answer just a point in the right direction).</p>
<p>thanks in advance!</p> | 1,104 | [
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] |
<p>In the MSUGRA breaking scenario, the stop particle typically appears at energies reachable at the LHC. Other sfermions, notably the partners of up, down, strange and charm are assumed to be degenerate in mass, and also heavier than the stop. Something similar holds for the stau.</p>
<p>Why is the third generation different in MSUGRA (not degenerate as the first two), and why is the mass hierarchy inverted wrt. the Standard Model sector (3rd generation sparticles lighter)?</p>
<p>(I guess these features are not neccessarily specific to MSUGRA, but might apply to more general models as well.)</p> | 1,105 | [
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<p>I know that if one astronaut falls into a black hole, then a distant observer will see him take an infinite amount of time to reach the event horizon (provided the observer can see light of arbitrarily large wavelengths).</p>
<p>But the falling astronaut will only take a finite amount of time to reach the horizon.</p>
<p>My question is: What will the falling astronaut see if he "looks backwards" while falling.
Will he see the distant observer growing old and all the stars dying by the time he reaches the horizon ?</p> | 1,106 | [
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<p>According to my calculations, it is a lot skinnier than Airy’s photon, but still a whole lot fatter than a straight line.</p>
<p>So, how does a photon get from point A to Point B?
The ray optics approximation treats a photon as an infinitely skinny straight line. Feynman, on the other hand, says a photon takes every possible path to get from Point A to point B, traversing the entire universe including all of space-time in the process. That’s very fat.
However, Feynman also says the photon interferes with itself enough to wipe out almost all traces of its passing except for a very narrow tube surrounding that classical straight line.
My question is, “How fat is that tube?”</p>
<p>Even though this is not a homework problem, I will show my first try. Considering only self interference, paths that differ from the shortest one by more than a half or quarter wavelength will be wiped out by interference. Those within say a tenth of a wavelength will mostly interfere constructively. So the tube will be an ellipsoid of paths from A to B that is defined by the length approximately equal to the distance from A to B plus one-fifth of a wavelength. (Replace “one-fifth” with any similar factor of your choosing, if you want to.) The width of this ellipse is the square root of twice its length (times that order one constructive interference factor), which is the distance from A to B expressed in photon wavelengths. </p>
<p>So the “width” or “fatness” of the photon increases with the square root of the distance from the source, unlike Airy’s refraction/diffraction disk, which increases linearly with the distance from the last obstruction.
Is this right? </p>
<p>This makes for some fairly fat photons. For instance using visible light, approximately 2 10^6 wavelengths per meter, if A and B are a meter apart, the width is 1000 wavelengths or about a millimeter. For light from the moon, it is about 20 meters wide. From the nearest star, about 200 kilometers wide. From across the universe, about 10 million kilometers, or about 30 light seconds wide.</p>
<p>I have not yet found any references to this ellipse other than Feynman’s own brief reference to an ellipse in the Feynman Lectures on Physics Volume 1, chapter 26.</p>
<p>Is there any experimental evidence that would support or refute this calculation? Of course, there are other causes of beam spreading, including misalignment, diffraction and the uncertainty principle. At least diffraction and misalignment could dominate the Feynman broadening discussed above. (I am going to promote this experimental issue to a separate question.)</p> | 1,107 | [
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<p>Why is equal time commutation relation used in canonical quantization of free fields?</p> | 1,108 | [
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<p>I'm a software developer, and I need to calculate the estimated amount of force expended typing stored text. Preferrably in some interesting way. (i.e. the force exerted on keys thus far is enough to push a car 5 miles) (or: equivalent to 100 kg of TNT)</p>
<p>Assumptions:</p>
<ul>
<li>We're not going to worry about deletes or moving the cursor or anything, I'm just counting characters of stored text.</li>
<li>I don't really care if the space requires more force or not, this is more of a "fun fact" than anything.</li>
<li>From what I've found online, <strong>the mean force required for a keystroke is about 12.9N</strong> <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2665300/" rel="nofollow">(source)</a>.</li>
<li>Hundreds of millions of characters have been typed.</li>
</ul>
<p>Questions</p>
<ul>
<li>What is a good way to make this something people can relate to?</li>
<li>How can I calculate it?</li>
</ul>
<p>Thank you all in advance for your valuable time and input.</p>
<p>EDIT: I thought my original post would make this pretty clear: I realize I only have the <em>force</em> required to push each keystroke. I'm looking for a way to demonstrate that force <em>applied</em> to something to help people quantify it, hypothetical energy in terms people can understand.</p> | 1,109 | [
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<p>We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is </p>
<p>$$\nabla^2\frac{q}{r}=-4\pi q\cdot \delta^3(\vec{r}).$$ </p>
<p>My question is, what is the Laplacian of $\frac{1}{r^2}$ (at the origin!)? Is there a charge distribution that would cause this potential? </p> | 1,110 | [
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<p>In my poor understanding of quantum physics, quantum entanglement means that certain properties of one of two 'entangled' quantum particles can lead to change over infinitely large distances when the other particles' properties are changed.</p>
<p>Disregarding this already mind-boggling event taking place over say 10 meters distance; how have physicists been able to demonstrate, beyond reasonable doubt, that this can take place over infinitely large distances?</p>
<p>For instance: have they done some of these tests between <a href="http://en.wikipedia.org/wiki/ISS" rel="nofollow">ISS</a> and Earth perhaps?</p>
<p>How can they be so sure?</p> | 1,111 | [
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<p>When looking at the wave function of a particle, I usually prefer to write</p>
<p>$$
\Psi(x,t) = A \exp(i(kx - \omega t))
$$</p>
<p>since it reminds me of classical waves for which I have an intuition ($k$ tells me how it moves through space ($x$) and $\omega$ tells me how it moves through time ($t$), roughly speaking). However, I noticed that you can translate this from the $(k,\omega)$ into the $(p,E)$ space by extracting $\hbar$:</p>
<p>$$
\Psi(x,t) = A \exp(i(px - Et)/\hbar)
$$</p>
<p>When looking at this representation, I couldn't help but be remembered of the uncertainty relations</p>
<p>$$
\Delta p \Delta x \geq \hbar\\
\Delta E \Delta t \geq \hbar
$$</p>
<p>This <em>cannot</em> be a coincidence, but the derivation of these relations (as described e.g. on Wikipedia) are simply over my head and my textbook merely motivates them with an argument based on wave packets.</p>
<p><strong>What is the connection here? Is there an intuitive explanation for the recurrence of both uncertainty relations in the wave function of a matter particle?</strong></p> | 1,112 | [
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<p>I have, for instance, a problem with a spherically symmetric charge distribution. I deduce here, in order to solve the problem easily, that the corresponding electric field must be symmetric. How is this type of deduction rigorously justified?</p> | 911 | [
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<p>Say I hook a 1KW steam engine to the steam heat in my apartment, and generate 1KW of electricity (the engine is 1KW mechanical, not 1KW thermal) from it, so I don't have to pay ConEd. </p>
<p>That means I am getting 1KW less heating power for the same amount of steam. Does it take the landlord's heating plant 1KW more to keep my room at the same temperature?</p> | 1,113 | [
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<p>I have a question about tidal forces on the far side of a body experiencing gravitational attraction from another body. </p>
<p>Let's assume we have two spherical bodies $A$ and $B$ whose centers are $D$ apart, and who have radiuses $R_{A}$ and $R_{B}$, much smaller than $D$. </p>
<p>Gravitation force has a law in $1/distance^{2}$. On the line $AB$, a mass $m$ at the point point of $B$ closest to $A$ experiences a pull towards the center of $A$ of magnitude $Km/(R_{B}-D)^{2}$, and the point farthest from $A$ experiences a pull of magnitude $Km/(R_{B}+D)^{2}$. </p>
<p>Note that I did not mention the direction of the force at that second point. It seems that the centers of mass being all on one side of that point, the force should point towards the center of $A$ on the line $AB$. However, the tidal bulge on this "far side" suggests that some force (?) is pulling on matter <em>away</em> from the center $A$. </p>
<p>How can we explain this tidal bulge on the far side? - I am specifically interested in a clear derivation using classical mechanics. </p> | 1,114 | [
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<p>Suppose a light wave with wavelength 3m. What happens if one tries to contain that wave within a 1m container? If I'm going about this entirely the wrong way or have wrong conceptions about light (which might be the case because I'm not a professional physicist), please tell me that instead.</p> | 1,115 | [
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<p>What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics, why should imaginary (or complex) time make sense?</p>
<hr>
<p>By imaginary I mean a multiple of $i$, and by complex I mean having a real and an imaginary part, i.e., $\alpha + i\beta$, where $\alpha, \beta \in {\mathbb R}$.</p> | 77 | [
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<blockquote>
<p>Related: <a href="http://physics.stackexchange.com/questions/21456/tubelightspower-lines-pictures/21513#comment50077_21513">Tubelights+power lines pictures?</a></p>
</blockquote>
<p>I would've edited this into the above question, but I realized that there' enough to it to qualify as a new one. Plus this seems to be a confusion of many people (including me).</p>
<p>I've already been able to explain part of this, so I'll put that explanation first just to make the question clearer.</p>
<p>The original question is as follows: In a circuit, you can have potential, but what is the origin of the field associated with it? Conducting wires have no net field, just some tiny opposing fields popping in and out of existence which facilitate current.</p>
<p>Explaining this for a simple capacitor is easy: With a charged (parallel plate) capacitor, one can see that we have a field between the plates but not in the rest of the circuit (for simplicity we can assume that the capacitor is just being shorted. So, traversing a different path in the line integral for potential gives a zero p.d, clearly a contradiction. After thinking about it, I realized that there is a significant fringe field near the terminals of the capacitor, which contributes heavily to the potential. SO the rest of the wire is equipotential without a field, and we have a p.d. only at the terminals due to the fringe fields. IMO there will be similar fringe fields that explain where the battery's potential comes from.</p>
<p>Now, I can't seem to get a similar explanation for current through a resistor. I fail to see any field being formed.</p>
<p>And my main issue is this: Let's say we set up a power line parallel to the ground. It may carry AC or DC current. Either way, it has a p.d with the ground at every point in time, which may vary. P.d. $\implies$ field, but I don't see any. In the AC situation, it <em>could</em> be from EMI, though I doubt that EMI is strong enough to produce the required field strength. In DC, @akhmateli mentioned charge being distributed on the wire surface, but I doubt that as well.</p>
<p>So where does the field come from? Is my explanation for a capacitor correct?</p>
<p>Oh, and in your explanation, I'd prefer no "this comes from the potential" or "these charges move due to the p.d.". I want one that talks about fields and charges only. I've had too many explanations of this which talk about "the field comes from the potential", which is IMO cheating.</p> | 1,116 | [
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<p>I'm working on a homework problem in Mathematica. We have to graph the height and the velocity of a function given an initial height and initial velocity. However, when I do the graph for the <a href="http://en.wikipedia.org/wiki/Velocity" rel="nofollow">velocity</a> with an initial velocity of 0 and an initial height of 100, the entirety of the velocity part of the graph is negative. Shouldn't the velocity be increasing as the object falls?</p>
<p><img src="http://s17.postimage.org/b7r7rhfqn/Screen_shot_2012_02_28_at_9_23_46_PM.png" alt=""></p>
<p>That is the graph and functions I am using. I have a feeling either something is typed wrong that I have overlooked in my numerous attempts to find it, or that there's something I'm just not understanding about what the graph is actually telling me. </p> | 1,117 | [
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<p>Currently I am working on my masters thesis about dualities in QFT and their geometric realizations.
As of now, I am trying to understand the article 'N=2 Dualities" by Davide Gaiotto. On the internet I found some exercises related to the article (<a href="http://www.sns.ias.edu/pitp2/2010files/Gaiotto-Problems.pdf" rel="nofollow">http://www.sns.ias.edu/pitp2/2010files/Gaiotto-Problems.pdf</a>). My questions are about some of these exercises.</p>
<p>I will shortly summarize the exercise and then put my question forward. The full exercise is reachable via the link above.</p>
<p>Exercise 1:
We first look at degree k meromorphic differentials with poles of order $k$ at $n$ points $z_i$ on the Riemann sphere: $\phi_k(z)=F(z)dz^k$. Here $F(z)$ is a rational function on the complex plane. If we want to know the behaviour of $\phi_k(z)$ at $\infty$ we change coordinates to $z'=1/z$ under which the k-differential transforms as $\phi_k(z')=F(1/z')(-dz'/(z')^2)^k$. Furthermore, these differentials are required to have fixed residues $\alpha_i$ on each $z_i$. The question then is how big the dimension is of the space of these k-differentials.</p>
<p>First of all it is unclear to me what precisely is meant with $$\phi_k(z) \approx \frac{\alpha_i}{(z-z_i)^k} dz^k +...$$. My interpretation is that $\phi_k(z)$ may be written as a fraction of two polynomials $f(z)/g(z)$ where $g(z)$ has $k^{th}$ order zeroes at $n$ points $z_i$ and $f(z)$ has $k(n-2)$ zeroes (to get the correct degree of the divisor of a k-differential on the Riemann Sphere, namely $-2k$). The zeroes of $f(z)$ we may choose freely (as long as we satisfy the fixed residues $\alpha_i$).</p>
<p>First I attempted to solve this with Riemann Roch. This led me to a counting of $k(n-2)+1$ free parameters, however this doesn't account for the fixed residues I think.
Then, with fixed residues, I reasoned it should be $(k-1)(n-2)$ by counting the free parameters for a k-differential. For n k'th order poles one has $(n-2)k$ zeroes to freely choose (in order that the degree of the divisor of the k-differential is $-2k$) and one constant $c$ multiplying $f(z)$ . </p>
<p>However, for fixed residues, one has to subtract $n-1$ parameters (not $n$ since the residues sum to zero), which leads to the total of $(k-1)(n-2)$ free parameters. T
his would also be the dimension of the vector space of k-differentials with n k'th order poles with fixed residues, since we can look at all linearly independent $F(z)'s$, ie different degrees of the polynomial $f(z)$ which may look like $\prod^l_{i=1}c(z-u_i)$ for $l\in {1,2,..,n}$, $u_i$ a zero and $c$ a constant.</p>
<p>Does anybody know if this counting and way of looking at the $F_i(z)$'s is correct?</p>
<p>3i)I guess my problems with this question depend very much on the definitions in question 1.</p>
<p>I tried to solve a simple example with $k=2$ and $n=3$: </p>
<p>$$x^2 + F_1(z) x + F_2(z) = 0$$</p>
<p>with</p>
<p>$$F_1(z)=\frac{c(z-u)}{(z-z_1)(z-z_2)(z-z_3)}$$</p>
<p>and</p>
<p>$$F_2(z)=\frac{d(z-v)(z-w)}{(z-z_1)^2(z-z_2)^2(z-z_3)^2}$$</p>
<p>According to my calculation in 1ii) it follows that only $w$ is a free parameter in this equation; $c$ and $u$ are completely determined by the fixed residues of $F_1$ and $d$ and $v$ are determined by the fixed residues of $F_2$ (and are functions of $w$).</p>
<p>EDIT:
When I try to solve this equation (with a change of variables to $y = x(z-z_1)(z-z_2)(z-z_3))$ with Mathematica, the expressions become very complicated and it does not follow that that $\frac{\partial\lambda}{\partial v}$ is a holomorphic one form on the curve, where $\lambda = xdz$ as asked in exercise 3ii. The way to see this, I think, is that the residue of $\lambda$ at the points $z_i$ seems not to be independent of the parameters $v,w$. </p>
<p>Does someone has an idea what mistakes I am making? Thanks,
Sam</p> | 1,118 | [
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<p>ever since I begun calculating thermodynamical cycles, I've had problems with determining the sign of the work along a particular bit of the cycle. Of course, I guess that an arbitrary cycle is 'bendy' and the sign of the work differential depends strongly on the coordinates, but usually the cycles I deal with consist of a couple of 'parts', for instance, adiabatics, isothermals, isochorics, etc. And whenever asked to calculate the total work done in a cycle (for instance to find the efficiency), I just kind of guessed the sign and managed to get by, but now when trying to understand this on a deeper level, this is coming back to haunt me.</p>
<p>So, suppose I have a thermodynamic cycle like this: <a href="http://i.imgur.com/lkTDsqp.png" rel="nofollow">LINK</a></p>
<p>Where: </p>
<p>(1)-adiabatic</p>
<p>(2)-isobaric</p>
<p>(3)-isothermal</p>
<p>(4)-isochoric</p>
<p>How do I know the sign of the work along each of these paths?</p> | 1,119 | [
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<p>I need assistance (or experience feedback) in estimating if a Venturi pump would work at low pressure.
A boiler under vacuum (10% atmosphere) will produce steam (at a given speed depending on the boiler power) and the steam flow will be used to suck water at the same temperature (of boiling water under this vacuum, about 50 degrees C) and pressure in a <a href="http://en.wikipedia.org/wiki/Venturi_pump" rel="nofollow">Venturi style setup</a>.</p>
<p>From the wiki articles I read about the Venturi tube, the usual equations giving the pressure difference between the constriction and the larger tube are for incompressible fluids. I do worry that 10% ATM steam would be very compressible though. So in short I need an equation for tying up the tube diameters, the steam feed speed, and the suction generated under such conditions</p> | 1,120 | [
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<p>Classically, black holes can merge, becoming a single black hole with an horizon area greater than the sum of both merged components.</p>
<p>Is it thermodynamically / statistically possible to split a black hole in multiple black holes? If the sum of the areas of the product black holes would exceed the area of the original black hole, it seems to be a statistically favorable transition by the fact alone that would be a state with larger entropy than the initial state</p> | 5 | [
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<p>Does anybody know a good reference that works out the <a href="http://www.google.com/search?q=chargino+neutralino+production+cross+section+susy+equations" rel="nofollow">equations</a> for the Chargino/Neutralino production cross section in SUSY? I'm trying to understand if there are any tricks for boosting the production cross section.</p>
<p>So far, I have just been testing using <a href="http://arxiv.org/abs/hep-ph/9611232" rel="nofollow">Prospino</a> and discovered a couple patterns which seem counter-intuitive. For one, the production cross section seems to increase as I increase the masses of the squarks. This is strange because I would expect there to also be a contribution from strong production of chargino/neutralino so lower squark masses should boost the production.</p>
<p>What am I missing?</p> | 1,121 | [
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<p>The Hamiltonian of tight binding model reads $H=-|t|\sum\limits_{<i,j>}c_i^{\dagger}c_j+h.c.$, why is there a negative sign in the hopping term?</p> | 1,122 | [
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<p>In nucleon-nucleon interactions of n-n, p-p, n-p how do you determine which pion is the mediator?</p> | 1,123 | [
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